Sometimes I am amazed by the level of unwarranted effort going into this thread! Those folks at Inmarsat know a thing or two about how their hardware and software work, and don't need references to articles about Doppler correction.
Nor do they need to fly another aircraft by the same route: they validated their calculations using the handshakes from a number of Malaysian aircraft flying in that area. That's part of what they were doing in the days between the aircraft vanishing and their conclusions being published.