Image enhancement theory

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 migration.

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  

$$a_h=\arctan\left(\frac{k_h}{k_z}\right).$$ (1)

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 (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.

 

pick
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.