Abstract
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.
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References
Baeschlin CF (1948) Lehrbuch der Geodäsie. Orell Füssli, Zürich
Brandstätter G, Wunderlich T (1982) Die Bestimmung der höchsten Gipfelkote Österreichs. Allg Vermessungs Nachr 89(8–9): 309–320
Bretterbauer K (1986) Das Höhenproblem in der Geodäsie. Österreichische Zeitschrift für Vermessungswesen und Photogrammetrie 74(4): 205–215
Flury J (2002) Schwerefeldfunktionale im Gebirge—Modellierungsgenauigkeit, Messpunktdichte und Darstellungsfehler am Beispiel des Testnetzes Estergebirge. Deutsche Geodätische Kommission C 557
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
Flury J, Peters T, Schmeer M, Timmen L, Wilmes H, Falk R (2007) Precision gravimetry in the new Zugspitze calibration system. In: Proceed IGFS06
Flury J, Rummel R (2008) The contribution of topographic masses to the geoid–quasigeoid separation. Deutsche Geodätische Kommission, in preparation
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)
Heiskanen W, Moritz H (1967) Physical geodesy. Freeman, San Francisco
Helmert FR (1884) Die mathematischen und physikalischen Theorien der höheren Geodäsie, vol 2. Teubner, Leipzig
Helmert FR (1890) Die Schwerkraft im Hochgebirge, insbesondere in den Tyroler Alpen. Veröff. des Königl. Preuss. Geod. Inst
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
Hofmann-Wellenhof B, Moritz H (2005) Physical geodesy, 2nd edn. Springer, Berlin
Höggerl N, Ruess D (2006) A new height system for Austria. Presentation DGK Seminar Vertical Datum, Bonn
Jin Y, McNutt MK, Zhu Y (1994) Evidence from gravity and topography data for folding of Tibet. Nature 371: 669–674
Kingdon R, Vaníc̆ek P, Santos M, Ellman A, Tenzer R (2005) Toward an improved height system for Canada. Geomatica 59(3): 241–250
Ledersteger K (1968) Astronomische und Physikalische Geodäsie (Erdmessung). In: Jordan W, Eggert E, Kneissl M (eds) Handbuch der Vermessungskunde, vol V. Metzler, Stuttgart
Mader K (1954) Die orthometrische Schwerekorrektion des Präzisions-Nivellements in den Hohen Tauern. Österreichische Zeitschrift für Vermessungswesen, special issue 15
Marti U (1998) Geoid der Schweiz 1997. Geodätisch-Geophysikalische Arbeiten in der Schweiz, vol 56. Swiss Geodetic Commission
Marti U (2005) Comparison of high precision geoid models in Switzerland. In: Tregonig P, Rizos C (eds) Dynamic planet. Springer, Berlin
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
Niethammer T (1932) Nivellement und Schwere als Mittel zur Berechnung wahrer Meereshöhen. Swiss Geodetic Commission
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 Commission
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: 282–289
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
Sünkel H (1986) . 74(2): 77–93
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 47–75
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
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
Torge W (2001) Geodesy, 3rd edn. de Gruyter, Berlin
Tsoulis D (2001) Terrain correction computations for a densely sampled DTM in the Bavarian Alps. J Geod 75: 291–307
Wirth B (1990) Höhensysteme, Schwerepotentiale und Niveauflächen. Geodätisch-Geophysikalische Arbeiten in der Schweiz, vol 42. Swiss Geodetic Commission
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Flury, J., Rummel, R. On the geoid–quasigeoid separation in mountain areas. J Geod 83, 829–847 (2009). https://doi.org/10.1007/s00190-009-0302-9
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DOI: https://doi.org/10.1007/s00190-009-0302-9