Examples of prestack imaging of overturned reflections

To illustrate the use of the proposed method to image overturned reflections I created a simple synthetic data set that contains such events. I immersed a thin high-velocity segment in a layered medium with a strong vertical velocity gradient (.97 s^-1). The reflector is dipping 44 degrees with respect to the vertical and extends from a surface coordinate of 1 km to a surface coordinate of 1.35 km.

I modeled and migrated 20 shots spaced 50 m apart, starting from the surface coordinate of 4.5 km. The receivers were in a symmetric split-spread configuration with maximum offset of 6.4 km. Because of the relative position of the reflector with respect to the shots, only the overturned reflections illuminate the reflector.

Figure  is the image obtained by applying the conventional imaging principle; that is; evaluating equation () at 251#251.The dark segment superimposed onto the image shows the position of the reflector in the model. The reflector is properly focused and positioned correctly. Figure  shows the image at 252#252 (top) and the image at 253#253 (bottom). As in Figure , the dark segment superimposed onto the images shows the position of the reflector in the model. In these two panels the reflector is almost as well focused as in Figure , but it is slightly shifted along its normal. As expected from the theoretical discussion above, the reflector is slightly lower for the negative 22#22 (top) than for the positive 22#22 (bottom).

Figure  shows an example of CIG computed by evaluating equation () at 251#251.The panel on the left (a) shows the offset-domain CIG, and the panel on the right (b) shows the angle-domain CIG. The energy is correctly focused at zero offset in a), and the event is flat in b), though the angular coverage is narrow because of the short range of shot locations (1 km).

 

Shot-Refl-Image-vover Figure 11 Image of the synthetic data set containing the overturned reflections migrated with the correct velocity and at 251#251.The dark segment superimposed onto the images shows the position of the reflector in the model.

 

Shot-Refl-Image-mov-vover Figure 12 Images of the synthetic data set containing the overturned reflections migrated with the correct velocity; at 249#249 (top) and at 250#250 (bottom) The dark segment superimposed onto the images shows the position of the reflector in the model. Notice the slight downward shift of the imaged reflector in a) and the slight upward shift of the imaged reflector in b).

 

Shot-Cig-Ang-vover Figure 13 Offset-domain CIG (left) and angle-domain CIG (right) corresponding to the image in Figure . Notice the focusing at zero offset in a), and the flatness of the moveout in b), though the angular coverage is narrow because of the short range of shot locations (1 km).

The second obstacle to image overturned reflections is the estimation of a velocity model that focuses and positions them correctly. To investigate this issue, I migrated the same data set with two inaccurate velocity functions. The first is 1% slower than the correct one, and the second is 1% faster than the correct one. Figure  shows the ``stacked'' images produced by these two migrations. The panel on the top (a) shows the image when the velocity is too low, and the panel on the bottom (b) shows the image when the velocity is too high. As in the previous figures, the dark segment superimposed onto the images shows the position of the reflector in the model. As expected the reflector is mispositioned and not as well focused as in Figure .

Figure  shows the CIG gathers taken at the same location as in Figure  for the migration with the low velocity and Figure  shows the CIG gathers for the migration with the high velocity. Notice that the velocity errors have caused a shift along the offset direction of the focal point in the offset-domain gathers. Towards positive offsets for the low velocity (Figure ) and towards negative offsets for the high velocity (Figure ). The angular coverage is too limited to notice a clear pattern in the angle-domain CIGs. In principle, a lateral shift in offset-domain gather should correspond to a tilt in the angle-domain gathers. These results seems to indicate that the residual moveout is asymmetric for overturned reflections, contrary to the symmetric moveout caused by velocity errors for regular reflections (see Figure ). A more definitive analysis requires the migration of a survey with wider angular coverage; that is, with wider shot range. However, this characteristic suggests that for updating the velocity from overturned waves, the migrated CIGs should be scanned using a different family of residual moveouts than the parabolic moveouts used for standard reflections. The shift in the focal point in the offset-domain gathers could be also directly used for updating the velocity along the path of the overturned reflections.

 

Shot-Refl-Image-vover-slow-fast Figure 14 Image of the synthetic data set containing the overturned reflections migrated with a velocity function 1% lower than the correct one (top) and with a velocity function 1% higher than the correct one (bottom). Notice the misfocusing and mispositioning of the reflector.

 

Shot-Cig-Ang-vover-slow Figure 15 Offset-domain CIG (left) and angle-domain CIG (right) corresponding to the image in Figure a. Notice the positive shift along the offset direction of the focal point in the offset-domain gathers.

 

Shot-Cig-Ang-vover-fast Figure 16 Offset-domain CIG (left) and angle-domain CIG (right) corresponding to the image in Figure b. Notice the negative shift along the offset direction of the focal point in the offset-domain gathers.