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The rank deficiency in estimation theory and the definition of reference systems

  • Athanasios Dermanis
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 127)

Abstract

The concept of reference frame is examined from the viewpoint of both geophysical and geodetic applications. The concept of parameter estimability in linear models is related to the deterministic concept of determinability in linear or nonlinear improper models without full rank. The geometry of such models is investigated in its linear and nonlinear aspects with emphasis on the common invariance characteristics of observable and estimable parameters and is applied to the choice of datum problem in geodetic networks. The time evolution of the reference frame is investigated and optimal choices are presented from different equivalent points of view. The transformation of a global geodetic network into an estimate of a geocentric Tisserand frame for the whole earth is investigated and a solution is given for the rotational part. The translation to a geocentric frame poses the problem of the estimability of the geocenter coordinates and the more general problem of estimability of coefficients of an unknown function of position, having as domain the frame-dependent coordinates.

Keywords

Reference frames estimable parameters earth rotation. 

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References

  1. Altamini, Z., P. Sillard, C. Boucher (2002): ITRF 2000: A new release of the International Terrestrial Reference Frame for earth science applications. Journal of Geophysical Research, vol. 107, No. B10, p. 2214.Google Scholar
  2. Baarda, W. (1995): Linking up spatial models in geodesy. Extended S-Transformations. Netherlands Geodetic Corn-mission, Publ. in Geodesy, New Series, no. 41, Delft.Google Scholar
  3. Bjerhammar, A. (1951): Rectangular reciprocal matrices with special emphasis to geodetic calculations. Bulletin Géodésique, 1951, 188–220.CrossRefGoogle Scholar
  4. Dermanis, A. (1991): A Unified Approach to Linear Estimation and Prediction. Presented at the 20th IUGG General Assembly, Vienna, August 1991.Google Scholar
  5. Dermanis, A. (1995): The Nonlinear and Space-time Geodetic Datum Problem. Intern. Meeting “Mathematische Methoden der Geodäsie”, Oberwolfach, 1–7 Oct. 1995.Google Scholar
  6. Dermanis, A. (1998): Generalized inverses of nonlinear mappings and the nonlinear geodetic datum problem. Journal of Geodesy, 72, 71–100.CrossRefGoogle Scholar
  7. Dermanis, A. (2000): Establishing global reference frames. Nonlinear, temporal, geophysical and stochastic aspects. In: M. Sideris (ed.) “Gravity, Geoid and Geodynamics 2000”, p. 35–42, IAG Symposia, Vol. 123, Springer, Heidelberg 2001.Google Scholar
  8. Dermanis, A. (2001): Global reference frames: Connecting observation to theory and geodesy to geophysics. IAG Scientific Assembly, 2–8 Sept. 2001, Budapest, Hungary.Google Scholar
  9. Dermanis, A. (2002): On the maintenance of a proper reference frame for VLBI and GPS global networks. In: E.Grafarend, F.W. Krumm, V.S. Schwarze (eds) “Geodesy — the Chalenge of the 3rd Millenium”, p. 61–68, Springer, Heidelberg.Google Scholar
  10. Grafarend, E. and B. Schaffrin (1976): Equivalence of estimable quantities and invariants in geodetic networks. Zeitschrffir Vemessungswesen, 101, 11, 485–491.Google Scholar
  11. Meissl, P. (1965): Über die innere Genauigkeit dreidimensionaler Punkthaufen. Zeitschrift fir Vemessungswesen, 90, 4, 109–118.Google Scholar
  12. Meissl, P. (1969): Zusammenfassung und Ausbau der inneren Fehlertheorie eines Punkthaufens. Deutsche Geodät. Komm., Reihe A, Nr. 61, 8–21.Google Scholar
  13. Moritz, H. and I.I. Mueller (1987): Earth Rotation. Theory and Observation. Ungar, New York.Google Scholar
  14. Munk, W.H. and G.J.F. MacDonald (1960): The Rotation of the Earth. A Geophysical Discussion. Cambridge University Press.Google Scholar
  15. Penrose, R (1955): A generalized inverse for matrices. Proc. Cambridge Philos. Soc., 51, 406–413.Google Scholar
  16. Schaffrin, B. and E. Grafarend (1986): Generating classes of equivalent linear models by nuisance parameter elimination–applications to GPS observations. Manuscripta geodaetica, 11 (1986), 262–271.Google Scholar
  17. Schaffrin, B. (2003): Some generalized equivalence theorems for least-squares adjustment. This volume.Google Scholar
  18. Sillard, P. and C. Boucher (2001): A review of algebraic constraints in terrestrial reference frames datum definition. Journal of Geodesy, 75, 63–73.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Athanasios Dermanis
    • 1
  1. 1.Department of Geodesy and SurveyingThe Aristotle University of ThessalonikiGreece

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