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Earthquakes on a specified fault (or fault segment) with magnitudes greater than a specified value have a statistical distribution of recurrence times. The mean recurrence time can be related to the rate of strain accumulation and the strength of the fault. Very few faults have a recorded history of earthquakes that define a distribution well. For hazard assessment, in general, a statistical distribution of recurrence times is assumed along with parameter values. Assumed distributions include the Weibull (stretched exponential) distribution, the lognormal distribution, and the Brownian passage-time (inverse Gaussian) distribution. The distribution of earthquake waiting times is the conditional probability that an earthquake will occur at a time in the future if it has not occurred for a specified time in the past. The distribution of waiting times is very sensitive to the distribution of recurrence times. An exponential distribution of recurrence times is Poissonian, so there is no memory of the last event. The distribution of recurrence times must be thinner than the exponential if the mean waiting time is to decrease as the time since the last earthquake increases. Neither the lognormal or the Brownian passage time distribution satisfies this requirement. We use the "Virtual California" model for earthquake occurrence on the San Andreas fault system to produce a synthetic distribution of earthquake recurrence times on various faults in the San Andreas system. We find that the synthetic data are well represented by Weibull distributions. We also show that the Weibull distribution follows from both damage mechanics and statistical physics. [ABSTRACT FROM AUTHOR]

Virtual California is a topologically realistic simulation of the interacting earthquake faults in California. Inputs to the model arise from field data, and typically include realistic fault system topologies, realistic long-term slip rates, and realistic frictional parameters. Outputs from the simulations include synthetic earthquake sequences and space-time patterns together with associated surface deformation and strain patterns that are similar to those seen in nature. Here we describe details of the data assimilation procedure we use to construct the fault model and to assign frictional properties. In addition, by analyzing the statistical physics of the simulations, we can show that that the frictional failure physics, which includes a simple representation of a dynamic stress intensity factor, leads to self-organization of the statistical dynamics, and produces empirical statistical distributions (probability density functions: PDFs) that characterize the activity. One type of distribution that can be constructed from empirical measurements of simulation data are PDFs for recurrence intervals on selected faults. Inputs to simulation dynamics are based on the use of time-averaged event-frequency data, and outputs include PDFs representing measurements of dynamical variability arising from fault interactions and space-time correlations. As a first step for productively using model-based methods for earthquake forecasting, we propose that simulations be used to generate the PDFs for recurrence intervals instead of the usual practice of basing the PDFs on standard forms (Gaussian, Log-Normal, Pareto, Brownian Passage Time, and so forth). Subsequent development of simulation-based methods should include model enhancement, data assimilation and data mining methods, and analysis techniques based on statistical physics. [ABSTRACT FROM AUTHOR]

Aftershock sequences present a unique opportunity to study the physics of earthquakes. Important questions concern the fundamental origin of three widely applicable scaling laws: (1) Gutenberg--Richter frequency--magnitude scaling, (2) Omori's law for aftershock decay rates, and (3) Båth's law for the difference between the magnitude of the largest aftershock and a mainshock. The high-resolution Parkfield seismic network provided the opportunity for detailed studies of the aftershock sequence following the 28 September 2004, M 6.0 Parkfield earthquake. In this article it is shown that aftershocks satisfy the Gutenberg--Richter scaling relation only for relatively large times after the mainshock. There is a systematic time delay for the establishment of this scaling law. The temporal evolution of the rates of occurrence of aftershocks is quantified using the generalized Omori's law. This scaling law contains two characteristic times c and τ. The analysis suggests that the parameter c plays the role of a characteristic time for the establishment of Gutenberg--Richter scaling. This time increases systematically with a decreasing lower magnitude cutoff. The systematic time delay is attributed to a cascade of energy from long wavelengths to short wavelengths. The parameter τ is a measure of the average time until the first aftershock occurs. We find that τ slightly varies with the lower magnitude cutoff of the sequence. We also note that the largest aftershock inferred from an extrapolation of Gutenberg--Richter scaling, M 5.0, is equal to the largest observed aftershock. This scaling associated with the universal applicability of Båth's law is attributed to a constant partitioning of energy between a mainshock and its associated aftershock sequence. We also give in this article the distribution of interoccurrence times between successive aftershocks. We show that this distribution is well approximated by a nonhomogeneous Poissons process driven by the modified Omori's law. The self-consistency between interoccurrence statistics and decay rates is taken as further evidence for the applicability of our studies. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]