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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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.
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.
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).
Hadamard J., 1923. Lectures on the Cauchy Problem in Linear Partial Differential Equations. Yale University Press, New Haven.
Heiskanen W.A. and Moritz H., 1969. Physical Geodesy. Freeman & Co., San Francisco, CA.
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.
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.
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.
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.
MacMillan W., 1930. The Theory of Potential. Dover Publications Inc., New York.
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.
Martinec Z., 1998. Boundary-Value Problems for Gravimetric Determination of a Precise Geoid. Lecture Notes in Earth Sciences 73. Springer-Verlag, Berlin, Germany.
Martinec Z., Vaníček P., Mainville A. and Véronneau M., 1995. The effect of lake water on geoidal heights. Manus. Geod., 20, 193–203.
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).
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.
Ralston A., 1965. A First Course in Numerical Analysis. McGraw-Hill, New York.
Stokes G.G., 1849. On the variation of gravity on the surface of the Earth. Trans. Cambridge Phil. Soc., 8, 672–695.
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.
Vaníček P. and Krakiwsky E.J., 1986. Geodesy: the Concepts. 2nd Revised Edition. North-Holland, Amsterdam, The Netherlands.
Vaníček P. and Martinec Z., 1994. The Stokes-Helmert scheme for the evaluation of a precise geoid. Manus. Geod., 19, 119–128.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vaníček, P., Novák, P., Sheng, M. et al. Does Poisson’s downward continuation give physically meaningful results?. Stud Geophys Geod 61, 412–428 (2017). https://doi.org/10.1007/s11200-016-1167-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11200-016-1167-z