Homotopy solutions of the acoustic eikonal equation for strongly attenuating transversely isotropic media with a vertical symmetry axis

Abstract

Many applications in seismology involve the modeling of seismic-wave traveltimes in anisotropic media. We have developed homotopy solutions of the acoustic eikonal equation for P-wave traveltimes in attenuating transversely isotropic media with a vertical symmetry axis. Instead of the commonly used perturbation theory, we use the homotopy analysis method to express the traveltimes by a Taylor series expansion over powers of an embedding parameter. For the derivation, we first perform homotopy analysis of the eikonal equation and derive the linearized ordinary differential equations for the coefficients of the Taylor series expansion. Then, we obtain the homotopy solutions for the traveltimes by solving the linearized ordinary differential equations. Results of our investigation with approximate formulas demonstrate that the analytical expressions are efficient methods for the computation of traveltimes from the eikonal equation. In addition, these formulas are also effective methods for benchmarking approximate numerical solutions in strongly attenuating anisotropic media.