STATIC AND DYNAMIC SCALING RELATIONS FOR EARTHQUAKES AND THEIR IMPLICATIONS FOR RUPTURE SPEED AND STRESS DROP

Abstract

We investigate the relation between a static scaling relation, M0 (seismic moment) versus f 0 (spectral corner frequency), and a dynamic scaling relation be- tween M0 and ER (radiated energy). These two scaling relations are not independent. Using the variational calculus, we show that the ratio eER/M0 has a lower bound, emin, for given M0 and f 0. If the commonly used static scaling relation (M0 ) 3 f0 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 efor 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 f 0 scaling relation to M0 (e 1), which is allowable within the observa- (3e) f0 tional uncertainties. This modification leads to a scale-dependent emin, emin , and a scale-dependent DrsV 3 (Drs static stress drop, V rupture 1.5M e/(3e) w 10 speed), DrsV 3 , and it can accommodate the range of presently avail- 1.5M e/(3e) w 10 able data on these scaling relations. We note that the scaling relation, DrsV 3 , suggests that even if eis scale independent and M0 (i.e., e 0), 1.5M e/(3e) 3 w 10 f0 Drs is not necessarily scale independent, although such scale independence is often implied. Small and large earthquakes can have significantly different Drs and V ;i f evaries with Mw, as suggested by many data sets, the difference can be even larger, which has important implications for rupture physics.