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Introduction

Anellipticity (deviation from ellipse) is an important characteristic of elastic wave propagation. One of the simplest and yet practically important cases of anellipticity occurs in transversally isotropic media with the vertical axis of symmetry (VTI). In this type of media, the phase velocities of $ qSH$ waves and the corresponding wavefronts are elliptic, while the phase and group velocities of $ qP$ and $ qSV$ waves may exhibit strong anellipticity (Tsvankin, 2001).

The exact expressions for the phase velocities of $ qP$ and $ qSV$ waves in VTI media involve four independent parameters. However, it has been observed that only three parameters influence wave propagation and are of interest to surface seismic methods (Alkhalifah and Tsvankin, 1995). Moreover, the exact expressions for the group velocities in terms of the group angle are difficult to obtain and too cumbersome for practical use. This explains the need for developing practical three-parameter approximations for both group and phase velocities in VTI media.

Numerous different successful approximations have been previously developed (Byun et al., 1989; Stopin, 2001; Dellinger et al., 1993; Alkhalifah, 2000b; Alkhalifah and Tsvankin, 1995; Zhang and Uren, 2001; Alkhalifah, 1998; Schoenberg and de Hoop, 2000). In this paper, I attempt to construct a unified approach for deriving anelliptic approximations.

The starting point is the anelliptic approximation of Muir (Dellinger et al., 1993; Muir and Dellinger, 1985). Although not the most accurate for immediate practical use, this approximation possesses remarkable theoretical properties. The Muir approximation correctly captures the linear part of anelliptic behavior. It can be applied to find more accurate approximations with nonlinear dependence on the anelliptic parameter. A particular way of ``unlinearizing'' the linear approximation is the shifted hyperbola approach, familiar from the isotropic approximations in vertically inhomogeneous media (Malovichko, 1978; de Bazelaire, 1988; Sword, 1987; Castle, 1994) and from the theory of Stolt stretch (Stolt, 1978; Fomel and Vaillant, 2001). I show that applying this idea to approximate the phase velocity of $ qP$ waves leads to the known ``acoustic'' approximation of Alkhalifah (1998,2000a), derived in a different way. Applying the same approach to approximate the group velocity of $ qP$ waves leads to a new remarkably accurate three-parameter approximation.

One practical use for the group velocity approximation is traveltime computations, required for Kirchhoff imaging and tomography. In the last part of the paper, I show examples of finite-difference traveltime computations utilizing the new approximation.


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Next: Exact expressions Up: On anelliptic approximations for Previous: On anelliptic approximations for

2013-03-02