Bayesian inference provides a method to use seismic intensity data or instrumental locations, together with geologic and seismologic data, to make quantitative estimates of the probabilities that specific past earthquakes are associated with specific faults. Probability density functions are constructed for the location of each earthquake, and these are combined with prior probabilities through Bayes' theorem to estimate the probability that an earthquake is associated with a specific fault. Results using this method are presented here for large, preinstrumental, historical earthquakes and for recent earthquakes with instrumental locations in the San Francisco Bay region. The probabilities for individual earthquakes can be summed to construct a probabilistic frequency–magnitude relationship for a fault segment. Other applications of the technique include the estimation of the probability of background earthquakes, that is, earthquakes not associated with known or considered faults, and the estimation of the fraction of the total seismic moment associated with earthquakes less than the characteristic magnitude. Results for the San Francisco Bay region suggest that potentially damaging earthquakes with magnitudes less than the characteristic magnitudes should be expected. Comparisons of earthquake locations and the surface traces of active faults as determined from geologic data show significant disparities, indicating that a complete understanding of the relationship between earthquakes and faults remains elusive.

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