High-resolution imaging of salt flanks

The high-resolution imaging of salt-dome flanks is an important application of the theory described in the previous two sections. In this case we can often assume that in the proximity of the salt flanks the data contains no, or little, energy dipping in the direction opposite to the reflections from the flanks. According to the theory developed in the previous section, this assumption enables the imaging of the salt flanks with higher resolution than otherwise possible. However, it is important to be aware that when increasing the image frequency content by applying the constraints in equation () in place of the constraints in equation (), we run the risk of aliasing the image. As discussed in the previous section, the conditions that avoid operator aliasing do not guarantee avoidance of image aliasing. Therefore, the constraints to avoid image aliasing [equation ()] must also be taken into account, and the image sampling must be reduced to achieve both goals of avoiding image aliasing and preserving high-frequency components. Because of image aliasing considerations, the images of the salt flanks from the Gulf of Mexico data that are shown in the following Figures are sampled twice as densely $\left(\Delta x_\xi= \Delta y_\xi= 18 {\rm m}\right)$,as the zero-offset data $\left(\Delta x_{D}= \Delta y_{D}= 36 {\rm m}\right)$.

The first step to apply the high-resolution imaging method presented in this paper is to determine the appropriate values for the bounds on the data dips. For the sake of simplicity, for this example I choose constant bounds; that is, $p^{\min}_x=-.082~{\rm s/km}$, $p^{\max}_x=.48~{\rm s/km}$,$p^{\min}_y=-.8~{\rm s/km}$, $p^{\max}_y=.8~{\rm s/km}$.In more geologically-complex cases it may be advantageous to allow the data-dip bounds to vary both in time and space. Figure  shows the same data spectrum as Figure ; the dashed line superimposed onto the spectrum cover the areas determined by the inequalities of equation (), according to the chosen bounds for $p^{\min}_x$ and $p^{\max}_y$.The areas that honor all anti-aliasing constraints, and thus that represent data components that are used by the imaging, are covered by crossing dashed lines. A large swath of the aliased energy with positive time dips is used by the high-resolution imaging, whereas it would be discarded if standard anti-aliasing methods were used.

The improvements in image resolution that are made possible by the proposed anti-aliasing method are demonstrated in Figure - Figure . Figure a shows the results of 3-D post-stack migration without using any anti-aliasing. Figure b shows the results when the new anti-aliasing constraints in equation () are applied. And Figure c shows the results when the standard anti-aliasing constraints in equation () are applied. The differences between the results are better appreciated by comparing windows zooming into smaller parts of the sections. Figure  shows the comparison for the shallower part of the section. The image obtained without anti-aliasing is uninterpretable because of the strong aliasing noise. The image obtained with the proposed method shows better resolution of several dipping reflectors and of the steep salt flank. Figure  demonstrates (same area shown in Figure ) that the high-frequency sediment truncation against the salt flank, (CMP X=700 m and Time=2.2 s) is well resolved in the image obtained using the proposed method, whereas it is poorly resolved in the image obtained using the traditional methods.

 

Wind-spec-ann
Figure 14 Frequency-wavenumber spectrum of a data window (same as in Figure ). The dashed line superimposed onto the spectrum cover the areas determined by the inequalities of equation (). The areas that respect all the operator anti-aliasing constraints, and thus that represent data components that are used by the imaging, are covered by dashed lines along both direction.

 

Comp-WL-anti
Figure 15 3-D migrations of a salt-dome flank in the Gulf of Mexico: (a) migration obtained without any anti-aliasing filter, (b) migration obtained with the application of the proposed ``high-resolution'' anti-aliasing filter, (c) migration obtained with the application of a ``standard'' anti-aliasing filter.

 

Comp-WT-anti
Figure 16 Zoom into the shallower part of the 3-D migrations of a salt-dome flank in the Gulf of Mexico shown in Figure : (a) migration obtained without any anti-aliasing filter, (b) migration obtained with the application of the proposed ``high-resolution'' anti-aliasing filter, (c) migration obtained with the application of a ``standard'' anti-aliasing filter.

 

Comp-WB-anti
Figure 17 Zoom into the deeper part of the 3-D migrations of a salt-dome flank in the Gulf of Mexico shown in Figure : (a) migration obtained without any anti-aliasing filter, (b) migration obtained with the application of the proposed ``high-resolution'' anti-aliasing filter, (c) migration obtained with the application of a ``standard'' anti-aliasing filter.