Bayesian modelling of the temporal evolution of seismicity using the ETAS.inlabru package

The Epidemic Type Aftershock Sequence (ETAS) model is widely used to model seismic sequences and underpins Operational Earthquake Forecasting (OEF). However, it remains challenging to assess the reliability of inverted ETAS parameters for a range of reasons. For example, the most common algorithms just return point estimates with little quantification of uncertainty. At the same time, Bayesian Markov Chain Monte Carlo implementations remain slow to run, do not scale well and few have been extended to include spatial structure. This makes it difficult to explore the effects of stochastic uncertainty. Here we present a new approach to ETAS modelling using an alternative Bayesian method, the Integrated Nested Laplace Approximation (INLA). We have implemented this model in a new R-Package called ETAS.inlabru, which builds on the R packages R-INLA and inlabru. Our work extends these packages, which provided tools for modelling log-Gaussian Cox processes, to include the self-exciting Hawkes process of which ETAS is a special case. Whilst we present the temporal component here, the model scales to a spatio-temporal model and may include a variety of spatial covariates. This is a fast method which returns joint posteriors on the ETAS background and triggering parameters. Using a series of synthetic case studies, we explore the robustness of ETAS inversions using this method of inversion. We also included runnable notebooks to reproduce the figures in this paper as part of the package's GitHub repository. We demonstrate that reliable estimates of the model parameters require that the catalogue data contains periods of relative quiescence as well as triggered sequences. We explore the robustness of the method under stochastic uncertainty in the training data and show that the method is robust to a wide range of starting conditions. We show how to precondition the model with historic earthquakes beyond the model domain. Finally, we show that rate dependent incompleteness of earthquake catalogues after large earthquakes has a significant and detrimental effect on the ETAS posteriors. We believe that the speed of the inlabru inversion, which include a rigorous estimation of uncertainty, will support the robust use of ETAS for seismicity modelling and OEF.