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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?

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    $\begingroup$ 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? $\endgroup$ Commented Jun 19, 2023 at 5:34
  • $\begingroup$ @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? $\endgroup$ Commented Jun 20, 2023 at 19:40
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    $\begingroup$ 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. $\endgroup$ Commented Jun 23, 2023 at 12:53

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As far as I am able to understand the physical background from the description of the problem, you don't know the emission time, so you can't calculate the distances between the emitter and the receivers. You can basically only calculate the lower bounds for pairwise distances between the receivers?

In this case, this is obviously not sufficient to determine the location of the receivers. Even using brute force algorithm or any kind of approximation method you can't come anywhere close to the solution by using only lower bounds for pairwise distances between receivers.

The problem is more interesting, but still very hard (because dimension is more than one), if you can somehow calculate all pairwise distances between the receivers.

And the problem becomes easy, if you can somehow calculate the distances between the emitter and the receivers. And also, if you can move the emitter or have multiple emitters in different locations. Then, using trilateration, or better a 4-lateration (4-ball intersection), you can simply determine the location of every receiver separately.

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