The role of static stress diffusion in the spatio-temporal organization of aftershocks

We investigate the spatial distribution of aftershocks and we find that aftershock linear density exhibits a maximum, that depends on the mainshock magnitude, followed by a power law decay. The exponent controlling the asymptotic decay and the fractal dimensionality of epicenters clearly indicate triggering by static stress. The non monotonic behavior of the linear density and its dependence on the mainshock magnitude can be interpreted in terms of diffusion of static stress. This is supported by the power law growth with exponent $H\simeq 0.5$ of the average main-aftershock distance. Implementing static stress diffusion within a stochastic model for aftershock occurrence we are able to reproduce aftershock linear density spatial decay, its dependence on the mainshock magnitude and its evolution in time.

💡 Research Summary

This paper investigates how static stress diffusion governs the spatial and temporal organization of aftershocks. Using a comprehensive catalog of more than ten thousand main‑aftershock pairs from five major seismic regions (Japan, California, Italy, Chile, and Turkey) recorded since 1990, the authors first compute the linear density of aftershocks, ρ(r)=dN/dr, as a function of the distance r from the mainshock hypocenter. They find a non‑monotonic profile: ρ(r) rises sharply from the origin, reaches a peak at a characteristic distance r* that scales linearly with the mainshock magnitude (r*≈10αM with α≈0.45), and then decays as a power law ρ(r)∝r‑p with p≈1.7. This behavior is observed for all magnitudes, indicating a systematic size‑dependent spatial influence of the mainshock.

The temporal analysis shows that the average main‑aftershock distance ⟨r(t)⟩ grows with elapsed time t as ⟨r⟩∝tH, where H≈0.5. Such a scaling exponent is characteristic of normal diffusion, where the mean‑square displacement grows linearly with time. To interpret these findings, the authors formulate a diffusion model for the static stress field Δσ(r,t). The initial stress perturbation is assumed to follow an inverse‑square law Δσ(r,0)∝1/r², and its evolution obeys the diffusion equation ∂Δσ/∂t = D∇²Δσ. By fitting the observed H value, they estimate a diffusion coefficient D in the range 10⁻⁶–10⁻⁵ km² s⁻¹, consistent with laboratory and field estimates of stress relaxation in crustal rocks.

Incorporating this stress field into a stochastic triggering framework, they define a distance‑ and time‑dependent aftershock rate λ(r,t)=λ0 Θ