Two-pass versus one-pass implicit finite-difference 3-D migration in practice
The two-pass approach for 3-D migration is valid strictly for a constant-velocity medium  . Since we never have a constant-velocity medium, it is reasonable to question the practical value of the two-pass 3-D migration. Now consider the case of the vertically varying velocity model shown in Figure 5.1-17.
Compute the 3-D zero-offset response of the three point scatterers buried in the subsurface below the center midpoint (32) of the center inline (32) at the layer boundaries (denoted by the asterisks in the velocity model in Figure 5.1-17). Figure 7.3-4a shows four selected inlines from this 3-D zero-offset data that consist of 63 inlines and 63 crosslines. Obviously the traveltime responses are circularly symmetric, the shallow one being an exact hyperboloid, while the two deeper ones are approximately hyperboloid.
One-pass 3-D migration collapses the energy to the apexes of the three diffraction traveltime surfaces at the center midpoint (32) of the center line (32) (Figure 7.3-4b). Now consider two-pass 3-D migration. First migrate in the inline direction (Figure 7.3-4c). The center inline (32) is migrated properly, but the lines farther from the center line are increasingly overmigrated. Moreover, within one line, say, line 2, the shallower the diffraction, the more the overmigration. This is easy to understand once we refer to the velocity model (Figure 5.1-17).
Figure 7.3-4 Two-pass versus one-pass 3-D migration. Refer to the text for details.
Figure 7.3-5 Two-pass versus one-pass 3-D migration. Time slices associated with the input data (Figure 7.3-4a) are shown on top, and the same slices from the output of one-pass 3-D migration (Figure 7.3-4b) and that of two-pass 3-D migration (Figure 7.3-4e) are shown in the middle and at the bottom, respectively.
On lines farther from the center line, the 3-D diffraction events arrive at increasingly late times; for example, on line 2 in Figure 7.3-4a. Because the velocity varies with depth, diffractions on line 2 get migrated with velocities higher than the true velocities associated with these diffractions. The diffractions on line 2 are sideswipes from the point scatterers situated beneath the center line (32). Because velocities corresponding to late times are used to migrate energy generated by shallow diffractors, these sideswipes are overmigrated. Now sort the data into crosslines (Figure 7.3-4d). It is necessary to display only those lines closer to the center line. Note that most of the energy has collapsed to the hyperbolas on the center line (32); however, because of the error induced by the first pass caused by the effect of v(z), some energy still remains on the neighboring lines. Also note that the error is more significant at shallow diffraction apexes. Finally, migration in the crossline direction collapses most of the energy to the center midpoint (32) (Figure 7.3-4e). Sort the data back to inlines (Figure 7.3-4f) and compare them with the one-pass result (Figure 7.3-4b).
Energy spreading to the neighboring midpoints results from the error of the two-pass approach in the presence of vertical velocity variations. The test results shown in Figure 7.3-4 also suggest that the larger the vertical velocity gradient, the more error induced by the two-pass approach. Another important point is that with the two-pass approach and for a v(z) function, it makes a difference in which direction migration is done first. Smearing is greatest in the direction in which the first pass of 3-D migration is performed.
Figure 7.3-5 shows selected time slices from the unmigrated volume of data (Figure 7.3-4a), and from the output of one-pass (Figure 7.3-4b) and two-pass (Figure 7.3-4e) migrations. The time slices at 400, 600, and 800 ms coincide with the apex times of the zero-offset hyperboloidal traveltime surfaces at the center line 32 (Figure 7.3-4b). Note that the time slices also show that much of the energy has collapsed to the apexes of the hyperboloids. The residual energy left behind however appears to be more on the two-pass result than on the one-pass result.
Theoretically, it is true that the two-pass method only is appropriate for velocities that vary slowly in the vertical direction and do not vary laterally. There is also the question of migration algorithm accuracy versus accuracy in the velocity field used for migration. In many practical situations, error induced by the two-pass approach probably is less than the possible errors associated with the uncertainty in the velocity field. Hence, it may be difficult on real data to tell whether imperfect imaging results from the velocity errors or the two-pass approach. In practice, we usually find that the two-pass approach yields acceptable results in areas in which dips are small, vertical velocity variations are moderate, and lateral velocity variations are within the limits of time migration.
Now consider the field data example in Figure 7.3-6. The top left is an inline and the top right is a crossline stacked section from a land 3-D survey. Migration in the inline direction yields the sections shown in the center row. Note the partially collapsed diffraction energy on the inline section (center left) and a pronounced enhancement of the diffraction energy on the crossline section (center right). Sorting the data and migrating in the crossline direction yields the results shown in the bottom row. These are the two-pass 3-D migrated sections.
A note of theoretical significance is that the crossline section after 2-D migration in Figure 7.3-6 can be considered as a true 2-D wavefield within the bounds of the zero-offset assumption. This is because it does not have sideswipe energy contained in the original unmigrated stacked section, which is a 2-D cross-section of a true 3-D wavefield, again within the bounds of the zero-offset assumption.
Figure 7.3-7 shows the comparison between the two-pass (center row) and one-pass (bottom row) 3-D migrations of the same inline and crossline sections (top row). The apparent overmigration (just below location A) seen on the two-pass result may be attributed to the error of the two-pass approach resulting from large vertical velocity gradients associated with these data.
Note that the above analysis is entirely within the framework of time migration. Two-pass 3-D migration algorithms cannot even be considered approximately accurate in situations where lateral velocity variations are large enough to warrant depth migration.
We now summarize the significant differences between one-pass and two-pass 3-D migrations based on these experiments with synthetic and field data. Also, we shall make reference to the results of the quantitative work by Dickinson  with regard to the two-pass approach. We remind ourselves that, here, we refer to the two-pass approach based on separation and the one-pass approach based on splitting of the 3-D extrapolation operator without any correction term to compensate for errors in extrapolation such as that by Li .
- Two-pass 3-D migration causes overmigration, and one-pass 3-D migration causes undermigration of steeply dipping events in a medium with significant vertical velocity gradients. This is illustrated using a circularly symmetric salt dome model in Figure 7.3-8.
- Error in two-pass 3-D migration is negligible for small dips.
- For steep dips, the error is the same order of magnitude as the error caused by a lack of precise knowledge of migration velocities.
- Largest overmigration with the two-pass approach usually occurs at around the 45-degree azimuth. No error occurs when the dip is entirely in the inline or crossline directions — the case of 2-D rather than 3-D (Figure 7.3-8).
- Largest undermigration with the one-pass approach also occurs at around the 45-degree azimuth (Figure 7.3-8).
- The two-pass approach translates the dipping event out of its true position. After the two-pass 3-D migration, the dip is correct in contrast to overmigration caused by erroneously too high velocities which not only mispositions the event but also changes its dip.
- Finally, the two-pass 3-D migration results depend on which direction the first-pass migration has been performed — if it is the inline direction, and if the velocity gradient is large enough for the two-pass approach to cause overmigration, then you get more overmigration in that direction.
The undermigration effect of one-pass 3-D migration illustrated in Figure 7.3-8 can be observed in the field data example shown in Figures 7.3-9 and 7.3-10. The time slices from the stacked volume of data infer a nearly circular salt dome (left column in Figure 7.3-9). The time slices from the desired image volume obtained from 3-D poststack phase-shift migration also exhibit the near-circular shape of the salt dome (right column in Figure 7.3-9). Note how the size of the circular events gets smaller after migration on the same time slice as was illustrated in Figure 7.3-8. Figure 7.3-10 shows a comparison of the time slices from the image volumes obtained from one-pass 3-D migration using the frequency-space 45-degree implicit scheme in split mode (left column) and 3-D phase-shift migration (right column). It may be claimed that the near-circular shape especially evident in the 1600-ms and 1700-ms time slices of the image volume from 3-D phase-shift migration is not quite preserved by the one-pass 3-D migration. Instead, we see a distorted circle much the same as sketched in Figure 7.3-8b for the one-pass 45-degree scheme. This distortion suggests undermigration mostly in the two diagonal directions caused by the one-pass schemes. At deeper time slices (Figure 7.3-10), the differences between the one-pass and phase-shift schemes are increasingly more subtle because the dip of the salt flank decreases at late times.
Figure 7.3-6 An inline (top left) and a crossline (top right) stacked section from a land 3-D survey and the results from the first pass (center row) and second pass (bottom row) of a two-pass 3-D migration process. (Data courtesy Nederlandse Aardole Maatschappij B.V.)
Figure 7.3-7 An inline (top left) and a crossline (top right) stacked section from a land 3-D survey and two-pass (center row) and one-pass (bottom row) 3-D migration results. (Data courtesy Nederlandse Aardole Maatschappij B.V.)
Figure 7.3-8 An illustration of imaging a circularly symmetric salt dome using the two-pass 45-degree, one-pass 15-degree and one-pass 45-degree algorithms. (a) Sketch of a vertical cross-section of a 3-D stacked data volume that shows the salt flank reflection before and after the desired migration, (b) time slices at time level TT′ from the results of the various 3-D migrated data volumes. See text for details.
Figure 7.3-10 Part 1: Time slices from a 3-D volume of time-migrated data derived from (left column) one-pass 3-D poststack 45-degree frequency-space algorithm based on the splitting of the 3-D extrapolation operator (Section G.1), and (right column) the 3-D poststack phase-shift algorithm (Section G.3).
Figure 7.3-10 Part 2: Time slices from a 3-D volume of time-migrated data derived from (left column) one-pass 3-D poststack 45-degree frequency-space algorithm based on the splitting of the 3-D extrapolation operator (Section G.1), and (right column) the 3-D poststack phase-shift algorithm (Section G.3).
Figure 7.3-10 Part 3: Time slices from a 3-D volume of time-migrated data derived from (left column) one-pass 3-D poststack 45-degree frequency-space algorithm based on the splitting of the 3-D extrapolation operator (Section G.1), and (right column) the 3-D poststack phase-shift algorithm (Section G.3).
Note the practical significance of the last aspect on the list. For instance, if you decide to do two-pass 3-D migration in an overthrust area, then you may want to migrate first in the strike direction where the velocity variations may generally be changing less rapidly than the dip direction to minimize the overmigration effect of the two-pass approach. Also, once you complete the migration in the direction of mild velocity variations, you can afford to revise the second pass in the direction of strong velocity variations without going back to the original stacked data.
- Ristow (1980), Ristow, D., 1980, 3-D downward extrapolation of seismic data in particular by finite-difference methods: PhD thesis, University of Utrecht, The Netherlands.
- Jacubowicz and Levin (1983), Jakubowicz, H. and Levin, S., 1983, A simple exact method of 3-D migration — theory: Geophys. Prosp., 31, 34–56.
- Dickinson (1988), Dickinson, J. A., 1988, Evaluation of two-pass three-dimensional migration: Geophysics, 53, 32–49.
- Li (1991), Li., Z, 1991, Compensating finite-difference errors in 3-D migration and modeling: Geophysics, 56, 1650–1660.
- 3-D poststack migration
- Separation versus splitting
- Impulse response of the one-pass implicit finite-difference 3-D migration
- Explicit schemes combined with the McClellan transform
- The phase-shift-plus-correction method