Nonlinear hypocentre determination

Petr Bulant & Ludek Klimes


We consider the robust nonlinear approach to hypocentre determination proposed by Tarantola and Valette, consisting in direct evaluation of the nonnormalized 3-D marginal a posteriori density function which describes the relative probability of the seismic hypocentre. The nonnormalized 3-D marginal a posteriori density function is discretized at the gridpoints of a sufficiently dense 3-D spatial grid of points. This approach takes into account the inaccuracy of the velocity model and the corresponding influence on the hypocentre determination, estimates the uncertainty of the hypocentre position, and allows for testing the model covariance function describing the uncertainty of the velocity model. The model covariance function is projected onto the uncertainty of the hypocentral position through the geometrical covariances of theoretical travel times calculated in the velocity model.

For the sake of simplicity and rapid numerical implementation, we consider just the diagonal elements of the geometrical travel-time covariance matrix in this paper. We discuss the distortion of the nonnormalized 3-D marginal a posteriori density function caused by this simplification, and present a numerical example.


Hypocentre determination, velocity model, accuracy of the velocity model, model covariance function, geometrical travel-time covariance matrix, marginal a posteriori density function of hypocentral coordinates, arrival-time residuals, arrival-time misfit.

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Seismic Waves in Complex 3-D Structures, 25 (2015), 17-36 (ISSN 2336-3827, online at