Modeling aftershock rates using simulations of spontaneous earthquake nucleation on rate and state faults

Large earthquakes are followed by increased seismic activity, usually referred to as aftershock sequences, that decays to the background rate over time. The decay of aftershocks is well-described empirically by Omori's law. Dieterich (1994) proposed that Omori's law could result from perturbing, by static stress steps, a population of nucleation sites governed by laboratory-derived rate and state friction. He used one-degree-of-freedom spring-slider system to represent elastic interactions and made a simplified assumption about frictional behavior during nucleation. The model was further explored in a number of studies (i.e., Gomberg et al., 2000) and used to interpret observations (i.e., Toda et al., 1998). In this study, we explore the consequences of Dieterich's approach using models of faults embedded in elastic continuum, where the nucleation process can be more complicated than assumed in Dieterich's model. Our approach is different from previous studies of aftershock rates with rate and state friction in that here, nucleation processes are simulated as a part of spontaneously occurring earthquake sequences in continuum fault models. We use two 2D models of a vertical strike-slip fault, the depth-variable model (Rice, 1993; Lapusta at el., 2000) and the crustal-plane model (Myers et al., 1996). We find that nucleation processes in continuum models and the resulting aftershock rates are well-described by the model of Dieterich (1994) when Dieterich's assumption that the state variable of the rate and state friction law is significantly behind its steady-state value holds during the entire nucleation process. On the contrary, aftershock rates in models where the state variable assumption is violated for a significant portion of the nucleation process exhibit behavior different from Dieterich's model. The state variable assumption is significantly violated, and hence the aftershock rates are affected, when stress heterogeneities are present within the nucleation zone. Aftershock rates are more affected by heterogeneities for larger values of the ratio a/b of rate and state parameters, a and b. This is consistent with the recent study by Rubin and Ampuero (AGU, 2004) who show that if the ratio a/b is close to one, the state variable assumption will eventually be violated in the middle portions of the nucleation zone. The aftershock rates we obtain for nucleation processes that occur at transitions between creeping and locked behavior differ significantly from the Dieterich's model and from Omori's law. For a spatially uniform shear stress step, we observe delayed peaks of aftershock activity followed by seismic quiescence. To understand how this might be related to observations, we model the response of a population of nucleation sites located at the rheological transition to non-uniform stress steps due to the varying distance from the mainshock. The resultant aftershock rates follow power-law decay for some time (several years for the parameters studied) and then drop to values much lower than the background seismicity rate. This result may explain previously observed turnoff of aftershock activity at the base of the seismogenic zone near the 1984 Morgan Hill earthquake (Tian and Rubin, Chapman meeting abstract, 2005).