![]() |
![]() |
![]() |
![]() | First-break traveltime tomography with the double-square-root 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.
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
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
A direct substitution from equations A-6 and A-7 results in
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
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:
![]() |
---|
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 ![]() ![]() ![]() ![]() |
![]() ![]() |
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:
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 | ![]() |
![]() |