Directional Statistics, Bayesian Methods of Earthquake Focal Mechanism Estimation, and Their Application to New Zealand Seismicity Data

A focal mechanism is a geometrical representation of fault slip during an earthquake. Reliable earthquake focal mechanism solutions are used to assess the tectonic characteristics of a region, and are required as inputs to the problem of estimating tectonic stress. We develop a new probabilistic (Bayesian) method for estimating the distribution of focal mechanism parameters based on seismic wave polarity data. Our approach has the advantage of enabling us to incorporate observational errors, particularly those arising from imperfectly known earthquake locations, allowing exploration of the entire parameter space, and leads to natural point estimates of focal mechanism parameters. We investigate the use of generalised Matrix Fisher distributions for parameterising focal mechanism uncertainties by minimising the Kullback-Leibler divergence. 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 for the most part are 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, enabling the posterior PDF to be succinctly summarised by reporting the estimated parameters of the fitted distribution.