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Walden Consulting
2 Wynn Road
Nashua, NH
03062
(M.V.M)
U.S. Geological Survey
345 Middlefield Road
Menlo Park,
CA 94025
(W.L.E., P.A.R)
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 = 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
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.
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