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# Examples

The first example, Figures  and , is a fairly simple synthetic model populated with reflectors at various dips and a few diffractors. One goal of this model is to demonstrate the ability of the transformation described in this paper to distinguish reflections from diffractions.

Figure  shows on the left a typical ADCIG. Clearly, this panel does not allow us to distinguish reflectors from diffractors. However, the dual angle decomposition presented in the right panels for two particular depths at the same location show a clear distinction between reflectors and diffractors: either the event concentrates around the dip direction (reflections), or it maps uniformly to all dips (diffractions).

imag1
Figure 4
Model image with perfectly imaged reflectors and diffractors.

 comb1 Figure 5 Image decomposition by scattering angle (left). Image decomposition by scattering angle and dip angle (right) at the depth levels marked on the left panel. The top-right panel corresponds to a diffractor, and the bottom-right panel corresponds to a reflector dipping at 30 degrees.

Figure  is a summary of the multi-angle decomposition for the entire synthetic model. Each rectangle represents an image point located roughly at the same location of the rectangle. In each box, the horizontal axis corresponds to the dip angle , and the vertical axis corresponds to the scattering angle .

chck1
Figure 6
Multi-angle decomposition for the image in Figure . In each box, the horizontal axis corresponds to the dip angle , and the vertical axis corresponds to the scattering angle .

The second example concerns the well known Marmousi model. Figures  depicts the migrated image, and the vertical line indicates the location of the ADCIG under analysis. Figure  shows on the left a standard 1-D ADCIG, and on the right the multi-angle decomposition for two particular depths at the same location. Each one of the right panels shows events dipping in opposite directions at different angles. In this example, the dip angle decomposition is not as sharp as in the preceding model due to the limited resolution of the space-domain slant-stacks. Nevertheless, the selected events are clearly located at different structural dip angles.

imagC
Figure 7
Marmousi model: migrated image.

 combC Figure 8 Image decomposition by scattering angle (left). Image decomposition by scattering angle and dip angle (right) at the depth levels marked on the left panel.

The next example corresponds to a 3-D seismic image (Figure ) obtained by narrow-azimuth migration Biondi (2003). The figure depicts the zero offset hx=0,hy=0 image, although in reality the cube has the full 5 dimensions of 3-D prestack data.

 imag3d Figure 9 3-D narrow-azimuth migrated image. The figure depicts the zero offset (hx=0,hy=0) image out of a cube with 5 dimensions.

I decompose the reflectivity at mx=1000,my=800 function of the four angles introduced in the preceding section. This 4-D image decomposition is practically impossible to fully illustrate on paper. A reasonable alternative is to display subsets of the decomposition, by appropriate summation over some of the angles. For example, Figure  shows an usual angle gather, obtained by summation over the reflector dip angles , and acquisition azimuth . The left panel depicts an angle gather computed using the 2-D formula, and the right panel shows an angle gather at the same location obtained by summation of the three angles after the 3-D image decomposition. As also seen by Tisserant and Biondi (2003), not much changes for the upper flat reflector, but the event for the bottom reflector shrinks according to the crossline structural dip.

The more interesting plots, however, concern other partial summations over the four angles. Figure  shows two such partial summations: on the left, the panels correspond to summation over the scattering and azimuth angles ( and ); on the right, the panels correspond to summation over the projections of the normal to the reflector ( and ). From top to bottom, each pair of panels corresponds to different depths z=1000,1150,1380 at the same location in the image mx=1000,my=800.

The panels on the left simply indicate the dip and azimuth direction of the normal to the reflection plane, measured by its projections on the Cartesian coordinate system. The right panels, however, show the reflectivity variation with scattering angle and acquisition azimuth. The energy is unevenly distributed function of scattering angle and azimuth. Furthermore, the energy decays at higher scattering angles. The effect is similar to that observed by Sava et al. (2001) for traditional 2-D angle-gathers. We can also observe amplitudes decaying with depth, which is consistent with smaller angular coverage for a fixed acquisition geometry.

Next: Conclusions Up: Sava: Image decomposition Previous: 3-D theory
Stanford Exploration Project
7/8/2003