Abstract
The geometrical approach to the least-squares, based on differential geometry with tensor structure and notations, describes the adjustment theory in a simple and plausible manner. The development relies heavily on orthonormal space and surface vectors, and on the extrinsic properties of surfaces linking the two kinds of vectors. In order to relate geometry to adjustments, the geometrical concepts are extended to an n-dimensional space and u- or r-dimensional surfaces, where n is the number of observations, u is the number of parameters in the parametric method and r is the number of conditions in the condition method, with n=u+r. Connection is made to Hilbert spaces by demonstrating that the tensor approach to the least-squares is a classical case of the Hilbert-space approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. MARUSSI:Fondamenti di Geodesia Intrinseca. Publication by the Italian Geodetic Commission, No. III 7. Milano, 1951.
M. HOTINE:Mathematical Geodesy. Monogr. Ser. Vol. 2, Envir. Sci. Serv. Admin., Washington, D.C., 1969.
P. VANÍCEK: “Tensor Structure and the Least Squares”,Bulletin Géodésique, Vol. 53, No. 3, pp. 221–225, 1979.
M. MOHAMMAD-KARIM:Diagrammatic Approach to Solve Least-Squares Adjustment and Collocation Problems. Technical Report No. 83, Department of Surveying Engineering, University of New Brunswick, Fredericton, N.B., Canada, September 1981.
J.M. TIENSTRA: “An Extension of the Technique of the Methods of Le≸st Squares to Correlated Observations”.Bulletin Géodésique, No. 6, 1947.
J.M. TIENSTRA: “The Foundation of the Calculus of Observations and the Method of Le≸st Squares”.Bulletin Géodésique, No. 10, 1948.
J.M. TIENSTRA:Theory of the Adjustment of Normally Distributed Observations. Edited by his friends. N. V. Uitgeverij, Argus, Amsterdam, 1956.
W. BAARDA:Statistical Concepts in Geodesy. Publications on Geodesy by the Netherlands Geodetic Commission, New Series, Vol. 2, No. 4, Delft, 1967.
W. BAARDA:Testing Procedure for Use in Geodetic Networks. Publications on Gecdesy by the Netherlands Geodetic Commission, New Series, Vol. 2, No. 5, Delft, 1968.
W. BAARDA:Vereffeningstheorie. Vol. 1 & 2. Computing Centre of the Delft Geodetic Institute, Technische Hogeschool, Delft, 1967 & 1970.
A.H. KOOIMANS: “Principles of the Calculus of Observations”.Rapport Spécial, Neuvième Congrès International des Géomètres, Pays-Bas, pp. 301–310, 1958.
W.M. KAULA:Theory of Satellite Geodesy. Blaisdell Publ. Co., Waltham, Mass., 1966.
H. SCHEFFÉ:The Analysis of Variance, John Wiley & Sons, Inc., New York, N.Y., 1967.
G. BLAHA:Analysis Linking the Tensor Structure to the Least-Squares Method. AFGL Technical Report—in press, Air Force Geophysics Laboratory, Hanscom AFB, Mass., 1984.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
BLAHA, G. Tensor structure applied to the least-squares method, revisited. Bull. Geodesique 58, 1–30 (1984). https://doi.org/10.1007/BF02521753
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02521753