# Crossline migration

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | 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

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{x^2}{(vt/2)^2} + \frac{y^2}{(vt/2)^2 - h^2} + \frac{z^2}{(vt/2)^2 - h^2} = 1,}****(**)

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

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x = x^\prime \cos\theta - y^\prime \sin\theta}****(**)

and

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y = x^\prime \sin\theta + y^\prime \cos\theta.}****(**)

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:

- 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-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.

**Figure 7.4-30**(a) A cross-section from the unmigrated 3-D stacked volume, (b) a cross-section from the migrated 3-D stacked volume, along the same inline traverse, (c) an inline stacked section derived from crossline migration, (d) 2-D migration of the section in (c). (Data courtesy Shengli Oil Field of China National Petroleum Corp.)**Figure 7.4-32**Velocity analyses at four locations along Inline 151 as in Figure 7.4-31 after 2-D prestack migration of the crossline-migrated data.**Figure 7.4-34**2-D prestack time migration of the stacked sections after crossline migration shown in Figure 7.4-33.

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:

- 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.
- 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.
- 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.
- Apply NMO and 2-D DMO corrections to the gathers from step (b) using the velocity field derived from the analysis in step (c).
- Perform 2-D common-offset migration using an appropriately smoothed form of the velocity field derived from the analysis in step (c).
- 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.
- 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.
- 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.

- 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).
- 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

- ↑
^{1.0}^{1.1}Berryhill, 1991, Berryhill, J. R., 1991, Kinematics of crossline prestack migration: Geophysics, 56, 1674–1676. - ↑ 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.
- ↑ 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.
- ↑ 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.

## See also

- 3-D prestack time migration
- 3-D DMO correction combined with 3-D common-offset migration
- 3-D migration velocity analysis
- Aspects of 3-D prestack time migration — a summary