Theoretical basis of some empirical relations in seismology

Empirical relations involving seismic moment Mₒ, magnitude MS, energy ES and fault dimension L (or area S) are discussed on the basis of an extensive set of earthquake data (MS ≧ 6) and simple crack and dynamic dislocation models. The relation between log S and log Mₒ 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 MS ∼ (2/3) log Mₒ ∼ 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 MS ∼ (1/3) log Mₒ ∼ log L ∼ (1/3) log ES. For very small earthquakes MS ∼ log Mₒ ∼ 3 log L ∼ log ES. Scaling rules are assumed and justified. This model predicts log ES ∼ 1. 5 MS ∼ 3 log L which is consistent with the Gutenberg-Richter relation. Since the static energy is proportional to σ̅L³, 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 ∼ MS consistent with the empirical relation. This relation together with a simple geometrical argument explains the magnitude-frequency relation log N ∼ − MS.