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

doi:10.11947/j.AGCS.2025.20240361

Abstract

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

CLC Number:

P223

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.

Figures/Tables 9

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References 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