A Bayesian approach to determining and parametrizing earthquake focal mechanisms

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 generalized Matrix Fisher distributions for parametrizing 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) allows the resulting posterior PDF to be well approximated by generalized 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 generalized 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.