Parameterization of anisotropic media by A-parameters

Abstract

Most common parameterization of anisotropic media is by twenty one independent elements a_ijkl of the density-normalized stiffness tensor or by twenty one independent elements A_αβ of the density-normalized matrix of elastic parameters in the Voigt notation. These parameters are commonly of significantly different sizes, are dimensional, in (km/s)², often appear in combinations. We are offering an alternative parameterization by twenty one A-parameters (anisotropic parameters), which removes the mentioned disadvantages and possesses some additional useful properties. For example, axes or planes of coordinate systems, in which A-parameters are defined, need not be related to symmetry axes or planes of the considered anisotropy symmetry as required in other similar parameterizations. In combination with the first-order weak-anisotropy approximation, in which anisotropy is considered as the first-order perturbation of reference isotropy, parameterization by A-parameters yields insight into the role of individual A-parameters in the wave propagation problems. For example, it turns out that in the first-order weak-anisotropy approximation, P- and S-wave velocities are each controlled by fifteen A-parameters. A set of six of them appears only in the expression for P-wave velocity, a set of other six A-parameters appears only in S-waves velocity expressions. Remaining set of nine A-parameters is common for both waves. We present transformation of A-parameters, analogue to Bond transformation, and useful formulae for the weak-anisotropy approximation for anisotropy of any symmetry and arbitrary tilt.