## 4 Permeability Coefficient Inversion Based on LSSVM
### Training Effect and Inversion Predicted Values  
The training samples data were imported into LSSVM, PSO-LSSVM, and WOA-LSSVM for learning and training. Finally, Table 5 shows the penalty factor C and kernel parameter \(\sigma^{2}\) determined after the optimization by each algorithm. The parameters of the standard LSSVM model were obtained by cross-validation and grid search, whereas the other two methods used swarm intelligence algorithms for 50 iterations of optimization. The graph of the relationship between iteration number and best fitness (Figure 5) shows that the best fitness after the 25 th iteration was almost 0.0004 using the PSO optimization; with the WOA optimization, the fitness remained at approximately 0.0003 after only the 3 rd iteration. Thus, the WOA is more efficient for this LSSVM parameter optimization problem. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sample No.** & **S1 (m)** & **S3 (m)** & **S8 (m)** & **k1 (cm/s)** & **k2 (cm/s)** & **k3 (cm/s)** \\ \hline
1 & 156.8044 & 163.6996 & 182.0443 & \(1.1764\times 10^{-2}\) & \(7.8283\times 10^{-3}\) & \(6.4522\times 10^{-4}\) \\
2 & 155.3094 & 161.1166 & 177.1096 & \(1.2511\times 10^{-2}\) & \(7.6024\times 10^{-3}\) & \(4.2058\times 10^{-4}\) \\
3 & 153.0786 & 156.3668 & 182.4155 & \(1.5508\times 10^{-2}\) & \(8.1640\times 10^{-3}\) & \(1.8685\times 10^{-4}\) \\
4 & 158.4942 & 170.7296 & 184.8027 & \(1.8487\times 10^{-2}\) & \(3.6028\times 10^{-3}\) & \(8.7967\times 10^{-4}\) \\
5 & 157.5185 & 168.5769 & 182.6820 & \(2.0629\times 10^{-2}\) & \(3.8116\times 10^{-3}\) & \(7.0408\times 10^{-4}\) \\
6 & 156.0721 & 163.2500 & 181.7591 & \(2.2267\times 10^{-2}\) & \(8.5654\times 10^{-3}\) & \(7.1884\times 10^{-4}\) \\
7 & 157.3836 & 168.2721 & 182.5879 & \(2.4760\times 10^{-2}\) & \(4.0420\times 10^{-3}\) & \(7.3432\times 10^{-4}\) \\
8 & 156.7038 & 165.9498 & 180.9569 & \(2.7730\times 10^{-2}\) & \(5.8726\times 10^{-3}\) & \(7.6078\times 10^{-4}\) \\
9 & 156.6105 & 165.4943 & 180.5551 & \(2.8513\times 10^{-2}\) & \(6.9004\times 10^{-3}\) & \(8.2702\times 10^{-4}\) \\
10 & 154.8529 & 161.2191 & 177.2296 & \(3.0916\times 10^{-2}\) & \(9.4636\times 10^{-3}\) & \(5.8060\times 10^{-4}\) \\
37 & 159.5657 & 175.4843 & 190.0068 & \(9.2359\times 10^{-2}\) & \(1.1962\times 10^{-3}\) & \(6.1696\times 10^{-4}\) \\
38 & 157.6833 & 170.5481 & 185.1042 & \(9.3799\times 10^{-2}\) & \(3.1825\times 10^{-3}\) & \(8.8813\times 10^{-4}\) \\
39 & 156.5085 & 168.0160 & 182.7083 & \(9.7588\times 10^{-2}\) & \(4.8241\times 10^{-3}\) & \(9.4834\times 10^{-4}\) \\
40 & 156.2111 & 167.6389 & 182.5911 & \(9.9811\times 10^{-2}\) & \(3.2608\times 10^{-3}\) & \(6.2821\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulated GWL and corresponding permeability coefficient sample data. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Optimal Parameters for LSSVM** & **LSSVM** & **PSO-LSSVM** & **WOA-LSSVM** \\ \hline Penalty Factor \(\mathcal{C}\) & 32 & 69 & 100 \\ Kernel Parameter \(\sigma^{2}\) & 0.0039062 & 0.0049889 & 0.0024701 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Optimal parameters for LSSVM based on the three methods. The relationship between the training values and predicted values of k1, k2, and k3 obtained through the PSO-LSSVM method is depicted in Figure 6, indicating excellent training performance. We used the coefficient of determination (R2) and the root mean square error (RMSE) as performance indicators [11] to evaluate the fitting effectiveness of sample data for the model. As shown in Table 6, the R2 values of the three methods were all very close to 1 and the RMSE values were relatively small, which indicates good fitting effects. Among them, the WOA-LSSVM prediction model achieved the best fitting effect.