Asperity model of an earthquake: Static problem

We develop an earthquake asperity model that explains previously determined empirical scaling relationships for repeating earthquakes along the San Andreas fault in central California. The model assumes that motion on the fault is resisted primarily by a patch of small strong asperities that interact with each other to increase the amount of displacement needed to cause failure. This asperity patch is surrounded by a much weaker fault that continually creeps in response to tectonic stress. Extending outward from the asperity patch into the creeping part of the fault is a shadow region where a displacement deficit exists. Starting with these basic concepts, together with the analytical solution for the exterior crack problem, the consideration of incremental changes in the size of the asperity patch leads to differential equations that can be solved to yield a complete static model of an earthquake. Equations for scalar seismic moment, the radius of the asperity patch, and the radius of the displacement shadow are all specified as functions of the displacement deficit that has accumulated on the asperity patch. The model predicts that the repeat time for earthquakes should be proportional to the scalar moment to the 1/6 power, which is in agreement with empirical results for repeating earthquakes. The model has two free parameters, a critical slip distance d(c) and a scaled radius of a single asperity (r) over tilde (o). Numerical values of 0.20 and 0.17 cm, respectively, for these two parameters will reproduce the empirical results, but this choice is not unique. Assuming that the asperity patches are distributed on the fault surface in a random fractal manner leads to a frequency-size distribution of earthquakes that agrees with the Gutenberg-Richter formula and a simple relationship between the h-value and the fractal dimension. We also show that the basic features of the theoretical model can be simulated with numerical calculations employing the boundary integral method.