Chapter 4 - Anisotropic Anelastic Media

Abstract

In isotropic media there are two well-defined relaxation functions, describing purely dilatational and shear deformations of the medium. The problem in anisotropic media is to obtain the time dependence of the relaxation components with a relatively reduced number of parameters. Fine layering has an “exact” description in the long-wavelength limit. The concept of eigenstrain allows us to reduce the number of relaxation functions to six; an alternative is to use four or two relaxation functions when the anisotropy is relatively weak. Fracture-induced anisotropic attenuation is studied, and harmonic quasi-static numerical experiments are designed to obtain the stiffness components of anisotropic anelastic media. The analysis of SH waves suffices to show that in anisotropic viscoelastic media, unlike the lossless case: the group-velocity vector is not equal to the energy-velocity vector, the wavevector is not normal to the energy-velocity surface, the energy-velocity vector is not normal to the slowness surface, etc. However, an energy analysis shows that some basic fundamental relations still hold: for instance, the projection of the energy velocity onto the propagation direction is equal to the magnitude of the phase velocity. The analysis is extended to qP–qS wave propagation and expressions of the wave velocities, wave surfaces and quality factors are given. It is also shown how to implement the memory-variable approach to recast the equation for motion in full differential form.