# Prestack Kirchhoff migration

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

A practical alternative to prestack Stolt migration that involves all of the prestack data at all times during the computation is to migrate prestack data using the Kirchhoff summation method. The summation of amplitudes is done along the table-top trajectories inferred by equation (15), and the necessary amplitude and phase treatment of the summed amplitudes are given by the integral solution to the scalar wave equation (migration principles).

 ${\displaystyle T_{0}={\rm {ln}}\ \tau _{0},}$ (15a)

 ${\displaystyle T_{n}={\rm {ln}}\ t_{n},}$ (15b)

Prestack Kirchhoff migration provides the flexibility to output selected CMP gathers in their migrated positions. Such an option is attractive if one wants to pick rms velocity functions from the selected output gathers to create a migration velocity field. This is not possible with prestack Stolt migration which generates a volume of prestack migrated data. Another advantage of prestack Kirchhoff migration is its ability to accommodate irregular topography.

Figure 5.4-5 is a CMP-stacked section associated with a seismic traverse over an overthrust structure. The line direction is approximately orthogonal to the thrust planes, and thus yields minimal 3-D effects. This data set is quite deceptive — at first, it gives the impression that time migration is not a valid strategy for imaging the subsurface. Although the reflector geometries are quite complex, velocities vary from 4000 m/s at the surface to 6500 m/s at the bottom of the section with not much refraction occurring at layer boundaries.

The stacking velocity field shown in Figure 5.4-6 does not appear to be geologically plausable. Nevertheless, use it without much editing for poststack time migration. The resulting section shown in Figure 5.4-7 already exhibits a fairly accurate image of the structurally complex zone above 2 s. We observe imbricate structures, folds, and reverse faults that accompany the thrust faults. The sole thrust — zone of detachment between the competent rock layers below 2 s and the incompetent rock layers above, can be traced along a trajectory that starts at approximately 1.75 s on the left-edge of the section, climbs upward and emerges on the surface at approximately midpoint 510.

We now perform prestack Kirchhoff migration using a range of constant velocities and obtain a set of migration panels as shown in Figure 5.4-8. To use the constant-velocity migration panels for picking rms velocities, they were split into vertical stripes each comprising 100 midpoints. By grouping the constant-velocity stripes with the same center midpoint, we obtain the velocity panels shown in Figure 5.4-9. Just as a constant-velocity-stack (CVS) panel (Figure 3.2-10) is used to pick a stacking velocity function, the constant-velocity-migration panels in Figure 5.4-9 can be used to pick an rms velocity function at each center CMP location. These functions can then be combined to create a migration velocity field as shown in Figure 5.4-10. Compare with Figure 5.4-6 and note that the migration velocity field indeed exhibits a pattern that conforms with the subsurface structure. By using the velocity field in Figure 5.4-10, the prestack time migrated section shown in Figure 5.4-11 was obtained. Compare with the poststack time migrated section (Figure 5.4-7) and observe that, in this case, the differences are marginal. The primary reason for this is that velocities are generally quite high, making even DMO correction unnecessary.