Deriving the course from the Inmarsat pings
Given a number of recent posts on the methods of interpreting the INMARSAT ‘ping’ data, here is my understanding of how this will have been done by the NTSB.
Each ping specifies a distance from the INMARSAT F1 sub-satellite point at 64degrees East on the equator. On the Earth’s surface this is a circle (but not a great circle). A model track can be specified from the last reported position of the aeroplane off the West coast of Malaysia to any point on the circle described by the last ping. The length of that track specifies the speed of the aeroplane (since the times are known) and hence where it would be when the intermediate (every hour) pings were exchanged. The distances of these ping positions from the sub-satellite point can be compared to the actual data (held by NTSB but not us). The point on the final ping circle (the arcs) can be moved until the model track matches the data. If there is no good match, change parameters like the final turning point, smooth changes in speed along the track etc. to get a good match to the data (there is a limit to how many things can be changed before the model can fit any data). Unless the aeroplane was performing extreme and random manoeuvres during its flight (in which case it would have crashed earlier) a course and speed of quite high accuracy can be obtained, I think.
The fuel exhaustion point is a different calculation and more uncertain IMHO.
A longer and even more boring post is at #5911 of this thread.