Scattering attenuation and the fractal geometry of fracture systems

Abstract

Scattering of seismic waves can be shown to have a frequency dependenceQ ⁻¹ ∝ ω^3−v if scattering is produced by arrays of inhomogeneities with a 3D power spectrumW _3D(k) ∝k ^−v. In the earth's crust and upper mantle the total attenuation is often dominated by scattering rather than intrinsic absorption, and is found to be frequency dependent according toQ ⁻¹ ∝ ω^γ, where −1<γ≤−0.5. IfD ₁ is the fractal dimension of the surface of the 3D inhomogeneities measured on a 2D section, then this corresponds respectively to 1.5<D ₁≤1.75, since it can be shown that γ=2(D ₁−2). Laboratory results show that such a distribution of inhomogeneities, if due to microcracking, can be produced only at low stress intensities and slow crack velocities controlled by stress corrosion reactions. Thus it is likely that the earth's brittle crust is pervaded by tensile microcracks, at least partially filled by a chemically active fluid, and preferentially aligned parallel to the maximum principal compressive stress. The possibility of stress corrosion implies that microcracks may grow under conditions which are very sensitive to pre-existing heterogeneities in material constants, and hence it may be difficult in practice to separate the relative contribution of crack-induced heterogeneity from more permanent geological heterogeneities.

By constrast, shear faults formed by dynamic rupture at critical stress intensities produceD ₁=1, consistent with a dynamic rupture criterion for a power law distribution of fault lengths with negative exponentD. The results presented here suggest empirically thatD ₁∼-1/2(D+1), thereby providing the basis for a possible framework to unify the interpretation of temporal variations in seismicb-value (b∼-D/2) and the frequency dependence of scattering attenuation (γ).