Abstract
SUMMARY 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) angle.
Highlights
Earth media are generally both attenuating and anisotropic, and conventional elastic isotropic approximations prove to be inadequate for fully exploiting the information contained in modern seismic data (Tsvankin & Thomsen 1994; Carcione 2007)
The underlying mathematics is well understood, the problem per se must be regarded as unresolved (Nechtschein & Hron 1996; Cerveny & Psencık 2005; Ruud 2006; Krebes & Daley 2007). This is primarily due to the ambiguities related to the signs of the complex-valued square roots involved in the expression of the vertical slownesses, which result in a set of mathematically correct but physically unreasonable planewave reflection coefficients
The result obtained by using the standard sign convention again yields seemingly non-physical effects beyond the equivalent elastic critical (EEC) angle, whereas following the Riemann-surface criterion and choosing the negative solution for the vertical slowness of the transmitted P-wave, as indicated in Fig. 5, leads to reasonable changes compared with the isotropic case and an analytical solution that is consistent with its numerical counterpart (Fig. 8)
Summary
Earth media are generally both attenuating and anisotropic, and conventional elastic isotropic approximations prove to be inadequate for fully exploiting the information contained in modern seismic data (Tsvankin & Thomsen 1994; Carcione 2007). The underlying mathematics is well understood, the problem per se must be regarded as unresolved (Nechtschein & Hron 1996; Cerveny & Psenc ́ık 2005; Ruud 2006; Krebes & Daley 2007) This is primarily due to the ambiguities related to the signs of the complex-valued square roots involved in the expression of the vertical slownesses, which result in a set of mathematically correct but physically unreasonable planewave reflection coefficients. In a recent study, Krebes & Daley (2007) compare SH-wave reflection coefficients of anelastic media with the elastic equivalent, localize the incidence angles where non-physical jumps or discontinuities occur and explore three different approaches for choosing the sign of the vertical slowness, which they apply to the P-SV case. We explore and attempt to resolve the ambiguities present in the analytical solution on the basis of the physically correct solution obtained by numerical modelling
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