# Estimation of Litho-Fluid Facies Distribution from Zero-Offset Acoustic and Shear Impedances
## 4 Methodology
### Zero-Offset Impedance Modeling  
#### 4.3.1 Statistical Modeling  
The idea of the model is to fit a linear relationship between the zero-offset impedance, which is the response variable, and the partially inverted impedances, which are the explanatory variables. The output of the model consists of the posterior means of the coefficients of the three partially inverted impedances. The general form of the model is:  
\[Imp_{0}=b_{1}Imp_{\theta 1}+b_{2}Imp_{\theta 2}+b_{3}Imp_{\theta 3} \tag{38}\]  
Figure 3: Plots of the zero-offset P-impedance (\(Z_{P0}\)) versus each of the partially inverted P-impedances (**a**) \(Z_{p}\)-Near, (**b**) \(Z_{p}\)-Mid, and (**c**) \(Z_{p}\)-Far, while the zero-offset S-impedance (\(Z_{S0}\)) is plotted versus each of the partially inverted S-impedances (**d**) \(Z_{s}\)-Near, (**e**) \(Z_{s}\)-Mid, and (**f**) \(Z_{s}\)-Far. where \(Imp_{0}\) is the zero-offset \(Z_{P}\) or \(Z_{S}\), while \(b_{1}\), \(b_{2}\), and \(b_{3}\) are the model's coefficients and \(Imp_{\theta 1}\), \(Imp_{\theta 2}\), and \(Imp_{\theta 3}\) are the impedances at the near, mid, and far angles, respectively. The model's parameters are estimated based on Bayesian theory and MCMC simulation. The Bayesian model consists of three components: the likelihood, priors, and posterior. The priors of the model were the parameters of the proposed distribution of the model coefficients. The posterior component was considered the probability distribution function of all realizations of the simulated coefficients. The linear expression of the \(Z_{P0}\) model was obtained from [PERSON]'s equation in (9), which can be rewritten as follows:  
\[Z_{P0}=bZ_{P}(\theta) \tag{39}\]  
where:  
\[b=\frac{1}{1+\delta sin^{2}(\theta)cos^{2}(\theta)+esin^{4}(\theta)} \tag{40}\]  
By substituting the near, mid, and far P-impedances, \(Z_{P1}\), \(Z_{P2}\), and \(Z_{P3}\), respectively, to Equation (39), three equations were obtained as follows:  
\[Z_{P0}=b_{1}Z_{P1} \tag{41}\]  
\[Z_{P0}=b_{2}Z_{P2} \tag{42}\]  
\[Z_{P0}=b_{3}Z_{P3} \tag{43}\]  
By adding the three Equations (41)-(43), the linear expression will be as follows:  
\[Z_{P0}=0.33b_{1}Z_{P1}+0.33b_{2}Z_{P2}+0.33b_{3}Z_{P3} \tag{44}\]  
Similarly, the linear expression of the \(Z_{S0}\) model was obtained based on [PERSON]'s equation in (10), which can be rewritten as shown below:  
\[Z_{S0}=b.Z_{S}(\theta) \tag{45}\]  
where  
\[b=\frac{1}{1+\left(\frac{Z_{P0}}{Z_{S0}}\right)^{2}(\epsilon-\delta)sin^{2}( \theta)cos^{2}(\theta)} \tag{46}\]  
After adding the three equations of the near, mid, and far angles, the linear expression will be as follows:  
Figure 3:0  
The likelihood function of the \(Z_{P}\) or \(Z_{S}\) models can be written as shown below:  
Figure 3:1  
Given the coefficients vector \(b\), the explanatory variables vector \(Imp_{\theta}\), and the variance \(\sigma^{2}\), the zero-offset impedance \(Imp_{0}\) follows a normal distribution with a mean equal to the linear expression of the covariates and coefficients, where (\(i\)) is the number of observations. The next layer of the model is the coefficients vector \(b_{j}\), which consists of \(b_{1}\), \(b_{2}\), and \(b_{3}\) such that the three coefficients follow a common normal distribution. The mean and variance of this distribution were obtained from Equation (40) for the \(Z_{P}\) model and Equation (46) for the \(Z_{S}\) model. Based on the weak anisotropy assumption, the mean of the anisotropy parameters was considered zero and the standard deviation was 0.2. Moreover, the squared ratio of \(Z_{P}\) and \(Z_{S}\) was set, according to the wells' data, to have a mean equal to 6 and a standard deviation (SD) equal to 3. The prior function for the \(Z_{P}\) and \(Z_{S}\) models can be written as follows:  
Figure 3:2  
where \(\mu\) and \(\tau\) are, respectively, the mean and variance of the normal distribution of the coefficients vector \(b\), which has an index (\(j\)) from 1 to 3.