## 2 Research Content and Methodology
### Methodology
#### 2.2.1 Precision Measurement Optimization Model
When the survey vessel is positioned at the center of the sea area, the relationship between the coverage width \(W\) and the angle \(\beta\) can be derived using the sine theorem and trigonometric functions. For a detailed derivation process, see Appendix A.1:  
\[\mu=\arccos(\frac{\sin\alpha\sin\beta}{\sqrt{(\cos\alpha\sin\beta)^{2}+(\cos \alpha\cos\beta)^{2}+(\sin\alpha\sin\beta)^{2}}}) \tag{1}\]  
\[W=\frac{D\cdot\sin\left(\frac{\theta}{2}\right)}{\sin\left(180^{\circ}-\frac{ \theta}{2}-\mu\right)}+\frac{D\cdot\sin\left(\frac{\theta}{2}\right)}{\sin\left( 180^{\circ}-\frac{\theta}{2}-(180^{\circ}-\mu)\right)} \tag{2}\]  
Here, \(\mu\) represents the angle between the center beam's line and the line along which the coverage width \(W\) is measured. By applying Equations (1) and (2), we find that the survey line's coverage width \(W\) is significantly influenced by \(\beta\). As shown in Figure 6, when \(\beta=90^{\circ}\), the line representing the coverage width coincides precisely with the major axis of the ellipse, which is the longest segment within the ellipse [23]. Therefore, when considering the direction of the survey lines, choosing \(\beta=90^{\circ}\)  
Figure 5: A spatial Cartesian coordinate system is established with the intersection point E of the survey vessel’s emitted multibeam’s symmetrical axis and the seabed slope as the origin. The horizontal plane of the coordinate system is the XOY plane, where the x-axis and y-axis are parallel to the length and width of the horizontal plane, respectively. The z-axis represents the central beam axis, indicating the vertical direction of the survey vessel. The red line segment represents the coverage width \(W\) of the MBES system, while the green rays show the survey line direction. As an inclined smooth plane, the seabed slope has its normal projected onto the XOY plane, forming an angle \(\beta\), with a slope \(\alpha\). The figure also marks the water depth \(D\) at the center of the sea area and the opening angle \(\theta\) of the multibeam transducer. can achieve the maximum coverage width, which is crucial for ensuring measurement efficiency at a given depth. After determining the initial paths and directions of the survey lines, we proceed to the final solution phase. By employing trigonometric functions, we can precisely calculate the MBES system's coverage width and overlap rate at various depths. In the design, we set a target overlap rate of 10%, aimed at maximizing the coverage area while reducing the total length of the survey lines, thereby ensuring the cost-effectiveness and efficiency of the measurement work. Based on the requirement that the coverage area must meet or exceed the total area of the marine region, we used Formulas (1) and (2) to calculate various parameters, as shown in the survey line layout projection in Figure 7. Through these calculations, we have obtained the final results, which indicate that under the given sea area conditions, the optimal number of survey lines can be determined precisely through the following set of equations:  
\[\left\{ \begin{aligned} x=D_{x}\cdot\tan\biggl{(}\frac{ \theta}{2}\biggr{)}\\ \frac{D_{x}-D_{1}}{x}=\tan\alpha\\ \frac{D_{i}-D_{i+1}}{d_{i}}=\tan\alpha\\ a=\frac{180^{\circ}-\theta}{2}-\alpha\\ b=180^{\circ}-\theta-a\\ W_{i}=D_{i}\cdot\sin\biggl{(}\frac{\theta}{2}\biggr{)}\biggl{(} \frac{1}{\sin a}+\frac{1}{\sin b}\biggr{)}\\ d_{i}^{\prime}=\frac{d_{i}\cdot\sin\Bigl{(}\frac{180^{\circ}- \theta}{2}\Bigr{)}}{\sin\Bigl{(}180^{\circ}-\frac{180^{\circ}-\theta}{2}- \alpha\Bigr{)}}\\ W_{1}+\sum_{i=1}^{n}d_{i}^{\prime}\geq\frac{h}{\cos\alpha}\end{aligned} \right. \tag{3}\]  
For more details, see Appendix A.2. Using the PMOM model-designed survey lines, the most efficient and accurate measurement plan can be obtained for such marine areas. Figure 6: The illustration represents the geometric diagram of the seabed coverage area by an MBES system.