Abstract
When height networks are being adjusted, many geodesists advocate the approach where the adjustment should be done by using geopotential numbers rather than the orthometric or normal heights used in practice. This is based on a conviction that neither orthometric nor normal heights can be used for the adjustment because these height systems are not holonomic, meaning–among other things–that height increments (orthometric or normal) when summed around a closed loop do not sum up to zero. If this was the case, then the two height systems could not be used in the adjustment; the non-zero loop closure would violate the basic, usually unspoken, assumption behind the adjustment, namely that the model claiming that height differences are observable is correct. In this paper, we prove in several different ways that orthometric and normal heights are theoretically just as holonomic as the geopotential numbers are, when they are obtained from levelled height differences using actual gravity values. This disposes of the argument that geopotential numbers should be used in the adjustment. Both orthometric and normal heights are equally qualified to be used in the adjustment directly.
Similar content being viewed by others
References
Andersen OB, Knudsen P (1982) Global marine gravity field from the ERS1 and GEOSAT geodetic mission altimetry. J Geophys Res 103(B4):8129-8137
Grafarend EW (1975) Cartan frames and a foundation of physical geodesy. Methoden und Verfahren der Mathematischen Physic/Band 12, Brosowski B, Martensen E (eds), BI-Verlag, Mathematical Geodesy, Mannheim, pp 179-208
Grafarend EW, Suffys R, You RJ (1995) Prospective heigts in geometry and gravity space. Allgemeine Vermessungs-Nachrichten 102:382-403
Grafarend EW (1997) Field lines of gravity, their curvature and torsion, the Lagrange and the Hamilton equations of the plumbline. Annali di Geofisica 40(5):1233-1247
Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San Francisco, 364pp
Hotine M (1969) Mathematical geodesy. ESSA Monograph 2, US Department of Commerce, Washington, 416 pp
Kaplan W (1991) Advanced calculus, 4th ed., Addison- Wesley, Reading, MA, 746 pp
MacMillan WD (1958) The theory of the potential. Dover Publications, New York
Marussi A (1985) Intrinsic geodesy. Springer-Verlag, Berlin, 240 pp
Miranda C (1970) Partial differential equations of elliptic type. second revised edition. Translated from Italian by Zane C. Motteler. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York
Sansò F (1977) The geodetic boundary value problem in gravity space. Memorie Scienze Fisiche, Accademia Nazionale dei Lincei
Vanícek P, Krakiwsky EJ (1986) Geodesy: the concepts, second edition. North-Holland Publishing, Amsterdam
Vanícek P (1982) To the problem of holonomity of height systems Letter to the editor. The Canadian Surveyor 36:122-123
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sansò, F., Vanícek, P. The orthometric height and the holonomity problem. J Geodesy 80, 225–232 (2006). https://doi.org/10.1007/s00190-005-0015-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-005-0015-7