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

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# Appendix C: Adjoint-state tomography with upwind finite-differences

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

 (41)

where the sign corresponds to the two neighbors of in direction. An upwind scheme (Franklin and Harris, 2001) picks the sign by
 (42)

The above strategy can be applied to and straightforwardly. For the shot-indexed eikonal equation 9, we approximate with the same upwind method while in this case is indexed for and .

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

 (43)

where
 (44)

Definitions of , and follow their upwind approximations, respectively. In equation C-3, are located in the same row as that of in . While sits on the diagonal, has a column index equals to the row of in . At , there is no upwind neighbor and the corresponding row contains all zeros.

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

 (45)

After regrouping the terms, we get
 (46)

Equation C-6 means the inverse of the operator does not need to be computed by an explicit matrix inversion. Instead, we can perform its application to a vector by a single sweep based on causal upwind ordering. The same conclusion can be drawn for operator B-4.

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 .

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

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

2013-10-16