EXAMPLES

To test the residual constant-offset migration and residual NMO+DMO operators, I made a synthetic constant-offset survey over a series of point diffractors. The constant-offset data were migrated with a slowness equal to .8333 times the correct slowness. I applied residual constant-offset migration at $\gamma=1.2$ and successfully recovered the images of the point diffractors as shown in Figure 8. I also applied residual NMO+DMO to the migrated constant-offset section to obtain the lower plot in Figure 9, which should differ from the upper plot in Figure 8 by only a residual zero-offset migration. The upper plot in Figure 9 is the result of applying residual zero-offset migration to the lower plot. As hoped, the images of the point diffractors are recovered.

There are some differences between the top plots in Figures 8 and 9. The most important difference is the reduction in amplitude of the top point diffractor in the NMO+DMO+residual zero-offset migrated image. Near the cusps of the operators (see Figure 7), the stationary phase approximation breaks down because there is a discontinuity in the derivative of the slope of the summation curve. The kinematic summation operators have amplitudes that drop to zero immediately at the edge of the cusp, while in reality, energy spreads out away from the cusp. Wave-theoretic residual NMO+DMO would not have this problem, but it is more expensive to apply.

One advantage of residual constant-offset migration over residual NMO+DMO followed by residual zero-offset migration is that residual constant-offset migration has cusps only for shallow depths and large offsets. The residual NMO+DMO operator has cusps for all offsets (except zero-offset) and all depths. For large depth-offset ratios, the cusps aren't a problem however, because the operator effectively reduces to a point, namely residual NMO. This is where residual NMO+DMO has a great advantage over residual constant-offset migration. The size of the aperture of the residual constant-offset migration operator grows without bound for increasing depths, while the residual NMO+DMO operator shrinks to a point; so residual NMO+DMO is much more economical to apply.

To test the residual constant-offset migration and residual NMO+DMO methods on real data and to see if residual NMO+DMO can be used as a velocity analysis tool after migration, I migrated 180 constant-offset sections from a survey in the Gulf of Mexico. Figure 10 shows a constant-offset section from the data; this offset was at about one-half the total cable length.

I chose a velocity model that varies only in depth and made the velocities increase with depth but purposefully too slow. After prestack migration, I stacked together every 12 migrated constant-offset sections to increase signal to noise, remove aliasing, and reduce the data volume. Figure 11 shows one partial stack of nearby migrated constant-offset sections.

After migration and partial stacking, residual NMO+DMO was applied to the migrated constant-offset sections for the range $0.645 < \gamma < 1.15$. Figures 12, 13, and 14 show stacked sections after residual NMO+DMO.

At the lower values of $\gamma$, corresponding to the largest increase in velocity, the deeper events stack well. At the higher values of $\gamma$,closer to the original velocity, the shallower events stack well. Note that events of all dips at a given depth stack well at the same value of $\gamma$. Also note that events stay fixed as their migration slowness changes. It is easier to analyze velocities using residual NMO+DMO than by scanning migration velocity with many full prestack migrations which forces events to move as the migration slowness changes.

If we wish to see the reflectors in their proper spatial locations, we can apply residual zero-offset migration to the stacks of Figures 12-14. Figures 15-17 show the result of residual zero-offset migration applied to each stack at its appropriate value of $\gamma$.

Figure 18 shows sample velocity analysis panels. Since residual NMO+DMO correctly accounts for residual common-reflection-point smear, these panels can be interpreted just as residual stacking velocity panels are interpreted for horizontal reflectors. Strictly speaking, these panels allow one to pick the best residual prestack time migration after prestack depth migration.

I plan to use residual NMO+DMO to analyze errors in interval velocity models used for prestack depth migration. My previous papers (Etgen, 1988, 1989) describe an operator that relates changes in a residual slowness measure to changes in the interval velocity model. The residual-slowness semblance panels obtained using residual NMO+DMO described in this paper are ideal for measuring residual slowness after prestack depth migration. Events don't move as residual slowness changes, the points in the images correspond to true common-reflection points and the residual zero-offset migration can be calculated and applied after velocity analysis.