Partial ping
From what I can infer from the comm protocol, the partial ping was initiated by the aircraft. I won't speculate on why. The satellite did not know to initiate the ping.
IMHO:
The partial ping can be analyzed for doppler and that might give some indication of position. However, since only one satellite is involved, this would produce a curve of possible positions, with an associated uncertainty, not a point.
Unless there was a complete handshake with the satellite (ie. aircraft sends ping to satellite, satellite replies, aircraft replies) timing data would not be generated, so one can not calculate a corresponding curve for aircraft position due to timing.
I have not analyzed the time and doppler information (don't have the proper tools in place and would take to long to generate - would rather work on my boat), but from experience on other systems, these might create somewhat orthogonal or independent curves which at their intersection would produce a position. Just speculating.