A fractal model for crustal deformation

It is hypothesized that crustal deformation occurs on a scale-invariant matrix of faults. For simplicity, a two-dimensional pattern of hexagons on which strike-slip faulting occurs is considered. The behavior of the system is controlled by a single parameter, the fractal dimension. Deformation occurs on all scales of faults. The fractal dimension determines the fraction of the total displacement that occurs on the first-order or primary faults. The value of the fractal dimension can be obtained from the frequency-magnitude relation for earthquakes. The results are applied to the San Andreas fault system in central California. Earthquake studies give D = 1.90. The main strand of the San Andreas fault is associated with the primary faults of the fractal system. It is predicted that the relative velocity across the main strand is 2.93 cm/yr. The remainder of the relative velocity of 5.5 cm/yr between the Pacific and North American plates occurs on higher-order faults. The predicted value is in reasonably good agreement with the value 3.39 + or - 0.29 cm/yr obtained from geological studies.