We can derive the expression for the common-azimuth Jacobian by simply applying the chain rule to Equation (32):
However, computing the reflection operator weight, Equation (2), and the WKBJ amplitudes, Equation (11), requires us to evaluate separately the source and receiver components of the dispersion relation. Therefore, for the ``true-amplitude'' migration weights, Equation (12), we need to use the more complicated expression for the common-azimuth dispersion relation as introduced by Biondi and Palacharla (1996).
In addition to the amplitude terms we have discussed in the preceding sections, ``true-amplitude'' common-azimuth migration requires an additional correction that takes into account that its dispersion relation is obtained by a stationary-phase approximation of the full 3-D DSR equation. We, therefore, need to augment the amplitude term in Equation (9) by another factor, which results from stationary-phase approximation theory Bleistein and Handelsman (1975):
This additional correction factor includes both a phase shift component and an amplitude component which increases with depth. It has thus a behavior similar to the correction term that is often used to transform 2-D data recorded with point sources and receivers to 2-D data recorded with line sources and receivers Clayton and Stolt (1981). The physical explanation is also analogous: Common-azimuth migration assumes that the data were recorded for all values of the crossline offset (yh) and then stacked along yh. The inverse of transforms the data recorded at zero crossline offset into the data ``expected'' by common-azimuth migration.
Figures 14-16 demonstrate the effects of applying the different weights in common-azimuth migration. These images are obtained by migrating a synthetic data set containing five dipping reflectors with dips from 0 to 60 degrees Biondi (2001).
Figure 14 shows a subset of the migration results. The front face of the cube displayed in the figure is an in-line section stacked over ph. The other two faces are sections through the prestack image as a function of the offset ray parameter (ph). Those images are obtained by migration without any weighting factors. Figure 15 shows the same subset as in Figure 14, but obtained by applying all the appropriate weights and the phase shift related to the stationary-phase approximation. The weights balance the amplitudes along the reflector, so that the dipping reflectors are comparable with the flat one.
Figure 16 shows one particular ADCIG, detailing the effects of each type of weighting: no weights (Figure 16a), Jacobian and modeling weights (Figure 16b), Jacobian, modeling and WKBJ weights (Figure 16c), and Jacobian, modeling, WKBJ and phase-shift weights (Figure 16d).