text:
# Proper acceleration

1. Jul 18, 2008

### yuiop

Hi,

Imagine a rocket accelerating in flat space. Let's say for the sake of argument it measures its acceleration by using an onboard accelerometer as (a).

(E1)What would a momentarily comoving inertial observer measure the acceleration of the rocket to be using his own local rulers and clocks as the rocket passes him?

(E2)What would an observer on the rocket measure the acceleration of a mass released from a somewhere near the nose of the rocket falling over a short distance? This measurement should be the same as the apparent acceleration (as in change of velocity per unit time) of the inertial observer in (E1) as measured by the observer in the rocket. This should also be similar to the average acceleration of a ball tossed up into the air by an observer onboard the rocket as measured by that same observer.

Using the equivalence principle, is the acceleration calculated by placing a mass on some weighing scales in a gravitational field exactly the same (in relativistic terms) as the acceleration measured by timing the fall time over a short distance so that the change in gravitational radius and velocity is infinitesimal?

Which of the above practical measurements is closest to the formal definition of proper acceleration?

2. Jul 18, 2008

### Al68

Both sound like measurements of coordinate acceleration instead of proper acceleration to me, since both would measure the second derivative of position wrt time (d^2 x/dt^2, or dv/dt), which is the definition of coordinate acceleration. I think the official definition of proper acceleration is the acceleration "felt" by an observer. (Change in momentum per unit mass/time). But they should be equal unless the coordinate acceleration is measured by an observer with relative velocity a significant fraction of c, wrt the observer being accelerated. In which case they would be related by sqr[1 + (at')^2/c^2].

Maybe someone else can post the equations in latex. And/or correct me if I got them wrong.

Al

Last edited: Jul 18, 2008
3. Jul 18, 2008

### Mentz114

If the momentarily comoving inertial observer always shares the proper time of the accelerating frame, then the acceleration will be measured as (a) at all times.

(a). Same as the icm observer.

Given all your caveats, I'm inclined to believe ( back of envelope) that they will give the same answer.

I can't work out how to use scales to measure acceleration. But a spring balance will give the same result as dropping and timing ( over a short enough distance).

[caveat]There are three ways known to define distance in Rindler space, but they converge for very short distances.

Last edited: Jul 18, 2008
4. Jul 19, 2008

### Jorrie

Hi Kev, interesting questions! Here's my view for what it's worth.
If one takes the rocket as pushed from the rear (as is usual), then the proper acceleration differs over the length of the rocket (even after transients have damped out), so you have to specify where the acceleration is measured. If you say "a rocket accelerating at a", one may perhaps assume that you mean the acceleration of the center of mass (COM) of the rocket.

Assuming the COM interpretation, a momentarily comoving inertial observer will measure the proper acceleration of the COM.

Again, you will have to specify where on the rocket the observer resides. My understanding is that the proper acceleration of the nose (in geometric units) will be a(x) = (1-ax)a, where x is the distance from the COM (or wherever you have measured a) to the nose. So, for an observer residing in the nose, the acceleration of the mass will then be be -a(x).

Does this make any sense in your scenario?

5. Jul 20, 2008

### yuiop

Hi Al68, Mentz114 and Jorrie! Thanks for all the thoughtful and helpful answers. :)

As always I have more questions :P

I agree that if the relative velocity is a significant fraction of c then the acceleration measured by an inertial observer would be different. I am however trying to analyse the situation where the relative velocity is an insignificant fraction of c and hence the specification of a momentarily comoving inertial observer. An example would be tossing a ball one meter into the air. The initial and final velocity of the ball would be insignificant relative to the the speed of light. The difference in gravity at the surface of the Earth and one meter above the Earth would also be almost insignificant as the radius of the Earth is much larger than one meter. For all practical purposes the kinematic acceleration of a ball thrown one meter into the air is a constant 1g going up and coming back down and even at the apogee where the ball is momentarily almost stationary.

Usually in descriptions of proper acceleration there is a assumption of a infinite amount of momentarily comoving observers, one for each "moment" and so your observation would seem to apply here.

That would be my intuition too. At the bottom of this post are some quotes from the mathpages website that seem to contradict this view (or more likely... I have misinterpreted what mathpages is saying)

Good point. I should have said spring balance rather than scales. ;)

Another good point. Assume measurement at the COM or a very short infinitesimal rocket.

Again, assume dx is much smaller than x where dx=x2-x1 and x2~x1~x where ~ means aproximately equal.
.
.
.
The real reason behind the questions in the OP is some statements and equations in the mathpages website which seem to contradict the assumption that acceleration measured by a momentarily comoving acceleration (Method E1) would not be the same as the proper acceleration (as measured by a spring balance type accelerometer onboard the rocket)

---------------------------------------------------------
From http://www.mathpages.com/rr/s7-03/7-03.htm

(Eq A): $$\frac{d^2 r}{d\tau^2}=\frac{-4m}{R^2(1+cos(\theta))^2}$$

"At q = 0 the path is tangent to the hovering worldline at radius R, and so the local gravitational acceleration in the neighborhood of a stationary observer at that radius equals -m/R^2, which implies that if R is approximately 2m the acceleration of gravity is about -1/(4m). Thus the acceleration of gravity in terms of the coordinates r and tau is finite at the event horizon, and can be made arbitrarily small by increasing m."

----------------------------------------------------------

It should be noted that he using a mixture of coordinate distance and proper time (tau) here. This method of measuring acceleration is interesting because it gives answers that are finite and real below the event horizon and suggests a particle can be momentarily stationary (apogee) at or even below the event horizon.

At the apogee the above equation can be expressed as:

(Eq A) $$\frac{d^2 r}{d\tau^2}=\frac{-m}{R^2}$$

It is not clear to me who would actually measure the acceleration to be -m/R^2 because it is not the proper acceleration measured by the hovering observer and it is not the measurent of a coordinate inertial observer at infinity who would measure the hovering rocket to be stationary.
---------------------------------------------------------
From the same page http://www.mathpages.com/rr/s7-03/7-03.htm

"However, this acceleration is expressed in terms of the Schwarzschild radial parameter r, whereas the hovering observer’s radial distance r' must be scaled by the “gravitational boost” factor, i.e., we have dr' = dr/sqrt(1-2m/r). Substituting this expression for dr into the above formula gives the proper local acceleration of a stationary observer

(EQ B): $$\frac{d^2 r'}{d\tau^2}=\frac{-m}{r^2\sqrt{1-2m/r}}$$

This value of acceleration corresponds to the amount of rocket thrust an observer would need in order to hold position, and we see that it goes to infinity as r goes to 2m."

---------------------------------------------------------

This last equation is what I would normally assume is meant by proper acceleration and is what would assume a spring balance type accelerometer would indicate. For values of R that are less than the event horizon radius the answers are infinite or imaginary suggesting the impossibility of having a stationary particle hovering below the horizon although it ws shown above that a particle can be momentarily stationary at an apogee, below the event horizon.

Ordinarily we would normally expect the acceleration indicated by a mass on a spring balance at the surface of the Earth to be the same as the acceleration indicated by dropping a mass a short distance of less than a meter and the equivalence principle indicates similar measurements would be made in an rocket accelerating with a constant proper acceleration of 1g. Mathpages suggests otherwise:

------------------------------------------------
From http://www.mathpages.com/rr/s6-07/6-07.htm

"At the apogee of the trajectory, when r = R, this reduces to

(Eq C): $$\frac{d^2 r}{dt^2}=\frac{-m}{R^2}\left(1-\frac{2m}{R}\right)$$

as expected. If R is infinite, the coordinate acceleration reduces to

(Eq D): $$\frac{d^2 r}{dt^2}=\frac{-m}{R^2}\left(1-\frac{2m}{R}\right)\left(1-\frac{6m}{R}\right)$$

"

------------------------------------------------

Both the above equations are the coordinate measurements of acceleration according to an observer at infinity (because they are derived directly from the Schwarzschild metric) and R is the radius where a projectile initially launched upwards away from the gravitational source momentarily comes to a stop, before falling back towards the gravitational source.

To try and summerise all the above as I see it (and I welcome clarifications) the proper acceleration measured by the accelerometer of a hovering observer is proportional to

(EQ B): $$\frac{-m}{r^2\sqrt{1-2m/r}}$$

and the local acceleration of a free falling particle that happens to be at apogee at the same radius as the hovering observer as measured by the hovering observer is:

(Eq A) $$\frac{-m}{R^2}$$

and the coordinate acceleration of that same particle according to an observer at infinity is:

(Eq C): $$\frac{-m}{R^2}\left(1-\frac{2m}{R}\right)$$

The problem is that the gist of the replies to this thread do not seem to agree that (Eq A) should be any different to (Eq B). Any ideas?

---------------------------------------------------------
"I have made this letter longer than usual because I lack the time to make it shorter." Blaise Pascal

6. Jul 20, 2008

### Jorrie

As I see it, Eqs. A and C are coordinate transforms of the proper acceleration (Eq. B) in curved spacetime. This does not quite agree with a rocket that accelerates in flat spacetime (Rindler coordinates)

Eq. A uses the local (static) observer's time, but the distant (infinity) observer's (Schwarzschild) radial parameter. Eg. B uses local time and radial parameters, while Eq. C uses distant time and radial parameters.

Last edited: Jul 20, 2008
7. Jul 20, 2008

### Al68

I think I just realized the point of your post, maybe. The object released in (E2) would actually be at rest wrt a momentarily comoving inertial observer (E1), and the proper acceleration of that object would equal zero, but the coordinate acceleration of the object (and the inertial observer in E2) as measured by the ship's observer should equal the proper acceleration of the ship.

But it seems from your last post that the coordinate acceleration of the ship as measured by an inertial observer may not equal the coordinate acceleration of that inertial observer (and a released object at rest with same observer) as measured by an observer on the ship. That seems very strange to me, too, unless I misinterpreted your point.

I think I should look closer at that page on mathpages and see what others have to say about it.

Al

8. Jul 20, 2008

### Jorrie

I think it must be a "locally static" observer, somehow using Schwarzschild coordinate measuring rods, but neither said observer nor rods can exist at or inside the event horizon.

9. Jul 20, 2008

### Mentz114

Kev,
any remarks I made may only apply to a uniformly accelerating frame. I wasn't thinking about curved spacetime. Rethink needed.

M

10. Jul 20, 2008

### Staff: Mentor

Hi kev,

As you know, I am not a GR expert, so take the following with a grain or two of salt.
a

a assuming that the length of the rocket is small enough to neglect time dilation etc. between the front and back. (or equivalently that the accelerometer is also in the nose next to the "falling" mass.

Yes.

I think that E1, the instantaneous acceleration in the momentarily co-moving inertial frame, is the definition of proper acceleration.

11. Jul 21, 2008

### yuiop

Hi Al, Mentz and Jorrie

It took me a while to figure out your point about the released object being at at rest with the momentarily comoving observer but it appears you are absolutely right. I am also coming to the same conclusion as you, that observers measuring the acceleration using methods E1 and E2 would obtain the same result as the proper acceleration indicated by the accelerometer in the accelerating rocket (or by a stationary hovering observer in a gravitational field as per the equivalence principle).

I think the confusion (in my head anyway) came from this equation from mathpages:

(Eq A) [tex]\frac{d^2 r}{d\tau^2}=\frac{-m}{R^2}[tex]

but because it a measurement made from a mix of proper time of the falling object and coordinate length it is not the actual sensible measurement of acceleration measured by any single observer and is probably best ignored.

As I mentioned above I do not think the measurement made by a mix of proper and coordinate measuring devices is a sensible measurement.

As mentioned above, I tend to agree, after further reflection, that proper acceleration measured by an onboard accelerometer and by methods E1 and E2 all yield the same answer (a).


text:
1800 102 5661

Title Real Analysis (Code-09050501)

 S.N Content of the topics Learning Objectives Teaching Guidelines Methodology Time (hours) 1 ·        Riemann integral, integrability of continuous and monotonic function, ·        The Fundamental theorem of integral calculus, ·        Mean value theorems of integral calculus, Student will be able to understand the study of Riemann integral, Improper integral and their convergence Lecture should be effective so that student will be able to grasp the topics easily Assignments/ Seminars/ Class tests/ Presentations 15 2 ·       Improper integral and their convergence, ·       Comparison test, Abel’s test and Dirichlet’s test, ·       Frullani’s integrals ·       Integral as a function of parameters ·       Continuity, Differentiablity and integrability of an integral of a function of parameter. Student will be able to understand the study of Metric space Lecture should be effective so that student will be able to grasp the topics easily Assignments/ Seminars/ Class tests/ Presentations 15 3 ·       Definition and examples of Metric spaces, ·       Neighborhoods, limit points, Interior points, ·       Open and closed sets, closure and interior, boundary points, ·       Subspace of a metric space, equivalent metrics, ·       Cauchy sequences, Completeness, ·       Cantor’s intersection theorem, ·       Baire’s Category theorem, contraction principle. Student will be able to understand the study of Compactness and connectedness Lecture should be effective so that student will be able to grasp the topics easily Assignments/ Seminars/ Class tests/ Presentations 15 4 ·        Continuous functions, uniform continuity, ·        Compactness for metric spaces, ·        Sequential compactness, Bolzano- Weierstrass property, ·        Total boundedness, Finite intersection property, ·        Continuity in a relation with compactness, connectedness, components, continuity in relation with connectedness Student will be able to understand the study of Compactness and connectedness Lecture should be effective so that student will be able to grasp the topics easily Assignments/ Seminars/ Class tests/ Presentations 15

format: plain


text:
# A Question about Hypothesis Testing

1. Dec 11, 2012

### Artusartos

http://www.math.harvard.edu/~phorn/362/362assn3-solns.pdf [Broken]

Why isn't the significane level for number 5 just 0.01? Because we know that if the mean is 30,000 (the same as the mean for the null hypothesis), the value of the power function is 0.01...


text:
# Tag Info

0

Look at a free body diagram. With red and the contact normal forces, with pink the friction forces, and with gray the gravity forces. If in the end any of the friction forces come up to being negative, then flip the orientation.

1

The definitions are equal: Sum of external forces zero, sum of external torques is zero. This comes from classical mechanics. For a perfect ideal fluid, the external force density is the pressure gradient: $\mathbf f = -\nabla p$, and therefore, uniform pressure in a fluid means no external force on it, and then it is in mechanical equilibrium. So, its more ...

4

Okay, the system is in equilibrium so all the forces must be balanced. First consider the weights We have $W$ downwards from the centre of each of the 3 balls (assuming their centres of mass are at their geometric centres). We also have $N_A = N_B = \frac{3}{2}W$ upwards from the points of contact between the plane and balls $A$ and $B$ respectively. ...

0

Be careful - you are applying the reasoning about a static situation to a situation that is not static. This rod will be rotationally and linearly accelerating, so you can no longer assume the net force or net torque is zero - $\Sigma F_y = m a_y$, and $a_y \ne 0$. If the rod is instantaneously horizontal and at rest, it's not too hard to find $F_n$ at ...

2

You idea is exactly right. With that diagram, you need to figure out where the center of gravity is. The position of the center of gravity is a direct function of the number of bricks. In general, when you have $n$ bricks with equal offsets, the center of gravity is exactly midway between bricks $1$ and $n$ (by symmetry). The first brick is in contact with ...

1

Take for example, the case when the charge Q at the centre is negative, so the force is attractive. If it gets displaced even by a small value $\delta r$ towards either, the separation with this particular vertex would be $(\frac{l}{\sqrt 2}-\delta r)$, while it is slightly larger for the others. Due to the inverse dependence of the force on this ...

0

Consider the square to be in the x-y plane. Then see what happens when you displace the central charge in the z-direction. Of course, the answer will depend upon whether Q is positive or negative and possibly also on the direction of displacement.

Top 50 recent answers are included


text:
## Morozan, Toader

Compute Distance To:
 Author ID: morozan.toader Published as: Morozan, Toader; Morozan, T.
 Documents Indexed: 121 Publications since 1959, including 4 Books Reviewing Activity: 94 Reviews Co-Authors: 15 Co-Authors with 53 Joint Publications 560 Co-Co-Authors
all top 5

### Co-Authors

 68 single-authored 43 Drăgan, Vasile 11 Halanay, Aristide 9 Stoica, Adrian-Mihail 3 Tudor, Constantin 2 Freiling, Gerhard 2 Ungureanu, Viorica Mariela 1 Aberkane, Samir 1 Damm, Tobias 1 Hochhaus, Andreas 1 Mukaidani, Hiroaki 1 Popa, Ioan-Lucian 1 Răsvan, Vladimir B. 1 Shi, Peng 1 Stoic, Adrian 1 Tiba, Dan 1 Vârsan, Constantin
all top 5

### Serials

 34 Revue Roumaine de Mathématiques Pures et Appliquées 14 Stochastic Analysis and Applications 8 Studii și Cercetări Matematice 6 Annals of the Academy of Romanian Scientists. Mathematics and its Applications 5 Mathematical Reports 4 International Journal of Control 4 Journal of Differential Equations 3 Journal of Mathematical Analysis and Applications 3 IMA Journal of Mathematical Control and Information 2 Optimal Control Applications & Methods 2 Dynamics of Continuous, Discrete and Impulsive Systems 2 European Journal of Control 2 Comunicările Academiei Republicii Populare Române 1 International Journal of Systems Science 1 Applied Mathematics and Optimization 1 Automatica 1 Časopis Pro Pěstování Matematiky 1 IEEE Transactions on Automatic Control 1 Integral Equations and Operator Theory 1 Results in Mathematics 1 SIAM Journal on Control and Optimization 1 Stochastics and Stochastics Reports 1 International Journal of Robust and Nonlinear Control 1 Electronic Journal of Differential Equations (EJDE) 1 Journal of Difference Equations and Applications 1 Electronic Journal of Qualitative Theory of Differential Equations 1 Dynamics of Continuous, Discrete & Impulsive Systems. Series B. Applications & Algorithms 1 Mathematical Concepts and Methods in Science and Engineering 1 ROMAI Journal 1 Preprint. “Babeș-Bolyai” University. Faculty of Mathematics and Physics. Research Seminars
all top 5

### Fields

 86 Systems theory; control (93-XX) 39 Probability theory and stochastic processes (60-XX) 18 Ordinary differential equations (34-XX) 10 Calculus of variations and optimal control; optimization (49-XX) 4 Difference and functional equations (39-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Operator theory (47-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Statistics (62-XX) 1 History and biography (01-XX) 1 Operations research, mathematical programming (90-XX)

### Citations contained in zbMATH Open

73 Publications have been cited 489 times in 263 Documents Cited by Year
Mathematical methods in robust control of discrete-time linear stochastic systems. Zbl 1183.93001
Dragan, Vasile; Morozan, Toader; Stoica, Adrian-Mihail
2010
Stability and robust stabilization to linear stochastic systems described by differential equations with Markovian jumping and multiplicative white noise. Zbl 1136.60335
Dragan, Vasile; Morozan, Toader
2002
Mathematical methods in robust control of linear stochastic systems. Zbl 1101.93002
Dragan, Vasile; Morozan, Toader; Stoica, Adrian-Mihail
2006
Optimal stabilizing compensator for linear systems with state-dependent noise. Zbl 0808.93066
Drăgan, V.; Morozan, T.; Halanay, A.
1992
Stabilization of some stochastic discrete-time control systems. Zbl 0517.93061
Morozan, T.
1983
The linear quadratic optimization problems for a class of linear stochastic systems with multiplicative white noise and Markovian jumping. Zbl 1365.93541
Dragan, Vasile; Morozan, Toader
2004
Observability and detectability of a class of discrete-time stochastic linear systems. Zbl 1095.93004
Dragan, Vasile; Morozan, Toader
2006
Almost periodic solutions of affine Ito equations. Zbl 0692.60045
Morozan, T.; Tudor, C.
1989
Global solutions to a game-theoretic Riccati equation of stochastic control. Zbl 0888.93055
Dragan, V.; Morozan, T.
1997
Stability and control for linear systems with jump Markov perturbations. Zbl 0818.93075
Morozan, T.
1995
Periodic solutions of affine stochastic differential equations. Zbl 0583.60055
Morozan, T.
1986
$$H^{2}$$ optimal control for linear stochastic systems. Zbl 1060.93107
Dragan, Vasile; Morozan, Toader; Stoica, Adrian
2004
Optimal stationary control for dynamic systems with Markov perturbations. Zbl 0525.93070
Morozan, T.
1983
Stability of linear systems with random parameters. Zbl 0153.12201
Morozan, T.
1967
Mean square exponential stability for some stochastic linear discrete time systems. Zbl 1293.93762
Dragan, Vasile; Morozan, Toader
2006
Mathematical methods in robust control of linear stochastic systems. 2nd ed. Zbl 1296.93001
Dragan, Vasile; Morozan, Toader; Stoica, Adrian-Mihail
2013
Exponential stability for discrete time linear equations defined by positive operators. Zbl 1094.39008
Dragan, Vasile; Morozan, Toader
2006
Global solutions of a class of discrete-time backward nonlinear equations on ordered Banach spaces with applications to Riccati equations of stochastic control. Zbl 1273.93044
Ungureanu, V. M.; Dragan, V.; Morozan, T.
2013
Parametrized Riccati equations for controlled linear differential systems with jump Markov perturbations. Zbl 0920.93039
Morozan, T.
1998
Stability of some linear stochastic systems. Zbl 0166.13902
Morozan, T.
1967
Stability of stochastic discrete systems. Zbl 0188.49102
Morozan, T.
1968
The method of V. M. Popov for control systems with random parameters. Zbl 0156.10202
Morozan, T.
1966
Mixed input-output optimization for time varying Itô systems with state-dependent noise. Zbl 0879.93058
Dragan, V.; Morozan, T.
1997
Asymptotic properties of input-output operators norm associated with singularly perturbed systems with multiplicative white noise. Zbl 1027.93036
Dragan, Vasile; Morozan, Toader; Shi, Peng
2002
Stochastic observability and applications. Zbl 1060.93019
Dragan, Vasile; Morozan, Toader
2004
Exponential stability in mean square for a general class of discrete-time linear stochastic systems. Zbl 1147.93044
Dragan, Vasile; Morozan, Toader
2008
Differential equations with positive evolutions and some applications. Zbl 1117.34053
Dragan, Vasile; Damm, Tobias; Freiling, Gerhard; Morozan, Toader
2005
Stability radii of some stochastic differential equations. Zbl 0858.60055
Morozan, T.
1995
Stochastic $$H^2$$ optimal control for a class of linear systems with periodic coefficients. Zbl 1293.93777
Dragan, Vasile; Morozan, Toader
2005
Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping. Zbl 1039.34052
Dragan, Vasile; Morozan, Toader
2003
Criteria for exponential stability of linear differential equations with positive evolution on ordered Banach spaces. Zbl 1222.34066
Dragan, Vasile; Morozan, Toader
2010
Stability radii of some discrete-time systems with independent random parameters. Zbl 0898.93035
Morozan, T.
1997
Game-theoretic coupled Riccati equations associated to controlled linear differential systems with jump Markov perturbations. Zbl 1050.93076
Dragan, Vasile; Morozan, Toader
2001
Parametrized Riccati equations associated to input-output operators for discrete-time systems with state-dependent noise. Zbl 0920.93042
Morozan, T.
1998
A class of discrete time generalized Riccati equations. Zbl 1185.93074
Dragan, Vasile; Morozan, Toader
2010
Parametrized Riccati equations for controlled linear discrete-time systems with Markov perturbations. Zbl 1056.93644
Morozan, T.
1998
A class of nonlinear differential equations on the space of symmetric matrices. Zbl 1053.34006
Dragan, Vasile; Freiling, Gerhard; Hochhaus, Andreas; Morozan, Toader
2004
Stability radii of some time-varying linear stochastic differential systems. Zbl 0898.93034
Morozan, T.
1997
Discrete-time Riccati equations connected with quadratic control for linear systems with independent random perturbations. Zbl 0763.93053
Morozan, T.
1992
Stochastic uniform observability and Riccati equations of stochastic control. Zbl 0810.93069
Morozan, T.
1993
Stability and control of some linear discrete-time systems with jump Markov disturbances. Zbl 0463.93074
Morozan, T.
1981
Stability and control for linear discrete-time systems with Markov perturbations. Zbl 0862.93068
Morozan, T.
1995
Robust stability and robust stabilization of discrete-time linear stochastic systems. Zbl 1284.93249
Dragan, Vasile; Morozan, Toader
2010
Bounded, periodic and almost periodic solutions of affine stochastic discrete-time systems. Zbl 0663.93060
Morozan, T.
1987
Iterative procedure for stabilizing solutions of differential Riccati type equations arising in stochastic control. Zbl 1071.93051
Dragan, Vasile; Morozan, Toader; Stoica, Adrian M.
2003
Linear quadratic, control and tracking problems for time-varying stochastic differential systems perturbed by a Markov chain. Zbl 1078.93574
Morozan, T.
2001
An $$H_{2}$$-type norm of a discrete-time linear stochastic system with periodic coefficients simultaneously affected by an infinite Markov chain and multiplicative white noise perturbations. Zbl 1302.93246
Morozan, Toader; Dragan, Vasile
2014
Iterative algorithm to compute the maximal and stabilising solutions of a general class of discrete-time Riccati-type equations. Zbl 1209.93091
Dragan, Vasile; Morozan, Toader; Stoica, Adrian-Mihail
2010
Boundedness properties for stochastic systems. Zbl 0284.93032
Morozan, Toader
1972
Asymptotic almost periodic solutions for Riccati equations of stochastic control. Zbl 0831.93067
Morozan, T.
1994
Periodic solutions of stochastic discrete-time systems. Zbl 0644.93056
Morozan, T.
1987
Exponential stability for a class of singularly perturbed Itô differential equations. Zbl 0971.34034
Dragan, V.; Morozan, T.
2000
Almost-periodic solutions for Riccati equations of stochastic control. Zbl 0806.93061
Morozan, T.
1994
Exponential stability for a class of linear time-varying singularly perturbed stochastic systems. Zbl 1023.34044
Dragan, Vasile; Morozan, Toader
2002
On the absolute stability of stochastic sampled-data control system. Zbl 0195.16201
Morozan, T.
1969
Stability of differential systems with random parameters. Zbl 0227.93028
Morozan, T.
1968
$$H_2$$ optimal control for a wide class of discrete-time linear stochastic systems. Zbl 1175.93239
Dragan, Vasile; Morozan, Toader
2009
Foundations of the dynamic programming approach for discrete-time stochastic control problems. Zbl 0383.49016
Morozan, T.
1977
Tracking almost periodic signals under white noise perturbations. Zbl 0693.93071
Halanay, A.; Morozan, T.
1989
Performance estimates in tracking under white-noise perturbations. Zbl 0694.93116
Dragan, V.; Halanay, A.; Morozan, T.
1989
Tracking discrete almost-periodic signals under random perturbations. Zbl 0642.93068
Halanay, A.; Morozan, T.; Tudor, C.
1988
Bounded solutions of affine stochastic differential equations and stability. Zbl 0606.60056
Halanay, A.; Morozan, T.; Tudor, C.
1986
Optimal stabilizing compensator for linear discrete-time systems under independent random perturbations. Zbl 0776.93059
Halanay, Aristide; Morozan, Toader
1992
On the Riccati equations of stochastic control. Zbl 0788.93091
Morozan, Toader
1992
Affine stochastic differential equations with almost periodic coefficients. Zbl 0810.60053
Morozan, T.
1991
Stochastic stability and control for discrete-time systems with jump Markov disturbances. Zbl 0401.93048
Morozan, T.
1979
Dynamic programming for some discrete-time control systems with random perturbations. Zbl 0458.49027
Morozan, T.
1980
Dual linear quadratic problem for linear control systems with jump Markov perturbations. Zbl 0862.93069
Morozan, T.
1996
Some Lyapunov type positive operators on ordered Banach spaces. Zbl 1284.34103
Dragan, Vasile; Morozan, Toader; Ungureanu, Viorica
2013
Output-based $$H_2$$ optimal controllers for a class of discrete-time stochastic linear systems with periodic coefficients. Zbl 1328.93280
Dragan, Vasile; Morozan, Toader; Stoica, Adrian-Mihail
2015
Liapunov and Riccati equations connected with stability and quadratic control for time-varying linear systems with jump Markov perturbations. Zbl 0856.93104
Morozan, T.
1995
Parameterized Riccati equations and input-output operators for time-varying stochastic differential equations with state-dependent noise. Zbl 1026.34065
Morozan, T.
1999
Linear quadratic optimization problems for some discrete-time stochastic linear systems. Zbl 1212.93324
Dragan, Vasile; Morozan, Toader
2009
Output-based $$H_2$$ optimal controllers for a class of discrete-time stochastic linear systems with periodic coefficients. Zbl 1328.93280
Dragan, Vasile; Morozan, Toader; Stoica, Adrian-Mihail
2015
An $$H_{2}$$-type norm of a discrete-time linear stochastic system with periodic coefficients simultaneously affected by an infinite Markov chain and multiplicative white noise perturbations. Zbl 1302.93246
Morozan, Toader; Dragan, Vasile
2014
Mathematical methods in robust control of linear stochastic systems. 2nd ed. Zbl 1296.93001
Dragan, Vasile; Morozan, Toader; Stoica, Adrian-Mihail
2013
Global solutions of a class of discrete-time backward nonlinear equations on ordered Banach spaces with applications to Riccati equations of stochastic control. Zbl 1273.93044
Ungureanu, V. M.; Dragan, V.; Morozan, T.
2013
Some Lyapunov type positive operators on ordered Banach spaces. Zbl 1284.34103
Dragan, Vasile; Morozan, Toader; Ungureanu, Viorica
2013
Mathematical methods in robust control of discrete-time linear stochastic systems. Zbl 1183.93001
Dragan, Vasile; Morozan, Toader; Stoica, Adrian-Mihail
2010
Criteria for exponential stability of linear differential equations with positive evolution on ordered Banach spaces. Zbl 1222.34066
Dragan, Vasile; Morozan, Toader
2010
A class of discrete time generalized Riccati equations. Zbl 1185.93074
Dragan, Vasile; Morozan, Toader
2010
Robust stability and robust stabilization of discrete-time linear stochastic systems. Zbl 1284.93249
Dragan, Vasile; Morozan, Toader
2010
Iterative algorithm to compute the maximal and stabilising solutions of a general class of discrete-time Riccati-type equations. Zbl 1209.93091
Dragan, Vasile; Morozan, Toader; Stoica, Adrian-Mihail
2010
$$H_2$$ optimal control for a wide class of discrete-time linear stochastic systems. Zbl 1175.93239
Dragan, Vasile; Morozan, Toader
2009
Linear quadratic optimization problems for some discrete-time stochastic linear systems. Zbl 1212.93324
Dragan, Vasile; Morozan, Toader
2009
Exponential stability in mean square for a general class of discrete-time linear stochastic systems. Zbl 1147.93044
Dragan, Vasile; Morozan, Toader
2008
Mathematical methods in robust control of linear stochastic systems. Zbl 1101.93002
Dragan, Vasile; Morozan, Toader; Stoica, Adrian-Mihail
2006
Observability and detectability of a class of discrete-time stochastic linear systems. Zbl 1095.93004
Dragan, Vasile; Morozan, Toader
2006
Mean square exponential stability for some stochastic linear discrete time systems. Zbl 1293.93762
Dragan, Vasile; Morozan, Toader
2006
Exponential stability for discrete time linear equations defined by positive operators. Zbl 1094.39008
Dragan, Vasile; Morozan, Toader
2006
Differential equations with positive evolutions and some applications. Zbl 1117.34053
Dragan, Vasile; Damm, Tobias; Freiling, Gerhard; Morozan, Toader
2005
Stochastic $$H^2$$ optimal control for a class of linear systems with periodic coefficients. Zbl 1293.93777
Dragan, Vasile; Morozan, Toader
2005
The linear quadratic optimization problems for a class of linear stochastic systems with multiplicative white noise and Markovian jumping. Zbl 1365.93541
Dragan, Vasile; Morozan, Toader
2004
$$H^{2}$$ optimal control for linear stochastic systems. Zbl 1060.93107
Dragan, Vasile; Morozan, Toader; Stoica, Adrian
2004
Stochastic observability and applications. Zbl 1060.93019
Dragan, Vasile; Morozan, Toader
2004
A class of nonlinear differential equations on the space of symmetric matrices. Zbl 1053.34006
Dragan, Vasile; Freiling, Gerhard; Hochhaus, Andreas; Morozan, Toader
2004
Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping. Zbl 1039.34052
Dragan, Vasile; Morozan, Toader
2003
Iterative procedure for stabilizing solutions of differential Riccati type equations arising in stochastic control. Zbl 1071.93051
Dragan, Vasile; Morozan, Toader; Stoica, Adrian M.
2003
Stability and robust stabilization to linear stochastic systems described by differential equations with Markovian jumping and multiplicative white noise. Zbl 1136.60335
Dragan, Vasile; Morozan, Toader
2002
Asymptotic properties of input-output operators norm associated with singularly perturbed systems with multiplicative white noise. Zbl 1027.93036
Dragan, Vasile; Morozan, Toader; Shi, Peng
2002
Exponential stability for a class of linear time-varying singularly perturbed stochastic systems. Zbl 1023.34044
Dragan, Vasile; Morozan, Toader
2002
Game-theoretic coupled Riccati equations associated to controlled linear differential systems with jump Markov perturbations. Zbl 1050.93076
Dragan, Vasile; Morozan, Toader
2001
Linear quadratic, control and tracking problems for time-varying stochastic differential systems perturbed by a Markov chain. Zbl 1078.93574
Morozan, T.
2001
Exponential stability for a class of singularly perturbed Itô differential equations. Zbl 0971.34034
Dragan, V.; Morozan, T.
2000
Parameterized Riccati equations and input-output operators for time-varying stochastic differential equations with state-dependent noise. Zbl 1026.34065
Morozan, T.
1999
Parametrized Riccati equations for controlled linear differential systems with jump Markov perturbations. Zbl 0920.93039
Morozan, T.
1998
Parametrized Riccati equations associated to input-output operators for discrete-time systems with state-dependent noise. Zbl 0920.93042
Morozan, T.
1998
Parametrized Riccati equations for controlled linear discrete-time systems with Markov perturbations. Zbl 1056.93644
Morozan, T.
1998
Global solutions to a game-theoretic Riccati equation of stochastic control. Zbl 0888.93055
Dragan, V.; Morozan, T.
1997
Mixed input-output optimization for time varying Itô systems with state-dependent noise. Zbl 0879.93058
Dragan, V.; Morozan, T.
1997
Stability radii of some discrete-time systems with independent random parameters. Zbl 0898.93035
Morozan, T.
1997
Stability radii of some time-varying linear stochastic differential systems. Zbl 0898.93034
Morozan, T.
1997
Dual linear quadratic problem for linear control systems with jump Markov perturbations. Zbl 0862.93069
Morozan, T.
1996
Stability and control for linear systems with jump Markov perturbations. Zbl 0818.93075
Morozan, T.
1995
Stability radii of some stochastic differential equations. Zbl 0858.60055
Morozan, T.
1995
Stability and control for linear discrete-time systems with Markov perturbations. Zbl 0862.93068
Morozan, T.
1995
Liapunov and Riccati equations connected with stability and quadratic control for time-varying linear systems with jump Markov perturbations. Zbl 0856.93104
Morozan, T.
1995
Asymptotic almost periodic solutions for Riccati equations of stochastic control. Zbl 0831.93067
Morozan, T.
1994
Almost-periodic solutions for Riccati equations of stochastic control. Zbl 0806.93061
Morozan, T.
1994
Stochastic uniform observability and Riccati equations of stochastic control. Zbl 0810.93069
Morozan, T.
1993
Optimal stabilizing compensator for linear systems with state-dependent noise. Zbl 0808.93066
Drăgan, V.; Morozan, T.; Halanay, A.
1992
Discrete-time Riccati equations connected with quadratic control for linear systems with independent random perturbations. Zbl 0763.93053
Morozan, T.
1992
Optimal stabilizing compensator for linear discrete-time systems under independent random perturbations. Zbl 0776.93059
Halanay, Aristide; Morozan, Toader
1992
On the Riccati equations of stochastic control. Zbl 0788.93091
Morozan, Toader
1992
Affine stochastic differential equations with almost periodic coefficients. Zbl 0810.60053
Morozan, T.
1991
Almost periodic solutions of affine Ito equations. Zbl 0692.60045
Morozan, T.; Tudor, C.
1989
Tracking almost periodic signals under white noise perturbations. Zbl 0693.93071
Halanay, A.; Morozan, T.
1989
Performance estimates in tracking under white-noise perturbations. Zbl 0694.93116
Dragan, V.; Halanay, A.; Morozan, T.
1989
Tracking discrete almost-periodic signals under random perturbations. Zbl 0642.93068
Halanay, A.; Morozan, T.; Tudor, C.
1988
Bounded, periodic and almost periodic solutions of affine stochastic discrete-time systems. Zbl 0663.93060
Morozan, T.
1987
Periodic solutions of stochastic discrete-time systems. Zbl 0644.93056
Morozan, T.
1987
Periodic solutions of affine stochastic differential equations. Zbl 0583.60055
Morozan, T.
1986
Bounded solutions of affine stochastic differential equations and stability. Zbl 0606.60056
Halanay, A.; Morozan, T.; Tudor, C.
1986
Stabilization of some stochastic discrete-time control systems. Zbl 0517.93061
Morozan, T.
1983
Optimal stationary control for dynamic systems with Markov perturbations. Zbl 0525.93070
Morozan, T.
1983
Stability and control of some linear discrete-time systems with jump Markov disturbances. Zbl 0463.93074
Morozan, T.
1981
Dynamic programming for some discrete-time control systems with random perturbations. Zbl 0458.49027
Morozan, T.
1980
Stochastic stability and control for discrete-time systems with jump Markov disturbances. Zbl 0401.93048
Morozan, T.
1979
Foundations of the dynamic programming approach for discrete-time stochastic control problems. Zbl 0383.49016
Morozan, T.
1977
Boundedness properties for stochastic systems. Zbl 0284.93032
Morozan, Toader
1972
On the absolute stability of stochastic sampled-data control system. Zbl 0195.16201
Morozan, T.
1969
Stability of stochastic discrete systems. Zbl 0188.49102
Morozan, T.
1968
Stability of differential systems with random parameters. Zbl 0227.93028
Morozan, T.
1968
Stability of linear systems with random parameters. Zbl 0153.12201
Morozan, T.
1967
Stability of some linear stochastic systems. Zbl 0166.13902
Morozan, T.
1967
The method of V. M. Popov for control systems with random parameters. Zbl 0156.10202
Morozan, T.
1966
all top 5

### Cited by 321 Authors

 33 Drăgan, Vasile 19 Morozan, Toader 17 Zhang, Weihai 12 Costa, Oswaldo Luiz V. 11 Ma, Hongji 10 Hou, Ting 8 Aberkane, Samir 8 Gershon, Eli 8 Shaked, Uri 8 Stoica, Adrian-Mihail 8 Ungureanu, Viorica Mariela 7 Ivanov, Ivan Ganchev 6 Keel, Lee H. 6 Mao, Xuerong 5 do Val, João B. R. 5 Fragoso, Marcelo Dutra 5 Freiling, Gerhard 5 Yuan, Chenggui 4 Jia, Yingmin 4 Lin, Yan 4 Lin, Zhongwei 4 Liu, Zhenxin 4 Palamarchuk, E. S. 4 Shi, Peng 3 Beane, Carlos 3 Benites, Guilherme R. A. M. 3 Gil’, Michael Iosif 3 Halanay, Aristide 3 Huang, Ran 3 Ichikawa, Akira 3 Jiang, Xiushan 3 Knap, Michael Jason 3 Liu, Yueying 3 Lukashiv, Taras O. 3 Mahmoud, Magdi Sadik Mostafa 3 Mellah, Omar 3 Mukaidani, Hiroaki 3 Raynaud de Fitte, Paul 3 Todorov, Marcos Garcia 3 Vargas, Alessandro N. 3 Wang, Yuhong 3 Yaesh, Isaac 2 Bedouhene, Fazia 2 Bin, Ning 2 Bogdanova, Boryana C. 2 Boukas, El-Kébir 2 Chen, Guopei 2 Chen, Xiang 2 Czornik, Adam 2 Da Prato, Giuseppe 2 de Paulo, Wanderlei L. 2 Deng, Feiqi 2 Dokuchaev, Nikolai G. 2 Fan, Hung-Yuan 2 Feng, Yu 2 Gao, Ming 2 Ge, Shuzhi Sam 2 Ho, Daniel W. C. 2 Hochhaus, Andreas 2 Ismail, Abdulla 2 Ji, Min 2 King-wah Chu, Eric 2 Ladde, Gangaram S. 2 Lam, James 2 Levajković, Tijana 2 Liu, Jianzhou 2 Mahalanabis, A. K. 2 Mena, Hermann 2 Nichols, William G. 2 Niu, Yugang 2 Ponsart, Jean-Christophe 2 Popa, Ioan-Lucian 2 Qi, Weiwei 2 Sauter, Dominique D. J. 2 Sheng, Li 2 Socha, Leslaw 2 Świerniak, Andrzej P. 2 Tan, Cheng 2 Tsokos, Christos P. 2 Tudor, Constantin 2 Tuffaha, Amjad M. 2 Wang, Meijiao 2 Yang, Ying 2 Yavin, Yaakov 2 Yi, Jianqiang 2 Yi, Yingfei 2 Yurchenko, I. V. 2 Yurchenkov, Alexander V. 2 Zhang, Chengke 2 Zhang, Juan 2 Zhu, Huainian 1 Acho, Leonardo 1 Adetona, O. 1 Akeb, Tassadit 1 Akhmet, Marat Ubaydulla 1 Antonyuk, Svitlana V. 1 Appleby, John A. D. 1 Araujo, Michael V. 1 Ashrafi Bafghi, Mahdi 1 Attili, Basem S. ...and 221 more Authors
all top 5

### Cited in 73 Serials

 27 Automatica 22 Stochastic Analysis and Applications 16 International Journal of Control 13 Mathematical Problems in Engineering 12 Journal of Mathematical Analysis and Applications 10 International Journal of Robust and Nonlinear Control 9 International Journal of Systems Science 9 SIAM Journal on Control and Optimization 9 European Journal of Control 8 Automation and Remote Control 8 International Journal of Systems Science. Principles and Applications of Systems and Integration 7 Systems & Control Letters 6 Journal of the Franklin Institute 6 Optimal Control Applications & Methods 5 Journal of Differential Equations 5 Journal of Difference Equations and Applications 4 Journal of Systems Science and Complexity 4 Asian Journal of Control 3 Computers & Mathematics with Applications 3 Applied Mathematics and Optimization 3 Journal of Computational and Applied Mathematics 3 Stochastics and Dynamics 3 Nonlinear Analysis. Hybrid Systems 2 Stochastics 2 Information Sciences 2 Journal of Functional Analysis 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Numerical Algorithms 2 Linear Algebra and its Applications 2 Stochastic Processes and their Applications 2 Cybernetics and Systems Analysis 2 Random Operators and Stochastic Equations 2 Abstract and Applied Analysis 2 Discrete Dynamics in Nature and Society 2 Acta Mathematica Sinica. English Series 2 The ANZIAM Journal 1 Acta Mathematica Academiae Scientiarum Hungaricae 1 Journal of Mathematical Physics 1 Applied Mathematics and Computation 1 Czechoslovak Mathematical Journal 1 Journal of Optimization Theory and Applications 1 Mathematics and Computers in Simulation 1 Results in Mathematics 1 Chinese Annals of Mathematics. Series B 1 Optimization 1 MCSS. Mathematics of Control, Signals, and Systems 1 Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni 1 Journal of Global Optimization 1 International Journal of Computer Mathematics 1 SIAM Journal on Mathematical Analysis 1 SIAM Journal on Scientific Computing 1 Theory of Probability and Mathematical Statistics 1 Electronic Journal of Differential Equations (EJDE) 1 Numerical Linear Algebra with Applications 1 Computational and Applied Mathematics 1 Discrete and Continuous Dynamical Systems 1 Doklady Mathematics 1 Nonlinear Dynamics 1 Open Systems & Information Dynamics 1 Differential Equations 1 Nonlinear Analysis. Real World Applications 1 Dynamical Systems 1 Journal of Applied Mathematics and Computing 1 Mediterranean Journal of Mathematics 1 African Diaspora Journal of Mathematics 1 International Journal of Control, I. Series 1 Algorithms 1 International Journal of Stochastic Analysis 1 Axioms 1 Journal of the Operations Research Society of China 1 Journal of Mathematics 1 International Journal of Partial Differential Equations 1 Modern Stochastics. Theory and Applications
all top 5

### Cited in 24 Fields

 202 Systems theory; control (93-XX) 139 Probability theory and stochastic processes (60-XX) 33 Ordinary differential equations (34-XX) 32 Calculus of variations and optimal control; optimization (49-XX) 20 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 12 Partial differential equations (35-XX) 11 Operator theory (47-XX) 10 Numerical analysis (65-XX) 9 Dynamical systems and ergodic theory (37-XX) 8 Linear and multilinear algebra; matrix theory (15-XX) 8 Difference and functional equations (39-XX) 6 Operations research, mathematical programming (90-XX) 3 Statistics (62-XX) 2 Mathematical logic and foundations (03-XX) 2 Abstract harmonic analysis (43-XX) 2 Functional analysis (46-XX) 1 Real functions (26-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Integral equations (45-XX) 1 Mechanics of particles and systems (70-XX) 1 Optics, electromagnetic theory (78-XX) 1 Quantum theory (81-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Information and communication theory, circuits (94-XX)


text:
### Communities

tag:snake search within a tag
answers:0 unanswered questions
user:xxxx search by author id
score:0.5 posts with 0.5+ score
"snake oil" exact phrase
votes:4 posts with 4+ votes
created:<1w created < 1 week ago
post_type:xxxx type of post
Notifications
Q&A

# Universal property of quotient spaces

+0
−0

A typical textbook theorem about quotient space is as follows:

Theorem (Gamelin-Greene Introduction to Topology p.106): Let $f$ be a continuous function from a topological space $X$ to a topological space $Y$. Let $\sim$ be an equivalence relation on $X$ such that $f$ is constant on each equivalence class. Then there exists a continuous function $g: X/{\sim} \rightarrow Y$ such that $f=g \circ \pi$.

It is very easy to prove it by using the definition of continuous functions between topological spaces because $\pi^{-1}(g^{-1}(V))=f^{-1}(V)$ for any open set $V$ in $Y$ with the well-defined $g([x]):=f(x)$. One such proof is done in the mentioned textbook as follows:

Proof. Since $f$ is constant on each equivalence class, one can define $g:X/{\sim}\to Y$ with $g([x])=f(x)$. One can easily check that $f=g \circ \pi$. To show that $g$ is continuous, consider any open set $V$ in $Y$. Since $f^{-1}(V)$ is open in $X$ by continuity of $f$ and $\pi^{-1}(g^{-1}(V))=(g\circ \pi)^{-1}(V)=f^{-1}(V)$, by the definition of the quotient topology on $X/{\sim}$, $g^{-1}(V)$ is open and the proof is complete.

But when I try to use the definition of continuity at a point, which I think is supposed to be easy too, I got stuck.

Let $x\in X$ and denote $[x]=\pi(x)$. I want to show that $g$ is continuous at $[x]$. Now let $U\subset Y$ be a neighborhood of $g([x])$. The goal is to find a neighborhood $O$ of $[x]$ in $X/{\sim}$ such that $g(O)\subset U$.

Since $f(x)=g([x])$, and $f$ is continuous (at $x$), there exists a neighborhood $N\subset X$ such that $x\in N$ and $f(N)\subset U$.

It suffices to show that $\pi(N)$ is a neighborhood of $[x]$ since $g(\pi(N))=f(N)\subset U$. But now I don't see how to show that $[x]$ is an interior point of $\pi(N)$.

Question: How can I go on from here or do I need to change the argument completely?

I suppose I would either need to change $\pi(N)$ to something else or apply some property of $\pi$ that I have not used yet.

Why does this post require moderator attention?
You might want to add some details to your flag.
Why should this post be closed?

#### 1 comment thread

Where does the quotient topology come in? (2 comments)

## 1 answer

+0
−0

I figured out an answer after posting the question for a while. I would like to record it here.

The problem with the mentioned argument is that one attempted to prove the continuity of $g$ at one point $[x]$ using merely the continuity of $f$ at $x$ instead of using the global continuity of $f$ on $X$. One way that I figured out to save the mentioned argument is that one should exploit the fact that $\pi^{-1}(g^{-1}(V))=f^{-1}(V)$, which, of course, could have been used in the simple proof via continuity of functions between topological spaces.

Proof. Define $g:X/{\sim}$ as in the standard proof. Without loss of generality, suppose $U$ is an open neighborhood of $g([x])$. Then $\pi^{-1}(g^{-1}(U))=f^{-1}(U)$ is open by the continuity of $f$ and thus $g^{-1}(U)$ is an open neighborhood of $[x]$ in $X/\sim$ by the definition of quotient topology. Since $g(g^{-1}(U))\subset U$, we have shown that $g$ is continuous at $[x]$. Since $[x]$ is arbitrary, the proof is complete.

Why does this post require moderator attention?
You might want to add some details to your flag.

#### 0 comment threads

Featured
Hot Posts

This community is part of the Codidact network. We have other communities too — take a look!

You can also join us in chat!

Want to advertise this community? Use our templates!

Like what we're doing? Support us!


text:
# The Impact of Meteorological Factors on Fine Particulate Pollution in Northeast China

Special Issue on 2019 Asian Aerosol Conference (AAC)

Can Meng1,2, Tianhai Cheng 1, Fangwen Bao3, Xingfa Gu1, Jian Wang4, Xin Zuo1,2, Shuaiyi Shi 1

State Key Laboratory of Remote Sensing Science, Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing 100094, China
University of Chinese Academy of Sciences, Beijing 100049, China
Department of Ocean Sciences and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
Department of Geography, The Ohio State University, Columbus, OH 43210, USA

Revised: February 19, 2020
Accepted: February 21, 2020

Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

Meng, C., Cheng, T., Bao, F., Gu, X., Wang, J., Zuo, X. and Shi, S. (2020). The Impact of Meteorological Factors on Fine Particulate Pollution in Northeast China. Aerosol Air Qual. Res. 20: 1618–1628. https://doi.org/10.4209/aaqr.2019.10.0534

## HIGHLIGHTS

• Wind speed, HPBL and temperature inversion impact PM2.5 in NE China significantly.
• Harbin: 91.6% pollution days occur when wind < 20 knots, HPBL < 1500 m, TI_T > 8°C.
• Changchun: 61.7% pollution days occur when wind < 20 knots, HPBL < 600 m.
• Shenyang: 70.3% pollution days occur when wind < 20 knots, TI_D > 775 m.

## ABSTRACT

Due to biomass burning and coal combustion, heavy fine particulate matter (PM2.5) pollution frequently occurs in Northeast China, threatening the health of more than 117 million inhabitants. Although meteorological conditions have always been considered key factors in the accumulation and dilution of PM2.5 pollution, their exact contribution to particulate pollution in Northeast China is still highly uncertain. Applying multiple regression analysis to observational data, we identify the wind speed, temperature inversion, and height of the planetary boundary layer as the dominant meteorological factors affecting PM2.5 pollution (PM2.5 > 75 µg m–3) in the major cities of Northeast China, with the wind speed and the planetary boundary layer playing the primary roles in Harbin and Shenyang, and Changchun, respectively. Heavy pollution (PM2.5 > 150 µg m–3) in this region typically occurs when the wind speed is less than 20 knots, the planetary boundary layer is below 500 m, and the temperature inversion is greater than 6°C. These results suggest that reducing the PM2.5 pollution requires us to focus not only on anthropogenic emissions but also on special meteorological conditions that can affect the air pollution mechanisms in Northeast China.

Keywords: Fine particulate (PM2.5) pollution; Multiple regression method; Wind speed; Temperature inversion; Planetary boundary layer height.

## INTRODUCTION

Several extremely severe atmospheric haze events have occurred recently in Northeast China, a major crop straw burning region of China, with air pollution reaching up to nearly 2–3 times of the historical record (Chen et al., 2015; Wen et al., 2018). Fine particulate matter (PM2.5) has been proved harmful to human health and regarded as the main factor impairing visibility (He et al., 2001; Wang and Christopher, 2003; Jung et al., 2010; Heal et al., 2012; Kan et al., 2012). It exerts large influences on the development of planetary boundary layer (PBL) through the positive feedback loop between aerosol and PBL (Petäjä et al., 2016; Li et al., 2017b; Lou et al., 2019) which in turn deteriorates the air quality. Besides, aerosol tends to affect the formation and development of cloud and precipitation through directly altering the radiation reaching the ground surface (Lohmann and Feichter, 2001; Yang et al., 2016) or indirectly altering the structure of clouds by acting as cloud condensation nucleus (Twomey, 1977; Chen et al., 2016; Guo et al., 2016; Guo et al., 2019), albeit its inconclusive effect on weather and climate systems (Rosenfeld et al., 2014; Fan et al., 2016; Li et al., 2019). In view of its potential huge influence, fine particulate pollution has attracted great attention from the Chinese government. Many regulations and standards have been made to control the fine particulate pollution (Wen et al., 2018). By adopting these measures, a gradual decrease in the PM2.5 concentration over time has been witnessed recently (Wen et al., 2018). However, PM2.5 pollution in Northeast China is still non-negligible, especially around harvest time (Chen et al., 2017).

The pollution during the heating season in Northeast China is mainly caused by evening biomass burning and coal combustion (Fang et al., 2001; Zhang et al., 2011; Fang et al., 2017). In addition, meteorological factors are important to the variation of PM2.5 concentrations. Researches show that wind speed has a negative effect on PM2.5 concentration (Chen et al., 2015; Liang et al., 2015; Chen et al., 2017; Wen et al., 2018). Wind speed can facilitate plume spread and dilution, leading to lower PM2.5 concentrations. In such way, atmospheric horizontal mixing has been considered as one of the most efficient dilution mechanisms (Marcazzan et al., 2001). Studies show that lower height of the planetary boundary layer (HPBL) weakens the dispersion ability of the atmosphere (Seinfeld et al., 1998; Dickerson et al., 2007; Marcazzan et al., 2001; Zhang et al., 2015; Wen et al., 2018). HPBL, also known as the boundary layer top, refers to the thickness of the planetary boundary layer. The characteristic of the boundary layer height is that the turbulence coefficient at this height is zero and the free atmosphere is above it. The diffusion and transportation of pollutants in the lower atmosphere depends to a large extent on the boundary layer structure. Within the boundary layer, turbulence can fully mix particles and gases (Garratt, 1994). In addition, the inversion layer boosts the accumulation of pollutants in the atmosphere (Wang et al., 2014b; Fang et al., 2017). Temperature inversion is a reversal of the normal behavior of temperature in the troposphere, in which a layer of cool air at the surface is overlain by a layer of warmer air. When inversion occurs, upward movement of the lower atmosphere decreases and the pollutants are trapped in the lower atmosphere (Xu et al., 2019). In winter, the recurrent or continuous thermal inversions with fog at ground level are the major contributors to the massive amount of air pollutants accumulated in the lower layer of the atmosphere (Marcazzan et al., 2001). Moreover, haze weather can form in a faster way under high humidity conditions. Some researchers conclude that the relative humidity (RH) could well explain the daily variations of PM2.5 (Sun et al., 2013; Fang et al., 2017; Ma et al., 2017). Previous studies indicated that the synoptic scale meteorological conditions and multiple meteorological factors can have a complicated impact to the PM2.5 pollution (Chen et al., 2018; Liu et al., 2018; Miao et al., 2017; Wang et al., 2014a).

Three typical large cities of Northeast China (i.e., Harbin, Changchun, and Shenyang) were studied in this research, using statistical data and regression model analysis, to investigate the impact of different meteorological factors on major cities in Northeast China. The study result of this research is of great benefit to making suitable and effective strategies for PM2.5 control in Northeast China.

## DATA AND METHODS

### Study Area

As shown in Fig. 1, Northeast China is covered mostly by plain. The Northeast Plain is surrounded by mountains, alluvial plains and terraces, with an elevation of about 200 m. The Northeast Plain is located in a temperate and warm temperate zone with continental monsoon climate characteristics. Significant seasonal differences exist in this area. The summer is short, warm and rainy. Meanwhile the winter is long, cold, and snowy. And the monsoon alternates between winter and summer.

Fig. 1. Map of the study area: (a) the topographic map of China and the location of Northeast China; (b) the topographic map of Northeast China and the location of Shenyang, Changchun, and Harbin; and the location of PM2.5 studying sites in (c) Shenyang, (d) Changchun, and (e) Harbin.

Considering their meteorological conditions and urban development status, three typical large cities in Northeast China were selected in this study to reflect the major characteristics of PM2.5 pollution in Northeast China. Among the three cities, Harbin, the capital of Heilongjiang Province, is located in the highest latitude area, having the lowest temperature, with a flat terrain. Over 10.9 million people live in Harbin. The southeastern part of Harbin is located in the hills; meanwhile, mountainous area occupied the north part. The Songhua River flows through the middle part of Harbin. Changchun is located in the middle part of Northeast China with over 7.5 million people. The Changchun area, apart from a small area of low hills in the east, is mostly comprised of terraces. The terrain of Changchun is flat and convenient for transportation (Fang et al., 2001). In Changchun, from November to March, coal burning phenomenon for heating purpose is widespread. The heating area is scattered in one-fourth of the city area (Fang et al., 2001). Shenyang, the largest city in Northeast China, has over 8.3 million people (Xu et al., 2000; Han et al., 2010). It is located in the southeastern part of Northeast China, mainly in the plains, with mountains and hills concentrated in the southeast. Shenyang shows similar meteorological characteristics with the other two cities mentioned above: a monsoon-influenced humid continental climate, which has monsoon-caused hot, humid summers and dry, cold winters driven by the Siberian anticyclone (Han et al., 2010).

### Data Source and Treatment

The time period of the studied PM2.5 data ranges from 2015 to 2017, provided by the China Meteorological Administration. The continuous hourly PM2.5 concentration data has been monitored by the 33 stations: 12 sites in Harbin, 10 sites in Changchun, and 11 sites in Shenyang. As can be seen in Fig. 1, the sites are evenly distributed throughout the cities. Thus the hourly average value of these sites can reasonably represent the PM2.5 concentration of the three cities. To keep consistency with the meteorological data, the PM2.5 concentration was calculated at 08:00 and 20:00 local time. The PM2.5 data is classified into three states: the non-pollution state (PM2.5 ≤ 75 µg m3), pollution state (PM2.5 > 75 µg m3), and heavy pollution state (PM2.5 > 150 µg m3) (Ma et al., 2017; Tian et al., 2017).

Meteorological data comes from the radiosonde measurements provided by the Integrated Global Radiosonde Archive developed by the American National Climatic Data Center (NCDC) (Durre et al., 2006). The humidity data and wind data are extracted specifically with wind velocity and wind direction at 850 hPa. In general, temperature drops as the altitude increases, beneficial to the convective phenomenon which accelerates the pollutants in the lower layer moving and diffusing upward. However, when temperature inversion occurs, the upward movement of the lower atmosphere is weakened, causing the accumulation of atmospheric pollutants in the lower layer. Temperature inversion phenomenon can be detected by the temperature profile of the atmosphere. Two factors, namely the temperature difference of the inversion layer (TI_T) and the depth of the inversion layer (TI_D), are used to describe the intensity of temperature inversion. In this research, the two factors of the inversion layer are derived from the Integrated Global Radiosonde Archive (IGRA) soundings data. We first analyze the temperature data for different air layers and find if there exists a temperature sequence segmentation in the profile that has monotonic increasing trend with the increase of the layer heights. If exist, the temperature inversion is defined. Then the temperature difference is calculated by subtracting the temperatures at the top and bottom of the inversion layers. And the depth of the inversion layer is calculated by subtracting the heights at the top and bottom of the inversion layers.

HPBL data used in this research is provided by National Centers for Environmental Prediction (NCEP) and Final (FNL) Reanalysis data. It is collected at 06:00 and 18:00 UTC. The HPBL data is a grid dataset and the horizontal resolution of HPBL is 1 × 1° (Zang et al., 2017). In this research, the HPBL values of Harbin, Changchun and Shenyang are calculated using the inverse distance weighting method based on the values in four closest grid points.

## METHODS

In order to decouple the contribution of different meteorological factors, a multiple regression model was used to analyze the correlations between PM2.5 concentration and meteorological factors. Based on atmospheric dispersion equations, the particulate concentration is correlated with meteorological parameters in multiplicative form other than additive form (Hien et al., 2002; Chaloulakou et al., 2003; Gu et al., 2018). The multiple regression analysis model can be constructed as follows:

where B represents the monitored PM2.5 concentration, k represents autochthonic contribution to PM2.5. k is considered to remain unchanged during each season. f(τn) represents the meteorological impact on PM2.5. Furthermore, the regression model can be expressed as follows:

The exponent αi represents the response of particulate concentration B to the rate of change in meteorological parameter Pi. f(τn) was calculated using five meteorological factors (Pi) in this model: wind velocity (WV), HPBL, TI_T, TI_D and RH. As a result, the model can be expressed as follows:

Then, take the logarithm for both sides of this equation. Eq. (3) can be further transformed to a multiple linear regression model:

where ln k, αi and ε represent the intercept, regression coefficients, and the error term respectively. WV, RH, TI_T, TI_D, and HPBL were selected in this research as the studied meteorological parameters which influence the PM2.5 concentration characteristics in Northeast China. To find the determinant(s) of the PM2.5 concentrations among various indicators, the data were analyzed by a stepwise multiple regression method with a significance level of 95% (p = 0.05) set for the regression coefficients.

The regression coefficient αi can only represent the positive or negative relationship between the PM2.5 concentration and the meteorological factors. To find the most important factors that impact the PM2.5 concentration in different cities of Northeast China, we calculated the standardized coefficient of the regression model. The standardized coefficient can compare the relative importance of different kind of factors to the PM2.5 concentration. The bigger the absolute value is, the more important the meteorological factor will be (Newman and Browner, 1991).

## RESULT AND DISCUSSION

### Variation of the PM2.5 Concentrations Correlated with Different Meteorological Factors

Massive biomass burning and coal consumption together with meteorological conditions potentially cause the high level of PM2.5 pollution in Northeast China’s winter (Han et al., 2010; Segura et al., 2013). Studies show that PM2.5 pollution in Northeast China originates from anthropogenic activities, such as straw burning and coal combustion for both industrial and domestic purpose (Zhang et al., 2010; Guan et al., 2018). Meanwhile, individual meteorological factors such as relative humidity, wind speed and temperature inversion have non-negligible impacts on local PM2.5 concentrations (Cheng and Lam, 1998; Ruellan and Cachier, 2001; Wen et al., 2018). The distribution and variation of the source of pollution are usually stable during each season. However, the dilution and accumulation abilities impacted by the meteorological conditions can change significantly. Investigating the impacts of meteorological factors is beneficial to a better understanding of the variations of PM2.5 mass concentrations.

As shown in Fig. 2, the correlation between PM2.5 concentration, and wind direction and speed were similar in the three cities. PM2.5 concentration is negatively associated with wind speed (Chen et al., 2017; Wen et al., 2018). High PM2.5 concentration tends to appear in a low wind speed situation. In Harbin, Changchun, and Shenyang, 79.2%, 79.1%, and 80.4% of the heavy PM2.5 pollution occurs in the daywhen the wind speed is less than 20 knots (a knot is a unit typically used to indicate wind speed, and thus used in this research; 1 knot = 1.852 km h1). Biomass burning in spring and autumn and coal combustion for domestic heating in winter produce a great deal of air pollutants (Zhang et al., 2010; Bao et al., 2015). Higher wind speed is likely to promote the spread of pollutant, which is beneficial for the dilution of PM2.5 pollutants (Wen et al., 2018). The westerly wind prevails during pollution days in Northeast China. In Harbin, over 70% of pollution days (PM2.5 > 75 µg m3) are related with the wind which has a direction between 240–330°. As for Changchun and Shenyang, the dominant wind direction in pollution days is concentrated at 210–330°. In general, Harbin, Changchun, and Shenyang are mainly dominated by the southwest wind during the whole year; meanwhile the northwest wind is the prevailing wind in winter.

Fig. 2. Relationships between atmospheric pollutant concentrations, wind direction, and wind speed in (a) Harbin, (b) Changchun, and (c) Shenyang. Direction frequency and average PM2.5 concentrations in different direction during pollution days in (d) Harbin, (e) Changchun, and (f) Shenyang.

As shown in Fig. 3, in terms of statistics, there is a general negative correlation between HPBL and PM2.5 concentrations for Harbin, Changchun, and Shenyang. Heavy PM2.5 pollution usually occurs in the lower HPBL (< 500 m). HPBL plays a vital role in the variation of PM2.5 concentration. The barrier effect at the top of the planetary boundary layer hinders air pollutants from being transported to the free troposphere, so that aerosol particles are constrained in the PBL (You et al., 2015; Li et al., 2017a;). Lower HPBL is usually related with higher concentrations of PM2.5, and higher HPBL is usually associated with smaller ground level PM2.5 concentration as a result of large vertical mixing ability (Yao et al., 2012).

Fig. 3. Correlation between PM2.5 and the height of the planetary boundary layer in Harbin, Changchun, and Shenyang for (a) pollution days (> 75 µg m–3) and (b) non-pollution days (≤ 75 µg m3).

As can be seen in Fig. 4, PM2.5 concentration is significantly and positively correlated with the intensity of temperature inversion which is described by TI_T and TI_D. In Harbin, Changchun, and Shenyang, 60.6%, 44.8%, and 56.9% of PM2.5 pollution days occur with temperature inversion phenomenon, respectively. Only 14.4%, 18.4%, and 15.4% of PM2.5 pollution days have no temperature inversion. PM2.5 concentration rises significantly when TI_T is greater than 6°C in Harbin and Changchun. However, the relationship between TI_T and PM2.5 in Shenyang is not so significant in pollution days. The TI_D in Shenyang is generally thicker than that of other two cities. Heavy pollution usually occurs when TI_D is bigger than 600 m in Harbin and Shenyang. Compared to summer, the inversion layer in winter is thicker and has a longer duration. The atmosphere remains static once temperature inversion occurs. In static atmosphere, air pollutants are not easily dispersed vertically and, thus, accumulated near the surface. Additionally, a large temperature difference within a thin inversion layer can also cause high concentrations of PM2.5 in the ground layer.

Fig. 4. Statistics of PM2.5 concentrations with temperature inversion. Variation of the PM2.5 concentrations correlated with the temperature difference of the inversion layer (TI_T) during (a) pollution days (> 75 µg m3) and (b) non-pollution days (≤ 75 µg m3). Variation of the PM2.5 concentrations correlated with the depth of temperature inversion (TI_D) during (c) pollution days (> 75 µg m3) and (d) non-pollution days (≤ 75 µg m3).

As can be seen in Fig. 5, the relationship between RH and PM2.5 in Northeast China is not as significant as other air pollution areas. There is even a negative correlation between PM2.5 and RH. Possible reasons are that rainy and snowy weather accompanied with moist air can deposit the particulate pollutants so that air becomes clear, and when the air pollution is caused by the dust particles, the moisture in the air remains constantly low (Sun et al., 2013).

Fig. 5. Variation of the PM2.5 concentrations correlated with different RH condition during (a) pollution days (> 75 µg m3) and (b) non-pollution days (≤ 75 µg m3) in Harbin, Changchun, and Shenyang respectively. The RH is divided into 5 segments (i.e., 0–20, 20–40, 40–60, 60–80, and 80–100).

### Quantitative Multiple Regression Analysis between PM2.5 Concentration and Meteorological Factors

PM2.5 concentration has dropped continuously during the past three years in Northeast China. However, daily PM2.5 levels varied remarkably, rising and falling rapidly with alternating sharp peaks and deep troughs (Wen et al., 2018). In Table 1, the confidence coefficient for meteorological factors above 95% confidence level (p-value < 0.05) indicates the factors significantly impact the daily variation of PM2.5 concentration to some extent. Wind speed, HPBL, and temperature inversion are important meteorological factors affecting the PM2.5 pollution in Northeast China. During pollution days, the regression coefficients of wind speed and HPBL are negative values, which show a clear negative influence of wind speed and HPBL on fine particle concentrations. Meanwhile, the values for the coefficients of TI_T and TI_D are positive, which indicates that temperature inversion has a positive impact on the accumulation of PM2.5. A meteorological factor plays the dominant role in affecting PM2.5 pollution when its standardized coefficient has the biggest absolute value. By comparing the standardized coefficients of different factors in the regression model, the most important factors are concluded. Wind speed is the key meteorological factor impacting PM2.5 concentrations in Harbin and Shenyang. Meanwhile, HPBL is the key factor in Changchun.

Straw burning and pollutant emission, near-surface temperature inversion, and static wind together with other extreme meteorological conditions contribute to the PM2.5 pollution, especially in autumn and winter. The breeze hinders the dispersion and transport of the atmospheric pollutants released by straw burning. On the contrary, high wind speed is beneficial to the horizontal dilution of the pollutants. On the other hand, the turbulence is strengthened with the increase of wind speed, which is also beneficial to the pollutant dilution and spreading. There are significant differences between the situations with and without the existence of temperature inversion. Temperature inversion decreases the upward movement of the lower atmosphere and causes the pollutants trapped in the lower atmosphere. Frequent atmospheric conditions with temperature inversion in winter can cause the accumulation of atmospheric particles (He et al., 2001; Han et al., 2010). The convection between upper and lower layers decreases, so that the atmospheric pollutants in the near-surface layer can hardly be transported upward. At the same time, the pollutants never stop accumulating, which exacerbates air pollution. The diffusion and transportation of pollutants in the lower atmosphere depends to a large extent on the boundary layer structure. Within the boundary layer, turbulence can fully mix particles and gases. All the three factors have significant influence on the PM2.5 concentrations in Northeast China. As can be seen in the multiple regression result (Table 1), during pollution days, all the three cities have the wind speed as one of the major impact factors. However, different from other two cities, the temperature inversion is not a major impact factor for Changchun. Such result may be caused by the fact that the altitude of Changchun (236.8 m) is higher than Harbin (171.7 m) and Shenyang (41.6 m). And the temperature inversion has the characteristic that it occurs more obviously in the lowland (or valley) area than the highland area.

As can be seen from Table 2, during non-pollution days, HPBL and temperature inversion are important factors affecting PM2.5 in Harbin and Changchun. In Shenyang, TI_D is the only important determinant impacting PM2.5 concentration. In summer, the thickened mixing layer improves the dispersion of pollutants in the atmosphere. In winter, frequent and persistent thermal inversions at ground level drive considerable amount of air pollutants to accumulate in the lower layer of the atmosphere (Marcazzan et al., 2001). Although wind speed, HPBL, and TI_T are important factors of PM2.5 concentration, their correlations with PM2.5 concentration during non-pollution days are not significant.

Fig. 6 shows the occurrence of PM2.5 pollution and non-pollution days in different meteorological conditions from 2015 to 2017. The meteorological factors for different cities are selected based on the significance level of 95% in Table 1. For Harbin, HPBL, WV, and TI_T are major impact factors. PM2.5 pollution usually occurs, with a possibility of 91.6%, when the wind speed is under 20 knots, HPBL is lower than 1500 m and TI_T is higher than 8°C. On the contrary, the possibility of occurrence of PM2.5 pollution is only 27.2% for the whole year. For Changchun, PM2.5 pollution is majorly impacted by wind speed and HPBL, the possibility is 61.7% when the wind speed is under 20 knots and HPBL is lower than 600 m. Meanwhile, the possibility is only 22.7% for the whole year. As for Shenyang, the possibility of the occurrence of PM2.5 pollution rises from 24.9% to 70.3% when the wind speed is under 20 knots, and TI_D is greater than 775 m.

Fig. 6. The occurrence of PM2.5 pollution and non-pollution days in different meteorological conditions from 2015 to 2017 in (a) Harbin, (b) Changchun, and (c) Shenyang.

## CONCLUSIONS

To evaluate the effect of meteorological conditions on PM2.5 pollution in Northeast China, we applied a regression method to long-term ground-based PM2.5 data and radiosonde measurements. We found that the wind speed, HPBL, and temperature inversion were the primary factors influencing the fine particulate pollution in this region. Whereas the wind speed exhibited the strongest correlations with the PM2.5 levels in Harbin and Shenyang, the HPBL played the key role in Changchun, with the meteorological data showing significant negative effects from both of these factors on the fine particle concentration. By contrast, the temperature inversion, described using TI_T and TI_D, was positively correlated with the concentration.

In Harbin, the probability of PM2.5 pollution was 91.6% when the wind speed was less than 20 knots, the HPBL was below 1500 m, and TI_T was greater than 8°C. In Changchun, the probability was 61.7% when the wind speed was less than 20 knots and the HPBL was below 600 m. Finally, in Shenyang, the probability was 70.3% when the wind speed was less than 20 knots and TI_D was more than 775 m. Unlike other typical polluted areas, the variation in the PM2.5 concentration during polluted days in Northeast China little depended on the RH.

In addition to anthropogenic emissions, which necessitate local control strategies, the ambient meteorological conditions play a large role in PM2.5 pollution in Northeast China and must be considered in order to efficiently decrease air pollution in this region. For example, when temperature inversion occurs, reducing coal and petroleum consumption should be contemplated as a potential preventative measure.

Our study provides preliminary results on the major meteorological factors that affect the accumulation and dispersion of PM2.5 pollution. Further studies may examine additional meteorological parameters and expand the observation period. Furthermore, we used the nocturnal HPBL data which is usually dominated by the residual layer, and the relevancy between PM2.5 and HPBL may not be true in nighttime. This issue can also be addressed in the future.

## ACKNOWLEDGEMENTS

This work is supported by the National Key Research and Development Program of China (2017YFC0212302). We would like to thank the Ministry of Environmental Protection of China for providing the PM2.5 observation data (http://106.37.208.233:20035/) used in this study. We highly appreciate the National Oceanic and Atmospheric Administration (NOAA) as well as the Integrated Global Radiosonde Archive (IGRA) for giving us access to the historical radiosonde data. We would also like to show great gratitude to the National Center for Atmospheric Research (NCAR), associated with the University Corporation for Atmospheric Research (UCAR), for the generous provision of HPBL data. Without these institutions the experiments could not be conducted.

## REFERENCES

1. Bao, J.Z., Yang, X.P., Zhao, Z.Y., Wang, Z.K., Yu, C.H. and Li, X.D. (2015). The spatial-temporal characteristics of air pollution in China from 2001–2014. Int. J. Environ. Res. Public Health 12: 15875–15887.
2. Chaloulakou, A., Kassomenos, P., Spyrellis, N., Demokritou, P. and Koutrakis, P. (2003). Measurements of PM10 and PM2.5 particle concentrations in Athens, Greece. Atmos. Environ. 37: 649–660.
3. Chen, T., Guo, J., Li, Z., Zhao, C., Liu, H., Cribb, M., Wang, F. and He, J. (2016). A CloudSat perspective on the cloud climatology and its association with aerosol perturbations in the vertical over Eastern China. J. Atmos. Sci. 73: 3599–3616.
4. Chen, W.W., Tong, D., Zhang, S.C., Dan, M., Zhang, X.L. and Zhao, H.M. (2015). Temporal variability of atmospheric particulate matter and chemical composition during a growing season at an agricultural site in northeastern China. J. Environ. Sci. 38: 133–141.
5. Chen, W.W., Tong, D.Q., Dan, M., Zhang, S.C., Zhang, X.L. and Pan, Y.P. (2017). Typical atmospheric haze during crop harvest season in northeastern China: A case in the Changchun region. J. Environ. Sci. 54: 101–113.
6. Chen, Z., Xie, X., Cai, J., Chen, D., Gao, B., He, B., Cheng, N. and Xu, B. (2018). Understanding meteorological influences on PM2.5 concentrations across China: A temporal and spatial perspective. Atmos. Chem. Phys. 18: 5343–5358.
7. Cheng, S.Q. and Lam, K.C. (1998). An analysis of winds affecting air pollution concentrations in Hong Kong. Atmos. Environ. 32: 2559–2567.
8. Dickerson, R.R., Li, C., Li, Z., Marufu, L.T., Stehr, J.W., McClure, B., Krotkov, N., Chen, H., Wang, P., Xia, X., Ban, X., Gong, F., Yuan, J. and Yang, J. (2007). Aircraft observations of dust and pollutants over northeast China: Insight into the meteorological mechanisms of transport. J. Geophys. Res. 112: D24S90.
9. Durre, I., Vose, R.S. and Wuertz, D.B. (2006). Overview of the integrated global radiosonde archive. J. Clim. 19: 53–68.
10. Fan, J., Wang, Y., Rosenfeld, D. and Liu, X. (2016). Review of aerosol–cloud interactions: Mechanisms, significance, and challenges. J. Atmos. Sci. 73: 4221–4252.
11. Fang, C.S., Zhang, Z.D., Jin, M.Y., Zou, P.C. and Wang, J. (2017). Pollution characteristics of PM2.5 aerosol during haze periods in Changchun, China. Aerosol Air Qual. Res. 17: 888–895.
12. Fang, F.M., Wang, Q.C., Liu, R.H., Ma, Z.W. and Hao, Q.J. (2001). Atmospheric particulate mercury in Changchun City, China. Atmos. Environ. 35: 4265–4272.
13. Garratt, J.R. (1994). The atmospheric boundary layer. Earth Sci. Rev. 37: 89–134.
14. Gu, X.F., Bao, F.W., Cheng, T.H., Chen, H., Wang, Y. and Guo, H. (2018). The impacts of regional transport and meteorological factors on aerosol optical depth over Beijing, 1980-2014. Sci. Rep. 8: 5113.
15. Guan, Q.Y., Li, F.C., Yang, L.Q., Zhao, R., Yang, Y.Y. and Luo, H.P. (2018). Spatial-temporal variations and mineral dust fractions in particulate matter mass concentrations in an urban area of northwestern China. J. Environ. Manage. 222: 95–103.
16. Guo, J., Deng, M., Lee, S.S., Wang, F., Li, Z., Zhai, P., Liu, H., Lv, W., Yao, W. and Li, X. (2016). Delaying precipitation and lightning by air pollution over the Pearl River Delta. Part I: Observational analyses. J. Geophys. Res. 121: 6472–6488.
17. Guo, J., Su, T., Chen, D., Wang, J., Li, Z., Lv, Y., Guo, X., Liu, H., Cribb, M. and Zhai, P. (2019). Declining summertime local-scale precipitation frequency over China and the United States, 1981–2012: The disparate roles of aerosols. Geophys. Res. Lett. 46: 13281–13289.
18. Han, B., Kong, S.F., Bai, Z.P., Du, G., Bi, T., Li, X., Shi, G.L. and Hu, Y. (2010). Characterization of elemental species in PM2.5 samples collected in four cities of Northeast China. Water Air Soil Pollut. 209: 15–28.
19. He, K.B., Yang, F.M., Ma, Y.L., Zhang, Q., Yao, X.H., Chan, C.K., Cadle, S., Chan, T. and Mulawa, P. (2001). The characteristics of PM2.5 in Beijing, China. Atmos. Environ. 35: 4959–4970.
20. Heal, M.R., Kumar, P. and Harrison, R.M. (2012). Particles, air quality, policy and health. Chem. Soc. Rev. 41: 6606–6630.
21. Hien, P.D., Bac, V.T., Tham, H.C., Nhan, D.D. and Vinh, L.D. (2002). Influence of meteorological conditions on PM2.5 and PM2.5-10 concentrations during the monsoon season in Hanoi, Vietnam. Atmos. Environ. 36: 3473–3484.
22. Jung, K.H., Patel, M.M., Moors, K., Kinney, P.L., Chillrud, S.N., Whyatt, R., Hoepner, L., Garfinkel, R., Yan, B. and Ross, J. (2010). Effects of heating season on residential indoor and outdoor polycyclic aromatic hydrocarbons, black carbon, and particulate matter in an urban birth cohort. Atmos. Environ. 44: 4545–4552.
23. Kan, H.D., Chen, R.J. and Tong, S.L. (2012). Ambient air pollution, climate change, and population health in China. Environ. Int. 42: 10–19.
24. Li, L., Tan, Q., Zhang, Y., Feng, M., Qu, Y., An, J. and Liu, X. (2017a). Characteristics and source apportionment of PM2.5 during persistent extreme haze events in Chengdu, southwest China. Environ. Pollut. 230: 718–729.
25. Li, Z., Guo, J., Ding, A., Liao, H., Liu, J., Sun, Y., Wang, T., Xue, H., Zhang, H. and Zhu, B. (2017b). Aerosol and boundary-layer interactions and impact on air quality. Natl. Sci. Rev. 4: 810–833.
26. Li, Z., Wang, Y., Guo, J., Zhao, C., Cribb, M.C., Dong, X., Fan, J., Gong, D., Huang, J., Jiang, M., Jiang, Y., Lee, S.S., Li, H., Li, J., Liu, J., Qian, Y., Rosenfeld, D., Shan, S., Sun, Y., Wang, H., Xin, J., Yan, X., Yang, X., Yang, X.Q., Zhang, F. and Zheng, Y. (2019). East Asian Study of Tropospheric Aerosols and their Impact on Regional Clouds, Precipitation, and Climate (EAST-AIRCPC). J. Geophys. Res. 124: 13026–13054.
27. Liang, X., Zou, T., Guo, B., Li, S., Zhang, H.Z., Zhang, S.Y., Huang, H. and Chen, S.X. (2015). Assessing Beijing's PM2.5 pollution: Severity, weather impact, APEC and winter heating. Proc. R. Soc. London, Ser. A 471: 20150257.
28. Liu, L., Guo, J., Miao, Y., Liu, L., Li, J., Chen, D., He, J. and Cui, C. (2018). Elucidating the relationship between aerosol concentration and summertime boundary layer structure in central China. Environ. Pollut. 241: 646–653.
29. Lohmann, U. and Feichter, J. (2001). Can the direct and semi-direct aerosol effect compete with the indirect effect on a global scale? Geophys. Res. Lett. 28: 159–161.
30. Lou, M., Guo, J., Wang, L., Xu, H., Chen, D., Miao, Y., Lv, Y., Li, Y., Guo, X., Ma, S. and Li, J. (2019). On the relationship between aerosol and boundary layer height in summer in China under different thermodynamic conditions. Earth Space Sci. 6: 887–901.
31. Ma, Q.X., Wu, Y.F., Zhang, D.Z., Wang, X.J., Xia, Y.J., Liu, X.Y., Tian, P., Han, Z.W., Xia, X.G., Wang, Y. and Zhang, R.J. (2017). Roles of regional transport and heterogeneous reactions in the PM2.5 increase during winter haze episodes in Beijing. Sci. Total Environ. 599: 246–253.
32. Marcazzan, G.M., Vaccaro, S., Valli, G. and Vecchi, R. (2001). Characterisation of PM10 and PM2.5 particulate matter in the ambient air of Milan (Italy). Atmos. Environ. 35: 4639–4650.
33. Miao, Y., Guo, J., Liu, S., Liu, H., Zhang, G., Yan, Y. and He, J. (2017). Relay transport of aerosols to Beijing-Tianjin-Hebei region by multi-scale atmospheric circulations. Atmos. Environ. 165: 35–45.
34. Newman, T.B. and Browner, W.S. (1991). In defense of standardized regression coefficients. Epidemiology 2: 383–386.
35. Petäjä, T., Järvi, L., Kerminen, V.m., Ding, A., Sun, J., Nie, W., Kujansuu, J., Virkkula, A., Yang, X., Fu, C., Zilitinkevich, S. and Kulmala, M. (2016). Enhanced air pollution via aerosol-boundary layer feedback in China. Sci. Rep. 6: 18998.
36. Rosenfeld, D., Sherwood, S., Wood, R. and Donner, L. (2014). Climate effects of aerosol-cloud interactions. Science 343: 379.
37. Ruellan, S. and Cachier, H. (2001). Characterisation of fresh particulate vehicular exhausts near a Paris high flow road. Atmos. Environ. 35: 453–468.
38. Segura, S., Estellés, V., Esteve, A.R., Utrillas, M.P. and Martínez-Lozano, J.A. (2013). Analysis of a severe pollution episode in Valencia (Spain) and its effect on ground level particulate matter. J. Aerosol Sci. 56: 41–52.
39. Seinfeld, J.H., Pandis, S.N. and Noone, K. (1998). Atmospheric chemistry and physics: From air pollution to climate change. Wiley, New York.

40. Sun, Y.L., Wang, Z.F., Fu, P.Q., Jiang, Q., Yang, T., Li, J. and Ge, X.L. (2013). The impact of relative humidity on aerosol composition and evolution processes during wintertime in Beijing, China. Atmos. Environ. 77: 927–934.
41. Tian, M., Wang, H.B., Chen, Y., Zhang, L.M., Shi, G.M., Liu, Y., Yu, J.Y., Zhai, C.Z., Wang, J. and Yang, F.M. (2017). Highly time-resolved characterization of water-soluble inorganic ions in PM2.5 in a humid and acidic mega city in Sichuan Basin, China. Sci. Total Environ. 580: 224–234.
42. Twomey, S. (1977). The influence of pollution on the shortwave albedo of clouds. J. Atmos. Sci. 34: 1149–1152.
43. Wang, H., Xu, J., Zhang, M., Yang, Y., Shen, X., Wang, Y., Chen, D. and Guo, J. (2014a). A study of the meteorological causes of a prolonged and severe haze episode in January 2013 over central-eastern China. Atmos. Environ. 98: 146–157.
44. Wang, J. and Christopher, S.A. (2003). Intercomparison between satellite‐derived aerosol optical thickness and PM2.5 mass: Implications for air quality studies. Geophys. Res. Lett. 30: 267–283.
45. Wang, Y.G., Ying, Q., Hu, J.L. and Zhang, H.L. (2014b). Spatial and temporal variations of six criteria air pollutants in 31 provincial capital cities in China during 2013-2014. Environ. Int. 73: 413–422.
46. Wen, X., Zhang, P.Y. and Liu, D.Q. (2018). Spatiotemporal variations and influencing factors analysis of PM2.5 concentrations in Jilin Province, Northeast China. Chin. Geog. Sci. 28: 810–822.
47. Xu, T., Song, Y., Liu, M., Cai, X., Zhang, H., Guo, J. and Zhu, T. (2019). Temperature inversions in severe polluted days derived from radiosonde data in North China from 2011 to 2016. Sci. Total Environ. 647: 1011–1020.
48. Xu, Z., Yu, D., Jing, L. and Xu, X. (2000). Air pollution and daily mortality in Shenyang, China. Arch. Environ. Health 55: 115–120.
49. Yang, X., Zhao, C., Guo, J. and Wang, Y. (2016). Intensification of aerosol pollution associated with its feedback with surface solar radiation and winds in Beijing. J. Geophys. Res. 121: 4093–4099.
50. Yao, L., Lu, N. and Jiang, S. (2012). Artificial Neural Network (ANN) for multi-source PM2.5 estimation using surface, MODIS, and meteorological data. 2012 International Conference on Biomedical Engineering and Biotechnology, Macao, 2012, pp. 1228–1231.

51. You, W., Zang, Z.L., Pan, X.B., Zhang, L.F. and Chen, D. (2015). Estimating PM2.5 in Xi'an, China using aerosol optical depth: A comparison between the MODIS and MISR retrieval models. Sci. Total Environ. 505: 1156–1165.
52. Zang, Z.L., Wang, W.Q., You, W., Li, Y., Ye, F. and Wang, C.M. (2017). Estimating ground-level PM2.5 concentrations in Beijing, China using aerosol optical depth and parameters of the temperature inversion layer. Sci. Total Environ. 575: 1219–1227.
53. Zhang, G., Li, J., Li, X.D., Xu, Y., Guo, L.L., Tang, J.H., Lee, C.S.L., Liu, X.A. and Chen, Y.J. (2010). Impact of anthropogenic emissions and open biomass burning on regional carbonaceous aerosols in South China. Environ. Pollut. 158: 3392–3400.
54. Zhang, P.F., Dong, G.H., Sun, B.J., Zhang, L.W., Chen, X., Ma, N.N., Yu, F., Guo, H.M., Huang, H., Lee, Y.L., Tang, N.J. and Chen, J. (2011). Long-term exposure to ambient air pollution and mortality due to cardiovascular disease and cerebrovascular disease in Shenyang, China. PLoS One 6: e20827.
55. Zhang, Q., Quan, J.N., Tie, X.X., Li, X., Liu, Q., Gao, Y. and Zhao, D.L. (2015). Effects of meteorology and secondary particle formation on visibility during heavy haze events in Beijing, China. Sci. Total Environ. 502: 578–584.

Aerosol Air Qual. Res. 20:1618-1628. https://doi.org/10.4209/aaqr.2019.10.0534