超高阶重力场模型最小二乘快速实现

Fast Realization of Ultra-high-degree Geopotential Model by Improved Leastsquares Method

采用最小二乘方法解算超高阶重力场模型,不可避免会遇到大型矩阵的计算,直接求解是难以实现的。本文从重力场模型的基本观测方程出发,利用正余弦函数和面球谐因子的正交性,分析系数矩阵及法矩阵的特点,在法矩阵块对角化的基础上,利用系数矩阵求解法矩阵时"次m"递增的特点,对法矩阵求解方程进行约化、对Legendre函数的计算和存储方式进行了设计,结合缔合Legendre函数关于赤道的对称性,解决了大型矩阵存储及计算效率低下的难题,实现了超高阶重力场模型最小二乘方法的小存储、高效率的解算。通过试验模拟,改进后的方法相比传统块对角方法效率提高300倍,利用此方法可以在普通PC机上快速、高精度地解算2160阶次超高阶重力场模型,算法精度相比数值积分方法至少提高了5个数量级,并且在一定程度上可以评估原始观测数据的精度。

Using the least-squares method to solve the ultra-high-degree geopotential model is very difficult because this method needs to calculate massive data. In this paper, we first analyzed the characteristics of coefficient matrix and normal matrix based on the basic observation equation of geopotential model using the orthogonality of trigonometric function and surface coherence factor.Then,we reduced the matrix solution equation and designed the calculation and storage method of Legendre function based on normal matrix block diagonalization using the increment characteristic of ＂secondary M＂ during the process of solving normal matrix by coefficient matrix solution method.At last, the problems of storage and low computational efficiency of large matrix were solved by combining the equator symmetry characteristic of Legendre function,reduction of the matrix solution equation and design of the calculation and storage method of Legendre function.Through the experimental test,the improved method compared to traditional block diagonal method canimprove the efficiency by 300 times, by using this method we can solve ultra-high-degree geopotential model in ordinary PC, and the precision of the model has been improved by 5 orders of magnitude at least when compared with the numerical integral method.At the same time,the accuracy of the original data can be evaluated.