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Using Zonotopes for Overestimation-Free Interval Least-Squares–Some Geodetic Applications

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Reliable Computing

Abstract

Interval methods are very convenient to describe the uncertainty of measurements and derived parameters in engineering sciences. In this paper, the geodetic determination of points in the 2d or 3d Euclidean space by least-squares estimation is studied. Up to now, two problems limited the applicability of interval mathematics. First, due to overestimation, interval boxes are too pessimistic uncertainty measures for point positions. Second, the shape of the interval boxes depends on the orientation of the geodetic coordinate system to parametrise the configuration. It is shown that both problems can be overcome by zonotopes which describe directly the factual range of the leastsquares problem with interval-valued observations. The advantages of this concept are discussed using typical geodetic network scenarios. In addition, the possible benefit for other engineering applications is motivated.

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Authors and Affiliations

  1. Institute of Engineering Geodesy and Measurement Systems IGMS, Graz Technical University, Steyrergasse 30, A-8010, Graz, Austria

    Steffen Schön

  2. Geodätisches Institut, Universität Hannover (GIH), Nienburger Strasse 1, D-30167, Hannover, Germany

    Hansjörg Kutterer

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  1. Steffen Schön
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  2. Hansjörg Kutterer
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Correspondence to Steffen Schön.

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Schön, S., Kutterer, H. Using Zonotopes for Overestimation-Free Interval Least-Squares–Some Geodetic Applications. Reliable Comput 11, 137–155 (2005). https://doi.org/10.1007/s11155-005-3034-4

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  • DOI: https://doi.org/10.1007/s11155-005-3034-4

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