Application of Snell’s law in weakly anisotropic media

Abstract

Snell’s law describes the relationship between phase angles and velocities during the reflection or transmission of waves. It states that horizontal slowness with respect to an interface is preserved during reflection or transmission. Evaluation of this relationship at an interface between two isotropic media is straightforward. For anisotropic media, it is a complicated problem because phase velocity depends on the angle; in the anisotropic reflection/transmission problem, neither is known. Solving Snell’s law in the anisotropic case requires a numerical solution for a sixth-order polynomial. In addition to finding the roots, they must be assigned to the correct reflected or transmitted wave type. We show that if the anisotropy is weak, an approximate solution based on first-order perturbation theory can be obtained. This approach permits the computation of the full slowness vector and, thereby, the phase velocity and angle. In addition to replacing the need for solving the sixth-order polynomial, the resulting expressions allow us to prescribe the desired reflected or transmitted wave type. The method is best implemented iteratively to increase accuracy. The result can be applied to anisotropic media with arbitrary symmetry. It converges toward the weak-anisotropy solution and provides overall good accuracy for media with weak to moderate anisotropy.