Application to a synthetic data set

To test our proposed method for transforming HOCIGs and VOCIGs into DDOCIGs we modeled and migrated a synthetic data set with a wide range of dips. The reflector has spherical shape with radius of 500 m. The center is at 1,000 meters depth and 3,560 meters horizontal coordinate. The velocity is constant and equal to 2,000 m/s. The data were recorded in 630 shot records. The first shot was located at a surface coordinate of -2,000 meters and the shots were spaced 10 meters apart. The receiver array was configured with asymmetric split-spread geometry. The minimum negative offset was constant and equal to -620 meters. The maximum offset was 4,400 meters for all the shots, with the exception of the first 100 shots (from -2,000 meters to -1,000 meters), where the maximum offset was 5,680 meters in order to record all the useful reflections. To avoid boundary artifacts at the top of the model both sources and receivers were buried 250 meters. Some of the reflections from the top of the sphere were muted out before migration to avoid migration artifacts caused by spurious correlation with the first arrival of the source wavefield.

Figure a shows the zero-offset section (stack) of the migrated cubes with the correct velocity (2,000 m/s), and Figure b shows the zero-offset section obtained with 4% too low of a velocity (1,920 m/s). Notice that, notwithstanding the large distance between the first shot and the left edge of the sphere (about 5,000 meters), normal incidence reflections illuminate the target only up to about 70 degrees. As we will see in the angle-domain CIGs, the aperture angle coverage shrinks dramatically with the increase of the reflector dip. On the other hand, real data cases are likely to have a vertical velocity gradient that improves the angle coverage of steeply dipping reflectors.

Figures  and display sections of the full image cube in the case of the low velocity migration. Figure  displays the horizontal-offset image cube, while Figure  display the vertical-offset image cube (notice that the offset axis in Figure  has been reversed to facilitate its visual correlation with the image cube displayed in Figure ). The side face of the cubes display the CIGs taken at the surface location corresponding to the apparent geological dip of 45 degrees. Notice that the events in the two types of CIGs have comparable shapes, as expected from the geometric analysis presented in the previous section, but their extents are different. The differences between the two image cubes are more apparent when comparing the front faces that show the image at a constant offset of 110 meters (-110 meters in Figure ). These differences are due to the differences in image-point dispersal for the two offset directions [equation () and equation ()].

Figures  and show the image cubes of Figures  and after the application of the transformations to DDOCIG, described in equations () and (), respectively. The two transformed cubes are almost identical because both the offset stretching and the image-point dispersal have been removed. The only significant differences are visible in the front face for the reflections corresponding to the top of the sphere. These reflections cannot be fully captured within the vertical-offset image cube because the expression in equation () diverges as $\alpha$ goes to zero. Similarly, reflections from steeply dipping events are missing from the horizontal-offset image cube because the expression in equation () diverges as $\alpha$ goes to 90 degrees.

 

Cube-hx-ball-slow-4p Figure 9 Horizontal-offset image cube when the migration velocity was 4% too low. Notice the differences with the vertical-offset image cube shown in Figure .

 

Cube-hz-ball-slow-4p Figure 10 Vertical-offset image cube when the migration velocity was 4% too low. Notice the differences with the horizontal-offset image cube shown in Figure .

 

Cube-hx-par-ball-slow-4p Figure 11 Transformed horizontal-offset image cube. Notice the similarities with the transformed vertical-offset image cube shown in Figure .

 

Cube-hz-par-ball-slow-4p Figure 12 Transformed vertical-offset image cube. Notice the similarities with the transformed horizontal-offset image cube shown in Figure .

The previous figures demonstrate that the proposed transformation converts both HOCIGs and VOCIGs into equivalent DDOCIGs that can be constructively averaged to create a single set of DDOCIGs ready to be analyzed for velocity information. In the following figures, we examine the DDOCIGs obtained by averaging the HOCIGs and VOCIGS using the weights in equations (), and we compare them with the original HOCIGs and VOCIGs.

We start from analyzing the CIGs obtained when the migration velocity was correct. Figure  shows the HOCIGs corresponding to different apparent reflector dips: a) degrees, b) 30 degrees, c) 45 degrees, and d) 60 degrees. The quality of the HOCIGs degrades as dip angle increases. Figure  shows the VOCIGs corresponding to the same dips as the panels in Figure . In this case, the quality of the VOCIGs improves with the reflector dip.

Figure  shows the DDOCIGs corresponding to the same dips as the panels in the previous two figures. Notice that the quality of the DDOCIG is similar to the quality of the HOCIG for small dip angles, and it is similar to the quality of the VOCIG for large dip angles. The focusing of the dipping reflectors (e.g. 60 degrees) is worse than the focusing of the flatter reflectors (e.g. 30 degrees) because of incomplete illumination. In general, the quality of the DDOCIG is ``optimal,'' given the limitations posed by reflector illumination.

The next set of three figures (Figure -) shows the previous offset-domain CIGs transformed into angle domain. The effects of incomplete illumination are more easily identifiable in these gathers than the offset-domain gathers. As for the offset-domain gathers, the angle-domain DDCIGs have consistent quality across the dip range, while the angle-domain gathers obtained from both HOCIG and VOCIG degrade at either end of the dip range.

The next six figures display the same kind of gathers as the past six figures, but obtained when the migration velocity was too low by 4%. They are more interesting than the previous ones, since they are more relevant to velocity updating. Notice that the offset range is doubled with respect to the previous figures (from $\mp320$ meters to $\mp 640$ m) in the attempt to capture within the image cubes all the events, even the ones imaged far from zero offset. For shot profile migration, making the offset range wider is not a trivial additional computational cost.

Figure  shows the HOCIGs. The 60 degrees CIG [panel d)] is dominated by artifacts and the corresponding angle-domain CIG shown in Figure d would be of difficult use for residual velocity analysis. Figure  shows the VOCIGs. As before, the CIGs corresponding to the milder dips are defocused (the artifacts on the left of the panels are caused by the top boundary). The 60 degrees CIG [panel d)] is better behaved than the corresponding HOCIG (Figure d), but it is still affected by the incomplete illumination. The DDOCIGs (Figure ) are the best focused CIGs. Finally the comparison of all the angle-domain CIGs (Figures - ) confirm that the DDOCIGS provide the highest resolution and the least-artifact prone ADCIGs, and thus they are the best suited to residual moveout analysis.

 

Cig-hx-ball-allang
Figure 13 HOCIGs corresponding to different apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was correct.

 

Cig-hz-ball-allang
Figure 14 VOCIGs corresponding to different apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was correct.

 

Cig-hrot-ball-allang
Figure 15 DDOCIGs obtained with the proposed method. The panels correspond to the following apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was correct.

 

Ang-Cig-hx-ball-allang
Figure 16 Angle-domain CIGS obtained from the HOCIGs in Figure . The panels correspond to the following apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was correct.

 

Ang-Cig-hz-ball-allang
Figure 17 Angle-domain CIGS obtained from the VOCIGs in Figure . The panels correspond to the following apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was correct.

 

Ang-Cig-hrot-ball-allang
Figure 18 Angle-domain CIGS obtained from the DDOCIGs in Figure . The panels correspond to the following apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was correct.

 

Cig-hx-ball-allang-slow-4p
Figure 19 HOCIGs corresponding to different apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was 4% too low.

 

Cig-hz-ball-allang-slow-4p
Figure 20 VOCIGs corresponding to different apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was 4% too low.

 

Cig-hrot-ball-allang-slow-4p
Figure 21 DDOCIGs obtained with the proposed method. The panels correspond to the following apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was 4% too low.

 

Ang-Cig-hx-ball-allang-slow-4p
Figure 22 Angle-domain CIGS obtained from the HOCIGs in Figure . The panels correspond to the following apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was 4% too low.

 

Ang-Cig-hz-ball-allang-slow-4p
Figure 23 Angle-domain CIGS obtained from the VOCIGs in Figure . The panels correspond to the following apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was 4% too low.

 

Ang-Cig-hrot-ball-allang-slow-4p
Figure 24 Angle-domain CIGS obtained from the DDOCIGs in Figure . The panels correspond to the following apparent reflector dips: a) $0^{\circ}$, b) $30^{\circ}$, c) $45^\circ$, and d) $60^{\circ}$.The migration velocity was 4% too low.

We have introduced a novel transformation of offset-domain Common Image Gathers (CIGs) that applied to either horizontal-offset CIGs (HOCIGs) or vertical-offset CIGs (VOCIGs) transforms them into the equivalent CIGs that would have been computed if the offset direction were aligned along the local geological dip (DDOCIGs). Transformation to DDOCIGs improves the quality of CIGs for steeply dipping reflections by correcting the image cubes from the image-point dispersal. It is particularly useful for velocity analysis when events are not focused around zero offset. The creation of DDOCIGs enables the constructive averaging of HOCIGs with VOCIGs to form DDOCIGs that contain accurate information for all the geological dips. The angle-domain CIGs obtained from the DDOCIGs should provide the best residual moveout information for velocity updating.

We tested the method on a synthetic data set that contains a wide range of dips. The results confirm the theoretical predictions and demonstrate the improvements that are achievable by applying the transformation to DDOCIGs for reflections from steeply dipping reflectors.

We would like to thank Guojian Shan for helping in the computations of the examples of HOCIG and VOCIG from the North Sea data set. A PROOF THAT THE TRANSFORMATION TO DIP-DEPENDENT OFFSET COMMON IMAGE GATHERS (DDOCIG) CORRECTS FOR THE IMAGE-POINT DISPERSAL This appendix proves that by applying the offset transformations described in equations () and () we automatically remove the image-point dispersal characterized by equations () and (). The demonstration for the VOCIGs transformation is similar to the one for the HOCIGs transformation, and thus we present only the demonstration for the HOCIGs. HOCIGs are transformed into DDOCIGs by applying the following change of variable of the offset axis h_x, in the vertical wavenumber k_z and horizontal wavenumber k_x domain:  

$$h_x= \frac{h_0}{\cos\alpha} = {\rm sign}\left(\tan{\alpha}\... ...right) h_0\left({1+\frac{k_{x}^2}{k_z^2}}\right)^{\frac{1}{2}}.$$ (71)

For the sake of simplicity, in the rest of the appendix we will drop the ${\rm sign}$ in front of expression () and consider only the positive values of k_x/k_z.

We want to prove that applying () we also automatically shifts the image by  

$$\Delta_z I_{x_h}= h_0\tan{\gamma}\tan{\alpha}\sin{\alpha}$$ (72)

in the vertical direction, and  

$$\Delta_x I_{x_h}= - h_0\tan{\gamma}\tan{\alpha}\cos{\alpha}$$ (73)

in the horizontal direction.

The demonstration is carried out into two steps: 1) we compute the kinematics of the impulse response of transformation () by a stationary-phase approximation of the inverse Fourier transform along k_z and k_x, 2) we evaluate the dips of the impulse response, relate them to the angles $\alpha$ and $\gamma$, and then demonstrate that relations () and () are satisfied.