Problem 113 Ground Zero
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To problem 113 author:
The wave velocity need to be known for me to solve for ground zero and find time zero. But the problem says: "We assume that the "shake" of the ground travels (or propagates) with the constant (but unknown) speed through the soil." Speed is unknown.
To get p0=(x0,y0),t0, I will need to know v.
p-p0 = v*(t-t0)2
(dx^2 + dy^2 = (v * dt)^2)
Can you give clues how ay to get v? (v can be computed if t0 is known, but the problem is also asking for t0)
I'm not the author of the problem, but would like to point out that if the velocity was known, the problem would be elementary - and, also, you'd only need at most three sensors to triangulate the location. You are given more than three. The whole challenge of this task is that the signal velocity is unknown.
You say you can compute v if t0 is known.
Perhaps you can pretend that t0 is known...?
- It is seismic, so, to solve the problem trilateration is the most apt method and not triangulation.
- Either method, we need to find the distances from the source to the stations.
- To get those distances we need the know the velocity computed as: d[i]= v * (t[i] - t0)
- The system of equations then become: (x[1] - x0)^2 + (y[1] - y0)^2 = d[1]^2 = (v(t[1]-t0)^2) (x[2] - x0)^2 + (y[2] - y0)^2 = d[2]^2 = (v(t[2]-t0)^2) (x[3] - x0)^2 + (y[3] - y0)^2 = d[3]^2 = (v(t[3]-t0)^2) (x[4] - x0)^2 + (y[4] - y0)^2 = d[4]^2 = (v(t[4]-t0)^2)
- And by using Non-Linear Least Squares, I expect solve x0,y0,t0.
- But I need to find the velocity.
What can you do if you do not know the answer? You guess. But your guess is probably wrong. So you guess again. And again. And again. And maybe you can use lessons learnt from previous wrong guesses to get less wrong over time.
And one more thing: "It is seismic, so, to solve the problem trilateration is the most apt method and not triangulation." While that would apply to real-life seismic events, this statement is wrong for this problem. There is no notion of signal strength, and its degradation, here.
OK. The task then is to guess the distances and elapsed time wherein velocity is constant.