Kiethlaaks:
ReadMyACARS - Assuming the Inmarsat doppler values are accurate, I ran the math which gave me the track and speed.
Inmarsat Values
[[2014/3/7 19:40:00, -109],
[2014/3/7 20:39:59, -140],
[2014/3/7 21:40:00, -169],
[2014/3/7 22:40:00, -202],
[2014/3/8 00:11:00, -250]]
Assuming a straight track heading 182, mach 0.39 @ 12000, the predicted doppler values are very close: [-102.21237107180059, -140.52053354680538, -174.36227202415466, -204.76829959079623, -247.19930766150355].
No other combination of speed and track comes close.
The slow initial speed you have calculated for the first two hours are due to the aircraft flying tangentially to the ping-rings. The doppler-shift only records the aircraft flying to or from the central location of the satellite coverage (ie: radially to the ping-rings). If the aircraft is flying tangentially to the ping-rings, the doppler-shift will be much smaller.
See my post at No9412 for a graphic. The first two hourly-tracks will give very small doppler-speeds, because the aircraft is almost flying along the ping-ring. (NB, it may not have been flying exactly south like this, i just chose south as an easy example for this graphic. If it had been initially flying 160 degrees, it would have given a slow doppler-speed, just as you have calculated.)
You will also note that ReadMyACARS (post 9401) has calculated an average speed requirement of 360 kts TAS (aka: groundspeed). So even your last doppler speed was slightly tangential to the ping-rings.
Correct me if I am wrong (because circumferential math is not my strong point), but the tangential angles that the aircraft was flying - according to your data and the average 360 kt speed requirement - was:
Hourly track legs:
a. 17 degrees from the tangential.
b. 23 degrees from the tangential.
c. 28 degrees from the tangential.
d. 34 degrees from the tangential.
e. 44 degrees from the tangential.
So it would appear that the initial heading was about 160 degrees, while the final heading was probably around the same, about 160 degrees. The mag-variation will not change this much.
You could plot those values from the claimed start location, using an average track-length of 360nm, and see if the total track tallies with the current search location. But do check that the average 360 kt ground-speed is correct, I am only taking the other poster's word for this (but it sounds about right). Alternative, you could back-plot the route, to see where the start point is.
Updated:
Actually, the ping-rings veer to the southwest the further south you go, so the tangential angle naturally increases as the aircraft flies south. In which case, the aircraft might have been flying a constant heading/track, of about 160 or 165 degrees, with the tangential angles to the ping-rings increasing naturally as the aircraft goes south. Someone will have to plot a more accurate map, to see if those tangentials make a straight track. I suspect that they do.
If anyone has an accurate map, with the location of the ping-rings correctly represented, i can plot it out. Sorry for the error.