# Migration from topography

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

A CMP-stacked section is assumed to be equivalent to a zero-offset wavefield and usually is referenced to a flat datum. Figure 4.6-37 shows a migrated CMP stacked section associated with a seismic line that follows a traverse with severe topography from an area with overthrust tectonics. Migration was done from a flat reference datum. When migrating data recorded over such an irregular and severe topography, however, one needs to account for the difference between the elevation profile and the reference datum. Otherwise, events appear to a migration algorithm shallower than they actually are if the flat reference datum is below the elevation profile, and thus are undermigrated. If the flat reference datum is above the elevation profile, then events appear to a migration algorithm deeper than they actually are, and thus are overmigrated.

Figure 4.6-38 shows the same data as in Figure 4.6-37 with migration from the floating datum which is a smooth form the elevation profile. A sketch of key events between midpoints A and B from both sections is provided in Figure 4.6-39. The flat reference datum in this case is above the elevation profile. Hence, migration from the flat datum causes overmigration as denoted by the dotted interpretation segments. By migrating the data from the floating datum and interpreting the resulting section, we obtain the solid segments in Figure 4.6-39.

Migration algorithms, with the exception of Kirchhoff summation and the constant-velocity Stolt method, are all based on wave extrapolation from one flat depth level to another. To accommodate an irregular topography, the following formal approach can be used:

1. Stack the data referenced to the floating datum and assume it to be the zero-offset wavefield recorded along the floating datum profile.
2. Apply wave-equation datuming (layer replacement) to extrapolate the zero-offset wavefield as defined in (a) from the floating datum to a flat datum above using a velocity the same as that just below the floating datum.
3. Migrate the output wavefield from (b) using a preferred migration algorithm.

The stacking velocity field required in step (a) is referenced to the floating datum. The migration velocity field required in step (c) would need to be derived by redefining the stacking velocities with respect to the flat datum. To circumvent this tedious task, migration from an irregular topography is done either by the zero-velocity trick [1] or the zero-wavefield trick [2].

In the first approach by Beasley and Lynn [1], a zero velocity value is assigned to the region between the floating datum and the flat datum, which is specified above the floating datum. Just as it should be in conventional processing, the stacked section is referenced to the flat datum, and the velocities are referenced to the floating datum. Extrapolate the stacked section down one depth step using the zero velocity as part of the migration process. This amounts to a simple vertical time shift. If the depth level intersects the floating datum profile, then invoke the diffraction term (equation 16a) for the traces in the stacked section that coincide with the intersection points. Continue the extrapolation process from one depth level to the next while turning on the diffraction term for those traces which coincide with the intersection points of the depth levels and floating datum profile.

 ${\displaystyle {\frac {\partial ^{2}Q}{\partial z\partial t}}={\frac {v}{4}}{\frac {\partial ^{2}Q}{\partial x^{2}}},}$ (16a)

In the second approach by Reshef [2], to start with, a zero wavefield is assigned to the flat datum level z0 in Figure 4.6-40. The stacked section and the velocities are referenced to the floating datum. Extrapolate this wavefield down one depth level to z1 using a velocity the same as that just below the floating datum profile. Import the traces 1 and 2 from the stacked section at the intersection points of the depth level and the floating datum profile, and insert them to the section referenced to the depth level z1. Now, extrapolate down to the next depth level z2, and import traces 3, 4, 5, and 6 from the stacked section, and insert them into section referenced to the depth level z2. At each depth level, while new traces are imported from the original stacked section, the previously imported traces are subjected to wave extrapolation as symbolized by the small arcs associated with each time sample.

## References

1. Beasley and Lynn, 1992, Beasley, C. and Lynn, W. S., 1992, The zero-velocity layer: migration from irregular surfaces: Geophysics, 57, 1435–1443.
2. Reshef, 1991, Reshef, M., 1991, Migration from irregular topography: Geophysics, 50, 1333–1335.