Journal of Geodesy

, Volume 86, Issue 4, pp 271–285 | Cite as

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

Original Article

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.

Keywords

Associated Legendre functions Exponent extension Floating point number Spherical harmonics Underflow problem 

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References

  1. Bosch W (2000) On the computation of derivatives of Legendre functions. Phys Chem Earth 25: 655–659CrossRefGoogle Scholar
  2. Brendt RP (1978) A Fortran multiple-precision arithmetic package. ACM Trans Math Softw 4: 57–70CrossRefGoogle Scholar
  3. Casotto S, Fantino E (2007) Evaluation of methods for spherical harmonic synthesis of the gravitational potential and its gradients. Adv Space Res 40: 69–75CrossRefGoogle Scholar
  4. Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Math Comp 19: 297–301CrossRefGoogle Scholar
  5. Deprit A (1979) Note on the summation of Legendre series. Celest Mech Dyn Astron 20: 319–323Google Scholar
  6. Fantino E, Casotto S (2009) Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients. J Geod 83: 595–619CrossRefGoogle Scholar
  7. Gleason DM (1985) Partial sums of Legendre series via Clenshaw summation. Manuscr Geod 10: 115–130Google Scholar
  8. Goldberg D (1991) What every computer scientist should know about floating-point arithmetic. ACM Comput Surv 23: 5–48CrossRefGoogle Scholar
  9. Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman and Co, San FranciscoGoogle Scholar
  10. 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: 279–299CrossRefGoogle Scholar
  11. IEEE Comp Soc (2008) IEEE standard for floating-point arithmetic. IEEE Std 754 revGoogle Scholar
  12. Intel (2003) Intel hyper-threading technology technical user’s guide. Intel CorpGoogle Scholar
  13. Jekeli C, Lee JK, Kwon JH (2007) On the computation and approximation of ultra-high-degree spherical harmonic series. J Geod 81: 603–615CrossRefGoogle Scholar
  14. Kaula WM (2000) Theory of satellite geodesy: applications of satellites to geodesy. Dover, MineoraGoogle Scholar
  15. Kellog OD (1929) Foundations of potential theory. Springer, BerlinGoogle Scholar
  16. Lozier DW, Smith JM (1981) Algorithm 567 extended-range arithmetic and normalized Legendre polynomials. ACM Trans Math Softw 7: 141–146CrossRefGoogle Scholar
  17. Olver FWJ, Lozier DW, Boisvert RF, Clark, CW (eds) (2010) NIST handbook of mathematical functions. Cambridge University Press, Cambridge. http://dlmf.nist.gov/
  18. Paul MK (1978) Recurrence relations for integrals of associated Legendre functions. Bull Geod 52: 177–190CrossRefGoogle Scholar
  19. 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
  20. Smith JM, Olver FWJ, Lozier DW (1981) Extended-range arithmetic and normalized Legendre polynomials. ACM Trans Math Softw 7: 93–105CrossRefGoogle Scholar
  21. Tscherning CC, Poder K (1982) Some geodetic applications of Clenshaw summation. Boll Geofis Sci Aff 4: 351–364Google Scholar
  22. Tscherning CC, Rapp RH, Goad C (1983) A comparison of methods for computing gravimetric quantities from high degree spherical harmonic expansions. Manuscr Geod 8: 249–272Google Scholar
  23. 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, TriesteGoogle Scholar
  24. Wittwer T, Klees R, Seitz K, Heck B (2008) Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic. J Geod 82: 223–229CrossRefGoogle Scholar
  25. Wolfram S (2003) The mathematica book, 5th edn. Wolfram Research Inc./Cambridge University Press, CambridgeGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.National Astronomical ObservatoryTokyoJapan

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