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![]() | First-break traveltime tomography with the double-square-root eikonal equation | ![]() |
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Following Appendix A, we let in the DSR case be the traveltime at vertex
and
approximate
in equation 8 by a one-sided finite-difference
For a Cartesian ordering of the discretized , i.e. vector
, the discretized operators
with
are matrices. Thanks to upwind finite-differences, these matrices are
sparse and contain only two non-zero entries per row. For instance, suppose
has
its upwind neighbor in
at
, then
We can sort entries of by their values in an increasing order, which
equivalently performs column-wise permutations to
. The results are lower triangular matrices.
In fact, during FMM forward modeling, such an upwind ordering is maintained and updated by the priority
queue and thus can be conveniently imported for usage here.
Note that the summation and subtraction of two (or more)
matrices are still lower
triangular. These matrices are also invertible, except for a singularity at
where we may set the entries
to be zero. Naturally, the inverted matrices are also lower triangular. One example is the linearized eikonal
equation that gives rise to equation 10. Following notation C-4 and assuming the
upwind neighbors of
are
and
, the linearized equation 9 reads
Lastly, the adjoint-state calculations implied by equations 12 and
16 multiply the transpose of these inverse matrices with the data residual. The matrix
transposition leads to upper triangular matrices. Accordingly, we solve the linear system with anti-causal
downwind ordering that follows a decrease in values of .
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![]() | First-break traveltime tomography with the double-square-root eikonal equation | ![]() |
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