The Forensic Mathematics Behind the Desperate Search for the Malaysia Airlines Plane
On January 17, the governments of Malaysia, China, and Australia agreed to suspend the search for Malaysia Airlines MH-370, a Boeing 777-200ER aircraft that vanished in mysterious circumstances in March 2014.
The aircraft was on a scheduled flight from Kuala Lumpur to Beijing when it disappeared from air traffic controllers’ radar screens. Military radar continued to track the aircraft, which deviated from its planned route and eventually flew south, finally traveling beyond radar range. The aircraft was never seen or heard from again and the 242 people on board are assumed dead.
The aircraft has never been found because nobody knows where it landed or crashed. The best guess is that it flew south for seven hours and then ditched in the Indian Ocean, some 1,800 kilometers southwest of Perth, Australia. But an extensive search of the sea surface and seafloor in that area has found nothing.
All that raises an important question: have the authorities been looking in the right place?
Today, Ian Holland of the Australian Defence Science and Technology Group publishes some of the reasoning that has defined the search area. Holland has been an important member of the team that has analyzed the data relating to the flight. In particular, he has focused on the last known signals sent from the aircraft to an orbiting Inmarsat communications satellite. In the absence of any other information from the plane, investigators have used these signals to determine the search area—but is there any more that can be gleaned from this data?
First some background. MH370 was fitted with a satellite data unit capable of relaying voice conversations and routine data transmissions. It sent its information via an Inmarsat satellite that is geostationary over the Indian Ocean. Although the aircraft transmitted no voice communication, the satellite data unit continued to operate, acknowledging two telephone calls from the ground that went unanswered and making several routine broadcasts such as electronic handshakes and the like.
At first glance, it’s hard to imagine how these brief data transmissions can provide any information about the aircraft’s location. But Holland and his colleagues have used them to gather a remarkable amount of information.
The communications protocol requires a ground station to make contact with the aircraft’s satellite data unit at a specific time and frequency, regardless of where the plane is on the planet. However, the signal takes time to travel from the ground to the aircraft and back again. This time, known as the burst time offset, is determined by the distance the signal has to travel.
This distance is straightforward to calculate. It defines a circle centered on the position on the ground directly below the satellite. However, the calculation does not suggest where on this circle the plane might be, and investigators have had to use other clues to narrow down this position.
In total, MH370 sent seven signals from its satellite data unit, each defining a slightly different circle. It sent its final signal at 0019 UTC on March 8, 2014, having initiated a log on request just eight seconds earlier.
That’s an important clue. Log on requests only occur when the satellite data unit restarts after some kind of shutdown. Investigators have assumed this shutdown occurred when the plane ran out of fuel and the SDU restarted using power from a device called a ram air turbine, which is deployed in an emergency to generate power.
If that is correct, the last transmission must have been near the end of the flight. But how near? Could MH370 have glided many tens or hundreds of kilometers before it hit the ocean? If so, this significantly increases the potential search area.
Holland says he and colleagues are able to narrow down this area using another line of mathematical investigation. The satellite data unit broadcasts at a specific frequency, but the aircraft’s velocity toward or away from the satellite introduces a Doppler shift that changes this frequency. This is known as the burst frequency offset.
So in theory it’s possible that this shift in frequency can indicate the direction of flight at that instant. In practice, this calculation is hard to do and is much tougher than calculating the distance. Holland’s paper today is largely about this calculation. “The Burst Frequency Offset is a more complex measurement which is generally less well understood,” he says.
The calculation is tough because of the number of variables that can influence the frequency. The aircraft’s motion is just one of them. The motion of the satellite plays a role, creating a Doppler shift associated with the uplink and downlink between the satellite and ground station.
The ground station also attempts to compensate for any Doppler shift by changing the frequency. And the oscillators in the satellite and aircraft transmitters are not perfect. They vary, producing changes in broadcast frequency.
Holland and co attempted to understand all these sources of frequency change by analyzing the broadcasts from MH370 during 20 previous flights in the week before it was lost.
Holland goes on to show that if the plane was flying level when a call was made to the plane from the ground soon after contact was lost, then the burst frequency offsets suggest it must have been flying south. That’s important.
He also shows that Doppler shifts on the final two broadcasts from the plane’s satellite data unit, suggest that it was descending rapidly. “The downwards acceleration over the 8 second interval between these two messages was found to be approximately 0.68g,” says Holland. This is consistent with the plane being out of control and out of fuel.
That has important implications for the search area. If the plane was in an uncontrolled descent, it cannot have flown far after the last broadcast of the satellite data unit. And that means the plane must lie somewhere near the arc calculated from the burst timing offset data. “This suggests that 9M-MRO should lie relatively close to the 7th BTO arc,” concludes Holland. But exactly where on this arc isn’t clear.
That’s interesting work which Holland is now opening up to outside scrutiny. He clearly sets out many of the assumptions he and his colleagues have had to make in coming to their conclusion. An important question for the community is whether these assumptions are all justified and whether Holland and his team have overlooked anything.
In the meantime, the families of the victims are conducting their own search for wreckage associated with the plane. And until new evidence emerges, the search for MH370 will remain suspended.
Ref: arxiv.org/abs/1702.02432: The Use of Burst Frequency Offsets in the Search for MH370
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