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

Following the analysis in Appendix A, we consider an implicit Eulerian discretization. For forward modeling, we solve the DSR eikonal equation by a version of the fast-marching method (FMM) (Sethian, 1999). First, a plane-wave with at subsurface zero-offset is initialized. Next, in the update stage the traveltime at a grid point is computed from its upwind neighbors. A priority queue keeps track of the first-break wave-front, and the computation is non-recursive.

To properly handle the DSR singularity, we design an ordering of the combination of upwind neighbors during the update stage. Assuming that is the upwind neighbor of in the 's direction for , we summarize the ordering as follows:

- First try a three-sided update:
- Solve equation A-9, return if ;

- Next try a two-sided update: solve equations A-10, A-12 and A-13
and keep the results as , and , respectively.
- If and , return ;
- If and , return ;
- If and , return ;

- Finally try a one-sided update:
- Solve equation A-14, return .

For an implementation of linearized tomographic operators 12 and 16, we choose upwind approximations (Franklin and Harris, 2001; Li et al., 2011; Lelièvre et al., 2011) for the difference operators in equation 8. In Appendix C, we show that the upwind finite-differences result in triangularization of matrices 11 and 15. Therefore, the costs of applying and and their transposes are inexpensive. Moreover, although our implementation belongs to the family of adjoint-state tomographies, we do not need to compute the adjoint-state variable as an intermediate product for the gradient.

Additionally, the Gauss-Newton approach approximates the Hessian in equation 6 by
. An update
at current is found by
taking derivative of equation 6 with respect to
, which results in the
following normal equation:

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

2013-10-16