We construct a probability model for rupture times on a recurrent earthquake source. Adding Brownian perturbations to steady tectonic loading produces a stochastic load-state process. Rupture is assumed to occur when this process reaches a critical-failure threshold. An earthquake relaxes the load state to a characteristic ground level and begins a new failure cycle. The load-state process is a Brownian relaxation oscillator. Intervals between events have a Brownian passage-time distribution that may serve as a temporal model for time-dependent, long-term seismic forecasting. This distribution has the following noteworthy properties: (1) the probability of immediate rerupture is zero; (2) the hazard rate increases steadily from zero at t ‫ס‬ 0 to a finite maximum near the mean recurrence time and then decreases asymptotically to a quasi-stationary level, in which the conditional probability of an event becomes time independent; and (3) the quasi-stationary failure rate is greater than, equal to, or less than the mean failure rate because the coefficient of variation is less than, equal to, or greater than In addition, the model provides 1/ 2 Ϸ 0.707. Ί expressions for the hazard rate and probability of rupture on faults for which only a bound can be placed on the time of the last rupture. The Brownian relaxation oscillator provides a connection between observable event times and a formal state variable that reflects the macromechanics of stress and strain accumulation. Analysis of this process reveals that the quasi-stationary distance to failure has a gamma distribution, and residual life has a related exponential distribution. It also enables calculation of "interaction" effects due to external perturbations to the state, such as stress-transfer effects from earthquakes outside the target source. The influence of interaction effects on recurrence times is transient and strongly dependent on when in the loading cycle step perturbations occur. Transient effects may be much stronger than would be predicted by the "clock change" method and characteristically decay inversely with elapsed time after the perturbation.

Abstract. The recent expansion of permanent Global Positioning System (GPS)networks provides crustal deformation data that are dense in both space and time. While considerable effort has been directed toward using these data for the determination of average crustal velocities, little attention has been given to detecting and estimating transient deformation signals. We introduce here a Network Inversion Filter for estimating the distribution of fault slip in space and time using data from such dense, frequently sampled geodetic networks. Fault slip is expanded in a spatial basis set B•(x) in which the coefficients are time varying, s(x, t) -EkM=i ½k(t)]3k(X). The temporal variation in fault slip is estimated nonparameterically by taking slip accelerations to be random Gaussian increments, so that fault slip is a sum of steady state and integrated random walk components. A state space model for the full geodetic network is formulated, and Kalman filtering methods are used for estimation. Variance parameters, including measurement errors, local benchmark motions, and temporal and spatial smoothing parameters, are estimated by maximum likelihood, which is computed by recursive filtering. Numerical simulations demonstrate that the Network Inversion Filter is capable of imaging fault slip transients, including propagating slip events. The Network Inversion Filter leads naturally to automated methods for detecting anomalous departures from steady state deformation.

A physically-motivated model for earthquake recurrence based on the Brownian relaxation oscillator is introduced. The renewal process defining this point process model can be described by the steady rise of a state variable from the ground state to failure threshold as modulated by Brownian motion. Failure times in this model follow the Brownian passage time (BPT) distribution, which is specified by the mean time to failure, µ, and the aperiodicity of the mean, α (equivalent to the familiar coefficient of variation). Analysis of 37 series of recurrent earthquakes, M-0.7 to 9.2, suggests a provisional generic value of α = 0.5. For this value of α, the hazard function (instantaneous failure rate of survivors) exceeds the mean rate for times > µ⁄2, and is ∼ ∼2 ⁄ µ for all times > µ. Application of this model to the next M 6 earthquake on the San Andreas fault at Parkfield, California suggests that the annual probability of the earthquake is between 1:10 and 1:13.

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