# Estimation of Litho-Fluid Facies Distribution from Zero-Offset Acoustic and Shear Impedances
## 4 Methodology
### Isotropic Simultaneous Inversion  
Simultaneous inversion is a prestack seismic inversion that simultaneously inverts the angle gathers into the reflectivities of the \(Z_{P}\), \(Z_{S}\), and density based on the linear relationships between the \(Z_{P}\) and each of the \(Z_{S}\) and density. The methodology used in this study [92] directly inverted for the \(Z_{P}\), \(Z_{S}\), and density [93; 94].  
The Aki-Richards equation [95] is an approximation of the relationship between the P-wave reflectivity between two media and the incident angle as follows:  
\[R(\theta)=\frac{1}{2}(\Delta V/V+\Delta\rho/\rho)-2(W/V)^{2}(2\Delta W/W+ \Delta\rho/\rho)sin^{2}\theta+\frac{1}{2}(\Delta V/V)tan^{2}\theta \tag{12}\]  
where \(R(\theta)\) is the P-wave reflectivity at an incident angle (\(\theta\)), while \(V\), \(W\), and \(\rho\) are the average P-wave velocity, S-wave velocity, and density between the two media, respectively. The idea of [PERSON]'s equation [93] is to re-express the Aki-Richards equation in (12) in terms of the acoustic and shear impedances as follows:  
\[R(\theta)=\frac{1}{2}\frac{\Delta I}{I}(1+tan^{2}\theta)-4\bigg{(}\frac{W}{V} \bigg{)}^{2}\bigg{(}\frac{\Delta J}{I}\bigg{)}sin^{2}\theta-\bigg{[}\frac{1}{2 }\bigg{(}\frac{\Delta\rho}{\rho}\bigg{)}tan^{2}\theta-2\bigg{(}\frac{W}{V} \bigg{)}^{2}\bigg{(}\frac{\Delta\rho}{\rho}\bigg{)}sin^{2}\theta\bigg{]} \tag{13}\]  
where \(I\) is the acoustic impedance, \(J\) is the shear impedance, the term (\(\frac{1}{2}\frac{\Delta I}{I}\)) is the zero-offset P-wave reflectivity, and the term (\(\frac{1}{2}\frac{\Delta J}{I}\)) is the zero-offset S-wave reflectivity, such that:  
\[\frac{1}{2}\frac{\Delta J}{J}=\frac{1}{2}(\Delta V/V+\Delta\rho/\rho),\frac{1} {2}\frac{\Delta J}{J}=\frac{1}{2}(\Delta W/W+\Delta\rho/\rho) \tag{14}\]  
[PERSON]'s equation could be re-expressed by replacing the reflectivities by the rock properties as follows:  
\[s(\theta)=\frac{1}{2}C_{1}W_{\theta}Dln(Z_{P})+\frac{1}{2}C_{2}W_{\theta}Dln( Z_{S})+C_{3}W_{\theta}Dln(\rho) \tag{15}\]  
where \(s(\theta)\) is the angle trace, \(W(\theta)\) is the angle-dependent wavelet, \(D\) is a derivative operator, \(Z_{P}\) is the acoustic impedance, \(Z_{S}\) is the shear impedance, \(\rho\) is the density, and \(\theta\) is the incident angle. The terms \(C_{1}\), \(C_{2}\), and \(C_{3}\) are given by:  
\[C_{1}=1+tan^{2}\theta,C_{2}=-8\big{(}\frac{V_{S}}{V^{p}}\big{)}tan^{2}\theta, C_{3}=-0.5 tan^{2}\theta+2\big{(}\frac{V_{S}}{V_{P}}\big{)}^{2}sin^{2}\theta \tag{16}\]  
where \(C_{1}\), \(C_{2}\), and \(C_{3}\) are constants, \(\theta\) is the incident angle, \(V_{S}\) is the S-wave velocity, and \(V_{P}\) is the P-wave velocity.  
Next, the terms \(ln(Z_{P})\), \(ln(Z_{S})\), and \(ln(\rho)\) were coupled by the following equations:  
\[ln(Z_{s})=kln(Z_{P})+k_{c}+\Delta ln(Z_{s}) \tag{17}\]  
\[ln(\rho)=mln(Z_{p})+m_{c}+\Delta ln(\rho) \tag{18}\]  
where \(k\) and \(m\) are the intercepts of the linear relations and \(k_{c}\) and \(m_{c}\) are constants, while \(\Delta ln(Z_{S})\) and \(\Delta ln(\rho)\) are controlled by the deviations away from the best-fit lines of the linear relationships. \(\Delta L_{D}\) and \(\Delta L_{S}\) are the displacements of \(ln(\rho)\) and \(ln(Z_{S})\) due to the deviation from the best-fit lines.  
The next step was to apply the derivative operator (\(D\)) to Equations (3) and (4):  
\[D\ln Z_{S}=kDlnZ_{P}+D\Delta lnZ_{S} \tag{19}\]  
\[D\ln\rho=mDlnZ_{P}+D\Delta ln\rho \tag{20}\]  
where \(D\) is the derivative operator, while \(k\) and \(m\) are the inteccepts of the linear relations. Eventually, the algorithm was applied to an initial guess and then iterated until it reached a solution at which the initial model matched the real data with the least-squared error.