Residual migration

Seismic Data Analysis

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

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 $v_{2}=v^{2}-v_{1}^{2}=1802\ {\text{m/s}}$ (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 .

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 .

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.

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  Figure 4.4-25  Tests for velocity errors in frequency-space explicit migration: (a) a zero-offset section that contains a diffraction hyperbola with 2500-m/s velocity, (b) desired migration using the phase-shift method, and (c) phase-shift migration using 10 percent lower velocity; migrations using 10 percent lower velocity and frequency-space explicit schemes with (d) 30-degree, (e) 50-degree, and (f) 70-degree accuracy.

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  Figure 4.4-26  Tests for velocity errors in frequency-space explicit migration: (a) a zero-offset section that contains a diffraction hyperbola with 2500-m/s velocity, (b) desired migration using the phase-shift method, and (c) phase-shift migration using 10 percent lower velocity; migrations using 10 percent higher velocity and frequency-space explicit schemes with (d) 30-degree, (e) 50-degree, and (f) 70-degree accuracy.

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  Figure 4.4-27  Tests for velocity errors in frequency-space explicit migration: (a) a zero-offset section that contains dipping events with 3500-m/s velocity, (b) desired migration using the phase-shift method, and (c) phase-shift migration using 10 percent lower velocity; migrations using 10 percent lower velocity and frequency-space explicit schemes with (d) 30-degree, (e) 50-degree, and (f) 70-degree accuracy.

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  Figure 4.4-28  Tests for velocity errors in frequency-space explicit migration: (a) A zero-offset section that contains dipping events with 3500-m/s velocity, (b) desired migration using the phase-shift method, and (c) phase-shift migration using 10 percent higher velocity; migrations using 10 percent higher velocity and frequency-space explicit schemes with (d) 30-degree, (e) 50-degree, and (f) 70-degree accuracy.

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  Figure 4.3-3  A sketch of the diffraction D and steeply dipping event before (B) and after (A) desired migration from the sections in Figure 4.3-2. The diffraction and the dipping event after finite-difference migration using the parabolic equation are denoted by FD − D and FD − B, respectively.

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.

$\sin \theta ={\frac {vk_{x}}{\omega }}$

(D-8c)

$v_{2}={\sqrt {v^{2}-v_{1}^{2}}},$

(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).

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  Figure 4.4-29  From top to bottom, desired migration as in Figure 4.2-15b and frequency-space explicit migration with 30-degree, 50-degree, and 70-degree accuracy, and using interval velocities derived from 100 percent of the rms velocities. The input CMP stack is shown in Figure 4.2-15a.

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  Figure 4.4-30  From top to bottom, desired migration as in Figure 4.2-15b and tests for velocity errors in frequency-space explicit migration with 30-degree, 50-degree, and 70-degree accuracy, and using interval velocities derived from 90 percent of rms velocities. The input CMP stack is shown in Figure 4.2-15a.

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  Figure 4.3-19  (a) A zero-offset section that contains three diffraction hyperbolas with a vertically varying velocity, (b) 15-degree finite-difference migration using a depth step of 20 ms; the output from the last stage of cascaded application of the 15-degree migration using (c) 4 cascades with 80-ms depth step, (d) 10 cascades with 200-ms depth step, (e) 20 cascades with 400-ms depth step, and (f) desired migration using the phase-shift method.

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  Figure 4.3-20  (a) A zero-offset section that contains three diffraction hyperbolas with a vertically varying velocity; four-stage cascaded migration using the phase shift-method with 80-ms depth step: (b) first-stage, (c) second stage, (d) third stage, (e) fourth stage, and (f) desired migration using the phase-shift method only once with 20-ms depth step. Compare the fourth stage (e) with the output of the four-stage cascaded migration using a dip-limited algorithm as in Figure 4.3-19c.

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 ${\bar {v}}(z),$ which is obtained by averaging the spatially varying velocity field v(x, z) and modifying it to meet the requirement that ${\bar {v}}(z)<v\left(x,\ z\right).$
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 ${\sqrt {v^{2}\left(x,\ z\right)-{\bar {v}}^{2}\left(z\right)}}$ (Section D.8).

See also

• Maximum dip to migrate
• Depth step size
• Velocity errors
• Stolt stretch factor
• Wraparound

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