Examples of reverse time migration of a synthetic data set

To illustrate the use of the proposed method to image seismic data by reverse-time migration, I migrated the same simple synthetic data set (with a dipping and a flat layer) that I used in Biondi (2002). Since the data were modeled in shot gathers, I sorted the data set in midpoint-offset coordinates before migration. As in the other paper, I migrated the data with a constant velocity to avoid artifacts caused by velocity discontinuities in the migration velocity. Therefore, the deeper reflector is slightly undermigrated.

Figure 1 is the equivalent of the ``stacked'' image, obtained by applying equation (12). The dipping reflector is well imaged within the range that is illuminated by the shots. The flat reflector is slightly undermigrated, as discussed above. The quality of the image is comparable to the quality of the image obtained by shot-profile migration using the same data.

 

MidOff-Image-dip Figure 1 Image of the synthetic data set with the correct velocity function.

 

MidOff-Cig-Ang-dip Figure 2 Offset-domain CIG (left a)) and angle-domain CIG (right b)) corresponding to the image in Figure 1. Notice the focusing at zero offset in a), and the flatness of the moveout in b).

Figure 2 shows on the left the offset-domain CIG (a) and on the right the angle-domain CIG (b). The offset domain CIG was obtained by applying equation (13), and the angle-domain CIG was obtained by applying the offset-to-angle transformation presented in Sava et al. (2001). The CIGs are located at a surface location where both reflectors are well illuminated (1,410 m). As expected the image is well focused at zero-offset in the panel on the left and the events are flat in the panel on the right.

As for shot-profile migration, the CIGs obtained by midpoint-offset reverse-time migration can be easily used for velocity updating. To illustrate this capability, I have migrated the same data set with a lower velocity ( .909 km/s). Figure 3 is the ``stacked'' image obtained using the lower velocity. Both reflectors are undermigrated and shifted upward.

Figure 4 shows on the left the offset-domain CIG (a) and on the right the angle-domain CIG (b). Now in the panel on the left, the energy is not well focused at zero offset but is spread over an hyperbolic trajectory centered at zero offset. The corresponding angle-domain CIG (ADCIG) (right) shows the characteristic smile typical of undermigrated ADCIG. The velocity information contained in the panel on the right can be easily used for velocity updating and tomographic inversion in a similar way as the ADCIG obtained by downward-continuation migrations are used Clapp and Biondi (2000); Clapp (2001).

 

MidOff-Image-dip-slow Figure 3 Image of the synthetic data set with the incorrect velocity function.

 

MidOff-Cig-Ang-dip-slow Figure 4 Offset-domain CIG (left) and angle-domain CIG (right) corresponding to the image in Figure 3. Notice the lack of focusing at zero offset in a), and the smile in b).