# Crossline migration

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

In 3-D poststack migration, we saw that 3-D poststack migration can be performed by a two-stage summation over the surface of the 3-D zero-offset diffraction hyperboloid. In practice, the two-pass 3-D poststack migration involves successive 2-D poststack migrations of the inlines followed by 2-D poststack migrations of the crosslines. Similarly, 3-D prestack migration also may be performed in two stages. While two-pass 3-D migration is only an approximation to a one-pass 3-D migration (3-D poststack migration), the computational benefits of the former may encourage its use in practice under certain circumstances.

The 3-D nonzero-offset traveltime t (Section G.6) for a zero source-receiver azimuth represents an ellipsoid (Figure G-2) given by

 ${\displaystyle {\frac {x^{2}}{(vt/2)^{2}}}+{\frac {y^{2}}{(vt/2)^{2}-h^{2}}}+{\frac {z^{2}}{(vt/2)^{2}-h^{2}}}=1,}$ (19)

where x and y are inline and crossline directions, respectively, v is the medium velocity and h is the half-offset.

The horizontal cross-section of this ellipsoid at a depth z is an ellipse (Figure 7.4-29). For a 3-D recording geometry with an arbitrary source-receiver azimuth defined by angle θ, the inline-crossline Cartesian coordinates (x, y) in terms of the local Cartesian coordinates (x′, y′) are given by

 ${\displaystyle x=x^{\prime }\cos \theta -y^{\prime }\sin \theta }$ (20a)

and

 ${\displaystyle y=x^{\prime }\sin \theta +y^{\prime }\cos \theta .}$ (20b)

By substituting equations (20a,20b) into equation (19), it can be shown that the vertical cross-section of the ellipsoid along an arbitrary source-receiver azimuth θ is also an ellipse [1]. This characteristic of the traveltime ellipsoid makes it possible to perform 2-D migration of 3-D prestack data in one direction only, so as to create 2-D prestack data regularized in offset and azimuth [1] > [2] [3]. This one direction is most likely to be the crossline direction for most cases that warrant the two-pass migration strategy in practice — thus the term crossline migration. Because of the irregular offsets and source-receiver azimuths of 3-D prestack data, an appropriate choice for crossline migration algorithm is based on Kirchhoff summation.

The output gathers from crossline migration along one inline can be associated with a hypothetical 2-D line in that inline direction with no sideswipe energy. These gathers can be processed subsequently, and a 2-D migration in the inline direction can be performed by one of the following strategies:

1. 2-D prestack time migration based on a workflow that includes 2-D DMO correction and 2-D common-offset migration (prestack time migration) to generate common-reflection-point (CRP) gathers for amplitude variation with offset (AVO) analysis (analysis of amplitude variation with offset), or
2. 2-D earth modeling in depth (earth modeling in depth) to derive an accurate image in depth (earth imaging in depth) by 2-D prestack depth migration in the inline direction that is assumed to coincide with the dominant structural dip direction.

Shown in Figure 7.4-30a is an inline section from a 3-D unmigrated zero-offset volume of data. The latter was created as part of the 3-D common-offset migration strategy for 3-D prestack time migration discussed earlier in this section. Specifically, following 3-D DMO correction and 3-D common-offset migration, data were stacked and unmigrated using the velocity field that was used in 3-D common-offset migration and an algorithm applicable to 3-D zero-offset wavefields to obtain a 3-D zero-offset wavefield. The section in Figure 7.4-30a is, therefore, a 2-D inline cross-section from the 3-D zero-offset wavefield. The 3-D zero-offset wavefield is subsequently migrated using the velocity field derived from a velocity analysis of the data after 3-D common-offset migration, and again, an algorithm applicable to 3-D zero-offset wavefields. The 2-D cross-section from the remigrated data volume along the same inline traverse is shown in Figure 7.4-30b.

The 2-D zero-offset section derived from crossline migration along the same inline traverse is shown in Figure 7.4-30c, and the image from 2-D prestack time migration of the gathers from the crossline migration is shown in Figure 7.4-30d. The unmigrated section in Figure 7.4-30c represents a 2-D zero-offset wavefield and, as such, it does not contain any sideswipe energy. Whereas the unmigrated section in Figure 7.4-30a represents a cross-section from a 3-D zero-offset wavefield, and therefore, it contains sideswipe energy. The image quality of the migrated sections in Figures 7.4-30b and 7.4-30d is comparable.

A crossline migration workflow includes the following steps:

1. Perform velocity analysis of unmigrated 3-D prestack data to derive a 3-D velocity field that is appropriate for migration in the crossline direction. Often, a single, vertically varying velocity function is adequate to use in crossline migration.
2. Perform crossline migration and create common-cell gathers along selected inline traverses. The resulting gathers will have traces with regular offset distribution and uniform source-receiver azimuth that is coincident with the inline direction. From here onward, apply the 2-D prestack time migration sequence described in prestack time migration.
3. First, perform velocity analysis at selected locations along the inline traverses for which data have been migrated in the crossline direction. Figure 7.4-31 shows the common-cell gathers and the computed velocity spectra at four analysis locations along an inline traverse. The gathers are from crossline-migrated data, and as such, have traces with regular offset distribution and uniform azimuth.
4. Apply NMO and 2-D DMO corrections to the gathers from step (b) using the velocity field derived from the analysis in step (c).
5. Perform 2-D common-offset migration using an appropriately smoothed form of the velocity field derived from the analysis in step (c).
6. Apply inverse NMO correction using the same velocity field as in step (d) and repeat the velocity analysis to derive a velocity field from the data which now have been first crossline migrated then 2-D prestack migrated in the inline direction. Figure 7.4-32 shows the common-cell gathers and the computed velocity spectra at four analysis locations along the same inline traverse as in Figure 7.4-31 after 2-D prestack time migration.
7. Apply NMO correction using the velocity field from step (f), stack the data and unmigrate using the same velocity field that was used in common-offset migration in step (e). The resulting sections shown in Figure 7.4-33 represent 2-D zero-offset wave-fields, and thus contain no sideswipe energy.
8. Use the velocity field from step (f) to remigrate the stacked sections. The resulting migrated sections shown in Figure 7.4-34 represent the final product from a 3-D prestack time migration based on the two-pass strategy outlined above.

Aside from being an integral component of a two-pass 3-D prestack time migration workflow described above, crossline migration has two other applications.

1. By restricting the aperture for Kirchhoff summation to crossline trace spacing, crossline migration can be used to perform azimuth moveout (AMO) correction of 3-D prestack data (processing of 3-D seismic data). The resulting prestack data still require DMO correction, but only as a 2-D process. As a bonus byproduct of this application, crossline migration yields prestack data with regular offset distribution which can be used as input to a 3-D prestack depth migration that requires uniformly sampled data (3-D prestack depth migration).
2. Crossline migration can be used to re-orient 3-D prestack data along a desired direction. This application is useful in merging data from a number of neighboring or partially overlapping 3-D surveys which have been conducted using different recording directions. Specifically, data from one survey can be crossline migrated to generate gathers along the inline direction associated with another survey.

## References

1. Berryhill, 1991, Berryhill, J. R., 1991, Kinematics of crossline prestack migration: Geophysics, 56, 1674–1676.
2. Canning and Gardner, 1996, Canning, A. and Gardner, G. H. F., 1996, A two-pass approximation to 3-D prestack depth migration: Geophysics, 61, 409–421.
3. Devaux et al., 1996, Devaux, V., Gardner, G. H. F., and Rampersad, T., 1996, 3-D prestack depth migration by Kirchhoff operator splitting: 66th Ann. Internal. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 455–458.
4. Baysal and Kosloff, 1998, Baysal, E. and Kosloff, D., 1998, Constructing 2-D prestack data from 3-D volume via two-pass 3-D Kirchhoff migration: Open-file technical document, Paradigm Geophysical.