Problem.
I have a radio transmitter, with known location $\langle t_x, t_y, t_z \rangle$, embedded in a medium whose index of refraction varies as a known function of depth:
$$n(\text{depth}) = a-be^c$$
...where $a,b$ and $c$ are known constants
The transmitter emits identical signals periodically at unknown emission times. These signals are received at $n > 4$ receivers, whose locations may be approximately known, or entirely unknown. The spacing between these receivers is small relative to the distance between the receivers and the transmitter. Given only the time of signal arrival at the $n$ receivers, I wish to determine their "true" locations.
What I've done so far.
Standard TDoA (Time Difference of Arrival)/multilateration algorithms cannot be used, as these are designed to determine the location of a single receiver given multiple transmitters (in practice, $>>4$ transmitters when working in $3$-dimensions).
Since the true emission time is unknown, all I have to work with are the differences in arrival times between pairs of antennae. What I'm doing now is to assume that the receiver positions are approximately known. I use these as initial conditions for a minimizer, 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:
$$\chi^2 = \sum_{all\;antennae\;pairs} \bigg[\Delta t_{ij} - (\text{ray tracing prediction for } \Delta t_{ij})\bigg]^2$$
While this method does improve slightly upon the approximately known receiver locations in testing, it does not come as close as needed, and yields larger errors than desired. Moreover, it is entirely uncapable of determining the receiver locations without an initial estimate.
Minimization is performed using the Minuit2 algorithm, and errors are extracted using the built-in Hessian functionality.
How else can this problem be approached, or how may this method be improved? How can one solve for the receiver positions without an initial estimate?
Some ideas.
Since the 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 determine the location of multiple receivers), perhaps the problem could be "reversed" somehow, treating the receivers as transmitters, and the transmitter as a receiver. We than have $n > 4$ "transmitters" trying to locate a single receiver... a problem which is relatively well understood. I can't seem to flesh this out, however (and it might be entirely impossible).
If the relative receiver positions were known, then it would perhaps be as simple as reconstructing the transmitter and performing a coordinate transformation.
Not an answer - just sharing experience
I faced a similar problem years ago for TDoA in deep water : that is to say that there are the effects of depth on temperature, speed of sound and salinity (known formulae). Moreover, I had many radio transmitters.
What I found is that these variations of properties in the medium were causing a lot of instability in the calculations.
AFter a while, what we decided was to use in a first step average values; solve the problem and iterate including gradually the changes of physical properties. This worked quite well and the method became a standard in the oil & gas company I was working for at that time.
What would be interesting is to see if such an approach could make your problem better conditioned from a numerical point of view.
Please, let me know since this is a very interesting problem.