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, M-0 (seismic moment) versus f(0) (spectral corner frequency), and a dynamic scaling relation between M-0 and ER (radiated energy). These two scaling relations are not independent. Using the variational calculus, we show that the ratio e = E-R/M-0 has a lower bound, e(min), for given M-0 and f(0). If the commonly used static scaling relation (M-0 proportional to f(0)(-3)) holds, then e(min) must be scale independent and should not depend on the magnitude, M-w. The observed values of e for large earthquakes [e.g., e(M-w 7)] are close to e(min). The observed values of e for small earthquakes are controversial, but the reported values of e(M-w 3) range from 1 to 0.1 of e(M-w 7), suggesting that e(min) may decrease as M-w decreases. To accommodate this possibility, we need to modify the M-0 versus f(0) scaling relation to M-0 proportional to f(0)(-3)), which is allowable within the observational uncertainties. This modification leads to a scale-dependent e(min), e(min) proportional to 10(1.5Mwepsilon/(3+epsilon)), and a scale-dependent Deltasigma(s)V(3) (Deltasigma(s) = static stress drop, V = rupture speed), Deltasigma(s)V(3) proportional to 10(1.5Mwepsilon/(3+epsilon)), and it can accommodate the range of presently available data on these scaling relations. We note that the scaling relation, Deltasigma(s)V(3) proportional to 10(1.5Mwepsilon/(3+epsilon)), suggests that even if e is scale independent and M-0 proportional to f(0)(-3) (i.e., epsilon = 0), Deltasigma(s) is not necessarily scale independent, although such scale independence is often implied. Small and large earthquakes can have significantly different Deltasigma(s) and V; if e varies with M-w, as suggested by many data sets, the difference can be even larger, which has important implications for rupture physics.