# Crack Models of Repeating Earthquakes Predict Observed Moment-Recurrence Scaling

Camilla Cattania, Paul Segall

### Abstract

Small repeating earthquakes are thought to represent rupture of isolated asperities loaded by surrounding creep. The observed scaling between recurrence interval and seismic moment, $T_r$ ~ $M^{1/ 6}$, contrasts with expectation assuming constant stress drop and no aseismic slip ($T_r$ ~ $M^{1/ 3}$). Here we demonstrate that simple crack models of velocity-weakening asperities embedded in a velocity-strengthening fault predict the $M^{1/ 6}$ scaling; however, the mechanism depends on asperity radius, $R$. For small asperities ( $R_{\infty} < R < 2R_{\infty} $, where $R_{\infty}$ is the nucleation radius) numerical simulations with rate-state friction show interseismic creep penetrating inwards from the edge, with earthquakes nucleating in the center and rupturing the entire asperity. Creep penetration accounts for ~$25%$ of the slip budget, the nucleation phase takes up a larger fraction of slip. Stress drop increases with increasing $R$; the lack of self-similarity due to the finite nucleation dimension. For $2 R_{\infty}<R\lesssim 4.3 R_{\infty}$ simulations exhibit simple cycles with ruptures nucleating from the edge. Asperities with $R\gtrsim 4.3R_{\infty}$ exhibit complex cycles of partial and full ruptures. Here $T_r$ is explained by an energy criterion: full rupture requires that the energy release rate everywhere on the asperity at least equals the fracture energy, leading to the scaling $T_r$ ~ $M^{1/ 6}$. Remarkably, in spite of the variability in behavior with source dimension, the scaling of $T_r$ with stress drop $\Delta\tau$, nucleation length and creep rate $v_{pl}$ is the same across all regimes: $T_r$ ~ $\sqrt{R_{\infty}}\Delta\tau^{5/ 6}~M_0^{1/ 6}/v_{pl}$. This supports the use of repeating earthquakes as creepmeters, and provides a physical interpretation for the scaling observed in nature.