基于多波束测线问题的数学建模分析
Mathematical Modeling Analysis of Multibeam Sounding Line Problem

本研究主要针对多波束测线在海洋测深任务中的应用进行探讨。多波束系统在海洋深度测量和海底地形研究中起着重要作用。研究中关注测线间隔、覆盖宽度和重叠率的设计。通过正弦定理和三角函数推导出海域中心点处的海水深度表达式，进而得到多波束测深的覆盖宽度和相邻条带之间的重叠率的数学模型。利用MATLAB软件对给定参数进行计算和模拟，得出不同测线距中心点处的距离时的海水深度、覆盖宽度和重叠率。进一步研究了测线方向和海底坡面法向之间的角度关系，将二维图形转化为三维图形，求解了坡度与测线方向的关系。在此基础上，通过穷举法确定最优测线布局，得出最短测线总距离为118,528米，设计出最优的测线布局。
This study mainly discusses the application of multibeam sounding lines in ocean depth measurement tasks. Multibeam systems play a crucial role in ocean depth measurement and seafloor topography research. The study focuses on the design of line spacing, coverage width, and overlap rate. By deriving the expression of water depth at the center of the sea area using the sine theorem and trigonometric functions, a mathematical model for the coverage width and overlap rate between adjacent bands in multibeam sounding is obtained. Using MATLAB software to calculate and simulate given parameters, the water depth, coverage width, and overlap rate at different distances from the central point of the sounding lines are determined. Further research is conducted on the angle relationship between the direction of the sounding lines and the normal to the seafloor slope, converting two-dimensional graphs into three-dimensional graphs to solve the relationship between slope and the direction of the sounding lines. Based on this, the optimal layout of sounding lines is determined through exhaustive enumeration, with the shortest total distance of sounding lines found to be 118,528 meters, resulting in the optimal layout of sounding lines being designed.