﻿ 基于双精度与四精度的重力场解算精度分析
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 大地测量与地球动力学  2020, Vol. 40 Issue (1): 94-97, 110  DOI: 10.14075/j.jgg.2020.01.018

### 引用本文

ZHU Yongchao, WAN Xiaoyun, ZHOU Baoxing. Accuracy Analysis of the Double Precision and Quadruple Precision Gravity Field Solution[J]. Journal of Geodesy and Geodynamics, 2020, 40(1): 94-97, 110.

### Foundation support

Doctoral Scientific Research Foundation of Shandong Jiaotong University, No.50004934; Key Research and Development Program of Shandong Province, No.2019GGX101043; National Natural Science Foundation of China, No. 41674026; Fundamental Research Funds for the Central Universities, No.2652018027.

### Corresponding author

WAN Xiaoyun, PhD, senior engineer, majors in satellite gravimetry, E-mail: wanxy@cugb.edu.cn.

### About the first author

ZHU Yongchao, PhD, engineer, majors in satellite gravimetry, E-mail: zyc_20081989@126.com.

### 文章历史

1. 山东交通学院交通土建工程学院，济南市海棠路5001号，250357;
2. 中国地质大学(北京)土地科学技术学院，北京市学院路83号，100083

1 原理

 $\begin{array}{l} U = \frac{{GM}}{r}{\sum\limits_{n = 0}^\infty {\left( {\frac{R}{r}} \right)} ^n}\sum\limits_{m = 0}^n {({{\overline C }_{nm}}\cos m\lambda + } \\ \;\;\;\;\;\;{\overline S _{nm}}\sin m\lambda ){\overline P _{nm}}\left( {\cos \theta } \right) \end{array}$ (1)

 $\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\overline P _{n, m}} = {f_3}({f_4}{\text{cos}}\theta {\overline P _{n - 1, m}} - {f_5}{\overline P _{n - 2, m}}),\\ m < n - 1 \end{array}\\ \begin{array}{l} {\overline P _{n, n - 1}} = {f_2}{\text{cos}}\theta {\overline P _{n - 1, n - 1}}, \\ m = n - 1 \end{array}\\ \begin{array}{l} {\overline P _{n, n}} = {f_1}\sin \theta {\overline P _{n - 1, n - 1}}, \\ m = n \end{array} \end{array}} \right.$ (2)

 $\left\{ {\begin{array}{*{20}{c}} {{f_1} = \sqrt {\frac{{2n + 1}}{{2n}}} }\\ {{f_2} = \sqrt {2n + 1} }\\ {{f_3} = \sqrt {\frac{{2n + 1}}{{\left( {n - m} \right)\left( {n + m} \right)}}} }\\ {{f_4} = \sqrt {2n - 1} }\\ {{f_5} = \sqrt {\frac{{\left( {n - m - 1} \right)\left( {n + m - 1} \right)}}{{2n - 3}}} } \end{array}} \right.$ (3)

${\overline C _{nm}}$${\overline S _{nm}}$为完全正规化后的nm次引力位系数，其值通过地面重力或者卫星精密轨道进行反演计算。利用卫星精密轨道解算重力场球谐系数的计算方法主要有动力学积分法、短弧长积分法、能量法等，本文选用动力学积分法。动力学积分法的公式推导思路是基于牛顿运动方程将卫星轨道方程线性化，以此建立卫星精密轨道与球谐位系数的关系式，其微分方程可用式(4)表达，由此推导出动力学积分方法的基本公式(5)[14]

 $\left\{ {\begin{array}{*{20}{c}} {\frac{{{\rm{d}}\mathit{\boldsymbol{X}}}}{{{\rm{d}}t}} = F(t, \mathit{\boldsymbol{X}})}\\ {\mathit{\boldsymbol{X}}({t_0}) = {\mathit{\boldsymbol{X}}_0}} \end{array}} \right.$ (4)
 $\mathit{\boldsymbol{X}}(t) = {\mathit{\boldsymbol{X}}_{{\rm{ref}}}}(t) + \mathit{\boldsymbol{S}}(t)\delta \mathit{\boldsymbol{p}} + \mathit{\boldsymbol{Q}}(t)\delta \mathit{\boldsymbol{X}}({t_0})$ (5)

 $\begin{array}{l} \mathit{\boldsymbol{\dot \rho }}(t) = {{\mathit{\boldsymbol{\dot \rho }}}_0}(t) + {{\mathit{\boldsymbol{\dot Q}}}^{\rm{A}}}\left( t \right)\delta {\mathit{\boldsymbol{X}}^{\rm{A}}}\left( {{t_0}} \right) - \\ {{\mathit{\boldsymbol{\dot Q}}}^{\rm{B}}}\left( t \right)\delta {\mathit{\boldsymbol{X}}^{\rm{B}}}\left( {{t_0}} \right) + \left( {{{\mathit{\boldsymbol{\dot S}}}^{\rm{A}}}\left( t \right) - {{\mathit{\boldsymbol{\dot S}}}^{\rm{B}}}\left( t \right)} \right)\delta \mathit{\boldsymbol{p}} \end{array}$ (6)

2 算例 2.1 递推勒让德函数计算精度的比较

 $\begin{array}{l} {\varepsilon _n}\left( \theta \right) = \left| {2{{\left[ {{{\overline P }_n}\left( {{\text{cos}} \theta } \right)} \right]}^2} + } \right.\\ \left. {4{{\left[ {\sum\limits_{m = 1}^n {{{\overline P }_{n, m}}} ({\text{cos}})} \right]}^2} - \left( {2n + 1} \right)} \right| \end{array}$ (7)

 图 1 勒让德函数计算阶误差 Fig. 1 Degree errors of Legendre function computing

2.2 数值积分器在两种精度模式下的结果对比

 图 2 数值积分器计算误差 Fig. 2 Computational errors of numerical integrator
2.3 不同精度下解算重力场的对比

 图 3 不同精度下解算的重力场结果对比 Fig. 3 Comparison of the results of gravity field calculation with different accuracy

 图 4 不同精度下的累积大地水准面误差 Fig. 4 Accumulated geoid errors with different accuracy
3 结语

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Accuracy Analysis of the Double Precision and Quadruple Precision Gravity Field Solution
ZHU Yongchao1     WAN Xiaoyun2     ZHOU Baoxing1
1. School of Civil Engineering, Shandong Jiaotong University, 5001 Haitang Road, Jinan 250357, China;
2. School of Land Science and Technology, China University of Geosciences (Beijing), 83 Xueyuan Road, Beijing 100083, China
Abstract: In this paper, the precision of gravity field model under double precision and quadruple precision mode is compared and analyzed based on dynamics method, including the calculation of associated Legendre function, numerical integrator and gravity field inversion results. In terms of Legendre function calculation, part of the angle can be calculated in the double accuracy mode to the 1 900 order and overflow problems occur beyond this order. However, in quadruple precision mode any angle satisfies the accuracy requirement and the result is 8 orders of magnitude higher relative to the double precision mode. The quadruple precision model of numerical integrator Adams predictive correction method is 4 orders of magnitude higher than the double precision model. In terms of gravity field inversion, the inversion results of dynamic method are consistent in double and quadruple precision mode, and the accumulated geoid error calculating to order 60 is statistically 1.29×10-5 m, mainly because the linearized error of the dynamics method dominates the calculation error. The nonlinear dynamics method is 7 orders of magnitude higher than the double precision model in the quadruple precision model, and its accumulated geoid errors are respectively 8.92×10-15 m and 8.16×10-8 m.
Key words: quadruple precision; Legendre function; numerical integrator; dynamic integration method