Time-Dependent Renewal-Model Probabilities When Date of Last Earthquake is Unknown

We derive time-dependent, renewal-model earthquake probabilities for the case in which the date of the last event is completely unknown, and compare these with the time-independent Poisson probabilities that are customarily used as an approximation in this situation. For typical parameter values, the renewal-model probabilities exceed Poisson results by more than 10% when the forecast duration exceeds similar to 20% of the mean recurrence interval. We also derive probabilities for the case in which the last event is further constrained to have occurred before historical record keeping began (the historic open interval), which can only serve to increase earthquake probabilities for typically applied renewal models. We conclude that accounting for the historic open interval can improve long-term earthquake rupture forecasts for California and elsewhere.