Efficient traveltime solution of the acoustic TI eikonal equation

TL;DR

This paper introduces a perturbation-based numerical method for efficiently solving the transversely isotropic (TI) eikonal equation, significantly reducing computational costs while maintaining high accuracy even in complex models.

Contribution

The authors develop a first-order perturbation approach that simplifies the TI eikonal equation, enabling accurate and efficient traveltime calculations without restrictions on model complexity.

Findings

Achieves large computational cost reduction compared to exact solvers.

Provides higher accuracy than previous approximations.

Effective in complex models like Marmousi.

Abstract

Numerical solutions of the eikonal (Hamilton-Jacobi) equation for transversely isotropic (TI) media are essential for imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI eikonal equation, which requires solving a quartic polynomial for every grid point. Analytical solutions of the quartic polynomial yield numerically unstable formulations. Thus, we need to utilize a numerical root finding algorithm, adding significantly to the computational load. Using perturbation theory we approximate, in a first order discretized form, the TI eikonal equation with a series of simpler equations for the coefficients of a polynomial expansion of the eikonal solution, in terms of the anellipticity anisotropic parameter. Such perturbation, applied to the discretized form of the eikonal equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for models with complex distribution of velocity and anisotropic anellipticity parameter, such as that for the complicated Marmousi model. The formulation allows for large cost reduction compared to using the exact TI eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy in the proposed algorithm, without any addition to the computational cost.