Strictly speaking, Stolt prestack residual migration is a constant
velocity process. However, with this process we obtain images that
correspond to velocities that have a given *ratio* to the original
one. Therefore, if the original velocity is variable, the new
velocities are also not constant, but only slightly faster or slower
than the reference. The true relationship between the original and new
velocities is still not fully understood, and remains a subject for
future research.

One possible measure of the degree to which the prestack image is focused is the flatness of the angle-domain common-image gathers (CIG) Biondi (1999); Prucha et al. (1999). An accurate velocity model is a sufficient condition for the CIGs to be flat. Once the CIGs are flat, summation of the flat events along the aperture-angle axis yields high-energy stacks, while summation along the nonflat events yields lower energy stacks.

The angle-domain common-image gathers are representations of the
depth images in a coordinate system defined by depth, midpoint, and
aperture-angle. The aperture-angles can be computed in the wavenumber
domain as a function of the offset and depth wavenumbers
(*k*_{h}, *k*_{z}) as
^{}

(1) |

- The representation in aperture-angle contains valuable information
for velocity analysis through the strong moveout of the events
migrated with incorrect velocity Prucha et al. (1999), as shown
in Figure 7. This property is also true when we represent
the CIGs as a function of the offset ray-parameter (
*p*_{h}). - Angle-domain CIGs where the angle axis is described through the
offset ray-parameter (
*p*_{h}) require knowledge about the velocity field. This is fine if we compute the CIGs after wave-equation depth migration. However, it is much more difficult to assess the correct velocity of the images that have been obtained by residual migration. It is, at least for this application, better to replace*p*_{h}with*a*_{h}, as defined in the preceding equation.

We can use the information contained in the CIGs to generate better focused images from a suite of images obtained through residual migration for different ratios between the original and modified velocities Sava (1999). Since the velocity model is not constant, different ratios will flatten the events more or less in different regions of the image. It follows that the energy of the stack will also vary with the ratio at every location in the image. Therefore, we can pick a map of ratios that represents the highest energy stack, and implicitly the flattest CIGs. At the same time, we can also extract the image that corresponds to the highest energy. In the rest of the paper, I call this image the best focused image.

The full image-enhancement procedure is outlined in the flow-chart shown in Figure 2.

Figure 2

10/25/1999