On anelliptic approximations for qP velocities in TI and orthorhombic media

Introduction

Anellipticity is a well-known characteristic of elastic wave propagation in anisotropic media. The simplest, yet practically important case of anellipticity, occurs in transversely isotropic media (Grechka, 2009; Thomsen, 2014; Tsvankin, 2012). In recent years, it has been recognized that transverse isotropy may not be sufficient to characterize the actual media encountered in many regions of the world and as a result, orthorhombic anisotropy has become a significant topic of interest (e.g. Xu et al., 2005; Bakulin et al., 2000; Thomsen, 2014; Fowler et al., 2014; Tsvankin, 2012; Grechka, 2009; Vasconcelos and Tsvankin, 2006; Tsvankin, 1997). One important example of an orthorhombic medium is a sedimentary basin exhibiting parallel vertical cracks embedded in a background medium with vertical transverse isotropy (Grechka, 2009; Schoenberg and Helbig, 1997; Tsvankin, 2012,1997). In such media, three-dimensional anellipticity remains an important characteristic of elastic wave propagation.Tsvankin (2012,1997) pointed out that the elastic wave propagation in TI media resembles the elastic wave propagation in the symmetry plane of orthorhombic media. This observation enables an accurate description of orthorhombic anellipticity using only a limited number of parameters by extending the approach used to approximate anellipticity in TI media.

The exact expressions for qP phase and group velocities in TI media involve four independent parameters (Gassmann, 1964; Berryman, 1979). Alkhalifah and Tsvankin (1995) and Alkhalifah (1998) showed that three combinations of those four parameters are sufficient to describe qP wave propagation with high accuracy. Although the exact expression of phase velocity in terms of phase angle is known, the exact expression for group velocity in terms of group angle appears too complicated for practical use. Therefore, accurate approximations involving a small number of independent parameters are needed. In orthorhombic media, the exact expression for qP phase velocity can be derived as a solution of a cubic equation and involves nine parameters. However, only six combinations of those nine parameters are sufficient to accurately describe qP wave propagation (Tsvankin, 2012,1997). The exact expression of qP group velocity in orthorhombic media can be derived from phase velocity expressions, but this expression is again cumbersome and can only be expressed in terms of the phase angle instead of group angle. Therefore, this expression is not always convenient for practical applications such as ray tracing and moveout correction, where the expression in terms of the group angle (seismic ray direction) is often preferred.

Many approximations have been proposed previously for both phase and group velocities in TI media (e.g. Stopin, 2001; Farra and Pšencík, 2013; Fomel, 2004; Alkhalifah, 1998; Alkhalifah and Tsvankin, 1995; Ursin and Stovas, 2006; Schoenberg and de Hoop, 2000; Dellinger et al., 1993; Alkhalifah, 2000a; Fomel and Stovas, 2010; Stovas, 2010; Alkhalifah, 2000b; Zhang and Uren, 2001; Daley et al., 2004; Tsvankin, 1996). Fowler (2003) presented a comprehensive comparative review of many of these approximations. Accuracy comparison of several group-velocity approximations (in terms of moveout approximations) was also presented by Aleixo and Schleicher (2010) and Golikov and Stovas (2012). Among these different approaches, Fomel (2004) proposed an extension of the Muir-Dellinger approach (Muir and Dellinger, 1985; Dellinger et al., 1993) using the shifted-hyperbola functional form. The resultant three-parameter approximation for phase velocity is identical to the acoustic approximation of Alkhalifah (1998,2000a) and the empirical approximation of Stopin (2001). The corresponding three-parameter approximation for group velocity was new at the time and proved to be exceptionally accurate in comparison with other known approximations.

In the first part of this study, we revisit the anelliptic approximations by Fomel (2004) and further improve their accuracy by using an empirical relationship between the vertical and horizontal anelliptic parameters extracted from many laboratory measurements of stiffness tensor coefficients. We also modify the functional form of the approximations to improve their behavior at large angles.

Many studies of elastic wave propagation and velocity approximations in orthorhombic media have been reported in the literature, and several alternative six-parameter approximations for qP phase velocity have been proposed (Grechka, 2009; Song and Alkhalifah, 2013; Hao and Stovas, 2014; Tsvankin, 1997; Alkhalifah, 2003). Several group-velocity approximations for orthorhombic media have been proposed in the form of moveout approximations (Xu et al., 2005; Vasconcelos and Tsvankin, 2006). Using the fact that the elastic wave propagation in each of the three symmetry planes of orthorhombic media is controlled by the same Christoffel equation as in the case of TI media (Tsvankin, 2012,1997), we develop novel approximations for orthorhombic qP velocities by starting from our approximations in TI media. We extend our anelliptic TI approximations to a 3D form suitable for approximation of phase and group velocities of qP waves in orthorhombic media. Using a set of test models, we check the accuracy of the proposed approximations and verify that they provide more accurate alternatives to the previously known approximations. In some of the models, the improvement in accuracy is dramatic and reaches a factor of ten. The proposed approximations can readily be used in seismic data processing and imaging applications. We show examples of applying the proposed phase-velocity approximations for TI and orthorhombic media in wave extrapolation experiments.

On anelliptic approximations for qP velocities in TI and orthorhombic media