Renormalization of earthquake aftershocks

Assume that each earthquake can produce a series of aftershock independently of its size according to its “local” Omori's law with exponent 1 + θ . Each aftershock can itself trigger other aftershocks and so on. The global observable Omori's law is found to have two distinct power law regimes, the first one with exponent p − = 1‐ θ for time t < t* ∼ 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 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.