Approximate dispersion relations for qP–qSV-waves in transversely isotropic media

To decouple qP and qSV sheets of the slowness surface of a transversely isotropic (TI) medium, a sequence of rational approximations to the solution of the dispersion relation of a TI medium is introduced. Originally conceived to allow isotropic P-wave processing schemes to be generalized to encompass the case of qP-waves in transverse isotropy, the sequence of approximations was found to be applicable to qSV-wave processing as well, although a higher order of approximation is necessary for qSV-waves than for qP-waves to yield the same accuracy. The zeroth-order approximation, about which all other approximations are taken, is that of elliptical TI, which contains the correct values of slowness and its derivative along and perpendicular to the medium’s axis of symmetry. Successive orders of approximation yield the correct values of successive orders of derivatives in these directions, thereby forcing the approximation into increasingly better fit at the intervening oblique angles. Practically, the first-order approximation for qP-wave propagation and the second-order approximation for qSV-wave propagation yield sufficiently accurate results for the typical transverse isotropy found in geological settings. After only slight modification to existing programs, the rational approximation allows for ray tracing, ( f -k) domain migration, and split-step Fourier migration in TI media—with little more difficulty than that encountered presently with such algorithms in isotropic media.