MAGNITUDE-FREQUENCY RELATIONS FOR EARTHQUAKES USING A STATISTICAL-MECHANICAL APPROACH

At very small magnitudes, observations indicate that the frequency of occurrence of earthquakes is significantly smaller than the frequency predicted by simple Gutenberg-Richter statistics. Previously, it has been suggested that the dearth of small events is related to a rapid rise in scattering and attenuation at high frequencies (i.e., the controversial ''f(max)'' problem) and the consequent inability to detect these events with standard arrays of seismometers. However, several recent studies have suggested that instrumentation cannot account for the entire effect and that the decline in frequency may be real. Working from this hypothesis, we derive a magnitude-frequency relation for very small earthquakes that is based upon the postulate that the system of moving plates can be treated as a system not too far removed from equilibrium. As a result, it is assumed that in the steady state, the probability P[E] that a segment of fault has a free energy E is proportional to the exponential of the free energy P is-proportional-to exp[-E/E(N)]. In equilibrium statistical mechanics this distribution is called the Boltzmann distribution. The probability weight E(N) is the space-time steady state average of the free energy of the segment. Earthquakes are then treated as fluctuations in the free energy of the segments. With these assumptions, it is shown that magnitude-frequency relations can be obtained. For example, previous results obtained by the author can be recovered under the same assumptions as before, for intermediate and large events, the distinction being whether the event is of a linear dimension sufficient to extend the entire width of the brittle zone. Additionally, a magnitude-frequency relation is obtained that is in satisfactory agreement with the data at very small magnitudes. At these magnitudes, departures from frequencies predicted by Gutenberg-Richter statistics are found using a model that accounts for the finite thickness of the inelastic part of the fault zone. The inelastic thickness of the fault zone that is obtained is in general agreement with thicknesses found from field observations following earthquakes. Thus, departures from simple Gutenberg-Richter scaling are apparently due at very large magnitudes to the finite width of the brittle layer, and at very small magnitudes to the finite thickness of the inelastic fault zone.