Quantitative studies of the validity of first Born approximation have been made by various authors Beydoun and Tarantola (88); Hudson and Heritage (81); Malin and Phinney (1985), but we need to have a feel for the accuracy we may expect given the amplitude of the perturbations we have in our case.
To do this, results of the scattering method were compared with plane wave reflection coefficients calculated using Zoeppritz equations. The first set of models tested in this way were the three interfaces described by Rutherford and Williams 1989. These correspond to a variety of different shale/gas-sand AVO signatures. The relevant medium parameters are shown in Table 1, and the changes in elastic parameters across the interface are comparable with our case.
|Model 1||Model 2||Model 3|
Table 1. Elastic constants for Rutherford and Williams 1989 interfaces.
Figure 1 shows the results of modeling these interfaces. The upper panels show results of Born modeling of a layer 1 km deep; corrections have been made for geometrical spreading and geophone directionality. The lower panels show the reflection coefficients for the same interfaces. The horizontal axes are half-offset in both cases, so comparisons can be made.
The two modeling techniques are based upon totally different assumptions - Zoeppritz matches plane wave boundary conditions across a plane interface, and Born treats the interface as a series of independent weak scatterers: therefore it would be unrealistic the results to be in exact agreement. However, despite this, they show very good qualitative agreement: the phases are all correct, the phase reversals happen all at approximately the right offset, and the amplitudes increase or decrease in the right way. The choice to display one set of results as raster plots and the second set of results as graphs was made to emphasize that we were not expecting perfect agreement.
The second set of interfaces that were tested corresponded to boundaries between lithologies used in the reservoir modeling project. This gave understanding of the levels of accuracy we obtained for our seismic modeling, and it also provides insight into AVO signatures we may expect when analyzing the results.
Figure 2 shows Born modeling and Zoeppritz reflection coefficients for shale/gas-sand, shale/oil-sand and shale/water-sand contacts with elastic parameters taken from Biondi et al.1996. The same corrections have been applied as in Figure 1 to facilitate qualitative comparison. Again the two modeling techniques give good agreement.
Both the shale/oil-sand and shale/gas-sand interfaces show increasing reflection coefficient with angle; but as could be expected, this effect is significantly larger for the shale/gas-sand interface.