The synthetics

The original Marmousi data set consists of 240 shots with 96 traces per shot. The first offset is 200 m; both the shot and the receiver spacings are 25 m. The first shot is at the lateral position 3000 m, and the recording time is 2.9 s. Overall, I used the same survey geometry and settings used by IFP, the builders of the original Marmousi data set. This includes the same source and receiver locations, an identical sampling interval and recording time, and the same minimum offset.

However, I made two adjustments to the original Marmousi acquisition settings:

• The peak frequency of the new, anisotropic, data is 30 Hz (5 Hz higher than the original data), which is more consistent with survey frequencies nowadays, and corresponds a causal ricker-wavelet point source.
• The spread length was extended to include offsets up to 3575 m, as opposed to only 2575 m in the original data.

The second modification is important for anisotropic inversion, because a majority of the information on anisotropy inversion comes from large offsets. Although the general complexity of the Marmousi model makes it less than ideal for anisotropic inversion, parts of the model are potentially invertible, including the region under the surface location 3000 m. In addition, I implemented absorbing boundary conditions at all four boundaries of the model. Doing so eliminates the presence of surface multiples; however, other inter-bed multiples can exist.

 

syn-marmo3000
Figure 5 Common-shot gathers for the model in Figure 1 resulting from a source at the 3 km location and geophones that range from 0.2 to 2.8 km locations. Both sources and receivers are at depth 25 m from the surface. Top: Shot gather from the original Marmousi data set, based on an isotropic medium. Bottom: Shot gather computed using the new acoustic equation (2) for VTI media applied to the VTI velocity model.

Figure 5 shows a common-shot gather (the first shot of the survey) from the original Marmousi data set , on the top, and the corresponding common-shot gather for the new VTI Marmousi data, on the bottom. The moveout of most reflections at this shot point appears hyperbolic, overall, a direct indication of the smoothness of the model under this shot point location. The reflection at two seconds might be the only exception. Head waves, which are the first arrivals at moderate to large offsets, are faster for the VTI case than for the isotropic case. The difference of 0.2 s, or 12 percent, is directly attributable to the average $\eta$ value of about 0.15 near the surface in that shot-point area. In addition, all reflections at far offsets arrive much earlier in the anisotropic model than the isotropic one, despite the fact that the two models have the same NMO velocity. We can attribute these differences mainly to the nonhyperbolic moveout, commonly associated with anisotropy.

 

syn-marmo5000
Figure 6 Common-shot gathers, similar to those in Figure 5, but the source is placed at the 5 km surface location and the geophones range from 2.2 to 4.8 km. Top: Shot gather from the original Marmousi data set, based on isotropy. Bottom: Shot gather based on the VTI model.

Another common-shot gather, shown in Figure 6 and caused by a source at 5000 m, demonstrates additional complications. The moveout no longer appears hyperbolic, which directly reflects the complexity of the model under this shot point. Differences between the isotropic (top) and anisotropic (bottom) sections are apparent. The diffraction at 0.7 seconds has large amplitudes in the isotropic section, and low amplitudes in the anisotropic one. We attribute this difference to the lower velocity contrast between the diffractor, which is of a high velocity, and the medium above it in the anisotropic model. Such amplitude differences have large implications in amplitude-versus-offset (AVO) analysis of reflections. Well-log (vertical) velocities do not constrain the AVO behavior in anisotropic media; the parameter $\eta$ is also needed to do so.

 

syn-marmo7000
Figure 7 Common-shot gathers, similar to those in Figure 5, but the source is at the 7 km surface location and the geophones span from the 4.2 to 6.8 km locations. Top: Shot gather from the original Marmousi data set, based on an isotropic medium. Bottom: Shot gather based on the VTI model.

Another common-shot gather, shown in Figure 7 and resulting from a source at 7000 m, is recorded at a more isotropic, yet complex, region of the anisotropic model. The overall differences between the isotropic (top) and anisotropic (bottom) sections are smaller than those associated with Figures 5 and 6. Head waves for the isotropic and anisotropic models at far offsets arrive at less than 0.1 second apart in this region. As a result, we can expect isotropic imaging to provide more reasonable results in this region.