# Seismic AVOA Inversion for Weak Anisotropy Parameters and Fracture Density in a Monoclinic Medium
## 2 Methods
### Bayesian Inversion for Weak Anisotropy Parameters and Fracture Density  
Assuming that there are two sets of fractures, we produced a matrix of data equations for the case of \(m\) incidence angles and \(n\) azimuthal angles:  
\[\left[\begin{array}{c}d^{\text{obs}}(\theta_{1},\phi_{1})\\ \vdots\\ d^{\text{obs}}(\theta_{m},\phi_{n})\end{array}\right]=\left[\begin{array}{c}w( \theta_{1},\phi_{1})\\ \ddots\\ w(\theta_{m},\phi_{n})\end{array}\right]=\left[\begin{array}{c}w(\theta_{1}, \phi_{1})\\ \vdots\\ F_{1}(\theta_{m})\end{array}\right]\left[\begin{array}{c}R_{pp}^{iso}( \theta_{1})+R_{pp}^{adi}(\theta_{1})\\ \vdots\\ R_{pp}^{iso}(\theta_{m})+R_{pp}^{adi}(\theta_{m})\end{array}\right]+ \tag{12}\] \[\left[\begin{array}{cccc}w(\theta_{1},\phi_{1})\\ \ddots\\ w(\theta_{m},\phi_{n})\end{array}\right]\left[\begin{array}{cccc}F_{1}( \theta_{1})&F_{2}(\theta_{1})&a(\theta_{1},\phi_{1})&b(\theta_{1},\phi_{1})\\ \vdots&\vdots&\vdots&\vdots\\ F_{1}(\theta_{m})&F_{2}(\theta_{m})&a(\theta_{m},\phi_{n})&b(\theta_{m},\phi_{n })\end{array}\right]\left[\begin{array}{c}\delta^{psd}\\ \varepsilon\\ N_{V1}\\ N_{V2}\end{array}\right]+\left[\begin{array}{c}n(\theta_{1},\phi_{1})\\ \vdots\\ n(\theta_{m},\phi_{n})\end{array}\right]\]  
From the above equation, it can be seen that elastic parameters of the isotropic background should be known for the accurate inversion of weak anisotropic parameters and fracture density. The \(V_{p}\), \(V_{s}\), and \(\rho\) values for all layers of interest were obtained from well logs using Backus averaging [34]. The forward problem has the simple form:  
\[\textbf{d=WFm+n=Lm+n} \tag{13}\]  
where the forward operator **L=WF**, **n** is random noise added into clear data, and the model parameters are:  
\[\textbf{m}\ =\ \left[\delta^{psd},\varepsilon,N_{V1},N_{V2}\right]^{T} \tag{14}\]  
and the wavelet matrix is:  
\[\textbf{W}\ =\ \left[\begin{array}{cccc}w(\theta_{1},\phi_{1})\\ &\ddots\\ &&w(\theta_{m},\phi_{n})\end{array}\right]. \tag{15}\]  
In Bayesian theory, the posterior probability density function (PDF) is calculated by the prior PDF and the likelihood function [35]:  
\[\textbf{p}(\textbf{m}|\textbf{d})\propto\textbf{p}(\textbf{m})\textbf{p}( \textbf{d}|\textbf{m}) \tag{16}\]  
where \(\textbf{p}(\textbf{m}|\textbf{d})\) is the posterior PDF, \(\textbf{p}(\textbf{m})\) is the prior PDF, and \(\textbf{p}(\textbf{d}|\textbf{m})\) is the likelihood function.  
In the case of observed seismic data containing Gaussian random noise, the likelihood function \(\textbf{p}(\textbf{d}|\textbf  
where \(\mathbf{R}(\mathbf{m})\ =\ \frac{1}{2}\mathbf{m}^{T}C_{\mathbf{m}}^{-1}\mathbf{m}\), \(\sigma_{\mathbf{d}}^{2}\) is the variance of data. By solving the derivative of the objective function with respect to the model parameters, we could obtain  
\[\frac{\partial J(\mathbf{m})}{\partial\mathbf{m}}\ =\ \frac{1}{\sigma_{\mathbf{d}}^{2}} \mathbf{L}^{T}(\mathbf{L}\mathbf{m}-\mathbf{d})+\frac{\partial R(\mathbf{m})} {\partial\mathbf{m}} \tag{20}\]  
setting \(\frac{\partial J(\mathbf{m})}{\partial\mathbf{m}}\) to zero gives the model parameters  
\[\mathbf{m}\ =\ \Big{(}\mathbf{L}^{T}\mathbf{L}+\frac{1}{2}\mu_{h}C_{ \mathbf{m}}^{-1}\Big{)}^{-1}\Big{(}\mathbf{L}^{T}\mathbf{d}+\frac{1}{2}\mu_{h }C_{\mathbf{m}}^{-1}\Big{)} \tag{21}\]  
where super parameter \(\mu_{h}\) allowed us to estimate the model parameters along the tuning curve, mainly through experiments. In this paper, the derived P-wave reflection coefficient, after removing the background effect, was completely linear, thus allowing us to invert the model parameters from seismic data \(\mathbf{d}\).