﻿ 椭球谐和球谐系数之间一个简单的转换关系
 文章快速检索 高级检索

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

A simple transformation between ellipsoidal harmonic coefficients and spherical harmonic coefficients
LIANG Lei1,2 , YU Jinhai1,2 , 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
Foundation support: 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)
First author: LIANG Lei(1990-), male, PhD candidate, majors in physical geodesy, satellite orbits, satellite gravity.E-mail:lianglei14@mails.ucas.edu.cn
Corresponding author: YU Jinhai, E-mail: yujinhai@ucas.edu.cn
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

1 第二类Lengenre函数与Laurent展开式

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

2 椭球谐系数与球谐系数之间的转换关系

(9)

(10)

(11)

(12)

(13)
(14)

r=R时，u=b，根据式(12)的第3式，有

(15)
(16)

(17)
(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

3 算例

 图 1 阶方差比较 Fig. 1 Degree variances comparison

4 总结与分析

﻿

 [1] HEISKANEN W A, MORITZ H. Physical geodesy[M]. San Francisco: W. H. Freeman and Company, 1967. [2] JEKELI C. The exact transformation between ellipsoidal and spherical harmonic expansions[J]. Manuscripta Geodaetica, 1988, 13: 106–113. [3] CLAESSENS S J, FEATHERSTONE W E. The Meissl scheme for the geodetic ellipsoid[J]. Journal of Geodesy, 2008, 82(8): 513–522. DOI:10.1007/s00190-007-0200-y [4] CLAESSENS S J. Spherical harmonic analysis of a harmonic function given on a spheroid[J]. Geophysical Journal International, 2016, 206(1): 142–151. DOI:10.1093/gji/ggw126 [5] THONG N C, GRAFAREND E W. A spheroidal harmonic model of the terrestrial gravitational field[J]. Manuscripta Geodaetica, 1989, 14(5): 285–304. [6] SONA G. Numerical problems in the computation of ellipsoidal harmonics[J]. Journal of Geodesy, 1995, 70(1-2): 117–126. DOI:10.1007/BF00863423 [7] MARTINEC Z, GRAFAREND E W. Solution to the Stokes boundary-value problem on an ellipsoid of revolution[J]. Studia Geophysica et Geodaetica, 1997, 41(2): 103–129. DOI:10.1023/A:1023380427166 [8] GIL A, SEGURA J. A code to evaluate prolate and oblate spheroidal harmonics[J]. Computer Physics Communications, 1998, 108(2-3): 267–278. DOI:10.1016/S0010-4655(97)00126-4 [9] SEBERA J, BOUMAN J, BOSCH W. On computing ellipsoidal harmonics using Jekeli's renormalization[J]. Journal of Geodesy, 2012, 86(9): 713–726. DOI:10.1007/s00190-012-0549-4 [10] FUKUSHIMA T. Recursive computation of oblate spheroidal harmonics of the second kind and their first-, second-, and third-order derivatives[J]. Journal of Geodesy, 2013, 87(4): 303–309. DOI:10.1007/s00190-012-0599-7 [11] YU Jinghai, CAO Huasheng. Elliptical harmonic series and the original stokes problem with the boundary of the reference ellipsoid[J]. Journal of Geodesy, 1996, 70(7): 431–439. DOI:10.1007/BF01090818 [12] BUCHDAHL H A, BUCHDAHL N P, STILES P J. On a relation between spherical and spheroidal harmonics[J]. Journal of Physics A:Mathematical and General, 1977, 10(11): 1833–1836. DOI:10.1088/0305-4470/10/11/011 [13] DECHAMBRE D, SCHEERES D J. Transformation of spherical harmonic coefficients to ellipsoidal harmonic coefficients[J]. Astronomy & Astrophysics, 2002, 387(3): 1114–1122. [14] GLEASON D M. Comparing ellipsoidal corrections to the transformation between the geopotential's spherical and ellipsoidal spectrums[J]. Manuscripta Geodaetica, 1988, 13(2): 114–129. [15] HU Xuanyu, JEKELI C. A numerical comparison of spherical, spheroidal and ellipsoidal harmonic gravitational field models for small non-spherical bodies:examples for the Martian moons[J]. Journal of Geodesy, 2015, 89(2): 159–177. DOI:10.1007/s00190-014-0769-x [16] KONOPLIV A S, ASMAR S W, BILLS B G, et al. The Dawn gravity investigation at Vesta and Ceres[J]. Space Science Reviews, 2011, 163(1-4): 461–486. DOI:10.1007/s11214-011-9794-8 [17] MORITZ H. Advanced physical geodesy[M]. Karlsruhe: Herbert Wichmann, 1980. [18] PARK R S, KONOPLIV A S, ASMAR S W, et al. Gravity field expansion in ellipsoidal harmonic and polyhedral internal representations applied to Vesta[J]. Icarus, 2014, 240(6): 118–132. [19] PEARSON J. Computation of hypergeometric functions[D]. Oxford: University of Oxford, 2009. [20] SANSÒ F, TSCHERNING C C. Fast spherical collocation:theory and examples[J]. Journal of Geodesy, 2003, 77(1-2): 101–112. DOI:10.1007/s00190-002-0310-5 [21] VERSHKOV A N. Determination of the spherical harmonic coefficients from the ellipsoidal harmonic coefficients of the Earth's external potential[J]. Artificial Satellites, 2002, 37(4): 157–168. [22] WALTER H G. Association of spherical and ellipsoidal gravity coefficients of the Earth's potential[J]. Celestial Mechanics, 1970, 2(3): 389–397. DOI:10.1007/BF01235139 [23] HU Xuanyu. The exact transformation from spherical harmonic to ellipsoidal harmonic coefficients for gravitational field modeling[J]. Celestial Mechanics and Dynamical Astronomy, 2016, 125(2): 195–222. DOI:10.1007/s10569-016-9678-z [24] 于锦海, 曾艳艳, 朱永超, 等. 超高阶次Legendre函数的跨阶数递推算法[J]. 地球物理学报, 2015, 58(3): 748–755. YU Jinhai, ZENG Yanyan, ZHU Yongchao, et al. A recursion arithmetic formula for Legendre functions of ultra-high degree and order on every other degrees[J]. Chinese Journal of Geophysics, 2015, 58(3): 748–755. [25] 于锦海. 地球重力场椭球谐模型的建立[J]. 解放军测绘学院学报, 1994(4): 309–317. YU Jinhai. Elliptical harmonic model about the Earth's gravity field[J]. Journal of the PLA Institute of Surveying and Mapping, 1994(4): 309–317. [26] 张传定. 大地测量应用卫星的轨道设计——椭球谐引力场下卫星的运动[J]. 测绘学报, 2000, 29(z1): 80–85. ZHANG Chuanding. Orbital design of satellite for geodetic applications[J]. Acta Geodaetica et Cartographica Sinica, 2000, 29(z1): 80–85. DOI:10.3321/j.issn:1001-1595.2000.z1.017 [27] 于锦海. O(T2)精度下椭球界面Dirichlet边值问题的积分解[J]. 地球物理学报, 2004, 47(1): 75–80. YU Jinhai. The integral solution of the Dirichlet's boundary value problem on the ellipsoid interface with the accuracy of O(T2)[J]. Chinese Journal of Geophysics, 2004, 47(1): 75–80. DOI:10.3321/j.issn:0001-5733.2004.01.011 [28] YU Jinghai, WU Xiaoping. The solution of mixed boundary value problems with the reference ellipsoid as boundary[J]. Journal of Geodesy, 1997, 71(8): 454–460. DOI:10.1007/s001900050113
http://dx.doi.org/10.11947/j.AGCS.2019.20180222

0

#### 文章信息

LIANG Lei, YU Jinhai, WAN Xiaoyun

A simple transformation between ellipsoidal harmonic coefficients and spherical harmonic coefficients

Acta Geodaetica et Cartographica Sinica, 2019, 48(2): 185-190
http://dx.doi.org/10.11947/j.AGCS.2019.20180222