. Estimation of elevation dependent deformations of a parabolic reflector of a large radio telescope 1 Christoph Holst & Heiner Kuhlmann JISDM, 2-4 November 2011, Hong Kong, China
. Motivation Laser Scanner Measurements 2 Estimation of the Deformation Results of the Adjustment Discussion and Conclusion
Motivation Very Long Baseline Interferometry (VLBI) 3 goal: estimation of baseline between telescopes (σ = 1 mm) observations: time difference t of signal arrival times time difference t depends on (among other things) signal path focal length & surface of main reflector possible deformations influence baseline estimation
Motivation possible deformations of the main reflector 1. whole main reflector: variation of focal length 90 elevation: flattening = increase of focal length 0 elevation: in-folding = decrease of focal length 4 2. local deformation patterns on the surface systematic residuals from best-fit surface = detection by high-resolution point cloud of laser scanner
Motivation radio telescope of investigation 5 Effelsberg radio telescope, Germany diameter: 100 m, focal length: 30 m, weight: 3200 t one of the largest fully steerable radio telescopes on earth significant deformation expected
. Motivation Laser Scanner Measurements 6 Estimation of the Deformation Results of the Adjustment Discussion and Conclusion
Laser Scanner Measurements choice of sanner and observing station 7 Leica HDS 6100 σs = s 0.1 mm/m, σ β = σ t = 125 µrad mounted on subreflector standing axis upside-down at 90 elevation standing axis nearly horizontal at 7.5 elevation no control points visible (outside moving construction) variable geodetic datum
Laser Scanner Measurements measurement concept 7 different elevation angles (90, 75, 60, 45, 30, 15, 7.5 ) 30 min per elevation angle 8 mm 8 mm grid of point cloud 370 Mio points per elevation angle 8
. Motivation Laser Scanner Measurements 9 Estimation of the Deformation Results of the Adjustment Discussion and Conclusion
Estimation of the Deformation g(x i, p) = x i 2 + yi 2 4f z i = 0... rotational paraboloid x i = [x i, y i, z i ] T... coordinates inside paraboloid system f... focal length 10 x i = R ϕ,x R ϕ,y X i x v... transformation X i = [X i, X i, Z i ] T... coordinates inside scanner system ϕ x, ϕ y... rotation x v = [x v, y v, z v ] T... translation p = [x v, y v, z v, ϕ x, ϕ y, f ] T... extrinsic & intrinsic parameters
Estimation of the Deformation orthogonal distance regression (ODR) separation of variables in two groups parameters p orthogonal contacting points X c parameter estimation by two different iterations 1. inner iteration: calculation of orthogonal contacting points X c 2. outer iteration: minimization of distances between X and X c 11 Gauß-Helmert model vs. orthogonal distance regression place of linearization coordinates paraboloid discrepancies algebraic geometric estimation results biased unbiased
Estimation of the Deformation preprocessing reduction to average point spacing of 25 cm 25 cm (Cyclone) 1 Mio points remaining stepwise adjustment procedure 1. segmentation of paraboloid (distance & intensity), ODR 2. elimination of outliers (threshold in discrepancies), ODR 3. elimination of points corresponding to the outer ring, ODR 12
Estimation of the Deformation final point cloud 13 500 000 points per elevation only points corresponding to the surface sufficient for parameter estimation detection of (possible) local surface deformations
. Motivation Laser Scanner Measurements 14 Estimation of the Deformation Results of the Adjustment Discussion and Conclusion
Results of the Adjustment extrinsic parameters: variable geodetic datum Elevation x v y v z v ϕ x ϕ y [ ] [m] [m] [m] [rad] [rad] 90-0.0467 0.0171-29.9088 3.1384 0.0094 75-0.0652-0.0187-29.9088 3.1410 0.0106 60-0.0784-0.0490-29.9081 3.1431 0.0113 45-0.0896-0.0790-29.9070 3.1451 0.0121 30-0.0992-0.0976-29.9056 3.1465 0.0126 15-0.1035-0.1087-29.9040 3.1475 0.0130 7.5-0.1051-0.1064-29.9034 3.1475 0.0131 15 but: no conclusion about absolute movement possible
Results of the Adjustment extrinsic parameters: variable geodetic datum 16 whole construction moves surface (line), laser scanner (point) & vertex (triangle) no rigid control points observed only relative deformation detected main reflector subreflector no conclusion about absolute movement possible
Results of the Adjustment variation of focal length elevation f [ ] [mm] 90 0 75-1.2 60-4.5 45-7.0 30-9.8 15-12.6 7.5-11.3 σ f = 0.05 mm, σ f = 0.07 mm too optimistic due to neglect of correlations however: significant variation 17
Results of the Adjustment spatial analysis of possible deformation patterns 18 no significant systematic post-fit discrepancies w no deformation patterns = homologous deformation results are similar for the other elevation angles (15 shown)
. Motivation Laser Scanner Measurements 19 Estimation of the Deformation Results of the Adjustment Discussion and Conclusion
Discussion and Conclusion Effelsberg 100m radio telescope deforms variation of focal length of 12.6 mm relevance for VLBI analysis no detected deformation patterns on surface homologous deformation 20 laser scanner measurements are suitable for deformation analysis high-resolution point cloud can be used for... parameter adjustment: estimation of focal length variation spatial analysis of residuals: detection of surface deformations
. thanks for your attention Christoph Holst & Heiner Kuhlmann Institute of Geodesy and Geoinformation University of Bonn Nußallee 17, 53115 Bonn, Germany tel.: 0049 228/73-3570 mail: c.holst@igg.uni-bonn.de 21