Aftershock Production in Mean Field Avalanche Models with Static Fields

Abstract

Advanced seismic hazard assessment frameworks rely on stochastic models which include aftershock production in the form of branching or self-exciting point processes. Such empirical constructs are based on debated statistical laws observed across catalogs of natural seismicity, which lack a derivation from first principles. Here, we derive the statistics of aftershock production in a generalized mean-field model of avalanche dynamics with static random thresholds and bimodal relaxation. The number of direct aftershocks is statistically characterized as a renewal counting process accounting for Borel-distributed refractory intervals. At the large-number limit, the expected number of aftershocks is proportional to the size of the parent event with a characteristic scale linearly depending only on the branching parameter governing refractory intervals, whereas the variance follows a distinct parabolic dependence with the same parameter. This model provides a rationale for the overdispersion in aftershock production observed in field data with respect to the Poissonian offspring numbers of standard Hawkes models, but it cannot explain the ubiquity of self-similar aftershock production found in catalogs and lab experiments.