Gutenberg-Richter-type earthquake size distributions: maximum likelihood estimation, unbiased estimation, and Bayesian forecasting

Sander Osinga , Dirk Kraaijpoel, Frans Aben, Maarten Maarten Pluymaekers

Characterizing earthquake size distributions using the Gutenberg-Richter (GR) law is ubiquitous in seismology. According to the GR law, earthquake magnitudes follow an exponential distribution, with a rate parameter commonly represented by the b-value. For many applications, including seismic hazard and risk assessment, estimating the b-value is therefore a common procedure. However, the parameter estimation is usually only a means to an end, as ultimately, the earthquake size distribution itself is the quantity of interest. We show that the typical approach of estimating the b-value parameter and plugging the estimate into the GR law is suboptimal and yields a biased size distribution. As an alternative, we derive an unbiased estimator for the distribution itself that also has a lower variance than the plug-in distribution. Both estimation approaches may be characterized as point estimations. From a forecasting perspective, however, and for hazard and risk assessment in particular, using point estimates is inadequate, as that does not properly account for model uncertainty. Ideally, one would take into account all plausible size distributions to the extent that they are consistent with the observations. Therefore, in a forecasting context we advise to use Bayesian inference to obtain the posterior predictive size distribution. We show that if the prior belief for the b-value is expressed as a Gamma distribution, a conjugate Gamma posterior distribution and closed-form expression for the posterior predictive earthquake size distribution are available, negating the need for numerically evaluating likelihood expressions and integrating over a posterior.