Scaling relations for earthquake source parameters and magnitudes

A data set of 41 moderate and large earthquakes has been used to derive scaling rules for kinematic fault parameters. If effective stress and static stress drop are equal, then fault rise time, τ, and fault area, S, are related by τ = 16S^(1/2)/(7π^(3/2)β), where β is shear velocity. Fault length (parallel to strike) and width (parallel to dip) are empirically related by L=2W. Scatter for both scaling rules is about a factor of two. These scaling laws combine to give width and rise time in terms of fault length. Length is then used as the sole free parameter in a Haskell type fault model to derive scaling laws relating seismic moment to M_S (20-sec surface-wave magnitude), M_S to S and m_b (1-sec body-wave magnitude) to M_S. Observed data agree well with the predicted scaling relation. The “source spectrum” depends on both azimuth and apparent velocity of the phase or mode, so there is a different “source spectrum” for each mode, rather than a single spectrum for all modes. Furthermore, fault width (i.e., the two dimensionality of faults) must not be neglected. Inclusion of width leads to different average source spectra for surface waves and body waves. These spectra in turn imply that m_b and M_S reach maximum values regardless of further increases in L and seismic moment. The m_b versus M_S relation from this study differs significantly from the Gutenberg-Richter (G-R) relation, because the G-R equation was derived for body waves with a predominant period of about 5 sec and thus does not apply to modern 1-sec m_b determinations. Previous investigators who assumed that the G-R relation was derived from 1-sec data were in error. Finally, averaging reported rupture velocities yields the relation v_R = 0.72β.