Earthquake frequency and moment magnitude relations for mainshocks, foreshocks and aftershocks: Theoretical b values

摘要

The Gutenberg-Richter relation, log n = a - b M, where n is the cumulative number of earthquakes greater than the magnitude, M and a and b are constants determined empirically, is discussed using a mathematical model of a fault and the moment magnitude, M_W. A simple power law of M_0 = k n~l which relates the earthquake moment, M_0 and frequency, n was discussed and the meanings of constants k and l were derived. The relation of M_0 =Δσ (LA) using the stress drop, Δσ, fault length, L and area of a fault plane, A, gives M_0 = Δσ L~3 and M_0 = Δσ hL~2 for two cases of A = L~2 and A = Lh, respectively, where the depth of the brittle-ductile transition layer, h = 2 X 10~4 m and Δσ=5 X 10~6 N/m~2 are assumed. The frequency-magnitude relation is obtained considering different rupture mechanisms of earthquakes. Deviation of the empirical Gutenberg-Richter relation at a large magnitude is interpreted as the effect of the seismogenic layer in earth interior. Theoretical analysis from assumption of different rupture mechanisms indicated varied b values for small and large events, as well as the different types of earthquakes. Mainshocks gave b of 1.0 and 0.75, respectively for small and large events. The b values of 0.75 and 0.50 for aftershocks and 0.50 and 0.375 for foreshocks were also obtained. Some results from both statistic calibration of earthquake catalogue and experiments were in agreement with our predicted values.