7.1 Estimation of the Epicentral Coordinates of a Seismic Event

A seismic source was activated at time T=0 in an unknown location at the surface of Earth. The seismic waves produced by the explosion have been recorded at a network of six seismic stations whose coordinates in a rectangular system are

The observed arrival times of the seismic waves at these stations are

where ?=0.10 s, the symbol ? being a short notation indicating that experimental uncertainties are independent and can be modeled using a Gaussian probability density with a standard deviation equal to ?.

Estimate the epicentral coordinates (X, Y) of the explosion, assuming a velocity of ?=5 km/s for the seismic waves. Use the approximation of a flat Earth surface, and consider that the coordinates in equation (7.1) are Cartesian.

Discuss the generalization of the problem to the case where the time of the explosion, the locations of the seismic observatories, or the velocity of the seismic waves are not perfectly known, and to the case of a realistic Earth.

Solution:

The model parameters are the coordinates of the epicenter of the explosion,

and the data parameters are the arrival times at the seismic network,

while the coordinates of the seismic stations and the velocity of the seismic waves are assumed perfectly known (i.e., known with uncertainties that are negligible with respect to the uncertainties in the observed arrival times).

For a given (X, Y), the arrival times...