Migration and spatial aliasing

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Seismic Data Analysis
Seismic-data-analysis.jpg
Series Investigations in Geophysics
Author Öz Yilmaz
DOI http://dx.doi.org/10.1190/1.9781560801580
ISBN ISBN 978-1-56080-094-1
Store SEG Online Store


The concept of spatial aliasing is presented in the 2-D Fourier transform. Here, we shall examine the effect of spatial aliasing on migration. Figure 4.6-1 shows a zero-offset section that contains a diffraction hyperbola with 2500-m/s velocity and 12.5-m trace spacing. By discarding every other trace, obtain another zero-offset section with 25-m trace spacing. Repeat the procedure to obtain the zero-offset sections with 50-m and 100-m trace spacings (Figure 4.6-1).

Figure 1.1-26  Three zero-phase wavelets (top row) and their respective amplitude spectra (bottom row). (a) The steeply defined slopes of the passband cause ripples in the wavelet and the actual amplitude spectrum. (b) A moderate and (c) gentle slope help eliminate the ripples. Refer to the text for a discussion of corner frequencies A, B, C, and D.

The f − k spectra of the zero-offset sections with the four different trace spacings are displayed in Figure 4.6-2. The diffraction hyperbola with 12.5-m trace spacing maps onto an inverted triangular area in the f − k plane (migration principles). The Nyquist wavenumber is 40 cycles/km and the bandwidth is given by the corner frequencies 6, 12 - 36, 48 Hz for the passband region of the spectrum. (See Figure 1.1-26 for the definition of corner frequencies.) The red is associated with the flat part of the passband region and the blue is associated with the taper zone.

The f − k spectrum of the zero-offset section with 25-m trace spacing (Figure 4.6-1) indicates spatial aliasing beyond approximately 24 Hz (Figure 4.6-2). Consequently, the triangular shape of the passband region in the f − k plane that defines the diffraction hyperbola is corrupted around the edges near the Nyquist wavenumber of 20 cycles/km. At a coarser trace spacing of 50 m, which corresponds to a Nyquist wavenumber of 10 cycles/km, the triangular shape in the spectrum is preserved below the threshold frequency for aliasing, approximately 12 Hz, only. Finally, at trace spacing of 100 m, which corresponds to a Nyquist wavenumber of 5 cycles/km, the triangular shape is obliterated, completely (Figure 4.6-2).

Figure 4.6-3 shows the results of Kirchhoff migration of the zero-offset sections in Figure 4.6-1. Frequency components that are spatially aliased are perceived by migration with a dip different from the actual dip along the diffraction hyperbola. Normally, energy is moved in the up-dip direction along the diffraction hyperbola and is mapped onto the apex. However, in each migrated section, the spatially aliased part of the energy is split away from the flanks of the diffraction hyperbola and mapped onto the regions to the left and right of the flanks. The aliased energy is dispersed. Since each frequency component of the aliased energy is perceived to have a different dip by migration, the displacement of the energy after migration is frequency dependent. The unaliased portion of the energy is of course mapped onto the apex. The more frequency components are spatially aliased, the less energy at lower frequencies is mapped onto the apex.

As discussed in migration principles, the triangular area on the f − k plane associated with the diffraction hyperbola (Figure 4.6-2) is mapped onto the circular area on the f − k plane associated with the migrated section (Figure 4.6-4). Ideally the area in the f − k plane of the migrated section should be semicircular in shape. Because the diffraction hyperbola is defined within a finite spatial aperture (Figure 4.6-1), there is an implicitly imposed dip limit on migration. As a result, the semicircular area is notched on either side (Figure 4.6-4).

Spatial aliasing corrupts the semicircular shape of the f − k spectrum on both ends of the spectrum at the vicinity of the Nyquist wavenumber. In case of severe undersampling, spatially aliased frequency components invade much of the f − k plane as shown in Figure 4.6-4.

Aside from the spatial aliasing noise, dispersive noise also is seen on data migrated with a dip-limited finite-difference algorithm (finite-difference migration in practice). Figure 4.6-5 shows migration of a zero-offset section that contains a diffraction hyperbola using the 15-degree implicit finite-difference method. Note the undermigration of the diffraction hyperbola that is caused by the 15-degree dip limitation, the dispersive noise A that is caused by the finite-difference approximations, and the spatially aliased energy B that splits away from the unaliased part that collapses to the apex.

Figure 4.6-6 shows the results of migration of the zero-offset sections in Figure 4.6-1 using an implicit frequency-space finite-difference scheme (frequency-space migration in practice). Note the dispersive noise caused by the finite-difference approximations in the section with 12.5-m trace spacing. The dispersive noise in the sections with 25-m, 50-m, and 100-m trace spacings, however, is attributed largely to spatial aliasing.

It is instructive to note that the diffraction energy appears slightly undermigrated with 12.5-m trace spacing, but is overmigrated with 25-m and 50-m trace spacings. As discussed in finite-difference migration in practice and frequency-space migration in practice, the fidelity of migration by finite-difference schemes is dictated by an intricately complex interplay between the various parameters — spatial and temporal sampling intervals, dip, frequency, and velocity. Depending on the values of these parameters, one scheme may cause undermigration in one case and overmigration in another case.

Figure 4.6-7 shows the f − k spectra of the migrated sections in Figure 4.6-6. Note that implicit frequency-space migration can create high-frequency noise beyond the passband of the input data. Note also that spatial aliasing combined with the inherent dispersive effect of finite-difference schemes corrupt the semicircular shape of the f − k spectrum on both ends of the spectrum.

Figure 4.6-8 shows the results of the migration of the zero-offset sections in Figure 4.6-1 using an explicit frequency-space finite-difference scheme (frequency-space migration in practice). There appears to be no aliasing noise in either sections with 12.5-m and 25-m trace spacings. Also note that, compared to the results of Kirchhoff migration (Figure 4.6-2), there is less aliasing noise in the sections with 50-m and 100-m trace spacings. These observations can be verified by referring to the f − k spectra shown in Figure 4.6-9. Explicit schemes, by the design criterion, attenuate energy associated with wavenumbers kx above a specified cutoff wavenumber defined by a fraction of the Nyquist wavenumber. This effectively removes part of the aliased energy that maps onto the spectral region above the cutoff wavenumber associated with the extrapolation filter for the explicit scheme. Note from the f − k spectrum in Figure 4.6-9 that almost all of the aliased energy has been filtered out for the case of the 25-m trace spacing. This is why aliasing noise is absent in the corresponding migrated section in Figure 4.6-8. Despite the wavenumber filtering effect of the explicit scheme, however, much of the aliased noise remains in the sections with the 50-m and 100-m trace spacings.

Figure 4.6-10 shows the results of phase-shift migration of the zero-offset sections in Figure 4.6-1, and Figure 4.6-11 shows the corresponding f − k spectra. These results are used as a benchmark to evaluate the results obtained from the other migration algorithms (Figures 4.6-3 through 4.6-9). Except for the aliasing noise, phase-shift migration produces no artifacts.

The experiments described above clearly demonstrate that all migration algorithms suffer from saptial aliasing. We now examine the effect of spatial aliasing on migration using a dipping events model. Figure 4.6-12 shows a zero-offset section that contains a set of dipping events with 3500-m/s velocity and 25-m trace spacing. By discarding every other trace, obtain another zero-offset section with 50-m trace spacing. Repeat the procedure to obtain the zero-offset section with 100-m trace spacing (Figure 4.6-12).

The f − k spectra of the zero-offset sections with the three different trace spacings are displayed in Figure 4.6-13. The dipping events with 25-m trace spacing map onto a series of radial lines in the f − k plane (migration principles). The Nyquist wavenumber is 20 cycles/km and the bandwidth is given by the corner frequencies 6, 12 - 36, 48 Hz for the passband region of the spectrum. As for the diffraction hyperbola model (Figure 4.6-3), the red is associated with the flat part of the passband region and the blue is associated with the taper zone.

The f − k spectrum of the zero-offset section with 50-m trace spacing (Figure 4.6-12), which corresponds to a Nyquist wavenumber of 10 cycles/km, indicates spatial aliasing beyond approximately 24 Hz (Figure 4.6-13). Consequently, the aliased segments of the radial lines map onto the left quadrant of the f − k spectrum. At a coarser trace spacing of 100 m, which corresponds to a Nyquist wavenumber of 5 cycles/km, spatial aliasing occurs first at approximately 12 Hz. Then, some of the energy already aliased becomes aliased for the second time at approximately 36 Hz. Moreover, part of the aliased energy is remapped onto the right quadrant (Figure 4.6-13).

Figure 4.6-14 shows the results of Kirchhoff migration of the zero-offset sections in Figure 4.6-12. No aliasing noise is present on the migrated section with 25-m trace spacing. Next, consider the migrated section with 50-m trace spacing. Frequency components that are spatially aliased are perceived by migration to dip in the direction opposite to the actual dips of the events. Normally, energy is moved in the up-dip direction, in this case from right to left as seen in the migrated section with 25-m trace spacing. However, in the migrated section with 50-m trace spacing, the spatially aliased part of the energy is split away from the dipping events and moved from left to right. Note that the aliased energy is dispersed. As for the diffraction hyperbola model (Figure 4.6-2), each frequency component of the aliased energy is perceived to have a different dip by migration, the displacement of the energy after migration is frequency dependent. The unaliased portion of the energy is of course moved from right to left and positioned accurately. The more frequency components are spatially aliased, the less energy at lower frequencies is mapped to the correct position.

Finally, consider the case of the migrated section with 100-m trace spacing (Figure 4.6-14). Note that there exists aliasing noise not only to the right of the dipping events but also in the left-most portion of the section. The latter is associated with the energy that has been aliased twice (Figure 4.6-3). Because of the complexity of aliasing, the noise essentially disperses over the whole of the section.

An interesting observation on the migrated sections in Figure 4.6-14 relates to the energy in the region above 1 s. In Kirchhoff migration, amplitudes in the input section are summed along a hyperbolic summation path and placed at the apex of the hyperbola. Imagine a summation path whose apex is situated at a time less than 1 s. There will be some energy placed at this apex location since the flanks of the summation path under consideration will intersect through traces with nonzero sample values. This situation is encountered when migrating marine data using Kirchhoff summation. Normally, the migrated section is muted above the water bottom to remove the noise created by migration within the water layer.

Besides data aliasing, there is also the problem of operator aliasing. In particular, for a low-velocity hyperbola or for a hyperbola with its apex situated at shallow times, Kirchhoff summation may require more than one sample per trace. This results in some energy in the form of precursors above the migrated sea-bottom reflection, when only one point per trace is included in the summation.

Figure 4.6-15 shows the f − k spectra of the migrated sections in Figure 4.6-14. As discussed in migration principles, migration rotates the radial lines on the f − k plane associated with the dipping events.

Figure 4.6-16 shows migration of a zero-offset section that contains a set of dipping events using the 15-degree implicit finite-difference method. Note the undermigration of the steeply dipping events caused by the 15-degree dip limitation, the dispersive noise A caused by the finite-difference approximations, and the spatially aliased energy B that splits away from the unaliased part and moves in the opposite direction.

Figure 4.6-17 shows the results of migration of the zero-offset sections in Figure 4.6-12 using an implicit frequency-space finite-difference scheme (frequency-space migration in practice). Note the dispersive noise caused by the finite-difference approximations in the section with 25-m trace spacing. In the sections with 50-m and 100-m trace spacings, two sets of dispersive noise can be distinguished — one that is caused by the finite-difference approximations and the other caused by spatial aliasing. At coarse spatial sampling, the steeply dipping events are faintly detected on the migrated section.

Figure 4.6-18 shows the f − k spectra of the migrated sections in Figure 4.6-17. As in the case of the diffraction hyperbola (Figure 4.6-7), implicit frequency-space migration can create high-frequency noise beyond the passband of the input data.

Figure 4.6-19 shows the results of the migration of the zero-offset sections in Figure 4.6-12 using an explicit frequency-space finite-difference scheme (frequency-space migration in practice). There is no aliasing noise in the section with 25-m trace spacing. But there is precursive dispersion along the steeply dipping events because of the inherent nature of the explicit scheme used here. This dispersion is not as severe as that observed on the result from the implicit scheme (Figure 4.6-17).

The corresponding f − k spectra shown in Figure 4.6-20 explains why there is less aliasing noise on the migrated sections in Figure 4.6-19 compared to those from Kirchhoff summation (Figure 4.6-14). As for the case of the diffraction hyperbola (Figure 4.6-9), explicit schemes attenuate energy associated with wavenumbers kx above a specified cutoff wavenumber. This effectively removes part of the aliased energy that maps onto the spectral region above the cutoff wavenumber associated with the extrapolation filter for the explicit scheme. Note from the f − k spectrum in Figure 4.6-20 that a significant portion of the aliased energy in the left quadrant has been filtered out for the case of the 50-m trace spacing. Despite the wavenumber filtering effect of the explicit scheme, however, much of the aliased noise remains in the section with the 100-m trace spacing.

Figure 4.6-21 shows the results of the phase-shift migration of the zero-offset sections in Figure 4.6-12, and Figure 4.6-22 shows the corresponding f − k spectra. These results are used as a benchmark to evaluate the results obtained from the other migration algorithms (Figures 4.6-14 through 4.6-20). Except for the aliasing noise, phase-shift migration produces no artifacts.

The effect of spatial aliasing on migration of field data, to begin with, is demonstrated in Figures 4.6-23 and 4.6-24. We see the original stacked section and its resampled versions at coarser trace spacings. From the migrations of these four stacked sections with coarser trace spacings, note the loss of spatial resolution. The nearly flat events are not adversely affected by spatial aliasing, while the steeply dipping reflection off the right flank of the salt diapir can only be detected on the migrated section with very coarse sampling by a low-frequency, weak-amplitude event. The diffraction energy off the tip of the salt diapir is largely dispersed into the region with nearly flat events to the right of the diapir.

We now examine the response of the various migration algorithms to spatial alaising using the data shown in Figures 4.6-25 and 4.6-26. The stacked data are associated with the same line sampled at four different trace spacings — 12.5, 25, 50, and 100 m. Figure 4.6-27 shows migrations of the stacked sections using Kirchhoff summation. The section with 12.5-m trace spacing provides a crisp image of the salt diapir, while the sections with coarser trace spacings degrade gradually. Specifically, it is almost impossible to delineate the salt boundary on the section with 50-m trace spacing, and the section with 100-m trace spacing does not even provide an image of the gently dipping reflections. This is because of the aliasing noise associated with the steep flanks of the salt diapir corrupting the surrounding reflections. Spatial aliasing not only adversely affects the quality of the image associated with a dipping event that is aliased, but it also can obliterate other nonaliased events in the data.

Figure 4.6-28 through 4.6-30 show migrations of the stacked sections in Figures 4.6-25 and 4.6-26 using steep-dip frequency-space implicit and explicit schemes, and the phase-shift method. Similar conclusions are drawn for the Kirchhoff summation results shown in Figure 4.6-27. Differences in terms of delineation of the salt boundary and the surrounding strata are attributable to the manner in which these algorithms are implemented and how they treat the velocity field for migration. For instance, when examining the results from the frequency-space implicit scheme (Figure 4.6-28), one must keep in mind the effect of spatial aliasing combined with the effect of undermigration caused by the dip-limited nature of the algorithm and the effect of dispersion caused by finite-difference approximations.

What is the remedy for spatial aliasing noise in migration? Arrange the sequence of the sections in Figures 4.6-27 through 4.6-30 from a coarser to a finer trace spacing. Note that the deleterious effect of spatial aliasing in migration disappears as we go to finer trace spacings. To avoid spatial aliasing, we must record with sufficiently fine CMP trace interval or interpolate the data that have been recorded with coarse spatial sampling.

Most modern surveys are conducted using spatial sampling rates that are perfectly adequate to meet exploration and development objectives. If we are dealing with vintage data with coarse spatial sampling, there are two ways to circumvent the effect of spatial aliasing. The first approach would be to filter out the aliased frequencies. This is undesirable, since it severely limits vertical and lateral resolutions (seismic resolution). The second approach would be to do trace interpolation before migration. In processing of 3-D seismic data and G.5, we discuss interpolation of aliased data. Note from worldwide assortment of shot records that the smaller the trace interval, the higher the Nyquist in the spatial wavenumber direction (Figures 1.3-10 and 1.3-11), and thus, the less likelihood of aliasing high-frequency data.

A schematic illustration of the spatial aliasing phenomenon is shown in Figure 4.6-31. Start with the spectral bandwidth that spans COA in the spatial wavenumber axis, where A is the location of the Nyquist wavenumber, and ON in the temporal frequency axis, where N is the location of the Nyquist frequency. Dip components 1 and 2 are aliased beyond frequency values AT and AS, respectively. Extend the wavenumber axis to DOB by making the trace interval half of the original. Event 1 no longer is spatially aliased within the frequency bandwidth ON. Event 2 still is aliased beyond the frequency value BV. However, at this point and beyond, there may be no significant energy, so further extension of the wavenumber axis may not be necessary. Another important point is that if the temporal frequency band only extended up to OG to start with, extension of the wavenumber axis to DOB also would result in Event 2 being unaliased. Thus, the amount of trace interpolation that is required also depends on temporal bandwidth as well as on structural dip.

Trace interpolation often is necessary when dealing with 3-D data and old data recorded with a large group interval. In a typical 3-D survey, the inline trace interval may be as little as 12.5 m, while the trace interval in the crossline direction, for some old data, may be as much as 100 m. Therefore, interpolation is required before migration in the crossline direction. You do not necessarily interpolate down to the inline trace spacing; instead, depending on the maximum structural dip and velocity in the area, the optimum trace spacing for interpolation in the crossline direction can be computed using equation (1-8). processing of 3-D seismic data provides more information on trace interpolation in relation to 3-D migration.

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