## 3 Results  
The speed of sound in seawater varies with temperature, salinity, and depth, and it is challenging to express their interdependencies analytically. Typically, empirical formulas are used to represent the relationships between them. Empirical formulas are summaries of a large number of experimental data on underwater sound speed measurements. In practice, seawater is usually measured for temperature (T), salinity (S), and pressure (P), and then empirical formulas are employed to obtain the sound speed (c). Pressure in seawater increases with depth, and the density (\(\rho\)) of seawater is generally around 1025 kg/m\({}^{3}\), while the gravitational acceleration (\(g\)) is approximately 9.81 m/s\({}^{2}\) (at the Earth's surface). According to \(P=\rho gh\), the relationship between seawater pressure and depth can be approximated as \(P\approx 10\) h, where h is the depth. The temperature, salinity, and depth information used in this study are sourced from the China Argo Real-time Data Center, comprising a total of 10 sets of data with depths ranging from 0 m to 2000 m. The [PERSON] formula [23] was employed to calculate underwater sound speed (in m/s), and the formula is as follows:  
\[v=1449.2+4.6T-0.055T^{2}+0.00029T^{3}+(1.34-0.01T)(S-35)+0.016D \tag{6}\]  
where \(v\) is the underwater sound speed, \(T\) is the water temperature (in degrees Celsius), \(S\) is the salinity (expressed in practical salinity units, PSU), and \(D\) is the depth (in meters). In practical situations, the function describing the variation of sound speed with depth in a marine area cannot be directly obtained. Researchers can provide an approximate sound speed profile through data processing and approximation methods. Empirical orthogonal function expansion can effectively approximate complex sound speed profiles using a limited number of expansion coefficients. The literature suggests that obtaining the first three to six orders of empirical orthogonal functions is sufficient for accurate sound speed profile inversion. In this study, the first five orders of empirical orthogonal functions were selected, with a cumulative variance contribution rate of 95.29%. By selecting appropriate orthogonal functions, the high-dimensional sound speed profile parameter space can be mapped to a low-dimensional expansion coefficient space, reducing the computational cost and complexity of sound speed profile calculations. As shown in Figure 2, the sound speed profile curves based on depth and velocity, calculated using the [PERSON] sound speed empirical formula with the original 10 sets of data, are displayed. Additionally, the inverted sound speed profile curves obtained through empirical orthogonal function inversion of the 10 datasets are depicted in the sound speed profile plot. After stratifying the inverted sound speed profile data using the method based on maximum density and maximum distance, a secondary curve fitting method [24] was applied to the stratified data to obtain a function representing sound speed as a function of depth. The horizontal propagation distance of each sound ray in each layer was calculated using numerical integration and accumulated. Finally, the total horizontal distance of the sound rays was obtained. As shown in Figure 3, where \(z_{0}\) is the depth of the sound source, p represents the target source, \(z_{i}\) to \(z_{i+1}\) represents the i-th layer in the stratification, and z is the depth of the target source. Within the interval between \(z_{i}\) and \(z_{i+1}\), a sufficiently small acoustic ray element \(ds\) is taken. Applying [PERSON]'s law of refraction, the horizontal trajectory equation for this segment of the acoustic ray is obtained [14]:  
\[dr_{i}=\frac{dz}{tan\alpha}=\frac{c(z_{i})cos\alpha_{0}}{\sqrt{c_{0}^{2}-cos \alpha_{0}{}^{2}c(z_{i})^{2}}}dz \tag{7}\]  
\[r_{i}=\int_{z_{i}}^{z_{i+1}}\frac{c(z_{i})cos\alpha_{0}}{\sqrt{c_{0}^{2}-cos \alpha_{0}{}^{2}c(z_{i})^{2}}}dz \tag{8}\]  
\[r=\sum_{i=1}^{k}r_{i} \tag{9}\]  
where \(\alpha_{0}\) is the initial grazing angle.