Skip to main content
SpringerLink
Log in

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

  • Reference work entry
Handbook of Geomathematics

Abstract

All planetary bodies like the Earth rotate causing centrifugal effect! The result is an equilibrium figure of ellipsoidal type. A natural representation of the planetary bodies and their gravity fields has therefore to be in terms of ellipsoidal harmonics and ellipsoidal wavelets, an approximation of its gravity field which is three times faster convergent when compared to the “ruling the world” spherical harmonics and spherical wavelets. Freeden et al. (1998, 2004). Here, various effects are treated when considering the Earth to be “ellipsoidal”: > Sections 2 and > 3 start the chapter with the celebrated ellipsoidal Dirichlet and ellipsoidal Stokes (to first order) boundary-value problems. > Section 4 is devoted to the definition and representation of the ellipsoidal vertical deflections in gravity space, extended in > Sect. 5 to the representation in geometry space. The potential theory of horizontal and vertical components of the gravity field, namely, in terms of ellipsoidal vector fields, is the target of > Sect. 6. > Section 7 is concentrated on the reference potential of type Somigliana–Pizzetti field and its tensor-valued derivatives. > Section 8 illustrates an ellipsoidal harmonic gravity field for the Earth called SEGEN (Gravity Earth Model), a set-up in ellipsoidal harmonics up to degree/order 360/360. Five plates are shown for the West–East/North–South components of type vertical deflections as well as gravity disturbances refering to the International Reference Ellipsoid 2000. The final topic starts with a review of the curvilinear datum problem refering to ellipsoidal harmonics. Such a datum transformation from one ellipsoidal representation to another one in > Sect. 9 is a seven-parameter transformation of type (i) translation (three parameters), (ii) rotation (three parameters) by Cardan angles, and (iii) dilatation (one parameter) as an action of the conformal group in a three-dimensional Weitzenbäck space W(3) with seven parameters. Here, the chapter is begun with an example, namely, with a datum transformation in terms of spherical harmonics in > Sect. 10. The hard work begins with > Sect. 11 to formulate the datum transformation in ellipsoidal coordinates/ellipsoidal harmonics! The highlight is > Sect. 12 with the characteristic example in terms of ellipsoidal harmonics for an ellipsoid of revolution transformed to another one, for instance, polar motion or gravitation from one ellipsoid to another ellipsoid of reference. > Section 13 reviews various approximations given in the previous three sections.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 679.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 649.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Abramowitz M, Stegun IA (1970) Handbook of mathematical functions. Dover, New York

    Google Scholar 

  • Akhtar N (2009) A multiscale harmonic spline interpolation method for the inverse spheroidal gravimetric problem. Universität Siegen, Siegen

    Google Scholar 

  • Ardalan AA (1996) Spheroidal coordinates and spheroidal eigenspace of the earth gravity field. Universität Stuttgart, Stuttgart

    Google Scholar 

  • Ardalan AA (1999) High resolution regional geoid computation in the World Geodetic Datum 2000. Universität Stuttgart, Stuttgart

    Google Scholar 

  • Ardalan AA, Grafarend EW (2000) Reference ellipsoidal gravity potential field and gravity intensity field of degree/order 360/360 (manual of using ellipsoidal harmonic coefficients “Ellipfree.dat” and “Ellipmean.dat”). http://www.uni-stuttgart.de/gi/research/paper/coefficients/coefficients.zip

  • Ardalan AA, Grafarend EW (2001) Ellipsoidal geoidal undulations (ellipsoidal Bruns formula): case studies. J Geodesy 75:544–552

    Article  MATH  Google Scholar 

  • Arfken G (1968) Mathematical methods for physicists, 2nd ed. Academic, New York/London

    Google Scholar 

  • Balmino G et al (1991) Simulation of gravity gradients: a comparison study. Bull Géod 65:218–229

    Article  Google Scholar 

  • Bassett A (1888) A treatise on hydrodynamics. Deighton, Bell and Company, Cambridge; reprint edition in 1961 (Dover, New York)

    Google Scholar 

  • Bölling K, Grafarend EW (2005) Ellipsoidal spectral properties of the Earth’s gravitational potential and its first and second derivatives. J Geodesy 79:300–330

    Article  MATH  Google Scholar 

  • Cajori F (1946) Newton’s principia. University of California Press, Berkeley, CA

    Google Scholar 

  • Cartan EH (1922) Sur les petites oscillations d’une masse fluide. Bull Sci Math 46(317–352):356–369

    Google Scholar 

  • Cartan EH (1928) Sur la stabilité ordinaire des ellipsoides de Jacobi. Proceedings of the International Mathematical Congress, Toronto 1924, 2, Toronto, University of Toronto Press, pp 2–17

    Google Scholar 

  • Cayley A (1875a) A memoir on prepotentials. Phil Trans R Soc Lond 165:675–774

    Article  Google Scholar 

  • Cayley A (1875b) On the potential of the ellipse and the circle. Proc Lond Math Soc 6:38–55

    Article  Google Scholar 

  • Chandrasekhar S (1969) Ellipsoidal figures of equilibrium. Yale University Press, New Haven, CT

    MATH  Google Scholar 

  • Chandrasekhar S, Roberts PH (1963) The ellipticity of a slowly rotating configuration. J Astrophys 138:801–808

    Article  MathSciNet  Google Scholar 

  • Cruzy JY (1986) Ellipsoidal corrections to potential coefficients obtained from gravity anomaly data on the ellipsoid. Rep. 371, Dept. of Geodetic Science and Surveying, The Ohio State University, Columbus

    Google Scholar 

  • Darboux G (1910) Lecons sur les systemes orthogonaux et les cordonées curvilignes. Gauthier-Villars, Paris

    Google Scholar 

  • Darwin GH (1906) On the figure and stability of a liquid satellite. Phil Trans R Soc Lond 206:161–248; Scietific Papers 3, Cambridge University Press, Cambridge, 1910, 436

    Google Scholar 

  • Dedekind R (1860) Zusatz zu der vorstehenden Abhandlung. J Reine Angew Math 58:217–228

    Google Scholar 

  • Doob JL (1984) Classical Potential theory and its probabalistic counterpart. Springer, New York

    Google Scholar 

  • Dyson FD (1991) The potentials of ellipsoids of variable densities. Q J Pure Appl Math XXV:259–288

    Google Scholar 

  • Eisenhart LP (1934) Separable systems of Stäckel. Ann Math 35:284–305

    Article  MathSciNet  Google Scholar 

  • Ekman M (1996) The permanent problem of the permanent tide; what to do with it in geodetic reference systems. Mar Terres 125:9508–9513

    Google Scholar 

  • Engels J (2006) Zur Modellierung von Auflastdeformationen und induzierter Polwanderung. Technical Reports Department of Geodesy and Geoinformatics University Stuttgart, Report 2006.1, Stuttgart

    Google Scholar 

  • Engels J et al (1993) The geoid as an inverse problem to be regularized. In: Anger G et al (eds) Inverse problems: principles and applications in geophysics, technology and medicine. Akademie-Verlag, Berlin, pp 122–167

    Google Scholar 

  • Ferrers NM (1877) On the potentials of ellipsoids, ellipsoidal shells, elliptic harmonic and elliptic rings of variable densities. Q J Pure Appl Math 14:1–22

    Google Scholar 

  • Finn G (2001) Globale und regionale Darstellung von Lotabweichungen bezüglich des Internationalen Referenzellipsoids. Universität Stuttgart, Stuttgart

    Google Scholar 

  • Flügge S (1979) Mathematische Methoden der Physik. Springer, Berlin

    MATH  Google Scholar 

  • Freeden W et al (1998) Constructive approximation of the sphere. Clarendon, Oxford

    Google Scholar 

  • Freeden W, Michel V (2004) Multiscale potential theory. Birkhäuser, Boston–Basel

    Google Scholar 

  • Friedrich D (1998) Krummlinige Datumstransformation. Universität Stuttgart, Stuttgart

    Google Scholar 

  • Gauss CF (1867) Werke 5, Theoria attractionis corporum sphraedicorum ellipticorum homogeneorum. Königliche Gesellschaft der Wissenschaften, Göttingen

    Google Scholar 

  • Gleason DM (1988) Comparing corrections to the transformation between the geopotential’s spherical and ellipsoidal spectrum. Manuscr Geod 13:114–129

    Google Scholar 

  • Gleason DM (1989) Some notes on the evaluation of ellipsoidal and spheroidal harmonic expansions. Manuscr Geod 14:114–116

    Google Scholar 

  • Gradshteyn IS, Ryzhik IM (1980) Tables of integrals, series and products. Corrected and enlarged edition, transl. by A. Jeffrey, Academic Press, New York

    Google Scholar 

  • Grafarend EW (1988) The geometry of the earth’s surface and the corresponding function space of the terrestrial gravitational field. Festschrift R. Sigl, Deutsche Geodätische Kommission, Bayerische Akademie der Wissenschaften, Report B 287, pp 76–94, München

    Google Scholar 

  • Grafarend EW, Ardalan AA (1999) World Geodetic Datum 2000. J Geodesy 73:611–623

    Article  MATH  Google Scholar 

  • Grafarend EW, Awange JL (2000) Determination of vertical deflections by GPS/LPS measurements. Zeitschrift für Vermessungswesen, 125:279–288

    Google Scholar 

  • Grafarend EW, Engels J (1998) Erdmessung und physikalische Geodäsie, Ergänzungen zum Thema Legendrefunktionen. Skript zur Vorlesung WS 1998/99. Universität Stuttgart, Stuttgart

    Google Scholar 

  • Grafarend EW, Heidenreich A (1995) The generalized Mollweide projection of the biaxial ellipsoid. Bull Géod 69:164–172

    Article  Google Scholar 

  • Grafarend E, Thong NC (1989) A spheroidal harmonic model of the terrestrial gravitational field. Manuscr Geod 14:285–304

    Google Scholar 

  • Grafarend EW, Krumm F, Okeke F (1995) Curvilinear geodetic datum transformations. Zeitschrift für Vermessungswesen 7:334–350

    Google Scholar 

  • Grafarend EW, Engels J, Varga P (1997) The spacetime gravitational field of a deformable body. J Geodesy 72:11–30

    Article  MATH  Google Scholar 

  • Grafarend EW, Finn G, Ardalan AA (2006) Ellipsoidal vertical deflections and ellipsoidal gravity disturbance: case studies. Studia Geophysica et Geodaetica 50:1–57

    Article  Google Scholar 

  • Green G (1828) An essay on the determination of the exterior and interior attractions of ellipsoids of variable densities. In: Ferrers NM (ed) Mathematical papers of George Green. Chelsea, New York

    Google Scholar 

  • Groten E (1979) Geodesy and the Earth’s gravity field. Vol. I: Principles and Conventional Methods. Vol. II: Geodynamics and advanced methods. Dümmler Verlag, Bonn

    Google Scholar 

  • Groten E (2000) Parameters of common relevance of astronomy, geodesy and geodynamics. The geodesist’s handbook. J Geodesy 74:134–140

    Google Scholar 

  • Hackbusch W (1995) Integral equations. Theory and numerical treatment. Birkhäuser Verlag, Basel

    MATH  Google Scholar 

  • Hake G, Grünreich D (1994) Kartographie. Walter de Gruyter, Berlin

    Google Scholar 

  • Heck B (1991) On the linearized boundary value problem of physical geodesy. Report 407, Department of Geodetic Science and Surveying, The Ohio State University, Columbus

    Google Scholar 

  • Heiskanen WH, Moritz H (1967) Physical geodesy. W.H. Freeman, San Francisco, CA

    Google Scholar 

  • Heiskanen WA, Moritz H (1981) Physical geodesy. Corrected reprint of original edition from W.H. Freeman, San Francisco, CA, 1967), order from: Institute of Physical Geodesy, TU Graz, Austria

    Google Scholar 

  • Helmert FR (1884) Die mathematischen und physikalischen Theorien der höheren Geodäsie, Vol. 2. Leipzig, B.G. Teubner (reprinted in 1962 by Minerva GmbH, Frankfurt (Main))

    Google Scholar 

  • Hicks WM (1882) Recent progress in hydrodynamics. Reports to the British Association, pp 57–61

    Google Scholar 

  • Hobson EW (1896) On some general formulae for the potentials of ellipsoids, shells and discs. Proc Lond Math Soc 27:519–416

    Article  Google Scholar 

  • Hobson EW (1965) The theory of spherical and ellipsoidal harmonics. Second Reprint of the edition 1931 (Cambridge University Press), Chelsea, New York

    Google Scholar 

  • Holota P (1995) Classical methods for non-spherical boundary problems in physical geodesy. Symposium 114: Geodetic Theory today. The 3rd Hotine-Massuri Symposium on Mathematical Geodesy. F. Sansó (ed) Springer, Berlin/Heidelberg

    Google Scholar 

  • Honerkamp J, Römer H (1986) Grundlagen der klassischen theoretischen Physik. Springer, Berlin/Heidelberg/New York

    Google Scholar 

  • Hotine M (1967) Downward continuation of the gravitational potential. General Assembly of the International Assembly of Geodesy, Luceone

    Google Scholar 

  • Jacobi CGJ (1834) Über die Figur des Gleichgewichts. Poggendorf Annalen der Physik und Chemie 33:229–238; reprinted in Gesammelte Werke 2 (Berlin, G. Reimer, 1882), 17–72

    Google Scholar 

  • Jeans JH (1917) The motion of tidally-distorted masses, with special reference to theories of cosmogony. Mem Roy Astron Soc Lond 62:1–48

    Google Scholar 

  • Jeans JH (1919) Problems of cosmogony and stellar dynamics. Cambridge University Press, Cambridge, chaps. 7 and 8

    Google Scholar 

  • Jekeli C (1988) The exact transformation between ellipsoidal and spherical harmonic expansions. Manuscr Geod 13:106–113

    Google Scholar 

  • Jekeli C (1999) An analysis of vertical deflections derived from high-degree spherical harmonic models. J Geodesy 73:10–22

    Article  MATH  Google Scholar 

  • Kahle AB (1967) Harmonic analysis for a spheroidal earth. RAND Corporation Document P-3684, presented at IUGG Assembly, St. Gallen

    Google Scholar 

  • Kahle AB, Kern JW, Vestine EH (1964) Spherical harmonic analyses for the spheroidal earth. J Geomagn Geoelectr 16:229–237

    Google Scholar 

  • Kassir MK, Sih GC (1966) Three-dimensional stress distribution around elliptical crack under arbitrary loadings. ASME J Appl Mech 33:601–611

    Google Scholar 

  • Kassir MK, Sih GC (1975) Three-dimensional crack problems. Mechanics of fracture, vol 2. Noordhoff International Publishing, Leyden

    Google Scholar 

  • Kellogg OD (1929) Foundations of potential theory. Springer, Berlin/Heidelberg/New York

    Google Scholar 

  • Klapp M (2002a) Synthese der Datumtransformation von Kugel- und Sphäroidalfunktionen zur Darstellung des terrestrischen Schwerefeldes – Beispielrechnungen zu den Transformationsgleichungen. Universität Stuttgart, Stuttgart

    Google Scholar 

  • Klapp M (2002b) Analyse der Datumtransformation von Kugel- und Sphäroidalfunktionen zur Darstellung des terrestrischen Schwerefeldes – Herleitung der Transformationsgleichungen. Universität Stuttgart, Stuttgart

    Google Scholar 

  • Kleusberg A (1980) The similarity transformation of the gravitational potential close to the identity. Manuscr Geod 5:241–256

    Google Scholar 

  • Kneschke A (1965) Differentialgleichungen und Randwertprobleme. Teubner Verlag, Leipzig

    MATH  Google Scholar 

  • Lamé G (1859) Lecons sur les cordonnées curvilignes et leurs diverses applications. Mallet-Bachelier, Paris

    Google Scholar 

  • Lamp SH (1932) Hydrodynamics. Cambridge University Press, Cambridge, pp 722–723

    Google Scholar 

  • Lebovitz NR (1998) The mathematical developments of the classical ellipsoids. Int J Eng Sci 36:1407–1420

    Article  MathSciNet  Google Scholar 

  • Lejeune Dirichlet G (1860) Untersuchungen über ein Problem der Hydrodynamik. J Reine Angew Math 58:181–216

    Google Scholar 

  • Lejeune Dirichlet G (1897) Gedächtnisrede auf Carl Gustav Jacob Jacobi gehalten in der Akademie der Wissenschaften am 1. Juli 1852. Gesammelte Werke, 2 (Berlin, G. Reimer), 243

    Google Scholar 

  • Lense J (1950) Kugelfunktionen. Akademische Verlagsgesellschaft Geest-Portig, Leipzig

    MATH  Google Scholar 

  • Lyttleton RA (1953) The stability of rotating liquid masses. Cambridge University Press, Cambridge, chap. 9

    Google Scholar 

  • Maclaurin C (1742) A treatise on fluxions. Edinburgh

    Google Scholar 

  • MacMillan WD (1958) The theory of the potential. Dover, New York

    MATH  Google Scholar 

  • Mangulis V (1965) Handbook of series for scientists and engineers. Academic, New York

    MATH  Google Scholar 

  • Martinec Z (1996) Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains. J Geodesy 70:805–828

    Google Scholar 

  • Martinec Z, Grafarend EW (1997) Solution to the Stokes boundary value problem on an ellipsoid of revolution. Studia Geophysica et Geodaetica 41:103–129

    Article  Google Scholar 

  • Martinec Z, Vaniček P (1996) Formulation of the boundary-value problem for geoid determination with a higher-order reference field. Geophys J Int 126:219–228

    Article  Google Scholar 

  • Mikolaiski HW (1989) Polbewegung eines deformierbaren Erdkörpers. PhD thesis, Deutsche Geodätische Kommission, Bayerische Akademie der Wissenschaften, Reihe C, Heft 354, München

    Google Scholar 

  • Milne EA (1952) Sir James Jeans, a Biography. Cambridge University Press, Cambridge, chap. 9

    Google Scholar 

  • Molodenski MS (1958) Grundbegriffe der geodÃtischen Gravimetrie. VEB Verlag Technik, Berlin

    Google Scholar 

  • Moon P, Spencer DE (1953) Recent investigations of the Separation of Laplace’s equation. Ann Math Soc Proc 4:302–307

    MATH  MathSciNet  Google Scholar 

  • Moon P, Spencer DE (1961) Field theory for engineers. D. van Nostrand, Princeton, NJ

    MATH  Google Scholar 

  • Moritz H (1968a) Density distributions for the equipotential ellipsoid. Dept. of Geodetic Science and Surveying, The Ohio State University, Columbus

    Google Scholar 

  • Moritz H (1968b) Mass distributions for the equipotential ellipsoid. Bolletino di Geofisica Teorica ed Applicata 10:59–65

    Google Scholar 

  • Moritz H (1973) Computation of ellipsoidal mass distributions. Dept. of Geodetic Science and Surveying, The Ohio State University, Columbus

    Google Scholar 

  • Moritz H (1980) Geodetic Reference System 1980. Bull Géod 54:395–405

    Article  Google Scholar 

  • Moritz H (1984) Geodetic Reference System 1980. The geodesist’s handbook. Bull Géod 58:388–398

    Google Scholar 

  • Moritz H (1990) The figure of the earth. Wichmann Verlag, Karlsruhe

    MATH  Google Scholar 

  • Moritz H, Mueller I (1987) Earth rotation. Theory and Observation. Ungar, New York

    Google Scholar 

  • Morse PM, Feshbach H (1953) Methods of theoretical physics, part II. McGraw Hill, New York

    Google Scholar 

  • Natanson JP (1967) Theory of functions of a real variable. Frederick, New York

    Google Scholar 

  • Niven WD (1891) On ellipsoidal harmonics. Phil Trans R Soc Lond A 182:231–278

    Article  Google Scholar 

  • Otero J (1995) A uniqueness theorem for a Robin boundary value problem of physical geodesy. Q J Appl Math

    Google Scholar 

  • Panasyuk VV (1971) Limiting equilibrium of brittle solids with fractures. Management Information Sevices, Detroit, MI

    Google Scholar 

  • Pflaumann E, Unger H (1974) Funktionalanalysis I. Zürich

    Google Scholar 

  • Pick M, Picha J, Vyskočil V (1973) Theory of the Earth’s gravity field. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Pizzetti P (1894) Geodesia – Sulla espressione della gravita alla superficie del geoide, supposto ellissoidico. Atti Reale Accademia dei Lincei 3:166–172

    Google Scholar 

  • Poincaré H (1885) Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Acta Math 7:259–380

    Article  MathSciNet  Google Scholar 

  • Polya G (1965) Mathematical discovery. On understanding, learning and teaching problem solving. Wiley, New York

    Google Scholar 

  • Press WH et al (1989) Numerical recipes. The art of scientific computing. Cambridge University Press, Cambridge

    Google Scholar 

  • Rapp RH, Cruzy JY (1986) Spherical harmonic expansions of the Earth’s Gravitational Potential to Degree 360 Using 30′ Mean Anomalies. Report No. 376, Department of Geodetic Science and Surveying, The Ohio State University, Columbus

    Google Scholar 

  • Riemann B (1860) Untersuchungen über die Bewegung eines flüssigen gleichartigen Ellipsoides. Abhandlung der Königlichen Gesellschaft der Wissenschaften, 9, 3–36; Gesammelte Mathematische Werke (Leipzig, B.G. Teubner, 1892), 182

    Google Scholar 

  • Roche MEd (1847) Mémoire sur la figure d’une masse fluide (soumise á l’attraction d’un point éloigné. Acad des Sci de Montpellier 1(243–263):333–348

    Google Scholar 

  • Routh EJ (1902) A treatise on analytical statics, vol 2. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Rummel R et al (1988) Comparisons of global topographic isostatic models to the Earth’s observed gravity field. Report 388, Department of Geodetic Science and Surveying, The Ohio State University, Columbus

    Google Scholar 

  • Sansà F, Sona G (2001) ELGRAM, an ellipsoidal gravity model manipulator. Bolletino di Geodesia e Scienze Affini 60:215–226

    Google Scholar 

  • Sauer R, Szabo J (1967) Mathematische Hilfsmittel des Ingenieurs. Springer, Berlin

    MATH  Google Scholar 

  • Schäfke FW (1967) Spezielle Funktionen. In: Sauer R, Szabó I (eds) Mathematische Hilfsmittel des Ingenieurs, Teil 1. Springer, Heidelberg/Berlin/New York, pp 85–232

    Google Scholar 

  • Shah RC, Kobayashi AS (1971) Stress-intensity factor for an elliptical crack under arbitrary normal loading. Eng Fract Mech 3:71–96

    Article  Google Scholar 

  • Shahgholian H (1991) On the Newtonian potential of a heterogeneous ellipsoid. SIAM J Math Anal 22:1246–1255

    Article  MATH  MathSciNet  Google Scholar 

  • Skrzipek MR (1998) Polynomial evaluation and associated polynomials. Numer Math 79:601–613

    Article  MATH  MathSciNet  Google Scholar 

  • Smith JR (1986) From Plane to Spheroid. Landmark Enterprises. Pancho Cordova, California

    Google Scholar 

  • Sneddon IN (1966) Mixed boundary value problems in potential theory. Wiley, New York

    MATH  Google Scholar 

  • Somigliana C (1929) Teoria generale del campo gravitazionale dell’ ellisoide di rotazione. Mem Soc Astron Ital IV

    Google Scholar 

  • Sona G (1996) Numerical problems in the computation of ellipsoidal harmonics. J Geodesy 70:117–126

    Article  Google Scholar 

  • Stäckel P (1897) Über die Integration der Hamiltonschen Differentialgleichung mittels Sepa- ration der Variablen. Math Ann 49:145–147

    Article  MATH  MathSciNet  Google Scholar 

  • Stokes GG (1849) On the variation of gravity on the surface of the Earth. Trans Cambridge Phil Soc 8:672–695

    Google Scholar 

  • Thomson W, Tait PG (1883) Treatise on natural philosophie. Cambridge University Press, Cambridge, pt. 2, pp 324–335

    Google Scholar 

  • Thong NC (1993) Untersuchungen zur Lösung der fixen gravimetrischen Randwertprobleme mittels sphäroidaler und Greenscher Funktionen. Deutsche Geodätische Kommission, Bayerische Akademie der Wissenschaften, Reihe C 399, München 1993 (in German)

    Google Scholar 

  • Thong NC, Grafarend EW (1989) A spheroidal harmonic model of the terrestrial gravitational field. Manuscr Geod 14:285–304

    Google Scholar 

  • Todhunter I (1966) History of the mathematical theories of attraction and the figure of the earth from the time of Newton to that of Laplace. Dover, New York

    Google Scholar 

  • van Asche W (1991) Orthogonal polynomials, associated polynomials and functions of the second kind. J Comput Appl Math 37:237–249

    Article  MathSciNet  Google Scholar 

  • Vaniček P et al (1996) Downward continuation of Helmert’s gravity. J Geodesy 71:21–34

    Article  MATH  Google Scholar 

  • Varshalovich DA, Moskalev AN, Khersonskii VK (1989) Quantum theory of angular momentum. World Scientific, Singapore

    MATH  Google Scholar 

  • Vijaykumar K, Atluri SN (1981) An embedded elliptical crack, in an infinite solid, subject to arbitrary crack face tractions. ASME J Appl Mech 48:88–96

    Article  Google Scholar 

  • Webster AG (1925) The dynamics of particles and of rigid, elastic and fluid bodies. Teubner, Leipzig

    MATH  Google Scholar 

  • Whittaker ET, Watson GN (1935) A course of modern analysis, vol 2. Cambridge University Press, Cambridge

    Google Scholar 

  • Yeremeev VF, Yurkina MI (1974) Fundamental equations of Molodenskii’s theory for the gravitational references field. Studia Geophysica et Geodaetica 18:8–18

    Article  Google Scholar 

  • Yu JH, Cao HS (1996) Elliptical harmonic series and the original Stokes problem with the boundary of the reference ellipsoid. J Geodesy 70:431–439

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Geodesy and Geoinformatics, Stuttgart University, Stuttgart, Germany

    Erik W. Grafarend

  2. Asperg, Germany

    Erik W. Grafarend

  3. Dublin Institute for Advanced Studies, Dublin, Ireland

    Matthias Klapp

  4. Department of Geophysics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Zdeněk Martinec

Authors
  1. Erik W. Grafarend
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthias Klapp
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zdeněk Martinec
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Geomathematics Group, Technische Universität Kaiserslautern, 3049, 67653, Kaiserslautern, Germany

    Willi Freeden

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    M. Zuhair Nashed

  3. Computational Mathematics, Technische Universität Braunschweig, Pockelsstraße 14, 38106, Braunschweig, Germany

    Thomas Sonar

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this entry

Cite this entry

Grafarend, E.W., Klapp, M., Martinec, Z. (2010). Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01546-5_7

Download citation

Publish with us

Policies and ethics

Search

Navigation