Quasi-compressional group velocity approximations in a weakly anisotropic orthorhombic medium
P. F. Daley, Edward S. Krebes
Using modifications to the standard linearized approximation of the phase velocity for quasi-compressional (qP) wave propagation in a weakly anisotropic orthorhombic medium, two approximate eikonal equations are constructed. Corresponding expressions for the group velocities are then derived. In the first approximation, the degenerate (ellipsoidal) case of qP wave propagation in an orthorhombic medium is examined and an exact group velocity expression obtained, together with the exact expressions for the slowness vector components, for this simple case. This ellipsoidal group velocity is taken as the reference or background velocity surface and is employed as a trial solution in the first approximate eikonal equation, where the resultant group velocity surface is shown to be a perturbed ellipsoid. All formulae are in terms of angles related to the group velocity vector. As in the solution method used for the first approximate qP eikonal equation, the method of characteristics is employed in obtaining a group velocity approximate expression using another related linearized eikonal equation. The result is a more complex expression for the group velocity vector components. For completeness, analytic expressions for the exact components of the group velocity vector are presented. The group velocity expressions, approximate versus exact, are numerically compared for two orthorhombic anisotropic models that may be classified as weakly anisotropic or, possibly more accurately, weakly anellipsoidal, as the background group velocity is an ellipsoid. The extension of what is presented here to more complex anisotropic structures can be achieved in a similar manner.