基于Adams和KSG的短弧积分方程快速离散化方法

doi:10.11947/j.AGCS.2025.20240361

测绘学报 ›› 2025, Vol. 54 ›› Issue (8): 1416-1426.doi: 10.11947/j.AGCS.2025.20240361

基于Adams和KSG的短弧积分方程快速离散化方法

胡少彬¹(), 陈秋杰¹(), 沈云中¹, 张兴福²

^1.同济大学测绘与地理信息学院，上海　200092
^2.广东工业大学测绘工程系，广东　广州　510006

收稿日期:2024-09-01 修回日期:2025-06-27 出版日期:2025-09-16 发布日期:2025-09-16
通讯作者: 陈秋杰 E-mail:hu08180815@163.com;qiujiechen@tongji.edu.cn
作者简介:胡少彬（2001—），男，博士生，研究方向为重力卫星时变重力场解算。E-mail：hu08180815@163.com
基金资助:
国家自然科学基金(42174099);国家重点研发计划(2021YFB3900101)

An efficient discretization approach for the short-arc integral equation based on Adams and KSG integrators

Shaobin HU¹(), Qiujie CHEN¹(), Yunzhong SHEN¹, Xingfu ZHANG²

^1.College of Surveying and Geo-Informatics, Tongji University, Shanghai 200092, China
^2.Department of Surveying and Mapping, Guangdong University of Technology, Guangzhou 510006, China

Received:2024-09-01 Revised:2025-06-27 Online:2025-09-16 Published:2025-09-16
Contact: Qiujie CHEN E-mail:hu08180815@163.com;qiujiechen@tongji.edu.cn
About author:HU Shaobin (2001—), male, PhD candidate, majors in time-variable gravity field estimation for gravity satellites. E-mail: hu08180815@163.com
Supported by:
The National Natural Science Foundation of China(42174099);The National Key Research and Development Program of China(2021YFB3900101)

摘要/Abstract

摘要：

短弧积分法是卫星重力反演的一种常用方法，本质是基于边值条件的牛顿运动方程积分解法。鉴于Adams和KSG积分器分别是常用的一重和二重多步法积分器，本文提出了一种联合Adams和KSG的短弧积分公式的离散化方法，给出了积分离散化系数的计算公式，便于短弧积分方程的快速离散化求解。以GRACE-FO卫星仿真计算为例，分别从积分离散化系数计算、位置和速度向量积分运算、球谐系数偏导数求解，以及重力场反演等多个角度，与传统短弧积分法进行对比分析。结果表明：①两种方法求得的离散化系数矩阵特征十分吻合，位置和速度方程离散化系数矩阵差异的均方根（root mean square，RMS）量级分别为10^-9和10^-6，相比于传统方法，本文方法位置和速度方程的离散化系数矩阵求解效率分别提升了约80%和90%；②在相同弧长的情况下，本文方法相应的速度向量积分误差与传统方法相当，然而位置向量积分精度在较长弧段下略高于传统方法；③两种方法求得的位置和速度方程球谐系数偏导数总体上均吻合，由于高价信号能量小，高阶偏导数存在差异；④本文方法重力场反演结果与传统数值积分法精度相当，但解算效率提升了74.4%。

关键词: 短弧积分法, Adams, KSG, 离散化

Abstract:

The short-arc integral approach is a widely used technique for satellite-based gravity field recovery, which essentially provides an integral solution to Newton's equation of motion based on the boundary value condition. Considering that Adams and KSG integrators are multistep methods for single and double integrals respectively, this paper proposes a discretization approach for the short-arc integral formula by integrating both Adams and KSG integrators. Consequently, concise formulas have been derived to calculate the coefficients for discretizing the integral equations, thereby contributing to efficient discretization of the short-arc integral equations. Taking the simulation calculation of GRACE-FO satellites as a case study, the proposed approach is compared with the conventional short-arc integral approach from multiple perspectives: computation of discretization coefficients for integral equations, integration of position and velocity vectors, solution of partial derivatives with respect to spherical harmonic coefficients, and gravity field estimation. The results suggest that: ① The discretization coefficient matrices exhibit a high level of consistency between the two approaches, with the RMS (root mean square) of the differences for the position and velocity equations at the orders of 10^-9 and 10^-6, respectively. However, compared to the conventional approach, the proposed approach significantly enhances efficiency in calculating the discretization coefficient matrices by approximately 80% for position equation and 90% for velocity equation. ② The proposed approach exhibits comparable integration error for velocity vector to the conventional approach using the same arc length, while demonstrating slightly higher accuracy in position vector integration under longer arc lengths. ③ The partial derivatives of the position and velocity equations with respect to spherical harmonic coefficients obtained from both approaches are generally consistent; however, discrepancies arise at high degrees due to low signal energy at those degrees. ④ The accuracy of the resulting gravity field models remains comparable between the two approaches, while the proposed approach significantly enhances the efficiency of solving gravity field models by 74.4% compared with the conventional numerical integral approach.

Key words: short-arc integral approach, Adams, KSG, discretization

中图分类号:

P223

胡少彬, 陈秋杰, 沈云中, 张兴福. 基于Adams和KSG的短弧积分方程快速离散化方法[J]. 测绘学报, 2025, 54(8): 1416-1426.

Shaobin HU, Qiujie CHEN, Yunzhong SHEN, Xingfu ZHANG. An efficient discretization approach for the short-arc integral equation based on Adams and KSG integrators[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(8): 1416-1426.

图/表 9

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参考文献 42

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Kepler参数	TLE历时	轨道倾角/(°)	升交点赤经/(°)	轨道偏心率	近地点幅角/(°)	平近点角/(°)	平运动速度/(圈/d)
GRACE-FO(A)	19 350.78	89.00	162.51	0.001 5	243.05	116.92	15.24
GRACE-FO(B)	19 350.78	89.00	162.51	0.001 5	242.59	117.39	15.24

方程	指标	CM	IM
位置方程	Max	5.8×10^-5	5.8×10^-5
	Min	0	-9.0×10^-8
	RMS	3.8×10^-9
速度方程	Max	6.5×10^-4	4.1×10^-4
	Min	-6.5×10^-4	-3.2×10^-4
	RMS	2.1×10^-6

方程	标准差倍数	百分比
位置方程	1倍	99.49
	2倍	99.59
	3倍	99.62
速度方程	1倍	99.81
	2倍	99.84
	3倍	99.86

积分方法	位置和速度方程离散化系数矩阵求解时间
积分方法	1 h	2 h	3 h	4 h	5 h	6 h
CM-P	0.15	0.63	1.37	2.54	3.94	5.58
IM-P	0.02	0.09	0.19	0.33	0.51	0.89
提升百分比-P	86.7%	85.7%	86.1%	87.0%	87.1%	84.1%
CM-V	0.15	0.63	1.37	2.54	3.95	5.59
IM-V	0.01	0.05	0.10	0.17	0.27	0.47
提升百分比-V	93.3%	92.1%	92.7%	93.3%	93.2%	91.6%

弧长	IMk5-P/m	IMk7-P/m	IMk9-P/m	IMk11-P/m	IMk5-V/(m/s)	IMk7-V/(m/s)	IMk9-V/(m/s)	IMk11-V/(m/s)	CM-P/m	CM-V/(m/s)
1 h	5.455 3×10^-7	3.390 5×10^-7	3.390 3×10^-7	3.390 9×10^-7	6.074 2×10^-10	4.615 6×10^-10	4.615 8×10^-10	4.617 4×10^-10	3.395 7×10^-7	4.616 5×10^-10
2 h	6.326 3×10^-7	3.581 6×10^-7	3.581 3×10^-7	3.581 5×10^-7	6.214 1×10^-10	4.453 2×10^-10	4.453 4×10^-10	4.454 6×10^-10	3.608 9×10^-7	4.458 0×10^-10
3 h	6.812 6×10^-7	3.684 5×10^-7	3.684 1×10^-7	3.684 3×10^-7	6.241 1×10^-10	4.425 0×10^-10	4.425 2×10^-10	4.426 5×10^-10	3.737 4×10^-7	4.431 5×10^-10
4 h	6.145 2×10^-7	3.716 2×10^-7	3.715 9×10^-7	3.715 9×10^-7	6.224 4×10^-10	4.426 8×10^-10	4.427 3×10^-10	4.428 4×10^-10	3.807 3×10^-7	4.436 3×10^-10
5 h	6.746 3×10^-7	3.803 8×10^-7	3.804 1×10^-7	3.806 2×10^-7	6.245 3×10^-10	4.430 6×10^-10	4.430 8×10^-10	4.432 0×10^-10	3.937 1×10^-7	4.441 2×10^-10
6 h	6.821 1×10^-7	3.901 1×10^-7	3.901 6×10^-7	3.901 5×10^-7	6.248 3×10^-10	4.433 8×10^-10	4.434 0×10^-10	4.435 5×10^-10	4.102 5×10^-7	4.447 3×10^-10

基于Adams和KSG的短弧积分方程快速离散化方法

An efficient discretization approach for the short-arc integral equation based on Adams and KSG integrators

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