Skip to main content
Log in

Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers

  • Original Article
  • Published:
Journal of Geodesy Aims and scope Submit manuscript

Abstract

By extending the exponent of floating point numbers with an additional integer as the power index of a large radix, we compute fully normalized associated Legendre functions (ALF) by recursion without underflow problem. The new method enables us to evaluate ALFs of extremely high degree as 232 =  4,294,967,296, which corresponds to around 1 cm resolution on the Earth’s surface. By limiting the application of exponent extension to a few working variables in the recursion, choosing a suitable large power of 2 as the radix, and embedding the contents of the basic arithmetic procedure of floating point numbers with the exponent extension directly in the program computing the recurrence formulas, we achieve the evaluation of ALFs in the double-precision environment at the cost of around 10% increase in computational time per single ALF. This formulation realizes meaningful execution of the spherical harmonic synthesis and/or analysis of arbitrary degree and order.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Bosch W (2000) On the computation of derivatives of Legendre functions. Phys Chem Earth 25: 655–659

    Article  Google Scholar 

  • Brendt RP (1978) A Fortran multiple-precision arithmetic package. ACM Trans Math Softw 4: 57–70

    Article  Google Scholar 

  • Casotto S, Fantino E (2007) Evaluation of methods for spherical harmonic synthesis of the gravitational potential and its gradients. Adv Space Res 40: 69–75

    Article  Google Scholar 

  • Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Math Comp 19: 297–301

    Article  Google Scholar 

  • Deprit A (1979) Note on the summation of Legendre series. Celest Mech Dyn Astron 20: 319–323

    Google Scholar 

  • Fantino E, Casotto S (2009) Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients. J Geod 83: 595–619

    Article  Google Scholar 

  • Gleason DM (1985) Partial sums of Legendre series via Clenshaw summation. Manuscr Geod 10: 115–130

    Google Scholar 

  • Goldberg D (1991) What every computer scientist should know about floating-point arithmetic. ACM Comput Surv 23: 5–48

    Article  Google Scholar 

  • Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman and Co, San Francisco

    Google Scholar 

  • Holmes SA, Featherstone WE (2002) A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalized associated Legendre functions. J Geod 76: 279–299

    Article  Google Scholar 

  • IEEE Comp Soc (2008) IEEE standard for floating-point arithmetic. IEEE Std 754 rev

  • Intel (2003) Intel hyper-threading technology technical user’s guide. Intel Corp

  • Jekeli C, Lee JK, Kwon JH (2007) On the computation and approximation of ultra-high-degree spherical harmonic series. J Geod 81: 603–615

    Article  Google Scholar 

  • Kaula WM (2000) Theory of satellite geodesy: applications of satellites to geodesy. Dover, Mineora

    Google Scholar 

  • Kellog OD (1929) Foundations of potential theory. Springer, Berlin

    Google Scholar 

  • Lozier DW, Smith JM (1981) Algorithm 567 extended-range arithmetic and normalized Legendre polynomials. ACM Trans Math Softw 7: 141–146

    Article  Google Scholar 

  • Olver FWJ, Lozier DW, Boisvert RF, Clark, CW (eds) (2010) NIST handbook of mathematical functions. Cambridge University Press, Cambridge. http://dlmf.nist.gov/

  • Paul MK (1978) Recurrence relations for integrals of associated Legendre functions. Bull Geod 52: 177–190

    Article  Google Scholar 

  • Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An Earth gravitational model to degree 2160: EGM2008. Presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, April 13–18, 2008. http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/index.html

  • Smith JM, Olver FWJ, Lozier DW (1981) Extended-range arithmetic and normalized Legendre polynomials. ACM Trans Math Softw 7: 93–105

    Article  Google Scholar 

  • Tscherning CC, Poder K (1982) Some geodetic applications of Clenshaw summation. Boll Geofis Sci Aff 4: 351–364

    Google Scholar 

  • Tscherning CC, Rapp RH, Goad C (1983) A comparison of methods for computing gravimetric quantities from high degree spherical harmonic expansions. Manuscr Geod 8: 249–272

    Google Scholar 

  • Wenzel G (1998) Ultra-high degree geopotential models GPM98A, B, and C to degree 1800. Paper presented to the joint meeting of the International Gravity Commission and International Geoid Commission, 7–12 September, Trieste

  • Wittwer T, Klees R, Seitz K, Heck B (2008) Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic. J Geod 82: 223–229

    Article  Google Scholar 

  • Wolfram S (2003) The mathematica book, 5th edn. Wolfram Research Inc./Cambridge University Press, Cambridge

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Toshio Fukushima.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fukushima, T. Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers. J Geod 86, 271–285 (2012). https://doi.org/10.1007/s00190-011-0519-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00190-011-0519-2

Keywords

Navigation