椭球谐和球谐系数之间一个简单的转换关系

本文推导的椭球谐系数和球谐系数相互之间转换关系的核心思想是在ε~2量级下利用Legendre函数的正交性,从球谐系数求解的积分表示出发,将积分中的椭球坐标变量与球坐标变量相互转换,从而得出椭球谐系数与球谐系数之间的转换关系。本文导出的转换关系有以下优点:①对于第二类Legendre函数的计算采用Laurent级数表示,使计算第二类Legendre函数更为简单;②保留了ε~2量级下,导出的转换关系相比文献[2]的形式更简单,满足物理大地测量边值问题线性化的要求;③顾及了余纬和归化余纬的区别。

A simple transformation between ellipsoidal harmonic coefficients and spherical harmonic coefficients

In this paper, the core idea of the conversion relationship between the ellipsoidal harmonic coefficients and the spherical harmonic coefficients is derived from the orthogonality of the Legendre function and using another coordinate variable replace the former coordinate variable in the integral expression of spherical harmonic coefficients or ellipsoidal harmonic coefficients. Then the conversion relationship between the spherical harmonic coefficient and the ellipsoidal harmonic coefficient is obtained. In addition, all the derivation of this paper is based on the squared magnitude of the ellipsoid flattening. From the conversion relationship between the ellipsoidal harmonic coefficient and the spherical harmonic coefficient, we can see that:①Using Laurent series to calculate the second type of Legendre function, it is more easier to calculate the second type of Legendre function; ②With the ε² magnitude preserved, the derived conversion relationship is simpler than the form of literature[2] and satisfies the requirements of linearization of the physical geodetic boundary value problem; ③The difference between colatitude and reduced latitude is considered and the result is more reasonable.