## 1 Introduction
The paper emphasizes that there is no one-size-fits-all solution for underwater positioning and navigation, advocating for the selection of appropriate methods or the combination of multiple approaches based on specific underwater conditions to achieve reliable results. Ref. [18] introduces a fuzzy multi-sensor fusion method based on Correntropy to enhance the positioning performance of underwater vehicles. This method employs fuzzy logic to address nonlinear problems, utilizes Correntropy for handling non-Gaussian outliers, and incorporates Kalman filtering for real-time minimum error variance processing. In addressing the issue of drift in underwater estimation, Ref. [19] investigates the benefits of intermittently using high-precision ground-based sensors, such as conventional GPS, or assisting unmanned aerial vehicles tracking the underwater robot from above using a camera, to correct the position. Ref. [20] presents an underwater positioning system based on floating buoys and acoustic modulation-demodulation. The paper begins by describing the concept of the proposed system and provides equations enabling the underwater receiver to compute its position. The system's correct operation is verified through testing, and limitations are discussed. In certain situations, specific tasks may require tilting the transducer of a USBL system by 30\({}^{\circ}\). For instance, in underwater exploration or tasks performed by underwater robots, locating specific geological features such as underground rock layers or sediments may be necessary. In such cases, tilting the USBL system transducer changes the propagation path of the sound wave signal, aiding geologists or explorers in better understanding underwater geological features to support resource exploration or scientific research. Figure 1 illustrates the schematic diagram of this system, where \(\alpha\) represents the angle between the incident wave direction and the tilt angle of the transducer. Addressing the mentioned application scenario, this paper proposes an improved k-means clustering stratification method based on maximum density and maximum distance. The method utilizes k-means clustering based on maximum density and maximum distance for sound velocity profile stratification under reasonable layer numbers and obtains stratified sound velocity profiles through secondary fitting. Experimental results demonstrate that this method significantly improves positioning accuracy in the specific application scenario mentioned above. This method holds great potential for a wide range of applications in underwater positioning and acoustic exploration. For instance, when underwater robots engage in geological exploration tasks, tilting the transducer of the USBL system can enhance the accuracy of identifying underwater geological features by altering the acoustic propagation path. This proves particularly beneficial for resource exploration, marine scientific research, and the positioning and monitoring of underwater structures. The main content of this paper is outlined as follows: Section 2 analyzes an improved hierarchical method for k-means clustering based on maximum density and maximum distance, elucidating the design principles of the algorithm. In Section 3, a second-order fitting-based ray correction method is examined, experiments are conducted using Argo data, and the corresponding results are presented. Furthermore, the advantages and disadvantages of the proposed algorithm are analyzed, and a comparison is made with four different algorithms. Section 4 is dedicated to the discussion of the experimental results. Finally, Section 5 provides a summary of this paper.