Prediction probabilities from foreshocks

Abstract

When any earthquake occurs, the possibility that it might be a foreshock increases the probability that a larger earthquake will occur nearby within the next few days. Clearly, the probability of a very large earthquake ought to be higher if the candidate foreshock were on or near a fault capable of producing that very large mainshock, especially if the fault is towards the end of its seismic cycle. We derive an expression for the probability of a major earthquake characteristic to a particular fault segment, given the occurrence of a potential foreshock near the fault. To evaluate this expression, we need: (1) the rate of background seismic activity in the area, (2) the long‐term probability of a large earthquake on the fault, and (3) the rate at which foreshocks precede large earthquakes, as a function of time, magnitude, and spatial location. For this last function we assume the average properties of foreshocks to moderate earthquakes in California: (1) the rate of mainshock occurrence after foreshocks decays roughly as t−1, so that most foreshocks are within three days of their mainshock, (2) foreshocks and mainshocks occur within 10 km of each other, and (3) the fraction of mainshocks with foreshocks increases linearly as the magnitude threshold for foreshocks decreases, with 50% of the mainshocks having foreshocks with magnitudes within three units of the mainshock magnitude (within three days). We apply our results to the San Andreas, Hayward, San Jacinto, and Imperial faults, using the probabilities of large earthquakes from the report of the Working Group on California Earthquake Probabilities (1988). The magnitude of candidate event required to produce a 1% probability of a large earthquake on the San Andreas fault within three days ranges from a high of 5.3 for the segment in San Gorgonio Pass to a low of 3.6 for the Carrizo Plain.