# Finding coordinates of ground-zero with seismic sensors

At the unknown t0 time an explosion occurred at an unknown point X,Y on the 2D plane.

We have several seismic sensors at points X1,Y1, X2,Y2, ..., Xn,Yn - they register explosion at moments t1, t2, ... , tn.

Suppose that seismic wave propagates with unknown but constant speed V.

How can we then find out X,Y of the explosion?

I faced such problem about 6 years ago and solved it somehow (I believe it was iterative solution). Recently I added it to my programming-puzzles site (http://www.codeabbey.com/index/task_view/ground-zero) but later I was asked by several users how this problem could be solved precisely (i.e. analytically) - and I still do not know this.

My idea of solution was like the following:

We have 4 variables and 4 or more binding equations for them. Regretfully, the system is not linear.

Nevertheless, if we fix V to some constant, we have only 3 variables left and can solve any three of equations for them.

We can choose these three equations in several ways (e.g. we can make 4 triplets out of 4 equations) - so we can find solution for each triplet and then calculate some aggregated "difference" of these solutions (like area of minimum polygon to enclose all solutions for X,Y).

So we now may search through values of V with the goal to minimize this "difference" of solutions.

It works (as other iterative approaches also do). But this give not a clue for precise geometric / analytical solution.