Foreshock and aftershocks in simple earthquake models.

Many models of earthquake faults have been introduced that connect Gutenberg-Richter (GR) scaling to triggering processes. However, natural earthquake fault systems are composed of a variety of different geometries and materials and the associated heterogeneity in physical properties can cause a variety of spatial and temporal behaviors. This raises the question of how the triggering process and the structure interact to produce the observed phenomena. Here we present a simple earthquake fault model based on the Olami-Feder-Christensen and Rundle-Jackson-Brown cellular automata models with long-range interactions that incorporates a fixed percentage of stronger sites, or asperity cells, into the lattice. These asperity cells are significantly stronger than the surrounding lattice sites but eventually rupture when the applied stress reaches their higher threshold stress. The introduction of these spatial heterogeneities results in temporal clustering in the model that mimics that seen in natural fault systems along with GR scaling. In addition, we observe sequences of activity that start with a gradually accelerating number of larger events (foreshocks) prior to a main shock that is followed by a tail of decreasing activity (aftershocks). This work provides further evidence that the spatial and temporal patterns observed in natural seismicity are strongly influenced by the underlying physical properties and are not solely the result of a simple cascade mechanism.

New approach to Gutenberg-Richter scaling.

We introduce a new model for an earthquake fault system that is composed of noninteracting simple lattice models with different levels of damage denoted by q. The undamaged lattice models (q=0) have Gutenberg-Richter scaling with a cumulative exponent ß=1/2, whereas the damaged models do not have well defined scaling. However, if we consider the "fault system" consisting of all models, damaged and undamaged, we get excellent scaling with the exponent depending on the relative frequency with which faults with a particular amount of damage occur in the fault system. This paradigm combines the idea that Gutenberg-Richter scaling is associated with an underlying critical point with the notion that the structure of a fault system also affects the statistical distribution of earthquakes. In addition, it provides a framework in which the variation, from one tectonic region to another, of the scaling exponent, or b value, can be understood.

Ergodicity in natural earthquake fault networks.

Numerical simulations have shown that certain driven nonlinear systems can be characterized by mean-field statistical properties often associated with ergodic dynamics [C. D. Ferguson, W. Klein, and J. B. Rundle, Phys. Rev. E 60, 1359 (1999); D. Egolf, Science 287, 101 (2000)]. These driven mean-field threshold systems feature long-range interactions and can be treated as equilibriumlike systems with statistically stationary dynamics over long time intervals. Recently the equilibrium property of ergodicity was identified in an earthquake fault system, a natural driven threshold system, by means of the Thirumalai-Mountain (TM) fluctuation metric developed in the study of diffusive systems [K. F. Tiampo, J. B. Rundle, W. Klein, J. S. Sá Martins, and C. D. Ferguson, Phys. Rev. Lett. 91, 238501 (2003)]. We analyze the seismicity of three naturally occurring earthquake fault networks from a variety of tectonic settings in an attempt to investigate the range of applicability of effective ergodicity, using the TM metric and other related statistics. Results suggest that, once variations in the catalog data resulting from technical and network issues are accounted for, all of these natural earthquake systems display stationary periods of metastable equilibrium and effective ergodicity that are disrupted by large events. We conclude that a constant rate of events is an important prerequisite for these periods of punctuated ergodicity and that, while the level of temporal variability in the spatial statistics is the controlling factor in the ergodic behavior of seismic networks, no single statistic is sufficient to ensure quantification of ergodicity. Ergodicity in this application not only requires that the system be stationary for these networks at the applicable spatial and temporal scales, but also implies that they are in a state of metastable equilibrium, one in which the ensemble averages can be substituted for temporal averages in studying their spatiotemporal evolution.

A simulation-based approach to forecasting the next great San Francisco earthquake.

In 1906 the great San Francisco earthquake and fire destroyed much of the city. As we approach the 100-year anniversary of that event, a critical concern is the hazard posed by another such earthquake. In this article, we examine the assumptions presently used to compute the probability of occurrence of these earthquakes. We also present the results of a numerical simulation of interacting faults on the San Andreas system. Called Virtual California, this simulation can be used to compute the times, locations, and magnitudes of simulated earthquakes on the San Andreas fault in the vicinity of San Francisco. Of particular importance are results for the statistical distribution of recurrence times between great earthquakes, results that are difficult or impossible to obtain from a purely field-based approach.

Ergodic dynamics in a natural threshold system.

Numerical simulations suggest that certain driven, dissipative mean-field threshold systems, including earthquake models, can be characterized by statistical properties often associated with ergodic dynamics, in the same sense as stochastic Brownian motion. We applied a fluctuation metric proposed by Thirumalai and Mountain [Phys. Rev. E 47, 479 (1993)]] for statistically stationary systems and find that the natural earthquake fault system in California demonstrates similar ergodic dynamics.

Self-organization in leaky threshold systems: the influence of near-mean field dynamics and its implications for earthquakes, neurobiology, and forecasting.

Threshold systems are known to be some of the most important nonlinear self-organizing systems in nature, including networks of earthquake faults, neural networks, superconductors and semiconductors, and the World Wide Web, as well as political, social, and ecological systems. All of these systems have dynamics that are strongly correlated in space and time, and all typically display a multiplicity of spatial and temporal scales. Here we discuss the physics of self-organization in earthquake threshold systems at two distinct scales: (i) The "microscopic" laboratory scale, in which consideration of results from simulations leads to dynamical equations that can be used to derive the results obtained from sliding friction experiments, and (ii) the "macroscopic" earthquake fault-system scale, in which the physics of strongly correlated earthquake fault systems can be understood by using time-dependent state vectors defined in a Hilbert space of eigenstates, similar in many respects to the mathematics of quantum mechanics. In all of these systems, long-range interactions induce the existence of locally ergodic dynamics. The existence of dissipative effects leads to the appearance of a "leaky threshold" dynamics, equivalent to a new scaling field that controls the size of nucleation events relative to the size of background fluctuations. At the macroscopic earthquake fault-system scale, these ideas show considerable promise as a means of forecasting future earthquake activity.

Nonlinear network dynamics on earthquake fault systems.

Earthquake faults occur in interacting networks having emergent space-time modes of behavior not displayed by isolated faults. Using simulations of the major faults in southern California, we find that the physics depends on the elastic interactions among the faults defined by network topology, as well as on the nonlinear physics of stress dissipation arising from friction on the faults. Our results have broad applications to other leaky threshold systems such as integrate-and-fire neural networks.