About the article by G. Molchan published in Geophys. J. Int.,239 (2024).
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 Poisson (property B). The real data favor the geometric distributions F1 (Shebalin et al., 2018). The F2 distribution of the number of all events in a cluster with relative magnitude (relative to the main shock) greater than – Δ is often also associated with 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.
The work is devoted to analyzing and resolving the described contradictions. It includes 3 steps:
Stage 1. Generalization of the ETAS model, designed to use any F1 distribution. Selection of a 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 real errors in cluster identification and because of the ambiguity of the choice of the representativeness threshold.
Stage 2 involves theoretical analysis of the F2a distribution for the number of events in the cluster. The relative magnitude (relative to the mode in the distribution of the strongest aftershock) of such events is greater than the – Δa threshold. Under the conditions of Bath’s law, Δa = Δ – 1.2. The limiting distribution of F2a is found for clusters with sufficiently strong main shock, m>>1. It turns out that in the subcritical regime its type coincides with the type of F1, and the distribution itself depends only on the threshold – Δa.
Stage 3: comparison of the limit distribution of F2a, corresponding to the geometric distribution of F1, with the real analogs of F2a 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 F2 distribution. (see Fig. 1 and Table 1).
Table 1. The slopes of the curves above the dashed horizontal in Fig. 1.
| slopes | threshold, Δ | ||||
| 0.2 | 0.5 | 0.8 | 1.0 | 1.5 | |
| Theory | 0.21 | 0.12 | 0.06 | 0.04 | 0.015 |
| Fig. 1, (i) | 0.22 | 0.10 | 0.05 | 0.03 | 0.015 |
| (ii) | 0.22 | 0.10 | 0.04 | 0.03 | 0.015 |
| [Sh], (i) | 0.20 | 0.11 | 0.06 | ||
Notation: (i) slopes (under Ba°th’s law); (ii) slopes obtained from linear regression estimation of the mode in the distribution of the strongest aftershock; [Sh] slopes for clusters of main events m>6.5 according to [Shebalin et al., 2018].
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. The complete mathematical analysis of the limiting distribution F2a , performed for the first time even for the traditional ETAS model, is of independent interest.
Источник Molchan G., Peresan A., Number of Aftershocks in Epidemic-type Seismicity Models. Geophys. J. Int. (2024) 239, 314–328, DOI: 10.1093/gji/ggae261
Примыкающие работы:
Molchan G. and E. Varini. The strongest aftershock in seismic models of epidemic type. Geophys. J. Int. (2024), 236, 1440–1454 DOI: 10.1093/gji/ggae001
Molchan G, E. Varini and A. Peresan, Productivity within the epidemic-type seismicity model. Geophys. J. Int. (2022), 231, 1545–1557 DOI: 10.1093/gjiggac269