球諧分析衛星加速度求定地球重力場

Abstract

本文利用衛星之三度空間加速度資料來求定地球之重力場，藉由衛星所感受之擾動加速度，透過球諧分析求定地球之重力場。由於衛星加速度可以利用引力位的梯度來表示，則引入球諧函數及其梯度的正交性質，吾人可以分別推得兩種形式之球諧分析公式。公式推導之過程中並以不連續公式趨近積分計算，且搭配快速傅利葉轉換的運算技巧使用，可快速的分析求得地位係數。選用EGM96之地位係數展開至70階，模擬德國衛星CHAMP（高度460km）在太空中運行7天的加速度觀測資料，並利用3D內插程式將散佈之衛星觀測資料內插至相同半徑之球面網格資料，以此網格資料進行球諧分析求得地位係數；此反算之結果可回復地位係數至30階。最後加入一些觀測量之隨機誤差進行試驗，則使得球諧分析加速度的結果只能達到15~25階的效果。

In this study, we determined the geopotential coefficients by 3-D accelerations of a satellite. The perturbation accelerations of a satellite can be determined by using the tracking data of GPS. The gravitational field can then be determined using spherical harmonic analyses from the perturbation accelerations of satellite. Since the acceleration of the satellite can be represented by the gradient of the geopotential, two spherical analytical formulas were derived from the orthogonality properties of the spherical function and its gradient. In deriving the formulas, the integrals are approximated by discrete summation. The technique of Fast Fourier Transform is also employed to efficiently compute geopotential coefficients. By using EGM96 geopotential coefficients to degree 70, we made a 7-day simulated accelerations data of CHAMP (altitude of 460km), and interpolated the scattered data to grided points on a regular grid on a spherical surface. We used the grided data to determine the geopotential coefficients by spherical harmonic analysis, and the achievable maximum degree is 30. Adding random errors to the simulated accelerations results in a lower achievable maximum of 15 to 25.