Continuous and elastically consistent viscoelastic reflection and transmission coefficient curves for plane inhomogeneous P and S waves

For inhomogeneous plane waves in dissipative media, correctly selecting the vertical slowness (q) is the key requirement to obtain physically acceptable reflection/transmission (R/T) coefficients (e.g. both continuous curves as a function of incidence angle, and consistency with the elastic coefficients at very low attenuation). Based on the fundamental branch-cut theory, the continuity criterion ensures the continuity of the R/T coefficient curves. The elastic consistence is obtained due to the definition of the mixed principal values and the consistent assumptions of time dependence of the harmonic plane wave, the Z-direction of the spatial coordinate system and the particle motion (polarization) vectors. The elastic consistence of R/T coefficients is checked by both elastic models with zero dissipation factor (1/Q) and the equal-Q model in which the Q values of the incident wave and the R/T waves are non-zero but equal for the P and S waves on both sides of the interface. In particular, the previously reported 180° phase difference between the viscoelastic coefficients with arbitrarily small attenuation but the same ratio of Q across the interface and the corresponding elastic coefficients is circumvented by introducing the mixed principal value. The continuity criterion is implemented directly in the complex vertical slowness squared (q2) plane in which the branch-cut is defined. Furthermore, the loci of q2 and several 'critical angles' defined for anelastic waves are clearly investigated and clarified. The loci of q2 are expressed as a function of the incidence angle and the incident wave inhomogeneity parameter D. The plots of wave inhomogeneity parameter (D and/or attenuation angle γ) versus incidence angle are calculated from the slowness vectors. The resultant D values, especially their equality for all vertically propagating R/T waves, are examined using the complex Snell's law to provide extra support to the correctness of the slowness vectors and thus of the R/T coefficients. It is worth noting that phase angle of the R/T waves from an inhomogeneous incident wave may exhibit rapid changes at vertical incidence because near-vertical reflection is still the dominant application of the exploration seismic method. To the best of our knowledge, this research is the first time it has been possible to obtain continuous and elastically consistent viscoelastic R/T coefficients.