Alternative Geometric Determination of Altazimuthal‐Distance Covariance Matrices
Publication: Journal of Surveying Engineering
Volume 113, Issue 2
Abstract
Conventional equations for determining the variance‐covariance matrix of vertical angles, geodetic azimuths and distances are based on the standard law of propagation of covariance using the Jacobian matrix of the corresponding functional relationships. A conceptually simpler geometric approach exclusively dependent on the notion of rotation matrices is presented here. The method completely avoids the cumbersome requirement of taking partial derivatives of non‐linear expressions. As an added advantage the method contributes to clarify several points related to dimensional transformations between linear and angular units.
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References
1.
Altaian, W., and de Oliveira, A. M. (1977). “Physical components of tensors.” Tensor, 31, 141–148.
2.
Arnold, K. (1964). “Zur Bestimmung geodätischer Azimute aus Simultanbeobachtungen von Satelliten.” Gerlands Beiträdge Geophysik, 74, 441–430.
3.
Goad, C. C. (1985). “Precise relative position determination using global position system carrier phase measurements in a nondifference mode.” Proc. 1st Int. Symp. on Precise Positioning with the Global Positioning Syst., U.S. Dept. of Commerce, NOAA, NOS, Rockville, Md.
4.
Groten, E. (1969). “Zur definition des mittleren punktfehlers.” Zeitschr. f. Vermess., 94, 455–457.
5.
Halmos, F., and Szádeczky‐Kardoss, Gy. (1974). “Azimutberechnung aus simultanen Satelliten‐Beobachtungen und geographischen Koordinaten.” Acta Geod. Geophys. et Mont., 9, 11–27.
6.
Hein, G., and Landau, H. (1983). “A contribution to operational geodesy. Part 3: OPERA—A multi‐purpose program for operational adjustment of geodetic observations of terrestrial type.” Detsche Geodätische Kommission, Reihe B. Heft Nr., 264, Munich, W. Germany.
7.
Heiskanen, W. A., and Moritz, H. (1967). Physical geodesy, W. H. Freeman and Co., San Francisco. Reprinted 1979, Inst., of Physical Geodesy, Tech. Univ., Graz, Austria; available from Dept. of Geod. Science and Surveying, The Ohio State Univ., Columbus, Ohio.
8.
Kaniuth, K., and Zernecke, R. (1979). “The azimuth Hohenpeissenberg‐Tromsø derived from simultaneous direction measurements to satellites. (An orientation control for the European triangulation network).” Optimization of Design and Computation of Control Networks, F. Halmos and J. Somogyi, Eds., Akadémiai Kiadó, Budapest, Hungary.
9.
Köhr, J. (1969). “Uüber mittlere Punktfehler.” Zeitschr. f. Vermess., 94, 445–455.
10.
Lamé, G. (1840). “Mémoire sur les coordonnées curvilignes.” Journal de Mathématiques Pures et Appliquées, 5, 313–347.
11.
McConnell, A. J. (1931). Applications of the absolute differential calculus. Reprinted 1957, under title Applications of tensor analysis, Dover Publications, Inc., New York, N.Y.
12.
Meissl, P. (1982). “Least squares adjustment. A modern approach.” Mitteilungen der geodatischen, Folge 43, Institute der Technishen, Universitat Graz, Graz, Austria.
13.
Mikhail, E. M. (1976). Observations and Least Squares, Harper and Row Publishers, Inc., New York, N.Y.
14.
Molodenskii, M. S., Eremeev, V. F., and Yurkina, M. I. (1962). Methods for Study of the External Gravitational Field and Figure of the Earth, (in Russian), translation, Nat. Tech. Info. Service, Springfield, Va.
15.
Mueller, I.I. (1969). Spherical and Practical Astronomy as Applied to Geodesy. Frederic Ungar Publishing Co., New York, N.Y.
16.
Nassau, J. J. (1948). Practical Astronomy. McGraw‐Hill Publishing Co., Inc., New York, N.Y.
17.
Pope, A. J. (1971). “Transformation of covariance matrices due to changes in minimal control.” Paper presented at AGU Fall Meeting, San Francisco; available from Nat. Geodetic Info. Center, NOAA, NOS, Rockville, Md.
18.
Rapp, R. H. (1975). Geometric geodesy. Vols. I & II; Dept. of Geod. Science and Surveying, Ohio State Univ., Columbus, Ohio.
19.
Remondi, B. W. (1984). “Using the Global Position System (GPS) phase observable for relative geodesy: modelling, processing and results,” thesis presented to the University of Texas at Austin, Austin, Tex., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
20.
Roelofs, R. (1950). Astronomy applied to land surveying, Ahrend & Zoon, Amsterdam, Netherlands.
21.
Sigl, R. (1969). “Azimutbestimmung mittels simultaner, Satellitenbeobachtungen.” Zeitschr. f. Vermess., 94, 385–389.
22.
Soler, T. (1976). “On differential transformations between Cartesian and curvilinear (geodetic) coordinates.” Report No. 236, Dept. of Geodetic Science, Ohio State Univ., Columbus, Ohio.
23.
Soler, T., and Chin, M. (1985). “On transformation of covariance matrices between local Cartesian coordinate systems and commutative diagrams.” Technical Papers, 45th Annual Meeting, ACSM, Washington, D.C., 393–406.
24.
Soler, T., Hothem, L. D., and Fury, R. J. (1986). “Precise geodetic surveying with code and carrier phase tracking GPS receivers,” Proceedings, 4th Int. Geod. Symp. on Satellite Positioning. Apr. 28–May 2, Univ. of Texas at Austin, Austin, Tex.
25.
Uotila, U. A. (1967). Introduction to adjustment computations with matrices, Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.
26.
Vanicek, P., and Krakiwsky, E. J. (1982). Geodesy: the concepts. North‐Holland Publishing Co., New York, N.Y.
27.
Veress, S. (1974). Adjustment by least squares. American Congress of Surveying and Mapping, Washington, D.C.
28.
Wolf, H. (1963). “Die Grundgleichungen der dreidimensionalen Geodaüsie in elementarer Darstellung.” Zeitschr. f. Vermess., 88, 225–233.
29.
Wolf, H. (1975). Ausgleichungsrechnung. Formeln zur praktischen Anwendung. Ferd. Dümmler Verlag, Bonn, W. Germany.
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Copyright © 1987 ASCE.
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Published online: Jun 1, 1987
Published in print: Jun 1987
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