Quantitative relationship between slow earthquake migration speed and frictional properties

Owing to the development of computational calculation abilities, numerical simulations of multi-scale earthquake cycles on the basis of rate- and state-dependent friction laws (RSF laws) have been performed in recent studies. This helps us to better understand the complicated cycle such as effect of stress perturbations due to nearby earthquakes. To fit the occurrence time of the observed earthquakes, it is necessary to know the propagation speed of postseismic slip which can trigger nearby earthquakes. However, we have to perform many trial simulations to establish the relationship between frictional properties and the propagation speed. By applying the RSF laws, postseismic slip can be described as weakly frictionally stable, with the value of the frictional parameter (a-b) slightly positive surrounding the source region of an earthquake where the value of (a-b) is strongly negative. Previous numerical simulation studies have investigated the relationship between the frictional properties and the postseismic slip propagation, but lack a unified explanation. In this study, we describe a theoretical relationship between the postseismic slip propagation speed and the frictional parameters of the RSF law, applying some approximations based on simple test simulations of the earthquake cycle, which is summarized as follows: 1) Lower values of effective normal stress "σ" increase the propagation speed exponentially with enough amplitude of shear stress loading. Otherwise, the propagation speed is independent of σ. 2) The frictional parameter "a" has a similar relationship to σ, but it has a negative linear trend in case of low amplitude of shear stress loading. 3) The frictional parameter "b" has positive linear trend as b1 in case of step function for shear stress loading or small amplitude of the loading, and the trend decreases (b1/4) with large stress amplitudes in case of a ramp function. 4) The frictional parameter "dc" controls propagation speed with a negative linear trend as dc-1 to dc-1/4 similar to b. 5) From the results of 1) to 4), we can explain the reason why A=aσ is more effective in determining the propagation speed rather than B - A = (b-a) σ or (B-A) / dc = (b-a) σ / dc as seen in the previous simulation results.