Vere-Jones' Self-Similar Branching Model

TL;DR

This paper analyzes Vere-Jones' self-similar branching model for earthquake statistics, predicting a two-branched magnitude distribution and discussing implications for Earth's criticality based on empirical observations.

Contribution

It extends the Vere-Jones model by predicting the distribution of triggered event magnitudes and explores the model's implications for Earth's criticality and earthquake statistics.

Findings

Predicted two-branched magnitude distribution with exponents beta-h and beta+h.

Empirical absence of two-branched distributions suggests Earth is near criticality.

Full catalog magnitude distribution can be insensitive to triggered event exponents.

Abstract

Motivated by its potential application to earthquake statistics, we study the exactly self-similar branching process introduced recently by Vere-Jones, which extends the ETAS class of conditional branching point-processes of triggered seismicity. One of the main ingredient of Vere-Jones' model is that the power law distribution of magnitudes m' of daughters of first-generation of a mother of magnitude m has two branches m'<m with exponent beta-d and m'>m with exponent beta+d, where beta and d are two positive parameters. We predict that the distribution of magnitudes of events triggered by a mother of magnitude $m$ over all generations has also two branches m'<m with exponent beta-h and m'>m with exponent beta+h, with h= d \sqrt{1-s}, where s is the fraction of triggered events. This corresponds to a renormalization of the exponent d into h by the hierarchy of successive generations of triggered events. The empirical absence of such two-branched distributions implies, if this model is seriously considered, that the earth is close to criticality (s close to 1) so that beta - h \approx \beta + h \approx \beta. We also find that, for a significant part of the parameter space, the distribution of magnitudes over a full catalog summed over an average steady flow of spontaneous sources (immigrants) reproduces the distribution of the spontaneous sources and is blind to the exponents beta, d of the distribution of triggered events.