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Wolfgang Torge
Third completely revised and extended edition
Walter de Gruyter Berlin New York 2001
Dr.-Ing. Wolfgang Torge
Institut fr Erdmessung
Universitt Hannover
Schneiderberg 50
This book contains 184 figures.
@ Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence
and durability.
Library of Congress Cataloging-in-Publication Data
Torge, Wolfgang.
[Geodsie. English]
Geodesy / Wolfgang Torge. - 3rd completely rev. and extended ed.
p. cm.
1. Geodesy. I. Title.
QB281 .T58132001
526'.l-dc21
Die Deutsche Bibliothek Cataloging-in-Publication Data
Torge, Wolfgang:
Geodesy / Wolfgang Torge. - 3., completely rev. and extended ed.
Berlin ; New York: de Gruyter, 2001
Copyright 2001 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany
All rights reserved, including those of translation into foreign languages. No part of this book
may be reproduced or transmitted in any form or by any means, electronic or mechanical,
including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher.
The use of registered names, trade names, trade marks, etc. in this book, even without a specific
statement, does not imply that these names are not protected by the relevant laws and regulations. While the advice and information in this book is believed to be true and accurate at the
date of its going to press, neither the author nor the publisher can accept any legal responsibility
for any errors or omissions that may be made. The publisher makes no warranty, express or
implied, with respect to the material contained herein.
Printed in Germany
Cover design: Rudolf Hbler, Berlin
Printing: Werner Hildebrand, Berlin
Binding: Lderitz & Bauer-GmbH, Berlin
Hans-Georg Wenzel
Preface to the Third Edition
The first edition of this book was published in 1980 as an English translation of
the book "Geodsie," which was printed in the German language in 1975 within
the "Gschen Series" of Walter de Gruyter and Co., Berlin and New York. A
thorough revision of the original text was undertaken at that time, but the
fundamental structure of the book was retained. The second edition (1991)
added the remarkable development of space techniques and their effects on
positioning and gravity field determination. Though it represented an extension
of the first edition, the basic subdivision into six chapters was not altered.
In contrast to the relatively minor changes of the second edition, the present
edition has been completely revised and significantly extended. This was
necessary in order to account for the tremendous changes that geodesy, on the
whole, has experienced over the past thirty years, driven by the progress in
space techniques and data acquisition and evaluation in general. A particular
consequence is that geodesy has transitioned to a three-dimensional concept,
based on a global reference system, with far-reaching consequences for geodetic
practice. High data rates and improved accuracy require a more rigorous
consideration of time as a fourth dimension and has led to a growing
contribution to geodynamics research.
The significant extension of the third edition is demonstrated by increases in the
number of pages (416 versus 264), the number of figures (184 versus 137), and
the number of references. The references increased by about 50 percent to more
than 700, half of which are new entries. There are now eight chapters instead of
six, and their content more clearly mirrors the fundamental changes of geodesy.
The "Introduction " still contains the definition and the over 2000 years history
of geodesy. It now also includes the three and four-dimensional geodetic
concepts. The survey of geodetic organizations and literature has been updated
as well. The new chapter on "Reference Systems " comprises the fundamentals
of geodetic reference and time systems and their realization and mutual
transformation. This information was previously dispersed throughout the book.
The third chapter, "The Gravity Field of the Earth," is similar to the
corresponding chapter in the previous edition, with some additions to the
description of the gravity field geometry and the spherical harmonic expansion.
The introduction to the geoid is now included here, while the earth tides section
has been moved to the last chapter. The next chapter, "The Geodetic Earth
Model," was extracted from the chapter on geodetic reference systems in the
previous edition. A separate treatment appeared necessary, especially due to the
importance of the "Geodetic Reference System 1980" and the "World Geodetic
System 1984."
The chapters on measurement and evaluation methods comprise the core of the
book and have been completely reorganized and revised. The chapter "Methods
of Measurements" now starts with a detailed description of atmospheric
refraction, which affects all geometric methods of geodesy. The dominating role
of satellite geodesy is recognized by treating this subject early. Space for
classical methods such as Doppler positioning has been reduced, while the
Global Positioning System (GPS) has naturally received increased emphasis
(now with 10 pages compared to about 4 in the previous edition). Developing
techniques such as satellite-to-satellite tracking and satellite gravity gradiometry
have been given separate space. Very Long Baseline Interferometry is now
included in the section on geodetic astronomy. Gravimetry was updated with
respect to absolute and airborne techniques. Terrestrial geodetic measurements
concentrate on combined angle and distance observations over shorter distances
and on precise leveling. Discussion of tilt and strain measurements are also
found here. The former subdivision of "Methods of Evaluation" into
astrogeodetic, gravimetric, satellite and combined methods has been replaced by
an introductory section on the residual gravity field, two large blocks on
positioning and gravity field modeling, and a section on combination solutions.
Positioning starts from the three-dimensional model, followed by horizontal
positioning and height determination, after proper reductions to the ellipsoid
and the geoid, respectively. The effect of topography on gravity field modeling
is now discussed in more detail, and a clear distinction is made between global
and local modeling, where the astrogeodetic methods have also been included.
The combined methods comprise the functional approach by earth models and
the statistical approach through least squares collocation.
The seventh and the eighth chapters reflect the effects of the developments in
geodesy on national and continental networks and on the geosciences. The
chapter on "Geodetic and Gravimetric Networks" is now free of computational
formulas and concentrates on the design and establishment of networks, with
special emphasis on the present transition to three-dimensional networks and
absolute gravimetry. The final chapter, "Structure and Dynamics of the Earth, "
has been extracted from the previous edition's chapter on "Methods of
Evaluation, Global Geodesy." This chapter considers the increasing role which
geodesy plays within the geosciences and, consequently, includes an
introduction to the geophysical earth model, especially to the upper layers of the
earth. Geodetic contributions to geodynamics research are now described more
systematically and extensive, ranging from earth rotation over sea level
variations, crustal movements, and gravity variations to earth tides.
The text is again illustrated by a large number of figures. Many of the figures
depict fundamental geodetic relations, while others show examples of recent
instrumentation and geodetic data processing results. The latter case required a
thorough revision of the figures and a revision of the associated passages of the
text. Among the new examples for geodetic instruments we have geodetic GPS
receivers, altimeter satellites, absolute and relative gravimeters, total stations,
and digital levels. The results include the International Celestial and Terrestrial
Reference Frames, The World Geodetic System 1984 (status 2000), the
geopotential model EGM96, the gravimetric geoids for the U.S.A. and for
Europe, and the classical and modern networks in those regions, as well as
examples of recent geodetic contributions to the investigation of geodynamics.
The primary purpose of the book is to serve as a basic textbook oriented
towards students of geodesy, geophysics, surveying engineering, cartography,
and geomatics, as well as students of terrestrial and space navigation. The book
is also a valuable reference for geoscientists and engineers facing geodetic
problems in their professional tasks.
The contents of the book are based in part on lecture courses given at the
University of Hannover, Germany and on guest lectures given abroad. The
author is indebted to individuals and organizations for providing illustrations;
due credit is given in the figure captions. He thanks M.Sc. Kyle Snow, who
checked the English text with extreme care, included the formulas and figures,
and prepared the manuscript for printing. The majority of the figures have been
drawn by cand. geod. Anke Daubner, Dipl.-Ing. Andreas Lindau handled the
electronic mailing and text storage, and assisted in the final proof-reading. The
staff of the Institut fr Erdmessung assisted in manifold ways in the preparation
of the manuscript. All this help is gratefully acknowledged. The outstanding
cooperation of the publisher, proved over a nearly 30 years association,
continued and calls for a cordial thanks to Dr. Manfred Karbe and the staff at
Walter de Gruyter.
Hannover, February 2001
l. l
Definition of Geodesy
The Problem of Geodesy
Historical Development of Geodesy
The Spherical Earth Model
The Ellipsoidal Earth Model
The Geoid, Arc Measurements and National Geodetic Surveys
Three-dimensional Geodesy
Four-dimensional Geodesy
Organization of Geodesy, Literature
Reference Systems
Basic Units and Fundamental Constants
Time Systems
Atomic Time, Dynamical Time
Sidereal and Universal Time
International Earth Rotation Service
Celestial Reference System
Equatorial System of Spherical Astronomy
Precession and Nutation
International Celestial Reference Frame
Terrestrial Reference System
Global Earth-Fixed Geocentric System
Polar Motion, Length of Day, Geocenter Variations
International Terrestrial Reference Frame
Gravity Field Related Reference Systems
Orientation of the Local Vertical
Local Astronomic Systems
The Gravity Field of the Earth
Fundamentals of Gravity Field Theory
Gravitation, Gravitational Potential
Gravitation of a Spherically Symmetric Earth
Properties of the Gravitational Potential
Centrifugal Acceleration, Centrifugal Potential
Gravity Acceleration, Gravity Potential
Geometry of the Gravity Field
Level Surfaces and Plumb Lines
Local Gravity Field Representation, Curvatures
Natural Coordinates
Spherical Harmonic Expansion of the Gravitational Potential
Expansion of the Reciprocal Distance
Expansion of the Gravitational Potential
Geometrical Interpretation of the Surface Spherical Harmonics
Physical Interpretation of the Harmonic Coefficients
The Geoid
Mean Sea Level
The Geoid as Height Reference Surface
Temporal Gravity Variations
Gravitational Constant, Earth Rotation
Tidal Acceleration, Tidal Potential
Non-Tidal Temporal Variations
The Geodetic Earth Model
The Rotational Ellipsoid
Parameters and Coordinate Systems
Spatial Geodetic Coordinates
The Normal Gravity Field
The Level Ellipsoid, Level Spheroids
The Normal Gravity Field of the Level Ellipsoid
Geometry of the Normal Gravity Field
Geodetic Reference Systems
Methods of Measurement
Atmospheric Refraction
Tropospheric Refraction
Ionospheric Refraction
5.2 Satellite Observations
Undisturbed Satellite Motion
Perturbed Satellite Motion
Artificial Earth Satellites
Direction, Range and Range Rate Measurements:
Classical Methods
Laser Distance Measurements
Satellite Altimetry
Satellite-to-Satellite Tracking, Satellite Gravity Gradiometry
5.3 Geodetic Astronomy
Optical Observation Instruments
Astronomic Positioning and Azimuth Determination
Very Long Baseline Interferometry
5.4 Gravimetry
Absolute Gravity Measurements
Relative Gravity Measurements
Gravity Reference Systems
Gravity Measurements on Moving Platforms
Gravity Gradiometry
Continuous Gravity Measurements
5.5 Terrestrial Geodetic Measurements
Horizontal and Vertical Angle Measurements
Distance Measurements, Total Stations
Tilt and Strain Measurements
Methods of Evaluation
6.1 Residual Gravity Field
Disturbing Potential, Height Anomaly, Geoid Height
Gravity Disturbance, Gravity Anomaly, Deflection of the Vertical
Statistical Description of the Gravity Field, Interpolation
6.2 Three-dimensional Positioning
Observation Equations
Geodetic Datum
6.3 Horizontal Positioning
Ellipsoidal Trigonometry
Reductions to the Ellipsoid
Computations on the Ellipsoid
Height Determination
Heights from Geometric Leveling
Trigonometrical Heights
Heights from GPS
Fundamentals of Gravity Field Modeling
The Geodetic Boundary-Value Problem
Gravitation of Topography
Gravity Reductions to the Geoid
Orientation and Scale of Gravity Field Models
Global Gravity Field Modeling
Spherical Harmonic Expansions
Low-degree Gravity Field Models
High-degree Gravity Field Models
Local Gravity Field Modeling
Gravimetric Geoid Heights and Deflections of the Vertical
Gravimetric Height Anomalies and Surface Deflections of
the Vertical
The External Gravity Field
Astrogeodetic Geoid and Quasigeoid Determination
Combined Methods for Positioning and Gravity Field Determination
Earth Models and Optimum Earth Parameters
Least Squares Collocation
Geodetic and Gravimetric Networks
Horizontal Control Networks
Design and Observation
Computation and Orientation
Vertical Control Networks
Three-dimensional Networks
Global and Continental Networks
Gravity Networks
Structure and Dynamics of the Earth
8.1 The Geophysical Earth Model
8.2 The Upper Layers of the Earth
Structure of the Earth's Crust and Upper Mantle
Isostasy
Interpretation of the Gravity Field
8.3 Geodesy and Geodynamics
Changes in Earth Rotation
Sea Level Variations
Recent Crustal Movements
Gravity Variations with Time
Earth Tides
1.1 Definition of Geodesy
According to the classical definition of Friedrich Robert Helmert (1880),
''geodesy ( = earth, = I divide) is the science of the measurement and
mapping of the earth's surface" Helmert's definition is fundamental to
geodesy even today. The surface of the earth, to a large extent, is shaped by the
earth's gravity, and most geodetic observations are referenced to the earth's
gravity field. Consequently, the above definition of geodesy includes the
determination of the earth s external gravity field. The original focus of
geodesy has expanded to include applications in ocean and space exploration.
For example, Geodesy, in collaboration with other sciences, now includes the
determination of the ocean floor and the surfaces and gravity fields of other
celestial bodies, such as the moon (lunar geodesy) and planets (planetary
geodesy). Finally, the classical definition has to be extended to include
temporal variations of the earth's surface and its gravity field.
With this extended definition, geodesy is part of the geosciences and engineering
sciences, including navigation and geomatics (e.g., NATIONAL ACADEMY OF
SCIENCES 1978). Geodesy may be divided into the areas of global geodesy,
geodetic surveys (national and supranational), and plane surveying. Global
geodesy includes the determination of the shape and size of the earth, it's
orientation in space, and it's external gravity field. A geodetic survey is for the
determination of the earth's surface and gravity field over a region that typically
spans a country or a group of countries. The earth's curvature and gravity field
must be considered in geodetic surveys. In plane surveying (topographic
surveying, cadastral surveying, engineering surveying), the details of the earth's
surface are determined on a local level, and thus curvature and gravity effects are
generally ignored.
There is a close relation between global geodesy, geodetic surveying, and plane
surveying. Geodetic surveys are linked to reference frames (networks) established
by global geodesy; these surveys adopt the global parameters for the figure of the
earth and its gravity field. On the other hand, the results of geodetic surveys may
contribute to the work of the global geodesist. Plane surveys, in turn, are
generally referenced to control points established by geodetic surveys. Plane
surveys are used extensively in the development of national and state map-series,
cadastral information systems, and in civil engineering projects. The
measurement and data evaluation methods used in national geodetic surveys are
often similar to those used in global geodetic work. For instance, space methods
(satellite geodesy) have long been a dominant technique in global geodesy and
are now commonly used in regional and local surveys. This requires a more
detailed knowledge of the gravity field at the regional and local scales.
With the corresponding classification in the English and French languages, the
concept of "geodesy" (la geodesie, "hhere Geodsie" after Helmert) in this text
refers only to global geodesy and geodetic surveying. The concept of "surveying"
(la topometrie, Vermessungskunde or "niedere Geodsie" after Helmert) shall
encompass plane surveying.
In this volume, geodesy is treated only in the more restrictive sense as explained
above (excluding plane surveying), and limited to the planet earth. Among the
numerous textbooks on surveying we mention KAHMEN and FAIG (1988),
KAHMEN (1997), ANDERSON and MIKHAIL (1998), BANNISTER et al. (1998). For
lunar and planetary geodesy see BILLS and SYNNOTT (1987), KAULA (1993), and
NEREM (1995a); numerical values of astrometric and geodetic parameters are
given by Yoder (1995).
Based on the concept of geodesy defined in [1.1], the problem of geodesy may be
described as follows:
"The problem of geodesy is to determine the figure and external gravity field of
the earth and of other celestial bodies as a function of time, from observations on
and exterior to the surfaces of these bodies."
This geodetic boundary-value problem incorporates a geometric (figure of the
earth) and a physical (gravity field) formulation; both are closely related.
By th&gure of the earth we mean the physical and the mathematical surface of
the earth as well as a geodetic reference earth, e.g., MORITZ (1990).
The physical surface of the earth is the border between the solid or fluid masses
and the atmosphere. The ocean floor is included in this formulation, being the
bounding surface between the solid terrestrial body and the oceanic water masses.
The irregular surface of the solid earth (continental and ocean floor topography)
cannot be represented by a simple mathematical (analytical) function. It is
therefore described point wise by the coordinates of control points. Given an
adequately dense control network, the detailed structure of this surface can be
determined by interpolation of data from terrestrial topographic and hydrographic
surveying (KRAUS 1992/97, KRAUS and SCHNEIDER 1988/90, HAKE and
GRNREICH 1994, KAHMEN 1997). On the other hand, the ocean surfaces (70%
of the earth's surface) are easier to represent. If we neglect the effects of ocean
l .2 The Problem of Geodesy
currents and other "disturbances", the ocean surfaces form a part of a level or
equipotential surface of the earth's gravity field (surface of constant gravity
potential). We may think of this surface as being extended under the continents
and identify it as the mathematical figure of the earth, which can be described by
a condition of equilibrium (HELMERT 1880/1884). J.B. LISTING (1873)
designated this level surface as geoid.
The great mathematician and geodesist Carl Friedrich Gauss (1777 - 1855) had already referred to
this surface: "What we call surface of the earth in the geometrical sense is nothing more than that
surface which intersects everywhere the direction of gravity at right angles, and part of which
coincides with the surface of the oceans" (C.F. Gauss: "Bestimmung des Breitenunterschiedes
zwischen den Sternwarten von Gttingen und Altona," Gttingen 1828), see also MORITZ (1977).
The description of the geoid's properties is the physical aspect of the problem of
geodesy. In solving this problem, the earth's surface and the geoid are considered
as bounding surfaces in the earth's gravity field, the field to which geodetic
observations are referenced. Based on the law of gravitation and the centrifugal
force (due to the earth's rotation), the external gravity field of the earth, or any
other celestial body, can be modeled analytically and described by a large number
of model parameters. A geometric description is given by the infinite number of
level surfaces extending completely or partially (as the geoid) exterior to the
earth's surface.
OCEAN SURFACE
OCEAN FLOOR
Fig. 1.1. Physical surface of the earth, geoid, and ellipsoid
Reference systems are introduced in order to describe the orientation of the earth
and other bodies in space (celestial reference system) as well as their surface
geometry and gravity fields (terrestrial reference system). The definition and
realization of these systems has become a major part of global geodesy; the use of
three-dimensional Cartesian coordinates in Euclidean space is adequate in this
context. However, due to the demands of practitioners, reference surfaces are
introduced to distinguish between curvilinear surface coordinates for horizontal
positioning and heights above some zero-height surface for vertical positioning.
Because of its simple equation, an ellipsoid of rotation, flattened at the poles, is
well suited for describing horizontal positions, and consequently is used as a
reference surface in geodetic surveying. In plane surveying, the horizontal plane
is generally a sufficient reference surface. Because of its physical meaning, the
geoid (or any other reference defined in the earth's gravity field) is well suited as
a reference surface for heights. For many applications, a geodetic reference earth
(earth model, normal earth) is needed. It is realized through a mean-earth
ellipsoid that optimally approximates the geometry (geoid) and the gravity field
of the earth. Fig. 1.1 shows the mutual location of the surfaces to be determined
in geodesy.
The body of the earth and its gravity field are subject to temporal variations of
secular, periodic, and abrupt nature, which can occur globally, regionally, and
locally. These variations also influence the orientation of the earth. Modern
geodetic measurement and evaluation techniques are used to detect these
variations to a high level of accuracy. If time-independent results are required,
geodetic observations must be corrected for temporal variations. By determining
temporal variations, the science of geodesy contributes to the investigation of the
kinematic and dynamic properties of the terrestrial body. Accordingly, the figure
of the earth and the external gravity field must be considered as time dependent
quantities: "Four-dimensional geodesy" (MATHER 1973).
The formulation of the problem of geodesy as described in [1.2] did not fully
mature until the nineteenth century. However, the question of the figure of the
earth was contemplated in antiquity. In fact, geodesy together with astronomy and
geography are among the oldest sciences dealing with the planet earth.
Superseding the use of the sphere as a model for the earth [1.3.1], the oblate
rotational ellipsoid became widely accepted as the model of choice in the first
half of the eighteenth century [1.3.2]. The significance of the gravity field was
also recognized in the nineteenth century, leading to the introduction of the geoid
[1.3.3]. In the second half of the twentieth century, satellite techniques permitted
the realization of the three-dimensional concept of geodesy [1.3.4]. A drastic
increase in the accuracy of geodetic observations required that time variations be
taken into account. This led to the concept of four-dimensional geodesy [1.3.5].
Extensive material on geodetic history is found in TODHUNTER (1873), FISCHER
(1975), BIALAS (1982), SMITH (1986) and LEVALLOIS (1988).
Various opinions on the figure of the earth prevailed in the past, e.g., the notion
of an earth disk encircled by oceans (Homer's Iliad - 800 B.C., Tholes ofMilet ~
1.3 Historical Development of Geodesy
600 B.C.). Considering the sphere aesthetically appealing, Pythagoras (~ 580 500 B.C.) and his school proposed a spherical shaped earth. By the time of
Aristotle (384 - 322 B.C.), the spherical concept was generally accepted and even
substantiated by observations. For example, observers noted the round shadow of
the earth in lunar eclipses and the apparent rising of an approaching ship on the
horizon. In China the spherical shape of the earth was recognized in the first
century A.D.
Fig. 1.2. Arc measurement of Eratosthenes
The founder of geodesy is Eratosthenes of Alexandria (276 - 195 B.C.), who,
based on the assumption of a spherical earth, deduced the earth radius from
measurements (SCHWARZ 1975). The principle of the arc-measurement method
developed by Eratosthenes was still applied in modem time: from geodetic
measurements, the length AG of a meridian arc is determined; astronomical
observations furnish the associated central angle (fig. 1.2). The radius of the
earth is then given by
Eratosthenes found that at the time of the summer solstice the rays of the sun
descended vertically into a well in Syene (modern day Assuan). Whereas in
Alexandria (approximately on the same meridian as Syene), the sun's rays formed
an angle with the direction of the plumb line. From the length of the shadow of a
vertical staff ("gnomon") produced in a hemispherical shell ("skaphe"),
Eratosthenes determined this angle as 1/50 of a complete circle, i.e., = 7 12'.
From Egyptian cadastre maps, which were based on the information of
"bematists" (step counters), Eratosthenes estimated the distance from Syene to
Alexandria to be 5000 stadia. With the length of an Egyptian stadium assumed as
148.5 m, the earth radius is computed to be 5909 km from Eratosthenes
estimations. This value departs from the radius of a mean spherical earth (6371
km) by only -7%. Another ancient determination of the earth's radius is attributed
to Posidonius (135 - 51 B.C.). Using the meridian arc from Alexandria to
Rhodes, he observed the star Canopus to be on the horizon at Rhodes, while at a
culmination height of 7 30' at Alexandria, which again equals the central angle
between the cities.
During the middle ages in Europe, the question of the figure of the earth was not
pursued further. Documentation from China shows that an astronomic-geodetic
survey between 17 and 40 latitude was carried out by the astronomers Nankung
Y eh and I-Hsing around 725 A.D. in order to determine the length of a meridian.
A 1 arc length was measured directly with ropes by the Arabs (~ 827 A.D.)
northwest of Bagdad by the caliphate of Al-Mam n. At the beginning of the
modern age, the French physician Fernel (1525) observed the geographical
latitudes of Paris and Amiens using a quadrant. He computed the corresponding
surface distance from the number of rotations of a wagon wheel.
Later arc measurements based on the spherical earth model benefited from
fundamental advances in instrumentation technology (1611, Kepler telescope)
and methodology. After the initial application of triangulation by Gemma Frisius
(1508 - 1555) in the Netherlands, and by Tycho Brake (1546 - 1601) in
Denmark, the Dutchman Willebrord Snellius (1580 - 1626) conducted the first
triangulation to determine the figure of the earth, HAASBROECK (1968). Snellius
used triangulation to measure the arc between Bergen op Zoom and Alkmaar
(Holland). The hitherto inaccurate arc length estimate by direct measurement was
replaced by an indirect procedure. The angles of a triangulation network were
observed with high precision instruments, and the scale was accurately
determined by precise measurement of short baselines. With proper reduction of
the observations to the meridian, the length of arc could be accurately calculated.
A direct length measurement using a chain was employed by Norwood when
determining the meridian arc between London and York (1633 - 1635).
The method of reciprocal zenith angles is yet another technique that has been
used to determine the central angle between points on a meridian arc. The Italian
fathers Grimaldi and Riccioli used this method in 1645 between Bologna and
Modena (Fig. 1.3). The central angle may be computed from the zenith angles z\
and z-i observed at locations P\ and PI according to
= +2-.
Fig 1.3. Central angle and reciprocal zenith angles
This procedure does not yield satisfactory results due to the inaccurate
determination of the curvature of light rays (refraction anomalies).
Through the initiative of the Academy of Sciences (founded in Paris, 1666),
France assumed the leading role in geodesy in the seventeenth and eighteenth
centuries. In 1669 - 1670 the French abbey J. Picard measured the meridian arc
through Paris between Malvoisine and Amiens with the aid of a triangulation
network; he was the first to use a telescope with cross hairs. The value Picard
obtained for the radius of the earth (deviation of +0.01%) aided Newton in the
verification of the law of gravitation, which he had formulated in 1665 - 1666.
In the sixteenth and seventeenth centuries, new observations and ideas from
astronomy and physics decisively influenced the perception of the figure of the
earth and its position in space. N. Copernicus (1473 - 1543) achieved the
transition from the geocentric universe of Ptolemy to a heliocentric system (1543:
"De revolutionibus orbium coelestium"), which Aristarchus of Samos (~ 310 250 B.C.) had already postulated. J. Kepler (1571 - 1630) discovered the laws of
planetary motion (1609: "Astronomia nova...", 1619: "Harmonices mundi"), and
Galileo Galilei (1564 - 1642) established the fundamentals for mechanical
dynamics (law of falling bodies and law of pendulum motion).
In 1666, the astronomer J. D. Cassini observed the flattening of the poles of
Jupiter. On an expedition to Cayenne to determine martian parallaxes (1672 1673), the astronomer J. Richer discovered that a one-second pendulum regulated
in Paris needed to be shortened in order to regain oscillations of one second.
From this observation, and on the basis of the law of pendulum motion, one can
infer an ncrease in gravity from the equator to the poles. This effect was
confirmed by E. Halley when comparing pendulum measurements in St. Helena
to those taken in London (1677 - 1678).
Building on these observations and on his theoretical work on gravitation and
hydrostatics, Isaac Newton (1643 - 1727) developed an earth model based on
physical principles: "Philosophiae Naturalis Principia Mathematica" (1687).
Based on the law of gravitation, Newton proposed a rotational ellipsoid as an
equilibrium figure for a homogeneous, fluid, rotating earth. The flattening
(a = semimajor axis, b = semiminor axis of the ellipsoid) of Newton's ellipsoid
was 1/230. He also postulated an increase in gravity acceleration from the equator
to the poles proportional to sin2 ( = geographical latitude). At the same time,
the Dutch physicist Christian Huygens (1629 - 1695), after having developed the
principle of pendulum clocks and the law of central motion, calculated an earth
model flattened at the poles ("Discours de la Cause de la Pesanteur," 1690).
Shifting the source of the earth's attractive forces to the center of the earth, he
obtained a rotationally-symmetric equilibrium-surface with a meridian curve of
fourth order and flattening of 1/576.
Arc measurements at various latitudes were required to verify the proposed
ellipsoidal earth-models. Theoretically, the length of a one-degree arc (meridian
arc for a difference of 1 in latitude), in the case of flattened poles, should
increase pole-ward from the equator. The ellipsoidal parameters a, b or a,/can be
computed from two arc measurements.
Fig. 1.4. Latitude arc measurement
l .3 Historical Development of Geodesy
We distinguish between arc measurements along an ellipsoidal meridian (latitude
arc measurement), along a parallel (longitude arc measurement), and arc
measurements oblique to the meridian.
For the computations in a latitude arc-measurement (Fig. 1.4), the
angles = 2- and ^?' = '2- [ are formed from the observed geographic
latitudes 1,2,', and '2. The corresponding meridian arcs AGand AG are
obtained from triangulation networks. For short arcs, one can replace the
meridian ellipse by the osculating circle having the meridian radius of curvature
= (^)evaluated at the mean latitude = (/2}( +2), where is also a
function of the ellipsoidal parameters a, f. From AG = MA#? and
AG' = M'A^', a and / may be determined. The larger the latitude interval
'-, the more accurate the computed flattening; whereas, the accuracy of the
semi-major axis length a depends in particular on the lengths of the meridian arcs.
For longitude arc measurements, corresponding relations are used between the
arc lengths measured along the parallels and the difference of the geographical
longitudes observed at the end points of the arcs. Arc measurements oblique to
the meridian require a proper azimuth determination for reduction to the
meridian.
Initial evaluations of the older arc measurements (Snellius, Picard, among others)
led to an earth model elongated at the poles. The same result was obtained by La
Hire, J. D. and J. Cassini. They extended the arc of Picard north to Dunkirk and
south to Collioure (1683 - 1718), with a latitude difference of 820'. The
computations from two arc segments yielded a "negative" flattening of -1/95,
which can be attributed primarily to measurement errors in the astronomic
latitudes. The intense dispute between the supporters of Newton and those of the
Cassinis over the figure of the earth was resolved by two further arc measurement
campaigns sponsored by the French Academy of Sciences.
Maupertuis and Clairaut, among others, participated in the expedition to Lapland
(1736 - 1737). The results of the Lapland arc measurement (average latitude
6620'and latitude interval 57 '.5) confirmed the polar flattening. Using the arc
measurement of the meridian through Paris (revised by Cassini de Thury and La
Caille, 1739 - 1740) the flattening was computed as 1/304. On a second
expedition (1735 - 1744) to Peru (modern day Ecuador), an arc at an average
latitude 131' S and with 37' amplitude was determined by Bouguer, La
Condamine and Godin. Combining this information with the Lapland arc led to a
flattening of 1/210. The flattening of the earth at the poles was thereby
demonstrated by geodetic measurements.
A synthesis between the physical and the geodetic evidence of the ellipsoidal
shape of the earth was finally achieved by A. C. Clairaut (1713 - 1765). The
theorem (1743), which bears his name, permits the computation of the flattening
from two gravity measurements at different latitudes, cf. [4.2.2]. A first
application of Clairaut's theorem is due to P. S. Laplace (1799), who from only 15
gravity values derived a flattening of 1/330. The wider application of this
"gravimetric method" suffered from the lack of accurate and well distributed
gravity measurements and from the difficulty of reducing the data to the earth
ellipsoid. Such problems were not overcome until the twentieth century.
With the rotational ellipsoid commonly accepted as a model for the earth,
numerous arc measurements were conducted up to the twentieth century. These
measurements generally served as a basis for geodetic surveys, see [1.3.3]. The
meridian arc through Paris was extended by Cassini de Thury and was included in
the first triangulation of France (1733 - 1750). A geodetic connection between
the astronomical observatories in Paris and Greenwich (1784 - 1787) was the
beginning of the national survey of Great Britain, with the final extension of the
Paris meridian arc to the Shetland islands. Particular significance was attained by
a measurement on the meridian through Paris, between Barcelona and Dunkirk
(1792 - 1798), commissioned by the French National Assembly and carried out
by Delambre und Mechain. The results served for the definition of the meter as a
natural unit of length (1799). These observations, combined with the Peruvian
arc measurement, yielded an ellipsoidal flattening of 1/334 .
As recognized by P. S. Laplace (1802), C. F. Gauss (1828), F. W. Bessel (1837),
and others, the assumption of an ellipsoidal-earth model is no longer tenable at a
high level of accuracy. The deviation of the physical plumb line, to which the
measurements refer, from the ellipsoidal normal then can no longer be ignored.
This deviation is known as the deflection of the vertical. While adjusting several
arc measurements for determination of the ellipsoidal parameters, contradictions
were found which greatly exceeded the observational accuracy. An initial
adjustment of arc measurements was carried out in 1806 by A. M. Legendre in his
treatise "Sur la methode des moindres carrees". C. F. Gauss was the first to adjust
a triangulation network (in and around Brunswick, 1803 - 1807) by the method of
least squares (GERARDY 1977).
This led to the improved definition of the "figure of the earth" by Gauss and
Bessel, who clearly distinguished between the physical surface of the earth, the
geoid as the mathematical surface, and the ellipsoid as a reference surface
approximating it, cf. [1.2]. With the definition of geodesy [1.1], F. R. Helmer t
made the transition to the actual concept of the figure of the earth (MORITZ 1990).
Friedrich Robert Helmert (1843 - 1917), one of the most distinguished geodesists of modern times,
was professor of geodesy at the technical university at Aachen and later director of the Prussian
Geodetic Institute in Potsdam and of the Central Bureau of the "Internationale Erdmessung".
Through his work, geodesy has experienced decisive impulses, the effects of which are still felt
today. In his fundamental monograph (1880/1884), Helmen established geodesy as a proper science
(WOLF 1993, HARNISCH and HARNISCH 1993).
Despite the discrepancies found from the adjustments, arc measurements
continued to be used to determine the dimensions of the earth ellipsoid.
However, the deflections of the vertical, being a physically phenomena and hence
having systematic characteristics, were treated as random observational errors. As
a consequence, this method provided parameters for best-fitting ellipsoids,
approximating the geoid in the area of the triangulation chains. Many of these
ellipsoids have been introduced as "conventional" ellipsoids for calculating the
national geodetic surveys, and thus arc measurements increasingly became part of
the geodetic surveys. Established by triangulation, these national surveys
provided control points for mapping, which remain the basis for many national
geodetic reference systems today (TORGE 1997). Gravity observations were
carried out at most arc measurements and at dedicated campaigns, especially after
the foundation of the "Mitteleuropische Gradmessung", cf. [1.4.2].
We mention the historically important arc of Gauss (arc measurement between
Gttingen and Altona 1821 - 1824, invention of the Heliotrope, adjustment
according to the method of least squares) and its extension to the triangulation of
the kingdom of Hannover (until 1844). Initiated by the Danish astronomer H.
Chr. Schumacher, this arc should become part of a central European meridian arc,
running from Denmark to Bavaria (triangulation by J. G. Soldner, 1808 - 1828)
and further south. Bessel and Baeyer carried out an arc measurement oblique to
the meridian in East Prussia (1831 - 1838), which connected the Russian
triangulations (W. Struve, C. Tenner) with the Prussian and Danish networks and
finally with the French-British arc along the meridian of Paris. Some long arcs
linking national triangulation-chains were built up over a 100 year period. Some
of these were not completed until 1950's, while others were never finished, owing
to the replacement of classical geodetic observation techniques by satellite
surveying methods. These long arcs include the American meridian arc (Alaska Tierra del Fuego), the North American longitude arc along the 39 parallel
between the Atlantic and the Pacific Oceans, the West European-African arc
along the meridian of Paris (Shetlands - Algeria), the Arctic Ocean to
Mediterranean Sea meridian arc (Hammerfest - Crete) and the African 30 east
meridian arc tied to it (Cairo - Cape Town), the European-Asiatic longitude arc
measurements at 48 (Brest - Astrachan) and at 52 latitude (Ireland - Ural
Mountains), as well as the latitude and longitude arc measurements in India (G.
Everest, W. Lambdon).
Since the 1880's, vertical control networks were established by geometric
leveling within the frame of the national geodetic surveys but independently from
the horizontal control systems. Heights were referred to a level surface close to
the geoid and defined by the mean sea level as observed at a tide gauge. The
geoid was not needed in this separate treatment of horizontal position and height,
but it played a major role as a geometric representation of the earth's gravity field.
An inevitable presupposition for the evaluation of large-scale arc measurements,
triangulations, and leveling was the regulation of the measuring units. It was
nearly one century after the introduction of the meter in France that a large
number of countries agreed upon a definition for the meter at the International
Meter Convention in Paris in 1875. Following a recommendation of the
"Europ ische Gradmessung" in 1883, the International Meridian Conference
(Washington, D. C, 1884) adopted the Greenwich meridian as the initial
meridian for longitude and the universal day (mean solar day) as the time unit
referenced to the meridian.
The three-dimensional concept of geodesy consists of the common treatment of
horizontal and vertical positioning within the same mathematical model. This was
suggested already by BRUNS (1878), who proposed to determine the surface of
the earth pointwise using a spatial polyhedron together with all exterior level
surfaces. However, three-dimensional computations were not carried out in
practice due to the uncertainty of trigonometric derived height differences over
large distances and due to the absence of geoid heights above the ellipsoid, which
were required for the proper treatment of geometric leveling.
The concept of three-dimensional geodesy was revived by MARUSSI (1949) and
(1969), while in 1945 Molodenski demonstrated that the physical surface
of the earth and its external gravity field can be determined from surface
measurements only, without needing the geoid, MOLODENSKI (1958).
V IS L (1946) introduced Stellar triangulation from high altitude balloons as a
first step to realize the three-dimensional concept. This technique was followed
by electromagnetic distance measurements in the 1950's and 1960's, using both
terrestrial and airborne methods. Satellite geodesy provided a technological
breakthrough after the launch of the Russian satellite Sputnik I in 1957.
Observations to orbiting satellites were used to establish control points in a threedimensional system, and provided valuable gravity field information. Beginning
in the 1980's, the NAVSTAR Global Positioning System (GPS) now dominates
geodetic measuring techniques. The practical problems of geodesy today include
the connection of classical horizontal and vertical control networks to the global
reference system established by space methods. A particular problem is the
determination of the geoid with respect to a global reference ellipsoid.
Recently, kinematic methods have gained great importance, especially with the
use of GPS. The measuring systems, or their components, are carried on moving
l .4 Organization of Geodesy, Literature
platforms (e.g., satellite, airplane, ship, car) and, by continuous positioning
(navigation), provide data referring to the geodetic reference system.
The beginning of four-dimensional geodesy may be reckoned from the detection
of polar motion by F. Kstner (1884 - 1885) and first observations of the earth
tides by E.V. Rebeur-Paschmtz (1889 - 1893). Monitoring of crustal
deformations related to seismic activities began in Japan and the U.S.A. about
100 years ago. Interest in this phenomena was motivated by powerful seismic
events, such as the San Francisco earthquake of 1906. In Fennoscandia, precise
leveling and tide gauge registrations began in the 1880's and were used to
determine the region's large-scale vertical uplift caused by postglacial rebound.
Today, the variations of the earth's rotation and the movements of the tectonic
plates are regularly observed through global networks, and a number of regional
control networks have been set up, especially at tectonic plate boundaries.
Gravity variations with time are derived from the analysis of satellite orbits
(large-scale) and terrestrial networks (small-scale). The earth tides have also
been modeled successfully using terrestrial and satellite methods.
Large efforts continue worldwide to measure and analyze all types of geodynamic
phenomena by geodetic methods, for instance that described in the NASA (1983)
Geodynamics Program, see also MUELLER and ZERBINI (1989).
In the future, geodetic observations will experience a further increase of accuracy
and an increase in resolution in space and time. Longer observation series will
permit the detection of long-term changes of the earth and its gravity field. Global
geodesy and geodetic surveys must take these temporal changes into account.
With improved modeling, the four-dimensional aspect of geodesy will
increasingly be employed in the evaluation and presentation of geodetic products
(LAMBECK 1988, BRUNNER and Rizos 1990).
The problems of global geodesy may be solved only with the international
cooperation of institutions involved in nation-wide geodetic work, together with
international geodetic services, cf. [1.4.2]. University institutes and departments
(e.g., geodesy, geophysics, astronomy, and space sciences) pursue fundamental
research. In some countries, academy or governmental institutes are also engaged
in geodetic research (China: Institute of Geodesy and Geophysics, Wuhan;
Finland: Finnish Geodetic Institute; Germany: Deutsches Geodtisches
Forschungsinstitut, Mnchen, Geoforschungszentrum Potsdam; Russia: Institute
of Physics of the Earth, Moscow). The geodetic surveys are carried out according
to the guidelines of the official survey system, either by central agencies or by
decentralized institutions (Australia: Australian Surveying and Land Information
Group; Canada: Geodetic Survey Division, National Resources Canada, China:
National Bureau of Surveying and Mapping; France: Institut Geographique
National; Germany: State geodetic surveys in cooperation with the federal
Bundesamt fr Kartographie und Geodsie BKG; Great Britain: Ordnance
Survey; India: Survey of India; Japan: Geographical Survey Institute; Russia:
Federal Service of Geodesy and Cartography; South Africa: Surveys and Land
Information; U.S.A.: National Geodetic Survey, National Oceanic and
Atmospheric Administration NOAA, (formerly U.S. Coast and Geodetic Survey).
In addition to these, a number of non-geodetic institutions, in the course of their
special tasks and projects, are also concerned with geodetic problems. These
groups develop theory, measuring systems and methods, and in particular are
involved with the collection and evaluation of geodetic data. We mention space
agencies (e.g., Goddard Space Flight Center of NASA, Greenbelt, Md.; Centre
National dEtudes Spatiales, Toulouse), geologic and Hydrographie services
(China: State Seismological Bureau; France: Bureau des Recherches
Geographiques et Minieres; Germany: Bundesanstalt fr Geowissenschaften und
Rohstoffe, Bundesamt fr Seeschiffahrt und Hydrographie; Great Britain:
Institute of Geological Sciences, Institute of Oceanographic Sciences; U.S.A.:
U.S. Geological Survey, U.S. Naval Observatory), University departments (e.g.,
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Cal.;
Lament Doherty Earth Observatory, Columbia Univ., Palisades, N.Y.) and
military agencies (e.g., U.S.A.: National Imagery and Mapping Agency NBvIA,
formerly Defense Mapping Agency). More details may be found in Journal of
Geodesy 74 (2000): 142 - 154.
At the beginning of the arc measurements in the kingdom of Hanover (1821), C.
F. Gauss had already expressed his desire for international collaboration.
According to Gauss, this geodetic network would be connected to neighboring
triangulation networks, aiming toward an eventual merger of the European
observatories. Organized international collaboration originated with the
memorandum by the Prussian general J. J. Baeyer (1794 - 1885): "ber die
Gre und Figur der Erde, eine Denkschrift zur Begrndung einer Mitteleuropischen Gradmessung" (1861). In 1862, the "Mitteleuropische Gradmessung" was
founded in Berlin as the first international scientific association of significance;
Baeyer became its first president. After expanding to the "Europische
Gradmessung" (1867) and to the "Internationale Erdmessung" ("Association
Geodesique Internationale," 1886), the association engaged in fruitful activity,
which was especially inspired by the works of Helmert as director of the Central
Bureau (LEVALLOIS 1980, TORGE 1996).
After the dissolution of the "Internationale Erdmessung" during the first World
War, the "International Union of Geodesy and Geophysics" (IUGG) was founded
in 1919. Today this organization has a membership of 75 countries. It consists of
one geodetic and six geophysical associations. The "International Association of
Geodesy" (LAG) is directed by a President who is elected every four years, and
who is assisted by a Vice President and a General Secretary. The IUGG and IAG
meet at General Assemblies at four-year intervals. In addition, numerous
symposia and scientific conferences are organized to treat special themes; among
these are the IAG Scientific Assemblies, which are held between the General
Assemblies.
The IAG consists of five sections: Positioning, Advanced Space Technology,
Determination of the Gravity Field, General Theory and Methodology,
Geodynamics. Commissions are established for continuing problems, whereas
transient problems are treated by special study groups. In addition, the LAG, in
collaboration with other scientific organizations, maintains the following
permanent services: International GPS Service (IGS) with the Central Bureau at
the Jet Propulsion Laboratory, Pasadena, California; Bureau Gravimetrique
International (BGI), Toulouse; International Geoid Service (IGeS), Milano;
International Center for Earth Tides (ICET), Brussels; International Earth
Rotation Service (IERS) with the Central Bureau at the Bundesamt fr
Kartographie und Geodsie (BKG), Frankfurt, a.M. (and until December 31,
2000: Observatoire de Paris); Permanent Service for Mean Sea Level, Bidston
Observatory, Merseyside, U.K.; Bureau International des Poids et Mesures-Time
Section, Sevres; International Laser Ranging Service (BLRS) and International
VLBI Service for Geodesy and Astrometry (IVS), since 1999. The LAG also
maintains an information and a bibliographic service and organizes summer
schools in many parts of the world (LAG 1997). A restructuring of LAG is under
discussion and may lead to major changes (SCHWARZ 2000a).
References to textbooks for geodesy and related fields (mathematics, physics,
astronomy, geophysics, surveying engineering and mapping) will be found in the
running text. A list of geodetic and geodetically relevant publication series is
given in Journal of Geodesy 74 (2000): 155 - 162.
We mention in particular the Journal of Geodesy (formerly Bulletin Geodesique
and Manuscripta Geodaetica, Springer: Berlin-Heidelberg-New York), which is
the official journal of the LAG. The results of each General Assembly of the LAG
are compiled in the Travaux (Proceedings). National reports are collected and
stored at the Central Bureau of the IAG. The proceedings of TAG symposia are
published in a separate series (Springer).
Among the recent scientific-technical journals in the field of geodesy,
geophysics, navigation, and surveying we mention in particular:
Geodaetica, Geophysica et Montanistica Hungarica (Hungary), Acta Geodaetica
et Cartographica Sinica (China), Annali di Geofisica (Italy), Artificial Satellites
(Poland), Australian Journal of Geodesy, Photogrammetry and Surveying, The
Australian Surveyor, Bolletino de Geodesia e Scienze Affini (Italy), Allgemeine
Vermessungsnachrichten (Germany), Bolletino die Geofisica Teorica ed
Applicata (Italy), EOS Transactions (American Geophysical Union), Geodesia
(The Netherlands), Geomatica (Canada), Geodeziya i Aerosyemka, Geodeziya i
Kartografiya (Russia), Geophysical Journal International (U.K.), Geophysical
Research Letters (U.S.A.,), Geophysics (U.S.A.), GPS-World (U.S.A.), GPS
Solutions (U.S.A.), Journal of Earthquake Prediction Research (China/Russia),
Journal of Geodynamics (The Netherlands), Journal of the Geodetic Society of
Japan, Journal of Geophysical Research (U.S.A.), Journal of Surveying
Engineering (U.S.A.), Marine Geodesy (U.S.A.), The Journal of Navigation
(U.S.A.), sterreichische Zeitschrift f r Vermessungswesen und Geoinformation
(Austria), Physics and Chemistry of the Earth A: Solid Earth and Geodesy (The
Netherlands), Reviews of Geophysics and Space Physics (U.S.A.), Revista
Cartografica (Mexico), Surveying and Land Information Systems (U.S.A.), Studia
Geophysica et Geodaetica (Czech Republic), Survey Review (U.K.), Surveys in
Geophysics (The Netherlands), Tectonophysics (The Netherlands), Vermessung,
Photogrammetrie und Kulturtechnik (Switzerland), Zeitschrift f r Vermessungswesen (Germany).
Technical Reports are issued by university and research institutes, as well as by
some governmental agencies. We mention here: Bull. d'Inf. Bureau
Gravimetrique International, Toulouse; Bull. d'Inf. Marees Terrestres, Brussels;
Bull. Earthquake Research Inst., Univ. of Tokyo; Bull. Geograph. Survey Inst.,
Tokyo; Geod. Geophys. Arb. in der Schweiz, Schweiz. Geod. Komm., Z rich;
Geowiss. Mittl. Studieng. Verm.wesen, TU Wien; IERS Techn. Notes, Paris;
IGS Techn. Reports JPL, Pasadena, U.S.A.; Journal of Wuhan Technical
University of Surveying and Mapping; Mitt. Bundesamt Kart. u. Geod.,
Frankfurt a.M.; Mitt. Geod. Inst. Univ. Bonn; Mitt. Geod. Inst. TU Graz; Mitt.
Inst. Geod. Photogr. Z rich; NASA Goddard Space Flight Center Rep.,
Greenbelt, Md.; Nat. Survey and Cadastre, Geod. Div. Techn. Rep.,
Copenhagen; Nederlandse Comm. voor Geodesic Publ.; Techn. Rep.,
Washington B.C.; NOAA-NOS-National Geod. Survey Techn. Rep., Rockville,
Md.; Publ./Rep. Finnish Geod. hist. Helsinki; Publ. Division of Geomatics,
Univ. of Calgary; Rep. Dep. of Geodetic Science and Surveying, The Ohio
State Univ., Columbus, Ohio; Rep. on Geodesy, Inst. of Geodesy and Geod.
Astronomy, Warsaw Univ. of Technology; Math, and Phys. Geodesy, TH Delft;
Schriftenreihe d. Institute d. Fachber. Vermessungswesen, Univ. Stuttgart;
1.4 Organization of Geodesy, Literature
Schriftenr. Univ. Studiengang Vermessungswesen, Univ. der Bundeswehr,
Mnchen; Univ. Rep. School of Geomatic Engineering, Univ. of New South
Wales, Sydney; Verff. Bayer. Komm, fr die Internationale Erdmessung,
Mnchen; Verff. Deutsche Geod. Komm., Mnchen; Wiss. Arb. Fachr.
Vennessungswesen, Univ. Hannover.
2 Reference Systems
Reference systems are introduced in order to model geodetic observations as a
function of unknown parameters of interest. The coordinate systems are defined
in terms of orientation, metrics, and curvature; they are three-dimensional in
principle (HEITZ 1988). A fourth dimension, time, enters through the mutual
motion of the earth and other celestial bodies and through the earth's
deformations. As with the earth, reference systems can be defined for the moon
and the planets in the solar system.
Basic units and fundamental constants are foundational to the geodetic
measurement and modeling process [2.1]. Time systems are based either on
processes of quantum physics or on the daily rotation of the earth [2.2 ]. Global
reference systems are realized through reference frames, e.g., the ITRF, which
is established and maintained by the International Earth Rotation Service [2.3].
We distinguish between the space-fixed celestial reference system [2.4] and the
earth-fixed terrestrial reference system [2.5] (KOVALEVSKY et al. 1989). In
addition, gravity related reference systems have to be introduced, as most
geodetic observations refer to the earth's gravity field [2.6].
Length, mass, and time are basic quantities used in geodesy. The units for these
quantities are the meter (m), the kilogram (kg), and the second (s) respectively.
They are defined through the International System of Units (Systeme
International d'Unite"s SI), established in 1960 by the llth General Conference
of Weights and Measures (CGPM) in Paris (MARKOWITZ 1973, BIPM 1991).
The definitions are as follows:
The meter is the length of the path traveled by light in vacuum during a
time interval of 1/299792458 of a second (CGPM 1983).
The kilogram is the unit of mass; it is equal to the mass of the international
prototype of the kilogram (CGPM 1901).
The second is the duration of 9192631770 periods of the radiation
corresponding to the transition between the two hyperfine levels of the
ground state of the cesium-133 atom (CGPM 1967).
The establishment and maintenance of the reference standards for these units is
the task of the Bureau International des Poids et Mesures (BIPM), located in
Sevres, France. BIPM cooperates with the national laboratories of standards
under the guidelines of the International Meter Convention (1875). These
national laboratories include the National Institute of Standards and
2.1 Basic Units and Fundamental Constants
Technology, Gaithersburg, Md., U.S.A.; the National Physical Laboratory,
Teddington, U.K.; and the Physikalisch-Technische Bundesanstalt,
Braunschweig, Germany.
The realization of the meter is based on interferometric measurements (relative
uncertainty 10"12) using light with highly stable frequencies (stabilized lasers).
The international kilogram prototype has been kept in BIPM since 1889;
national prototypes are related to it with an uncertainty of 10"9. The BIPM
Time Section (until 1987: Bureau International de l'Heure BIH, Paris) defines
the second (relative uncertainty 10~14) and the atomic time scale, cf. [2.2.1].
Previous definitions of the meter and the second were based on natural measures. The meter was
intended to be one ten-millionth part of the meridian quadrant passing through Paris. Its length
was derived from an arc measurement, cf. [1.3.2], and realized in 1799 by a prototype meter bar
called "metre des archives" (legal meter). Following the International Meter Convention, a more
stable version (platinium-iridium bar) was manufactured (international meter). It has been
preserved since 1889 at the BIPM. This definition (uncertainty 10"7 ) was valid until 1960 when,
for the first time, the wavelength of a certain spectral line of light became the defining quantity.
Since ancient times, the natural measure for time has been the daily rotation of the earth about its
axis. The mean solar day, cf. [2.2.2], was determined by astronomic observations, and the second
was defined as 1/86400 part of that day. From the 1930's on, it became obvious that this
definition was uncertain by about 10"7 due to the irregularities of the earth's rotation, cf. [2.5.2].
As a supplementary SI unit, the radian (rad) is used for plane angles :
The radian is the plane angle between two radii of a circle subtended by an
arc on the circumference having a length equal to the radius.
Geodesy, astronomy, and geography also use the sexagesimal graduation with
1 full circle = 360 (degrees), 1 = 60' (minutes), and = 60" (seconds, also
arcsec). With 2 rad corresponding to 360, an angle a is transformed from
radian to degree by
= p()arad, = 180/.
Among the fundamental constants used in geodetic models is the velocity of
light in a vacuum, which is by definition (1983)
c = 299 792 458ms-1,
and the gravitational constant (CODATA system of physical constants 1986),
which is defined as
G = (6.672 5910.000 85)xlO-11m3 kg'1 s'2.
Cavendish carried out the first experimental determination of G in 1798 with a torsion balance.
Current work concentrates on increasing the relative accuracy of G to better than 10"4. This
includes investigations into dependence of G on material, external influences, distance and
direction, as well as non-inverse-square properties of gravitation (GILLIES 1987, FISCHBACH and
TALMADGE 1999).
Other units and constants used in geodesy, astronomy, and geophysics will be
introduced in the corresponding chapters, see also AHRENS (1995), BURA
(1995), GROTEN (2000).
Time plays a fundamental role in geodesy. This is due to the fact that most
measurement methods use the signal travel-time of electromagnetic waves for
positioning, and that a uniform time scale is also needed in order to model the
motion of artificial satellites. On the other hand, a time system is required for
describing the relative motion of the earth in the solar system with respect to
inertial space and for describing earth deformations due to internal and external
forces.
Time systems are defined by the unit for a time interval and by a time epoch.
They are based either on the definition of the SI second [2.2.1] or on the diurnal
rotation of the earth about its axis [2.2.2]. Fundamental descriptions of time
systems are found in MUELLER (1969), MORITZ and MUELLER (1987),
SEIDELMANN (1992).
A uniform time-scale of high accuracy is provided by the International Atomic
Time (Temps Atomique International: TAI). It corresponds to the definition of
the SI second, cf. [2.1], which has been made approximately equal to the
second of the formerly used ephemeris-time. The latter was defined by the
motion of the earth about the sun and determined through long-term astronomic
observations. The origin of TAI was chosen so that its epoch (January 1, 1958,
0 h) coincided with the corresponding epoch of Universal Time UT1, cf.
[2.2.2]. The TAI day comprises 86400s, and the Julian Century has 36525
TAI days.
TAI is realized by a large set (more than 200) of atomic clocks (mostly cesium
beam frequency standards providing long-term stability and a few hydrogen
2.2 Time Systems
masers) maintained at about 60 laboratories around the world. Clock
comparisons are performed at a number of timing centers, employing GPS
observations for time links, cf. [5.2.5]. From these local determinations, a
weighted mean is calculated at the BIPM Time Section. The relative frequencystability of TAI is between a few 10~l5(over minutes to days) and 10 (over
years). Due to relativistic effects, the readings of the atomic clocks are reduced
to a common height reference (SI second "on the geoid").
The motions of celestial bodies and artificial satellites have to be described by
a strictly uniform time scale (inertial time). This is provided by a dynamical
time, which is based on motions of bodies in the solar system. Dynamical time
scales refer either to the barycenter of the solar system (Barycentric Dynamic
Time TDB) or to the geocenter (Terrestrial Time TT). The TT unit is
practically equivalent to TAI, with a constant difference resulting from the
epoch definition of TAI:
TT = TAI+ 32.184s.
Dynamical time is used in celestial mechanics with Newton's equations of
motion, e.g., as an argument for the astronomical ephemeris of the moon and
the sun.
The diurnal rotation of the earth provides a natural measure for time.
Corresponding time systems are introduced in order to relate earth-based
observations to a space-fixed system: Sidereal and Universal (solar) Time.
Hereby, two periodic motions of the earth play a role (Fig. 2.1):
ECLIPTIC
EQUATORIAL
Fig. 2.1 Earth rotation, equatorial plane, and ecliptic plane
the diurnal rotation (spin) of the earth about its polar axis. This rotational
axis approximately coincides with the axis of maximum moment of inertia,
and it passes through the earth's center of mass, cf. [2.5.2]. The equatorial
plane is perpendicular to the axis of rotation,
the annual revolution of the earth around the sun. Following Kepler's laws,
the earth describes an ellipse with the sun at one of its focal points. Minor
perturbations arise due to the gravitation of the moon and other planets.
The plane of the earth's orbit is called the ecliptic plane; it has an obliquity
of about 23.5 with respect to the equatorial plane.
By circumscribing the unit sphere around the center of the earth, simple
geometric relations are obtained. The celestial equator and the ecliptic are
defined by the intersections of the sphere with the corresponding planes. The
vernal equinox (also first point of Aries) is the intersection of the ecliptic and
the equator where the sun passes from the southern to the northern hemisphere.
Sidereal time is directly related to the rotation of the earth. Local Apparent (or
true) Sidereal Time (LAST) refers to the observer's (local) meridian; it is equal
to the hour angle of the (true) vernal equinox (Fig. 2.2), cf. [2.4.1]. The vernal
equinox is affected by precession and nutation and experiences long and shortperiodic variations, cf. [2.4.2]. If nutation is taken into account, we obtain
Local Mean Sidereal Time (LMST), referring to the mean vernal equinox. For
the Greenwich meridian the corresponding hour angles are called Greenwich
Apparent Sidereal Time (GAST) and Greenwich Mean Sidereal Time (GMST).
The astronomic longitude is the angle between the meridian planes of the
observer and Greenwich. It is given by, cf. [2.6.2]
Fig. 2.2. Rectascension, sidereal time, hour angle, and longitude
A = LAST-GAST = LMST-GMST.
LAST is determined from astronomical observations to fixed stars and
extragalactic radio sources. The mean sidereal time scale is still affected by
precession (long-periodic). The mean sidereal day is the fundamental unit; it
corresponds to the time interval of two consecutive transits of the mean vernal
equinox through the meridian.
For practical reasons, solar time is used in everyday life. It is related to the
apparent diurnal motion of the sun about the earth. Since this revolution is not
uniform, a "mean" sun is introduced which moves with constant velocity along
the equator and coincides with the true sun at the vernal equinox. Mean solar
time is equal to the hour angle of the mean sun plus 12 hours. If referred to the
Greenwich mean astronomical meridian, cf. [2.5.1], it is termed Universal Time
(UT). Its fundamental unit is the mean solar day, being the interval between
two transits of the fictitious sun through the meridian.
The conversion of Universal Time to Greenwich Mean Sidereal Time is
rigorously possible and is given by a series development with time defined by
the International Astronomical Union (MORITZ and MUELLER 1987). Since the
orbital motion of the earth is about 1 per day (360/365d), the year has one
day more in sidereal days than in solar days. We have the following
approximation:
1 mean sidereal day = 1 mean solar day - 3m55.90s = 86164.10s . (2.6)
The earth's rotation rate is 15.041 0771 s , and its angular velocity is
= 2^/86 164. 10s =7.2921 15 xlO' 5 rads'.
Universal time is obtained from a network of stations operating within the
frame of the International Earth Rotation Service, cf. [2.3]. The observed local
time UTO refers to the instantaneous rotation axis, which is affected by polar
motion, cf. [2.5.2]. In order to compare the results of different stations,
reductions to a Conventional Terrestrial Pole are applied. The reduction in
astronomic longitude /. corresponds to a change in time, cf. [5.3.3]. It
transforms UTO to UT1, which refers to the conventional terrestrial system, cf.
[2.5.3]:
UT^UTO + ,.
The precision of UT1 is about 0.01 to 0.02 ms at a Id resolution.
UT1, as well as Greenwich Mean Sidereal Time, still contains the variations of the earth's rotation
with time, which are of secular, periodic, and irregular character, cf. [2.5.2]. An approximation to
a uniform time scale can be achieved by modeling the seasonal variations of annual and
semiannual type. With the corresponding reduction 5, we obtain
UT2 = UTO + , + M.s .
A practical time scale, as needed in navigation for instance, has to provide a
uniform unit of time and maintain a close relationship with UT1. This led to the
introduction of the Coordinated Universal Time (UTC). Its time interval
corresponds to atomic time TAI, cf. [2.2.1], and its epoch differs by not more
than 0.9s from UT1. In order to keep the difference
DUTl|=|UTl-UTC|<0.9s,
"leap seconds" are introduced to UTC when necessary. UTC is provided by the
BIPM Time Section and broadcasted by time signal stations, while DUT1 is
calculated by the BERS, cf. [2.3].
Among the continuously broadcasting time stations are DCF77/Mainflingen (77.5 kHz),
HBG/Prangins (75 kHz); MSF/Rugby (60 kHz) in Europe; WWV resp. WWVB/Ft. Collins,
Colorado (2500 to 20000 kHz resp. 60 kHz); and WWVH/Kauai, Hawaii (2500 to 15000 kHz).
The International Earth Rotation Service (IERS) is in charge of providing and
maintaining conventional celestial and terrestrial reference frames. These
frames are a realization of the reference systems recommended by the
International Astronomical Union (IAU) and the International Union of
Geodesy and Geophysics (IUGG). IERS is also responsible for the
determination of the orientation parameters of the earth as functions of time,
which relate the two frames to each other (SEIDELMANN 1992, REIGBER and
FEISSEL 1997).
Established by the IAU and IUGG, the IERS has operated since January 1,
1988. It collects, analyzes, and models observations of a global network of
astronomic and geodetic stations (about 300 sites in 1996), operating either
permanently or for a certain time span. Observation techniques include Very
Long Baseline Interferometry (VLBI), Lunar Laser Ranging (LLR), Global
Positioning System (GPS), Satellite Laser Ranging (SLR), and DORIS
(Doppler Orbit determination and Radio positioning Integrated on Satellite), cf.
[5.2] to [5.3].
2.4 Celestial Reference System
The different types of observations are evaluated at the respective IERS
coordinating centers and then combined by an adjustment at the IERS Central
Bureau. The results include the positions (coordinates) of both the extragalactic
radio sources and the terrestrial stations, the earth orientation parameters
(EOF), and other information. With respect to the EOF, VLBI provides
information about precession, nutation, polar motion, and UT1, cf. [2.4.2],
[2.2.2]. Satellite techniques contribute to the daily interpolation of UT and to
the determination of polar motion, cf. [2.5.2]. The results are disseminated
through bulletins, annual reports, and technical notes. The combined solutions
have an accuracy of 0.0003" for EOF and 0.01 m for the positions of the
terrestrial stations, cf. [2.4], [2.5]. The evaluation of the observations is based
on the IERS Conventions, which are consistent with the IAU and IUGG/IAG
recommendations for reference systems (MCCARTHY 1996), cf. [2.4.2], [4.3].
Among the early international agreements on positioning and time were the introduction of the
Greenwich zero meridian and Universal Time (1884). The first fundamental catalogue of selected
stars was published in the 1880's, which started a series of star catalogues providing the positions
of fixed stars.
International activities of monitoring the earth's rotation date back to 1899, when the
International Latitude Service (ILS) started to determine polar motion through latitude
observations at five observatories located around the globe on the 3908' northern parallel. After
extension to the International Polar Motion Service (IPMS), and in cooperation with the Bureau
International de l'Heure (BIH) established in 1912, about 50 astronomical observatories
contributed to the determination of polar motion and time. An accuracy of 0.02" resp. 1 ms
was reached for mean values over 5 days. IPMS and the earth rotation section of BIH have been
replaced by IERS, while the BIH activities on time are continued at the BIPM, cf. [2.2.1].
An inertial system is needed in order to describe the motions of the earth and
other celestial bodies in space, including those of artificial satellites. Such a
system is characterized by Newton's laws of motion; it is either at rest or in the
state of a uniform rectilinear motion without rotation. A space-fixed system
(celestial reference system) represents an approximation to an inertial system
and can be defined by appropriate conventions: Conventional Inertial System
(CIS). The coordinate frame for such a system is provided by spherical
astronomy [2.4.1]. The spatial orientation of this frame varies with time, and
therefore, modeling of the variations is required [2.4.2]. The International
Celestial Reference Frame represents the realization of the celestial reference
system [2.4.3], KOVALEVSKY et al. (1989), SEIDELMANN (1992).
The coordinates of the celestial reference-system are defined by the equatorial
system of spherical astronomy (MUELLER 1969, EICHHORN 1974). We
introduce a three-dimensional Cartesian coordinate system with the origin at
the center of mass of the earth (geocenter). The Z-axis coincides with the
rotational axis of the earth. The X and K-axes span the equatorial plane, with
the -axis pointing to the vernal equinox and the 7-axis forming a right-handed
system (Fig. 2.3), cf. [2.2.2].
PLANE OF THE
HOUR CIRCLE
Fig. 2.3. Astronomic equatorial system
In the sequel, we shall also shift the origin of this system to the position of an observer on the
earth (topocenter) or to the barycenter of the solar system. The directions to celestial bodies then
vary with different definitions of the origin (parallaxes), cf. [5.3.3]. Since the earth's radius is
negligibly small compared to the distances to stars and extragalactic radio sources, no distinction
is necessary between a topocentric and a geocentric system.
We circumscribe the unit sphere (celestial sphere) about the earth. The
rotational axis meets the sphere at the celestial north and south poles PN and PS.
The great circles perpendicular to the celestial equator, which contain the
celestial poles, are called hour circles, and the small circles parallel to the
equator are termed celestial parallels.
The right ascension a is the angle measured in the plane of the equator
between the planes of the hour circles passing through the vernal equinox and
the celestial body S; it is reckoned from the vernal equinox anticlockwise. The
declination is the angle measured in the plane of the hour circle between the
equatorial plane and the line OS (positive from the equator to PN and negative
to PS).
The position of a celestial body S can be described either by the Cartesian
coordinates ,,, or by the spherical coordinates a,S,r (r = distance from the
origin 0). We have the transformation
(JT
'cosacosc^
r = = r sin a cos <5
In geodesy, only directions are important for stars and extragalactic sources.
With r = 1, a and describe the position of S on the unit sphere. They can also
be expressed by the lengths of the corresponding arcs on the equator and the
hour circle.
ZENITH Z
VERTICALCIRCLE
.CELESTIAL M E R I D I A N
CELESTIAL EQUATOR
N O R T H POINT
CELESTIAL H O R I Z O N
H O U R CIRCLE
VERNAL E Q U I N O X
Fig. 2.4. Astronomic equatorial and horizon system
We introduce the local meridian plane of the observer, spanned by the local
vertical (direction of the plumb line) and the rotational axis, after a parallel
shift from the geocenter to the topocenter. The zenithal point is the
intersection of the vertical with the unit sphere, and the celestial meridian is the
great circle through and the poles (Fig. 2.4). The hour angle h is measured in
the equatorial plane between the celestial meridian through Z and the hour
circle of 5, reckoned from the upper meridian toward west. Because of the
earth's rotation, the hour angle system (h,) depends on time. The ,-system is
rotated, with respect to the a,<5-system, about the polar axis by the angle of
sidereal time LAST, cf. [2.2.2]. We have the relation (Fig. 2.2)
which is used with time determination, cf. [5.3.2].
The earth's axis of rotation, which has been introduced as the Z-axis, changes
its spatial orientation with time. As a consequence, the position (, ) of a
celestial body varies, with a superposition of long and short-periodic effects
(MORITZ and MUELLER 1987, SEIDELMANN 1992, DICKEY 1995).
(25800 a)
.9.2"
NUTATION
- ^ SPIN AXIS
+ NUTATION
Fig. 2.5. Precession and nutation
The lunisolar precession is a long-periodic effect caused by the gravitation of
the moon and the sun on the equatorial bulge of the earth. This creates a force
couple (torque) which tends to turn the equatorial plane into the plane of the
ecliptic (Fig. 2.5). In combination with the moment of the earth's rotation, the
earth's axis describes a gyration of a cone with a generating angle of 23.5
(corresponding to the obliquity of the ecliptic ), about the northern pole of the
ecliptic EN. The vernal equinox moves clockwise along the ecliptic at a rate of
50.3 "/year, making a complete revolution in about 25800 years. The gravitation
of the planets causes a slow dislocation of the earth's orbit and thereby an
additional migration of the vernal equinox along the equator and a change in .
planetary precession. The sum of the lunisolar and the planetary precession is
termed general precession.
The precession is superimposed by short-periodic effects known as nutation,
which has periods between 5 days and 18.6 years. These periods are mainly due
to the time variations of the inclination of the moon's orbit with respect to the
ecliptic (appr. 5). Other components have semiannual and semimonthly
periods and stem from the oscillations of the sun and moon between the earth's
northern and southern hemisphere.
Precession and nutation can be modeled as a function of time using the
ephemerides of the moon, the sun, and the planets. The IAU (1976) theory of
precession provides three time-dependent Eulerian rotation angles for reducing
the positions of celestial bodies to a common reference. For the reference epoch
J 2000.0 (Julian epoch January 1, 2000, 12h TDB, cf. [2.2.1]), we have the
fundamental constants "general precession in longitude at the ecliptic"
(5029.0965 "/century) and "obliquity of the ecliptic" (232621.412").
The JAU (1980) theory of nutation describes this effect by a rotation about the
cone of precession. The deviation of the true pole from the mean pole is
modeled by two time-dependent parameters. Hereby, the earth is regarded as an
elliptical, rotating, elastic, and ocean-free body with solid inner and liquid outer
cores (WAHR 1981, SEIDELMANN 1992). For the epoch J 2000.0 the constant of
nutation is 9.2025".
The IAU models for precession and nutation define the reference pole for the
international celestial reference frame (Celestial Ephemeris Pole CEP). CEP is
free of diurnal or quasidiumal nutation terms (amplitudes < 0.001") with
respect to the space- or earth-fixed coordinate systems. It is also referred to as
the pole of the instantaneous equatorial system, cf. [2.4.3].
The IAU models for precession and nutation provide a precision of 0.001" at 5
to 7 days resolution. An improved theory has been developed at the IERS based
on recent VLBI and LLR data. Larger offsets (< 0.02") of the celestial pole
from CEP have been found; these are published regularly by IERS (MCCARTHY
1996). GPS also contributes to the determination of the short-periodic nutation
terms (ROTHACHER et al. 1999).
According to an IAU recommendation, the IAU (1976/1980) models for precession and nutation
shall be replaced by January 1, 2003. The new model is the IAU 2000A (precision 0.0002"), as
published in the IERS conventions 2000. CEP will then be substituted by the Celestial
Intermediate Pole (CIP), as defined by the model for periods greater than two days together with
additional time-dependent corrections provided by IERS.
The instantaneous position of a celestial body, cf. [2.4.1], is called true position
at the epoch t. By accounting for nutation, we obtain the mean position at epoch
t, which refers to the mean celestial equator and the mean vernal equinox, cf.
[2.2.2]. If precession is also taken into account, we get the mean position at the
reference epoch J 2000.0.
The International Celestial Reference System (ICRS), as recommended by IAU,
is based on the general theory of relativity, with the time coordinate defined by
the international atomic time (ARIAS et al. 1995, MA and FEISSEL 1997). ICRS
approximates a space-fixed conventional inertial-system (CIS) with the origin
at the barycenter of the solar system. It is assumed that no global rotation of the
system exists. This implies that the defining sources are either free from proper
motion (component of spatial motion tangent to the celestial sphere) or that this
motion can be modeled. The coordinate axes are defined by the celestial
reference pole and the vernal equinox as provided by the IAU models for
precession and nutation. They are realized through mean directions to
extraterrestrial fiducial objects: stellar or radio source CIS (MUELLER 1988),
cf. [2.4.2].
The stellar system is based on the stars of the Fundamental Catalogue FK5
(FRICKE et al. 1988). It provides the mean positions (,) and the proper
motions (generally < 1 "/year) of 1535 fundamental stars for the epoch J 2000.0,
with precisions of 0.01...0.03" and 0.05"/century respectively. A supplement
to FK5 contains additional stars up to an apparent magnitude of 9.5. The mean
equator and the mean vernal equinox for J 2000.0 are realized by the FK5
catalogue, with an accuracy of 0.05". Due to refraction uncertainties, earthbased astrometry can hardly improve this accuracy.
Astronomic space missions have significantly improved the realization of a
stellar CIS. The fflPPARCOS astrometry satellite (ESA, 1989 - 1993) was
used to construct a network by measuring large angles between about 100000
stars (up to an apparent magnitude of 9) covering the entire sky. The reference
frame thus established provides an accuracy of 0.001" and 0.0005 "/year for
proper motion (HlPPARCOS 1995, KOVALEVSKY et al. 1997). From improved
FK5 data and HlPPARCOS results, an FK6 catalogue has been developed for a
small number of stars (340 "astrometrically excellent"), resulting in an
improvement of proper motion as compared to the HlPPARCOS catalogue
(WiELEN et al. 1999). Future astrometric space missions will employ optical
interferometry and thus increase the positional accuracy to 0.00001"
(BROSCHE and DICK 1996).
The radio source system is based on extragalactic radio sources (quasars and
other compact sources). It was adopted as ICRS by the IAU in 1997 and has
superseded the previous stellar system (FK5) since 1998. Due to the large
distances (> 1.5 billion light years), these sources do not show a measurable
proper motion. The system is realized through the International Celestial
Reference Frame (ICRF), established and maintained by IERS (MCCARTHY
1996, MA et al. 1998). ICRF contains the coordinates (equatorial system, epoch
J 2000.0) of more than 600 objects. About 200 of them are well observed
2.5 Terrestrial Reference System
Fig. 2.6. International Celestial Reference Frame (ITRF), from ffiRS (1995):
Missions and goals for 2000
"defining sources," and 100 more are used for densification and connection to
the stellar-fixed reference system (Fig. 2.6). The southern sky is not as well
covered, as the telescopes are concentrated in the northern hemisphere. The
coordinates of the radio sources are determined by radio astronomy, with a
precision of better than 0.001" on the average and 0.0003" for the most
precisely observed objects (MA and FEISSEL 1997, BROSCHE und SCHUH
The link between the stellar and the radio source CIS is given with an accuracy of 0.05...0.1",
which corresponds to the uncertainty of FK5. This connection will be improved by the results of
the astrometric space missions (optical signals from a limited number of radio sources) to 0.001"
or better for the epoch of observation.
An earth-fixed reference system is introduced for positioning and navigation on
and close to the earth's surface and for describing the earth's gravity field as
well as other physical parameters. It is defined by a three-dimensional
geocentric coordinate system [2.5.1]. The orientation of this system changes
with time and with respect to the solid earth's body as well as to the celestial
reference system [2.5.2]. The system is realized by the IERS International
Terrestrial Reference Frame, which includes its relation to the International
Celestial Reference Frame [2.5.3].
An earth-fixed (i.e., rotating with the earth) system of spatial Cartesiancoordinates ,, is used as the fundamental terrestrial coordinate system (Fig.
2.7). Its origin is at the earth's center of mass (geocenter), being defined for the
whole earth including hydrosphere and atmosphere. The Z-axis is directed
towards a conventional "mean" terrestrial (north) pole. The "mean" equatorial
plane is perpendicular to it and contains the X and 7-axes. A "mean" rotational
axis and equatorial plane has to be introduced because the rotation of the earth
changes with respect to the earth's body over time, cf. [2.5.3]. The AZ-plane is
generated by the conventional "mean" meridian plane of Greenwich, which is
spanned by the mean axis of rotation and the Greenwich zero meridian, to
which Universal Time refers, cf. [2.2.2]. The and Ajaxes are realized
indirectly through the coordinates of terrestrial "fiducial" stations, cf. [2.5.3].
The y-axis is directed so as to obtain a right-handed system.
MEAN MERIDIAN
PLANE OF
D^-, AYIC
MEAN ROTATIONAL AXIS
" A. /
UNIT SPHERE
MEAN EQUATORIAL
Fig. 2.7. Earth-fixed geocentric Cartesian system
The instantaneous axis of rotation is the common starting point for defining the Z-axes of the
space-fixed and the earth-fixed reference systems. By referring to a reference epoch, the
dependence on time is modeled, cf.[2.4.2], [2.5.2]. The directions of the Jf-axes of both systems
differ by the angle of Greenwich mean sidereal time GMST, cf. [2.2.2].
In order to describe analytically certain physical properties of the earth (gravity
field, magnetic field, topography, etc.), spherical coordinates ,, are
employed. Here r = radial distance from the geocenter, $ = polar distance
(colatitude), and = geocentric longitude. Instead of &, the geocentric latitude
the coordinates of stations on the earth's surface and the gravity vector. Polar
motion consists of several components:
A free oscillation with a period of about 435 days (Chandler period), with
an amplitude of 0.1" to 0.2", in a counterclockwise sense as viewed from
the north pole. The Chandler wobble is due to the fact that the spin axis of
the earth does not coincide exactly with a principal axis of inertia.
For a rigid earth, this would lead to a gyration of the rotational axis about the principal axis
of inertia with a period of /((7-) = 305 days (Euler period). Here C is the earth's polar
moment of inertia, and A = B is the mean equatorial moment (rotational symmetry assumed).
The difference between the Chandler and the Euler period results from the non-rigidity of
the earth. This consideration neglects the fact that there is a small deviation between the axis
of rotation and the axis of angular momentum, which is invariable in space. However, the
deviation is less than 0.001" with periods < 1 d.
The Chandler wobble is superposed by an annual oscillation forced by
seasonal displacements of air and water masses. It proceeds in the same
direction as the Chandler wobble with amplitudes of 0.05 " to 0.1".
A secular motion of the pole has been observed for more than 100 years.
The motion consists of an irregular drift of about 0.003 "/year in the
direction of the 80 W meridian. Secular motion is mainly due to the
melting of the polar ice and to large-scale tectonic movements; it attains
large amounts over geological epochs, polar wander.
More irregular variations occur at time scales from a few days to years
with amplitudes up to 0.02". They originate primarily from mass
redistributions within the atmosphere, but variations due to ocean volume
changes, ground water variations, and earthquakes also occur.
Fig. 2.9. Polar motion 1996 - 1999, and mean pole displacement 1890 - 1999,
from DERS Annual Report 1998
The superposition of these components results in a slightly perturbed spiral like
curve of the instantaneous pole with a slowly advancing mean position (Fig.
2.9). Over one year, the deviations from the mean position remain < 0.3",
corresponding to 9 m on the earth's surface.
The reference for describing the actual position of the pole with respect to the
solid earth is provided by the IERS reference pole. It agrees within 0.03" with
the Conventional International Origin (CIO), which was defined by the mean
position of the north pole as determined between 1900.0 and 1906.0. The
position of the instantaneous pole (Celestial Ephemeris Pole, cf. [2.4.2]) with
respect to the reference pole is given by the rectangular coordinates jc/>, yp,
which are defined in the plane tangential to the pole. The jc/>-axis is in the
direction of the Greenwich mean meridian (consistent with the previous BIH
zero meridian), and the y/>-axis is directed along the 90W meridian. These
plane coordinates are usually expressed as spherical distances (in units of
arcsec) on the unit sphere.
The angular velocity (O of the earth's rotation, as monitored from the earth,
changes with time. Relative changes may reach several 10"8, which
corresponds to several ms for one day. The variations are generally described
by the excess revolution time with respect to 86 400 s and then called Length
Of Day (LOD). They are derived by comparing astronomical time
determinations, which deliver Universal Time UT1, with the uniform time
scales TAI or UTC, cf. [2.2.2].
U T 1 - T A I (seconds)
Fig. 2.10. Difference between atomic time scale TAI and Universal time UT1
(1962-1998) and length of day (LOD 1979-1987), from IERS Inform. 1998
The following components of LOD variations have been observed (Fig. 2.10):
A secular decrease in the angular velocity of the earth's rotation is caused
mainly by tidal friction. It lengthens the day by about 2 ms/century
(STEPHENSON and MORRISON 1994).
Fluctuations over decades are due to motions in the earth's liquid core and
to slow climatic variations.
The tides of the solid earth and the oceans produce variations of about 1 ms
with long (annually) and short (monthly and less) periodic parts.
Seasonal effects are explained by atmospheric excitation, with
contributions from water and ice budget variations.
More irregular oscillations stem from different sources, such as terrestrial
mass displacements (earthquakes), solar activity, and atmospheric events,
e.g., El Nino.
While the effect of polar motion on observations is dependent on location,
LOD changes act uniformly on all points. The pole coordinates and LOD, as
well as , are provided as Earth Orientation Parameters (EOF) by the IERS
with daily resolution and accuracy of 0.0003" resp. 0.02 ms or better
(REIGBER and FEISSEL 1997).
The position of the geocenter (origin of the terrestrial reference system)
changes slightly in time with respect to the monitoring observatories. Annual
and semiannual variations have been found, with amplitudes of several
mm/year, from the analysis of satellite orbits. The variations are caused
primarily by mass redistributions in the atmosphere and the oceans and by
continental water variations. Through the coordinates of the ITRF stations, cf.
[2.5.3], the geocenter is given with an accuracy of a few mm (DONG et al. 1997,
RAY 1999).
The International Terrestrial Reference System is realized by the IERS through
a global set of space geodetic observing sites. The geocentric Cartesian
coordinates and velocities of the observing sites comprise the International
Terrestrial Reference Frame (ITRF). The stations participating in the ITRF
carry out observations either continuously or at certain time intervals (Fig.
2.11). Observations are made on twelve of the larger tectonic plates, which
permits the derivation of station velocities related to plate tectonics, cf. [8.2.3].
Annual realizations of the ITRF are published by the IERS. The ITRF97 is
comprised of the geocentric positions (,,) for more than 550 stations at
about 320 sites and corresponding site velocities (BOUCHER et al. 1999). The
accuracy of the results depends on the observation techniques and is maximum
for VLBI, SLR, and GPS observations (0.5...2 cm and +1...3 mm/year resp.).
Several time variable effects are taken into account, including displacements
due to the solid earth tides, ocean and atmospheric loading effects, and
postglacial rebound, cf. [8.2.2]. The ITRF solutions satisfy the condition of no
residual net-rotation relative to the plate tectonics model NNR-NUVEL1A;
vertical movements are not allowed at all, cf. [8.2.3]. The orientation of the
180* 210' 240' 270 300' 330'
9CT 120 150' 180'
.^ . ?/: , $&&**
$>. ^
7Vy
; ^>
-30'
Collocated techniques -> 49
F/g. 2.1 L International Terrestrial Reference Frame (ITRF) sites 1997, from
BOUCHER et al. (1999)
ITRF is given with respect to the IERS reference pole and reference meridian,
cf. [2.5.2]. The actual (time t) position vector r of a point on the earth's surface
is derived from its position at the reference epoch (to) by
r(f) = r 0 +r c ('-0
where r0 and r0 are the position and velocity respectively at to.
The relation between the celestial (ICRS) and the terrestrial (ITRS) reference
systems is given by spatial rotations, which depend on the earth rotation
parameters introduced in [2.2.2], [2.4.2], [2.5.2], SEEBER (1993), MCCARTHY
(1996), RICHTER, Bu. (1995). The complete transformation from the celestial
to the terrestrial system reads as
r(lTRS) = R2 (-,), (~yf)R, (GAST)N(f)P(/)r(lCRS).
The position vector as given in the ICRS is first transformed by the precession
matrix P(/) from the reference epoch t0 (J 2000.0) to the observation epoch /.
The nutation matrix N(f) then transforms from the mean to the instantaneous
true equator and vernal equinox. The Eulerian angles in these two rotation
matrices are given in the models for precession and nutation, cf. [2.4.2]. The
apparent Greenwich sidereal time GAST, cf. [2.2.2], is used to rotate the
system about the Z-axis:
cos(GAST) sin (GAST) 0
R3(GAST) = -sin (GAST) cos (GAST) 0
with GAST calculated from UT1. Finally, counterclockwise rotations about the
Xand 7-axes are computed as functions of the pole coordinates XP and (small
angles), cf. [2.5.2] .
0 y,
-yf
2 (-*!) =
-xf
0 xp]
Equations (2.17) and (2.18) provide the transformation from the instantaneous
space-fixed system to the conventional terrestrial system.
Most geodetic and astronomic observations on or close to the earth's surface
refer to the earth's gravity field by orientation along the local vertical.
Consequently, local gravity-field-related reference systems are introduced for
the modeling of these observations. The orientation of the local systems with
respect to the global reference system is given by astronomic latitude and
longitude [2.6.1]. These orientation parameters are used for transformation
from the local systems into the global system and back [2.6.2].
The direction of the plumb line (local vertical) with respect to the global
geocentric system is given by two angles (Fig. 2.12). The astronomic
(geographic) latitude is the angle measured in the plane of the meridian
between the equatorial plane and the local vertical through the point P. It is
reckoned positive from the equator northward and negative to the south. The
angle measured in the equatorial plane between the Greenwich meridian plane
and the plane of the meridian passing through is the astronomic (geographic)
longitude A; it is reckoned positive toward the east. The gravity potential W
locates P in the system of level surfaces W = const., cf. [3.2.1]. The local
astronomic meridian plane is spanned by the local vertical at P and a line
parallel to the rotational axis, cf. [2.4.1].
2.6 Gravity Field Related Reference Systems
LOCAL ASTRONOMIC
MERIDIAN PLANE
* V
Fig. 2.12. Astronomic latitude and longitude
We introduce the outer surface normal n (unit vector), which is normal to the
level surface W=WP and passes through P. It is directed to the zenith, which is
opposite of the direction of the gravity vector g. From Fig. 2.12, we see that
cos sin A
Latitude and longitude can be determined by the methods of geodetic astronomy, cf. [5.3].
Together with the potential W, they form a triple of three-dimensional coordinates defined in the
gravity field, cf. [3.2.3].
Geodetic and astronomic observations are tied to the direction of the plumb line
at the point of observation and thereby to the earth's gravity field. An exception
is distance measurements, which are independent of the reference system.
Thus, these observations establish local gravity-field related systems: Local
astronomic systems (Fig. 2.13). Their origin is at the point of observation P.
The z-axis coincides with the local vertical and points toward the zenith. The xaxis (north) and the >>-axis (east) span the horizontal plane, which is tangent to
the level surface W- Wp. This ,^-system is left-handed.
Observable geometric quantities include astronomic azimuths, horizontal
directions and angles, zenith angles, spatial distances, and leveled height
differences.
The astronomic azimuth A is the angle measured in the horizontal plane
between the astronomic meridian of P and the vertical plane spanned by the
vertical through P and the target point P,. It is positive as measured from the xaxis in a clockwise direction. Horizontal directions and angles may be regarded
as azimuths lacking orientation, or as azimuth differences. The zenith angle
(zenith distance) z is the angle measured in the vertical plane between the local
vertical and the line joining P and P/. It is positive as measured from the outer
surface normal. The spatial distance s is the length of the straight line joining P
and Pj, Geometric leveling also refers to the local vertical, providing a height
difference with respect to W = WP over a very short distance. It may be
regarded as the boundary case for trigonometric heighting, with a zenith angle
of 90. Gravity measurements and measurements of gravity gradients also refer
to the local astronomic system.
z ZENITH
Fig. 2.13. Local astronomic system
According to Fig. 2.13, the position vector between P and P, is given by
cos A sin z
= s sin A sin z
cos z
The local astronomic system is used for astronomic and geodetic applications.
Fig. 2.14. Astronomic triangle
In geodetic astronomy, only direction measurements (zenith angles and
azimuths) to celestial bodies are performed. The local system is called the
horizon system, and the origin is named topocenter. The points of intersection
of the plumb-line direction with the celestial sphere are known as the zenithal
point and the nadir point Z. The intersection of the horizontal plane with the
celestial sphere is the celestial horizon. The azimuth in astronomy is usually
reckoned from the south point and is considered positive westward to the north.
The relation between the horizon system and the equatorial hour angle system,
cf. [2.4.1], is given by the astronomic triangle (Fig. 2.14), see also Fig. 2.4. It is
formed on the celestial sphere by the vertices /V (north pole), (zenithal
point), and S (celestial body). The triangle contains the compliments to
declination (90-) and astronomic latitude (90-), the hour angle h, the
zenith angle z, the explement of the azimuth (360 -), and the parallactic
angle q. From spherical astronomy we obtain the transformations
cos A sin z = sin cos - cos cos h sin
sin A sin z = -cos S sin h
cos z = sin sin + cos cos h cos
Here the azimuth A is reckoned in the geodetic sense, i.e., positive from the
The transition to the ,^-system (a = right ascension) is given by the local
apparent sidereal time LAST, see (2.12):
a = LAST -h.
Astronomic longitude is obtained by comparing LAST with the Greenwich
sidereal time (2.5):
= LAST-GAST.
Equations (2.21) to (2.23) are the fundamental equations for determining ,
and A from measurements of and GAST at given a, S, cf. [5.3.2]. Equation
(2.21) also follows from (2.20) if we take (2.11) and (2.29) into account.
For geodetic applications, the observations carried out in the local astronomic
systems have to be transformed into the global geocentric system for further
use in establishing geodetic control networks.
Due to the non-parallelism of the plumb lines, the orientation of the local systems depends on
position and thus changes rapidly from place to place. Computations in one individual system are
therefore admissible only in very limited areas when applying formulas of plane geometry.
The plumb line direction can be referred to the global geocentric-system by
means of the "orientation" parameters astronomic latitude and longitude
(Fig. 2.15). After a parallel shift of the global system into the local one (Fig.
2.16), we transform the latter one to a right-handed system by applying the
reflection matrix
2 ZENITH
Fig. 2.15. Local astronomic and global geocentric system
z (ZENITH)
Fig. 2.16 Transformation between the local astronomic and the geocentric
We then rotate the local system by 90 - around the (new) y-axis and by
180 - around the z-axis with the rotation matrices
2 (90-) =
cos 0 sin
-cosA sin 0
3(180-) = -sin -cosA 0
Coordinates differences between P, and P in the geocentric system are thus
obtained by
= ,
with given by (2.20) and
The transformation matrix reads as
= 3 (80-) 2 (90-)8 2 =
-88 -sinA
-sin sin cos cos sin
The inversion of (2.26) is performed easily considering that A is orthonormal:
A-'=A T .
We obtain
-sin
coscos
sin sin
cos sin sin
Equations (2.27) to (2.30) are the basic equations for the evaluation of local
geodetic measurements within the three-dimensional reference frame, cf.
[6.2.1].
The external gravity field plays a fundamental role in geodesy. This is because
the figure of the earth has evolved under the influence of gravity, and most
geodetic observations refer to gravity. Modeling of the observations thus
requires knowledge of the gravity field. In addition, the analysis of the gravity
field yields information on the structure of the earth's interior; in this way
geodesy contributes to geophysics, cf.[8.2.4].
The fundamental quantities gravitation and gravity, together with their
corresponding potentials, are introduced in [3.1], where the main properties of
the gravity field are also described. The geometry of the gravity field is
especially important for local applications [3.2], while the spherical harmonic
expansion provides a powerful tool for a global gravity-field representation
[3.3]. The geoid, as a physically defined reference surface for heights, is of
basic interest in geosciences and engineering [3.4]. Gravity variations with time
can be modeled in part (tidal effects), but to a large degree this work is still in
the research stage [3.5].
The theory of the gravity field is extensively treated in geodetic and geophysical
literature (e.g., HEISKANEN and VENING-MEINESZ 1958, HEISKANEN and
MORITZ 1967, JEFFREYS 1970, PICK et al. 1973, GROTEN 1979).
A body rotating with the earth experiences the gravitational force of the masses
of the earth [3.1.1] to [3.1.3], and other celestial bodies as well as the
centrifugal force due to the earth's rotation [3.1.4]. The resultant is the force of
gravity [3.1.5]. In the case of artificial satellites, it is noted that a satellite does
not rotate with the earth; hence, only gravitation acts on the satellite.
According to Newton's law of gravitation (1687), two point masses mi and m2
attract each other with the gravitational force (attractive force)
/^ "*i"*2
= -OIT 7>
where G is the gravitational constant, cf. [2.1],
/I 1
1 \
3 The Gravity Field of the Earth
= 6.673xl(r u m 3 kg-V,
and / is the distance between the masses. The vectors K and 1 point in opposing
directions. The Si-unit of K is m kg s" 2 .
By setting the mass at the attracted point P to unity, (3.1) transforms into the
gravitational acceleration (henceforth also termed gravitation):
m\
P(X,Y,Z)
Fig. 3.1. Gravitation
b originates at and is directed towards the source point P'. The vector 1 may
be expressed by the position vectors r and r' (Fig. 3.1), e.g., in the global
Cartesian ,,- system:
= r-r,
, and r'1 =
The unit of the acceleration b is ms"2 .
As gravitation depends only on the distance between the attracting mass and the attracted point, it
does not depend on the choice of the coordinate system. While global applications require a
3.1 Fundamentals of Gravity Field Theory
geocentric coordinate system, local coordinate systems are useful for problems of limited spatial
extent.
The earth is composed of an infinite number of differential mass elements dm.
The gravitation on the unit mass at P results from the integral over the
individual contributions. Equation (3.3) correspondingly transforms to
eanhfr-r
The mass element dm can be expressed by the volume density p = p(rr) and the
volume element dv:
= pdv,
where p is expressed in kg m"3.
The representation of gravitational acceleration, the gravity field, and related
computations are simplified if the scalar quantity "potential" is used instead of
the vector quantity "acceleration." Because the gravitational field is invariant to
rotations:
curlb = 0,
the vector b may be represented as the gradient of a potential V (e.g.,
KELLOGG 1929, SIGL 1985):
b = gradf.
For Si point mass m, see (3.3), we have
S~^\/f
V = - .with limF = 0.
~***
For the earth, see (3.5) and (3.6), we obtain
eaith
The potential at P indicates the work that must be done by gravitation in order
to move the unit mass from infinity (V = 0) to P. The unit of potential is mV2.
If the density function p = fir1) was known for the earth, (3.5) resp. (3.10) would permit
calculation of the gravitation as a function of position. In reality, more detailed density
information is available for the upper layers of the earth only, while global models merely
consider radial density changes, cf. [3.1.2], [8.1]. Consequently, gravity-field observations have
to be used in order to model the exterior gravity field.
To a first approximation, the earth can be viewed as a sphere with a centrally
symmetric density structure, i.e. composed of spherical shells with constant
density, cf. [8.1]. We calculate the gravitation in the interior and exterior of this
model using the system of spherical coordinates r,&,A introduced in (2.14).
For this purpose, the orientation of the system is changed such that the & -axis
coincides with the line joining the coordinate origin and the calculation point
P (Fig. 3.2).
r'sinS'dX'
Fig. 3.2. Surface element of a spherical shell
The potential of a homogeneous spherical shell of radius r' with infinitesimal
thickness dr' and density is given in analogy to (3.10) by
Here, integration is over the surface of the shell/and
is the surface element. Inserting (3.12) into (3.11) gives
A distinction is now made in the integration as to whether the attracted point
is exterior or interior to the spherical shell.
For an attracted point lying in the exterior (r > r'), the potential is given by
integration of (3.13):
V,'=4aGpdr=G.
dm=4nprndr'
represents the mass of the spherical shell. The potential of the spherical earth
composed of concentric homogeneous shells is
Hence, it is equal to the potential of the entire mass of the earth concentrated
at the center of mass. The gravitation follows from
=:
^=-^
With GM= 398.6 12 mV2 and the radius of the earth R = 6371 km, the
value of the potential at the surface of the earth (r = R) amounts to V = 6.26
7 mV 2 , and the gravitation is b = 9.82 ms"2.
For a point in the interior (r < r'), we obtain from (3.13) for the potential of the
spherical shell:
Here, V' is constant; therefore, the gravitation is zero:
The potential inside an earth constructed of shells includes the contribution of
Ve (3.14) due to the masses interior to the sphere r = const. It is also comprised
of Vt (3.18), which is due to the spherical shell having thickness R - r:
For a homogeneous earth (p = const.) we have
Vi=-Gpr2+2Gp(R2-r2) = 2Gp\ R2- .
The gravitation of the earth composed of shells is
--G^-
r'2dr'
according to (3.15), which represents the mass inside the sphere r = const. The
masses outside this sphere have no effect on the gravitation. For a homogeneous
sphere, (3.22) and (3.23) transform to
bi=-Gpr.
"We now investigate the fundamental properties of the gravitational potential and
its first and second derivatives.
Starting from the potential (3.10)
gravitation is given by the gradient (3.8) through the components, see (3.5),
- = G (If
dm , etc.
The second derivatives read as
\ft \
We now have to distinguish between the cases where the attracted point P lies
exterior or interior to the earth's masses, cf. [3.1.2]. Here, we neglect the mass
of the atmosphere (about 6 of the total mass) and the variations of
gravitation with time (maximum relative effect about 10~7 ). The earth's surface
S then constitutes a boundary surface between the mass-free exterior space and
the earth's interior.
If P lies exterior to the surface 5, then / > 0 always. Then according to (3.25) (3.27), the potential and its first and second derivatives are single-valued, finite
and continuous functions, vanishing at infinity.
We now apply the Laplacian differential operator = div grad to V. In the
A",7,Z-system, this reads as
r+V+Va.
When introducing (3.27) into (3.28), the first and second terms cancel each
other. This leads to Laplace's differential equation of second order, which
governs the exterior gravitational field:
Continuous functions having continuous first and second order derivatives and
fulfilling (3.29) are called harmonic functions.
If the attracted body lies inside the body of the earth, then the case / = 0 is
possible. This requires special attention because of the discontinuity of III.
To this end, we consider enclosed by a sphere (center at P0, radius p),
SPHERE K
VARIABLE DENSITY
Fig. 3.3. Gravitational potential inside the earth
where is chosen sufficiently small so that the density = const, inside K (Fig.
3.3). The potential at is composed of the contributions from masses lying
interior and exterior to K. From (3.10) and (3.21) and using
we find
earth-K
In the limits /?0 and #>0, agreement is obtained with the expression for the
exterior potential (3.10). Differentiation yields
=- 111
As q>0, we also have X-X0=Y-Y0=Z-Z0 > 0 , so that once again we
obtain agreement with the exterior case (3.26). The second derivatives are given
= G
~ ill j
dm +3G
.ra-K V I
dm nGp etc
III ~i
^.k V
eanh-K
-\
' -
For q>0, the last term does not vanish:
tta.
The gravitational potential and its first derivatives are thus single-valued, finite,
and continuous in the interior as well. According to (3.30), the second
derivatives exhibit discontinuities at abrupt changes in density. Inserting (3.30)
into (3.28), we get Poisson's differential equation:
Hence, Fis not a harmonic function in the interior of the earth.
Finally, we mention Gauss' integral formula, which connects the normal
derivatives dV/dns on any boundary surface S (which in general is not an
equipotential surface) and the second derivatives contained in the Laplace
operator (3.28):
Fig. 3.4. Outer surface normal on the bounding surface and on the equipotential
Here, v is the volume of the body of surface S (Fig. 3.4). The left-side term may
be interpreted as "gravitational flux" through S. As shown in potential theory, it
is proportional to the total mass
rfrfrf
-dS = -4nGM.
Taking the limit at the source point P' in (3.34), equation (3.32) reduces to
Poisson's differential equation (3.31) and to Laplace's differential equation for
the exterior space (p = 0). Based on Gauss' formula, basic relationships can be
established between observations in the gravity field and parameters describing
the surface 5, cf. [6.5.1].
The centrifugal force acts on an object of mass on the earth's surface. It arises
as a result of the rotation of the earth about its axis. We assume here a rotation
of constant angular velocity about the rotational axis, with the axis assumed
fixed with the earth. The small effects of time variations of the rotational vector
can be taken into account by reductions, cf. [2.5.2]. The centrifugal acceleration
(3.35a)
acting on a unit mass is directed outward and is perpendicular to the spin axis
(Fig. 3.5). With the geocentric latitude , we have
and the magnitude
= 0) r cos #7.
(3.35b)
CENTER OF
Fig. 3.5. Gravitation, centrifugal acceleration, and gravity
The angular velocity
= 7.292115xl(Prads
is known with high accuracy from astronomy, cf. [2.2.2]. Consequently, z = z|
can be calculated if the position of P is known.
As the Z-axis of the earth-fixed Jf,7,Z-system coincides with the axis of rotation,
cf. [2.5.1], we have
P=
v0j
z=gradZ,
we introduce the centrifugal potential
Remark: Here, the symbols z and Z are used for the centrifugal acceleration and
potential respectively. They were introduced earlier for local and global
coordinates and will be employed again as such in later sections.
Differentiating twice and applying the Laplacian operator yields
= 2 2 .
Therefore, the analytic function Z, as opposed to F(3.29), is not harmonic.
For points on the equator of the earth, the centrifugal potential has a value of 1 . 1
10s mV2 , and the centrifugal acceleration is 0.03 ms"2 (~ 0.3% of gravitation).
At the poles, we have = 0 and = 0.
The gravity acceleration, or gravity g (Latin: gravitas), is the resultant of
gravitation b and centrifugal acceleration z (Fig. 3.5):
= b + z.
By multiplying with the mass m of the attracted point, we obtain the force of
F = mg.
The direction of g is referred to as the direction of the plumb line (vertical); the
magnitude g is called gravity intensity (often just gravity). With (3.10) and
(3.38), the gravity potential of the earth becomes
^v + ^V.
earth '
It is related to the gravity acceleration by
g = gradiF.
gT =(gradfF)T =(WX,WY,WZ).
In the A^Z-system, we have
Taking (2.19) into account, we obtain the components of gravity expressed by
the plumb line parameters ,:
cos cos
cos sin .
curl g = curl grad W = 0
follows from the corresponding properties of gravitation and centrifugal
acceleration and can also be expressed by the conditions
=W
W =W
*' XY
' YJ(l '' X7.
'' ZX > " Y7.
'' 7Y
W and its first derivatives are single-valued, finite, and continuous in the whole
space as a consequence of the characteristics of V and Z. Exceptions are the
uninteresting cases r>> (then also Z>) and g = 0 (direction of the plumb line
is not unique). Due to the behavior of V, the second derivatives of W are
discontinuous inside the earth at abrupt density changes. For geodesy, the most
3 .2 Geometry of the Gravity Field
important surface of discontinuity is the physical surface of the earth, with a
density jump from 1.3 kgm~ 3 (density of air) to 2700 kgm"3 (mean density of
the upper crust).
From (3.31) and (3.39), we obtain the generalized Poisson differential equation
In outer space (/? = 0), it becomes the generalized Laplace differential equation
With the conditions (3.47) and (3.48) resp. (3.49), the gravity potential W
possesses only five (out of nine) mutually independent second derivatives. They
are closely related to the curvature of the level surfaces and the plumb lines, cf.
Because of the flattening at the earth's poles and the centrifugal acceleration, g
depends on the latitude. The value of gravitation for a spherical model, 9.82
ms~ 2 , cf. [3.1.2], decreases at the equator and increases at the poles of an
ellipsoidal model. The centrifugal acceleration further diminishes the equatorial
value, while gravitation at the poles is not influenced by centrifugal
acceleration, cf. [3.1.4]. As a result, gravity varies between 9.78 ms"2 (equator)
and 9.83 ms'2 (poles), see also [4.3].
A geometrical representation of the gravity field is given by the level surfaces
and the plumb lines [3.2.1]. Local field properties are described by the
curvatures of level surfaces and plumb lines [3.2.2], and a system of "natural"
coordinates can be based on these properties [3.2.3].
The surfaces of constant gravity potential
= const.
are designated as equipotential or level surfaces (also geopotential surfaces) of
gravity. As a result of an infinitesimal displacement ds, and in view of (3.43),
the potential difference of differentially separated level surfaces (Fig. 3.6) is
PLUMB LINE
Fig. 3.6. Neighboring level surfaces
This means that the derivative of the gravity potential in a certain direction is
equal to the component of the gravity along this direction. Since only the
projection of ds along the plumb line enters into (3.51), dW is independent of
the path. Hence, no work is necessary for a displacement along a level surface
W= const.; the level surfaces are equilibrium surfaces.
If ds is taken along the level surface W= WP, then it follows from dW'= 0 that
cos(g,c/s) = cos 90 = 0: gravity is normal to W = WP. The level surfaces are
intersected at right angles by the plumb lines. The tangent to the plumb line is
called the direction of the plumb line and has been defined already in [3.1.5]. If
ds is directed along the outer surface normal n, then, because cos(g,n) = cos
180 = -1, the following important relationship exists:
dW = -gdn.
It provides the link between the potential difference (a physical quantity) and
the difference in height (a geometric quantity) of neighboring level surfaces.
According to this relation, a combination of gravity measurements and (quasi)
differential height determinations, as provided by geometric leveling, delivers
gravity potential differences, cf. [5.5.3].
If g varies on a level surface, then, according to (3.52), the distance dn to a
neighboring level surface also changes. Therefore, the level surfaces are not
parallel, and the plumb lines are space curves. As a consequence of the gravity
increase of 0.05 m s 2 from the equator to the poles, the level surfaces of the
earth converge toward the poles by 0.05ms"2/9.8 ms~ 2 , or 5xlO" 3 , in a relative
sense. Consequently, two level surfaces that are 100.0 m apart at the equator are
separated by only 99.5 m at the poles (Fig. 3.7).
3.2 Geometry of the Gravity Field
EQUIPOTENTIAL
W = const.
Fig. 3.7. Equipotential surfaces and plumb lines close to the earth
The level surfaces inside the earth and in the exterior space are closed spheroidal surfaces. The
geoid is the level surface that approximates mean sea level. Because of its importance as a
reference surface for heights, it will be treated separately in [3.4]. As an outer limit in the realm of
the definition of gravity, one may consider the level surface for which the gravitation and
centrifugal acceleration in the equatorial plane cancel each other. The equatorial radius of this
surface would be 42 200 km.
The concept of the level surface was introduced by MacLaurin (1742), whereas Clairaut (1743)
thoroughly discussed level surfaces and plumb lines as a whole. BRUNS (1878) included the
determination of the exterior level surfaces in their entirety in the fundamental problem of
geodesy.
From the properties of the potential function W = W(r), it follows that the level
surfaces which lie entirely in the exterior space are analytical surfaces; that is,
they have no salient or singular points, cf. [3.1.5], and can be expanded in
Taylor series. Level surfaces extending partially or completely inside the earth
exhibit discontinuities in the second derivatives where density jumps occur.
These surfaces can thus be constructed from pieces of different analytical
surfaces only.
Using the local astronomic x,y^-syslem introduced in [2.6.2], we develop the
potential W in the vicinity of the origin P into a series. This local representation
reads as
Here, W^dW/dx, =32/2, Wv=d2W/dxdy, etc. represent the first
and second order partial derivatives at P in the local system. If the calculation
point is located on the level surface through P, we have
W = WP, Wx=Wy=0, Wf=-g.
By solving for z, we get the equation of the level surface in the neighborhood of
Here, we have neglected terms of third and higher order, taking into account
that is of second order compared to and y, due to the small curvature of the
level surfaces.
The curvature of the level surface at along an azimuth A is described by the
curvature of the normal section (intersection of the vertical plane with the
surface), which is called normal curvature. The well-known formula for the
depression of a sphere (local approximation to the level surface) with respect to
the horizontal je^-plane gives
with 5 = distance from and RA = radius of curvature in the azimuth A (Fig.
3.8). We substitute x,y with the local polar coordinates
x = scosA, y = ssir\A
and introduce (3.55) into (3.54). The normal curvature then reads as
in2 A).
For the andy-directions (A = 0 and A = 180), we obtain the curvatures
l _1.
W i
k -_
'"R,
= _ = __ L.
s ' ' ,
s'
where Rx and Ry are the corresponding curvature radii. Analogously, the
geodetic torsion in the direction of the meridian (expressing the change of
z(ZENITH)
UMB LINE
x(NORTH)
W=WP
y(EAST)
Fig. 3.8. Curvature of level surfaces and plumb lines
direction normal to the meridian) is given by
W_
The normal curvature assumes its extreme values in the mutually perpendicular
directions of principal curvature A! and A2 = AI 90. By considering the
extrema, we find
tan 24, = 2 -2 .
Introducing (3.59) into (3.56) yields the corresponding principal curvatures
tan 4.2
The mean curvature of the level surface is given by
Outside the masses of the earth, the plumb lines can also be described
analytically. In the local astronomic system, the equation of the plumb line is
, y = y(s), z = z ( s ) ,
where s is the arc length reckoned in the direction of gravity (Fig. 3.8). The line
element along s thus differs from gravity only by the "scale factor" g:
with x' = ds/dx, etc. The curvature vector of the plumb line lies in the principal
normal through P and thus in the horizontal plane. It reads as
=K
where is the total curvature, and A is the azimuth of the principal normal.
Differentiating (3.63) with respect to s, and considering that at P: x' = y' = 0, z'
= -1, a substitution into (3.64) yields
gcosA
gsinA
= arctanW_
The curvatures of the projections of the plumb line on the ,-plane
(A = 0)and>,z-plane (A = 90)follow from (3.65):
K = wa Ky=
* ~~f' ~~f
From (3.56) to (3.67), we recognize that the curvatures of the level surfaces and
the plumb lines depend on the second derivatives of the gravity potential.
Consequently, they experience discontinuities at abrupt density changes, as
discussed for the potential function, cf. [3.1.5].
The gravity gradient tensor (Etvs tensor) is comprised of the second
derivatives of Was follows:
'wa wv wa^
] = grad g = grad (grad W ) =
y*
With (3.57), (3.58), (3.67), and W2 = -g, it can be transformed into the Marussi
l dg
fgdz
which completely describes the geometry of the gravity field (GRAFAREND
1986, MORITZ and HOFMANN-WELLENHOF 1993). As already stated in [3.1.5],
(3.68) resp. (3.69) only contain five independent elements.
The Etvs tensor (3.68) includes the gravity gradient
*>
gradg = -
dg/dx
dg/dy
dg/dz
which describes the variation of gravity in the horizontal plane and in the
vertical direction. The horizontal gradient is formed by the components dg/dx
and dg/dy, and points in the direction of maximum gravity increase in the
horizontal plane. The vertical component (often called vertical gradient)
dg/dz describes the gravity change with height. If we combine the generalized
Poisson equation (3.48) with the mean curvature (3.61), we get
- = -4
This relation was found by BRUNS (1878). It connects the vertical gradient with
the mean curvature of the level surface and offers a possibility to determine this
curvature from gravity measurements, cf. [5.4.5].
We introduce a system of non-linear "natural" coordinates <J>,A,W defined in the
gravity field. Astronomical latitude and astronomical longitude describe
the direction of the plumb line at the point P. They have been introduced
already in [2.6.1] as orientation parameters of the local gravity field system with
respect to the global geocentric system. The gravity potential W locates P in the
system of level surfaces W = const. (Fig. 2.12). Hence, P is determined by the
non-orthogonal intersection of the coordinate surfaces = const., = const.,
and the surface W = const. The coordinate lines (spatial curves) are called
astronomic meridian curve (, = const.), astronomic parallel curve (, =
const.), and isozenithal line (, = const.).
The natural coordinates can be determined by measurements. Astronomic
positioning provides latitude and longitude, cf. [5.3.2]. Although W cannot be
measured directly, potential differences can be derived from leveling and
gravity measurements and then referred to a selected level surface, e.g., the
geoid, cf. [5.5.3].
The relationship between the global .A^Z-system and the ,,^ is
obtained from (3.45):
)5>8
Solving for the natural coordinates yields the highly non-linear relations:
= arctan
-wz
= arctan Lr-
= W(X,Y,Z)
Differential relations between the local Cartesian coordinates x,y,z (local
astronomic system) and the global <I>,A,\V-system are given by
3 . . 3 ,
-dx + dy + -dz , etc.,
where dx, dy, and dz can be derived from local measurements, cf. [2.6.2].
The partial derivatives of and describe the change of the plumb line
direction when moving in the gravity field. This corresponds to the curvature of
the level surface (when moving in the horizontal plane) and of the plumb line
(when moving vertically). We have the following relations:
BW _
~Bz~
where the curvature and torsion parameters are given by (3.57), (3.58), and
(3.67). Introducing (3.74) into the differential relations leads to the
which again contains the elements of the Marussi tensor (3.69), see
GRAFAREND (1972), MORITZ and HOFMANN-WELLENHOF (1993).
As the orientation of the local systems changes from point to point, the
differentials dx, dy, dz are imperfect ones, with loop closures differing from
zero:
,,\, on the other hand, possess perfect differentials with
Equation (3.75) offers the possibility to transform local observable-quantities
(azimuths, horizontal directions and angles, zenith angles, distances, potential
differences from leveling and gravity) to the global gravity field system, where
the astronomic latitude and longitude coordinates are also observable.
A theory of "intrinsic geodesy" based on the differential geometry of the gravity field has been
developed by MARUSSI (1949,1985), see also HOTINE (1969). Using only observable quantities,
reductions to conventional reference systems are completely avoided. On the other hand, in order
to practically evaluate (3.75), a detailed knowledge of the curvature of the gravity field would be
necessary. This would require a dense survey of the second derivatives of the gravity potential, as
the curvature close to the earth's surface is rather irregular. Present gravity models do not
sufficiently provide this information, and gravity gradiometer techniques are either time
consuming or not yet operational, cf. [5.4.5]. But even with a better knowledge of the curvature of
the gravity field, the transfer of coordinates would hardly be made in the system of natural
coordinates, due to the complex structure of the gravity field.
Because the density function - /Xr') of the earth is not well known, the
gravitational potential V = V(r) cannot be computed by Newton's law of
gravitation using (3.10). However, a convergent series expansion of V is
possible in the exterior space as a special solution of Laplace's differential
equation (3.29). It can be easily derived from an expansion of the reciprocal
distance appearing in Newton's law [3.3.1], [3.3.2], see KELLOGG (1929), SlGL
(1985), BLAKELEY (1996). This solution corresponds to a spectral
decomposition of the gravitational field [3.3.3]. The coefficients of the series
expansion provide the amplitudes of the respective spectral parts [3.3.4], Any
observable functional of V can be evaluated for the determination of these
coefficients, thus allowing a global analytical representation of the gravitational
field, cf. [6.6.1].
3.3 Spherical Harmonic Expansion of the Gravitational Potential
Applying the law of cosines to the triangle OPT (Fig. 3.1), we obtain
(r'\
- = (/- 2 +/ 2 -2/r'cos^V 2 =- 1+ -2-cos^
for the reciprocal distance III appearing in (3.10), between the attracted point P
and the attracting point P'. Here, is the central angle between the directions
from 0 to and to P' respectively. If III is expanded in a series converging
for r'< r, and if the terms are arranged according to increasing powers of r'l r,
then it follows that
f 1=0
The P/(cos i//) terms represent polynomials of /* degree in cos^. They are
known as Legendre polynomials (zonal harmonics), and they are computed for
the argument / = cos by means of
2'x/! d
(3-80a)
A rapid calculation is possible with the recurrence formula (WENZEL 1985)
with P0 = 1 , P, = t .
We now introduce the unit sphere around the origin of the coordinates O (Fig.
3.9). The projections of OP and OP' on , together with the north pole
projection N, form a spherical triangle. It contains the spherical coordinates
, and ',' , and the central angle appears as a spherical distance on ,
see also [2.5.1]. Spherical trigonometry provides the following relationship
cos = cos $cos &' + sin $sin tf'cos (' - ) .
The corresponding decomposition of P/(cos ) leads to
(r=1)
Fig. 3.9. Spherical polar triangle on the unit sphere, spherical coordinates
P, (cos ) = , (cos u)P, (cos &')
+2 7-H?to (cos )cosm Plm
1 (l + m)\
+Plm
Again, the P/(t) are the Legendre polynomials with the argument t cos $ or
t = cos&'. The associated Legendre functions of the first kind, Pim(t) (I =
degree, m = order), are obtained by differentiating Pffy m times with respect to
Up to degree 3, the Legendre polynomials and the associated Legendre
functions are given as follows:
P0 =1, Pl =
P2 =-
P = sin &, P2l = 3 sin &cos &, P22 = 3 sin
P,,=sin&cos
.(3.83b)
A series development for the calculation of P/m(t) is given in HEISKANEN and
MORITZ (1967, p.24). By substituting (3.81) into (3.79), the expansion of III
into spherical harmonics is completed.
The functions
which appear in (3.81), are called Laplace's surface spherical harmonics. They
characterize the behavior of the developed function (here lit) on the unit sphere,
cf. [3.3.3]. The orthogonality relations are valid for these functions, i.e., the
integral over the unit sphere of the product of any two different functions is
for l, q m , or k * i. For the product of two equal functions Y,cm or Y^ ,
\\ YLda.
Jfor
21 + 1 (l-m)!
see HEISKANEN and MORITZ (1967, p.29).
We substitute the spherical harmonic expansion of I//, (3.79) and (3.81), into
the volume integral (3.10):
'=-*
1=0 m=0
+p,m
I for m =
2 form*0
In abbreviated form this development can be expressed as
where the Vt are called solid spherical harmonics, and the
combinations of the surface spherical harmonics (3.84).
are linear
For / = 0, the integration yields the potential of the earth's mass M concentrated
at the center of mass (3.16). We extract this term, introduce the semimajor axis
a of the earth ellipsoid as a constant, and denote the mass integrals by Cim, Sim
(spherical harmonic coefficients). The gravitational potential expanded in
spherical harmonics is then written as
The harmonic coefficients (also Stokes' constants) are given by
CIO=C, = r
.(3.90)
r \
,s w*>in/\,
\ ^m(cosz?')j
[dm
The following coefficients are sometimes used, particularly in satellite geodesy:
^=-Q,^=-C/m,^=-5/m.
Calculations in the gravitational field become more convenient with the (fully)
normalized spherical harmonic functions Plm (cos $). They are computed from
the conventional harmonics, (3.80) and (3.82), according to
for m=Q
form*
with t = cosz?, etc. Recursive formulas are also available for the calculation of
the normalized harmonics and its derivatives (PAUL 1978, TSCHERNING et al.
1983, WENZEL 1985):
(2/ + !)(/ + in -!)(/- IK -1)'
for / > m + 1
and the control
(3.93c)
m=0
In addition to the orthogonality relation (3.85) for the surface harmonics, we
now also have
_ icos/n/ll
P,m\
[sin \
over the unit sphere . According to (3.92), for an expansion of the gravitational
potential analogous to (3.89), the harmonic coefficients are now given by
1 for m = 0
2 for m
Equation (3.89), or the corresponding equations employing fully normalized harmonics ((3.92)
and (3.95)), represent a spherical solution of Laplace's differential equation (3.29). It can also be
derived straightforwardly by the method of variable separation, after substituting the Cartesian
coordinates with spherical coordinates (HEISKANEN and MORITZ 1967, pp.18 ff.).
The expansion converges outside a sphere of radius r = a, which just encloses the earth (Brillouin
sphere). After the theorem of Runge-Krarup, an expansion of V into converging spherical
harmonics can also be used in the interior of the earth, down to a sphere completely inside the
earth and close to its surface (Bjerhammar sphere), KRARUP (1969), MORITZ (1980, p.69). Such
an expansion represents an arbitrarily good approximation of the outer gravitational field.
Naturally, this analytical extension does not satisfy the Poisson equation (3.31) of the actual
gravitational field in the earth's interior.
With present accuracies of the determination of the earth's gravity field, the gravitation of the
atmosphere cannot be neglected. As the density of the atmosphere primarily depends on height,
corresponding models can be used to calculate the potential and the gravitation of the atmosphere
as a function of height. These calculations are based on the relations derived for the potential
inside an earth constructed of homogeneous spherical shells, cf. [3.1.2]. With an atmospheric
mass of 5.32 18 kg, we get a potential value of 55.6 m 2 s 2 for h = 0, and 54.8 m 2 s"2 for h =
100 km. This effect is taken into account by corresponding reductions, cf. [4.3].
The extension of the spherical harmonic expansion for V to the gravity potential
W is performed easily by adding the centrifugal potential Z (3.38). If we express
the distance p to the rotational axis by spherical coordinates (2.14), the
centrifugal potential reads as
Z=r2sin2#
or, after introducing the Legendre polynomial P2 according to (3.83a), as
By adding (3.96) to (3.89) we get the expansion for the gravity potential.
We now discuss the properties of the surface spherical harmonics (3.84), which
describe the behavior of the gravitational potential on the unit sphere. The zero
points of these functions divide the surface into regions with alternating signs,
bounded by meridians and parallels.
For the order m = 0, we obtain the Legendre polynomials P, (costf). Because of
their independence of the geographical longitude A, they divide the surface into
zones of positive and negative signs: zonal harmonics. These harmonics possess
/ real zeros in the interval 0 < # < . For even /, the sphere is divided
symmetrically with respect to the equator & - 90 , and the case for odd / results
in an asymmetric division. The /^(cosi?) for w^Ohave (l-m) zeros in the
interval 0<&<. Because of the multiplication by cosw/l or sinw/l, the
surface harmonics are longitude dependent, furnishing 2m zeros in the interval
0 < < : tesseral harmonics. Finally, for m = l , the dependence on &
disappears, and the sphere is divided into sectors of alternating signs: sectorial
harmonics (Fig. 3.10).
8 = 54.7
= 90
9 = 125.3"
9 = 180
P, (cos S)
P, (cos 9;
ZONAL HARMONICS
,, (cos ) sin 6
TESSERAL HARMONICS
P 77 (cos 9,) sin 7
SECTORIAL HARMONICS
Fig. 3.10. Spherical harmonics on the unit sphere, with alternating positive
(grey) and negative (white) sign
The amplitudes of the individual terms given by the surface harmonics are
determined by the harmonic coefficients. For example, the series has only zonal
harmonics for an earth rotationally symmetric with respect to the Z-axis; the
coefficients with m must all vanish. For a mass distribution symmetric with
respect to the equator, the zonal harmonic coefficients with odd / must be
absent.
Summarizing, we state that the spherical harmonic expansion of the
gravitational potential represents a spectral decomposition of the gravitational
field. The field is separated into structures of wave-length 360 //,
corresponding to a spatial resolution of 180// . With increasing height, the field
is smoothed by the factor (/r) .
The spherical harmonic expansion has transformed the single volume integral
over the earth's masses (3.10) into an infinite series. The harmonic coefficients
now carry mass integrals for the individual contribution of the corresponding
wave-length to the total potential. The lower degree harmonics have a simple
physical interpretation.
As already stated above, the zero degree term (/ = 0) represents the potential of
a homogeneous or radially layered spherical earth, see (3.16):
The terms of degree one and two (/ = 1,2) can be calculated from (3.90) by
introducing the harmonic functions Plm (3.83) and subsequently transforming the
spherical coordinates to Cartesian coordinates using (2.14). For / = 1 , this yields
As known from mechanics, the integrals divided by the mass M are the
coordinates of the center of mass of the earth. Since we have placed the origin
of the coordinate system at the center of mass, we have
C,=CU=SU=0.
For / = 2 , we obtain
dm.
These expressions are functions of the moments of inertia
A = j|{ (r2 +'2 ), 5 = JJJ (r2 + '2 )</m,
+Y'2)dm
and of the products of inertia
with respect to the axes of the global ^,,-system. If we neglect polar motion,
the Z-axis coincides with one principal axis of inertia (maximum moment of
inertia). Consequently, we have
F, on the other hand, would only become zero if the A'-axis coincided with one
of the equatorial principal-axes of inertia. Due to the conventional definition of
the^-axis (Greenwich meridian), Fdoes not vanish.
Using the above expressions for A, B, C, and F, the harmonic coefficients of
second degree may also be formulated as follows:
c2'2 -~
s 4a 2 M' 2'2 2a2M
J2 =-C2 is also known as dynamical form factor. It characterizes the polar
flattening of the earth's body by the difference between the mean equatorial
moment of inertia (with A ~ B) and the polar moment of inertia. As this is the
largest deviation from a spherical earth model, the numerical value for C2 is
three orders of magnitude larger than the values of the successive coefficients.
Slight contributions to the ellipsoidal form of the earth also come from the even
zonal harmonics of higher degree, mainly 1 = 4 and / = 6. The coefficients C2,2
and S2,2 describe the asymmetry of the equatorial mass distribution in relation to
the rotational axis (ellipticity of the equator) and the torsion of the
corresponding principal axes of inertia with respect to the conventional X and Ydirections, cf. also [4.2.1]. If the odd zonal harmonic-coefficients differ from
zero, the corresponding terms in the expansion of V represent an asymmetric
mass distribution with respect to the equatorial plane, cf. [3.3.3]. The main
contribution comes from Cj, and may be geometrically interpreted as a
difference in the flattening for the northern and the southern hemisphere.
Numerical values for the coefficients are given in [6.6.2].
The geoid is of fundamental importance for geodesy, oceanography, and
physics of the solid earth. Due to the demands for, and achievable accuracy of, a
modern geoid, the classical definition of the geoid must be reconsidered today
[3.4.1]. In geodesy and oceanography, the geoid serves as a height reference
surface for describing continental and sea surface topography [3.4.2], [3.4.3].
Geophysics exploits the geoid as a gravity field representation revealing the
distribution of deeper located masses, cf. [8.2.4].
The geoid has been introduced by C. F. Gauss as a refined model of the figure
of the earth, cf. [1.2]. It was defined by Gauss as the equipotential surface of the
earth's gravity field coinciding with the mean sea level of the oceans.
This physical definition considers the waters of the oceans as freely-moving
homogeneous matter, subject only to the force of gravity and free from
variations with time. Upon attaining a state of equilibrium, the surface of such
idealized oceans would assume a level surface of the gravity field. The ocean
surface may be regarded as being extended under the continents, e.g., by a
system of conducting tubes.
With the gravity potential value W0, the equation of the geoid reads as
W0.
It follows from the properties of the gravity potential W, that the geoid is a
closed and continuous surface, cf. [3.1.3]. As it extends partially inside the solid
3.4 The Geoid
earth (under the continents), its curvature will display discontinuities at abrupt
density changes. Thus, although not being an analytical surface in a global
sense, it may be sufficiently well approximated by a spherical harmonic
development, cf. [3.3.2].
The problem of downward continuation of the gravity field into the earth's masses presupposes
knowledge about the density distribution of the atmospheric and topographic masses. Geoid
calculations consequently depend on the corresponding assumptions (STRYKOWSKi 1998), cf.
As well known from oceanography, mean sea level is not an equilibrium surface
in the earth's gravity field, due to ocean currents and other quasi-stationary
effects. In addition, sea surface variations with time can only be partially
reduced by averaging over time or modeling. Hence, mean sea level still varies
over longer time spans, cf. [3.4.2].
Consequently, a refined geoid definition is needed at the "cm" accuracy level
(RlZOS 1982, RAPP 1983a). By applying a minimum condition on the deviations
between mean sea level and the geoid, the geoid could be defined as the
equipotential surface which best fits mean sea level at a certain epoch (MATHER
1978, RAPP 1995a).
Another choice would be to define the geoid as the level surface which optimally fits mean sea
level at a selected set of tide gauges used for defining the vertical datum of national or continental
height systems, cf. [3.4.3]. Such a definition would lead to only small corrections for the existing
height systems but not result in a best fit over the oceans.
The geoid definition also has to include the treatment of the permanent tidal
effects, cf. [3.5.2]. There are three different definitions possible. The mean
geoid includes the direct effect of attraction and the indirect effect of
deformation. It would coincide with an "undisturbed" sea level; hence it is of
interest for oceanography. For the non-tidal geoid, the total tidal-effect would
be eliminated. This would agree with the theoretical demand of geodesy to have
no masses outside the boundary surface "geoid", cf. [6.5.3]. As the response of
the earth to the permanent tidal part is not known, the zero-geoid is preferred in
geodesy. Here, the attraction part is eliminated but the permanent deformation
retained. This definition takes into account the fact that positioning also refers
to a deformed earth (LAG resolution, Gen. Ass. Hamburg 1983, EKMAN 1989).
With respect to a best-fitting reference ellipsoid, cf. [4.3], the geoid deviates by
30 m; maximum deviations reach about 100 m, cf. [6.6.3].
Finally, a refined geoid definition should also consider variations with time,
which result from the displacements of terrestrial masses, cf. [3.5.3]. Such a
definition then would refer to a certain epoch.
The ocean surface does not coincide with a level surface (e.g. the geoid) of the
earth's gravity field; the deviations are called sea surface topography (SST, also
dynamic ocean topography). Instantaneous SST is affected by temporal
variations of long-term, annual, seasonal, and short-term character, occurring at
different scales. Averaging the ocean surface over time (at least over one year)
or modeling ocean tides provides mean sea level (MSL) for the corresponding
time interval. Even after reducing all time-dependent parts, a quasi-stationary
SST would remain. It is caused by nearly constant oceanographic and
meteorological effects, which generate ocean currents and ocean surface slopes.
The r.m.s. variation is 0.6 to 0.7 m , and the maximum deviation from the
geoid is about 1 m or more (LISITZIN 1974).
Short term variations of the sea surface (waves) are averaged out in the mean value over time
(e.g., at tide gauge observations over one hour) or by the smoothing effect of the "footprints" in
satellite altimetry, cf. [5.2.7]. Tidal effects can deviate considerably between the open ocean and
shelf areas, adjacent seas, and coastal zones. This is due to unequal water depths and to the fact
that the continents impede the movement of water. On the open sea, the tidal amplitude is less
than one meter (r.m.s. variation 0.3 m), while it can amount to several meters in coastal areas
(Bay of Fundy, Nova Scotia: more than 15 m).
Oceanic tidal models are based on Laplace's tidal equations. These models take bathymetry, tide
gauge data, and in more recent solutions, satellite altimetry and satellite orbit analyses into
account. The models solve for about 10 to 12 partial tides (annual, semiannual, monthly,
fortnightly, daily, and half-daily). They are provided either in grid form (1 , 0.5x0.5 ) or
as a spherical harmonic expansion (SCHWiDERSKi 1980, 1983, ANDERSEN et al. 1995). Ocean tide
models have also been derived by global gravity modeling (SCHWiNTZER et al. 1997, LEMOINE et
al. 1998). The accuracy of the oceanic tidal models amounts to a few cm on the open oceans but is
less at shelf areas and close to the coast.
Fluctuations of annual, semiannual, and seasonal character are of meteorological origin
(atmospheric pressure, winds), of oceanographic nature (ocean currents, differences in water
density as a function of temperature, salinity and pressure), and are also due to a variable -water
budget (changing water influx as a result of meltwater, monsoon rains, etc.). The amplitude of
these variations is on the order of 0.1 to 1 m, and scales of a few 100 km are found, e.g., at
meandering ocean streams and eddies (NEREM 1995b). In addition, a secular rise of about 1 to 2
mm/a has been observed over the last 100 years. This trend is expected to increase, reaching 0.5
to 1 m over the next century, due to climate changes producing a thermal expansion of the water
masses, a melting of the polar ice caps and the glaciers, and isostatic movements (LAMBECK 1988,
HANNAH 1989).
Mean sea level can be derived from tide gauge records, satellite altimetry, and
oceanographic methods.
Tide gauges (mareographs) continuously record the height of the water level
with respect to a height reference surface close to the geoid, cf. [3.4.3].
Averaging the results over long time intervals (month, year) eliminates most
variations with time. In order to fully remove the tidal period of a complete
lunar cycle (nutation), the record should extend over 18.6 years, cf. [2.4.2]. The
precision of the mean monthly and annual values is better than 1 cm. These
results may be systematically disturbed if the tide gauge location is not directly
linked to the open ocean and data is thereby affected by local sea level
anomalies (swell in shallow waters, estuary effects at river mouths). In addition,
local or regional vertical crustal movements (sedimentary subsidence,
postglacial uplift, etc.) may act at the tide gauge location and systematically
influence (bias) the sea level registration. These movements may reach a few
mm/a (EKMAN 1993, MlTCHUM 1994).
Tide gauge data are available for over 1750 stations worldwide, but only a few
stations cover a time span of a few centuries (at Amsterdam registrations go
back to 1700), WOODWORTH (1997). In the open oceans, pressure tide gauges
contribute in monitoring sea surface variability but lack connection to
continental height systems (TOLKATCHEV 1996). Tide gauges along the
continental coasts generally have been connected to the geodetic height control
system, thus permitting the derivation of MSL from the geoid or a zero height
reference close to it. Sea level slopes up to several 0.1 m/1000 km and more
have been detected by this method, cf. [3.4.3].
Satellite altimetry directly provides sea surface topography with respect to an
ellipsoidal reference surface, cf. [5.2.7]. With the exception of the polar regions,
satellite altimeters cover the oceans with repeated tracks (e.g., with a 10 days
repetition rate at TOPEX/POSEIDON) and permit derivation of mean sea
surface heights to an accuracy of a few cm. The solutions are derived over a
certain time period (e.g., one year) and are provided in grids of a few minutes of
arc (ANZENHOFER et al. 1996, RAPP and Yi 1997). Sea surface topography is
obtained by referring these results to a geoid model. If the altimetric solutions
for different epochs are compared, sea surface variations with time can be
determined, MINSTER et al. 1995, cf. [8.3.2].
Oceanographic methods derive sea surface topography from measurements at
sea (LAMBECK 1988, RUMMEL and ILK 1995). Steric leveling assumes that
equipotential and isobaric surfaces coincide at a certain depth (e.g., 2000 m):
level of no motion. Using water density values derived from salinity,
temperature, and pressure data, the equation of hydrostatic equilibrium yields
the gravity potential difference (or the dynamic height) between the ocean
surface and the reference level of no motion. This method is applicable in the
deep oceans (STURGES 1974). Geostrophic leveling (dynamic leveling) is based
on the hydrodynamic equations and uses observed ocean current velocities. It
can also be applied in shelf areas.
Fig. 3.11. Sea surface topography, spherical harmonic expansion up to degree
20, contour line interval 0.1 m, after LEMOINE et al. (1998)
Oceanic sea surface topography models have been developed that correspond to
a spherical harmonic expansion up to degree 36 (or a minimum wave-length of
10) with an accuracy of a few cm to 0.1 m (LEVITUS 1982). Satellite altimetry
solutions, partly in combination with gravity field modeling, are available up to
degree 20, for time spans over 5 to 10 years, and with an accuracy of better than
0.1 m (SCHWINTZER et al. 1997, LEMOINE et al. 1998), Fig 3.11. Quasistationary sea surface topography is characterized by an increase from the polar
regions toward the equator and by strong inclinations along the main ocean
The geoid is used in geodesy, cartography, and oceanography as a reference
surface for heights and depths (continental and ocean bottom topography, as
well as sea surface topography). A point P can be attributed to a specified level
surface by its gravity potential W (Fig. 3.12). With respect to the geoid potential
W0, the "height" of P is given by the negative potential difference to the geoid,
which is called the geopotential number C. We get from (3.52)
The integral is independent of the path; hence, PQ is an arbitrary point on the
geoid. C can be determined from geometric leveling and gravity measurements
along any path between P0 and P.
3.4TheGeoid
SEA SURFACE TOPOGRAPH
EARTH CRUST
Fig. 3.12. Geoid, mean sea level, continental and sea surface topography
The geopotential number is an ideal measure for describing the behavior of
masses (e.g., water masses) in the gravity field. It could be used as a "height" in
several applications, as in hydraulic engineering and oceanography. A more
general use is limited by the potential unit m2 s~2 , which is in contradiction to
the obvious demand for a metric height system that employs the "meter" unit.
In order to achieve a certain agreement with the numerical value of the height in
meters, the geopotential unit (gpu) 10 m 2 s~ 2 , or kgalm, is also used for the
geopotential number. With g = 9.8 ms"2 , the values of C are about 2% smaller
than the corresponding height values.
The dynamic height ffr* is obtained by dividing the geopotential number
through a constant gravity value. Usually the normal gravity * calculated for
the surface of the level ellipsoid at 45 latitude is used, cf. [4.2.2]:
The surfaces /15"1 = const, remain equilibrium surfaces. Hence, points located
on the same level surface have the same dynamic height. Unfortunately, a
geometric interpretation of the dynamic heights is not possible, and larger
corrections are necessary in order to convert leveling results into dynamic
height differences, cf. [6.4.1]. Because of this, dynamic heights have not been
widely used in geodesy but are used in oceanography, cf. [3.4.2].
National or continental height systems, and terrain-data based on them
(topographic maps, digital terrain models), use either orthometric or normal
heights.
The orthometric height H is defined as the linear distance between the surface
point and the geoid, reckoned along the curved plumb line (Fig. 3.12). This
definition corresponds to the common understanding of "heights above sea
level". Expanding the right-hand side of (3.104) in Hand integrating along the
plumb line from P0 (H = 0) to P (H) we obtain
# ==, g=-]gdH.
H{
g is the mean gravity along the plumb line; gravity values inside the earth are
required for its calculation. This is performed by introducing a model of the
density distribution of the topographic masses. As this distribution is known
only imperfectly, the accuracy of computed orthometric heights depends on the
accuracy of the density model. In addition, points of equal orthometric height
deviate slightly from a level surface, which is due to the non-parallelism of the
level surfaces, cf. [3.2.1]. These drawbacks are compensated by the fact that
orthometric heights represent the geometry of the topographic masses.
Geometric leveling only needs small corrections for the transformation into
orthometric height differences, cf. [6.4.1].
In order to avoid any hypothesis on the distribution of the topographic masses,
normal heights H^ have been introduced and are used in a number of countries.
The mean gravity g in (3.106) is replaced by the mean normal gravity along
the normal plumb line, which is only slightly curved:
can be calculated in the normal gravity field of an ellipsoidal earth model.
The reference surface for the normal heights is the quasi-geoid, which is close
to the geoid but not a level surface. It deviates from the geoid on the mm to cmorder at low elevations and may reach one-meter deviation in the high
mountains. On the oceans, geoid and quasigeoid practically coincide, cf. [6.1.1].
The zero height surface (vertical datum) of national height systems generally is
defined by the mean sea level derived from tide gauge records over a certain
time interval. These reference surfaces only approximate the geoid, due to the
sea surface topography and local anomalies, with deviations up to one meter, cf.
3.5 Temporal Gravity Variations
The adjustment of continental-wide leveling networks provides heights that refer to one common
level surface. By comparing with the mean sea level obtained at tide gauges, sea level slopes that
partially agree with oceanic leveling results have been found, cf. [3.4.2]. MSL at the Pacific coast
of the U.S.A. is about l m higher than at the Atlantic coast (ZiLKOSKl et al. 1995), and the mean
Baltic Sea level is estimated to be about 0.5 m above MSL of the Mediterranean Sea (GRONWALD
1963). On the other hand, there are also larger discrepancies of a few 0. l m between the results of
geometric and oceanic leveling. These can be traced back to differently defined reference
surfaces, to the particular behavior of MSL along the coastlines, and to systematic errors in
geometric leveling over long distances (FISCHER 1977). It should be mentioned that older leveling
networks have often been adjusted without any gravity reduction, or by substituting actual gravity
with normal gravity. A unification of the different vertical datum systems to a world-wide
standard thus would require vertical shifts to a common reference (global geoid definition) and a
uniform treatment of the height measurements (RUMMEL and TEUNISSEN 1988, RAPP 1995b), cf.
[7.2].
Gravity changes with time may be divided into effects due to a time dependent
gravitational constant and variations of the earth's rotation [3.5.1], tidal
accelerations [3.5.2], and variations caused by terrestrial mass displacements
[3.5.3]. These changes are of global, regional, or local character and occur
either at well-known frequencies (tides) or at time scales ranging from secular
to abrupt (LAMBECK 1988, MUELLER and ZERBINI1989).
Based on cosmological considerations, DlRAC (1938) postulated a secular
decrease of the gravitational constant G, with relative changes of
G/G = -l<r10 to -"/ (0 = dG/dt). Even to this day, laboratory
experiments and the analysis of long-term observations to artificial satellites and
the moon have not supported the assumption G 0 (GILLIES 1987, BURSA
The earth's rotational vector is subject to secular, periodic, and irregular
variations, leading to changes of the centrifugal acceleration z, cf. [2.5.2]. In a
spherical approximation, the radial component of enters into gravity, cf.
[3.1.4]. By multiplying (3.35b) with cos^J ( = geocentric latitude), we obtain
z r =-6Trcos>.
Differentiation yields the effect of changes in latitude (polar motion) and
angular velocity (length of day) on gravity:
6zr = co2rsin2(pd<p-2G}rcos2ipda).
Polar motion does not exceed a few .'/a, and rotation changes are at the order
of a few ms. Hence, corresponding gravity variations on the earth's surface (r =
6371 km) remain less than 0.15~ and 0.018~2 respectively.
Tidal acceleration is caused by the superposition of lunisolar gravitation (and to
a far lesser extent planetary gravitation) and orbital accelerations due to the
motion of the earth around the barycenter of the respective two-body system
(earth-moon, earth-sun etc.). The periods of these orbital motions are about 28
days for the moon and 365 days for the sun, and the gravimetric tidal effect is
on the order of 10~7 g (MELCHIOR 1983, ZuERN and WILHELM 1984, WENZEL
1997a).
For a rigid earth, the tidal acceleration at a given point can be determined from
Newton's law of gravitation and the ephemerides (coordinates) of the celestial
bodies (moon, sun, planets). The computations are carried out separately for the
individual two-body systems (earth-moon, earth-sun etc.), and the results are
subsequently added, with the celestial bodies regarded as point masses.
We consider the geocentric coordinate system to be moving in space with the
earth but not rotating with it (revolution without rotation). All points on the
earth experience the same orbital acceleration in the geocentric coordinate
system (see Fig. 3.13 for the earth-moon system). In order to obtain
equilibrium, orbital acceleration and gravitation of the celestial bodies have to
cancel in the earth's center of gravity. Tidal acceleration occurs at all other
points of the earth. The acceleration is defined as the difference between the
gravitation b, which depends on the position of the point, and the constant part
b 0 , referring to the earth's center:
b,=b-b0.
The tidal acceleration deforms the earth's gravity field symmetrically with
respect to three orthogonal axes with origin at the earth's center. This tidal
acceleration field experiences diurnal and semidiurnal variations, which are due
to the rotation of the earth about its axis.
If we apply the law of gravitation to (3.110), we obtain for the moon (m)
Fig. 3.13. Lunar gravitation, orbital acceleration, and tidal acceleration
, _
GM r
Here, Mm = mass of the moon, and lm and rm = distance to the moon as
reckoned from the calculation point P and the earth's center of gravity
respectively. We have b, = 0 for lm=rm. Corresponding relations hold for the
earth-sun and earth-planet systems.
We now make the transition from the tidal acceleration to the tidal potential:
b,=gradr ( =grad(F-r o ).
In the geocentric system, using spherical coordinates rm,y/m, the law of
gravitation yields the potential of a point mass according to (3.9):
(3.113a)
(3.113b)
The potential of the homogeneous b0-field is given by multiplying bQ with
Inserting (3.113) and (3.114) into (3.112), and adding an integration constant so
that V, = 0 for r = 0 and lm=rm, we get for the tidal potential
The tidal potential, and functional thereof, can be calculated either from the
ephemerides of the celestial bodies or from a spherical harmonic expansion.
Tidal potential catalogues are based primarily on the latter method, as the series
expansions converge rapidly close to the earth's surface (r = R), with
l m =1/^0 for the moon and a corresponding relation of 1/23600 for the sun.
The results from calculations employing the ephemerides serve as a control for
the tidal potential catalogues.
We develop (3.113b) into a series according to (3.79). When inserting into
(3.1 15), the terms of degree zero and one cancel and we obtain
where P,(cos^) are the Legendre polynomials. The largest contribution
( 98%) stems from degree two. Restricting ourselves now to the degree 1 = 2,
and inserting P2 (3.83) in the form
cos2iy
=(cos
-i<
we get the main term of the tidal potential series
For r = R , and neglecting the slight variation of rm , the expression before the
parentheses is called Doodson's tidal constant. It is 2.628 m2s"2 for the moon
and 1.208m2 s"2 for the sun. Hence, the solar tides amount to 46% of the lunar
tides.
Differentiating (3.117) generates the tidal acceleration. The radial component
(positive outward) is found to be
The tangential component (positive in the direction toward the moon) is
Equations (3.117) to (3.119) permit calculation of the tidal effects on the level
surfaces, on gravity, and on the plumb line direction for a rigid earth.
Taking the relation (3.52) between a potential change and the vertical shift of a
level surface into account, we obtain a tidal-induced increase of the level
surface. This amounts to 0.36m for the moon and 0.16m for the sun at
i^ = 0and 180 respectively. At ^ = 90 and 270, we have a decrease of
0.18m and 0.08m respectively. For stationary systems, the level surfaces
would experience a corresponding deformation, and freely moving masses of
water would assume the form of one of these surfaces (equilibrium tide), Fig.
Fig. 3.14. Tidal acceleration and equilibrium tide
According to (3.118), gravity changes (opposite sign) would vary between
-1.15~2 (moon) and -0.55~2 (sun) for = 0 (zenithal position); and
+0.55~2 (moon) and +0.35~2 (sun) for ^ = 90 and 135. Changes in the
direction of the plumb line are given by bjg. There is no tidal effect at = 0
and 90. Maximum values occur at = 45 and 135, with fluctuations of
0.017" (moon) and 0.008" (sun).
Equation (3.117) provides the dependence of the tidal potential on the zenith
angle (and the distance) to the celestial body. The variation of the tidal potential
and acceleration is more easily recognized if we change to the earth-fixed
coordinate system (^,A) for the point of calculation and to the equatorial
system of astronomy (,) for the celestial body, cf. [2.4.1]. Following (2.21),
we have (for the moon) the relation
cos^m =sin0Jsin m +cosi?cos<5 m cos/z m
with the hour angle given by (2.22) and (2.23):
A m = L A S T - a m = / l + GAST-a m .
Substituting into (3.117) yields Laplace's tidal equation (for the moon):
sin 2 sin 2
coshm + cos2 ^cos2 8m cos2/zm
The quantities rm, Sm , and hm vary with time, having different periods. The first term, which is
independent of the earth's rotation, exhibits long-periodic variations (14 days for the moon, 1/2
year for the sun). It also contains a non-periodic part, which only depends on latitude, causing a
permanent deformation of the level surfaces including the geoid, cf. [3.4.1]. Using (3.52), and
taking the inclination of the ecliptic into account, the geoid is thus lowered by 0.19 m at the poles
and raised by 0.10 m at the equator (KMAN 1989).
The second term oscillates with diurnal periods because of the daily rotation of the earth as
expressed by the hour angle h, and the third term introduces semidiurnal periods. Long-periodic
terms enter through the declination S and the right ascension a. As seen from (3.122), longperiodic and semidiurnal tides are symmetric about the equator, while the diurnal tides are
antisymmetric. The diurnal tide has its maximum at = 45 and vanishes at the equator and the
poles, while the semidiurnal tide reaches its maximum at the equator and is zero at the poles. The
long-periodic tides have a maximum at the poles.
Each of the three tidal constituents in (3.122) varies in a complicated way, since
they contain products of different time varying functions. However, the
ephemerides of the moon and the sun can be expressed as harmonic functions of
five fundamental astronomic quantities, considering that these quantities
essentially change uniformly with time (MELCHIOR 1983). Introducing these
harmonic series into (3.122) yields a spectral analysis of the tidal potential, and
with (3.118) and (3.119) we get a spectral analysis of the tidal acceleration.
Thus, potential and acceleration are represented by the sum of time-dependent
cosine functions having constant periods and amplitudes and phases that depend
on latitude and height (partial tides). Tab. 3.1 gives the periods and amplitudes
of the main gravimetric partial tides for - 45.
A first expansion for the moon and the sun was carried out by DOODSON (1921). The development
by CARTWRIGHT and TAYLER (1971) and CARTWRIGHT and EDDEN (1973) contains 505 partial
tides (uncertainty less than lnms~ 2 ) and was recommended by IAG for the computation of the
tides of the rigid earth (RAPP 1983b). A recent tidal catalogue by HARTMANN and WENZEL (1995)
is based on a spherical harmonic development to degree 6 (moon) and degree 3 (sun) and includes
the effects of Venus, Mars, and Jupiter (4 orders of magnitude smaller than the tidal effects of
moon and sun). It also takes the flattening of the earth into account. This catalogue provides 12
935 partial tides with an accuracy of 0.001 runs' 2 , WENZEL (1996).
As the earth is not a rigid body, it reacts in a different way to the tidal force. The
solid earth behaves mainly as an elastic body: earth's body tides. In the oceans,
tidal oscillations depend on the ocean-bottom topography, with large differences
occurring at the coastlines and at the shelf areas: ocean tides, ZAHEL (1997).
While the measurement of tidal effects will be discussed in [5.4.6] and [5.5.4],
the theory of earth tides and results of earth-tide observations are given in
Tab. 3.1. Principal Gravimetric Partial Tides for = 45, h -= 0
(solar
(nms~ 2 )
hours)
long-periodic waves
const. / tide
const, s tide
declin. tide to SO
ellipt. tide to MO
declin. tide to MO
182.62 d
27.55 d
main diurnal / tide
main diurnal 5 tide
ellipt. tide to Ol
main diurnal /s decl.
semi-diurnal waves
main/tide
main s tide
ellipt. tide to M2
declin. tide to M2, S2
ter-diurnal waves
terdiurn. / tide
diurnal waves
The terrestrial gravity field is affected by a number of variations with time due
to mass redistributions in the atmosphere, the hydrosphere, and the solid earth.
These processes take place at different time scales and are of global, regional,
and local character (WAGNER and McADOO 1986, MUELLER and ZERBINI 1989,
TORGE 1993).
Long-term global effects include postglacial rebound, melting of the ice caps
and glaciers, as well as sea level changes induced by atmospheric warming;
slow motions of the earth's core and mantle convection also contribute.
Subsidence in sedimentary basins and tectonic uplift are examples of regional
effects. Groundwater variations are primarily of seasonal character, while
volcanic and earthquake activities are short-term processes of more local extent.
The magnitude of the resulting gravity variations depends on the amount of
mass shifts and is related to them by the law of gravitation. As seen from (3.5),
a shift of the observer also affects gravitation, especially where height changes
play a dominant role. Research and modeling of these variations is still in the
beginning stages. Large-scale variations have been found from satellite-derived
gravity field models, but small-scale effects can be detected only by terrestrial
gravity measurements. Simple models have been developed for the relation
between atmospheric and hydrological mass shifts and gravity changes, cf.
[5.4.1]. Generally, gravity changes produced by mass redistributions do not
exceed the order of 10~9 to 10~8g, while geoid changes remain less than 1
mm/a, cf. [8.3.4].
A geodetic earth model is used as a reference for the actual surface and external
gravity field of the earth. It should provide a good fit to the geoid and to the
gravity field, and thus allow the linearization of non-linear geodetic problems.
On the other hand, the mathematical formation of the model should be simple
and possibly permit calculations by closed formulas. The model should serve as
a standard for applications not only in geodesy and cartography but also in
astronomy and geophysics; it should satisfy the demands and needs of these
disciplines too.
Based on these considerations, the level ellipsoid has been introduced. It
possesses a simple geometry, and coordinate systems that refer to it
approximate the "natural" coordinate system sufficiently well [4.1]. The
ellipsoid's mass and rotation provide a "normal" gravity field, which can be
calculated if the ellipsoid surface is defined to be in equilibrium [4.2]. State of
the art earth models are recommended from time to time as a standard and are
given the name Geodetic Reference System [4.1.3].
The rotational ellipsoid was introduced as a geometrical figure of the earth in
the 18th century, cf. [1.3.2]. By fitting its dimension and orientation to the
geoid, it approximates this level surface within about 100 m. The geometry of
the ellipsoid can be described in a simple manner, together with ellipsoidal
surface coordinates and curvature [4.1.1], [4.1.2]. The use of global and local
three-dimensional ellipsoidal systems provides an approximation to the
corresponding systems of the actual earth and permits the separation between
horizontal position and height [4.1.3].
Geometry and coordinate systems of the ellipsoid are well documented in
geodetic literature, e.g., GROSSMANN (1976), BOMFORD (1980), HECK (1995),
The rotational ellipsoid is generated by rotating the meridian ellipse about its
minor axis. The shape of the ellipsoid is described by two geometric parameters,
the semimajor axis a and the semiminor axis b (Fig. 4.1). Generally, b is
replaced by a smaller quantity, describing the (small) polar flattening of the
4 The Geodetic Earth Model
Fig. 4.1. Meridian Ellipse
ellipsoid, which is more suitable for series expansions. We have the
(geometrical) flattening
f=
the first eccentricity
and the second eccentricity
e=
The following relations hold among those quantities:
From the geometric definition of the ellipse as the curve having a constant value
for the sum of the distances r/ and r2 to the focal points F:
we derive the linear eccentricity
4.1 The Rotational Ellipsoid
Fig. 4.2. Geodetic coordinates latitude and longitude
We now introduce a spatial ,, -Cartesian coordinate system (Fig. 4.2). The
origin of the system is situated at the center of the ellipsoid O. The -axis
coincides with the minor axis of the ellipsoid. The equation of the surface of the
ellipsoid is then given by
X2+Y2
The system of geodetic surface coordinates is defined by the ellipsoidal latitude
and longitude (also geodetic latitude and longitude). #J_is_the angle
measured in the meridian plane between the equatorial plane (, -plane) of
the ellipsoid and the surface normal at P. Longitude is the angle measured in
the equatorial plane between the zero meridian (X-axis) and the meridian
plane of P. Here, is positive northward and negative southward, and is
positive as reckoned toward the east. The ellipsoidal meridian plane is formed
by the surface normal and the -axis, and are defined to have angular
values, but they may also be considered as curvilinear surface coordinates. The
coordinate lines of this orthogonal system are the meridians ( = const.) and the
parallels, or circles of latitude, (<p= const.). With
we introduce the radius of the circle of latitude
Fig. 4.3. Geodetic, reduced, and geocentric latitude
as a new variable (Fig. 4.2). Substituting p into (4.4) and differentiating yields
the slope of the ellipsoidal tangent at P (Fig. 4.3):
b\p
- ==
By combining (4.4) and (4.7), and substituting p with (4.5), the parametric
representation of the meridian ellipse follows:
a2 cos cos
(a2 cos2 + b2 sin2 }
, =
a2 cos sin
(a2 cos2 + b2 sin2
1/2 '
#>j\V
Instead of , other latitude parameters are used for special applications. The
geocentric latitude has already been introduced together with the longitude
and the geocentric distance r as spherical coordinate, cf. [2.5.1]. From Fig. 4.3,
the corresponding equation of the ellipse is given by
= rcosq>, Z =
where p follows from (4.5).
The reduced latitude is obtained by projecting (parallel to the -axis) from
the ellipse to the concentric circle of radius a (Fig. 4.3). Since the ratio of the
elliptical to the circular ordinates is b/a (ellipse as the affine image of the
circle), we have
p = acos , Z=asin
= bsin
Using instead of formally transforms ellipsoidal formulas into spherical
formulas.
Comparing (4.9) and (4.10) with (4.7) provides the transformation between ,
, and :
^ J
tan/? = -1
(4.1 la)
A series expansion yields the differences in the angles:
The maximum difference occurs at = 45, with ( - ) = 690".
The meridians and parallels are the lines of curvature of the rotational ellipsoid.
The principal radii of curvature are therefore in the plane of the meridian and in
the plane of the prime vertical perpendicular to the meridian plane (Fig. 4.4).
The Curvature of_the meridian (M = curvature radius) as a plane curve
= Z(/?) in the Z,p -plane is given by
"'7.....
With (4.7) and its derivative, and taking (4.2) into account, we obtain the
meridian radius of curvature
a(l-e2)
'-^.
Fig. 4.4. Curvature of the rotational ellipsoid
The plane of a parallel circle (oblique section of the rotational ellipsoid) and the
vertical plane in the same tangential direction intersect at point P with the angle
. The theorem of Meusnier (regarding surface curvatures, see, e.g., STOKER
1969) provides the radius of curvature in the prime vertical:
Because of rotational symmetry, the origin of is on the spin axis. Inserting
(4.6) and (4.8) into (4.14), one obtains
A comparison of (4.13) and (4.15) shows that N>M
( = 90), the polar radius of curvature becomes
At the poles
At the equator ($7=0) the values are
4. 1 The Rotational Ellipsoid
0=,=.
The curvature of the normal section of an ellipsoidal, or geodetic, azimuth a is
computed according to Euler's formula by
Here, Ra is the radius of curvature. The geodetic azimuth a is defined as the
angle measured in the horizontal plane between the ellipsoidal meridian plane
of and the vertical plane containing the normal to and the target point; a is
reckoned from north in the clockwise direction. The mean curvature J is given
2(M
N)
The arc lengths of the coordinate lines of the ^>,/i-system are computed using
and N. For the arc elements of the meridian and the parallel respectively, we
obtain (Fig. 4.4)
= Md<p, dL = Ncosq>dA.
With (4.13), the length of the meridian arc (starting at the equator) becomes
Equation (4.21a) can be reduced to an elliptic integral of the second kind, which
cannot be evaluated in a closed form (KUTTERER 1998). Practical computations
may be based on numerical integration (e.g., by Simpson's rule) or on a
binomial expansion of the denominator. Subsequent term-by-term integration
Short arcs ( = 2 - 1 < 1) can be calculated by a rapidly converging Taylor
expansion. Expanding about the middle latitude = \( + 2 )/2 yields
.-,
0 + -.
According to (4.20), the arc length of a circle of latitude between the geodetic
longitudes and 2 is given by
With a = 6 378 137 m, b = 6 356 752 m, and e2 = 0.006 694 380 (for numerical
values see [4.3]), we get for the radii of curvature at the poles and at the equator
c = 6 399 594 m, M0 = 6 335 439 m, 7V0 = a.
The arc lengths along the meridian and the parallel for = 50 are
= l) = 111 229m,
( = 1) = 71 696m,
= 1')= 1853.8m,
( = 1') = 1194.9m,
= l " = 30.90m,
= 1" = 19.92m.
Local approximations to the ellipsoid use the Gaussian osculating sphere of
At the latitude 0 it has the same Gaussian curvature as the ellipsoid.
Global approximations can be based on a sphere with the mean radius
the radius derived from equality of volumes (i.e., volume of sphere = volume of
ellipsoid)
JL=lfcfb,
or the radius for a sphere having a surface area equal to that of the ellipsoid. The
latter one results from an integration over the ellipsoidal surface elements dG
and dL (4.20), which after a series expansion yields
The numerical values for these three approaches agree within a few meters,
which leads to a global value of R = 6371 km.
The ellipsoidal surface coordinate system (,) can be extended to a spatial
system by introducing the height h of the point P above the ellipsoid measured
along the surface normal (Fig. 4.5). The point Q on the ellipsoid thus is obtained
by projecting the point P along the ellipsoidal normal (Helmert projection). The
spatial coordinates ,, are designated as geodetic coordinates.
Fig. 4.5. Spatial geodetic coordinates
The coordinate surfaces ( = const., = const., h = const.) of this system are
orthogonal. The coordinate lines (-line = geodetic meridian, -line = geodetic
parallel, /z-line = ellipsoidal normal) represent planar curves.
In (4.8) we substitute the first eccentricity e2 for the semiminor axis b, taking
(4.15) into account; the coordinate vector for the point Q on the ellipsoid (4.8)
then transforms into
=N
cos0?sin/l
(l - e2 )sin
For the point P, we get according to Fig. 4.5
r = r + An ,
n = cos sin
with the surface normal
+ ) cos sin A
The inverse problem can be solved for and h only by iterative methods. From
(4.27) we get (HEISKANEN and MORITZ 1967, p. 183)
,h = Jx*+Y2
_.
._ _
TjX2+Yz
l-e
\-
The iteration process may start with h = 0, which results in a first approximation
for , and so on. Close to the earth's surface (h.N) the process converges
quickly. Efficient methods have also been developed for large heights
(BORKOWSKi 1989, SJ BERG 1999). Closed formulas with negligible residual
Fig. 4.6. Global and local ellipsoidal system
errors are given by BOWRING (1985). The transformation (4.28) is a standard
problem in satellite geodesy, cf. [6.2.1].
Local ellipsoidal (geodetic) systems are introduced in analogy to the local
astronomic systems, cf. [2.6.2], and represent an approximation to them (Fig.
4.6). With the origin at the point P, the local system is connected to the
ellipsoidal vertical (outer surface normal n to the ellipsoid) through the
geodetic latitude and longitude (4.26). The z-axis is directed towards the
ellipsoidal zenith, with the x,y -plane being perpendicular to it. The -axis
points to the ellipsoidal north (direction of the meridian), and the y -axis points
towards east (left-handed system).
A target point P, is described with respect to by the geodetic (ellipsoidal)
azimuth a, introduced already in [4.1.2], the ellipsoidal zenith angle ,, and the
straight distance s between P and P,. The zenith angle is measured in the
vertical plane between the ellipsoidal vertical and the connecting line and
reckoned positively from the zenith. These polar coordinates can be transformed
into the local Jc,y,z" -system by a relation corresponding to (2.20):
cosorsin
= s sin a sin
After applying the reflection matrix S2 (2.24), the local system is transformed
to the global ,, -system by the rotation matrices R 2 (90-p) and
R3 (180-/I), which correspond to (2.25) and (2.26):
, )
A = R3 (180 - A)R 2 (90 - p)S2
-sin07cosA -sin/, cc
- sin sin cos cos sin
The inversion of (4.32) gives
=-,
A-'=A =
-sin/I
-sin^sin/l
which corresponds to (2.29) and (2.30).
A "normal" gravity field may be referenced to the rotational ellipsoid by
considering the latter to be a "level" ellipsoid. This earth model is now generally
accepted as a geodetic reference system; higher order models do not offer any
advantage [4.2.1]. The external gravity field of the level ellipsoid can be
determined unambiguously from the parameters defining it [4.2.2]. The
geometry of the normal gravity field is of special interest for geodetic
applications [4.2.3].
4.2 The Normal Gravity Field
By adding the total mass M and the rotational angular velocity to the
geometric parameters a and / of the rotational ellipsoid, we introduce an
ellipsoidal gravity field composed of gravitation and centrifugal acceleration:
normal gravity field. In addition, we require the surface of this ellipsoid to be a
level surface of its own gravity field. According to the theorem of StokesPoincare, the gravity field then is uniquely defined in the space exterior to the
ellipsoid.
Theorem of Stokes-Poincare: if a body of total mass rotates with constant
angular velocity about a fixed axis, and if S is a level surface of its gravity
field enclosing the entire mass, then the gravity potential in the exterior space of
S is uniquely determined by M, u), and the parameters defining S.
The earth model thus defined is called a level (or equipotential) ellipsoid.
Instead of a,f, M and co, other sets of four independent parameters may be used
for its definition. If the parameters are given values which correspond to the real
earth, then an optimum approximation to the geometry of the geoid and to the
external gravity field is achieved: mean earth ellipsoid, cf. [6.8.1]. The theory
of the level ellipsoid has been developed by P. Pizetti (1894), C. Somigliana
(1929), and others (HEISKANEN and MORITZ 1967, p.64).
From the physical point of view, an earth model is required which is in
hydrostatic equilibrium. All its level surfaces then coincide with the surfaces of
equal density and equal pressure. Deviations from this model would indicate
stress in the earth's body, cf. [8.1]. The theory of equilibrium figures has been
discussed since the days of Newton and Clairaut, cf. [1.3.2], see LEDERSTEGER
(1956/1969), MORITZ (1990).
In the above definition of the level ellipsoid, nothing has been stated regarding
the interior mass distribution. But from the theory of equilibrium figures, it
follows that only the homogeneous ellipsoids of MacLaurin exist in
equilibrium. On the other hand, the surface of an equilibrium figure constructed
of shells of equal density, and thus corresponding more to the real structure of
the earth, is not an ellipsoid. Nevertheless, a layered structure of the interior
mass of the level ellipsoid that approximates the actual situation, and
sufficiently well reproduces the gravity field of the level ellipsoid, can be found
(MORITZ 1968a). The maximum deviation between the level surfaces and the
surfaces of equal density are on the order of f2 only, and the differences in
stress at the model remain considerably smaller than in the real earth. The level
surface thus can also serve as a bounding surface for a geophysical earth model
(MARUSSI et al. 1974).
There have been several attempts to construct earth models with a better fit to
the geoid and the external gravity field than that provided by the level ellipsoid.
A physical approximation consists of reference figures derived from truncated
spherical harmonic expansions of the gravity potential of the earth: level
spheroids. By assuming symmetry about the equator and truncating at the
degree 1 = 2 (Brims' spheroid) and 1 = 4 (Helmert's spheroid), we obtain
surfaces of the fourteenth and twenty-second order respectively. The deviations
from the rotational ellipsoid having the same axes are on the (/ 2 ) for / = 2
and (/3) for 1 = 4. From the harmonic coefficients C 22 and S 22 (3.102), a
triaxial ellipsoid can also be calculated as a geometrical approximation to the
geoid. This results in a difference of 70 m between the radii of the equatorial
principal axes of inertia (corresponding to an equatorial flattening of 1/90 000),
where the larger radius is directed to 345 longitude (BURSA 1995).
These higher order earth models do not significantly reduce the deviations
between the geoid and the level ellipsoid. On the other hand, computations
related to these surfaces and their gravity fields become more complicated.
Finally, they are not suitable as physical normal figures. Although, for instance,
triaxial rotational ellipsoids exist as equilibrium figures (homogeneous
ellipsoids of Jacobi), such an ellipsoid would yield a completely unnatural form
when using the actual values for the angular velocity and mass of the earth.
The external gravity field of the level ellipsoid (normal gravity field) can be
modeled by closed formulas in the system of ellipsoidal coordinates , ,u. The
reduced latitude and the geodetic longitude have been introduced in [4.1.1].
The third coordinate u is the semiminor axis of the ellipsoid with constant linear
eccentricity , see (4.3), which passes through the point P (Fig. 4.7). From
(4.20), and putting V"2 + 2 for the semimajor axis, the transformation from
the ellipsoidal coordinates to the Cartesian ones is given by
JI + (/M) cos cos
=u
I + (/M) cos sin
For = 0, the /?,/l,u-system with = 90 - and u = r degenerates into the
system of spherical coordinates (2.14).
Fig. 4.7. Level ellipsoid and ellipsoidal coordinates
We denote the vector of normal gravity by and the normal gravity potential
by U. In analogy to (3.43), we have
= grad U .
With respect to the global , , -system, is given in analogy to (3.72) by
cos sin
Corresponding to (3.42), U is composed of the gravitational potential VE and the
potential of the centrifugal acceleration ZE:
= VE+ZE.
The gravitational potential satisfies Laplace's differential equation (3.29) in the
space exterior to the ellipsoid that contains the total mass.
By expressing Laplace's equation in ellipsoidal coordinates, we get a solution
based on ellipsoidal harmonics. Adding the centrifugal potential (3.38), and
taking both rotational symmetry and the condition of the ellipsoid surface as a
level surface into account, we obtain a closed expression for the normal gravity
potential (HEISKANEN and MORITZ 1967, p.64):
arctan- + a2-^-f sin2 --)+ (u 2 +e 2 )cos 2 . (4.38)
Here, q is an auxiliary quantity depending only on the geometric parameters
and u. On the ellipsoid surface (u = ), it is denoted q0:
Hence, in agreement with Stokes' theorem, cf. [4.2.1], the normal gravity
potential is determined by four parameters (a, b, , ). It does not depend on
the geodetic longitude. If one puts u = b and q = q0 in (4.38), then the potential
of the level ellipsoid reads
The normal gravity is perpendicular to the level ellipsoid, so that in
accordance with (4.36), only the orthogonal component appears in the
derivative of U (4.38). If the geodetic latitude is used instead of the reduced
latitude , then for the normal gravity on the ellipsoid we obtain the formula of
Somigliana (1929):
ay. co.' + .*.'
a cos1 + b2 sin
For numerical computations, the form
., ,
Wlth
Yo = Ya 7r
(l-e 2 sin>)
^=ra
,. 4.,,41b
is more convenient (MORITZ 2000).
Here, the normal gravity, which depends on latitude only, is represented by the
four parameters a, b, ya (normal gravity at the equator), and yb (normal
gravity at the pole). The ellipsoidal parameters a, b, , ), , yb appearing in
(4.38) and (4.41) are interrelated according to the theorem of Pizetti
and the theorem of Clairaut
-'2
-1/2
- arctane'-
e sl
Thus, only four independent parameters are remaining. In (4.43), besides the
second eccentricity e1 and the geometric flattening f , there is also the gravity
Y-Ya
'_ /*
The abbreviation is used for both the reduced latitude and the gravity
flattening; confusion is not to be anticipated.
The normal gravity in the exterior space is obtained by partial differentiation of
(4.38). Near the ellipsoid, a Taylor series expansion with respect to the
ellipsoidal height is sufficient, see below.
Application of normal gravity field formulas, (4.38) to (4.43), is often facilitated
by series expansions with respect to/ or some other quantity that describes the
polar flattening.
We start with the spherical harmonic expansion of the gravitational potential.
Due to the symmetry with respect to the rotational axis (tesseral terms = 0) and
the equatorial plane (odd zonal terms = 0), we obtain, upon adding the
centrifugal potential (3.96a) expressed in spherical coordinates, the potential of
normal gravity in terms of Legendre polynomials, cf. [3.3.1],
If 2 is substituted from (3.83a), the expansion up to = 1 (corresponding to / =
2) yields
U= r
a\ T(3 220 1}
JA -cos tf +r
2GM
Solving for r and setting U = Uo gives the radius vector to the level ellipsoid,
where we have put r = a on the right side:
r =
cos2 z?
sin2??
The normal gravity /follows from the derivative of (4.46) with respect to r.
GM_
1-3 - v|-cos^--|GAf
If we substitute either & = 90 (equator) or 0 (pole) in (4.47) and (4.48), then
we obtain either the semimajor axis a and the equatorial gravity or the
semiminor axis b and the polar gravity of the ellipsoid. Using these values, the
geometric flattening f (4.la) and the gravity flattening (4.44) may be
computed according to
= --J22+2m.
m =
is the ratio of the centrifugal acceleration to the normal gravity at the equator.
From (4.48) and (4.49), we arrive at an approximation to the theorem of Pizetti
-/4,
and an approximation to Clairaut's theorem (4.43)
Substituting (4.49) and (4.50) into (4.48), we obtain Newton's gravity formula,
cf. [1.3.2] :
If two y0 gravity values are known on the ellipsoid (gravity reduction problem)
at different geographic latitudes , then and may be computed from (4.53).
With known values for the semimajor axis and the angular velocity <y, (4.50)
supplies the quantity m. Finally, Clairaut's theorem (4.52) yields the geometric
flattening/ which thus can be determined from gravity values. Application of
this principle to the real earth - that is, deriving geometric form parameters from
physical quantities - leads to the gravimetric method of physical geodesy, cf.
The relations above (linear in / , and m) may also be derived by series
expansions of the closed formulas. They had already been found by Clairaut
("Theorie de la Figure de la Terre" 1743). The expansion up to terms of the
order /2 yields (IAG 1971)
fJ = -J,+
+ -Ji+J
2 m +m ,
= -fJ + Lm-YLjm
+m\
< <^
m=-,
One of the first applications of Clairaut's theorem was made by Helmert (1901).
An adjustment to the gravity formula (4.57) of about 1400 free-air reduced
gravity values yielded the parameters = 9.7803ms"2 and = 0.005 302, with
a flattening of / = 1/298.3 .
The harmonic coefficients of second and fourth degree may be computed from/
and m as follows:
J22=-f----f2+
n, J44=--f+-frn.
For today's accuracy requirements, an expansion up to n = 3 (corresponding to /
= 6) is generally adequate. That is, the expansion has to include the terms of the
order /3 (COOK 1959). Developments up to the order /5 have been given by
CHEN (1982).
Near the earth's surface, a Taylor series expansion with respect to the ellipsoidal
height h is sufficient for the derivation of the normal gravity in the exterior
space :
(4 59)
The partial derivative dy/dh is obtained by applying Bums' equation (3.71) to
the exterior space:
where J is the mean curvature of the ellipsoid (4.19). A series expansion up to
the order of/leads to the vertical component of the normal gravity gradient
The second derivative can be derived from a spherical approximation of , see
(3. 17). With
_GM dy _ 0 GM
, ~3~-~2 - 2
Ywe obtain
- Inserting the above into (4.59), with r = a and Y = y0, leads to the normal
gravity as a function of and h:
+- h 2 .
For higher altitudes, has to be derived by differentiating the normal gravity
potential (4.38) either in the /?,A,w-system or, after corresponding
transformation, in the ,, or the #, A, r -system, cf. HEISKANEN and MORITZ
1967, p.228.
With = 9.81ms"2 and a = 6378 km, we get dy/dh= -3.(^ms2/mand
2 / 2 = 1.5xlO~^ms~ 2 /m 2 . More detailed numerical values are given in
[4.3]. In gravity reductions the value -3.0865~2/ is used conventionally.
The geometry of the normal gravity field is represented by the spheropotential
surfaces and the normal plumb lines.
The spheropotential surfaces are surfaces of constant normal gravity potential
/ = C/(r) = const.
With the exception of the surface of the level ellipsoid (U= U0), spheropotential
surfaces deviate from ellipsoids and are not parallel to each other. The normal
plumb lines intersect the spheropotential surfaces orthogonally. Due to the nonparallelism of the level surfaces, they are slightly curved in the plane of the
meridian (Fig. 4.8).
EQUATORIAL PLANE
Fig. 4.8. Spheropotential surfaces, normal plumb line, normal height
In order to describe the geometry of the normal gravity field, "normal" geodetic
coordinates ,,U are introduced. They are defined in analogy to the
"natural" coordinates ,, of the actual gravity field, cf. [3.2.3]. They refer
to the point Q, which is related to the surface point (,,) by the
UQ=WP.
The surface thus defined in a point-wise manner approximates the physical
surface of the earth, with deviations less than 100 m and less than one arcmin
respectively. This surface is called the telluroid (HlRVONEN 1960). It is not a
level surface of the normal gravity field, but it resembles the earth's surface.
The normal geodetic latitude <pN is the angle measured in the meridian plane
between the equatorial plane of the ellipsoid and the direction of the normal
plumb line. It differs from the geodetic latitude , introduced in [4.1.1], by the
small angle . This difference is a result of the plumb line curvature (Fig.
4.8), see below. The normal geodetic longitude is equal to the geodetic
longitude . The normal gravity potential U relates the point Q to the level
surface U = UQ.
Instead of U, the potential difference C/0 - UQ to the level ellipsoid may be
used. With the geopotential number C = W0-WP (3.104), and the condition
UQ = Wp, we also have U0=W0. From this, we may derive the normal height
// , already introduced in [3.4.3], by
Hence, HN is defined as the distance between Q and the level ellipsoid
measured along the normal plumb line. To a good approximation, HN may be
measured along the ellipsoidal normal passing through the surface point.
According to (3.107), is the mean normal gravity between the ellipsoid and
Q. Substituting /from (4.63) into (3.107) and integrating yields
Hence, may be computed rigorously in an iterative manner. Since C can be
derived from measurements, the normal height can be determined without any
hypothesis. Extending the normal heights downward from the earth's surface
yields the quasigeoid, which is used as a reference surface for heights, cf.
The normal geodetic coordinates " ,, have gained special importance
for the direct determination of the physical surface of the earth according to the
theory of Molodenski, cf. [6.5.1]. Normal heights have been introduced for a
number of national height systems, cf. [3.4.3], [7.2].
The curvature of the spheropotential surfaces is described by the second
derivatives of 7, in analogy to the actual gravity field, see (3.57), (3.58). In the
local ellipsoidal system, the curvatures in the direction of the meridian and the
parallel are given by
The geodesic torsion is zero due to the rotational symmetry of the level
ellipsoid:
t?=
^ = 0.
On the ellipsoid, the curvature is given by the principal radii of curvature A/and
TV, see (4.13), (4.14):
1 , *- .1.
Taking the rotational symmetry into account, and following (3.67), we get for
the curvature of the projections of the normal plumb line onto the ," and the
, -plane respectively:
On the level ellipsoid, we have with (4.20)
Introducing the derivative dy/d<p from (4.53) and inserting into (4.71) yields
with sufficient approximation
~ = sin 2^
with = gravity flattening (4.44). For the change of the normal gravity along
the meridian, we thus get
_ . . /* :.
which corresponds to 8.2 nms~ 2 /m at = 45. Together with the relation
U- = -dy/dz , (4.68) to (4.72) completely define the Marussi tensor (3.69) for
the normal gravity field. According to (3.75), the differential transformation
from the local to the global geodetic system is also provided by the curvature
parameters.
Finally, we derive the differences between the geodetic coordinates , and the
normal geodetic coordinates ,:
, = " .
From (4.71) and (4.72) we obtain
dtp" = - J K%dHN = ^-sin 2p/T.
And with /?= 0.0053 and = 6371 km we get
= -0.00017" si 2^? /"',
where HN is in meters.
Geodetic reference systems provide numerical values for the parameters of a
geodetic earth model. The systems are recommended by the International Union
of Geodesy and Geophysics (IUGG) and represent the best parameter values for
a designated epoch. The systems generally serve as a standard over a long time
span for geodesy and related disciplines such as astronomy, cartography,
geophysics, engineering, and navigation. Actual best values are determined at
shorter time intervals and published by the International Association of
Geodesy, cf. [6.8.1].
In the 19th and early 20th century, the geometric parameters of reference ellipsoids were derived
from various terrestrial data sets and then introduced as a reference for national geodetic surveys,
cf. [1.3.3], [7.1.2]. Normal gravity formulae referred to these ellipsoids have been derived since
about 1900 and used for national gravimetric surveys. These regional or local reference systems
4.3 Geodetic Reference Systems
may be regarded as precursors of the present global systems, based on the theory of the level
All reference systems are supposed to be geocentric, with the Z-axes coinciding with the earth's
axis of rotation and the direction of the A"-axis pointing to the Greenwich meridian. While the
earlier reference systems may have large deviations from the geocentric system, recent reference
systems agree at the "cm"-order. The orientation of geodetic systems with respect to the earth is
described by the "Geodetic Datum", cf. [6.2.2].
Geodetic reference systems based on the theory of the level ellipsoid were first
introduced in 1924/1930. At the IUGG General Assembly in Madrid 1924,
Hayford's ellipsoid was introduced as the International Ellipsoid, with the
a = 6378388m, / = 1/297.0.
The General Assembly in Stockholm (1930) adopted the gravity formula
established by G. Cassinis for Hayford's ellipsoid:
Y0 =9.780 49(1 + 0.005 2884sin>-0.000 0059sin2 2) ms' 2 . (4.75b)
This corresponds to the normal gravity formula (4.57), assuming a level
The geometric parameters a and/were calculated by J. F. Hayford (1909) from astrogeodetic
observations in the U.S.A. W. A. Heiskanen (1928) determined the equatorial gravity from an
adjustment of isostatically reduced gravity values. The international reference system of
1924/1930 is thus defined by the four parameters a,f,ya ,ca. The corresponding ellipsoid has been
applied in numerous geodetic surveys; also, the normal gravity formula has found broad
At the General Assembly of the IUGG in Luzern (1967), the 1924/1930
reference system was replaced by the Geodetic Reference System 1967
(GRS67), see IAG (1971). It was defined by the following parameters:
= 6 378160m, GM = 398603xl0 9 mV, J2 =1082.7x10^. (4.76a)
The angular velocity of the earth's rotation
<y = 7.2921151467xlO- 5 rads- 1 ,
not mentioned in the IUGG resolution, was adopted as the fourth parameter.
The reference ellipsoid corresponding to this definition was declared a level
The calculation of the semimajor axis was based on astrogeodetic observations collected over the
entire earth, which were transformed into a uniform system by the inclusion of gravimetric data.
Observations of space probes yielded the geocentric gravitational constant, which includes the
mass of the atmosphere. The dynamic form factor was derived from the orbit perturbations of
artificial satellites, and the angular velocity was adopted from astronomy. The GRS67 has been
used especially for scientific problems and for a number of recently established geodetic
networks.
At the IUGG General Assembly in Canberra (1979), the Geodetic Reference
System 1980 (GRSSO^) was introduced. It is also based on the theory of the
geocentric equipotential ellipsoid, with the defining constants (MORITZ 2000):
a = 6378137 m
equatorial radius of the earth
GM = 398 600.5x10 m s" geocentric gravitational constant of the earth
J/ 2 == l1082.63
usz.w x 10"
(including the atmosphere)
dynamical torm
form tactor
factor ot
of me
the eartn,
(excluding the permanent tidal deformation)
= 7.292 115 10"5 rad s"1
angular velocity of the earth.
With Mm = 0.88x 10"*M, we have GM^ =0.35xl0 9 mV.
The system is consistent with the 1976 IAU system of astronomical constants,
With respect to the orientation, it is stated that the minor axis of the reference
ellipsoid be parallel to the direction defined by the Conventional International
Origin and that the primary meridian be parallel to the zero meridian of the BIH
adopted longitudes, cf. [2.5.2].
The equatorial radius has been derived from laser distance measurements to satellites, satellite
altimetry, and Doppler positioning, with an uncertainty of 0.5 m. The calculation of the
geocentric gravitational constant was based on space probes and lunar and satellite laser data
(0.1xlO~ 9 ), while the value for the dynamic form factor was taken from global gravity models
(51().
Numerical values for derived parameters include (MORITZ 2000):
geometric constants, cf. [4.1.1], [4.1.2]:
b = 6 356752.3141 m
= 521 854.0097 m
c = 6 399 593.6259 m
e2 = 0.006 694 380 022 90
e'2 = 0.006 739 496 775 48
1.003352810681 18
/ = (
1/f == 298.257
G = 10 001 965.7293 m
semiminor axis
linear eccentricity (4.3)
polar radius of curvature (4.16)
first eccentricity (e) (4.1b)
second eccentricity (eO (4.1c)
flattening (4. la)
reciprocal flattening
meridian quadrant (4.21a)
physical constants, cf. [4.2.2]:
U0 = 62 636 860.850 m2 s'2
J4 = -0.00000237091222
J6= 0.00000000608347
Jt = -0.00000000001427
m = 0.003 449 786 003 08
ya= 9.7803267715ms2
Yb= 9.8321863685ms' 2
= 0.005302440112
k= 0.001931 851353
normal potential at ellipsoid (4.40)
spherical harmonic coefficient (4.45)
normal gravity at equator (4.41)
normal gravity at pole
Normal gravity can be computed by the closed formula (4.41) or the series
r0=r
l+ 0.005 279 0414sin> +0.000 023 2718sin>
+ 0.000 0001262 sin> +0.000 000 0007 sin>
which is accurate to 10 3 5 2 . The conventional series (4.57) has an accuracy
of only "2 :
0 = 9.780327(1 + 0.005 3024sin>-0.0000058sin22p) ms" 2 . (4.78b)
Inserting the values for the GRS80 into (4.63) yields the change of normal
gravity with height:
-4.3x10
with h in km. A development accurate to lOnms" 2 for heights up to 10 km is
given by WENZEL (1989).
According to the definition of GM, 0 refers to the total mass of the earth
including the atmosphere. If normal gravity values are required on the ellipsoid,
or within the range of the atmosphere, the effect of the air masses above the
calculation point must be subtracted from 0. The corresponding reduction
amounts to -8.78~2 (h = 0), -4.75~2 (/j = 5km), and -."2 ( =
30 km), ECKER and MITTERMAYER (1969).
5 Methods of Measurement
Modeling of geodetic parameters is based on observations taken on the earth's
surface and in it's exterior space. Different measurement methods are available,
delivering geometric or physical quantities. Geometric methods rely primarily
on electromagnetic waves and thus are influenced by atmospheric refraction
[5.1]. The measurement methods may be divided into
observations employing artificial satellites as targets (including the moon)
or sensors: satellite observations [5.2],
observations to fixed stars and extragalactic radio sources: geodetic
astronomy [5.3],
terrestrial gravity and gravity gradient measurements: gravimetry [5.4],
determination of coordinate differences between points on the surface of the
earth: terrestrial geodetic measurements [5.5].
The measurement methods depend on available technology, where electronics
governs data collection and online data-processing (KAHMEN 1978,
SCHLEMMER 1996). Satellite techniques now dominate global and regional
surveys, while terrestrial methods are used for interpolation in space and time.
Accuracy limits are governed by calibration errors and elimination or reduction
of disturbing influences such as temperature, atmospheric pressure, magnetic
field variations, microseismicity, atmospheric refraction, and local instabilities.
Thus the inherent precision of the respective technique may deteriorate by a
factor of two to three, or more.
5.1 Atmospheric Refraction
In practically all geodetic measurements, electromagnetic waves serve as signal
carriers; this includes the methods of satellite and terrestrial geodesy as well as
geodetic astronomy. From the broad spectrum of electromagnetic waves, the
visible light (380 to 780 nm, corresponding to 7.9 to 3.8-1014Hz), the near
infrared (up to 1 ), and the microwave parts (1mm to 1 m, corresponding to
300 GHz to 300 MHz) are used. When traveling through the atmosphere, the
waves experience changes in velocity and curvature of the path, depending on
the physical state of the atmosphere [5.1.1]. Signal propagation is different in
the troposphere and the ionosphere and has to be treated separately [5.1.2],
Different methods have been developed in order to eliminate or reduce the
effects of atmospheric refraction on geodetic measurements. These include
instrument design, observation methodology, and the use of atmospheric models
based on data collected on the earth's surface and in space. The individual
strategies will be discussed in the chapters that pertain to measurement
methods, see also BRUNNER (1984a), DE MUNCK and SPOELSTRA (1992).
Acoustic waves are employed for positioning on, and mapping of, the ocean floor and the bottom
of rivers and lakes. The propagation of these waves through water differs from their behavior in
vacuum and depends on water properties, cf. [5.5.2].
5.1.1 Fundamentals
According to Fermat's principle, the path s of an electromagnetic wave is
determined by the condition of a minimum travel time of the wave (MORITZ
and HOFMANN-WELLENHOF 1993, p. 158):
= J < # = J = min.
The velocity differs from the velocity in vacuum c (2.2) by the index of
refraction n (also called refractive index):
For a gaseous medium, > 1 is proportional to the density of the gas. If the
medium is dispersive for a certain spectral domain, n also depends on the
wavelength: dispersion. An average value for near the earth's surface is
1.0003. Instead of n, the refr activity
is frequently used.
Inserting (5.2) into (5.1) yields
At = \ nds = min .
rC J
n ds = ds ,
(5.4) can also be expressed as a minimum condition for the "electromagnetic"
path length (Fig. 5.1):
s = \ ds = \ nds = min .
Solving the variational problem (5.5) yields the path s , but requires the
knowledge of n along the path.
n = const.
n + dn = const.
grad n
Fig. 5.1. Ray bending in the atmosphere
The effect of refraction on distance is given by the difference between the
actual path length J and the straight-line s (chord):
~s -s = \ n ds\ds = \{n \)ds +
\ n ds \ n ds \.
The first term on the right side accounts for the difference in length due to the
longer travel time in the atmosphere, while the second term represents the effect
of the bending of the ray (JANES et al. 1991).
The refraction effect on curvature can be estimated by assuming that the air
density is stratified horizontally (Fig. 5.1). Snell's law describes the bending of
the ray as it passes through layers of varying refractive index, which
corresponds to Fermat's principle:
n sin z = const.
Or for two points Pl and P2:
(5 .7b)
, sin Zj = n2 sin z2 .
Under the above assumption, the angle between the normal to the surface n =
const, and the tangent to the ray with curvature 1/r is the zenith angle z.
Differentiation of (5.7) yields
sin z dn + ncosz dz = 0 .
dn = (grad n ) ds = |grad n\ cos z ds ,
we obtain the curvature
1 dz
|grad n\ .
- = = ---L smz.
r ds
By separating the horizontal and the vertical component of grad n, we get the
curvatures of the ray projected into the horizontal and the vertical planes. The
corresponding effects on horizontal and vertical angles are called horizontal
(lateral) and vertical refraction, respectively.
Horizontal refraction is about one to two orders of magnitude less than vertical
refraction. Neglecting it yields a simplified formula for the curvature of vertical
refraction:
1 dn .
- = --- -sinz,
n dn
.. . .
where h is the height, cf. [4.1.3].
In terrestrial geodetic measurements, we have n ~ 1 and z ~ 90 , which leads
=- .
(5.9b)
Instead of 1/r , the coefficient of refraction k is often used. It is defined as the
ratio between the radius of the earth R and the curvature radius r.
The vertical refraction angle is the effect of refraction on observed zenith
angles (Fig. 5.2). It results from integrating 1/r resp. dn/dh along the path:
If/
\dn ,
x = V(5-5 J)as.
LOCAL HORIZONTAL PLANE
Fig 5.2. Vertical refraction
Here the local vertical gradient of is weighted according to the distance from
the observer; values from closer distances receive a greater weight. For a
spherical arc (r = const.), and taking (5.10) into account, (5.1 la) reduces to
0 =s.
(5.11h)
In most geodetic applications, the signal is transferred by a modulation of the
carrier wave. This can be regarded as a superposition of a group of waves with
different frequencies. While phase velocity v ph , introduced in (5.2), refers to the
monochromatic carrier wave, the center of a short wave group (signal energy)
travels with the group velocity
In a dispersive medium, we have = () and /v ph /c//l*0 (LEICK 1995).
Taking (5.2) into account delivers the corresponding group refractive index
with/= frequency, cf. [5.1.2], [5.1.3].
For standard air (temperature 273.15 K, air pressure 1013.25 hPa, humidity 0.0
hPa, CO2 content 0.0375 %), the phase refractivity may be calculated as follows
(IAG resolution, Gen. Ass. Birmingham 1999):
where is the carrier wave length in , and ph is the corresponding phase
refractive index. The group refractivity is given by
with wgr = group refractive index.
According to (5.6) and (5.11), the refraction effect on distances and angles
depends on the index of refraction and its gradient along the path of the ray,
which behave differently in the troposphere than in the ionosphere.
5.1.2 Tropospheric Refraction
The troposphere is the lower layer of the atmosphere. It extends to a height of
about 9 km at the poles and 16 km at the equator. Weather processes take place
in tropospheric region, where nearly 90% of the atmospheric masses are
concentrated. The tropopause as a boundary layer separates the troposphere
from the stratosphere, which extends to about 50 km. The troposphere,
tropopause, and stratosphere are electronically neutral. The index of refraction
depends on temperature T, pressure and humidity e. For visible light, the
troposphere behaves as a dispersive medium, cf. [5.1.1]. The refractive index
decreases with height and becomes nearly 1 at about 40 km. Tropospheric
refraction is the combined effect from the ground to this "effective" height.
Above 50 km the atmosphere is ionized, cf. [5.1.3].
The meteorological parameters T, p, e not only depend strongly on height but
also on latitude, land/ocean distribution, topography, vegetation, and local
conditions. These variables produce large to small-scale anomalies of n.
Variations with time are of long-term, seasonal, daily, and turbulent character.
Rapid fluctuations are especially pronounced close to the earth's surface, up to
10 to 30 m above the ground.
Temperature decreases in the troposphere almost linearly with height h according to
dT/dh = -0.0065C/m ,
(5. 16a)
followed by a slight increase in the stratosphere. Horizontal temperature gradients may reach a
few "C/100 km. Close to the ground, temperature variations are pronounced, including
temperature inversion during night time and convection at noon.
Air pressure p decreases exponentially with height. Assuming hydrostatic equilibrium, the
vertical pressure gradient depends on density p and gravity g. Near the surface of the earth, this
leads to
dp/dh = -pg = -0.034- = -0. 12 h Pa/m
(5. 16b)
at standard conditions (7= 288 K,p = 1013 hPa).
Humidity is rather irregularly distributed and concentrated in a layer of a few km above ground,
where strong variations also occur with time. Humidity is measured by the water vapour pressure
e, which is about 10 to 20 hPa at mid-latitudes close to the surface. It tends to decrease with
height, with a mean value of
de/dh = -0.0035 h Pa/m
(5. 1 6c)
near the ground.
Empirical relations have been derived between the index of refraction and the
meteorological parameters for both light and microwaves (BOMFORD 1980,
p.42, LANGLEY 1998).
With atmospheric conditions different from the standard air, cf. [5.1.1], the
group refractivity of visible light and near infrared waves in ambient moist air
is (IAG resolution, Gen. Ass.Birmingham 1999):
1013.25 T
-11.27-,
with Tin Kelvin,/? and e in hPa. Equation (5.17) is also valid for unmodulated
light with the corresponding phase refractivity (5.14).
The refractivity of microwaves (independent of the wavelength) is given by the
formula of Thayer (1974)
#. =( -!)106 =77.60 -13 + 3.78- 105-^,
rji
yi2
which is practically identical with the formula of Essen and Froome (IAG
resolution, Gen. Ass. Berkeley 1963).
The first term on the right side of (5.18) represents a "dry" component of the refractivity. It
contributes about 90% to the total tropospheric refraction in the lower 15 km and can be modeled
from surface pressure, assuming hydrostatic equilibrium. The "wet" component, as expressed by
the terms depending on e (especially the last one), is extremely difficult to model but approaches
zero at around a height of 10 km.
In order to keep the error in the index of refraction less than 10"6, the meteorological parameters
in (5.17) and (5. 18) have to be determined to about 1C for temperature, 3.5 hPa for pressure,
and 25 hPa (light) resp. 0.2 hPa (microwaves) for humidity.
Differentiating (5.17) and (5.18) with respect to the height h yields the
dependence of the curvature on the meteorological parameters. Neglecting
minor terms and taking (5.16b) into account, we obtain for the surface near
(5.1*)
for light. For microwaves, the last term on the right side (wet component^
Thus according to (5.10), the coefficient of refraction k has to be determined for light with an
accuracy of 2C in temperature, 6 hPa in air pressure, and 0.0002C/m in the temperature
gradient in order to achieve a relative accuracy of 1%. For microwaves, the admissible errors may
be two times larger. The gradient of the water vapor pressure should be determined with 0.005
hPa/m for light and 0.0001 hPa/m for microwaves. Hence, the most critical parameters are the
vertical gradients of temperature and, especially for microwaves, of the water vapor pressure.
According to (5.1 Ib), an error of 1% in k would produce an error in the refraction angle of 0.2"
over a distance of 10 km and 0.4" over 25 km.
In the layers close to the ground, the strong variations of the meteorological
parameters in space and time lead to corresponding changes in the coefficient of
refraction, with pronounced seasonal and day/night variations (HOEPCKE 1966).
Under average daytime conditions with a clear sky, and for heights between 40
m and 100 m above the ground, we have for light
,=0.13 or /}=8A,
and for microwaves
*m=0.25orrm=4A.
Tropospheric models generally assume concentric spherical layers and
azimuthal symmetry and neglect variations with time. Global models are
provided by standard atmospheres in the form of vertical profiles for
temperature, pressure, and density. The U.S. standard atmosphere (1976)
approximates mean mid-latitude conditions for dry air. Latitudinal and seasonal
departures are given by supplements, NOAA (1966, 1976).
More refined refraction models have been developed for the reduction of
geodetic measurements; these models employ observed surface data as input
(HOPFIELD 1969, SAASTAMOINEN 1972/1973, NEILL 1996). A "mapping
function" takes into account the increase in refraction with larger zenith angles
z. A simple approximation is given by l/cos z for elevation angles that are not
too small.
The path delays that GPS (Global Positioning System) signals experience when passing through
the atmosphere, cf. [5.2.5], can be exploited for tropospheric and ionospheric monitoring (AVIS
et al. 1996). By separating the "dry" component from the tropospheric signal delay, the integrated
perceptible water vapor can be estimated from the "wet" component (Bevis et al. 1992).
Permanent GPS networks, cf. [7.3.1], supply this information on global and regional scales, with
high temporal resolution.
Atmospheric sounding by radio occupation is possible with spacebome GPS receivers installed
on Low Earth Orbiters (LEO's), YUNK and MELBOURNE (1996). Here, the GPS signal is tracked
after rising or before setting of the GPS satellite. The observed Doppler shift induced by
atmospheric bending is used for constructing profiles of atmospheric density, pressure,
temperature, and water vapor content, and evaluated for developing tropospheric models.
5.1.3 Ionospheric Refraction
As part of the higher atmosphere, the ionosphere is characterized by the
presence of free, negatively-charged electrons and positive ions. lonization is
caused primarily by the impact of solar ultraviolet radiation and consequently
depends on the density of the atmospheric gas and the intensity of the radiation.
The ionosphere covers the region between 50 km and 1500 km above the earth,
with a maximum electron density at a height of 200 to 300 km.
The ionosphere acts like a mirror at frequencies below 30 MHz. Radio waves of
higher frequencies pass through the ionosphere but experience frequencydependent effects (dispersive medium). Measurements to targets above the
ionosphere are also affected by the electron concentration in the plasmasphere,
which extends up to a height of several earth radii above the equator and does
not exist at the poles (KLOBUCHAR 1991, WANNINGER 1995).
The index of refraction depends primarily on the number Ne of electrons per
m3 : electron density. As a first order approximation, the phase refractive index
is given by
nph=l-K^,
with the constant K = 40.28 mV2 and/= frequency. Higher terms of the order
I//3 and I//4 also depend on the intensity of the earth's magnetic field and the
direction of the signal propagation. In daytime, Ne ( el/m3 ) varies between
about 108 ...1010 (60 to 90 km) over 1011 (105 to 160 km) to 11 ...1012 (160 to
180 km) and 1012 (300 to 400 km).
As seen from (5.2) and (5.21), the phase velocity is larger than the velocity of
light in vacuum, which corresponds to a larger wavelength of the signal
compared to vacuum. Since signal propagation follows the group velocity, we
insert (5.21) into (5.13) and obtain the group refractive index
ne=\ +
Inserting (5.21) resp. (5.22) into (5.6) delivers the difference between the
electromagnetic path length s" and the straight-line connection s. This yields for
carrier phase (ph ) and for range (ngr ) observations respectively:
, 2 J *.
where the small effect of the path's bending has been neglected.
The integral of the electron density along the path is called total electron
content (TEC):
It gives the number of electrons along the signal path in a cylindrical column
with a cross section of 1 m 2 . Its unit is 1 TECU = 1016 electrons/m2. For a
spherically-layered ionosphere, we may introduce the electron content along a
vertical column of height h and relate it to the TEC along the path by an oblique
factor F ("mapping function"):
For z < 70, we haveF ~ I/cos z( , with z, = zenith angle at the subionospheric
point P{ (Fig. 5.3). Pi is located at the "mean height" h, of the ionosphere
(single-layer model), with, e.g., ht = 350 km. z, can be calculated from h, and
the zenith angle z measured from the ground:
sin z, = - sin ,
R = radius of the earth.
SATELLITE,
IONOSPHERE
Fig. 5.3. Ionospheric refraction
The electron content in the ionosphere varies strongly with time. There are pronounced variations
of daily, seasonal, and about an 11-year (solar activity cycle) period. Superimposed on to these
more regular variations are irregular disturbances. Short-term scintillations occur primarily in the
equatorial zones but also in the polar and auroral regions (magnetic storms). Traveling
ionospheric disturbances of wavelike structure proceed with horizontal speeds between 100 and
1000 m/s at scales of some 10 to 1000 km and at periods from several minutes to a few hours.
Ionospheric models describe the distribution of Ne in space and time; they are
based on the dependency of the ionospheric state on the position of the sun.
Assuming a spherical shell distribution, they provide a smoothly varying TEC
along vertical profiles and a mapping function for inclined signal propagation.
Among these models are the International Reference Ionosphere (IRI) 1995 and
the GPS-broadcast model (KLOBUCHAR 1991). These models may deviate from
reality by some 10%, due to the ionospheric disturbances. Refined models
include observed ionospheric parameters and sunspot numbers (BROWN et al.
1991, KLEUSBERG 1998). The dispersion effect, on the other hand, allows us to
eliminate most of the ionospheric refraction effects by employing two
frequencies for signal propagation, cf. [5.2.4], [5.2.5], [5.2.7].
The ionospheric delay of GPS signals provides information on the structure and temporal
behavior of the ionosphere. By analyzing the two carrier waves used to eliminate ionospheric
refraction, cf. [5.5.2], the total electron content (TEC) along the line of sight from the receiver to
the GPS satellite can be measured. The results of global GPS ground networks are used to
generate global maps of TEC. Space-based GPS data, cf. [5.1.2], will significantly improve this
ionospheric imaging, especially the vertical resolution (YuNK and MELBOURNE 1996).
Satellite geodesy utilizes artificial satellites and the moon as extraterrestrial
targets or sensors. For a point mass earth model, the orbital motion of a satellite
is described by Kepler's laws [5.2.1]. The actual gravitational field and nongravitational forces create orbital "perturbations" [5.2.2]. Satellites used for
geodetic applications differ in design, equipment, and orbit according to the
respective observation technique and the mission purpose [5.2.3]. Classical
measurement methods, introduced and employed from the 1960's to the 1980's,
demonstrated the efficiency of satellite observations for establishing largeregion geodetic control networks and gravitational field determination [5.2.4].
Today, the Global Positioning System (GPS) governs three-dimensional
positioning at all scales [5.2.5], while laser distance-measurements primarily
contribute to global reference networks [5.2.6]. By monitoring the ocean
surface, satellite altimetry contributes to gravity field modeling [5.2.7], and
high-resolution global gravity field recovery is expected from future satellite-tosatellite tracking and gravity gradiometry missions [5.2.8].
The theory of satellite orbits and satellite measurement methods are treated in
textbooks and monographs on satellite geodesy, e.g., KAULA (1966),
SCHNEIDER (1988), SEEBER (1993).
5.2.1 Undisturbed Satellite Motion
After the satellite has separated from the carrier, it begins its unrestrained
revolution about the earth. We assume the gravitational point mass model, cf.
[3.1.2], and neglect the mass of the satellite with respect to the earth's mass. If
we also neglect perturbations of non-gravitational type and the effect of other
celestial bodies (two-body problem), Newton's second law of motion provides
the equation of motion in the gravitational field:
,_ _.
r is the geocentric position vector of the satelh'te and r = d2r/dt2 its
acceleration, M and V are the mass and the gravitational potential of the earth
respectively, cf. (3.16). The integration of the vectorial second-order differential
equation (5.27) introduces six integration constants, e.g., position and velocity
at a given epoch.
The fundamental theory on the two-body problem is given by celestial mechanics (BROUWER and
CLEMENCE 1961, KOVALEVSKY 1989, SCHNEIDER 1992/1993). Such works also address the
fundamentals of orbit perturbations, orbit computation, and the treatment of three and multi-body
Johannes Kepler (1571-1630) derived three laws of planetary motion from the
astronomic observations collected by Tycho de Brake (1546-1601). Keplerian
motion provides a geometric description of the equation of motion for an
undisturbed central motion.
According to Kepler's laws, the satellite moves in an elliptical orbit. One focal
point of the ellipse, with semimajor axis a and first eccentricity e (not to be
confused with the corresponding parameters of the earth ellipsoid), coincides
with the center of mass of the earth. In the orbital system (Fig. 5.4), the position
of the satellite is described by the distance r from the center of mass and the
true anomaly v. The true anomaly is the geocentric angle between the directions
to the satellite and perigee. Instead of v, the eccentric anomaly E can be used,
with the relations
cos E - e
With Kepler's third law, the mean (angular) velocity
is introduced, describing a mean orbital motion. The mean anomaly
M = n(t-T)
SATELLITE (t)
v(t)
PERIGEE
Fig. 5.4. Satellite orbital system
represents yet another parameter for describing the satellite's position in the
orbit. It is generally preferred because it increases linearly with time. T is the
epoch of the passage through the perigee, the closest approach to the earth.
From , can be computed iteratively using Kepler's equation:
M = E-esinE.
The orbital system is transformed into the space-fixed equatorial system, cf.
[2.4.1], by three rotations (Fig. 5.5). The right ascension of the ascending node
and the inclination i provide the orientation of the orbital plane in space. The
argument of perigee orients the ellipse in the orbital plane. From the result of
this transformation, we obtain the geocentric position vector (2.11) as a function
ELLIPSE (a,e)
Fig. 5.5. Orbital and equatorial system
of the six Keplerian elements a, e, , , , and (or equivalently , , or 7):
cos (co + )cos - sin (o> + v)sin
cos J sin
sin (5
cos /
,(5.32a)
1 + ecosv
The six Keplerian parameters completely describe the orbital motion of the
undisturbed satellite. They correspond to the integration constants of the
equation of motion (5.27) and are used for the approximation of satellite orbits.
5.2.2 Perturbed Satellite Motion
The actual orbit of a satellite departs from the Keplerian orbit due to the effects
of various "disturbing" forces. This includes the non-spherical parts of the
earth's gravitation, the gravitational effects of moon and sun, atmospheric air
drag, and solar radiation pressure, among others. These disturbing forces cause
variations in time in the orbital elements (orbital perturbations) of secular, long,
and short-periodic nature. The actual orbit can be viewed as the envelope of
Keplerian ellipses, which are given at each instant by the actual orbital elements
(osculating ellipses).
In order to account for the complete gravitation of the earth, the gravitational
potential of a spherically symmetric earth must be amended by the perturbing
potential R (not to be confused with the disturbing potential [6.1.1]):
According to equations (3.89) to (3.91), R can be expressed by a spherical
harmonic expansion of the gravitational potential V through the harmonic
coefficients Jlm, Klm (I > 2). By substituting (5.33) into (5.27), the equation of
motion now reads
- + grad R.
r2 r
The spherical coordinates ,&, (2.14) used in the expansion of V can be
replaced by the orbital elements according to (5.28) and (5.32), see also Fig.
5.4. R then is described by the time-variable Keplerian orbital parameters and
the harmonic coefficients:
The second-order differential equation (5.34) can be transformed into a system
of first-order differential equations. They represent the time rates of the orbital
parameters as a function of partial derivatives of the perturbing potential (i.e., of
the harmonic coefficients) with respect to them. These first-order differential
equations are known as Lagrange's perturbation equations (KAULA 1966, p.29,
SEEBER 1993, p.75):
2 dR
na dM
l-e2 dR
na2e dM
Vl-e 2 dR
na2e du)
cos/
dt " na2^di
dt na Jl-e sin / o(0
/1-e2 dR
na2 vl - e2 sin / "
na e\jl - e sin / d/
na2e de
na da
The influence of the gravitation of moon and sun on a satellite can be calculated
by corresponding extension of (5.27), which leads to the equation of motion for
a four-body problem. As a result, a satellite orbit experiences secular and longperiodic perturbations, which may reach 100 m and more. In addition, solid
earth and ocean tides, cf. [8.3.5], especially affect low-flying satellites.
Corresponding corrections are based on the ephemeris of the moon and sun and
on earth and ocean tide models.
Air drag is caused by friction of the satellite with atmospheric particles. It is
proportional to the velocity of the satellite and depends on atmospheric density
and the effective cross-sectional area to mass ratio. With increasing altitude, the
air drag decreases rapidly and approaches zero at about 1000 km. Air drag
effects are corrected using high-altitude atmospheric models such as CIRA72
(Cospar International Reference Atmosphere 1972), cf. [5.1.2].
High-altitude satellites are especially affected by solar radiation pressure due
to incident photons. The resulting perturbations depend on the attitude of the
satellite with respect to the sun, the area to mass ratio, the reflectivity, and the
solar flux. The earth-reflected solar radiation pressure (albedo) remains
significantly smaller than the direct effect. Modeling of the radiation pressure
effects is difficult, especially for satellites of complex structure.
Electromagnetic interactions with the magnetic field of the earth occur in the
ionosphere; however, they are small and can be considered by corrections. At
the present level of accuracy, relativistic effects also have to be taken into
Orbit determinations are based on analytical or numerical methods
(BOCCALETTI and PUCACCO 1996/1999). For analytical solutions, all acting
forces are expressed by rigorous relations and integrated in closed form. A first
order approximation is already provided by the solution of (5.35). The position
vector at any epoch t is given by the orbital elements at an initial epoch / 0 , the
parameters of the gravitational field, and other models of disturbing forces:
r = r(a0,e0&Q,i0,a)0,M0;GM,Jlm,Klm;...;/).
For numerical methods, all forces are calculated for a particular position of the
satellite and used as a starting condition for a step-wise integration. Classical
astronomic methods for orbit determination are used, as developed by Cowell
(integration of the total force) or Encke (integration of the difference to an
osculating Kepler ellipse). Conventional integration methods based on a
polynomial fit to consecutive points (e.g., the Runge-Kutta method) provide a
rigorous solution.
The analytical method is rather laborious, and difficulties arise at applying it to
non-gravitational forces. It is well suited for estimating the effects of perturbing
forces on the satellite's orbit and for the planning of satellite missions and
projects. Numerical methods are simple and generally applicable. They are used
nearly exclusively today.
With precise models for the perturbing forces, orbit predictions can be extended
to several revolutions of the satellite. Predicted orbits for individual satellites
are published by the responsible agencies, while navigation satellites transmit
their own orbital data, cf. [5.2.4], [5.2.5]. Precise orbit determinations are
necessary for positioning and satellite altimetry. On the other hand, an orbit
analysis can be used for deriving or improving the harmonic coefficients of the
gravitational field, cf. [6.6.2].
Under good satellite tracking, and with precise models of both the gravitational
field and the non-gravitational forces available, the ephemeris of laser and
altimeter satellites can be determined with an accuracy of a few cm for short
arcs (fractions of a day) and with dm-accuracy for long arcs (few days). The
orbital data of high-altitude navigation satellites are predicted with an accuracy
of a few m, and orbital post-processing using tracking data of dedicated
networks reaches sub-dm accuracy, cf. [5.2.5].
5.2.3 Artificial Earth Satellites
Since the launch of Sputnik I (1957), artificial earth satellites have been used
for geodetic purposes such as positioning and the determination of the earth's
gravity field and rotation parameters. Only a few satellite missions have been
designed exclusively for geodetic applications. However, a large number of
satellites developed for navigation, remote sensing, and geophysics were and
are used extensively in geodesy.
A satellite can be regarded as a moving target at high altitude and then used
mainly for positioning. Because the satellite's orbit is affected by the
gravitational field of the earth, the satellite may also serve as a sensor for
gravitation. Satellites may reflect incident light only (passive satellites), or they
may carry on board subsystems such as transmitters/receivers, different type
sensors, clocks, and computers (active satellites). In the latter case, an energy
supply is required, and lifetime is rather limited.
The mean orbital velocity of a satellite moving in an approximately circular
orbit (r = a) is given from (5.29) by
For a satellite close to the earth (h = 1000 km), we obtain, with r = R + h= 7370
km, a velocity of 7.4 km/s. Kepler's third law yields the period of revolution U
= 2xr/v = 104 min.
The intersection of the orbital plane with a non-rotating earth represents a great
circle on the earth's surface: subsatellite track. The rotation of the earth causes a
western displacement of subsequent satellite orbits (Fig. 5.6), with a shift on the
equator given by
sidereal day
= 15 0 /] = 0.25/].
Fig. 5.6. Subsatellite tracks
The latitude range of the subsatellite tracks is determined by the inclination of
the satellite.
The following aspects have to be considered during the design (choice of orbital
parameters) of satellite missions for geodetic applications:
For positioning, the network geometry of the ground stations and the satellites
plays a primary role. Simultaneous direction measurements from two ground
stations to a satellite form a plane, and the intersection of planes provides
positions. Range measurements utilize the intersection of spheres, whereas
range differences, derived from Doppler-frequency shifts, use the intersection of
hyperboloids. Satellites at high altitudes are less influenced by gravitational and
air drag perturbations and therefore preferred.
Low altitude satellites are required for the determination of the gravitational
field. This is mainly due to the attenuation factor (ajr) in the spherical
harmonic expansion of the gravitational potential (a e = semimajor axis of the
earth ellipsoid), cf. [3.3.2]. Consequently, the relative errors of the harmonic
coefficients increase rapidly at higher degrees /. For / = 20, the amplitude of the
corresponding structure of wavelength 2000 km is only 5% at a satellite altitude
of 1000 km, as compared to the earth's surface. This attenuation is increased by
the fact that the harmonic coefficients become smaller with increasing degree
and order, cf. [6.6.2]. In order to resolve the gravitational field with a certain
degree / at the equator, it follows from (5.38) that the satellite has to perform 21
revolutions/day, or a longer observation time is required. As seen from (5.35),
the recovery of the harmonic coefficients strongly depends on the satellite's
inclination. Corresponding satellite coverage is needed in order to avoid ill
conditioning; too small inclinations and eccentricities must be avoided.
Gravitational field determination can be based on range and range-rate
measurements between satellites and ground stations, as well as between
satellites. A high field resolution is achieved by gravity gradiometers carried on
board low-flying satellites. Satellite altimetry provides the distance to sea level
and thus a high-resolution approximation to the geoid, cf. [3.4.2]. Both an
orientation into the vertical and a high orbital accuracy is required in that case.
Non-gravitational perturbing effects can be reduced by a small cross-sectional
surface and large mass; a spherical shape offers special advantages.
Atmospheric drag and solar radiation pressure may also be compensated by a
drag-free system. In such a system, a proof mass is shielded by a shell attached
to the satellite. The mass is affected only by gravitation, while surface forces act
in addition on the shell. By continuously measuring the position changes
between proof mass and shell, a feedback system keeps the satellite centered on
the proof mass.
In order to detect variations with time, in position (point velocities), and in the
gravitational field, long-term observations have to be carried out, preferably by
the same satellite system.
Satellites used in geodesy may be equipped with the following techniques, of
which combinations are used in many missions:
Direction measurements have been made available by a sun-light reflecting
skin (early balloon satellites), by flashing light devices, and by mirror
arrays, cf. [5.2.4],
transmitters/receivers serve for the continuous emission/reception of
modulated radio waves used for distance and Doppler measurements, cf.
[5.2.4], [5.2.5],
retroreflector arrays of fused silica corner cubes reflect laser light pulses
and are employed for laser distance measurements, cf. [5.2.6],
vertical distance measurements to the ocean surface are performed by radar
altimeters, cf. [5.2.7],
gravitational gradiometers measure the gravitational gradient within the
body of the satellite, cf. [5.2.8].
Dedicated systems have been developed specifically for precise orbit determination. They consist
of space, ground, and control segments. The French DORIS (Doppler Orbitography and Radio
positioning Integrated by Satellite) system is based on a worldwide network of beacons emitting
radio signals (2 GHz) that are received and processed as Doppler frequency shifts on board the
satellite. The German PRARE (Precision Range and range Rate Experiment) is a space borne,
two-way, dual-frequency microwave system (2.2 and 8.5 GHz), where the double signal travel
time is calculated in the space segment. These systems have operated on different satellites since
the 1990's, cf. [5.2.7], and are also exploited for positioning of the ground stations. The radialorbit accuracy achieved a few cm to 1 dm, and velocities are derived to within 0.1 to 0.3 mm/s.
GPS receivers are increasingly installed on satellites for precise orbit determination; they deliver
similar accuracies to those discussed above (YUNCK and MELBOURNE 1996).
High demands are placed on the determination of time. At orbital velocities of
several km/s, the time epoch has to be determined to 1 $ in order to keep
orbital errors less than 1 cm. Distance measurements to satellites require time
interval measurements to 0. 1 ns in order to obtain cm-accuracy. Rubidium or
cesium frequency standards, which are tied to UTC by time signals, are capable
of this level of accuracy. Quartz oscillators can be used in satellite receivers if
an external control is provided, e.g., through the satellite system, cf. [5.2.5].
More details on satellites employed in geodesy are given in the following
chapters.
5.2.4 Direction, Range and Range Rate Measurements: Classical Methods
Satellite observations began in 1957. They were based partly on methods
developed for the observation of the moon and of balloons. While some of the
classical techniques applied up to the 1980's are no longer of importance, other
such methods have been developed further and are still used extensively today.
Some results of the early satellite missions are still of relevance for the
strengthening and orientation of geodetic networks and for the calculation of
earth models, cf. [6.8.1].
Direction measurements to satellites prevailed until about 1970 and led for the
first time to global and regional three-dimensional networks. They also
provided the low-degree harmonic coefficients of the gravitational potential.
For optical direction measurements, an illuminated satellite is photographed on
film or glass plates together with the fixed stars.
Balloon satellites reflecting both sunlight (e.g., Pageos, 1966-1972, diameter of 30 m, / = 87, h =
2800 to 5600 km) and light flashes from active satellites were used. Ballistic cameras (e.g., Wild
BC4) mounted azimuthally were easy to operate and had a large field of view. Equatoriallymounted astronomic cameras could follow the motion of the stars. Using a large focal length,
even faint stars could be detected and observed with high precision. Orbital cameras were
designed so that they could also follow the motion of a satellite.
Upon developing the photographs and identifying the stars, the satellite and star image points
were measured on a precision comparator ( 1 ). The plate coordinates of the satellite were
then transformed to the spatial directions right ascension and declination, whereby the
transformation parameters were derived from the known directions to the stars, cf. [2.4.1]. Since
the satellite travels in the atmosphere, astronomic refraction reduction as applied to the stars, cf.
[5.3.3], had to be reduced by the satellite refraction (Fig. 5.28), with
Az=0.48"
= = .
/z[lOOOkm]
The accuracy of the direction measurements was 0.2" to 2". An increase in
accuracy was not possible mainly due to scintillation effects and comparator
measuring errors.
The Japanese satellite EGS (also Ajisai, h = 1500 km) is a recent example of
direction measurements. The satellite is equipped with both mirror and laser
reflectors. By rotation of the satellite about its axis, incident sunlight is reflected
periodically with two flashes per second.
Microwave distance-measurements were employed in the 1960's and still play a
fundamental role today.
The Secor (Sequential Collation of Range) system used modulated microwaves (two frequencies)
emitted from ground stations and re-transmitted from transponders on board the satellites.
Distances were derived from signal's travel time, using the phase comparison method, cf. [5.5.2].
Using this method and employing spatial trilateration, isolated geodetic networks were connected
to a global reference system. The precision obtained was a few m, although systematic errors up
to several 10 m also occurred.
From the 1970's to the 1980's, Doppier positioning became an operational tool
for establishing or improving geodetic networks. By orbital analysis, it also
delivered improved geopotential models and earth rotation parameters.
;?
Fig. 5.7. Satellite Doppler positioning
With Doppler measurements, a transmitter aboard a satellite S continuously
emits a stable frequency fs (Fig. 5.7). A signal is received at the ground station
with the frequency fg and a time shift At with respect to the transmission time
/. / is shifted against /, due to the relative velocity s = ds/dt between the
sateflite and the observer (Doppler effect):
The Doppler frequency shift is proportional to s ; a reversal in sign occurs at
the time of the closest approach of the satellite to the observer ( s = 0). In
principle, a range difference (range rate) can be determined from (5.40) by
integration over time. In practice, fg is compared with a constant reference
frequency / generated within the Doppler receiver, with //,. Integration
over a time interval yields the Doppler count
(/-/,)<*
With (5.40), we obtain the observation equation
which provides the range rate Sj - si from the Doppler counts.
As with any microwave technique, Doppler measurements do not depend on
weather conditions, and they allow large amounts of data to be accumulated
within short time intervals. Today, Doppler measurements are used with several
satellite missions and with the DORIS positioning system, cf. [5.2.3].
The Navy Navigation Satellite System (NNSS) or Transit System was an important application of
Doppler measurements (KouBA 1983, ANDERLE 1986). Developed as a navigation system for the
U.S. Navy, it was opened to civil use early on and operated successfully between 1964 and 1996.
Positioning was based on 4 to 7 Transit satellites in orbit ( = 1100 km, i = 90), which
continuously transmitted frequencies of 150 and 400 MHz. With the orbital planes of the satellites
evenly distributed in longitude, and due to earth rotation, a satellite became visible at least every
two hours along the equator. The satellite orbits were determined by Doppler measurements from
four tracking stations located in the U.S.A. Initially they referred to the World Geodetic System
1972 (WGS72) and later to WGS84, cf. [5.2.5]. The satellites transmitted their "broadcast
ephemeris" at two-minute intervals, together with UTC time signals, with an accuracy of 10 to
20 m. "Precise ephemeris" ( 1 to 2 m) were later made available to authorized users. A number
of Doppler receivers were built as portable units for geodetic purposes. These contained a
reference oscillator, a microprocessor, a data-recording unit, the antenna, and an energy supply.
The influence of ionospheric refraction was practically eliminated by the use of two frequencies,
cf. [5.1.3], and tropospheric refraction was computed using atmospheric models and observed
meteorological data, cf. [5.1.2]. The positional error of 10 to 30 m, obtained at the observation
of a single satellite pass, could be reduced to 2 to 5 m (broadcast ephemeris) with 30 to 50
passes and to 0.5 to l m using precise ephemeris. Relative positioning employed simultaneous
observations on two or more stations, which strongly reduced orbital and refraction errors and led
to baseline accuracies of 0.2 to 0.5 m (SEEBER et al. 1982).
With the GPS-system becoming operational, the NNSS system was no longer maintained.
5.2.5 Global Positioning System (GPS)
The NAVSTAR/GPS (Navigation System with Time and Range/ Global
Positioning System) is a radio navigation system based on satellites. It is
operated by the U.S. Department of Defense (DOD), which started development
of the system in 1973. The first GPS satellites were launched in 1978, and the
system became fully operational in 1993. GPS provides real time navigation
and positioning by one-way microwave distance measurements between the
satellites and the GPS receivers. Early on, the use of GPS for geodetic
applications was investigated (BOSSLER et al. 1980), and GPS positioning is
now extensively employed in geodesy at all scales, including the determination
of variations with time. Several textbooks on GPS are available, e.g., WELLS
(1986), LEICK (1995), HOFMANN-WELLENHOF et al. (1997), see also SEEBER
(1993), and for the actual state, TEUNISSEN and KLEUSBERG (1998a),
MCDONALD (1999).
The basic idea of GPS is to have at least four satellites above the horizon
available 24 hours everywhere on the earth. In principle, the three-dimensional
coordinates of the receiver's antenna can be derived from three observed
distances. The computation is based on the known ephemeris of the satellites
and the intersection of spherical shells. As the clocks of the satellite and the
receiver are not synchronized, a fourth distance measurement is necessary in
order to determine the clock synchronization error. Therefore, the original
distances derived from the travel time of a signal are called pseudoranges (Fig.
5.8).
We distinguish between the space, the control, and the user segment of GPS.
The space segment consists of 21 satellites (plus 3 additional spares) arranged
in six nearly circular orbits (i = 55, 12 h period of revolution) at an altitude of
about 20 200 km (Fig. 5.9, Fig. 5.10). Due to the limited lifetime of a satellite
(10 years on the average), a regular replacement is scheduled in blocks, which
are intended to correspond to improvements in satellite technology (block /
consisted of 28 satellites launched between 1989 and 1998).
Atomic clocks (2 rubidium and 2 cesium clocks per satellite) provide a highprecision frequency standard, with 10"12 to 10"13 relative long-term stability, cf.
[2.2.1]. They produce the fundamental frequency of 10.23 MHz. By
SAT B
SAT D
SAT A
Fig. 5.8. GPS positioning
Fig. 5.9. GPS (Global Positioning
System) satellite, courtesy The
Aerospace Corporation, El Segundo,
CA, U.S.A.
Fig. 5.10. GPS orbit constellation
multiplication, the LI (1575.42 MHz = 19.05 cm wavelength) and the L2
(1227.60 MHz = 24.45 cm) carrier waves are derived and continuously
emitted. LI and L2 serve as carriers for two code modulations and for a data
signal (navigation message). The codes are given as binary signals (+1 and -1
sequence) in a pseudo-random noise (PRN) form (Fig. 5.11). The C/A-code
(coarse/acquisition code) is modulated on LI only, with a frequency of 1.023
MHz (= 293 m wavelength) and a repetition rate of 1 ms. The P-code (precise
code) is modulated on LI and L2 and has a frequency of 10.23 MHz (= 29.3 m
wavelength) and a repetition sequence of 266 days. It primarily serves for
PRNCODE
Fig. 5.11. GPS signals
precise navigation and is reserved to authorized users. The navigation message
is transmitted on LI and L2. It contains the satellite's ephemeris (Keplerian
elements and certain time derivatives and orbital correction terms), a clock
correction with respect to GPS-time, the coefficients of a ionospheric refraction
model, and information on the status of the GPS system. The broadcast
ephemeris are accurate to a few m and approach sub-meter accuracy.
With GPS being fully operational, an accuracy deterioration for civil users has
been introduced by DOD. The Precise Positioning Service (PPS) provides all
GPS signals, and thus the full accuracy of the system ( 10 to 20 m or better) to
authorized users (mainly U.S. military). The Standard Positioning Service
(SPS) only delivers the C/A-code, while the P-code is encrypted by AntiSpoofing (AS), which results in the non-accessible Y-code.
Between 1990 and 2000, the fundamental frequency of the satellite clocks was destabilized, and
the ephemeris data were manipulated by Selective Availability (SA). This introduced errors in the
pseudoranges, which degraded the absolute accuracy to about 100 m in horizontal position and
150 m in height. For geodetic purposes, these effects were largely eliminated by differential
techniques. The suspension of SA led to an accuracy improvement in positioning and time
transfer, reaching the accuracy available for authorized users.
The GPS control segment is responsible for maintaining the operation of the
GPS satellites. It consists of the master control station (Colorado Springs, CO,
U.S.A.) and five globally distributed monitoring stations. The stations are
equipped with cesium standards and GPS receivers. They continuously measure
pseudoranges to all satellites and transfer the results to the master station. After
computation of the satellite orbits and the clock correction, the (extrapolated)
broadcast ephemeris and GPS time are transmitted to the satellites for storage
and retransmission by three ground antennas collocated with monitoring
stations. This operational control system is supplemented by a number of
additional monitoring stations (in 1999: 7 stations) operated by the U.S.
National Imagery and Mapping Agency (NIMA).
The GPS ephemerides refer to an earth-fixed system, as defined by the
coordinates of the monitoring stations: World Geodetic System (WGS).
WGS has been developed by DOD since the end of the 1950's (realizations
WGS 60, WGS 72, and WGS 84). The system is intended to serve for mapping,
charting, positioning and navigation, following international standards for
geodetic reference systems. The defining parameters of the latest version of
WGS 84 are as follows (SLATER and MALYS 1998, NIMA 2000):
semimajor axis a = 6378137m,
reciprocal flattening I// = 298.257 223 563,
geocentric gravitational constant GM = 398600.4418x10'mV2, which
includes the atmospheric part GMatm =0.35 109 mY 2 ,
angular velocity of the earth's rotation = 7.292115xlO"5rads'.
WGS 84 thus practically coincides with the Geodetic Reference System 1980,
cf. [4.3]. The associated gravity field is given by the global geopotential model
EGM 96, complete to degree and order 360, cf. [6.6.3].
The coordinates of the monitoring stations are given for the epoch 1997.0,
taking earth tides (tide-free system), cf. [3.4.1], and plate tectonic motions into
account. The accuracy of the WGS 84 coordinates is 0.05 m, which is also the
level of agreement between WGS 84 and the International Terrestrial Reference
Frame (ITRF), cf. [2.5.3].
GPS-time (unit SI second) is defined by the cesium clocks of the control
segment stations. It agreed with UTC on January 5, 1980 and differs from it by
a nearly integer number of seconds. This difference is increasing because no
"leap seconds" are introduced to the GPS time scale, cf. [2.2.2]. The actual
difference between GPS-time and UTC is part of the GPS navigation message
( 0.1 ). GPS is a very efficient system of time dissemination. A clock
comparison is possible with an accuracy of 100 ns and 10 ns and better with
differential techniques. Special GPS-time receivers are available as C/A onechannel instruments and operate in an automatic mode (LARSON and LEVINE
A similar satellite navigation system has been developed in the former Soviet Union since the
1970's. GLONASS (Global Navigation Satellite System) also operates as a one-way ranging
system. It has been operational since 1996 (LANGLEY 1997a, ZARRAO et al. 1998). Its space
segment comprises 24 (including 3 spares) satellites, arranged at a regular spacing of 45 in three
nearly circular orbits (/' = 64.8, h = 19 100 km, revolution time about 11.2 h). The orbits are
spaced 120 apart from each other, with five satellites always being visible. The satellites transmit
on two frequency bands (1602-1615 MHz, 1246-1256 MHz) with different frequencies for each
satellite (contrary to GPS). On the other hand, the C/A- and P-code modulations are the same for
all satellites. There is no degradation of the GLONASS signals. The satellites are equipped with
laser retroreflectors for laser tracking. The control segment consists of a master control station
near Moscow and secondary stations distributed over the territory of the former Soviet Union.
The GLONASS results refer to the reference system PZ90, which is based on the Soviet geodetic
reference system SGS85. Transformation parameters to WGS84 are known with an accuracy of a
few m (MiSRA and ABBOT 1994). GLONASS uses its own time system (UTC + 3 hours) which is
synchronized to UTC within 1 by the use of leap-seconds.
A European navigation satellite system called GALILEO has been discussed since 1999. It may
employ four carrier frequencies, and the intended on-line positional accuracy is 10 m (95%), once
every second (DIVIS 1999).
The user segment is composed of the many GPS receivers operating in
navigation, geodesy, and surveying. The main components of a receiver include
the antenna, the receiver electronics, the microprocessor, the oscillator, the
memory, the user interface, and a power supply. An additional telemetry unit
can be used for data transfer between different receivers (differential mode).
The signals transmitted from the satellites are received and amplified by the
antenna. After identification, the signals are processed to pseudoranges in the
channels of the receiver electronics. One channel is generally responsible for
the tracking of one satellite. Hence, a minimum of four channels is necessary to
determine position and time. Modern receivers contain 12 or more channels for
each frequency (multi-channel technique). The microprocessor controls the
operation of the receiver and calculates the three-dimensional position of the
antenna in WGS84 and the velocity and azimuth of moving objects. A quartz
oscillator is used for generating the reference frequency, which is approximately
synchronized with GPS-time. All data (pseudoranges, phases, time, navigation
message) are stored in receiver memory for post-processing, which is typical for
multi-station observation sessions employed in geodesy, cf. [6.2.1]. The user
interface includes a keyboard and a display, which provides a communication
link between the user and receiver. Power is provided by internal, rechargeable
nickel-cadmium batteries.
Geodetic GPS receivers deliver high accuracies in the static and kinematic
mode (BOUCHER 1987, LANGLEY 1997b). This is achieved by dual-frequency
(LI and L2) multi-channel instruments with the P-code and the full carrier wave
available on LI and L2. Other characteristics include low receiver-noise in code
and carrier phase, a high data rate (> 1 Hz), and a large memory capacity. The
antenna phase center should be stable and protected against multipath effects.
The Macrometer (1982) was the first GPS receiver for geodetic applications (code-free, single
frequency, 6 parallel channels). The Texas Instruments TI4100 (1984) provided all geodetically
relevant observables (P and C/A-code pseudoranges, carrier phases on LI and L2) in a multiplex
channel technique (SEEBER et al. 1985). The efficiency of geodetic receivers is discussed in the
literature (e.g., COORS et al. 1998). Fig. 5.12 to 5.14 show some examples. Combined
GPS/GLONASS receivers are also available (Fig. 5.15). They are able to track any combination
of GPS and GLONASS satellites (12 to 21 will be visible with fully deployed systems). This will
Fig. 5.12. GPS continuously
operating reference station (CORS)
with receiver (GPS total station
4700) and choke ring antenna,
courtesy Trimble Navigation Ltd.,
Sunnyvale, CA, U.S.A.
Fig. 5.13. Geodetic GPS dualfrequency receiver (GPS total station
SR530), with terminal TR500 and
telemetry antenna, courtesy Leica
Geosystems AG, Heerbrugg,
Fig. 5.14. Geodetic GPS dualfrequency receiver (GePoS
Experience) with antenna, courtesy
Zeiss/Spectra Precision AB,
Danderyd, Sweden
Fig. 5.15. GPS/GLONASS receiver
with antenna, courtesy JAVAD
Positioning Systems/TOPCON,
Paramus, NJ, U.S.A
improve the satellites- receiver geometry and the availability of satellites, as well as the integrity
of the solution, and lead to a higher accuracy for absolute positioning with broadcast ephemeris.
A distinction is made between code and carrier phase measurements for GPSpositioning.
Code measurements employ the travel time / of a signal between the satellite
and the antenna of the receiver. The time difference is determined by crosscorrelating an arriving code sequence with a code copy generated in the
receiver. Multiplication of with the velocity of light c gives the distance
between the satellite and the antenna, cf. [5.1.1]. Considering the clock
synchronization error 5t , the observation equation for the pseudoranges R
The distance is given by
-x,
where XS,YS,ZS and Xp,Yp,Zp are the geocentric coordinates of the satellite
and the ground stations respectively. Using the transmitted broadcast ephemeris,
the receiver clock correction and the coordinates of the ground station can be
derived by simultaneous measurements to four satellites. Equation (5.43)
presupposes that atmospheric refraction effects are taken into account by proper
reductions (see below).
The measurement noise governs the accuracy limit for this method. Accuracy
depends on the wavelength and is at the m-level for C/A-code and at the dmlevel for P-code. If used as a standard method in navigation, a positional
accuracy of about 10 m is achieved.
Pseudorange differences can be derived from integrated Doppler frequency
shifts (Doppler counts) of the carrier frequency according to (5.41) and (5.42).
The differences are used for the determination of velocity in navigation.
Doppler solutions also play a role in the removal of ambiguities, which occur
with carrier phase measurements. The method is not suitable for real-time
positioning due to the long observation time required.
Relative (differential) GPS methods determine differences of coordinates between two or more
stations. This is accomplished either by simultaneous observations on the stations, including at
least one with known coordinates, or by using correction data provided by permanently operating
reference stations. Corresponding DGPS-Services for navigation deliver range corrections derived
from the differences between observed and calculated pseudoranges at reference stations.
Accuracies of a half to a few m are achieved over distances between about 50 and several 100 km.
Geodesy and surveying require accuracies of at least two orders of magnitude
better than that required for navigation. This is achieved by carrier phase
measurements. Due to the shorter wavelength of carrier phases, the
measurement noise is only in the mm to sub-mm-range (BEUTLER et al. 1987).
The carrier phase is detected by comparing the received carrier signal with the
reference frequency generated in the receiver after subtraction of the code. In
order to reconstruct L2 under AS conditions (P-code encryption), different
techniques have been developed, such as squaring of L2 (eliminates the code
signal) and cross-correlation of LI and L2. The measured phase difference
= -0
(,0 = phase of the carrier and reference waves respectively) is related to the
distance s by
=(s-NA + cSt),
which is well known from terrestrial distance measurements, cf. [5.5.2]. is an
integer number of complete carrier cycles within the range j, and Si is the
receiver clock synchronization error. The ambiguity introduced by poses a
primary problem for the evaluation of (5.45). Among the algorithms available
for ambiguity determination, we have the inclusion of ambiguity-free Doppler
solutions, the combination of code and carrier phases, and statistical search
methods applied to combinations of LI and L2. Difficulties arise when the
phase lock is lost due to signal obstruction. Such sudden jumps of the carrier
phase by an integer number of cycles are called cycle slips. They are either
removed during preprocessing or taken into account by introducing an
additional ambiguity for the affected pseudorange.
The GPS signals experience path delays while traveling through the
atmosphere,. Ionospheric refraction, cf. [5.1.3], causes range errors at the m to
100 m order of magnitude (KLOBUCHAR 1991). Using the satellite message, a
reduction can be applied to the travel time when only a single-frequency
receiver is available. Two-frequency receivers allow application of (5.23) to the
frequencies /J and / 2 , which leads to the distance (code measurement)
In (5.46) the ionospheric refraction is eliminated, and sl and s2 are the
observed distances on LI and L2 respectively. A corresponding equation can be
derived for carrier phase observations. Tropospheric refraction may cause
delays on the order of one m (zenith direction) to 10 m (close to the horizon).
Refraction is accounted for by tropospheric models and observed surface
weather data, cf. [5.1.2]. The "wet" component remains a critical part of these
reductions; an improved determination is possible by water vapor radiometers,
which measure the water vapor content along the path.
Multipath effects result from signal reflection near the antenna. They affect code
and carrier measurements and may produce errors on the order of cm to dm. A
reduction of these effects is possible by a proper design of the antenna and by
careful site selection. The antenna's phase center (close to the geometric center)
varies in dependence on the satellite's elevation and azimuth; this must be taken
into account by calibration (MENGE et al. 1998).
The error budget of GPS is composed of orbital and satellite clock errors,
residual refraction effects, and receiver errors. Orbital errors are at the order of a
few m (Standard Positioning Service).
High-quality GPS data and orbits are available for post-processing through the International GPS
Service (IGS), which operates under the auspices of IAG, cf. [7.3.1]. IGS provides, among other
things, a precise and a "rapid" GPS ephemeris. The precise ephemeris is available after 7 days
and has a precision of about +3.. .5 cm. The "rapid" ephemeris is available after 24 hours and is
only slightly less precise. Satellite clock parameters are given with a precision of 1 ns. The
accuracy of these products should be close to the precision.
Clock errors are about 1 m. The residual errors of ionospheric refraction reach
only a few cm with two-frequency receivers, while tropospheric model errors
may attain one dm. The receiver noise is roughly about 0.1% of the signal's
wavelength, which corresponds to a dm-order of magnitude with the C/A- code
and 1 to 2 mm with the P-code (LANGLEY 1997b). Phase center variations of
the antenna may reach a few cm.
The accuracy of GPS positioning also depends on the geometric constellation of
the satellites with respect to the receivers and on the duration of the observation
time. A longer observation time increases the accuracy, especially for long
baselines and for the height component. The accuracy of the kinematic mode
(moving GPS receiver) is lower than that of the static mode.
The accuracy of an observed pseudorange can be expressed by its standard deviation (also user
equivalent range error). The strength of the satellite geometry is characterized by a quantity called
"Positional Dilution of Precision" (PDOP). It is defined as the ratio between the standard
deviation of a position derived from a certain satellite constellation and the standard deviation of
an observed pseudorange (LANGLEY 1999). The numerator of this quotient follows from the trace
of the coordinates covariance matrix, which depends on the design of the network, e.g., SEEBER
(1993, p.292). PDOP values can be calculated in advance and serve for the planning of
observations and for rapid information on expected positioning quality. For instance, a PDOP
value of 3 means that the accuracy of positioning is three times worse than the accuracy of the
pseudorange observation. If separated into the horizontal and the vertical components, it turns out
that the determination of heights is less accurate than horizontal positioning by a factor of about
two. This results from the fact that all observed satellites are above the receiver but distributed
over the total horizon.
Relative positioning (differential GPS) significantly reduces the errors that
occur in absolute positioning, especially orbital and refraction errors. This is
especially valid for shorter baselines, due to the correlation of these errors at the
two ground stations. For orbital errors, a (pessimistic) rule of thumb allows an
estimate of the error to be expected in a baseline (length b) from the orbital
error dr:
where s is the distance between the satellite and the receiver (maximum 25 000
km). Hence, if an accuracy of 1 cm is required for the baseline, the orbital error
should not exceed 2.5 m at b = 100 km and 0.25 m at b = 1000 km. When the
precise ephemerides from the IGS are used, orbital errors no longer play a large
role. The obtainable accuracy of relative positioning with good PDOP
conditions and in the static mode is about
(3...5mm + 0.5...1xlO-*s)
for the horizontal coordinates. In the kinematic mode, this reduces to
(10.. . 20 mm + 1... a x i o m s ) .
Height determination is less accurate by about a factor of two.
The Global Positioning System has drastically changed surveying methods in
geodesy, navigation, and other applications. This is mainly due to the high
accuracy achieved with static and kinematic positioning, real-time evaluation,
and operational flexibility. Direct visibility between the ground stations is not
necessary; only visibility to the satellites is required. The system is weather
independent and usable day and night. The use of GPS is still increasing and is
strongly supported by global and regional GPS services, cf. [7.3.1], [7.3.2].
5.2.6 Laser Distance Measurements
Laser distance measurements may be made to satellites equipped with corner
cube reflectors: Satellite Laser Ranging (SLR). This method offers high
accuracy due to the favorable propagation of laser light in the atmosphere. It
also offers a low-cost, long-lifetime space segment. On the other hand, SLR
depends on weather conditions and requires a considerable operational effort at
the ground segment.
At the ground station, laser pulses are emitted at epoch t, reflected at the
satellite, and received again at epoch t + . If refraction effects are taken into
account by corresponding reductions, the distance is obtained by
j =-Af.
Nowadays, travel time of the photons can be measured with an accuracy of 10
to 100 ps, which corresponds to a ranging accuracy of 1.5 to 15 mm (BENDER
Many satellites (about 30 in 1999) are equipped with laser reflector arrays for precise orbit
determination, e.g., satellites carrying radar altimeters for ocean height determination, cf. [5.2.7].
Dedicated SLR missions for positioning and geodynamic research include the satellites Starlette
(France, 1975, h = 800 to 1100 km, / = 50), Lageos 1 and 2 (U.S.A., 1976/1992, h 5900 km,;
= 110/52, COHEN et al. 1985), Ajisai (Japan, 1986, h = 1500 km, / =50), and Etalon 1 and 2
(USSR, 1989, h * 19 000 km, / = 65). These satellites are spherical in shape (diameter 0.2 to 2
m) and possess a favorable surface-to-mass ratio (Fig. 5.16). The GLONASS satellites and a few
GPS satellites also carry laser retroreflectors on board.
Fig. 5.16. Laser satellite LAGEOS, courtesy National Aeronautics and Space
Administration (NASA)
Laser distance measuring systems have developed rapidly since the 1960's.
They consist of the laser unit (Nd: Yag-laser = Yttrium-aluminum garnet crystal
doped with neodymium ions), the transmitting and receiving optics (telescopes),
and the receiver electronics (secondary-electron photomultiplier). The
mechanical mounting provides an automatic tracking of the satellite's pre-
calculated orbit. Due to atmospheric disturbances and reflection, only a few
laser pulses (photons) return. The travel time is measured by a time-interval
counter and an atomic clock, regularly compared with UTC. A process
computer controls the complete measurement and registration.
About 50 laser-satellite-systems are operating worldwide (as of 1999), either in the stationary or
mobile mode. The Wettzell laser ranging system employs a Nd:Yag laser (532 nm) and a 75 cm
telescope. It operates with high-energy short pulses (pulse length 180 ps, pulse energy 100 mJ) at
a pulse repetition rate of 1 to 10 Hz in the single-shot mode. Visible and /or infrared light is used,
allowing a day-and-night operation to satellites at altitudes between 800 and 40 000 km and to the
earth's moon (DASSING et al. 1992), Fig. 5.17. Further developments are directed to reduce the
pulse length to some 10 ps and to reduce the pulse energy. The emission of short-pulse trains
increases the probability of single photon detection.
Fig. 5.17. 75 cm telescope, Wettzell Laser Ranging System (WLRS),
Fundamentalstation Wettzell, Germany, courtesy Bundesamt fr Kartographie
und Geodsie (BKG), Frankfurt a.M., Germany
Mobile systems have been developed in the U.S.A. (NASA) and in Germany/Netherlands, among
others (SiLVERBERG 1978). Operating with low energy and single-photon detection, these systems
are employed mainly for the investigation of recent crustal movements, cf. [8.3.3]. A
transportable, integrated geodetic-observatory has been developed by BKG (Bundesamt fr
Kartographie und Geodsie, Germany). In addition to the laser unit, it includes a VLBI module
and GPS unit. It is employed for strengthening fundamental reference networks, especially in the
southern hemisphere.
The accuracy of laser distance measurements depends on the pulse length, the
stability of the photomultiplier, and the time resolution. Atmospheric refraction
effects may reach 2 m at the zenith and 10 m at an elevation of 10; they are
corrected with standard atmospheric models. Depending on the satellite's
altitude, some 100 to 1000 distances are measured during one passage, with a
single shot precision of 5 to 10 mm. By compressing the data to "normal
points" (average over 30 s to 120 s at Lageos), sub-cm precision is achieved.
Laser reflectors have been placed on the moon by the Apollo 11 (1969), 14, and
15 missions and by Luna 217 Lunokhod 2. These reflectors provide targets for
lunar laser ranging (BENDER et al. 1973). Pulsed lasers with a tightly bundled
beam and a powerful telescope are necessary in order to recapture the weak
returning signal (single photon technique). The tracking system must provide a
2 arcsec pointing accuracy.
Observations to the moon have been carried out regularly since 1969 by the McDonald
Observatory, University of Texas, and by systems operating in Hawaii and Grasse/France. Results
have also been provided by laser stations in Australia and Germany (Wettzell). The accuracy of
these measurements is about 1 cm. They especially contribute to investigations of the dynamics of
the earth-moon system and on terrestrial geodynamic processes (SMITH et al. 2000).
5.2.7 Satellite Altimetry
Satellite altimetry is based on a satellite-borne radar altimeter that transmits
pulses in the vertical direction to the earth's surface. The ocean surface reflects
the pulses perpendicularly, and the measurement of the travel time ? furnishes
the height of the satellite above the instantaneous sea surface (Fig. 5.18):
(The denotation a should not be confused with the abbreviation for the
semimajor axes of the satellite's orbit and the earth ellipsoid). A proper
reduction of atmospheric refraction effects is presupposed in (5.49).
SEA SURFACE
GEOID
Fig. 5.18. Satellite altimetry
In spherical approximation, this observation can be expressed as
where rs and rp are the geocentric distances to the satellite and to the
subsatellite point P on the ellipsoid; N is the geoid height and SST the height of
the surface topography. Satellite tracking provides rs and positioning gives rp .
Altimetry thus delivers information on the geoid and on sea surface topography.
Ellipsoidal formulas are given by GOPALAPILLAI (1974).
Radar altimeters operate in the 14 GHz frequency range with short (a few ns)
pulses and a high-pulse frequency (e.g., 100 pulses/s). The effects of beam
divergence and finite pulse length result in measurements that refer to a "mean"
sea surface within a circular "footprint" (few km diameter); short-wavelength
features of the ocean (waves) are thereby smoothed out. For example, by
averaging the measurements over 1 s, the along-track resolution is about 7 km.
Satellite altimetry missions are designed to provide either an exact repetition of
ground tracks (days to weeks) or a dense pattern of profiles. The different
modes are achieved by orbital maneuvers (KNUDSEN 1993). The latter is for
determination of the altimetric geoid and the former for investigation of ocean
variability (Fig. 5.19).
Fig. 5.19. TOPEX/Poseidon subsatellite tracks covered within the 9.9 days
repeat cycle, from BOSCH (2001)
The first global survey with a radar altimeter was accomplished by the Geos-3 satellite (U.S.A.,
1975-1978). The oceanographic satellites SEASAT (1978) and GEOSAT (U.S.Navy, 1985-I989)
carried improved altimeter systems and operated in heights close to 800 km with 108 inclination
and repetition rates of 3 and 17 days (TAPLEY et al. 1982, MCADOO and SANDWELL 1988). The
European Remote Sensing Satellites ERS-1 (1991) and ERS-2 (1995) operate at similar heights
with 98 inclination (Fig. 5.20). Repetition rates are 35 and 168 days respectively, and ground
track distances at the equator are 80 km and 8 km respectively for geodetic missions (KOCH 1989,
JGR 1998). The NASA/CNES (French space agency) TOPEX/Poseidon satellite (1992, 5.3 and
13.6 GHz) is placed in a circular orbit at an altitude of 1340 km and an inclination of 63.
Repetition time is 10 days, and the equatorial ground track interval 316 km (Fu et al. 1994,
CHENEY et al. 1995), see Fig. 5.21.
Fig. 5.20. European Remote
Sensing (altimeter) Satellite ERS,
courtesy European Space Agency
(ESA)
Fig. 5.21. TOPEX/Poseideon
(altimeter) satellite, courtesy
JPL/NASA, Pasadena, CA, U.S.A.
High radial-orbit accuracy is required for satellite altimetry. Consequently, the
satellites are equipped with laser reflectors, with some having active systems
such as Doppler transmitters, C-band radar transponders, and GPS. The spaceborne orbit determination systems DORIS and PRARE have been installed on
the TOPEX/Poseidon and ERS-2 respectively (TAPLEY et al. 1995, ANDERSEN
et al. 1998), cf. [5.2.3]. Further orbital improvements have been achieved by
"tailored" gravitational field models developed for a dedicated altimeter mission
(TAPLEY et al. 1996). The orbital error thus has been reduced to about 0.5 m
for the GEOS-3 mission, to 3 cm for TOPEX/Poseidon, and to a range of 5 to
10 cm for ERS-1. For the evaluation of a region sampled multiple times with
high spatial-resolution altimeter profiles, the track crossover discrepancies are
adjusted by minimum conditions with time-dependent, low-order polynomial
models (e.g., shift and inclination) applied on the individual tracks (van GYSEN
and COLEMAN 1997). Residual orbit and other long-term errors are further
reduced in this way, cf. [6.6.3].
The altimeter instrumental noise is now less than 2 to 3 cm (TOPEX/Poseidon,
ERS). Corrections have to be applied for systematic instrumental (bias and
drift) and sea-state effects as well as for tropospheric and ionospheric path delay
(modeling, use of two frequencies for TOPEX/Poseidon), cf. [5.1.3]. After
reduction of the ocean tides and large-scale air pressure effects, the altimetric
results refer to the quasi-stationary sea surface and yield its height with an
accuracy of 5 to 10 cm.
5.2.8 Satellite-to-Satellite Tracking, Satellite Gravity Gradiometry
High-resolution gravity-field determination from space requires low-orbiting
satellites and highly sensitive sensors, cf. [5.2.3]. This is achieved by satelliteto-satellite tracking and satellite gravity gradiometry (COLOMBO 1989, SNEEUW
and ILK 1997, ILK 2000).
Satellite-to-satellite tracking (SST) employs microwave systems for measuring
range rates between two satellites. High-low (one high and one low-flying
satellite) and low-low (two low-flying satellites at the same altitude)
configurations have been designed (RUMMEL et al. 1978), Fig. 5.22. The basic
observables are the range rates (radial velocities) and changes of the range rates,
which are due to gravitational and non-gravitational "disturbing" forces, cf.
[5.2.2]. The gravitational field parameters (harmonic coefficients) can be
derived after proper compensation of the surface forces. In order to achieve a
gravity field resolution of 100 km, the orbital altitude of the lower satellite must
not exceed a few 100 km. The relative velocity between the satellites has to be
determined with an accuracy of 1 to 10 /s, and precise tracking should be
guaranteed by high-altitude satellite systems (GPS) and ground stations.
GPS-SATELLITES
Fig. 5.22. Satellite-to-satellite tracking: a) low-low mode, b) high-low mode
SST experiments started in the 1970's, for example, between GEOS-3 and the geostationary
satellite ATS-6 (nearly circular equatorial orbit, h = 36 000 km). The CHAMP (Challenging
Mini-Satellite Payload for Geophysical Research and Application) satellite (Germany, launch in
2000) began at an altitude of 450 km, which will be reduced to 300 km within the mission time of
5 years. It moves in a nearly circular polar-orbit (i=83). The satellite (dimensions 4 m l m x
1.6 m) carries a GPS receiver (high-low mode tracking), laser reflectors, a three-axes
accelerometer (10~* ms 2 , for measuring and later compensation of the surface forces), and
magnetometer systems (Fig. 5.23). A gravity field recovery up to degree and order 50 is expected
(REiGBERetal. 2000).
Fig. 5.23. Champ satellite, courtesy Geoforschungszentrum (GFZ) Potsdam,
The GRACE (Gravity Recovery and Climate Experiment) mission (U.S.A./Germany, launch
scheduled for 2001, mission duration 5 years) will employ two satellites of the CHAMP type in
the low-low mode about 200 to 300 km apart at an initial orbit altitude of 450 km (i = 89.5). The
relative range measurements between the satellites will be integrated by GPS tracking. A gravity
field resolution up to degree and order 150 tolSO will be achieved, with accuracies of 10 to 30
ms2 for the gravity anomalies and 0. l m for the geoid.
An essential mission goal is the detection and monitoring of large-scale gravity field variations
with time, cf. [8.3.4].
Satellite gravity gradiometry determines the components of the gravitational
gradient tensor (second derivatives of the gravitational potential). On the earth's
surface, gravity gradiometry has been employed since about 1900 with sensor
pairs (accelerometers) sensitive to local changes of the gravity field in a certain
direction. By different orientation of the sensors, different components of the
gravity gradient can be determined, cf. [5.4.5]. For space-borne applications, the
attenuation of the gravity field with height (at a few 100 km height, the offdiagonal elements of the gradient tensor are a few 10"9 s"2 only) requires a high
accuracy for the second derivatives (at the order of 10~n to 10"13 s 2 ), which
can be achieved with conventional or superconducting electronics (WELLS
1984, PAIK et al. 1988). High demands are posed on the attitude control of the
sensor pairs and on the drift of the accelerometers. Surface forces cancel when
the output of an accelerometer pair is differenced (MORITZ 1968c).
ESA is planning a dedicated gradiometry mission GOCE (launch 2004?), with a drag-free satellite
(h = 270 km, i = 97) in a sun-synchronous near circular orbit. Tracking will be performed by
GPS, and earth-pointing orientation is provided by star trackers. Accelerometer pairs (10~ 12 s" 2 )
will be arranged over three mutually orthogonal directions, with baselines of 70 cm. A mission
duration of 12 months should provide a gravity field resolution up to degree and order 300, with
accuracies of about 10 //ms"2 (gravity anomalies) and 1 cm (geoid). The low-degree potential
terms will be determined primarily through GPS tracking, while the higher degrees will result
from gradiometry (RUMMEL 1986, RUMMEL et al. 2000b).
Classical geodetic astronomy is concerned with the determination of astronomic
latitude, longitude, and azimuth from ground-based optical direction
measurements to fixed stars, which also requires time determination (MUELLER
1969, SCHDLBAUER 2000). Several types of observational instruments are
available for this purpose [5.3.1], and different methods of observation have
been developed [5.3.2]. A number of reductions are necessary in order to refer
the observations to the celestial reference frame [5.3.3].
Geodetic astronomy is based on spherical astronomy (EICHHORN 1974,
KOVALEVSKY 1995). Its importance has decreased since the development of
efficient satellite positioning and gravimetric methods and is now restricted to
more local applications of gravity field (plumb line direction) and azimuth
determinations. On the other hand, radio waves emitted from extragalactic
sources are used extensively in order to derive base-line vectors between
fundamental terrestrial stations and to determine earth-rotation parameters:
Very Long Base Line Intetferometry [5.3.4].
5.3.1 Optical Observation Instruments
Optical observations to fixed stars are carried out in the local astronomic
(horizon) system. The direction to a star is determined by the astronomic
azimuth A and the zenith angle z (sometimes the altitude or elevation angle 90
- z is used), cf. [2.6.2]. Due to the movement of the observer with respect to the
stars, simultaneous time measurements are required.
Astronomic instruments are either permanently installed in observatories or
constructed as transportable devices for field operation.
Observations of highest precision with stationary instruments have been utilized by the former
International Time Service and International Polar Motion Service. Among other instruments, the
photographic zenit tube has been employed. In this case, stars near the zenith are photographed
symmetrically with respect to the meridian. The zenith angles and the hour angles can be
determined from the tracks of the stars. The direction of the vertical is established by a mercury
pool. Comparable precision (0.05") was achieved by the Danjon prism astrolabe. Thereby, the
time is determined when a star crosses the horizontal circle of a given zenith angle. These
astronomic observatory measurements are no longer of significance for geodesy, as celestial and
terrestrial reference-frames are now established by other space techniques, cf. [2.3].
The universal instrument was employed for field measurements of first-order
precision (0.1" to 0.3"). It consists of a high-precision theodolite, cf. [5.5.1],
of very stable design with a few attachments for astronomic observations.
An angled telescope with a horizontal eyepiece permits observations near the
zenith. To eliminate personal errors, the movable thread of the registering
micrometer is driven to follow the star so that impulses are generated and
recorded at uniform intervals. The suspension level serves to measure the tilt of
the horizontal axis. The Horrebow level, mounted at right angles to the
horizontal axis, registers any changes in the tilt of the telescope. The Kern
DKM3-A and the Wild T4 universal theodolites were used widely (Fig. 5.24).
Fig. 5.24. Universal instrument
Kern DKM3-A, courtesy Kern/Leica
Fig. 5.25. ZEISS Ni2 level with prism
astrolab, courtesy Carl Zeiss,
Oberkochen, Germany
The prism astrolab is used for the simultaneous observation of astronomic
latitude and longitude. With this device, one measures the transit times of those
stars that cross the same small-circle parallel to the horizon (almucantar). The
constant zenith distance (usually ~ 30) is realized by a prism placed in front of
the telescope, and the direction of the vertical is defined by the surface of a pool
of mercury or by a compensator pendulum. Astrolabe attachments are
particularly common. These are mounted either on a theodolite (e.g., the Wild
T3 astrolabe with a mercury pool) or on an automatic level (Zeiss Ni2
astrolabe), DEICHL (1975), Fig. 5.25.
Transportable zenith cameras are also used for rapid determination of latitude
and longitude. Such an instrument consists of a camera oriented in the direction
of the plumb line (focal length 300 to 1000 mm, relative aperture ~ 1:5), which
can be rotated around the plumb-line axis in any azimuth. In addition to a
timing device, the instrument is also furnished with two levels that are arranged
at right angles to each other (Fig. 5.26).
Fig. 5.26. Transportable zenith camera, Institut fr Erdmessung, Universitt
The field of stars near the zenith is photographed for two positions of the
camera differing by 180 in azimuth. After the photographic plates are
developed (exposure time 1 s), the star tracks and the intersection of the lines
connecting the fiducial marks are measured on a comparator. The
transformation of the plate coordinates into the a, -system provides the
rectascension and the declination of the zenith point, after averages are taken
and corrections are made for the reading of the levels. The observations (several
plates) require one half to one hour, including instrumental setup and
disassembly. The attainable accuracy is 0.5" or better (BlRARDl 1976,
SEEBER and TORGE 1985).
Recent developments in the CCD (charge-coupled device) technique allow creation of an
electronic image of the stars. The principle of a CCD is based on the photoelectric effect
produced on a semiconductor plate. The number of collected photoelectrons is proportional to the
light received. A CCD matrix is composed of a number of linear arrays. CCD's with 2000 2000
pixels and more are available with pixel dimensions of 10 and less and a resolution of 0.1
pixels. Using an image processing unit and a PC, an automatic evaluation of star observations is
achieved, reaching similar accuracies as with photographic methods (KOVALEVSKY 1995,
GERSTBACH 1996).
For time determination, an accuracy of 1 ms is needed in optical geodetic
astronomy. This is provided by quartz clocks, which are based on quartz crystal
oscillators (frequency stability 10~8...1(9 over a few hours) and are
synchronized by time signals, cf. [2.2.2]. A simple time measurement is
possible with a GPS receiver, cf. [5.2.5]. In order to record the time of a star
transit through the horizontal or vertical thread of a telescope, or the time of
taking a star image, a registration device (chronograph) has to be implemented
in the observation system.
5.3.2 Astronomic Positioning and Azimuth Determination
The determination of astronomic latitude, longitude and azimuth is based on the
relations given in [2.6.2], where the star positions (#,<?) are given by star
catalogues, cf. [2.4.3]. We mention here only a few of the many methods
developed in geodetic astronomy.
In determining the astronomic latitude , it is required, according to (2.21), to
ascertain the zenith angle and the hour angle h. The latter quantity has to be
derived from the rectascension a and the measured time, see (2.22). By
differentiating (2.21) with respect to the observed quantities, we obtain
differential formulas for estimating the effect of observational errors on the
results and for finding optimum configurations for the observations. For the
latitude we obtain
cos A
An error in has a minimum effect for meridian transits of the star, and an error
in h has no effect on in that case.
For an upper culmination (the smaller zenith angle) of a northern star (A = 0)
or a southern star (A = 180), the latitude is given by (see also Fig. 2.4)
= SN - ZN and = Ss + zs
respectively. Measuring the meridian zenith angle (e.g., to Polaris) is therefore
most suitable for the determination of the latitude.
If one observes a pair of stars consisting of both a northern and a southern star
having approximately the same zenith angle, averaging (5.51) significantly
reduces the uncertainties in z due to refraction (Sterneck method):
In the Horrebow-Talcott method, the small difference between the meridian
zenith angle of the northern and southern stars in a star pair is measured by a
registering micrometer. The optical axis in each case is adjusted to the same
zenith angle using the Horrebow level that is mounted on the horizontal axis.
Since accurate circle and time readings are not required, this method provides
very precise results (0.1" when about 20 star pairs are observed).
The astronomic longitude is given by the difference between the local
sidereal time LAST and the Greenwich sidereal time GAST (2.23):
= LAST -GAST,
where l s corresponds to 15". According to (2.22), LAST is related to the hour
angle h:
LAST = /2 + a.
h can be computed from the zenith angle according to (2.21). If the latitude is
known, we have:
, cosz-sin<sin
cos/z = ---- .
Converting measured universal time UT to GAST allows the determination of
By differentiating (2.21), we get
dh =
dz ----
sin A cos
cot A /.
The effect due to errors in is minimum when observations are made on the
prime vertical (A = 90), while the effect here is zero for errors in . The
influence of refraction is largely eliminated when observing east and west stars
of the same altitude that are symmetric with respect to the meridian. On the
other hand, by observing the time of transit across the meridian (h = 0) we get
LAST = a. An accuracy of 0.01 to 0.02 s is obtained from approximately 30
transits.
The accuracy of the determination of longitude depends primarily on the
systematic errors of the observer, the instrument, and the time comparison. If
the determinations are made by the same observer, using the same instrument
and the same time signal transmitting station, as well as the same stars, then
longitude differences are essentially free from these errors. Longitude
determinations of high accuracy have thus been carried out as measurements of
differences with respect to a reference station.
An economical method to determine the latitude and longitude simultaneously
is known as the method of position lines.
Fig. 5.27. Method of position lines
The zenith angles z,, z2 of two stars ^(,,), 2(2,(52) are observed at
sidereal times LAST,, LAST2 and at azimuths ,, A^. If 5,, S2 are projected
on the earth's surface, then the intersections of the circles centered at the
projections 5,", S'2 having radii z t , z2 represent two geometric positions and
(P) for the point of observation (Fig. 5.27). Near P, the circles can be replaced
by their tangent lines (position lines). The intersection of these lines yields an
approximation to P. Computationally, one obtains the corrections = - 0
and = - 0 upon introducing an approximate position ^( 0 , 0 ). When
observations are made with the prism astrolabe, cf. [5.3.1], the zenith angle
predetermined by the prism is treated as an additional unknown. One obtains
accuracies of a few 0.1" from about 20 stars evenly distributed above the
horizon.
A zenith camera also permits the simultaneous determination of latitude and
longitude. From the coordinates a2, St of the zenith, derived from the
observations, we obtain
= c*r-GAST,
cf. [5.3.1], see also Figs. 2.2, 2.4.
If the latitude is known, then the azimuth A can be obtained, according to
(2.21), from the hour angle h derived from the sidereal time and the
rectascension (5.54):
tan A
=-.
sin cos h - cos tan
fC C0\
Differentiation of (2.21) yields
^-dh + cotzsmAd^,
where q is the parallactic angle, cf. [2.6.2]. An error in h has a minimum effect
for = 90 (stars near the pole). For some 10 observations, we get an accuracy
of 0.3" to 0.5". The azimuth of a terrestrial target is obtained by additionally
measuring the angle between the directions to the star and the target.
5.3.3 Reductions
In order to refer the "observed" positions (epoch t) of fixed stars to the system
of the star catalogue (mean positions at the reference epoch t0), several
reductions have to be applied.
Astronomic refraction causes an apparent increase in the star's altitude (Fig.
5.28). The true zenith angle is obtained from the observed quantity by
adding the astronomic refraction angle ^ :
= ' + ..
According to (5.10) and (5.11), the refraction angle depends on the vertical
gradient of the refractive index along the path. For a standard atmosphere
(temperature 288.15 K, atmospheric pressure 1013.25 hPa), we obtain for
/<70:
0 =57. 08 " ' -0.067 "tow3 '.
For actual conditions (temperature T, pressure /?), we have the
F/'g. 5. 25. Astronomic and satellite refraction
= 0 -- ,
(SAASTOMAINEN 1972/1973, KOVALEVSKY 1995). The accuracy of .
varies between a few 0.01" and a few 0.1" and depends strongly on
systematic deviations from the atmospheric model (turbulences, slope of the
atmospheric layers).
The diurnal aberration is an apparent displacement in direction resulting
from the finite velocity of light and the relative velocity of the observer
with respect to the stars, due to the earth's rotation. The corresponding
reduction reaches a maximum of 0.3 " at the equator.
The geocentric (or diurnal) parallax represents the difference between the
topocentric and the geocentric direction; it can be neglected for star
observations, cf. [2.4.1].
Through these reductions, the "observed" position is transformed to the
"apparent" position (apparent place) at epoch t. The reduction from the "mean"
position (epoch f 0 ) to the apparent position (epoch t) involves the following
Applying precession and proper motion for the time interval t-t0
transforms the mean position (f 0 ) to the mean position at epoch t, cf.
[2.4.2], [2.4.3].
Accounting for nutation transforms the mean position (f) to the true position
(t). The origin of the system is still at the barycenter of the solar system.
The transition to the (geocentric) apparent position (t) is performed by
accounting for the apparent directional changes arising from the orbital
motion of the earth around the sun (annual aberration, up to 20") and the
difference between the heliocentric and the geocentric directions (annual
parallax, less than 1").
Instead of reducing from the mean position ( t0 ) to the apparent position (f),
astronomic almanacs that contain the apparent places of fundamental stars for a
particular year (Apparent Places of Fundamental Stars, Astronomisches
Recheninstitut Heidelberg) can be used.
Finally, we must consider that the results (astronomic positions and azimuths)
refer to the instantaneous spin-axis of the earth. The results must be transformed
into the IERS reference pole (ICRF) by applying reductions for polar motion,
cf. [2.5.2]. Multiplying the polar motion rotation matrix (2.18) with the unit
vector of the local vertical (2.19) gives the reductions for latitude, longitude,
and azimuth (MUELLER 1969, p.86):
= <DICRF - = - (xp cos - yp sin )
= AICRF - = - (xp sin + yp cos A^tan ,
= AlCRF-A =
where xp , yp are the pole coordinates with respect to ICRF.
5.3.4 Very Long Baseline Interferometry
Extragalactic radio sources (quasars = quasi-stellar radio sources, radio
galaxies) emit waves in the cm to dm range, which can be detected by large
antennas (radio telescopes) used in radio astronomy. The approximate angular
resolution of such a telescope is given by the wavelength/diameter ratio, and
thus it is limited to a few arcmin for telescope diameters less than 100 m. By
employing a receiving system of two widely (a few 1000 to 10 000 km)
separated radio telescopes (baseline), the resolution can be increased to 0.001 "
and better: Very Long Baseline Interferometry (VLBI), MORITZ and MUELLER
(1987, p.381).
The wave train from an extragalactic radio source arrives at the telescope P2
with a phase difference with respect to the telescope ^ . is related to the
time delay , the time the wave requires to travel the path difference CT (c =
velocity of light in vacuum), Fig. 5.29. Due to the earth's rotation, and
depend on time t. The following relations are valid:
DIRECTION TO RADIO SOURCE
Fig. 5.29. Very Long Baseline Interferometry
where and are the respective wavelength and the frequency of the received
radio wave. We introduce the baseline vector
rrRS = 2 ~ ri =
Y2-Yt
za-z,
described in the terrestrial geocentric system (2.14) and the unit vector to the
quasar:
ICRS ~
cos a cos
sinacosS
given in the celestial reference system (2.11). After transformation of s to the
terrestrial system according to (2.16), we obtain, see Fig. 5.29,
The negative sign takes the direction of s into account, which is opposite to the
direction of wave propagation. By comparing the two wave trains received at
.Pj and P2, interferences are obtained. The frequency of the interference fringes
(maxima and minima) changes due to the earth's rotation:
With (5.64) and (5.67), the fringe frequency can be expressed as
s = ds/dt . Equations (5.67) and (5.69) represent the VLBI observation
equations (CAMPBELL and WITTE 1978, CARTER et al. 1985).
The VLBI observables are the time delay (or the phase delay, respectively)
and the delay rate dT/dt . They are derived by a comparison of the signals,
which are recorded on magnetic tape, along with the precise time provided by
hydrogen maser frequency standards. Typically, 10 to 20 sources are tracked
over periods of 3 to 6 minutes several times over a 24-hour observation session.
The data tapes are sent to a correlator center where they are replayed and the
signals processed (MARK and recently MARK IV correlator, CLARK et al.
1987). The correlation function furnishes the delay and the delay rate between
the two stations. The observable represents a group delay, while is a phase
delay, cf. [5.1.1], which involves the problem of ambiguity resolution, cf.
[5.2.5]. The fringe frequency (delay rate) observation is free of this problem
but of less importance due to its lower accuracy as compared to the delay
observation and since it allows only the determination of a reduced set of
parameters (SEEBER 1993, p.430).
Reductions are applied for the daily aberration, cf. [5.3.3], for systematic clock
differences (clock synchronization), for the effects of the tropospheric
refraction, cf. [5.1.2], and for relativistic effects. The effect of the ionosphere is
compensated for by observing in two frequency-bands, namely 2.2 and 8.4
GHz, cf. [5.1.3]. Main error sources result from timing (1 ps) and frequency
instabilities (10~15 over a few days) as well as from tropospheric models. The
determination of the "wet" component plays a major part in this aspect, cf.
[5.1.2]. Attempts have been made to measure the water vapor content along the
signal path by water vapor radiometers in order to estimate the wet component
with cm-accuracy. The vertical baseline component is mainly affected by the
uncertainty of the wet component.
Among the parameters to be estimated from (5.67) and (5.69) are the
components of the baseline vector in the terrestrial reference system (cm
accuracy). Global solutions for a given epoch deliver a precision of a few mm
for the coordinates and annual station velocities accurate to a mm/a. They also
yield earth rotation parameters, cf. [2.5.3]. These parameters enter into the
models through the transformation between the terrestrial and the celestial
system. They can be determined with an accuracy better than 0.001" (pole
coordinates) and 0.1 ms (UT1), with a one-day resolution, cf. [2.5.2]. In
addition, corrections to the precession and nutation models can be derived from
VLBI (ROBERTSON et al. 1985, CAMPBELL et al. 1992).
Fig. 5.30. 20m-radio telescope, Fundamentalstation Wettzell, courtesy
Bundesamt fr Kartographie und Geodsie (BKG), Frankfurt a.M., Germany
Today, around 20 stationary radio telescopes participate in international programs in order to
establish the reference frames, to determine the earth rotation parameters, and to derive station
velocities, especially within the International Earth Rotation Service. The stations are mainly
located in North America, Europe, and Japan, Fig. 5.30. Mobile radio telescopes have been
developed for rapid surveying of regions with recent crustal movements, such as California or the
Eastern Mediterranean , and to fill gaps of the terrestrial reference system in the southern
hemisphere.
Gravimetry deals with the measurement of the gravity intensity (gravity) and
the gravity gradient by terrestrial methods on or close to the earth's surface
(MARSON and FALLER 1986, TORGE 1989). "Absolute" gravity measurements
refer directly to the standards of length and time [5.4.1], while "relative"
measurements use a counterforce for the determination of gravity differences
[5.4.2]. A global gravity reference system is needed in order to refer local and
regional gravity networks to a common standard [5.4.3], Gravity measurements
on moving platforms are valuable for areas difficult to access [5.4.4]. Local
gravity-field information can be obtained by the measurement of the gravity
gradient [5.4.5]. The continuous record of gravity provides gravity variations
with time, especially the gravimetric earth tides [5.4.6].
The unit of gravity in the Si-system is ms"2. The units mGal = 10~5 ms"2 and
3 = 1088'2 =10nms" 2 are still in widespread use in geodesy and
geophysics and derived from the unit Gal (after Galileo) of the former cgssystem.
5.4.1 Absolute Gravity Measurements
An "absolute" gravity measurement determines the gravity g from the
fundamental acceleration quantities length and time. We distinguish between
the pendulum and the free-fall method, both introduced by Galileo Galilei (1564
- 1642), FALLER and MARSON (1988), TORGE (1991).
The pendulum method is no longer applied today but governed gravimetry for
about 300 years. Because of its fundamental importance, and because recent
results are still part of some gravity networks, a short introduction is given here.
The pendulum method is based on the measurement of the period and the length
of a freely swinging pendulum. For a mathematical pendulum (point mass m
suspended on a weightless wire of length /) we have the equation of oscillation
ml<p + mgsii\(p = Q,
with the phase angle = <p(t}, and the angular acceleration ip = d*<pjdt* (Fig.
5.31). Integration over a full period leads to an elliptical integral. After
expansion into a series, we obtain the period of oscillation
Fig. 5.31. Absolute pendulum method: a) mathematical pendulum, b) reversible
where the amplitude <pQ is kept small. Thus, gravity is derived from the
measurement of T and /.
A mathematical pendulum is difficult to realize. Equations (5.70) and (5.71)
also hold for a physical pendulum if / is replaced by the reduced pendulum
Here, J is the moment of inertia with respect to the axis of rotation O, m the
total mass, and a the distance between and the center of mass. The reversible
pendulum is characterized by two axes of rotation, for which, after a
corresponding adjustment, the same oscillation time is achieved. The distance
between the two axes is equal to the reduced pendulum length, thus avoiding
the direct determination of J, m, and a (Fig. 5.31).
The reversible pendulum was introduced by Kater (1818), and a limited number of observations
were carried out, primarily after the I860's (transportable devices by Repsold and others). After
the fundamental gravity determination in Potsdam, cf. [5.4.3], only a few more experiments were
performed. Due to errors in determining the length of the swinging pendulum, the accuracy
achieved remained at a few ums 2 (SCHLER et al. 1971).
The free-fall method is based on the equation of motion
of a freely falling body. Here m is mass; z is along the local vertical axis, and
z = d2z/dt2 (Fig. 5.32). Assuming a homogeneous gravity field along the
falling distance, double integration of (5.73) yields the free-fall equation
Equation (5.74) relates the position z of the falling body at the time / to gravity.
The integration constants z0 and z0 represent z and z = dz/'dt at the starting
time of the experiment (t = 0).
z(m)
t(s)
Fig. 5.32. Distance-time diagram: a) free-fall method, b) symmetrical rise and
z0 and z0 slightly deviate from zero due to problems in accurately defining the starting position
and small microseismic accelerations. The gravity change along the falling distance (nonhomogeneous gravity field) is taken into account by introducing the vertical gradient dg/dz into
(5.74) and by referring the final gravity value to a fixed reference height, e.g. l m (NIEBAUER
If at least three position/time pairs are measured, z0 and z0 can be eliminated
in (5.74), and gravity is given by
fe-Ofe-Ofe-O
For the symmetrical rise and fall, the test mass is thrown vertically upward and
falls back after having reached the apex (Fig. 5.32). It is sufficient to measure
time at the same two positions during rise and fall. Evaluation of (5.74) yields
g=
/22-/,2
with = z2 - Zj, /, = r3 -1 2 , 2 = r4 - f,.
Generally, more than the necessary number of position/time pairs is measured.
Equation (5.74) then serves as observation equation for a least squares
adjustment of g, z0, and z 0 .
Accuracy demands for absolute gravimetry are at the order of 9 g or
10~~8 ms . Hence, for a falling distance of 0.2 m (falling time 0.2 s), accuracies
of 0.2 nm and 0.1 ns are required for position and time respectively. This is
achieved by interferometric distance measurements and simultaneous electronic
timing.
CORNER CUBE
LONG PERIOD
MIRROR N
PHOTODETECTOR
Fig. 5.33. Michelson interferometer principle
For recent free-fall gravimeters, a polarization or iodine stabilized He-Ne gas
laser ( = 633 nm) serves as the length standard and an atomic (rubidium)
frequency normal as the time standard. A Michelson interferometer is used for
the distance measurement, with two corner cube reflectors as the main
components (Fig. 5.33). One of the reflectors is fixed and serves as a reference;
the other one represents the falling body. By splitting the laser light into a
measurement and a reference beam, and superimposing them again after parallel
reflection, light interferences occur. The zero crossings of this fringe signal
have a distance of /2, and the fringe frequency increases with time due to the
velocity increase according to z(t} = gt (Fig. 5.34). The zero crossings are
sensed by a photodiode, converted to an electronic signal, amplified, triggered,
and counted. A time measurement (atomic clock and time interval counter) is
carried out after a preset number n of zero crossings, which corresponds to a
falling distance of
The experiments are performed in vacuum (10"4 Pa) in order to eliminate air
resistance. Microseismicity is to a large part absorbed by long-period (T> 10 s)
compensation devices. A further reduction is achieved by randomization,
performing a large number (several 100 to a few 1000) of drops per station. For
the rise and fall method, systematic errors that are proportional to the falling
body's velocity (residual air drag, timing errors) cancel.
FRINGE SIGNAL
FRINGE PULSES
SCALED FRINGE
lm n
|;,
CLOCK PULSES
If5t n
Fig. 5.34. Timing of scaled fringe pulses, after ZUMBERGE (1981)
Several reductions have to be applied to the observed gravity values. The
gravimetric earth tides can be reduced with an accuracy of a few 0.01 //ms 2 in
most parts of the world, cf. [8.3.5]. The polar motion reduction, according to
(3.109) and (5.63), is given by
sin 2( cosA - yf sin ),
with = rotational velocity of the earth, R = earth radius, and xp and yp =
coordinates of the instantaneous pole with respect to the IERS reference pole.
The geodetic coordinates , sufficiently approximate astronomic latitude and
longitude. The factor ' =1.2 accounts for the earth's elasticity, cf. [5.4.6].
The direct (gravitation) and indirect (deformation) effect of air pressure
variations is taken into account by a reduction
with = actual air pressure, and pn = normal air pressure as given by a standard
atmosphere, both in hPa (NffiBAUER 1988). The reduction of the adjusted
gravity value from the reference height to ground is performed by relative
gravity measurements, with an accuracy of 0.01 to 0.02 //ms"2, cf. [5.4.5].
The long-term stability of the length and time standards is controlled by
calibration of the laser (10~9 to 10~10 frequency stability) and the atomic clock
(10~10). The repeatability of the gravimeter system can be checked by regular
measurements at a reference station (Fig. 5.35), while the accuracy should be
estimated by comparisons with other instruments, cf. [5.4.3].
DEVIATION FROM g
(Ums'2)
0.150.1 0.05-
MEAN: g = 9811157.34 MTTIS'!
Fig. 5.35. Long-term stability control of JILAG-3 absolute gravimeter at the
reference station Clausthal, Germany
The accuracy of absolute gravity measurements depends strongly on site
conditions. Stable sites (hard bedrock, low man-made noise) provide better
results than locations in sediments, close to the coast, or in urban environment.
The drop-to-drop scatter (0.05 to a few //ms"2) is reduced by a large number
of measurements. The adjusted station gravity-value is generally derived with a
standard deviation of 0.01 to 0.03 //ms"2. The accuracy is at the order of a few
0.01 //ms"2, due to unmodeled systematic effects of the instrument (e.g., floor
recoil) or the environment (atmospheric loading, ground water variations).
Systematic discrepancies between different instruments may reach 0.05 //ms"2
and more, cf. [5.4.3],
The free-fall method was developed in the 1950's (Volet and Sakuma at the BIPM Sevres, Cook
at the National Physical Laboratory, Teddington). A transportable instrument was designed and
employed by Faller in 1968. Among the presently operating transportable gravimeters are the
JILA (Joint Institute for Laboratory Astrophysics, Boulder, CO, U.S.A.) and the FG5 (Micro-g
Solutions Inc., U.S.A.) free-fall instruments (FALLER et al. 1983, NIEBAUER et al. 1995). Around
200 position/time data pairs are collected over one drop, evenly distributed in distance over the
drop length of 20 cm, and adjusted on-line to a fitting parabola. The falling object moves in a coaccelerated 'drag-free' chamber. The chamber eliminates residual air drag and serves, by
adequate acceleration, for dropping and catching the comer cube as well as for transporting it
back to the initial position. The reference corner cube is isolated from ground motions by a
"super-spring", which by a feed-back system electronically generates effective free-oscillation
periods between 30 and 60 s. While the JILA gravimeters have a horizontal interferometer basis,
the FG5 instruments employ a vertical basis (Figs. 5.36, 5.37). The vertical basis eliminates the
influence of floor vibration and tilt on the optical path length. The instruments are disassembled
for transportation (FG5: 240 kg in 8 containers). Setting up at a station requires about two hours,
and observations are generally carried out over one to two days, depending on local noise (ToRGE
et al. 1987, KLOPPING et al. 1997). A portable modification of the FG5 gravimeter can be used in
outdoor environment on quiet sites, providing a precision of 0.1 &* in 10 minutes.
FREE-FALLING
INTERFEROMETER
DRAG-FREE
ION PUMP
SUPPORT SPRINGS
SUPERSPRING
H- MAIN SPRING
INTERNAL SERVO COIL
5.36. Free-fall gravimeter FG5
5.37. Free-fall gravimeter FG5 view,
principle, courtesy Micro-g Solutions, courtesy Micro-g Solutions, Inc.,
Inc., Arvada, CO, U.S.A.
Arvada, CO, U.S.A.
Transportable rise-and-fall instruments have been developed by the Istituto di Metrologia
"G.Colonnetti", Torino, Italy (ALASIA et al. 1982) and by Jaeger S.A., France (SAKUMA 1983).
An option for the rise-and-fall mode is also available for the FG5 gravimeter.
An ocean-bottom absolute gravimeter has been constructed primarily for monitoring long-term
gravity variations in geodynamically active zones (ZUMBERGE and CANUTESON 1995).
5.4.2 Relative Gravity Measurements
A "relative" gravity measurement yields the gravity difference between two
stations or variations of gravity with time, cf. [5.4.6]. Either time or length is
measured, keeping the other quantity fixed. As a consequence, relative
measurements can be performed more easily than absolute ones.
For the pendulum method, the oscillation periods T} and 2 of the same
pendulum are measured at two stations P{ and P2. From (5.71) we obtain
f =f
or, after simple transformation, the gravity difference
Agu = g2 ~g, = -2g,-^L + g ,
( -,)y2
The relative pendulum method has been employed extensively since v. Sterneck (1887) developed
a transportable device (pendulum length 25 cm, two pendulums swinging on the same support in
opposite phase in order to eliminate floor recoil effects). Although the systematic effects that are
independent of time and position cancel with this differential method, the accuracy could not be
increased over a few //ms~2 due to problems in keeping the pendulum length constant during a
field survey. The method was superseded in the 1930's by elastic spring gravimeters. The
pendulum method was used until the I960's for establishing gravimeter calibration lines,
exploiting the fact that pendulum results are given in the unit of acceleration and do not need to
be calibrated.
Relative gravity meters use a counterforce in order to keep a test mass in
equilibrium with gravity. Gravity changes in space or time are monitored by
corresponding changes of the counterforce, which are transformed to the gravity
unit by a calibration function. An elastic counterforce is used nearly
exclusively, but magnetic counterforces are also employed in instruments
operating on moving platforms and in the stationary mode, cf. [5.4.4], [5.4.6].
The elastic spring gravimeter is based on the principle of a spring balance. If
gravity changes, the spring length will also change in order to maintain static
equilibrium between gravity and the elastic force. According to Hooke's law,
the strain is proportional to the stress for small elongations. We distinguish
between translational systems (seldom used) and rotational systems.
In a translational system (vertical spring balance), the condition of equilibrium
is given by (Fig. 5.38a)
mg-k(l-l0) = 0,
where k is the spring constant and / (resp. /0 ) is the length of the spring with
(resp. without) a load. Applying (5.81) on a gravity difference Ag furnishes a
linear relationship between Ag and the observed difference in length / :
= -/ = -^-/.
An undamped spring generates a harmonic oscillation with the proper frequency
> 0 =V*
=2^=2\'-^-.
and the oscillation time
By differentiation, we obtain the mechanical sensitivity
In order to assess gravity changes with a relative accuracy of 10~8, length
changes of a 0. l m long spring would have to be determined to 1 nm.
Rotational systems (lever spring balance) consist of a lever that supports a mass
m and rotates about an axis O. Equilibrium can be produced through a
horizontal torsion spring or through a vertically or obliquely acting helical
spring. The equilibrium of the torques for the lever torsion spring balance (Fig.
5.38b) yields
where a = length of the lever, a = angle between the horizontal and the lever,
= torsion constant, and a0 = pretension angle of the spring. This non-linear
system becomes a linear one for a = 0, with
(0 + ,)
*"
Fig. 5.38. Elastic spring gravimeter principle: a) vertical spring balance, b) lever
torsion spring balance, c) general lever spring balance
For the general lever spring balance, the spring counterforce acts under an
arbitrary angle on the lever carrying the mass. The line connecting the rotation
axis with the point where the spring is mounted deviates by an angle from
the vertical (Fig. 5.38c). With the distance
h = (bd/l)sina
(5.88a),
the equilibrium condition for the torques reads
mgasin(a+ S}-
sina =
The sensitivity of this non-linear system can be significantly increased by
approximating the torques of gravity and of the elastic spring (astatization).
With a zero-length spring (/ = 0), we have the sensitivity
g sin
High sensitivity is achieved at a small angle and a ~ 90 . For a = 0.1 m,
a + = 90, and = 100", displacements have to be measured with 2 in
order to obtain a relative accuracy of 10"8. Compared to the linear system, the
sensitivity is thus increased by a factor of 2000.
The required accuracies of 0.1 //ms"2, or better, place high demands on the
reading systems as well as on the stability of the counterforce with time.
5.4Gravimetry
Optical and/or electrical reading systems are used to observe the position of the
test mass. A capacitive position-indicator is usually employed and is connected
to a digital readout unit. The zero-method is preferred for the measurement of
the displacement, with a compensation device for restoring the zero position.
Mechanical compensation is performed by a measurement screw, while
electronic feedback systems are not affected by screw errors (R DER et al.
The elasticity of the spring should exhibit a time stability of 10"8 over several
hours, which is the time interval required for transporting the gravimeter
between the stations of a large-scale network, cf. [7.4]. Spring materials include
NiFe alloys (small thermoelastic coefficient) and fused quartz (large but linear
thermoelastic coefficient, small coefficient of thermal expansion). In addition,
the measurement system has to be protected against changes in temperature
(thermostat), air pressure (air-tight sealing), and magnetic field (shielding of
metal alloy springs). The effects of mechanical shocks and vibrations can be
reduced by a damping device, in addition to air-damping.
Spring gravimeters have been developed since the 1930's for use in geophysical exploration.
From the 1950's, instruments were available which could be used for establishing large-scale
gravity networks. Most of these early gravimeters had a limited measuring range (e.g. 2000
\52) and required a reset screw for changing to another gravity range (Askania gravimeter:
torsion spring balance, metal alloy; Worden gravimeter: fused quartz system with horizontal
beam and vertical counter spring). The LaCoste and Romberg astatized gravimeters employ a
metal zero-length spring, acting at 45 inclination on the horizontal beam (model G: 70 000
&'2 range, measuring screw with 10 $2 per one rotation), KRIEG (1981), KANNGIESER
(1983), Fig. 5.39. Recently developed instruments are microprocessor-controlled and are highly
automated. They employ capacitive transducers and electronic feedback systems with worldwide
HOUSING^
DIALGEAR
--BOX
'SCREW
SHOCK ELIMIN; SPRING \EVER
F/g. 5.39. LaCoste and Romberg gravimeter principle, courtesy LaCoste and
Romberg, Inc., Austin, Texas, U.S.A.
Fig. 5.40. LaCoste and Romberg
gravimeter (Graviton-EG), courtesy
LaCoste and Romberg, Inc., Austin,
TX, U.S.A
Fig. 5.41. Scintrex Autograph CG-3M
gravimeter, courtesy Scintrex, Concord,
range. Self-leveling, a high data acquisition rate, and on-line evaluation (data compression and
analysis, earth tides reduction, drift control) are further characteristics (Scintrex CG-3 Autograph:
Worden type quartz system; LaCoste and Romberg model -meter), FALK (1995), BONVALOT et
al. (1998), Figs. 5.40, 5.41.
Options of conventional land gravimeters include underwater and bore-hole instruments. After
sealing in a pressure and water protected diving bell, an underwater gravimeter is lowered to the
sea bottom and remotely operated from on board a survey vessel. The underwater gravimeters are
used mainly in the shelf areas, BEYER et al. (1966). Bore-hole gravimeters are characterized by
small dimensions and remote-controlled operation at high temperatures.
Air/sea gravimeters will be descibed in [5.4.4] and recording (earth tides) gravimeters in [5.4.6].
Despite all measures to protect the gravimeter's measuring system against
environmental disturbances, the zero reading changes with time: drift and tares.
The drift is caused by aging of the spring material (approximating zero after
some years) and short-term changes which occur during a field survey. This
"transportation" drift results from reactions of the spring to vibrations and small
shocks, uncompensated temperature fluctuations, and elastic aftereffects after
unclamping the lever. It depends on the spring material and on measurement
conditions and can reach a few //msVday. Larger mechanical shocks may
produce sudden tares of the same order of magnitude or more. The drift is
determined by repeated station occupations during one day and subsequent
modeling. Different methods have been developed depending on the
instrumental behavior and the network structure; among them are the profile,
the star, and the step method (Fig. 5.42).
After reducing the gravimeter reading for the earth tides, cf. [8.3.5], the drift
function can be modeled by a low-order polynomial with time (Fig. 5.43):
Fig. 5.42. Drift determination methods: a) profile method, b) star method, c)
step method
with t0 = starting time (e.g., beginning of the survey) and </,, d2 = drift
parameters. The network adjustment delivers the drift parameters, based on
repeated observations, cf. [7.4].
The gravimeter reading z (in counter units) is converted to the gravity unit by
means of the calibration function:
F(z) depends on the physical and geometrical parameters of the measuring
system, see (5.82), (5.87), (5.89), which cannot be determined with the desired
0 '
1> time (h)
Fig. 5.43. Drift determination (profile method)
accuracy. Therefore, the calibration function is derived by comparing reading
differences with known differences of gravity. A low degree polynomial (linear
"scale factor" and eventually small non-linearities due to the properties of the
feed-back system) provides the transformation from the readings (stations /, /)
to the gravity difference:
with Yl, Y2 = calibration coefficients. Periodic calibration terms may be added
for taking cyclic errors of a measuring screw into account.
Laboratory and field methods are available for determining the coefficients of the calibration
function. In the laboratory, gravity changes can be simulated and compared with the
corresponding gravimeter readings. The tilt-table method uses the inclination by a known angle
for producing an apparent gravity variation, and the mass method uses the defined change of the
gravimeter mass. Special methods have been developed for recording gravimeters, cf. [5.4.6].
Calibration lines provide gravity differences, determined by absolute gravimeters, and may be
densified by relative gravimetry. The calibration-line surveys exploit the fact that gravity varies
with latitude and height (KANNGIESER et al. 1983). The limited gravity range of these lines only
allows determination of an approximate value for the linear calibration factor; an improved
estimate must be based on a global gravity-reference-system, cf. [5.4.3]
The accuracy of gravity differences (# < 1000 to 2000 //ms'2) observed with
well calibrated and drift-controlled instruments is 0.1 to 0.2 ,ums 2 . Repeated
measurements and the use of several instruments increase the accuracy to 0.05
to 0.1 //ms'2 and 0.02 //ms'2 for local ties (TORGE 1984, BECKER et al. 2000).
5.4.3 Gravity Reference Systems
Gravity reference systems provide homogeneity of gravimetric surveys by
realizing a gravity standard through the gravity values of a selected number of
The need to establish a global reference system arose at the end of the 19th century when larger
sets of absolute and relative pendulum measurements had to be combined. The Potsdam Gravity
System was introduced in 1909 by IAG. It was based on reversible pendulum measurements
carried out in the Geodetic Institute Potsdam by K hnen and Furtw ngler (1898 - 1904). Relative
pendulum ties to national base stations transferred the absolute value to other parts of the world.
Since the 1930's, new absolute and relative gravity measurements revealed that the Potsdam
gravity value was 140 ^ms~2 too high and that transfer errors of several 10 2 had occurred.
The Potsdam Gravity System was superseded by the International Gravity
Standardization Net 1971 (IGSN71), recommended by the I.U.G.G. (MORELLI
et al. 1974). This network contains 1854 gravity stations (among them about
Fig. 5.44. International Gravity Standardization Net 1971 (IGSN71): absolute
gravity stations and selected network ties, after MORELLI et al. (1974)
500 primary stations) determined by 10 absolute and about 25 000 relative
measurements, including 1200 relative pendulum ties (Fig. 5.44). The mean
uncertainty of the adjusted gravity values is less than l//ms 2 . High relativeaccuracy is provided at gravimeter calibration lines, which extend in the northsouth direction in America, Europe and Africa, and in the western Pacific.
Meanwhile the IGSN71 has been extended to previously uncovered parts of the
world. Regional networks have been connected to IGSN71, or transformed
(shift and scale factor) to it, with the help of identical stations. The IGSN71
gravity values can be used to derive the linear calibration factor of relative
gravimeters with a relative accuracy of some 10"5.
FROM MEAN
' l-h 1 I i
JILAG
FG5
Fig. 5.45. Absolute gravimeter comparison, BIPM Sevres 1997: Deviations
from mean value and standard deviations for JILAG-, FG5- and other
gravimeters, after ROBERTSSON et al. (2001).
With the increasing availability of transportable absolute gravimeters of
accuracy of 0.05 //ms'2 and better, the gravity reference can be established
independently from a global system with any gravity survey (TORGE 1998). The
linear calibration factors of relative gravimeters then are derived from the
absolute values available in the survey area. The IGSN71 consequently does not
need a readjustment but rather is improved continuously by networks based on
absolute gravimetry. This strategy requires a regular quality control of the
absolute gravimeter systems, cf. [5.4.1]. International comparisons have been
carried out since the 1980's at the BIPM, Sevres (Fig. 5.45). For advanced
absolute gravimeters, both the r.m.s. scatter around the BIPM mean value and
the long-term stability is a few 0.01 //ms 2 , which characterizes the present
state of the realization of the gravity standard (ROBERTSSON et al. 2001).
5.4.4 Gravity Measurements on Moving Platforms
Kinematic methods have been developed for rapid and high-resolution
gravimetric surveys in areas of having challenging environmental conditions,
such as oceans, the polar regions, high mountains, and tropical forests. Ships
and airplanes are predominantly used as platforms, but helicopters and land
vehicles also have been employed for local surveys. Compared to stationary
gravimetry, additional difficulties arise in kinematic gravimetry, including
problems with orientation of the gravity sensor and with separating gravity from
non-gravitational accelerations (EHLINGER 1978, BROZENA and PETERS 1995,
CANNON and LACHAPELLE 1997).
The principle of kinematic gravimetry is based on Newton's equation of motion.
In the local astronomic system (also called local level system), cf. [2.6.2], the
gravity vector is expressed by:
r/,
where r = d2r/dt2 is the platform acceleration, r the platform velocity (r =
position vector), and f is the vector of measured acceleration (also called
specific force). It is assumed that the accelerometers are fixed to the vehicle,
which requires a transformation from the vehicle's body frame b to the local
level system /. The corresponding rotation matrix R contains the orientation
angles between the two frames. As the platform moves with r with respect to
the earth, inertial accelerations arise. These accelerations are taken into account
by the last term in (5.93). o>'ie and ^, are vectors of angular velocities of the
earth's rotation with respect to the inertial frame / and of the platform's rotation
with respect to the earth-fixed frame e, respectively.
Operational sea and airborne gravimetry employs modified land gravimeters
mounted on a damped two-axes stabilized platform. Stabilization occurs in the
local-level frame by two gyroscope/accelerometer pairs operating in a feed-back
mode. For this "scalar" gravimetry, only the magnitude of gravity is
determined, and (5.93) reduces to
Here, fz and z are the vertical components of the specific force and the
platform acceleration, (u = angular velocity of the earth rotation,
- geodetic
latitude, a = geodetic azimuth, = platform velocity with respect to the earth,
and r = distance to the earth's center. For the static case, (5.94) transforms into
the equilibrium conditions of relative gravimetry, cf. [5.4.2].
vsma
Fig. 5.46. Etvs effect
The velocity dependent terms on the right side of (5.94) represent the Etvs
reduction. From Fig. 5.46, it can be identified as the Coriolis acceleration,
which increases (for a west-east directed course) the angular velocity of the
earth rotation, and the centrifugal acceleration arising from the platform's
angular velocity v/r around the center of the earth. Close to the earth (r = R =
6371 km) the Etvs reduction amounts to
with v in km/h. The second term is small for sea gravimetry but attains large
values with airborne applications. As velocities can be determined by GPS
navigation with an accuracy of 0.05 m/s, the uncertainty of the Etvs reduction
is now less than 10 //ms 2 .
Instead of using a stabilized platform, the gravity sensor can be rigidly connected to a vehicle.
GPS supported inertial navigation systems (INS) are employed with this strapdoyvn inertial
gravimetry, characterized by a high digital data rate. In the scalar mode, only one approximatelyvertical accelerometer is used, while in the vector mode three orthogonally mounted
accelerometers determine the specific force vector. According to (5.93), the orientation between
the body and the local level frame is needed continuously and is calculated from the output of the
INS gyros. Due to the high demands on attitude control, vector gravimetry is still in the
experimental stage. Demands are less stringent for scalar gravimetry, especially if the output of
the accelerometer triad is used for the determination of the magnitude of gravity (rotation
invariant scalar gravimetry), JEKELI (1995), TIMMEN et al. (1998), WEI and SCHWARZ (1998).
The methods for separating gravity from non-gravitational accelerations depend
on the frequency of the accelerations and differ for sea and airborne gravimetry.
High-frequency vibrations can be strongly reduced by the damping of the
measuring system. For sea gravimetry (stabilized platform), "disturbing"
accelerations occur with periods between 2 and 20 s, and they may reach
amplitudes of 0.1 g. Due to low ship velocity (10 to 20 km/h) and the nearly
constant reference surface (sea level), low-pass filtering sufficiently suppresses
the vertical accelerations. By averaging the recorded data over time intervals of
1 to 5 minutes, mean gravity values over some 0.1 to 2 km are thus obtained.
The effect of horizontal accelerations remains small because of the stabilization.
Off-leveling effects generally can be neglected (attitude accuracy about 10").
More critical are cross-coupling effects, which occur with horizontal lever
spring gravimeters between the horizontal and the vertical component of the
disturbing acceleration. They may reach 50 //ms 2 or more and must be
corrected using the horizontal acceleration records. Vertical line gravimeters are
free from these errors.
For airborne gravimetry, accelerations vary with periods from 1 to 300 s (longperiodic eigenmotion of the airplane) and with amplitudes up to 0.01 g and
more. Large airplane velocities (250 to 450 km/h) prevent an effective filtering,
and thus with a long filter-length (several minutes) only mean gravity values
over some 10 km are obtained. Consequently, accelerations have to be
determined independently by differential GPS (carrier phase measurements)
through the second time derivative of position or the first time derivative of
velocity. Over water and ice areas, radar and laser altimetry can also be
employed for height determination. Heights are needed in order to reduce the
gravity data to a common reference level, by the free air reduction, cf. [6.5.3].
In 1923, F.A.Vening-Meinesz constructed a three-pendulum instrument for gravity measurements
in a submerged submarine; world-wide cruises followed until the 1960's. Since that time, sea
gravimeters (e.g., Askania, LaCoste and Romberg) mounted on gyro-stabilized platforms have
operated on board surface vessels, helicopters (1980's), and airplanes (1990's). Since the 1990's,
Fig. 5.47. Gravity sensor Gss30 and gyrostabilized platform KT 30, courtesy
Bodenseewerk Geosystem, berlingen, Germany
these campaigns have also been supported by GPS. A vertical spring is employed in the
Bodenseewerk straight-line gravimeter (Fig. 5.47).
Force-balanced accelerometers, as developed for inertial navigation, are small and robust with
respect to strong dynamics but have less resolution and larger drift rates than conventional land
gravimeters. For a linear system, the proof mass is constrained to move in only one direction and
maintained at the zero position by an electromagnetic field. The electrical current needed to
maintain zero is proportional to the acceleration. Depending on the direction of the sensitive axis,
dedicated components of the specific force are measured (BELL and WATTS 1986, NEUMAYER and
HEHL 1995), Fig. 5.48. Force-balanced accelerometers are especially suited for use under rough
conditions on sea and in air; they have also been employed on board deep sea vessels (COCHRAN
et al. 1999). Similar properties are found with vibrating string gravimeters. Here, gravity
measurements are based on the fact that the vibrational frequency of a string under tension is
proportional to the square root of g (BowiN et al. 1972).
CASE^
/DETECTOR
Fig. 5.48. Force-balanced accelerometer principle: translational (suspended
mass) system (left) and rotational (pendulum) system (right)
Sea and airborne gravimetric surveys generally are carried out along parallel
tracks. Orthogonal tracks serve for control and accuracy improvement by
adjustment of the cross-over discrepancies. Sea gravimetry has been
concentrated in regions of geological interest and in shelf areas, while recent
airborne surveys have covered areas lacking in gravity field information
(Greenland, arctic ocean, Antarctica, Switzerland), FORSBERG and BROZENA
(1993), KLINGELE et al. (1997), Fig. 5.49.
Fig. 5.49. Sea gravimetry profiles (1965-1972), Western Mediterranean Sea,
Osservatorio Geofisico Sperimentale, Trieste, after FlNETTI and MORELLI
The accuracy of sea and airborne gravity measurements (data recording
generally with l s average) depends on the survey conditions (sea state, air
turbulence, ship and aircraft properties, flight altitude and velocity), on attitude
errors, and, for airborne gravimetry, on the separation between gravity and
disturbing accelerations. Accuracies of 5 to 20 //ms"2 are achieved with sea
gravimetry with a resolution of about 1 km along track (track distances 5 to 10
km). Airborne gravimetry generally is carried out at flight heights of a few km ,
but low speed and elevation (several 100 m) surveys are also performed,
especially with helicopters (HAMMER 1983). A resolution of 5 to 10 km is
routinely obtained now (helicopter 1 km), with accuracies of 20 to 50 //ms"2
(helicopter 5 //ms 2 ). An increase in resolution and accuracy by a factor of two
is expected (GUMERT 1995). It must be remembered that the attenuation of the
gravity field with height, cf. [3.3.3], prevents a high frequency resolution at
high flight altitudes.
5.4.5 Gravity Gradiometry
The gravity gradient tensor (3.68) contains local gravity field information, and
thus is of interest for high-resolution gravity field determination. It is generally
expressed in the local astronomic (local level) system, cf. [3.2.2], The unit of
the components of grad g is s 2 . In view of the magnitude of the components
and the measuring accuracy, they are generally expressed in 10"9 s = n s 2 ,
traditionally called E tv s unit (E).
A gravity gradiometer determines the components of grad g, either all or
several or linear combinations of them, by exploiting the different reaction of
neighboring proof masses to the gravity field. A gradiometer unit consequently
consists of two gravity sensors (mostly accelerometers) rigidly connected and
generally orientated in the local level system. Taylor expansions of gravity in
the two sensors 1 and 2, with respect to the center of mass C of the system, and
differences in the output of the sensors (specific force f) yields in the stationary
C,-fi=(gnidgX(ra-riy,
with r,, r2 = position vectors of the sensors in the local level system. A
gradiometer system is composed of several gradiometer units orientated in
different directions in order to derive the corresponding components (Fig. 5.50).
Rotation of the gradiometer units in the gravity field provides another means for
the determination of different components.
On the earth's surface, gravimeters can be used to approximate the components
of grad g by measuring gravity differences between adjacent stations. The
horizontal gradient (Wa,Wv) can be derived with a precision of 10 ns"2 from
gravity profiles or area surveys, with station distances of 10 to 100 m (HAMMER
Wxx*
Fig. 5.50. Gravity gradiometer translational system principle with longitudinal
(left and center) and transverse (right) constellation
1979). The vertical component fFs can be determined with the same precision
by repeated relative gravity measurements on tripods, with heights up to 3 m
(RDER et al. 1985).
The torsion balance, developed by R.v.Etvs around 1900, was the first dedicated gravity
gradiometer. It consists of two equal masses situated at different heights and rigidly connected by
a beam system. At the center of mass the system is suspended by a torsional thread. Equilibrium
of the torques acting on the masses is achieved by horizontal rotation, which depends on the
components Ww -Wa, Wv, Wa, W^. These quantities and the zero position of the beam are
determined by observing the beam direction at five different azimuths. A precision of 1 to 3 ns"2
was obtained (MUELLER et al. 1963). The torsion balance was widely employed in applied
geophysics between 1920 and 1940.
Terrestrial gravity-gradiometry in the stationary mode is time consuming and
strongly affected by small local mass anomalies. Topographic reductions have
to be taken into account even in the immediate surrounding (within 100 m),
which limits the application to flat or moderate hilly areas.
Measurements on moving platforms allow rapid data collection (e.g., in a l s
rate or more). With airborne and satellite applications, cf. [5.2.7], topographic
effects are significantly reduced. In the kinematic mode, the gradiometer system
is mounted on a gyro-stabilized platform. The effects of the platform's rotation
about the earth have to be taken into account, cf. [5.4.4]. Due to differencing,
non-gravitational forces of linear type cancel with this method.
A gravity gradiometer survey system for automobile and airplane applications has been developed
at Bell Aerospace, with a precision of 10 ns"2 and a resolution of a few km (JEKELI 1988a,
VASCO and TAYLOR 1991). It consists of three gradiometer units, each equipped with two
accelerometer pairs, mounted orthogonally on a slowly rotating disk (Fig. 5.51). The gradiometer
units are combined under different orientation on a gyro-stabilized platform. Superconducting
technology may greatly reduce vibrational and long-term effects and provide a precision of
1 ns" 2 . Airborne or shipborne surveys could then resolve the gravity field down to 1 km or less
and by integration give an accuracy of 1 //ms'2 for gravity (PAIK et al. 1997).
Fig. 5.51. Rotating gravity gradiometer principle, after JEKELI (1988a)
5.4.6 Continuous Gravity Measurements
Continuous gravity records contain information on earth tides and ocean tidal
loading, free oscillations and the nearly diurnal free wobble of the earth, inner
core translation, polar motion, and volcanic and earthquake activity (ZRN et al.
1997, CROSSLEY et al. 1999), cf. [8.3.5]. These effects occur at time scales
between seconds and several years and have amplitudes of about 1 to 1000
(tides) nms'2. Hence, a recording gravimeter should provide a resolution of
0.01 to 1 nms"2 and a high stability with time (low drift rates). It should be
time-controlled within 10 ms. In order to reduce environmental effects
(temperature changes, microseismicity, local inclinations), recording
gravimeters generally are installed at underground sites (basement, tunnel).
Recording gravimeters operate in an electronic feedback mode, cf.[5.4.2], over
a limited measuring range, e.g., 10 //ms 2 . The voltage output is proportional to
gravity and first undergoes an analogue filtering in order to reduce the highfrequency noise. It is then digitized by an A/D converter. Digital filtering
delivers a data set (1 to 10 s samples), which is stored on a PC. Further
numerical filtering and data reduction may be appropriate as well as the
reduction of spikes due to earthquakes and the interpolation of data gaps
(WENZEL 1996). An analogue output offers a convenient on-line control of the
data acquisition.
Spring-type and superconducting gravimeters are used for gravity recording
(MELCHIOR 1983).
Elastic spring gravimeters can be employed if supplemented with a low-pass
filter, a recording unit, and a quartz clock. Special earth tide gravimeters have
also been developed and are characterized by long-term stability (e.g., by a
double thermostat). Recent land gravimeters offer the option of an earth tides
mode through increased sensitivity, large memory, and computer-controlled
remote operation. The long-term drift of these instruments has to be removed by
filtering. Consequently, only short-period effects (e.g., diurnal and higherfrequency tides) can be determined, at a noise level of a few 0.1 to 1 nms 2 .
For the superconducting gravimeter, the gravity acting on the proof mass (Nb
sphere) is compensated by a magnetic counterforce (Fig. 5.52). The magnetic
field is generated by superconducting coils and thus is extremely stable with
time. The position of the mass is monitored by a capacitive detector and held
fixed by a feedback coil. Liquid helium provides the superconducting state at a
temperature of 4.2 K. The measuring system is kept in an insulating Dewar
vessel (Fig. 5.53). For long-term remote operation, a reduced size option is
available with no need for liquid helium refilling (GOODKIND 1991, RICHTER
Fig. 5.52. Superconducting
gravimeter principle, after GWRInstruments information and RICHTER
Fig. 5.53. RSG remote controlled
tidal superconducting gravimeter
view, courtesy of GWR-Instruments
Inc., San Diego, CA, U.S.A.
1995, WARBURTON and BRINTON 1995). The instrumental drift of a
superconducting gravimeter is very small (on the order of 10 nms"2/a), and
may be modeled by an exponential function. Absolute gravity measurements
can be used to control the drift at longer time intervals. From tidal analysis, a
noise level between 0.01 to 0.1 nms"2 (short periods) and 1 to 10 nms"2 (very
long time intervals) has been found (HINDERER et al. 1994, NEUMEYER and
DITTFELD 1997).
Calibration of a recording gravimeter is performed by relative and absolute
methods. A relative calibration is realized by parallel registration with a
"calibrated" gravimeter or by recording on a station with well-known tidal
parameters. Absolute calibration is generally performed on a vertical-gravimeter
calibration line. Laboratory calibration procedures include the artificial periodic
acceleration on a vertically-oscillating platform, the controlled vertical
displacement of large external masses, and the parallel registration with an
absolute gravimeter. Accuracies of a few 0.1 % are achieved (RICHTER et al.
1995, FRANCIS 1997). The instrumental phase shift is determined by recording
the gravimeter's response to a defined impulse.
The gravity signal is strongly correlated with atmospheric pressure. A linear regression with local
air pressure (between -2.5 and -3.5 nms"2/hPa ) reduces the main part of this effect. More refined
models are available which also take the air pressure around the station and the elastic response of
the earth's crust into account (MERRIAM 1992). Variations of groundwater level and precipitation
also affect the gravity record (order of some 10 nms"2) but are difficult to model (VIRTANEN
Longer (several months and more) gravity records can be subjected to a tidal
analysis. It is based on the spectral decomposition of the observed signal into a
number of partial tides. By comparing the observations (hourly samples,
atmospheric pressure effects reduced) with the gravimetric tides for a rigid
earth, cf. [3.5.2], deviations in amplitude and phase are found, which depend on
the earth's elastic response to the tidal forces. For the partial tide /', this is
expressed by the amplitude factor
with 4 = observed resp. calculated (solid earth) amplitude, and the phase shift
. = . (obs) - , (theor) ,
with Oy = observed resp.calculated phase. The observation equation for a least
squares spectral analysis then reads
/ (/) = (theor )cos (coj + , (theor ) + , ) ,
/=!
with /(/) = recorded gravity value at time /, and . = circular frequency for the
partial tide / (WENZEL 1976, 1997).
Elastic spring gravimeters allow the determination of 10 to 20 partial tides (mainly diurnal,
semidiurnal, terdiurnal), with an observation time of 4 to 6 months. Superconducting gravimeters
can resolve up to 40 tides (including semiannual and annual) by registration over several years.
The gravimetric factor for polar motion has also been derived from long-term series. As an
example, a 158 days registration with a LaCoste and Romberg feedback gravimeter at Hannover
( = 52.387'N, = 9.7 13, = 50 m) yielded for the lunar diurnal tide Ol
(1) = 1.151 0.001, (1) = 0.16 0.08
and for the semidiurnal tide M2
(2) = 1.188 0.0005, (2) = 1.70 0.03
(TiMMEN and WENZEL 1994a). The factor for Ol is close to the observed global value 1.155,
while the M2 result differs due to ocean load and attraction, cf. [8.3.5].
A global data bank for tidal gravity measurements (more than 300 stations) is maintained at the
International Center for Earth Tides in Brussels (MELCHIOR 1994).
Terrestrial Geodetic Measurements
Terrestrial geodetic measurements are carried out by directly observing
geometric quantities between points on the earth's surface. The majority of the
observations refer to the gravity field of the earth through orientation in the
local astronomic systems. The measurement of horizontal and zenith angles
[5.5.1] and distances [5.5.2] allows relative positioning, where combined
instruments (total stations) are generally used. Very precise height differences
are provided by leveling [5.5.3]. Strain and tilt measurements serve for
detecting local changes of distances and inclination with time [5.5.4].
Due to the high accuracy and economy of satellite-based measurement
techniques, terrestrial geodetic measurements are used primarily for
interpolating satellite-derived results or in areas where satellite methods fail or
need terrestrial support (underground and underwater positioning, forests, urban
areas, engineering surveys, local geodynamics).
Inertial surveying provides kinematic three-dimensional positioning by evaluating accelerometer
measurements on a moving gyro-stabilized platform. The influence of the gravity field has to be
taken into account by reductions. The method is mainly used for station densification in control
networks, especially for mapping purposes. The accuracy achieved depends on distance and is
a b o u t ( 0 . 1 m + 10x10"* s), SCHWARZ(1986), JEKELI (2001).
Terrestrial methods are treated in textbooks on surveying, e.g., KAHMEN and
FAIG (1988), BANNISTER et al. (1998). Classical surveying instruments are
described by DEUMLICH (1988), while the actual state is dealt with in JOECKEL
und STOBER (1999), and DEUMLICH and STAIGER (1999). KAHMEN (1978) and
SCHLEMMER (1996) concentrate on the fundamentals of electronics employed
in geodetic instruments, while Brunner (1984b) deals with the effects of
atmospheric refraction.
5.5.1 Horizontal and Vertical Angle Measurements
The horizontal angle is defined as the angle measured in the horizontal plane of
the local astronomic system between two vertical planes. It is formed by the
difference in horizontal directions to the target points which define the vertical
planes. The vertical angle is the angle measured in the vertical plane between
the horizontal plane and the direction to the target point. Generally the zenith
angle, the complement to 90, is introduced instead of the vertical angle, cf.
A theodolite is used for measuring horizontal and vertical angles. The principal
components of this instrument are a horizontal and a vertical circle with
graduation, a telescope capable of being rotated about the vertical and the
horizontal axes, and a mechanism for reading the circles. In order to orientate
the theodolite with respect to the plumb line direction, it is equipped with spirit
or electronic levels.
Regarding the reading of the circle graduation, we distinguish between optical
and electronic or digital theodolites.
Optical theodolites of highest precision were developed since the second half of the 18th century
and were then used until the 1960's for first order triangulation (with the Kern DKM3 and the
Wild T3 theodolites being the latest developments) at station distances of 30 to 60 km. They were
characterized by very stable construction and circle diameters of 100 to 250 mm. Circle
graduation errors were less than 0.5", and reading accuracy reached 0.1" by using a coincidence
microscope with micrometer screw. The lens aperture of the telescope was 60 to 70 mm and the
magnification 30 to 40 or more. Standard deviations of 0.2" to 0.4" have been achieved for an
adjusted horizontal direction, cf. [7.1.1].
Today, horizontal angles are measured only over maximum distances of a few
km for densification surveys or dedicated local networks. Electronic theodolites
(lens aperture 40 to 45 mm, magnification of 30 or more) are available for this
purpose and have superseded the optical analogue instruments (Fig. 5.54).
Generally, the electronic theodolite is combined with a distance meter to
produce a total station, cf. [5.5.2].
The horizontal and vertical circles of an electronic theodolite are either coded or
carry an incremental graduation. Reading is microprocessor-controlled and
performed by optical-electronic scanning and subsequent interpolation
(electronic micrometer). Electronic levels and a dual-axes compensator serve
for leveling the instrument; a residual tilt correction may also be applied
automatically. Collimation and horizontal axis errors are either eliminated by
measurement at both positions of the telescope or corrected internally. An
accuracy of 0.5" to 1" is obtained for a direction observed in two positions.
Gyrotheodolites have been developed for the determination of astronomic
azimuths by combining a theodolite with a gyroscope. The principle of the
gyroscope is based on the fact that a rapidly rotating gyro with horizontal spin
axis swings into the north direction due to the mutual effects of the gyro's spin,
the earth's gravity, and the earth's rotation. An accuracy of 3" can be obtained
from 20 individual measurements. Gyrotheodolites are employed primarily for
mining and tunnel surveys, Fig. 5.55.
For the measurement of zenith angles, the theodolite is equipped with a reading
index for the vertical circle. By leveling the index either manually (spirit level)
or automatically, the local vertical is realized with an accuracy of a few 0.1".
With electronic theodolites (circle diameter about 70 mm, electronic levels with
2"/2 mm) an accuracy of 0.5" to 2" is obtained for an observed zenith angle.
Fig. 5.54. Electronic theodolite
(Leica T 1800), courtesy Leica
Fig. 5.55. Gyrotheodolite (Gyromat
2000), courtesy Deutsche Montan
Technologie (DMT), Essen, Germany
Further processing of zenith angles, cf. [6.4.2], requires the consideration of
errors due to vertical refraction.
The refraction angle depends on the coefficient of refraction (5.11) and thereby on the
meteorological conditions along the path of light, particularly the vertical gradient of temperature,
see (5.19). Generally, the refraction angle is derived from meteorological data taken at the
endpoints of the observation line. This easily leads to errors of a few arcsec and more for
distances larger than a few km. According to (5.lib), the effect of this error on the height
difference increases with the square of the distance and reaches the order of a few decimeters over
a few km. The error of the observed zenith angle, on the other hand, only propagates with
distance and thus remains at the order of a few cm.
An approximately symmetric behavior of refraction is to be expected for simultaneous
observations at the endpoints, especially with cloudy weather and prior to the isothermal
conditions of the evening and if the ray of light is more than 15 to 20 m above the ground. The
uncertainty of the refraction angle remains less than 1" for distances below 10 to 25 km. This has
led to the method of observing reciprocal simultaneous-zenith-angles.
For the direct determination of the refraction angle, the dispersion of the light may be utilized, cf.
[5.1.1]. As the light path at shorter wavelengths has a greater curvature than at longer ones, the
use of two different wavelengths causes a difference between the two angles of refraction at the
target point. This dispersion angle depends on the effective coefficients of refraction along the
light path. Neglecting the small influence of water vapor pressure, it is proportional to the
refraction angle but about two orders of magnitude smaller. The main error source of the
measurement of the dispersion angle is atmospheric turbulence. Experiments at distances less
than 20 km gave uncertainties of 1 to 2" for the refraction angle. An eventual application of
dispersometers probably will be restricted to distances of a few km (WILLIAMS and KAHMEN
5.5.2 Distance Measurements, Total Stations
Terrestrial distances represent important geometric quantities for positioning.
They provide geometric relations between points and establish the scale of
classical geodetic networks.
Until about I960, the scale of triangulation networks, constructed from angle measurements, was
derived from base lines having lengths of 5 to 10 km. Measuring rods and, since 1900, wires or
tapes served to measure the base line length. With the J derin (1880) method, freely hanging
invar (NiFe alloy) wires 24 m in length were used, characterized by a small coefficient of thermal
expansion. The relative accuracy of the more recent base lines amounts to " 6 , which
corresponds to 1 mm/1 km.
For field calibrations of wires and tapes, several international calibration lines were established
by interferometric methods. Starting from the length of a standard meter, the V is l light
interference comparator provided an optical multiplication up to base line lengths of 864 m
(relative accuracy 10"7).
Electromagnetic distance measurements started at the end of the 1940's. They
either use light waves ( = 0.4 to 0.8 ) and the near infrared (up to = 1 )
or microwaves ( = 1 to 10 cm) as carriers of the measuring signal (RUEGER
1997, JOECKEL und STOBER 1999). The travel time of the signal serves as a
measure for the distance.
Microwaves are hardly absorbed by the atmosphere and allow the measurement
of large distances (50 km and more) even under unfavorable weather
conditions. The effect of humidity on refraction, on the other hand, is great and
may significantly deteriorate the results. Distances measured by light waves are
about one order of magnitude more accurate, but their use may be limited by
visibility disturbances (haze, fog), cf. [5.1.2].
Time measurement is performed by the pulse or the phase comparison method.
For the pulse method, the transmitter emits a pulse which is reflected at the
target and observed at the receiver. An electronic timer measures the time /
that the signal requires to travel forth and back along the distance s. We have
where we assume that refraction effects have been taken into account by
corresponding corrections, cf. [5.1.1]. If the uncertainty in distance is to remain
less than 5 mm, the time of propagation must be obtained to an accuracy of
0.03 ns. This high accuracy demand can be fulfilled by short (a few ns) laser
pulses, electronic counting with a high frequency oscillator, and averaging the
results of a large amount (e.g., 1000) of individual measurements, cf. also
For the phase comparison method, a high-frequency carrier wave is sent out by
the transmitter and modulated continuously (amplitude or frequency
modulation), with modulation frequencies between about 10 and 100 MHz. The
corresponding half wavelength (because of the double distance traveled by the
signal) serves as a "yard stick" (about 1 to 10 m) in surveying the distance (Fig.
5.56). The phase shift found between the transmitted and the received signal
represents the residual part of the distance above an integral number of
complete wavelengths.
Fig. 5.56. Phase comparison method principle
Travel time and phase shift are related through
TV + /2
with TV = number of complete periods, and the modulation frequency
= refractive index, cf. [5.1.1]. Substituting (5.101) and (5.102) into (5.100),
and assuming that refraction is corrected separately, delivers the distance
With the residual part of a wavelength
=^,
the distance can also be expressed by
* = ^4 + ^
(5-105)
Hence, a distance measuring unit consists of an oscillator and transmitter, a
receiver, a phase meter, and a microprocessor. With a digital phase detector, the
measuring process can be fully automatized; the resolution achievable is 10"3 to
10"4, which corresponds to a mm-precision. The number is determined
automatically by applying several slightly different modulation frequencies
generated by frequency division.
Terrestrial microwave distance measurements started with the development of the tellurometer by
T. L. Wadley (1956). Here, the master station emitted a modulated (modulation frequencies
between 7.5 and 150 MHz) carrier wave ( = 8 mm to 10 cm), which was retransmitted from the
active transponder (receiver and transmitter). Measurement of ranges up to 70 km and more were
obtained. The accuracy strongly depended on refraction uncertainties and could reach
(l0...15mm + 3xl0^s).
Electro-optical distance measurements trace back to the first geodimeter developed by E.
Bergstrand (1948). Long-range distance meters used laser light (He-Ne gas laser) with
modulation frequencies between 15 and 50 MHz and were able to measure distances up to 60 km
on clear days, with an accuracy of
Microwave and long-range electro-optical distance measurements have been carried out
extensively from the 1950's to the 1970's. The measurements were primarily for establishing first
order control networks and for strengthening existing horizontal control, cf. [7.1.1]. Long-range
distance meters have become obsolete, as large-scale positioning is carried out nowadays almost
exclusively by satellite methods.
Today, terrestrial distance measurements are restricted to ranges of a few km
and about 10 km maximum. Visible light and (mostly) near infrared are used,
either in the pulse or (mainly) in the phase comparison method. Pure distance
meters are available for special applications (partly suitable to be mounted on a
theodolite), Fig. 5.57, but generally electronic total stations are employed,
measuring distances, horizontal directions and zenith angles simultaneously.
Fig. 5.57. Pulsed infrared laser distance meter (Leica Di 3000S), courtesy Leica
Geosystems AG, Heerbrugg, Switzerland
The measuring range of a distance measuring unit depends on the number of
prisms posted on the target station. With pulsed infrared laser light (gallium
arsenide diode), distances up to 5 km (1 prism) or 10 to 20 km (multiple prisms)
can be observed with an accuracy of
(3...5mm + lxlO- 6 s)
and within a measuring time of a few seconds.
With the phase comparison method, using modulated infrared light, distances
up to 2 to 3 km can be measured with one prism with an accuracy of
(l...2mm + l... 2x10^).
When integrated into a total station, the microprocessor-controlled operation
also includes the (partly) automatic searching and pointing of the target,
leveling by a two-axes compensator (range 5', precision 0.3"), measurement of
horizontal direction and zenith angle, data storage in the internal memory,
refraction corrections with standard or actual meteorological data,
transformation from polar to local Cartesian coordinates, and a graphic display
(FEIST et al. 1998), Figs. 5.58, 5.59.
The calibration of electronic distance meters includes the control of the
modulation frequency by a frequency meter and the determination of the
instrumental constants (zero point correction and possible cyclic errors) on a
comparator or on a short (about 1 km) calibration line. Calibration lines are
usually partitioned into several sections and determined by laser interferometry
Fig. 5.58. Total station (Trimble
ITS 500), courtesy Trimble
Navigation Ltd., Sunnyvale, CA,
Fig. 5.59. Total station (Geodimeter
System 600), courtesy Spectra
Precision AB, Danderyd, Sweden
or with a short-range distance meter of high precision.
The error budget of distance measurements contains a constant part that
depends on uncertainties in timing or phase measurement and on the zero point
stability. The distance dependent part is determined by errors of the modulation
frequency and by refraction effects.
The meteorological parameters generally are measured only at the instrument
and at the target point, and the arithmetic mean is introduced with the refraction
reduction. This value may not be representative for the entire distance, and thus
a limiting factor for precise distance measurements is set (ILIFFE and DODSON
1987). Measurements conducted under the same atmospheric conditions may be
highly correlated, but this correlation can be significantly reduced if
observations are carried out under different conditions (HPCKE 1965).
Special distance meters have been developed in order to measure shorter base
lines with very high precision (calibration lines, local geodynamic control).
These developments are based either on instrumental refinements and better
determination of the refraction effects or on the use of two or three different
wavelengths.
The Mekometer ME 5000 uses a He-Ne laser as a light source, with a polarization modulated
wavelength of 0.6 m (FROOME 1971, MEIER and LOSER 1986), Fig. 5.60. The modulation
frequency is automatically adjusted such that the distance becomes an integer number of
wavelengths and is measured with a frequency meter. The Geomensor (COM-RAD, England)
uses a Xenon flash tube as light source (SCHERER 1985). Meteorological data are provided by a
special sensor unit. Distances of 5 km can be measured with one prism, and a precision of
(0.1...0.2mm + 0.2xlOmay be obtained. However, refraction effects may increase the distance-dependent contribution to
0.5x 1CT6 s (DoosoN and FLEMING 1988).
Fig. 5.60. Distance meter Mekometer ME 5000, courtesy Leica Geosystems
AG, Heerbrugg, Switzerland
If the distance is measured with different wavelengths, the effect of dispersion can be exploited,
cf. [5.1.1]. For light waves, the difference in the distances obtained with "blue" and "red" light
particularly depends on the influence of temperature and atmospheric pressure on the index of
refraction. If an additional microwave measurement is performed, the effect of humidity is then
reflected mainly in the deviation of the microwave results from the light results. From (5.14),
(5.15), (5.17), and (5.18), a relation between the geometric length of the path and the differences
between the distances obtained with different light waves and a microwave can be established. A
resolution of a few 0.01 mm is required at these small differences in order to obtain mm-accuracy
for the distance. The terrameter has been developed as a two-wave instrument (He-Cd laser 441.6
nm, He-Ne laser 632.8 nm) to measure distances with a precision of 0.1 mm (LANGBEIN et al.
1987). By adding a microwave ( = 3 cm) and employing a light modulation frequency of 3 GHz,
the relative accuracy of this method could be increased to 10"7 and better (HUGGETT and SLATER
The refraction corrections which have to be applied to the observed distances
before further processing can be split up into three parts (H PCKE 1966). The
distance J0 read on the instrument is based on a standard value n0 for the
refractive index calculated from standard temperature and air pressure (5.17). If
a more realistic value is available from local meteorological measurements,
the relation
Jn =J0n0
provides a first velocity correction
As seen from (5.20a), the radius of the light curve differs from the earth's radius
(r ~ 8 R). Hence, the light passes through atmospheric layers with a greater
density and refractive index than estimated by the mean value , calculated
from the endpoint data. With (5.10) and the coefficient of refraction k = 0.13,
which can be used to derive a second velocity correction
(5.108b)>
This correction is less than 1 mm over a distance of 15 km and can be neglected
For three-dimensional computations, the chord distance s is required. By
assuming a spherical arc with radius r we have
or after a series expansion
= 2r -- +
2r 6(2r)
Introducing (5.10) yields the curvature reduction
which is part of the reduction formula (5.6). This reduction is less than 0.1 mm
for a distance of 15 km and can be neglected. By adding (5.107) to (5.109), we
obtain the total reduction from the observed distance to the chord:
2k k2
The reduction from the chord distance to the length of the normal section and
the geodesies on the ellipsoid will be given in [6.3.2].
Fig. 5.61. Acoustic positioning on the sea bottom (T = transponder, S
transmitter)
Control points have been established locally on the ocean floor mainly for geodynamic
investigations (e.g., sea floor spreading at active ridge zones). Control points are usually arranged
in arrays of 3 to 4 stations, with station separation of 5 to 10 km. Acoustic -waves (velocity about
1500 m/s in water) are used exclusively for relative positioning, with acoustic transponders
powered by batteries or sources of nuclear energy. Acoustic signals (5 to 20 kHz) are emitted
from a ship-borne transducer and sent back by the transponders. The slant range is calculated
from the propagation time of the signal traveling forth and back. Spatial trilateration then
provides the relative positions (RINNER 1977), Fig. 5.61. With microsecond timing, the accuracy
of the acoustic wave method is determined by the sound velocity, which depends on temperature,
salinity, and water pressure, cm-accuracy can be achieved over a few km (CHADWELL et al. 1998).
The relation to the global reference system is established by GPS-positioning on board of the
surface vessel.
5.5.3 Leveling
In geometric leveling, differences in height are determined using horizontal
lines of sight between points in close proximity to each other. Leveling is
conducted with a leveling instrument (level) and two vertically posted leveling
rods (Fig. 5.62). The leveled height difference between the rods is given by
the difference between the backsight (b) and the foresight (f) reading:
Sn-b-f.
The leveling instrument consists primarily of a measuring telescope capable of
rotation about the vertical axis. The line of sight is brought into the horizontal
either by a coincidence bubble in conjunction with a tilting screw or, for most
modern instruments, automatically by a compensator that is comprised mainly
of a gravity pendulum (first introduced with the ZEISS Ni2 in 1950). A setting
accuracy of 0.1" to 0.4" is achieved by both methods. The use of a
compensator increases the speed of leveling, but spirit levels offer advantages if
high-frequency oscillations (traffic, wind) occur, due to better damping.
w=w,
Fig. 5.62. Geometric leveling principle
High-precision levels employ telescopes with an aperture of 40 to 50 mm and a
magnification of 30 to 40. Distances between the level and the rods are kept
within 30 to 40 m. Setting-up the instrument in the middle of two subsequent
rod positions is typical, as it eliminates errors due to non-parallelism of the
collimation and the bubble axes as well as symmetric refraction effects. In the
analogue mode, the leveling rods carry two graduation lines on invar tape,
displaced against each other and numbered differently in order to detect reading
errors. The line of sight is adjusted to the closest graduation mark by means of a
parallel plate mounted in front of the objective's lens. The amount of the
displacement is measured by a micrometer (Fig. 5.63).
Digital levels were introduced with the Wild NA 2000 (INGENSAND 1990).
They are used in connection with invar staffs that carry a binary code (Fig.
5.64). A code section around the horizontal sight is projected on a CCD sensor
in the image plane of the telescope. A subsequent processing of the image by a
microprocessor includes electronic scanning (A/D conversion) and correlation
with a digital reference signal, whereby the automatically measured distance has
to be taken into account (MEiER-HiRMER 1997, RUEGER and BRUNNER 2000).
F/g. 5.63. Bubble level (Wild/ Leica
N3), courtesy Leica Geosystems
Fig. 5.64. Digital level (Zeiss
DiNi 12), courtesy Zeiss/Spectra
In order to transfer heights over larger distances, the individual leveled
differences are summed. For one set-up, the non-parallelism of the level
surfaces may be neglected (quasi-differential method). The observed difference
then corresponds to the height difference of the level surfaces passing
through the rod sites. Summing the individual differences between two bench
marks Pl and P2 yields the "raw" leveled height difference
At longer distances, the effect of the non-parallelism may reach the cm-order of
magnitude and more, cf. [3.2.1]. Hence depends on the path taken and does
not provide a unique height in any height system. A unique height
determination can be achieved only by considering gravity g, that is by referring
to potential differences kW. According to (3.52) we have
Thus potential differences can be determined without any hypothesis from
leveling and surface gravity. In order to obtain height differences in any specific
height system from the raw leveling results, gravity reductions have to be
applied, cf. [6.4.1].
The accuracy of precise leveling depends on many influences. Some of the
leveling errors behave in a random manner and propagate with the square root
of the number of individual set-ups. Other errors are of systematic type and may
propagate with distance in a less favorable way. Hence particular attention must
be afforded to reduce them, by instrumental measures and modeling or by
employing dedicated measurement methods.
Important error sources include ( 1980):
Misleveling of the instrument. This random type error is of the order of a few 0.1" and
results in an error of a few 0.01 mm for an individual height difference. Systematic effects
arise from a residual adjustment error of the bubble or an imperfect operation of the
compensator and cause an "obliquity of the horizon". These effects cancel when measuring
in two opposite positions of the compensator.
Magnetic effects on the compensator. These are kept small generally, but a regular control is
advisable.
Rod graduation errors. This includes a "mean" scale error, errors of the individual graduation
marks, and effects of thermal expansion. Routine calibrations deliver the corresponding
corrections, with a remaining random part less than 5 to 10 (SCHLEMMER 1984).
Rod inclination errors. These errors can be kept sufficiently small by properly adjusting the
rod bubble and carefully holding the rod in the vertical position.
Vertical refraction. This depends mainly on the vertical temperature gradient, cf. [5.1.2]. The
irregular part (shimmer) acts as a random error (0.01 mm under cloudy skies). Systematic
influences occur particularly with measurements made in terrain with steep slopes or those
made close to the ground. The effects may reach 0.01 to 0.1 mm per l m height difference.
They can be modeled in part by a refraction correction, with the vertical temperature gradient
as a function of height (KUKKAM KI 1938). More refined models are available, which
include other meteorological and hydrological data, terrain slope, and orientation with
respect to the sun (ANGUS-LEPPAN 1984).
Vertical movements of the instrument and the rods. These errors depend on the stability of
the ground and on the manner in which the instrument is set up; movements of 0.01 to 0.1
mm per station are possible. Movements proportional to time are cancelled by an
observation succession , , / , , / , bu (I, II = scales of the rod). In addition, from the mean
of forward and backward leveling runs, those rod movements are eliminated which are
proportional to time and which occur while the instrument is brought to the next station.
Earth tides effects. The tidal effect on the plumb line causes periodic inclinations of the line
of sight. The inclinations can be modeled by the horizontal tidal component acting in the
azimuth of the leveling line. Starting from (3.119) and taking the elasticity of the earth into
account, the tidal reduction for the moon reads
<J,(m) =0.06sin2 cos(am - or)s mm/km .
Here, am and a are the azimuths of the moon and of the leveling line, and y/m the
geocentric central angle between the directions to the moon and the computation point, s is
the length of the leveling line in km. The same equation is valid for the sun, with an effect
of 46% of that for the moon (KUKKAMAKI 1949).
In order to eliminate or reduce systematic errors, precise leveling is always
carried out with equal back and foresights ("leveling from the middle") at less
than 50 m distance. Observations should be performed during cloudy weather,
preferably in the morning and in the evening hours. Line of sights very close to
the ground (0.5 m or less) should be avoided. Leveling is always conducted
twice, in opposing directions and possibly under different meteorological
conditions. For a 1 km double-run leveling, one can attain a standard deviation
of 0.2 to 1.0 mm. Residual systematic effects (especially those due to local
vertical movements and refraction) may cause small correlations between the
height differences obtained from the forward and backward runs and affect the
results of large network adjustments (LUCHT 1972).
The time needed for precise leveling can be significantly reduced by employing
a motorized procedure, whereby the instrument and the rods are carried in and
operated from an automobile. This mode also reduces time dependent errors and
partially eliminates asymmetric refraction effects, as the line of sight is more
remote from the ground (PESCHEL 1974, BECKER 1987).
For leveling across broad waterways and inlets of the sea, several methods have
been developed:
In reciprocal leveling, approximately horizontal sights to specially designed
targets are taken simultaneously with a precise level from both sides of the
waterway. For longer series of observations including a change of the
instruments, height differences over 1 to 2 km can be determined with a
precision of 1 to 2 mm (JELSTRUP 1955, KAKKURI 1966).
Hydrostatic leveling is based on the principle of communicating tubes. A
hose filled with water (free of air bubbles, uniform temperature) is laid
between the shores of the watercourse to be bridged. The water level
observed at the vertical ends of the hose belong to the same level surface
(GRABOWSKI 1987). The method may be applied for ranges up to 20 km
(Fehmarn-Belt/Baltic Sea) and delivers mm-precision (ANDERSEN 1992).
In the Netherlands, it is used in an operational mode (WAALEWUN 1964).
In hydrodynamic leveling (geostrophic leveling), the height is transferred
over the waterway utilizing water level records, which have to be reduced
for the effects of sea surface topography, cf. [3.4.2]. This implies the
development of a hydrodynamic model, which takes water velocity, wind
drag, water depth and bottom friction, atmospheric pressure, water density,
and gravity and Coriolis force into account. The method has been applied,
for instance, for a height transfer over the British channel (70 km) and over
the Fehmarn-Belt (CARTWRIGHT and CREASE 1963, WBBELMANN 1992)
with a precision of about 1 cm.
5.5.4 Tilt and Strain Measurements
Tilt and strain observed on the surface of the earth indicate the response of the
earth's crust to external and internal forces such as earth tides, tectonic
processes, and seismic and volcanic activities. Tilt and strain are dimensionless
quantities and are given in radian or arcsec and extension (positive sign) or
compression per distance, respectively.
Over time intervals of years to decades, long-term tilt and strain can be
determined from repeated observations of geodetic control networks using
satellite and terrestrial techniques, cf. [8.3.3]. Tiltmeters and strainmeters (also
called extensometers), on the other hand, have been developed in order to
monitor continuously local deformations (MELCHIOR 1983, AGNEW 1986,
ZRN 1997a).
Short-term (up to one day) tilt and strain is dominated by tidal deformations and
is at the order of 10~8 to 10~7, which correspond to inclinations of 0.002" to
0.02" and lengths changes of 0.01 to 0.1
/ . Long-term effects of tectonic
origin generally are only at the order of a few 10~7 /year. Episodic effects related
to seismic or volcanic events may reach the same order of magnitude and more
over a few hours to a few weeks and months. Consequently, the instrumental
sensitivity of tilt and strainmeters should be at least about 10"9 to 10~', and the
stability with time should be better than 10"7/year.
Tiltmeters measure the inclination of the earth's surface with respect to the local
vertical. Two mutually perpendicular sensors are needed in order to completely
determine the tilt, which are usually orientated in the NS and EW-directions.
Tiltmeters have been designed as horizontal and vertical pendulums, bubble
levels, and long water tubes (ZRN et al. 1986).
Horizontal pendulums consist of two nearly vertical threads that support an approximately
horizontal beam with an attached mass (Zllner suspension), Fig. 5.65. Because of the small
inclination of the rotational axis with respect to the vertical, a tilt of the support (basis about 30
cm) or a plumb line variation cause a strongly amplified angular deflection (astatization), which
may be further enlarged optically. Fused quartz (Verbaandert-Melchior pendulum) ore metallic
alloys are used as pendulum material in order to keep thermal effects small. Calibration is
performed by controlled tilting of the instrument (VAN RUYMBEKE 1976). Among the vertical
pendulums is the Askania borehole instrument, which may operate in depths of 20 to 60 m. The
pendulum's (length 60 cm) suspension allows it to swing freely, and the deflections are sensed by
two three-plate capacitive transducers installed at right angles to each other. The pendulum is
calibrated by displacing a small mass over a known distance (FLACH 1976). Short-base bubble
levels with electric sensors and -water-tube tillmeters with lengths of several 100 m have also been
used. For the water-tube tiltmeters, water level variations at the endpoints of the tube are
measured by capacitive or interferometric methods (KRIINEN 1979, SLATER 1983).
ROTATIONAL AXIS
INCLINATION
DEFLECTION-O ANGLE
Fig. 5.65. Zllner horizontal pendulum principle
Strainmeters measure relative displacements of the earth's crust (KING and
BILHAM 1973). For a complete determination of the strain tensor, which
contains six independent components, a strain meter array should be arranged
with orientation in different spatial directions, although in most cases only
horizontal Strainmeters have been installed.
Short-baseline Strainmeters (baseline length 10 to 30 m and more) are represented by invar -wires
and fused quartz tube rods. One end is fixed to a rock, and the crustal displacement is measured at
the other end by inductive or capacitive transducers (KING and BILHAM 1976). Laser Strainmeters
apply the Michelson interferometer principle. Operated in an evacuated tube, they can measure
distances up to 1 km with nm resolution (BERGER and LEVINE 1974, WYATT et al. 1990).
Among the instrumental errors of tiltmeters and Strainmeters are the
uncertainties of the calibration (about 0.1 to 1 %) and the direct effects of
temperature and air pressure variations, which are kept small by the selection of
the material and appropriate shielding. Long-term drift effects are at the order of
10"6 to 10~7/year and to a large part are due to problems inherent with the
sensor-rock coupling. Effects induced by meteorological and hydrological
variations (air temperature, air pressure, solar radiation, rainfall, groundwater)
pose severe problems in interpreting the results, especially for tiltmeters. These
disturbances have pronounced daily and seasonal periods but also happen at
other time scales. Modeling of these effects is still in its infancy, but their
influence can be reduced by installing the instruments below the earth's surface,
in tunnels, mines, natural caves, and boreholes.
Geologic, topographic, and cavity effects may cause large local distortions of
the tiltmeter and strainmeter data. This is due to variable rock properties
including local fractures, irregular topography, and different cavity reactions to
deformation. Local distortion may reach 10 to 15% and more, leading to non-
representative results (HARRISON 1976). Attempts to model these effects have
been only partly successful (SATO and HARRISON 1990, WYATT et al. 1990,
KOHL and LEVINE 1995). Consequently, in addition to carefully selecting the
observation site, preference now is given either to short-base tiltmeters and
strainmeters operating in deep boreholes or to long-baseline instruments. In the
latter case, local effects are reduced by integrating over a large distance of some
10 to some 100 m.
Tiltmeter and strainmeter results primarily contribute to earth tide research in
the short-periodic part and to the detection of anomalous tilt and strain related to
coseismic and volcanic activity, cf. [8.3.3].
6 Methods of Evaluation
Geodetic evaluation methods may be separated into positioning and gravity
field determination. This separation is possible because positioning only
requires an approximate knowledge of the gravity field, and gravity field
modeling needs only approximate positions. The linearization of the gravity
field is essential in both cases, and it permits a statistical description [6.1].
Nowadays, positioning is based on three-dimensional models [6.2], while
classical strategies distinguished between horizontal positioning [6.3] and
height determination [6.4]. Gravity field modeling utilizes some basic
observables and can be formulated as a boundary-value problem of potential
theory [6.5]. Global models are derived mainly from the results of space
geodesy [6.6], whereas local gravity field estimation is primarily based on
terrestrial data [6.7]. Special methods have been developed for combining
positioning and gravity field determination [6.8].
Geodetic evaluation methods are described in textbooks on geodesy, e.g.,
LEDERSTEGER (1956/1969), HEISKANEN and MORITZ (1967), GROTEN (1979),
MORITZ (1980), VANICEK and KRAKIWSKY (1986), HECK (1995).
The actual gravity field can be sufficiently well approximated by the normal
gravity field of the level ellipsoid, cf. [4.2.2], which results in linear relations
between the observations and the unknown parameters. The fundamental
quantity of the residual gravity field is the disturbing potential. It is closely
related to the height anomaly and the geoid height [6.1.1]. The residual gravity
vector generally is expressed by the gravity anomaly and the deflection of the
vertical [6.1.2]. The residual gravity field may be treated by statistical methods,
which is of interest in gravity field interpolation and modeling [6.1.3].
Approximation of the actual gravity potential W (3.42) by the normal gravity
potential U (4.37) leads to the disturbing potential (also anomalous potential) T
defined at the point P:
TP=WP-UP.
W and U contain a gravitational and a centrifugal part. As the centrifugal
acceleration of the earth is known with high accuracy, cf. [3.1.4], we may
6. l Residual Gravity Field
assume that the centrifugal parts of W and U are identical. The disturbing
potential then is formed by the difference of the gravitation of the earth and of
the level ellipsoid, and thus is a harmonic function outside the earth's masses.
Hence, it obeys Laplace 's differential equation (3.29)
= 0,
where = Laplace operator. can be expanded into spherical harmonics, in
analogy to the corresponding developments for the gravitational potentials
(3.89) and (4.45). Expressed in spherical coordinates r,#,/l, the disturbing
potential as a spatial function reads, in abreviated form, see (3.88):
In its full form the disturbing potential is given by
Comparing (6.3) and (6.4) yields the surface spherical harmonics
/""* A//
T, = - j(AC,mcosmA + M, m sintfa)/} m (costf).
Due to the properties of /, only the even zonal harmonic coefficients AC, differ
from C, , while all the other coefficients are identical with the coefficients of the
actual gravity field, cf. [4.2.2]. The development of (6.3) and (6.4) starts ail = 2
since equality of the masses of the earth and the ellipsoid is assumed, as is
coincidence of the center of the earth's masses with the center of the ellipsoid,
The disturbing potential is closely related to the vertical distance between and
the point Q located on the spheroidal surface U = UQ, cf. [6.5.1]. Q is
associated with by the condition
UQ=Wf,
cf. [4.2.3], Fig. 6.1. This distance is called height anomaly . Geometrically, it
is the difference between the ellipsoidal height h and the normal height H"
(3. 107), (4.66):
W=Wf
TELLUROID U0 = Wf
QUASIGEOID
GEOID IV =W 0
~~~ ELLIPSOID U = U
Fig. 6.1. Ellipsoidal, normal and orthometric height
= h-H\
where we have neglected the slight curvature of the normal plumb line. The
surface for which (6.6) holds at every point is called the telluroid (HIRVONEN
1960). A corresponding relation holds at any point in the exterior space.
The telluroid represents an approximation to the physical surface of the earth.
By extending HN downward from P we obtain the quasigeoid, and ^becomes
the distance between the level ellipsoid U = U0 and the quasigeoid, which is
often used as a zero height surface, cf. [3.4.3]. is also called quasigeoid
height.
If is located on the geoid, we obtain the geoid height N (also called geoid
undulation) as the vertical distance between the ellipsoid and the geoid. A
geometric definition follows by differencing the ellipsoidal height h and the
orthometric height #(3.106):
N = h-H,
where the effect of the plumb line curvature has been neglected.
The difference between the geoid height and the height anomaly is equal to the
difference between the normal height and the orthometric height and follows
from (3.106) and (3.107):
The difference depends on the height and thus is zero on the oceans. It also
depends on the "mean" gravity anomaly g-y, which corresponds to the
Bouger anomaly Ag B , cf. [6.5.3 ].
6.1.2 Gravity Disturbance, Gravity Anomaly, Deflection of the Vertical
The gravity vector g at P can be approximated by the vector of normal gravity ,
which leads to the gravity disturbance
ogp=gp-jp,
Fig. 6.2. Neglecting the small angle between the directions of g and
(deflection of the vertical), we obtain the magnitude
Fig. 6.2. Actual and normal gravity
g can be measured on the earth's surface and in the exterior space. The
calculation of , on the other hand, presupposes the knowledge of the geodetic
coordinates of P, which is not assumed in the classical geodetic boundary value
problem, cf. [6.5.1]. Consequently, gravity field modeling generally employs
the gravity anomaly
Ag^gp-Yp,
with the magnitude
^gP=gf-YQ.
Again, is related to by the condition (6.6). y can be calculated by (4.63),
starting from normal gravity y0 on the ellipsoid (4.41) and replacing h by HN
(3.107) as determined by leveling, cf. [3.4.3]. This free-air reduction is given by
(4.61):
(6.14))
The free-air gravity anomaly, defined on and outside the earth's surface
according to Molodenski, reads
Ag?=g+<5g;-7o.
In many applications, /dH" is approximated by - 3.086 8 2 /m .
The determination of the geoid requires that the gravity anomalies are given
everywhere on that level surface. In order to apply the Laplace equation, the
masses outside the geoid have to be removed. Several types of gravity
reductions are available for this purpose. The methods differ by the manner in
which the topographical masses are displaced, cf. [6.5.3]. The gravity anomaly
on the geoid is defined as the difference between the gravity on the geoid g0 ,
and the normal gravity y0 on the ellipsoid (Fig. 6.2):
The difference (vector quantity) between the directions of the actual plumb line
and a reference direction in the normal gravity field is called deflection of the
vertical (Fig. 6.3). With respect to the reference direction, we distinguish
between (JEKELI 1999):
The deflection of the vertical QN defined on the surface or the exterior of
the earth, with the direction of the normal plumb line at Q as a reference
(Molodenski definition). The reference direction practically coincides with
the surface normal to U = Up at P,
The deflection of the vertical at the earth's surface, referring to the
ellipsoidal normal at P (Helmen definition). It differs from QN only by the
slight curvature (effect of a few 0.1") of the normal plumb line, cf. [4.2.3].
This definition is preferred generally, as the ellipsoidal normal is provided
by the geodetic coordinates,
The deflection of the vertical 0 defined on the geoid (Pizetti definition). It
is given by the difference between the actual plumb line on the geoid and
the ellipsoidal normal. It differs from the previous definitions by the
curvature of the actual plumb line and is of importance for the
determination of the geoid.
w=wa
Qo
Fig. 6.3. Deflection of the vertical according to a) Molodenski, b) Helmert, c)
Pizetti
The deflection of the vertical is expressed either by its magnitude and its
azimuth ae or, more generally, by its components in the north-south and eastwest directions. A geometric derivation follows from spherical trigonometry on
the unit sphere around the definition point P (Fig. 6.4). Here, we assume that
the minor axis of the reference ellipsoid is parallel to the Z-axis of the global
reference system and that the ellipsoidal initial meridian is parallel to the A'-axis.
These conditions are practically fulfilled with modern reference systems and
well approximated by classical geodetic systems, cf. [6.2.2]. After parallel
displacement, we identify as the point of intersection of the Z-axis with the
unit sphere and Zfl and Zg as the directions to the astronomic and the geodetic
zenith respectively. The deflection of the vertical represents the spherical
distance between Za and Z g ; its azimuth is denoted by . The deflection is
decomposed into the meridional component (positive when Za is north of
Z g ) and the component in the prime vertical TJ (positive when Za is east of
Zg). Along the azimuth to a target point Pt, we have the vertical deflection
component .
From spherical trigonometry we get
and with
Fig. 6.4. Vertical deflection components
COST;!, sin??;;, si
the components are given by (linear approximation)
According to Fig. 6.4, the component in the azimuth a is composed by two
= & + &.
These relations can also be derived by differencing (3.45) and (4.36), after
linearization, cf. [6.2.2].
Equations (6.17) and (6.18) are valid for all definitions of the deflection of the
The residual gravity field quantities depend on the geodetic earth model used for approximating
the gravity field and on its orientation with respect to the earth, cf. [6.2.2]. If referred to a
geocentric mean earth ellipsoid, they are designated as absolute quantities, otherwise they are
relative only. The r.m.s. scatter of absolute height anomalies and geoid heights is 30 m
(maximum values about 100 m). The free-air gravity anomalies vary by about 400 5~2
(maximum values of a few 1000 8~2 ) and the deflections of the vertical by 7" (maximum
30" to in the high mountains).
6.1.3 Statistical Description of the Gravity Field, Interpolation
The residual gravity field can be viewed as a realization of a stochastic process
and treated by statistical methods (MORITZ 1962, 1970). The gravity anomaly is
used as a fundamental gravity-field parameter in this aspect, as gravity data are
available with high resolution on the continents and on the oceans.
We assume that the mean value of the gravity anomalies Ag over the earth is
zero, cf. [6.5.4]:
where M{ } is the mean value operator and represents the unit sphere. The
surface element can be expressed in spherical coordinates &,A by
The statistical behavior of Ag is described by the covariance function
It is calculated as the mean value of all products of gravity anomalies at the
points and P', having constant spherical distance i/son the unit sphere. C(y)
shall only depend on and not depend on the position (homogeneity of the
anomalous gravity field) and the azimuth (isotropy), GRAFAREND (1976).
C(^) describes the mutual correlation of the gravity anomalies, which
decreases with increasing distance. For = 0, we have Ag = Ag', and the
covariance transforms into the variance
2 (Ag) = M{Ag2}=-M/Ag2 da.
From the theory of stochastic processes, the statistical properties should be derived from an
infinite number of process realizations. As only one realization of the gravity field is available,
the hypothesis of ergodicity is necessary, which states that the statistical quantities may also be
calculated from mean values over one realization (MORITZ 1980, p.269).
As shown in [6.6.1], Ag as a functional of can be expanded into spherical
harmonics. On the earth's surface (r = R) the abbreviated form of this expansion
with Ag, = Laplace's surface harmonics, cf. [3.3.2]. Again the terms of degree
0 and 1 are missing. As a consequence of (6.23), C(y) also can be expanded
into spherical harmonics in the definition range 0<<:
with P, (cos i/) = Legendre polynomials. Because of isotropy, only zonal terms
exist in (6.24). The harmonic coefficients c, are derived by inversion
according to potential theory. Equation (6.25) can be solved for a known
covariance function by numerical integration. By inserting (6.21) into (6.25),
and taking (6.23) into account, we obtain
Hence, the coefficients are given by the anomaly degree variances defined as
mean values over the squares of Ag, .
Based on a global set of point free-air anomalies, KAULA (1959) first gave an
estimate for the covariance function. TSCHERNING and RAPP (1974) derived a
covariance function and a degree variance model that are often applied. The
model is based on satellite-derived harmonic coefficients for the degrees 2 to 10
and a set of 1 equal area anomalies (approximately quadratic compartments
with constant area llOkmxllOkm). The variance of the point anomalies is
and that of the mean anomalies
The correlation of the anomalies approaches zero at a spherical distance of
about 30 to 40 (Fig. 6.5). The anomaly degree variance model reads
COV (Sg, Ag)1'E.A. (10
180 ()
Fig. 6.5. Global covariance function of gravity anomalies, model TSCHERNING
and RAPP (1974)
for/=0,1
for / = 2
(//ms'2 )2 ,
1:2 f o r / > 3
with A = 42 528 and = 24. 0 = (RB/R)2 = 0.999 617 is the ratio between the
radii of the Bjerhammar sphere, cf. [3.3.2], and the earth, Fig. 6.6.
For regional applications, a covariance function may be derived by subtracting
the long-wave part of the gravity field, which results in a decrease in the
variance and in the correlation length. Local interpolation is even possible by a
plane covariance function, e.g., the Gauss function
Fig. 6.7, TORGE et al. (1984).
It should be noted that covariance functions have to be positive definite. This
condition is fulfilled for (6.24), as all coefficients according to (6.26) are nonnegative.
ANOMALY DEGREE
VARIANCES (10/52)2
, DEGREE I
Fig. 6.6. Anomaly degree variances, model TSCHERNING and RAPP (1974) and
geopotential model EGM96 (LEMOINE et al. 1998)
COV (10 '2)2
cov = 1100 'e*
3(')
Fig. 6.7. Local covariance functions of trend-removed 6'xlO' mean free air
anomalies (TORGE et al. 1984)
An important application of the anomaly covariance function is the
interpolation of gravity anomalies at points not surveyed. Simple interpolation
methods such as the manual construction of isoanomaly maps or the geometric
interpolation using adjacent data are not ideal and do not deliver optimum
results. Least squares prediction utilizes the statistical information inherent in
the covariance function and takes the errors of the observations into account.
In the usual linear prediction, the gravity anomaly at the point P is estimated by
a linear function of the anomalies observed at the points Pi (/ = !,...,). We
assume that, in addition to the covariance function of the anomalies, an error
covariance function is also available. Such a function can be derived from a
priori error and error correlation estimates, although generally only error
variances are known. The following covariances, for any distance , can be
calculated:
CPi = M{Agp Ag. } : crosscovariance of Agp with the observation Ag, ,
j : autocovariance of the observations,
Di} = M\n. rtj }: autocovariance of the observational errors (noise n),
and combined into
c=
C,,
\. (6.29)
Now, the prediction error is introduced, being the difference between the true
gravity anomaly and the predicted value Ag. The requirement of a minimum
prediction error variance, in analogy to least squares adjustment, leads to the
result of least squares prediction
where the observed anomalies have been collected in the vector
... Ag,, ... Ag).
Under the assumption that the gravity anomalies and their errors are not
correlated, the corresponding matrices C and D can be added element by
element, leading to
C = C + D.
The prediction of point free-air anomalies is successful only for very densely surveyed areas, as
these anomalies strongly depend on height. A smoother gravity anomaly field is obtained by
averaging over larger surface elements (e.g., 5'x5', 30'x30') or reduction of the effect of the
topographic masses, where the Bouguer anomalies are especially well suited for interpolation, cf.
[6.5.3]. As well known from least-squares adjustment theory, the predicted values of the gravity
anomalies are relatively independent from the choice of the covariance function, while the error
estimates strongly depend on it. Realistic prediction results can be expected only within the
correlation length defined by a covariance of 1/22
Three-dimensional positioning is generally carried out in a Cartesian coordinate
system; the use of ellipsoidal coordinates results in more complicated models
[6.2.1]. The Geodetic Datum provides the orientation of a three-dimensional
model with respect to the global geocentric system [6.2.2]. Three-dimensional
modeling has been investigated by WOLF (1963a, b), GRAFAREND (1978), and
TORGE (1980a) among others.
The observation equations relate the observed quantities to the unknown station
coordinates and other parameters (functional model). By "observations," we
mean geometric quantities that result from preprocessing of the original
measurements, such as signal travel time, phase and frequency, and readings on
graduated circles or staffs. We also assume that instrumental corrections have
been applied (e.g., calibration) and that influences from the atmosphere
(refraction) and the gravity field (earth tides) have been taken into account. The
observations equations are mostly non-linear. They have to be linearized for the
subsequent least squares adjustment, which also includes the errors of the
observations (stochastic model). The theory of errors and adjustment methods
are discussed in textbooks by GRAFAREND and SCHAFFRIN (1993), WOLF
(1997), and KOCH (1999).
The observations used for three-dimensional positioning may be divided into
satellite and terrestrial measurements.
Satellite observations include
Spatial directions, cf. [5.2.4],
Ranges derived from GPS and laser distance measurements, cf. [5.2.5],
[5.2.6],
Range rates from Doppler measurements, cf. [5.2.4].
We add
Baseline vectors obtained from VLBI, cf. [5.3.4].
Satellite and VLBI networks generally are adjusted separately. The resulting
Cartesian coordinates and
Cartesian coordinate differences
can then be introduced as "observed" parameters into subsequent combined
adjustments, together with other satellite networks or with terrestrial data.
Fig. 6.8. Satellite tracking principle
Satellite observations provide the components of the topocentric observation
vector s directed from the station P to the satellite S (Fig. 6.8). The vector s is
related to the geocentric station vector r,, and the radius vector rs of the
satellite by
r p +s-r s = 0 ,
The observation vector is formed by the distance and the spatial direction to the
satellite given in the hour angle system:
cos hGr cos
sin hGr cos
hGr = GAST-a
where J is the declination and
is the Greenwich hour angle, cf. [2.4.1].
Inserting (6.34) into (6.33) and solving for the components of s yields the
=arctan= arctan-
Range rates to the satellite positions (/',/) are given by
- s=
; = X. - Xp etc., -; = Xt - Xp , etc.
If the directions to the quasars are known, VLBI observations deliver the
baseline vector between two terrestrial stations P: and P2 (5.65):
The relations of the original measurements to the geometrical quantities
introduced in (6.36) to (6.38) are given by (5.42) for Doppler counts, (5.43) and
(5.45) for GPS, (5.48) for satellite laser ranging, and (5.65) for VLBI.
The observation equations contain a large number of parameters in addition to the station
coordinates. Among them are the earth orientation parameters, parameters describing earth tide
effects and crustal deformation, as well as the satellite coordinates and the coefficients of the
gravitational field and other "disturbing" forces. There are two strategies to handle the large
amount of data and unknown parameters.
The first one consists in estimating most of the unknowns in one adjustment and to adopt only a
few parameters (e.g., the earth orientation parameters) from other sources. This leads to "satelliteonly" earth models, which provide a global set of station coordinates and the harmonic
coefficients of the gravitational field, cf. [6.6.2].
In positioning, only a limited number of parameters, mainly the station coordinates, are
introduced as unknowns, while numerical values for the other parameters are taken from models.
Highly accurate orbital parameters are available for GPS and laser satellites and are disseminated
by operational services. Parameter corrections taken from models are small and can easily be
introduced into the adjustment. Examples include the orbital improvement for short arcs (less than
one revolution) through corrections to the six Keplerian elements, the introduction of a local
tropospheric "scale factor," and the determination of clock biases.
Several error influences are eliminated or greatly reduced when simultaneous
measurements are carried out from two or more ground stations to the same
satellites (relative positioning). This is due to the high error correlation between
neighboring stations. By differencing the measurements, a large part of the
errors cancel. Since the absolute orientation is lost with this strategy, the
absolute coordinates of at least one station have to be provided.
Exact simultaneous measurements lead to a purely geometric method of evaluation. The satellite
is regarded as a high-altitude target, and short-arc orbital fitting can improve the results (WOLF
1967, SCHWARZ 1969). Geometric satellite networks were established by stellar triangulation and
by trilateration from the 1960's to the 1970's. The global network of the U.S. National Geodetic
Survey (1966-1970, 45 stations) was determined by direction measurements with Wild BC4
cameras to the Pageos satellite, cf. [5.2.4]. The scale of this network was derived from long
traverses measured with electronic distance meters; an accuracy of 4 to 5 m was achieved
(SCHMID 1974).
Different methods have been developed for the evaluation of simultaneously
observed GPS networks by forming linear combinations of the observations
(WELLS et al. 1987, TEUNISSEN and KLEUSBERG 1998b). As a standard, "single
differences " s* - s* between the distances measured simultaneously from two
receivers /J, P2 to the satellite A at epoch tl are formed (Fig. 6.9). Thereby, the
satellite clock errors are eliminated and refraction and orbital errors are reduced.
"Double differences " are formed by differencing two single differences taken
to different satellites A, B at the same epoch i, : (sf -s^)-(sf -s" V With
double difference, the receiver clock errors vanish and refraction ana orbital
errors are further reduced. "Triple differences " are constructed by differencing
two double differences taken at the epochs tt and tj . The ambiguity which
enters into carrier phase measurements cancels in the observation equation. Due
to the loss of information, triple differences allow reduced-accuracy positioning
only but are useful for the removal of cycle slips, cf. [5.2.5].
Aft)
Fig. 6.9. GPS single, double and triple differences
Sophisticated software packages are available for the adjustment of satellite
networks; for GPS networks we mention W BBENA (1989), BEUTLER et al.
(1996a), WEBB and ZUMBERGE (1997). The adjustment delivers the Cartesian
coordinates or coordinate differences of the ground stations and their full
variance-covariance matrix, which has to be taken into account for further
processing, e.g., for the combination with terrestrial data. If the orientation of
the networks to be combined is not identical, a corresponding transformation
(datum shift) has to be included in the adjustment, cf. [6.2.3].
Terrestrial measurements include
Astronomic azimuths, latitudes, and longitudes, cf. [5.3.2],
Horizontal directions (azimuths without orientation) and horizontal angles
(differences of azimuths), cf. [5.5.1],
Zenith angles, cf. [5.5.1],
Distances, cf. [5.5.2],
Leveled height differences, cf. [5.5.3].
By substituting (2.20) into (2.29) and taking (2.30) into account, we obtain the
observation equations for azimuths A, zenith angles z, and distances s:
-sinOcosA ' -sinOsinA 7+ cosO
cos cos AY" + cos sin 7 + sin
= arccos-
= 2 -,.
Astronomic latitude and astronomic longitude A enter as orientation
parameters in (6.39). They relate the local astronomic to the global geocentric
system and are treated as additional unknown parameters. If observed latitudes
and longitudes are available, they are introduced as observed parameters in the
adjustment.
Least squares adjustment requires linear relations between the observations and
the unknowns. Corresponding differential relations are derived by numerical or
analytical differentiation of (6.39). Analytical expressions for the partial
derivatives dA/dX , etc. are found in WOLF (1963b) and HEISKANEN and
MORITZ (1967, p.217).
Fig. 6.10. Geometric-astronomic leveling
Geometric leveling can be incorporated into three-dimensional computations
after transformation of the leveled height difference -dn (5.111) into the
geocentric coordinate system. An ellipsoidal height difference dh is obtained by
reducing dn for the effect of the (surface) deflection of the vertical (Fig. 6.10).
In the azimuth of the leveling line, the component (6.18) is effective, which
dh=dn-eds.
The negative sign prefixing ds is based on the sign definitions inherent in
(6.17) and (6.7) resp. (6.8). A differential relationship between the ellipsoidal
height and^r,7,Zis provided by (4.26a) and (4.27). With
and replacing , with ,, we obtain
dh = dr = cos Ocos dX + cos sin dY + sin dZ .
Again, we have assumed that the axes of the ellipsoidal and the geocentric
system are parallel:
= dY, dZ = dZ.
Integration of (6.40) yields the ellipsoidal height difference
which can be included as an "observation" in spatial computations: Geometricastronomic leveling (HEITZ 1973). It is noted that both integrals in (6.42) have
to be formed over the same path. The differential relation for (6.42) follows
from differencing (6.41) for the points Pl and P2:
The first integral in (6.42) can be easily computed by summing the leveled height differences.
The evaluation of the second integral poses difficulties, as vertical deflections are generally only
available at larger station distances (several 10 km in classical networks). This introduces the
problem of the interpolation of deflections of the vertical, cf. [6.7.4]. In flat and hilly areas, with
an average distance of the vertical deflection stations of 25 km, an accuracy of 0.1
m/100km can be achieved for the ellipsoidal height differences (B UMKER 1984), while "cm"
accuracy requires station distances of 5 to 10 km and sophisticated interpolation methods
(TORGE 1977).
The three-dimensional concept was already introduced by VILLARCEAU (1868) and BRUNS (1878).
Bruns suggested a pointwise determination of the earth's surface by a spatial polyhedron
constructed from terrestrial measurements and orientated by astronomical observations. The
feasibility of this concept was demonstrated in some test networks (TORGE and WENZEL 1978,
HRADILEK 1984), but large-scale application was prevented due to the uncertainties of
trigonometrical height transfer over larger distances and the problems with reducing geometric
leveling.
Ellipsoidal coordinates ,, and sometimes h, cf. [4.1.3], are used for many
applications in geodesy, cartography, and navigation. They can be easily
derived from the Cartesian coordinates by the transformation (4.28). Network
adjustments in the ^AA-system are more complicated and restricted to special
cases. Nevertheless, differential relations between the observations and the
ellipsoidal coordinates are useful for certain problems, e.g., deriving reductions
onto the ellipsoid and for two-dimensional ellipsoidal calculations, cf. [6.3.2],
Equations (4.27) provide the fundamental relations between the ,, and the
X,Y,Z-systems. Differentiation yields
h)d<p
dY = A
where A is given by (4.32). Again we assume parallelism of the ellipsoidal and
the Cartesian coordinate systems. Equation (6.44) can be immediately used if
satellite derived coordinates or coordinate differences are to be adjusted in the
, , -system. Differential formulas for the terrestrial observations A, z, s are
obtained by inserting (6.44) into the differential relations for Cartesian
coordinates and reordering (WOLF 1963b, HEISKANEN and MORITZ 1967), cf.
We finally mention the straightforward transformation from the "natural"
coordinates ,,// (orthometric height) or HN (normal height) to ellipsoidal
coordinates <p,A,h, where //and HN are derived from the gravity potential W by
(3.106) resp. (3.107). According to (6.17), the deflection of the vertical (,)
transforms from the plumb line direction to the ellipsoidal normal:
= -,
If normal geodetic coordinates , (4.73) are required, the curvature of the
normal plumb line has to be taken into account by (4.74):
) = -, = = \ 5_.
cos 0J
The relation between h and //resp. HN is given by, see Fig. 6.2:
h = H + N = HN+,
where we have neglected the small (sub mm-order of magnitude) effect of the
plumb line curvature. Gravimetric evaluation techniques allow calculation of
the deflection of the vertical and the geoid height or the height anomaly from
gravity data, cf. [6.7]. Equations (6.45) and (6.46) thus would allow us to
establish a geocentric system of ellipsoidal coordinates (HEISKANEN 1951).
Since and can be determined only with an accuracy of 0.1" (corresponding to 3 m in
horizontal position) or less, this method is no longer of interest for horizontal positioning. The
height transformation (6.46), on the other hand, is most useful, as it permits connection of GPSderived ellipsoidal heights with heights determined by geometric leveling, cf. [6.4.3].
The geodetic datum describes the orientation of any geodetic system with
respect to the global geocentric system, cf. [2.5.1]. Nowadays, networks
established by satellite methods are tied to the ITRF, cf. [2.5.3], and hence
geocentric. Larger deviations occur with classical geodetic networks, which
were orientated by astronomical observations, cf. [6.3.3].
In the most general case, a non-geocentric ,, -system is transformed to the
.Y,7,Z-system by a similarity transformation in space, i.e., by three translations,
three rotations, and a change in scale (Fig. 6.11):
Here, rT = (,,) and 7T =(X,Y,Z} are the position vectors injhe two
systems, and r0T =(^f 0 ,7 0 ,Z 0 ) contains the coordinates of the origin of the
X, F^Z^s^stem with respect to the geocenter O. We assume that the scale of
the , , -system differs only slightly from the scale of the global reference
frame and that the axes of the two systems are approximately parallel.
Consequently, m is a small scale correction, and the rotation matrix is composed
of three small Eulerian angles; it takes the form
Fig. 6.11. Transformation between 3D-Cartesian coordinate systems
In order to determine the seven parameters of the transformation (6.47), at least
three points with coordinates given in both systems are required.
After converting the Cartesian coordinates into ellipsoidal ones, (6.47) can be
expressed in ellipsoidal coordinates ,,. The datum parameters in that case
also have to include the geometric parameters of the ellipsoid, i.e., the
semimajor axis a and the flattening/ Of practical interest are the changes of the
ellipsoidal coordinates that result from translation, rotation, change in scale,
and change of the parameters of the ellipsoid. We insert (4.27) into (6.47) and
take the total differential. As the real position of does not change, we have
dr = 0. Neglecting the linear scale factor and substituting the differentials by
(small) differences, a spherical approximation (M + h = N+h = a, f= 0) yields
(MERRY and VANlCEK 1974, Torge 1980b):
os /
f f_
*',
where A"1 is given by (4.34) and
C = a - sin cos
-cos/l
- sin sin
. (6.49b)
Equation (6.49) can be used for estimating the changes of the coordinates if the
changes of the parameters of the geodetic datum are known. All differences are
formed in the sense geocentric - non-geocentric, e.g., = a(geocentric) a(non-geocentric). Formulas which take the flattening into account are given by
EHLERT (1991), and ABD-ELMOTAAL and EL- (1995).
Classical geodetic networks have been orientated by the ellipsoidal coordinates
of an initial (fundamental) point PF and by condition equations for the
parallelism of the axes with respect to the geocentric system. If we apply (6.49)
at a running point and at the fundamental point P F , the translation can be
expressed in changes F,F,hF of the fundamental point (TORGE 1980b).
By differentiating (6.45) and (6.46), and considering that - dA = dH = 0,
= -, = -cos<p, dN = dh .
Corresponding equations hold for the "normal" geodetic coordinates, cf. [4.2.3].
Hence, the coordinate changes at any point also can be expressed as changes of
the deflection of the vertical and the geoid height (or height anomaly), in
dependence on the corresponding changes in the fundamental point (VENINGMEINESZ 1950). Spherical approximation yields (HEISKANEN and MORITZ
1967, p.208)
- (cos <pF cos + sin sin #?cos ( - ))^F
- sin sin ( - ,, }
- (sin F cos - cos sin cos ( - ))
= sin sin ( -
+ cos ( - }
( dNF da . 2
- + + sm <pF df
{ a
dN = -a
sin - sin cos <pcos (A
in ( -
.(6.51 continued)
(dNF + da + a sin2 df} - da
+ (sin2 - 2 sin <pF sin <p}a df
These relations have played a role in the optimum fitting of horizontal control
networks to the geoid, cf. [7.1.2].
Fig. 6.12. Rotations between the local ellipsoidal and the local astronomic
We now investigate how the (approximate) parallelism of the axes of classical
geodetic networks with respect to the geocentric system has been achieved.
We describe the deviation between the local astronomic x,^-system (2.20) and
the local ellipsoidal ~x,y,J-system (4.29) by three Eulerian angles, after
reflection of the y and y -axes (producing right-handed systems), Fig. 6.12:
with the rotation matrix
The Eulerian angles are the components of the deflection of the vertical , cf.
[6.1.2], in the meridian (), in the prime vertical (), and in the horizontal plane
(). If the axes of the global ,, and , , -systems are not parallel, the
following relations hold, according to (2.26), (4.20), (6.47), and (6.52):
= = AR(^,^,^)x = R^.fTp,^ )Ax
After inserting A (4.32) and A (2.28), the evaluation of (6.54) results in nine
equations. Three of the nine equations are independent from each other
(orthogonality relations). After Taylor expansion of the trigonometrical
functions (,) inherent in (2.28) at the point (,), we obtain the components
(linear approximation) of the deflection of the vertical if the axes of the global
systems are not parallel:
= - + sin % - cos A ?
= ( - /l)sin + cos (cos + sin ) + sin ^
where the equations for and (f depend on each other.
We also generalize the equations for the azimuth and the zenith angle given in
the local astronomic and the local ellipsoidal system. From (6.53) we have
Ax = R (?, ?, =} .
Inserting (2.20) and (4.29) yields, after linearization of the trigonometrical
functions , at , :
-a = (\ - A)sin + (( - ^?)sin a - cos ^( - A)cosor)cot
, (6.56a)
=- (cos a sin -sin a. sin cos )^
+ (cosacos
+ sir\ocsm<psm }E7
-cos sin ^
With the global rotations
we obtain condition equations for the parallelism of the axes, which was
presupposed when introducing the deflections of the vertical, cf. [6.1.2]:
After inserting (6.57) into (6.56), the condition equations for the azimuth and
the zenith angle read
Equation (6.58) is known as Laplace 's equation of orientation, while (6.59)
furnishes the component of the deflection of the vertical in the azimuth a
(6.18), VANICEK and WELLS (1974).
A geometric interpretation of the condition equations (6.58) and (6.59) reveals that they prevent
rotations about the vertical and the horizontal axis of a theodolite. In addition, a rotation about the
line of sight must be forestalled to guarantee the parallelism of the global and the ellipsoidal
system. This can be accomplished if, in addition to (6.58) and (6.59), another zenith angle
equation is introduced at the fundamental point, possibly with an azimuth which differs by 90.
Due to the problems of vertical refraction, classical networks generally employed only the
Laplace equation for orientation. A three-dimensional network then would need at least three
Laplace azimuths at points well distributed over the network.
Two-dimensional positioning utilizes (6.58) for the reduction of observed azimuths and horizontal
directions onto the ellipsoid, cf. [6.3.2], and (6.59) enters into trigonometric height determination,
In classical geodetic networks, horizontal positioning has been separated from
height determination. This is due to the fact that ellipsoidal height differences
derived from geometric leveling or zenith angles are affected by greater
uncertainties, which enter through the reductions of the gravity field and
vertical refraction, but especially because users require heights defined in the
gravity field.
With the ellipsoid as a reference surface, two-dimensional positioning requires
ellipsoidal trigonometry [6.3.1] and reductions of the observed positional
quantities onto the ellipsoid [6.3.2]. Ellipsoidal "observation" equations allow
the calculation of coordinates, after proper orientation [6.3.3]. Ellipsoidal
geodesy is found in textbooks such as GROSSMANN (1976), BOMFORD (1980),
and HECK (1995).
In order to carry out computations on the ellipsoidal surface, points on the
ellipsoid are connected by surface curves. The arc of the normal section and the
geodesic are employed primarily.
The normal section is defined by the curve of intersection of the vertical plane
with the ellipsoid. Azimuths and distances, after being reduced to the ellipsoid,
refer to normal sections. Since the surface normals of two points on the
ellipsoid are in general skewed to each other, the reciprocal normal sections
from P{ to P2 and from P2 to /f do not coincide (Fig. 6.13). The differences in
azimuth (less than 0.1" for S = 100 km) and in distance (less than 1 for S =
100 km) are small and can be taken into account easily (BOWRING 1971).
Fig. 6.13. Normal sections and geodesic
Normally, because of its favorable properties, the geodesic is used for
ellipsoidal calculations. It is uniquely defined as the shortest connection
between two points and is generally bounded by the two normal sections. This
definition is equal to the condition that the geodetic curvature (curvature of the
normal projection of a surface curve onto the tangential plane) equals zero.
As known from differential geometry, the geodetic curvature kg is represented
by the triple scalar product
* g =(r'xr>.
Here, r' = dr/dS is the tangent vector and r* = d2r/dS2 is the curvature vector.
Also, S = arc length of the geodesic, and n designates the normal vector to the
surface. With kg = 0, we obtain a second-order vectorial differential equation
for the geodesic:
(r'xr')-n=0;
the local projection of the geodesic onto the ellipsoid is a straight line. We
introduce the ^,/l-system of ellipsoidal coordinates (Fig. 6.14) and express the
geodesic by = (<}. Corresponding evaluation of (6.61) yields the scalar
d<p
_ dp _
2p = 0
d<p d<p
d(p\d(p
with/? = cos = radius of the circle of latitude (4.6) and the principal radii of
curvature and (4.13), (4.15). From Fig. 6.14, we take the relations
cos (
+ dX = const.
Fig. 6.14. Ellipsoidal (geodetic) surface coordinates
_ cosa __ sina
~dS~ ' ~dS~
which are valid for any surface curve. We form / and the second
derivative and insert them into (6.62). Integration yields Clairaut 's equation
N cos sin a = const .
(6. 64)
The constant corresponds to the radius of the parallel circle at which the
geodesic has a 90 azimuth. By differentiation with respect to S, and taking
(6.63) into account, (6.64) is transformed into
sinortan^?
Equations (6.63) and (6.64) resp. (6.65) form a system of first-order differential
equations for the geodesic. The solution of this system leads to elliptic integrals
which cannot be solved elementarily, cf. [6.3.3].
Two-dimensional positioning on the ellipsoid implies the solution of ellipsoidal
triangles, bounded by geodesies. As the curvature of surface curves changes
according to (4.18), the solution of a triangle not only requires three geometric
elements (angles, distances) but also the orientation parameters latitude and
azimuth.
At point distances less than 100 km (classical terrestrial networks) the ellipsoid
may be approximated by the Gaussian osculating sphere (4.23), with a latitude
calculated as an arithmetic mean of the latitudes of the triangle vertices. Closed
spherical formulas then can be used for the solution of triangles (HECK 1995,
p.366), e.g., the spherical law of sines
sin (b/R)
with the spherical angles ex, and sides a, b, and R = radius of the Gaussian
sphere. Errors due to the spherical approximation remain less than 0.002" for
the angles and less than 1 mm for the distances. We finally mention the
spherical excess which plays a role in spherical trigonometry. It is defined as
the surplus over 180 of the angle sum of a spherical triangle. The excess is
proportional to the area F of the triangle:
=- 2
For an equilateral triangle with S = 50 km, the excess amounts to 5.48".
In order to carry out a transfer of ellipsoidal coordinates on the ellipsoid,
observed azimuths and distances have to be reduced to the ellipsoidal quantities
a and S, which refer to the geodesic.
The reduction of the astronomic azimuth A is composed of three parts.
Laplace's equation (6.58) takes the effect of the deflection of the vertical into
account. The corresponding reduction to the normal section reads
The first term in (6.68) is the azimuthal component of the deflection of the vertical. It does not
depend on the azimuth and corresponds to a twist in the observed directions; horizontal angles are
not affected. The second term can be viewed as an "error" in setting up the theodolite, by
orientating it along the plumb line direction instead of the ellipsoidal normal. While the first term
reaches the order of magnitude of the vertical deflections and more, a strong reduction takes place
in the second term due to cot . Hence, this direction dependent term is of the order of a few 0.1"
in flat areas only but may reach some arcsec in the mountains.
If the target point is not situated on the ellipsoid but at a height h2, a further
reduction has to be applied. Namely, the vertical plane formed by the ellipsoidal
normal at /J and the target point P2, in general, does not contain the ellipsoidal
Fig. 6.15. Skew-normal reduction of the azimuth
normal through P2 (Fig. 6.15). Therefore, the ensuing normal section does not
pass through the footpoint Q2, but through Q2, which requires a reduction by
the angle Qf2PjQ2. This skew-normal reduction can be derived from the partial
derivative ///^ , formed for the adjustment in the ^,/l,/z-system, cf. [6.2.1]; it
where e is the first eccentricity and b the semiminor axis of the ellipsoid, cf.
[4.1.1]. For = 0 and a = 45, the reduction attains only 0.11" at h2 = 1000 m.
Finally, the azimuth has to be reduced from the normal section to the geodesic:
ag-an=
e*
cos2 sin 2 S2.
(6.70).
At = 0 and a = 45, this reduction reaches only 0.028" for S = 100 km.
Fig. 6.16. Reduction of the spatial distance on the ellipsoid
The reduction of the chord distance s to the ellipsoid is a purely geometric
problem, as distances do not depend on the gravity field. We assume that the
effects of atmospheric refraction have been reduced beforehand, cf. [5.5.2].
From Fig. 6.16 we read
s2 = (R + /z, )2 + (R + h2 )2 - 2(R + h, )(R + hi )cos
S = Rus
sn=2Rsin,
which gives closed formulas for the reduction to the normal section
sin-^_,
where A is taken from Euler's formula (4.18). After series expansion, the
different contributions to the reduction become apparent:
S-S = -2^LS-^
" + -34..
"'"'
The first term in (6.7Ib) corresponds to a reduction from the mean height to the ellipsoid. It
reaches the meter-order of magnitude in the mountains at distances of several km. The second
term takes the inclination of the distance into account. It generally remains below the meter-order
of magnitude in the lowland but may attain large values in the high mountains. The magnitude of
these reduction terms does not allow substitution of the ellipsoidal heights by orthometric or
normal heights, otherwise relative errors of 10"s have to be expected. The last term in (6.7 Ib)
provides the transition from the ellipsoidal chord to the normal section and reaches the cm-order
only at larger distances.
The reduction from the normal section to the geodesic is given by
S 8 -S =
:cos
q>sin22aS5.
The magnitude reaches the meter-order only at distances of several 1000 km
and can be neglected in classical network computations.
The importance of ellipsoidal calculations has decreased, since the results of
three-dimensional positioning in Cartesian coordinates can easily be
transformed into ellipsoidal coordinates, see (4.28). Azimuths and distances
derived from Cartesian coordinates, see (6.39), can be reduced to the ellipsoid
according to [6.3.2]. Nevertheless, computations on the ellipsoid are still of
significance in navigation, and they have been the basis for the calculation of
classical horizontal control networks, which are still in use today.
The geodetic datum of two-dimensional systems has been established by the
deflection of the vertical and the geoid height at a fundamental point, cf. [6.2.2],
and the parameters of the reference ellipsoid. Equation (6.57) then allows
transformation of observed astronomical latitude and longitude into the
corresponding ellipsoidal quantities, and (6.46) provides the height relation
between the geoid and the ellipsoid. Having reduced the horizontal directions
and the distances onto the ellipsoid, only one rotation about the ellipsoidal
normal in the fundamental point is possible. This rotation is fixed by applying
the Laplace equation on the astronomic azimuth observed at the fundamental
point. Equations (6.57) and (6.58) thus provide the parallelism of the axes of the
ellipsoidal system with respect to the global geocentric system.
The accuracy of the parallelism of axes depends on the accuracy of the astronomical observations
entering into (6.57) and (6.58), and hence is about one to two arcsec or better. The origin of the
classical systems may deviate by some 100 m from the geocenter. This is due to the fact that one
arcsec in astronomical positioning corresponds to 30 m in horizontal position, but is especially
due to the use of only relative deflections of the vertical and geoid heights, cf. [7.1.21.
Having fixed the ellipsoidal coordinates and one geodetic azimuth at the
fundamental point, ellipsoidal coordinate transfer can be carried out with the
ellipsoidal "observations" referring to the geodesic. Here, it is assumed that the
ellipsoidal network is errorless; for adjustment strategies see [7.1.2].
We distinguish between
The direct problem, i.e., to compute the ellipsoidal coordinates 2,2 of the
point P2, as well as the azimuth a2, given the coordinates {,^ of point
P}, the azimuth a,, and the distance S,
The inverse problem, i.e., to compute the azimuths ,, a2 and the distance
S, given the coordinates , and 2,2 of /^ and P2.
The direct and the inverse problem correspond to the solution of the ellipsoidal
polar triangle P^NP2 (Fig. 6.17), EHLERT (1993). With the geodesic introduced
as a surface curve connecting P} and P2, elliptic integrals appear in the
solutions, cf. [6.3.1], hence no closed solutions are available.
Numerous solutions of the direct and the inverse problem have been developed.
They are based either on series expansions or on numerical integration
(KNEISSL 1958/1959, SCHN DELBACH 1974).
A Taylor series expansion of latitude, longitude, and azimuth as a function of
the arc length was given early on by Legendre (1806):
Fig. 6.17. Ellipsoidal polar triangle
4-4 =
where the first-order derivatives are provided by (6.63) and (6.65). The higherorder derivatives are calculated according to
_ d (\
~d^~l)jdSdS
(d<p\da
dadSdS'
where the equation of the geodesic enters through da/dS .
Since the Legendre series are expanded with respect to S, they converge slowly. An expansion to
the fifth (,) resp. fourth-order (of) is necessary at mid-latitudes in order to provide an
accuracy of 0.0001" and 0.001" resp., at distances of 100 km. Developments up to the tenthorder are available (KRACK 1982), and modifications of (6.73) improve the convergence and
allow an efficient solution for lengths of several 100 km and more.
Series expansions suitable for very long geodesies have been developed by
Bessel (1826) and improved by Helmert (1880). For this method, the ellipsoidal
polar triangle is projected onto a concentric sphere with radius (Fig. 6.18).
The latitude (p\s transformed to the reduced latitude (4.10). Due to Clairaut's
equation (6.64), which now reads
Fig. 6.18. Transfer of ellipsoidal coordinates: Bessel-Helmert solution
cos sin a = const.,
the ellipsoidal azimuths are preserved if the azimuth a, is transferred to the
sphere. The relation between the ellipsoidal distance S and the spherical
distance , as well as those between the ellipsoidal and the spherical longitude
differences , and /, are given by differential formulas which correspond to
' sin
= cosa,
da cos
By combination with (6.63), the following relations are obtained:
- e2 cos2
The elliptic integrals resulting from (6.76) are solved either by expanding the
square roots in series and subsequent integration term by term or by numerical
methods. As the series expansions are controlled by the (small) numerical
eccentricity, they converge rapidly. After projection onto the sphere, the
coordinate transfer is carried out by formulas of spherical trigonometry, and the
results are then transformed back to the ellipsoid by inverse relations.
Computations around the globe are possible with mm-accuracy (VINCENTY
1975, KLOTZ 1993).
Numerical methods are based either on a polynomial approximation of the
integrals and subsequent integration (methods of Newton, Gauss, Simpson and
others), SCHMIDT (1999), or an numerical integration. In the latter case, the
length of the geodesic is subdivided into small increments, which are calculated
using the differential formulas (6.63) and (6.64), Fig. 6.19. Clairaut's equation
serves for keeping the increments on the direction of the geodesic. Summing the
6.4 Height Determination
N cos sin = const.
Fig. 6.79. Transfer of ellipsoidal coordinates: numerical solution
line elements provides a first approximation. By iteration with smaller
increments and comparison of the results, a given error limit can be obtained.
With increments of 100 m length, sub-cm accuracy can be achieved with
geodesies of 10 000 km length (DORRER 1966, KlViOJA 1971).
Inversion of the direct-problem solutions provides formulas for the inverse
problem and is available for all methods. The adjustment of ellipsoidal
networks would require differential relations da/, da/ and dS/ ,
dS/dX. These relations can be derived from the corresponding differentials of a
three-dimensional model reduced to the ellipsoid or by differentiating the
solutions of the inverse problem (KNEISSL 1958, p.615, WOLF 1963b).
Precise height determination is based on the relative methods of geometric
leveling, triangulation using zenith angles, and GPS heighting. Geometric
leveling after applying small reductions provides heights defined in the gravity
field [6.4.1], Zenith angles deliver ellipsoidal heights if the influence of the
gravity field is taken into account [6.4.2]. GPS heighting directly results in
ellipsoidal heights, which have to be reduced to gravity field related quantities
The raw results of geometric leveling Sn dn are transformed into potential
differences by taking surface gravity g along the leveling line into account
(5.113). If connected to the zero height surface (geoid, quasigeoid) geopotential
numbers (3.104) are obtained:
C = W0-Wp=fgdn.
In principle, the evaluation of (6.77) requires a gravity value at each leveling
point, i.e., at distances of 100 m or less for precise leveling. This requirement
can be diminished if we postulate the same relative accuracy for the influence of
leveling and gravity on the geopotential differences:
Assuming a leveling accuracy of 0. 1 mm, gravity would only be needed with an
accuracy of 100 8~2 for = 10 m, and 10 3~2 for = 100 m.
Consequently, gravity values can be interpolated from gravity measurements
carried out at station distances of 5 to 20 km in flat areas, and at 1 to 2 km in the
mountains. Preferably, gravity stations should be established at sites where the
gravity changes depart from linearity (variations in slope or direction of the
leveling line, gravity anomalies). An average gravity value taken over a larger
distance then can be used for transforming the leveled height difference into
potential units.
Precise leveling in fundamental networks is carried out in closed loops, cf.
[7.2]. The calculation of heights is performed by adjusting potential differences,
with the condition, cf. [3.2.3],:
Subsequent transformation yields metric heights, cf. [3.4.3].
An alternative approach, used classically, is to first convert the raw leveling
results (with <I>dn ) into differences of the respective height system and then
adjust height differences.
Dynamic height differences are obtained by differencing (3.105):
UJ% = H^ - H^ = Ania + df ,
with the dynamic height reduction
For orthometric heights, we expand (3.106) by dynamic heights:
//( 2 =2-,= A//,df + (//2 - //2dyn )-(//,- tf,dy
This leads to
// 2 => 12 + 12 ,
with the orthometric height reduction
/O
^dn + ^J^H
^^H
(6.8ib)
An analogue equation is valid for the transformation of leveled height
differences into normal height differences, where g is replaced by and by
HN (3.107). The normal height reduction reads:
_ 45
_ y45
H -^-^H
The dynamic height reduction only depends on the path. It attains values between a few mm (flat
terrain) and some cm to dm (mountains). Hence, it has to be taken into account even in. local
surveys if the vertical reference system is based on dynamic heights. The orthometric and the
normal height reduction include the dynamic reduction, and, in addition, contain two terms with
the mean gravity values at the end points of the leveling line. The different effects substantially
cancel each other, with the consequence that these reductions are below one mm in flat areas and
only reach a few cm in the mountains.
Mean normal gravity can be calculated by (4.67) with the spherical
approximation
Mean gravity g, on the other hand, requires the knowledge of g along the
plumb line between the geoid and the earth's surface. At any point Pf with
height //, we have
H'
where g is the surface gravity at P. The actual vertical gravity gradient is given
by (3.71), where we may introduce (4.60) for the free-air part and 2670 kg/m3
as a mean crustal density; this yields
g =g + 0.848 10^ (H-H')ms-2.
After inserting into (3.106) and integration between ff = 0 and if = H/2, we
g = g + QA24xlQ-6Hms-2 .
Orthometric heights based on this estimate are also called Helmeri-heights.
As shown in [6.5.3], the second term on the right side of (6.84) can be interpreted as a reduction
of the surface gravity to actual gravity at H/2, with the Bouguer plate as a model of the
topography. The effect of model errors remains small in flat terrain, but improved models that
take actual topography and density into account are needed in the mountains (S NKEL 1988).
Zenith angles can be used for a trigonometrical height transfer if refraction
effects have been reduced sufficiently, cf. [5.5.1], Fig. 6.20. The ellipsoidal
zenith angle is obtained from the observed quantity z' by
= ' + + = + ,
where is the angle of refraction (5.11) and the vertical deflection component
in the azimuth of the line of sight (6.18). Using spherical trigonometry, the
ellipsoidal height difference is given by (KNEISSL 1956, p.358):
(6.86,
S is the length of the ellipsoidal normal section, R the radius of curvature (4.18),
andALl=(Ai+Al)/2.
Fig. 6.20. Trigonometrical height transfer
The use of reciprocal zenith angles offers significant advantages. With the
taken from Fig. 6.20, we apply the law of tangents on the triangle PtOP2 . In
connection with (6.85), the height difference is obtained:
Here, only differences in and appear. Symmetric refraction conditions may
be expected with simultaneous observations, cf. [5.5.1], thus refraction effects
will mostly cancel with the use of simultaneous reciprocal-zenith-angle
measurements (KUNTZ and SCHMITT 1995). They also offer a possibility to
determine the coefficient of refraction. Combining (5.11), (6.85), and (6.87),
and neglecting the deflections of the vertical, yields
An average value of k = 0.13 0.04 was obtained by Gauss for the arc measurements in the
kingdom of Hannover. This value was confirmed by other surveys for lines of sight with a large
ground clearance. Close to the ground, k may vary between -1 and +1. Network adjustment
models may be extended by introducing individual refraction coefficients for each station or for
each line (URADILEK 1984), and observed meteorological parameters may also contribute to the
determination of actual values for k, cf. [5.1.2], [5.5.1]. In spite of these refinements, refraction
irregularities strongly limit the application of trigonometrical heighting. A cm-order of accuracy
can be obtained over distances of a few km, but errors of the dm-order of magnitude have to be
expected with larger distances.
Trigonometric leveling significantly diminishes the errors of a trigonometrical
height transfer, by reducing the lines of sight to 100 to 300 m (REGER and
BRUNNER 1982). Height differences are determined by measuring zenith angles
and slope distances with a total station, cf. [5.5.2]. Either simultaneous
reciprocal-observations are carried out using two reflector-equipped total
stations or the method of leveling "from the middle" is applied, in analogy to
geometric leveling, cf. [5.5.3]. Refraction effects are substantially reduced using
this strategy. The reductions are largely due to the short distances and the fact
that the lines of sight run approximately parallel to the earth's surface. In this
way, a favorable error propagation over larger distances is obtained (accuracies
of 1 to 2 mm per km). The efficiency of the method can be increased by
motorized procedures (BECKER 1987). By using short lines of sight,
trigonometric leveling resembles the procedure employed in geometric leveling.
Hence, the results represent a reasonable approximation to leveled height
Trigonometric height determination with long lines played an important role in the establishment
of classical horizontal networks by triangulation, as it simultaneously provided heights for
reduction onto the ellipsoid and for the construction of topographical maps. Today, it is restricted
to special applications, e.g., to the height determination of inaccessible sites. Trigonometric
leveling, on the other hand, has been employed successfully for surveying vertical control
networks of large extension (WHALEN 1985).
The Global Positioning System (GPS), cf. [5.2.5], provides global Cartesian
coordinates and, more accurately, coordinate differences, which can easily be
transformed to ellipsoidal coordinates, see (4.28), DODSON (1995). Ellipsoidal
height differences can be achieved with cm-accuracy over distances up to some
100 km or more, applying differential methods and long (e.g., 24 hours)
observation times. For short distances, even sub-cm accuracy can be obtained
(GRRES and CAMPBELL 1998). Consequently, GPS heighting may support or
substitute time-consuming geometric leveling, at least for distances larger than
about 10 km. This strategy requires that the reduction of ellipsoidal heights to
normal or orthometric heights can be performed with a similar accuracy (TORGE
1990). According to (6.46), quasigeoid or geoid differences provide the
necessary reductions:
//"=/-", // = -.
Today, global quasigeoid or geoid models have relative accuracies of a few
decimeters, and regional models may reach the cm to decimeter accuracy, cf.
[6.6], [6.7]. The use of these models for the reduction of GPS heights
presupposes that the model describes the same surface as used as zero height
surface for the height system, cf. [3.4.3]. This demand can be fulfilled by fitting
the model to quasigeoid or geoid heights at control points where GPS and
leveled heights are available. The fitting also reduces long and medium-wave
model errors.
Following the method of least squares collocation, cf. [6.8.2], the discrepancies
found at the control points can be modeled as follows (MiLBERT 1995, DENKER
et al. 2000):
hgps-HN-Cmaa=t + s + n,
where t describes a trend component, s is a stochastic signal part, and n
represents the random noise of all types of observations involved (GPS,
leveling, geopotential model). A simple trend function may consist of a threeparameter datum shift according to (6.41):
t = cos <pcos " + cos ^sin 7 + sin .
Equivalently, the function may describe a change in the ellipsoidal coordinates
of some initial point, which corresponds to a vertical shift and tilts in the NS
and the EW-direction. The signal part is derived from an empirical covariance
function of the de-trended residuals and modeled e.g., by an exponential
function, cf. [6.1.3].
Using trend reduction only, the r.m.s. discrepancies between regional and local quasigeoid models
and GPS/leveling control points have been reduced to the dm-order of magnitude over a few 1000
km and to a few centimeter over several 100 km (TORCH et al. 1989). By including also a signal
part, cm-accuracy may be obtained over distances up to 1000 km, which corresponds to the
accuracy of classical leveling networks (DENKER 1998).
For local (several 10 km) applications, gravity field related heights may be estimated by purely
mathematical interpolation between GPS/leveling control points, employing, e.g., low-order
surface polynomials or splines (!LLNER and J GER 1995). If a dense net of control points is
available, and if the gravity field is sufficiently smooth, cm-accuracy can be achieved (COLLIER
and CROFT 1997). The accuracy strongly decreases in rough topography, and a topographic
reduction is necessary then, cf. [6.5.3].
With the further improvement of quasigeoid/geoid models, "GPS leveling" will
become even more important.
6.5 Fundamentals of Gravity Field Modeling
Gravity field modeling is part of the geodetic boundary-value problem [6.5.1].
Topography plays an important role in the solution of this problem [6.5.2].
Gravity reductions serve for reducing observed gravity field data onto the geoid
and also provide different kind of gravity anomalies for field interpolation and
geophysical interpretation [6.5.3]. While the orientation of the gravimetrically
derived geoid is uniquely defined, the scale has to be determined by distance
measurements [6.5.4].
6.5.1 The Geodetic Boundary-Value Problem
The geodetic boundary-value problem comprises the determination of the
surface of the earth and of its external gravity field from observations on or
close to the earth's surface (SANS and RUMMEL 1997). The surface to be
determined is either the geoid (Stokes' problem) or the physical surface of the
earth (Molodenski's problem), SANS (1995).
Based on Green's third identity and the generalized Laplace equation (3.49), the
problem may be formulated by a non-linear integral equation of the second kind
in the gravity potential fF(MOLODENSKi 1958, MOLODENSKI et al. 1962):
+22(2+2)+22$ = 0
Here ns is the outer surface normal to the surface S, v is the volume of the
earth and its rotational velocity, and / denotes the distance between the source
point and the point of calculation. If W were known on 5, then the geometry of
the surface would remain as the only unknown quantity. After the determination
of S, an upward continuation of W would deliver the external gravity field.
The problem is linearized by approximating the earth's surface by the telluroid
(physical surface) or the ellipsoid (geoid) and by approximating the actual
potential W by the normal potential U, cf. [6.1.1]. As the centrifugal part is
well-known, (6.92) then transforms into an integral equation for T
The integration is performed over the known telluroid . As the surface normal
deviates from the direction of the plumb line, dT/dnz not only depends on
the gravity anomaly but also on the deflection of the vertical and the slope of
the terrain, cf. [6.7.2]. If applied to the geoid as a boundary surface, this
dependence reduces to the gravity anomaly.
Instead of the integral equation (6.93), the geodetic boundary-value problem
can be formulated by Laplace 's differential equation (6.2):
The residual gravity field parameters observed on the earth's surface, or
reduced to the geoid, then enter into boundary conditions for the solution of
(6.94). The primary "observables" are the height anomalies resp. the geoid
heights, and the gravity anomalies. Deflections of the vertical and gravity
gradient components play a role only in local calculations.
A Taylor development of U in the telluroid point Q gives
-y
_ , +.
{on )Q
with n = normal to U = UQ and with normal gravity
Yo=-\^r\
l "k
Inserting (6.95) into (6.1) yields the height anomaly
-U0)
^.
With the condition (6.6), we obtain
Equation (6.96) is called Bruns' theorem. If applied to the geoid it delivers the
geoid height
,_T-(W0-U0)
Under the condition f/0 = Wa, it reads
now refers to the geoid, and 0 is normal gravity on the ellipsoid (4.41).
The deflection of the vertical is the horizontal derivative of ^resp. TV (Fig. 6.21).
Taking (6.96) and (6.97) into account, the components in the direction of the
meridian and the prime vertical, cf. [6.1.2], are given by
where the ellipsoidal arc elements are provided by (4.20). The negative sign
follows from the sign conventions for the quasigeoid (geoid) and the vertical
deflection, see (6.40).
PRIME VERTICAL
(M + h)d<?
(N + h)cos(p
Fig. 6.21. Deflection of the vertical components and height anomaly
In spherical approximation we obtain
c =
yr d<p'
yrcos<pdA
The gravity disturbance (6.11) is related to by
( du}
- -=-where again we have neglected the deflection of the vertical. We develop at
the telluroid point Q:
Inserting into (6.99) and taking (6.96b) into account yields the gravity anomaly
^_, = _
(9n)
Y (dn)
(6,ola)
With the spherical approximation, cf. [4.2.2],
r'
Equations (6.97), (6.98), and (6.101) represent boundary conditions for the
solution of the Laplace equation. Because of the importance of (6.101), this
first-order partial differential equation in T is known as the fundamental
equation of physical geodesy.
High-precision gravity-field modeling (e.g., geoid determination with cmaccuracy) requires some refinements in the formulation and solution of the
geodetic boundary-value problem (MORITZ 1974, HECK 1991). This includes
the transition to an ellipsoidal approximation by developing the potential in
ellipsoidal harmonics or by applying ellipsoidal corrections to the spherical
approximation (JEKELI 1988b, WANG 1999). The mass of the atmosphere has to
be taken into account by a corresponding reduction, cf. [4.3], and the
topography has to be smoothed by a terrain correction, cf. [6.5.3].
The geodetic boundary-value problem resembles the third boundary-value problem of potential
theory, namely to determine a harmonic function given a linear combination of the function and
its normal derivative on a bounding surface. It differs from the classical problem, as the bounding
surface is supposed to be unknown: free boundary-value problem. At the physical surface of the
earth, gravity data do not provide potential derivatives along the surface normal but rather refer to
the plumb line: free and oblique boundary-value problem (GRAFAREND and NIEMEIER 1971). The
horizontal components of the surface position vector cannot be determined with sufficient
accuracy from gravimetric data, cf. [6.2.1], consequently the geometric part of the problem
generally is restricted to the determination of heights: scalar free gravimetric boundary-value
problem (HECK 1997).
With the rapid progress in satellite positioning, the geometry of the earth's surface can be
assumed to be known with increasing accuracy; the only remaining unknown then is the external
gravity potential. Hence, a fixed boundary-value problem can be formulated, which employs
gravity disturbances as boundary values (KocH and POPE 1972, BJERHAMMAR and SVENSSON
1983). This corresponds to the second (Neumann) boundary-value problem of potential theory,
which is to determine a harmonic function from its derivative on the bounding surface. Finally, a
mixed altimetric-gravimetric boundary-value problem may be set up, taking into account that
altimetric geoid heights and gravity anomalies are the main data sets available on a global scale
(SANS 1981, MARTINEC 1998).
6.5.2 Gravitation of Topography
The short wavelength part of the gravitational field is dominated by the effect of
the topographical masses. By reducing this effect, the gravity field is smoothed
significantly, which simplifies gravity field interpolation and transformation.
For the determination of the geoid, the topography has to be removed in order to
establish the geoid as a boundary surface (FORSBERG and TSCHERNING 1997).
The effect of the topographic masses on gravity field parameters is calculated
by the law of gravitation. The evaluation of (3.10) and corresponding integrals
for other parameters poses problems, as topography is rather irregular in
geometry (heights) and, to a lesser extent, also in density. Therefore, the
topographic masses are subdivided into elementary bodies for which closed
solutions of the mass integrals exist. Rectangular prisms of constant density are
especially appropriate, as the heights of the topography nowadays are provided
in gridded form by digital elevation models. Vertical cylindrical columns with
constant density and height have been used extensively in the past.
In a system of three-dimensional Cartesian coordinates, the gravitational
potential of the topography is expressed by
with / = Jx2 +y2 +z2 . The topographic effects on the deflection of the vertical,
the gravity disturbance, and the gravity anomaly follow from differentiation,
according to (6.98), (6.99), and (6.101). The integration over a rectangular
Fig. 6.22. Gravitation of topography: rectangular prism method
prism delivers closed formulas for the potential and its derivatives (FORSBERG
1984, NAGY et al. 2000). As an example, for a point located at the origin of the
,^,-system, the vertical component of the gravitation is given by (NAGY
1966), Fig. 6.22:
y^
b2=Gp xln (y + /) + y\u (x + /)- zarctan
The total effect of topography results from the sum over the gravitation of the
individual elementary bodies:
(6.104).
For heights given in a regular grid, Fast Fourier Transform (FFT) techniques
provide a powerful tool for the efficient calculation of topographic effects
(FORSBERG 1985, SIDERIS 1985, SCHWARZ et al. 1990).
Digital elevation models (DEM), also called digital terrain models, are available on global and
regional scale. They are primarily based on digitized topographic and bathymetric maps. Space
techniques contribute increasingly to the improvement of OEM's. Satellite radar altimetry serves
for the height determination of the Greenland and Antarctica ice sheets and provides bathymetric
information, due to the correlation between the ocean surface and the ocean bottom (SMITH and
SANDWELL 1994). Space and airborne interferometric synthetic aperture radar (INSAR) has
become an efficient method for the development of high resolution OEM's (ZEBKER and
GOLDSTEIN 1986, NIELSEN et al. 1998).
Among the global OEM's are the ETOPO5 (5'x5'gridded land and seafloor elevations) and the
GLOBE (global land one-kilometer base elevation) models provided by the NOAA National
Geophysical Data Center. GLOBE is given in a 30"x30" grid; the accuracy depends on the data
quality and varies between 20 m and a few 100 m (HASTINGS et al. 2000). The NASA/NIMA
Shuttle Radar Topography Mission (SRTM, February 2000) collected a global (between 60
latitude) INSAR data set, which will be used for a DEM with 30 m mean resolution, accurate to
10 to 20 m. Regional OEM's have been developed in many countries with resolutions down to
one arcsec (SMITH and ROMAN 2000).
For large-region applications, a mean density value of 2670 kgrrT3 (corresponding to the density
of granite) generally is introduced for the topographic masses. More refined density models have
been used for local studies, where the density values are estimated from geological information,
rock samples, and gravity profiles exploiting the density-dependent relation between gravity and
height (Nettleton method, e.g., TORGE 1989).
Spherical harmonic models of the topography have been derived from global OEM's (PAVUS and
RAPP 1990). They serve for the development of topographic-isostatic models, cf. [8.2.2], which
may be used for the prediction of mean gravity anomalies in unsurveyed areas, cf. [6.5.3].
In order to determine the geoid as a boundary surface in the gravity field, the
topographic masses have to be removed, and the observed gravity values have
to be reduced to the geoid. This is done by gravity reductions, which provide
gravity anomalies on the geoid.
Depending on how the topographic masses are displaced, different types of gravity anomalies are
obtained. Uncertainties of the topographic model propagate into the calculation of the geoid. This
has to be taken into account in the calculation of orthometric heights (3.106), where the same
density model has to be employed.
GRAPHY
COGEOID
COMPENSATING
Fig. 6.23. Geoid and cogeoid
The displacement of the topographical masses changes the gravitational field of
the earth, including the potential of the geoid. The level surface which possesses
the geoid potential after the displacement is called cogeoid: indirect effect of the
gravity reductions (WiCHiENCHAROEN 1982, MARTINEC 1998). The following
steps may be distinguished in the calculation of the geoid (Fig. 6.23):
Reduction of the direct effect of the topography on gravity and adding of
the direct effect of the dislocated masses if necessary, both to be calculated
by the law of gravitation, cf. [6.5.2],
Calculation of the primary indirect effect on the potential, caused by the
removal of the topography and the dislocation of the topographic masses.
The calculation is performed according to some rule of compensation:
SV = Vt0f-Vc,
with = potential of the topography and Vc = potential of the
compensating masses,
Calculation of the vertical distance between the geoid and the cogeoid
according to (6.97b):
Reduction of the gravity values from the geoid to the cogeoid: secondary
indirect effect. Here, a free-air reduction (6.101) is sufficient:
Calculation of the heights Nc of the cogeoid above the ellipsoid, cf. [6.6],
[6.7],
Calculation of the geoid heights according to
N = NC+SN.
In principle, every kind of gravity reduction could serve for the calculation of
the geoid according to this scheme. Naturally, the indirect effect should be
small in order to avoid laborious and error susceptible computations. Other
criteria for the selection of gravity reductions include the smoothness of the
resulting gravity anomalies, which facilitates interpolation, and their
geophysical significance, which would allow interpretation. Under these
aspects, we may distinguish between the effects of (homogeneous) topography,
density anomalies within the topography and the earth's crust, and isostatic
compensation masses, cf. [8.2.2].
The free-air anomaly is generally used for the calculation of the geoid, with the
assumption that no masses exist above the geoid. The free-air reduction
(// = orthometric height) provides the reduction of the surface gravity to the
geoid, and the simple free-air anomaly on the geoid is given by
*gF=g + ogF-Y0.
According to (6.109), the correct reduction to the geoid requires the knowledge of the actual
vertical gravity gradient. Splitting the gradient into a normal and an anomalous part gives
_=_
a/7 dH
The actual and the normal part may differ by 10% or more. The normal gravity gradient can be
calculated by (4.61). The calculation of the anomalous part corresponds to the downward
continuation of a harmonic function. It can be formulated by Poisson's integral, which is a
solution of the first (Dirichlet) boundary-value problem of potential theory, and solved by an
integration over the surface gravity anomalies, cf. [6.7.2]. If the gravity anomaly depends linearly
on elevation, the anomalous gradient corresponds to the terrain correction ( 1980, p.421),
see below. Due to the assumptions made in (6.1 10), it equals the direct effect of the topographic
and the dislocated masses on gravity (SiDERis 1990). A discrete solution of this problem has been
given by BJERHAMMAR (1969, 1985), which takes into account that gravity data are given only at
discrete points. This solution satisfies all given data and generates missing data, and it is
harmonic down to an internal sphere (Bjerhammar sphere), cf. [3.3.2].
The free-air anomaly on the geoid should be clearly distinguished from the free-air anomaly
defined on the surface of the earth (6. 1 5) where the normal gravity gradient is used for reduction.
The terrain correction removes irregularities of the topography. It creates a
plate (spherical or planar) of constant thickness and density (Bouguer plate) by
filling mass deficits below and removing excess masses above the plate (Fig.
6.24). For planar approximation (Bouguer plate extending to infinity), both
measures increase gravity at P; the terrain correction then is always positive. It
can be calculated from digital elevation models, cf. [6.5.2], and attains values of
1 to 10 5~2 in flat areas, reaching several 100 5~2 in the mountains. From
(6.102) we obtain
_ jj
i=H
+00+00 l=H
6gT=Gpll J ~
dxdydz.
For small surface slopes, the distance
- xp )2 + (y - yp )2 + (z - zp )2
may be approximated by
-*,> )2 + (y - )2
The linear approximation of the terrain correction then reads
where resp. Hp are the orthometric heights of the terrain and the point of
calculation (FORSBERG and TSCHERNING 1997).
/ >-"
TERRAIN^CORR.
00 *-
(CORR.
BOUGUER PLATE
Fig. 6.24. Bouguer plate and terrain correction
By including the terrain correction into (6.110), we obtain the terrain-corrected
free-air anomaly, called Faye-anomaly. The shift of the topographic masses
now corresponds to a compensation of the Bouguer plate on the geoid
(Helmert's compensation method). Here, the surface density
' =
= pH
asJ '
replaces the volume density p and takes the height of topography into account.
As the mass displacement is slight, the indirect effect of the free-air and the compensation
reduction remains small. It reaches at most a few meters in the absolute sense and is of the cm to
dm-order for geoid differences. As the height-dependent effect of the topography has not been
removed, free-air anomalies are strongly correlated with height. Therefore, point free-air
anomalies are not suited for interpolation and cannot be geophysically interpreted. For limited
areas, the height-dependence can be described by linear regression, which corresponds to the
application of the Bouguer plate reduction (see below). Interpolation and calculation of mean
gravity anomalies is then facilitated ( ROTEN and REINHART 1968).
By removing the effect of topography explicitly through a topographic
reduction Jgtop , we obtain the Bouguer gravity anomaly Aga . After the
removal of the masses the surface gravity again is reduced to the geoid by the
free-air reduction and compared with the normal gravity (Fig. 6.24):
A&^g-tf&op+tfgf-JO
The topographic reduction is calculated from digital elevation models, cf.
In most cases, the topographic reduction is decomposed into the Bouguer plate
reduction Sgp and the terrain correction SgT . The Bouguer plate reduction
accounts for the gravitation of an infinitely extended horizontal plate with
constant density. Its thickness is given by the height of the computation point.
The gravitational effect of the Bouguer plate is derived from the attraction of a
circular cylinder on a point located on the cylinder axis (GARLAND 1979,
MILITZER and WEBER 1984). By extending the cylinder radius to infinity, one
obtains
ogp = 2 = 0.000 419 pH '2 ,
(6.1 15)
where is taken in kg/m3 and H in m. The terrain correction reduces the
topography to the Bouguer plate (see above). The Bouguer anomaly then reads
bgB=S-SgP+SgT+SgF-Yt.
For large-region applications (e.g., national gravimetric surveys), a spherical Bouguer plate and a
corresponding terrain correction is used frequently, with a calculation extending 170 km from the
computation point and conventional density being 2670 kg/m3 .
Due to the removal of the height-dependent part of topography, Bouguer anomalies display
smooth long-wave variations only. Hence, they are well suited for interpolation. Revealing
density anomalies below the geoid, the Bouguer anomalies are of considerable significance in
geophysics and geology, cf. [8.2.4]. On the other hand, since the topographic masses are
completely removed and not restored (i.e. shifted to infinity), the indirect effect is very large
(several 100 m). Hence, Bouguer anomalies are not used for geoid computations.
The Bouguer plate model allows a simple calculation of the mean gravity g
along the plumb line required for the computation of the orthometric height H
(3.106), (6.81). If we assume a linear change of g along the vertical, g will be
found at the height H/2. Hence it can be derived from surface gravity by
removing a Bouguer plate of thickness H/2, a free-air reduction from H to H/2,
and a subsequent restoration of the Bouguer plate above H/2. Removing and
restoring the Bouguer plate has the same (negative) effect on gravity; so we
Evaluation with (6.115) and (6.109), and introducing the density value 2670
kg/m3 , leads to (6.84).
Equation (6.117) also provides an important interpretation of the difference
between the heights of the geoid and the quasigeoid resp. the normal and the
orthometric height. With the mean normal gravity
the mean gravity anomaly (6.9) is identified as the Bouguer anomaly (terrain
correction neglected)
This fact permits a simple transformation from the quasigeoid to the geoid, cf.
Isostatic anomalies are formed by not only removing the effect of topography
but by also restoring compensation masses in the earth's crust below the geoid,
according to some isostatic model. In this way, the crust is regularized,
obtaining constant thickness and density, cf. [8.2.2]. The gravitational effect of
the compensating masses is taken into account by an isostatic reduction Sg, to
be calculated from the isostatic model according to [6.5.2]. The isostatic
anomaly then is given by
; ^g-S^+Sg, +SgF-Yo.
The largest part of the earth's topography is isostatically compensated. Hence, isostatic anomalies
are small and vary smoothly about zero, with the exception of uncompensated areas. They may be
successfully employed for gravity prediction, and they are of great value for geophysical and
geodynamic interpretation, cf. [8.2.4]. The compensating masses are arranged more remote from
topography than in the free-air reduction (see above). Therefore, the indirect effect is larger and
may reach 10 meters. Because of this, isostatic anomalies are rarely used for geoid calculations.
The following assumptions were made for the spherical harmonic expansion of
the disturbing potential and the transformation to the quasigeoid resp. geoid, cf.
[6.1.1], [6.5.1]:
The level ellipsoid and the earth have the same mass:
Mm=M .
Hence, no zero-degree term T0 appeared in the expansion (6.4),
The center of the ellipsoid and the earth's center of mass (origin of the
global coordinate system) coincide; no first-degree term entered into (6.4),
cf. also [3.3.4],
The normal potential U said the actual potential Ware related by
UQ=WP,
which corresponds to the condition
U0=W9
between the potential of the level ellipsoid and the geoid.
A first-degree term in the spherical harmonic expansion would not affect the
gravity anomaly, as the corresponding expansion (6.135) contains the factor
( / - I ) . Hence, the ellipsoid may be positioned in the geocenter without
changing the gravity field: the gravimetric method yields "absolute" results.
Because of residual uncertainties in the determination of the mass and the
potential, small differences between the values for the geoid and the ellipsoid
have to be admitted:
W0-U0.
The spherical harmonic expansion of then must be extended by
and Bruns' formula must take 0 and the potential difference into account:
GSM SW
For spherical approximation, the constant NQ corresponds to a change in scale
of the geoid. The expansion of the gravity anomaly (6.135), according to
(6.101), has to be extended by (HEISKANEN and MORITZ 1967, p. 100 ff)
=2.
+SW
G SM
+ SW.
With the usual assumption of equality of mass and potential, the gravimetric
solution of the boundary-value problem delivers results which refer to a bestfitting ellipsoid, where the equatorial radius ("scale") remains unknown by N0.
f^
s w- ^\_
I,
BEST riTTINIG
\\ ELLIPSOID U = W
GEOID W =W0
Fig. 6.25. Gravimetric geoid and geodetic reference ellipsoid
The zero-degree undulation N0 can be determined by comparing
gravimetrically computed geoid heights with geoid values derived by
differencing distances that refer either to the geoid or to the ellipsoid. The
primary data sets used are from satellite altimetry on the oceans and GPS
heights on the continents (RAPP and BALASUBRAMANIA 1992, JEKELI 1998),
Fig. 6.25). The geoid height from these satellite techniques is given by
#,=*-#.
where /isat stands for the ellipsoidal height of the altimeter or the GPS height,
and H is the height of the altimeter above the geoid (result of the altimeter
measurements) or the orthometric height derived from leveling. The zerodegree term is obtained by
N^N^-N^.
It can be used to derive the semimajor axis of the best-fitting ellipsoid to which
the gravimetric geoid heights refer:
<W=*U+tf.
Generally, the ellipsoid adopted as part of a geodetic reference system, cf. [4.3],
is kept unchanged. The gravimetrically determined geoid heights are then
corrected in order to refer to that reference ellipsoid (e.g., the GRS80 ellipsoid):
Nat=Nilt=Np+N0.
Classically, best-fitting ellipsoids have been determined by comparing horizontal distances
measured on the geoid with the ellipsoidal distances obtained by calculation, cf. [7.1.2]. A
separation between and SW in principle would be possible using (6.126), together with
(6.125), using the condition #0 = 0 (6.19). However, deficiencies in the global gravity coverage
still prevent use of this method. On the other hand, the geocentric gravitational constant GM is
known today with high accuracy from space probes and high-flying satellites ( ROTEN 2000).
The determination of N0 thus provides the potential of the geoid. From Topex/Poseidon altimeter
data, the equatorial radius of an ellipsoid that best fits the zero-tide geoid, cf. [3.4.1], has been
estimated to 6 378 136.49 0.1 m (RAPP 1995c).
6.6 Global Gravity Field Modeling
Global gravity field modeling is required for large-scale problems including the
determination of satellite orbits, inertial navigation, and development of
geophysical and geodynamic models. The geoid is required for establishing a
global vertical reference system and for deriving sea surface topography.
Finally, global models provide the long wavelength part of the gravity field for
local gravity field approximation, cf. [6.7].
Global gravity models are based on spherical harmonic expansions [6.6.1]. The
low frequency part of these series expansions stems from the analysis of
satellite orbits [6.6.2]. Expansions of higher degree are achieved by combining
the low-degree models with the results of terrestrial gravimetry and satellite
altimetry [6.6.3], BOUMAN (1997), NEREM et al. (1995), RAPP (1998).
Equations (6.3) to (6.5) provide the development of the disturbing potential T
into spherical harmonics. A gravity field model thus is represented by the
spherical harmonic coefficients. The functional relations between T and other
relevant gravity field parameters, cf. [6.5.1], also allow spherical harmonic
expansions for the height anomaly, the geoid height, and the gravity anomaly.
These expansions generally employ fully normalized spherical harmonics, cf.
By inserting (6.4) into Bruns' theorem (6.96b), we obtain the spherical
harmonic expansion for the height anomaly
Here, the fully normalized coefficients and spherical harmonics are indicated by
bars. Equation (6.97b) delivers the corresponding expansion for the geoid
height, which can be also be derived from (6.131), taking (6.9) into account:
Differentiation of (6.4) with respect to r gives the spherical harmonic expansion
for the gravity disturbance (6.99):
Inserting (6.4) and (6.133) into (6.101b) yields the expansion of the gravity
'=2
\r )
Substituting T, from (6.5) gives
(l ~l{-} (^cosm/l + ^ sin )* (cos
By comparing the abbreviated form
" / V+1
with (6.134), we obtain the relation between the surface spherical-harmonics of
and Ag:
Corresponding equations can be derived for the deflection of the vertical and for higher
derivatives of T (WENZEL 1985). Due to the scarcity of terrestrial data of these types, these
expansions are of limited importance. Second-order derivatives of obtained from satellite
gravity gradiometry, on the other hand, will be of great importance for future gravity field
modeling (RUMMEL et al. 1993).
Equations (6.131) to (6.135) permit the harmonic coefficients to be determined
from "observations", by least-squares adjustment, cf. [6.6.2], [6.6.3]. The
coefficients can also be determined by quadrature over the observations.
Taking the orthogonality relations and the properties of the fully normalized
harmonics into account, the inversion of (6.131) and (6.135) yields (JEKELI
1998):
-""[ = -[{rr\-\NP l m (cos&)\ C O S m \da
M,J 4 {)
[sin m/l J
IAS,.
da. (6.138b)
The integration is extended over the unit sphere . In contrast to least squares
adjustment, the quadrature approach only allows exploitation of one type of
data for the determination of the harmonic coefficients.
Global data sets do not cover the earth homogeneously, and they have a limited
spatial resolution. Consequently, mean geoid heights and mean gravity
anomalies over surface compartments are introduced in gravity field modeling,
where the surface blocks generally are bounded by meridians and parallels. The
mean values are calculated according to
UN da, Ag =
ll^gda.
The block size depends on the data distribution, cf. [6.6.3]. The maximum
gravity field resolution which can be achieved is ^fK. This corresponds to a
maximum degree of the spherical harmonic expansion /^ = 180/resolution,
cf. [3.3.3]. The introduction of mean values causes a local smoothing of the
gravity field, which also leads to smoothed harmonic coefficients. This fact has
to be taken into account by damping factors (< 1), which depend on the degree
and the dimension of the compartment (KATSAMBALOS 1979).
The truncation of the spherical harmonic expansion at / produces an
omission error due to the neglected part of the gravity field. This error can be
estimated from the degree variances of the gravity field parameters. Degree
variance models have been derived for gravity anomalies, see (6.27). They are
based on the covariance function of the gravity anomalies (6.25), but can also
be calculated from Laplace's surface harmonics Ag, (6.26). In spherical
approximation (r = a = R), (6.137) gives
Inserting (6.140) into (6.26) and evaluating yields the anomaly degree- variances
as a function of the harmonic coefficients:
This relation can especially be used to calculate the low-degree variances from
the harmonic coefficients obtained from satellite orbit analysis.
The functional relations between the gravity field parameters, cf. [6.5.1], also
permit the calculation of degree variances for geoid heights, deflections of the
vertical, and higher-order derivatives (TSCHERNING and RAPP 1974,
TSCHERNING 1976). With = GM/r2, a comparison between (6.131) and (6.135)
gives the geoid degree variance
(m, 10 4 ms 2 , arcsec)
0.01- 0.001
1000 5000 10000 50000
Fig. 6.26. Omission error for geoid heights (height anomalies), vertical
deflections, and gravity anomalies (degree variance model TSCHERNING and
RAPP 1974)
The omission error for geoid heights, gravity anomalies, and deflections of the
vertical is shown in Fig. 6.26. For the presently usual truncation at degree 360,
it amounts to 0.23 m, 253 8~2, and 3.8". An expansion to / = 7200,
corresponding to a spatial resolution of 2.8 km, would reduce these errors to
0.002 m, 27 '2, and 0.4".
Low-degree gravity field models are required for precise satellite orbit
calculation, positioning of tracking stations, and long-wave geoid
representation. They are primarily based on satellite tracking data, which are
analyzed with respect to the deviations of the actual orbit from a precalculated
reference orbit, cf. [5.2.2]. "Satellite-only" models use tracking data
exclusively. In some cases, they have been utilized for the precise orbit
determination of a specific satellite, especially by taking into account
observations to that satellite ("tailored" models).
The "perturbing" gravitational potential (5.34) changes the Keplerian orbit
elements with time (5.35). As the gravitational potential is modeled by a
spherical harmonic expansion, the harmonic coefficients enter as unknowns into
the observation equations of satellite geodesy, cf. [6.2.1]. In order to resolve the
gravitational field to a certain degree, satellites in different altitudes and with
different inclinations are required, in addition to a good global distribution of
the tracking stations and a sufficiently long observation time. The attenuation of
the gravity field with height and the decrease of the harmonic coefficients at
higher degrees, together with deficiencies in the geometry of the satellite orbits
and the ground stations, limits the resolution of "satellite-only" models. With
the satellites presently available (altitudes of 800 km and more), developments
up to degree 70 have been carried out. Satellite-to-satellite tracking and satellite
gravity gradiometry at orbital altitudes of about 300 km will extend this range
up to degree 150 to 200, cf. [5.2.8], RUMMEL 1979.
We now regard the influence of the gravitational field on the satellite orbit. We
distinguish between secular (linear), long-periodic (few days to months), and
short-periodic (periods less than one day or one satellite revolution)
perturbations (KAULA 1966, SEEBER 1993). The relationship between the
perturbations of the orbital elements and the harmonic coefficients is obtained
by transforming the spherical coordinates of the expansion (3.89) into the
Keplerian elements, forming the derivatives of the perturbing potential with
respect to the elements, and inserting them into (5.35).
Determination of the secular and long-periodic perturbations requires the
integration over long arcs (several days). Here, short-periodic perturbations with
periods of one or several revolutions are already eliminated. Furthermore, the
influence of the tracking station coordinates becomes smaller with longer
integration intervals, so that the stations may be even introduced as known. For
the low-degree zonal coefficients, the integration over one satellite revolution
yields the following variations for the orbital elements of main interest:
lsin'/ \J +
= -3;r(l-?2) -f-
l--sin 2 / sin/cosiyJ 3 +
-7. Y f ,1sin
;&/,+
4 > cos/cos
with = (l - e 2 ) and ae = semimajor axis of the earth ellipsoid.
/2 and higher even-zonal-coefficients cause secular perturbations in and (.
For / < 90, decreases in time (westward regression of the nodal line). The
change in corresponds to a rotation of the orbital ellipse in the orbital plane
(Fig. 5.5). This rotation produces long-periodic perturbations in the quantities e
and /, as they depend on 0). The even zonals thus can be determined primarily
from the perturbations in and , while the odd zonals are obtained from / and
e. If the perturbations are added to the orbital elements of the initial epoch, one
obtains the orbital elements at a specific epoch as a function of the zonal
harmonics. Substitution into (6.33) provides a system of observation equations
for the determination of the zonal harmonic coefficients. As mentioned earlier,
the coefficients depend particularly on the inclination but also on the semimajor
axis and the eccentricity.
The tesseral harmonics are responsible for small-amplitude (a few 100 m)
short-periodic perturbations, especially in the elements /, , . They can be
determined only from dense observations over short arcs. Weather-independent
microwave measurements are particularly suitable for this purpose.
Several tesseral terms of higher degree and order can be determined by
resonance effects, arising after days to weeks. These effects occur if the ratio of
the mean angular velocity of the satellite to the rotational velocity of the earth is
an integer number, which produces an enhancement of perturbation in a repeat
orbit.
Recent "satellite-only" models employ several million records of tracking data. Laser distance
and microwave range and range-rate measurements form the bulk of the data, but optical
directions still are included and assist in stabilizing the solutions. Spacebome positioning systems,
including GPS, increasingly contribute to the data set. The observations to more than 30 satellites
are generally used, with altitudes varying between 800 and 20 000 km and inclinations between
40 and 110.
While limitations in computational facilities previously forced zonal, tesseral
and resonant terms to be computed separately, the harmonic coefficients can
now be rigorously determined by a common least squares adjustment, together
with the coordinates of the tracking stations, ocean tidal parameters, polar
motion and earth rotation. Sometimes the horizontal station velocities and the
variations of the low-degree coefficients with time are also included in the
adjustment (BOUMAN 1997).
Weighting of data and modeling of systematic effects poses a special problem for these complex
adjustments. As the accuracy estimates generally are too optimistic, the calculated standard
deviations often are "scaled", e.g., by a factor of 5 (SCHWINTZER et al. 1997).
In most cases, "satellite-only" models are improved and stabilized by combining
them with satellite altimetry and surface gravity data. Mean values for lxl
blocks are introduced as "observations" and calculated either from point or
smaller block values, cf. [6.6.3]. The observation equations for satellite
altimetry are given by (5.49), after inserting (6.131) for the altimetric geoid
heights and taking a low-degree spherical harmonic model of sea surface
topography into account. The observation equation for the gravity anomalies is
given by (6.135). The normal equations of these data sets are added to the
normal equations of the satellite-only model (RAPP 1997). Tab. 6.1 gives the
values for the low degree-and-order harmonic coefficients, as derived from a
recent combination solution.
Tab. 6.1. Fully normalized harmonic coefficients (xlO 6 ), GRIM4-C4 model
(SCHWINTZER et al. 1997)
-484.1656
The relative accuracy of the harmonic coefficients derived from low-degree
spherical harmonic expansions decreases with increasing degree, from about
10"7 at I = 2 to about 1% at / = 10 and approaches 100% at 7 = 30 to 40
(satellite-only models) resp. at / = 65 to 70 (combination models). The longwave part of the marine geoid is accurate to about 0.8 m (satellite-only) and
0.25 m (combination). Orbit determination for dedicated satellites
(TOPEX/POSEIDON, ERS-1) is possible with an accuracy of a few cm with
these models.
The first gravity field information from space came from Sputnik I (1957), with the dynamic form
factor J2 (polar flattening); and from Vanguard I (1958), with the coefficient J} (unequal
flattening at the north and south pole). The Smithsonian Astrophysical Observatory (SAO)
Standard Earth I provided a model complete to degree and order 8 (LUNDQUIST and VEIS 1966).
Among the recent low-degree solutions are the NASA Goddard Space Flight Center (GSFC) earth
model GEM T3 (satellite-only and combination with GEOS-3, Seasat, and GEOSAT altimetry
and lxl surface gravity anomalies, complete to degree and order 50, resolved also for the
coordinates of about 400 tracking stations and ocean tides, polar motion and earth rotation
parameters, LERCH et al. 1994); the joint (GSFC, University of Texas, Ohio State University,
CNES France) gravity model JGM-2S (satellite-only, complete to degree and order 70, NEREM et
al. 1994); and the JGM-3 (70,70, including Doris and GPS-tracking, TAPLEY et al. 1996), which
was developed primarily for the TOPEX/POSEIDON mission. The GRIM4 models
(Geoforschungszentrum Potsdam and Groupe de Recherches de GeOde"sie Spatiale, Toulouse)
include a satellite-only (60,60) and a combination (72,72) solution and also solve for a large
number of additional parameters (SCHWINTZER et al. 1997), see also BOUMAN (1997), RAPP
6.6.3 High-degree Gravity Field Models
Parameters of high-degree gravity-field models are restricted to the harmonic
coefficients and do not include other unknowns: geopotential models. Degree
and order of the spherical harmonic expansion depend on the spatial resolution
of the gravity field data available and on their global distribution. Of particular
interest is the high-resolution quasigeoid resp. geoid needed for the
establishment of a global vertical datum, cf. [3.4.3], as a reference for satellite
altimetry in the determination of sea surface topography, cf. [3.4.2], and for the
transformation of GPS-heights to normal or orthometric heights, cf. [6.4.3].
The models combine low-degree gravity-field models with surface gravity
anomalies and altimetric geoid heights. Hence, they utilize:
The harmonic coefficients of a "satellite-only" model (sometimes a
combination model is used), with the full error covariance matrix,
Mean free-air anomalies from terrestrial gravimetry on land and sea,
Mean geoid heights from satellite altimetry. Some solutions use mean
gravity anomalies derived from the geoid heights, with a transformation
based either on an integral formula, cf. [6.7.1], or on least squares
collocation, cf. [6.8.2].
Mean free-air anomalies are calculated from point anomalies collected e.g., at the Bureau
Gravimetrique International or the U.S. National Imagery and Mapping Agency (NIMA). Least
squares prediction is employed for interpolation to gridded values, where Bouguer anomalies are
used as intermediate gravity field quantities, cf. [6.5.3], RAPP (1997). lxl mean values based
on measurements are available for a large part of the world (about 80%). On the continents,
30'x30' mean anomalies have also been derived for large regions, but great gaps exist on the
oceans and in antarctica. The accuracy of the mean anomalies amounts to about + 50 to 200
5~2 ( and RAPP 1990, KEN YON 1998).
Mean altimetric geoid heights are derived from individual measurements to the sea surface, after
least squares adjustment of the crossover discrepancies and reduction of the sea surface
topography by an oceanographic model. Data sets of high accuracy and resolution are available
from the GEOSAT and ERS-1 geodetic missions, which cover the oceans between 72 resp. 82
latitude with an equatorial track spacing of 4 resp. 8 km. Mean geoid heights, and by inversion
30*30' mean gravity anomalies, have been calculated from mean values derived for smaller
block sizes, cf. [6.7.1], where the accuracy of the altimetric gravity anomalies is about 20
urns"2 and more homogeneous than surface gravity.
By combining the mean anomalies from surface gravimetry and altimetry, only a few percent of
the earth's surface remain uncovered. These gaps are filled either by some low-degree model
values and/or isostatic anomalies calculated from topographic-isostatic models (PAVLIS and RAPP
High-degree geopotential models are calculated either by least squares
adjustment or by quadrature methods (RAPP 1997).
A least squares adjustment, in principle, would utilize all available data in order
to determine the full set of potential coefficients (130 321 coefficients at l,m =
360,360; 3.24 million coefficients at 1800,1800), together with the error
variance/covariance matrix. Currently, computational limitations still prevent a
rigorous adjustment of these quantities. Consequently, non-optimum solutions
have been developed, which, for instance, presuppose a complete data coverage
on a grid and homogeneous and uncorrelated errors. Special arrangements of
the normal equation matrix (block-diagonal technique) then allow an efficient
computation by iterative procedures (COLOMBO 1981, WENZEL 1985, PAVLIS et
al. 1996).
The quadrature approach employs the integration over the gravity anomalies
according to (6.138). As a global and homogeneous data set is required,
altimetric geoid heights have to be transformed into gravity anomalies and data
gaps have to be filled by model values. After the calculation of the harmonic
coefficients from the gravity anomalies, they are combined by adjustment with
the coefficients of a low-degree gravity model (COLOMBO 1981, RAPP and
PAVLIS 1990).
The long-wave structures of the free-air anomalies and the geoid, as derived
from a recent geopotential model, are shown in Figs. 6.27 and 6.28. The free air
anomalies vary rather irregularly about zero, but a correlation with extended
mountain chains (Cordilleras, Himalaya) can be recognized. The principal
features of the geoid include the maxima near New Guinea (+80 m), in the
North Atlantic, the southwestern Indian Ocean, and in the Andes, as well as the
minima at Sri Lanka (-105 m), in Antarctica, to the west of California, and near
Puerto Rico.
Early spherical harmonic expansions are due to Jeffreys (1941-1943), Zhongolovich (1952), and
Uotila (1962). Sparse surface gravity data coverage limited these expansions to the low-degree
harmonics. Kaula (1959) introduced a constraint from satellite orbit analysis in order to develop a
Fig. 6.27. EGM96 gravity anomalies, spherical harmonic expansion truncated at
degree and order 36, contour line interval 2 (LEMOINE et al. 1998)
Fig. 6.28. EGM96 geoid heights, spherical harmonic expansion truncated at
degree and order 36, contour line interval 5 m (LEMOINE et al. 1998)
8,8 geoid model (RAPP 1998). Among the recently developed geopotential models are the Ohio
State University (OSU) model OSU91 (combination of the GEM-T2 36,36 model with
and 30x30 gravity anomalies from surface gravimetry and altimetry, quadrature method, RAPP
et al. 1991) and the Geoforschungszentrum Potsdam (GFZ) model GFZ96 (combination of the
GRIM-4 60,60 and 72,72 models with terrestrial anomalies and ERS-1 geoid heights, least
squares iteration, GRUBER et al. 1997), both complete to degree and order 360.
Most complete sets of satellite tracking data (including altimetry in the form of tracking), gravity
anomalies, and recent satellite altimetry (GEOSAT, ERS-1 in the polar regions) have been
utilized in the NASA/GSFC and NIMA joint geopotential (360,360) model EGM96 (LEMOINE
et al. 1997, 1998). The model is composed of a 70,70 combination model (least squares
adjustment resolving for tidal parameters and a 20,20 model of sea surface topography, as well as
for station coordinates) and 30'x30' mean anomalies. Up to degree and order 359, a blockdiagonal solution was performed, and the degree 360 coefficients were calculated by quadrature.
Geoid accuracy estimates are between 0.5 m (oceans) and l m (land).
A 1800,1800 ultra-high spherical harmonic model GPM98 was developed by WENZEL (1999) by
combining EGM96 with 5'x5' mean gravity anomalies collected from surface gravity and
altimetry for about 75% of the earth's surface (the remaining areas being filled by larger block
size values), where integral formulas were applied for the calculation of the coefficients. In areas
well covered by high-resolution data, this solution provides a relative geoid accuracy of a few cm
and gravity anomalies accurate to several 10
Tailored geopotential models have been developed in order to better approximate the gravity field
in a certain region (WEBER and ZOMORRODIAN 1988, KEARSLEY and FORSBERG 1990). Here, a
global high-resolution spherical harmonic expansion is used as a starting model, and higher
degree coefficients are modified and extended so as to better reproduce high resolution gravity
anomalies in the region. Among these models are a 360,360 solution for Europe (BASit et al.
1989), a 500,500 solution for Canada (Li and SIDERIS 1994), and a 720,720 solution for China
(Lu et al. 2000).
Local gravity field approximation is especially useful for the determination of
geoid or quasigeoid heights with high accuracy, as needed for the reduction of
GPS heights, cf. [6.4.3]. Integral formulas allow a pointwise calculation of
gravity field quantities, and thus provide the possibility of an arbitrarily high
gravity field resolution which depends only on data coverage and quality
(SANSO and RUMMEL 1997).
Utilizing gravity anomalies as the primary data set, classical solutions aim at the
determination of geoid heights and geoidal deflections of the vertical [6.7.1].
Reduction to the geoid is avoided in the calculation of the corresponding
surface quantities, where the quasigeoid plays a special role [6.7.2]. Once the
gravity field is known on the geoid or on the physical surface of the earth, an
upward continuation provides gravity field quantities in space [6.7.3].
Astronomically determined deflections of the vertical furnish differences of
geoid or quasigeoid heights and may locally support or substitute gravimetric
solutions [6.7.4].
An alternative approach to the integral formulas is least squares collocation
which will be discussed under the more general aspect of "integrated geodesy"
in [6.8.2].
The series expansion (6.4) for the disturbing potential T can also be represented
by a surface integral. By inserting (6.137) into (6.3), this expansion reads
As known from potential theory, the surface spherical harmonics Ag, are
derived by inversion of (6.136), as a surface integral of the gravity anomalies
over the unit sphere :
where P, (cos^) are the Legendre polynomials. Inserting into (6.144) yields the
disturbing potential on thegeou/in spherical approximation:
where the integral kernel
W = -^/
(cos ^ )
;=2 ~ 1
<6147a>
can be expressed in closed form:
S(w}
+ l-5cosiif-6sin
-3cosi^ln
V) =
f ^ 2sin +2sJi n^ .(6.147b)
This integral formula has been derived by STOKES (1849); it is called Stokes'
formula. If an accuracy of the cm-order of magnitude is required, ellipsoidal
corrections have to be applied (S NKEL 1997).
By inserting (6.146) into Bruns' theorem (6.97b), we obtain the geoid height
6.7 Local Gravity Field Modeling
where ym is a mean gravity value over the earth. Stokes' formula can also be
derived as a solution of the integral equation (6.93), if applied to the geoid.
Stokes 'function S () acts as a weighting function on the gravity anomalies. It
depends on the spherical distance ^between the point of computation and the
surface element i/<rwith the gravity anomaly #. S (} decreases with until a
first zero at = 39, and then oscillates with large values until = 180 (Fig.
6.29). The neighborhood of the computation point requires particular attention,
as S () becomes infinite at = 0. The contribution of the innermost zone can
be estimated in planar approximation (e.g., with a radius st = 5 km), by
expanding Ag in a Taylor series and performing integration term by term. To a
first approximation, the effect of the inner zone on the geoid height depends on
the gravity anomaly in the computation point:
The components of the deflection of the vertical are obtained by differentiating
T in north-south and east-west direction (6.98). This can be realized by
expressing \ (6.146) in spherical coordinates of the computation point and
the source point (the corresponding formulas of spherical trigonometry are
taken from the spherical polar triangle, see Fig. 2.14), differentiation, and
resubstitution of . We obtain
-JJ
[cos a]
Ur,
[sin a J
Fig. 6.29. Original and modified
Stokes' function
Fig. 6.30. Vening-Meinesz function
where a is the azimuth of the great circle from the computation point to the
source point. Equation (6.150) was derived by VENING-MEINESZ (1928).
The Vening-Meinesz function
cos(^/2)
^7pr + 8siny-6cos(^/2)-3
2 sin ( 12)
l-sin (/2)
sin^
is infinite at = 0 and then decreases rapidly, attaining only small values after
= 50 to 60 (Fig. 6.30). The contribution of the innermost zone depends
primarily on the horizontal gradient of the gravity anomalies:
Stokes' and Vening-Meinesz' formulas allow a pointwise calculation of the
geoid height and the deflection of the vertical, by integrating the gravity
anomalies given on the surface of the geoid, cf. [6.5.3]. The properties of
Stokes' function require high resolution gravity data all over the earth, while the
effect of remote zones is small in the calculation of vertical deflections and can
be estimated by low-degree global gravity field models. The inner zone may
contribute some cm to the geoid height, this is well accounted for at gravity
station distances of 1 to 5 km, depending mainly on the roughness of
topography. The effect of the inner zone on the deflection of the vertical can
reach several arcsec, especially in the mountains. A dense gravity survey and/or
the calculation of the effect of topography is needed in order to achieve an
accuracy better than one arcsec.
In practice, the integrals (6.148) and (6.150) are solved by a summation of finite
surface elements. For this purpose, either a set of gridded point anomalies is
formed from the observed data, using e.g., least squares prediction or spline
interpolation, or mean values over surface blocks delineated by meridians and
parallels are calculated. The latter case also requires the integration of the
Stokes or Vening-Meinesz function over the block. After gridding, a very
efficient solution is obtained in the spectral domain using Fast Fourier
Transform (FFT) techniques. The convolution required in (6.148) and (6.150)
becomes a simple multiplication, and the results are easily retransformed to the
space domain by the inverse FFT (SCHWARZ et al. 1990, HAAGMANS et al.
1993, S NKEL 1997).
-150 -120
Fig. 6.31. Global distribution of terrestrial gravity data, from IAG (1997)
Point gravity anomalies (or mean gravity anomalies for some countries) are collected and kept at
a few global gravity data bases (U.S. National Imagery and Mapping Agency NIMA: about 30
mill, point values, Bureau Gravimetrique International: about 13 mill, data freely available, Fig.
6.31). The accuracy of point anomalies derived from land, sea and airborne gravimetry varies
between 5 and 50 5~2, cf. [5.4.4]. In order to avoid long-wave systematic errors in
gravity field modeling, the anomalies have to refer to the same gravity (IGSN71), horizontal
(ITRF, GRS80) and vertical reference systems, where the global vertical datum poses a special
problem, cf. [3.4.3], HECK (1990).
The data gaps still existing in large parts of the oceans can be filled by the
results of satellite altimetry, cf. [5.2.7]. For this purpose, the altimetrically
derived geoid heights are transformed to gravity anomalies either by least
squares collocation, cf. [6.8.2], or by a surface integral resulting from the
inversion of Stokes' formula (6.148), taking (6.101b) into account
(LELGEMANN 1976):
Here, and Np are the altimetric geoid heights at a particular surface element
do and at the computation point P, respectively. Due to the properties of the
integral kernel, the influence of the more remote zones on Ag decreases rapidly.
Hence, by combining the altimetric geoid heights with a global geopotential
model, the integration can be restricted to a radius of a few degrees.
A Itimetric gravity anomalies have been calculated for block sizes down to 5'x5' and 2'x2 r from
high resolution altimetry as provided by the geodetic missions of Geosat and ERS-1 (SANDWELL
and SMITH 1997, ANDERSEN and KNUDSEN 1998).
Purely gravimetric or gravimetric/altimetric calculations of geoid heights and
deflections of the vertical suffer from the data gaps at the polar caps, in some
continental areas, and at coastal zones. They are also hampered by long-wave
systematic data errors and by inhomogeneous spatial resolution and accuracy of
the gravity data. Free-air anomalies on land, in addition, are rather irregular due
to the gravitation of topography.
Global geopotential models, on the other hand, provide the long-wave part of
the gravity field, cf. [6.6.3], and dense gravity data with station distances down
to 1 to 3 km are available in many regions, usually together with high-resolution
digital elevation models, cf. [6.5.2]. This leads to solutions that combine the
global model with the gravity data in a limited region, where data smoothing
techniques are used by considering the terrain effect.
Combination solutions apply the remove-restore technique (FORSBERG and
TSCHERNING 1981, DENKER et al. 1986), which includes the following steps:
Reduction of the gravity anomalies Ag by the anomaly part of the global
model Ag M ,
Smoothing of the anomalies by some kind of terrain reduction Agr (see
below),
Gridding of the residual gravity anomalies
Ag^Ag-Ag^-Agr,
Application of Stokes' formula (6.148) on the residual gravity anomalies,
resulting in residual geoid heights 7Vres,
Restoration of the effects of the global model and the terrain to the residual
geoid heights:
N = NKS+NM+NT.
The remove-restore technique can also be applied on the deflections of the vertical or any other
gravity field quantity. It has been used successfully also with least squares collocation, but for
large-scale geoid modeling, integral formulas together with FFT is the only practicable technique
to date.
Since the residual gravity anomalies no longer contain the long and short-wave parts of the
gravity field, they are considerably smaller and smoother than the original data, and they possess
(approximately) homogeneity and isotropy properties, cf. [6.1.3]. With global models complete to
degree and order 360, the integration area can be restricted to the region with dense data coverage
and a narrow edge zone around it.
As only a small radius of integration is required for the remove-restore
technique, a planar approximation of Stokes' formula is permitted. Stokes'
function then reduces to
sin(y/-/2)
with / = MX -,,} + (y ~ yp ) and xp,yp = plane coordinates of the
computation point. The spherical surface element is replaced by the planar
dS = R2da.
Inserting (6.156a) and (6.156b) into (6.148) yields Stokes ' formula in planar
(6 156c)
which is very convenient to evaluate by FFT techniques; the integration can be
restricted to spherical distances of a few degrees.
Different strategies can be pursued for the application of terrain reductions
(FORSBERG and TSCHERNING 1997).
When Helmert 's condensation method is employed, terrain-corrected Bouguer
anomalies are used for gridding, cf. [6.5.3]. After restoring the Bouguer plate
term ("condensation"), Faye anomalies reduced by the effect of the global
model serve for the calculation of residual geoid heights.
TERRAIN CORRECTION
RESIDUAL TERRAIN
-^MEAN HEIGHT*"'
Fig. 6.32. Bouguer plate with terrain correction and residual terrain correction
With the residual terrain correction, only the high-frequency part of
topography is taken into account in the remove-restore process, as the longwave part has been subtracted already with the global model (FORSBERG and
TSCHERNING 1981). It is calculated from a residual terrain model (RTM) which
refers to a reference topography, as provided by a global topographic model
(spherical harmonic expansion of the same degree and order as the geopotential
model, RUMMEL et al. 1988) or the moving average over mean heights of
15'xl5' or 30'x30' blocks. This procedure produces a balanced set of positive
and negative anomalies (Fig. 6.32). The prism method is used generally for the
calculation of the residual terrain correction, cf. [6.5.2].
If applied to Stokes' formula, the remove-restore technique implies that the
complete spectrum of the geoid heights is computed from the gravity anomalies
in the integration area, augmented by the values of the global model outside this
region. In the case of long-wave discrepancies between the terrestrial gravity
data and the global model, this leads to an distortion of the long wavelengths of
the geoid. This problem is avoided by least squares spectral combination
(SJBERG 1979, WENZEL 1982). Here, the long-wave spectral components of
the global model and of the gravity anomalies are combined within the area of
integration, using a least squares adjustment with spectral weights
w,=-
Fig. 6.33. Gravimetric Geoid99 of the U.S.A., contour line interval 2 m (SMITH
and ROMAN 2001), courtesy D.A. Smith, National Geodetic Survey/NOAA
The error degree variances ] ( ) of the potential coefficients of the global
model are estimated in analogy to (6.142), and the error degree variances of the
terrestrial anomalies /^^j are derived from an error covariance function
(WEBER 1984), in analogy to (6.25). Here, Stokes' function is extended by the
spectral weights to form an optimum integral kernel
w cos
W = ZT-r
^( ^)
1=2 I I
(6-157b)
This function is no longer infinite at = 0, and it converges to zero more
rapidly than the original Stokes' function, see Fig. 6.33.
Early gravimetric geoid calculations with Stokes' formula are due to Hirvonen (1934) and Tanni
(1948). Based on isostatic anomalies, the "Columbus Geoid" was calculated at the Ohio State
University (HHSKANEN 1957). A combination of a low-degree satellite model (Goddard GEM-6,
degree and order 16) with lxl free-air gravity anomalies (integration radius 10) was
presented by MARSH and VINCENT (1974). Among the recent regional solutions is the gravimetric
geoid for the U.S.A.. It is based on a 2'x2' grid of Faye anomalies, constructed from Bouguer
anomalies, and employs the EGM96 model as a global reference, cf. [6.6.3], Fig. 6.33. This
geocentric geoid, accurate to about 0.2 m, also has been fitted to the North American vertical
datum, cf. [7.2], SMITH and MILBERT (1999). Least squares spectral combination has been applied
in the calculation of a European quasigeoid, cf. [6.7.2].
Gravimetric Height Anomalies and Surface Deflections of the
The geodetic boundary-value problem for the physical surface of the earth has
been formulated by M.S Molodenski, through the integral equation (6.92),
MOLODENSKI et al. (1962). By introducing the telluroid as an approximation
to the earth's surface, an integral equation for the disturbing potential was
obtained (6.93). In contrast to the derivative dT/dn (n = normal to the level
surface), the derivative dT/dn^. ( = normal to the telluroid) now not only
depends on the gravity anomaly but also on the deflection of the vertical and the
inclination of the terrain. Inserting 3/3 into (6.93) yields (HEISKANEN and
MORITZ 1967, p.299)
with x, y = angle of terrain inclination in NS and EW-direction,
maximum inclination, and , = vertical deflection components.
= angle of
A simpler integral equation can be derived by expressing T as the potential of
an infinitely thin surface layer condensed on the telluroid. With the surface
density (6.113), the law of gravitation (3.10) yields
T = G Z-.
As the potential of a surface layer is harmonic outside the surface, Laplace's
equation is fulfilled, and we may introduce (6.159) and its normal derivative
into (6.101). This strategy results in an integral equation which only depends on
Ag and on the terrain inclination. It can be solved by successive approximation,
leading to a series expansion for T(MORITZ 1971). With Bruns' formula (6.96),
the corresponding development for the height anomaly is obtained. In
spherical approximation (= unit sphere, R = earth's radius), and limiting the
series to its first two terms, the height anomaly is given by
where S(^) is Stokes' function (6.147). The main term corresponds to Stokes'
formula as applied to the surface gravity anomalies (6.14). The first correction
term, in close approximation, is
It depends on the terrain inclination (HN = normal height) and on gravity
anomalies. Assuming a linear correlation of the gravity anomalies with height,
Gj can be approximated by the gravimetric terrain correction (6.112), MORITZ
(1968b), SlDERIS (1990). Hence, Faye anomalies, cf. [6.5.3], are well suited for
the computation of height anomalies.
Since the integral kernel in (6.160b) decreases rapidly with increasing spherical distance , the
integration can be restricted to a limited area. Higher-order terms in (6.160) contain the tangent of
the terrain inclination and can be neglected generally. In order to ensure convergence of
Molodenski's series expansion, extreme inclinations and singularities (steep slopes) need to be
removed by some smoothing procedure. The Molodenski correction terms reach the dm-order of
magnitude in the high mountains and remain at the cm-order in the lowlands. If the removerestore technique, cf. [6.7.1], is applied, the corrections reduce by about one-order of magnitude
and the series convergence is significantly improved.
Molodenski's problem has been thoroughly investigated by MORITZ (1971) and others, and the
existence and uniqueness of the solution was proved by H RMANDER (1976) and SANS (1988).
A very efficient method for calculating the height anomaly is provided by the
"gradient solution" (MORITZ 1980). Here, the surface gravity anomalies are
first reduced to sea level (geoid or quasigeoid); then Stokes' integral is applied,
leading to height anomalies on sea level. An upward continuation finally gives
the surface height anomaly:
The radial derivative of Ag is given by
and again can be approximated by the terrain correction (see above). The
vertical gradient of ^results from (6.96) and (6.101):
(6.161c)
where # is the surface gravity anomaly. If the surface anomalies are reduced to
the level of the computation point P, H in (6. 161 a) has to be substituted by
H-HP. The last term in (6.161a) then vanishes, as outside the integral
means Hp
These solutions are particularly well suited for FFT techniques (FORSBERG and
TSCHERNING 1997).
The surface deflection of the vertical (see Molodenski's definition in [6.1.2]) is
derived from (6.160) by differentiation according to (6.98):
IstaaJ
The principal term is Vening-Meinesz' formula (6.150), and the Molodenski
correction terms again take the effect of the terrain into account, reaching a few
arcsec.
Molodenski's problem is characterized by the fact that no assumptions on the density distribution
within the earth are necessary, in contrast to the geoid determination using Stokes' formula. By
the relation (6.9) between the geoid and the quasigeoid, a simple method is available to derive
geoid heights from height anomalies:
Data reductions onto the geoid and calculations of indirect effects are avoided by this strategy,
and density hypotheses enter only through the Bouguer anomaly Ags . On a large scale, the
Bouguer anomalies are negative on the continents, cf. [8.2.4], hence the quasigeoid generally is
above the geoid. The differences between the geoid and the quasigeoid are of the cm to dm-order
of magnitude in flat and hilly regions but may assume one meter and more in the mountains. As
the quasigeoid is highly correlated with the height, the quasigeoid mirrors the topography.
Fig. 6.34. European gravimetric quasigeoid EGG97 (DENKER and TORGE
1998): contour line interval 2 m
One example for a large-region quasigeoid determination is the European Gravimetric Quasigeoid
EGG97 (DENKER and TORGE 1998). It is based on a high-resolution data set of point and mean
gravity anomalies derived from surface gravimetry and satellite altimetry in the marine areas. The
remove-restore technique was applied using the EGM96 geopotential model and a 15'xl5'
moving average reference topography. Gridded l'xl.5' residual gravity anomalies were
transformed to height anomalies by least squares spectral combination, cf. [6.7.1]. The main part
of EGG97 (Fig. 6.34) stems from the global model, but the contributions from terrestrial gravity
data and topography still amount to 0.4 m (maximum 4 m) and 0.03 m (maximum 0.8 m),
respectively. The error estimates ( a few cm/100 km to 0.1 m/1000 km) were confirmed by
comparisons with GPS/leveling control points, e.g., on the European NS-GPS-traverse between
Munich and Troms (TORGE et al. 1989).
The gravity field outside the earth is modeled by upward continuation from the
boundary surface, either globally or locally.
Global modeling employs the spherical harmonic expansion of the disturbing
potential (6.4) and the corresponding expansions for the height anomaly (6.131)
and the gravity anomaly (6.135). These expansions converge outside a sphere
enclosing the earth, cf. [3.3.2]. They are especially useful for the orbit
determination of satellites, but the present limited spatial resolution makes them
less suited for low altitudes, e.g., for airborne applications.
For local modeling, we start with the free-air anomaly Ag, where according to
(6. 134) the function
is harmonic in space. The calculation of gravity anomalies in space from
boundary values thus corresponds to the first boundary-value problem
(Dirichlet problem) of potential theory. The solution is given by Poisson 's
integral. In spherical approximation we have
where Agp refers to the point P in space and Ag are the surface anomalies on
the spherical boundary surface. The integral kernel decreases rapidly with ,
which allows restriction of the integration area on a limited zone around the
computation point.
By inserting (6.164) into (6.101b) we obtain
= -- --- ,
which now represents a differential equation in space. The solution leads to a
spatial extension of Stokes' formula for the disturbing potential, which was
derived by Pizetti:
tf*
3RI
cos^ f5 + 31n
,1N
, (6.166b)
is the extended Stokes' function.
Bruns' theorem finally gives the separation between the geopotential surface W
= W p and the spheropotential surface U = UQ = Wp (height anomaly in space):
Corresponding relations can be derived for the gravity disturbance and the
deflection of the vertical, using (6.98) and (6.99).
Geoid and quasigeoid height differences can be obtained from deflections of the
vertical, determined according to (6.17) from astronomic and geodetic latitudes
and longitudes.
In astronomic leveling, the deflections of the vertical are integrated along the
path, either on the geoid or on the earth's surface (Fig. 6.35). On the geoid, we
dN = -0 ds,
where 0 is the vertical deflection component in the azimuth of the path (6.18),
reduced to the geoid according to Pizetti's definition, cf. [6.1.2]. Integration
between /J and P2 yields the geoid height difference
The negative sign follows from the sign conventions for the geoid height (6.8)
and the deflection of the vertical (6.17).
II GEOID
I ELLIPSOID
GEOID W=Wa
II ELLIPSOID
Fig. 6.35. Astronomic leveling
The geoidal deflection of the vertical is obtained by reducing the observed
astronomic latitude and longitude onto the geoid:
where 0 and 0 are the astronomic coordinates on the geoid (Fig. 6.36). The
reductions follow from the integration of the plumb line curvature Kf, Ky,
between the earth's surface and the geoid (3.74):
H= orthometric height. Inserting (3.67) and (3.70) yields
(6.i70b)
R = mean radius of the earth. With (6.17), the NS and EW-components of the
vertical deflection are given by
\,
and the azimuthal component reads
= 0 cos a + 0 sin a .
(6. 17 Ib)
In order to evaluate (6.170), the gravity and the horizontal gravity gradient along the plumb line
are required. Digital terrain models allow estimation of these quantities with an accuracy between
0.1" and 1", but errors may be larger in the high mountains. The angle of plumb line curvature
itself attains values of a few 0.1" in the lowlands and may reach 10" and more at high mountain
Instead of integrating the deflections of the vertical on the geoid, the surface
vertical deflections (definitions from Helmert or from Molodenski) may be
used. Helmen deflections of the vertical are given by
= 0-,
where the components of are obtained from (6.170b). Inserting into (6.168b)
As seen from Fig. 6.36, the second term on the right side equals the orthometric
height reduction known from geometric leveling (6.81b); the angle of plumb
line curvature is the horizontal derivative of E. We thus have
1.2 = -J ds - E12.
For height anomalies, the difference follows from the differential (MORITZ
II EQUATOR
Fig. 6.36. Plumb line curvature in the meridian plane
The first term describes the effect of Molodenski's vertical deflection. The
second term enters because the physical surface of the earth is not a level
surface. Using (6.161c) and integration along the path yields Molodenski's
astronomic leveling of height anomalies
The relation between geoid and quasigeoid height differences follows from
(6.9) and (6.81):
or when taking (6.173b) into account
where E"2 is the normal height reduction.
As the small corrections in (6.173), (6.175), and (6.176) can be derived easily from surface
gravity and a digital terrain model, the integration of surface vertical deflections is of advantage
compared to the tedious reductions onto the geoid required in (6.168).
The line integrals of astronomic leveling presuppose that the deflections of the
vertical are given continuously along the path. In reality, vertical deflections are
available only at larger distances (several 10 km or more), which is due to the
time-consuming astronomic observations. This poses the problem of
interpolation between the vertical deflection points (in the following we do not
distinguish between the vertical deflections on the geoid and on the earth's
surface). Interpolation can be carried out by purely mathematical methods or
supported by additional information on the gravity field behavior.
In the simplest case of mathematical interpolation, a linear change of the
deflections of the vertical
between the stations Pt and P2 is assumed. The
integration then yields
where s is the distance between Pl and P2 .
The linear interpolation model is adequate in flat areas and where the distances between the
vertical deflection points are not too large (some km to some 10 km). An area with sufficient
control point coverage can be evaluated either by forming triangles and adjusting the geoid height
misclosures or by a surface polynomial approximation to the geoid (VANICEK and MERRY 1973).
The polynomial coefficients are determined from the corresponding series expansions of the
vertical deflection components, which result from (6.98) and Brans' formula
1 dN
= --3-, n = ~-- TT.
cos$?
where the condition of integrability of a potential field, cf. [3.1.5] must be heeded:
Least squares prediction, cf. [6.1.3], offers another efficient interpolation method ( 1969),
while least squares collocation, cf. [6.8.2], even allows direct estimation of geoid heights from the
vertical deflections, thus providing an alternative to the integral formulas.
Additional gravity field information between the vertical deflection points can
be supplied by terrain models, gravity anomalies, and zenith angles.
A digital terrain model (possibly also taking density variations into account)
can be used to calculate the effect of topography on the deflections of the
vertical. For more extended calculation areas, the effect of isostatically
compensating masses should also be considered, cf. [6.5.2]. By subtracting
these effects from the observations, the vertical deflection field is smoothed,
and mathematical interpolation methods may be applied. The interpolated
residual deflections of the vertical are augmented by the effects of topography
and isostasy, leading to a densified network of vertical deflection points. This
remove-restore method has proved to be efficient especially in mountainous
areas; an interpolation accuracy of 1 to 2 arcsec can be achieved.
If a dense field of gravity stations around the vertical deflection points is
available, it can be utilized for a gravimetric interpolation. Here, gravimetric
deflections of the vertical are computed according to (6.150) by integrating the
gravity anomalies over a limited area (e.g., three times the distance between the
vertical deflection points). This gravimetric part is then removed from the
observations, and a systematic difference between the astrogeodetic and the
gravimetric vertical deflections has to be taken into account (different reference
systems, effect of the zones neglected in the calculation of the gravimetric
vertical deflections). The residual deflections of the vertical thus obtained are
smooth and easy to interpolate. This method has been extended to
astrogravimetric leveling, with a gravimetric "correction" to quasigeoid
differences obtained from linear interpolation of surface deflections of the
vertical (MOLODENSKI et al. 1962). Gravimetric interpolation allows calculation
of geoid or quasigeoid height differences with cm to dm accuracy, even at
larger spacing of the vertical deflection points.
Reciprocal zenith angles deliver differences of the vertical deflection
components in the line of sight. According to (6.85), the observed zenith
angles and the ellipsoidal quantities ^are related by (Fig. 6.20)
= z , + , , 2=2+2.
Inserting the above relations into (6.87), and taking sign conventions into
account, yields
2-=+2 --- ,
where S = spherical distance between and P2 . Starting at a vertical deflection
point, these differences can be used for the interpolation of deflections of the
vertical. The method has found limited application in mountainous areas, where
an interpolation accuracy of about one arcsec has been achieved (GLEINSVIK
The advantage of the astrogeodetic method of geoid determination consists in
its independence from data outside the area of calculation, in contrast to the
gravimetric method where a global coverage with gravity data is needed. In
addition, the demands on the accuracy of the point heights are less stringent as
with the formation of gravity anomalies. On the other hand, the establishment of
a vertical deflection point requires substantially more time than a gravity
measurement. A station spacing of 10 to 20 km is available only in few regions,
and even distances of up to 30 to 50 km are limited to well surveyed countries.
Large parts of the continents are covered only sparely, with concentration on
profiles along first-order triangulation chains, cf. [7.1.1]. Thus, the accuracy of
astronomic leveling depends entirely on the quality of interpolation. In densely
surveyed areas, an accuracy of a few cm to 0.1 m over some 100 km can be
achieved (B UMKER 1984, ELMIGER and GURTNER 1983). The accuracy can be
increased by topographic-isostatic reductions, including geological information
on rock densities. By fitting the results to GPS/leveling control points, the geoid
heights are referred to the national vertical datum and systematic effects are
reduced, which increases the accuracy to 0.01 to 0.02 m over several 100 km
(GROTE 1996).
The superior efficiency of gravimetric methods has greatly reduced the
application of astronomic leveling. It is now restricted to areas or profiles which
are not well covered by gravity data or where the coverage is not representative,
as in the mountains where gravity stations are typically concentrated along the
roads.
Astronomic leveling was introduced by HELMERT (1884) and first applied in the Harz mountains,
Germany. From the 1950's to the 1970's, astrogeodetic geoid determinations were carried out in a
number of countries, using astronomic observations on the first-order triangulation points (
1969). Deflections of the vertical, and the resulting geoid, referred to the national geodetic datum
and served for the reduction of horizontal angles and chord distances onto the national reference
ellipsoid, cf. [6.3.2], [7.1.2]. Large-scale solutions included the "Bomford" geoid for Europe
(LEVALLOIS and MONGE 1975) and the continent-wide geoid determination by FISCHER et al.
(1968), with an average accuracy of a few meters. High precision astrogeodetic geoid models
have been developped in Switzerland and Austria, based on a densified net of vertical deflection
points and high-resolution digital terrain models and employing remove-restore techniques
(ERKER 1987, MARTI 1997).
Combined Methods for Positioning and Gravity Field
Combined evaluation methods permit simultaneous determination of positions
and the gravity field. These quantities are eventually combined with systematic
model parameters within one mathematical model. By introducing geometrical
and gravity field observations, the information content of the data is more
completely exhausted. On the other hand, the large amount of data and
6.8 Combined Methods for Positioning and Gravity Field Determination
unknown parameters raises serious problems with respect to proper weighting
of the observations, modeling of systematic effects, and data processing. Hence,
the application of the combination methods is still restricted to a limited number
of parameters and observations.
A functional combination model is based on the observation equations for all
relevant data and uses least squares adjustment for the determination of earth
models and optimum earth parameters [6.8.1]. In least squares collocation, a
stochastic model for gravity field estimation is added, which leads to a very
general method of combination [6.8.2].
"Earth models" are determined by rigorous least squares adjustment from
satellite tracking data, either alone or in combination with satellite altimetry and
surface gravity. They contain the coordinates of the tracking stations and station
velocities, the parameters of a low-degree gravity field model, as well as ocean
tidal and earth rotation parameters, cf. [6.6.2].
Using the latest supercomputers, present earth models include about 5000 geopotential
coefficients and about 10000 parameters in total. The coordinates of the tracking stations are
determined with cm-accuracy, and station velocities are accurate to a few mm/a. The long-wave
part of the geoid (wavelength > 550 km) is accurate to about 0.9 m on the continents and 0.3
m on the oceans.
Combination solutions also provide actual values of a model earth, as
represented by a level ellipsoid with optimum approximation to the geoid and
the external gravity field, cf. [4.2.1]. These actual values must not be confused
with the parameters of a geodetic reference system, recommended as a standard
for a longer time interval, cf. [4.3]. The best estimates (1999) for the defining
parameters of a level ellipsoid are as follows (GROTEN 2000):
The geocentric gravitational constant
GM = (398 600.4420.00l)xl09 mY2
including the mass of the atmosphere. This value is derived from orbit
analyses of satellites and space probes. A variation of GM with time has not
been found,
The equatorial radius of the earth
tf = 6378136.60.1m.
The radius is determined by an optimal fit between ellipsoidal heights and
orthometric heights, as obtained for satellite tracking stations on the
continents, and from satellite altimetry on the oceans, applying the
minimum condition
on the geoid heights. According to [6.5.4], the geoid potential WQ could
also be introduced as a defining parameter if GM and a are known:
WQ = 62 636 856.0 0.5 m
a and W0 are given in the zero-tide system, which contains the indirect tidal
distortion, cf. [3.4.1]. The equatorial radius and the geoid potential can be
assumed constant with time.
The second-degree zonal harmonic coefficient (dynamical form factor)
J2 =(1082.6267 0.000l)x 10^
in the tide-free system and
in the zero-tide system. J^ is obtained from global gravity models, cf.
[6.6]. A long-term temporal change of J2 was found from the analysis of
satellite orbits over long time intervals:
The mean angular velocity of the earth 's rotation
<y = 7.2921 15 xlOis provided by the International Earth Rotation Service, cf. [2.2.2]. It
experiences a long-term variation
= du)/ dt = (-4.5 O.l)xl0~22 rad s"2 ,
which may be decomposed into a component due to tidal dissipation
(-o.lxlCT22 rads"2) and a non-tidal part (+1.6xl(T22 rads' 2 ) probably
reflecting the postglacial rebound of the earth's crust and mantle.
Among the quantities derived from the defining parameters we have
The reciprocal flattening
l/f = 298.25642 0.000 01
The normal gravity at the equator
= 9.780 327 0.000 001 ms'2 .
The minimum conditions for the deflections of the vertical
and the gravity anomalies
in principle, could also serve to derive the parameters of a best approximating
ellipsoid (HEISKANEN and MORITZ 1967, p.216). While (6.182) has been used
for the determination of local reference ellipsoids in classical national geodetic
surveys, cf. [7.1.2], (6.183) has not been exploited due to lacking global gravity
Least squares collocation combines the calculation of station coordinates and
other deterministic unknowns (harmonic coefficients, earth ellipsoid and earth
orientation parameters, calibration and drift coefficients, etc.) with the
estimation of gravity field quantities of unsurveyed points, utilizing many types
of observables (KRARUP 1969, MORITZ 1973). The general form of the
observation equation reads
where 1 is the linearized vector of observations. It is composed of the
deterministic part Ax (x = parameter vector, A = design matrix containing the
differential relations between observations and parameters) and two random
parts. The signal vector s contains the residual gravity field quantities at any
point, either observed or to be predicted. It may include, in contrast to least
squares prediction, any kind of gravity field quantities such as residual
harmonic coefficients, geoid or quasigeoid heights, gravity anomalies,
deflections of the vertical, etc. The noise vector n represents the observational
errors. Each of the random quantities is supposed to have a mean value of zero.
The statistical behavior is described by the covariance matrix C of the signals
and the covariance matrix D of the noise, where mutual independence of signal
and noise is assumed, cf. [6.1.3].
After linearization with a geodetic earth model, all geodetic observations can be expressed in the
form (6.184). The observation equations for leveled heights, gravity, astronomic latitudes,
longitudes, and azimuths, satellite tracking data, VLBI observations, zenith angles, and spatial
distances are found in the corresponding sections of [6]. While all observations depend on
position (described by the geodetic coordinates in connection with the parameters of the
ellipsoid), a dependence on the gravity field does not exist with VLBI observations and spatial
distances. A complete set of observation equations is given by GRAFAREND (1978).
Least squares collocation is an overdetermined problem with respect to the
parameters (number of observations exceeds the number of parameters) and an
underdetermined problem with respect to the gravity field signal (more signals
have to be predicted than have been observed). It is solved by applying a least
squares minimum condition on the weighted quadratic sum of the signal part
and the noise, thus combining least squares adjustment with least squares
prediction (MORITZ 1980). The solution for the parameter vector is given by
x = A - ' A ATcrli ,
with C = C + D . The component of the signal vector predicted in an unsurveyed
point results in
where the covariance vectors and matrices are explained in [6.1.3] but now may
include heterogeneous signals.
The simultaneous determination of station coordinates and gravity field
quantities has been designated as "integrated" or "operational" geodesy (EEG
and KRARUP 1973, HEIN 1986). The main problem in it's practical application
is the solution of a system of equations with a dimension equal to the number of
observations for the inversion of the covariance matrix C. By separation into
subsystems with low correlation (e.g., horizontal positioning on the one hand,
and height respectively gravity field determination on the other), the dimension
of the system of equations can be reduced significantly. This strategy leads to
least squares adjustment for the deterministic part, and offers an efficient and
flexible solution for gravity field determination, which is used extensively in
various applications.
If applied to gravity field estimation, the elements of the signal covariance
matrix C are required, describing the correlation between heterogeneous
residual gravity field quantities. Since all these quantities belong to the same
gravity field, the covariances have to be derived from a basic covariance
function through covariance propagation. The covariance function of the
disturbing potential T is selected for this purpose, as all residual gravity field
quantities are related to in a simple manner. This covariance function is
defined in analogy to the covariance function of the gravity anomalies (6.21). It
is considered to be the mean value of the products of in the points and Pf for
a constant spherical distance and assumes homogeneity and isotropy. The
function is given by:
() = cov (, ',) = { '}.
Covariance propagation is well known from the theory of errors and is applied
here to gravity field signals. As demonstrated by (6.96) to (6.101), the residual
gravity field quantities (either observed or to be predicted) can be expressed as a
linear functional of T. For an observation /, we thus have
/,=Z,f'r(P'),
where Lt is the functional to be applied to in order to transform into the
observed quantity. The covariance between and /, is obtained by applying Z,
on the covariance function (}=(,"), which can be expressed as a
function of the coordinates of and P' in space:
For different types of observation at and P', the covariance results from a
subsequent application of the fimctionals L valid for the transformation of T into
the respective observation:
The same rules have to be followed if heterogeneous signals shall be estimated.
A statistical description of the earth's gravity field is available by anomaly
degree variance models, cf. [6.1.3]. The relation of these models to the basic
covariance function can be derived by applying the mean value operator (6.187)
on the spherical harmonic expansion of the disturbing potential (6.4). This
where the potential degree variances are defined in analogy to (6.26):
Equation (6.142) provides the relation between the degree variances of the
disturbing potential and the gravity anomalies:
where we have taken Bruns' formula = / into account. Inserting (6.192)
into (6. 191) yields
/=2(/-l)
which enables the calculation of the basic covariance function from an anomaly
degree variance model. For local applications, the covariance function has to be
fit to the gravity field behavior in the area of calculation, cf. [6.1.3].
The advantage of least squares collocation is the flexibility in estimating any
kind of gravity field quantity from different types of gravity field observations,
at surveyed and unsurveyed points. The data can be processed as discrete values
and need not be continuous, as required for the application of integral formulas,
cf. [6.7]. Neither gridding of the data nor reduction to some reference level is
required. For homogeneous and continuously distributed data, least squares
collocation transforms into the integral formulas (MORITZ 1976). On the other
hand, the amount of data that can be handled is still limited by computing
facilities. Applications have therefore been restricted to limited areas and data
sets. By introducing some restrictions on the data, computing time can be
reduced and larger data sets evaluated: fast collocation (BOTTONI and
BARZAGHI 1993, GROTE 1996).
Least squares collocation has been applied especially for local and regional geoid determination
(RAPP 1978, ARABELOS 1980, TSCHERNING and FORSBERG 1986, DENKER 1988), but also for the
estimation of gravity anomalies from altimetric geoid heights and for downward continuation
problems. Remove-restore techniques are used generally, cf. [6.7.1], which reduces the data
collection area.
7 Geodetic and Gravimetric Networks
Geodetic and gravimetric networks consist of monumented control points that
provide the reference frames for positioning and gravity-field determination at
all scales. Global networks allow realization of the reference systems defined by
international conventions. Regional networks form the fundamental basis for
national or supranational (continental) geodetic and gravimetric surveys, which
are the basis of geo-information systems and map series. Local networks are
typically established for engineering and exploration projects and for
geodynamic investigations.
In the sequel we concentrate on regional networks, which are increasingly
integrated into global reference-frames established for positioning, navigation,
cf. [2.5.3], and gravity, cf. [5.4.3]. Horizontal and vertical control networks
have been established separately, following the classical treatment of
positioning and heighting; these networks still are the basis of national geodetic
systems [7.1], [7.2]. Geodetic space methods allow establishment of threedimensional networks within the geocentric reference system and are now
superseding the classical control networks [7.3]. Gravity networks serve the
needs of geodesy and geophysics, with the reference provided either by a global
network or by absolute gravimetry [7.4].
If reobserved after a certain time span, geodetic and gravimetric networks can
be utilized for the investigation of long-term changes of position and gravity
with time. Specially designed networks have been set up on global, regional,
and local scale in order to monitor geodynamic processes of different origins,
The establishment of geodetic networks is treated in textbooks on geodesy and
geodetic surveying (BOMFORD 1980, KAHMEN and FAIG 1988, HECK 1995,
LEICK 1995); for gravity networks see TORGE (1989, 1998).
7.1 Horizontal Control Networks
National horizontal control networks were established from the 18th century
until the 1960's, with the networks' design, observation, and computation
changing with the available techniques [7.1.1]. Computations were carried out
on a reference ellipsoid fitted to the survey area. Since the 1960's, spatial
geodetic methods have allowed orientation of the classical networks with
respect to the global reference system [7.1.2].
Horizontal control networks are realized by trigonometric (triangulation) points,
which in principle should be distributed evenly over the country. One
distinguishes between different orders of trigonometric points, from first order
or primary (station separation 30 to 60 km) to second order (about 10 km) to
fourth or even fifth order (down to one to two km) stations. The maximum
distance between first-order points was determined by the terrestrial
measurement methods, which required intervisibility between the network
stations. Consequently, first and some second-order stations were established on
the top of hills and mountains; observation towers (wooden or steel
constructions with heights of 30 m and more) were erected especially in flat
areas. The stations have been permanently marked by underground and surface
monuments (stone plates, stone or concrete pillars, bolts in hard bedrock).
Eccentric marks have been set up in order to aid in the recovery and verification
of the center mark.
Classical first and second-order horizontal control networks have been observed
by the methods of triangulation, trilateration, and traversing.
^<f
EXTENSION NET
Fig. 7.1. Triangulation with baseline
extension net and Laplace azimuth
(principle)
TRILATERATION
Fig. 7.2. Trilateration with Laplace
azimuth (principle)
In triangulation, all angles of the triangles formed by the trigonometric points
are observed with a theodolite (Fig.7.1). The instrument is set up on the
observation pillar or tower; at large distances the targets are made visible by
light signals. Either directions (successive observation of all target points) or
angles (separate measurement of the two directions comprising one angle) are
observed in several sets (i.e., in both positions of the telescope), distributed over
the horizontal circle of the theodolite. The scale of a triangulation network is
obtained from the length of at least one triangulation side, either derived from a
short base line through a base line extension net or measured directly by a
distance meter, cf. [5.5.2]. Astronomic observations provide the orientation of
the network, cf. [7.1.2], whereas an astronomic azimuth is needed for the
horizontal orientation according to Laplace's equation (6.58): Laplace station.
In extended networks, base lines and Laplace stations often were established at
distances of a few 100 km in order to control the error propagation through the
network with respect to scale and orientation (effects of lateral refraction).
Trilateration employs electromagnetic distance meters in order to measure the
lengths of all triangle sides of a network, including diagonals (Fig.7.2). At least
one Laplace azimuth is needed for the orientation of the net. Electromagnetic
distance measurements put less demands on the stability of observation towers
as compared to angular measurements, and the use of laser light and
microwaves makes the method more independent from weather conditions.
Traverses combine distance and angular measurements, where the traverse
stations are arranged along a profile (Fig.7.3). Again, a Laplace azimuth is
required in order to orientate the traverse. Traversing represents a very effective
and flexible method for establishing horizontal control networks, with no more
need to establish stations on hilltops. It has been employed primarily for the
densification of higher order networks, with the coordinates of the existing
control points providing orientation.
-TERRESTRIAL-
- ORIENTATION-
Fig. 7.3. Traverse connecting two control points (principle): a) without
additional orientation, b) with orientation by Laplace azimuths, c) with
orientation by directions to control points
7. l Horizontal Control Networks
Horizontal control networks can also be formed by combining the methods of
triangulation, trilateration, and traversing. Such networks are very stable in
design, and allow establishment of first and second-order control
simultaneously. Optimization methods have been developed for the design and
survey of trigonometric networks. Starting from the demands on accuracy and
reliability, these methods provide information on the optimum point
configuration and the distribution of the measurements in the network given the
limitations of available equipment and the maximum allowable cost of the
survey (BOSSLER et al. 1973, GRAFAREND and SANS 1985).
Triangulation as part of a national geodetic survey started in France (1733-1750: Carte
gome"trique de la France, under the direction of Cassini de Thury) and in Great Britain (since
about 1780 triangulation by the Ordnance Survey, under W. Roy and others). It continued to be
the method for establishing horizontal control networks in the 19th and 20* century until the
introduction of electromagnetic distance measurements. Triangulation often started with chains
(in many cases established along meridians and parallels) composed of triangles or quadrilaterals
with diagonals tied together every few 100 km. The meshes of this framework then were filled by
first or second-order areal triangulation. The triangulations of Bavaria (1808-1828, J.G. Soldner)
and of Prussia (since 1875, O. Schreiber) brought significant improvements in measurement and
calculation techniques, which also influenced other national geodetic surveys. Large-scale
networks (chains and filling nets) were developed in the U.S.A. (since about 1830, J.F. Hay ford,
W. Bowie) and in the former Soviet Union (since the 1930's, T.N. Krassovski), cf. [7.1.2].
Trilateration was applied from the 1950's to the 1970's for strengthening, extending, and
densifying triangulation networks. Airborne microwave methods were employed for the rapid
survey of regions with difficult access and for bridging water areas (a few m to 10m accuracy
over some 100 km). Traversing has been used mainly for network densification since the 1960's,
but first-order geodimeter traverses also strengthened continental networks (U.S.A.) or even
established them (Australia). From the 1960's to the 1980's, satellite methods were utilized to
control the quality of horizontal control networks and especially to determine the orientation of
the ellipsoidal systems with respect to the global geocentric system (Doppler positioning).
Classical first and some second-order horizontal control networks have been
calculated on a reference ellipsoid within the system of ellipsoidal coordinates,
cf. [4.1]. Lower order networks are primarily calculated in planar Cartesian
coordinates, after conformal mapping of the ellipsoid onto the plane (LEE 1974,
GROSSMANN 1976, KUNTZ 1990, HOOIJGBERG 1997).
The observed horizontal angles, directions, and spatial distances were reduced
to the ellipsoid first, cf. [6.3.2]; the gravity-field-related reductions (deflections
of the vertical, geoid height) were not considered during earlier surveys. The
network adjustment was carried out either by the method of conditions or by
variation of the coordinates, with redundancy resulting from triangle
misclosures, diagonals in trilateration quadrilaterals, and additional base lines
and Laplace azimuths. For the method of conditions, the geometry of the
network was adjusted first. The coordinates transfer from an initial point (see
below) was then carried out using the solutions of the direct problem on the
ellipsoid; while for the method of variation of the coordinates, ellipsoidal
"observation equations" were derived from the solution of the inverse problem,
cf. [6.3.3]. Among the deficiencies of this classical "development method" are
the neglecting of the deflections of the vertical, the inadequate reduction of
distances on the ellipsoid, and especially the step by step calculation of larger
networks, with junction constraints when connecting a new network section to
an existing one. This led to network distortions of different type, with regionally
varying errors in scale (10~5) and orientation ( a few arcsec). Relative
coordinate errors with respect to the initial point increased from a few
decimeters over about 100 km to about one meter over several 100 km and
reached 10 meters and more at the edges of extensively extended networks.
The geodetic datum of a horizontal control network comprises the parameters of
the reference ellipsoid and of the network's orientation with respect to the
earth's body, cf. [6.3.3].
Conventional ellipsoids, as computed by the adjustment of several arc
measurements, were introduced during earlier geodetic surveys, cf. [1.3.3].
Some more recent surveys refer to locally best-fitting ellipsoids, as derived from
a minimum condition on the vertical deflections, according to (6.182):
using the equations (6.51). Topographic-isostatically reduced deflections of the
vertical have been used for larger networks in order to eliminate short and
medium-wave effects. The ellipsoid parameters of a geodetic reference system,
cf. [4.3], have been used recently. Tab. 7.1 gives the parameters of some
Tab. 7.1. Parameters of reference ellipsoids (rounded values), NIMA (2000)
Name, year
semimajor axis a (m)
reciprocal flattening \lf
Airy 1830
Bessel 1841
Clarke 1866
Hayford 1909
= Int.E11.1924
Krassovski 1940
reference ellipsoids that have been used for national geodetic surveys
(STRASSER 1957, NTMA 2000).
The ellipsoids of Airy (applied in Great Britain), Everest (India etc.), Bessel (Germany, Austria,
Japan etc.), Clarke 1866 (U.S.A., Canada etc.), and Clarke 1880 (France etc.) are based on the
adjustment of arc measurements distributed over the continents. The Hayford ellipsoid fits best to
the vertical deflection field in the U.S.A., it has been introduced in a number of countries. The
Krassovski ellipsoid resulted from a fit to the Russian triangulation, with additional data from
western Europe and the U.S.A. The ellipsoids of the geodetic reference systems GRS67 (Australia
etc.) and GRS 80 represent optimum approximations to the geoid.
The orientation of the ellipsoid was realized by defining the ellipsoidal
coordinates of a fundamental (initial) point, also called network origin, and by
conditions for the parallelism of the axes of the ellipsoidal and the global
geocentric systems, cf. [6.3.3].
In earlier surveys, the coordinates of the fundamental point were fixed by
postulating equality between observed astronomic latitude, longitude, and
orthometric height and the corresponding ellipsoidal quantities. This is identical
to setting the deflection of the vertical and the geoid height of the fundamental
point to zero:
This strategy provides a good approximation of the ellipsoid to the geoid close
to the origin, but may lead to larger deviations at more remote areas (Fig.7.4). If
Fig. 7.4. Locally fitting
"conventional" ellipsoid
Fig. 7.5. Regionally best fitting
a sufficient number of vertical deflection points were available and well
distributed over the area of calculation, the minimum condition (7.1) was used.
This condition permits determination of the deflection of the vertical in the
fundamental station and, eventually, the parameters of a best-fitting ellipsoid.
This procedure led to an optimum fitting over the whole area, and kept the
deflections of the vertical small. In many cases, the geoid height of the origin
point was defined indirectly by reducing the base lines onto the geoid and
treating them as ellipsoidal quantities (Fig. 7.5). The minimum condition for the
geoid heights (6.181)
2 =.
was occasionally applied using relative geoid heights calculated from
astronomic leveling, cf. [6.7.4], and utilizing the last equation of (6.51).The
parallelism of the axes was achieved by the condition equations (6.57) and
(6.58) for the deflection of the vertical and the azimuth (Laplace equation);
these were applied to the observations at the fundamental point and eventually
at additional Laplace stations. Some recent geodetic datums utilize the ellipsoid
parameters of a Geodetic Reference System, cf. [4.3], with optimum
approximations to the geoid (mean earth elupsoid), Fig. 7.6. Tab. 7.2 lists the
ellipsoids and the origin points used for some geodetic datums (HooiJBERG
1997, NIMA 2000).
Fig. 7.6. Mean earth ellipsoid
Tab. 7.2. Reference ellipsoids and origin points of some geodetic datums
Australian Geodetic 1984
(AGD84)
Deutsches
Hauptdreiecksnetz
(DHDN), Germany
European Datum 1950
(ED50)
North American 1927
(NAD27)
Ordnance Survey of
Great Britain 1936
(OSG36)
Pulkovo 1942, former
South American 1969
(SAD69)
Name of Origin
Latit.
Longit.
-2557'
13313'
Rauenberg/B erlin
Intern.Ellipsoid
Potsdam, Helmertturm
Kalianpur
Meades Ranch, Kansas
geocentric
Pulkovo
Chua, Brazil
The horizontal control network of the U.S.A. was constructed by triangulation chains with mesh
sizes of about 500 km. Nodal nets at the junctions of the chains were treated as constraints in the
adjustment, and areal networks later filled the meshes (BAKER 1974). A conventional ellipsoid
(Clarke 1866) was introduced, with the orientation obtained from the minimum condition for
vertical deflections (7.1): North American Datum 1927 (NAD27), see Tab.7.2. This system has
been replaced by the North American Datum of 1983 (NAD83), which combines the horizontal
control networks of the U.S.A., Canada, Greenland, Mexico, and Central America by a rigorous
adjustment (SCHWARZ 1989, SCHWARZ and WADE 1990), Fig.7.7. The observations include
terrestrial data (horizontal directions, azimuths, distances), Doppler stations (large-scale control
and reference to the geocenter), and VLBI baselines (scale and orientation). The adjustment
(about 1.8 million observations, more than 900 000 unknowns and 266 000 stations) was
performed by Helmert-blocking (WOLF 1978) and utilized a height-constrained three-dimensional
procedure, cf. [6.2], (VINCENTY 1980, 1982). The two-dimensional results refer to the Geodetic
Reference System 1980, with a geocentric position accurate to about 2 m. The relative station
uncertainty is a few cm to a few dm for distances between 10 and 300 km (SNAY 1990).
Deflections of the vertical and geoid heights were calculated for all stations by gravimetric
methods, cf. [6.7.1].
Since the 1950's, attempts have been made in Europe to establish a unified European
triangulation network (RETrig). A central European network formed by selected triangulation
Fig. 7.7. Horizontal control network of the U.S.A. (NAD83), with first and
second order triangulation, and traverses, courtesy National Geodetic Survey,
National Ocean Service, NOAA
chains was calculated first and later augmented by blocks in the southwest, southeast, and north
(WHITTEN 1952): European Datum 1950 (ED50), see Tab.7.2. The International Ellipsoid 1924
served as the reference surface and was orientated by the minimum condition (7.1), which
provided vertical deflection components ( = 3.36", 77 = 1.78") for the origin Potsdam,
Helmertturm. The base lines, reduced to the geoid, indirectly provided the geoid height (N = 0.4
m) at the origin (WOLF 1987). For western Europe, a rigorous readjustment of all first-order
triangulation nets was carried out later, including electromagnetic distance measurements, VLBI,
satellite laser ranging, and Doppler observations: European Datum 1987 (ED87), PODER and
HORNIK (1989). The system is more homogeneous than ED50, with improved parallelism of the
axes (better than 0.1 arcsec) and scale (0. IxlO" 6 ). The International Ellipsoid 1924 was
preserved as a reference surface, and the translation was taken from ED50 by adopting the ED50
coordinates for the station Munich, Frauenkirche (" 0 =-83, K0 = -97m, Z0 = -117m).
While ED50 was introduced in several countries, and served as the NATO military system until
the 1990's, no practical application was found for ED87. In the former Soviet Union, the
triangulation chains were adjusted on the Krassovski 1940 ellipsoid: Pulkovo Datum 1942,
IZOTOV (1959). Parameters and orientation of the ellipsoid were derived from the minimum
condition (7.1), with the geoid height set to zero at the origin Pulkovo. Reductions to the ellipsoid
became possible by observed and gravimetrically interpolated deflections of the vertical and
quasigeoid heights from astrogravimetric leveling, cf. [6.7.4]. After extension to the eastern
European countries and inclusion of new measurements, a new adjustment was carried out
leading to the astrogeodetic net 1983.
The primary triangulation net of Germany, developed since the 1870's, is an example of a local
horizontal control network. The northwestern part, between the rivers Elbe and Main, was
covered by triangulation chains and densification nets between 1870 and 1895 as part of the
geodetic survey of Prussia. A conventional ellipsoid (Bessel 1841) was introduced for the
calculation and orientated by the condition (7.2) at the origin Rauenberg/Berlin, whereby the
geoid height was fixed indirectly through the reduction of five base lines, one of them located
near Berlin. The Laplace azimuth from Rauenberg to Berlin, Marienkirche provided the
network's orientation on the ellipsoid. After the network adjustment, the ellipsoidal coordinates
were calculated according to the development method, where the horizontal directions were not
reduced for the vertical deflections. The eastern parts of Prussia and the triangulations of the
southern German states were later tied to this "Schreiber 's block " utilizing common points at the
networks' margins: Deutsches Hauptdreiecksnetz (DHDN), see Tab.7.2. In western Germany, the
DHDN has been locally improved since the 1950's by additional horizontal directions and
electromagnetic distance measurements, without changing the original datum. While the relative
accuracy over some 10 to 100 km is at the dm order of magnitude, distortions up to l m have
been found between different parts of the network. In eastern Germany, a complete
retriangulation was carried out after 1950 and calculated within the common adjustment of the
eastern European triangulations (see above): Staatliches Trigonometrisches Netz 1942/1983
(STN42/83), IHDE und LINDSTROT (1995). Due to the different datum definitions, DHDN and
STN42/83 differ systematically by about 2" in latitude and 4" in longitude. The DHDN90 thus
consists of three blocks observed over a time span of more than 100 years with different
orientation (SCHMIDT 1995), Fig.7.8. Local transformations between the two systems have been
performed for practical applications, and a complete recalculation will be carried out together
with the transformation to the three-dimensional system DREF 1991, which is part of the
European Reference Frame EUREF, cf. [7.3.2].
Fig. 7.8. Primary triangulation net of Germany (DHDN90), courtesy
Bundesamt fr Kartographie und Geodsie (BKG), Frankfurt a.M, Germany
Satellite positioning was employed early to control the quality of horizontalcontrol networks and to determine the datum transformation parameters with
respect to the geocentric system, ASHKENAZI et al. (1988). Equations (6.47)
were used to calculate translation, rotation, and scale factor from identical
stations coordinated in the global system (especially in the WGS84) and in the
local or regional geodetic datum. The results are mean values for the respective
geodetic datum, depending strongly on the distribution of the identical points.
Tab. 7.3 provides the translations, rotations, and scale factors for some geodetic
datums.
Tab. 7.3. Translation, rotation, and scale factors for some geodetic datums
(mean values), DMA (1987), NIMA (2000), DHDN and Pulkovo 1942
(STN42/83): fflDE and LINDSTRQT (1995)
Geodetic
Datum (see
Tab. 7.2)
AGD84
DHDN
OSG36
Pulkovo 42
SAD69
7-Parameter-Transformation
Translation (m)
Rotation (arcsec)
mxlO 6
3-Parameter-Transf.
Translation only (m)
Remark: Pulkovo 42 7-Parameter-Transformation values are valid for eastern
Germany (STN42/83); 3-Parameter-Transformation values are valid for Russia.
The translation values are at the order of magnitude of the deflections of the
vertical. They are larger when conventional ellipsoids have been used and
decrease with best-fitting ellipsoids. The rotation angles mirror the accuracy of
the astronomic observations and are usually not significant. While the scale
errors for older networks reach 10~5 and more; they attain only 10"6 and less in
recent systems, characterizing the progress in length determination. Due to
network distortions, local datum-shift values may differ significantly from the
mean values, e.g., by 5 to 10 m and more for the translation parameters. If a
three-parameter solution (translation only) is carried out, the translation values
for older networks experience considerable changes.
The transformation from a local geodetic datum to the geocentric system can be
done either by a complete readjustment including spatial observations
(example: NAD83) or by mean or local (more accurate) transformation
parameters. The regional behavior of the translation parameters may also be
modeled by low-order polynomials and provided in the form of datum shift
contour charts (DMA 1987). Of special interest is the transformation of
ellipsoidal coordinates, which includes the transition from a conventional or
best-fitting ellipsoid to a geocentric one. From (6.49) we obtain the changes in
the ellipsoidal latitude, longitude, and height (spherical approximation, rotations
neglected, ellipsoidal formulas are given by DMA 1987, EHLERT 1991):
0 -si
Here, the sign of the translation vector
z-z
has been changed (reduction!). All differences are formed in the sense
"geocentric - local system", resulting in the transformation
, = + AA, h =
(7.4c)
Again, the changes in , , h can be modeled and given in contour charts,
provided a sufficiently large number of identical points have been used. The
accuracy of these transformations depends on the area and the number of points
available in both systems. A few m accuracy has been achieved for continentwide networks.
Classical horizontal control networks form the basis for many applications in
surveying and mapping. They suffer from network distortions, also they are not
geocentric and typically do not use modern geodetic standard-ellipsoids. Within
the near future, they will be transformed and integrated into the global
geocentric three-dimensional reference frame, cf. [7.3.2].
7.2 Vertical Control Networks
Vertical control networks have been established separately from horizontal
control nets. This is due to the demand that heights have to refer to the gravity
field rather than the ellipsoidal system used for horizontal positioning.
Vertical control networks are determined by geometric leveling and
occasionally by hydrostatic leveling, cf. [5.5.3], the control points being
designated as bench marks. According to the leveling procedure and the
accuracy achieved, national geodetic surveys distinguish between different
orders of leveling. First-order leveling is carried out in closed loops (loop
circumferences of some 100 km) following the rules for precise leveling. The
loops are composed of leveling lines connecting the nodal points of the network
(Fig.7.9). The lines, in turn, are formed by leveling runs that connect
neighboring benchmarks (average spacing 0.5 to 2 km and more). The firstorder leveling network is densified by second to fourth-order leveling.
" * **"
LEVELING NETWORK
Fig. 7.9. Leveling network (principle)
Leveling lines generally follow main roads, railway lines, and waterways. The
benchmarks consist of bolts in buildings, bedrock, or on concrete posts. Long
pipes are set up in alluvial regions. Underground monuments are established in
geologically stable areas in order to secure the network stability with respect to
variations with time. An accuracy of about 1 mm per km is achieved for firstorder networks. They should be reobserved at time intervals of some 10 years,
as regional and local height changes can reach 1 mm/year and more, especially
in areas which experience crustal motion, cf. [8.3.3].
Prior to the adjustment of a leveling network, the observed raw height
differences have to be transformed either to geopotential differences or to
differences of normal or orthometric heights by considering surface gravity, cf.
[6.4.1]. The adjustment then utilizes the loop misclosure condition of zero and
is carried out either by the method of condition equations or, preferably, by the
method of parameter variation.
First-order leveling networks were established in many countries between the I860's and the
1930's when surface gravity along the leveling lines was not available. Hence, gravity reductions
were either neglected completely or actual gravity was approximated by normal gravity, leading
to normal or spheroidal orthometric heights. These path-dependent heights may differ from
normal or orthometric heights by some mm (flat terrain) to some dm (mountains). With gravity
data being generally available now, gravity can be taken into account in calculations of modern
leveling networks (WOLF 1974).
The vertical datum (zero height surface) is defined by mean sea level (MSL) as
derived from tide gauge records. National height systems may differ by some
dm to 1 m, and more, between each other. They may also differ from the geoid
as a global reference surface, which is due to the effect of sea surface
topography, cf. [3.4.3]. Network distortions arise if the vertical datum is
constrained to MSL of more than one tide gauge.
Estimates of the vertical datum differences between different height systems are available from
satellite positioning and global geoid models and from continental-wide leveling connected to tide
gauges (RAPP 1995b, IHDE et al. 1998). For instance, the zero height surface of the North
American Vertical Datum of 1988 is about 0.7 m below MSL at Amsterdam, which was used for
the European leveling network. In Europe, the national vertical datums have been derived from
MSL records in the Mediterranean, the North Sea, and the Baltic Sea. Taking MSL in Amsterdam
(used, e.g., in the Netherlands and in Germany) as reference, MSL in Kronstadt (near
St.Petersburg, Russia) is about 15 cm higher, and tide gauges along the Mediterranean (Marseille:
France, Genoa: Italy, Trieste: Austria) give a MSL that is about 0.4 to 0.5 m lower (SACHER et al.
In North America, a previous adjustment of first-order leveling provided the North American
Vertical Datum of 1929 (NAVD29), which was constrained to the MSL of 26 tide gauges in the
U.S.A. and Canada (BAKER 1974). After replacing destroyed bench marks and extensive
releveling, a new adjustment of the leveling data of the U.S.A., Canada, Mexico, and Central
America was carried out in geopotential numbers: North American Vertical Datum of 1988
(NAVD88). Heights are given as orthometric heights according to Helmert (6.84) and refer to
MSL of one primary bench mark (Father Point/ Rimouski, Quebec, Canada), ZILKOSKI et al.
The United European Leveling Net (UELN) is formed by first-order leveling lines of the
European countries (Fig.7.10). Periodic adjustments have been carried out since 1954, with
continuous improvement and extension by including new data. The adjustments are performed in
geopotential numbers, and normal heights are available. The average accuracy (UELN95) of the
leveling is 1.1 mm/km, and the standard deviations related to Amsterdam remain less than 0.1
F/g. 7.70. United European
Leveling Net (UELN95), courtesy
BKG, Frankfurt a.M., Germany
Fig. 7.11. Primary leveling network of
Germany (DHHN92), courtesy BKG,
Frankfurt a.M., Germany
m. The vertical datum is taken from MSL of the North Sea as determined in the period 1940 to
1958 at Amsterdam (Normal Amsterdamsch Peil NAP of 1950). The UELN is connected to a
large number of tide gauges, which permits determination of sea surface topography around
Europe. In the future, UELN will be extended to a kinematic height system by including vertical
point velocities (ADAM et al. 2000a).
First-order leveling in Germany started around 1865 and first led to a height system calculated
from the raw leveling data. Complete resurveys were carried out from 1912 to 1960 and from
1980 to 1985 (western Germany, including gravity measurements). These networks were adjusted
as normal-orthometric heights: Deutsches Haupthhennetz (DHHN). The vertical datum was
derived by leveling from the Amsterdam normal tide gauge, representing MSL for the period
1683/1684 (\VAALEWIJN 1986). The zero height surface (Normal-Null, N.N.) was fixed by a
standard bench mark, established 37.000 m above N.N. at the former Berlin observatory and,
since 1912, by underground marks about 40 km east of Berlin. In eastern Germany, releveling
was carried out in the 1970's and adjusted as normal heights. The vertical datum was taken from
MSL at the tide gauge in Kronstadt: Hhennull ( ). Due to the different definitions of the
vertical datum and the heights, systematic height differences between 8 and 16 cm occurred at the
boundary between western and eastern Germany. A readjustment was performed in geopotential
numbers, where the vertical datum is defined by the geopotential number of the nodal point
Wallenhorst taken from the UELN86 adjustment and thus refers to MSL at Amsterdam
(Fig.7.11): DHHN92. Normal heights have been introduced as official heights, referring to the
reference surface (quasigeoid) Normalhhennull (NHN), WEBER (1995).
7.3 Three-dimensional Networks
Leveling networks are characterized by high accuracy, but systematic errors
may accumulate over large distances. Due to the time-consuming measurement
procedure, repetition surveys are feasible only after long time intervals. A more
rapid establishment of vertical control networks can be achieved by
trigonometric leveling, cf. [6.4.2], and a drastic change is to be expected by
GPS-heighting in connection with precise geoid or quasigeoid models, cf.
[6.4.3]. In this way, vertical control networks will be integrated into threedimensional control systems, cf. [7.3.2]. Geometric leveling will maintain its
importance over short distances, and especially in areas of recent crustal
movements, such as regions of land subsidence and zones of earthquake or
volcanic activity, cf. [8.3.3].
Geodetic space methods deliver three-dimensional coordinates with cmaccuracy in the global geocentric system on global, regional, and local scale.
Global networks realize the terrestrial reference system and provide reference
stations for the establishment of continent-wide networks, wherein GPS plays a
fundamental role [7.3.1]. Further network densification is carried out nearly
exclusively by GPS, with subsequent integration of existing classical control
nets [7.3.2].
The International Terrestrial Reference Frame (ITRF) represents the global
basis for three-dimensional positioning. It is defined by the geocentric Cartesian
coordinates and the horizontal velocities of a global set of space geodetic
observing sites, given for a certain epoch, with an accuracy of about 1 cm and
1 to 3 mm/year respectively, cf. [2.5.3]. The ITRF results are based on
observations obtained by global networks employing different space techniques,
which are combined within the frame of the International Earth Rotation
Service (IERS), cf. [2.3].
Nowadays, the Global Positioning System (GPS) is the primary technique used
in all scales for positioning and navigation. The GPS results refer to the World
Geodetic System 1984 (WGS84). The underlying global network is comprised
of a limited number of GPS tracking stations, determined with an accuracy of
about 5 cm. At this accuracy level, WGS84 also agrees with ITRF, cf. [5.2.5].
The International GPS Service (IGS), established by IAG, has been operational
since January 1, 1994. It consists of a global network of more than 200
permanently operating GPS stations equipped with geodetic, two-frequency
GPS receivers and several data and analysis centers (BEUTLER et al. 1996b).
IGS products include highly accurate ( a few cm) GPS satellite ephemerides,
clock corrections, earth rotation parameters provided on a daily basis, and
weekly solutions for the station coordinates (sub-cm accuracy in horizontal
position, 1 to 2 cm in height) and velocities. In addition, the raw GPS tracking
data (phase and pseudorange observations), station clock, ionospheric, and
tropospheric information are available. By using ITRF positions and velocities
for a subset of IGS stations, the IGS network is connected to ITRF and
contributes significantly to it. On the other hand, IGS stations and products
serve for densifying ITRF on continental scale.
The establishment of continent- wide fundamental networks with GPS began at
the end of the 1980's; an approximate homogeneous station-coverage is
generally the goal. Station distances are several 100 km, and at least three
stations per country are generally selected as a reference for further
densification and for the transformation of existing control networks, cf. [7.1.2],
[7.3.2]. The station sites are selected according to the requirements of GPS
observations (no visibility obstructions between 5 to 15 and 90 elevation,
absence of multipath effects, no radio wave interference). Generally the stations
are monumented by concrete pillars, providing a forced centering for the GPS
antenna and a height reference mark. Eccentric marks are established in order to
locally control horizontal-position and height. Existing first-order control points
may be used if they fulfill the GPS requirements, otherwise the GPS stations
should be connected to the existing control networks by local surveys.
Dedicated GPS campaigns are carried out for the determination of the station
coordinates, employing relative positioning, cf. [5.2.5]. This strategy requires
the inclusion of at least one reference station with coordinates given in the
ITRF. Practically all ITRF and IGS stations in the survey region are introduced
as reference ("fiducial") stations. Temporary reference stations may be
established through baseline connections to IGS stations in order to improve the
connection of the network to the ITRF. Depending on the number of stations
and available GPS receivers (two-frequency geodetic type), either all stations
are observed simultaneously or the network is divided into blocks that are
observed sequentially (Fig.7.12). All observations made simultaneously during
a given time interval are called a "session" (SNAY 1986). The duration of one
session is typically 24 hours, which permits determination of the ambiguity
unknowns and a simultaneous solution for the station coordinates and
tropospheric correction parameters. The results of one session are highly
correlated. Consequently, several sessions are generally carried out, leading to a
total observation time of some days to one week.
By referencing the network to IGS stations and applying the IGS precise orbital
data, the effect of reference station and orbital errors on the station coordinates
is only at the few mm level. When different type GPS receivers are employed in
one campaign, corrections have to be applied for antenna phase-center
Fig. 7.12. GPS network constructed
from individual blocks (principle) from baselines to permanent GPS
stations (principle)
differences. In addition, phase-center variations have to be taken into account
by calibration (SEEBER et al. 1998). Longer observation periods increase the
accuracy of the results, due to the changing satellite geometry and the reduction
of residual tropospheric, multipath, and antenna effects. This is especially valid
for the height component, where small satellite elevations improve the
geometry of the solution but introduce larger tropospheric errors. Continentwide GPS networks can be determined with an accuracy of about 1 cm.
Permanent ("active ") GPS stations have increasingly been established since the
1990's at regional and local scales. They are equipped with geodetic GPS
receivers that continuously track all available satellites. Undisturbed visibility to
the satellites is achieved by installing the antennas several m to 10m above the
ground on concrete pillars, steel grid masts, etc., or on the top of buildings.
Permanent stations provide the raw GPS tracking data, clock corrections,
atmospheric information, and daily to weekly solutions for the station
coordinates with a precision of a few mm. They serve primarily for the
maintenance of the GPS reference system realized by the fundamental network,
by controlling variations with time due to plate tectonics and other geodynamic
activities. The establishment of GPS control points is facilitated by differential
GPS using baselines to the permanent stations and exploiting the data made
available for these stations. Thus, a single receiver can be used for precise
positioning (Fig. 7.13).
Fundamental GPS networks with permanent stations densify ITRF and IGS on a
continental scale and provide the reference for national systems. Following the
ITRF strategy, reference epochs are defined for the station coordinates of the
fundamental networks, which may differ from the epoch of the ITRF stations
introduced and from the time of the observation campaign. Consequently,
reductions have to be applied which take the station velocities between the
different epochs into account.
A European Reference Frame (EUREF) has been built up since the end of the 1980's, as
realization of a European Terrestrial Reference System (ETRS). The EUREF stations were
determined by successive GPS campaigns that included ITRF and IGS stations and generally
covered several countries (SEEGER et al. 1998, ADAM et al. 2000b). The Reference System was
defined for the epoch 1989.0 and realized through the European Terrestrial Reference Frame 1989
(ETRF89), which agrees with the definition of ITRF89. In 1999, the ETRF comprised about 200
stations, with distances between 300 and 500 km (Fig.7.14). ETRF89 shall rotate with the stable
part of the European tectonic plate, which allows the frame to remain unchanged over a longer
time interval. About 90 permanent GPS stations serve for maintaining ETRS and densifying the
IGS network. Further applications include investigations of recent geodynamics, sea level
monitoring, and contributions to weather prediction, utilizing the tropospheric zenith path delays
(BRUYNINX 1999), cf. [5.1.2].
Fig. 7.14. European Reference Frame (EUREF89/97) with permanent GPS
stations (status 1999), courtesy BKG, Frankfurt a.M., Germany
A European Vertical Reference Network (EUVN) is under construction in order to unify, at a
few-cm accuracy level, the different height systems used in the European countries. The network
comprises about 200 stations determined in dedicated GPS campaigns (1997) using one-week
observation times. It includes EUREF sites, nodal points of the European leveling net UELN, tide
gauges, and a number of permanent GPS stations. Three-dimensional coordinates, geopotential
numbers, and normal heights will be calculated; hence EUVN will provide fiducial points in order
to fit the European quasigeoid to a unified European height system, cf. [6.4.3]. By extension to a
kinematic height system, EUVN shall be used for monitoring large-scale, vertical crustalmovements and sea level changes (ADAM et al. 2000a).
In South America, a Sistema de Referenda Geocentrico para America del Sur (SIRGAS) was
established in 1995 by simultaneous GPS observations (10 days) on some 60 stations well
distributed over the sub-continent (HOYER et al. 1998)). Referring to ITRF94, SIRGAS provides
the basis for the transformation of the national networks to the global reference system. Within
the frame of SIRGAS, another GPS campaign was carried out in 2000 (including Central and
North America) on SIRGAS stations, leveling benchmarks, and tide gauges in order to derive a
unified vertical height system.
National geodetic control-networks have been established by GPS since the
1990's and now supersede the classical horizontal (and possibly vertical)
control networks (AUGATH 1997). Although the strategies for establishing and
maintaining these three-dimensional reference networks are still under
discussion and differ from country to country, the following directions clearly
can be identified:
establishment of a large-scale, three-dimensional network with station
distances between 10 and 50 km,
installation of permanent GPS stations with distances of 50 to 100 km,
transformation of the existing horizontal control network, eventually using
additional GPS control.
A three-dimensional network is established by dedicated GPS campaigns using
relative positioning, either as a densification of existing continental reference
networks or by connection to ITRF and IGS stations, cf. [7.3.1]. Station sites
are selected so that they coincide with fust and second-order trigonometric
points, or are located close to them. GPS observation criteria have to be taken
into account, and monuments set below ground are beneficial for the long-term
preservation of the network. A separation into a fundamental network with
station distances of several 10 km (corresponding to the first-order
trigonometric points) and densification nets with distances down to 10 km
(second-order triangulation) may be useful for larger countries. Precise leveling
benchmarks and tide gauges may be included into these networks in order to
incorporate the existing vertical control system. While all stations can be
observed simultaneously in smaller networks, a separation into observation
blocks with subsequent GPS campaigns will generally be necessary. Each
station is occupied at least two times on different days, with session durations
between 8 and 24 hours. IGS precise orbital data is used in the network
adjustment. Nearby global and continental reference stations are used as
"fiducial stations". A "multi-station" adjustment utilizes all data collected
during one session, while the "multi-session" adjustment combines the results
of several sessions. The national network is transformed to a certain ITRS
epoch. Accuracies of 1 cm and better are achieved for the horizontal
components, and 1 to 2 cm for the heights.
Permanent GPS stations are established at distances of 50 to 100 km or more.
The permanent stations are used to maintain the national reference frame with
respect to recent crustal movements, but they also serve as reference for all
types of GPS surveys which employ differential GPS (DGPS), WANNINGER
(1996). The stations are equipped with geodetic GPS receivers; they track all
visible GPS satellites at a high data rate (e.g., 1 s). Site selection and delivery of
data products (raw tracking data, station coordinates, atmospheric corrections,
etc.) follows the rules developed for permanent stations in continental networks,
By connecting the three-dimensional network stations to first and second-order
trigonometric points, the existing horizontal control networks can be
transformed into the three-dimensional reference frame. Additional GPS control
points may be needed if the classical networks contain larger distortions; the
selection of the control points depends on the network peculiarities, cf. [7.1.2].
A 7-parameter transformation may suffice for homogeneous networks of high
precision, but usually more sophisticated transformation models will be
necessary, including polynomial, least squares, or spline approximation
(MORITZ 1978). In this way, the local cm-accuracy of classical networks will be
kept, and the effect of the network distortions can be reduced to the order of a
few cm to dm over distances of some 10 to 100 km.
After the introduction of a three-dimensional reference frame, GPS-positioning
can be carried out in the relative mode by connection to the reference stations in
the survey area. Two or more GPS receivers are required; one (or more)
generally serve as a temporary reference with continuous satellite tracking. If a
network of permanent GPS stations is operated by a national GPS service, with
telemetry data transfer to the users, DGPS can be applied (SEEBER 1993,
WANNINGER 2000). Positioning is now possible with one receiver only,
depending on the baselines between the new station and the permanent stations.
For short (few to 10 km) baselines, a relative cm-accuracy can be achieved in
quasi real-time after proper ambiguity solution. For longer baselines, the results
are degraded by the distance-dependent errors of GPS, cf. [5.2.5]. By
connecting to an array of at least three permanent stations, the known geometry
can be used to rapidly determine the ambiguities and to calculate baseline
corrections for ionospheric, tropospheric, and orbit effects. Post-processing of
DGPS can yield relative sub-cm accuracy. After the completion of the
transformation to a three-dimensional reference frame, the classical horizontal
networks of lower order will no longer be maintained.
By including first-order leveling benchmarks in the three-dimensional network,
the differences between the ellipsoidal heights and the heights of the national
height system can be determined (geoid heights, quasigeoid heights). These
GPS/leveling control points allow the national height system to be fit to a geoid
or quasigeoid model, and they can be used to derive gravity-field related heights
(orthometric heights, normal heights) for all three-dimensional reference
stations, cf. [6.4.3]. This will lead to a large-scale vertical control system
identical with the three-dimensional reference frame and not restricted to the
leveling lines. Depending on the accuracy of the geoid-/quasigeoid "reduction"
of GPS heights, the application of geometric leveling will be reduced to more
local problems where mm-accuracy is required.
Fig. 7.75. German GPS Reference Net 1991(DREF91) with EUREF stations
(black circles), courtesy BKG, Frankfurt a.M., Germany
In Germany, a three-dimensional network was established in 1991: Deutsches Referenznetz 1991
(DREF91). It consists of 109 stations that are mostly collocated with first or second-order
trigonometric points (Fig.7.15). The network comprises the ITRF and EUREF stations and
represents the national reference frame embedded in the ETRF, cf. [7.3.1]. Two independent GPS
campaigns were carried out (8-hour observation time per session), and the adjusted coordinates
(accuracy 1 cm for the horizontal component and 1 to 2 cm for the height) were transformed to
ETRF89 (LiNDSTROT 1999). The network is maintained by about 20 permanent stations operated
by BKG and the state survey agencies. DREF densification nets (station distances 15 to 25 km)
are under construction by the state survey agencies, which also operate a DGPS service:
Satellitenpositionierungsdienst der deutschen Landesvermessung (SAPOS). About 200 permanent
GPS stations are planned, which will store the GPS signals and provide them to users in RINEX
(receiver independent exchange) format, together with other products. Different accuracy levels
are available for real-time and post-processing applications (HANKEMEIER 1996). With SAPOS
being fully operational, the lower order trigonometric control networks will not be further
maintained. Other examples of national three-dimensional reference networks are found in
GUBLER and HORNIK (1999) and GUBLER et al. (1999).
Permanent GPS networks play an increasing role in regions of earthquake activity, cf. [8.3.3], In
Japan, a country-wide network of more than 1000 stations is operated by the Geographical
Survey Institute, which continuously monitor strain accumulation and tectonic movements.
7.4 Gravity Networks
Gravity networks provide the frame for gravimetric surveys on global, regional,
or local scales. They consist of gravity stations where gravity has been
determined by absolute or relative methods. On a global scale, the gravity
standard is realized by the International Gravity Standardization Net 1971
(IGSN71), but absolute gravimeters now allow independent establishment of
the gravity standard, cf. [5.4.3]. A global, absolute-gravity base-station network
has been established primarily for the investigation of long-term gravity
variations with time, cf. [8.3.4].
National gravimetric surveys are based on a primary or base network, which in
most cases is densified by lower order nets. The gravity base-network stations
should be evenly distributed over the area; station distances of a few 100 km are
typical for larger countries. The station sites should be stable with respect to
geological, hydrological, and microseismic conditions, and they should be
located at permanent locations (observatories etc.). Eccentric stations serve for
securing the center station and for controlling local height and mass changes.
The stations of the subsequent densification networks may be collocated with
horizontal and vertical control points. Horizontal position and height of the
network stations should be determined with m and mm to cm-accuracy,
respectively.
Absolute gravimeters are increasingly employed for the establishment of base
networks, partly in combination with relative gravity-meters. Densification
networks are observed primarily with relative instruments. Relative gravimeters
need to be calibrated, and repeated measurements are necessary in order to
determine the instrumental drift. The use of several instruments reduces residual
systematic effects, cf. [5.4.2]. Relative gravimetry requires at least two absolute
stations in order to derive the gravity "datum" and corrections to the linear
calibration coefficient that was determined by the manufacturer or on a
calibration line. A network optimization is beneficial for obtaining maximum
accuracy and reliability for given restrictions (WENZEL 1977).
The establishment of gravity networks for geophysical and geodynamic
applications follows the same rules, but the distribution of the gravity stations is
determined by the geological structures or the geodynamic processes under
investigation, cf. [8.3.4].
Gravity networks are generally adjusted by the method of parameter variation.
Gravity values and - for relative gravimeters - drift and calibration coefficients
are the parameters to be determined. Absolute gravity data and relative
gravimeter readings, or reading differences, are introduced as observations. The
observation equation for an absolute gravity measurement on the station i reads
*, = A ,
where z~ represents the station mean value corrected for polar motion and earth
tides and reduced to ground level, cf. [5.4.1]. For relative gravity
measurements, the observation equation follows from (5.90) and (5.92). If we
restrict ourselves to the linear terms of the drift and the calibration function, the
relative gravimeter reading z. (after approximate calibration and application of
the earth tides reduction) on the station / (time tt) is transformed to gravity by
Ylz, - d^ + N 0 = g f ,
which leads to the observation equation
Here, dl = linear drift coefficient, ^ = 1 + yl = linear calibration coefficient,
yl = scale correction, N0 = level unknown. In most applications, reading
differences between the stations / andy are introduced as observations, hereby
eliminating the level unknowns (TORGE 1993):
te, = Zj -z,=gj -gi -y, (z. -z,) + 4 (tj -/,.).
The accuracy of primary gravity networks, established by absolute gravimeters
or by a combination of absolute and relative gravimetry, is about 0.1 (ims~2
and may reach 0.05
In Germany, gravity base networks and densifying networks were established beginning in the
1930's. The first base-network was observed with relative pendulum instruments; relative
gravimeters have contributed since the 1950's. These networks were tied to the Potsdam absolute
gravity value, cf. [5.4.3], accuracy increased from some Hums' 2 to a few 0. lums"2 . Absolute
gravimetry was introduced in 1976/1977 in order to establish a combined absolute/relative basenetwork in western Germany: Deutsches Schweregrundnetz 1976 (DSGN76), SIGL et al. (1981).
This base network was extended to eastern Germany and completely remeasured in 1994/1995:
DSGN94. The network consists of 30 stations (one center and at least two eccentric points) that
were observed with an absolute gravimeter FG5. Repeated observations and relative ties (several
gravimeters of type LaCoste and Romberg and Scintrex) served for investigations of accuracy
PAYSANDU.
Fig. 7.16. Primary gravity net of
Germany (DHSN96), with German
gravity base net (DSGN94) stations
(black circles), courtesy BKG,
Fig. 7.17. Gravity network of Uruguay,
with absolute stations (black triangles),
relative stations, and earth tide station
(black circle), after SUBIZA et al.
(0.05urns" 2 ) and reliability (TORGE et al. 1999a). A first-order densification net (average station
distance 30 km) was observed with several relative gravimeters (1978-1982, 1994, accuracy
better than 0.018"2), Fig.7.16. Further densification down to a few km has been realized or is
in progress (WEBER 1998). The gravity network of Uruguay is an example of a combined
adjustment of a few absolute stations and relative measurements carried out mainly along leveling
lines, Fig.7.17 (SuBiZA et al. 1998).
In the U.S.A., absolute gravity measurements have primarily been carried out within the frame of
dedicated projects (investigation of recent vertical crustal-movements, establishment of
gravimeter calibration lines, etc.) by NOAA/NGS and other agencies, covering the country by
more than 150 stations in 2000. A gravity network established in 1976/1979 by LaCoste and
Romberg gravimeters (more than 200 sites) has been constrained to the absolute standard (PETER
et al. 1989).
Geodesy, within the frame of the geosciences (geophysics, geology, petrology,
mineralogy, geochemistry), deals with the determination of the figure and the
external gravity field of the earth. The results of geodetic work provide
boundary conditions for the development of static and dynamic geophysical
earth models. These models, in turn, provide significant information for the
planning and implementation of geodetic operations.
Geophysical earth models are based on a radial structure of physical properties,
and presuppose hydrostatic equilibrium [8.1]. These assumptions are not valid
for the upper layers of the earth, where geodynamic processes play an important
role [8.2]. Geodesy contributes to geodynamic research by monitoring the
actual variations of the orientation, the physical surface, and the gravity field of
the earth [8.3].
From the extensive geophysical literature we mention the classical work of
JEFFREYS (1970) and the recent textbooks by BERCKHEMER (1990), STACEY
(1992), LOWRIE (1997) and FOWLER (1999). The interrelations between
geodesy and geophysics are treated especially in HEISKANEN and VENINGMEINESZ (1958), LAMBECK (1988), and MORITZ (1990). The actual state of
geophysical data collection and parameter estimation is given in AHRENS
Various observations show that the earth does not posses a homogeneous
The mass M of the earth as derived from the geocentric gravitational
constant GM, cf. [6.8.1], and the constant of gravitation G, cf. [2.1],
amounts to M = 5.974xl0 24 kg. With the volume of the earth ellipsoid
1083 xlO18 m 3 , we obtain the mean density
/? m =5.515xl0 3 k g m 3 .
As the density of the earth's crust only amounts to 2.7...2.9xl0 3 kgm 3 ,
density must increase toward the interior of the earth.
Astronomic observations of the lunisolar precession, cf. [2.4.2], deliver the
dynamic (mechanical) ellipticity
8 Structure and Dynamics of the Earth
with a value of 1/305.45. With the dynamical form factor provided by
satellite geodesy, cf. [6.8.1],
(C and A being the polar and the mean equatorial moment of inertia,
respectively) we obtain the moment of inertia with respect to the rotational
C = 0.3307a 2 M.
If the earth were a homogeneous sphere, we would have C - 0.4a2M . This
again indicates a density increase with depth.
Seismology shows that the earth has a shell-like construction, with the shell
boundaries being defined by discontinuities of the seismic waves velocities.
With the velocities of the seismic waves being known, and under the
assumption of hydrostatic equilibrium, density, gravity, and pressure inside a
spherically layered earth model can be calculated as a function of the radial
distance from the earth's center of mass.
The assumption of hydrostatic pressure in the earth's interior is justified by the fact that the earth
originally existed in a liquid state. In which case, the pressure depended only on the weight of the
masses lying above, and it increased toward the center of the earth.
Seismology determines the velocities of the primary (compression) and the
secondary (shear) seismic waves, vp and v} . From these velocities the seismic
is derived, where is the bulk modulus (compressibility) and is density. is
defined as the ratio between the hydrostatic pressure and the dilation
experienced by a body under this pressure. The relationship between changes of
pressure and density is given by
dp=dp.
Under hydrostatic equilibrium, the increase of pressure with depth depends on
the weight of the additional vertical mass column. With the radial distance r, the
fundamental hydrostatic equation reads
= -g(r)p(r)dr,
where the minus-sign indicates that pressure decreases with increasing radius.
Finally, from (8.4) and (8.5) we obtain the relation between height and density
changes (Adams-Williamson equation):
According to (3.52), the radial change of the gravity potential W is given by
= -g(r)dr.
6000 DEPTH (Km)
Fig. 8.1. Density p (3 kg/m 3 ),
gravity g (m/s 2 ), and pressure p
(10" Pa) inside a spherically symmetric earth model, after DZIEWONSKI
and ANDERSON (1981)
Fig. 8.2. Spherical earth model, with
homogeneous shells separated by
discontinuity zones (D.), not to scale,
after DZIEWONSKI and ANDERSON (1981)
dp = p(r)dW.
Hence, the surfaces of equal pressure (isobaric surfaces) coincide with
equipotential surfaces and, after (8.4), also with surfaces of equal density.
Starting from density and gravity values on the earth surface, density, pressure
and gravity inside the earth can be calculated iteratively, using (8.3), (8.4) and
(3.22), (3.23). Here we use the total mass and the polar moment of inertia as
boundary conditions. Corresponding spherically symmetric earth models
consist of several layers characterized by chemical and physical properties
(composition, pressure, temperature), BULLEN (1975), DZIEWONSKI and
ANDERSON (1981). The velocities, or the velocity gradients, of the seismic
waves change abruptly at the boundaries of the layers (discontinuity surfaces or
zones), as does density (Fig.8.1). Inside one layer, density increases smoothly
and reaches about 13000kgm"3 at the earth's center. Gravity remains nearly
constant within the earth's mantle and decreases almost linearly to zero in the
core. Pressure increases continuously with depth.
The layered structure of the earth is shown in Fig.8.2. The earth's crust
(average thickness over the continents and the oceans about 24 km) is the
uppermost layer and is characterized by a complex structure. It is separated
from the upper mantle by the A/o/iorov/c/c-discontinuity. Lateral density
variations are pronounced in the crust, but are found also in the upper mantle,
cf.[8.2.1]. The crust and the uppermost part of the mantle (also called lid of the
low-velocity layer beneath it) behave rigidly and are affected by plate tectonic
motions, cf. [8.2.3]. The lower mantle starts at a depth of 650 to 670 km and is
separated from the core at 2890 km depth, through the Wiechert-Gutenbergdiscontinuity. The liquid outer core extends to 5150 km, followed by the solid
inner core.
The density distribution of these earth models can also be tested by comparing
its elasticity parameters with the results obtained from the observation of natural
oscillations and (to a limited extent) earth tides, cf. [8.3.5]. The free oscillations
especially provide an important constraint on the models.
More refined models have to take the deviations from spherical symmetry into
account, as well as departures from hydrostatic equilibrium. These deviations
are clearly indicated by the odd zonal and the tesseral harmonic coefficients of
the gravity potential, cf. [3.3.4], and by other geophysical observations. Seismic
tomography especially revealed large-scale lateral variations of the P and Swave velocities in the mantle, correlated with zones of density and temperature
anomalies (DZIEWONSKI and WOODHOUSE 1987). A global deviation from
8. 2 The Upper Layers of the Earth
hydrostatic equilibrium is indicated by the flattening of a rotating spheroidal
body in equilibrium, composed of density layers that are approximately
ellipsoidal. According to Clairaut, the following relation holds between the
polar moment of inertia C, the mass M, the ellipsoidal quantities a and m (4.50),
and the hydrostatic flattening fh :
Inserting the observed values results in hydrostatic flattening values of about
1/299.7, which differ significantly from 1/298.25 derived directly from satellite
orbit analyses, cf. [6.8.1], DENIS (1989).
The larger actual flattening may be attributed to a "fossil" flattening of the lower mantle, which
developed when the earth's rotational velocity was larger and which is not yet compensated. The
still incomplete recovery of the ancient ice loads at the polar caps may be another explication.
Refined geophysical earth models that take the ellipsoidal form and the rotation
of the earth into account may use the level ellipsoid as a good approximation for
the external boundary surface, cf. [4.2.1].
Large deviations from the spherically symmetric earth model are found in the
earth's crust and upper mantle [8.2.1]. Topographic mass excesses (mountains)
and deficiencies (oceans) are, to a large part, compensated by the underlying
masses, which leads to isostatic equilibrium [8.2.2]. The theory of plate
tectonics introduces (nearly) rigid lithospheric plates that move against each
other, causing crustal deformations, especially at the plate boundaries [8.2.3].
Since the gravity field reflects the distribution of the terrestrial masses, it
provides an essential constraint in the development of crust and mantle models
The heterogeneous structure of the uppermost layers of the earth is recognized
directly by the distribution and composition of the topographic masses. There is
a pronounced difference between the mean elevation of the continents (about
0.5 km) and the mean depth of the oceans (about 4.5 km). Ocean depths
increase with growing distance from the ocean ridges (mean depth around 2.5
km) due to thermal cooling and contraction of the oceanic lithosphere with seafloor spreading, cf. [8.2.3]. Consequently, the age of the oceanic crust is at most
200 million years, while the continental crust dates back 4 billion years
(CATENA VE 1995).
The crust is composed of a variety of sedimentary, igneous (effusive and
intrusive), and metamorphic rocks. Density changes occur primarily between
different types of rock, but larger density variations are also found within the
same rock material, especially in sediments. Density estimates are based on
surface rock samples, borehole probes, and the relationship between density and
seismic wave velocities (ST. MUELLER 1974). The upper mantle has been
investigated by seismic methods especially, and three-dimensional models are
now available from seismic tomography (WOODHOUSE and DZIEWONSKI 1984).
. 200q...2200
GRANITE 2500...2800
jEDIMENTS 2200...2500
CONRAD-DISCONTINUITY
"v p > 6.5 Kms" 1
LOWER CRUST
BASALI.GABBRO 2700..3100
IOHOROVICIC-DISCONT1NUITY
V p >7.8 Kms' 1
Fig. 8.3. Structure of the earth's crust with rock densities (kg/m 3 ) and
velocities of seismic primary waves, after TORGE (1989)
The structure of the earth's crust and upper mantle may be briefly described as
follows (Fig.8.3):
A sediment layer with highly varying thickness is found in the uppermost
stratum in large parts of the crust. Seismic wave velocities and rock densities
(average value 2400 kg m"3 for consolidated sandstone) vary considerably in
this zone. In continental areas, the next lower layer of the upper crust consists
mainly of acidic rocks such as granite (mean density 27 00 kg m 3 ); primary
wave velocities vary between 5.9 and o.Skms"1. The lower crust is composed
of basic rocks such as basalt and gabbro (mean density 2700 kg m"3). Wave
velocities exceed o.Skms'1 and gradually increase to more than 7kms" 1 . The
boundary between the upper and the lower crust (Conrad - discontinuity) is
marked only in some parts of the continents, at depths of 10 to 20 km. Beneath
the oceans, consolidated sediments and basalt lava flows are found in the upper
part of the crust above a basaltic layer of 6 to 7 km thickness (TANIMOTO 1995).
A sharp (over a few km) change in seismic velocity (v p > 7.8kms~ 1 ) is found at
an average depth of 35 km on the continents and 10 km on the oceans. This
Mohorovicic-disconunuity (Moho) defines the boundary between crust and
mantle. Ultrabasic rocks (peridotite, with olivine as the main mineral
constituent) are assumed to be located below the Moho, with a density of 3300
to 3400 kg m 3 . The depth of the Moho is closely related to topography. On the
continents, it may be less than 20 km (Afar hotspot), reaching about 30 to 40
km at shields and platforms. Cenozoic mountain belts (Alps, Rocky Mountains,
Himalaya) are characterized by a crustal thickness of 60 to 80 km. On the
oceans, the crustal thickness is more constant. An extremely thin crust of a few
km is found at slow spreading and fracture zones, while a thick crust of about
20 km appears where hotspots (mantle plumes) are located under ridge axes, as
in southern Iceland. These variations of crustal thickness are mainly due to
isostasy and plate tectonics, cf. [8.2.2], [8.2.3].
A global description of the Moho-depth is provided by a spherical harmonic expansion up to
degree and order 20 (SOLLER et al. 1982). More detailed studies on the depth of the Moho are
available for the U.S.A. (ALLENBY and SCHNETZLER 1983) and for Europe (MEISSNER et al.
1987), among others. Three-dimensional models are available for the crust (MOONEY et al. 1998)
and for the upper mantle.
When considering the topographic masses and ocean waters as deviations from
hydrostatic equilibrium, the removal of topography and the filling of the oceans
should create an equilibrium figure, with a gravity field coinciding with the
normal gravity field, cf. [4.2.2]. However, from the systematic behavior of the
residual gravity field quantities, it follows that the visible mass excesses and
deficiencies are, to a large part, compensated by a corresponding mass
distribution in the interior of the earth (HEISKANEN and VENING-MEINESZ
During the arc measurement in Peru, cf. [1.3.2], Bouguer discovered that the deflections of the
vertical as computed from the masses of the mountains were larger than the observed values. In
the 19th century, the Survey of India (G. Everest) revealed significant differences between
observed and calculated deflections of the vertical caused by the Himalaya Mountains, the
computed values being several times larger than the observed ones. This observation was the
basis for the theory of isostasy developed by Airy and Pratt (see below).
The large-scale behavior of the Bouguer anomalies, cf. [6.5.3], is another indication for the
compensation of the visible mass anomalies. In mountainous areas, the Bouguer anomalies are
generally negative, reaching values as low as -2000urns"2, while positive values (up to
4000urns" 2 ) are common over the oceans. A correlation with the mean height or depth (mean
over several 100 km) can be demonstrated and approximated by a regression of
-1000 s"2/l000m height, and +1000 urns"2/1000m depth.
Finally, the geoid heights produced by the topographic masses would reach values of up to about
500 m (HELMERT 1884), whereas the observed values remain below 100 m.
The model of isostasy is used to explain these observations. It postulates that
the topographic masses are compensated in such a way that hydrostatic pressure
is achieved at a certain depth of compensation. The compensation depends on
the dimension of the topographic load and may be achieved by different
mechanisms. Loads of several 10 to 100 km dimensions are supported by the
strength of the lithosphere and are not isostatically compensated. Larger loads
may lead to an elastic flexure of the lithosphere. Large-scale topographic
features of several 100 km dimension, and more, are generally in isostatic
equilibrium, at least for the major part of the earth, the exceptions being areas of
postglacial isostatic rebound (see below) and the deep-sea trenches, cf. [8.2.3].
t -fj t
cor ist.
i"
VARIABLE p
|PW|
Fig. 8.4. Isostatic model of Airy
Fig. 8.5. Isostatic model of Pratt
The classical isostasy models of Airy and Pratt are based on the assumption that
isostatic compensation takes place locally in vertical columns only. Utilizing the
hydrostatic equation (8.5), the condition of isostasy then reads (with g = const.)
\ p dz = const.,
H+T
with z = depth, H = height of topography, T = depth of compensation. The
model developed by G.B. Airy in 1855 (also designated Airy-Heiskanen model)
is based on a crust of constant density p0 and varying thickness, where the
"normal" column of height H = 0 has the thickness T0 (Fig.8.4). The
continental topography ( H > 0) forms mountain "roots" (thickness d^ ), while
8 .2 The Upper Layers of the Earth
"antiroots" (thickness dx ) are found beneath the oceanic columns. The ocean
depth is denoted as t. In this way, the crust penetrates with varying depths into
the upper mantle: floating equilibrium. Neglecting the earth's curvature, the
following equilibrium conditions hold for the continental and the oceanic
(A, -A, K =(p 0 -/vX
with p0 = density of the crust, pM = density of the upper mantle, pw = sea
water density. With the conventional values
p0 = 2670kgm 3 ,
pM =3270kgm , pw = 1030kg m' , the thickness of the root and the antiroot is
dmDt=4A5H,doc=2.13t.
The thickness TQ of the normal column can be estimated from isostatic gravity
anomalies calculated on the basis of a certain depth of compensation, cf. [6.5.3].
For T0 = 30 to 40km , these anomalies generally do not depend on the height of
the topography. Hence, the depth of compensation is in good agreement with
the depth of the Moho as obtained from seismology, cf. [8.2.1].
The isostatic model of J.H. Pratt (1855, also called Pratt-Hayford model)
assumes a crustal layer of constant thickness TO and allows lateral changes in
density in order to obtain isostatic equilibrium (Fig.8.5). With the density p0
for the normal column ( H - 0 ), continental columns generate densities smaller
than p0 , while oceanic columns are denser. The equilibrium conditions for the
continents and the oceans are:
with /o = 2670kgm~3 and pw =1030kgm 3 the densities of the continental and
the oceanic columns are given by
The depth of compensation can be estimated from the behavior of residual gravity field quantities
calculated with depth dependent assumptions. By utilizing topographic-isostatically reduced
deflections of the vertical in the U.S.A., Hayford obtained minimum values for a compensation
depth of 1 13.7 km. This value is close to the thickness of the continental lithosphere, cf. [8.2.3].
A refined isostatic model was proposed by VENING-MEINESZ (1931). It admits
regional isostatic compensation by assuming that the upper layer behaves like
an elastic plate overlying a weak fluid. A surface load causes a flexure of the
plate, with a regional bending over a horizontal distance wider than the load
dimension (Fig.8.6). The amount of flexure depends on the distance from the
load and can be calculated from the load, the density contrast between the plate
and the substratum, and the elastic parameters (Young's modulus, Poisson's
ratio) of the plate (ABD-ELMOTAAL 1995).
I;:.. ' . '
Fig. 8.6. Regional isostatic model of Vening-Meinesz
Globally, the models of Airy respectively Vening-Meinesz describe the dominating isostatic
features, but lateral density variations (Pratt model) also contribute to isostasy in some regions. In
the compensated parts of the earth, isostatic anomalies vary non-uniformly about zero (maximum
values of about 500^ms 2 ), and they clearly indicate areas which are not in isostatic equilibrium.
On the other hand, they are not very sensitive with respect to a change of the model or variations
of the model parameters, which makes discerning of different models and estimating the absolute
depth of the compensation level difficult. Utilizing global models of the topography, cf. [6.5.2],
spherical harmonic expansions of the topographic-isostatic potential have been developed
(SNKEL 1936b). These models can be used for gravity field interpolation and for the analysis of
the observed gravity field (RUMMEL et al. 1988).
Incomplete isostatic compensation is found in some mountainous regions and in
the areas of strong pleistocene glaciation (Canada, Fennoscandia, Greenland,
Antarctica). Mountains may become overcompensated by large erosion, and as
a consequence a vertical uplift will occur. The melting of the ice masses after
the pleistocene glaciation has caused an isostatic imbalance, which has since
been compensated by postglacial rebound. This phenomenon has been
especially well investigated in Fennoscandia, where recent uplift rates reach 1
cm/year (KAKKURI 1986). As the rate of uplift depends on the viscosity of the
mantle, corresponding observations contribute to the determination of this
geophysical parameter (LAMBECK et al. 1998), cf. [8.3.3].
The theory of plate tectonics synthesizes a multitude of individual observations
of geological and geophysical nature. The theory integrates the concepts of
continental drift (WEGENER 1915) and sea-floor spreading (DffiTZ 1961, HESS
1962). According to this model, new oceanic crust is formed by uprising
basaltic magma at the axes of the mid-oceanic ridges, and it spreads out to both
sides of the rift system. The spreading sea-floor is characterized by stripes of
interchanging positive and negative magnetic anomalies aligned parallel to the
ridges, which indicate the reversal of the earth's magnetic field occurring
irregularly at intervals of tens of thousands to tens of million years (VINE and
MATTHEWS 1963). Radiometrie age determinations of the oceanic rocks show
that the age of the ocean floor increases with the distance from the ridge axes
and does not exceed 200 million years.
Before that time (Permian and Triassic), the supercontinent Pangaea, postulated by Wegener,
united all present land masses. Break up started during the Jurassic period, when Pangaea rifted
into Laurasia (today North America and Eurasia) and Gondwana (today South America, Africa,
India, Antarctica and Australia), with the Tethys Sea between them. This rifting process finally
led to the present distribution of the continents and oceans.
The spreading rates of the ocean floor (referring to geological time spans) can
be derived from the spacing of the magnetic anomalies and the rock age. They
vary between 2 cm/year (e.g., at the Reykjanes Ridge south of Iceland) and 15
cm/year at the East Pacific Rise, MINSTER and JORDAN (1978).
Plate tectonics (McKENZiE and PARKER 1967, MORGAN 1968) postulates seven
larger (Pacific, North and South American, Eurasian, African, IndianAustralian, Antarctic) and more than 20 smaller, nearly rigid lithospheric plates
which move against each other on the asthenosphere. The lithosphere includes
the earth's crust and the uppermost part of the mantle; it possesses a thickness
of 70 to 100 km under the deep oceans and 100 to 150 km under the continents.
The asthenosphere is characterized by low viscosity (resistance to flow within a
fluid), which allows a viscous flow on geological time scales (LE PlCHON et al.
1973). The plate boundaries can be identified by an accumulation of seismic
(earthquakes) and volcanic activity, where the boundary zones vary in width
between some 10 to some 100 km and more (Fig.8.7).
The movement of the tectonic plates can be described as follows (LE PICHON et
al.1973, LOWRIE 1997), Fig.8.8:
Fig. 8.7. Main lithospheric plates and direction of plate movements, AN
Anatolian, AR = Arabian, CA = Caribbean, CO = Cocos, NA = Nazca, PH
Philippines, SO = Somalia plate, after TORGE (1989)
The mid-ocean ridges represent diverging (constructive) plate boundaries where
new lithospheric material is formed from magma uprising from the
asthenosphere and pressed apart. When colliding with a another plate, the
cooled, heavier oceanic plate is forced to sink into the upper mantle
(subduction) where it is consumed at depths of around 700 km: converging
(destructive) plate boundary. This process creates deep-sea trenches and island
arcs (e.g., at the western and northern Pacific, subduction rate of about 9
cm/year at the Japan trench) or mountain ranges (e.g., the Andes). The collision
of two continental plates leads to the formation of mountain chains (e.g., the
Himalaya and the Alps). Transform faults with relative motion parallel to the
strike of the fault are usually found between ridge segments, but also occur
where two plates meet with shear movements (e.g., San Andreas Fault,
California, shear movements of several cm/year): conservative plate boundary.
About 85 percent of the earth's surface is covered by the (nearly) rigid plates,
while deformations are concentrated on the plate boundary zones. Thermic
convection (heat transfer by movement of molecules) flows in the mantle are
assumed to be the driving mechanism for the plate movements (RUNCORN
1962). Different theories exist on the size and the location of the convection
cells (whole-mantle or layered mantle convection).
The motions of the lithospheric plates on the spherical earth can be described as
a rotation of a spherical cap about an axis through a fixed point (pole of
rotation) with a certain angular velocity (GORDON 1995). From these
parameters, the relative plate motion (direction and magnitude) can be
calculated for any location. Geologically-recent (average over the last few
million years) plate velocities have been estimated from the spacing of the
MID-OCEANIC
Dinrr
SPREADING,
100150DEPTH
^UTHOSPHERE
ASTHENOSPHERE
MANTLE CONVECTION
Fig. 8.8. Motion of lithospheric plates at diverging and converging plate
boundaries, vertical scale exaggerated, after TORGE (1989)
magnetic anomalies across the mid-ocean ridges and from the azimuths of
submarine transform faults and slips from large earthquakes. Recent models
such as NUVEL-1 (DE METS et al.1990) and NUVEL-1A (DE METS et al.1994)
include the major plates (about 15), keeping either one plate fixed (NUVEL-1:
Pacific plate), or referencing the motions to a rotation-free system coupled with
the earth: no-net rotation (NUVEL-1 A). An absolute plate motion can be
derived by reference to the hotspots (SOLOMON and SLEEP 1974). Here, hot
material is rising from deep mantle plumes that do not participate in the plate
tectonic motions. Hotspots are characterized by surface volcanism and high heat
flow; examples are Hawaii, Iceland, and Afar (Ethiopia).
It has to be noted that the plate velocities derived from these models represent
the average over geological time spans. Furthermore, they suffer from the
choice of the plates used, from strong deformations occurring at the converging
plate boundaries, and from the existence of intra-plate deformations, as well as
from the assumption of non-moving hotspots. Geodetic measurements allow
determination of the actual plate motions and identification of local and regional
deformations at the plate boundaries and inside the plates, cf. [8.3.3].
The observed gravity field reflects the integrated effect of the mass distribution
inside the earth and reveals deviations from spherical symmetry and hydrostatic
equilibrium, cf. [8.1]. Static and dynamic geophysical earth models must fulfill
the constraints imposed by the gravity field. The inverse problem, i.e., the
determination of the density distribution from the external gravity field, on the
other hand, cannot be solved uniquely (SKEELS 1947, MARTINEC 1994). This
fact is seen in Stokes' theorem, where the external gravity field of an
equipotential surface is completely determined without knowing the distribution
of the internal masses, cf. [4.2.1], Consequently, gravity field interpretation
requires additional information through geophysical and geological data, where
seismically derived depths of bounding surfaces and the composition and
density of the masses play the major role.
Gravity field interpretation is based on residual gravity field quantities obtained
by reducing the effect of the normal gravity field, cf. [6.1], and eventually also
on the well-known gravitation of the uppermost layers of the earth, cf. [6.5.3].
The primary gravity-field parameters used for interpretation are gravity
anomalies and, to a limited extent, deflections of the vertical, as well as geoid
heights. The effect of the masses on gravity anomalies and vertical deflections
is inversely proportional to the square of the distance, while geoid heights
depend on the reciprocal distance to the masses. Consequently, gravity
anomalies and deflections of the vertical are more suited for investigating the
density distribution in the upper layers of the earth. Gravity anomalies react
primarily to vertically extended masses, while vertical deflections reflect the
effect of horizontal layers, hence they especially support local investigations.
Geoid heights reveal deeper seated mass anomalies, which generally have large
The spectral decomposition of the gravity field as provided by the spherical
harmonic expansion, cf. [6.6.1], is especially appropriate for global and regional
interpretation. Degree variance models for the gravity anomalies (6.27) and the
geoid heights (6.142) show that about 95 percent of the geoid variance is
concentrated in degrees 2 to 10 (corresponding to wavelengths of 20000 to
4000 km), while this long-wave spectral part attains only 9 percent of the
gravity anomalies. Medium (degree 2 to 180) and short (degree 181 to 2000)
wavelengths, on the other hand, each contribute more than 40 percent to the
anomaly variance. Nearly 10 percent of the anomaly variance still stem from
wavelengths less than 20 km (degrees > 2000), reflecting small structures in the
upper crust (e.g., salt domes). Deflections of the vertical show a spectral
distribution similar to that of the gravity anomalies.
Hence, the interpretation of the geoid concentrates on the long and mediumwave part of the spectrum. Density and/or temperature anomalies are thought to
produce the low degrees of the spherical harmonic expansion, while mantle
convection and lithospheric structures are seen in wavelengths of thousands of
kilometers (PHILLIPS and LAMBECK 1980, BOWIN 2000). Shorter wavelengths
of a few 100 to 1000 km can be correlated with diverging and converging plate
boundaries and with hotspots (CAZENAVE 1994). Areas of postglacial rebound
or significant crustal thinning are also reflected in this spectral part.
Slow-spreading ridges and hotspots may exhibit relative geoid maxima of several meters. Deepsea trenches are characterized by narrow zones of geoid depression up to 5 to 20 m, followed by a
geoid rise along the island arcs (Fig.8.9). Postglacial land uplift areas show a geoid depression (up
to 10 m in Fennoscandia), which is correlated with present uplift rates (BJERHAMMAR 1981),
Fig.8.10. The Ivrea body (western Alps) is an example of a local geoid rise (up to 9 m) due to the
Fig. 8.9. Geoid structures at the Japan
subduction zone, EGM96 geoid
model, spectral part degree 11 to 360,
contour line interval 1 m, after
LEMOiNEetal. (1998)
Fig. 8.10. Geoid structure at the
Fennocandian land uplift area,
EGM96 geoid model, spectral part
degree 11 to 360, contour line interval
1 m, after LEMOINE et at. (1998)
shifting of lower crustal/upper mantle material to a shallow level ( 1989).
Modelling of the long-wave geoid structures can be based on equivalent point
masses arranged primarily around the bounding surfaces as determined by
seismology, cf. [8.1], BowiN (1994). The solutions strongly depend on the
choice of the spectral part to be modeled, the distribution of the masses, and the
introduced density differences.
The interpretation of gravity anomalies uses either the spherical harmonic
expansion (global and large-scale investigations) or local models based on
observed or gridded data. Different types of gravity anomalies can be used for
regional and local investigations.
Point free-air anomalies depend strongly on height and are not suited for
interpretation. The long and medium-wave part provided by global models, or
corresponding mean anomalies, on the other hand, can be exploited, due to the
smoothing of the high frequencies. The free-air anomalies then may be
interpreted as isostatic anomalies with a compensation depth of zero. Structures
of plate tectonics (e.g., subduction zones) and postglacial rebound can be
identified, where proper filtering again may be necessary.
Bouguer anomalies are employed for regional and local investigations, as they
are free from the effect of topography, cf. [6.5.3]. They mainly reflect density
anomalies in the crust and upper mantle and can be correlated with tectonic
structures such as ocean ridges, deep-sea trenches, continental grabens, youngfolded mountains, and with upper mantle structures (KOGAN and McNuir
1993). Isostatic compensation is indicated by the large-scale systematic
behavior of the Bouguer anomalies, with negative values in the mountains and
positive values in the oceans, cf. [8.2.2], Fig.8.11. Bouguer anomalies play an
important role in geophysical prospecting (DOBRIN 1976).
Fig. 8.11. Regional Bouguer gravity anomaly map of the United States,
composed of wavelengths longer than 250 km, referred to IGSN71 and GRS67,
rock density 2670 kg/m3 , contour line interval 200 /s2, after KANE and
More detailed investigations of the deeper regions of the crust are made
possible by further reducing the effects of known or assumed mass
distributions. Isostatic anomalies take the effect of the compensating masses
into account; deviations from zero indicate areas of isostatic imbalance and are
often correlated with geological features (SIMPSON et al. 1986). The reduction
of geologically known structures ("crustal stripping") allows, among others, the
estimation of the depth of sedimentary basins and of the crust-mantle boundary
(HAMMER 1963, TOMODA and FUJIMOTO 1981).
Gravity anomalies have been used for modeling a variety of crust and mantle
structures of local and regional extent. Here, a starting model is iteratively
improved by varying the geometry and densities of the masses, taking the
constraints from seismic data and geology into account. We mention the
Aglpms )
Fig. 8.12. Bouguer and free-air
anomalies across the Mid-Atlantic
ridge and crustal density model, with
oceanic layers (2600 and 2900
kg/m 3 ), low-density zone (3150
kg/m 3 ), and upper mantel (3400
kg/m 3 ), after TALWANI et al.
Fig. 8.13. Generalized free-air
anomalies (Bouguer anomalies on
land) across the the Puerto Rico
trench and crustal density model, with
unconsolidated (2000 kg/m 3 ) and
consolidated (2400 kg/m 3 ) sediments,
upper (2700 kg/m 3 ) and lower (3000
kg/m 3 ) crust, and upper mantle
(3400 kg/m 3 ), after TALWANI et al.
following from the relationships found between the gravity field and crustal
structures (KAULA 1992, NEREM et al. 1995):
Oceanic ridges show negative Bouguer anomalies (up to -2000//ms"2) due
to high anomalous mantle material, while free-air anomalies deviate only
slightly from zero, Fig.8.12.
Deep-sea trenches are characterized by strong, negative free-air anomalies
(up to -4000 s 2 ) , which are explained in part by thick sedimentary
layers and sea floor topography. Further inland, large positive anomalies
occur due to cooling of the subtracted material, Fig.8.13.
Continental grabens are correlated with strong negative Bouguer anomalies
due to sedimentary layers and/or anomalous mantle material; local highs
may occur through crustal thinning.
Young, folded-mountains arising at continental collision zones exhibit
negative Bouguer anomalies, indicating isostatic compensation. Isostatic
anomalies may differ from zero due to recent tectonic processes and
incomplete compensation.
Postglacial rebound areas are characterized by negative free-air anomalies.
8.3 Geodesy and Recent Geodynamics
From repeated measurements, geodetic and gravimetric techniques allow
detection of temporal variations of the orientation, surface, and gravity field of
the earth. The observed changes are used for modeling the effects on geodetic
data and for referring the data to common reference epochs. The variations
contain, on the other hand, valuable information on geodynamic processes of
global, regional, and local scales. Geodesy thus contributes to research in
geodynamics, in collaboration with astronomy, oceanography, meteorology,
solid earth geophysics, and geology.
In order to obtain significant results for geodynamic modeling, the temporal and spatial resolution
of the observations as well as their accuracy must be tuned to the frequencies and amplitudes of
the signals to be detected. For this reason, observation stations and platforms must remain stable
in position over time, or their changes accurately monitored.
Space-geodetic observations provide changes of the earth rotation due to the
redistribution of masses in the atmosphere, the hydrosphere, and the solid earth
[8.3.1]. Satellite altimetry, together with tide gauge measurements, represents a
powerful tool for the study of sea level variations [8.3.2]. Recent crustal
movements are observed by space and terrestrial techniques; they serve as
constraints for the modeling of geodynamic processes and as precursor
phenomena [8.3.3]. Observed gravity variations contain the integral effect of
terrestrial mass redistributions and support and supplement the geodetic
observations [8.3.4]. Continuous records of geodetic and gravimetric data are
especially useful for earth tide research [8.3.5].
Extensive treatises on the contributions of geodesy to geodynamics research are
found in LAMBECK (1988) and MUELLER and ZERBINI (1989).
Temporal changes of the earth rotation vector with respect to the earth's body
are described by polar motion and variations of the length of the day (LOD).
These changes proceed at different time scales and reach the 0.1" respectively
ms order of magnitude. The integral effect of the LOD variations over time
leads to larger deviations of the rotational time UT1 from the atomic time TAI
(DICKEY 1995), cf. [2.5.2]. VLBI and satellite techniques, including GPS,
allow determination of the earth rotation changes with a resolution of a few
hours and an accuracy of better than 0.001" respectively 0.01ms
(ROTHACHER et al. 1999), cf. [2.3].
The observed rotation changes mirror the combined effects of terrestrial mass
redistributions, which affect the inertia tensor of the earth, and of the related
motion, which acts on the respective angular momentum of the layer (e.g.,
atmosphere, oceans, continental waters, mantle, core). The law of the
conservation of the total angular momentum of the earth then requires
corresponding changes of the rotational vector. The integral effect of the
redistribution of masses can be recovered by repeated gravity field observations,
From the many sources of rotational changes, only a limited number have been
clearly identified. The mass redistribution mainly affects polar motion, while
LOD variations are dominated by the mass motions (CHAO 1994).
Secular effects have been found in polar motion (redistribution of ice masses
and postglacial rebound) and in LOD (tidal friction in the oceans). Decadal
effects could be due to core/mantle processes but also to long-term atmospheric
variations. Interannual variations are especially pronounced in LOD; they are
produced by changes in the atmospheric angular momentum. The Chandler
wobble in polar motion is excited from atmospheric mass redistributions, but
strong earthquakes may also play a role (CHAO et al. 1996). Annual,
semiannual, and seasonal effects stem mainly from mass shifts in the
atmosphere and in the oceans, together with changes in the groundwater layers,
in ice/snow coverage, and in the water level of large lakes.
An outstanding example of a large-scale interannual effect is the El Nino phenomenon. It
manifests itself by strong disturbances of the atmospheric and oceanic states in the tropical Pacific
(atmospheric pressure, wind, ocean temperature, sea level inclination in west-east direction due to
ocean warming and shift of water masses), with effects reaching far beyond the Pacific. It recurs
after two to seven years with varying intensity; the last strong occurrence happened in 1997/1998.
El Nifio effects have been clearly identified in LOD (increase of several 0.1 to 1 ms) and in
irregular perturbations of polar motion (SALSTEIN et al. 1999).
Diurnal and subdiurnal effects, as well as more irregular variations in earth
rotation, are excited primarily by processes in the atmosphere and hydrosphere
and by earthquakes as well. Tidal deformations of the solid earth and the oceans
cause maximum rotation changes of a few 0.001" respectively some 0.1 ms;
they can be modeled with sufficient accuracy.
By using reasonable models to reduce the effects (tidal effects, atmospheric
circulation, ocean circulation, etc.), the residual part of polar motion and LOD
can be analyzed, providing information on further geophysical processes and
model deficiencies (WILSON 1995), Fig.8.14.
The nearly diurnal free -wobble (also free-core nutation) is a case of special interest (ZRN
1997b). Because the rotational axes of the fluid outer core and the earth's mantle are slightly
TOTAL VARIATION; ZONAL TIDES
ARE REMOVED
SEASONAL OSCILLATION
RESIDUAL OSCILLATION
iktft^ft^^
EFFECT OF ZONAL TIDES, PERIODS
GREATER THAN 35 DAYS
WWVWWW^^
Fig. 8.14. Length of day (LOD) variations, and separation of tidal and seasonal
effects, after IERS annual report 1998
misaligned, restoring forces occur at the elliptic core/mantle boundary that try to realign the axes.
This leads to a damped wobble of the rotational axis around the figure axis, with nearly diurnal
frequency in retrograde (opposite to the earth's rotation) direction and an amplitude of 0.00017".
In space, this corresponds to a nutation of the rotational axis around the axis of total angular
momentum. The effect has been observed with VLBI (HERRING and DONG 1994) and is also
recognized through resonant terms in the tidal analysis of gravity observations, cf. [8.3.5]. The
results are utilized for the improvement of the nutation theory, cf. [2.4.2], and for modeling the
earth's deep interior.
Variations of sea level with time are relevant in geodesy for the definition and
realization of height reference surfaces, especially the geoid, cf. [3.4.3]. The
determination and interpretation of sea level changes, on the other hand, is of
fundamental importance in order to better understand the ocean-atmosphere
interactions and their influence on climate. An acceleration of the global sea
level rise would especially indicate climate changes related to global warming.
The study of sea level variations thus requires close collaboration between
oceanography, meteorology, and geodesy.
Sea level changes occur at a wide range of temporal and spatial scales, with
amplitudes at the 0.1 to 1m order of magnitude (LisixziN 1974). Tide gauge
measurements and satellite altimetry allow a direct determination of these
changes, while oceanographic modeling utilizes meteorological and
oceanographic data, cf. [3.4.2]. Observed variations of the earth rotation and of
the earth's gravity field contain indirect information about the redistribution of
the oceanic water masses, cf. [8.3.1], [8.3.4].
Tide gauge records can be evaluated for the determination of the oceanic tides
and other short and medium-term phenomena, including the effects of
atmospheric pressure changes, storm surges, and meltwater inflow. Averaging
over long time intervals reveals long-term water level variations. For the 20
century, an average global rise of 0.1 to 0.2 m/100 years has been found, with
large regional and local scatter and decadal variations (WOODWORTH 1997),
Fig. 8.15.
Fig. 8.15. Sea level records for San Francisco and Brest (vertical offset
applied),and trend for the 20th century, after WOODWORTH (1997)
It must be emphasized that tide gauge data only provide relative water level changes, by a
superposition of absolute water level changes and vertical movements of the mareograph. Vertical
crustal movements are at the mm/year order of magnitude and may reach several mm/year and
more in areas of postglacial land uplift, cf. [8.3.3]; hence a long-term height control with mmaccuracy is required for tide gauges. This is achieved locally by leveling between bench marks
and globally by repeated GPS heighting. GPS sampling rates of a half to one year are typical, and
connection should be made to the International Terrestrial Reference Frame (ITRF). Repeated
absolute gravimetry represents an independent method for the determination of height changes
and delivers additional information about the internal mass redistribution, cf. [8.3.4], CARTER et
al. (1989).
Satellite altimetry provides a nearly global and quasi-continuous monitoring of
the sea level with cm-accuracy. Recent ocean tide models allow reduction of the
tidal effects with cm-accuracy (SHUM et al. 1997).The analysis of long-term
(several years) observation series improved the ocean tide models (SMITH et al.
2000b) and revealed a number of other phenomena, with variations of 0.1 to
0.3m (NEREM et al. 1997). This includes ocean basin-wide decadal and
interannual fluctuations, probably due to the shift of water masses (BOSCH
2001), Fig. 8.16. The annual cycle has a 180 phase shift between the northern
and the southern hemisphere caused by thermal expansion and contraction.
Interannual and seasonal variations can be correlated with the variability of
ocean currents such as the Gulf Stream meandering and eddies and the El Nino
phenomenon (sea level rise in the eastern and subsidence in the western
Pacific), TAPLEY et al. (1994).
*)'
2-tO'
Fig. 8.16. Mean drift rate (mm/year) of sea level change determined between
1992 and 2000 from TOPEX/Poseidon altimetry, from BOSCH (2001)
The combination of tide gauge data and satellite altimetry leads to significant
improvements. Here, the tide gauge data serve as ground truth and can be used
to remove biases in the altimetric results, which may be affected by altimeter
drift, systematic orbit errors, and differences between the tracking station's
reference systems. By combining the results of different altimetry missions the
spatial resolution is improved. Oceanographic and meteorological data such as
sea surface temperature, salinity, current velocities, and air pressure may also be
integrated.
Recent crustal movements (horizontal and vertical) are determined by repeated
geodetic measurements carried out at certain epochs, or continuously. The
observation sites have to be monumented, and their local stability over time,
with respect to changes caused of geological, hydrological, and man-made
phenomena, must be controlled. Present-day sub-centimeter accuracies require
reductions for the influence of body tides and ocean tide loading (DUCARME
and JANSSEN 1990), atmospheric pressure loading (VAN DAM and WAHR
1987), and earth's rotation changes (GlPSON and MA 1998).
On a global scale, space-geodetic networks (VLBI, SLR, GPS, DORIS) provide
geocentric coordinates for certain epochs and station velocities with mm/year to
cm/year accuracy (PLAG et al. 1998), where GPS plays an increasing role
(LARSON et al. 1997). The network solutions are combined by the IERS, and the
annual station velocities are then part of the International Terrestrial Reference
Frame ITRF, cf. [2.5.3]. The horizontal velocities are primarily due to the
motions of the tectonic plates (ROBBINS et al. 1993), while the detection and
interpretation of vertical motions is still in its infancy (SOUDARIN et al. 1999).
For stations located in the interior region of the tectonic plates, the recently
observed horizontal velocities generally agree well with the velocities given by
geological/geophysical models as mean values over the past 3 to 10 million
years, cf. [8.2.3]. Larger discrepancies are found at the plate boundaries, as the
local deformations are not taken into account in the models, Fig.8.17.
-SO'
2 cnVa APKIM10.0
2cm/aNNRNUVEL-1A
Fig. 8.17. Recent plate motions as determined from geodetic observations
(APKIM 10.0 model) and the geophysical model NNR NUVEL-1A, courtesy
Deutsches Geodtisches Forschungsinstitut (DGFI), Mnchen
By combining global geodetic data sets, geodetic Actual Plate Kinematic Models (APKIM) have
been developed (REWES et al. 1992). These models assume the plates' interior as being rigid and
rotating on the earth's surface. They admit deformation zones at the plate boundaries that result
from the forces exerted by the adjacent plates. By interpolation, the actual velocity field is
provided in a lxl grid, with the condition that the integrated velocities over the whole earth's
surface are zero.
Regional crustal movements are derived from the repeated survey of national or
continental control networks and from dedicated networks set up in areas of
geodynamic activity.
Triangulation and leveling networks established in large parts of the world
between the end of the 19 and the second half of the 20th century are valuable
for the detection of long-term crustal movements (KAKKURI 1993b).
Systematically repeated surveys of these classical networks have been carried
out in high earthquake risk areas such as California and Japan. Higher repetition
rates became possible with Laser distance measurements and mobile VLBI and
SLR-systems; this led to the detection of medium-term crustal movements
(STEIN 1987). GPS receivers are now the primary tool for investigations of
recent crustal movements, where measurement campaigns are carried out at
different epochs or permanent stations are operated (BEVis et al. 1997). After
strain analysis, the geodetic results can be correlated or combined with
geological and geophysical data (HOLT et al. 2000) and thus contribute to the
development of global or regional stress maps (ZOBACK 1992).
We mention some examples of large-scale monitoring of plate-boundary and intraplate
deformation. Triangulation began in Iceland (diverging plate boundary) in 1938, followed by
repeated angular and distance measurements, and GPS epoch observations have been carried out
since 1986 (VLKSEN 2000). In Japan (converging plate boundary), a continuously operating GPS
network with an average station distance of 25 km has been established (Tsuji et al. 1995), and
permanent GPS arrays are also in operation in California (shear movements at the San Andreas
Fault), BOCK et al. (1997). The strain field in the plate collision zones of southeast Asia (Eurasian,
Pacific/Philippine, and Australian plates, WILSON et al. 1998) and the eastern Mediterranean
(Eurasian, African, and Arabian plates, KAHLE et al. 2000) have been derived by repeated GPS
campaigns.
Large-scale vertical crustal movements are found in areas of postglacial
rebound, recent mountain building, continental erosion, and sedimentary
subsidence. While geometric leveling only allowed repetition rates of several
decades, GPS-heighting (epoch measurements or permanent stations) offers the
possibility to determine elevation changes with high temporal resolution.
Repeated gravity measurements can also be employed, cf. [8.3.4].
Leveling and GPS-heighting refer to different reference surfaces (geoid and ellipsoid,
respectively). As the geoid is affected by mass redistributions, the leveling results should be
io u
20 u
Fig. 8.18. Land uplift (relative to sea level) in mm/year, as determined from
repeated geodetic leveling, after KAKKURI (1986)
reduced accordingly, applying the formulas for gravity field modeling on estimated or observed
gravity variations with time, cf. [6.6.1], [6.7.1]. Even with large mass shifts, as in Fennoscandia,
this reduction remains significantly below 10% of the height changes and can generally be
neglected (EKMAN 1993).
Fennoscandia is among the examples of well-surveyed areas of recent vertical movement, where
the postglacial land uplift has been investigated by leveling, sea level data, and gravimetry. The
apparent land uplift reaches a maximum of 9 mm/year with respect to mean sea level (Fig.8.18)
and is associated with a viscous inflow of mass in the upper mantle (EKMAN and MKINEN 1996),
cf. [8.2.4]. Repeated leveling in the Swiss Alps revealed mountain uplift rates of 1 mm/year and
more. These uplifts were caused partly by isostatic rebound after erosion and partly by
compression at the European/African plate boundary (UBLER et al. 1981). From leveling along
the German North Sea coast, a land subsidence of 0.5 to 1 mm/year is suspected, with significant
local effects at the river estuaries (LEONHARD 1987).
Local investigations in earthquake and volcanic risk areas have generally
employed a multitude of techniques (terrestrial distance measurements,
leveling, GPS, strain and tilt measurements, gravimetry) in order to detect
precursor phenomena and to monitor surface deformations during and after
activity phases (RlKITAKE 1982). In seismotectonically active zones, the data
provide information on the accumulation of strain energy, its release during an
earthquake, and the relaxation that follows the quake (LARSON 1995, HUDNUT
1995). Significant deformations have been found in connection with larger
earthquakes (BOCK et al. 1993). Preseismic uplift has been explained as the
result of the generation of dilatancies, which occur through the opening of
micro-cracks and the subsequent filling with underground water. With
volcanoes, crustal deformation due to magma injection and outflow is easier to
detect, and the detection plays an important role in forecasting volcanic
eruptions. Continuous strain and tilt measurements could indicate precursor
effects shortly before a phase of activity but may be strongly deteriorated by
meteorological and hydrological disturbances (TAKEMOTO 1991). The
combination of pointwise measurements (GPS) and interferograms generated
from synthetic aperture radar (SAR) images taken at different times (e.g., at the
35 days repeat cycle of the ERS satellites) is a promising tool for obtaining
surface information related to vertical ground deformation (MASSONNET et al.
5cm /-Icm }
65'N
20'W
ia'W
Fig. 8.19. Horizontal crustal deformations 1987 - 1990 in northwestern Iceland,
as determined from repeated GPS measurements , with (assumed) non-movable
stations in the Krafla fissure swarm, A = Askja volcanoe, K = Krafla volcano,
after VLKSEN (2000)
The San Andreas Fault, California, is among the earthquake areas where co and inter-seismic slip
rates are regularly determined by geodetic methods (SAVAGE et al. 1987, GAN et al. 2000). The
Krafla (northern Iceland) rifting episode (1975-1984) and the following period of relaxation has
been monitored by terrestrial measurements, GPS, and gravimetry (TRYGGVASSON 1994). The
rifting process was triggered by the in and outflow of magma in a shallow magma reservoir,
which caused repeated inflation and deflation of the Krafla volcano (BJRNSSON 1985). During
the rifting episode, large horizontal (several m) and vertical (up to 1m and more) crustal
movements have been observed, which decreased to a few cm/year along a narrow zone around
the Krafla fissure swarm during the stress relaxation phase that followed. Eventually, the motions
approached the average plate-spreading rate of 2 cm/year (JAHN et al. 1994), Fig.8.19. Satellite
radar interferometry identified a post-rifting subsidence (several mm to 2 cm/year) above the
magma chamber and along the spreading segment, due to cooling contraction and ductile flow of
material away from the spreading axis (SIGMUNDSSON et al. 1997). Geodetic monitoring of
volcanoes is described by DVORAK (1995). Among the routinely surveyed active volcanoes are
the Kilauea and Mauna Loa, Hawaii (OWEN et al. 2000), Long Valley, California (RuNDLE and
WHITCOMB 1986), and Mount Etna, Italy (BONACCORSO et al. 1995).
Man-made vertical crustal movements are related to the exploitation of natural
gas, oil, and geothermal fields, the withdrawal of groundwater, mining, and load
changes in water reservoirs. They are of more local character and generally
result in surface subsidence. Monitoring is carried out by leveling and GPSheighting, and gravimetry also contributes, cf. [8.3.4].
8.3.4 Gravity Variations with Time
Gravity variations with time result from a multitude of sources, cf. [3.5]. Here
we consider only the variations that are caused by the redistribution of
terrestrial masses and thus provide information on geodynamic processes, cf.
[3.5.3]. Gravimetric tidal effects will be discussed within the frame of earth
tides, cf. [8.3.5]. More details are found in TORGE (1989).
Global gravity variations can be determined by comparing the harmonic
coefficients of the gravitational field obtained from the analysis of satellite
orbits of different epochs, cf. [6.6.2]. Changes of the low degree zonal
harmonics (up to degree 5) have been found from laser satellites (Lageos,
Starlette, Ajisai) over time intervals of 10 to 20 years (NEREM and KLOSKO
1996). The change of the dynamical form factor J2 =dJ2/dt = -2.7xlO~"/e is
especially pronounced and suspected to result from postglacial rebound. Annual
and semiannual variations of the second-degree harmonics are related to mass
redistributions in the atmosphere, the oceans, and the continental groundwater
(CHENG and TAPLEY 1999). If the harmonics of degree one are included in the
evaluation, variations indicate the movement of the geocenter with respect to
the terrestrial reference frame realized by the coordinates of the tracking
stations, cf. [2.5.2], [3.3.4].
A terrestrial network of absolute gravity stations well distributed over the earth's surface and
repeatedly observed at time intervals of 10 years or more could also provide long-term global
gravity changes (MATHER et al. 1977). This is intended by the International Absolute Gravity
Baseslation Network (IAGBN) proposed by IAG (BoEDECKER and FRITZER 1986). The majority
of the 36 scheduled stations have been installed, but only a few observation have been repeated.
Terrestrial absolute and relative gravity measurements are required in order to
detect regional and local gravity changes with time. Corresponding
investigations concentrate on areas where recent height changes occur, caused
by postglacial uplift, mountain building, earthquake and volcanic activity, and
man-made land subsidence, cf. [8.3.3].
In order to detect and analyze the small gravity changes associated with
geodynamic phenomena, a high observational accuracy and the adequate
reduction of tidal effects and local disturbances is required. Present gravimetric
earth tide models allow sufficient reduction of the tidal effects for most parts of
the continents, cf. [8.3.5]. Polar motion only affects absolute gravimetry and
can be reduced. This is also valid for the effect of atmospheric pressure
changes, which can be modeled by a Bouguer plate, modified through the
deformation caused by the load of the atmospheric masses (SPRATT 1982), cf.
[5.4.1]. For relative gravity measurements, atmospheric pressure effects
generally cancel, but they must be taken into account for continuous gravity
recording, cf. [8.3.5]. Temporal variations of the ground water level and soil
moisture mainly occur seasonally, with superimposed short-term fluctuations of
a few hours to a few days. They may produce maximum gravity changes of 50
to 100 nms"2 (seasonal) and several lOOnms 2 (strong rainfall). For simple
hydrological structures, a reduction is possible by applying the Bouguer plate
with P = pore volume (%), and = change of groundwater level in m. A
corresponding relation holds for the soil moisture reduction. The lack of
groundwater and soil moisture data, as well as deficiencies of this simple
model, generally prevent an adequate reduction of these hydrological effects
(M KINEN and TATTARI 1988).
Observed gravity changes contain the combined effect of internal mass
redistributions and a vertical shift of the observer, provided that earth tides,
atmospheric pressure, and groundwater/soil moisture effects have been reduced
properly. Hence, they can be transformed to height changes if a reasonable
model for the internal mass redistribution is available (STRANG VAN HEES
1977, BIRO 1983). Assuming a linear relationship between gravity and height
changes, the conversion factor may vary between -1.5 and -3.58~2/
(JACHENS 1978). The free-air relation of -38"2/ is often found locally and
corresponds to a vertical shift without mass changes (e.g., dilating sphere). For
larger areas, the Bouguer-plate relation of -28~2/8 more typical,
indicating internal mass displacements according to the Bouguer plate model,
cf. [6.5.3]. Simultaneous measurements of gravity and height variations allow
the conversion factor to be determined. Gravity data can then be used for the
determination of height changes, and the geophysical model for the internal
mass redistribution can be verified (TORGE 1986).
Postglacial uplift has been monitored in Fennoscandia for some decades by relative gravimetry in
connection with leveling, resulting in a mean conversion factor of -2nms"2/mm (EKMAN and
GRAVITY (prns
-0.1 -0.2 -0.3
-! YEAR
Fig. 8.20. Absolute gravity variations from 1987 to 1995 at Churchill,
Manitoba, Canada, after LAMBERT et al. (1996)
M KINEN 1996), cf. [8.3.3], In Canada, repeated absolute gravity measurements yielded gravity
changes of -10 to -20nms~2/year , in agreement with geophysical rebound models (LAMBERT et
al. 1996), Fig. 8.20.
Gravimetric networks established in the Central Andes (subduction of the Nazca plate) and in the
Venezuelan Andes (Caribbean/South American plate boundary) are examples of investigations at
tectonic plate boundaries (BECKER and GROTEN 1998, DREWES et al. 1991). While local effects
(subsidence of sedimentary basins, variations related to the rainy and the dry season) have been
identified, long-term gravity changes caused by plate tectonic processes are still difficult to
extract over time spans of several years.
Gravimetry is extensively employed in areas of seismoleclonic and volcanic activity, as in
California (JACHENS et al. 1983) and in Japan (SATOMURA et al. 1986); the results are, in many
cases, well correlated with changes in elevation and strain. An absolute gravity network
established in Yunnan (collision zone of the Indian and the Eurasian plates) and observed several
times between 1990 and 1995 is intended to monitor long-term changes at this region of high
earthquake risk (TORGE et al. 1999). The Krafla rifting phenomena in northern Iceland mirrored
itself in gravity and height variations with significant correlation, but forerunners are hard to
detect (TORGE et al. 1992), Fig. 8.21, cf. [8.3.3]. Occasionally, gravity changes of a few 52
have been observed before strong earthquakes, one example being the 1976 Tangshan/China
(M=7.8) earthquake (Li et al. 1989). Volcano monitoring, in many cases, includes gravimetry,
which has proved to be an efficient tool for detecting magma inflation and deflation and to
contribute to eruption forecasting and observation of the phase of activity (RYMER and LOCKE
Repeated gravity measurements also contribute to the investigation of land subsidence caused by
man-made activities. This includes the exploitation of geothermal fields (ALLIS and HUNT 1986),
of natural gas and oil (VAN GELDEREN et al. 1999), the withdrawal of groundwater, and the
effects of mining (LYNESS 1985).
6H(m)
Fig. 8.21. Long-term gravity and height changes along an EW-profile ( =
6540') in northern Iceland, related to the Krafla rifting episode, after TORGE et
al. (1992)
Tidal effects on a rigid earth can be calculated from the ephemerides of the
moon, the sun, and the planets, cf. [3.5.2]. In reality, the solid earth reacts to the
tidal forces like an elastic body: Earth tides (also body tides). The ocean tides
follow the law of hydrodynamics, with strong disturbances occurring in the
adjacent seas and along the coast lines. Tidal phenomena, especially earth tides,
are discussed extensively in MELCHIOR (1983), BAKER (1984), and WILHELM
et al. (1997).
The tidal response of a spherically symmetric, non-rotating, oceanless earth can
be described by applying the dimensionless Love numbers h, k, l (I also called
Shida number) to the individual terms of the series expansion of the tidal
potential vt (3.116), WAHR (1995).
The vertical deformation of the earth's surface is given by (Fig.8.22)
where the shift of the level surface follows from (3.52):
The horizontal displacement in NS and EW-direction is obtained by
' g cos0Jd/t
with , = geocentric latitude and longitude.
W+ V,+ Va= const.
W + V, = const.
DEFORMED
UNDEFORMED
W= const.
Fig. 8.22. Vertical shift of a level surface and the physical surface of the earth
caused by the tides
The tidal-induced mass shift causes an additional deformation potential which
is proportional to the tidal potential:
Vd=k.V,.
The Love numbers depend on density and elastic properties (compressibility,
rigidity) of the earth's body, and they are functions of the radial distance from
the earth's center. Restricting ourselves to the dominant (about 98%) term of
degree two in the spherical harmonic expansion, we have for a spherical earth
model with solid mantle, liquid outer and solid inner core, cf. [8.1]:
h = 0.60, / = 0.08, k = 0.30.
The gravitational potential on the earth's surface experiences a tidal-induced
change which is composed of the direct attraction (3.117), the deformation part
(8.18), and the potential change due to the vertical shift of the surface (8.16):
Differentiating with respect to the radial distance r delivers the vertical
component of the tidal acceleration. According to (3.117), (3.118) we have
Expressing the deformation potential through a spherical harmonic expansion of
degree two (3.89), and with (8.18) we obtain
-^- = --Vd= ~-kVt.
Inserting (8.20) into (8.19) and taking (3.118) and (8.16) into account yields the
vertical acceleration component
with the gravimetric (amplitude) factor
already introduced in (5.97).
The horizontal component of the tidal acceleration and its relation to the
corresponding component on a rigid earth (3.119) follows from
with the //// (amplitude)4/o;ctor
r = l + k-k.
With the above model values for the love numbers h and k we obtain
<J = 1.15, y = 0.69.
Hence, the gravity change observed on an elastic earth is larger than on a rigid
one, which is due to the vertical shift of the observer. The tilt factor mirrors the
flexibility of the earth's surface with respect to the tidal force.
Rotation and ellipticity of the earth cause a slight latitude dependence of the
tidal parameters (WAHR 1981). Mantle anelasticity results in a frequency
dependent amplitude increase and a phase delay of the tidal response (DEHANT
1987); lateral inhomogeneities should be visible, with a dependence on position.
The resonance effect caused by the liquid outer core (nearly diurnal free
wobble, cf. [8.3.1]), is reflected in the diurnal tidal band by a frequency
dependence of the Love numbers. The deviations from a spherically symmetric
earth model remain below 0.1%.
Earth tide observations include the effect of the ocean tides, which is composed
of the direct attraction and the loading effect of the water masses (JENTZSCH
1997). The ocean tide effects are especially pronounced in the semidiurnal tides
and reach several percent at stations located in the continents' interior. Close to
the coast, they may assume up to 10% in the gravimetric tidal signal, several
10% in strain, and 100% and more in tilt, with corresponding phase shifts.
The gravitational and the loading part of the oceanic tidal effect can be calculated from ocean tide
models, cf. [3.4.2]. Here, the ocean load is considered as a surface layer and expanded into
spherical harmonics where a development up to degree 10000 is required. The reaction of the
earth's surface to the load is described by load Love numbers, which depend on the underlying
earth model (FARRELL 1972). A convolution of the ocean tidal model and Green's functions (load
Love numbers weighted according to the distance from the load) provides the loading effects in
gravity, strain, and tilt for each partial tide.
Tidal perturbations on the earth's surface reach 1 to 28 2 in gravity, 0.01" to
0.02" in tilt, and 10~7 to 10~8 in strain, see the estimates for a rigid earth in
[3.5.2]. Hence, they are clearly visible in geodetic data series and, after an earth
tide analysis, can be evaluated in order to
derive tidal reductions for geodetic (VLBI, SLR, LLR, GPS, precise
leveling etc.) and gravimetric measurements,
verify and improve global earth models with respect to deviations from
spherical symmetry, after subtraction of the oceanic tidal effects,
verify ocean tide models after subtraction of the body tides,
investigate local changes in the tidal parameters associated with
earthquakes and volcanic activity.
While a relative accuracy of about 1% and 0.5 generally suffices for tidal
reductions, 0.1% respectively 0.05 and better is required for model
verification and geodynamic investigations.
Gravimetric earth tide observations especially contribute to tidal analysis
(WENZEL 1997b). Here, elastic-spring type gravimeters provide the shortperiodic partial tides (Fig.8.23), and superconducting instruments also deliver
long-periodic tides and the pole tide (Fig.8.24), cf. [5.4.6].
earth tide station Hannover no. 709
gravity Bieter LaCoste-Rbmberg no. 298
Fig. 8.23. Gravimetric earth tide record, obtained with Lacoste and Romberg
gravimeter G298, Institut fr Erdmessung (IfE), University of Hannover
Fig. 8.24. GWR superconducting gravimeter drift, with (top to bottom) raw
gravimeter signal, earth tides reduced signal, atmospheric pressure, and gravity
residuals after removal of earth tides and atmospheric pressure effects, courtesy
GWR-Instruments, Inc., San Diego, CA, U.S.A.
As an example, Tab.8.1 contains the gravimetric tidal parameters for a selected number of partial
tides (out of a total number of 57 analyzed wave groups) from a long-term observation with a
superconducting gravimeter (DITTFELD 2000), cf. [3.5.2].
The main results of the gravimetric earth tide analysis are (WENZEL 1997b):
The average noise level amounts to a few O.Olnms 2 for the short-periodic
tides and may reach lOnms"2 for the annual tide, which characterizes the
high observational precision.
The separation of the small SI wave mirrors the quality of the analysis, as
the diurnal tides are strongly contaminated by meteorological effects.
Tab. 8.1. Adjusted Gravimetric Earth Tide Parameters (selection, rounded
values), Earth Tide Station No.765, GFZ/Potsdam ( = 52.38, = 13.07E,
// = 81m), Superconducting Gravimeter GWR TT70 No.018, recording time
June 1992 to October 1998 (2250 days), after DITTFELD (2000)
Ampl. Factor
Phase Lead
nms 2
Long-periodic
Semidiurnal
Terdiurnal
Standard deviation (short and long-periodic tides adjusted): 9nms 2 , only short-periodic tides:
0.8nms 2 . Air pressure regression coefficient -2.776nms"2/hPa , pole-tide -factor 1.13.
The standard deviations of the adjusted tidal parameters are approximately
inversely proportional to the amplitude of the waves. The amplitude factor
of the principal waves (Ol, PI, Kl, M2, S2) can be obtained with a relative
accuracy of about 0.01% and the phase lead with 0.01, and better. The
long-periodic tides (Mm, Mf) are accurate to a few % and a few degrees.
After reduction of the oceanic tidal effects, the tidal parameters are in close
agreement with advanced geophysical earth models. A dependence on
latitude, lateral heterogeneities, or heat flow has not been clearly
recognizable (BAKER et al. 1996, Z RN 1997a),
The resonance effect of the liquid outer core can be clearly identified in the
diurnal tides (^^) and used for model verification.
Ocean tide effects manifest themselves in the semidiurnal waves and
provide useful constraints for oceanic tidal models (DUCARME and
MELCHIOR 1983),
Free oscillations (vertical component) of the earth with periods between 10
s and 54 min. as excited by strong earthquakes have been analyzed from
high resolution records and can be used for the support of global seismic
World-wide synthetic gravity tide parameters have been calculated for a lx 1 grid, based on the
WAHR (1981)/DEHANT (1987) body tide and the SCHWIDERSKI (1980) ocean tide models. With the
exception of strongly disturbed coastal zones, the synthetic parameters agree well with the
gravimetric earth tide observations and provide the gravimetric tidal reductions with the desired
accuracy (see above), TIMMEN and WENZEL (1994b).
The Global Geodynamics Project (GGP, 1997 - 2003) comprises about 20 stations equipped with
superconducting gravimeters that collect gravity data with a sampling rate of 1 to 10 s and an
accuracy of lnms~ 2 . The results will be used for improving earth and ocean models, should
provide information on geodynamic phenomena, and serve as ground truth for the GRACE
mission, cf. [5.2.8], CROSSLEY et al. (1999).
Strain and tiltmeter measurements fail to reliably determine the global or
regional tidal response of the earth, which is due to strong distortion by local
heterogeneities. But these measurements contribute significantly to local
investigations, cf. [5.5.4]. Long-term space-geodetic observations such as
VLBI, SLR, and GPS allow for direct monitoring of tidal displacements of the
station sites. Thus they can be used to directly derive the Love numbers h and 1,
the resonance parameters in the diurnal tidal band, and the ocean loading effects
(MATHEWS et al. 1997, SCHUH and HAAS 1998).
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Boll. Geof. teor. appl. = Bolletino di Geofisica teorica ed applicata. Osservatorio
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Bull. G6od. = Bulletin Geodesique. Springer, Berlin-Heidelberg-New York.
DGK = Verffentlichungen der Deutschen Geodtischen Kommission bei der
Bayerischen Akademie der Wissenschaften, Mnchen und Frankfurt a.M.
EOS = Transactions American Geophysical Union, Washington, D.C.
J. Geod. = Journal of Geodesy. Springer, Berlin-Heidelberg-New York.
JGR = Journal of Geophysical Research. American Geophysical Union, Washington,
Man. Geod. = Manuscripta Geodaetica. Springer, Berlin-Heidelberg-New York.
Mitt. BKG = Mitteilungen de Bundesamtes fr Kartographie und Geodsie, Frankfurt
OSU Rep. = Reports of the Department of Civil and Environment Engineering and
Geodetic Science, Geodetic Science and Surveying, The Ohio State University,
Columbus, Ohio.
Unisurv G = Univ. of New South Wales, Geodesy Reports, Kensington, N.S.W.
Wiss. Arb. Univ. Hannover = Wissenschaftliche Arbeiten der Fachrichtung
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ZfV = Zeitschrift fr Vermessungswesen. K. Wittwer, Stuttgart.
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