Estimation of Focal Length Variations of a 100-m Radio Telescope’s Main Reflector by Laser Scanner Measurements
Publication: Journal of Surveying Engineering
Volume 138, Issue 3
Abstract
Due to gravitation, the main reflector of a radio telescope underlies a deformation that causes a change in focal length depending on the variations of the elevation angle of the telescope. To estimate these gravity dependent deformations of the main reflector of the 100-m radio telescope at Effelsberg, Germany, this study proposes a measurement concept based on a laser scanner being mounted upside down on the subreflector. The measurements that have been performed at seven different elevations between 90 and 7.5° are used to estimate the focal length variation of the main reflector parameterized by a rotational paraboloid. To guarantee reliability of the adjustment, this study uses an orthogonal distance regression (ODR) rather than a classical least squares adjustment in a Gauss-Helmert model and formulates the independence of the focal length estimation from the absolute position and orientation of the main reflector in space as a requirement for a reliable adjustment approach. This investigation attests that the ODR has superior reliability with regard to this criterion. A three-step adjustment procedure based on an alteration of the ODR and several outlier eliminations is used to determine the variations of the focal length due to gravitation. The estimated focal length decreases by a maximum of 12.6 mm when tilting the reflector from 90 to 7.5° elevation angle. The postfit discrepancies between the best-fit paraboloid and the reflector’s surface are Gaussian distributed within the accuracy of the measurements. This face supports the assumption of a homologous deformation of the main reflector.
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Acknowledgments
The writers thank Hans-Peter Helfrich for suggestions and discussions regarding the orthogonal distance regression and Ernst-Martin Blome for taking care of the measurements. Additionally, the writers thank the reviewers for constructive comments that helped to improve the manuscript.
References
Abbondanza, C., and Sarti, P. (2010). “Effects of illumination functions on the computation of gravity-dependent signal path variation models in primary focus and Cassegrainian VLBI telescopes.” J. Geodes.JOGEF8, 84(8), 515–525.
Ahn, S. J. (2008). “Geometric fitting of parametric curves and surfaces.” J. Inf. Process. Syst., 4(4), 153–158.
Ahn, S. J., Rauh, W., Cho, H. S., and Warnecke, H.-J. (2002). “Orthogonal distance fitting of implicit curves and surfaces.” IEEE Trans. Pattern Anal. Mach. Intell.ITPIDJ, 24(5), 620–638.
Ahn, S. J., Rauh, W., and Warnecke, H.-J. (2001). “Least-squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola and parabola.” Pattern Recogn.PTNRA8, 34(12), 2283–2303.
Clark, T. A., and Thomsen, P. (1988). “Deformations in VLBI antennas.” NASA Technical Memorandum 100696, NASA, Greenbelt, Md.
Deutsches Institut für Normung (DIN). (1984). “Begriffe, Kurzzeichen und Formelzeichen im Vermessungswesen—Teil 4: Ausgleichungsrechnung und Statistik.” DIN 18709, Beuth Verlag, Berlin.
Deutsches Institut für Normung (DIN). (2010). “Concepts, abbreviations and symbols in geodesy—Part 4: Adjustment of observations and statistics.” DIN 18709, Beuth Verlag, Berlin.
Dutescu, E., Heunecke, O., and Krack, K. (2009). “Formbestimmung bei Radioteleskopen mittels terrestrischem Laserscanning.” Allgem. Verm. Nachr., 6, 239–245.
Hachenberg, O. (1968). Beiträge zur Radioastronomie. Max-Planck-Institut für Radioastronomie Bonn, Studien zur Konstruktion des 100-m-Teleskops, Vol. 1, Dümmler, Bonn.
Helfrich, H.-P., and Zwick, D. (1993). “A trust region method for implicit orthogonal distance regression.” Numer. AlgorithmsNUALEG, 5(10), 535–545.
Helfrich, H.-P., and Zwick, D. (1995). “Trust region algorithms for the nonlinear least distance problem.” Numer. AlgorithmsNUALEG, 9(1), 171–179.
Helfrich, H.-P., and Zwick, D. (1996). “A trust region algorithm for parametric curve and surface fitting.” J. Comput. Appl. Math., 73(1–2), 119–134.JCAMDI
Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., and Stuetzle, W. (1992). “Surface reconstruction from unorganized points.” Comput. Graph., 26(2), 71–78.CGRADI
Kupferer, S. (2004). “Zur korrekten Linearisierung von nichtlinearen GH-Modellen.” Allgem. Verm. Nachr., 11–12, 394–396.
Leica Geosystems. (2011). “Leica HDS 6100, latest generation of ultra-high speed laser scanner.” 〈www.leica-geosystems.com/hds〉 (May 11, 2011).
Lenzmann, L., and Lenzmann, E. (2004). “Strenge Auswertung des nichtlinearen Gauß-Helmert-Modells.” Allgem. Verm. Nachr., 2, 68–73.
Lösler, M. (2009). “New mathematical model for reference point determination of an azimuth-elevation type radio telescope.” J. Surv. Eng., 135(4), 131–135.JSUED2
Martínez-Morera, D., and Sarlabous, E. (2003). “On the distance from a point to a quadric surface.” Revista Investigacion Operacional, 24(2), 153–161.
Mikhail, E. M., and Ackermann, F. (1976). Observations and least squares, Dun-Donelly, New York.
Moré, J. (1978). “The Levenberg-Marquardt algorithm, implementation and theory.” Numerical analysis, Watson, G. A., ed., Lecture Notes in Mathematics 630, Springer Berlin, Heidelberg, Germany, 105–116.
Max Planck Institute for Radio Astronomy (MPIfR). (2011). “Radio-Observatorium Effelsberg.” 〈http://www.mpifr-bonn.mpg.de/public/eff_d.html〉 (Nov. 5, 2011).
Neitzel, F. (2010). “Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation.” J. Geodes.JOGEF8, 84(12), 751–762.
Niemeier, W. (2008). Ausgleichungsrechnung. Statistische Auswertemethoden, 2nd Ed., de Gruyter, Berlin/New York.
Rouhani, M., and Sappa, A. D. (2009). “A novel approach to geometric fitting of implicit quadrics.” Lect. Notes Comput. Sci., 5807, 121–132.LNCSD9
Sampson, P. D. (1982). “Fitting conic sections to ‘very scattered’ data: An iterative refinement of the Bookstein algorithm.” Comput. Graph. Image Process.CGIPBG, 18(1), 97–108.
Sarti, P., Abbondanza, C., Petrov, L., and Negusini, M. (2011). “Height bias and scale effect induced by antenna gravitational deformations in geodetic VLBI analysis.” J. Geodes.JOGEF8, 85(1), 1–8.
Sarti, P., Vittuari, L., and Abbondanza, C. (2009a). “Laser scanner and terrestrial surveying applied to gravitational deformation monitoring of large VLBI telescopes’ primary reflector.” J. Surv. Eng., 135(4), 136–148.JSUED2
Sarti, P., Abbondanza, C., and Vittuari, L. (2009b). “Gravity-dependent signal path variation in a large VLBI telescope modelled with a combination of surveying methods.” J. Geodes.JOGEF8, 83(11), 1115–1126.
Shankar, N. U. et al. (2009). “Photogrammetric measurements of a 12-metre preloaded parabolic dish antenna.” Proc., National Workshop on the Design of Antenna and Radar Systems (DARS), ISRO Telemetry Tracking and Command Network (ISTRAC), Bangalore, India.
Sovers, O. J., Fanselow, J. L., and Jacobs, C. S. (1998). “Astrometry and geodesy with radio interferometry: experiments, models, results.” Rev. Mod. Phys., 70(4), 1393–1454.RMPHAT
Subrahmanyan, R. (2005). “Photogrammetric measurement of the gravity deformation in a cassegrain antenna.” IEEE Trans. Antennas Propag.IETPAK, 53(8), 2590–2596.
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Published In
Journal of Surveying Engineering
Volume 138 • Issue 3 • August 2012
Pages: 126 - 135
Copyright
© 2012. American Society of Civil Engineers.
History
Received: Aug 11, 2011
Accepted: Jan 30, 2012
Published online: Feb 1, 2012
Published in print: Aug 1, 2012
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