Physics of Earthquake Aftershocks: Governing Laws and Application to Morocco

Aftershocks are smaller seismic events that follow a major earthquake and reflect the gradual redistribution of stress within the Earth’s crust. Their frequency, magnitude distribution, and temporal decay are commonly described by three foundational empirical relations: the Gutenberg–Richter law, Båth’s law, and the modified Omori law. This paper makes two contributions. First, it presents a unified analytical treatment of these laws by deriving an integrated form of the modified Omori law for the cumulative number of aftershocks. The result is obtained through a variable-separable first-order differential equation with integration carried out up to an arbitrary future time T following the mainshock, yielding closed-form expressions under different decay regimes. Second, the paper applies the model to the September 8, 2023 Mw 6.8 Morocco earthquake using aftershock data obtained from the European-Mediterranean Seismological Centre (EMSC). The catalog contains several hundred aftershocks recorded over nine months. Parameter estimates from the modified Omori formulation indicate a decay exponent p = 1.56 ± 0.02 and characteristic time c = 0.28 ± 0.07, suggesting a relatively rapid decline in aftershock activity over time. The empirical patterns are broadly consistent with established aftershock decay behavior. The analysis provides a structured framework for interpreting regional aftershock sequences and contributes to ongoing efforts to understand seismic hazard dynamics in tectonically active regions.