Introduction to mathematical modeling of earthquakes

We first overview statistics and kinematics, necessities in modeling of earthquakes. In statistics, size-frequency distributions of earthquakes and temporal changes of aftershock activities are main subjects. We pay attention not only to power-law behaviors but to non-power-law behaviors as well. In particular, comparison of the two size-frequency distributions, the one by Gutenberg-Richter and the other based on the characteristic earthquake scheme which assumes periodic generation of earthquakes similar in size, is important from a viewpoint of earthquake prediction. Kinematically, a framework is presented which treats earthquakes as generation of dislocations, discontinuities in the displacement over fault surfaces. As static models, we discuss percolation models on a tree and a two-dimensional square lattice. Here the size-frequency generally decays exponentially or stretched exponentially as earthquakes become large, and does algebraically (Gutenberg-Richter law) only at the critical point. Time depe...