Advertisement

Studia Geophysica et Geodaetica

, Volume 61, Issue 3, pp 412–428 | Cite as

Does Poisson’s downward continuation give physically meaningful results?

  • Petr Vaníček
  • Pavel NovákEmail author
  • Michael Sheng
  • Robert Kingdon
  • Juraj Janák
  • Ismael Foroughi
  • Zdeněk Martinec
  • Marcelo Santos
Article

Abstract

The downward continuation (DWC) of the gravity anomalies from the Earth’s surface to the geoid is still probably the most problematic step in the precise geoid determination. It is this step that motivates the quasi-geoid users to opt for Molodenskij’s rather than Stokes’s theory. The reason for this is that the DWC is perceived as suffering from two major flaws: first, a physically meaningful DWC technique requires the knowledge of the irregular topographical density; second, the Poisson DWC, which is the only physically meaningful technique we know, presents itself mathematically in the form of Fredholm integral equation of the 1st kind. As Fredholm integral equations are often numerically ill-conditioned, this makes some people believe that the DWC problem is physically ill-posed. According to a revered French mathematician Hadamard, the DWC problem is physically well-posed and as such gives always a finite and unique solution. The necessity of knowing the topographical density is, of course, a real problem but one that is being solved with an ever increasing accuracy; so sooner or later it will allow us to determine the geoid with the centimetre accuracy.

Keywords

inverse problem convergence Jacobi iterations Poisson integral equation convexity of equipotential surfaces physical constraints 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Duquenne H., 2007. A data set to test geoid computation methods. Proceedings of the 1st International Symposium of the International Gravity Field Service (IGFS), Istanbul, Turkey. Harita Dergisi, Special Issue 18, 61–65.Google Scholar
  2. Ellmann A. and Vaníček P., 2006. UNB application of Stokes-Helmert’s approach to geoid computation. J. Geodyn., 43, 200–213, DOI: 10.1016/j.jog.2006.09.019.CrossRefGoogle Scholar
  3. Förste C., Bruinsma S.L., Abrikosov O., Lemoine J.-M., Marty J.C., Flechtner F., Balmino G., Barthelmes F. and Biancale R., 2014. EIGEN-6C4 The latest combined global gravity field model including GOCE data up to degree and order 2190 of GFZ Potsdam and GRGS Toulouse. GFZ Data Services (http://doi.org/10.5880/icgem.2015.1).Google Scholar
  4. Hadamard J., 1923. Lectures on the Cauchy Problem in Linear Partial Differential Equations. Yale University Press, New Haven.Google Scholar
  5. Heiskanen W.A. and Moritz H., 1969. Physical Geodesy. Freeman & Co., San Francisco, CA.Google Scholar
  6. Huang J., Vaníček P., Pagiatakis S. and Brink W., 2001. Effect of topographical mass density variation on geoid in the Canadian Rocky Mountains. J. Geodesy, 74, 805–815.CrossRefGoogle Scholar
  7. Hwang C. and Hsiao Y., 2003. Orthometric corrections from leveling, gravity, density and elevation data: a case study in Taiwan. J. Geodesy, 77, 279–291, DOI: 10.1007/s00190-003-0325-6.CrossRefGoogle Scholar
  8. Kingdon R. and Vaníček P., 2011. Poisson downward continuation solution by the Jacobi method. J. Geodet. Sci., 1, 74–81, DOI: 10.2478/v10156-010-0009-0.CrossRefGoogle Scholar
  9. Kingdon R., Vaníček P. and Santos M., 2009. Modeling topographical density for geoid determination. Can. J. Earth Sci., 46, 571–585, DOI: 10.1139/E09-018.CrossRefGoogle Scholar
  10. MacMillan W., 1930. The Theory of Potential. Dover Publications Inc., New York.Google Scholar
  11. Martinec Z., 1996. Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains. J. Geodesy, 70, 805–828, DOI: 10.1007 /BF00867158.CrossRefGoogle Scholar
  12. Martinec Z., 1998. Boundary-Value Problems for Gravimetric Determination of a Precise Geoid. Lecture Notes in Earth Sciences 73. Springer-Verlag, Berlin, Germany.Google Scholar
  13. Martinec Z., Vaníček P., Mainville A. and Véronneau M., 1995. The effect of lake water on geoidal heights. Manus. Geod., 20, 193–203.Google Scholar
  14. Molodenskij MS, Eremeev VF, Yurkina MI (1960) Methods for study of the external gravitational field and figure of the Earth. Translated from Russian by the Israel programme for scientific translations, Office of Technical Services, Department of Commerce, Washington, DC (1962).Google Scholar
  15. Pavlis N.K., Holmes S.A., Kenyon S.C. and Factor J.K., 2012. The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J. Geophys. Res., 117, B04406, DOI: 10.1029/2011JB008916.CrossRefGoogle Scholar
  16. Ralston A., 1965. A First Course in Numerical Analysis. McGraw-Hill, New York.Google Scholar
  17. Stokes G.G., 1849. On the variation of gravity on the surface of the Earth. Trans. Cambridge Phil. Soc., 8, 672–695.Google Scholar
  18. Sun W. and Vaníček P., 1996. On the discrete problem of downward Helmert’s gravity continuation. Reports of the Finnish Geodetic Institute, 96(2), 29–34.Google Scholar
  19. Vaníček P. and Krakiwsky E.J., 1986. Geodesy: the Concepts. 2nd Revised Edition. North-Holland, Amsterdam, The Netherlands.Google Scholar
  20. Vaníček P. and Martinec Z., 1994. The Stokes-Helmert scheme for the evaluation of a precise geoid. Manus. Geod., 19, 119–128.Google Scholar
  21. Vaníček P., Sun W., Ong P., Martinec Z., Vajda P. and ter Horst B., 1996. Downward continuation of Helmert’s gravity. J. Geodesy, 71, 21–34.CrossRefGoogle Scholar
  22. Vaníček P., Tenzer R., Sjöberg L.E., Martinec Z. and Featherstone W.E., 2004. New view of the spherical Bouguer gravity anomaly. Geophys. J. Int., 159, 460–472.CrossRefGoogle Scholar

Copyright information

© Institute of Geophysics of the ASCR, v.v.i 2017

Authors and Affiliations

  • Petr Vaníček
    • 1
  • Pavel Novák
    • 2
    Email author
  • Michael Sheng
    • 1
  • Robert Kingdon
    • 1
  • Juraj Janák
    • 3
  • Ismael Foroughi
    • 1
  • Zdeněk Martinec
    • 4
  • Marcelo Santos
    • 1
  1. 1.Department of Geodesy and Geomatic EngineeringUniversity of New BrunswickFrederictonCanada
  2. 2.New Technologies for the Information Society, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic
  3. 3.Department of Theoretical GeodesySlovak University of TechnologyBratislavaSlovakia
  4. 4.Geophysics SectionDublin Institute for Advanced StudiesDublinIreland

Personalised recommendations