Synthetic Examples

This section consists of two main, simple synthetic examples to illustrate two of the main concepts described in the previous section. Throughout this section, we will refer to two different methodologies, first, the conventional methodology, and second, the proposed methodology. The conventional methodology consists of the transformation from ODCIGs into ADCIGs with the transformation for the single-mode case Sava and Fomel (2003). The proposed methodology is the one discussed in the previous section. All the examples are a single-shot gather experiment.

The first example consists of three sections, all of them displayed in Figure . From left to right we have a flat-layer, a $10^\circ$ dipping layer, and a $30^\circ$ dipping layer. All the sections consist of, from top to bottom, a single shot gather, an image from one shot gather, an angle-domain common-image gather obtained with the conventional methodology, and the angle-domain common-image gather obtained with the proposed methodology. The data space consists of only one shot gather; therefore, the representation in an ADCIG should have non-zero energy for only one angle. Table 1 shows the survey details and the computed angle values for this experiment. In both angle-domain gathers, the solid line corresponds to the expected angle value; as expected the angle gathers obtained with the proposed methodology perfectly match the values in the table.

 

ps-all-x
Figure 3 First synthetic example. From left to right, a flat-layer case, a $10^\circ$ dipping layer, and a $30^\circ$ dipping layer. From top to bottom, a single shot gather, an image from a single shot gather, an angle-domain common-image gather obtained with the conventional methodology, and the angle-domain common-image gather obtained with the proposed methodology.

 

$$2\theta (^\circ)$$ Refl. dip Shot-loc. CIG-Loc

$$40.45^\circ$$ Flat layer 500 2000

$$10^\circ 38.7^\circ$$ 500 1750

$$30^\circ 35^\circ$$ 500 1000

Our second experiment is also a single-shot experiment. The intention of this exercise is to present the separation of the full-aperture angle ($\theta$) into its P-incidence ($\phi$), and its S-reflection ($\sigma$) components. Figure  shows, from top to bottom, the flat-layer case, a $10^\circ$ dipping layer case, and a $30^\circ$ dipping layer case for this experiment. Each case consists of, from left to right, the image of a single-shot gather; the corresponding angle-domain common-image gather, which is taken at the location marked in the image; the corresponding P-angle-domain common-image gather; and the corresponding S-angle-domain common-image gather. Table 2 shows the corresponding values for this experiment. The solid lines in each of the angle-gathers represent the computed value in table 2.

 

ps-ind-ang
Figure 4 Second synthetic example, independent angle separation: From top to bottom, a flat-layer case, a $10^\circ$ dipping layer, and a $30^\circ$ dipping layer. From left to right: the image of a single shot gather, the corresponding PS-ADCIG, the independent P-ADCIG, and the independent S-ADCIG. The solid lanes represent the corresponding values in table 2.

 

Refl. dip Shot-loc. CIG-Loc. P-angle S-angle

$$26^\circ 14^\circ$$ Flat layer 500 1000

$$10^\circ 48^\circ 22.3^\circ$$ 500 1500

$$30^\circ 47^\circ 22^\circ$$ 500 1000

These two synthetic examples clearly show that the proposed methodology accurately transforms SODCIGs into ADCIGs for the converted-mode case. Moreover, we are able to compute the specific incidence and reflection angles.