Computing the Earth Gravity Field with Spherical Harmonics

  • Michael Gerstl


The expensive evaluation of the spherical-harmonic series expansion of the earth gravity field is optimised by transition to 4-dimensional coordinates. That makes disappear square roots and trigonometric functions. The singularity at the poles inherent in spherical coordinates is removed by the increase of dimension. Instead of the associated Legendre functions we obtain a basis of hypergeometric Jacobi polynomials that reproduces under derivation. Thus, the calculation of their derivatives cancels in the Forsythe summation technique; for the Clenshaw summation, the recursions of function value and derivatives are decoupled.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Milton Abramowitz and Irene A. Stegun. Handbook of mathematical functions. Dover Publications, New York, 10 edition, 1972.MATHGoogle Scholar
  2. [2]
    Michael Gerstl. DOGS-Manual, volume X, Mathematische Grundlagen. Deutsches Geodätisches Forschungsinstitut, München, 1999.Google Scholar
  3. [3]
    Robert Sauer and István Szabó. Mathematische Hilfsmittel des Ingenieurs, volume I. Springer, Berlin, Heidelberg, New York, 1967.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Michael Gerstl
    • 1
  1. 1.Bayerische Akademie der WissenschaftenDeutsches Geodätisches ForschungsinstitutMünchen

Personalised recommendations