[Paper Review] Power law distribution of seismic rates: theory and data

А. И. Саичев, Didier Sornette|ArXiv.org|Dec 7, 2004

Geotechnical and Geomechanical Engineering19 citations

TL;DR

The paper proposes that the power-law tail in seismic rate distributions across space-time bins in Southern California arises from cascading earthquake triggering in the ETAS model, with a stable exponent μ ≈ 1.6. By incorporating pre-existing frozen heterogeneity in fault networks, the extended theory successfully predicts empirical distributions across time windows from 1 to 1000 days using parameters calibrated on the largest window, challenging Poisson-based forecasting and declustering methods.

ABSTRACT

We report an empirical determination of the probability density functions P(r) of the number r of earthquakes in finite space-time windows for the California catalog, over fixed spatial boxes 5 x 5 km^2 and time intervals dt =1, 10, 100 and 1000 days. We find a stable power law tail P(r) ~ 1/r^{1+mu} with exponent mu \approx 1.6 for all time intervals. These observations are explained by a simple stochastic branching process previously studied by many authors, the ETAS (epidemic-type aftershock sequence) model which 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. We develop the full theory in terms of generating functions for describing the space-time organization of earthquake sequences and develop several approximations to solve the equations. The calibration of the theory to the empirical observations shows that it is essential to augment the ETAS model by taking account of the pre-existing frozen heterogeneity of spontaneous earthquake sources. This seems natural in view of the complex multi-scale nature of fault networks, on which earthquakes nucleate. Our extended theory is able to account for the empirical observation satisfactorily. In particular, the adjustable parameters are determined by fitting the largest time window $dt=1000$ days and are then used as frozen in the formulas for other time scales, with very good agreement with the empirical data.

Motivation & Objective

• To explain the empirically observed power-law tail in seismic rate distributions across space-time bins in Southern California.
• To test whether the heavy-tailed distribution of seismic rates is intrinsic to triggered seismicity models rather than statistical artifacts.
• To improve the ETAS model by incorporating pre-existing frozen heterogeneity in spontaneous earthquake sources to better match empirical data.
• To challenge the use of Poisson statistics in seismic forecasting and declustering, which may misrepresent the true distribution of seismic rates.

Proposed method

• The authors use generating functions to model the space-time organization of earthquake sequences within the ETAS branching process framework.
• They develop approximations for the generating function and probability distribution, including a factorization approach for finite space windows.
• The theory incorporates a spatially heterogeneous, pre-existing distribution of spontaneous earthquake sources, modeled as a frozen stress/fault network.
• Numerical solutions of nonlinear integral equations are used to validate analytical approximations.
• The model is calibrated on the largest time window (dt = 1000 days), and the same parameters are used for smaller windows to test consistency.
• Theoretical predictions are compared with empirical data from the California seismic catalog across four time windows: 1, 10, 100, and 1000 days.

Experimental results

Research questions

• RQ1 Why does the empirical distribution of seismic rates in space-time bins exhibit a stable power-law tail with exponent μ ≈ 1.6 across multiple time scales?
• RQ2 Can the ETAS model of triggered seismicity explain the observed heavy-tailed distribution of seismic rates without relying on Poisson statistics?
• RQ3 How does the inclusion of pre-existing frozen heterogeneity in spontaneous earthquake sources improve the fit to empirical data?
• RQ4 To what extent do standard declustering and likelihood scoring methods based on Poisson statistics fail due to the intrinsic heavy-tailed nature of seismic rates?

Key findings

• The empirical probability density function of seismic rates in California exhibits a stable power-law tail P_data(r) ∼ 1/r^{1+μ} with exponent μ ≈ 1.6 across all time windows from 1 to 1000 days.
• The ETAS model with cascading aftershocks naturally generates a power-law tail in seismic rates, explaining the observed heavy-tailed distribution.
• Incorporating pre-existing frozen heterogeneity in spontaneous earthquake sources is essential to quantitatively match empirical data, especially for finite space-time windows.
• The model parameters calibrated on the 1000-day window successfully predict the distribution for shorter windows (1, 10, 100 days) without re-fitting, demonstrating robustness and scale invariance.
• The factorization approximation for finite domains provides a semi-quantitative match to exact numerical solutions, with the main discrepancy being a faster decay in the true distribution for large r.
• The findings imply that Poisson-based likelihood scores and declustering methods may be fundamentally flawed due to the intrinsic heavy-tailed nature of seismic rates.