Static and Dynamic Scaling Relations for Earthquakes and Their Implications for Rupture Speed and Stress Drop

We investigate the relation between a static scaling relation, M0 (seismic moment) versus f0 (spectral corner frequency), and a dynamic scaling relation between M0 and ER (radiated energy). These two scaling relations are not independent. Using the variational calculus, we show that the ratio e= ER/M0 has a lower bound, emin, for given M0 and f0. If the commonly used static scaling relation (M0(alpha)f0^-3) holds, then emin must be scale independent and should not depend on the magnitude, Mw. The observed values of efor large earthquakes [e.g., e(Mw 7)] are close to emin. The observed values of emin. The observed values of e for small earthquakes are controversial, but the reported values of e(Mw 3) range from 1 to 0.1 of e(Mw 7), suggesting that emin may decrease as Mw decreases. To accommodate this possibility, we need to modify the M0 versus f0 scaling relation to M0(alpha)f0^-(3+(epsilon), ((epsilon) =<1), which is allowable within the observational uncertainties. This modification leads to a scale-dependent emin, emin (alpha)101.5Mw(epsilon)/(3+(epsilon)), and a scale-dependent (delta)(sigma)sV3 ((delta)(sigma)s = static stress drop, V = rupture speed), (delta)(sigma)s V3 (alpha) 101.5Mw(epsilon)/(3+(epsilon)), and it can accommodate the range of presently available data on these scaling relations. We note that the scaling relation, (delta)(sigma)s V3 (alpha) 101.5Mw(epsilon)/(3+(epsilon))), suggests that even if e is scale independent and M0(alpha)f0^-3 (i.e., (epsilon)= 0), (delta)(sigma)s is not necessarily scale independent, although such scale independence is often implied. Small and large earthquakes can have significantly different (delta)(sigma)s and V; if e varies with Mw, as suggested by many data sets, the difference can be even larger, which has important implications for rupture physics.