Journal of Geodesy

, Volume 92, Issue 2, pp 123–130 | Cite as

Transformation between surface spherical harmonic expansion of arbitrary high degree and order and double Fourier series on sphere

Original Article


In order to accelerate the spherical harmonic synthesis and/or analysis of arbitrary function on the unit sphere, we developed a pair of procedures to transform between a truncated spherical harmonic expansion and the corresponding two-dimensional Fourier series. First, we obtained an analytic expression of the sine/cosine series coefficient of the \(4 \pi \) fully normalized associated Legendre function in terms of the rectangle values of the Wigner d function. Then, we elaborated the existing method to transform the coefficients of the surface spherical harmonic expansion to those of the double Fourier series so as to be capable with arbitrary high degree and order. Next, we created a new method to transform inversely a given double Fourier series to the corresponding surface spherical harmonic expansion. The key of the new method is a couple of new recurrence formulas to compute the inverse transformation coefficients: a decreasing-order, fixed-degree, and fixed-wavenumber three-term formula for general terms, and an increasing-degree-and-order and fixed-wavenumber two-term formula for diagonal terms. Meanwhile, the two seed values are analytically prepared. Both of the forward and inverse transformation procedures are confirmed to be sufficiently accurate and applicable to an extremely high degree/order/wavenumber as \(2^{30}\,{\approx }\,10^9\). The developed procedures will be useful not only in the synthesis and analysis of the spherical harmonic expansion of arbitrary high degree and order, but also in the evaluation of the derivatives and integrals of the spherical harmonic expansion.


Associated Legendre function Fourier series expansion Recurrence formula Spherical harmonic expansion Wigner d function 



The author appreciates valuable suggestions and fruitful comments by anonymous referees to improve the readability of the article.

Supplementary material

190_2017_1049_MOESM1_ESM.pdf (320 kb)
Supplementary material 1 (pdf 320 KB)


  1. Alpert B, Rokhlin V (1989) A fast algorithm for the evaluation of Legendre expansions. Yale U/Department of Computer Science Report 671Google Scholar
  2. Cheong HB (2000) Double Fourier series on a sphere: applications to elliptic and vorticity equations. J Comput Phys 157:327–349CrossRefGoogle Scholar
  3. Cheong HB, Park JR, Kang HG (2012) Fourier-series representation and projection of spherical harmonic functions. J Geod 86:975–990CrossRefGoogle Scholar
  4. Clenshaw CW (1955) A note on the summation of Chebyshev series. Math Tables Other Aids Comput 9:110–118Google Scholar
  5. Colombo OL (1981) Numerical methods for harmonic analysis on the sphere. OSU/Department Geodesy Science Survey Report 310Google Scholar
  6. Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Math Comput 19:297–301CrossRefGoogle Scholar
  7. Dilts GA (1985) Computation of spherical harmonic expansion coefficients via FFT’s. J Comput Phys 57:439–453CrossRefGoogle Scholar
  8. Driscoll JR, Healy D (1989) Computing Fourier transforms and convolutions on the 2-sphere. In: Proceedings of the 34th IEEE FOCS, pp 344–349Google Scholar
  9. Driscoll JR, Healy D (1994) Computing Fourier transforms and convolutions on the 2-sphere. Adv Appl Math 15:202–250CrossRefGoogle Scholar
  10. Driscoll JR, Healy D, Rockmore D (1997) Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs. SIAM J Comput 26:1066–1099CrossRefGoogle Scholar
  11. Egersdorfer R, Egersdorfer L (1936) Formeln und Tabellen der zugeordneten Kugelfunktionen 1. Art von \(n=1\) bis \(n=20\). Reichs fur Wett Wiss I:18–47Google Scholar
  12. Edmonds AR (1957) Angular momentum in quantum mechanics. Princeton Univ Press, PrincetonCrossRefGoogle Scholar
  13. Elovitz M (1989) A test of a modified algorithm for computing spherical harmonic coefficients using an FFT. J Comput Phys 80:506–511CrossRefGoogle Scholar
  14. Foldvary L (2015) Sine series expansion of associated Legendre functions. Acta Geod Geophys 50:243–259CrossRefGoogle Scholar
  15. Fukushima T (2011) Efficient parallel computation of all-pairs N-body acceleration by do loop folding. Astron J 142:18–22CrossRefGoogle Scholar
  16. Fukushima T (2017) Rectangular rotation of spherical harmonic expansion of arbitrary high degree and order. J Geod. doi: 10.1007/s00190-017-1004-3
  17. Ghobadi-Far K, Sharifi MA, Sneeuw N (2016) 2D Fourier series representation of gravitational functionals in spherical coordinates. J Geod 90:871–881CrossRefGoogle Scholar
  18. Gooding RH (1971) A recurrence relation for inclination functions and their derivatives. Celest Mech 4:91–98CrossRefGoogle Scholar
  19. Gooding RH, Wagner CA (2008) On the inclination function and a rapid stable procedure for their evaluation together with derivatives. Celest Mech Dyn Astron 101:247–272CrossRefGoogle Scholar
  20. Gooding RH, Wagner CA (2010) On a Fortran procedure for rotating spherical harmonic coefficients. Celest Mech Dyn Astron 108:95–106CrossRefGoogle Scholar
  21. Gruber C (2011) A study on the Fourier composition of the associated Legendre functions; suitable for applications in ultra-high resolution. Scientific technical report STR11/04, GFZ. PotsdamGoogle Scholar
  22. Gruber C, Abrykosov O (2014) High resolution spherical and ellipsoidal harmonic expansions by fast Fourier transform. Stud Geophys Geod 58:595–608CrossRefGoogle Scholar
  23. Gruber C, Abrykosov O (2016) On computation and use of Fourier coefficients for associated Legendre functions. J Geod 90:525–535CrossRefGoogle Scholar
  24. Gruber C, Novak P, Sebera J (2011) FFT-based high-performance spherical harmonic transformation. Stud Geophys Geod 55:489–500CrossRefGoogle Scholar
  25. Healy D, Kostelec P, Rockmore D (2004) Towards safe and effective high-order Legendre transforms with applications to FFTs for the 2-sphere. Adv Comput Math 21:59–105CrossRefGoogle Scholar
  26. Healy D, Rockmore D, Kostelec P, Moore S (2003) FFTs for the 2-sphere: improvements and variations. J Fourier Anal Appl 9:341–385CrossRefGoogle Scholar
  27. Healy D, Rockmore D, Moore S (1996) An FFT for the 2-sphere and applications. J Fourier Anal Appl 9:341–385CrossRefGoogle Scholar
  28. Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San FranciscoGoogle Scholar
  29. Hobson EW (1931) The theory of spherical and ellipsoidal harmonics. Cambridge Univ Press, CambridgeGoogle Scholar
  30. Hofsommer DJ, Potters ML (1960) Table of Fourier coefficients of associated Legendre functions. Proc KNAW Ser A Math Sci 63:460–466Google Scholar
  31. Ito T, Fukushima T (1997) Parallelized extrapolation method and its application to the orbital dynamics. Astron J 114:1260–1267CrossRefGoogle Scholar
  32. Kellogg OD (1929) Foundations of potential theory. Springer, BerlinCrossRefGoogle Scholar
  33. Kostelec PJ, Rockmore DN (2008) FFTs on the rotation group. J Fourier Anal Appl 14:145–179CrossRefGoogle Scholar
  34. Maslen D (1998) Efficient computation of Fourier transform on compact groups. J Fourier Anal Appl 4:19–52CrossRefGoogle Scholar
  35. Moazezi S, Zomorrodian H, Siahkoohi HR, Azmoudeh-Ardalan A, Gholami A (2016) Fast ultrahigh-degree global spherical harmonic synthesis on nonequispaced grid points at irregular surfaces. J Geod 90:853–870CrossRefGoogle Scholar
  36. Mohlenkamp MJ (1999) A fast transform for spherical harmonics. J Fourier Anal Appl 5:159–184CrossRefGoogle Scholar
  37. Moritz H (1980) Advanced physical geodesy. Herbert Wichmann, KarlsruheGoogle Scholar
  38. Moritz H (2000) Geodetic reference system 1980. J Geod 74:128–162CrossRefGoogle Scholar
  39. Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2012) The development and evaluation of the Earth gravitational model 2008 (EGM2008). J Geophys Res 117:B04406CrossRefGoogle Scholar
  40. Rexer M, Hirt C (2015a) Ultra-high degree surface spherical harmonic analysis using the Gauss–Legendre and the Driscoll/Healy quadrature theorem and application to planetary topography models of Earth, Mars and Moon. Surv Geophys 36:803–830CrossRefGoogle Scholar
  41. Rexer M, Hirt C (2015b) Spectral analysis of the Earth’s topographic potential via 2D-DFT: a new data-based degree variance model to degree 90,000. J Geod 89:887–909CrossRefGoogle Scholar
  42. Ricardi LJ, Burrows ML (1972) A recurrence technique for expanding a function in spherical harmonics. IEEE Trans Comput 21:583–585CrossRefGoogle Scholar
  43. Risbo T (1996) Fourier transform summation of Legendre series and D-functions. J Geod 70:383–396CrossRefGoogle Scholar
  44. Rokhlin V, Tygert M (2006) Fast algorithms for spherical harmonic expansions. SIAM J Comput 27:1903–1928CrossRefGoogle Scholar
  45. Schuster A (1902) On some definite integrals and a new method of reducing a function of spherical coordinates to a series of spherical harmonics. Proc R Soc Lond 71:97–101CrossRefGoogle Scholar
  46. Schuster A (1903) On some definite integrals and a new method of reducing a function of spherical coordinates to a series of spherical harmonics. Philos Trans R Soc Lond A 200:181–223CrossRefGoogle Scholar
  47. Smith B, Sandwell D (2003) Accuracy and resolution of shuttle radar topography mission data. Geophys Res Lett 32:L21S01Google Scholar
  48. Sneeuw NJ, Bun R (1996) Global spherical harmonic computation by two-dimensional Fourier methods. J Geod 70:224–232CrossRefGoogle Scholar
  49. Stacey FD, Davis PM (2008) Physics of the Earth, 4th edn. Cambridge Univ Press, CambridgeCrossRefGoogle Scholar
  50. Swarztrauber PN (1979) On the spectral approximation of discrete scalar and vector functions on the sphere. SIAM J Numer Anal 16:934–949CrossRefGoogle Scholar
  51. Swarztrauber PN (1993) The vector harmonic transform method for solving partial differential equations in spherical geometry. Mon Weather Rev 121:3415–3427CrossRefGoogle Scholar
  52. Tachikawa T, Hato M, Kaku M, Iwasaki A (2011) Characteristics of ASTER GDEM version 2. In: Proceedings of the IEEE international geoscience and remote sensing symposium, pp 3657–3660Google Scholar
  53. Tscherning CC, Poder K (1982) Some geodetic applications of Clenshaw summation. Boll Geod Sci Aff 41:349–375Google Scholar
  54. Wandelt BD, Gorski KM (2001) Fast convolution on the sphere. Phys Rev D 63:123002CrossRefGoogle Scholar
  55. Wigner EP (1931) Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren. Vieweg Verlag, BraunschweigCrossRefGoogle Scholar
  56. Zucker PA (1991) Smoothing and desmoothing in the Fourier approach to spherical coefficient determination. Proc IAG Symp 107:533–542Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.National Astronomical Observatory/SOKENDAIMitakaJapan

Personalised recommendations