Fractional Models in Seismology: Dynamics, Statistics and Predictions

Dear Colleagues, 

The active study and modeling of aftershock series are carried out in many world centers. Based on the phenomenological laws (of Omori‒Utsu, Gutenberg‒Richter, Grassberger, etc.), as seismology cornerstones, the classic ETAS and Utsu‒Ogata models were derived, whose modifications are actively "tested for strength", using current information on the observation data. The seismic data bank is growing and the total power of computing tools involved in the problem of predictions is increasing, but the predictive efficiency, as recent events show, is still not high enough. Earth is too complex an object, the spread of space‒time scales is too large, and local (infinitesimal) operators of standard continuum mechanics no longer correspond to the scales of inhomogeneities, just as a medical scalpel does not correspond to the scales of ruptures in Earth's crust. The dynamics of such systems evolving through a sequence of catastrophic events requires a nonlocal theory for its description, the most developed chapter of which is the fractional analysis. The kernels of fractional operators are of the power type, and this explains their link with fractals—self-similar structures.  In the framework of probability theory, the self-similarity is realized in the family of the Levy-stable distributions, whose limit theorem forms the base of superstatistics, covering the main seismology phenomena. 

This Issue reveals subtle aspects of these concepts in their connection with seismological phenomenology.

Prof. Dr. Vladimir V. Uchaikin
Guest Editor

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