Number of Aftershocks in Epidemic-Type Seismicity Models and in Reality

The widely used ETAS seismicity model describes the clustering of seismic events as an epidemic-type process (property A), assuming that the F1 distribution of the number of direct aftershocks is Poissonian (property B). The real data favor the geometric distribution F1 (e.g. Shebalin et al., 2018, Dokl. Akad. Nauk, 481 (3), 963–966). The F2 distribution of the number of cluster events with main shock m and magnitudes greater than m-Δ is also often attributed to geometric type. However, the coincidence of distribution types F1 and F2 turns out to be in contradiction with the A-property, and the geometric type F1 is in contradiction with the B-property. This study, which is analyzing and resolving the described contradictions, develops in the following three steps.

Step 1- Generalization of the ETAS model, designed to use any F1 distribution, and selection of special class of F1 distributions, including both Poisson and geometric distributions. The class of F1 models is united by a common property inherent in the Poisson distribution: the number of events with F1 distribution at random thinning of sample elements changes the mean, but retains the F1 type. This requirement is relevant because of errors in real clusters identification and because of the ambiguity in the choice of the representativeness magnitude threshold.

Step 2- Instead of the F2 distribution, we consider its more natural analog F2a. It refers to cluster events with magnitude greater than m_a-Δ_a , where m_a is the mode in the theoretical distribution of the strongest aftershock with main shock m. Under the conditions of Bath's law, namely m_a = m -1.2, both distributions coincide if  Δ_a = Δ-1.2. The limiting distribution of F2a is found for clusters with sufficiently strong main shock, m>>1. Remarkably, in the subcritical regime its type coincides with the type of F1, and the distribution itself depends only on the relative threshold Δ_a. In practice, this asymptotic result applies to the magnitude range where we expect self-similarity in seismicity.

Step 3- Comparison of the limit distributions of F2a corresponding to the geometric distribution of F1, different Δ_a and b-value b=1, with its real analogs obtained from the global ANSS catalog for large events with magnitude m>6. In the absence of any fitting, we obtained surprisingly good agreement of the distributions within the principal values (0-0.95) of the theoretical limit F2a distribution.

Thus, the important structural A-property of the ETAS model and the consistency of the choice of the geometric distribution F1 in its generalization are confirmed. This justifies the use of such a model to simulate seismicity. The complete mathematical analysis of the limiting distribution F2a, performed for the first time even for the traditional ETAS model, is of independent interest (for further details see Molchan and Peresan, 2024, Geophys. J. Int. 239, 314-328 and references therein).