A recent study of repeating earthquakes on the San Andreas Fault in central
California by Nadeau and Johnson [1998] found that the smallest events occ
urred on patches having a linear dimension of the order of 0.5 m, displacem
ents of about 2 cm, and stress drops of the order of 2000 MPa, roughly 10 t
imes 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 m
oment 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 asper
ity 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 str
ess drop and increase in displacement with increasing event size. The spati
al distribution of hypocenters in the Parkfield area is shown to be consist
ent with this simple fractal model and with a hierarchical clustering of as
perities having a fractal dimension of D=1 and discrete rescaling factor of
about 20.