Operator aliasing

  The previous section analyzed aliasing in the image space, that occurs when the spreading representation of the imaging operator is aliased. Data-space aliasing, the dual of image-space aliasing, occurs when the gathering representation of the imaging operator is aliased in the data space. This data-space aliasing is commonly called operator aliasing . When operator aliasing occurs, noise is actually added to the image, in contrast to the image-space aliasing case. The next set of figures will illustrate the issues related to operator aliasing by analyzing a simple example in which the data are plane waves and the summation operator is a slant stack.

Figure 9 and Figure 10 show two plane waves. The plane waves are adequately sampled when the waveform is a 30 Hz sinusoid (Figure 9a), but the one with positive time dip is aliased when the waveform is a 60 Hz sinusoid (Figure 10a). The data aliasing can be observed both in the time-space domain, where the data appears to be dipping in the opposite direction, and in the wavenumber domain. The corresponding spatial spectra are shown at the bottom of the Figures. The solid lines correspond to the positive-dip plane wave, and the dotted line to the negative-dip plane wave. The spectrum for the positive-dip plane wave (solid line) at the bottom of Figure 9a shows two spikes at $\pm.75 k_{N}$ replicated at $\pm 1.25 k_{N}{~\rm and~} \pm 2.75 k_{N}$.Because of the doubling of the temporal frequency, in the spectrum at the bottom of Figure 10a the aliased spikes at $\pm 1.25 k_{N}$ moved into the central band to $\pm .5 k_{N}$.

Data summation along a given trajectory is equivalent to a two-step process; first the data are shifted to align the events along the desired trajectory. Second, the traces are stacked together. In the case of the slant-stack operator, the summation trajectories are lines and the first step is equivalent to the application of linear moveout (LMO) with the desired dip. Figure 9b and Figure 10b show the results of applying LMO with the slowness of the positive-dip plane wave to the corresponding data in Figure 9a and Figure 10a. The traces on the right side of the sections are the results of stacking the corresponding data. At 30 Hz no aliasing occurs, and after LMO only the original plane wave stacks coherently, as desired. In contrast, at 60 Hz both plane waves stack coherently after LMO as well as the original plane wave. In general, artifacts are generated when data that are not aligned with the summation path stack coherently into the image. This phenomenon is the cause of the aliasing noise  that degrades the image when operator aliasing occurs. To avoid adding aliasing noise to the image we could lowpass filter the input data according to the operator dips. The resulting anti-aliasing constraints are:

$$\begin{eqnarray} \omega^{D}_t& \leq & \frac{\pi}{\Delta x_{D}p^{{\rm op}}_x }, \... ...r \\ \omega^{D}_t& \leq & \frac{\pi}{\Delta y_{D}p^{{\rm op}}_y };\end{eqnarray}$$ (3)

where $\Delta x_{D}$ and $\Delta y_{D}$are the sampling rates of the data axes, and $p^{{\rm op}}_x$ and $p^{{\rm op}}_y$are the operator dips. These, or equivalent, relationships have been presented by a number of authors Abma (1998); Bevc and Claerbout (1992); Gray (1992); Lumley et al. (1994). Although, these constraints may be correct for several important cases, they do not take into account the fact that operator aliasing depends on the presence of conflicting dips in the data, as shown by the previous example. More precisely, it depends on the dip bandwidth in the data.

 

Fig-30-sec
Figure 9 Two plane waves with dips of .5 s/km and .166 s/km and 30 Hz waveform, a) before linear moveout and b) after linear moveout.

 

Fig-60-sec
Figure 10 Two plane waves with dips of .5 s/km and .166 s/km and 60 Hz waveform, a) before linear moveout and b) after linear moveout.

 

Fig-60-gath
Figure 11 Two plane waves with dips of .5 s/km and 0 s/km and 60 Hz waveform, a) before linear moveout and b) after linear moveout.

To further examine the idea of operator aliasing depending on the dip bandwidth in the data, we consider the two plane waves shown in Figure 11. In this case the two plane waves have a 60 Hz waveform, as in Figure 10, but with the second plane wave flat instead of dipping with a negative time dip. The two plane waves have conflicting dips; but the additional plane wave does not interfere with the stacking of the original plane wave even with a 60 Hz waveform.

The last two examples demonstrate that the limits on the dip range for unaliased summation paths are a direct function of the expected dips in the data along the summation axes. If $\left(p^{\min}_x,p^{\min}_y\right)$ and $\left(p^{\max}_x,p^{\max}_y\right)$ are respectively the minimum and maximum dips expected in the data, then, to avoid operator aliasing, the operator dip must fulfill the following inequalities:

$$\begin{eqnarray} p^{\max}_x - \frac{2 \pi}{\omega^{D}_t\Delta {x}} \leq p^{{\rm ... ...rm op}}_y \leq p^{\min}_y + \frac{2 \pi}{\omega^{D}_t\Delta {y}}.\end{eqnarray}$$ (4)

The inequalities expressed in equation (4) can be easily recast as anti-aliasing constraints on the maximum frequency in the data as:

$$\begin{eqnarray} \omega^{D}_t\leq \frac{2 \pi}{\Delta {x}\left(p^{{\rm op}}_x - ... ... \frac{2 \pi}{\Delta {y}\left(p^{\max}_y - p^{{\rm op}}_y\right)}.\end{eqnarray}$$ (5)

The constraints expressed in equation (5) can be used as alternatives, or in conjunction with the constraints expressed in equation (3) to antialias summation operators. Examining the inequalities expressed in equation (4), we can notice that the two sets of constraints are equivalent when, for each frequency, the data dip limits ${p^{\min}}$ and ${p^{\max}}$are set to $k_{Nyq}/\omega^{D}_t$, where k_Nyq is the Nyquist wavenumber along each axis. That is, if we assume that there is no spatial aliasing in the data, the constraints expressed in equation (5) are equivalent to the constraints expressed in equation (3).

The data-dips limits ${\bf p^{\min}}$ and ${\bf p^{\max}}$can be both spatially and time varying according to the expected local dips in the data. Therefore, the anti-aliasing filtering applied to the data as a consequence of the constraints in equation (5) can be fairly complex, and dependent on: local dips, time, and spatial coordinates. If no a priory knowledge on the local dips is available, and the summation is carried out along the midpoint axes, twice the inverse of propagation velocity is a reasonable bound on the absolute values of both ${\bf p^{\min}}$ and ${\bf p^{\max}}$.In contrast, in the case that the summation is performed along the offset axes, as for CMP stacks, ${\bf p^{\min}}$ can be safely assumed to be positive, and at worst equal to zero. In practice the bounds on the data's expected dips should take into account all types of events, and not only the dips of the reflections that we aim to image. For example, in CMP gathers recorded on land, ${p^{\max}}$ should take into account low-velocity events such as ground roll.

The most substantial benefits of applying the more general constraints expressed in equation (5) are achieved when asymmetric bounds on the dips in the data enable imaging without aliasing high-frequency components that are present in the data as aliased energy, and consequently would be filtered out if the constraints in equation (3) were applied. An important case when asymmetric bounds on the data dips are realistic is the imaging of steep salt-dome flanks, as in the Gulf of Mexico data set shown above. In this case, we can assume that the negative time dips in the data are small. According to the equations in (4), the increase in ${p^{\min}}$ raises the limit on the maximum positive operator dip. In practice, the application of the generalized constraints in equation (5), when ${\bf p^{\min}} \neq {\bf p^{\max}}$cause the migration operator to be asymmetric, with dip bandwidth dependent on reflector direction. Figure 12 and Figure 13 show an example of the effects of asymmetric dip bounds on the migration operator. For both images, the image sampling is the same as in Figure 6 $\left(\Delta x_\xi= \Delta y_\xi= 20 {\rm m}\right)$,but the data sampling is assumed to be coarser than the image sampling by a factor of two; that is, $\Delta x_\xi= \Delta y_\xi= 40 {\rm m}$.When the constraints in equation (3) are applied (see Figure 12), the operator has lower resolution than in Figure 6. But if we assume that $p^{\min}_x = 0$,and apply the constraints in equation (5) (see Figure 13), the positive time dips are imaged with the same resolution as in Figure 6.

 

Imp-antialias-nodirect Figure 12 Image obtained by applying Kirchhoff migration with ``standard'' anti-aliasing. Sampling rates are: $\Delta x_\xi=\Delta y_\xi=20 {\rm m}$and $\Delta x_{D}=\Delta y_{D}=40 {\rm m}$.

 

Imp-antialias-direct Figure 13 Image obtained by applying Kirchhoff migration with ``directed'' anti-aliasing assuming $p^{\min}_x = 0$.Sampling rates are: $\Delta x_\xi=\Delta y_\xi=20 {\rm m}$and $\Delta x_{D}=\Delta y_{D}=40 {\rm m}$.