Effective attenuation anisotropy of thin-layered media

Abstract

One of the well-known factors responsible for the anisotropy of seismic attenuation is interbedding of thin attenuative layers with different properties. Here, we apply Backus averaging to obtain the complex stiffness matrix of an effective medium formed by an arbitrary number of anisotropic, attenuative constituents. Unless the intrinsic attenuation is uncommonly strong, the effective velocity function is controlled by the real-valued stiffnesses (i.e., independent of attenuation) and can be determined from the known equations for purely elastic media. Attenuation analysis is more complicated because the attenuation parameters are influenced by the coupling between the real and imaginary parts of the stiffness matrix. The main focus of this work is on effective transversely isotropic models with a vertical symmetry axis (VTI) that include isotropic and VTI constituents. Assuming that the stiffness contrasts, as well as the intrinsic velocity and attenuation anisotropy, are weak, we develop explicit first-order (linear) and second-order (quadratic) approximations for the attenuation-anisotropy parameters [Formula: see text], [Formula: see text], and [Formula: see text]. Whereas the first-order approximation for each parameter isgiven sim-ply by the volume-weighted average of its interval values, the second-order terms include coupling between various factors related to both heterogeneity and intrinsic anisotropy. Interestingly, the effective attenuation for P- and SV-waves is anisotropic even for a medium composed of isotropic layers with identical attenuation, provided there is a velocity variation among the constituent layers. Contrasts in the intrinsic attenuation, however, do not create attenuation anisotropy, unless they are accompanied by velocity contrasts. Extensive numerical testing shows that the second-order approximation for [Formula: see text], [Formula: see text], and [Formula: see text] is close to the exact solution for most plausible subsurface models. The accuracy of the first-order approximation depends on the magnitude of the quadratic terms, which is largely governed by the strength of the velocity (rather than attenuation) anisotropy and velocity contrasts. The effective attenuation parameters for multiconstituent VTI models vary within a wider range than do the velocity parameters, with almost equal probability of positive and negative values. If some of the constituents are azimuthally anisotropic with misaligned vertical symmetry planes, the effective velocity and attenuation functions may have different principal azimuthal directions or even different symmetries.