## 3 Results
### Horizontal Distance Positioning Results Comparative Analysis  
Using an incident angle of 45\({}^{\circ}\) as an illustration, Table 2 provides a comparison of the localization performance between the improved k-means clustering method and the traditional k-means clustering method at various layer numbers. The chosen number of layers, K, is determined as \(\lfloor T\rfloor\times i\%,i=10,\,20,\ldots 80\), where \(\lfloor T\rfloor\) represents the floor of the length of dataset T.  
The comparison results between the two stratification methods at different clustering layers are illustrated in Figure 6. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Incidence Angle (\(\alpha\))** & **35\({}^{\circ}\)** & **40\({}^{\circ}\)** & **45\({}^{\circ}\)** & **50\({}^{\circ}\)** & **55\({}^{\circ}\)** & **60\({}^{\circ}\)** \\ \hline R(m) & & 2732.32 & 2274.36 & 1913.97 & 1611.33 & 1348.49 & 1114.48 \\ \hline Sound line correction based on & r & 2705.23 & 2255.44 & 1900.36 & 1601.33 & 1341.04 & 1108.91 \\ quadratic fitting & \(e_{r}\) & 0.9\% & 0.8\% & 0.7\% & 0.6\% & 0.5\% & 0.4\% \\ \hline Sound ray correction based on constant & r & 2426.60 & 2032.12 & 1731.83 & 1471.97 & 1242.57 & 1036.26 \\ sound velocity method & \(e_{r}\) & 11.1\% & 10.6\% & 9.5\% & 8.6\% & 7.8\% & 7.0\% \\ \hline Sound ray correction based on constant & r & 2378.87 & 2026.69 & 1727.91 & 1469.08 & 1240.42 & 1034.65 \\ gradient method & \(e_{r}\) & 12.9\% & 10.8\% & 9.7\% & 8.8\% & 8.0\% & 7.1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Horizontal distance positioning effects of three sound ray correction methods. Figure 5: Error diagram of three sound ray correction methods. When the number of clustering layers is relatively low, there is a potential risk of neglecting or losing detailed and complex information. This may result in the localization algorithm's inability to accurately capture the features and variations in the underwater environment, consequently reducing localization accuracy. As the number of layers increases, localization accuracy may reach an optimal level. This is because an increased number of layers can better capture subtle changes and features in the underwater environment, providing more accurate localization information. However, as the number of layers continues to increase, it may introduce more noise and unnecessary complexity, thereby diminishing localization accuracy. The sound velocity profile in the underwater environment is a crucial factor in localization. With an increase in the number of layers for sound velocity profile stratification, it becomes possible to model and represent changes in the sound velocity profile more accurately. However, an excessive number of layers may introduce too much noise and unnecessary complexity, thereby affecting localization accuracy. Therefore, in clustering-based stratification methods, there is a need to strike a balance in the selection of the number of layers to achieve optimal localization accuracy. Considering that higher layer numbers increase computational complexity, leading to longer runtimes and decreased computational efficiency, based on the comprehensive evaluation depicted in Figure 6, this study opts for a layer number of K=10 for subsequent experimental analysis and research. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**K** & **5** & **9** & **13** & **17** & **21** & **26** & **30** & **34** \\ \hline \(r_{1}\) & 1871.92 & 1890.65 & 1911.68 & 1927.55 & 1946.70 & 1969.76 & 1978.29 & 1996.15 \\ \(e_{1}\) & 2.2\% & 1.2\% & 0.1\% & 0.7\% & 1.7\% & 2.9\% & 3.3\% & 4.2\% \\ \(t_{1}\) & 0.389 & 0.410 & 0.440 & 0.485 & 0.526 & 0.579 & 0.610 & 0.707 \\ \hline \(r_{2}\) & 1873.88 & 1891.81 & 1909.36 & 1927.14 & 1948.45 & 1971.15 & 1992.93 & 2007.38 \\ \(e_{2}\) & 2.1\% & 1.2\% & 0.2\% & 0.7\% & 1.8\% & 3.0\% & 4.1\% & 4.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison results between the improved method and the traditional method.