Interpretation of the Omori Law

The known Omori law is presented in the form of differential equation that describes the evolution of the aftershock activity. This equation is derived hypothetically with taking into account deactivation of the faults in epicentral zone of the main shock. A generalization of the Omori law is proposed.

💡 Research Summary

This paper, “Interpretation of the Omori Law,” by Anatol V. Guglielmi, offers a novel physical reinterpretation and a mathematical generalization of the classic empirical law describing the decay of aftershock activity following a main earthquake.

The author begins by reframing the standard hyperbolic Omori formula, n(t) = k/(c+t), not merely as an empirical fit but as the solution to a specific differential equation. He demonstrates that this decay law is mathematically equivalent to the solution of a first-order, nonlinear differential equation: dn/dt + σn² = 0, where σ is a constant. This reformulation is crucial because it shifts the perspective from a static descriptive formula to a dynamic equation governing the “evolution” of aftershock activity.

The core intellectual contribution lies in the physical interpretation of this differential equation. Guglielmi draws an insightful analogy with a well-known process in ionospheric physics: the radiative recombination of oppositely charged particles (e.g., O₂⁺ and e⁻), where the loss rate of charge density is proportional to the product of the densities (i.e., ~n²). To translate this analogy to the Earth’s crust, the author proposes a conceptual counterpart to the “pair of charges.” He suggests that the two adjacent sides of a fault under shear stress constitute an “active fault” (symbolically ↭), which has the potential to rupture and generate an aftershock. The interaction and stress release between these sides, leading to a “deactivated” or passive fault (||), is hypothesized to be the seismic equivalent of the recombination process. Consequently, the decay of aftershock activity is interpreted as the macroscopic manifestation of the collective deactivation of numerous small-scale active faults within the main shock’s epicentral zone.

Building on this interpretive model, the paper naturally progresses to a generalization of the Omori law. The author argues that the deactivation rate coefficient σ is unlikely to be constant in a complex, relaxing earthquake source. By allowing σ to be a function of time, σ(t), the differential equation yields a more general solution: n(t) = n₀ /