SPDE–ETAS: Fast and Accurate Bayesian Inference for the Spatio-Temporal Epidemic-Type Aftershock Sequence (ETAS) Model

We propose a stochastic partial differential equation (SPDE) formulation of the Epidemic-Type Aftershock Sequence (ETAS) model for efficient Bayesian inference of spatially varying background seismicity. While recent Bayesian ETAS formulations already model the background rate using Gaussian Process priors, their application to large earthquake catalogs is limited by the associated dense covariance structure. Using synthetic earthquake catalogs, we demonstrate that the proposed SPDE–ETAS model accurately recovers both background and triggering parameters, achieving estimation performance comparable to previous Gaussian Process–based Bayesian ETAS models and superior stability relative to kernel-based approaches. The sparse precision matrix induced by the SPDE representation leads to substantial reductions in computational cost and memory usage, enabling scalable inference without compromising accuracy. Application to the Italian earthquake catalog (1960–2025) reveals spatially coherent background seismicity patterns aligned with major tectonic features, and provides robust and well-constrained Bayesian estimates of ETAS triggering parameters. These results establish the SPDE–ETAS framework as a computationally efficient and flexible alternative for Bayesian earthquake modeling, particularly suited for large and high-resolution seismic catalogs.