Non-singular Formulas for Computing Gravity Vector and Vertical Deviation

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.