The approximation is then evaluated as

(11) |

It is also easy to show that the approximation of by the sum of its columns, equation (10), is equivalent to the choice of a constant vector as in equation (11). Therefore, it will bias the imaging process towards model that are constant. In the case of stacking, it encourages flat reflectors, that is consistent with the flat reflector assumptions underlying the stacking process. In the case of a more complex imaging operator aimed at imaging complex structure, this bias towards flat reflectors may be less appropriate.

Fold normalization is effective when the geometry
is irregular but without sizable data gaps.
However, when these gaps are present the
normalization weights tend to become large.
Even if instability can be easily avoided by the
weights modification expressed in
equation (7),
gaps are going to be left in the uniformly sampled data.
These gaps are likely to introduce
artifacts in the image because migration operators
spread them as *migration smiles*.
The gaps should be filled using the information from
nearby traces before migration.
In the next section we discuss how that can be done
within the context of inverse theory.

9/18/2001