Weak elastic anisotropy by perturbation theory

Abstract

We demonstrate the advantages of adopting a wave-vector-based coordinate system (WCS) for the application of perturbation theory to derive and display approximate expressions for qP- and qS-wave polarization vectors, phase velocities, and group velocities in general weakly anisotropic media. The advantages stem from two important properties of the Christoffel equation when expressed in the WCS: (1) Each element of the Christoffel matrix is identical to a specific stiffness component in the WCS, and (2) the Christoffel matrix of an isotropic medium is diagonal in the WCS. Using these properties, one can easily identify the small components of the Christoffel matrix in the WCS for a weakly anisotropic medium. Approximate solutions to the Christoffel equation are then obtained by straightforward algebraic manipulations, which make our perturbation theory solution considerably simpler than previously published methods. We compare and contrast our solutions with those discussed by other workers. Numerical comparisons between the exact, first-order, and zero-order qS-wave polarization vectors illustrate the accuracy of our approximate formulas. The form of the WCS phase-velocity expressions facilitates the derivation of closed-form, first-order expressions for qP- and qS-wave group-velocity vectors, providing explicit formulas for the direction of propagation of seismic energy in general weakly anisotropic media. Numerical evaluation of our group-velocity expressions demonstrates their accuracy. We discuss problems with the approximate qS-wave group velocities and polarizations in neighboring directions of singularities. Standard methods are used to transform our solutions from the WCS to the acquisition coordinates, as illustrated by application to orthorhombic symmetry.