Transversely isotropic (TI) media is a more realistic model for processing seismic data, for example, fractured media with preferred fracture direction, or composite by periodic thin layers. In particular, TI media with vertical symmetry axis (VTI) are widely used as models for P-wave propagation in shales, abundant rock in hydrocarbon reservoirs. However, the P-wave propagation in homogeneous media VTI have as their main characteristics, depend on four parameters of rigidity and also to possess: complicated algebraically phase velocity equation, difficult group velocity equation to explain and moveout equation nonhyperbolic. Therefore, several authors have presented parameterization and obtained approximations to these equations depending on three parameters only. Among these, the moveout approximations have been widely used in inverse methods to estimate lithological parameters in homogeneous media VTI. Such methods have generally been successful in estimated stacking velocity vn and the anellipticity parameter η, since these are the only ones required for generating initial models for the steps of seismic processing in the time domain. One of the most used methods for estimating parameters is the basedsemblance velocity analysis, though, because this method is limited to sections with small offset-depth ratio, adaptations for anisotropic media, considering nonhyperbolic moveout approximatios are required. In this paper, based on anelliptical approximation shifted hyperbola, anelliptical rational approximations are presented for: phase velocity, group velocity and moveout nonhyperbolic in homogeneous VTI horizontally layered media. The validity of these approximations is made by calculating their relative errors by comparing with other known approximations in the literature. To semblance-based velocity analysis is performed to measure the accuracy of the rational moveout approximations to estimate parameters in VTI media. The results demonstrate the great potential of rational approximations in inverse problems. In order to adapt to VTI media, we modify two coherence measurements by semblance which are sensitive to amplitude and phase variations. The accuracy and robustness of the adapted coherence measurements are validated by estimation of in anisotropic parameters in VTI media.
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