HIGH-ORDER PERTURBATIONS OF THE PHASE VELOCITY AND POLARIZATION OF QP AND QS WAVES IN ANISOTROPIC MEDIA

Abstract

An approximate expression of the eikonal equation and the polarization vector may be obtained in weakly anisotropic media from first-order perturbation theory. The advantage of this approximation for qP wave is that the squared phase velocity is linear in the elastic parameters. For qS waves, the first-order approximation is more complicated and can be expressed in terms of the square root of a quadratic function in the elastic parameters. Higher order perturbations can be obtained by an iterative procedure which improves the accuracy of the approximations. Explicit analytic formulae of the approximate squared phase velocities and polarizations are given for orthorhombic and transversely isotropic symmetries. Numerical comparisons between the exact and the approximate phase velocities and polarization vectors obtained at different orders illustrate the accuracy of the approximate formulae presented. For realistic anisotropy, the second-order expressions of the squared phase velocities are accurate approximations which do not cost much more with respect to the first-order computations. Third order expressions of the squared phase velocities are very accurate and need only computation of the first-order approximations. Second order expressions should be used to have good approximations of polarization vectors outside the vicinity of singularities. Higher order approximations of the qS-waves eigenvectors should be applied in neighbouring directions of singularities.