The anti-aliasing dilemma

Image-space aliasing can be corrected during processing without losing resolution by decreasing the spatial sampling of the image, because the image sampling is a processing parameter. In contrast, operator aliasing can only be avoided by suppressing some high-frequency components from the image, because the data sampling depends on acquisition parameters. Unfortunately this lowpass filtering decreases the image resolution. We therefore face a dilemma when imaging aliased data: on one side we need to apply anti-aliasing filters to avoid aliasing noise, on the other side we do not want to lose image resolution. Figure  exemplifies the problem. It shows the results of 3-D zero-offset time migration of a salt-dome flank in the Gulf of Mexico. The section on the left (Figure a) was 3-D migrated without applying any anti-aliasing filter, while the section on the right (Figure b) was obtained by applying a standard anti-aliased migration. Whereas the image obtained without anti-aliasing is much noisier than the anti-aliased one, it also has higher resolution. In the shallow part of the section shown in Figure a, the aliasing noise is so strong that is impossible to appreciate the higher resolution of Figure a compared with Figure b. But when comparing zooms into the deeper part of the sections (Figure ), it is apparent that by applying anti-aliasing we lose resolution. In particular, the high-frequency dipping event at about CMP X=700 m and Time=2.2 s is poorly resolved in the anti-aliased migration (Figure b). The anti-aliased migration misses a whole wavelet cycle of the sediment truncation against the salt flank. If we consider that hydrocarbon reservoirs are often located at the sediment-salt interfaces, we appreciate the potential advantages of improving the resolution of events such as the sediment truncation shown in Figure .

 

Comp-WL-intro Figure 1 3-D migrations of a salt-dome flank in the Gulf of Mexico: (a) migration obtained without any anti-aliasing filter, (b) migration obtained with the application of a ``standard'' anti-aliasing filter.

 

Comp-WB-intro Figure 2 Zoom into the 3-D migrations of a salt-dome flank in the Gulf of Mexico shown in Figure : (a) migration obtained without any anti-aliasing filter, (b) migration obtained with the application of a ``standard'' anti-aliasing filter.

Migration without anti-aliasing achieves higher resolution than the anti-aliased one because it images data components that are aliased in the data space. In particular, the steeply dipping energy reflected from the salt flanks visible in the data window shown in Figure  are aliased. Figure  shows the frequency-wavenumber spectrum of the data window in Figure . In addition to the central unaliased band of the spectrum, Figure  shows also the two spatially aliased bands on either side. The vertical black lines correspond to the Nyquist wavenumbers. The aliased dipping events correspond to a ``cloud'' in the spectrum that starts in the main band but crosses the positive Nyquist line and trespasses upon the aliased band. However, because there are no events dipping with negative time dips, the aliased components are still recoverable by the simple anti-aliasing method presented in this paper.

 

Wind-data Figure 3 Data window containing aliased reflections from the salt flanks.

 

Wind-spec Figure 4 Frequency-wavenumber spectrum of the data window shown in Figure . Notice that the aliased events correspond to a ``cloud'' in the spectrum that starts in the main band but crosses the positive Nyquist line and trespasses upon the aliased band.