Theory of earthquake recurrence times

Abstract

The statistics of recurrence times in broad areas have been reported to obey universal scaling laws, both for single homogeneous regions and when averaged over multiple regions. These unified scaling laws are characterized by intermediate power law asymptotics. On the other hand, Molchan (2005) has presented a mathematical proof that if such a universal law exists, it is necessarily an exponential, in obvious contradiction with the data. First, we generalize Molchan's argument to show that an approximate unified law can be found which is compatible with the empirical observations when incorporating the impact of the Omori-Utsu law of earthquake triggering. We then develop the theory of the statistics of interevent times in the framework of the Epidemic-Type Aftershock Sequence (ETAS) model of triggered seismicity and show that the empirical observations can be fully explained. Our theoretical expression well fits the empirical statistics over the whole range of recurrence times, accounting for different regimes by using only the physics of triggering quantified by the Omori-Utsu law. The description of the statistics of recurrence times over multiple regions requires an additional subtle statistical derivation that maps the fractal geometry of earthquake epicenters onto the distribution of the average seismic rates in multiple regions. This yields a prediction in excellent agreement with the empirical data for reasonable values of the fractal dimension d ≈ 1.8, the average clustering ratio n ≈ 0.9, and the productivity exponent α ≈ 0.9 times the b value of the Gutenberg-Richter law. Copyright 2007 by the American Geophysical Union.