# Estimation of Litho-Fluid Facies Distribution from Zero-Offset Acoustic and Shear Impedances
## 4 Methodology
### Partial-Log-Constrained Inversion
The term \(\frac{\Delta\rho}{\rho}\) can be obtained by [PERSON]'s equation as follows [98]:  
\[\frac{\Delta\rho}{\rho}=\frac{1}{4}\left(\frac{\Delta V_{P}}{V_{P}}\right) \tag{28}\]  
According to Equations (24)-(28), the reflectivity will be as follows:  
\[R(\theta)=\left(\frac{5}{8}-\frac{1}{2}\frac{V_{S}^{2}}{V_{P}^{2}}sin^{2} \theta+\frac{1}{2}tan^{2}\theta\right)\frac{\Delta V_{P}}{V_{P}}-4\frac{V_{S}^ {2}}{V_{P}^{2}}sin^{2}\theta\frac{\Delta V_{S}}{V_{S}} \tag{29}\]  
After inverting the partial stacks separately into \(V_{P}\) and \(V_{S}\) volumes at the near, mid, and far angles, the zero-offset velocities Epsilon and Delta were solved by Equations (21) and (22). The current study applied the same methodology to \(Z_{P}\) and \(Z_{S}\) by replacing the velocities of Equations (9) and (10) by impedances, as shown below:  
\[Z_{P}(\theta)=Z_{P0}\big{[}1+\delta sin^{2}(\theta)cos^{2}(\theta)+\epsilon sin ^{4}(\theta)\big{]} \tag{30}\]  
\[Z_{S}(\theta)=Z_{S0}\big{[}1+\left(\frac{Z_{P0}}{Z_{S0}}\right)^{2}(\epsilon- \delta)sin^{2}(\theta)cos^{2}(\theta)\big{]} \tag{31}\]  
Partial inversion is sensitive to the quality of seismic data because the magnitude of the noise may be more than that of the anisotropy effect. Hence, the zero-offset impedances were estimated from the impedances at the near, mid, and far incident angles by using statistical modeling and MLFN. Seismic data consisted of three 3D volumes of angle ranges, from 5\({}^{\circ}\) to 15\({}^{\circ}\), 15\({}^{\circ}\) to 25\({}^{\circ}\), and 25\({}^{\circ}\) to 40\({}^{\circ}\). Accordingly, three partial stacks were created at the near, mid, and far central incident angles: 10\({}^{\circ}\), 20\({}^{\circ}\), and 32.5\({}^{\circ}\), respectively. The advantage of impedance modeling is that it normalizes the error and reduces the anisotropy effect simultaneously. An important step prior to the modeling process is to calculate Epsilon and Delta at the well locations in order to have an idea about the degree of anisotropy within the study area and to specify the model's assumptions accordingly. The anisotropy parameters were calculated by the methodology of [PERSON] and [PERSON] [79], who calculated the layer-induced stiffness parameters from the averaged properties of the isotropic elastic layers and then obtained Epsilon and Delta as functions of those parameters. The stiffness parameters were obtained from the following equations:  
\[a=\langle\frac{\lambda}{\lambda+2\mu}\rangle^{2}\langle\frac{1}{\lambda+2\mu} \rangle^{-1}+4\langle\frac{\mu(\lambda+\mu)}{\lambda+2\mu}\rangle \tag{32}\]  
\[R(\theta)=R(0)+Asin^{2}(\theta)+Btan^{2}(\theta) \tag{24}\]0  
\[R(\theta)=R(0)+Asin^{2}(\theta)+Btan^{2}(\theta) \tag{24}\]1  
\[R(\theta)=R(0)+Asin^{2}(\theta)+Btan^{2}(\theta) \tag{24}\]2  
where \(\langle\rangle\) is the Backus averaging symbol, \(\mu\) is the shear modulus, and \(\lambda\) is [PERSON]'s constant, while \(a,c,f\), and \(l\) are the stiffness parameters. Next, Epsilon and Delta were obtained from the following equations:  
\[R(\theta)=R(0)+Asin^{2}(\theta)+Btan^{2}(\theta) \tag{24}\]3\[R(\theta)=R(0)+Asin^{2}(\theta)+Btan^{2}(\theta) \tag{24}\]4  
Another premodeling step is to explore the relations between the zero-offset impedances, which are the \(Z_{P}\) and \(Z_{S}\) logs, and the angle impedances, which are the composite traces obtained from the partial-stack inversion. Figure 3 shows a fair linear relationship based on the data of wells (\(\times\)1) and (\(\times\)2), which consist of 1292 observations for \(Z_{P}\) 1582 observations for \(Z_{S}\). The linearity between variables is proven visually through the best-fit lines (in blue). In addition, the Pearson correlation coefficient (R-value) was calculated in Python using NumPy. The R-values show strong positive correlations at the near and mid angles, while the correlation is moderate at the far angle. This is matching with [PERSON]'s equations in (9) and (10), which consider a linear relationship between the zero-offset velocity and the velocity at an angle theta.