Two approaches and the subsequent results of approximations for phase velocities in transversely isotropic (TI) media are examined in the context of what has been termed weak anisotropy. What is addressed is the range of anisotropic complexity from general or exact, to mild or weak anelliptic and finally to linearized weak and the limits within which the approximations can be assumed to be applicable. In what follows, the term weak anelliptic anisotropy will not be synonymous with linearized weak anisotropy. The phase (wavefront normal) velocity is the quantity selected for this study, as it is perhaps that which is most often chosen as a candidate for approximation, from which other relevant, associated quantities may be derived or computed - including group (ray) velocity, polarization vectors and intermediate values associated with amplitude computations, reflection and transmission coefficients. The quasi-compressional, qP , and quasi-shear, qS V , are approximated in such a manner that they are dependent on two parameters - ellipticity and anellipticity. Both of these quantities are physically realizable and measurable.
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