I modeled and migrated 100 shots spaced 10 m apart, starting from the surface coordinate of 1,000 m. The receivers were in a symmetric split-spread configuration with maximum offset of 2,550 m. I assumed two reflectors: one dipping 10 degrees and the other flat. The dipping reflector is shallower than the flat one. To avoid artifacts caused by velocity discontinuities in the migration velocity the reflectors were modeled as thin high-velocity (1.5 km/s) layers in a constant velocity medium (1 km/s). The migration velocity was set to constant, and equal to the background velocity. Therefore, the deeper reflector is slightly undermigrated.
Figure 1 is the image obtained applying the conventional imaging principle [equation (1)]. The dipping reflector is properly imaged within the range that is illuminated by the shots. The flat reflector is slightly undermigrated, as mentioned above.
Figure 2 shows on the left the offset-domain CIG (a) and on the right the angle-domain CIG (b). The CIGs are located at a surface location where both reflectors are illuminated well (1,410 m). As expected, the image is nicely focused at zero-offset in the panel on the left and the events are flat in the panel on the right.
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Shot-Image-dip
Figure 1 Image of the synthetic data set with the correct velocity function. |  |
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Shot-Cig-Ang-dip
Figure 2 Offset-domain CIG (left a)) and angle-domain CIG (right b)) corresponding to the image in Figure 1. Notice the focusing at zero offset in a), and the flatness of the moveout in b). |  |
To illustrate the usefulness for velocity updating of the proposed method to compute ADCIGs, I have migrated the same data set with a lower velocity ( .909 km/s). Figure 3 is the ``stacked'' image obtained using the lower velocity. Both reflectors are undermigrated and shifted upward.
Figure 4 shows on the left the offset-domain CIG (a) and on the right the angle-domain CIG (b). Now in the panel on the left, the energy is not focused at zero offset, but it is spread over an hyperbolic trajectory centered at zero offset. The corresponding ADCIG (right) shows the characteristic smile typical of an undermigrated ADCIG. The velocity information contained in the panel on the right can be easily used for velocity updating and tomographic inversion in a similar way as the ADCIG obtained by downward-continuation migrations are used Clapp and Biondi (2000); Clapp (2001).
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Shot-Image-dip-slow
Figure 3 Image of the synthetic data set with the incorrect velocity function. |  |
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Shot-Cig-Ang-dip-slow
Figure 4 Offset-domain CIG (left) and angle-domain CIG (right) corresponding to the image in Figure 3. Notice the lack of focusing at zero offset in a), and the smile in b). |  |