Shot illumination examples

As described above, normalization by shot illumination only approximates an inverse operation if the illumination problems come from poor sampling on the shot axis (or reciprocal shot) rather than on both shot and receiver axes. As a best case scenario, to test the validity of this approach, I generated a dataset consisting of 20 split-spread shots over the Marmousi velocity model, using a paraxial Fourier finite-difference wavefield extrapolation algorithm Ristow and Ruhl (1994). The receiver axis was well sampled, as each shot-gather contained 200 traces.

The top panel in Figure  shows the result of migrating the 20 shots with the correct velocity model, and the center panel shows the shot illumination. Clearly areas of high illumination correspond to bright spots in the migrated image. Panel (c) shows the migrated image after normalization by the shot illumination. Although only 20 shots were migrated into this image, its clarity is comparable with the full migration of the entire dataset (about 240 shots). It is also encouraging that while the shot locations are visible on Figure  (a), they are hidden in Figure  (c): the normalized image appears to be unbiased by the recording geometry.

 

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Figure 7 Normalization by shot illumination: the panel (a) shows shot profile migration with the correct velocity model of 20 split-spread shots over the Marmousi model, panel (b) shows shot illumination, and panel (c) shows the normalized image.

Compensation for shot illumination in Figure  worked so well partly because the modeling and migration procedures were true adjoints. To test how robust this normalization would be in the more realistic situation where the modeling and migration procedures are not true adjoints, I migrated the same 20 shots with an incorrect velocity model. Panels (a) and (b) of Figure  show the migrated image, and the corresponding shot illumination respectively. Panel (c) shows the migrated image after normalization. Despite the incorrect velocity, the amplitude artifact visible in Figure  (a) are largely gone from Figure  (c). For example in the top 500 m of the section, the footprint of the acquisition geometry is removed from Figure  (c). The vertical amplitude streaks lower in the section are also reduced -- illumination of the target reflectors at 2600 m depth is more uniform after normalization, especially in the highlighted area.

 

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Figure 8 Normalization by shot illumination: the panel (a) shows shot profile migration with the incorrect velocity model of 20 split-spread shots over the Marmousi model, panel (b) shows shot illumination, and panel (c) shows the normalized image.