Field data examples
The deconvolution parameters discussed in Oz Yilmaz's Seismic Data Analysis are examined using field data examples below. Application of statistical deconvolution to pre- and poststack data is discussed. Additionally, application of deterministic deconvolution to marine data to convert the recorded source signature to its minimum-phase equivalent, and to land data recorded using a vibroseis source to convert the autocorrelogram of the sweep signal to its minimum-phase equivalent are addressed in the pages below.
Velocity analysis and statics corrections
We shall analyze field data with three different nearsurface characteristics. Specifically, near-surface models with combinations of irregular topography and refractor geometry are examined. Refraction statics solutions are based on the variable-thickness scheme based on equation (52) and residual statics solutions are based on equation (25), both solved by the generalized linear inversion schemes (Sections C.4 and C.8).
The first field data example is from an area with nearly flat topography and presumably irregular base of weathering. Shown in Figure 3.4-29a is a CMP-stacked section based on elevation statics corrections that involved a flat datum and constant weathering velocity. Note the presence of traveltime distortions along the major reflections down to 2 s caused by the unresolved long-wavelength statics anomalies. We also note very short-wavelength traveltime distortions, much less than a cable length. This latter component of the statics can be resolved by surface-consistent residual statics corrections as shown in Figure 3.4-29b. Although the CMP stacking quality has been improved after the residual statics corrections, the long-wavelength statics anomalies remain unresolved.
Figure 3.4-30 shows plots of the first-break picks from the far-offset arrivals associated with the refracted energy. While most of the first-break picks consistently follow a linear moveout from shot to shot, note that there are some local deviations that indicate a moderate degree of complexity in the near surface. Figure 3.4-31 shows the CMP-stacked section after the application of refraction statics using the generalized reciprocal method. Compare with Figure 3.4-29a and note the significant elimination of long-wavelength statics. Also plotted are the intercept time anomalies at all shot-receiver stations. Recall that equations (50a, 50b) yield multiple values of intercept time estimates at each station. These multiple values need to be reduced to unique intercept time values at each station so as to be able to estimate the thickness of the weathering layer at each station, uniquely. The statics solution at all shot and receiver stations shows that the generalized reciprocal method can correct for all wavelengths of statics caused by undulations along the base of the weathering layer. Any remaining (residual) very short-wavelength statics should be corrected for by using a reflection-based method (Residual statics corrections).
Figure 3.4-32a shows the CMP-stacked section after the application of refraction statics corrections based on the variable-thickness, least-squares scheme (equation 52a). Compare this result with Figure 3.4-29a and note that the long-wavelength statics anomalies have been removed. Also, note that both the generalized reciprocal method (Figure 3.4-31a) and the least-squares method (Figure 3.4-32a) yield comparable results. The section in Figure 3.4-32a can further be improved by applying residual statics corrections to remove the short-wavelength statics components (Figure 3.4-32b).
The results of the least-squares statics estimates are summarized in Figure 3.4-33. The weathering velocity was assumed to be 450 m/s. Frame 1 shows the estimated intercept times as a function of the shot/receiver station number. Frame 2 shows the pick fold, namely the number of picks in each shot and receiver gather. Note the tapering of the pick fold at both ends of the line.
A quantitative measure of the accuracy of the least-squares solution is the sum of the differences between the observed picks tij and the modeled traveltimes (equation 52a) over each shot and receiver gather. These cumulative residual time differences over each shot and receiver are plotted in frame 3 of Figure 3.4-33. Large residuals often are related to bad picks. Nevertheless, even with good picks, there may be large residuals attributable to the inappropriateness of the model assumed for the near-surface.
Figure 3.4-33 also shows the estimated thickness of the weathering layer at all shot-receiver stations (frame 4). Finally, the computed statics and the near-surface model are shown in frames 5 and 6, respectively.
The next field data example is from an area with irregular topography associated with a sand dune and presumably a near-flat base of weathering. Figure 3.4-34a shows the CMP-stacked section with elevation statics corrections. Note the severe distortions of the geometry of shallow reflections and a very poor signal-to-noise ratio in the central part of the section. Residual statics corrections (Figure 3.4-34b) cannot improve the interpretation, especially in the center of the line where the first breaks show significant departures from a consistent linear moveout (Figure 3.4-35).
Figure 3.4-36 shows the CMP-stacked section with the application of refraction statics corrections using the generalized reciprocal method. Again, note the multiple-valued intercept time values at shot/receiver stations. The statics solution based on the reduced intercept times shows a significant medium- to long-wavelength variations. After the application of these statics corrections, the near-surface effects on the reflector geometries have been largely removed (compare with Figure 3.4-34a).
Figure 3.4-29 (a) A CMP stack with elevation statics corrections, (b) same as in (a) with residual statics corrections. (Data courtesy Nederlandse Aardolie Maatschappij B.V.)
Figure 3.4-31 (a) A CMP stack with refraction statics applied using the generalized reciprocal method (compare with Figures 3.4-29a and Figure 3.4-32a), (b) estimated multiple intercept times at each shot-receiver station and the computed shot (open squares) and receiver (solid dots) statics.
Figure 3.4-33 Results of the least-squares refraction statics estimated from and applied to the data in Figure 3.4-32. In each frame, shot attribute is denoted by × and receiver attribute is denoted by the vertical bars. R is the refractor that represents the base of the weathering layer. See text for details.
Figure 3.4-34 (a) A CMP stack with elevation statics corrections, (b) same as in (a) with residual statics corrections.
Figure 3.4-37a shows the CMP-stacked section after the application of refraction statics corrections based on the least-squares method. Compare with Figure 3.4-34a and note the significant improvement in the center of the line. This section can be improved further by applying residual statics corrections and thus removing the short-wavelength statics components (Figure 3.4-37b).
The results of the least-squares statics estimates are summarized in Figure 3.4-38. The weathering velocity was assumed to be 800 m/s. (The description of the frames in Figure 3.4-38 is the same as that of Figure 3.4-33.)
The third field data example is from an area with an abrupt change in topography and presumably surface-following the base of weathering. The CMP-stacked section with elevation statics corrections is shown in Figure 3.4-39a. Residual statics corrections significantly improve the stacking quality (Figure 3.4-39b); but the long-wavelength statics anomalies remain on the section and appear as spurious structural anomalies.
Figure 3.4-40 shows the first-break picks from the far-offset arrivals associated with the refracted energy. Figure 3.4-41 shows the CMP-stacked section with the refraction statics applied using the generalized reciprocal method and the first-break picks in Figure 3.4-40. Compare with Figure 3.4-39b and note the removal of the spurious structural discontinuity along the strong reflection just above 2 s on the left half of the section.
By using the first-break picks shown in Figure 3.4-40, the variable-thickness least-squares parameters for the near-surface were computed (equation 52a). The corresponding CMP-stacked section is shown in Figure 3.4-27a. Note the elimination of the spurious structural discontinuities seen in Figure 3.4-39b between 1 and 2 s. The CMP stacked section can be improved further by applying residual statics corrections (Figure 3.4-42b).
The results of the least-squares statics estimates are summarized in Figure 3.4-43. For the variablethickness estimate, the weathering velocity was assumed to be 900 m/s. (The description of the frames in Figure 3.4-43 is the same as that of Figure 3.4-33.)
Figure 3.4-44 is a stacked section with only the field statics applied. The pull-up at midpoint location A probably is caused by a long-wavelength statics anomaly. Start with CMP gathers (Figure 3.4-45a) and apply linear-moveout (LMO) correction (Figure 3.4-45b). Assuming that the first breaks correspond to a near-surface refractor, we use the estimated velocity from the first breaks (usually from a portion of the cable) to apply the LMO correction. The CMP-refraction stack of the shallow part of the data after the LMO correction is shown in Figure 3.4-45c. This section is the equivalent of the pilot trace section that is associated with the reflection-based statics corrections. (An example of this is shown in Figure 3.3-37.)
Traveltime deviations are estimated from the LMO-corrected gathers (Figure 3.4-45b) and are decomposed into shot and receiver intercept time components based on equation (52a). These intercept times are used to compute shot and receiver static shifts, which are then applied to the CMP gathers shown in Figure 3.4-45a. A comparison of the CMP-refraction stack section with (Figure 3.4-45d) and without (Figure 3.4-45c) refraction statics corrections clearly indicates removal of the significant long-wavelength statics anomaly centered at midpoint location A (Figure 3.4-44). The CMP-stacked section after the refraction statics corrections shown in Figure 3.4-46 no longer contains the false structure (compare with Figure 3.4-44). This long-wavelength anomaly cannot be removed by reflection statics corrections alone (Figure 3.4-47). Nevertheless, the residual statics corrections resolved the short-wavelength statics components that were present in the data. By cascading the two corrections — refraction and residual statics, we get the improved section in Figure 3.4-48.
The last field data example for refraction and residual statics corrections is from an overthurst belt with irregular topography and large elevation differences along the line traverse. Figures 3.4-49 and 3.4-50 show selected shot records. Note that the first breaks are very distinct, and the first arrivals do not manifest significant departures from linear moveout. Nevertheless, there are significant distortions along the reflection traveltime trajectories; these are largely attributed to the subsurface complexity associated with the overthrust tectonism in the area.
Figure 3.4-51 shows selected CMP gathers with elevation corrections applied and the data referenced to a flat datum of 1800 m above the topographic profile of the line. Following the normal-moveout correction (Figure 3.4-52), note that the CMP gathers exhibit short-wavelength deviations less than a cable length along the reflection traveltime trajectories. Velocity analysis and moveout correction were performed from a floating datum — a smoothed version of the topographic profile. The CMP stack with elevation corrections is shown in Figure 3.4-53.
The same CMP gathers as in Figures 3.4-51 and 3.4-52 with refraction statics applied are shown in Figures 3.4-54 and 3.4-55. A comparison of these sets of gathers indicates that the statics problem is primarily of residual nature — differences between refraction and elevation statics are not significant. In other words, long-wavelength statics, in this case, are associated for the most part with irregular topography. Differences between the CMP stack with refraction statics (Figure 3.4-56) and the CMP stack with elevation statics (Figure 3.4-53) are marginal.
Figure 3.4-36 (a) A CMP stack with refraction statics applied using the generalized reciprocal method (compare with Figures 3.4-34a and Figure 3.4-37a), (b) estimated multiple intercept times at each shot-receiver station and the computed shot (open squares) and receiver (solid dots) statics.
Figure 3.4-38 Results of the least-squares refraction statics estimated from and applied to the data in Figure 3.4-37. In each frame, shot attribute is denoted by × and receiver attribute is denoted by vertical bar. R is the refractor that represents the base of the weathering layer. See text for details.
Figure 3.4-39 (a) A CMP stack with elevation statics corrections, (b) same as in (a) with residual statics corrections.
Figure 3.4-41 (a) A CMP stack with refraction statics applied using the generalized reciprocal method (compare with Figures 3.4-39a and Figure 3.4-42a), (b) estimated intercept times and the computed shot (open squares) and receiver (solid dots) statics.
Figure 3.4-43 Results of the least-squares refraction statics estimated from and applied to the data in Figure 3.4-42. In each frame, shot attribute is denoted by × and receiver attribute is denoted by vertical bar. R is the refractor that represents the base of the weathering layer. See text for details.
Short-wavelength traveltime deviations observed on the CMP gathers in Figures 3.4-54 and 3.4-55 have been resolved by residual statics corrections as shown in Figures 3.4-57 and 3.4-58. Reflection traveltimes in Figure 3.4-57 are much like hyperbolic and those in Figure 3.4-58 are reasonably flat after moveout correction. The corresponding CMP stack shown in Figure 3.4-59, when compared with Figure 3.4-56, clearly demonstrates the improvement attained by residual statics corrections.
In areas with severely irregular topography and large elevation changes along line traverses, one may consider extrapolating the recorded data from the topographic surface to a flat datum above the topography by using the wave-equation datuming technique (further aspects of migration in practice).  applied this technique to the data as in Figures 3.4-49 and 3.4-50. You still will need to apply residual statics corrections to account for short-wavelength statics not associated with topography, but related to the near-surface layer geometry.
Finally, in the presence of a permafrost layer or a series of lava flows at the near-surface, the problem inherently is dynamic in nature. Specifically, under such circumstances, rays through the near-surface do not follow near-vertical paths, and thus the near-surface problem cannot be posed as a statics problem. Instead, one needs to estimate accurately a velocity-depth model that accounts for the near-surface complexity so as to honor ray bending through the near-surface layer.
Figure 3.4-60 shows a CMP-stacked section from an area with a permafrost layer at the near-surface. Note that refraction statics followed by residual statics corrections (Figure 3.4-61) yield a section with improved event continuity. Nevertheless, there still exist a number of spurious structural features that have to be accounted for. Figure 3.4-62 shows a CMP-stacked section from an area with lava flows at the near-surface. Although residual statics corrections have improved event continuity, spurious faults are troublesome (Figure 3.4-63). The traveltime distortions on the stacked sections in Figures 3.4-61 and 3.4-63 strongly suggest that they cannot be resolved by statics corrections alone. Additional work, such as velocity-depth modeling (earth modeling in depth) and imaging in depth, is required to account for lateral velocity variations associated with near-surface complexities that result from lava flows and a permafrost layer.
Figure 3.4-61 The CMP-stacked section in Figure 3.4-60 with residual statics corrections.
Figure 3.4-63 The CMP-stacked section in Figure 3.4-60 with residual statics corrections.
The radon transform
We shall now consider field data examples of the Radon transform. Figure 6.4-16a shows a deep-water CMP gather that contains strong multiples below 3.5 s. The conventional velocity-stack gather (Figure 6.4-16b) shows the familiar amplitude smearing, whereas the velocity-stack gather based on the Radon transform (Figure 6.4-16c) shows better focusing of hyperbolic events. The reconstructed CMP gather (Figure 6.4-16d) contains all the hyperbolic events present in the original CMP gather (Figure 6.4-16a) and excludes noise. The amplitudes on the reconstructed CMP gather appear to be faithfully restored to their original values.
A noisier CMP gather from a shallow-water survey is shown in Figure 6.4-17a. Note the strong-amplitude, low-frequency bursts of energy similar to ground roll. A spike in the offset domain maps along a curved trajectory in the velocity domain; the larger the offset and the shallower the time at which this spike is located, the more the curvature of the trajectory (Figures 6.4-11 through 6.4-15). Note the presence of curved trajectories in the conventional velocity-stack gather in Figure 6.4-17b. Although the velocity-stack gather based on the Radon transform (Figure 6.4-17c) also contains these features, it does not have the amplitude smearing that dominates the conventional velocity-stack gather. The CMP gather reconstructed from the velocity-stack gather shown in Figure 6.4-17c contains all the hyperbolic events and excludes the random noise and coherent noise with linear moveout present in the original CMP gather (compare Figures 6.4-17a and d).
The result shown in Figure 6.4-17d suggests that velocity-stack transformation, when implemented as a special form of the discrete Radon transform, can be used to attenuate random and coherent linear noise on CMP data .
Another potential application of velocity-stack processing is in the construction of high-resolution constant-velocity stacks. Figure 6.4-18 shows constant-velocity-stack panels constructed from about 100 CMP gathers. These panels were generated twice — by computing conventional velocity-stack gathers and sorting them to CVS panels, and by computing velocity-stack gathers using the Radon transform and sorting the results into CVS panels. The multiples contaminate the primaries on the panels based on conventional velocity-stack gathers even though the constant-velocity range used is associated with the primaries. This is because of the lateral smearing of amplitudes associated with multiples and primaries in the velocity space as illustrated in Figure 6.4-2d. The multiples appear to be significantly attenuated on the panels based on the velocity-stack gathers computed by using the Radon transform. The lateral smearing has been reduced in the velocity space by the Radon transform as illustrated in Figure 6.4-4d.
Figure 6.4-16 (a) A deep-water CMP gather with strong multiples; (b) the conventional velocity-stack gather; (c) the proper velocity-stack gather; (d) the CMP gather reconstructed from (c). Compare with (a) and note the preservation of amplitudes along hyperbolic events. (Data courtesy Norwegian Petroleum Directorate.)
Figure 6.4-17 (a) A shallow-water CMP gather with strong linear noise; (b) conventional velocity-stack gather; (c) proper velocity-stack gather; (d) the CMP gather reconstructed from (c). Compare with (a) and note that hyperbolic events are enhanced, and random and linear noise are suppressed. (Data courtesy Abu Dhabi National Oil Company.)
Although f − x deconvolution usually is applied to stacked data, it also may be applied to moveout-corrected common-offset sections or CMP gathers. Figure 6.5-1 shows a CMP-stacked section before and after f − x deconvolution. Both the input and output sections have been displayed using the same display gain. Note the significant reduction of the noise and enhancement of the coherent signal in the section.
A way to assess the effectiveness of noise attenuation is by examining the f − k spectrum of the data before and after f − x deconvolution as shown in Figure 6.5-2. The f − k spectrum of the input section shows that the bandwidth of the data is approximately 10-70 Hz. It also shows the presence of random noise in the stacked section which maps over a rectangular area in the f − k plane. Specifically, band-limited random noise contains energy at all wavenumbers for all frequencies within the passband. After f − x deconvolution, note that the energy in the f − k spectrum is limited to the region of coherent signal in the vicinity of the frequency axis.
If a time-variant scaling is applied to the output of f − x deconvolution, the residual noise in the data is boosted. Figure 6.5-3 shows the stacked section as in Figure 6.5-1 after f − x deconvolution and AGC scaling. Compare the sections in Figures 6.5-1b and 6.5-3b and note that AGC has scaled up the random noise. Nevertheless, by comparing sections in Figures 6.5-3a,b, note that the relative signal-to-noise ratio of the data has increased after f − x deconvolution.
Figure 6.5-1 A CMP-stacked section (a) before, and (b) after f − x deconvolution.
Figure 6.5-2 The f − k spectra of the stacked section as in Figure 6.5-1 (a) before, and (b) after f − x deconvolution.
Figure 6.5-3 (a) The CMP-stacked section as in Figure 6.5-1a, and (b) after f − x deconvolution and AGC scaling.
Figure 6.5-4 A portion of a CMP-stacked section (a) before, and (b) after f − x deconvolution with no addback, (c) the difference between (a) and (b).
Figure 6.5-5 A portion of a CMP-stacked section (same as in Figure 6.5-4a) (a) before, and (b) after f − x deconvolution with 20 percent add-back, (c) the difference between (a) and (b).
Figure 6.5-6 A portion of a CMP-stacked section (same as in Figure 6.5-4a) (a) before, and (b) after f − x deconvolution with 40 percent add-back, (c) the difference between (a) and (b).
Figure 6.5-7 A portion of a CMP-stacked section (same as in Figure 6.5-4a) (a) before, and (b) after f − x deconvolution with 80 percent add-back, (c) the difference between (a) and (b).
Figure 6.5-8 A CMP-stacked section (a) before, and (b) after f − x deconvolution.
Figure 6.5-9 A CMP-stacked section (a) before, and (b) after f − x deconvolution.
We now test f − x deconvolution for noise attenuation in case of a CMP stack that contains near-horizontal reflections as shown in Figure 6.5-4a. The section after f − x deconvolution is shown in Figure 6.5-4b, and the difference between the input (Figure 6.5-4a) and the output (Figure 6.5-4b) is shown in Figure 6.5-4c. The difference section represents the error in the prediction process, and as such, it contains the noise uncorrelated from trace to trace. All three sections in Figure 6.5-4 are shown with the same display gain.
Note in the shallow portion of the filtered section in Figure 6.5-4b above 1 s the zipper effect of the coherent linear noise which has remained in the data after stacking. The spatial prediction filter predicts coherent signal which includes primary and multiple reflections, diffractions, and coherent linear noise. Therefore, it should not be surprising to observe a pronounced coherent linear noise trend in the data after f − x deconvolution.
To circumvent the smeared appearance of the data after f − x deconvolution, sometimes a portion of the difference section is added back to the output. Figures 6.5-5, 6.5-6 and 6.5-7 show the results of 20, 40, and 80 percent add-back. As the percent add-back is increased, the texture of the output from f − x deconvolution resembles that of the input section more closely. In practice, typical add-back value varies between 0-40 percent.
Now consider a stacked data set with moderate structural complexity as shown in Figure 6.5-8. Note that a significant portion of the random noise has been attenuated by f − x deconvolution, while diffractions at fault locations and reflections have been preserved.
Finally, Figure 6.5-9 shows a stacked section with a complex structure. Following f − x deconvolution, coherent signal — reflections and diffractions, has largely been preserved. Hence, f − x deconvolution for noise attenuation is a robust process even in the presence of a complex pattern of coherent signal as in Figure 6.5-9. The application of f − x deconvolution to land data from areas with complex structures such as those associated with overthrust tectonics can improve the stack quality significantly. Also, land data from areas with near-surface complexity that causes poor penetration of the source energy into the subsurface can benefit from application of f − x deconvolution.
2-D prestack depth migration
We now review results of shot-profile migration of a 2-D land and a 2-D marine line. Figure 8.3-22 shows the CMP-stacked section of the land line over an imbricate structure associated with overthrust tectonics. The topographic high is the surface expression of the imbricate structure. Refraction and residual statics corrections were applied to the data before shot-profile migration.
Figure 8.3-22 also shows the poststack time-migrated section. The main thrust plane starts from the surface at CMP 340 and dips down to the right. To account for the severe elevation differences, migration was performed from the irregular topography (further aspects of migration in practice).
The recording geometry for this land line has severe irregularities with duplicate shots and missing shots. Here, we only want to discuss results of prestack depth migration and leave out the velocity-depth model estimation. Selected common-receiver gathers sorted from the migrated shot records are shown in Figure 8.3-23. Most events exhibit flat character. Note also the symmetry of some events with respect to the receiver location as seen on gathers in the top row — they are associated with flat or gently dipping reflectors. The asymmetry of some events with respect to the receiver location is seen on gathers in the bottom row in Figure 8.3-23.
Stacking of the common-receiver gathers yields the depth image obtained from shot-profile migration (Figure 8.3-24). For comparison, the depth image from poststack depth migration using the same velocity-depth model is shown also in Figure 8.3-24. Trace spacing in the prestack depth-migrated section corresponds to the average shot spacing, whereas that in the poststack depth-migrated section corresponds to the CMP spacing. Note from the prestack depth-migrated section (Figure 8.3-24) the improved image below 2 km of the imbricate structure to the right of the main thrust plane which starts at the surface in the vicinity of CMP 350 and dips down to the right.
Figure 8.3-25 shows the CMP-stacked section from the marine line over two salt diapirs. The target zone is base-salt and the subsalt region. For comparison, Figure 8.3-25 also shows the poststack time-migrated section. Note the velocity pull-up along the base-salt reflection at 1.9 s below CMP 400 and CMP 800 — typical of salt diapirs.
Selected common-receiver gathers sorted from the migrated shot records are shown in Figure 8.3-26. Note that most events exhibit flat character; this suggests that the velocity-depth model used in prestack depth migration is fairly accurate. Flat or gently dipping reflectors manifest themselves as events on the common-receiver gathers that are symmetric with respect to the receiver locations. Dipping reflectors, on the other hand, manifest themselves as events that are asymmetric with respect to the receiver locations.
Stacking of the common-receiver gathers yields the depth image obtained from shot-profile migration (Figure 8.3-27). For comparison, the depth image from poststack depth migration using the same velocity-depth model also is shown in Figure 8.3-27. Trace spacing in the prestack depth-migrated section corresponds to the shot spacing, whereas that in the poststack depth-migrated section corresponds to the CMP spacing. Note from the prestack depth-migrated section the improved image of the base-salt reflector and the subsalt region. The base-salt reflector appears to be faulted and shows a structural closure at approximately 3.5 km below CMP 630.
Figure 8.3-22 (Top) A CMP-stacked section from a land line shot over a topographic high; (bottom) time migration of the CMP-stacked section. (Data courtesy Lasmo International.)
Figure 8.3-25 (Top) A CMP-stacked section from a marine line over two salt diapirs; (bottom) time migration of the CMP-stacked section. (Data courtesy Mobil North Sea Limited.)
- Bevc (1997), Bevc, D., 1997, Flooding the topography: Wave-equation datuming of land data with rugged topography: Geophysics, 62, 1558–1569.
- Hampson, 1987, Hampson, D., 1987, The discrete Radon transform: A new tool for image enhancement and noise suppression: 57th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 141–143.
- Deconvolution exercises
- Mathematical foundation of deconvolution
- First breaks
- Field statics corrections
- Flat refractor
- Dipping refractor
- The plus-minus method
- The generalized reciprocal method
- The least-squares method
- Processing sequence for statics corrections
- Model experiments
- Velocity analysis and statics corrections exercises
- Topics in moveout and statics corrections
- Velocity-stack transformation
- The discrete radon transform
- The parabolic radon transform
- Practical considerations
- Impulse response of the velocity-stack operator
- Radon-transform multiple attenuation
- Design of spatial prediction filters
- 2-D prestack depth migration
- Shot-geophone migration
- Shot-profile migration
- Sensitivity of image accuracy to velocity errors