Numerical analysis

First, we analyze which one of the two solutions for $\tan{\gamma}$ is appropriate. For this, we plot both solutions for different values of the velocity ratio $\phi$. Figure  presents such result. The right panel presents the positive solution surface, the left panel presents the negative one. The positive solution is more stable than the negative solution. Note that the solution for the quadratic system (7) is singular when $\phi=1$. Thus, system (7) reduces to the known relation for single-mode case. The solid blue line at $\phi=1$ represents this case The negative solution is not well behaved for any of the values of $\phi$.

 

ang_cwv_2sols
Figure 3 Both solutions for $\tan{\gamma}$ on equation (7). Left: Positive solution, the blue line at $\phi=1$ corresponds to the single-mode case. Right: Negative solution.

Figure  shows the first of equations (4), that is the expression for the full-aperture angle as a function of $\alpha$ and the velocity ratio, $\phi$. Remember that for the converted-mode case, $\alpha$ is related to the geologic dip (equation (12), but it's not the dip itself. In order to understand better the previous plot, we take a look at Figure . This figure is a cut along $\phi=2$ on Figure  (dotted line) and it's compared against the conventional approach, which is $\tan{\gamma}$ equals the partial derivative of depth with respect to offset. If we omit the contribution of $\alpha$ and $\phi$, we introduce a considerable error in the transformation from SODCIGs into ADCIGs for the converted-mode case.

 

ang_cwv_alpha Figure 4 Full-aperture angle ($\gamma$) as a function of $\alpha$ and $\phi$, from the first of equation (4).

 

ang_cwv_diff Figure 5 Difference between the conventional approach for $\tan{\gamma}$ (solid line) versus the transformation with the correction for $\alpha$and $\phi$ (dotted line). This is a cut for $\phi=2$ on Figure .

The first of equations (4) establishes a relationship between $\tan{\gamma}$and the partial derivative of depth with respect to offset. We derive this relation following a wave-equation approach. Appendix A shows that we can arrive at the same conclusion following an integral summation approach. Figure  summarizes both approaches. This figure presents two surfaces, both of them correspond to $\partial z / \partial h$. The color surface represents the computation with the integral summation approach (Appendix B); and the black surface represents the computation with the wave-equation approach (equation (4)). Both surfaces have a perfect match, this strongly suggests that equations (4) is accurate, and must be followed for an appropriate transformation from SODCIGs into ADCIGs for converted-mode seismic.

 

ang_cwv_wei_surf_comp Figure 6 Wave-equation approach compared with Integral summation approach. Both of them arrive to the same surface.

 

• Synthetic model