CMT data inversion using a Bayesian information criterion to estimate seismogenic stress fields

We developed an inversion method to estimate the stress fields related to earthquake generation (seismogenic stress fields) from the centroid moment tensors (CMT) of seismic events by using Akaike's Bayesian information criterion (ABIC). On the idea that the occurrence of an earthquake releases some part of the seismogenic stress field around its hypocentre, we define the CMT of a seismic event by a weighted volume integral of the true but unknown seismogenic stress field. Representing each component of the seismogenic stress field by the superposition of a finite number of 3-D basis functions (tri-cubic B-splines), we obtain a set of linear observation equations to be solved for the expansion coefficients (model parameters). We introduce prior constraint on the roughness of the seismogenic stress field and combine it with observed data to construct a Bayesian model with hierarchic, highly flexible structure controlled by hyper-parameters. The optimum values of the hyper-parameters are objectively determined form observed data by using ABIC. Given the optimum values of the hyper-parameters, we can obtain the best estimates of model parameters by using a maximum likelihood algorithm. We tested the validity of the inversion method through numerical experiments on two synthetic CMT data sets, assuming the distribution of fault orientations to be aligned with the maximum shear stress plane in one case and to be random in the other case. Then we applied the inversion method to actual CMT data in northeast Japan, and obtained the pattern of the seismogenic stress field consistent with geophysical and geological observations.