Theoretical Basis of Some Empirical Relations in Seismology by Hiroo Kanamori and Don

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 Aa; Atr = 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 0L 3, where ~ is the average stress, this relation suggests a constant apparent stress ~/¢i where r/ is the efficiency. The earthquake data suggest r/0 ~ 1⁄2 Atr. 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 N M s .