Vertically inhomogeneous media

For the v(z) medium case, I will use the reflector model shown in Figure , which includes reflectors with dips ranging from 0 to 90 degrees. In v(z) media, unlike in homogeneous media, dips up to and beyond 90 degrees can be recorded with a limited aperture due to the ray bending. However, since our equation is based on the equivalent medium assumption some of the restrictions of the homogeneous case will hold in the v(z) approximation as well.

 

modelvz
Figure 10 A reflector model consisting of reflectors dipping at 0, 30, 45, 60, 75, and 90 degrees in a v(z) transversely isotropic model with v=1.5+0.6z km/s and ${\eta=0.2}$.

Figure  shows four synthetic seismograms generated using the model in Figure  for offsets of (a) 0, (b) 1, (c) 2, and (d) 3 km.

 

synvz
Figure 11 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 prestack time migration of the synthetic data given in Figure  plotted in depth for offsets, from top to bottom, of zero, 1, 2, and 3 km, respectively. The migrated sections overall agree well with the model used to generate the synthetic seismograms. However, such an agreement in this v(z) example is less evident compared to what we obtained in the homogeneous medium case. This is some what expected since equivalent medium derivations are approximations. As dip increases slight under migration is apparent, however, these results are preliminary; improvements are expected in a follow up paper.

 

migvzm
Figure 12 Prestack 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.