# Estimation of Litho-Fluid Facies Distribution from Zero-Offset Acoustic and Shear Impedances
## 5 Results and Discussion
Another visual comparison is made by the histogram plots of the inverted \(Z_{P}\) for the three models, as shown in Figure 18, where the \(Z_{P}\), obtained from the simultaneous inversion, has extreme low and high values of impedance with relatively high standard deviation (1445) and low mean (6195). The extreme values of simultaneous inversion appear in purple and green colors in Figure 16a. On the other hand, the least standard deviation (1068) belongs to the \(Z_{P}\) of the MLFN. Moreover, the means of the anisotropic models are higher than those of simultaneous inversion, which show extremely low \(Z_{P}\) values down to 2000 ((m/s)-(g/cc)). Similarly, Figure 19 shows the histogram plots extracted from the the S-impedance sections, where the isotropic method has extremely low \(Z_{S}\) values, down to 1000 ((m/s)-(g/cc)), high values, up to 9000 ((m/s)-(g/cc)), and relatively high standard deviation (1671). On the other hand, the statistical model and MLFN have lower standard deviations, which are 821 and 886, respectively. Another comparison is to evaluate the residuals' plots of all models by calculating the difference between the impedance logs and the predicted impedance for each model. Figure 20 shows the plots of the calculated residuals versus the observation indexes for the \(Z_{P}\) models (Figure 20a-c) and \(Z_{S}\) models (Figure 20d-f). It can be noticed that the residuals of the MLFN model are centered around zero and have low magnitude compared to those  
Figure 14: The performance plots of the (**a**) \(Z_{P}\) and (**b**) \(Z_{S}\) MLFN models, 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. Figure 15: The error histograms of the (**a**) \(Z_{P}\) and (**b**) \(Z_{S}\) MLFN models, where the error value is on the X-axis and the error frequency (instances) is on the Y-axis. The errors of the training, validation, and testing data appear in blue, green, and red colors, respectively. of the simultaneous inversion and statistical modeling. In order to determine the best \(Z_{P}\) model, the R-value was calculated for each model, as shown in the Table 2.  
Figure 16: Arbitrary sections extracted from the zero-offset acoustic impedance volumes obtained from (**a**) isotropic inversion, (**b**) statistical modeling, and (**c**) MLFN. The inserted curves are the acoustic impedance logs of the wells \(\times 1\) and \(\times 2\). Figure 17: Arbitrary sections extracted from the zero-offset shear impedance volumes obtained from (**a**) isotropic inversion, (**b**) statistical modeling, and (**c**) MLFN. The inserted curves are the shear impedance logs of the wells \(\times 1\) and \(\times 2\). The simultaneous (isotropic) inversion shows low R-values for \(Z_{P}\) and \(Z_{S}\) (84% and 73%, respectively), which indicates high inversion errors. The statistical model has enhanced the correlations to be 88% and 85% for \(Z_{P}\) and \(Z_{S}\), respectively. However, the statistical models' results are still close to the isotropic case because the random error has been reduced by using only one value of each coefficient for the entire volume. Therefore, the error normalization is not as much as desired. On the other hand, the MLFN method has the highest R-values, 94% and 92% for \(Z_{P}\) and \(Z_{S}\), respectively. This shows how ML can reduce the randomness in the data and turn it into recognized patterns. It can be concluded that the statistical model has resulted in more accurate impedance volumes than the simultaneous inversion; however, the statistical model has only one value for each coefficient, which is not enough to normalize the random error as desired. The model can be more hierarchical if there are more facies data in several wells. On the other hand, the MLFN model could estimate the zero-offset impedance efficiently by turning the randomness in data into identified patterns.