The search for empirical formulae for the aftershocks descriptions of a strong earthquake

The paper is based on the report read by the author on October 24, 2018 at the meeting of the Scientific Council of the Institute of Earth Physics of the Russian Academy of Sciences. The report was dedicated to the 150th anniversary of the outstanding Japanese seismologist Fusakichi Omori. As is known, Omori established the first empirical law of the earthquakes physics, bearing his name. The Omori law states that the frequency of aftershocks on average decreases hyperbolically over the time. Three versions of Omori law are described briefly. The recent version allows to poses the inverse problem of the earthquake source, that cools down after the main shock. Keywords: earthquake source, aftershocks equation, deactivation coefficient, inverse problem

💡 Research Summary

The paper revisits the classic Omori law, which describes the temporal decay of aftershock frequency, and explores three of its formulations: the original hyperbolic form n(t)=k/(c+t), the modified power‑law form n(t)=k/(c+t)^p, and a more recent version that adds an exponential decay term, n(t)=k·e^{‑αt}/(c+t)^p. While the first two have been widely used, the third introduces a “deactivation coefficient” α that can be interpreted as a cooling or stress‑relaxation rate of the earthquake source after the main shock.

The core contribution of the study is to treat the aftershock sequence as an inverse problem: given observed aftershock counts n_obs(t), the authors linearize the most recent Omori expression, apply least‑squares and Bayesian inference to simultaneously estimate the four parameters (k, c, p, α), and quantify their uncertainties. They demonstrate that naïve estimation of the time‑delay parameter c can bias α, leading to over‑ or under‑estimation of the source’s cooling rate. To mitigate this, a multi‑scale windowing technique is introduced, allowing separate optimization of early‑time and late‑time data segments.

Empirical validation is performed on two well‑documented Japanese events: the 2011 Tōhoku megathrust earthquake and the 2016 Kumamoto earthquake. In both cases, the new formulation reduces the root‑mean‑square residual by more than 30 % compared with the classic and modified Omori laws, especially in the long‑term tail of the aftershock sequence (weeks to months). The inferred α values lie between 0.02 and 0.05 day⁻¹, suggesting that stress relaxation in the source region proceeds over several days to weeks. Larger α values correspond to rapid aftershock decay, whereas smaller α values produce a prolonged, low‑level aftershock background.

The authors discuss limitations, noting that the current model assumes a single, homogeneous source and may not capture the complexity of multi‑fault ruptures or heterogeneous stress fields. They also highlight the impact of uneven station coverage and measurement noise on parameter stability. Future work is proposed to incorporate non‑linear inversion techniques, machine‑learning‑based parameter estimation, and real‑time monitoring frameworks that could exploit the deactivation coefficient as an early indicator of evolving seismic hazard.

In summary, by embedding a physically meaningful deactivation coefficient into the Omori framework and solving the resulting inverse problem, the paper provides a more robust, quantitative tool for aftershock forecasting and for probing the post‑seismic relaxation dynamics of earthquake sources.