# Velocity errors

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

## Kirchhoff migration in practice

We now examine the response of Kirchhoff migration to velocity errors. Figure 4.2-10 shows the diffraction hyperbola and migrations using the 2500 m/s medium velocity, and 5, 10, and 20 percent lower velocities. With increasingly lower velocities, the diffraction hyperbola is collapsed less and less taking the shape of a frown — it is undermigrated.

Figure 4.2-11 shows the same diffraction hyperbola and migrations using the 2500 m/s medium velocity, and 5, 10, and 20 percent higher velocities. With increasingly higher velocities, the the diffraction hyperbola is inverted more and more taking the shape of a smile — it is overmigrated.

The under-and overmigration effects resulting from the use of erroneously low or high velocities on the dipping events model are seen in Figures 4.2-12 and 4.2-13, respectively. Label the correct position of the event with the steepest dip from the desired migration on the results of migrations with different velocities and note the event mispositioning caused by erroneously low and high velocities. Compare with the desired migration and also note that the steeper the dip the more the under- and overmigration effect. Sensitivity of migration to velocity errors can be measured quantitatively via equations 1 and 2.

 ${\displaystyle d_{x}={\frac {v^{2}t}{4}}{\frac {\Delta t}{\Delta x}}}$ (1)

 ${\displaystyle d_{t}=t\left[1-{\sqrt {1-\left({\frac {v\Delta t}{2\Delta x}}\right)^{2}}}\,\right]}$ (2)

From the migrated sections in Figure 4.2-14, note that the bow tie becomes increasingly less resolved at lower velocities; this indicates undermigration.

Figure 4.2-15 shows a CMP-stacked section and the desired migration. The steep left flank of the salt dome has been imaged with acceptable accuracy. The accuracy of the imaging of the slightly overturned right flank can only be inferred by the lateral positioning of the gently dipping reflections in the vicinity of the salt flank.

Figure 4.2-16 shows results of Kirchhoff migration of the stacked section in Figure 4.2-15 using velocities lower than what may be optimum for imaging. While the undermigration of the left flank of the salt dome is not so evident, the steeply dipping reflection off the right flank intersects the gently dipping reflections associated with the surrounding strata, thus providing a clue for undermigration.

Figure 4.2-17 shows results of Kirchhoff migration of the stacked section in Figure 4.2-15 using velocities higher than what may be optimum for imaging. While overmigration effects may be marginal on the section with a small velocity error (105 percent of optimum velocities), migration with higher velocity errors (110 and 120 percent of optimum velocities) shows signs of overmigration in the form of crossing events along the left flank of the salt diapir. Under- and overmigration effects caused by large velocity errors often are detectable; nevertheless, small velocity errors can cause subtle effects making it difficult to judge whether there is under- or overmigration. Uncertainties in migration velocities inevitably cause uncertainties in the interpretation made from migrated sections or volumes of data. For instance, the shape of the salt diapir inferred from the results shown in Figures 4.2-16 and 4.2-17 varies significantly depending on the percent velocity errors.

## Finite-difference migration in practice

Figure 4.3-10 shows the diffraction hyperbola and its migration using the 2500 m/s medium velocity and 5, 10, and 20 percent lower velocities. When velocities lower than medium velocity are used, the diffraction hyperbola gets undermigrated by the 15-degree algorithm more than it would be by an algorithm with no dip limitation (compare with Figure 4.2-10).

Figure 4.3-11 shows the diffraction hyperbola and its migration using the 2500 m/s medium velocity and 5, 10, and 20 percent higher velocities. When velocities higher than medium velocity are used, the diffraction hyperbola gets overmigrated less by the 15-degree algorithm than it would be by an algorithm with no dip limitation (compare with Figure 4.2-11). Moreover, note the increase in dispersive noise as a result of overmigration.

Figure 4.3-12 is the dipping events model with migrations using the 3500 m/s medium velocity, and 5, 10, and 20 percent lower velocities. For comparison, label the correct position of the event with the steepest dip from the desired migration on the results of migration with different velocities. As in any other migration method, velocity errors cause events to be mispositioned at increasingly steeper dips. The undermigration effect of lower velocities is reinforced by the inherently undermigrating nature of the 15-degree algorithm. As a result, dipping events are undermigrated more by the 15-degree algorithm in contrast than by an algorithm with no dip limitation (compare with Figure 4.2-12).

Figure 4.3-13 is the dipping events model with migrations using the 3500 m/s medium velocity, and 5, 10, and 20 percent higher velocities. For comparison, again, label the the correct position of the event with the steepest dip from the desired migration on the results of migration with different velocities. The overmigration effect of higher velocities is counteracted by the inherently undermigrating nature of the 15-degree algorithm. As a result, dipping events are not overmigrated by the 15-degree algorithm as much as they would be by an algorithm with no dip limitation (compare with Figure 4.2-13).

Velocity error test results on field data using the 15-degree implicit scheme are shown in Figures 4.3-14 and 4.3-15. Figure 4.3-16 is a sketch of the under- and overmigration effects. As noted above with the dipping events model (Figures 4.3-12 and 4.3-13), when using velocities greater than medium velocities, overmigration is not as pronounced with the 15-degree finite-difference migration as it is with a 90-degree algorithm, such as the Kirchhoff or phase-shift method. On the other hand, when using velocities lower than medium velocities, the undermigration effect is more pronounced with the 15-degree finite-difference migration in comparison with a 90-degree algorithm (compare Figure 4.3-16 with Figure 4.5-13).

At first, it may appear to be sensible to compensate for the undermigration caused by a low-dip algorithm by adjusting migration velocities. For example, the best match between the desired migration and the 15-degree finite-difference results for the dipping event in Figure 4.3-16 occurs when 10 percent higher velocities are used in the finite-difference migration. While for one dip this adjustment may be acceptable, for another dip it may not be. Therefore, deficiencies of a migration algorithm should not be compensated for by making modifications to the velocity field for migration.

## Frequency-space migration in practice

We now examine the response of a steep-dip implicit algorithm to velocity errors. Figure 4.4-12 shows the constant-velocity diffraction hyperbola and its migrations using the medium velocity, and 5, 10, and 20 percent lower velocities. When velocities lower than medium velocity are used, the diffraction hyperbola is undermigrated (Figure 4.4-12d,e,f), but not as much as in the case of the 15-degree equation (Figure 4.3-10d,e,f). On the other hand, the diffraction hyperbola gets undermigrated by the steep-dip algorithm more than it would be by an algorithm with no dip limitation (compare with Figure 4.2-10).

Figure 4.4-13 shows the constant-velocity diffraction hyperbola and its migrations using the medium velocity, and 5, 10, and 20 percent higher velocities. When velocities higher than medium velocity are used, the diffraction hyperbola is overmigrated (Figure 4.4-13d,e,f), more so than with the 15-degree equation (Figure 4.3-11d,e,f). On the other hand, the diffraction hyperbola gets overmigrated by the steep-dip algorithm less than it would be by an algorithm with no dip limitation (compare with Figure 4.2-11). Whatever the velocity used for migration, dispersive noise is present persistently in finite-difference results (Figures 4.4-12 and 4.4-13).

Figure 4.4-14 shows the constant-velocity dipping events model with migrations using the medium velocity, and velocities that are 5, 10, and 20 percent lower. For comparison, label the correct position of the event with the steepest dip from the desired migration on the results of migration with different velocities. As in any other migration method, velocity errors cause events to be increasingly more mispositioned at steeper dips. The undermigration effect of lower velocities is reinforced by the inherently undermigrating nature of the steep-dip algorithm, although not as much as in the case of the 15-degree algorithm (Figure 4.3-12). As a result, dipping events are more undermigrated by the steep-dip algorithm in contrast to an algorithm with no dip limitation (compare with Figure 4.2-12).

Figure 4.4-15 is the dipping events model with migrations using the 3500 m/s medium velocity, and 5, 10, and 20 percent higher velocities. For comparison, again, label the correct position of the event with the steepest dip from the desired migration on the results of migration with different velocities. The overmigration effect of higher velocities is counteracted by the inherently undermigrating nature of the steep-dip algorithm, although not as much as in the case of the 15-degree algorithm (Figure 4.3-13). As a result, dipping events are not overmigrated by the steep-dip algorithm as much as they would be by an algorithm with no dip limitation (compare with Figure 4.2-13).

The nature of the precursive and postcursive dispersion along steeply dipping events in the output from finite-difference migration, theoretically depends on the type of differencing scheme used in approximating the differential operators associated with the scalar wave equation. In principle, an implicit finite-difference migration implemented in the frequency-space domain yields less dispersion compared to an implicit scheme implemented in the time-space domain. This is because the former requires differencing in x and z, while the latter requires differencing in t in addition to x and z. In practice, however, dispersive noise contaminates migration results from implicit schemes almost without exception.

Velocity error tests on field data using the 65-degree implicit scheme are shown in Figures 4.4-16 and 4.4-17. With velocities erroneously too low, note the pronounced undermigration of the steep salt flank. With velocities erroneously too high, note the overmigration of the steep salt flank and the crossing of reflections at the vicinity of the crest of the salt structure.

Figure 4.4-25 shows migrations of a zero-offset section that contains a diffraction hyperbola using a frequency-space explicit algorithm based on 30-degree, 50-degree and 70-degree extrapolation filters and a velocity that is 90 percent of the medium velocity. For comparison, desired migration using the phase-shift method with the medium velocity and 90 percent of the medium velocity are also shown in the same figure. Migration with an erroneously low velocity yields the undermigrated form of the diffraction hyperbola as seen in Figure 4.4-25c. Note, however, the dip-limited explicit schemes appear to cause less undermigration compared to the phase-shift method with 90-degree accuracy. This behavior is in contradiction to intuition — the undermigration effect of an erroneously low velocity is reinforced by a dip-limited algorithm. In fact, this intuitive effect was demonstrated by the steep-dip implicit scheme (Figure 4.4-12).

The deceptive behavior of the explicit schemes that contradicts our intuition can be explained by the fact that these schemes filter out the energy at high wavenumbers (Figures 4.4-19 and 4.4-20). As a result, the steep limbs of the undermigrated diffraction hyperbola are truncated (Figure 4.4-25). This in turn makes the result of migration using lower velocity appear less undermigrated in case of an explicit scheme compared to the case of the phase-shift method with 90-degree accuracy. In fact, if we apply a wavenumber filter to reject the high wavenumbers from the output of phase-shift migration (Figure 4.4-25c), the result would resemble the output of an explicit scheme (Figures 4.4-25d,e,f). Also note from Figure 4.4-25 that the low-dip explicit scheme manifests the effect of undermigration much less than the steep-dip explicit scheme. This is because the wavenumber filtering effect is more severe for the low-dip explicit scheme (Figure 4.4-20).

Figure 4.4-26 shows migrations of a zero-offset section that contains a diffraction hyperbola using a frequency-space explicit algorithm based on 30-degree, 50-degree and 70-degree extrapolation filters and a velocity that is 110 percent of the medium velocity. Again, for comparison, desired migration using the phase-shift method with the medium velocity and 110 percent of the medium velocity are also shown in the same figure. Migration with an erroneously high velocity yields the overmigrated form of the diffraction hyperbola as seen in Figure 4.4-26c.

We make the following observations from Figure 4.4-26:

1. The dip-limited explicit schemes, much like the implicit schemes (Figure 4.4-13), cause less overmigration compared to the phase-shift method with 90-degree accuracy.
2. The low-dip explicit scheme manifests the effect of overmigration much less than the steep-dip explicit scheme.
3. The wavenumber filtering effect (Figure 4.4-20) further truncates the steep limbs of the overmigrated hyperbola.

The interplay of the three factors results in the response to velocity errors by the explicit schemes as shown in Figure 4.4-26.

Tests for velocity errors are repeated for a zero-offset section that contains a set of dipping events as shown in Figures 4.4-27 and 4.4-28. For comparison, label the correct position of the event with the steepest dip from the desired migration on the results of migration with different velocities. The residual diffractions off the end of the dipping reflectors on the migrated sections from the explicit schemes have been truncated by the wavenumber filtering effect of the extrapolation filters. This filtering effect is most prominent in the case of the explicit scheme with the low-dip limit and erroneously high velocity (Figure 4.4-28d).

Field data examples for tests of velocity errors for the explicit schemes are shown in Figures 4.4-29, 4.4-30, and 4.4-31. First, note the better imaging of the salt flanks by the steep-dip explicit scheme compared to the low-dip explicit scheme (Figure 4.4-29). The undermigration effect of erroneously low velocities (Figure 4.4-30) and the overmigration effect of erroneously high velocities (Figure 4.4-31) may be compared with the results of migration using optimum velocities (Figure 4.4-29).

We shall complete this section by reviewing the performance of Kirchhoff summation, finite-difference frequency-space implicit, and frequency-space explicit schemes with various dip limits. Figure 4.4-32 shows the compilation of the results of migration of a zero-offset section that contains a diffraction hyperbola. For comparison, desired migration using the phase-shift method is included in the panel. The dip-limited nature of the implicit and explicit schemes is manifested by the incomplete focusing of the energy at the apex of the diffraction hyperbola (Figures 4.4-32c,g,h). The undermigration effect caused by the dip limitation is alleviated by using a steep-dip explicit scheme (Figure 4.4-32j). A steep-dip implicit scheme, on the other hand, can actually overshoot in the opposite direction and cause overmigration (Figure 4.4-32f). The differencing approximations are manifested by the dispersive noise (Figure 4.4-32c).

Figure 4.4-33 shows the compilation of the results of migration of a zero-offset section that contains a set of dipping events. Again, for comparison, desired migration using the phase-shift method is included in the panel. The dip-limited nature of the implicit and explicit schemes is manifested by the undermigration of the steeply dipping events (Figures 4.4-33c,g,h). This effect is alleviated by using a steep-dip explicit scheme (Figure 4.4-33j). A steep-dip implicit scheme, on the other hand, can actually overshoot in the opposite direction and cause overmigration (Figure 4.4-33f). The differencing approximations are manifested by the dispersive noise accompanying the steeply dipping events (Figure 4.4-33c,d,e,f).

## Frequency-wavenumber migration in practice

We now examine the response of phase-shift migration to velocity errors. Figure 4.5-7 shows the diffraction hyperbola and phase-shift migrations using the 2500 m/s medium velocity, and 5, 10, and 20 percent lower velocities. The lower the velocity, the more the diffraction hyperbola is undermigrated. These results are used as a benchmark to evaluate the response of the other migration algorithms discussed in this chapter. Specifically, compare Figure 4.5-7 with Figures 4.2-10 (the case of Kirchhoff migration), 4.3-10 (the case of implicit finite-difference migration), 4.4-12 (the case of implicit frequency-space migration), and 4.4-25 (the case of explicit frequency-space migration), note how the various algorithms respond to velocity errors with significant differences.

Figure 4.5-8 shows the same diffraction hyperbola and phase-shift migrations using the 2500 m/s medium velocity, and 5, 10, and 20 percent higher velocities. The higher the velocity, the more the diffraction hyperbola is overmigrated. These results are used to benchmark the response to velocity errors by the other migration algorithms — Figures 4.2-11 (the case of Kirchhoff migration), 4.3-11 (the case of implicit finite-difference migration), 4.4-13 (the case of implicit frequency-space migration), and 4.4-26 (the case of explicit frequency-space migration).

The under- and overmigration effects caused by the use of erroneously low or high velocities on the dipping events model are seen in Figures 4.5-9 and 4.5-10, respectively. Label the correct position of the event with the steepest dip from the desired migration on the results of migrations with different velocities, and note the event mispositioning caused by erroneously low and high velocities. Recall that sensitivity of migration to velocity errors can be measured quantitatively via equations 1 and 2. The test results shown in Figures 4.5-9 and 4.5-10 are used to benchmark the response to velocity errors by the other migration algorithms — Figures 4.2-12 and 4.2-13 (the cases of Kirchhoff migration), 4.3-12 and 4.3-13 (the cases of implicit finite-difference migration), 4.4-14 and 4.4-15 (the cases of implicit frequency-space migration), and 4.4-27 and 4.4-28 (the cases of explicit frequency-space migration).

 ${\displaystyle d_{x}={\frac {v^{2}t}{4}}{\frac {\Delta t}{\Delta x}}}$ (1)

 ${\displaystyle d_{t}=t\left[1-{\sqrt {1-\left({\frac {v\Delta t}{2\Delta x}}\right)^{2}}}\,\right]}$ (2)

An aspect of phase-shift migration uniquely different from others is its exceptional quality of output. As noted from Figures 4.5-7, 4.5-8, 4.5-9, and 4.5-10, phase-shift migration produces no dispersive noise since it does not involve any differencing of differential operators. Instead, the entire design of extrapolation operator and application of migration are in the frequency-wavenumber domain. The results do not suffer from any dip limitation since the phase-shift method is based on an extrapolation filter that is exact for all dips up to 90 degrees (Figure 4.5-1). Of course, we must also remind ourselves of the fact that the phase-shift method is limited to velocities that vary only in the vertical direction.

Figures 4.5-11 and 4.5-12 are field data examples of phase-shift migration using erroneously low and high velocities, respectively. Figure 4.5-13 shows a sketch of the combined results of these migrations. Clearly, velocities that are too low cause undermigration of the steeply dipping event that defines the flank of the salt diapir and incomplete collapse of the diffraction off the tip of the salt diapir. Velocities that are too high cause overmigration as manifested by the crossing events at the vicinity of the crest of the salt diapir.

Figure 4.5-14 shows the results of phase-shift migration of the stacked section in Figure 4.2-15 using velocities lower than what may be optimum for imaging. The undermigration of the left flank of the salt dome is not so evident. However, the steeply dipping reflection off the right flank intersects the gently dipping reflections associated with the surrounding strata — an indication of undermigration.

Figure 4.5-15 shows results of phase-shift migration of the stacked section in Figure 4.2-15 using velocities higher than what may be optimum for imaging. Migration with erroneously high velocities (110 and 120 percent of optimum velocities) shows signs of overmigration in the form of crossing events along the left flank of the salt diapir.

## Principles of dip-moveout correction

What happens if the NMO correction that precedes the DMO correction were applied with the wrong velocity? The DMO process requires an input that is NMO corrected using the medium velocity (equation 2). Thus, we try to pick a vertically varying velocity function from the flattest part of the section for NMO correcting the data. The optimum stacking velocities are not used because they depend on dip. However, it is the stacking velocities that are picked from conventional velocity analyses. There is always the possibility that an accurate dip-independent velocity function will not be determined for NMO correcting the input data before DMO correction. The constant-velocity model in Figure 5.1-3 is used to examine this problem.

 ${\displaystyle d_{t}=t\left[1-{\sqrt {1-\left({\frac {v\Delta t}{2\Delta x}}\right)^{2}}}\,\right]}$ (2)

Assume that the velocity used for NMO correction is 20 percent higher than the velocity that should be used — the 3000-m/s medium velocity. Start with the CMP gathers in Figure 5.1-5b and apply the NMO correction using the incorrect velocity (3600 m/s). The results are shown in Figure 5.1-14a. Note the undercorrection at some gathers due to the high velocity used. Follow the DMO processing sequence described earlier. Note that events no longer are aligned after the first NMO and DMO corrections (Figure 5.1-14d). Therefore, the stack obtained from these gathers is not expected to be any better than the conventional stack derived from the gathers in Figure 5.1-14a. The stacked sections are shown in Figure 5.1-15.

Perhaps the CMP stack can be improved by repicking the velocities after DMO correction. To test this idea, consider the following procedure. First, apply inverse NMO correction (Figure 5.1-14e) to the gathers with the velocity function that was used in the first NMO correction step (Figure 5.1-14a). Then, assuming we pick the correct velocity function, use it on the second NMO correction (Figure 5.1-14f). Significant improvement is seen when these gathers are stacked (Figure 5.1-16c). For a fair comparison, refer to the conventional stack with the 3000 m/s repicked velocity in Figure 5.1-16b. Although not shown, similar conclusions are reached from tests using velocities that are too low for NMO correction before DMO processing.