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 ẽ = ER/M₀ has a lower bound, ẽ_ (min), for given M₀ and f₀. If the commonly used static scaling relation (M₀ ∝ f₀^ (-3) ) holds, then ẽ_ (min) must be scale independent and should not depend on the magnitude, Mw. The observed values of ẽ for large earthquakes e. g. , ẽ (Mw 7) are close to ẽ_ (min). The observed values of ẽ for small earthquakes are controversial, but the reported values of ẽ (Mw 3) range from 1 to 0. 1 of ẽ (Mw 7), suggesting that ẽ_ (min) may decrease as Mw decreases. To accommodate this possibility, we need to modify the M₀ versus f₀ scaling relation to (M₀ ∝ f₀^ (- (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) Mwϵ/ (3+ϵ), and a scale-dependent ΔσₛV³ (Δσₛ = static stress drop, V = rupture speed), ΔσₛV³ ∝ 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, ΔσₛV³ ∝ 10^ (1. 5M) wϵ/ (3+ϵ), suggests that even if ẽ is scale independent and M₀ ∝ f₀^ (-3) (i. e. , ϵ = 0), Δσₛ is not necessarily scale independent, although such scale independence is often implied. Small and large earthquakes can have significantly different Δσₛ and V; if ẽ varies with Mw, as suggested by many data sets, the difference can be even larger, which has important implications for rupture physics.