How strong is an asperity?

A recent study of repeating earthquakes on the San Andreas Fault in central California by Nadeau and Johnson [1998] found that the smallest events occurred on patches having a linear dimension of the order of 0.5 m, displacements of about 2 cm, and stress drops of the order of 2000 MPa, roughly 10 times larger than rock strengths measured in the laboratory. The stress drop for larger events was observed to decrease as a power law of the seismic moment reaching the commonly observed value of 10 MPa at about magnitude 6. These large strengths are shown here to be consistent with laboratory data if the preexisting microcracks are all healed. A hierarchical fractal asperity model is presented, which is based on recent laboratory observations of contact distributions in sliding friction experiments. This “Cantor dust” model is shown to be consistent with the observed power law decrease in stress drop and increase in displacement with increasing event size. The spatial distribution of hypocenters in the Parkfield area is shown to be consistent with this simple fractal model and with a hierarchical clustering of asperities having a fractal dimension of D=1 and discrete rescaling factor of about 20.