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A 3-D reflection dataset resides in a five dimensional space. Typically
we describe this space in terms of () or () where
*t* is time and are the source, receiver, midpoint, and
offset vectors.
In Biondi and Vlad (2001) the dimensionality of the dataset was
decreased by one by describing offset by a scalar rather than a vector.
This is far from an ideal solution, especially in the case of a land dataset
and/or data over complicated geology. In both cases the earth being
sampled at different azimuths can vary significantly.
By stacking we are making an implicit assumption that there isn't any variation (or
at least significant variation) due to azimuth. This can affect both
our amplitudes and our ability to accurately estimate the model velocity Clapp and Biondi (1995).
If we use a five dimensional model space we must modify our estimation procedure.
The solution is to perform individual estimations at
different *h*_{y} or azimuths. Solving independent problems is not generally a workable
solution. First we aren't imposing any smoothness over *h*_{y} or azimuth, something
that we know should physically exist. Our estimation procedure is likely to produce
an answer far from smooth over the added axes. Figure
and Figure show fold maps for a portion of a 3-D land and marine
datasets. The left panel shows the *cmp*_{x},*cmp*_{y},*h*_{x} cube, the latter the
*cmp*_{x},*cmp*_{y},*h*_{y} cube at the same midpoint location. Note how in both cases the coverage
varies significantly as a function of *h*_{y}. Take the marine case for example.
Standard acquisitions techniques would lead to *cmp*_{y} (cross-line direction) locations to be banded along different
*h*_{y} locations (caused by the multiple towed cables) and few large *h*_{x} at small *h*_{y}
(due to cable feathering).

**fold.land
**

Figure 5 The fold from a portion of a 3-D land dataset.
The left panel is a subset at a constant *h*_{y}. Three
panels from the subset are shown. The right panel is a subset at a constant *h*_{x}.
Both panels show the same *cmp*_{x} ,*cmp*_{y} location.

**fold.elf
**

Figure 6 The fold from a portion of a 3-D marine dataset.
The left panel is a subset at a constant *h*_{y}. Three
panels from the subset are shown. The right panel is a subset at a constant *h*_{x}.
Both panels show the same *cmp*_{x} ,*cmp*_{y} locations.

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** Previous:** Approximate inverses
Stanford Exploration Project

10/14/2003