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

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