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# A practical subsurface model

Figure 7 shows a velocity model and the corresponding wavefield for a source excited near the surface at lateral distance 1850 m from the origin. The wavefield is calculated using the finite-difference method applied to equation (20) with absorbing boundary conditions. The curves correspond to solutions of the eikonal equation (16) for the VTI model (solid curve) and an equivalent isotropic model (dashed curve). Both models have the same vertical and NMO velocities, and as a result the two wavefront curves coincide at the zero angle from the vertical. The biggest difference between the two wavefronts occurs near horizontal wave propagation, where the influence of the different values affects the wavefront the most. The corresponding elastic curves (plotted in gray, but indistinguishable) exactly coincide with the acoustic ones. The model was constructed so that the source and receivers are placed in the water layer, which conveniently, is isotropic. As a result, no wave artifacts, similar to the one in Figure 2 appear.

fddatamodel
Figure 7
Top: A velocity model consisting of five layers with velocity equal 1500, 1900, 1700, 2400, and 3000 m/s from top to bottom. The corresponding for the VTI model are 0, 0.1, 0.2, 0.15, and 0.05 from top to bottom. Bottom: The wavefield at 1 s caused by a source at distance 1850 m and depth 50 m for the VTI model. The solid black curve is the solution of the eikonal equation for the VTI model, and the dashed curve is the solution for the corresponding isotropic model.

Figure 8 shows common-shot gathers corresponding to the model in Figure 7 computed using geophones placed near the surface. The geophones cover the whole 4-km lateral distance. The top gather in Figure 8 corresponds to the VTI model, and the bottom gather corresponds to the isotropic model. The differences (indicated by arrows) are concentrated at later times, because the largest anisotropies are at depth. Figure 8 also demonstrates the importance of anisotropy in processing; such differences in traveltimes, as well as amplitudes, will considerably hamper isotropic processing when anisotropy similar to that modeled here is ignored.

synmodelf
Figure 8
Common-shot gathers for the model in Figure 7 using geophones that span the 4-km distance, buried at depth 50 m from the surface. On top is a shot gather corresponding to the VTI model; on the bottom, a shot gather corresponding to the isotropic model. Both synthetic gathers were calculated using the acoustic wave equation (20). The arrows point to some of the differences in energy arrival times between the two media.

Next: Finite difference applied to Up: Acoustic anisotropic wave equation: Previous: Eliminating the artifact
Stanford Exploration Project
10/9/1997