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I've recently asked a similar question aiming at the same problem on Math.SE, for which I've set a bounty.

Problem.

I have a radio calibration transmitter with known fixed location $\langle t_x, t_y, t_z\rangle$ embedded in a medium whose index of refraction varies as a known function of depth. That function is of the form:

$$n(\text{depth}) = a-be^c$$

...where $a,b$ and $c$ are known constants

The transmitter emits $N$ identical signals at unknown emission times. These signals are received at $n > 4$ (in my case, typically $8$) TDoA nodes, whose positions may be approximately known. The spacing between the receivers is small relative to the distance between a given receiver and the transmitter.

Using only the times of arrival at each station (or, really, the differences in arrival times), I wish to determine the "true" locations of the $n$ receivers.

What I have so far.

Notice that standard TDoA localization techniques cannot be applied, as these are designed to locate a single source given arrival times at several (in practice, $n >> 4$) receivers with well known locations.

The TDoA self-calibration problem is less studied. I've combed a wealth of papers, but have found none that are applicable to my particular circumstances. Thrun, for example, describes a solution in the case of multiple transmitters, Kuang describes a solution for recovering both the transmitter and receiver locations for the minimal case of $6$ receivers and $8$ transmitters, and various other authors offer solutions for other specific circumstances. Moreover, none cover the case of anisotropic media.

Since the true emission time is unknown, all I have to work with are the time differences. What I'm doing now is to assume that the uncertainty in the receiver positions is small. I use the approximately known locations as initial conditions for a minimizer (Minuit2), which minimizes the sum of the squared difference between the measured $\Delta t$'s (where $\Delta t_{ij}$ is the difference in arrival time between receiver $i$ and receiver $j$) and the "predicted" $\Delta t$ (determined by inputting the known transmitter and iterated receiver locations into a "ray tracer", which uses Fermat's principle to compute the light propagation time) for all combinations of two antennae (in the combinatoric sense). That is, I minimize the objective:

$$\chi^2 = \sum_{all\; antennae\; pairs} \bigg[\Delta t_{ij} - (\text{ray tracing prediction for } \Delta t_{ij})\bigg]^2$$

While this method does improve slightly upon the approximately known receiver locations in testing, it does not come close as needed, and yields larger errors than desired. Moreover, in many cases the minimizer "shoots off", giving unreasonable and large estimates for the receiver locations (I've tried setting software limits just larger than the assumed uncertainty in receiver coordinates, however the solution often simply ends up at a limit).

Question: How else can this problem be approached, or how may this method be improved?


Some Ideas.

Since the anisotropic medium in question has a refractive index which varies as an exponential function of depth (by which I mean a function of the form $a-be^c$, as previously mentioned), light rays are "reversible". That is, a ray sent from the transmitter to a receiver follows the same path as a ray sent from that receiver to the transmitter.

Although standard TDoA algorithms cannot be applied directly to the problem as only a single transmitter is available, and is attempting to recover the location of multiple receivers in $\mathrm{3D}$, perhaps the problem could be "reversed" somehow, treating the receivers as transmitters, and the transmitter as a receiver. We then have $n >> 4$ "transmitters" trying to locate a single "receiver"... a problem which is relatively well understood.

If the relative receiver positions were known (which they aren't), for instance, then it would perhaps be as simple as fixing the receiver positions, reconstructing the transmitter, and performing a coordinate transformation.

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