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 E_R (radiated energy). These two scaling relations are not independent. Using the variational calculus, we show that the ratio ẽ = E_R/M_0 has a lower bound, ẽ_(min), for given M_0 and f_0. If the commonly used static scaling relation (M_0 ∝ f_0^(-3)) holds, then ẽ_(min) must be scale independent and should not depend on the magnitude, M_w. The observed values of ẽ for large earthquakes [e.g., ẽ(M_w 7)] are close to ẽ_(min). The observed values of ẽ for small earthquakes are controversial, but the reported values of ẽ(M_w 3) range from 1 to 0.1 of ẽ(M_w 7), suggesting that ẽ_(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 ∝ f_0^(-(3+ϵ), (ϵ ≤ 1), which is allowable within the observational uncertainties. This modification leads to a scale-dependent ẽ_(min), ẽ_(min) ∝ 10^(1.5)M_wϵ/(3+ϵ), and a scale-dependent Δσ_sV^3 (Δσ_s = static stress drop, V = rupture speed), Δσ_sV^3 ∝ 10^(1.5M)_w^(ϵ/(3+ϵ)), and it can accommodate the range of presently available data on these scaling relations. We note that the scaling relation, Δσ_sV^3 ∝ 10^(1.5M)_wϵ/(3+ϵ), suggests that even if ẽ is scale independent and M_0 ∝ f_0^(-3) (i.e., ϵ = 0), Δσ_s is not necessarily scale independent, although such scale independence is often implied. Small and large earthquakes can have significantly different Δσ_s and V; if ẽ varies with M_w, as suggested by many data sets, the difference can be even larger, which has important implications for rupture physics.

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