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Published October 1975 | Published
Journal Article Open

Theoretical basis of some empirical relations in seismology


Empirical relations involving seismic moment M_o, magnitude M_S, energy E_S and fault dimension L (or area S) are discussed on the basis of an extensive set of earthquake data (M_S ≧ 6) and simple crack and dynamic dislocation models. The relation between log S and log M_o is remarkably linear (slope ∼ 2/3) indicating a constant stress drop Δσ; Δσ = 30, 100 and 60 bars are obtained for inter-plate, intra-plate and "average" earthquakes, respectively. Except for very large earthquakes, the relation M_S ∼ (2/3) log M_o ∼ 2 log L is established by the data. This is consistent with the dynamic dislocation model for point dislocation rise times and rupture times of most earthquakes. For very large earthquakes M_S ∼ (1/3) log M_o ∼ log L ∼ (1/3) log E_S. For very small earthquakes M_S ∼ log M_o ∼ 3 log L ∼ log E_S. Scaling rules are assumed and justified. This model predicts log E_S ∼ 1.5 M_S ∼ 3 log L which is consistent with the Gutenberg-Richter relation. Since the static energy is proportional to σ̅L^3, where σ̅ is the average stress, this relation suggests a constant apparent stress ησ̅ where η is the efficiency. The earthquake data suggest ησ̅ ~ 1/2 Δσ. These relations lead to log S ∼ M_S consistent with the empirical relation. This relation together with a simple geometrical argument explains the magnitude-frequency relation log N ∼ − M_S.

Additional Information

Copyright © 1975, by the Seismological Society of America. Manuscript received April 8, 1975. We thank Tom Hanks and Bob Geller for useful discussions. This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Air Force Office of Scientific Research under Contract F44620-72-C-0078, and Earth Sciences Section of the National Science Foundation under Grant DES74-22489.

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