Bayesian Predictive Distribution for the Magnitude of the Largest Aftershock

Abstract:

Aftershock sequences, which follow large earthquakes, last hundreds of days and are characterized by well defined frequency-magnitude and spatio-temporal distributions. The largest aftershocks in a sequence constitute significant hazard and can inflict additional damage to infrastructure. Therefore, the estimation of the magnitude of possible largest aftershocks in a sequence is of high importance. In this work, we propose a statistical model based on Bayesian analysis and extreme value statistics to describe the distribution of magnitudes of the largest aftershocks in a sequence. We derive an analytical expression for a Bayesian predictive distribution function for the magnitude of the largest expected aftershock and compute the corresponding confidence intervals. We assume that the occurrence of aftershocks can be modeled, to a good approximation, by a non-homogeneous Poisson process with a temporal event rate given by the modified Omori law. We also assume that the frequency-magnitude statistics of aftershocks can be approximated by Gutenberg-Richter scaling. We apply our analysis to 19 prominent aftershock sequences, which occurred in the last 30 years, in order to compute the Bayesian predictive distributions and the corresponding confidence intervals. In the analysis, we use the information of the early aftershocks in the sequences (in the first 1, 10, and 30 days after the main shock) to estimate retrospectively the confidence intervals for the magnitude of the expected largest aftershocks. We demonstrate by analysing 19 past sequences that in many cases we are able to constrain the magnitudes of the largest aftershocks. For example, this includes the analysis of the Darfield (Christchurch) aftershock sequence. The proposed analysis can be used for the earthquake hazard assessment and forecasting associated with the occurrence of large aftershocks. The improvement in instrumental data associated with early aftershocks can greatly enhance the analysis and facilitate better forecasting and hazard mitigation. Aftershocks occur in other relaxation phenomena, for example, in fracture experiments on porous materials and acoustic emissions, in solar flares, after stock market crashes to mention a few, therefore, the results of this study can also be applicable to those problems.