Figure shows the comparison of the space-domain impulse response to the Fourier equivalent developed above for the zero dip reflector. Both seem to provide identical results when viewed at subsurface offset h=0 at the x-z plane. The smile shapes are due to the limited extent of the acquisition along the surface. However, Figure compares the algorithms across the x-h plane at the depth of a correctly migrated image point. While at zero offset the two images are the same, the Fourier-domain implementation has obvious replications at the end of the dog-bone shaped energy distribution. Aliased energy is also introduced into the upper-right and lower-left corners of the image space. This periodicity is not encountered with the space-domain implementation.
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Figure 1 Migration impulse response through a constant velocity medium of both space-domain and Fourier domain implementation of the shot-profile imaging condition viewed at h=0 on the x-z image plane. |
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Figure 2 Migration impulse response through a constant velocity medium of both space-domain and Fourier domain implementation of the shot-profile imaging condition viewed at the depth of a focused image point on the x-h image plane. |
By adding dip to the reflection point, the dog-bone shape in the x-h plane becomes skewed. Figures and are produced with a reflector with 20^{o} and 40^{o} dip respectively. While the zero-offset image of the x-z plane remains the same, the periodicity of the Fourier-domain algorithm is again visible as compared to the computation of the imaging condition in the space-domain. Notice that as dip increases, the saddle point of the dog-bone has moved away from zero offset and the image gives the sense of tipping into the page.
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Figure 3 20^{o} dip reflector image comparison. Left is Fourier-domain implementation. Right is space-domain implementation. Image is extracted at the depth of a focused image point on the x-h image plane. |
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Figure 4 40^{o} dip reflector image comparison. Left is Fourier-domain implementation. Right is space-domain implementation. Image is extracted at the depth of a focused image point on the x-h image plane. |
The onset of the replications are at precisely half the offset and surface location axes. To remove this type of artifact in the Fourier domain, interpolation of both axes by a factor of two would be required. From equation 7, we note that the wavefields have already required a factor of two interpolation to facilitate the algorithm. Interpolating the image by a factor of two again substantially increases memory consumption and computational effort for this implementation.
The first shot from the Marmousi synthetic data was migrated to examine the effects of the above periodicity injected into the image using the Fourier-domain imaging condition. Figure compares the image from the first shot in the data computed with both space-domain and Fourier-domain imaging conditions. At great depth, the images are largely comparable, while at less than 1000 meters, the images are completely different. This is due to the combined effects of periodicity of the Fourier computed image and the steep dips of the model.
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Figure 5 Images from the first shot in the Marmousi data set. Left panel computed with the Fourier-domain imaging condition, and the right panel with the conventional space domain algorithm. |
Deeper in the section, the problem is less apparent and the Fourier domain imaging condition is much closer to the space-domain result. Figure demonstrates how the zeroing of the evanescent waves through the course of the migration effectively limits the range of wavenumber energy allowed into the image. After the evanescent limitations are more restrictive than the effects of the Fourier domain periodicity, the artifacts begin to diminish.
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Figure 6 Images from the first shot in the Marmousi data set. Depth slices, computed with the Fourier domain imaging condition, are extracted from z=250m and z=1188m. |
However, the images for a complex medium are definitely not strictly equivalent for the two alternative imaging condition implementations. By interpolating the k_{x} and k_{h} axes, it is possible to remove the limitation imposed by the periodicity at the memory/disk cost of 4n_{x}^{2} per depth level.