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Although it is not easy to find irregularly sampled zero-offset
non-synthetic reflection datasets, such a case was chosen for
investigation because of its simplicity; any results should be
directly applicable to the prestack case.
The Appendix shows a derivation for the 444#444 (parabolic) wave
equation () and for the 429#429 one
(). Both can be expressed as:
|
a Qxxz + Qxx + b Qz = 0,
|
(180) |
where the subscripts denote partial derivatives along the
corresponding axes. Plugging in the templates in (),
() (), we get:
In the case of the 444#444 equation, a=0, and for the 429#429one, 446#446. In both cases 447#447. They are the same as for the regular sampling
case, which is simply a particular case of this equation (with
particular values of ki). The stability of the
downward continuation undertaken in this manner is proven for all
practical purposes by the results in Figs. and
. This means that the special stability precautions
taken by () are an unnecessary
complication.
The resulting tridiagonal system is solved and the values of
448#448 are found. The lens term () which
is applied
after each downward continuation step with the above described
equations does not depend in any way on the sampling of the x-axis and
is therefore the same as in 2#2 - x migrations of evenly sampled data.
Unfortunately, the so-called 1/6 trick [(),
section 4.3] cannot be straightforwardly applied when the spatial axis
is unevenly sampled. With a bit of work, an equivalent formula
can also be deduced for the irregular sampling case.
The proof in the Appendix ensures that no hidden regular sampling
assumption has been incorporated in the 444#444 and 429#429 wave
equation approximations.
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Stanford Exploration Project
6/7/2002