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## Dimensionality

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 hy or azimuths. Solving independent problems is not generally a workable solution. First we aren't imposing any smoothness over hy 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 cmpx,cmpy,hx cube, the latter the cmpx,cmpy,hy cube at the same midpoint location. Note how in both cases the coverage varies significantly as a function of hy. Take the marine case for example. Standard acquisitions techniques would lead to cmpy (cross-line direction) locations to be banded along different hy locations (caused by the multiple towed cables) and few large hx at small hy (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 hy. Three panels from the subset are shown. The right panel is a subset at a constant hx. Both panels show the same cmpx ,cmpy 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 hy. Three panels from the subset are shown. The right panel is a subset at a constant hx. Both panels show the same cmpx ,cmpy locations.

Next: Data size Up: COMPLICATIONS Previous: Approximate inverses
Stanford Exploration Project
10/14/2003