Continuous reflection and transmission coefficient curves and ray tracing at an interface of viscoelastic anisotropic media based on complex Snell’s law

ABSTRACT The interaction of incident waves with interfaces in viscoelastic anisotropic media generates reflected and transmitted waves. Incorrect signs in square-root terms can lead to phase velocity errors or nonphysical jumps in reflection/transmission (R/T) coefficients. To address this, we introduce two square-root terms in the Christoffel matrices: the JKM term, with subterms J, K, and M (linked to the wave mode) and the BAC term, with subterms B, A, and C (related to the normal slowness component). Their signs are determined using a continuity criterion (CC) to avoid nonphysical behavior. Building upon this, we apply complex Snell’s law (CSL) to compute accurate R/T coefficients. Our analysis indicates that the BAC term can cause singularities in phase velocity under postcritical incidence. To clarify the differences between CSL and the approximate real slowness direction (RSD) method, we define three incidence angles: the critical angle (for pure elastic waves), the JKM branch-cut angle, and the BAC singular point, where the locus of the JKM or the BAC term crosses a branch cut in the complex plane. Numerical experiments indicate consistent results across all methods under precritical incidence. Under postcritical incidence, RSD can produce nonphysical jumps. Beyond the JKM branch-cut angle, CSL, using the principal value, also indicates discontinuities, whereas the CC maintains smooth physical curves. Thus, combining CSL with CC provides a more stable and accurate computational framework. Comparing the three ray velocities under CSL reveals that the energy velocity is more reliable than either the real part of the ray velocity or that computed using RSD. All ray velocities for scattered waves are calculated without using stationary slowness vectors, ensuring the continuity of real and imaginary tangential slowness components across the interfaces and, thus, the accurate R/T coefficients.