## 3 Methodology
### SVM Classification Model  
#### 3.2.1. \(\mathrm{SVM}\)  
The SVM is the core concept, and its accuracy is directly related to the accuracy of the classification results. Taking a classical binary classification task as an example, the basic principle of the SVM is introduced [32; 33]. The mechanism of an SVM is to find an optimal classification hyperplane that meets the classification requirements, so that the hyperplane can maximize the blank areas on both sides of the hyperplane while ensuring classification accuracy. In theory, an SVM can achieve optimal classification of linear separable data. The training sample set is given as \((x_{i},y_{i})\), \(i=1,2,3,\ldots,1\), \(x\in\mathrm{R}\), \(y\in\{\pm 1\}\), and the segmentation hyperplane is marked as \((w\cdot x)+b=0\). To classify all samples correctly and obtain classification intervals, the following constraints must be satisfied:  
\[y_{i}[(w\cdot x_{i})+b]\geq 1\quad i=1,\ 2,\ 3,\ldots,l \tag{6}\]  
The objective of optimization is to find a straight line (\(w\) and \(b\)), to make the nearest point to the line as far as possible. The distance from point to line can be expressed as the following:  
\[d=\frac{y_{i}[(w\cdot x_{i})+b]}{\|w\|} \tag{7}\]  
The optimization objective changes to Equation (8),  
\[\underset{w,b}{\mathrm{argmax}}\{\frac{1}{\|w\|}\underset{i}{\mathrm{min}}y_{ i}[(w\cdot x_{i})+b]\} \tag{8}\]  
Since \(y_{i}[(w\cdot x_{i})+b]\geq 1\), only \(\underset{w,b}{\mathrm{argmax}}\frac{1}{\|w\|}\) should be considered. In addition, the solution of the maximum value of \(\frac{1}{\|w\|}\) is equivalent to the solution of the minimum value of \(\frac{1}{2}\|w\|^{2}\), so the objective function can be expressed as follows:  
\[\left\{ \begin{array}{cl}&\underset{w,b}{\mathrm{min}}\frac{1}{\|w\|}\|^{2} \\ \text{s.t.}&y_{i}[(w\cdot x_{i})+b]\geq 1\quad i=1,\ 2,\ 3,\ldots,l\end{array} \right. \tag{9}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Subfacies**Characteristics & \(G\) & \(S\) & \(A_{\mathbf{m}}\) & \(A_{\mathbf{s\text{-}T}}\) & \(A_{\mathbf{s\text{-}B}}\) & \(\overline{K_{p}}\) & \(\Lambda_{\mathbf{L}}\) & \(\lambda_{\mathbf{R}}\) \\ \hline Delta front & 0.51 & 13.78 & 105.09 & 53.72 & 17.20 & 0.31 & 1.10 & 2.81 \\ \hline Shore-shallow & 0.52 & 11.79 & 119.26 & 35.20 & 47.49 & 0.96 & 1.39 & 2.66 \\ \hline Delta front & 0.55 & 19.27 & 110.43 & 20.59 & 43.87 & 0.93 & 1.06 & 1.97 \\ \hline Delta plain & 0.47 & 32.33 & 109.83 & 83.92 & 114.18 & \(-\)0.23 & 0.77 & 2.70 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sedimentary characteristics of log curves and corresponding subfacies in the study area (ARY-01). A Lagrange multiplier can be used to solve constrained optimization problems. The Lagrange relaxation method can be expressed as follows:  
\[L(w,b,\infty)=\frac{1}{2}\|w\|^{2}-\sum\limits_{i=1}^{l}\alpha_{i}\left\{y_{i}[ \left(w\cdot x_{i}\right)+b]-1\right\},\ \infty_{i}\geq 0 \tag{10}\]  
The solution to the constrained optimization problem is determined by the saddle point of the Lagrange function, and the solution to the optimization problem satisfies its partial derivative of \(w\) and \(b\) at the saddle point of 0. In addition, according to the dual property, the above optimal classification surface problem can be transformed into solving the maximum value of Equation (11) under the constraints of \(\infty_{i}\geq 0\) and \(\sum\limits_{i=1}^{n}\alpha_{i}\ y_{i}=0\ (i=1,\ 2,\ 3,\ldots,n;j=1,\ 2,\ 3, \ldots,n)\). \[\sum\limits_{i=1}^{n}\alpha_{i}-\frac{1}{2}\sum\limits_{i=1}^{n}\sum\limits_{ j=1}^{n}\alpha_{i}\infty_{j}\ y_{i}y_{j}\left(x_{i}\cdot x_{j}\right) \tag{11}\]  
The optimal solution of Equation (11) can be expressed as follows:  
\[\alpha^{*}=\left(\alpha_{1}^{*},\alpha_{2}^{*},\cdots,\alpha_{l}^{*}\right)^{T} \tag{12}\]  
When the optimal solution is obtained, the optimal weight vector \(w^{*}\) and the optimal bias \(b^{*}\) are, respectively:  
\[\left\{