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Static and Dynamic Scaling Relations for Earthquakes and Their Implications for Rupture Speed and Stress Drop

Seismological Laboratory
California Institute of Technology
Pasadena, California 91125
(H.K.)

EOST-IPGS
Université Louis Pasteur
5, Rue René Descartes
F67084 Strasbourg, France
(L.R.)

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 E_R (radiated energy). These two scaling relations are not independent. Using the variational calculus, we show that the ratio = E_R/M₀ has a lower bound, _min, for given M₀ and f₀. If the commonly used static scaling relation () holds, then _min must be scale independent and should not depend on the magnitude, M_w. The observed values of for large earthquakes [e.g., (M_w 7)] are close to _min. The observed values of for small earthquakes are controversial, but the reported values of (M_w 3) range from 1 to 0.1 of (M_w 7), suggesting that _min may decrease as M_w decreases. To accommodate this possibility, we need to modify the M₀ versus f₀ scaling relation to , ( 1), which is allowable within the observational uncertainties. This modification leads to a scale-dependent _min, _min 10^1.5M_w/(3+), and a scale-dependent _sV³ (_s = static stress drop, V = rupture speed), _sV³ 10^1.5M_w/(3+), and it can accommodate the range of presently available data on these scaling relations. We note that the scaling relation, _sV³ 10^1.5M_w/(3+), suggests that even if is scale independent and (i.e., = 0), _s is not necessarily scale independent, although such scale independence is often implied. Small and large earthquakes can have significantly different _s and V; if 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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