Rational approximation of P-wave kinematics — Part 2: Orhorhombic media
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Orthorhombic anisotropy is a modern standard for 3D seismic studies in complex geologic settings. Several seismic data processing methods and wave propagation modeling algorithms in orthorhombic media rely on phase-velocity, group-velocity, and traveltime approximations. The algebraic simplicity of an approximate equation is an important factor in these media because the governing equations are more complicated than transversely isotropic media. To approximate the P-wave kinematics in acoustic orthorhombic media, we have developed a new 3D general functional equation that has a simple rational form. Using the general form, we adopt two versions of rational approximations for the phase velocity, group velocity, and traveltime. The first version uses a simpler functional form and parameter definition within the orthorhombic symmetry planes. The second version is more accurate, using one parameter that is defined out of the symmetry planes. For the phase velocity, we obtain another approximation that is no longer rational but is still algebraically simple, exact for 3D transversely isotropic media, and it is exact within the symmetry planes of orthorhombic media. We find superior accuracy in our approximations compared with previous ones, using numerical studies on multiple moderately anisotropic orthorhombic models. We investigate the effect of the negative anellipticity parameters on the accuracy and find that, in models in which the error of the existing most accurate approximations exceeds 2%, the error of the new approximations remains below 0.2%. The adopted approximations are algebraically simpler and stably more accurate than existing approximations; therefore, they may be considered as attractive alternatives for the existing approximations in many practical applications. We extend the applicability of our approximations by using them to obtain the equations of group direction as a function of phase direction and vice versa, which are useful in wave propagation modeling methods.