Rational approximation of P-wave kinematics — Part 1: Transversely isotropic media

In seismic data processing and several wave propagation modeling algorithms, the phase velocity, group velocity, and traveltime equations are essential. To have these equations in explicit form, or to reduce algebraic complexity, approximation methods are used. For the approximation of P-wave kinematics in acoustic transversely isotropic media, we have developed a new flexible 2D functional equation in a continued fraction form. Using different orders of the continued fraction, we obtain different approximations for (1) phase velocity as a function of phase direction, (2) group velocity as a function of group direction, and (3) traveltime as a function of offset. Then, we use them in the approximation of the group direction as a function of phase direction, and phase direction as a function of group direction. The proposed approximations have a rational form, which is considered algebraically simple and computationally efficient. The used continued fraction form rapidly converges to exact kinematics. By introducing the optimal ray into our approximations and using it for parameter definition, the convergence becomes faster, so the accuracy of the existing most accurate approximations is available by the third order, and new most accurate approximations are obtained by the fourth order of the proposed general form. The error of the most accurate version of the proposed approximations is below 0.001% for moderate anisotropic models with an anellipticity parameter up to 0.3. This high accuracy is considered to be attractive in practical implementations that use the kinematic equations and their derivatives.