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Proceedings of the National Academy of Sciences of the United States of America
quake (B) and we cannot tell which will be followed by a characteristic earthquake (C) is then the ratio of the number of foreshocks to the total number of events or
Assuming the rate of foreshocks is the rate at which foreshocks precede mainshocks [P(F|C)] times the mainshock rate, P(F) =P(F|C)*P(C), then
Thus, the probability of a characteristic earthquake on the fault after a potential foreshock is a function of the rate of background earthquakes [P(B)], the rate of mainshocks [P(C)], and the rate at which foreshoeks preceded mainshocks [P(F|C)]. For other precursors, the probability of an earthquake after the phenomenon has occurred depends on the long-term probability of the mainshock, the false alarm rate of the phenomenon, and the rate at which that phenomenon precedes the mainshock. Collecting the data to determine the last two quantities will require much effort beyond demonstrating a correlation with the mainshock.
Phenomena related to earthquake prediction can be broken into three classes: (i) phenomena that provide information about the earthquake hazard useful to the public, (ii) precursors that are causally related to the failure process of a particular earthquake, and (iii) the intersection of these two classes, predictive precursors that are causally related to a particular earthquake and provide probabilities of earthquake occurrence greater than achievable from a random distribution. In the long term, probabilities derived from geologic rates and historic catalogs are predictors, while conditional probabilities are precursors. Aftershock and foreshock probabilities derived from time-decaying rates are predictors, whereas all other investigated phenomena are precursors. At this time, no phenomenon has been shown to do better than random, and we have no predictive precursors so far. The data necessary to prove a better than random success are much greater than that needed to show a causal relationship to an earthquake.