Aftershocks are an extreme example of short-term earthquake clustering that appears to be quite distinct from the long-term regional clustering of large earthquakes discussed above. Aftershocks can temporarily increase the local seismicity rates to more than 10,000 times the pre-mainshock level. Although Coulomb stress interactions provide an explanation for many aftershock patterns, those models alone do not account for either the rates of seismicity that occur in response to the stress changes or the subsequent decay of rates inversely proportional to time, as expressed in Omori’s aftershock decay law (Equation 2.8). The most fully developed explanation for these and other properties of aftershocks is based on the rate- and state-dependent fault frictional properties observed in laboratory experiments (see Section 4.4). These frictional properties require that the initiation of earthquake slip (earthquake nucleation) be a delayed instability process in which the time of an earthquake is nonlinearly dependent on stress changes (35). This approach has resulted in a state-dependent model for earthquake rates that provides quantitative explanations for observed aftershock rates in response to a stress change, the Omori decay law, and various other features of aftershocks (Box 5.1).

Aftershocks can also be generated by dynamic stresses during the passage of seismic waves. At large epicentral distances, these transients are much greater than the static Coulomb stresses, although they act only over short intervals. Short-term dynamic loading was responsible for triggering seismicity across the western United States after the 1992 Landers, California earthquake (36). Immediately following the Landers earthquake, bursts of seismicity were observed at locations more than 1000 kilometers from the mainshock (Figure 5.2). The mechanisms for after-

BOX 5.1 State-Dependent Seismicity

A physically based method for quantitative modeling of the relationships between stress changes and earthquake rates is provided by the rate- and state-dependent representation of fault friction. This approach treats seismicity as a sequence of earthquake nucleation events and specifically includes the time and stress dependence of the earthquake nucleation process as required by rate- and state-dependent friction. The result is a general state-dependent formulation for earthquake rates:1


where R is earthquake rate (in some magnitude interval), ? is a state variable, t is time, and S is Coulomb stress. The normalizing constant r is defined as the steady-state earthquake rate at the reference stressing rate . A is a dimensionless fault

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