A Comparison of Bayesian Hierarchical Space-Time Models for Earthquake Data

Abstract

Stochastic earthquake models are often based on a marked point process approach as for instance presented in Vere-Jones (1995). This gives a fine resolution both in space and time making it possible to represent each earthquake with corresponding foreshocks and aftershocks separately. However, it is not obvious that this microscopic approach is advantageous when aiming at earthquake predictions. In the present paper we take a macroscopic point of view considering grid cells of 0.5°x 0.5°, or about 50 km x 50 km, and time periods of 4 months, which seems suitable for predictions. Hereby, also the effects of foreshocks and aftershocks are circumvented. More specifically, we will discuss different alternative Bayesian hierarchical space-time models in the spirit of Wikle et al. (1998). For each time period the observations are the magnitudes of the largest observed earthquake within each grid cell. In our models these largest observed earthquakes are represented by hidden system state variables called potentials. The potentials at each time period and grid point are decomposed into a time independent term and various alternative time dependent terms with spatial description. As data we apply parts of an earthquake catalog provided by The Northern California Earthquake Data Center where we limit ourselves to the area 32°-37° N, 115°-120°W for the years 1981 through 1999 containing the Landers and Hector Mine earthquakes of magnitudes respectively 7.3 and 7.1 on the Richter scale. Based on the various space-time models earthquake predictions for the next two time periods for all grid cells are arrived at. The models are implemented within an MCMC framework in Matlab. The model that gives the overall best predictions is claimed to give valuable and new knowledge on the spatial and time related dependencies between earthquakes.