## 2 AVO Attributes  
AVO attributes are attributes that are derived from pre-stack seismic data that are calculated through AVO equation analysis. The following is the written form of Equation (1):  
\[\mathrm{R}(\theta)=\mathrm{A}+\mathrm{B}\mathrm{sin}^{2}\theta+\mathrm{C} \mathrm{sin}^{2}\theta\tan^{2} \tag{1}\]  
where  
\[ \begin{array}{c}\mathrm{A}=\frac{1}{2}\left(\frac{\mathrm{A}\mathrm{V} \mathrm{p}}{\mathrm{V}\mathrm{p}}+\frac{\mathrm{A}\mathrm{p}}{\mathrm{p}} \right),\\ \mathrm{B}=\frac{\mathrm{A}\mathrm{V}\mathrm{p}}{2\mathrm{V}\mathrm{p}}-4 \mathrm{k}\left(\frac{\mathrm{A}\mathrm{V}\mathrm{s}}{\mathrm{V}\mathrm{s}} \right)-2\mathrm{k}\left(\frac{\mathrm{A}\mathrm{p}}{\mathrm{p}}\right),\\ \mathrm{C}=\frac{\mathrm{A}\mathrm{V}\mathrm{p}}{2\mathrm{V}\mathrm{p}}\\ \mathrm{k}=\left(\frac{\mathrm{V}\mathrm{s}}{\mathrm{V}\mathrm{p}}\right)^{2} \end{array} \]  
A is referred to as the intercept, B is the gradient, and C is the curvature term. If the third term (curvature) is removed, then:  
\[ \begin{array}{c}\mathrm{R}(\theta)=\mathrm{A}+\mathrm{B}\sin^{2}\theta\\ \mathrm{If}\theta_{\mathrm{near}},\ \theta_{\mathrm{mid}},\ \mathrm{and}\ \theta_{\mathrm{far}}\ \mathrm{are}\ \mathrm{known},\\ \mathrm{R}(\theta_{\mathrm{near}})=\mathrm{A}+\mathrm{B}\sin^{2}\theta_{ \mathrm{near}}\\ \mathrm{R}(\theta_{\mathrm{mid}})=\mathrm{A}+\mathrm{B}\sin^{2}\theta_{ \mathrm{mid}}\\ \mathrm{R}(\theta_{\mathrm{far}})=\mathrm{A}+\mathrm{B}\sin^{2}\theta_{ \mathrm{far}}\end{array} \]Then, it can be turned into matrix and A and B are obtained through over-determined inversion as follows:  
\[\left[ \begin{array}{c}\text{R}(\theta_{\text{near}})\\ \text{R}(\theta_{\text{mid}})\\ \text{R}(\theta_{\text{far}})\end{array} \right]=\left[ \begin{array}{cc}1& \sin^{2}\theta_{\text{near}}\\ 1&\sin^{2}\theta_{\text{mid}}\\ 1&\sin^{2}\theta_{\text{far}}\end{array} \right]\left[ \begin{array}{c}\text{ A}\\ \text{B}\end{array} \right]\]  
\[\left[ \begin{array}{c}\text{A}\\ \text{B}\end{array} \right]=\left(\left[ \begin{array}{cc}1&\sin^{2}\theta_{ \text{near}}\\ 1&\sin^{2}\theta_{\text{mid}}\\ 1&\sin^{2}\theta_{\text{far}}\end{array} \right]^{\top}\left[ \begin{array}{cc }1&\sin^{2}\theta_{\text{near}}\\ 1&\sin^{2}\theta_{\text{mid}}\\ 1&\sin^{2}\theta_{\text{far}}\end{array} \right]^{\top}\left[ \begin{array}{c }\text{R}(\theta_{\text{near}})\\ \text{R}(\theta_{\text{mid}})\\ \text{R}(\theta_{\text{far}})\end{array} \right]\right] \tag{2}\]  
Generally, AVO cross-plotting is being conducted to perform the analysis where intercept and gradient are defined as the X and Y axis. From the cross-plotting technique, it can also be used to classify AVO response to the gas saturated reservoir (see Figure 1). Numerous AVO attributes are derived from the combination of intercept and gradient; some of them are dot product of intercept and gradient. For positive intercept value condition examples, as the angle of incident increases, the product is negative if the amplitude is decreasing and positive if the amplitude is increasing and vice versa. This method could produce a positive value to the class 3 AVO anomalies and strengthen the indication of it [12]. The statement can be explained further in the Equation (3) below:  
\[\text{Product}=\text{A}*\text{B} \tag{3}\]  
This Equation (3) can be further elaborated by utilizing sign function that extracts the property of value of intercept or gradient attributes whether positive, negative, or zero. The sign function is defined as follows:  
\[\text{sign}\text{x}\ :=\left\{ \begin{array}{c}-1\text{ if x}<0\\ 0\text{ if x}=0\\ 1\text{ if x}>0\end{array} \right.\]  
Then, it is applied to product attributes as follows:  
\[\text{Intercept}-\text{sign Gradient Product}=\text{A}*\text{sign}(\text{B}) \tag{4}\]  
t can also be used to create scaled Poisson's ratio change where it could give a better sight to class 2 AVO anomalies. The attributes can also be expected to show a decreasing value into a reservoir zone and increasing value into a water filled porosity zone [12].