BOAC and Gysbreght.
From the Inmarsat filed data you have for each ping: a time H, a BTO (a length of time), a BFO (a frequency value). Please, reread the first page of the "raw data".
The BTO is only used to know at which distance R is the a/c from the satellite at the time H. From this distance R you can know that the a/c is somewhere on a circle. This circle is the intersection of:
- The sphere centered on the satellite at the time H and with a radius R,
and
- the "sphere" centered at the Earth center and which has the radius of the Earth plus the a/c altitude. This "sphere" is not a true sphere because the Earth is not perfectly spherical, so the circle is not a true circle.
The BFO comes from:
-1- the speed of the a/c relatively to the satellite,
-2- the speed of the satellite relatively to the a/c,
-3- the speed of the satellite relatively to the Earth station,
-4- an "error" from the a/c component which communicate with the satellite,
-5- the position of the satellite relatively to its theoretical fixed position above the equator.
#3 and #5 are perfectly known at whatever time. #4 is deduced from the data of the begining of the flight and from older flights of the a/c. So the global effect of #1 and #2 could be known from the filed BFO.
Theoreticaly, the a/c compensate the BFO from its own speed relatively to the satellite (#1). But this compensation is based on a fixed satellite. So the compensation the a/c mades is wrong when the satellite is not at its theoretical position. Say this value is F1.
The other useful component of the BFO is the one from the satellite speed relatively to the a/c (#2). Say this component value is F2.
Only the sum F1 + F2 is known. It results from the measured BFO after two corrections: 1) for BFO due to the satellite speed relatively to the Earth station and 2) for the "error" from the a/c component deduced from BFOs at times when the a/c position was known before take off.
For a given point on the circle, you can know F2 (because the satellite speed value and direction are perfectly known at the time H). So you can deduce F1.
But, you can't deduce a speed of the a/c from F1, only a "couple" [speed + direction of flight]. There are an infinity of couples which are in accordance with F1, thus with the BFO, but you have a minimum speed (flight direction exactly away from the satellite) and a maximum angle between the direction of flight and the line of sight of the satellite given by the maximum speed of the a/c. This error is also very small, so the incertitude about the "couple" is pretty large.
Fortunately, there are two other constraints:
- you have to reach the next circle at the right time (the time of the next ping),
- the point reached has to be in accordance with the new BFO at this time.
If you knows the first point (from primary radar position extrapolation), it is easy to find a flight path in accordance with all data, after making suppositions about the speed of the a/c (for each one, different headings...).
If you dont't know the first point, you have to compute many hypothesis (about the first point) to find credible flight pathes.