SPDE-ETAS: Fast and accurate Bayesian inference for the spatio-temporal epidemic type aftershock sequence (ETAS) model

Abstract

The Epidemic-Type Aftershock Sequence model is a well-established point process framework for characterizing earthquake occurrences. Traditionally, ETAS parameters are estimated using maximum likelihood methods, where the background rate is smoothed by a Gaussian kernel density. More recently, a Bayesian formulation has been proposed, in which the background rate is modeled as a Gaussian Process prior, enabling a flexible semi-parametric estimation. However, this approach can become computationally demanding for large datasets due to the dense covariance structure of the Gaussian Process.An alternative representation of the Gaussian Process is the Gaussian Markov Random Field, which offers a sparse precision matrix formulation that significantly reduces computational complexity. Building on this connection, we propose the Stochastic Partial Differential Equation ETAS model, where the Gaussian Process covariance matrix is replaced by a sparse precision matrix within a Log-Gaussian Cox Process framework. The precision matrix is constructed using the finite element method on a triangular mesh, and the log-prior of the Gaussian Markov Random Field is approximated via the Laplace approximation to accelerate computation.The model is first validated on synthetic datasets to assess its accuracy and computational efficiency compared to the kernel-based and Gaussian Process epidemic-Type Aftershock Sequence approaches. It is then applied to the Italian earthquake catalog (1960–2025) to estimate the stationary background seismicity with its uncertainties, relevant for seismic hazard assessment.