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

Next: Appendix B: Frechét derivative Up: Li et al.: DSR Previous: Acknowledgments

# Appendix A: Causal discretization of DSR eikonal equation

To simplify the analysis, we consider first the DSR branch as shown in Figure 1 and described by equation 1. We assume a rectangular 2-D velocity model and thus a cubic 3-D prestack volume with and axes having the same dimension as . After an Eulerian discretization of both and , we denote the grid spacing in as , and in , and as .

update1
Figure 17.
An implicit discretization scheme. The arrow indicates a DSR characteristic. Its root is located in the simplex .

In Figure A-1, we study the traveltime at grid point and its relationship with neighboring grid points , and with a semi-Lagrangian scheme. According to the geometry in Figure 1, in the space the DSR characteristic (Duchkov and de Hoop, 2010) is straddled by .

In order to compute , we could continue along this characteristic up until its intersection with the simplex . Suppose the intersection point is and 's are its barycentric coordinates, i.e.

 (18)

This leads to the following discretization:
 (19)

Here we further assume that and are locally constant and that ray-segments between and are well approximated by straight lines. This also means that a linear interpolation in within the simplex is exact, i.e. , where for . The minimization over all possible intersection points in equation A-2 guarantees a first-arrival traveltime.

Defining the ratio in grid spacing as and denoting and , equation A-2 can be re-written with the barycentric coordinates in A-1 as

 (20)

Figure A-1 is based on a particular direction of the diving wave: rightward from the source and leftward from the receiver, as in Figure 1. This yields the above positions of and , and the formula A-3 becomes an update from the quadrant. Since in general the direction of a diving wave is not a priori known, we compute one such update from each of the lower quadrants and take the smallest amongst them as a value of .

To explore the causal properties of equation A-3, we first assume that the minimum is attained at some such that for . From the Kuhn-Tucker optimality conditions (Kuhn and Tucker, 1951), there exists a Lagrange multiplier such that

 (21)

 (22)

Taking a linear combination of A-4 and A-5 to match the right-hand side of A-3, we find that and thus
 (23)

 (24)

This means that if defined by equation A-3 depends on then for , or
 (25)

and a Dijkstra-like method (Dijkstra, 1959) is applicable to solve the discretized system.

A direct substitution from equations A-6 and A-7 results in

 (26)

If , and are known, then can be recovered by solving the 4th degree polynomial equation A-9 and choosing the smallest real root that satisfies condition A-8. This gives a three-sided update at . The discretization is implicitly causal and provides unconditional consistency and convergence.

If there is no real root or none of the real roots satisfy A-8, the minimizer can not lie in the interior of simplex and at least one of the s must be zero. If , it is easy to show that one of the other barycentric coordinates is also zero and equation A-3 simplifies to

 (27)

which is a causal discretization of the DSR singularity in equation 3. On the other hand, if and but , i.e. the slowness vector at is vertical, then
 (28)

A similar Kuhn-Tucker-type argument shows that equation A-11 is also causal: if , then . In this case, can be computed by solving
 (29)

Equation A-12 is equivalent to setting in equation 1. Analogously, when and but , we have and
 (30)

with the causal solution satisfying . Equations A-12 and A-13 provide a two-sided update at . Finally, if but and , i.e. and , we obtain the simplest one-sided update:
 (31)

We note that the one-sided update A-14 could be considered a special case of two-sided updates: if (or ), then A-14 becomes equivalent to A-12 (or, respectively, A-13). Similarly, the two-sided updates can be viewed as special versions of the three-sided one: e.g., if , then A-9 becomes equivalent to A-13. This means that the causal criteria for formulas A-9, A-12 and A-13 can be relaxed (the inequalities do not have to be strict). This relaxation is used to streamline the update strategy in the Numerical Implementation section.

In Figure A-1 and the corresponding semi-Lagrangian discretization A-2, the ray-path is linearly approximated up to its intersection with the simplex at a priori unknown depth . An alternative explicit semi-Lagrangian discretization can be obtained in the spirit of Figure 1 by tracing the ray up to the pre-specified depth . In Figure A-2, we consider the DSR characteristic being straddled by , where , and . Denoting for the intersection point between DSR characteristic and simplex , we obtain the following discretization:

 (32)

One could perform the same analysis of A-3 through A-9 to equation A-15. For the sake of brevity, we omit the derivation and show the resulting explicit discretization scheme:
 (33)

where for . More generally, to account for various possible directions of the diving wave, we can set and .

update2
Figure 18.
An explicit discretization scheme. Compare with Figure A-1. The arrow again depicts a DSR characteristic with its root confined in the simplex .

Compared with equation A-9, equation A-16 does not require solving a polynomial equation. Moreover, depends only on values in lower z-slices, which means that the system of equations can be solved in a single sweep in the direction. Unfortunately, despite this efficiency on a fixed grid, the explicit discretization has a major disadvantage stemming from the requirement that the characteristic should be straddled by . This imposes an upper bound on based on the slope of the diving wave. Moreover, since every diving ray is horizontal at its lowest point, the convergence is possible only if under mesh refinement. In practice, this means that the results are meaningful only if is significantly smaller than . We note that restrictive stability conditions also arise for time-dependent Hamilton-Jacobi equations of optimal control, where sufficiently strong inhomogenieties can make nonlinear/implicit schemes preferable to the usual linear/explicit approach (Vladimirsky and Zheng, 2013).

The above analysis also applies to the first branch of the DSR eikonal equation in Figure 2. However, in the discretized domain, the slowness vectors at and are always aligned in the direction, either upward or downward. For this reason, there is no DSR characteristic that accounts for the second and third scenarios. We will refer to the first and last branches in Figure 2 as causal branches of DSR eikonal equation, and the left-over two as non-causal branches.

search
Figure 19.
When slowness vectors at and are pointing in the opposite directions, the ray-path must intersect with line at certain point .

Note that when the slowness vectors at and are pointing in opposite directions, there must be at least one intersection of the ray-path with the depth level in-between. As shown in Figure A-3, ray segments between these intersections fall into the category of causal branches. Thus a search process for the intersections is sufficient in recovering the non-causal branches during forward modeling. Moreover, because we are interested in first-breaks only, the minimum traveltime requirement allows us to search for only one intersection, such as denoted in Figure A-3:

 (34)

Other possible intersections in intervals and have already been recovered when computing and , as long as we enable the intersection searching from the beginning of forward modeling. The traveltime of non-causal branches from equation A-17 is then compared with that from causal branches, and the smaller one should be kept.

Unfortunately, this search routine induces considerable computational cost. Moreover, we note that, under a dominant diving waves assumption, the first DSR branch, despite being causal, becomes useless if A-17 is turned-off.

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

Next: Appendix B: Frechét derivative Up: Li et al.: DSR Previous: Acknowledgments

2013-10-16