Homogeneous media

For the homogeneous medium case, I will use the reflector model shown in Figure , which includes a syncline structure with flanks dipping at about 50 degrees. Since we are forced to have a finite aperture coverage, it will be hard to migrate large dip angles in a homogeneous medium. Another dipping event at a shallower depth is also included in the model.

 

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Figure 5 A simple reflector with a syncline structure in the middle embedded in a homogeneous transversely isotropic model with v=2.0 km/s and $\eta=0.3$.

The synthetic seismograms are generated considering a VTI medium with velocity of 2 km/s and a realistic $\eta$ of 0.3. Figure  shows four synthetic seismograms generated using the model in Figure  for offsets of (a) 0, (b) 1, (c) 2, and (d) 3 km. The limited recording aperture has cut off some of the energy of the reflection from the right flank of the syncline. As a result, the right flank is expected to be weaker after migration, due to the missing energy. Also, the appearance of under migration usually accompanies dipping reflections that have not been totally recorded at the surface.

 

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Figure 6 Synthetic seismograms for the model in Figure  for (a) coincident source and receiver (zero-offset), (b) an offset of 1 km, (c) an offset of 2 km, and (d) an offset of 3 km.

Figure  shows the prestack time migration of the synthetic data given in Figure  for, again, an offset of (a) 0, (b) 1, (c) 2, and (d) 3 km. All the migrated sections for the various offsets seem accurate and the reflections are well positioned.

 

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Figure 7 Prestack time migration of the synthetic seismograms shown in Figure , again, for (a) zero-offset, (b) offset of 1 km, (c) offset of 2 km, and (d) offset of 3 km. The lower energy of the right flank of the syncline is the result of the limited aperture.

One way to test the accuracy of the migrated sections is to convert them to depth and overlay the depth model in Figure  over these migrated sections. Figure  shows the migrated sections in depth, converted using the velocity of 2 km/s, from top to bottom having offsets of 0, 1, 2, and 3 km, respectively. All migrated sections agree well with the model used to generate the synthetic seismograms. Since the synthetic seismograms were generated using exact (within the limit of ray theory) traveltimes, the accuracy of the migration is attributable to the accuracy of the midpoint-offset traveltime equation, derived in this paper. Again, an appearance of under migration of the right flank of the syncline is the result of the limited recording aperture, that has cut of some the energy associated with this flank.

 

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Figure 8 The time migrated sections converted to depth for an offset, from top to bottom, of zero, 1, 2, and 3 km, respectively. The reflector shape in Figure  is overlaid on the migration results.

Looking at the moveout of the dipping events after migration, shown in Figure , clearly demonstrates the accuracy of the midpoint-offset traveltime equation for dipping events. Therefore, any moveout misalignment can only be attributable to inaccurate medium parameters used in the migration, not the equation used.

 

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Figure 9 Detail wiggle plots of the migrated sections sorted in common gather format, where the different offsets. are plotted next to each other.