# Solving the inverse GPS/Multilateration problem

Problem.

I have a set of $$N \geq 16$$ light receivers with unknown locations $$\langle x_n, y_n, z_n\rangle$$, and a light emitter with known location $$\langle x_e, y_e, z_e\rangle$$. The light emitter emits a signal at an unknown time, and each receiver records the time of signal arrival at that location, $$t_n$$. The light propagates in a medium whose index of refraction is a function of depth, of the form $$A + Be^{Cx}$$ (where $$A,B$$ and $$C$$ are some constants), hence the light signal takes a curved path through the medium (as per Fermat's principle).

Using only the times of signal arrival (or, really, the differences in signal arrival times between stations, as I do not know the emission time), the signal velocity, and the coordinates of the transmitter, I wish to reconstruct the coordinates of all $$N$$ receivers.

What I've tried.

I've tried a simple least squares minimization algorithm, wherein I minimize the squared difference between the measured and predicted arrival time differences (the latter computed according to Fermat's principle, so as to account for the depth-dependent refractive index). This (perhaps unsurprisingly) failed on both simulated and real-life data (the former produced by simply picking some points in space at random, computing the arrival time differences according to Fermat's principle, feeding these times to the algorithm and trying to reconstruct the randomly chosen points). I've also tried a simple Tikhonov regularization approach (simply adding a small, constant regularization parameter).

I've spent some time combing the literature, and haven't found anything that quite matches my specific circumstances.

How might this be approached?

• No, you can't. Even in the most simple case where light propagation is isotropic, you only get a one-parameter family of spheres. It would be more interesting if all the stations could be active, so that for instance you could bounce a signal, after some station-specific delay. // You already solved the reversed problem, the 16 stations are known and the position of station $e$ is to be determined? Commented Jun 19, 2023 at 5:34
• @LutzLehmann Correct, the position of the 16 stations are known (by nature of these positions being precisely measured when the receivers were installed). I don't quite follow your comment on "bouncing a signal". Could you elaborate? Commented Jun 20, 2023 at 19:40
• I don't understand: "the position of the 16 stations are known". So the positions of the 16 stations / receivers are known and you wish to find the positions of other $N - 16$ stations / receivers? Please correct the problem description in this case. Commented Jun 23, 2023 at 12:53