Hierarchical Bayesian calibration of regional moment magnitude predictive models

Onsite Earthquake Early Warning (EEW) applications require robust predictive models that can relate the early part of the recorded seismic waves to seismicity and/or ground shaking characteristics, the latter representing the variables used to make decisions about raising or not an alarm. The use of limited information to establish these models within a real-time estimation setting, typically corresponding to the first few seconds after the identification of the seismic wave arrival, introduces significant uncertainty (variability) in the predictions. To increase the number of historical observations available for the model calibration, data from multiple seismic stations within the same geographic region are utilized for the latter task. Unfortunately, this introduces addi-tional sources of variability in the predictive model development, originating from local geographical effects and site conditions that affect differently the underlying physics at each station. This paper introduces a hierarchical Bayesian model updating framework to address this challenge. To better focus the research effort, the calibration of earthquake magnitude nonlinear regression models based on the seismic maximum predominant period is investigated, though underlying principles can be extended to other type of decision variables and intensity measures used in EEW applications. Within the hierarchical updating framework, the predictive models for each station are separately calibrated (first level), while sharing information across the stations by selecting common calibration hyper-parameters (second level). This facilitates a reduction of the variability of the estimates originating from regional wave propagation effects while simultaneously providing sufficient amount of data for calibrating the predictive models even for stations with scarce available records. To support computational ef-ficiency in the hierarchical calibration, conjugate priors are utilized, while for accommodating nonlinear regression relationships samples from the posterior distribution, representing the model calibration, are obtained using Metropolis-within-Gibbs sampling. As illustrative case study, the application to the Sichuan region of Southwestern China is examined, with extensive comparisons performed for the accuracy and robustness of the resultant probabilistic predictive models.