On anelliptic approximations for qP velocities in VTI media

A unified approach to approximating phase and group velocities of qP seismic waves in a transversely isotropic medium with vertical axis of symmetry (VTI) is developed. While the exact phase-velocity expressions involve four independent parameters to characterize the elastic medium, the proposed approximate expressions use only three parameters. This makes them more convenient for use in surface seismic experiments, where the estimation of all four parameters is problematic. The three-parameter phasevelocity approximation coincides with the previously published ‘acoustic’ approximation of Alkhalifah. The group-velocity approximation is new and noticeably more accurate than some of the previously published approximations. An application of the group-velocity approximation for finite-difference computation of traveltimes is shown. I N T R O D U C T I O N Anellipticity (deviation from an ellipse) is an important characteristic of elastic wave propagation. One of the simplest and yet practically important cases of anellipticity occurs in transversely isotropic media with a vertical axis of symmetry (VTI). In this type of medium, the phase velocities of qSH-waves and the corresponding wavefronts are elliptic, while the phase and group velocities of qPand qSV-waves may exhibit strong anellipticity (Tsvankin 2001). The exact expressions for the phase velocities of qPand 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 in 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 are 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 successful approximations have been previously developed (Byun, Corrigan and Gaiser 1989; Dellinger, Muir and Karrenbach 1993; Alkhalifah and Tsvankin 1995; Alkhalifah 1998, 2000b; Schoenberg and de Hoop 2000; Stopin 2001; Zhang and Uren 2001). A comparative review of different VTI approximations is provided by Fowler (2003). In this paper, an attempt is made to construct a unified approach to deriving anelliptic approximations. The starting point is the anelliptic approximation of Muir (Muir and Dellinger 1985; Dellinger et al. 1993). 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 behaviour. It can be applied to find more accurate approximations with non-linear 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; Sword 1987; de Bazelaire 1988; Castle 1994) and from the theory of Stolt stretch (Stolt 1978; Fomel and Vaillant 2001). It is shown here that applying this idea to approximate the phase velocity of qP-waves leads to the ‘acoustic’ approximation of ∗E-mail: [email protected] C © 2004 European Association of Geoscientists & Engineers 247 gpr413 GPR-xml.cls Apr l 14, 2004 12:48