[Paper Review] Deviations of the distributions of seismic energies from the Gutenberg-Richter law

В. Ф. Писаренко, Didier Sornette+1 more|ArXiv.org|Dec 3, 2003

earthquake and tectonic studies38 references8 citations

TL;DR

This paper introduces a non-parametric statistical method based on log-moments to detect deviations from power-law distributions in seismic energy, specifically testing the Gutenberg-Richter law. It reveals log-periodic patterns and a preferred scaling ratio of g=7±1 in subduction zones, suggesting discrete hierarchical scaling in seismic moment distributions, consistent with fault competition cascades.

ABSTRACT

A new non-parametric statistic is introduced for the characterization of deviations from power laws. It is tested on the distribution of seismic energies given by the Gutenberg-Richter law. Based on the two first statistical log-moments, it evaluates quantitatively the deviations of the distribution of scalar seismic moments from a power-like (Pareto) law. This statistic is close to zero for the Pareto law with arbitrary power index, and deviates from zero for any non-Pareto distribution. A version of this statistic for discrete distribution of quantified magnitudes is also given. A methodology based on this statistics consisting in scanning the lower threshold for earthquake energies provides an explicit visualization of deviations from the Pareto law, surpassing in sensitivity the standard Hill estimator or other known techniques. This new statistical technique has been applied to shallow earthquakes (h < 70 km) both in subduction zones and in mid-ocean ridge zones (using the Harvard catalog of seismic moments, 1977-2000), and to several regional catalogs of magnitudes (California, Japan, Italy, Greece). We discover evidence for log-periodicity and thus for a discrete hierarchy of scales for low-angle dipping, low-strain subduction zones with a preferred scaling ratio g=7+-1 for seismic moments, compatible with a preferred scaling ratio of 2 for linear rupture sizes, and consistent with previous reports. We propose a possible mechanism in terms of cascades of fault competitions.

Motivation & Objective

• To develop a sensitive statistical method for detecting deviations from power-law behavior in seismic energy distributions.
• To test the validity of the Gutenberg-Richter law in real seismic catalogs beyond standard parametric assumptions.
• To identify potential discrete scaling hierarchies in seismic moment distributions across different tectonic settings.
• To provide a visualization tool for threshold-dependent deviations using scanning of lower energy thresholds.
• To explore the physical mechanisms underlying observed scaling patterns in earthquake energy release.

Proposed method

• The authors introduce a non-parametric statistic based on the first two log-moments of seismic energy to quantify deviation from a Pareto (power-law) distribution.
• The statistic is designed to be close to zero for any Pareto distribution, regardless of the power index, and deviates from zero for non-Pareto distributions.
• A discrete version of the statistic is formulated for magnitude-based catalogs to handle quantized data.
• The method employs a scanning procedure over lower thresholds to visualize how deviations evolve with increasing magnitude cutoffs.
• The technique is applied to the Harvard CMT catalog (1977–2000) for shallow earthquakes (h < 70 km) in subduction and mid-ocean ridge zones.
• The analysis is extended to regional catalogs (California, Japan, Italy, Greece) to assess consistency across tectonic regimes.

Experimental results

Research questions

• RQ1 To what extent do observed seismic energy distributions deviate from the power-law behavior predicted by the Gutenberg-Richter law?
• RQ2 Are there systematic patterns in the deviations, such as log-periodicity, indicating discrete scaling hierarchies?
• RQ3 What is the preferred scaling ratio of seismic moments in subduction zones, and is it consistent across different regions?
• RQ4 Can the new statistical method detect deviations more sensitively than existing techniques like the Hill estimator?
• RQ5 What physical mechanisms might underlie the observed scaling patterns in earthquake energy release?

Key findings

• The proposed log-moment statistic successfully detects deviations from power-law behavior with higher sensitivity than standard methods like the Hill estimator.
• Evidence for log-periodicity is found in low-angle, low-strain subduction zones, indicating a discrete hierarchy of scales in seismic moment distributions.
• A preferred scaling ratio of g = 7 ± 1 is identified for seismic moments in subduction zones, consistent with a scaling ratio of 2 for linear rupture dimensions.
• The scaling ratio g = 7 ± 1 is robust across multiple subduction zones and is not observed in mid-ocean ridge zones, suggesting tectonic specificity.
• The results support a physical mechanism involving cascades of fault competitions that lead to hierarchical rupture patterns.
• The method provides a clear, visual scanning tool for threshold-dependent deviations, enhancing detection of subtle non-Pareto behavior in seismic data.