The Sigsbee data set was modeled by simulating the geological setting found on the Sigsbee escarpment in the deep-water Gulf of Mexico. The model exhibits the illumination problems due to the complex salt shape, characterized by a rugose salt top (see Figure ). We choose a target zone ( see Figure ) to see the effects of illumination on imaging close to the salt.
Figure shows the shot-profile migration image (using cross-correlation imaging condition) corresponding to the portion of Sigsbee model shown in figure . Notice how the amplitudes of the reflectors fade away as they get closer to the salt.
Figure shows a coefficient filter (target-oriented Hessian) at constant depth as the x coordinate moves from the sediments to the salt boundary. Figure a shows point 1, with coordinates (far from the salt). Figure b shows point 2, with coordinates . Figure c shows point 3, with coordinates . Figure d shows point 4, with coordinates .
The shape of the filter is not dependent only on the acquisition geometry but the subsurface geometry (presence of the salt body). In the area unaffected by the salt the filter looks the same as is the constant velocity case, but as we get closer to the salt the illumination varies (in intensity and angle) and the filter behaves differently. This is due to a focusing and defocusing effect created by the salt. To correct this effect we computed the least-squares inverse image.
Figure shows a comparison between the migration and the postack inversion images in the target area. The reflection coefficients are shown in Figure a. Notice the position of the faults. Figure b shows the illumination, which is the diagonal of the Hessian matrix. Notice the decrease in the illumination as it gets closer to the salt with the exception of a narrow strip where energy focuses close to the salt. The migration result is shown in Figure a. The reflectors dim out as they get closer to the salt. In contrast, Figure b shows the postack inversion result, the resolution increases and the section looks more balanced. The fault can be followeded and interpreted closer to the salt body.
The salt, in the inversion image looks distorted. This is due to the fact that data values (migration) in the salt boundary are bigger than everywhere else (Figure a), and so are the data residuals. Thus, the solver expends most of the time decreasing the residuals in that area. A residual weight designed to decrease the salt contribution should improve the image. Figures and compare the migration and the prestack inversion images in the reduced target area. The resolution increases, and the section looks more balanced in the prestack inversion result. The fault can be follow and interpreted closer to the salt body in the inversion image.