[Paper Review] Scaling law for seismic hazard after a main shock

Stefano Lise, Maya Paczuski+1 more|arXiv (Cornell University)|Mar 1, 2004

Earthquake Detection and Analysis3 references3 citations

TL;DR

This paper proposes a unified scaling law that integrates the Omori law for aftershock decay and the Gutenberg-Richter law for magnitude distribution, showing that the rate of aftershocks after a main shock depends universally on time, mainshock magnitude, and aftershock threshold. The key finding is a universal short-time rate independent of magnitude, validated by data collapse in Southern California, which challenges current hazard models that assume magnitude-dependent early aftershock rates.

ABSTRACT

After a large earthquake, the likelihood of successive strong aftershocks needs to be estimated. Exploiting similarities with critical phenomena, we introduce a scaling law for the decay in time following a main shock of the expected number of aftershocks greater than a certain magnitude. Empirical results that support our scaling hypothesis are obtained from analyzing the record of earthquakes in California. The proposed form unifies the well-known Omori and Gutenberg-Richter laws of seismicity, together with other phenomenological observations. Our results substantially modify presently employed estimates and may lead to an improved assessment of seismic hazard after a large earthquake.}

Motivation & Objective

• To unify the empirically observed Omori law for aftershock decay and the Gutenberg-Richter law for magnitude distribution into a single scaling framework.
• To test whether the rate of aftershocks greater than a threshold magnitude depends systematically on the magnitude of the main shock and the aftershock threshold.
• To provide a data-driven, empirically validated scaling hypothesis that improves upon existing seismic hazard forecasting models.
• To resolve inconsistencies in the reported values of the Omori law's parameter c by linking it to a universal time scale t₀ and magnitude-dependent scaling.
• To offer a new theoretical basis for understanding seismicity as a hierarchical, scale-invariant process governed by critical phenomena principles.

Proposed method

• Proposed a unified scaling form for the rate of aftershocks R_{M,m}(t), where M is the mainshock magnitude and m is the aftershock threshold magnitude.
• Used a rescaling of time t by A·10^{-bM} and rate R_{M,m}(t) by (t₀ + 10^{b(m−M)}·t) to test for data collapse onto a universal function G(x).
• Applied the data collapse method to Southern California earthquake catalog (1984–2002) with 3.5×10⁵ events, isolating mainshocks with 5.5 ≤ M ≤ 6.5 and aftershocks with 2.5 ≤ m ≤ M−1.5.
• Excluded sequences with secondary mainshocks within 4 months to avoid correlation effects, ensuring isolated aftershock sequences.
• Derived the hazard rate λ_{M,m}(t) by subtracting background seismicity R_m(∞) and a magnitude derivative term, providing a corrected forecast model.
• Used the scaling hypothesis to relate the Omori law parameter c to the universal time scale t₀ and the magnitude difference (M−m), explaining its variability.

Experimental results

Research questions

• RQ1 Can the Omori law for aftershock decay and the Gutenberg-Richter law for magnitude distribution be unified under a single scaling framework?
• RQ2 Is the initial rate of aftershocks after a main shock independent of the mainshock magnitude M and the aftershock threshold m, as predicted by the scaling hypothesis?
• RQ3 Does the observed data collapse onto a universal scaling function G(x) when time and rate are rescaled according to the proposed law?
• RQ4 How does the parameter c in the Omori law vary with magnitude threshold m, and can this be explained by the scaling model?
• RQ5 To what extent does the proposed model improve upon current seismic hazard forecasting methods that assume magnitude-dependent early aftershock rates?

Key findings

• The short-time aftershock rate R₀ ≈ 10⁻² sec⁻¹ is universal and independent of both mainshock magnitude M and aftershock threshold m, indicating a universal initial response to large quakes.
• Data collapse onto a single universal function G(x) was achieved using t₀ ≈ 6 sec (range 4–12 sec), confirming the scaling hypothesis with statistical significance.
• The duration of correlated aftershock activity scales as t_d ≈ (x₀/A)·10^{bM}, with x₀ ≈ 10⁻¹, implying about two months of excess activity after a magnitude 6+ mainshock.
• The Omori law parameter c is not constant but scales as c = t₀·10^{b(M−m)}, explaining its reported variability across studies and linking it to the universal time scale t₀.
• The derived hazard rate λ_{M,m}(t) differs fundamentally from current models: it predicts no magnitude dependence in the early rate, contradicting the 10^{b(M−m)} dependence used in Reasenberg’s model.
• The consistency check via R₀·c ≈ R_{M,m}(t_d)·t_d yields t₀ ≈ 10 sec, within the estimated uncertainty range, validating the self-consistency of the model.