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

The DSR eikonal equation can be derived by considering a ray-path and its segments between two depth levels.
Figure 1 illustrates a diving ray (Zhu et al., 1992) in 2-D with velocity . We denote
as the total traveltime of the ray-path beneath depth , where and are *sub-surface*
receiver and source lateral positions, respectively.

raypath
A diving ray and zoom-in of
the ray segments between two depth levels.
Figure 1. |
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At both source and receiver sides the traveltime satisfies the eikonal equation, therefore

The boundary condition for DSR eikonal equation is that traveltimes at the subsurface zero-offset plane, i.e. , are zero: .

Equation 2 has a *singularity* when
, in which case the slowness
vectors at and sides are both horizontal and equation 2 reduces to

Note that equations 2 and 3 describe in full prestack domain
by allowing not only receivers but also sources to change
positions. In contrast, the eikonal equation

root
All four branches of DSR
eikonal equation from different combination of upward or
downward pointing of slowness vectors. Whether the slowness
vector is pointing leftward or rightward does not matter
because the partial derivatives with respect to and
in equation 2 are squared. Figure 1
and equation 1 belong to the last situation.
Figure 2. |
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Similarly to the eikonal equation, the DSR eikonal equation is a nonlinear first-order partial differential equation. Its solutions include in general not only first-breaks but all arrivals, and can be computed by solving separate eikonal equations for each sub-surface source-receiver pair followed by extracting the traveltime and putting the value into prestack volume. However, such an implementation is impractical due to the large amount of computations. Meanwhile, for first-break tomography purposes, we are only interested in the first-arrival solutions but require an efficient and accurate algorithm. In this regard, a finite-difference DSR eikonal solver analogous to the fast-marching (Sethian, 1999) or fast-sweeping (Zhao, 2005) eikonal solvers is preferable.

In upwind discretizations of the DSR eikonal equation on the grid in domain, one has to make a decision
about the -slice, in which the finite differences are taken to approximate
and
. In Figure 1, it appears natural to approximate these partial
derivatives in the -slice below . We refer to the corresponding scheme as *explicit*, since
it allows to directly compute the grid value based on the already known values from the
next-lower . An alternative *implicit* scheme is obtained by approximating
and
in the same -slice as , which results in a coupled system of
nonlinear discretized equations. In Appendix A, we prove the following:

- The explicit scheme is very efficient to use on a fixed grid, but only conditionally convergent. This property is also confirmed numerically in Synthetic Model Examples section.
- The implicit scheme is
*monotone causal*, meaning depends on the smaller neighboring grid values only. This enables us to apply a Dijkstra-like method (Dijkstra, 1959) to solve the discretized system efficiently. Importantly, the DSR singularity requires a special ordering in the selection of upwind neighbors, which switches between equations 1 and 3 when necessary. We provide a modified fast-marching (Sethian, 1999) DSR eikonal solver along with such an ordering strategy in Numerical Implementation section. - The causality analysis in Appendix A applies only to the first and last
*causal branches*out of all four shown in Figure 2. Additional post-processings, albeit expensive, can be used to recover the rest two non-causal branches as they may be decomposed into summations of the causal ones.

In practice, we find that, for moderate velocity variations, the first-breaks correspond only to causal branches. An example in Synthetic Model Examples section serves to illustrate this observation. Therefore, for efficiency, we turn off the non-causal branch post-processings in forward modeling and base the tomography solely on equations 1 and 3.

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

2013-10-16