| 椭球谐和球谐系数之间一个简单的转换关系 |
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| 引用本文： | 梁磊,于锦海,万晓云.椭球谐和球谐系数之间一个简单的转换关系[J].测绘学报,2019,48(2):185-190. |
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| 作者姓名： | 梁磊 于锦海 万晓云 |
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| 作者单位： | 中国科学院大学地球与行星科学学院,北京 100049;中国科学院计算地球动力学重点实验室,北京100049;中国地质大学(北京)土地科学技术学院,北京,100083 |
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| 基金项目： | 国家重点研发计划（2016YFB0501702）；国家自然科学基金（41774089；41504018；41674026）；CAS/CAFEA国际创新团队项目（KZZD-EW-TZ-19） |
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| 摘 要： | 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。
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| 关 键 词： | 球谐系数 椭球谐系数 第二类Legendre函数 椭球改正 Laplace方程 |
| 收稿时间： | 2018-05-22 |
| 修稿时间： | 2018-09-14 |
A simple transformation between ellipsoidal harmonic coefficients and spherical harmonic coefficients |
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| LIANG Lei,YU Jinhai,WAN Xiaoyun.A simple transformation between ellipsoidal harmonic coefficients and spherical harmonic coefficients[J].Acta Geodaetica et Cartographica Sinica,2019,48(2):185-190. |
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| Authors: | LIANG Lei YU Jinhai WAN Xiaoyun |
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| Institution: | 1. College of Earth and Planetary Science, University of Chinese Academy of Sciences, Beijing 100049, China;
2. Key Laboratory of Computational Geodynamics, Chinese Academy of Sciences, Beijing 100049, China;
3. School of Land Science and Technology, China University of Geosciences(Beijing), Beijing 100083, China |
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| Abstract: | 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 literature2] 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. |
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