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# 2D Synthetic data example

To illustrate the migration of the multiples, I created a simple 2D synthetic model consisting of a water-bottom reflector dipping at 15 degrees. A diffractor sits on top of this reflector at a horizontal coordinate of 5600 m. The coordinate origin is the point at which the dipping reflector intercepts the surface. Two hundred CMP gathers were generated with 100 traces in an offset range from 0 to 3000 m. The first CMP corresponds to a horizontal position of 5000 m and the CMP interval is 10 m. Four events were simulated: primary reflection, water-bottom reflection, diffraction and diffracted multiple using the traveltime equations presented before. I used a Ricker wavelet with peak frequency of 20 Hz.

 zsoff_mig_all Figure 8 Zero subsurface-offset section extracted from the image obtained by migrating the data with a two constant-velocity layer model: 1500 m/s above the reflector and 2800 m/s below the reflector.

cigs1_all
Figure 9
Subsurface-offset common-image gathers extracted form the image obtained by migrating the data with a two constant-velocity layer model. (a) at primary location (CMP_X=6280 m). (b) at the location of the diffractor (CMP_X=5800 m). (c) at a diffracted multiple location (CMP_X=4600 m) and (d) at a water-bottom multiple location (CMP_X=3600 m).

 zsoff_mig_all_const Figure 10 Zero subsurface-offset section extracted from the image obtained by migrating the data with constant water velocity.

Figure  shows the zero subsurface offset section extracted form the image data migrated with a velocity model that consists of two constant velocity layers: 1500 m/s above the reflector and 2800 m/s below the reflector. Two reference velocities were used for the migration at each depth step. As expected, the water-bottom (primary) reflector and the diffraction are properly imaged since they were migrated with a velocity close to the true velocity. The water-bottom multiple and the diffracted multiple, on the other hand, have both been migrated. This can be seen more clearly in Figure , which shows four subsurface-offset common image gathers at four different horizontal locations: (a) CMP_X=6280 m, a primary location; (b) CMP_X=5800 m, diffractor location; (c) CMP_X=4600 m, a diffracted multiple location; and (d) CMP_X=3600 m, a water-bottom multiple location. The primary and the diffraction are well focused at zero subsurface offset, but not the multiples, since they were migrated with the wrong velocity.

There is, however, an important difference between the image of the water-bottom multiple and that of the diffracted multiple. This can be seen in Figure  that shows the zero-subsurface offset section but for the image obtained by migrating thye data with constant water velocity (1500 m/s). The primary and the diffractor are both perfectly imaged since they were migrated with the exact velocity. The water-bottom multiple is also imaged perfectly as a primary with twice the dip of the real primary. The diffracted multiple, on the other hand, is still poorly imaged and does not show as a diffractor at all, since its kinematics do not match those of a primary reflection as explained before. Figure  shows the same image gathers as those in Figure . Notice the good focusing of the water-bottom multiple.

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Figure 11
Subsurface-offset common image gathers extracted from the image obtained by migrating the data with constant water velocity. (a) at a primary location (CMP_X=6280 m). (b) at the location of the diffractor (CMP_X=5800 m). (c) at a diffracted multiple location (CMP_X=4600 m) and (d) at a water-bottom multiple location (CMP_X=3600 m).

Since the ``natural'' prestack domain of data migrated with wave-equation migration for velocity analysis is the aperture angle rather than the subsurface offset, it is worth looking at the results of the migration in this domain. Figure  shows angle-domain common-image gathers extracted from the image obtained by migrating the data with the two-velocity layer model at the same spatial locations as in Figure . The primary shows a good coverage of aperture angles. The diffraction samples even more aperture angles, since it is not restricted by Snell's law whereas the water-bottom multiple shows the characteristic overmigration. The diffracted multiple shows the expected, complicated apex-shifted moveout.

Finally, Figure  shows the angle-domain common-image gathers at the same spatial locations as in Figure  but corresponding to the data migrated with constant velocity. Notice how the moveout of then water-bottom multiple is flat like that of the primary but its range of aperture angles is much smaller as is intuitively obvious. The diffracted multiple shows focusing at its apex, located at an aperture angle of about 15 degrees.