Residual migration

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

The constant-velocity Stolt algorithm has useful applications in residual migration as described here and migration velocity analysis as described in dip-moveout correction and prestack migration. Consider a zero-offset section that contains a set of dipping events as shown in Figure 4.5-25a. The desired migration is obtained by using the medium velocity of v = 3500 m/s as shown in Figure 4.5-25b. Suppose, instead, that you migrate using a velocity of 3000 m/s. The resulting migrated section is shown in Figure 4.5-25c. Label the event with the steepest dip (AB) from the desired migration on the migrated section with v1 = 3000 m/s, and note the undermigration of the dipping events. By migrating the already migrated section using a velocity of ${\displaystyle v_{2}=v^{2}-v_{1}^{2}=1802\ {\text{m/s}}}$ (Section D.8), we get the section shown in Figure 4.5-25d. Note that this section obtained from the two-stage migration using velocities 3500 m/s and 1802 m/s is equivalent to the one-stage migration using a velocity of 3500 m/s. The second stage of the two-stage migration using a velocity of 1802 m/s is called residual migration [1].

So, what is the practical use of residual migration? It can improve upon the result of migration using a dip-limited finite-difference algorithm. Figure 4.5-26a shows a zero-offset section that consists of three point scatterers in a layered earth model with vertically varying velocity field. A 15-degree dip-limited finite-difference migration has difficulty collapsing these diffractions (Figure 4.5-26b). Now, first migrate the zero-offset section with the constant-velocity Stolt algorithm using the lowest value, 2000 m/s, in the vertically varying velocity function. The result is shown in Figure 4.5-26d. Then, take this section and migrate it again (Figure 4.5-26e) using the appropriate residual velocity (Section D.8) and the 15-degree finite-difference algorithm. When compared with the single-stage finite-difference migration (Figure 4.5-26b), note the superior performance of the residual migration. Also compare this with the desired migration using the phase-shift method (Figure 4.5-26c). The important point to keep in mind is that input to residual migration (the second stage) must be data which have been migrated (first stage) using a constant velocity [1].

A field data example is shown in Figure 4.5-27 with a sketch of the migration results in Figure 4.5-28. The single-stage 15-degree finite-difference result shows the typical undermigrated character (Figure 4.3-3). The 1500-m/s constant-velocity Stolt migration followed by the finite-difference migration seems to produce an output that is reasonably close to the desired migration.

A limitation of residual migration is that an adequate migration is not always achieved since the first-stage migration requires constant velocity which may be far off from the velocity field associated with the data. This is the case in Figure 4.5-27, since after residual migration, the dipping event still is slightly undermigrated (see the sketch in Figure 4.5-28). Undermigration occurs because the apparent dip perceived by the second-stage migration still may be too large to be handled accurately. From equation (D-8c) note that the lower the velocity used in migration, the smaller the dip that is perceived by migration. If the residual velocity function given by equation (D-96b) is not too different from the original velocity function because of a large vertical gradient, then residual migration may not be adequate.

 ${\displaystyle \sin \theta ={\frac {vk_{x}}{\omega }}}$ (D-8c)

 ${\displaystyle v_{2}={\sqrt {v^{2}-v_{1}^{2}}},}$ (D-96b)

Residual migration is different from cascaded migration that is discussed in finite-difference migration in practice. The latter involves application of a dip-limited algorithm, such as an implicit finite-difference scheme, repeatedly. Whereas residual migration involves the application of a dip-limited algorithm only once to data which already have been migrated using a constant-velocity Stolt migration. In practice, Stolt migration can be replaced with phase-shift migration and a vertically varying velocity function with a gently varying gradient to accommodate the constant-velocity requirement for the first-stage migration.

Whether it is residual or cascaded migration, the theoretical requirement is for constant velocity to be used at each stage preceding the last stage. Departure from this restriction will always limit the implementation of residual and cascaded migration. Figure 4.5-29 shows a zero-offset section that contains a set of dipping events with 3500-m/s constant velocity. First, migrate with a constant velocity of 2500 m/s; then, perform a cascade of four migrations with appropriate residual velocities (Section D.8) to obtain the accurate image that is equivalent to the result from the desired migration using the phase-shift method applied only once with the medium velocity. When the same exercise is repeated using a dip-limited implicit finite-difference algorithm, results are not satisfactory even with constant velocity (Figure 4.5-30). When a variable velocity is used at each stage preceding the last stage the dip-limited algorithm causes overmigration (Figure 4.3-19), while the phase-shift method with no dip limit yields accurate result (Figure 4.3-20).

Despite the limitations mentioned above, residual migration is used in practice in the following mode:

1. Perform phase-shift migration using a vertically varying velocity function ${\displaystyle {\bar {v}}(z),}$ which is obtained by averaging the spatially varying velocity field v(x, z) and modifying it to meet the requirement that ${\displaystyle {\bar {v}}(z)
2. Follow with a residual migration using a dip-limited implicit or explicit frequency-space finite-difference migration with a residual velocity field equal to ${\displaystyle {\sqrt {v^{2}\left(x,\ z\right)-{\bar {v}}^{2}\left(z\right)}}}$ (Section D.8).

References

1. Rothman et al., 1985, Rothman, D., Levin, S., and Rocca, F., 1985, Residual migration: Applications and limitations: Geophysics, 50, 110–126.