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.