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![]() | On anelliptic approximations for ![]() | ![]() |
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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
waves and the corresponding wavefronts are elliptic, while
the phase and group velocities of
and
waves may exhibit strong
anellipticity (Tsvankin, 2001).
The exact expressions for the phase velocities of
and
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
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
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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![]() | On anelliptic approximations for ![]() | ![]() |
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