Journal of Geodesy

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

2D Fourier series representation of gravitational functionals in spherical coordinates

  • Khosro Ghobadi-Far
  • Mohammad Ali Sharifi
  • Nico Sneeuw
Original Article

Abstract

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.

Keywords

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

Notes

Acknowledgments

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.

References

  1. Albertella A, Sacerdote F (1995) Spectral analysis of block averaged data in geopotential global model determination. J Geod 70(3):166–175. doi: 10.1007/BF00943692 CrossRefGoogle Scholar
  2. Antoni M, Keller W (2013) Closed solution of the Hill differential equation for short arcs and a local mass anomaly in the central body. Celest Mech Dyn Astron 115(2):107–121. doi: 10.1007/s10569-012-9454-7 CrossRefGoogle Scholar
  3. Betti B, Sansò F (1989) The integrated approach to satellite geodesy. In: Sansò F, Rummel R (eds) Lecture notes in earth sciences. Theory of satellite geodesy and gravity field determination, vol 25. Springer, Berlin, pp 373–416. doi: 10.1007/BFb0010557
  4. Brovelli M, Sansò F (1990) Gradiometry: the study of the \({V}_{yy}\) component in the BVP approach. Manuscr Geod 15(4):240–248Google Scholar
  5. Casotto S, Fantino E (2009) Gravitational gradients by tensor analysis with application to spherical coordinates. J Geod 83(7):621–634. doi: 10.1007/s00190-008-0276-z CrossRefGoogle Scholar
  6. Cheong HB, Park JR, Kang HG (2012) Fourier-series representation and projection of spherical harmonic functions. J Geod 86(11):975–990. doi: 10.1007/s00190-012-0558-3 CrossRefGoogle Scholar
  7. Colombo OL (1981) Numerical methods for harmonic analysis on the sphere. Tech. Rep. No. 310, Ohio State University, ColumbusGoogle Scholar
  8. Colombo OL (1989) Advanced techniques for high-resolution mapping of the gravitational field. In: Sansò F, Rummel R (eds) Lecture notes in earth sciences. Theory of satellite geodesy and gravity field determination, vol 25. Springer, Berlin, pp 335–369Google Scholar
  9. Dilts GA (1985) Computation of spherical harmonic expansion coefficients via FFT’s. J Comput Phys 57(3):439–453. doi: 10.1016/0021-9991(85)90189-5 CrossRefGoogle Scholar
  10. EGG-C (2010) GOCE Level 2 product data handbook. Issue 4, GO-MA-HPF-GS-0110Google Scholar
  11. Fantino E, Casotto S (2009) Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients. J Geod 83(7):595–619. doi: 10.1007/s00190-008-0275-0 CrossRefGoogle Scholar
  12. Giacaglia GEO (1980) Transformations of spherical harmonics and applications to geodesy and satellite theory. Stud Geophys Geod 24(1):1–11. doi: 10.1007/BF01628375 CrossRefGoogle Scholar
  13. Gooding RH, Wagner CA (2008) On the inclination functions and a rapid stable procedure for their evaluation together with derivatives. Celest Mech Dyn Astron 101(3):247–272. doi: 10.1007/s10569-008-9145-6 CrossRefGoogle Scholar
  14. Gooding RH, Wagner CA (2010) On a Fortran procedure for rotating spherical-harmonic coefficients. Celest Mech Dyn Astron 108(1):95–106. doi: 10.1007/s10569-010-9293-3 CrossRefGoogle Scholar
  15. Gruber C, Abrykosov O (2014) High resolution spherical and ellipsoidal harmonic expansions by fast Fourier transform. Stud Geophys Geod 58(4):595–608. doi: 10.1007/s11200-013-0578-3 CrossRefGoogle Scholar
  16. Gruber C, Novák P, Sebera J (2011) FFT-based high-performance spherical harmonic transformation. Stud Geophys Geod 55(3):489–500. doi: 10.1007/s11200-011-0029-y CrossRefGoogle Scholar
  17. Gruber C, Novák P, Flechtner F, Barthelmes F (2014) Derivation of the topographic potential from global DEM models. In: Rizos C, Willis P (eds) International association of geodesy symposia, earth on the edge: science for a sustainable planet, vol 139. Springer, Berlin, pp 535–542. doi: 10.1007/978-3-642-37222-3_71
  18. Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman and Company, San FranciscoGoogle Scholar
  19. Hofsommer D, Potters M (1960) Table of Fourier coefficients of associated Legendre functions. Proc KNAW Ser A Math Sci 63(5):460Google Scholar
  20. Holmes SA, Featherstone WE (2002) A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions. J Geod 76(5):279–299. doi: 10.1007/s00190-002-0216-2 CrossRefGoogle Scholar
  21. Hwang C (1991) Orthogonal functions over the oceans and applications to the determination of orbit error, geoid and sea suface topography from satellite altimetry. Tech. Rep. No. 414, Ohio State University, ColumbusGoogle Scholar
  22. Jeffreys B (1965) Transformation of tesseral harmonics under rotation. Geophys J Int 10(2):141–145. doi: 10.1111/j.1365-246X.1965.tb03057.x CrossRefGoogle Scholar
  23. Jekeli C (1996) Spherical harmonic analysis, aliasing, and filtering. J Geod 70(4):214–223. doi: 10.1007/BF00873702 CrossRefGoogle Scholar
  24. Kaula WM (1966) Theory of satellite geodesy. Blaisdel Publishing company, WalthamGoogle Scholar
  25. Koop R (1993) Global gravity field modelling using satellite gravity gradiometry. Tech. Rep. New Series, 38, The Netherlands Geodetic Commission, DelftGoogle Scholar
  26. Kostelec PJ, Rockmore DN (2008) FFTs on the rotation group. J Fourier Anal Appl 14(2):145–179. doi: 10.1007/s00041-008-9013-5 CrossRefGoogle Scholar
  27. Meissl P (1971) A study of covariance functions related to the Earth’s disturbing potential. Tech. Rep. No. 151, Ohio State University, ColumbusGoogle Scholar
  28. Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2012) The development and evaluation of the Earth gravitational model 2008 (EGM2008). J Geophys Res Solid Earth (1978–2012) 117(B4). doi: 10.1029/2011JB008916
  29. Petrovskaya MS, Vershkov AN (2006) Non-singular expressions for the gravity gradients in the local north-oriented and orbital reference frames. J Geod 80(3):117–127. doi: 10.1007/s00190-006-0031-2 CrossRefGoogle Scholar
  30. Ricardi LJ, Burrows ML (1972) A recurrence technique for expanding a function in spherical harmonics. IEEE Trans Comput 6:583–585CrossRefGoogle Scholar
  31. Risbo T (1996) Fourier transform summation of Legendre series and \(D\)-functions. J Geod 70(7):383–396. doi: 10.1007/BF01090814
  32. Rummel R, Van Gelderen M (1992) Spectral analysis of the full gravity tensor. Geophys J Int 111(1):159–169. doi: 10.1111/j.1365-246X.1992.tb00562.x
  33. Rummel R, Van Gelderen M (1995) Meissl scheme-spectral characteristics of physical geodesy. Manuscr Geod 20(5):379–385Google Scholar
  34. Schuster A (1902) On some definite integrals, and a new method of reducing a function of spherical co-ordinates to a series of spherical harmonics. Philos Trans R Soc Lond Ser A 200:181–223CrossRefGoogle Scholar
  35. Sharifi MA, Ghobadi-Far K (2015) Representation of gradients of a scalar field on the sphere using a 2D Fourier expression. In: Arefi H, Motagh M (eds) The international archives of photogrammetry, remote sensing and spatial information sciences, vol XL-1/W5. Copernicus GmbH, Gottingen, pp 689–694Google Scholar
  36. Sneeuw N (1992) Representation coefficients and their use in satellite geodesy. Manuscr Geod 17:117–123Google Scholar
  37. Sneeuw N (2000) A semi-analytical approach to gravity field analysis from satellite observations. C-527, Deutsche Geodätische Kommission, MünchenGoogle Scholar
  38. Sneeuw N, Bun R (1996) Global spherical harmonic computation by two-dimensional Fourier methods. J Geod 70(4):224–232. doi: 10.1007/BF00873703 CrossRefGoogle Scholar
  39. Šprlák M, Novák P (2014) Integral formulas for computing a third-order gravitational tensor from volumetric mass density, disturbing gravitational potential, gravity anomaly and gravity disturbance. J Geod 89(2):141–157. doi: 10.1007/s00190-014-0767-z CrossRefGoogle Scholar
  40. Tóth G, Földváry L (2005) Effect of geopotential model errors on the projection of GOCE gradiometer observables. In: Jekeli C, Bastos L, Fernandes L (eds) International association of geodesy symposia. Gravity, geoid and space missions, vol 129. Springer, Berlin, pp 72–76. doi: 10.1007/3-540-26932-0_13
  41. Wessel P, Smith W (1998) New, improved version of generic mapping tools released. Eos Trans Am Geophys Union 79(47):579–579. doi: 10.1029/98EO00426 CrossRefGoogle Scholar
  42. Wigner E (1959) Group theory and its application to the quantum mechanics of atomic spectra. Academic Press, New YorkGoogle Scholar
  43. Yi W, Rummel R, Gruber T (2013) Gravity field contribution analysis of GOCE gravitational gradient components. Stud Geophys Geod 57(2):174–202. doi: 10.1007/s11200-011-1178-8 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Khosro Ghobadi-Far
    • 1
  • 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

Personalised recommendations