FINITE DIFFERENCE METHOD

Finite differences are used to model wave propagation through an anisotropic medium. The method is described in more detail in this report Dellinger et al. (1992). In a recent report Cunha 1991 described a modeling scheme which takes discontinuities in medium properties into account by applying different FD operators to the wavefield and the medium. Different implementations of a staggered grid or derivatives can influence the accuracy of reflection responses. In this paper, we are not employing his method, but rather use cascaded derivative operators. As can be seen from Figure , the reflection amplitudes produced by our modeling scheme are in nearly perfect agreement with analytical solutions to the reflection amplitudes. Figures and show the analytically calculated responses compared with the modeled extracted responses.

 

ppcalibrate Figure 2 PP calibration curve for the FD modeling response. Reflection amplitudes were extracted using the dual experiment. Zoeppritz plane-wave energy reflection coefficients are calculated analytically.

 

iso1.ref Figure 3 Analytically calculated energy reflection and transmission coefficients from Zoeppritz equations.

 

extraiso
Figure 4 Extracted amplitudes from the FD modeling experiment and the dual experiment. The dual experiment estimates all propagation effects but omits reflection at the boundary. For that reason the two signals have different phases.

Such a test is clearly necessary, if we want to make sure the modeling algorithm produces correct reflection coefficients at medium interfaces. The above test gives us confidence in the modeling algorithm for this type of blocky model.

Using S&M equivalent medium theory, we ``add'' fractures to the chalk-rock matrix. The cracks are vertical and have a constant azimuth along the inline direction. We generally compare inline and crossline amplitude effects produced by the fracturing. Amplitude analysis is carried out in the slant-stack domain.