Malaysia Airlines Flight 370 disappeared in the early hours of March 8 en route to Beijing from Kuala Lumpur with 239 people on board. As this issue of SIAM News goes to press, no confirmed remnants of the plane and no survivors have been found. This great tragedy has saddened people the world over. We sympathize with the family members of the passengers, crew, and pilots of MH370.
On March 28, the search area for MH370 was refocused. The new region is in the southern Indian Ocean, about 1850 km west of Perth, Australia; it is 1100 km northeast of the area of the Indian Ocean initially searched, starting on March 17. Both search areas were identified by a team of engineers from the British satellite company Inmarsat, in collaboration with the US National Transportation Safety Board (NTSB) and Britain’s Air Accident Investigation Branch (AAIB). An enormous amount of engineering expertise—much more than can be addressed in this article—has been devoted to solving the mystery. As stated in an article in The New York Times on Sunday, March 23,
"By Sunday afternoon [March 9], a team of Inmarsat engineers set to work using the principles of trigonometry to determine the distance between the satellite and the plane at the time of each ping, and then to calculate two rough flight paths."
In this article I consider some of the mathematics used by the Inmarsat engineering team to deduce the flight path of MH370. In addition, using a simulated fictitious flight (XX123), I illustrate some of the inherent difficulties of the team’s search.
Publicly Available Data
Figure 1. Map of the path of fictitious flight XX123. For a departure from Colombo, Sri Lanka, at 12 noon and arrival in Perth, Australia, at 7:48 PM, the three red arcs show the location of the plane at 3 PM, 5 PM, and 7 PM. Based only on this information, the blue curve shows the path of the plane traveling at 400 knots deduced by the methods described in this article. The green curve shows how the deduced flight path changes when the starting location is assumed to be 220 km due east of Colombo. A plane traveling along the green path would have a speed of 416 knots and would be 970 km from Perth at 7:48 PM.
The only publicly available reliable data for MH370 are (1) the location of the INMARSAT 3-F1 satellite
in a geostationary orbit at an elevation of 35,800 km over the point on the earth at 1.5º N, 64.5º E; (2) the data released by AAIB on March 25
, which includes the times at which the satellite pinged the aircraft, together with the measured burst frequency offsets (i.e., the Doppler-shift frequencies) at those times; and (3) a map released by the Malaysian government
showing that the angle between the 8:11 AM arc and the satellite position was about 40º. The initial search area is accurately shown on a map posted on March 17 on the website of The Washington Post
. That map also includes the approximate location of the satellite and four red arcs marked with the times 5:11 AM, 6:11 AM, 7:11 AM, and 8:11 AM. Although the map is qualitatively correct, the data it seeks to represent does not agree quantitatively with the three reliable data sets described previously.
Because the methodology used by the Inmarsat team to deduce possible flight paths depends on accurate knowledge of the locations of the arcs, it is not currently possible for their calculations to be accurately reproduced by outsiders. Instead, in this article I illustrate the mathematical methods used by the Inmarsat team by considering fictitious flight XX123, starting at 12 noon from Colombo, Sri Lanka, and moving toward Perth, Australia, at a constant speed of 400 knots (741 km/h) along an arc of a great circle. The only data assumed to be reliable are that the plane was in the southern hemisphere somewhere on three known arcs at 3 PM, 5 PM, and 7 PM, and that the starting location was at 7º N, 80º E. The three arcs, which are shown in red in Figure 1, go through (92.6º E, 8.5º S), (101.3º E, 18.5º S), and (111.6º E, 28.4º S). The goal, using only this data, is to deduce the speed and flight path of the fictitious aircraft, assuming that it was traveling on a great circle.
Question 1: Did the Plane Head North or South?
The Inmarsat team used a sophisticated analysis of Doppler-shift data to conclude that MH370 headed south into the Indian Ocean, rather than north into central Asia. To understand why Doppler data could be used to reach this conclusion, let’s perform the following thought experiment:
Your friend holds a basketball at eye level, with her hands outstretched. Her eye is the satellite; the basketball is the earth. Your friend is looking straight at a point on the equator of the basketball. This is the position of the satellite, indicated by the open black circle in the top left corner of Figure 1. (In reality, this point is just above the equator, but that doesn’t matter for the purposes of our thought experiment.)
We represent the flight of the plane by the motion of an ant walking on the basketball. The ant starts just above the equator. You want to see whether the ant is getting closer to or farther from your friend’s eye. Suppose that the ant is walking due south on the basketball. While it is still in the northern hemisphere, the distance between the ant and your friend’s eye decreases. The ant is closest to your friend’s eye at the instant it crosses the equator. Once it passes into the southern hemisphere, the ant starts to move away from your friend’s eye.
The Doppler effect is a property of an electromagnetic signal that is sent by one object (the aircraft) and received by another (the satellite). If the aircraft is moving toward the satellite, the light received by the satellite is more blue than the light that was sent; if the aircraft is moving away from the satellite, the received light is more red. The change in the frequency of the transmitted light, whether toward (high-frequency) blue or (low-frequency) red, is called the Doppler shift. The Inmarsat engineers were able to calculate the Doppler shift at each ping. As I explain in the Epilogue, their results showed that the plane took the southerly route. Before releasing this finding to the public, they validated their results by comparison with satellite data gathered by other planes that were flying in a known direction.
Question 2: How Did Inmarsat Deduce Possible Flight Paths?
Where in the Southern Hemisphere did MH370 go? The shape and location of the four red arcs in the Washington Post map of March 15 provide an important clue. As in the map shown in Figure 1 for fictitious flight XX123, the Washington Post map for MH370 suggests that the arcs are on concentric circles whose common center is given by the position of the satellite.
This fact can be established from the following quote from an AAIB press report of March 25:
"The position of the satellite is known, and the time that it takes the signal to be sent and received, via the satellite, to the ground station can be used to establish the range of the aircraft from the satellite. This information was used to generate arcs of possible positions from which the Northern and Southern corridors were established."
In simpler terms, every point on a given arc is at the same distance from the satellite. If you think of the satellite as being located at the tip of an ice cream cone, then the arcs are on the intersection of the cone with a spherical ball of ice cream in the cone.
Let’s imagine that the satellite antenna looks like the familiar dish antennas we often see here on earth. When the satellite antenna points straight down to earth, it points at the position of the satellite designated by the little black circle in the top left corner of Figure 1. For the purposes of this discussion, let’s call this point on the earth the satellite pole. The role it will play is similar to that of the North Pole on familiar maps of the earth. We can also talk about longitude and latitude measured with respect to the satellite pole. We refer to these notions of longitude and latitude as being in the satellite coordinate system. The red arcs are circles of latitude measured in the satellite coordinate system.
To estimate the flight path of MH370, the Inmarsat team initially assumed that after a certain time the plane switched to auto-pilot and was therefore traveling along the arc of a great circle at close to constant speed and at a constant elevation. From radar data, we know that the speed of the plane an hour after takeoff was 540 mph. In the approach presented here, by contrast, rather than specifying the speed of the plane at the beginning of the calculation, we infer it in the course of the calculation.
Before going into technical details, let’s do a second thought experiment, also using a basketball and, because we don’t know the exact locations of the arcs for MH370, using instead the data for flight XX123. For this experiment:
Begin by drawing the three XX123 arcs on the basketball with a red marker. Now put the basketball in a hoop that fits snugly around the ball. Pin the rim onto the basketball at the starting location of the plane at 12 noon, and put another pin at the exact opposite point on the ball. Now we consider all the possible great circles through this point. Just as we can tell which line of longitude we are on by specifying how far east or west of the Greenwich meridian we are, we can tell which great circle we are on using an angle that I call the angle of the great circle. The rim of the hoop can now define any of the great circles that pass through the starting point. Imagine that the rim has a ruler marked on it.
To continue, you need to pick a speed for the plane. Because you can’t assume any knowledge of the actual speed of the plane, you need to choose a reasonable range of speeds and do the experiment for each speed in that range. For each choice of speed, draw marks on the rim that are equally spaced and that correspond to the distance the plane would travel in two hours at the speed you picked. Now rotate the rim around the basketball. Do the three marks on the ruler line up exactly with the three red arcs? If so, you have a match. If you find a match, you have found a possible flight path.
You could actually do this experiment with a real basketball and rim. Unless someone had told you ahead of time that there was an answer to be found, and had shown you roughly where to look for it, you might find it extremely challenging to find a match. This is what makes the original work of the Inmarsat team so impressive. For XX123, I worked out some mathematical equations that enabled me to use a computer simulation to search for a match. The derivation of the equations uses mathematics—trigonometry, vector algebra, and appropriate choices of coordinates on the sphere—that could easily be explained to high school students. I presume that the Inmarsat team made similar calculations. I then coded my formula up in Matlab and ran the code to find the blue flight path for XX123 shown in Figure 1.
Figure 2. Direction of XX123’s flight path as a function of speed. For each speed on the horizontal axis, the three heavier curves show the angle on the vertical axis required for fictitious flight XX123 to cross the three arcs in Figure 1 at the required times, assuming that the plane starts from Colombo. A match can be detected at a speed of 740 km/h (400 knots). If the starting location is moved 220 km east, however, the thin solid curves indicated with a W in the legend exhibit an approximate match at a speed of 770 km/h (416 knots).
Back in our thought experiment, if you didn’t pin the rim on at exactly the right starting place, you might have come up with a very different solution. This, in essence, is the problem faced by the Inmarsat engineers, who didn’t know the exact location of the plane at the time it went on auto-pilot. In fact, the March 28 move of the search area was motivated by a more careful analysis of Malaysian radar data; the analysis suggested that before going on auto-pilot, the plane had traveled faster and hence farther west than thought initially. This sensitivity to initial conditions is illustrated by the green flight path shown in Figure 1, which starts 220 km due east of the “true” starting location of XX123.
Recall that to perform this experiment you had to pick a speed for the plane. If you chose the speed incorrectly, you may not have found a match. Figure 2 illustrates the matches I found for flight XX123. The horizontal axis of the plot shows the range of speeds I used in my search for a match. For each of the three arcs and for each speed, I computed the angle of the great circle along which the plane would have to fly to get from its starting location to that arc in the required time. The vertical axis shows those angles. You have a match when there is an angle and a speed at which all three curves meet.
From a given starting location there are two possible flight paths that cross the arcs at the correct times: one headed south and the other north.
On March 25, AAIB released the results of a sophisticated analysis of Doppler-shift data for the relative motion of MH370 and the satellite, which provides strong confirmation that the aircraft took the southerly route. As I will explain in a forthcoming article, the Inmarsat engineering team was able to reach this conclusion only by taking into account a slight elliptical rotation of the satellite about its nominal location above the equator.
This new technique developed by the Inmarsat team—a tour de force—is central to their ongoing investigation of the fate of MH370. Recently, I analyzed the technique mathematically and derived a formula for the velocity of a plane at each ping time in terms of the Doppler shift and the angles of the arcs. (Formulae like this are surely well known to experts in the important field of Doppler-based tracking.) In a future pedagogical article, I plan to demonstrate how, with arc angles and Doppler shifts measured at suitably chosen ping times, this formula can be used to determine the flight path of a plane, even if it is not flying along a great circle or at constant speed. It has been challenging for the Inmarsat team to use such a method to completely eliminate the uncertainty in the flight path of MH370 because, at times during the first three hours, the path of the plane was extremely erratic and the plane was not pinged at sufficiently short intervals during that time.
Acknowledgments: Thanks to Joel Achenbach, Alan Boyle, Matthew Goeckner, Brian Marks, and Yannan Shen for helpful conversations. Yanping Chen has produced movies illustrating the thought experiments in this article; see http://www.utdallas.edu/~zweck/.