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, (*k*_{x},*k*_{h}), we can view the
spatial energy components of the image at a reflector depth.

Figure 1

The shaded box in the center of each diagram represents the energy
delimited by the Nyquist wavenumbers. The thick line at *k*_{x} = 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
(*k*_{s},*k*_{r})-plane, the replicated *k*_{x}=0 energy is now coincident with
the Nyquist boundaries on the (*k*_{x},*k*_{h})-plane. However, in the
presence of dipping energy () 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 (*k*_{x},*k*_{h})-plane is,

(1) |

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:

(2) |

7/8/2003