# Estimation of Litho-Fluid Facies Distribution from Zero-Offset Acoustic and Shear Impedances
## 5 Results and Discussion
The number of iterations is shown on the X-axis and the value of each coefficient is shown on the Y-axis. Each coefficient was simulated in R by three chains, which appear in green, black, and red colors. For both the \(Z_{P}\) and \(Z_{S}\) models, the chains reached their stationary distribution in less than 5000 iterations. However, the chains of the \(Z_{P}\) model seem more converged than those of the \(Z_{S}\) model due to the high random error of the \(Z_{S}\) inversion. The posterior means and standard deviations of the coefficients were finally obtained for each model, as shown in Table 1. It is clear from the posterior means of the coefficients that the rate of change of the zero-offset impedance with the angle of incidence is almost constant at the near and mid incident angles (10\({}^{\circ}\) and 20\({}^{\circ}\)), while the rate of change decreases at the far angle (32.5\({}^{\circ}\)). In other words, the zero-offset \(Z_{P}\) changes with the far \(Z_{P}\) at a lower rate compared to the near- and mid-angle impedances. Figure 11: A plot of the velocity versus incident angle, obtained from ultrasonic measurements on shale cores. Figure 12: The trace plots of the Markov chains that simulate the coefficients of the variables for the \(Z_{P}\) statistical model: b1, b2, and b3. After applying the posterior means of the coefficients to Equation (38), the \(Z_{P}\) and \(Z_{S}\) equations will be:  
Figure 6:0 Figure 6:1  
The next step is to estimate the zero-offset impedances by the MLFN. The training, testing, and validation data were the same as those used for the statistical model. Two networks were trained for the \(Z_{P}\) and \(Z_{S}\) models by the Levenberg-Marquardt algorithm such that the best performance was reached at 21 iterations for the \(Z_{P}\) model and 58 for the \(Z_{S}\) model, as shown in Figure 14, where the number of iterations is on the X-axis and the mean square error is on the Y-axis. The error reduction trends of the training, validation, and testing data appear in blue, green, and red colors, respectively. The three trends are close to each other, which means that there is no over-fitting. The networks reached the least mean square error (MSE) error values at 92,840 for the \(Z_{P}\) model and 36,893 for the \(Z_{S}\) model. In addition, the error values were plotted in a histogram, as shown in Figure 15, where the errors of the training, validation, and testing data appear in blue, green, and red colors, respectively. The error distribution for the \(Z_{P}\) and \(Z_{S}\) models is centered around the zero-error (yellow) line. The quality check of the resulting zero-offset \(Z_{P}\) and \(Z_{S}\) starts with color matching with the impedance logs, as shown in Figures 16 and 17, respectively. It is clear that the MLFN's impedances (Figures 16c and 17c) better match the impedance logs at the wells (\(\times 1\)) and (\(\times 2\)) than those of simultaneous inversion (Figures 16a and 17a) and statistical modeling (Figures 16b and 17b). In addition, there are extreme impedance values shown in isotropic inversion compared to the anisotropic methods that have more normalized inversion error values. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Coefficient** & \multicolumn{2}{c}{\(\mathbf{b_{1}}\)} & \multicolumn{2}{c}{\(\mathbf{b_{2}}\)} & \multicolumn{2}{c}{\(\mathbf{b_{3}}\)} \\ \hline
**Property** & **Mu** & **SD** & **Mu** & **SD** & **Mu** & **SD** \\ \hline \(Z_{P}\) Model & 0.3532 & 0.01 & 0.3555 & 0.01 & 0.286 & 0.01 \\ \(Z_{S}\) Model & 0.3672 & 0.01 & 0.364 & 0.01 & 0.266 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The statistical properties, mean (Mu) and standard deviation (SD), of the coefficients of the zero-offset \(Z_{P}\) and \(Z_{S}\) statistical models. Figure 13: The trace plots of the Markov chains that simulate the coefficients of the variables for the \(Z_{S}\) statistical model: b1, b2, and b3.