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Journal of Geodetic Science

Editor-in-Chief: Eshagh, Mehdi

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On the topographic effects by Stokes’ formula

L.E. Sjöberg
Published Online: 2014-10-17 | DOI: https://doi.org/10.2478/jogs-2014-0014


Traditional gravimetric geoid determination relies on Stokes’ formula with removal and restoration of the topographic effects. It is shown that this solution is in error of the order of the quasigeoid-to-geoid difference, which is mainly due to incomplete downward continuation (dwc) of gravity from the Earth’s surface to the geoid. A slightly improved estimator, based on the surface Bouguer gravity anomaly, is also biased due to the imperfect harmonic dwc the Bouguer anomaly. Only the third estimator,which uses the (harmonic) surface no-topography gravity anomaly, is consistent with the boundary condition and Stokes’ formula, providing a theoretically correct geoid height. The difference between the Bouguer and no-topography gravity anomalies (on the geoid or in space) is the “secondary indirect topographic effect”, which is a necessary correction in removing all topographic signals.

Keywords: Bouguer gravity anomaly; geoid; notopography gravity anomaly; secondary indirect topographic effect; topographic correction


  • Abbak R., Sjöberg L. E., Ellmann A., Ustun A., 2012, A precise gravimetric geoid model in amountainous areawith scarce gravity data: a case study in central Turkey, Stud Geophys Geod 56: 909-927Google Scholar

  • Ågren J., Sjöberg L. E., Kiamehr R., 2009,The new gravimetric geoid model KTH08 over Sweden, J Appl. Geod. 3: 143-153Google Scholar

  • Ellmann A., Vanicek P., 2007, UNB application of Stokes-Helmert‘s approach to geoid computation, J Geodyn 43: 200-2013Web of ScienceGoogle Scholar

  • Forsberg R. 2001, Developments of a Nordic cm-geoid with basics of geoid determination. In: Harson B G (ED.) Lecture notes for autumn school organized by the NKG, Fevik, Norway, 28 August-2 September, 2000, Statens Kartverk, Hoenefoss, Norway.Google Scholar

  • Heiskanen A.W., Moritz H., 1967, Physical Geodesy,WH Freeman and Co., San Francisco and LondonGoogle Scholar

  • Hofmann-Wellenhof B., Moritz H., 2005, Physical Geodesy. Springer, Wien and New York.Google Scholar

  • Sanso F., Sideris M., 2013, Geoid determination - Theory and Methods, Lecture Notes in Earth System Sciences, SpringerGoogle Scholar

  • Kiamehr R., 2006, Precise Gravimetric Geoid Model for Iran Based on GRACE and SRTM Data and the Least-Squares Modification of Stokes’ Formula:with Some Geodynamic Interpretations. Doctoral Thesis in Geodesy, Royal Institute of Technology, Stockholm, SwedenGoogle Scholar

  • Molodensky M.S., Eremeev V.F., Yurkina M.I., 1962, Methods for Study of the External Gravitational Field and Figure of the Earth. Tranls. from Russian (1960), Israel program for Scientific Translations, Jerusalem, IsraelGoogle Scholar

  • Sideris M., 1994, Regional geoid determination. In: Vanicek P, Christou N T (Eds.) The geoid and its geophysical interpretation. CRC Press, Boca Raton, FL, pp. 77-94Google Scholar

  • Sjöberg L. E., 2003a, A computational scheme to model the geoid by the modified Stokes’s formula without gravity reductions. J Geod 77: 423-432Google Scholar

  • Sjöberg L. E., 2003b, A general model of modifying Stokes’ formula and its least-squares solution, J Geod 77: 459-464Google Scholar

  • Sjöberg L. E., 2007, The topographic bias by analytical continuation in physical geodesy. J Geod 81: 345-350Google Scholar

  • Sjöberg L. E., 2010, A strict theory for geoid-to-quasigeoid separation, J Geod 84: 699-702Web of ScienceGoogle Scholar

  • Sjöberg L. E., 2013, On the isostatic gravity anomaly and disturbance and their applications to Vening Meinesz-Moritz inverse problem of isostasy, Geophys J Int 193: 1277-1282Web of ScienceGoogle Scholar

  • Stokes G. G., 1849, On the variation of gravity on the surface of the earth, Trans Cambridge Phil Soc, 8: 672-695Google Scholar

  • Tziavos I. N., Sideris M. , 2013, Topographic reductions in gravity and geoid modeling, In: Sanso and Sideris, M (Eds.) Geoid determination - Theory and Methods, Lecture Notes in Earth System Sciences, Springer, Ch. 8.Google Scholar

  • Ulotu P.E., 2009, Geoid model of Tanzania from sparse and varying gravity data density by the KTH method. Doctoral Thesis in Geodesy, Royal Institute of Technology, StockholmGoogle Scholar

  • Vajda P., Vanicek P., Meurers B., 2006, A new physical foundation for anomalous gravity. Stud Geophys et Geod 50(2): 182-216Google Scholar

  • Vajda P., Vanicek P., Novak P., Tenzer R., Ellmann A, 2007, Secondary indirect effects in gravity anomaly data inversion or interpretation. JGR 112, B06411: 1-11Web of ScienceGoogle Scholar

  • Vanicek P., Tenzer R., Sjöberg L. E., Martinec Z., Featherstone W. E., 2004, New views of the spherical Bouguer gravity anomaly, Geophys J Int 159: 460-472Google Scholar

  • Vanicek P., et al., 2013, Testing Stokes-Helmert geoid model computation on a synthetic gravity field: experiences and shortcomings, Stud Geophys Geod 57: 369-400 Web of ScienceGoogle Scholar

About the article

Received: 2014-06-06

Accepted: 2014-10-01

Published Online: 2014-10-17

Citation Information: Journal of Geodetic Science, Volume 4, Issue 1, ISSN (Online) 2081-9943, DOI: https://doi.org/10.2478/jogs-2014-0014.

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©2014 L.E. Sjöberg. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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Lars Sjöberg
Geosciences, 2018, Volume 8, Number 4, Page 143

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