Hey there.. got to this topic from electronics-lab.com, via Fileark sound localization sensor.
I am doing very similar product like you - sound localization based on impact of metal BB pellet on metal target (square).
Long story short, so I am not boring - most probably, math won't be your biggest problem, in fact, it is/was the simplest problem of my setup.
So you have four sensors like in your image
From your input, you get 4 values. In example on picture, let's say that they were:
1: 1200 (A..1)
2: 2000 (B..2)
3: 2500 (C..3)
(the unit is not important for now.. in this case, it could be CPU cycles or whatever)
my default, most simple algorithm did this:
Imagine four circles, with their center in microphones. Lets call those circles M0, M1, M2, M3.
You know delta times.. so let's do this:
M0 radius = 0000
M1 radius = 1200
M2 radius = 2000
M3 radius = 2500
Now you have four imaginary circles (one is in fact point - zero radius) and they are touching big circle on your picture.
What you have to do is start adding number one to each of those circle radii.
M0 = 1, M1 = 1201, M2 = 2001, M3 = 2501..
Four imaginary circles are now a bit bigger and closer to impact zone.
After each addition, you have to calculate intersections of neighbour circles. In ideal case, they will all end up in the same point - point of impact. In real world, there will be some error.. then you have to calculate when where all intersection closest and pick a point between them.
*EDIT*: i forgot that you need to know distance between neighbour microphones in our imaginary unit (miliseconds, CPU cycles, whatever) to calculate the intersection - simple triangle.
Also - this is a solution for 2D, not for 3D. If you want to minimize error, mic's should be as close to edges of measured plane as possible. If they are not, then you have to use 3D coordinates for microphones, edges of target and do lots of triangle math in 3D.
I hope it was understandable - English is not my native language
And in real world, there will be many problems, as I have said, math is nothing. Problem is with intensity of signals at microphones - it goes down with distance squared, also with precise measuring the difference between signal arrival at microphones. Don't even think about using piezoelectric disks clamped onto surface of the target. Speed of sound in steel is nearly 6000 m/s, there will be various reflections and therefore it is virtually impossible to calculate with cross-corelation, because the signal from each sensor is different. A lot! It is possible to try triggering with first sine wave coming from sensor, but it is about milivolt levels, very picky on environment and if you are shooting, it will register shooting sound before impact sound - not good.