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?
