AS A physicist, I was excited to see how a complicated problem involving the search for Malaysia Airlines Flight MH370 was solved using very limited data.
Scientists thinking out of the box formulated a solution using well-known scientific principles in a fine example of scientific creativity. In the case of MH370, which disappeared on March 8, the radar transponder and the aircraft's Communication Addressing and Reporting System (ACARS) were mysteriously switched off about an hour after take-off, when it was over the South China Sea.
ACARS is the system that feeds information about the aircraft's location, altitude, speed and heading to a ground station, either directly by radio waves or via a satellite link.
Without this information, an aircraft cannot normally be located once it is out of range of civilian radar.
In this case, however, military radar detected MH370 changing course and heading west, with its last known radar contact somewhere between the Strait of Malacca and the Andaman Sea.
Six "pings", or handshakes, one every hour, between the aircraft and a satellite known as Inmarsat 3-F1 also suggested that the aircraft remained in the air for another five hours before its fuel was depleted.
Assuming a maximum cruising speed of 450 knots, this produced an enormous search area with a radius of more than 4,000km.
Inmarsat 3-F1 is a geostationary communication satellite located at longitude 64.5 degrees east, 35,800km above the equator.
The aircraft's response to the satellite's hourly "pings" would normally contain at least the aircraft's position and heading. But in this case, the signal was empty because ACARS was not active.
How then, were the scientists and engineers at Inmarsat, the company that operates the communication satellite, able to determine that MH370 headed towards the southern Indian Ocean?
They began by making use of the time delay between the satellite sending out the ping and the response from the aircraft to figure out the distance of the aircraft from the satellite.
However, timing alone could not pinpoint the aircraft's location. At most, an arc on the earth's surface equidistant from the satellite could be drawn.
Together with the estimation of the minimum and maximum speed of the aircraft, the possible final location of the plane was narrowed down to two "corridors".
One extended northwards into Central Asia, and the other one went south to the Indian Ocean. This information was released to the public on March 15.
But scientists did not give up and tried to squeeze as much information out of the six pings, or data points, as possible.
They came up with a mathematical model that involved the Doppler effect.
Discovered in 1842, the Doppler effect has to do with the way the frequency of a wave (light, sound or electromagnetic pulse) appears to change when the source of an object emitting a wave moves relative to an observer.
By measuring the frequency shift, the speed of the object can be estimated.
The effect can be heard when a police car or emergency vehicle travels towards you on the highway. As the car approaches with its siren blasting, the pitch of the sound is high. But as the car passes by, the pitch drops suddenly. A radar speed gun uses the same principle to catch speeding offenders on the road.
As the location of the last radar contact of MH370 was known, the trajectory of the aircraft could, in principle, be plotted.
The concept appeared simple, but actually implementing it on the six data points was a not straightforward matter because of the many uncertain parameters.
Furthermore, as the satellite was located above the equator, any southern trajectory it implied would have an equally probable mirror image in the north. It seemed that Doppler frequency shift alone would not be able to resolve the ambiguity.
Fortunately for the scientists, Inmarsat 3-F1 was an old satellite that had drifted slightly out of the perfectly geostationary orbit. Its orbit was inclined at about 1.5 degrees from the equatorial plane. And it was not exactly geostationary either.
To someone on the ground, it appeared to wobble about the geostationary position.
The satellite motion therefore broke the symmetry. This meant there was a possibility the north-south ambiguity could be resolved.
Instead of trying to derive the speed and location from the frequency shift directly, the scientists built a mathematical model that predicted the frequency shift, considering the motion of the satellite and the aircraft.
By comparing the measured frequency shift with the predicted shift, they found that the measured shift of the southern trajectory was consistent with the shift predicted by the model.
The discrepancy for the northern trajectory, however, was larger in comparison. They were therefore able to conclude that the aircraft headed south towards an area in the southern Indian Ocean west of Australia.
But the scientists did not stop at that. They checked the model using actual data from aircraft travelling along similar routes. The model worked as expected, and the results were announced on Monday.
The job of narrowing the search down further has fallen to those who analyse satellite images that cover the southern Indian Ocean. Depending on their functions, the imaging instruments carried by these satellites have different resolutions and spatial coverage.
For example, meteorological satellites need to observe a wide area, but they do not require a high resolution. On the other hand, there are satellites capable of acquiring very high resolution images, but have a limited spatial coverage.
These latter satellites can detect objects as small as half a metre wide. But each scene covers only about 15km or less.
They also travel in their respective fixed orbits governed by gravity, and cannot be steered at will.
Then, there is the problem of cloud cover. To look for debris over such a vast area is sometimes described as looking for a needle in a haystack. In this case, however, no one knew where the haystack was.
New radar data showing that the plane was going faster between the South China Sea and the Strait of Malacca, and therefore running out of fuel more quickly than was previously believed, has led to a further modification of the search zone.
Hopefully, some of the debris can be recovered soon.
After that, yet another group of experts, this time from the aviation industry, will work on the mystery of Flight MH370's disappearance.
Only then will the world know what really happened to the ill-fated plane, and the relatives of those on board may finally have some sense of closure.
The writer is principal research scientist and head of research at the Centre for Remote Imaging, Sensing and Processing, National University of Singapore.
