# Estimation of Litho-Fluid Facies Distribution from Zero-Offset Acoustic and Shear Impedances
## 5 Results and Discussion
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **Isotropic Inversion** & **Statistical Modeling** & **MLFN** \\ \hline \(Z_{P}\) & 84\% & 88\% & 94\% \\ \(Z_{S}\) & 73\% & 85\% & 92\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The correlation coefficients of the predicted \(Z_{P}\) and \(Z_{S}\) obtained from isotropic inversion, statistical modeling, and MLFN at the wells (\(\times\)1) and (\(\times\)2). Figure 18: The histogram plots of the zero-offset P-impedance obtained from (**a**) isotropic inversion, (**b**) statistical modeling, and (**c**) MLFN. A previous method [90] aimed at calculating the zero-offset velocities and anisotropy parameters empirically by inverting the partial stacks into velocities at different incident angles and solving [PERSON]'s anisotropy equations [61]. However, this method is applicable to high-resolution seismic data. On the other hand, the current study uses statistical modeling and neural networks (MLFN) to reduce the anisotropy effect and normalize the error simultaneously. [PERSON] estimated the P- and S-impedances from the P- and S-wave reflectivities by using a joint inversion [130]. The author reduced the nonuniqueness of the inversion output by adding the \(V_{S}/V_{P}\) ratio as a lithology constraint. Compared to simultaneous inversion, joint inversion follows the background lithology trend and hence yields more accurate impedances. The current study also follows a lithology constraint by estimating the zero-offset impedances from the impedances at different incident angles. In other words, the impedances at non-normal angles add facies information to the model and yield a unique solution for the zero-offset impedance. The \(Z_{P}\) and \(Z_{S}\) volumes, obtained from the MLFN method, were applied to Equations (58) and (59) in order to solve for the anisotropy parameters Epsilon and Delta. Figure 21 shows the resulting Epsilon and Delta sections plotted against those obtained from the methodology of Liner and Fei [79] at the well (\(\times 1\)). Figure 19: The histogram plots of the zero-offset S-impedance obtained from (**a**) isotropic inversion, (**b**) statistical modeling, and (**c**) MLFN. Figure 21: Cross-sections extracted from the anisotropy parameters’ volumes Epsilon (**a**) and Delta (**b**) calculated by applying the MLFN-based \(Z_{P}\) and \(Z_{S}\) to the impedance-domain Thomsen’s anisotropy equations. The inserted curves are the Epsilon and Delta calculated at the well (\(\times\)1). Figure 20: The residual plots of the isotropic inversion, statistical modeling, and MLFN for each of the zero-offset \(Z_{P}\) (**a**–**c**) and \(Z_{S}\) (**d**–**f**), where the number of samples of each model is on the X-axis and residuals are on the Y-axis. Figure 22 shows the histograms of Epsilon and Delta sections, where the values of Epsilon range from \(-0.2\) to \(0.2\) with a mean value of \(0.002\), while the Delta ranges from \(-0.075\) to \(+0.075\) with a mean value of \(0.001\). These values are within the weak-anisotropy ranges, according to [PERSON]'s model [53]. The composite traces of the resulting Epsilon and Delta were extracted from their volumes and plotted against the facies log at the well (\(\times 1\)), as shown in Figure 23. The tops of the gas horizons, A and B, show high positive Epsilon values (blue rectangles) due to the reduction in the P-wave velocity. The positive sign is reasonable because the vertical velocity decreases relative to the horizontal velocity, which is in the same direction as the bedding. Figure 23: The anisotropy parameters, Epsilon and Delta, against the facies log at the well (\(\times 1\)), where A, B, C, and D are sand zones. Figure 22: The histograms of the anisotropy parameters (**a**) Epsilon and (**b**) Delta sections, which were calculated by applying the MLFN-based \(Z_{p}\) and \(Z_{S}\) to the impedance-domain [PERSON]’s anisotropy equations. Another observation is that the anisotropy degree is directly proportional to the contrast of the elastic properties between layers.