Using Seismic Waves detected at 4 Seismometers to determine the Epicenter

[Ira_anabelle]

Homework Statement

Earthquake:
You know the geographical position of four different seismometers, the speed of the s-wave and p-wave, the level of the focus under the earths surface and the times of arrival of both of the waves at each station. Determine a way to find the exact location of the epicenter.

Note that the primary (p)-wave arrives before the secondary(s)-wave.

Relevant Equations

N/A

My first attempt was to work with the the difference in arrival times, but that didnt account for the focus to be under the epicenter. So I tried again in combination with the angle between the stations but have not arrived at a clear solution.

 

[berkeman]

Welcome to PF.

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[haruspex]

There does seem to be a surfeit of information. Knowing the difference in p/s arrival times and their speeds at three stations should be enough to locate the focus. The fourth station and given depth seem unnecessary. Hard to know whether you are expected only to use whatever sufficient set of data you choose, or you are supposed to use some likelihood maximisation.

But I don't understand how you can have computed the epicentre except by locating the focus. Please post the details of your attempt.

Mind, I am assuming that the stations are not too far apart, or you will have to worry about the core.

 

[Ira_anabelle]

To be honest I didnt find my notes with the angle calculation of the center of the earth. Just these ones. I tried to get a formula for the epicentral distance first, to combine three of them an then find the Epicenter. I don't think it‘s right since I am not using the sphere/geographical coordinates.
There is btw the information given that the speeds are set to be constant and the distance are small. I forgot to write it into the question.

 

[haruspex]

tried to get a formula for the epicentral distance first

I see no way to do that. The waves come from the focus, not the epicentre. You can calculate the distance from the focus to each station. That gives you a pyramid (inverted) on a quadrilateral base.