We propose that a controlling parameter of static stress drop during an earthquake is related to the scaling properties of the fault-surface topography. Using high resolution laser distance meters, we have accurately measured the roughness scaling properties of two fault surfaces in different geological settings (the French Alps and Nevada). The data show that fault-surface topography is scale dependent and may be accurately described by a self-affine geometry with a slight anisotropy characterized by two extreme roughness exponents (HR), H||=0.6 in the direction of slip and H⊥=0.8 perpendicular to slip.
Disregarding plastic processes like rock fragmentation and focusing on elastic deformation of the topography, which is the dominant mode at large scales, the stress drop is proportional to the deformation, which is a spatial derivative of the slip. The evolution of stress-drop fluctuations on the fault plane can be derived directly from the self-affine property of the fault surface, with the length scale (λ) as stdΔσ(λ)∝λHR-1.
Assuming no characteristic length scale in fault roughness and a rupture cascade model, we show that as the rupture grows, the average stress drop, and its variability should decrease with increasing source dimension. That is for the average stress drop Δσ(r)∝rHR-1, where r is the radius of a circular rupture. This result is a direct consequence of the elastic squeeze of fault asperities that induces the largest spatial fluctuations of the shear strength before and after the earthquake at local (small) scales with peculiar spatial correlations.