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f-k domain

Images - Chapter 8 Energy distribution of seismic signal in the f-k domain. (a) Minimum apparent velocity determines region in which energy is present. (b) If there is a maximum frequency, then there is also a maximum wavenumber.

Aliasing for a one-dimensional signal

Images - Chapter 8 Aliasing for a one-dimensional signal. (a) Nonaliased signal and its amplitude spectrum. (b) The same signal as (a), but at a coarse sampled interval which produces aliasing. Notice how the overlapping of aliased energy has changed the amplitude in (b).

For the one-dimensional case (the single variable signals), aliasing causes duplicates of the input spectrum in multiples of 1/Δt, where Δt is the sampling interval, to overlap. If this overlapping does occur, the aliasing effects cannot be removed without additional information about the sampled spectrum from sources other than the sampled spectrum itself. A trivial example would be if it was known that a 4-millisecond dataset contained frequencies between 0 and 90 Hz along with an unknown amount of aliased 150 Hz. Since 150 Hz produces an alias at 125-(150-125) Hz = 100 Hz, the correct spectrum for the 4-millisecond sampling interval could be reconstructed by zeroing out the 100-Hz contribution. Another illustration of the effect of aliasing for a one-dimensional signal is given in this figure.

Dipping events

Images - Chapter 8 Data consisting of four dipping events. The amplitude spectra of these data are given in the next figure for three spacing intervals between traces: 2.5, 5.0, and 12.5 m.

f-k spectra

Images - Chapter 8 f-k spectra for (a) 2.5-m, (b) 5-m, and (c) 12.5-m spacing between traces.

Spatial aliasing can also occurs for multidimensional signal, like 2D signals. However, with regard to the removal of the spatial aliasing effects, things are very different, because we are generally dealing with two (or more) dimensions. Let us start by looking at examples of $f-k$ plots of aliased seismic data. Consider a 2D signal made of four dipping events, as depicted in the previous figure. This 2D signal represents a zero-offset section. This figure shows the f-k amplitude spectrum of this signal for three spacing intervals between traces: 2.5 m, 5 m, and 12.5 m. The data are considered spatially aliased if some of the energy of the data is wrapped around in the f-k amplitude spectrum plot. For 2.5-m spacing, the data are not aliased, and the different dipping events in the t-x map onto straight lines in the f-k domain with inverse slopes, as discussed earlier. For 5-m spacing, we notice that event D is aliased. However, contrary to what we discussed in Chapter 4 for one-dimensional signals (single variable signals), a wrap-around caused by spatial aliasing may sometimes not include any overlapping with the nonaliased signal, as illustrated in (b). In this case, the effect of aliasing can be corrected by zeroing the aliased region in the f-k domain (see the section on dip filtering for more details).
By increasing the spacing between traces to 12.5 m, we can see that events D and C are now severely aliased and overlap with the nonaliased signal.

f-k dip filtering

Images - Chapter 8 f-k dip filtering for multiple attenuation.

F-k dip filtering can help discriminate against multiples on the basis of moveout. This figure shows this f-k filtering procedure. The data are transformed from the t-x domain to the f-k domain. F-k dip filtering distinguishes between primary and multiple reflections based on residual moveout after NMO corrections have been applied. Multiple events are flattened, with approximately no residual moveout, whereas the primary events will have some residual moveout. The data with NMO corrections are transformed in the f-k domain, in which primary and multiple events are now well separated based on their dips
(or equivalently apparent velocities). The zero dip corresponds to multiples which can be zeroed out. The following steps consist of performing the inverse Fourier transform and removing the NMO correction. We can see that f-k dip filtering can be effective for multiple removal, especially in the far offsets, but can leave multiples in the near offsets, where the moveout is minimal.

Model

Images - Chapter 8 Synthetic model and parameters.

f-k dip filtering

Images - Chapter 8 f-k dip filtering for a VC shot close to the cable. (a) Raw shot gather for a VC cable located at 362.5 m from the shot point. (b) The f-k spectrum of the raw shot gather. (c) and (d) Separation of the f-k spectrum of the raw shot gather into negative and positive wavenumber. (e) and (f) Separation of the raw shot gather into upgoing wavefield and downgoing wavefield.

Again, this figure shows an example of a vertical cable shot gather for one cable using a receiver spacing of 6.25 m. The model used to generate these data is shown in the previous figure. We can see that most of the events are linear and have opposite gradients, just as in VSP data. In the f-k spectrum, the events in the vertical cable data are arranged as a function of the gradients only. The downgoing wavefield is located in the positive wavenumbers, and the upgoing wavefield is located in the negative wavenumbers. As we can see in this figure, this separation is very clear. The up/down separation based on $f-k$ filtering consists of zeroing energy in the $f-k$ domain corresponding to the negative wavenumbers to extract the downgoing field and zeroing energy in the f-k domain corresponding to the positive wavenumbers to find the upgoing field, as illustrated in this figure.

One may expect that the f-k dip-filtering method is more difficult to apply for shot points located far away from the cable because the arrivals of seismic events is no longer linear function of receiver positions. It turns out that this is not actually the case. For a shot located at 1 km from the cable, except for the direct wave, all the rest of the events remain linear, with a positive gradient for downgoing waves and a negative gradient for upgoing waves. We can see that the up/down separation is quite effective, even in this case.

Note that this procedure is generally applied on shot gather domain, in which the dip separation of events is more effective. However, the effectiveness analysis of the up/down separation can also be carried out in the receiver gathers, as illustrated in the next figure. Although the downgoing wavefields contain only multiples, the up/down separation is not totally equivalent to multiple attenuation because the upgoing wavefield also includes also multiple events, as illustrated in the next figure.

Up/down separation

Images - Chapter 8 Receiver gathers after up/down separation using f-k dip filtering.

f-k spectra

Images - Chapter 8 f-k spectrums considering different vertical cable samplings.

We have simulated the situation in which the vertical cable receiver spacing is 6.25 m, 12.50 m, 18.75 m, and 25 m. This figure shows the $f-k$ spectrum of some shot gathers for the different receiver spacing considered. We notice that aliasing occurs for a receiver spacing of 18.5 m and greater. As we can see, at 18.5-m and 25-m receiver spacing, some downgoing event aliases are shifted to negative wavenumbers at higher frequencies instead of staying in positive wavenumbers. The same problem happens for upgoing event aliases, which are shifted in positive wavenumbers instead of staying in negative wavenumbers at higher frequencies. So due to aliasing, it is now difficult to achieve a proper up/down separation. A trade-off has to be made between having a good up/down separation and keeping the whole signal spectrum. One possible way for improving the f-k dip filtering is to use an interpolation method, like the f-x interpolation, to create the data required to satisfy the sampling theorem.

Resampling steps

Images - Chapter 8 Resampling steps.

The principle for selecting a temporal sampling interval is exactly the same as that for selecting the spacing interval between receivers and/or the spacing interval between shot points. The usual temporal sampling interval used today is 2 ms. For a typical seismic record of 75 Hz, the sampling theorem requires that the temporal sampling interval be less than 6 ms to ensure that the recorded wavefield can be reconstructed at any time with maximum fidelity. With a 2-ms sampling interval, we are well within the requirement of the sampling theorem.

Moreover, the data at a 2-ms sampling interval can be resampled to the commonly used 4-ms sampling interval after an antialiasing filter has been applied, as illustrated in this figure. Note that by collecting data at a 2-ms sampling interval, we avoid possible distorting of the desired spectrum with noise which may contain frequencies higher than 125 Hz [i.e., 1/(2Δt), with Δt = 4 ms].

The situation is quite different regarding our usual selection of the spacing interval for receivers, for instance. One typical spacing between receivers used in seismic acquisition today is 25 m. Based on the sampling criterion, all seismic events with an absolute value of the apparent velocity of less than 2750 m/s (in the frequency ranges of 0 to 75 Hz) will be aliased; in other words, the recorded wavefield alone is not enough to properly reconstruct these events at any point in the space other than the point where the measurement is recorded. Seismic events with an absolute value of the apparent velocity of less than 2750 m/s include direct waves, groundroll, air waves, and even some desired reflection primaries. All these events will be aliased. As described above, the spacing interval between receivers must ideally be of the order of 1 m to avoid aliasing. With about 25-m spacing between receivers, a typical land-seismic acquisition requires picking up, putting down, and maintaining 150,000 geophones. Multiplying this number of geophones by a factor of 25 to achieve a 1-m spacing is not yet economically viable. On the other hand, we cannot rely on seismic processing to solve this problem, either. As we will see in the next figure, sometimes the aliased energy overlaps so much with the desired signal that f-k dip filtering becomes impractical. The solution to this aliasing problem, adopted by the oil and gas industry in the 1930s, when reflected seismic experiments were first used for petroleum exploration on land, is known as hard-wired array recording (sometimes called array recording).

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