Nonspherical wavefronts in anisotropic media are often assumed to be approximately elliptical. However, in transversely isotropic media only the wavefront of SH waves is always an oblate ellipsoid. The wavefront of SV waves is never an ellipsoid, and the wavefront of P waves is an oblate ellipsoid if and only if the expression (c 11 - c 44 ) (c 33 - c 44 ) - (C 13 + c 44 ) 2 vanishes. This cannot happen if the anisotropy is due to lamellation. The occurrence of elliptical wavefronts cannot be detected on the basis of surface observations of times alone, since the complete wave field can be transformed into one with spherical wavefronts everywhere by simple stretching of layers. All concepts and algorithms applicable to spherical wavefronts are applicable to elliptical wavefronts, in particular the determination of a velocity as the zero-offset limit of the stacking velocity. Since the resulting velocity is invariant under the stretching transformation, it can represent only a velocity that is itself an invariant, viz., the velocity at right angles to the direction of stretching. However, amplitude observations of SH waves give an independent indication of elliptical anisotropy. Representation of ellipsoid wavefronts allowas a simple determination of the ratio "zero-offset limit of stacking velocity/vertical velocity." Constraints on the parameters of the thin layers that constitute a lamellated medium can be translated into constraints of the above velocity ratio. For P waves this ratio is centered around unity for a wide range of constituent parameters.--Modified journal abstract.

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