Impact of Di?erent Spherical Harmonic Coe?cients Structure on the Earth's Gravity Field Recovery
Yong Su*, Bing-xin Li, Xin-yu Xu
1. School of Civil Engineering and Geomatics, Southwest Petroleum University, Chengdu 610500, China
2. School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China
Abstract：
The global gravitational model can be expressed as a series of spherical harmonic coefficients computed up to a certain degree and order. The main ordering characteristic can depend on the degree, order, or type of coefficient. When determining the spherical harmonic coefficients using the least squares method, it is essential to analyze the ordering pattern of these coefficients and their positions (e.g., indices) in the vector or matrix. A systematic analysis on the type of coefficient arrangement is presented in this paper. Moreover, the index algorithm for each coefficient ordering pattern is provided. Additionally, the structure of the normal equation matrix with different coefficient arrangement patterns is analyzed. Based on the analysis and the algorithm presented in this paper: (1) we can calculate the index of each coeffi cient in the vector or matrix of the normal equation; (2) we can change the structure of the normal equation matrix of the Earth’s gravity fi eld from one type of coefficient arrangement to another. Furthermore, (3) we can directly combine different structures of the normal equation matrix, calculated from different types of gravity satellite missions, to form combined normal equations. This approach is beneficial for the determination of Earth’s gravitational model using multi-type observation data.
2025, Vol. 22 
