On the evaluation of plane-wave reflection coefficients in anelastic media

Analytical evaluations of the reflection coefficients in anelastic media inherently suffer from ambiguities related to the complex square roots contained in the expressions of the vertical slowness and polarization. This leads to a large number of mathematically correct but physically unreasonable solutions. To identify the physical solution, we compute full-waveform synthetic seismograms and use a frequency-slowness method for evaluating the amplitude and phase of the corresponding reflection coefficient. We perform this analysis for transversely isotropic media. The analytical solution space and its ambiguities are explored by analysing the paths along the Riemann surfaces associated with the square roots. This analysis allows us to choose the correct sign. Although this approach is generally effective, there are some cases that require an alternative solution, because the correct integration path for the vertical slowness does not exist on the corresponding Riemann surface. Closer inspection then shows that these ‘pathological' cases, which are essentially characterized by a higher-attenuation layer overlying a lower-attenuation layer, can readily be resolved through an appropriate change of direction on the Riemann sheet. The thus resulting recipe for the analytical evaluation of plane-wave reflection coefficients in anelastic media is conceptually simple and robust and provides correct solutions beyond the equivalent elastic critical (EEC) angl