A simple transformation between ellipsoidal harmonic coefficients and spherical harmonic coefficients

doi:10.11947/j.AGCS.2019.20180222

Acta Geodaetica et Cartographica Sinica ›› 2019, Vol. 48 ›› Issue (2): 185-190.doi: 10.11947/j.AGCS.2019.20180222

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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)

Abstract

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

CLC Number:

P223

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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A simple transformation between ellipsoidal harmonic coefficients and spherical harmonic coefficients

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