Publication Date:
2004-02-01
Description:
We investigate the relation between a static scaling relation, M (sub 0) (seismic moment) versus f (sub 0) (spectral corner frequency), and a dynamic scaling relation between M (sub 0) and E (sub R) (radiated energy). These two scaling relations are not independent. Using the variational calculus, we show that the ratio e = E (sub R) /M (sub 0) has a lower bound, e (sub min) , for given M (sub 0) and f (sub 0) . If the commonly used static scaling relation (M (sub 0) varies as f (sub 0) (super -3) ) holds, then e (sub min) must be scale independent and should not depend on the magnitude, M (sub w) . The observed values of e for large earthquakes [e.g., e(M (sub w) 7)] are close to e (sub min) . The observed values of e for small earthquakes are controversial, but the reported values of e(M (sub w) 3) range from 1 to 0.1 of e(M (sub w) 7), suggesting that e (sub min) may decrease as M (sub w) decreases. To accommodate this possibility, we need to modify the M (sub 0) versus f (sub 0) scaling relation to M (sub 0) varies as f (sub 0) (super -(3+epsilon )) (epsilon 〈 or =1), which is allowable within the observational uncertainties. This modification leads to a scale-dependent e (sub min) , e (sub min) varies as 10 (super 1.5Mwepsilon /(3+epsilon )) , and a scale-dependent Delta sigma (sub s) V (super 3) (Delta sigma (sub s) = static stress drop, V = rupture speed), Delta sigma (sub s) V (super 3) varies as 10 (super 1.5Mwepsilon /(3+epsilon )) , and it can accommodate the range of presently available data on these scaling relations. We note that the scaling relation, Delta sigma (sub s) V (super 3) varies as 10 (super 1.5Mwepsilon /(3+epsilon )) , suggests that even if e is scale independent and M (sub 0) varies as f (sub 0) (super -3) (i.e., epsilon = 0), Delta sigma (sub s) is not necessarily scale independent, although such scale independence is often implied. Small and large earthquakes can have significantly different Delta sigma (sub s) and V; if e varies with M (sub w) , as suggested by many data sets, the difference can be even larger, which has important implications for rupture physics.
Print ISSN:
0037-1106
Electronic ISSN:
1943-3573
Topics:
Geosciences
,
Physics
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