引用本文: 朱永超, 万晓云, 于锦海. 引力和垂线偏差的非奇异公式[J]. 武汉大学学报 ( 信息科学版), 2017, 42(12): 1854-1860.
ZHU Yongchao, WAN Xiaoyun, YU Jinhai. Non-singular Formulas for Computing Gravity Vector and Vertical Deviation[J]. Geomatics and Information Science of Wuhan University, 2017, 42(12): 1854-1860.
 Citation: ZHU Yongchao, WAN Xiaoyun, YU Jinhai. Non-singular Formulas for Computing Gravity Vector and Vertical Deviation[J]. Geomatics and Information Science of Wuhan University, 2017, 42(12): 1854-1860.

## Non-singular Formulas for Computing Gravity Vector and Vertical Deviation

• 摘要: 基于\frac\barP_nm\left( \cos \theta \right)\sin \theta \left( m>0 \right)的非奇异递推公式，给出了基于球坐标的引力矢量和垂线偏差非奇异计算公式；针对极点λ可任意取值引起的地方指北坐标系方向的不确定性问题，证明了引力矢量在转换到地心直角坐标系后不随λ的变化而变化，即与λ的取值无关。最终的数值计算结果表明，直角坐标系下的非奇异计算公式与本文提出的球坐标下的非奇异计算公式所得计算结果绝对值差异小于10-16m/s2，证明了该非奇异公式的正确性。最后总结了所有引力位球函数一阶导、二阶导非奇异性计算的一般思路。

Abstract: When computing gravity vector and vertical deviation using spherical harmonic function, singular problem exists in the formulas expressed by spherical coordinates. This will cause some errors in gravity vector and vertical deflection data and influence their application. This paper aims at proposing an alternative method to solve this problem. Based on the non-singular expression of \frac\barP_nm\left( \cos \theta \right)\sin \theta (m > 0), the paper gives the non-singular formulas expressed by spherical coordinates for computing gravity vector and vertical deviation. At North and South Poles, the paper proves that even values of λ are arbitrary, the values of gravity vector are sole when the values are transferred to Earth fixed rectangular coordinate system. In order to show the validity of our method, the paper computes gravity vectors at points θ=0, λ= \fraci360 2π(i=0, 1...359) using the former 100 degrees and orders of EGM2008. The absolute differences between the computing results by our method and the non-singular formula expressed in Cartesian coordinates are smaller than 10-16 m/s2, which show the validity of our method. The non-singular expression based on spherical function derived by the paper can make full use of the high accurate algorithms of Legendre function, so the proposed method has better generality ability compared with the non-singular formula expressed by Cartesian coordinates. Finally the methods for non-singular computing of all the first or second derivations of gravity field potential are summarized. The method of this paper can also be directly applied to the non-singular calculation of the spherical harmonic model of magnetic field, and the basic idea is similar to that of this paper.

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