Time-Based Multilateration

In 2D space solution seems deceptively simple, because you have only 1 parameter to optimize, that is T0, i.e. time of radiation release. When this is known, one can compute distances, buffer points accordingly and accurately locate 'point' where all 3 buffers intersect. This is rather simple function for GIS in projected coordinate system:

Note that stations are labelled by time stamps in seconds and I used velocity equal 343 m/sec. Length of edges expressed in metres.

The tricky part is defining search interval for T0. I set upper interval very close to time minimum (6) and defined lower as 6 - (1036/343), i.e. used longest edge between stations expressed in seconds. These are results of test run within this range:

Values in column DISTANCE computed as minimum total of distances to two nearest neighbors for each intersection point, i.e. value to be minimized. GIS function also returns number of intersection pairs - third column. As one can see function has potentially more than 1 minimum, which makes it hard for optimization. However we can safely dismiss cases, where only one pair of intersections found, because it means that only 2 out of 3 buffers intersect each other. It makes function uni-modal. So I used colored values for search limits and golden section search solved task in 10 seconds. Minimum with centimetre positional accuracy was found at T0 = 5.2620 seconds.