Dynamical System Analysis and Forecasting of Deformation Produced by an Earthquake Fault

We present a method of constructing low-dimensional nonlinear models describing the main dynamical features of a discrete 2-D cellular fault zone, with many degrees of freedom, embedded in a 3-D elastic solid. A given fault system is characterized by a set of parameters that describe the dynamics, rheology, property disorder, and fault geometry. Depending on the location in the system parameter space, we show that the coarse dynamics of the fault can be confined to an attractor whose dimension is significantly smaller than the space in which the dynamics takes place. Our strategy of system reduction is to search for a few coherent structures that dominate the dynamics and to capture the interaction between these coherent structures. The identification of the basic interacting structures is obtained by applying the Proper Orthogonal Decomposition (POD) to the surface deformation fields that accompany strike-slip faulting accumulated over equal time intervals. We use a feed-forward artificial neural network (ANN) architecture for the identification of the system dynamics projected onto the subspace (model space) spanned by the most energetic coherent structures. The ANN is trained using a standard back-propagation algorithm to predict (map) the values of the observed model state at a future time, given the observed model state at the present time. This ANN provides an approximate, large-scale, dynamical model for the fault. The map can be evaluated once to provide a short-term predictions or iterated to obtain a prediction for the long-term fault dynamics.