Contribution of dynamics of cascading to the earthquake statistics: Dependency of b-value on fault properties and stress in the multiscale circular patch model

The Gutenberg-Richter law is one of the most well-known laws in seismology. Various theoretical studies explained the magnitude-frequency statistics of power-law [e.g., Aki, 1980; Andrews, 1980; Rice and Ben-Zion, 1996; Ide and Aochi, 2005; Dieterich and Richards-Dinger, 2010; Langenbruch and Shapiro, 2014], mainly through the fractal nature of a fault system or the characteristics of self-organization.

In the present work, we explore a contribution of dynamics of cascading process [Ellsworth and Beroza, 1995; Okuda and Ide, 2018; Ellsworth, 2018; Ide, 2019] to the b-value of the G-R law. Rupture should encounter numerous energy barriers during its multiscale growth from a microfracture to a large slip, and the ability of the rupture to cascade beyond the barriers should affect the b-value.

We review the multiscale circular patch model [Ide and Aochi, 2005], which assumes patch-like fractal distribution of a slip-weakening distance of friction on a fault system and produces power-law statistics. The b-value, obtained by 100000 independent simulations with random patch distribution in the two-dimensional model, is dependent on frictional fault properties and has negative dependency on the background shear stress. The result is consistent with the studies of rock experiments and seismological observations [Goebel et al., 2017, references in Scholz, 2015]. We present a mathematical model to explain and reproduce the result, which integrates an energetic threshold of cascading and the probability distribution of patches.