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| Traveltime approximations for transversely isotropic media with an inhomogeneous background | |
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Next: Appendix B: Expansion in
Up: Alkhalifah: TI traveltimes in
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To derive a traveltime equation in terms of perturbations in , we first establish the form for the
governing equation for TI media given by the eikonal representation.
The eikonal equation for -waves in TI media in 2D (for simplicity) is
given by
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(25) |
To
solve equation A-1 through perturbation theory, we assume that is small, and thus, a trial solution can be expressed
as a series expansion in given by
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(26) |
where , and are coefficients of the expansion given in units of traveltime, and, for practicality,
terminated at the
second power of .
Inserting the trial solution, equation A-2, into equation A-1 yields a long formula, but by setting ,
I obtain the zeroth-order term given by
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(27) |
which is the eikonal formula for VTI anisotropy. By equating the coefficients of the powers of the
independent parameter , in succession,
we end up first with the coefficients of first-power in
, simplified by using equation A-3, and given by
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(28) |
which is a first-order linear partial differential equation in .
The coefficient of
, with some manipulation, has the
following form
which is again a first-order linear partial differential equation in
with an obviously
more complicated source function given by the right-hand side.
Though the equation seems complicated, many of the variables of the source function (right-hand side) can be evaluated during the evaluation of
equations A-3 and A-4 in a fashion that will not add much to the cost.
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| Traveltime approximations for transversely isotropic media with an inhomogeneous background | |
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Next: Appendix B: Expansion in
Up: Alkhalifah: TI traveltimes in
Previous: Bibliography
2013-04-02