Up: Sava: Image enhancement
Residual migration has proved to be a useful tool in imaging
and in velocity analysis. Recent publications show that Stolt residual
migration can be applied in the prestack domain Stolt (1996), and,
furthermore, that it can be posed as a velocity-independent process
Sava (1999). Consequently, we can use Stolt
residual migration in the prestack domain to obtain a better-focused
image without making any assumption about the velocity.
This is why Stolt residual migration in the prestack domain appears
to be a good choice for image enhancement after wave-equation
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
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
(kh, kz) as
There are two major reasons for the representation of the angle-domain
CIGs with the aperture-angle as the ``offset'' axis:
- 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 (ph).
- Angle-domain CIGs where the angle axis is described through the
offset ray-parameter (ph) 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 ph with
ah, 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
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 Image enhancement flowchart. We start with
the prestack original image, Fourier transform it on all axes, apply
residual migration with a given ratio, sum over the offset axis to
obtain the zero-offset (ZO) image, and convert it to the original
depth. We then repeat the same procedure to obtain ZO images for a
suite of ratios and pick the maximum energy. Finally, we compare the
ZO section before residual migration to the best focused ZO section
after residual migration and optimal picking.
Up: Sava: Image enhancement
Stanford Exploration Project