# Efficient traveltime solutions of the acoustic TI eikonal equation

## 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, it requires 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 anisotropy 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 direct 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.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22382180

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 282; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; ANALYTICAL SOLUTION; ANISOTROPY; COMPARATIVE EVALUATIONS; EIKONAL APPROXIMATION; GAIN; HAMILTON-JACOBI EQUATIONS; ISOTROPY; MATHEMATICAL MODELS; NONLINEAR PROBLEMS; NUMERICAL SOLUTION; PERTURBATION THEORY; POLYNOMIALS

### Citation Formats

```
Waheed, Umair bin, E-mail: umairbin.waheed@kaust.edu.sa, Alkhalifah, Tariq, E-mail: tariq.alkhalifah@kaust.edu.sa, and Wang, Hui, E-mail: hui.wang@kaust.edu.sa.
```*Efficient traveltime solutions of the acoustic TI eikonal equation*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2014.11.006.

```
Waheed, Umair bin, E-mail: umairbin.waheed@kaust.edu.sa, Alkhalifah, Tariq, E-mail: tariq.alkhalifah@kaust.edu.sa, & Wang, Hui, E-mail: hui.wang@kaust.edu.sa.
```*Efficient traveltime solutions of the acoustic TI eikonal equation*. United States. doi:10.1016/J.JCP.2014.11.006.

```
Waheed, Umair bin, E-mail: umairbin.waheed@kaust.edu.sa, Alkhalifah, Tariq, E-mail: tariq.alkhalifah@kaust.edu.sa, and Wang, Hui, E-mail: hui.wang@kaust.edu.sa. Sun .
"Efficient traveltime solutions of the acoustic TI eikonal equation". United States.
doi:10.1016/J.JCP.2014.11.006.
```

```
@article{osti_22382180,
```

title = {Efficient traveltime solutions of the acoustic TI eikonal equation},

author = {Waheed, Umair bin, E-mail: umairbin.waheed@kaust.edu.sa and Alkhalifah, Tariq, E-mail: tariq.alkhalifah@kaust.edu.sa and Wang, Hui, E-mail: hui.wang@kaust.edu.sa},

abstractNote = {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, it requires 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 anisotropy 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 direct 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.},

doi = {10.1016/J.JCP.2014.11.006},

journal = {Journal of Computational Physics},

number = ,

volume = 282,

place = {United States},

year = {Sun Feb 01 00:00:00 EST 2015},

month = {Sun Feb 01 00:00:00 EST 2015}

}