Renormalization of the ETAS branching model of triggered seismicity from total to observable seismicity

Abstract

Several recent works point out that the crowd of small unobservable earthquakes (with magnitudes below the detection threshold $m_d$) may play a significant and perhaps dominant role in triggering future seismicity. Using the ETAS branching model of triggered seismicity, we apply the formalism of generating probability functions to investigate how the statistical properties of observable earthquakes differ from the statistics of all events. The ETAS (epidemic-type aftershock sequence) model assumes that each earthquake can trigger other earthquakes (``aftershocks''). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. The triggering efficiency of earthquakes is assumed to vanish below a lower magnitude limit $m_0$, in order to ensure the convergence of the theory and may reflect the physics of state-and-velocity frictional rupture. We show that, to a good approximation, the ETAS model is renormalized onto itself under what amounts to a decimation procedure $m_0\to m_d$, with just a renormalization of the branching ratio from $n$ to an effective value $n(m_d)$. Our present analysis thus confirms, for the full statistical properties, the results obtained previously by one of us and Werner, based solely on the average seismic rates (the first-order moment of the statistics). However, our analysis also demonstrates that this renormalization is not exact, as there are small corrections which can be systematically calculated, in terms of additional contributions that can be mapped onto a different branching model (a new relevant direction in the language of the renormalization group).

41 pages including 8 figures

References in corpus (6)

• Scale free networks of earthquakes and aftershocks
• Apparent Clustering and Apparent Background Earthquakes Biased by Undetected Seismicity
• Constraints on the Size of the Smallest Triggering Earthquake from the ETAS Model, Baath's Law, and Observed Aftershock Sequences
• Magnitude-Dependent Omori Law: Empirical Study and Theory
• Vere-Jones' Self-Similar Branching Model
• Power law distribution of seismic rates: theory and data

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• Revisiting Seismicity Criticality: A New Framework for Bias Correction of Statistical Seismology Model Calibrations
• High Resolution Long- and Short-Term Earthquake Forecasts for California
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