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Renormalization of earthquake aftershocks

Together with the Gutenberg-Richter distribution of earthquake magnitudes, Omori&#39;s law is the best established empirical characterization of earthquake sequences and states that the number of smaller earthquakes per unit time triggered by a main shock decays approximately as the inverse of the time ($1/t^p$, with $p \approx 1$) since the main shock. Based on these observations, we explore the theoretical hypothesis in which each earthquake can produce a series of aftershock independently of its size according to its ``local&#39;&#39; Omori&#39;s law with exponent $p=1+θ$. In this scenario, an aftershock of the main shock produces itself other aftershocks which themselves produce aftershocks, and so on. The global observable Omori&#39;s law is found to have two distinct power law regimes, the first one with exponent $p_-=1 - θ$ for time $t < t^* \sim κ^{-1/θ}$, where $0<1-κ<1$ measures the fraction of triggered earthquakes per triggering earthquake, and the second one with exponent $p_+=1 + θ$ for larger times. The existence of these two regimes rationalizes the observation of Kisslinger and Jones [1991] that the Omori&#39;s exponent $p$ seems positively correlated to the surface heat flow: a higher heat flow is a signature of a higher crustal temperature, which leads to larger strain relaxation by creep, corresponding to fewer events triggered per earthquake, i.e. to a larger $κ$, and thus to a smaller $t^*$, leading to an effective measured exponent more heavily weighted toward $p_+>1$.