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Tieman's sequence of transformation equations ideally applies to
a world defined by continuous infinite functions of space and time.
The use of two dimensional Fourier transforms in this method, both in
the forward and inverse sense, indicate that there may be resolution
questions and artifacts to deal with when this transform is applied
to finite spaces, not to mention boundary effects inevitable with the
slant stack. Precision of the two Fourier transformed axis, the retarded
time axis, , and the midpoint axis, *y*, is important for
the Tieman method because the transformation involves a shift of the
third axis, the Snells parameter axis, *p*, as a function of the Fourier
transformed axis. This is evident in the relation of Snells parameter
in the shot domain to Snells parameter in the midpoint domain. The
following is the Tieman transformation where the represents a Fourier
transform and *Y* and *S* represent the slant stack of the the cmp
gather and the slant stack of the shot gather respectively. This notation
follows the convention of Harlan and Claerbout1996.

| |
(2) |

Dense sampling of the time axis is often achieved in real data
situations, but the midpoint axis and offset axis will, in most cases
be sampled much less. Coarse sampling in the midpoint axis
could be problematic, especially if the sampling is such that dips
evaluated by the factor in the Tieman
transformation are aliased. If the transformation
is applied to unmigrated data, diffraction events could create confusing
dips which could be significant to the shift equation. Aliasing due
to coarse sampling along the offset axis of the cmp gather will
also limit the accuracy of the transform, though not to the degree that
aliasing would effect the direct application of a slant stack to the
corresponding shot gather
Harlan and Claerbout (1996).

** Next:** THE SIMULATION
** Up:** Holden: Tieman & synthetic
** Previous:** PLANE WAVES & SNELL
Stanford Exploration Project

11/12/1997