Synthetic data test

The second synthetic example illustrates the importance of computing true PS-ADCIGs. For this purpose, I use the polarity-flip characteristic of converted-wave data. In true PS-ADCIGs, the correct representation of the polarity flip should happen at zero-angle, since the zero-angle represents normal incidence, and also there is no conversion from P to S energy at normal incidence. The normal incidence location is the point in the image space that distinguish opposite particle motion; hence, the separation between positive and negative polarities.

The model depicted in Figure  includes both gentle and steep dips [panel (a)]. Both the P-velocity and the S-velocity models, panels (b) and (c), respectively, consist of a vertical gradient for a non-constant $\gamma$ value. The data was created with an analytical Kirchhoff modeling scheme. The synthetic data consists of 200 shots with a shot spacing of 50 m and 400 receivers with a receiver spacing of 25 m. Figure  shows a single common-shot gather for this dataset. The left panel exhibits the PP component and the right panel the PS component.

 

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Figure 5 Synthetic model. From left to right: (a) layer distribution. (b) P-velocity model. (c) S-velocity model.

 

shot-baina Figure 6 Single common-shot gather for the synthetic model in Figure . (a) PP gather, (b) PS gather.

I migrated the synthetic data using a wave-equation shot-profile migration scheme. Figure  shows the final migration result together with four selected common-image gathers. The top panels represent the zero subsurface-offset section for the PP migration (left), and the PS migration (right). Four solid lines at 3.5, 4.5, 5.5, and 6.5 km are superimposed into both migrations. These lines represent the locations for the four common-image gathers underneath each migration result. The middle panels on Figure  represent the SODCIGs taken at the position indicated by the solid lines on the migration result. The bottom panels are the ADCIGs, both the PP-ADCIGs and the PS-ADCIGs were obtained with the conventional method.

The result for the PP image is accurate; all the energy is focused at zero subsurface offset, and the angle gathers are completely flat. This is the expected result, since we performed the migration with the correct velocity model. The PS results are the most interesting. First the migration section at zero subsurface offset has positive and negative amplitudes along the first reflection. The flat reflector has vanished because there is no conversion from P to S energy at normal incidence. The SODCIGs are focused at zero, and the polarity changes across the zero value. The PS-ADCIGs are obtained with the conventional method; therefore, they represent only the pseudo-opening angle. I follow the method previously described and combine the PS-ADCIGs with the image dip information (Figure ) to obtain true PS-ADCIGs.

 

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Figure 7 Migration result for the synthetic example. Result for the PP section (left), and result for the PS section (right). The top panel on each side presents the image at zero subsurface offset, the center panel presents the SODCIGs taken at the position marked on the image, and the bottom panel shows the ADCIGs.

 

step-baina Figure 8 Local image-dip field for the third synthetic.

Figure  shows the true PS-ADCIGs. The left panel presents the PS result, the same result that is in Figure . The top-left panel is the image at zero subsurface offset, and the bottom-left panel shows the PS-ADCIGs. The right panel presents the final PS result. The top-right panel is the result of stacking in the angle domain of the true PS-ADCIGs after correcting the polarity flip Rosales and Rickett (2001). The bottom-right panel shows the true PS-ADCIGs, which are taken at the locations marked by the solid lines in the final image.

Observe the areas marked with an oval in both the PS-ADCIGs and the true PS-ADCIGs in Figure . The marked areas are taken at different reflectors for different dip values. The CIG at 3.5 km shows the most significant change; the polarity flip is completely corrected at zero angle, and there is larger angle coverage, since equation  stretches the events horizontally for an accurate representation of the half-aperture angle. Although the changes in the other CIGs, with respect to the polarity flip, are not that obvious, the polarity flip is now located at zero angle. Additionally, the events are stretched horizontally and they do not have any residual moveout.

 

image-ps-ant2
Figure 9 Final result for the second synthetic example. (a) Result for the PS section before correction, (b) result for the PS section after correction. The top-left panel is the image at zero subsurface offset, the bottom-left panel represent four PS-ADCIGs. Top-right panel is the image after stacking in the angle domain. Bottom-right panel are the true PS-ADCIGs.