Anelliptic rational approximations of traveltime P-wave reflections in VTI media

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Transversely isotropic media with a vertical symmetry axis (VTI) is the model of the subsurface suitable for processing seismic reflection data surface of sedimentary basins formed by shale. The propagation of P waves in VTI media is characterized by four independent elastic parameters and complex algebraic equations for the phase and group velocities. Therefore, there is a need to obtain approximations accurate to the phase and group velocities in VTI homogeneous media. Several authors have described approaches to the phase and group velocities with only three parameters, in homogeneous horizontally layered VTI media through hypotheses as weak anisotropy of the medium and anellipticity wavefront. In this work, we have used rational approximants in shifted-hiperbola approximation and obtained rational anelliptitic approximations of the phase and group velocities in homogeneous horizontally stratified VTI media. We have verified the accuracy of the approximations, compared with other approximations in the literature. As an application, we converted the group velocity approximations in nonhyperbolic moveout approximations and performed parameter estimation by means of semblancebased velocity analysis. The results show the validity of anelliptic rational approximations in inverse processes. Introduction Due to the limitations of isotropic models in more complex lithologies, such as sedimentary basins formed by shales, the seismic reflection survey is considered as a model of subsurface anisotropic media, especially the VTI media. In homogeneous media TIV, the wavefront of the SH phase velocity is elliptical and has exact equation that depends only two elastic parameters. However, the Pand SV-waves have: strongly anelliptical wavefront for both phase velocity and group velocity; algebraically complicated exact equations for the phase velocity; and are characterized by five independent elastic moduli tensors ( ). Moreover, even in TIV media, it is difficult to explain exact equations for the group velocity. Another remarkable feature of anisotropic media is the nonhyperbolic behavior of moveout curve. Thus, it is necessary to obtain approximations for the phase and group velocities and thus for moveout curves, which have precision and are practical to perform the steps of seismic data processing. Thomsen (1986), using the physical characteristics of the elastic parameters and the properties in the vertical direction, introduced a parameter of elastic moduli which facilitates the study of the effects of wave propagation in homogeneous anisotropic media VTI. Alkhalifah and Tsvankin (1995) found that only three of these parameters influence the propagation of P-waves in TIV media. Authors such as Muir and Dellinger (1985), Thomsen (1986), Dellinger (1993), Alkhalifah and Tsvankin (1995), Alkhalifah (1998), Fomel (2004), Psensic (2013), among others, have shown approaches the phase velocity in homogeneous VTI media, which depend explicitly only three elastic parameters. Dellinger, and Muir (1985) and Dellinger (1993) showed the anelliptical approximation of the phase velocities, using the properties of the elliptical anisotropy. Assuming analogy in the form of approximations obtained elliptical approximation of group velocity and consequently the moveout approximations for TI media. Fowler (2003) defined the elliptical component of the phase velocity, and through convenient parameterization of elastic parameters obtained anelliptical approximation for phase velocity in VTI media, equivalent to those obtained by other authors. However, using heuristics pure, converted them to approaches: by dispersion relations, group velocities and time equations. Fomel (2004) inspired by the anelliptical approximation (Dellinger et al,1993) used the shifted-hyperbola approximation (Malovichko 1978; Sword 1987; de Bazelaire 1988; Castle 1994) and Stoltstretch correction (Stolt 1978; Fomel and Vaillant 2001 ) to obtain, separately, the acoustic phase velocity approximation (Alkhalifah, 1998) of the P-wave in VTI media. However, by analogy form, obtained group velocity approximation and non-hyperbolic traveltime, very accurate. In this work, we obtained anelliptical approximation for the phase velocity of the compressional wave in vertical media such as rational approximating the shiftedhyperbola approximation (Fomel, 2004). Using the conversion technique by similarity of form (Dellinger, 1993), we obtained anelliptical approximations to group velocity towards these; we obtain new nonhyperbolic moveout approximations. To prove the accuracy of such approximations, we calculated the relative errors of these compared to other approximations. We also conducted semblance-based velocity analysis, to show the robustness of rational approximations of traveltime in estimating parameters. Phase velocity in VTI media The wave propagation in VTI media is characterized by the independent elastic parameters, density-normalized: