Journal of Geodesy

, Volume 93, Issue 6, pp 849–868 | Cite as

Sub-centimetre geoid

  • Ismael ForoughiEmail author
  • Petr Vaníček
  • Robert W. Kingdon
  • Mehdi Goli
  • Michael Sheng
  • Yosra Afrasteh
  • Pavel Novák
  • Marcelo C. Santos
Original Article


This paper represents a milestone in the UNB effort to formulate an accurate and self-consistent theory for regional geoid determination. To get the geoid to a sub-centimetre accuracy, we had to formulate the theory in a spherical rather than linear approximation, advance the modelling of the effect of topographic mass density, formulate the solid spherical Bouguer anomaly, develop the probabilistic downward continuation approach, incorporate improved satellite determined global gravitational models and introduce a whole host of smaller improvements. Having adopted Auvergne, an area in France as our testing ground, where the mean standard deviation of observed gravity values is 0.5 mGal, according to the Institute Geographique Nationale (Duquenne in Proceedings of the 1st international symposium of the international gravity field service “gravity field of the earth”, International gravity field service meeting, Istanbul, Turkey, 2006), we obtained the standard deviation of the gravity anomalies continued downward to the geoid, as estimated by minimizing the L2 norm of their residuals, to be in average 3 times larger than those on the surface with large spikes underneath the highest topographic points. The standard deviations of resulting geoidal heights range from a few millimetres to just over 6 cm for the highest topographic points in the Alpine region (just short of 2000 m). The mean standard deviations of the geoidal heights for the whole region are only 0.6 cm, which should be considered quite reasonable even if one acknowledges that the area of Auvergne is mostly flat. As one should expect, the main contributing factors to these uncertainties are the Poisson probabilistic downward continuation process, with the maximum standard deviation just short of 6 cm (the average value of 2.5 mm) and the topographic density uncertainties, with the maximum value of 5.6 cm (the average value of 3.0 mm). The comparison of our geoidal heights with the testing geoidal heights, obtained for a set of 75 control points (regularly spaced throughout the region), shows the mean shift of 13 cm which is believed to reflect the displacement of the French vertical datum from the geoid due to sea surface topography. The mean root square error of the misfit is 3.3 cm. This misfit, when we consider the estimated accuracy of our geoid, indicates that the mean standard deviation of the “test geoid” is about 3 cm, which makes it about 5 times less accurate than the Stokes–Helmert computed geoid.


Geoid Downward continuation by least squares Stokes–Helmert method Gravimetric geoid 



University of New Brunswick


Downward continuation


Upward continuation


Least squares


Earth gravitational model


Primary indirect topographic effect


Primary indirect atmospheric effect


Primary indirect density effect


Least-squares downward continuation


Global navigation satellite system


Direct topographic effect


Direct density effect


Direct atmospheric effect


Secondary indirect topographic effect


No-topography anomaly (spherical complete Bouguer gravity anomaly)


Standard deviation


Near-zone (contribution of close gravity data to geoidal heights)


Far-zone (contribution of distant gravity data to geoidal heights)


Root mean square error


Digital terrain model



The authors wish to acknowledge that the final stages of their work were supported by the NSERC Discovery grant to P. Vaníček. P. Novák was supported by the project 18-06943S of the Czech Science Foundation. As the research reported here is based on a hard work of scores of researchers we should mention at least those who contributed the most during the past 25 years. We feel that at least Z. Martinec, J. Huang, M. Najafi, W. Featherstone, S. Wenke, J. Janák and A. Ellmann should be thanked in particular. This research was built on their shoulders.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Geodesy and Geomatics EngineeringUniversity of New BrunswickFrederictonCanada
  2. 2.Department of Civil EngineeringShahrood University of TechnologyShahroodIran
  3. 3.School of Surveying and Geospatial EngineeringUniversity of TehranTehranIran
  4. 4.NTIS – New Technologies for the Information SocietyUniversity of West BohemiaPlzeňCzech Republic

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