Reflection and transmission coefficients of spherical waves at an interface separating two dissimilar viscoelastic solids

Spherical wave reflection and transmission (R/T) coefficients at an interface are not only of theoretical significance but also play an important role in the amplitude variation with offset (AVO) analysis of wide-angle reflection seismic data and cross-borehole surveys. For sources close to the interface the resulting wavefields cannot be adequately described in terms of a single incident plane wave. Rather, the spherical waves must be viewed as the superposition of an infinite number of plane waves. Moreover, the R/T coefficients for each individual plane wave in viscoelastic media have proven to be more complicated than expected due to the difficulty in selecting the correct vertical slowness. In such attenuating media the R/T coefficients cannot be properly determined by simply replacing the real elastic parameters with their complex viscoelastic counterparts. In this study, the reflection and transmission coefficients of spherical waves at a plane interface separating two dissimilar viscoelastic solids are rigorously investigated. The difficulty in selecting the vertical slowness is shown to be circumvented if the spherical wavefields are calculated from the plane wavefields using the Sommerfeld integral appropriate for the dissipative materials. However, some resulting phase curves of the complex spherical wave R/T coefficients tend to be of opposite sign to the corresponding phase curves of plane waves due to non-uniqueness of the latter for post-critical wave incidence. In this contribution we propose a new definition of spherical wave R/T coefficients for viscoelastic media which differs from the conventional one. Its advantages are that it is not explicitly expressed as a function of the R/T angles, it is valid for both P and S waves, yet it is consistent with the existing definitions of spherical wave R/T coefficients but is more robust. By way of examples we compute both spherical wave reflection coefficients (SWRC) and spherical wave transmission coefficients (SWTC) for two different viscoelastic models. Unlike plane waves, both the SWRC and the SWTC of converted PS waves are found to be non-zero at vertical incidence and may be drastically affected by the existence of longitudinal PS waves which are confirmed by full waveform calculations for the converted PS waves.