A Bayesian Framework for Earthquake/Aftershock Forecasting and Testing

Robert Shcherbakov

The occurrence of a significant mainshock presents an opportunity to test different existing or novel statistical approaches to model the evolution of the corresponding sequences of earthquakes that precede and follow the mainshock. Among several statistical measures, the computation of the probability to have the magnitude of the largest expected aftershock to be above a certain value during a future time interval is of critical importance. In this respect, the 2019 Ridgecrest, California, earthquake sequence represents the latest highly productive and non-standard sequence to be analyzed in detail. In this work, the parametric modelling of the frequency-magnitude statistics and the earthquake occurrence rate are carried out. In addition, the problem of constraining the magnitude of the largest expected aftershocks to occur during the evolution of the sequence is addressed using two approaches. The first one is based on the extreme value theory, whereas the second one employs the Bayesian predictive framework. The latter approach has allowed to incorporate the complex earthquake clustering through the Epidemic Type Aftershock Sequence (ETAS) process and the uncertainties associated with the model parameters into the computation of the corresponding probabilities. To compute the Bayesian predictive distribution, the Markov Chain Monte Carlo method is used to sample the posterior distribution of the model parameters (Shcherbakov et al., 2018, 2019). The results indicate that the inclusion of the foreshock sequence into the analysis produces higher probabilities for the occurrence of the largest expected aftershocks after the M7.1 mainshock compared to the approach based on the extreme value distribution combined with the Omori-Utsu law for the earthquake rate. A critical aspect of any earthquake forecasting methods is their retrospective testing and validation. Several statistical methods are used to test the consistency of the employed forecasting schemes to reproduce the observed number of earthquakes and their magnitude distributions during the forecasting time interval. As a result, the suggested approach based on the Bayesian predictive distribution with the ETAS model outperformed more traditional approaches based on the extreme value distribution combined with the Omori based rate functions.