The reference figure of the rotating earth in geometry and gravity space and an attempt to generalize the celebrated Runge–Walsh approximation theorem for irregular surfaces

Original Paper


First, we discuss the Reference Figure of the Earth of type plane, sphere, ellipsoid, regular topography, irregular topography, fractal geometry. The Standard Reference of the Earth, the Telluroid, is derived from the anharmonic Somigliana–Pizzetti gravity field, also called World Geodetic Datum, strongly influenced by the Rotation of the Earth in terms of spheroidal coordinates/spheroidal gravity field. In detail, we treat the Telluroid Mapping based on a two step procedure of type (astronomic longitude/astronomic latitude, modulus of gravity). The result is an algebraic equation of degree ten. Second, we discuss the important Runge–Walsh Approximation Theorem in the context of singularities of the Earth Geometry and Gravity Field as mentioned by Bocchio–Livieratos–Grafarend. The example by Sanso that a grain of sand blows up the series expansion of the external gravity field is of key importance.


Singular points in the Earth topography Singular points in the Earth gravity field Equilibrium figures Anholonomity Somigliana–Pizzetti gravity field Transformation of differential manifolds Runge–Walsh approximation theorem: limits 

Mathematics Subject Classification

86A30 31A05 35J56 51H25 53A05 


  1. Ballani, L., Engels, J., Grafarend, E.: Global base functions for the mass density in the interior of a massive body (Earth). Manuscripta Geodaetica 18, 99–114 (1992)Google Scholar
  2. Baranov, W.: La formulle de Stokes est—elle correcte? Bull. Geod. 49, 27–34 (1975)CrossRefGoogle Scholar
  3. Bielanski, R., Gorniewicz, L., Plaskacz, S.: Topological approach to differential inclusions on closed subset of \(R^n\). In: Jones, C., Kirchgraber, U., Walter, H.O. (eds.) Dynamics reported, pp. 225–250. Springer, Berlin-Heidelberg (1992)Google Scholar
  4. Bochhio, F.: The holonomity problem in geophysics. Boll. Sci. Aff. 34, 456 (1975)Google Scholar
  5. Bochhio, F.: Geodetic singularities. Rev. Geophys Space Res. 20, 399–409 (1981)CrossRefGoogle Scholar
  6. Bochhio, F.: An inverse geodetic singularity problem. J. R. Atr. Soc. 72 (1981)Google Scholar
  7. Bochhio, F.: An existance theorem for the abwolute geodetic singularities. J. R. Atr. Soc. 72 (1983)Google Scholar
  8. Bochhio, F.: Tidal and rotational perturbations of Earth’s gravity gradients. Geophys. Geod. Singul. J. R. Atr. Soc. 81, 463–467 (1985)CrossRefGoogle Scholar
  9. Bode, A., Grafarend, E.: The space like Molodenski problem including the rotational term of the gravity potential. Manuscr. Geod. 6, 33–61 (1981)MATHGoogle Scholar
  10. Bode, A., Grafarend, E.: The telluroid mapping based on anormal gravity potential including the centrifugal term. Boll. Geod. 41, 21–56 (1982)Google Scholar
  11. Broher, H.W., Vogler, G.: Bifurcational aspects of parameter resonance. In: Jones, C., Kirchgraber, U., Walter, H.O. (eds.) Dynamics reported, pp. 1–53. Springer, Berlin-Heidelberg (1992)Google Scholar
  12. Bruns, E.H.: Die Figur der Erde. Königt Preuss. Geod. Inst., P. Stankiewich Buchdruckerei, Berlin, Pubt (1878)Google Scholar
  13. Colombo, O.L.: The convergence of the external spherical harmonic expansion of the gravitational potential. Bolletino di Geodesia e Science Affini 42, 221–239 (1983)Google Scholar
  14. Colombo, O.L., Kleusberg, A.: Application of an Earth orbiting gravity gradiometer. Boll. Geod. 57, 83–101 (1983)CrossRefGoogle Scholar
  15. Cusmann, H.: A survey of nominalization techniques applied to perturbe a Keplerian Systems. In: Jones, C., Kirchgraber, U., Walter, H.O. (eds.) Dynamics reported, pp. 54–112. Springer, Berlin-Heidelberg (1992)Google Scholar
  16. Dermanis, A., Livieratos, E.: Applications of deformation analysis in geodesy and geodynamics. Rev. Geophys. Space Phys. 21, 41–50 (1983)CrossRefGoogle Scholar
  17. Dermanis, A., Sansorn, P.: A geodynamic boundary value problem. Boll. Geod. 41, 65–87 (1982)Google Scholar
  18. Dieudonne, J.: Foundation of modern analysis. Academic Press, New York (1960)Google Scholar
  19. Ekman, M.: The stationary effect of moon and sun upon the gravity of the earth, and some aspects of the definition of gravity, Rep. 5, Geodetic Institute, Uppsala University (1979)Google Scholar
  20. Ekman, M.: On the definition of gravity. Bull. Geod. 55, 167–168 (1981)CrossRefGoogle Scholar
  21. Erwe, F.: Differential und Integralrechnung. Band 1. B.I. Hochschultaschenbücher, Bd. 30. Bibliograph. Inst. (RI., Mannheim) (1962)Google Scholar
  22. Forward, R.L.: Flattening spacetime near the Earth. Phys. Rev. D. 26, 735–744 (1982)CrossRefGoogle Scholar
  23. Freeden, W., Gerhards, C.: Geomathematically oriented potential theory. CRC Press, Taylor & Frances, Boca Raton (2013)MATHGoogle Scholar
  24. Freeden, W., Michel, V.: Multiscale potential theory with applications to geoscience, pp. 128–270. Birkhänser, Bustan-Basel-Berlin (2004)CrossRefMATHGoogle Scholar
  25. Freeden, W., Schneider, F.: Runge–Walsh approximation for the Helmholtz equation. J. Math. Anal. Appl. 235, 533–566 (1999)CrossRefMATHMathSciNetGoogle Scholar
  26. Grafarend, E.: Threedimensional geodesy I—the holonomity problem. In: Vermessungswessen, Z. (ed.) (1975)Google Scholar
  27. Grafarend, E.W.: The definition of the telluroid. Bull. Geod. 52, 25–37 (1978)CrossRefMathSciNetGoogle Scholar
  28. Grafarend, E.W.: Dreidimensionale geodätische Abbildungsgleichungen und die Näherungsfigur der Erde. Zeitschrift für Vermessungswesen 103, 132–140 (1978)MATHGoogle Scholar
  29. Grafarend, E.W.: Six lectures on geodesy and global geodynamics. In: Moritz, H., Sünkel, H., Graz (eds.) Mitteilungen der geodätischen Institute der Technischen Universität Graz, Folge 41, pp. 531–685 (1978)Google Scholar
  30. Grafarend, E.W.: The Bruns Transformation and a dual setup of geodetic observational equations. Report U.S. Department of Commerce National Oceanic and Atmospheric Administration, National Geodetic Survey, Rockville. Report NOS 85 NGS 16 (1979), (1980)Google Scholar
  31. Grafarend, E.W.: Kommentar eines Geodäten zu einer Arbeit E.B. Christoffel. In: Christoffel, E.B., Butzer, P.L., Feher, F. (ed.) Aachen, Birkhäuser, Basel (1981)Google Scholar
  32. Grafarend, E.W.: Die Beobachtungsgleichungen der dreidimensional Geodäsie im Geometrie- und Schwereraum, ein Beitrag zur operationellen Geodäsie. Z. Vermessungswesen 106, pp. 411–429 (1981)Google Scholar
  33. Grafarend, E.W.: The Bruns transformation and a dual setup of geodetic observational equations. NOAA Techn. Rep. NOS 85 NGS 16, Washington (1982)Google Scholar
  34. Grafarend, E.W., Lohse, P.: The minimal distance mapping of the topographic surface onto the reference ellipsoid of revolution. Manuscripta Geodaetica 16, 92–110 (1991)Google Scholar
  35. Grafarend, E.W., Sanso, F.: The multibody space-time geodetic boundary value problem and the Honkasalo term. Geophys. J. R. Astr. Soc. 78, 255–275 (1984)CrossRefGoogle Scholar
  36. Groten, E.: On the determination and applications of gravity gradients in geodetic systems. In: Proceedings of Sixth Symposium on Mathematical Geodesy, pp. 17–55. Commissione Geodaetica Italiana, Firenze (1976)Google Scholar
  37. Groten, E.: A remark on M. Heikkinen’s paper ”On the Honkasalo term in tidal corrections to gravimetrie observations”. BulI. Geod. 54, 221–223 (1980)CrossRefGoogle Scholar
  38. Groten, E.: a) Reply to M. Ekman’s ”On the definition of gravity”. Bull. Geod. 55, 169 (1981)CrossRefGoogle Scholar
  39. Groten, E.: b) Local and global gravity field representation. Rev. Geophys. Space Phys. 19, 407–414 (1981)CrossRefGoogle Scholar
  40. Heck, B.: A numerical comparison of some telluroid mappings. In: Proceedings I Hotijne-Marussi Symposion on Math. Geodesy, pp. 18–38. Milano (1981)Google Scholar
  41. Heck, B.: Combination of leveling and gravity data for detecting real crustal movements. In: Proceedings of International Symposium Geodetic Network and Computations, pp. 20–30, cd. Sigl, R., Rep. 258/VIl, Deutsche Geodätische Kommission, Bayerische Akademie der Wissenschaften, München (1982)Google Scholar
  42. Heikkinen, M.: On the Honkasalo term in tidal corrections to gravimetric observations. Bull. Geod. 53, 239–245 (1979)CrossRefGoogle Scholar
  43. HeImert, F.R.: Die mathematischen und physikalischen Theorien der höheren Geodäsie, Teubner, Leipzig, 1. Teil, Teil, II (1880), (1884)Google Scholar
  44. Hirvonen, R.: New theory of gravimetriy geodesy. Ann. Acad. Sci. Fennicae, A III (56) Helsinki (1960)Google Scholar
  45. Hirvonen, R.: The reformation of geodesy. J Geophys. Res. 66(5), 1471–1478 (1961)CrossRefMATHGoogle Scholar
  46. Honkasalo, T.: On the tidal gravity correction. Boll. Geod. Sei. Affini 21, 34–36 (1964)Google Scholar
  47. Hotine, M.: Mathematical Geodesy, Washington (1969)Google Scholar
  48. Hörmander, L.: An introduction to complex analysis in several variables. North Holland Publishing Company, Amsterdam (1973)MATHGoogle Scholar
  49. Koch, K.R.: Die geodätische Randwertaufgabe bei bekannter Erdoberfläche. Z. VermessWes 96, 218–224 (1971)Google Scholar
  50. Krarup, T.: A contribution to the mathematical foundation of physical geodesy. Internal report of Geodetic Institute Copenhagen, Copenhagen (1969)Google Scholar
  51. Krarup, T.: A contribution to the mathematical foundation of physical geodesy. Danish Geodetic Institute, Report No. 44, Copenhagen (1969)Google Scholar
  52. Krarup, T.: Letter 111 on Molodenski’s problem. Internal report of Geodetic Institute Copenhagen, Copenhagen (1973)Google Scholar
  53. Krarup, T.: The mathematical foundation of Molodenski’s problem. Internal report of Geodetic Institute Copenhagen, Copenhagen (1973)Google Scholar
  54. Krarup, T.: Last letter to the SSG /10. 4.31”, Mathematica1 Techniques in Physical Geodesy”. La forrnule de Stokes est - elle corrcte? Qui, certainment!, Copenhagen (1975)Google Scholar
  55. Krarup, T.: La formule de Stokes est—elle corrccte? Commentaires sur le papier de W. Baranov. Bull. Geod. 50, 251–252 (1976)CrossRefGoogle Scholar
  56. Kreyszig, E.: Advanced engineering mathematics, 5th edition. Wiley, New York (1983)Google Scholar
  57. Kühnel, W.: Differential geometry: curves, surfaces, manifolds. AMS Vol. 16, Student Mathematical Library (2002)Google Scholar
  58. Lang, S.: Analysis I. Addison Wesley Publ Co, Reading Mass. USA (1968)Google Scholar
  59. Livieratos, E.: On the geodetic singularity problem. Manuscripta Geodaetica 1, 269–292 (1976)Google Scholar
  60. Love, A.E.H.: The yielding of the earth to disturbing forces. Proc. R. Astr. Soc. A. 82, 13–88 (1990)Google Scholar
  61. Love, A.E.H.: Some problems on geodynamics. Dover, New York (1991)Google Scholar
  62. Marussi, A.: Fondamenti di geodesia intrinseca. Mem , Com. Geod. Ital 7 (1951)Google Scholar
  63. Mikolaiski, H.W., Braun, P.: Dokumentation der Programme zur Multiplikation nach Kugelfunktionen entwickelter Felder, Institute of Geodesy, University of Stuttgart, Report 10, Stuttgart (1988)Google Scholar
  64. Paok, H.J.: Superconducting tensor gravity gradiometer for satellite geodesy and inertial navigation. J. Ast. Sci. 29, 1–18 (1981)Google Scholar
  65. Pick, M.: Solution of the inner Stokes problem for a sphere. Studia Geophys. Geod. 20, 193–196 (1916)CrossRefGoogle Scholar
  66. Pick, M., Picha, J., Vyskoeil, V.: Theory of the Earth’s Gravity Field. Elsevier, Amsterdam (1973)MATHGoogle Scholar
  67. Pizzetti, P.: Sopra il calcolo terico delle deriviazioni del geoide dall’ ellissoide. Ätti Accad. Sci. Torino. 46, 331–350 (1911)MATHGoogle Scholar
  68. Rummel, R., Tenunissen, P.: A connection between geometric and gravimetric geodesy—some remarks on the role of the gravity field, Feestbundel ter gelegenheid van de 65ste verjaardag van Professor Baarda, Deel, pp. 603–623. Delft University, Department of Geodetic Science (1982)Google Scholar
  69. Runge, C.D.: Zur Theorie der eindeutigen analytischen Funktionen. Acta Mathematica 6, 229–244 (1885)CrossRefMATHMathSciNetGoogle Scholar
  70. Sanso, F.: The geodetic boundary value problem in gravity space. Atti Accad. naz. Lincei, Serie VIII, XIV (3) Roma, pp. 39–97 (1977)Google Scholar
  71. Sanso, F.: The gravity space approach to the geodetic boundary value problem including rotational effects. Manuscr. Geod. 4, 207–244 (1979)MATHGoogle Scholar
  72. Sanso, F.: A discutission of natural representations of the geodetic boundary problem including rotational effects. Intitutto diTopografia, Fotogrammetria e Geofisica, Circular Letter, Milanno (1980)Google Scholar
  73. Sanso, F.: Dual relations in geometry and gravity space. Z. VermessWes. 105, 279–290 (1980)Google Scholar
  74. Sanso, F.: A note on density problems and the Runge Krarup’s Theorem. Bolletino di Geodesia e Science Affini 41, 422–477 (1982)Google Scholar
  75. Sanso, F.: Three lectures on mathematical theory of elasticity. In: Moritz, H., Sünkel, H. (eds.) GrazMitteilungen der geodätischen Institute der Technischen Universität Graz, 4t, pp. 461–530 (1982)Google Scholar
  76. Shida, T.: Horizontal pendulum observations of the change of plumbline at Kamigano” Kyeta. Mem. Coll. Sci. Eng. Tokyo 4, 23–174 (1912)Google Scholar
  77. Simen, Z.: A comment on Honkasalo’s correction. Studia Geophys. Geod. 24, 92–96 (1980)CrossRefGoogle Scholar
  78. Stokes, G.G.: On the variation of gravity on the surface of the earth. Trans. Camb. Phil. Soc. 8, 672–695 (1849)Google Scholar
  79. Wahr, J.: Computing tides, nutations and tidally induced variations in the earth’s rotation rate for a rotating, elliptical earth. In: Moritz, H., Stinke!, H., Graz (eds.) Mitteilungen der geodätischen Institute der Technischen Universität Graz, 41, pp. 327–379 (1982)Google Scholar
  80. Walsh, J.L.: The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions. Bull. Am. Math. Soc. 35, 499–544 (1929)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Geodesy and GeoinformaticsStuttgart UniversityStuttgartGermany

Personalised recommendations