Elastic-wave velocities in anisotropic media of arbitrary symmetry-generalization of Thomsen's parameters epsilon, delta and gamma

We derive the explicit analytical expressions of the phase velocities of the three bulk waves (qP, qS(1) and qS(2)) propagating in an arbitrary direction in a homogeneous strongly anisotropic medium of arbitrary symmetry (triclinic). The mathematical formulation is concise and symmetrical with respect to all the wave types. A simple geometrical interpretation of the formulae is proposed. The conciseness of the formalism hides the great complexity of the solutions when they are explicitly expressed in terms of the elastic constants of the media and of the direction cosines of the unit propagation vector, which makes further analytical developments daunting. We use perturbation techniques to derive simplified approximate expressions of these solutions. In a continuation of the work of Thomsen (1986) in weakly transversely isotropic media, we extend the work of Sayers (1994) in arbitrary weakly anisotropic media, but restricted to qP waves, by deriving the complete directional dependence of the velocities of the three bulk waves (qP, qS(1) and qS(2)). Furthermore, contrary to the previous references, we do not need to assume that the anisotropy is weak since our model is based on the perturbation of a reference model which can exhibit strong S-wave birefringence. We do not use spherical harmonic decomposition, as Sayers (1994) did, but only elementary rotations of the components of the elastic tensor of the considered medium. The forms of the proposed solutions naturally introduce generalizations of Thomsen's classical parameters epsilon, delta and gamma, but they are relevant for arbitrary symmetry. Numerical comparisons between the exact and the approximate velocities using perturbation theory in transversely isotropic and triclinic media validate the proposed solutions, and show that the first-order approximations are reasonable even in the presence of a moderate amount of anisotropy.