Spherical Harmonic Transforms Using Quadratures and Least Squares

  • J. A. R. Blais
  • M. A. Soofi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3993)


Spherical Harmonic Transforms (SHTs) which are essentially Fourier transforms on the sphere are critical in global geopotential and related applications. For analysis purposes, discrete SHTs are difficult to formulate for an optimal discretization of the sphere, especially for applications with requirements in terms of near-isometric grids and special considerations in the polar regions. With the enormous global datasets becoming available from satellite systems, very high degrees and orders are required and the implied computational efforts are very challenging. Among the best known strategies for discrete SHTs are quadratures and least squares. The computational aspects of SHTs and their inverses using both quadrature and least-squares estimation methods are discussed with special emphasis on information conservation and numerical stability. Parallel and grid computations are imperative for a number of geodetic, geophysical and related applications, and these are currently under investigation.


Spherical Harmonic Spectral Coefficient Spherical Harmonic Analysis Spherical Topology Quadrature Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Adams, J.C., Swarztrauber, P.N.: SPHEREPACK 2.0: A Model Development Facility (1997), http://www.scd.ucar.edu/softlib/SPHERE.html
  2. 2.
    Blais, J.A.R., Provins, D.A.: Spherical Harmonic Analysis and Synthesis for Global Multiresolution Applications. Journal of Geodesy 76(1), 29–35 (2002)MATHCrossRefGoogle Scholar
  3. 3.
    Blais, J.A.R., Provins, D.A.: Optimization of Computations in Global Geopotential Field Applications. In: Sloot, P.M.A., Abramson, D., Bogdanov, A.V., Gorbachev, Y.E., Dongarra, J., Zomaya, A.Y. (eds.) ICCS 2003. LNCS, vol. 2658, pp. 610–618. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Blais, J.A.R., Provins, D.A., Soofi, M.A.: Optimization of Spherical Harmonic Transform Computations. In: Sunderam, V.S., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds.) ICCS 2005. LNCS, vol. 3514, pp. 74–81. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Colombo, O.: Numerical Methods for Harmonic Analysis on the Sphere. Report no. 310, Department of Geodetic Science and Surveying, The Ohio State University (1981)Google Scholar
  6. 6.
    Driscoll, J.R., Healy Jr., D.M.: Computing Fourier Transforms and Convolutions on the 2-Sphere. Advances in Applied Mathematics 15, 202–250 (1994)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Healy Jr., D., Rockmore, D., Kostelec, P., Moore, S.: FFTs for the 2-Sphere - Improvements and Variations, To appear in Advances in Applied Mathematics (June 1998), Preprint from: http://www.cs.dartmouth.edu/~geelong/publications
  8. 8.
    Heiskanen, W.A., Moritz, H.: Physical Geodesy, 363 p. W.H. Freeman and Company, New York (1967)Google Scholar
  9. 9.
    Mohlenkamp, M.J.: A Fast Transform for Spherical Harmonics. PhD thesis, Yale University (1997)Google Scholar
  10. 10.
    Mohlenkamp, M.J.: A Fast Transform for Spherical Harmonics. The Journal of Fourier Analysis and Applications 5(2/3), 159–184 (1999), Preprint from: http://amath.colorado.edu/faculty/mjm
  11. 11.
    Mohlenkamp, M.J.: Fast spherical harmonic analysis: sample code (2000), http://amath.colorado.edu/faculty/mjm
  12. 12.
    Moore, S., Healy Jr., D., Rockmore, D., Kostelec, P.: SpharmonKit25: Spherical Harmonic Transform Kit 2.5 (1998), http://www.cs.dartmouth.edu/~geelong/sphere/
  13. 13.
    Provins, D.A.: Earth Synthesis: Determining Earth’s Structure from Geopotential Fields, Unpublished PhD thesis, University of Calgary, Calgary (2003)Google Scholar
  14. 14.
    Sneeuw, N.: Global Spherical Harmonic Analysis by Least-Squares and Numerical Quadrature Methods in Historical Perspective. Geophys. J. Int. 118, 707–716 (1994)CrossRefGoogle Scholar
  15. 15.
    Soofi, M.A., Blais, J.A.R.: Parallel Computations of Spherical Harmonic Transforms. In: Oral presentation at the Annual Meeting of the Canadian Geophysical Union, Banff, Alberta, Canada (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • J. A. R. Blais
    • 1
    • 2
  • M. A. Soofi
    • 1
  1. 1.Department of Geomatics Engineering 
  2. 2.Pacific Institute for the Mathematical SciencesUniversity of CalgaryCalgaryCanada

Personalised recommendations