大地测量计算机代数分析及可视化研究

There are many complicated and fussy mathematical analysis processes in geodesy, such as the power series expansions of the ellipsoid’s eccentricity, high order derivation of complex and implicit functions, operation of trigonometric function, expansions of special functions and integral transformation. Traditional algorithms derived by hand mainly have following problems: (1) The expressions are complex and lengthy, which makes the computation process very complicated and time-consuming. (2) Some approximate disposal is adopted, which influences the computation accuracy.(3) Some formulas are numerical and only apply to a specific reference ellipsoid, which are not convenient to be generalized. Taking all kinds of complex mathematical analysis processes in geodesy, the computer algebra analysis and visualization of geodesy are systematically carried out to bread, deep and detailed extent with the help of computer algebra analysis method and the powerful ability of mathematical analysis of computer algebra system. The new algorithms and mathematical models in symbolic form are established, which have more concise form, stricter theory basis and higher accuracy compared to traditional ones. The rules and characteristics hiding in all kinds of complicated mathematical models and massive data of geodesy. The breakthrough and innovation of some mathematical analysis problems in the special field of geodesy can be realized, which will further enrich and perfect the theoretical system of geodesy. The research results could be widely applied in such fields as geographic information system, photogrammetry, remote sensing, cartography, navigation, and etc.

大地测量涉及大量的椭球偏心率幂级数展开、隐函数高阶导数求取、三角函数运算、特殊函数展开、积分变换等复杂繁琐的数学分析过程，传统算法主要依靠人工推导完成，存在以下问题：（1）表达式复杂冗长，计算繁琐费时；（2）存在一定的近似处理，影响了计算精度；（3）有些公式表现为适用于特定参考椭球的数值形式，不便于推广使用。本项目以大地测量各种复杂数学分析过程为研究对象，利用计算机代数分析方法，借助计算机代数系统强大的数学分析能力，从广度、深度和精细程度上系统开展大地测量计算机代数分析及可视化研究，推导和建立理论上更为严密、形式上更为简单、精度上更为精确的符号化的算法模型，揭示大地测量各类复杂数学模型和海量数据背后隐藏的规律，实现大地测量特定数学分析问题的突破和创新，进一步丰富和完善大地测量学的理论体系。研究成果可广泛应用于地理信息系统、摄影测量、遥感、地图制图、导航等领域。

传统大地测量数学分析过程主要依靠人工推导完成，存在表达式复杂冗长、近似处理、数值公式不便于推广使用等问题。本项目利用计算机代数分析方法，深入开展了几何大地测量、物理大地测量、卫星大地测量以及三轴分层地球自转和地震简正模理论计算机代数分析研究。主要成果有：（1）推导了椭球各纬度间正反解符号表达式、高斯投影复变函数表达式、圆柱投影和圆锥投影直接变换公式等符号化新公式，构建了斜轴高斯投影、极区不分带高斯投影等算法模型，为长线工程建设和极区海图编制提供了理论基础；（2）围绕一定密度分布质体/天体外部引力场球谐、椭球谐或圆谐谱表达，研究了常密度、线性变化密度和一般多项式密度分布多面体模型的引力场球谐级数展开，二维质体引力场圆谐级数展开，以及天体/质体外部引力场椭球谐级数展开等计算方法，完善了球谐域正演方法；（3）提出了地形引力曲率和各地层引力曲率的概念，建立了笛卡尔坐标积分核的引力曲率模型，推导了球坐标下tesseroid块体的优化引力曲率的3D和2D积分公式，基于GPS、InSAR和GRACE等时变重力场观测数据，分析和解释了青藏高原3D地壳形变、尼泊尔区域垂直地壳构造运动和华北地区地表沉降等问题，丰富和拓展了物理大地测量理论体系；（4）提出了基于卫星双向时间频率传递确定重力位、基于GNSS共视法确定高程差和重力位和利用GNSS IPPP技术测定重力位等一系列新的重力位获取方法，推动了大地测量与卫星导航学科交叉融合发展；（5）构建了核幔边界地形耦合的参数化模型，研究了地形耦合对地球自转本征模的周期和耗散系数的影响，完善了三轴三层地球自转理论，提出了基于宽频带耦合的最优序列估计法，克服了传统迭积剥离法在高频高阶简正模探测上的明显劣势。累计发表学术论文160篇，SCI/EI收录78篇，出版专著7部，相关成果获国家科技进步一等奖和湖北省科技进步二等奖等奖励7项。