Bayesian Estimation Of Hypocenter With Origin Time Eliminated

Abstract

In this study a new approach to hypocenter location is formulated from a Bayesian point of view. Observational data for hypocenter location are the arrival times of phases at seismic stations. Unknown parameters are the spatial coordinates and origin time of a quake focus. In general, we have some prior information about the spatial coordinates, but not about the origin time. Then, according to Bayes’ theorem, the probability density function (pdf) of hypocenter parameters posterior to observed data is proportional to a product of the likelihood and the prior pdf of the spatial coordinates. Integrating the posterior pdf over the whole range of origin time, we can eliminate the origin time from the location problem, and obtain the marginal pdf of only the spatial coordinates. The best estimate of a hypocenter is given by a set of the spatial coordinates which maximizes the marginal pdf. Supposing Gaussian errors in both observational and prior data, we obtain a simple and robust algorithm to invert arrival time data for a hypocenter. Estimation errors of parameters are evaluated by an asymptotic covariance matrix, which gives a good approximation to exact covariance when the estimate is linearly close to a true hypocenter. The advantages of this approach are as follows: i) the use of prior data resolves unknown parameters completely within finite errors, and ii) the elimination of origin time suppresses the instability of estimates. © 1984, The Seismological Society of Japan, The Volcanological Society of Japan, The Geodetic Society of Japan. All rights reserved.