The 2-D Fourier transform
Multichannel processing operations can be loosely defined as those that must operate on several data traces, simultaneously. Multichannel processes can be useful in discriminating against noise and enhancing signal on the basis of a criterion that can be distinguished from trace to trace, such as dip or moveout. The 2-D Fourier transform is a basis for both analysis and implementation of multichannel processes.
Consider the six zero-offset sections in Figure 1.2-1. The trace spacing is 25 m with 24 traces per section. All have monochromatic events with 12-Hz frequency, but with dips that vary from 0 to 15 ms/trace. From the discussion on the 1-D Fourier transform, we know about frequency, particularly temporal frequency, or the number of cycles per unit time. This is the Fourier dual for the time variable. However, a seismic wavefield is not only a function of time, but also a function of a space variable (offset or midpoint axis). The Fourier dual for the space variable is defined as spatial frequency, which is the number of cycles per unit distance, or wavenumber. Just as the temporal frequency of a given sinusoid is determined by counting the number of peaks within a unit time, say 1 s, the wavenumber of a dipping event is determined by counting the number of peaks within a unit distance, say 1 km, along the horizontal direction. Just as the temporal Nyquist frequency is defined as in equation (1), the Nyquist wavenumber is defined as
where Δx is the spatial sampling interval. For all of the sections in Figures 1.2-1 through 1.2-6, the Nyquist wavenumber is 20 cycles/km, since the trace interval is 25 m.
To compute the wavenumber that is associated with the section corresponding, say, to the 15-ms/trace dip in Figure 1.2-1, follow a peak or trough across the section. First compute the total time difference along the selected peak or trough across the section:
Then convert this to number of cycles by dividing by the (temporal) period:
The spatial extent of the section is 575 m; therefore, the wavenumber associated with the 15-ms/trace dip and the 12-Hz frequency is
To continue this discussion, we will map these sections to the plane of temporal frequency versus spatial wavenumber, then look at two quadrants of this plane. The following convention will be used: Events with downdip to the right are assigned positive dip, while events with updip to the right are assigned negative dip. Additionally, positive dips map onto the right quadrant, which corresponds to positive wavenumbers, while negative dips map onto the left quadrant, which corresponds to negative wavenumbers.
The plane of frequency-wavenumber (the f – k plane) appears at the bottom of each section in Figure 1.2-1. The section with zero-dip events maps onto a single point on the frequency axis at 12 Hz. Zero dip is equivalent to zero wavenumber. The magnitude of the spike corresponds to the peak amplitude of the sinusoids that make up the traces in the section. Therefore, the f – k plane actually represents the 2-D amplitude spectrum of the section in the t – x domain. These data have been transferred from the time-space domain to the frequency-wavenumber domain. This process is described mathematically by the 2-D Fourier transform.
There is a practical relationship between the four variables: time-space (t – x), and their Fourier duals, frequency-wavenumber (f – k). Measure the inverse of the stepout in the 15-ms/trace section in Figure 1.2-1 by following a peak, trough, or zero crossing from trace to trace. Stepout is defined as the slope Δt/Δx. In this case, the inverse of the stepout is
Now, compute the ratio:
From this, the inverse of the stepout measured in the t – x domain along a constant phase is equal to the ratio of the frequency to the wavenumber associated with the event
Therefore, while retaining fixed stepout, doubling the frequency means doubling the wavenumber.
Note that all sections in Figure 1.2-1 have the same frequency component. However, from 0 to 15 ms/trace, the number of peaks increases horizontally across each section. That is, for a given frequency, higher dips are assigned to higher wavenumbers, as seen on the f – k plots.
From Figures 1.2-1 through 1.2-6, consider the same dip components, but at different frequencies. Map each individual section to the f – k plane. Nothing unusual happens until the section with 15-ms/trace dip at 36 Hz is reached in Figure 1.2-3. Here there is no positive dip. In fact, as a whole, the section displays a checkerboard character making it difficult to determine whether the dip is positive or negative.
At 48 Hz (Figure 1.2-4), the correct dip direction is observed in the first four sections. However, the fifth section, which corresponds to the 12-ms/trace positive dip, shows a negative dip. Therefore, it is mapped onto the negative quadrant, which is the wrong quadrant for this section. This dip component (12 ms/trace) at this frequency (48 Hz) is spatially aliased. In fact, any dip greater than 12 ms/trace is spatially aliased at this frequency.
In the next set of sections in Figure 1.2-5, spatial aliasing occurs at 60 Hz for a 9-ms/trace dip. Spatial aliasing not only causes mapping to the wrong quadrant, but also causes mapping with the wrong dip. One obvious example of this is mapping a 15-ms/trace dip at 60 Hz (Figure 1.2-5). Finally, at 72 Hz (Figure 1.2-6), the 6-ms/trace dip component is on the verge of spatial aliasing. Moreover, the 15-ms/trace dip component is spatially aliased twice; it folds back to the positive-dip quadrant and appears at a lower dip.
This same analysis can be used for the negative-dip components. From Figures 1.2-1 through 1.2-6, note that each section as a whole was mapped onto a single point in the frequency-wavenumber domain. Each section has an associated unique frequency and wavenumber assigned to it. These zero-offset sections can be considered representations of plane waves that propagate at a unique angle from the vertical and carry a monochromatic signal. The wavefront is defined as the line of constant phase, while the direction of propagation is perpendicular to the wavefront. Since a seismic wave-field is a superposition of many dips and frequencies, it is equivalent to the synthesis of many plane-wave components. In this respect, the physical meaning of the 2-D Fourier transform is important, for it is an analysis (decomposition) of a wavefield into its plane-wave components.
A recorded wavefield is a composite of many dip and frequency components, such as those shown in Figures 1.2-1 through 1.2-6. Suppose that sections with the same dip, but with different frequencies, are superimposed. The composite sections are shown in Figure 1.2-7 with the composite amplitude spectra below each section. For a given dip, all frequency components map onto the f – k plane along a straight line that passes through the origin. The higher the dip, the closer the radial line in the f – k domain is to the wavenumber axis. The zero-dip components map along the frequency axis. From the 9-, 12-, and 15-ms/trace dips, note that the spatially aliased frequencies are located along the linear segments that wrap around to the opposite quadrant in the amplitude spectrum. The steeper the dip, the lower the frequency at which spatial aliasing occurs.
So far, a discrete number of frequencies was considered. For a continuum of frequency components associated with a single dip, we anticipate that they would map along a straight, continuous line in the f – k domain, as shown in Figure 1.2-8. While the dipping event in Figure 1.2-8 is not aliased, the dipping event in Figure 1.2-9 is spatially aliased beginning at approximately 21 Hz.
Examination of the monochromatic single-dip sections in Figures 1.2-1 through 1.2-6 shows that each section maps onto a single point in the f – k domain. An extension of this observation is made in Figure 1.2-10. Events with the same dip in the t – x domain, regardless of their location, map onto a single radial line in the f – k domain. When events are spatially aliased, the radial line wraps around at the Nyquist wavenumber (Figure 1.2-11). These concepts have important practical implications, for they lead to f – k dip filtering of coherent linear noise (frequency-wavenumber filtering). Events with different dips that may interfere in the t – x domain can be isolated in the f – k domain.
The numerical computation of the 2-D Fourier transform involves two 1-D Fourier transforms. Figure 1.2-12 shows the steps that are involved. A brief mathematical formulation of the 2-D Fourier transform is given in Section A.2.
In summary, 2-D Fourier transformation is a way to decompose a wavefield into its plane-wave components. Each plane wave carries a monochromatic signal that propagates at a certain angle from the vertical. Events with the same dip in the t – x domain, regardless of location, are mapped onto a single line in the radial direction in the f – k domain. In migration, we shall discuss migration methods in the f – k domain, and in noise and multiple attenuation, we shall demonstrate use of f – k filtering to remove coherent linear noise and attenuate multiples.
Spatial aliasing has serious effects on the performance of multichannel processes such as f – k filtering (frequency-wavenumber filtering) and migration (Further aspects of migration in practice|migration in practice]]). Because of spatial aliasing, these processes can perceive events with steep dips at high frequencies as different from what they actually are and, hence, do not treat them properly. For example, migration moves the spatially aliased frequency components in the wrong direction and generates a dispersive noise that degrades the quality of the migrated section.
How is spatial aliasing avoided? Compare the sections in Figures 1.2-8 and 1.2-9. Both have the same frequency content, 6 to 42 Hz. The data in Figure 1.2-9 are spatially aliased because the dipping event is steeper than in Figure 1.2-8. Some ways to avoid spatial aliasing follow:
- Apply time shifts so that the steep events appear to have lower dips. Although this could change the dips that were low to higher dips, making them spatially aliased, it often is a feasible solution for certain situations. For instance, a linear moveout can be applied to CMP gathers to circumvent spatial aliasing of shallow events in prestack migration . Also, a hyperbolic moveout can be applied to CMP gathers before multiple attenuation in the f – k domain (frequency-wavenumber filtering).
- If a low-pass filter were applied to the traces in Figure 1.2-9 so that the frequencies up to 21 Hz were retained, then the segment that is wrapped around to the negative quadrant of the amplitude spectrum is removed. Although spatial aliasing is eliminated, a significant part of the recorded frequency band is lost. This approach is not desirable.
- Figure 1.2-13 shows a single dipping event recorded with three different trace spacings. The 2-D amplitude spectra suggest a third approach to solving the spatial aliasing problem. Note that the coarser the trace spacing, the more frequencies are spatially aliased. The same frequency bandwidth is kept in all three cases. The 12.5-m trace spacing provides a frequency band with no spatial aliasing. For a 25-m trace spacing, frequencies beyond 36 Hz are spatially aliased; while for a 50-m trace spacing, frequencies beyond 18 Hz are spatially aliased. For this latter case, spatial aliasing is so severe that the aliased frequencies wrap around the wave-number axis twice. We see that spatial aliasing can be avoided by selecting a sufficiently small trace spacing. This approach requires either a data-dependent interpolation scheme (processing of 3-D seismic data) to generate extra traces or modification of the field recording geometry. If the latter approach were taken, more shots and/or more recording channels are needed.
To circumvent spatial aliasing, data often are recorded with twice as many number of channels per shot record as that would be used in processing. Prior to dropping every other trace at the start of a processing sequence, a wavenumber filter is applied to remove the wavenumber components for all frequencies beyond the Nyquist wavenumber that corresponds to the trace spacing of the data after dropping alternating traces from the shot record. This wavenumber filter actually is a spatial high-cut antialias filter akin to the high-cut antialias frequency filter discussed in the 1-D Fourier transform.
Consider the shot record shown in Figure 1.2-14a recorded with 368 channels at a group interval of 12.5 m. The record which is intended for input to a processing sequence comprises 192 channels with 25-m group interval. Just as dropping every other sample in a time series can result in frequency aliasing of some of the high frequencies, dropping alternating traces could result in spatial aliasing (Figure 1.2-14b). Note that spatial aliasing in the original record (Figure 1.2-14a) occurs starting at about 75 hz. Trace decimation without spatial antialias filter causes spatial aliasing to occur starting at about 40 Hz (Figure 1.2-14b).
To circumvent spatial aliasing as a result of dropping every other trace in the record, a prior application of a spatial high-cut antialias filter is needed (Figure 1.2-15). Apply a wavenumber filter to remove the energy within the f – k region that is between 50 and 100 wavenumber in both quadrants of the f – k plane (Figure 1.2-15a). Since the trace spacing of the original record in Figure 1.2-14a is 12.5 m, the corresponding Nyquist wavenumber is 40 cycles/km. Following trace decimation, the Nyquist wavenumber is 20 cycles/km, corresponding to 25-m trace spacing. Note in the f – k spectrum in Figure 1.2-15a that the wavenumber filter has removed the energy between 20 and 40 cycles/km for all frequencies in both quadrants. Compare the f – k spectra of the decimated data without (Figure 1.2-14b) and with antialias filtering (Figure 1.2-15b) and note that the wavenumber filter has removed the spatially aliased high-frequency components of the steeply dipping events in the original gather.
The spatial antialias filter design and application involves normal-moveout correction of the data (Section 3.1). Consider the 240-channel shot record shown in Figure 1.2-16a with 6.25-m group interval. We want to apply spatial antialias filter and reduce the number of channels by half with 12.5-m group interval. Follow the steps below:
- Apply normal-moveout correction to the input gather (Figure 1.2-16a) that needs to be decimated to unalias the aliased energy.
- Apply 2-D Fourier transform to map the data onto the f – k plane. Following the normal-moveout correction, the energy maps in the f – k plane closer to the frequency axis, thus alleviating the spatial aliasing (Figure 1.2-17a).
- Design a spatial antialias filter by specifying reject zones for all frequencies between 50 and 100 wavenumber in both quadrants of the f – k plane.
- Apply the wavenumber filter to the moveout-corrected data in the f – k domain (Figure 1.2-17b).
- Apply inverse Fourier transform of the wavenumber-filtered data (Figure 1.2-16b).
- Now drop every other trace to obtain the decimated data (Figure 1.2-16c). The Nyquist wavenumber of the gather following the application of the spatial antialias filter is one-half ot the Nyquist wavenumber of the original gather (Figure 1.2-17c).
- Apply inverse moveout correction (Figure 1.2-16d). The resulting shot record contains one-half the number of channels as in the original record (Figure 1.2-16a), and its energy is within the f – k region defined by one-half the Nyquist wavenumber associated with the original record (compare Figures 1.2-17a and 1.2-17d).
The original moveout-corrected shot record (Figure 1.2-16a) after dropping every other trace, but without the application of the antialias wavenumber filter, is shown in Figure 1.2-16e with the f – k spectrum of the resulting record shown in Figure 1.2-17e. This decimated shot record (Figure 1.2-16e) should be compared with the antialiased-filtered record (Figure 1.2-16c) after inverse moveout correction (Figures 1.2-16d and f). Examine the f – k spectra of the data after trace decimation with (Figure 1.2-17d) and without antialias filtering (Figure 1.2-17f) and note that, for this data set, wavenumber filtering has not been successful in removing spatially aliased energy. Wavenumber filtering for trace decimation is successful if the aliased energy in the original record is confined to the reject zones for all frequencies between 50 and 100.
Figure 1.2-13 A single, isolated dipping event sampled at three different trace spacings with the corresponding f – k spectra. No spatial aliasing occurs with the 12.5-m trace spacing (left). Frequencies beyond 36 Hz are aliased with the 25-m trace spacing (center). Double aliasing occurs with the 50-m trace spacing (right). Although events on the f – k spectra appear to have different dips, all three have the same dip on the t – x gathers (top). This deceptive character is because of the different horizontal scales used in displaying the f – k spectra.
Figure 1.2-14 (a) A shot record with its 2-D amplitude spectrum displayed on top; trace spacing is 12.5 m and the Nyquist wavenumber is 40 cycles/km. Minimum and maximum offsets are 140 m and 4727.5 m, respectively. (b) The same record as in (a) with every other trace dropped; hence, trace spacing is 25 m and the Nyquist wavenumber is 20 cycles/km. Note that little energy is spatially aliased above 75 Hz in the original record (a), while spatial aliasing occurs beginning at 45 Hz in the record with every other trace dropped (b).
Figure 1.2-15 (a) The same shot record as in Figure 1.2-14a after the application of a wavenumber filter to remove the energy beyond half the Nyquist wavenumber for all frequencies. (b) The same record as in (a) after dropping every other trace. Note that spatial aliasing is circumvented first by applying a high-cut spatial antialiasing wavenumber filter (a) prior to dropping every other trace. In contrast, dropping every other trace without the prior application of an antialias filter causes spatial aliasing of events with large moveout (Figure 1.2-15b).
Figure 1.2-16 (a) A 240-channel shot record with 6.25-m group interval; (b) after normal moveout correction; (b) after the application of wavenumber filter to remove the energy between one-half the Nyquist and the Nyquist wavenumbers for all frequencies; (c) the same record as in (b) after dropping every other trace so that the resulting shot record comprises 120 channels with 12.5-m group interval; (d) the same record as in (c) after inverse moveout correction; (e) the same record as in (b) after dropping every other trace without wavenumber filtering; (f) the same record as in (e) after inverse moveout correction.
So far, only the synthesis of a single dipping event from a discrete number of frequency components has been considered. This analysis now is extended to a range of dips. Figure 1.2-18 shows a section with dips that vary from 0 to 45 degrees and the corresponding 2-D amplitude spectrum. These same dips, but with higher frequency content, also are seen in Figure 1.2-19. Events with 0-, 5-, 10-, and 15-degree dips are not spatially aliased. The 20-degree dip is aliased at nearly 72 Hz, the 30-degree dip at nearly 48 Hz, and the 45-degree dip at nearly 36 Hz. Again, the steeper the dip, the lower the frequency at which spatial aliasing occurs.
Given a dip value, how is the maximum unaliased frequency determined? Consider the 20-degree dipping event in Figure 1.2-19. First, measure the dip in milliseconds per trace. There are 256 traces in the t – x model with 25-m trace spacing. The 20-degree dip is equivalent to 7 ms/trace. Frequency components with periods less than twice the dip are spatially aliased. Thus, given the dip in milliseconds per trace, the threshold frequency at which spatial aliasing begins is 500 per dip. In the present case, the threshold frequency is 500/7 ≈ 72 Hz. This is verified by examining the amplitude spectrum in Figure 1.2-19.
Figure 1.2-20 shows three field records and their 2D amplitude spectra, known as f – k spectra. By now, it is easy to recognize and relate various events on the shot gathers to those on the f – k spectra. Event A is the high-amplitude dispersive coherent noise with very low group velocity. When the spatial extent of these waves broadens, bandwidth in the wavenumber direction becomes smaller. Conversely, when the spatial extent becomes smaller, the event, such as G, spans a wider wavenumber bandwidth in the f – k spectrum (compare events A, F, and G). Events B and C are parts of the guided wave packet. Event C contains aliased energy above 42 Hz (indicated by D on the f – k spectrum). Primaries and associated multiples are mapped into region E between the frequency axis and event C.
Spatial aliasing not only is a concern in a prestack application of a multichannel filter, such as f – k filtering, but also during poststack processing. Specifically, migration of CMP stacked data suffers from spatial aliasing if the trace interval is too coarse. An optimum CMP trace interval to avoid spatial aliasing can be computed as follows. Consider a dipping reflector with a dip angle of θ (Figure 1.2-21). Also consider a normal-incidence plane wave with a dominant period T recorded at the surface with a trace separation Δx. (This is the zero-offset case where Δx is the CMP trace interval.) From the geometry in Figure 1.2-21, we write
where Δt is the two-way time separation between the arrival times of the plane wave at the two receiver locations, A and B, and ν is the medium velocity. Spatial aliasing occurs when the wavefront separation in time Δt equals half the dominant period T. When this criterion is applied to equation (6), we get the following expression for the maximum threshold frequency, fmax = 1/T, that is not aliased for a given dip, velocity, and CMP trace interval:
Table 1-12 shows the evaluation of equation (7) for a particular numerical example. Equation (7) also can be expressed in terms of receiver group interval 2Δx. Suppose the maximum dip is 30 degrees. If the sampling interval is 4 ms, then the Nyquist frequency is 125 Hz. After antialias filtering, the frequency band extends up to 90 Hz, provided the high-cut filter is at three-quarters of the Nyquist. For a bandwidth without spatial aliasing, we have to select a 12.5-m CMP trace interval.
To circumvent the deleterious effect of spatial aliasing on migrated data, trace interpolation is applied to attain a trace interval appropriate for the bandwidth of the CMP-stacked data. As stated in the 1-D Fourier transform, 1-D interpolation of a time series, such as a seismic trace, from a coarse to a finer sampling interval does not recover the frequencies lost by the original sampling; but, it only generates extra samples. On the other hand, interpolation of 2-D data, such as a seismic section, from a coarse to a finer trace interval, is possible for all signal frequencies, including those that are aliased. This is because of the fact that one can detect and measure the dip as defined by the stepout of equation (5) on a stacked section for the unaliased frequencies and use the dip information to interpolate not only the unaliased but also the aliased frequencies. Modern data acquisition geometries for 2-D data often do not require trace interpolation. Nevertheless, trace interpolation usually is required in the crossline direction prior to 3-D migration (processing of 3-D seismic data).
|Threshold Frequency (Hz) for CMP Trace Interval (m)|
|Dip Angle (deg)||12.5||25||37.5||50|
- Li et al., 1991, Li, Z., Lynn, W., Chambers, R., Larner, K., and Abma, R., 1991, Enhancements to prestack frequency-wavenumber (f – k) migration: Geophysics, 56, 27–40.