The implementation of the extended split-step migration for 2-D and 3-D prestack data sets have been tested on different synthetic models. Figure 2 we show the 2-D impulse response of the migration operator for zero-offset migration, using a velocity field with a strong lateral and vertical gradient equal to 0.5 s-1 (the direction of the gradient is in the same direction that the coordinate increases).
Figure 2 shows the impulse responses using 10 reference velocities for the zero-offset case. Figure 2 (left) plots the impulse response of the extended split-step migration with a real vertical wavenumber. This impulse response shows that the migration is dip limited.
The impulse response in Figure 2 (right) shows how the evanescent energy helps to increase the maximum dip handled by the split-step migration. Moreover, the evanescent energy is more important in the low velocity areas of the velocity model (small in-line coordinates).
When evanescent energy is used during the interpolation of the downward continued wavefields with different reference velocities, the extended split-step migration can handle steeper events. Therefore, extending both the range in the vertical wavenumber domain, kz, and the number of reference velocities improves the image obtained with the extended split-step migration in areas with a strong velocity lateral gradient.
Figure 10 shows a geological model in the North Sea used in a 2-D finite-difference modeling provided by ELF and CCG. This geological model represents the rise of a salt diapir, characterized by a strong lateral velocity gradient in the seismic cable length. The seismic synthetic data set has the following acquisition parameters: source spacing 50m; receiver spacing 25m; final CMP spacing 12.5m; near-offset 170m; far-offset 3350m; maximum recorded time 5s; and time sampling 4ms.
Figure 6 shows the extended split-step prestack migration with evanescent energy using 5 reference velocities and a maximum offset 800m. Using fewer offsets it was possible to image the very steep event located below the shallow salt body. Comparing this image with the prestack image using all the offsets (Fig. 6), the amplitude of the steep events is very low relative to the amplitude of the other reflectors, and this event disappears in the final zero-offset depth migrated section.
The other important imaging targets in this salt dome synthetic model are both
the deeper flank of the salt dome (dipping at 
) and the
reflector just below the salt. In the prestack images (figs.
7 and 4), the bottom of the salt dome was not imaged using all the offsets. In contrast, this reflector was imaged in the prestack migration with a maximum offset of 800m because the energy of those reflectors exists on the near-offsets in migrated CDPs (Fig. 6 and 8). The
coherent noise at the end of the depth section, should be attenuated by
padding zeros at the end of the data set.
The real seismic data set of the mode in Figure (10) is a 3-D seismic survey. I extracted the corresponding line to Figure (7) and applied a bandpass filter and stacked the traces of the two parallel lines in order to replace the missing or edited traces. Using a velocity field, different to the velocity in 10, we migrated it with the migration algorithm presented in this paper. The resulting prestack migrated image is plotted in Figure 9; it was migrated with 5 reference velocities. This migrated image shows that the modeling was used for just the important features of the geologic model (Figure 10). It is observed that the mean salt dome features that were discussed with the synthetic imaging results are imaged in the real data migrated image. For example, the steep event just below the small salt body is imaged, what it is missing is the base of the salt dome. We are working with the 3-D prestack data set of this real seismic data, in order to obtain a migrated image using the common-azimuth approximation.
The SEG-EAEG 3-D synthetic model was the third synthetic data set used to verify that our 3-D extended split step migration was correctly implemented. Figure 13 shows a cross-line section of the velocity data cube. This synthetic data set is characterized by a salt dome embedded in a linear vertical gradient, where salt and small salt features introduce a strong lateral gradient that affect the underlying horizontal reflectors.
Figures 12 and 14 show the 3-D extended split-step post-stack migration of the zero-offset cube SEG-EAEG model. The migration used 10 reference velocities to obtain this image. The bottom of the salt dome is well imaged below the salt irregularities in the top of the salt. Moreover, small lateral features and steep fault planes on the bottom of the salt dome are correctly imaged (see Fig. 13).
In order to improve this migration algorithm we are considering implementing a bilinear interpolation in the source and receiver axis of the downward continued wavefield. Using this bilinear interpolation we expect to reduce the number of reference velocities necessary to image steep reflectors in a velocity field with a lateral velocity gradient. In the case of diapirs, we want to hold the salt velocity constant for the downward continuation and use the linear interpolation of the wavefield elsewhere.