MH370: Hunt depending on complex mathematics

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Details of investigators’ analytical techniques to determine the fatal track of Malaysia Airlines’ missing flight MH370 have illustrated the mathematical complexity of the inquiry.

But close alignment between predicted and known tracks from other aircraft have reinforced confidence in the modelling techniques.

Central to narrowing the search area has been derivation of possible flightpaths using timing and frequency offset of signals between the missing Boeing 777, the Inmarsat-3F1 satellite and a ground station in Perth.

Newly-disclosed details of the analysis from the Australian Transport Safety Bureau reveal it has even taken into account temperature effects of a 67min eclipse of the satellite as it passed through the Earth’s shadow.

While the Boeing 777’s satcom terminal adjusts its transmitting frequency to compensate for Doppler shifts created by the jet’s heading and ground speed, it does not account for vertical motion of the aircraft or movement of the satellite away from its nominal location.

“These small errors are immaterial to the communications performance,” says the inquiry. “But [they] do affect the [frequency offset].”

Calculations also have to include error biases in the aircraft and satellite equipment as well as automatic frequency compensation by hardware at the ground station.

Analysts checked the frequency offset calculations using data from nine previous flights of the vanished 777, as well as 87 other aircraft – fitted with the same satcom equipment – which were airborne at the same time as MH370.

Comparisons of known flightpaths, including those of Malaysia Airlines services to Europe, showed good correlation between predictions from the investigators’ models and the actual tracks flown.

As part of the verification, frequency offsets were predicted for random flightpaths generated in all directions from the last known radar plot of MH370, off the tip of Sumatra. All the paths which matched the investigators’ measured frequency offsets corresponded to tracks ending in the southern Indian Ocean.

“This was able to confirm that the southern corridor was the only valid solution,” the inquiry states.

While the investigators have assumed that the aircraft turned south sometime after 18:25UTC, the location of this turn is unknown.

Satellite data analysis has been carried out using a range of assumed locations. But analysis has also been performed with no assumptions, says the inquiry, in order to check possibilities which fit “realistic” times and distances.

Uncertainty over MH370’s airspeed and the heading followed by the aircraft after the turn has generated a range of possible tracks constrained by the 777’s performance.

But there is also a lack of detailed knowledge about the behaviour of the aircraft following the probable exhaustion of its fuel.

Investigators state that there is evidence of long periods without en route manoeuvring and a steadily-maintained cruise altitude, indicating that the aircraft could have been under autopilot control for an extended period.

After fuel exhaustion the aircraft could have glided for over 100nm if it had been under pilot control. Given that this would make the search area “impractically large”, the inquiry has considered the possibility – without conclusive evidence – that there was no pilot input.

If the autopilot had been engaged it could have remained active after the first engine flamed out. But it would have disconnected after the loss of power from the second engine. Natural asymmetric forces would have acted on the aircraft as it lost altitude and the 777 would have entered a spiral descent.