The strongest aftershock in seismic models of epidemic type

Abstract: We consider an epidemic-type aftershock model (ETAS($F$)) for a large class of distributions $F$ determining the number of direct aftershocks. This class includes Poisson, Geometric, Negative Binomial distributions and many other. Assuming an exponential form of the productivity and magnitude laws, we find a limiting distribution of the strongest aftershock magnitude $m_a$ when the initial cluster event $m_0$ is large. The regime can be either subcritical or critical; the initial event can be dominant in size or not. In the subcritical regime, the mode of the limiting distribution is determined by the parameters of productivity and the magnitude laws; the shape of this distribution is not universal and is effectively determined by $F$. For example, the Geometric $F$-distribution generates the logistic law, and the Poisson distribution (studied earlier) generates the Gumbel type 1 law. The accuracy of these laws for moderate initial magnitudes is tested numerically. The limit distribution of the Bath's difference $m_0$-$m_a$ is independent of the initial event size only if the regime is critical, and the ratio of exponents in the laws of magnitude and productivity is contained in the interval $(1,2)$. Previous studies of the $m_a$-distribution have dealt with the traditional Poisson $F$ model and with arbitrary (not necessarily dominant) initial magnitude $m_0$.