Abstract

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 = ER/M0 has a lower bound, min, for given M0 and f0. If the commonly used static scaling relation (

\(M_{0}{\propto}f_{0}^{-3}\)
) holds, then min must be scale independent and should not depend on the magnitude, Mw. The observed values of for large earthquakes [e.g., (Mw 7)] are close to min. The observed values of for small earthquakes are controversial, but the reported values of (Mw 3) range from 1 to 0.1 of (Mw 7), suggesting that min may decrease as Mw decreases. To accommodate this possibility, we need to modify the M0 versus f0 scaling relation to
\(M_{0}{\propto}f_{0}^{-(3+{\varepsilon})}\)
, (ϵ ≤ 1), which is allowable within the observational uncertainties. This modification leads to a scale-dependent min, min ∝ 101.5Mwϵ/(3+ϵ), and a scale-dependent ΔσsV3 (Δσs = static stress drop, V = rupture speed), ΔσsV3 ∝ 101.5Mwϵ/(3+ϵ), and it can accommodate the range of presently available data on these scaling relations. We note that the scaling relation, ΔσsV3 ∝ 101.5Mwϵ/(3+ϵ), suggests that even if is scale independent and
\(M_{0}{\propto}f_{0}^{-3}\)
(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 varies with Mw, as suggested by many data sets, the difference can be even larger, which has important implications for rupture physics.

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