Bayesian hierarchical space-time modeling of earthquake data

Stochastic earthquake models are often based on a marked point process approach as for instance presented in Vere-Jones (Int. J. Forecast., 11:503-538, 1995). This gives a fine resolution both in space and time making it possible to represent each earthquake. However, it is not obvious that this approach is advantageous when aiming at earthquake predictions. In the present paper we take a coarse point of view considering grid cells of 0.5 x 0.5 degrees, or about 50 x 50 km, and time periods of 4 months, which seems suitable for predictions. More specifically, we will discuss different alternatives of a Bayesian hierarchical space-time model in the spirit of Wikle et al. (Environ. Ecol. Stat., 5:117-154, 1998). For each time period the observations are the magnitudes of the largest observed earthquake within each grid cell. As data we apply parts of an earthquake catalogue provided by The Northern California Earthquake Data Center where we limit ourselves to the area 32-37 degrees N, 115-120 degrees W for the time period January 1981 through December 1999 containing the Landers and Hector Mine earthquakes of magnitudes, respectively, 7.3 and 7.1 on the Richter scale. Based on space-time model alternatives one step earthquake predictions for the time periods containing these two events for all grid cells are arrived at. The model alternatives are implemented within an MCMC framework in Matlab. The model alternative that gives the overall best predictions based on a standard loss is claimed to give new knowledge on the spatial and time related dependencies between earthquakes. Also considering a specially designed loss using spatially averages of the 90th percentiles of the predicted values distribution of each cell it is clear that the best model predicts the high risk areas rather well. By using these percentiles we believe that one has a valuable tool for defining high and low risk areas in a region in short term predictions.