Abstract
Spherical harmonic series, commonly used to represent the Earth’s gravitational field, are now routinely expanded to ultra-high degree (> 2,000), where the computations of the associated Legendre functions exhibit extremely large ranges (thousands of orders) of magnitudes with varying latitude. We show that in the degree-and-order domain, (ℓ,m), of these functions (with full ortho-normalization), their rather stable oscillatory behavior is distinctly separated from a region of very strong attenuation by a simple linear relationship: \(m = \ell \sin \theta\), where θ is the polar angle. Derivatives and integrals of associated Legendre functions have these same characteristics. This leads to an operational approach to the computation of spherical harmonic series, including derivatives and integrals of such series, that neglects the numerically insignificant functions on the basis of the above empirical relationship and obviates any concern about their broad range of magnitudes in the recursion formulas that are used to compute them. Tests with a simulated gravitational field show that the errors in so doing can be made less than the data noise at all latitudes and up to expansion degree of at least 10,800. Neglecting numerically insignificant terms in the spherical harmonic series also offers a computational savings of at least one third.
Similar content being viewed by others
References
Abramowitz M, Stegun IA (1970 Handbook of mathematical functions. Dover Publications Inc., New York
Colombo OL (1981) Numerical methods for harmonic analysis on the sphere. Report no.310, Department of Geodetic Science and Surveying, Ohio State University, Columbus
Flury J (2006) Short-wavelengthspectralproperties of the gravity field from a range of regional data sets. J Geod 79:624–640. DOI 10.1007/s00190-005-0011-y
Garmier R, Barriot JP, Konopliv AS, Yeomans DK (2002) Modeling of the Eros gravity field as an ellipsoidal harmonic expansion from the NEAR Doppler tracking data. Geophys Res Lett 29(8). DOI 10.1029/2001GL013768
Gleason DM (1985) Partial sums of Legendre series via Clenshaw summation. Manus Geodaetica. 10:115–130
Gradshteyn IS, Ryzhik IM (1980) Table of integrals. series, and products. Academic, New York
Haagmans RRN (2000) A synthetic earth for use in geodesy. J Geod 74:503–511
Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman and Co., San Francisco
Hobson EW (1965) The theory of spherical and ellipsoidal harmonics. Chelsea Publ. Co., New York
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
Jeffreys H (1943) The determination of the Earth’s gravitational field. Mon Not Roy Astron Soc Geophys Suppl 5:55–66
Jekeli C (1996) Spherical harmonic analysis, aliasing, and filtering. J Geod 70(4):214–223
Jekeli C (1998) The world of gravity according to Rapp. In: Forsberg R, Feissl M, Ditriech R (eds) Geodesy on the move: gravity, geoid, geodynamics, and antarctica, IAG Symposia, vol 119. Springer, Berlin Heidelberg New York, pp 79–91
Konopliv AS, Banerdt WB, Sjogren WL (1999) Venus gravity: 180th degree and order model. Icarus 139:3–18
Koop R, Stelpstra D (1989) On the computation of the gravitational potential and its first and second derivatives. Manus Geodaetica 14:373–382
Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96, NASA Technical Paper NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt
Lerch FJ, Nerem RS, Putney BH, Felsentreger TL, Sanchez BV, Marshall JA, Klosko SM, Patel GB, Williamson RG, Chinn DS, Chan JC, Rachlin KE, Chandler NL, McCarthy JJ, Luthcke SB, Pavlis NK, Pavlis DE, Robbins JW, Kapoor S, Pavlis EC (1994) A geopotential model from satellite tracking, altimeter and surface gravity data: GEMT3. J Geophys Res 99(B2):2815–2839
Paul MK (1978) Recurrence relations for integrals of associated Legendre functions. Bull Géodésique 52:177–190
Pavlis NK, Rapp RH (1990) The development of an isostatic gravitational model to degree 360 and its use in global gravity modeling. Geophys J Inter 100:369–378
Pavlis NK, Holmes SA, Kenyon S, Schmidt D, Trimmer R (2004) A preliminary gravitational model to degree 2160. In: Jekeli C, Bastos L, Fernandes J (eds), Proceedings of the IAG international symposium on gravity, geoid, and satellite missions, 30 August – 3 September 2004, Porto, Portugal. Springer, Berlin Heidelberg New York, pp 18–23
Rapp RH, Pavlis NK (1990) The development and analysis of geopotential coefficient models to spherical harmonic degree 360. J Geophys Res 95(B13):21885–21911
Rappaport NJ, Konopliv AS, Kucinskas AB, Ford PG (1999) An improved 360 degree and order model of the Venus topography. Icarus 139:19–30
Reigber C, Schwintzer P, Barth W, Massmann FH, Raimondo JC, Bode A, Li H, Balmino G, Biancale R, Moynot B, Lemoine JM, Marty JC, Barlier F, Boudon Y (1993) GRIM4-C1-C2p: combination solutions of the global earth gravity field. Surv Geophys 14:381–393
Smith DE, Zuber MT, Frey HV, Garvin JB, Head JW, Muhleman DO, Pettengill GH, Phillips RJ, Solomon SC, Zwally HJ, Banerdt WB, Duxbury TC, Golombek MP, Lemoine FG, Neumann GA, Rowlands DD, Aharonson O, Ford PG, Ivanov AB, Johnson CL, McGovern PJ, Abshire JB, Afzal RS, Sun X (2001) Mars Orbiter Laser Altimeter: Experiment summary after the first year of global mapping of Mars. J Geophys Res 106:23 689–23 722
Tapley B, Watkins M, Ries J, Davis G, Eanes R, Poole S, Rim H, Schutz B, Shum CK, Nerem R, Lerch F, Marshall JA, Klosko SM, Pavlis N, Williamson R (1996) The joint gravity model 3. J Geophys Res 101(B12):28029–28049
Young RGE (1970) Combining satellite altimetry and surface gravimetry in geodetic applications. Report TE-37, Massachusetts Institute of Technology, Cambridge
Yuan DN, Sjogren WL, Konopliv AS (2001) Gravity field of Mars: a 75th degree and order model. J Geophys Res Planets, 106(E10):23377–23401
Zhongolovich ID (1952) The external gravitational field of he Earth and the fundamental constants related to it. (in Russian) Acad Sci Publ Inst Teor Astron Leningrad (translation to English by Aeronautical Charting and Information Service, gov. ac. no. AD-733840, 1971)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jekeli, C., Lee, J.K. & Kwon, J.H. On the computation and approximation of ultra-high-degree spherical harmonic series. J Geod 81, 603–615 (2007). https://doi.org/10.1007/s00190-006-0123-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-006-0123-z