Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Geodetic Science

Editor-in-Chief: Eshagh, Mehdi

Open Access
See all formats and pricing
More options …

On the geoid and orthometric height vs. quasigeoid and normal height

Lars E. Sjöberg
Published Online: 2018-12-31 | DOI: https://doi.org/10.1515/jogs-2018-0011


The geoid, but not the quasigeoid, is an equipotential surface in the Earth’s gravity field that can serve both as a geodetic datum and a reference surface in geophysics. It is also a natural zero-level surface, as it agrees with the undisturbed mean sea level. Orthometric heights are physical heights above the geoid,while normal heights are geometric heights (of the telluroid) above the reference ellipsoid. Normal heights and the quasigeoid can be determined without any information on the Earth’s topographic density distribution, which is not the case for orthometric heights and geoid. We show from various derivations that the difference between the geoid and the quasigeoid heights, being of the order of 5 m, can be expressed by the simple Bouguer gravity anomaly as the only term that includes the topographic density distribution. This implies that recent formulas, including the refined Bouguer anomaly and a difference between topographic gravity potentials, do not necessarily improve the result. Intuitively one may assume that the quasigeoid, closely related with the Earth’s surface, is rougher than the geoid. For numerical studies the topography is usually divided into blocks of mean elevations, excluding the problem with a non-star shaped Earth. In this case the smoothness of both types of geoid models are affected by the slope of the terrain,which shows that even at high resolutions with ultra-small blocks the geoid model is likely as rough as the quasigeoid model. In case of the real Earth there are areas where the quasigeoid, but not the geoid, is ambiguous, and this problem increases with the numerical resolution of the requested solution. These ambiguities affect also normal and orthometric heights. However, this problem can be solved by using the mean quasigeoid model defined by using average topographic heights at any requested resolution. An exact solution of the ambiguity for the normal height/quasigeoid can be provided by GNSS-levelling.

Keywords: ambiguous quasigeoid; geoid; geoidquasigeoid difference; resolution; vertical datum; quasigeoid


  • Ågren J., 2004, Regional geoid determination methods for the era of satellite geodesy. PhD thesis in geodesy, Royal Institute of Technology, StockholmGoogle Scholar

  • Duquenne H., 2007, A data set to test geoid computation methods. Proc. 1st Int. Symp. of the Int. Gravity Field Services, Istanbul, Harita Dergisi, Special Issue 18: 61-65Google Scholar

  • Ellmann A., Vanicek P., 2007, UNB application of Stokes’s-Helmert’s approach to geoid computation. J Geodyn 43:200-213Web of ScienceCrossrefGoogle Scholar

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

  • Flury J., Rummel R., 2009, On the geoid-quasigeoid separation in mountainous areas. J Geod 83: 829-847.Google Scholar

  • Foroughi I., Tenzer R., 2017, Comparison of different methods for estimating the geoid-to-quasigeoid separation. Geophys J Int 2010: 1001-1020CrossrefGoogle Scholar

  • Foroughi I., Vanicek P., Sheng M., Kingdon R. W., Santos M. C., 2017, In defence of the classical height system. Geophys J Int 211(2): 1154-61.Google Scholar

  • Molodensky M. S., Eremeev V. F., Yurkina M. I., 1962, Methods for study of the external gravitational field and figure of the earth, Transl. From Russian (1960), Israel program for Scientific Translations, Jerusalem, IsraelGoogle Scholar

  • Sjöberg L.E., 1995, On the quasigeoid to geoid separation, Manuscr Geod 20: 182-192Web of ScienceGoogle 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(2003): 459-464Google Scholar

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

  • Sjöberg L. E., 2009a, The terrain correction in gravimetric geoid determination - is it needed? Geophys J Int 176:14-18Google Scholar

  • Sjöberg L. E., 2009b, Solving the topographic bias as an Initial Value Problem.Art. Sat. 44(3): 77-84Google Scholar

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

  • Sjöberg L. E. 2012, The geoid-to-quasigeoid difference using an arbitrary gravity reduction model. Stud Geophys Geod 56(2012): 929-933CrossrefWeb of ScienceGoogle Scholar

  • Sjöberg L E, 2013. The geoid or quasigeoid- which reference surface should be preferred for a national height system? J Geod Sci 3: 103-109Google Scholar

  • Sjöberg, L. E., 2015a, Rigorous geoid-from-quasigeoid corrections using gravity disturbances. J Geod Sci 5:115-118Google Scholar

  • Sjöberg L. E., 2015b, The topographic bias in Stokes’ formula vs. the error of analytical continuation by an Earth Gravitational Modelare they the same? J Geod Sci 5:171-179Google Scholar

  • Sjöberg L. E., 2018, On the topographic bias and density distribution in modelling the geoid and orthometric heights, J Geod Sci 8: 30-33Web of ScienceGoogle Scholar

  • Sjöberg L. E., Bagherbandi M., 2017, Gravity Inversion and Integration- Theory and Applications in Geodesy and Geophysics, Springer Int. Publ. AG, Cham, SwitzerlandGoogle Scholar

  • Vanicek P., Kingdon R., Santos M., 2012, Geoid versus quasigeoid: a case of physics versus geometry, Contr. Geophys. & Geod. 42(1): 101-117.Google Scholar

About the article

Received: 2018-08-06

Accepted: 2018-11-06

Published Online: 2018-12-31

Published in Print: 2018-12-01

Citation Information: Journal of Geodetic Science, Volume 8, Issue 1, Pages 115–120, ISSN (Online) 2081-9943, DOI: https://doi.org/10.1515/jogs-2018-0011.

Export Citation

© by Lars E. Sjöberg, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in