On anelliptic approximations for qP velocities in TI and orthorhombic media

Discussion

Our choice of Muir-Dellinger parametrization leads naturally to a four-parameter velocity approximation in TI media and a nine-parameter approximation in orthorhombic media. The approximations are improved by shifted-hyperboloid functional form. Although highly accurate, these approximations require the same number of parameters as the exact expressions. The benefits of their introduction may not be apparent in the case of phase velocity but are apparent in the consideration of group-velocity approximations because the exact expressions for group velocity in both types of media can be prohibitively complex and cannot be expressed easily in terms of group angle. As observed by previous researchers, the sufficient number of parameters to describe qP wave propagation in TI and orthorhombic media is smaller: three and six respectively. Therefore, we apply the novel relationships between anisotropic parameters summarized in Figure 1 to effectively reduce the number of parameters from the proposed four- and nine-parameter approximations to three- and six-parameter approximations respectively.

Apart from the functional form proposed in this paper, many other forms of phase-velocity and group-velocity approximations, especially in TI media, have been extensively investigated in the past (e.g. Stopin, 2001; Farra and Pšencík, 2003,2013; Alkhalifah, 1998; Ursin and Stovas, 2006; Mensch and Rasolofosaon, 1997; Fomel and Stovas, 2010; Pšencík and Gajewski, 1998; Zhang and Uren, 2001; Alkhalifah, 2000b; Daley et al., 2004; Hao and Stovas, 2014; Alkhalifah and Tsvankin, 1995; Farra, 2001; Alkhalifah, 2000a; Stovas, 2010; Tsvankin, 1996; Aleixo and Schleicher, 2010). While some of them are based on physical assumptions, others are derived purely from mathematical arguments. Our proposed approximation is an alternative, which provides both accuracy and connection with the physical wave phenomena. They are based on the original Muir-Dellinger approximatons (Muir and Dellinger, 1985; Dellinger et al., 1993), which were derived on the basis of perturbation from ellipitical phase-velocity surfaces. The primary advantage of the Muir-Dellinger parameterization is the ease of conversion between the phase- and group-velocity approximations (e.g. equations 23 and 24), which provides practical convenience. Alternative highly accurate form for phase- and group-velocity approximations is the generalized moveout approximation (Fomel and Stovas, 2010), which was recently applied to anisotropic velocity approximations by Hao and Stovas (2014) and Sripanich and Fomel (2015).

Our approximations are readily applicable to approximate phase and group velocities in the case of transversely isotropic and orthorhombic media whose symmetry axis is aligned with the coordinate axis, e.g., VTI, HTI, and VOR. In the case of TTI (tilted tranversly isotropic) and TOR (tilted orthorhombic), the coordinates simply need to be rotated via Bond transformation before applying the proposed approximations.

On anelliptic approximations for qP velocities in TI and orthorhombic media