描述地球引力場中的球諧函數在許多理論及實際應用中相當重要，尤其是在全球大地測量領域，如引力位、大地水準面的決定，且通常定義在以地球為中心的地心地固坐標系統(Earth-centered Earth-fixed system, ECEF)上。近年來，衛星觀測數據如重力梯度資料等，提供了高精度且完整的資料供解算地球重力場，然而這些資料大多是在軌道坐標系統(Orbital system)上所觀測而得，為了使用這些衛星觀測資料來確定球諧函數係數，必須將這些資料由衛星軌道系統透過旋轉方法轉至地心地固坐標系統，因此，球諧函數的旋轉是一項重要的研究課題。
除此之外，Wigner-d函數被廣泛應用在處理球諧函數的旋轉問題中，而本研究即基於Wigner-d函數的應用下，透過Pseudospectral演算法將球諧函數旋轉至其他坐標系中，而由於這些旋轉方法皆須計算Wigner-d函數，故Wigner-d函數的解算亦影響了旋轉成果。在本研究中，透過Pseudospectral演算法實驗模擬旋轉成果。藉由本研究的成果顯示，球諧函數的旋轉藉由Pseudospectral演算法有快速且穩定的計算流程，並可得到精確的成果。
然而，Wigner-d函數的各項解算方法中，其中之一為遞迴函式，計算d函數時，包含了三個方向的遞迴關係，其解算精度亦受限於球諧展開階數，故本研究實驗成果中亦透過Risbo提出的三方向遞迴函式配合快速傅立葉(FFT)解算Wigner-d函數，並將此成果視為參考真值，再透過Gimbutas精度模型，來分析球諧展開階數對Pseudocpectral演算法之計算成果精度的影響。本實驗成果顯示了在相同旋轉角時，其相對精度在低階展開階數時的表現佳，另一方面，展開階數為固定時，計算成果在低旋轉角時的精度表現亦較佳，且符合文獻預期成果。

Spherical harmonics of the Earth’s gravitational field are of great importance in many theoretical and practical applications, particularly in Global Geodesy. Space measurements from satellite missions like gradiometric measurements from GOCE mission provide highly precise data for the determination of the Earth’s gravitational field. These highly precise data are measured in the satellite orbit. In order to use the data to determine the spherical harmonic coefficients, a rotation from the satellite orbital coordinate system to the Earth-centered Earth-fixed coordinate system is therefore necessary. Consequebtly, the spherical harmonic expression involves the rotation parameters which are expressed by the so-called Wigner-D function.

Furthermore, the Wigner-d function is widely used in the computation of the spherical harmonics, if the coordinate system is rotated. In this research, we simulate the rotations through Pseudospectral algorithm. The results of our simulation show that the rotation of spherical harmonics can be carried out fast and stably via the Pseudospectral algorithm.

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