RESIDUAL NMO+DMO

Figure 5 shows the positions of a point ``P_orig" for several different values of residual slowness $\gamma$ for one initial dip and 3 different offsets. The new positions P₀, P_h, and P_H for different offsets move along different trajectories as migration slowness changes. If, for a fixed $\gamma$, the new points were above one another vertically, a residual NMO correction could convert constant-offset sections to zero-offset sections for a change in migration slowness. Residual constant-offset migration would be equivalent to residual NMO followed by residual zero-offset migration. For zero dip, as expected this is the case. Residual constant-offset migration is just residual NMO followed by residual time to depth conversion. For nonzero dip (as shown in Figure 5), the new positions do not line up vertically. Furthermore, the amount of the vertical movement as a function of offset is not the same as it is for zero dip. Using analogy to full prestack migration, call residual DMO the movement required by residual constant-offset migration and not described by residual NMO or residual zero-offset migration. Figure 6 shows how residual constant-offset migration can be built from three processes: residual DMO, residual NMO, and residual zero-offset migration.

The previous section gave the necessary equations to compute residual constant-offset migration operators; subtracting the residual zero-offset migration part of residual constant-offset migration, obtained from the same equations, gives residual NMO+DMO. Amplitudes of the residual NMO+DMO operators are obtained with the same method used to obtain the amplitudes of residual constant-offset migration operators. Figure 7 shows the residual NMO+DMO operators for a series of depth points for a large offset for $\gamma=1.2$ and $\gamma=.833$.

Residual constant-offset migration has the desired property that it performs residual common-reflection-point gathering. For any residual slowness, any point on the output images of different constant-offset sections all correspond to the same reflection point. This is a necessary step for correct migration velocity analysis. Velocity analysis uses the traveltime of reflections from a single point in the earth; thus, when performing migration velocity analysis by comparing traveltimes versus offset, the relevant part of residual constant-offset migration is residual NMO+DMO. The residual zero-offset migration part of residual constant-offset migration confuses velocity analysis by moving the image of a fixed reflection event around the image as the migration slowness changes (Fowler, 1988; Etgen, 1989). It is preferable to ignore the residual zero-offset migration term and keep a fixed reflection event at a fixed location in the image. This is analogous to conventional velocity analysis where NMO and stacking are performed at a fixed t₀. The true-depth position of the reflector can be calculated and stored, and residual zero-offset migration can be applied later, after velocity analysis. The kinematic residual zero-offset migration is needed by tomographic velocity analysis methods.