Sorry if this is not the proper place to ask the question, couldn't find any other relevant forum (perhaps dsp exchange).
I am looking into finding a most likely position of a source given recordings from 4 microphones with unknown locations. I have the speech data recorded on the microphones. Cross-correlation gives me an estimate of the delay between the recordings and knowing the sound velocity in air, I can find the relative distances between the microphones. So, I get something like 5m between microphones 1 and 2, 3.5m between 2 and 3 and so on (6 distance estimates in all). That means f.ex. microphone 2 could be anywhere on a circle with a radius of 5m and centered around 1st microphone and so with the other pair combinations. Is there a way (optimization method) to give an estimate of the most probable location of the source relative to the microphone positions ? It looks a very under-determined system, but I was wondering whether one can arrive at a likelihood estimate ?
Appreciate your help and sorry again if I am posting in the wrong forum.
Edit:
Thank to @John Omielan for pointing out the error of 'under-determined vs. over-determined'. Maybe I should add that the distance measurements are prone to small errors and not absolute.