The common-cell sorting problem is not solved completely by restoring uniformity in the fold of coverage. As mentioned earlier, the centroid of the midpoints may not coincide with the center of the cell. Theoretically, if the centroid departs significantly from the center, then placing the stacked trace at the centroid, rather than at the cell center, may be considered. This destroys equal spacing of the stacked traces, primarily in the crossline direction. In principle, however, 3-D poststack migration based on the Kirchhoff integral method may be used to produce migrated data volume with uniform trace distribution.
Before we embark on searching for a way to alleviate the problems associated with 3-D binning as a result of cable feathering as described above, first, we shall investigate the effect of cable feathering itself on stacking the data. Figure 7.1-10 shows a single cell that is associated with the field geometry sketched in the same figure. The cell is 12.5 m in the inline (labeled as shot line) direction and 50 m in the crossline direction. Different symbols represent the midpoints that are associated with different shot lines. This cell contains midpoints from six different shot lines. In Figure 7.1-10, the midpoint distribution corresponds to the ideal case of a constant feathering angle of 10 degrees over the entire survey. The shooting direction is assumed to be the same for all the shot lines that contribute midpoints to this cell. Assume a simple earth model with a single, 30-degree dipping event. Also assume a constant-velocity medium above the dipping interface.
With common-cell sorting, the traveltimes of the arrivals in a single cell may not follow a single hyperbolic moveout curve  . Figure 7.1-11 shows the traveltime curves for the cell under consideration (Figure 7.1-10) for three different shooting directions:
- Strike-line shooting — no dip perceived along the inline direction (maximum cross-dip case).
- Shooting direction at a 45-degree azimuth with respect to dip direction.
- Dip-line shooting — no dip perceived along the crossline direction.
Figure 7.1-11 Traveltimes and the least-squares-fit hyperbolic moveout curves associated with (a) strike-line shooting, (b) shooting in the 45-degree direction from the downdip azimuth, and (c) dip-line shooting. These traveltimes were derived from a single planar interface with a 30-degree dip in a constant-velocity medium. The feathering angle is 10 degrees and midpoint distribution in the cell is shown in Figure 7.1-10. Numbers along the moveout curves denote the lines that contribute midpoints to the cell under study .
Figure 7.1-12 (a) The stacking operator associated with midpoint scatter along the crossline direction. This operator was derived from traveltime deviations from the best-fit hyperbolic moveout curve in Figure 7.2-3a. (b) The amplitude spectrum suggests that high frequencies are attenuated (causing smearing of stacked amplitudes) because of midpoint scatter .
Figure 7.1-13 Factors that influence crossline smearing. Arrows in circle indicate an increase or decrease in the adjacent quantity. For instance, decrease in velocity causes increase in crossline smear .
Note that data from different shot lines contribute to different portions of the traveltime curves. For a single dipping event, the 2-D recording geometry normally yields a hyperbolic moveout curve. As the line orientation becomes more parallel to the reflector strike, cross-dip increases and traveltimes deviate from the ideal hyperbolic moveout curve. This ideal hyperbola corresponds to the case of no cable feathering, in which all midpoints in the cell coincide with the cell center. The traveltime deviation is worse when shooting in the strike direction (Figure 7.1-11a). The traveltime deviation increases with increasing feathering angle, cross-dip, and cell dimension in the crossline direction. It is also more significant in the case of large moveout that occurs at lower velocities and shallow depths. If a common-cell stack were done along the best-fit hyperbolic path, then a loss of high frequencies would be expected.
How significant is the high-cut action caused by midpoint scatter in the crossline direction? Measure the time differences between the ideal hyperbolic path and the actual moveout curve in Figure 7.1-11a to derive the stacking operator plotted with its amplitude spectrum in Figure 7.1-12. Note the high-cut filtering action of the stacking operator; the -6 dB amplitude level is at about 70 Hz. Whether or not the effects of midpoint scatter are significant on a particular data set depends on the amount of cross-dip, cable feathering, and desired bandwidth of the stacked data.
A summary of the cause-and-effect relationship for cable feathering is given in Figure 7.1-13. Crosscurrents cause cable feathering, which makes the midpoints scatter within a cell in the crossline direction. If the shooting direction is such that a dipping interface has a cross-dip component, then the traveltimes associated with that interface in a common-cell gather deviate from a single hyperbolic moveout curve. This causes amplitude smearing during stacking, which acts as a high-cut filter as illustrated in Figure 7.1-12b. The cutoff frequency primarily is a function of the amount of cross-dip, reflection time, and velocity. The lower the velocity, that means usually the shallower the event, the larger the cross-dip (maximum along the strike line direction), and the larger the cell size the more likelihood of crossline smearing.
The most effective way to avoid crossline smear is to have the cell dimension in the crossline direction sufficiently small, which means shooting with close line spacing. The multicable recording geometry used in modern marine 3-D surveys provides the means for achieving sufficiently small crossline spacing to minimize crossline smearing on stacked data.
- Levin, 1983, Levin, F. K., 1983, The effects of streamer feathering on stacking: Geophysics, 48, 1165–1171.
- Levin 1984, Levin, F. K., 1984, The effect of binning on data from a feathered streamer: Geophysics, 49, 1386–1387.
- Bentley and Yang, 1982, Bentley, L. and Yang, M., 1982, Scatter of midpoints grouped in cells and its effects on stacking: unpublished technical document, Western Geophysical Company.
- Migration aperture
- Spatial sampling
- Other considerations
- Marine acquisition geometry
- Cable feathering
- 3-D binning
- Strike versus dip shooting
- Land acquisition geometry