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The clock advance description of the Epidemic Type Aftershock Sequence (ETAS) model

Matthias Holschneider

TL;DR

The paper addresses the limitation of the Epidemic Type Aftershock Sequence (ETAS) model in incorporating microseismicity due to infinite counts at small magnitudes. It introduces a clock-advance interpretation, recasting triggering as time-deformed, independent background processes and formulating ETAS as an infinite superposition of Levy jump processes with productivity Y=F(m). A dyadic, band-limited construction yields a convergent, finite mean productivity D=E(Y), and the critical condition D<1 guarantees almost-sure convergence, enabling microseismicity to be modeled without a lower magnitude cutoff. The framework extends to space by additively incorporating a spatial kernel and a Markov-chain formulation in infinite dimensions, providing a principled basis for estimating ETAS parameters across scales. This approach removes artificial cutoffs, supports cross-scale seismology, and lays the groundwork for spatial-temporal modeling of microseismicity using Levy-process tools.

Abstract

In this short paper we propose to extend the ETAS model to micro-seismic events. For that we interpret the triggered events in an ETAS model as individual local clock advances of an independent background process. The solution of the ETAS model thus becomes the sum of an infinite Markov chain of independent time adjusted background processses. This allows the incorporation of events at all scales. No artificial small magnitude cutoff is needed.

The clock advance description of the Epidemic Type Aftershock Sequence (ETAS) model

TL;DR

The paper addresses the limitation of the Epidemic Type Aftershock Sequence (ETAS) model in incorporating microseismicity due to infinite counts at small magnitudes. It introduces a clock-advance interpretation, recasting triggering as time-deformed, independent background processes and formulating ETAS as an infinite superposition of Levy jump processes with productivity Y=F(m). A dyadic, band-limited construction yields a convergent, finite mean productivity D=E(Y), and the critical condition D<1 guarantees almost-sure convergence, enabling microseismicity to be modeled without a lower magnitude cutoff. The framework extends to space by additively incorporating a spatial kernel and a Markov-chain formulation in infinite dimensions, providing a principled basis for estimating ETAS parameters across scales. This approach removes artificial cutoffs, supports cross-scale seismology, and lays the groundwork for spatial-temporal modeling of microseismicity using Levy-process tools.

Abstract

In this short paper we propose to extend the ETAS model to micro-seismic events. For that we interpret the triggered events in an ETAS model as individual local clock advances of an independent background process. The solution of the ETAS model thus becomes the sum of an infinite Markov chain of independent time adjusted background processses. This allows the incorporation of events at all scales. No artificial small magnitude cutoff is needed.
Paper Structure (7 sections, 72 equations, 3 figures)

This paper contains 7 sections, 72 equations, 3 figures.

Table of Contents

  1. The ETAS model
  2. The ETAS model in moment space
  3. The clock advance picture
  4. A Markov chain in infinite dimensional state space
  5. Extension to spatial processes
  6. Conclusion and outlook
  7. Campbell's theorem

Figures (3)

  • Figure 1: The jump distribution can be cut into dyadic blocks.
  • Figure 2: In the upper figure is depicted a random sample the Levy path for the first three moment blocks. The blue line is an approximation to the limit by using only the first $3$ blocks.
  • Figure 3: The triggered events can be realized by modifying the local time of an independent background process. Here we illustrate this by an event of unit size $Y=1$. The clock modified catalog is displayed horizontally whereas the original background catalog is along the vertical axis. Only the events colored in read will be in the locally-clock-advanced catalog.