Figure 4 shows a simple 2-D acoustic model based on one used by Dong and Keys 1997, as shown in Figure 4. The only difference is that all the layers here have a 10 degree dipping angle.
For the first interface, there is no velocity change and only density change. According to equation r_coeff, the reflection coefficient is constant, 0.05. Similarly, we can reach the same result from the acoustic AVO approximation. The second layer has changes in velocity and density, but in opposite signs. Therefore, these two changes cancel each other out and give a zero-valued intercept. Slope B is equal to 0.05. Reflection coefficient R increases from zero to nonzero value with the increase of the incident angle. The third interface has only a velocity change and no density change. The velocity drops across the interface and results in a negative intercept and slope.
We use an acoustic modeling program developed by Dong, which is based on the reflectivity method Müller (1985). For such kind of layer model, the modeling result is not only kinematically, but also dynamically exact. As shown in the following result, such an accurate modeling program is very helpful for verifying the performance of our inversion program. Figure 5 is a common-shot gather. The first two events have a similar pattern, except that the second one goes to a zero-valued amplitude in the near offset. However, the third event shows an opposite pattern.
Figure 6 shows an image gather from the inversion result. Since the correct velocity model has been used in calculating the WKBJ Green's function, the three events have been flattened in the image gather. However, due to the NMO stretching effect, the events broaden from near to far offset.
One way to check the accuracy of our inversion result is to pick the peak amplitude along the three events and then compare it with the theoretical solution. Figure 7 shows that the numerical results match the theoretical ones very accurately.
Figure 8 and 9 shows the intercept A and slope B estimated from the inversion result. Compared with the theoretical results under acoustic approximation, our solution matches the theoretical one very well. These two figures also show the stretching effect very clearly. How to remove this stretch effect efficiently is our next research topic.