Long-Term b Value and Seismic Rate from Earthquake Catalogs with Time-Varying Completeness and Uncertain Magnitudes

Abstract Seismic hazard assessment relies on the statistical analysis of seismic catalogs. In this context, a robust estimation of the long-term b value and seismic rate is required. This assumes that the Magnitude-Frequency Distribution parameters are considered constant over the entire period of the catalog, providing a single set of values for both historical and instrumental records. However a major challenge arises from the differences in size, quality, and completeness between these records. In addition, magnitude uncertainties are often neglected, leading to biased parameter estimates. We propose an approach to jointly estimate the long-term b value and seismic rate while explicitly accounting for magnitude uncertainties and temporal variations of completeness through a detection function. The inversion is performed using a Bayesian framework providing a full probabilistic solution, from which posterior distributions, parameter uncertainties and trade-offs can be derived. After illustrating the benefits of the method on synthetic catalogs, we apply it to the seismic catalog of the Alps. We jointly invert for one b value and one seismic rate while allowing separate detection functions for the historical and instrumental periods, each with its own level of uncertainty. Our results demonstrate that accounting for both magnitude uncertainties and time-varying detectability leads to improved estimates of seismic parameters. The resulting probabilistic framework provides reliable inputs for probabilistic seismic hazard assessment.