## 2 Theory and Method
### Derivation of the Inversion Equation  
Azimuth AVO inversion is one of the important means of obtaining reservoir fracture parameters. In this process, the convolution model can be used to effectively establish the mathematical relationship between the reflection coefficient and the seismic data, as follows:  
\[d_{nml\times 1}=G_{nml\times 4n}m_{4n\times 1} \tag{6}\]  
where \(d\) is the seismic data and \(G\) is the observation matrix with the following equation:  
\[G= \begin{bmatrix}W_{\theta_{1}}K_{M}(\theta_{1})&W_{\theta_{1}}K_{\mu}(\theta _{1})&W_{\theta_{1}}K_{\delta_{N}}(\theta_{1},\varphi_{1})&W_{\theta_{1}}K_{ \delta_{T}}(\theta_{1},\varphi_{1})\\ \vdots&\vdots&\vdots&\vdots\\ W_{\theta_{1}}K_{M}(\theta_{1})&W_{\theta_{1}}K_{\mu}(\theta_{1})&W_{\theta_{ 1}}K_{\delta_{N}}(\theta_{1},\varphi_{m})&W_{\theta_{L}}K_{\delta_{T}}(\theta _{l},\varphi_{m})\end{bmatrix} \]  
where \(W_{\theta_{i}}\) is the matrix representing the mathematical relations of the wavelets in the convolution process as follows:  
\[W_{\theta_{i}}= \begin{bmatrix}w_{\theta_{i}}^{1}&0&0&\cdots\\ w_{\theta_{i}}^{2}&w_{\theta_{i}}^{1}&0&\ddots\\ w_{\theta_{i}}^{3}&w_{\theta_{i}}^{2}&w_{\theta_{i}}^{1}&\ddots\\ \vdots&\ddots&\ddots&\ddots\end{bmatrix} \]  
and  
\[K_{M}(\theta_{1})= \begin{bmatrix}a(\theta_{1})&0&\cdots&0\\ 0&a(\theta_{1})&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&0&a(\theta_{1})\end{bmatrix} _{n\times n},K_{\mu}(\theta_{1})= \begin{bmatrix} b(\theta_{1})&0&\cdots&0\\ 0&b(\theta_{1})&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&0&b(\theta_{1})\end{bmatrix} _{n\times n},\]  
Figure 3: Relative error of reflection coefficient. \[K_{\delta_{N}}(\theta_{1},\varphi_{1})= \begin{bmatrix}c(\theta_{1},\varphi_{1})&0& \cdots&0\\ 0&c(\theta_{1},\varphi_{1})&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&0&c(\theta_{1},\varphi_{1})\end{bmatrix} _{n\times n},\]  
\[K_{\delta_{T}}(\theta_{1},\varphi_{1})= \begin{bmatrix}d(\theta_{1},\varphi_{1} )&0&\cdots&0\\ 0&d(\theta_{1},\varphi_{1})&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&0&d(\theta_{1},\varphi_{1})\end{bmatrix} _{n\times n}\]  
During the inversion process, significant differences in the order of magnitude among the parameters to be inverted can cause instability, as the large numerical disparity negatively impacts the convergence and accuracy of the inversion algorithm. To address this issue effectively, it is essential to normalize all inversion parameters to a common scale, thus mitigating any potential interference from these magnitude differences. In this study, seismic data from different azimuths were first superimposed to enhance the effect of the isotropic background. This approach allowed for the normalization of the background isotropic parameters by dividing them by the mean value of the isotropic background. For the normalization of fracture parameters, the seismic data at various azimuths were differentiated to remove the influence of the isotropic background. The fracture parameters were then normalized by dividing by the mean value of fracture weakness, ensuring consistent standardization. This methodology effectively controls the variability in the inversion process, leading to more reliable and accurate results. By implementing this approach of standardization, the final inversion equation not only mitigates the instability caused by the significant order-of-magnitude differences between the various parameters but also enhances the overall accuracy of the inversion results. This allows the inversion model to more effectively capture the characteristics of fractures, ultimately ensuring the extraction of more reliable reservoir fracture parameters in practical applications. As a result, the normalized inversion equation can be formulated as follows:  
\[d^{*}_{nml\times 1}=G_{nml\times 4n}m^{*}_{4n\times 1} \tag{7}\]  
where  
\[d^{*}_{nml\times 1}= \begin{bmatrix}\frac{S_{2}-S_{1}}{M^{0}}+\frac{S_{2}-S_{1}}{ \beta_{N}^{0}}\\ \vdots\\ \frac{S_{X}+S_{X-1}}{M^{0}}+\frac{S_{X}-S_{X-1}}{\beta_{N}^{0}}\end{bmatrix}  m^{*}_{4n\times 1}=