Anelliptic approximations for phase and group velocities of qP waves in transversely isotropic (TI) media have been widely applied in various seismic data processing and imaging tasks. We have revisited previously proposed approximations and suggested two improvements. The first improvement involves finding an empirical connection between anelliptic parameters along different fitting axes based on laboratory measurements of anisotropy of rock samples of different types. The relationship between anelliptic parameters observed was strongly linear suggesting a novel set of anisotropic parameters suitable for the study of qP-wave signatures. The second improvement involves suggesting a new functional form for the anelliptic parameter term to achieve better fitting along the horizontal axis. These modifications led to improved three-parameter and four-parameter approximations for phase and group velocities of qP-waves in TI media. In several model comparisons, the new three-parameter approximations appeared to be more accurate than previous approximations with the same number of parameters. These modifications also served as a foundation for an extension to orthorhombic media, where qP velocities involved nine independent elastic parameters. As determined by previous researchers, qP-wave propagation in orthorhombic media could be adequately approximated using just six combinations of those nine parameters. We have developed novel six-parameter approximations for phase and group velocities for qP-waves in orthorhombic media. The orthorhombic phase-velocity approximation provides a more accurate alternative to previously known approximations and can find applications in full-wave modeling, imaging, and inversion. The group-velocity approximation is also highly accurate and can find applications in ray tracing and velocity analysis.

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