Earthquake populations have recently been postulated to be an example of a self-organized critical (SOC) phenomenon, with fractal spatial and temporal correlations and a power-law distribution of seismic energy or moment corresponding to the Gutenberg-Richter (G-R) frequency-magnitude law. In fact, strict SOC behavior is not seen in all models and is confined to those with weak annealed (permanent) heterogeneity and an intermediate tectonic driving velocity or strain energy rate. Depending on these conditions, distributions may also occur that are subcritical, where the largest events have a reduced probability of occurrence compared to the G-R trend, or supercritical, where the largest events have an elevated probability of occurrence, leading to “characteristic” earthquakes. Here we show type examples of all three types of behavior, lending support to a generalization of the Gutenberg-Richter law to a modified gamma distribution (a power law in energy or moment with an exponential tail with positive, zero, or negative argument). If earthquakes are an example of a critical phenomenon, then the a priori assumption of the G-R law in probabilistic hazard analysis is no longer valid in all cases. The appropriate distribution may also depend systematically on the size of the area, with smaller areas concentrating on individual fault segments more likely to produce a characteristic distribution. This previously unexpected effect of Euclidean zoning is an example of the degree of preconditioning inherent in some of the fundamental assumptions of seismic hazard analysis. Other assumptions, such that of stationarity in the process over long time periods, are borne out by SOC. The assumption of a random Poisson process is firmly at odds with SOC as an avalanche process involving strong local and weaker long-range interactions between earthquakes. The gamma distribution for the case of subcritical behavior predicts a maximum “credible” magnitude that may be independently determined from long-term slip rates, defined as the magnitude where the contribution to the total moment or intensity is negligible though finite. This soft maximum replaces the need to independently impose a hard, though somewhat artificial, maximum in distributions such as the G-R law. The same approach can be taken for the overall seismic hazard expressed by negligible contribution to ground motion, with similar results.

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