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![]() | First-break traveltime tomography with the double-square-root eikonal equation | ![]() |
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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:
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:
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![]() | First-break traveltime tomography with the double-square-root eikonal equation | ![]() |
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