Journal of Geodesy

, Volume 90, Issue 9, pp 871–881 | Cite as

2D Fourier series representation of gravitational functionals in spherical coordinates

  • Khosro Ghobadi-FarEmail author
  • Mohammad Ali Sharifi
  • Nico Sneeuw
Original Article


2D Fourier series representation of a scalar field like gravitational potential is conventionally derived by making use of the Fourier series of the Legendre functions in the spherical harmonic representation. This representation has been employed so far only in the case of a scalar field or the functionals that are related to it through a radial derivative. This paper provides a unified scheme to represent any gravitational functional in terms of spherical coordinates using a 2D Fourier series representation. The 2D Fourier series representation for each individual point is derived by transforming the spherical harmonics from the geocentric Earth-fixed frame to a rotated frame so that its equator coincides with the local meridian plane of that point. In the obtained formulation, each functional is linked to the potential in the spectral domain using a spectral transfer. We provide the spectral transfers of the first-, second- and third-order gradients of the gravitational potential in the local north-oriented reference frame and also those of some functionals of frequent use in the physical geodesy. The obtained representation is verified numerically. Moreover, spherical harmonic analysis of anisotropic functionals and contribution analysis of the third-order gradient tensor are provided as two numerical examples to show the power of the formulation. In conclusion, the 2D Fourier series representation on the sphere is generalized to functionals of the potential. In addition, the set of the spectral transfers can be considered as a pocket guide that provides the spectral characteristics of the functionals. Therefore, it extends the so-called Meissl scheme.


2D Fourier series representation Representation coefficients Gravitational functionals Spectral transfers Meissl scheme 



Figure 2 of the paper was produced using the generic mapping tools (GMT) (Wessel and Smith 1998). Helpful comments of the three anonymous reviewers are acknowledged. The authors would like to also thank the editor-in-chief J. Kusche and responsible editor S. Bettadpur for handling the manuscript.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Khosro Ghobadi-Far
    • 1
    Email author
  • Mohammad Ali Sharifi
    • 1
    • 2
  • Nico Sneeuw
    • 1
    • 3
  1. 1.School of Surveying and Geospatial Engineering, University College of EngineeringUniversity of TehranTehranIran
  2. 2.Research Institute of Geoinformation Technology (RIGT)University College of EngineeringTehranIran
  3. 3.Institute of GeodesyUniversität StuttgartStuttgartGermany

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