Earthquake size‐frequency relations from an analytical stochastic rupture model

Abstract

An analytical stochastic model of earthquake rupture is presented. Although the general formulation makes no a priori self‐similarity assumption, scale independent earthquake size‐frequency (ESF) statistics emerge from the interaction between fault strength heterogeneity and rupture mechanics. The model also gives theoretical derivations for scale dependent ESF statistics, including gradual frequency saturation for small earthquakes, and a continuum of possible transitions in the statistics of large earthquakes. A model fault is characterized as a surface composed of a large number of asperities, defined as small, discrete contact surfaces of finite strength. An earthquake rupture grows via successive asperity‐breaking subevents. The probability of rupture to a certain size earthquake or greater is the total product of incremental rupture growth probabilities. These are governed by a new asperity failure probability distribution, derived as a mixture of Weibull distributions. By including fracture mechanics, the resulting mixture distribution, characterized by the scale and shape parameters Λ and Γ, is written in terms of the rupture size. The important difference between the mixture distribution and the Weibull distribution is that the mixture results in power law ESF statistics. Self‐similarity occurs when material heterogeneity and rupture mechanics are mutually “tuned” to give Λ = Γ = 1. Frequency saturation for small earthquakes is shown to be a simple consequence of a general aseismic nucleation process. The nucleation magnitude mν defines the roll‐off point in ESF statistics that occurs when the area of seismic rupture equals the area of precursory, aseismic rupture growth. By fitting the model to the Parkfield earthquake data set, mν = 0.9 is obtained. The change in ESF scaling for the transition from moderate to large earthquakes depends on the changes in stress at the growing rupture front. The Northern California Earthquake Data Center seismicity data set is fitted for a transition at magnitude m ≃ 5 with Λ′ = 0.4 and Γ′ = 0.75. This result indicates that the change in the ESF relation associated with the transition from moderate to large earthquakes in California and Nevada is caused by surface rupture accompanied by partial width and slip saturation.