Doppler pings from MH370 Inmarsat : how was the data processed?
I would not think arrival time. The sat has to work with ground stations that it is approaching and receding from, so it can work with the Doppler shifted frequencies and probably uses the shift to calibrate position and orbital elements.

Depending on satellite position and orbit, one of the paths will give a different shift than the other. Slant angles would be different and if the sat is in between the arcs, one would cause blue shifting the the other red. Probably pretty easy for the them to figure out.

The police Dopplers can easily measure a car speed within 1 MPH. Consider the effect from the speed of a 777.