We implemented an AVO inversion algorithm in V(x,z) media. Our approach is a two-step inversion scheme:
Since the velocity model used in AVO analysis is relatively smooth, the finite-difference forward modeling result is accurate enough in Kirchhoff inversion. Both the synthetic and field data example can verify the accuracy of the finite-difference scheme.
On the basis of Bleistein et al. (1987), we proposed another pair of Kirchhoff inversion operators that have a more obvious physical meaning. One is the specular reflection coefficient R, and the other is R multiplied by the cosine of half of the specular incident angle, . The reflection coefficient R, organized into common-image gathers, is not only necessary in estimating the AVO intercept A and slope B, but also essential to update the velocity model. Through checking common-image gathers, we can update the velocity model and produce a more accurate image. This feature will also prevent the velocity error from propagating into the final AVO coefficients.
One of the fundamental differences between Kirchhoff inversion and Kirchhoff depth migration is that Kirchhoff inversion has an extra weighting function varying along the integral curves. We investigated the relationship between the weighting function and double-square-root (DSR) equation in the homogeneous medium. It is interesting to see that the weighting function has double peaks in the common-offset configuration. This observation tells us that the largest contribution to the image is not from the middle of the integral curve, but from the two flanks. Therefore, it is very important to include the locations of the two peaks in order to recover a true-amplitude image.
We applied our algorithm to both synthetic and field datasets. The synthetic example shows that this new scheme is very accurate in calculating the reflection coefficient and the specular incident angle. When applying our approach to the Mobil AVO dataset, we updated the velocity model according to the common-image gathers. Furthermore, we estimated the AVO coefficients, intercept A and slope B, and then created a fluid-line expression of Vp/Vs anomaly. Our result shows that there is a strong Vp/Vs anomaly in the middle section that suggests a potential hydrocarbon indicator.