Gutenberg-Richter and characteristic earthquake behavior in simple mean-field models of heterogeneous faults

Abstract

The statistics of earthquakes in a heterogeneous fault zone is studied analytically and numerically in a mean field version of a model for a segmented fault system in a three-dimensional elastic solid [1,2]. The studies focus on the interplay between the roles of disorder, dynamical effects, and driving mechanisms. A two-parameter phase diagram is found, spanned by the amplitude of dynamical weakening (or "overshoot") effects ǫ and the normal distance L of the driving forces from the fault. In general, small ǫ and small L are found to produce Gutenberg-Richter type power law statistics with an exponential cutoff, while large ǫ and large L lead to a distribution of small events combined with characteristic system-size events. In a certain parameter regime the behavior is bistable, with transitions back and forth from one phase to the other on time scales determined by the fault size and other model parameters.The implications for realistic earthquake statistics are discussed.

Statistical physics of earthquakes: Comparison of distribution exponents for source area and potential energy and the dynamic emergence of log‐periodic energy quanta

“…For example, constitutive mechanical rules are used which are known to be much simpler than those implied by the observation of rate and state dependence of frictional slip in laboratory sliding experiments [Deiterich, 1979]. However, we have seen that the scaling properties of other near-critical systems are robust with respect to the details of the assumed local dynamics.…”

“…Using the same model, Ceva [1995] showed that the nature of quenched disorder can have a strong effect on the observed behavior, with point defects having more impact than more extended heterogeneities. Dahmen et al [1998] examined the behavior of a three-dimensional dynamic earthquake model including inertial terms, and calculated the position of the phase boundary between a "small earthquake" phase with a finite correlation length and a "metastable" phase with large characteristic earthquakes, depending on the values of the characteristic cut off size and the slipweakening amplitude. Their model predicts a first-order phase transition between the two phases, in the sense that there is a finite gap between the size of the largest events and the rest of the population.…”

Simulated dynamical weakening and Abelian avalanches in mean-field driven threshold models

“…The magnitude of the stochastic term must overcome this deficit for a rupture to propagate. A previous approach [8] is to linearize the stress excess about κ = 0 (not including terms of and compute the first crossings of the random walk with the line. The exact result is given by the Bachelier-Levy formula [17].…”

“…However, the limiting case of mean-field models are beginning to yield to analytical understanding [8,9,10,11]. Mean field model simulations exhibit complex event histories and regimes of behavior, including a power law magnitude-frequency relation.…”

“…Despite their simplicity, mean-field models remain sensitive to the choice of update rules and details of implementation. This is especially evident in models that impose a particular form of frictional weakening, where different modes of behavior appear as the strength of the weakening is varied [8].…”