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To demonstrate how operator aliasing arises, it is informative to study an example where improper sampling generates artifacts in the image. When analyzing this problem, it is common to superpose the shot-receiver ,sr, and location-offset, xh, axes. The definition of the transformation between the coordinate pairs dictates how the movement of energy in one frame is related to changes in the other. Even though there is only rotation between the two coordinate systems, the act of overlaying them on the same Cartesian grid inappropriately applies a stretch of $\sqrt{2}$ and gives rise to confusion especially when trying to interpret the Fourier duals of these dimensions. Therefore, these axis are treated separately in this analysis.

Figure [*] explains how we will visualize distributions of energy in the image-space. By transforming the surface location and CIG-offset axes, (x,h), to the Fourier domain variables, (kx,kh), we can view the spatial energy components of the image at a reflector depth.

Figure 1
Cartoon illustrating a sample energy distribution in the image-space and the hypothetical position of Fourier domain replications (dotted lines) outside of Nyquist boundaries (shaded region). Dot-dashed lines represent replicated energy from the kx=0 band of energy corresponding to a single flat reflector at this image depth. Panels (a) and (c) depict the energy components of the data-space. Panels (b) and (d) depict the energy components of the image-space. The bottom row shows the result of a compression along the ks-axis (indicated by the filled arrows) as a result of migrating every other available shot-location.

The shaded box in the center of each diagram represents the energy delimited by the Nyquist wavenumbers. The thick line at kx = 0 is the 2-D Fourier transform in the (x,h)-coordinate system of the image at the depth of a flat reflector. Outside of the shaded region the periodicity of Fourier spectrum, due to discrete sampling, is depicted by two of an infinite number of possible Nyquist replicas (dashed lines). Note that the juxtaposition of all of the replications gives rise to the lines of continuous horizontal energy (dot-dash lines).

In panels a and b, which are the data-space and image-space coordinate representations of energy spectra, sampling requirements are honored and no aliasing occurs. The first set of horizontal energy replications will be the first aliased energy to enter the shaded region if an axis is compressed by decimating the data. In our experiments, we have focused on subsampling the shot-axis as indicated by the filled arrows pointing toward the origin. Panels c and d depict the effects of subsampling the shot axis by a factor of two. The replicated energy bands are compressed toward the origin by a factor of two relative to the overlying panels. The energy in the right and left columns moves to the same wavenumber values on their respective axes, despite the rotation of the Nyquist boundaries. Although aliased energy has now moved below Nyquist on the (ks,kr)-plane, the replicated kx=0 energy is now coincident with the Nyquist boundaries on the (kx,kh)-plane. However, in the presence of dipping energy ($k_x \ne 0$) aliasing will arise with this degree of subsampling.

We consider two approaches to control the aliasing problems associated with the acquisition and subsampling situations mentioned above. First, wavenumbers from the source and receiver wavefields are band-limited to prevent the entry of aliased duplications into the imaging condition. This does not require eliminating these components from the propagating wavefields, as we can save appropriate portions of the wavefields in disposable buffers for imaging. Second, a band-limited source function is propagated for the duration of each shot-gather migration which effectively zeros energy in the aliased band during imaging. This method manufactures a source function with a wavenumber spectrum limited to the cutoff frequencies imposed by the resampled data axis. There is no additional computational overhead when anti-aliasing with a thick source function, though anti-aliasing by restricting the wavenumbers requires two additional Fourier transforms.

The analytical band-limit is required to appropriately delimit non-aliased wavenumbers for either method. After compression of either data axis, the stretch factor to show the movement of wavenumbers in the (kx,kh)-plane is,  
\mbox{BandLimit}(k_x) = \frac{max(\Delta r, \Delta s)}{max(a \Delta r,b \Delta
 s)} min(Ny_r,Ny_s)\end{displaymath} (1)
where ($\Delta r,\Delta s$) are the original receiver and source increments, (a,b) are the down-sampling ratios from the original to the new receiver and source grids, and (Nyr,Nys) are the original Nyquist limits of the data-space. The max functions in the expression are required because the maximum grid-spacing along either shot or receiver axis alone dictates the aliasing criteria for the kx-axis.

The band-limit value calculated with equation (1) is the criteria for eliminating all energy external to the new rigorously defined Nyquist limit. However, if the dipping energy in the data is limited to less than Nyquist, the maximum dip, |p|max, may be used to relax the band-limiting criteria resulting in the definition:
\mbox{BandLimit}(k_x,\vert p\vert _{max})= \frac{Ny_x-\vert p\vert _{max}/2}{max(a\Delta r,b\Delta s)}.\end{displaymath} (2)
Note that if the energy distribution is asymmetric (i.e. $p_{min}
\ne -p_{max}$), two one-sided limits can be implemented. In the case of only zero dip, the band-limit wavenumber restriction is twice the rigorous definition from equation (1).

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