Abstract
The Bruns formula is generalized to three dimensions with the derivation of equations expressing the height anomaly vector or the geoid undulation vector as a function of the disturbing gravity potential and its spatial derivatives. It is shown that the usual scalar Bruns formula provides not the separation along the normal to the reference ellipsoid but the component of the relevant spatial separation along the local direction of normal gravity. The above results which hold for any type of normal potential are specialized for the usual Somigliana-Pizzetti normal field so that the components of the geoid undulation vector are expressed as functions of the parameters of the reference ellipsoid, the disturbing potential and its spatial derivatives with respect to three types of curvilinear coordinates, ellipsoidal, geodetic and spherical. Finally the components of the geoid undulation vector are related to the deflections of the vertical in a spherical approximation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. BLAHA: “Second-Order Derivatives in a Local Frame Developed in Spheroidal and Spherical Coordinates”. Bull. Géod., Vol.54, No. 1, pp. 119–135 (1980).
F. BOCCHIO: “First-Order Numerical Computations of the Three-Dimensional Ellipsoidal Field”. Geophys. J. R. astr. Soc., Vol.49, pp. 633–644 (1977).
A. BODE and E. GRAFAREND: “The spacelike Molodenski problem including the rotational term of the gravity potential”. Manus. Geod., Vol.6, No. 1, pp. 33–61 (1981).
K. DANAS and A. DERMANIS: “Numerical Realization of Isoparametric Telluroid Mapping”. Quater. Geod., Vol.4, No. 2, pp. 149–168 (1983).
A. DERMANIS: “The Geodetic Boundary Value Problem Linearized with Respect to the Somigliana-Pizzetti Normal Field”. Manus. Geod., Vol.9, No. 2, pp. 77–92 (1984).
A. DERMANIS and E. LIVIERATOS: “Deformation Analysis of Isoparametric Telluroid Mappings”. Boll. di Geod. e Sci. Affini, Vol.43, No. 4, pp. 301–312 (1984).
E. GRAFAREND: “The definition of the Telluroid”. Bull. Géod., Vol.52, No. 1, pp. 25–38 (1978a).
E. GRAFAREND: “Dreidimensionale geodätische Abbildungsgleichungen und die Näherungsfigur der Erde”, ZfV, Vol.103, pp. 132–140 (1978b).
E. GRAFAREND: “The Bruns Transformation and a Dual Setup of Geodetic Observational Equations”. NOAA Tech. Rep. NOS 85 NGS 16 (1980).
E. GRAFAREND: “Spacetime telluroid versus spacetime geoid or the Bruns-Love dialogue”. Commemorative volume on the occasion of Erik Tengström's 70th birthday, Report No. 19, Department of Geodesy, Uppsala University, pp. 105–127 (1983).
E. GRAFAREND, B. HECK and E.H. KNICKMEYER: “The free versus fixed geodetic boundary value problem for different combinations of geodetic observables”. Bull. Géod., Vol.59, No. 1, pp. 11–32 (1985).
H. GUGGENHEIMER: “Differential Geometry”. Mc Graw-Hill, (1963), Dover reprint, 1981.
W. HEISKANEN and H. MORITZ: “Physical Geodesy”. Freeman Publ. Co., San Francisco (1967).
M. HOTINE. “Mathematical Geodesy”. ESSA Monograph No. 2, U.S. Dept. of Commerce Washington D.C. (1969).
T. KRARUP. “Letters on Molodensky's Problem I–IV”. Communications to the members of IAG Special Study Group 4.31. Unpublished. (1973).
E.H. KNICKMEYER: “Eine approximative Lösung der allgemeinen linearen Geodätischen Randwertaufgabe durch Reihenentwicklungen nach Kugelfunktionen”. Deutsche Geod. Komm, Reihe C, Heft Nr. 304 (1984).
E.H. KNICKMEYER: “Eine Lösung der Grundgleichung der Physikalischen Geodäsie für beliebege Ränder und Normalfelder durch Kugelfunktionsentwicklungen”. ZfV, Vol.111, pp. 70–82 (1986).
A. MARUSSI: “Principi di Geodesia intrinseca applicati al campo di Somigliana”. Boll. di Geod. e Sci. Affini, Vol.2, pp. 1–8 (1950). English translation in Marussi (1985).
A. MARUSSI: “Fondamenti di Geodesia Intrinseca”. Comm. Geod. Italiana, Memorie No. 7. (1951). English translation in Marussi (1985).
A. MARUSSI: “Intrinsic Geodesy”. Springer Verlag (1985).
H. MORITZ: “Advanced Physical Geodesy”. Wichmann Verlag, Karlsruhe (1980).
F. SANSÒ: “Dual relations in geometry and gravity space”. ZfV, Vol.105 pp. 279–290 (1980).
F. SANSÒ: “Recent advances in the theory of the geodetic boundary value problem”. Reviews of Geophysics and Space Physics, Vol.19, pp. 437–449 (1981).
C.C. TSCHERNING: “Computation of the Second-Order Derivatives of the Normal Potential Based on the Representation by a Legendre Series”. Manus. Geod., Vol.1, No. 1, pp. 71–92 (1976a).
C.C. TSCHERNING: “On the Chain-Rule Method for Computing Potential Derivatives”. Manuscr. Geod., Vol.1, No. 2, pp. 125–141 (1976b).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dermanis, A. The Bruns formula in three dimensions. Bull. Geodesique 61, 297–309 (1987). https://doi.org/10.1007/BF02520556
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02520556