Point process models for quasi-periodic volcanic earthquakes

Long period (LP) earthquakes are common at active volcanoes, and are ubiquitous at persistently active andesitic and dacitic subduction zone volcanoes. They provide critical information regarding the state of volcanic unrest, and their occurrence rates are key data for eruption forecasting. LPs are commonly quasi-periodic or 'anti-clustered', unlike volcano-tectonic (VT) earthquakes, so the existing Poisson point process methods used to model occurrence rates of VT earthquakes are unlikely to be optimal for LP data. We evaluate the performance of candidate formulations for LP data, based on inhomogeneous point process models with four different inter-event time distributions: exponential (IP), Gamma (IG), inverse Gaussian (IIG), and Weibull (IW). We examine how well these models explain the observed data, and the quality of retrospective forecasts of eruption time. We use a Bayesian MCMC approach to fit the models. Goodness-of-fit is assessed using Quantile-Quantile and Kolmogorov-Smirnov methods, and benchmarking against results obtained from synthetic datasets. IG and IIG models were both found to fit the data well, with the IIG model slightly outperforming the IG model. Retrospective forecasting analysis shows that the IG model performs best, with the initial preference for the IIG model controlled by catalogue incompleteness late in the sequence. The IG model fits the data significantly better than the IP model, and simulations show it produces better forecasts for highly periodic data. Simulations also show that forecast precision increases with the degree of periodicity of the earthquake process using the IG model, and so should be better for LP earthquakes than VTs. These results provide a new framework for point process modelling of volcanic earthquake time series, and verification of alternative models.