## 3 Results
### Sound Ray Correction Method Comparison  
As illustrated in Figure 4, the schematic diagram depicts three principles for handling sound speed profiles. In the diagram, \(z_{i}\) to \(z_{i+1}\) represent the i-th layer in the stratification, \(C_{0}\) is the original sound speed profile, \(C_{1}\) is the sound speed profile obtained from the secondary curve fitting method, and \(C_{2}\) is the sound speed profile obtained from the constant sound speed method. The constant sound speed method considers the sound speed profile in each layer as a constant sound speed region for sound speed profile processing. In this experiment, the constant sound speed adopted is the sound speed point where the center of each layer is located. Although this method is computationally simple, it can lead to significant positioning deviations when the number of layers is large. \(\mathbf{C_{3}}\) represents the sound speed profile obtained from the constant gradient method, which approximates the sound speed profile in each layer by directly connecting the sound speeds at the beginning and end of the profile layer. However, when faced with slightly complex profiles in the layers, this method may introduce substantial deviations.  
This study employs a sound ray correction method based on secondary curve fitting to more accurately fit the data of the sound speed profile, thereby improving the accuracy of sound ray positioning. In theory, when the sound speed profile is infinitely layered, the accuracy of the three methods is almost the same. However, in practical applications, it is necessary to ensure accuracy in positioning under certain computational complexity. Clearly, under the condition of an equal number of layers, the sound ray correction method based on secondary curve fitting is closer to the original sound speed profile, thereby enhancing the accuracy of sound ray positioning.  
Using the stratification method based on maximum density and maximum distance clustering, subsequent application of the secondary curve fitting sound ray correction method, the constant sound speed sound ray correction method, and the constant gradient sound ray correction method was performed. This was done to obtain horizontal distances for different incident angles (\(\mathbf{\alpha}\)) ranging from 35\({}^{\circ}\) to 60\({}^{\circ}\). The horizontal distance positioning results for the three sound ray correction methods were then calculated using the error formula from Equation (10), as shown in Table 1.  
\[e_{r}=\frac{|R-r|}{R}\times 100\% \tag{10}\]  
Here, \(e_{r}\) is the error, \(R\) is the true horizontal distance, and \(r\) is the horizontal distance obtained by different methods.  
Figure 4: Principle diagram of three sound velocity profile processing methods.  
The different errors in Table 1 are plotted using Matlab 9.10.0.1602886 (R2021a). The horizontal axis represents different incident angles, and the vertical axis represents the corresponding error values under different incident angles. The generated error comparison chart of the three sound ray correction methods is shown in Figure 5.  
From Figure 5, it can be observed that, compared to the constant sound speed ray correction method and the constant gradient ray correction method, the combination of the sound speed profile stratification method proposed in this paper and the quadratic curve fitting ray correction method results in smaller errors in horizontal distance positioning in the application scenario with a 30\({}^{\circ}\) tilted transducer.