Surface-wave ray tracing equations and Fermat's principle in an anisotropic earth

Abstract

Summary. Ray tracing equations for surface waves in an anisotropic earth are derived in two ways: first, from the Hamilton’s canonical equations, and secondly, from Fermat’s principle. Phase velocity, including its azimuthal variation, is required to solve the equations, but group velocity is eliminated from the equations. The difference of direction between the wave vector and the ray path is one of the features of wave propagation in an anisotropic media and the equations explicitly show dependence upon such an angle. By putting that angle to be zero, ray tracing equations in a transversely isotropic (or simply isotropic) medium are obtained. In an isotropic medium, phase traveltime, which is an integration of phase slowness along the ray path, is stationary. In an anisotropic medium, phase travel-times is not stationary. Instead, phase slowness projected onto the ray path and integrated along the ray path is stationary. Ray tracing by the bending method in an anisotropic media should utilize such a stationary quantity. In a weakly anisotropic medium, however, the angle ($a) between the wave vector and the ray path is small and cos $, - 1 up to first order in $a. Thus phase traveltime is approximately stationary in a weakly anisotropic medium.