“Zoeppritz” rationalized, and generalized to anisotropic media

This paper is meant to be a guide through a labyrinth of reflectivity and transmissivity at interfaces between anisotropic elastic media. The Zoeppritz equations for isotropic plane-wave reflection and transmission coefficients are cast in a form from which explicit elementary expressions for the reflectivity and transmissivity matrices are derived in terms of four submatrices of the Zoeppritz coefficient matrix, two submatrices associated with each of the two media. The two submatrices associated with a medium are also the building blocks for a simple construction of the propagator matrix for a layer of that medium. All expressions are applicable to anisotropic media, subject to the requirement that the media involved must be at least monoclinic with a mirror plane of symmetry parallel to the reflecting plane. The submatrices associated with a medium depend only on the elastic properties of that medium and are functions of the eigenvalues and eigenvectors of the Christoffel equations governing plane-wave propagation in the medium. The advantages of this approach to reflectivity are (a) heuristic and (b) it allows a programmer to further modularize computer codes needed for computation of reflection coefficients and propagator matrices in anisotropic elastic layered media programs. Some curious results concerning reflectivity between isotropic and orthorhombic media are presented.