On the effective dimension and dynamic complexity of earthquake faults

We measure the effective dimensionality of a driven, dissipative fault model as its dynamics explore a wide parameter range from a crack like model to a dislocation model. The dynamics of each fault model are probed by recording (a) the first and second order moments of the stresses and slips defined in the fault plane, and (b) the surface deformations that indirectly reflect the brittle processes of the fault and which are observable by InSAR and GPS techniques. In order to study the asymptotic attractors of the model we identify the coherent structures (dominant modes) present in the surface deformation fields and project the model dynamics onto the principal directions defined by these coherent structures. The projection is based on the Karhunen–Loève procedure for the determination of an optimal set of basis functions based on second order statistics. We estimate the effective dimensionality of the dynamics by computing the number of modes needed to capture a certain fraction of the statistical variation of the surface deformation fields. We detect a sharp transition in the number of effective degrees of freedom as we vary the dynamic weakening toward larger and dynamically more significant values. This transition is also associated with a separation of the dynamics in slow and fast degrees of freedom and with the presence of multiple length and time scales in the dynamics. This conclusion is also supported by direct dimension estimates using the correlation dimension. We finally compute the significance of evidence for nonlinearity using the method of surrogate data on the correlation dimension statistics.