Journal of Geodesy

, Volume 83, Issue 9, pp 829–847 | Cite as

On the geoid–quasigeoid separation in mountain areas

  • Jakob FluryEmail author
  • Reiner Rummel
Original Article


The separation between the reference surfaces for orthometric heights and normal heights—the geoid and the quasigeoid—is typically in the order of a few decimeters but can reach nearly 3 m in extreme cases. The knowledge of the geoid–quasigeoid separation with centimeter accuracy or better, is essential for the realization of national and international height reference frames, and for precision height determination in geodetic engineering. The largest contribution to the geoid–quasigeoid separation is due to the distribution of topographic masses. We develop a compact formulation for the rigorous treatment of topographic masses and apply it to determine the geoid–quasigeoid separation for two test areas in the Alps with very rough topography, using a very fine grid resolution of 100 m. The magnitude of the geoid–quasigeoid separation and its accuracy, its slopes, roughness, and correlation with height are analyzed. Results show that rigorous treatment of topographic masses leads to a rather small geoid–quasigeoid separation—only 30 cm at the highest summit—while results based on approximations are often larger by several decimeters. The accuracy of the topographic contribution to the geoid–quasigeoid separation is estimated to be 2–3 cm for areas with extreme topography. Analysis of roughness of the geoid–quasigeoid separation shows that a resolution of the modeling grid of 200 m or less is required to achieve these accuracies. Gravity and the vertical gravity gradient inside of topographic masses and the mean gravity along the plumbline are modeled which are important intermediate quantities for the determination of the geoid–quasigeoid separation. We conclude that a consistent determination of the geoid and quasigeoid height reference surfaces within an accuracy of few centimeters is feasible even for areas with extreme topography, and that the concepts of orthometric height and normal height can be consistently realized and used within this level of accuracy.


Geoid Quasigeoid Orthometric height Normal height Topographic masses Mean gravity along plumbline 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Baeschlin CF (1948) Lehrbuch der Geodäsie. Orell Füssli, ZürichGoogle Scholar
  2. Brandstätter G, Wunderlich T (1982) Die Bestimmung der höchsten Gipfelkote Österreichs. Allg Vermessungs Nachr 89(8–9): 309–320Google Scholar
  3. Bretterbauer K (1986) Das Höhenproblem in der Geodäsie. Österreichische Zeitschrift für Vermessungswesen und Photogrammetrie 74(4): 205–215Google Scholar
  4. Flury J (2002) Schwerefeldfunktionale im Gebirge—Modellierungsgenauigkeit, Messpunktdichte und Darstellungsfehler am Beispiel des Testnetzes Estergebirge. Deutsche Geodätische Kommission C 557Google Scholar
  5. Flury J (2006) Short wavelength spectral properties of gravity field quantities. J Geod 79(10–11): 624–640 doi: 10.1007/s00190-005-0011-y CrossRefGoogle Scholar
  6. Flury J, Peters T, Schmeer M, Timmen L, Wilmes H, Falk R (2007) Precision gravimetry in the new Zugspitze calibration system. In: Proceed IGFS06Google Scholar
  7. Flury J, Rummel R (2008) The contribution of topographic masses to the geoid–quasigeoid separation. Deutsche Geodätische Kommission, in preparationGoogle Scholar
  8. Flury J, Gerlach C, Hirt C, Schirmer U (2008) Heights in the Bavarian Alps: Mutual validation of GPS, levelling, gravimetric and astrogeodetic quasigeoids. Proceedings GRF2006 (submitted)Google Scholar
  9. Heiskanen W, Moritz H (1967) Physical geodesy. Freeman, San FranciscoGoogle Scholar
  10. Helmert FR (1884) Die mathematischen und physikalischen Theorien der höheren Geodäsie, vol 2. Teubner, LeipzigGoogle Scholar
  11. Helmert FR (1890) Die Schwerkraft im Hochgebirge, insbesondere in den Tyroler Alpen. Veröff. des Königl. Preuss. Geod. InstGoogle Scholar
  12. Hirt C, Flury J (2007) Astronomical-topographic levelling using high-precision astrogeodetic vertical deflections and digital terrain model data. J Geod doi: 10.1007/s00190-007-0173-x
  13. Hofmann-Wellenhof B, Moritz H (2005) Physical geodesy, 2nd edn. Springer, BerlinGoogle Scholar
  14. Höggerl N, Ruess D (2006) A new height system for Austria. Presentation DGK Seminar Vertical Datum, BonnGoogle Scholar
  15. Jin Y, McNutt MK, Zhu Y (1994) Evidence from gravity and topography data for folding of Tibet. Nature 371: 669–674CrossRefGoogle Scholar
  16. Kingdon R, Vaníc̆ek P, Santos M, Ellman A, Tenzer R (2005) Toward an improved height system for Canada. Geomatica 59(3): 241–250Google Scholar
  17. Ledersteger K (1968) Astronomische und Physikalische Geodäsie (Erdmessung). In: Jordan W, Eggert E, Kneissl M (eds) Handbuch der Vermessungskunde, vol V. Metzler, StuttgartGoogle Scholar
  18. Mader K (1954) Die orthometrische Schwerekorrektion des Präzisions-Nivellements in den Hohen Tauern. Österreichische Zeitschrift für Vermessungswesen, special issue 15Google Scholar
  19. Marti U (1998) Geoid der Schweiz 1997. Geodätisch-Geophysikalische Arbeiten in der Schweiz, vol 56. Swiss Geodetic CommissionGoogle Scholar
  20. Marti U (2005) Comparison of high precision geoid models in Switzerland. In: Tregonig P, Rizos C (eds) Dynamic planet. Springer, BerlinGoogle Scholar
  21. Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74(7–8): 552–560 doi: 10.1007/s001900000116 CrossRefGoogle Scholar
  22. Niethammer T (1932) Nivellement und Schwere als Mittel zur Berechnung wahrer Meereshöhen. Swiss Geodetic CommissionGoogle Scholar
  23. Niethammer T (1939) Das astronomische Nivellement im Meridian des St. Gotthard, Part II, Die berechneten Geoiderhebungen und der Verlauf des Geoidschnittes. Astronomisch-Geodätische Arbeiten in der Schweiz, vol 20. Swiss Geodetic CommissionGoogle Scholar
  24. Rapp RH (1997) Use of potential coefficient models for geoid undulation determinations using a spherical harmonic representation of the height anomaly/geoid undulation difference. J Geod 71: 282–289CrossRefGoogle Scholar
  25. Santos M, Vaníc̆ek P, Featherstone WE, Kingdon R, Ellmann A, Martin BA, Kuhn M, Tenzer R (2006) Relation between the rigorous and Helmert’s definitions of orthometric heights. J Geod 80(12): 691–704 doi: 10.1007/s00190-006-0086-0 CrossRefGoogle Scholar
  26. Sünkel H (1986) . 74(2): 77–93Google Scholar
  27. Sünkel H, Bartelme N, Fuchs H, Hanafy M, Schuh WD, Wieser M (1987) The gravity field in Austria. In: Austrian Geodetic Commission (ed) The gravity field in Austria. Geodätische Arbeiten Österreichs für die Intenationale Erdmessung, Neue Folge, vol IV, pp 47–75Google Scholar
  28. Tenzer R (2004) Discussion of mean gravity along the plumbline. Stud Geophys Geod 48(2): 309–330 doi: 10.1023/B:SGEG.0000020835.10209.7f CrossRefGoogle Scholar
  29. Tenzer R, Vaníc̆ek P, Santos M, Featherstone WE, Kuhn M (2005) The rigorous determination of orthometric heights. J Geod 79: 82–92 doi: 10.1007/s00190-005-0445-2 CrossRefGoogle Scholar
  30. Torge W (2001) Geodesy, 3rd edn. de Gruyter, BerlinGoogle Scholar
  31. Tsoulis D (2001) Terrain correction computations for a densely sampled DTM in the Bavarian Alps. J Geod 75: 291–307CrossRefGoogle Scholar
  32. Wirth B (1990) Höhensysteme, Schwerepotentiale und Niveauflächen. Geodätisch-Geophysikalische Arbeiten in der Schweiz, vol 42. Swiss Geodetic CommissionGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Center for Space ResearchUniversity of TexasAustinUSA
  2. 2.Institute for Astronomical and Physical GeodesyTechnische Universität MünchenMunichGermany

Personalised recommendations