Bayesian methods of earthquake focal mechanism estimation and their application to New Zealand seismicity data
Authors: David Walsh, Richard Arnold, John Townend
Paper number: 3739 (EQC U536)
Abstract
We investigate a new probabilistic method of estimating earthquake focal mechanisms — which describe how a fault is aligned and the direction it slips during an earthquake — taking into account observational uncertainties. Robust methods of estimating focal mechanisms are required for assessing the tectonic characteristics of a region and as inputs to the problem of estimating tectonic stress. We make use of Bayes’ rule, a probabilistic theorem that relates data to hypotheses, to formulate a posterior probability distribution of the focal mechanism parameters, which we can use to explore the probability of any focal mechanism given the observed data. We then attempt to summarise succinctly this probability distribution by the use of certain known probability distributions for directional data. The advantages of our approach are that it (1) models the data generation process and incorporates observational errors, particularly those arising from imperfectly known earthquake locations; (2) allows exploration of all focal mechanism possibilities; (3) leads to natural estimates of focal mechanism parameters; (4) allows the inclusion of any prior information about the focal mechanism parameters; and (5) that the resulting posterior PDF can be well approximated by generalised statistical distributions. We demonstrate our methods using earthquake data from New Zealand. We first consider the case in which the seismic velocity of the region of interest (described by a velocity model) is presumed to be precisely known, with application to seismic data from the Raukumara Peninsula, New Zealand. We then consider the case in which the velocity model is imperfectly known, with application to data from the Kawerau region, New Zealand. We find that our estimated focal mechanism solutions are for the most part consistent with all available polarity data, and correspond closely to solutions obtained using established methods. Additionally, the distribution of focal mechanism parameters can be accurately and succinctly summarised by the parameters of the probability distributions we have examined.
Technical Abstract
We develop a new probabilistic (Bayesian) method for estimating the distribution of focal mechanism parameters based on seismic wave polarity data. We investigate the use of generalised Matrix Fisher distributions for parameterising focal mechanism uncertainties. The advantages of our approach are that it (1) models the data generation process and incorporates observational errors, particularly those arising from imperfectly known earthquake locations; (2) allows exploration of the entire parameter space; (3) leads to natural point estimates of focal mechanism parameters; (4) allows the inclusion of a priori information about the focal mechanism parameters; and (5) that the resulting posterior PDF can be well approximated by generalised Matrix Fisher distributions. We present here the results of our method in two situations. We first consider the case in which the seismic velocity of the region of interest (described by a velocity model) is presumed to be precisely known, with application to seismic data from the Raukumara Peninsula, New Zealand. We then consider the case in which the velocity model is imperfectly known, with application to data from the Kawerau region, New Zealand. We find that our estimated focal mechanism solutions are for the most part consistent with all available polarity data, and correspond closely to solutions obtained using established methods. Further, the generalised Matrix Fisher distributions we examine provide a good fit to our Bayesian posterior PDF of the focal mechanism parameters. Finally, we demonstrate how informative prior distributions on focal mechanism parameters can be incorporated into our model.
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