# A Stochastic Earthquake Recurrence Model with Temporal and Spatial Dependence

Consideration of the dependence between the occurrences of earthquakes is an important aspect of seismic hazard analysis which has drawn much Attention. CurrentIy used techniques for harzard evaluation are based on Poisson assumptions for earthquake occurrences, such as spatial and temporal independence betwe.en events. While this is an adequate description for some cases, it is not representstive of all data. In addition it is not consistent with any geophysical description of the earthquake-generating process. A model which permits better estimates of the probabilities of earthquake occurrence is desirable in seismic hazard analysis. Assumptions of independence between earthquakes in space and time have been shown to lead to the overestimation of seismic design parameters. Hodeling of dependence can lead to more realistic policy decisions and cost-effective design.

Recent studies have indicated that a positive correlation exists between an earthquake recurrence interval and the size of the preceding event. This correlation is used to develop a stochastic model of earthquake occurrence for use in seismic hazard evaluation. Temporal and spatial dependence are included in the model. The model is based on the Time-Predictable Recurrence Hodel of Sh fmazakd and Nakata (1980) and their basic assumptions are adopted in the formulation of the stochastic model of seismic events. Stress is accumulated at a constant rate along a fault. Earthquakes occur when the accumulated stress reaches a threshold value at some location on the fault. The size of an earthquake is measured by the change in stress level. Larger events correspond to larger stress releases. After an earthquake, the amount of time required for stress to build up to the threshold state determines the time to the next earthquake. This time is related to the size of the recent earthquake and the rate of stress accumulation. The size of the stress releases are assumed to be independent identically distributed sandom variables. Spatial dependence is included by introducing a probability distribution for the location of a future earthquake conditional on the location of the epicenter of the previous event.

The model is tested against observed data and comparisons are made with other existing models. Examples are run for sections of the Calaveras fault zone and the San Andreas fault ·zone. These study zones were chosen because historical data for these areas suggest that the time-predictable hypothesis is a good description of the pattern of earthquake occurrences. Comparisons with observed data on the Calaveras fault show that the cumulative slip and the cumulative number of events in a fixed interval generated from the model are a good match to the data. Comparisons with the Poisson model suggest that the Poisson model may underestimate occurrence probabilities when a long gap has elapsed since the last earthquake or when the last earthquake was not very large.