- I)
- The normal shift corrects for the effects of the offset direction
on the location of the image point;
that is,
the transformation to the angle domain
shifts the image points
from different locations in the offset domain
(
*I*_{xh},*I*_{zh}and*I*) to the same location in the angle domain ()._{0} - II)
- The image location in the angle domain ()lies on the normal to the apparent geological dip
passing through the crossing point of the source and receiver rays
().
is located at the crossing point of
the lines passing through
*S*and_{0}*R*and orthogonal to the source ray and receiver ray, respectively. The shift along the normal to the reflector, caused by the transformation to angle domain, is thus equal to:_{0}(13) (14)

Figure illustrates Properties I and II. These properties are far from obvious and their demonstration constitutes one of the main results of this paper. They also have several important consequences; the three results most relevant to migration velocity analysis are:

- 1.
- ADCIGs obtained from HOCIGs and VOCIGs can be constructively averaged, in contrast to the original HOCIGs and VOCIGs. We will exploit this property to introduce a robust algorithm for creating a single set of ADCIGs that is insensitive to geological dips, and thus is ready to be analyzed for velocity information.
- 2.
- The reflector-point dispersal that negatively affects offset-domain CIGs is corrected in the ADCIGs, at least at first order. If we assume the raypaths to be stationary, for a given reflecting segment the image points for all aperture angles share the same apparent dip, and thus they are all aligned along the normal to the apparent reflector dip.
- 3.
- From equation (14),
invoking Fermat's principle and applying simple trigonometry,
we can also easily derive a relationship
between the total normal shift
and the total
traveltime perturbation caused by
velocity errors as follows:
(15) *S*is the background slowness around the image point and is defined as the difference between the perturbed traveltime and the background traveltime. We will exploit this relationship to introduce a simple and accurate expression for measuring residual moveouts from ADCIGs.

7/8/2003