# Estimation of Litho-Fluid Facies Distribution from Zero-Offset Acoustic and Shear Impedances
## 4 Methodology
### Partial-Log-Constrained Inversion  
The separate inversion of angle stacks is based on a partial inversion approach that obtains the zero-offset P- and S-wave velocities from the inverted P- and S-waves at different angular ranges [90]. The idea is based on [PERSON]'s equations in (9) and (10), which relate the vertical P-wave (\(V_{P}\)) and S-wave (\(V_{S}\)) velocities to the anisotropy parameters, Epsilon and Delta, in VTI media. Partial-log-constrained inversion requires a wavelet to be extracted from each angle stack so that the wavelet is centered at the average angle of the angle stack [90]. An initial model was created from logging data to guide the inversion. This method required S-wave velocity logs in the initial model to constrain the inversion. Each partial seismic volume was inverted into a velocity volume at the central angle of its angle range. Next, the velocity-domain [PERSON]'s equations in (9) and (10) were used to solve for the zero-offset P- and S-wave velocities as well as the Epsilon and Delta parameters, as shown in the following equations:  
\[ \begin{bmatrix}1&sin^{2}(\theta_{1})cos^{2}(\theta_{1})&sin^{4}(\theta_{1})\\ 1&sin^{2}(\theta_{2})cos^{2}(\theta_{2})&sin^{4}(\theta_{2})\\ \vdots&\vdots&\vdots\\ 1&sin^{2}(\theta_{n})cos^{2}(\theta_{n})&sin^{4}(\theta_{n})\end{bmatrix} \begin{bmatrix}V_{P0}\\ V_{P0}\\ V_{P0}\\ V_{P0}\\ \end{bmatrix} = \begin{bmatrix}V_{qP}(\theta_{1})\\ V_{qP}(\theta_{2})\\ \vdots\\ V_{qP}(\theta_{n})\end{bmatrix}  \tag{21}\]  
where \(V_{P0}\) is the zero-offset P-wave velocity, \(\epsilon\) and \(\delta\) are the anisotropy parameters, \(V_{qP}\) is the quasivertical P-wave velocity, \(\theta\) is the incident angle, and \(n\) is the number of partial stacks. \[ \begin{bmatrix}1&sin^{2}(\theta_{1})cos^{2}(\theta_{1})\\ 1&sin^{2}(\theta_{2})cos^{2}(\theta_{2})\\ \vdots&\vdots\\ 1&sin^{2}(\theta_{n})cos^{2}(\theta_{n})\end{bmatrix}\begin{bmatrix}V_{s0}\\ V_{p0}\\ \frac{V_{p0}^{2}(\epsilon-\delta)}{V_{s0}}\end{bmatrix} = \begin{bmatrix}V_{qSV}( \theta_{1})\\ V_{qSV}(\theta_{2})\\ \vdots\\ V_{qSV}(\theta_{n})\end{bmatrix}  \tag{22}\]  
where \(V_{50}\) is the zero-offset P-wave velocity, \(\epsilon\) and \(\delta\) are the anisotropy parameters, \(V_{qSV}\) is the quasivertical S-wave velocity, \(\theta\) is the incident angle, and n is the number of the partial stacks. The log-constrained inversion of partial stacks was adopted from the nonlinear optimization inversion [90], such that the objective function is obtained by:  
\[f(V)=\parallel S-D\parallel\rightarrow min \tag{23}\]  
where \(V\) is the inverted P- or S-wave velocity, \(D\) is the angle-domain seismic data, and \(S\) is the corresponding prediction, which can be expressed according to the convolution model [96]. The reflectivity function is based on the [PERSON] and [PERSON] approximation of the angle-dependent reflectivity [97], as shown below:  
\[R(\theta)=R(0)+Asin^{2}(\theta)+Btan^{2}(\theta) \tag{24}\]  
where \(R(\theta)\) is the reflectivity at an incident angle \(\theta\), while \(R(0)\) is the zero-offset reflectivity, which is given by:  
\[R(0)=\frac{1}{2}\bigg{(}\frac{\Delta V_{P}}{V_{P}}+\frac{\Delta\rho}{\rho}\bigg{)} \tag{25}\]  
\[A=-2\bigg{(}\frac{V_{5}}{V_{P}}\bigg{)}^{2}\bigg{(}\frac{2\Delta V_{S}}{V_{S}}+ \frac{\Delta\rho}{\rho}\bigg{)} \tag{26}\]  
\[B=\frac{1}{2}\bigg{(}\frac{\Delta V_{P}}{V_{P}}\bigg{)} \tag{27}\]where \(V_{P}\), \(V_{S}\), and \(\rho\) are, respectively, the average P-wave velocity, \(S\)-wave velocity, and density of the underlying and overlying layers, while \(\Delta V_{P}\), \(\Delta V_{S}\), and \(\Delta\rho\) are the differences between these properties at the underlying and overlying strata.