A fractal model for crustal deformation

We hypothesize that crustal deformation occurs on a scale-invariant matrix of faults. For simplicity we consider a two-dimensional pattern of hexagons on which strike-slip faulting occurs. 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. Our results are applied to the San Andreas fault system in central California. Earthquake studies give D = 1.90. We associate the main strand of the San Andreas fault with the primary faults of our fractal system. We predict 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 ± 0.29 cm/yr obtained from geological studies.