## 3 Results
### Horizontal Distance Positioning Results Comparative Analysis
Figure 6: The two hierarchical methods compare the results at different numbers of clustering levels. In the subsequent analysis, we will undertake a comparative assessment of the horizontal distance localization results obtained through various methodologies. Figure 7 presents the outcomes derived from the enhanced k-means clustering hierarchical method employed in this study, while Figure 8 displays results from traditional k-means clustering, hierarchical clustering, gradient-based clustering, and equidistant tangent-based clustering methods. Figure 8: (**a**) Layered renderings based on traditional k-means clustering algorithm; (**b**) layered renderings based on hierarchical clustering algorithm; (**c**) layered method rendering based on gradient difference algorithm; (**d**) layered method rendering based on equidistant algorithm. Figure 7: Layered renderings of the method in this paper. These methodologies were applied to stratify previously acquired profiles of inverted sound velocity. Subsequently, we individually employed each of the five methods to perform quadratic curve fitting on the data. This multifold approach allowed us to discern nuanced patterns in the stratification results. By utilizing these techniques, we effectively stratified the inverted sound velocity profiles and subsequently applied each method for a secondary quadratic curve fitting of the data. Table 3 compares the differences in horizontal distance positioning accuracy under a 30\({}^{\circ}\) tilted transducer for the improved k-means clustering sound speed profile stratification method proposed in this paper, the traditional k-means clustering sound speed profile stratification, the hierarchical clustering sound speed profile stratification, the minimum gradient difference-based sound speed profile stratification, and the equidistant sound speed profile stratification. All these methods utilize the quadratic curve fitting method for intra-layer fitting. Using the error formula in Equation (10), the table illustrates the discrepancies in horizontal distance results between different stratification methods at various incident angles and the ground truth. Based on the Table 3, it can be observed that there are differences in the positioning accuracy of different stratification methods at different incident angles compared to the true values. In the case of a 30\({}^{\circ}\) tilted transducer, the improved k-means clustering sound speed profile stratification method proposed in this paper exhibits relatively better horizontal distance positioning accuracy, while the accuracy of the gradient-based sound speed profile stratification method is lower. The positioning accuracy of the hierarchical clustering method, traditional k-means clustering, and the equidistant-based sound speed profile stratification methods falls in between. To assess the method's applicability under varying water depths, it is essential to test its performance at different depths. Water depth can influence the speed of sound propagation; hence, the method's performance was experimentally compared at different depths. Considering the complexity of underwater environments, including factors like ocean currents and temperature variations, it is necessary to investigate the method's robustness and stability under diverse environmental conditions. Therefore, the following data comprise four sets retrieved from Argo: 29 October 2003, at coordinates \"23.171,129.430\"; 21 October 2008, at coordinates \"19.593,121.142\"; 21 September 2012, at coordinates \"11.415,138.064\"; and 2 January 2023, at coordinates \"23.007,133.196\".