First-break traveltime tomography with the double-square-root eikonal equation |

**Siwei Li ^{}, Alexander Vladimirsky^{}and Sergey Fomel^{}**

^{}Bureau of Economic Geology

John A. and Katherine G. Jackson School of Geosciences

The University of Texas at Austin

University Station, Box X

Austin, TX 78713-8924
^{}Department of Mathematics

Cornell University

430 Malott Hall

Ithaca, NY 14853-4201

First-break traveltime tomography is based on the eikonal equation. Since the
eikonal equation is solved at fixed shot positions and only receiver positions can move along the ray-path,
the adjoint-state tomography relies on inversion to resolve possible contradicting information between
independent shots. The double-square-root eikonal equation allows not only the receivers but also the shots
to change position, and thus describes the prestack survey as a whole. Consequently, its linearized tomographic
operator naturally handles all shots together, in contrast with the shot-wise approach in the traditional
eikonal-based framework. The double-square-root eikonal equation is singular for the horizontal waves, which
require special handling. Although it is possible to recover all branches of the solution through
post-processing, our current forward modeling and tomography focus on the diving wave branch only. We consider
two upwind discretizations of the double-square-root eikonal equation and show that the explicit scheme is only
conditionally convergent and relies on non-physical stability conditions. We then prove that an implicit upwind
discretization is unconditionally convergent and monotonically causal. The latter property makes it possible to
introduce a modified fast marching method thus obtaining first-break traveltimes both efficiently and
accurately. To compare the new double-square-root eikonal-based tomography and traditional eikonal-based
tomography, we perform linearizations and apply the same adjoint-state formulation and upwind
finite-differences implementation to both approaches. Synthetic model examples
justify that the proposed approach converges faster and is more robust than the traditional one.

- Introduction
- Theory

- Numerical Implementation
- Synthetic Model Examples
- Discussion
- Conclusions
- Acknowledgments
- Appendix A: Causal discretization of DSR eikonal equation
- Appendix B: Frechét derivative of DSR tomography
- Appendix C: Adjoint-state tomography with upwind finite-differences
- Bibliography
- About this document ...

First-break traveltime tomography with the double-square-root eikonal equation |

2013-10-16