The Paradox of the expected Time until the Next Earthquake

TL;DR

This paper analytically investigates how the expected time until the next earthquake depends on the elapsed time since the last one, revealing that this relationship varies with the statistical distribution of earthquake intervals.

Contribution

It provides a comprehensive analytical framework showing how different statistical models predict increasing or decreasing expected times to the next earthquake based on elapsed time.

Findings

Expected time decreases with elapsed time for certain distributions like Weibull with exponent > 1.

Expected time increases with elapsed time for distributions like log-normal and power-law.

Finite sampling from Poisson intervals can bias estimates, showing increasing expected times.

Abstract

We show analytically that the answer to the question, "The longer it has been since the last earthquake, the longer the expected time till the next ?" depends crucially on the statistics of the fluctuations in the interval times between earthquakes. The periodic, uniform, semigaussian, Rayleigh and truncated statistical distributions of interval times, as well as the Weibull distributions with exponent greater than 1, all have decreasing expected time to the next earthquake with increasing time since the last one, for long times since the last earthquake; the log-normal and power-law distributions and the Weibull distributions with exponents smaller than 1, have increasing times to the next earthquake as the elapsed time since the last increases, for long elapsed times. There is an identifiable crossover between these models, which is gauged by the rate of fall-off of the long-term tail of the distribution in comparison with an exponential fall-off. The response to the question for short elapsed times is also evaluated. The lognormal and power-law distributions give one response for short elapsed times and the opposite for long elapsed times. Even the sampling of a finite number of intervals from the Poisson distribution will lead to an increasing estimate of time to the next earthquake for increasing elapsed time since the last one.