椭球谐和球谐系数之间一个简单的转换关系

doi:10.11947/j.AGCS.2019.20180222

测绘学报 ›› 2019, Vol. 48 ›› Issue (2): 185-190.doi: 10.11947/j.AGCS.2019.20180222

椭球谐和球谐系数之间一个简单的转换关系

梁磊^1,2, 于锦海^1,2, 万晓云³

1. 中国科学院大学地球与行星科学学院, 北京 100049;
2. 中国科学院计算地球动力学重点实验室, 北京 100049;
3. 中国地质大学(北京)土地科学技术学院, 北京 100083

收稿日期:2018-05-22 修回日期:2018-09-14 出版日期:2019-02-20 发布日期:2019-03-02
通讯作者: 于锦海 E-mail:yujinhai@ucas.edu.cn
作者简介:梁磊(1990-),男,博士生,研究方向为物理大地测量、卫星轨道、卫星重力。E-mail:lianglei14@mails.ucas.edu.cn
基金资助:
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）

A simple transformation between ellipsoidal harmonic coefficients and spherical harmonic coefficients

LIANG Lei^1,2, YU Jinhai^1,2, WAN Xiaoyun³

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

Received:2018-05-22 Revised:2018-09-14 Online:2019-02-20 Published:2019-03-02
Supported by:
The State's Key Project of Research and Development Plan (No. 2016YFB0501702);The National Natural Science Foundation of China (Nos. 41774089;41504018;41674026);The Project of CAS/CAFEA International Partnership for Creative Research Teams (No. KZZD-EW-TZ-19)

摘要/Abstract

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ε²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ε²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。

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

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 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.

Key words: spherical harmonic coefficients, ellipsoidal harmonic coefficients, second Legendre function, ellipsoidal correction, Laplace equation

中图分类号:

P223

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.

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.

参考文献

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椭球谐和球谐系数之间一个简单的转换关系

A simple transformation between ellipsoidal harmonic coefficients and spherical harmonic coefficients

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