b-a Probabilist : a Bayesian Estimation of the Gutenberg-Richter Parameters (b,a) for non-Truncated Catalogs with Temporal Detectability Changes

Seismic hazard assessment relies on the statistical analysis of seismic catalogs. A significant challenge arises from the strong differences in completeness and quality between historical and instrumental catalogs. Combining these catalogs thus necessitates statistical methods capable of integrating and addressing the biases introduced by these differences.

Two key parameters to estimate in statistical seismology are the b-value and the seismic rate, which define the Gutenberg Richter distribution. Usually, their estimation requires to truncate the catalog above an arbitrarily defined completeness magnitude. Here instead, we introduce a detection function that models the varying detectability of earthquakes over time. The detection function is characterized by two additional parameters, which we jointly estimate with the b-value and the seismic rate through a Bayesian approach. This method enables the use of all observed events to produce a full probabilistic solution, delivering posterior distributions for each parameter while also quantifying uncertainties and correlations between them. The estimation process is implemented using a Markov chain Monte Carlo (McMC) method to efficiently explore the parameter space.

We show an application to a seismic catalog in metropolitan France. We assume that the b-value and the seismic rate remain constant over time, and we invert for two distinct detection functions—one for the historical catalog and another for the instrumental catalog. Our results demonstrate that the b-value and the seismic rate are better estimated when accounting for these variations in detectability and catalog completeness. Finally, by producing full probabilistic solutions, this method provides b-value and seismic rate estimates with their uncertainties, thus providing valuable information for PSHA calculations.