Geoid, Cauchy’s Problem and Displacement

  • Petr Holota
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 117)

Abstract

The classical definition of the geoid is exposed first. Then its tie to the Cauchy problem with initial conditions on a spacelike surface is discussed. In this connection an example shows a process indicating a maximum mass concentration which generates an approximation of the external gravity potential. Finally, the D’Alembert and Euler’s spatial equation for density is expressed within Truesdell’s concepts and related to the temporal deformation of the Earth.

Keywords

Manifold Europe Geophysics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bers, L., John, F. and Schechter, M. (1964). Partial differential equations. John Wiley and Sons, Inc., New York—London—Sydney.Google Scholar
  2. Brovelli, M.A. and Sansò, F. (1993). Is the determination of the geoid a reasonable problem? Bull. No. 2, Int. Geoid Service. D.I.I.A.R. — Politecnico di Milano, pp. 53 —65.Google Scholar
  3. Burša, M. (1974). Gravity anomalies, geoid heights and deflections of the vertical from satellite observations. Report No. 8 of the Res. Inst. of Geod., Topog. and Cartog., Series 3, Prague, 1974, pp. 33–48 (in Czech).Google Scholar
  4. Courant, R. (1964). Partial differential equations. Mir Publishers, Moscow (in Russian).Google Scholar
  5. Engels, J., Grafarend, E., Keller, W., Martinec, Z., Sansò, F. and Vaní(ek, P. (1993). The geoid as an inverse problem to be regularized. Anger, G., Gorenflo, R., Jochmann, H., Moritz, H. and Webers, W. (eds.): Inverse problems: Principles and applications in geophysics, technology and medicine. Akad. Vlg., Berlin, pp. 122 —166.Google Scholar
  6. Gaposchkin, E.M. and Lambeck, K. (1970). 1969 Smithsonian Standard Earth (II). SAO Spec. Report. 315.Google Scholar
  7. Grafarend, E.W. (1989). The geoid and the gravimetric boundary value problem. Report No. 18, Dept. of Geod., The Royal Inst. of Technology, Stockholm.Google Scholar
  8. Grafarend, E.W. (1993). Gauge theory, field equations of gravitations, the definition and computation of the spacetime deforming geoid. Montag, H. and Reigber, Ch. (eds.). Geodesy and Physics of the Earth. Int. Assoc. of Geodesy Symposia No. 112, SpringerVlg., Berlin etc., pp. 226–232.Google Scholar
  9. Grafarend, E.W. (1994). What is a geoid? Vanícek, P. and Christou, N.T. (eds.): Geoid and its geophysical interpretation. CRC Press, Inc., Boca Raton etc.Google Scholar
  10. Grafarend, E.W., Engels, J. and Varga, P. (1996). Temporal variation of the terrestrial gravity field due to internal/external volume and surface forces: Functional relations between generalized Love-Shida Functions. Rapp, R.H., Cazenave, A.A. and Nerem, R.S. (eds.). Global gravity field and its temporal variations. Int. Assoc. of Geodesy Symposia No. 116, Springer-Vlg., Berlin etc., pp. 174–175.Google Scholar
  11. Heck, B. (1992). Some remarks on the determination of the geoid in the framework of the internal geodetic boundary value problem. Holota, P. and Vermeer, M. (eds.). Proc. First continental workshop on the geoid in Europe, Prague, 1992. Res. Inst. of Geod., Topog. and Cartog., Prague in co-operation with IAG–Subcommis. for the Geoid in Europe, Prague, pp. 458–471.Google Scholar
  12. Holota, P. (1979). Representation of the Earth’s gravity field by the potential of point masses. Obs. of Artif. Sat. of the Earth, No. 18, Warszawa-Lodz, pp. 159–169.Google Scholar
  13. Holota, P. (1988). On the theory of the geoid determination in continental areas. Proc. 6th Int. Symp. Geod. and Phys. of the Earth, Potsdam, 1988. Veröff. d. Zentralinst. f. Phys. d. Erde., Nr. 102, Potsdam, 1989, pp. 218–228.Google Scholar
  14. Holota, P. (1995). Two branches of the Newton potential and geoid. Siinkel, H. and Marson, I. (eds.). Gravity and Geoid. Int. Assoc. of Geodesy Symposia No. 113, Springer-Vlg., Berlin etc., pp. 205–214.Google Scholar
  15. Hörmander, L. (1975). The boundary problems of physical geodesy. The Royal Inst. of Technology, Division of Geodesy, Stockhlom; also: Archive for Rational Mechanics and Analysis 62 (1976), pp. 1–52.Google Scholar
  16. Kellogg, O.D. (1953). Foundations of potential theory. Dover Publications, Inc., New-York.Google Scholar
  17. Listing, J.B. (1873). Über unsere jetzige Kenntnis der Gestalt und Grösse der Erde. Nachrichten von d. Kgl. Ges. d. Wiss. u. d. G.A. Universität zu Göttingen. Vlg. d. Dieterichschen Buchhandlung, Göttingen.Google Scholar
  18. Lundquis, C.A. and Giacaglia, C.E.O. (1972). Geopotential representation with sampling functions. Henriksen, S.W., Mancini, A. and Chovitz, B.H. (eds.). The use of artificial satellites for geodesy. Geophys. Monog. 15, Am. Geophys. Union, Washington, D.C.Google Scholar
  19. Moritz, H. (1990). The figure of the Earth- Theoretical geodesy and the Earth’s interior. H. Wichmann Vlg., Karlsruhe.Google Scholar
  20. Oden, J.T. (1973). Finite element applications in mathematical physics. Whiteman, J.R. (ed.). The mathematics of finite elements and applications. Academic Press, London and New York, pp. 239–282.Google Scholar
  21. Sansò, F. and Dermanis, A. (1982). A geodynamic boundary value problem. Anno XLIBoll. di Geod. e Sci. Affini-N. 1, pp. 65–88.Google Scholar
  22. Schulze, B.W. (1977). Interpretation of an inverse problem of geophysics by the potential theory. Gerl. Beitr. Geophys. 86, pp. 291–302.Google Scholar
  23. Schulze, B.W. and Wildenhain, G. (1977). Metoden der Potentialtheorie far elliptische Differentialgleichungen beliebiger Ordnung. Akademie-Verlag, Berlin.Google Scholar
  24. Truesdell, C. (1966). The elements of continuum mechanics. Springer Vlg., New York etc.Google Scholar
  25. Vanícek, P. and Christou, N.T. (eds., 1994 ). Geoid and its geophysical interpretations. CRC Press, Inc., Boca Raton etc.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Petr Holota
    • 1
  1. 1.Topography and CartographyResearch Institute of GeodesyPraha -východCzech Republic

Personalised recommendations