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Problem.

I have a set of $16$ light receivers with entirely unknown locations, and a pair of light transmitters with exactly known locations. One light transmitter is stationary, and located near to the receivers. The second light transmitter is far from the receivers, and has fixed $x,y$-coordinates but varying depth.

The stationary light transmitter emits a signal, and the time of signal arrival at each receiver is recorded. The moving light transmitter also emits a signal, but at various depths (for a total of $>=10$ signals), and the time of signal arrival at each receiver is again recorded. The light signals propagate in a medium whose index of refraction is a function of depth, meaning that the transit time between a point $\langle x_0,y_0,z_0 \rangle%$ and a point $\langle x_1,y_1,z_1\rangle$ is given by:

$$\frac{1}{c} \int_{z_0}^{z_1} \sqrt{1+\bigg(\frac{dr}{dz}\bigg)^2}\times n(z)\; dz$$

...where $r$ is the $xy$-distance between the two points.

The signals are distinguishable, by nature of being sufficiently separated in time. I do not know the true arrival times, as I do not know the signal emission times. I do, however, know the difference in arrival times between any two points (to high accuracy and precision). Using only the known transmitter locations, form of the refractive index, and arrival time differences, I wish to reconstruct the locations of the receivers.


What I've Tried.

I've previously solved the reverse problem in a similar context (that is, the "GPS problem", finding the unknown location of a transmitter using many receivers), however I'm quite stuck on this one. I began by trying the obvious, which is minimizing the sum of squared differences between the measured and "predicted" arrival time differences over the receiver coordinates. I found this to be quite sensitive to the initial guess in simulation, and I can't think of a good way to obtain a sufficiently "good" starting point.

I tried various objective functions (a simple sum of the absolute differences, a variation of the "Huber loss", etc.), and also tried adding various regularization terms to the $\chi^2$ error, to no avail. I'm working in Python using SciPy, and I've tried using it's basinhopping and differential evolution global minimization algorithms, as well.

A conversation with one researcher yielded the suggestion that I begin by trying to recover the receiver configuration (that is, relative positions) and/or the distances between pairs of receivers by using multidimensional scaling (MDS), and then adding that in some manner to the objective function, however I found MDS unreliable in accurately estimating the distances between receivers. Another conversation yielded the suggestion that I formulate this as a convex optimization problem, however I have little familiarity with such techniques and can't seem to formulate this in a way that would open it up to convex optimization algorithms.

Since I have several transmitters neither along a line nor in a plane, it seems to me that I should have sufficient information to reconstruct the locations. It occurs to me that if I could somehow "reverse" the problem such that it becomes the so-called "GPS problem", it would be relatively easily solvable.

If I am unable to reconstruct the actual locations, it seems to me that I should at least be able to recover their relative configuration (that is, recover their coordinates up to e.g. reflection symmetry). If this is the case, I'm open so solutions for this, as well.

How else might I approach this?

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  • $\begingroup$ The problem is similar to what the GPS system does to measure the location of the receiver. It uses at least four transmitters, but it doesn't suffer from the problem that three of those transmitters are co-linear like in your case. If we formulate this in cylindrical coordinates, then it should be obvious that the co-linear measurements can't determine the angle. The fourth measurement can determine the angle, but only up to reflection symmetry, as far as I can tell, so you can never get an unambiguous result. Maybe I am missing something. $\endgroup$ Commented May 27, 2023 at 19:56

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This is just a suggestion but maybe try doing a plane wave reconstruction first, see e.g the added figure. Plane wave reconstruction Illustration

The function $$\eta(\theta) = (\Delta t - \Delta t') = (\Delta t - n\frac{\cos(\theta)\Delta z}{c})$$ with $\Delta t$ denoting the observed time difference, $\Delta t'$ the actual time difference and n the index of refraction inbetween the detectors will be minimal at the right angle, maybe you can play around with the distance $\Delta z$ and the angle (take e.g the angle to be the direct connection between the detectors and the closest emittor as there the signal will not bend much, thus making the angle a proxy for the location).

Then when you have this initial guess for the distance between and location of the detector you can optimize this by moving them around and looking how the simulated result changes.

Or is this what you meant with the obvious minimization? If not I hope this might help in some way

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