Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Applied Geodesy

Editor-in-Chief: Kahmen, Heribert / Rizos, Chris


CiteScore 2018: 1.61

SCImago Journal Rank (SJR) 2018: 0.532
Source Normalized Impact per Paper (SNIP) 2018: 1.064

Online
ISSN
1862-9024
See all formats and pricing
More options …
Volume 13, Issue 2

Issues

The stochastic model for Global Navigation Satellite Systems and terrestrial laser scanning observations: A proposal to account for correlations in least squares adjustment

Gael KermarrecORCID iD: https://orcid.org/0000-0001-5986-5269 / Ingo Neumann / Hamza Alkhatib / Steffen Schön
Published Online: 2019-01-24 | DOI: https://doi.org/10.1515/jag-2018-0019

Abstract

The best unbiased estimates of unknown parameters in linear models have the smallest expected mean-squared errors as long as the residuals are weighted with their true variance–covariance matrix. As this condition is rarely met in real applications, the least-squares (LS) estimator is less trustworthy and the parameter precision is often overoptimistic, particularly when correlations are neglected. A careful description of the physical and mathematical relationships between the observations is, thus, necessary to reach a realistic solution and unbiased test statistics. Global Navigation Satellite Systems and terrestrial laser scanners (TLS) measurements show similarities and can be both processed in LS adjustments, either for positioning or deformation analysis. Thus, a parallel between stochastic models for Global Navigation Satellite Systems observations proposed previously in the case of correlations and functions for TLS range measurements based on intensity values can be drawn. This comparison paves the way for a simplified way to account for correlations for a use in LS adjustment.

Keywords: stochastic model; terrestrial laser scanner; GNSS; variance model; correlation model

References

  • [1]

    Alkhatib H, Kutterer H. Estimation of Measurement Uncertainty of kinematic TLS Observation Process by means of Monte-Carlo Methods. Journal of Applied Geodesy 2013, 7, 2, 125–134.Google Scholar

  • [2]

    Alkhatib H, Neumann I, Kutterer H. Uncertainty modeling of random and systematic errors by means of Monte Carlo and fuzzy techniques. J Appl Geodesy 2009, 3, 67–79.Google Scholar

  • [3]

    Alkhatib H, Paffenholz J-A, Kutterer H. Sequential Monte Carlo filtering for nonlinear GNSS trajectories. In: Sneeuw N, Novák P, Crespi M und Sansò F (Hg.) VII Hotine-Marussi Symposium on Mathematical Geodesy. Proceedings of the Symposium in Rome 2009, International Association of Geodesy (IAG). 1st Edition. Springer Verlag, Berlin, Heidelberg (International Association of Geodesy Symposia, 137:81–86).

  • [4]

    Amiri-Simkooei AR, Teunissen PJG, Tiberius C. Application of Least-Squares variance component estimation to GPS observables. J Surv Eng 2009, 135, 4, 149–160.CrossrefGoogle Scholar

  • [5]

    Andrews LC, Phillips RL. Laser beam propagation through random media. Second Edition. SPIE – The international society for optical engineering, Washington, USA, 2005.Google Scholar

  • [6]

    Bass M, Decusatis C, Enoch JM. Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments. Third Edition. McGraw Hill Book Co, 2009.Google Scholar

  • [7]

    Bischoff W, Heck B, Howind J, Teusch A. A procedure for testing the assumption of homoscedasticity in Least-Squares residuals: a case study of GPS carrier-phase observations. J Geodesy 2005, 78, 7–8, 397–404.CrossrefGoogle Scholar

  • [8]

    Blaskow R, Schneider D. Analysis and correction of the dependency between laser scanner intensity values and range. ISPRS Archives 2014, 40, 5, 107–112.Google Scholar

  • [9]

    Elosegui P, Davis JL, Jaldehag RTK, Johansson JM, Niell AE, Shapiro II. Geodesy using the Global Positioning System: The effects of signal scattering on estimates of site position. JGR 1995, 100, B7, 9921–9934.Google Scholar

  • [10]

    Boehler W, Marbs A. 3D Scanning instruments. In: Proc. of the CIPA WG6 Int. Workshop on scanning for cultural heritage recording 2002. http://www.isprs.org/commission5/workshop/.Google Scholar

  • [11]

    Bona P. Precision, cross correlation, and time correlation of GPS phase and code observations. GPS Sol 2000, 4, 2, 3–13.CrossrefGoogle Scholar

  • [12]

    Bureick J, Alkhatib H, Neumann I. Robust spatial approximation of laser scanner points clouds by means of free-form curve approaches in deformation analysis. J Appl Geodesy 2016, 10, 1, 27–35.Google Scholar

  • [13]

    Carrea D, Abellan A, Humair F, Matasci B, Derron MH, Jaboyedoff M. Correction of terrestrial LiDAR intensity channel using Oren–Nayar reflectance model: An application to lithological differentiation. ISPRS J. Photogramm. Remote Sens. 2016, 113, 17–29.CrossrefGoogle Scholar

  • [14]

    Coren F, Sterzai P. Radiometric correction in laser scanning. Int. J. Remote Sens., 2006, 27, 15, 3097–3104.CrossrefGoogle Scholar

  • [15]

    Craymer MR, Wells DF, Vanicek P, Davlin RL. Specifications for urban GPS surveys. Surveying and Land Information Science 1990, 50, 4, 251–259.Google Scholar

  • [16]

    Dach R, Lutz S, Walser P, Fridez P. Bernese GNSS Software Version 5.2. User manual, Astronomical Institute, University of Bern, Bern Open Publishing. doi:10.7892/boris.72297, ISBN: 978-3-906813-05-9. 2015.Google Scholar

  • [17]

    El-Rabbany AE-S. The effect of Physical Correlations on the Ambiguity Resolution and Accuracy Estimation. PhD thesis, Department of Geodesy and Geomatics Engineering, University of New Brunswick, Canada, 1994.Google Scholar

  • [18]

    Euler HJ, Goad CC. On optimal filtering of GPS dual frequency observations without using orbit information. Bulletin Geodesique 1991, 65, 2, 130–143.CrossrefGoogle Scholar

  • [19]

    Hug C, Wehr A. Detecting and identifying topographic objects in imaging laser altimeter data. Int. Archives Photogramm. Remote Sens. 1997, 32, 19–26.Google Scholar

  • [20]

    Goad CC. Precise positioning with the GPS. Applied Geodesy, Lecture Notes in Earth Sciences 1987, 12, 17–30.CrossrefGoogle Scholar

  • [21]

    Griffiths DJ, Introduction to electrodynamics. 3e éd. Prentice-Hall, Upper Saddle River, NJ [u. a.], 1999.Google Scholar

  • [22]

    Hebert M, Krotkov E. 3D measurements from imaging laser radars: how good are they? Image and Vision Computing 1992, 10, 3, 170–178.CrossrefGoogle Scholar

  • [23]

    Heinz E, Mettenleiter M, Kuhlmann H, Holst C. Strategy for Determining the Stochastic Distance Characteristics of the 2D Laser Scanner Z+F Profiler 9012A with Special Focus on the Close Range. Sensors 2018, 18, 2253.CrossrefGoogle Scholar

  • [24]

    Hofmann-Wellenhof B, Lichtenegger H, Collins J. GPS Theory and Practice. Fifth revised edition. Springer, Wien, New York, 2001.Google Scholar

  • [25]

    Holst C, Kuhlmann H. Challenges and Present Fields of Action at Laser Scanner Based Deformation Analysis. J Appl Geodesy 2016, 10, 1, 17–25.Google Scholar

  • [26]

    Höfle B, Pfeifer N. Correction of laser scanning intensity data: Data and model driven approaches. ISPRS 2007, 62, 6, 415–433.Google Scholar

  • [27]

    Ishimaru A. Wave propagation and scattering in random media. IEEE Press and Oxford University Press, New York, 1994.Google Scholar

  • [28]

    Jelalian AV. Laser Radar Systems. Artech House Inc, 1992.Google Scholar

  • [29]

    Joseph A. What is the difference between SNR and CN0. Inside GNSS November December 2010, 20–25.Google Scholar

  • [30]

    Jurek T, Kuhlmann H, Host C. Impact of spatial correlations on the surface estimation based on terrestrial laser scanning. J Appl Geodesy 2017, 11, 3, 143–155.Google Scholar

  • [31]

    Kauker S, Holst C, Schwieger V, Kuhlmann H, Schön S. Spatio-temporal correlations of terrestrial laser scanning. AVN 2016, 6, 170–182.Google Scholar

  • [32]

    Kauker S, Schwieger V. A synthetic covariance matrix for monitoring by terrestrial laser scanning. J Appl Geodesy 2017, 11, 2, 77–87.Google Scholar

  • [33]

    Kermarrec G, Schön S. On the Matérn covariance family: a proposal for modelling temporal correlations based on turbulence theory. J Geodesy 2014, 88, 11, 1061–1079.CrossrefGoogle Scholar

  • [34]

    Kermarrec G, Schön S. Taking correlation into account with a diagonal covariance matrix. J Geodesy 2016, 90, 9, 793–805.CrossrefGoogle Scholar

  • [35]

    Kermarrec G, Schön S. Apriori fully populated covariance matrices in Least-Squares adjustment – case study: GPS relative positioning. J Geodesy 2017a, 91, 5, 465–484.CrossrefGoogle Scholar

  • [36]

    Kermarrec G, Schön S. Fully populated VCM or the hidden parameter. Journal of Geodetic Science 2017b, 7, 1, 151–161.Google Scholar

  • [37]

    Kermarrec G, Schön S, Kreinovich V. Possible explanation of empirical values of the Matérn smoothness parameter for the temporal covariance of GPS measurements. Applied Mathematical Science 2017c, 11, 35, 1733–1737.CrossrefGoogle Scholar

  • [38]

    Kermarrec G, Schön S. Taking correlations into account: a diagonal correlation model. GPS sol 2018, 21, 4, 1895–1906.Google Scholar

  • [39]

    Kermarrec G, Alkhatib H, Neumann I. On the Sensitivity of the Parameters of the Intensity-Based Stochastic Model for Terrestrial Laser Scanner. Case Study: B-Spline Approximation. Sensors, 2018, 18, 9, 2964.CrossrefGoogle Scholar

  • [40]

    Koch KR. Parameter estimation and hypothesis testing in linear models. Springer, Berlin, 1999.Google Scholar

  • [41]

    Koivunen AC, Kostinski AB. The feasibility of Data Whitening to Improve Performance of Weather Radar. AMS 1999, 38, 6, 741–749.Google Scholar

  • [42]

    Lambertus T, Belton D, Helmholz P. Empirical Investigation of a stochastic model based on intensity values for terrestrial laser scanning. AVN 2018, 125, 43–48.Google Scholar

  • [43]

    Langley R. GPS receiver system noise. GPS World 1997, 8, 40–45.Google Scholar

  • [44]

    Luati A, Proietti T. On the equivalence of the weighted Least-Squares and the generalised Least-Squares estimators, with applications to kernel smoothing. Annals of the Institute of Statistical Mathematics 2011, 63, 4, 673–686.Google Scholar

  • [45]

    Luo X, Mayer M, Heck B. Analysing Time Series of GNSS Residuals by Means of ARIMA Processes. In: Sneeuw N, Novák P, Crespi M, Sansò F (eds.) VII Hotine-Marussi Symposium on Mathematical Geodesy, International Association of Geodesy Symposia, vol. 137. Springer, Berlin, Heidelberg, 2012.Google Scholar

  • [46]

    Luo X, Mayer M, Heck B, Awange JL. A realistic and easy-to-implement weighting model for GNSS phase observations. IEEE Transactions on Geoscience and Remote Sensing 2014, 52, 10, 6110–6118.CrossrefGoogle Scholar

  • [47]

    Misra P, Enge P. Global Positioning System. Revised Second Edition. Ganga-Jamuna Press, 2012.Google Scholar

  • [48]

    Ozendi M, Akca D, Topan H. A generic point error model for TLS derived point clouds. In: Proc. SPIE 10332, Videometrics, Range Imaging and Applications XIV, 2017, doi:.CrossrefGoogle Scholar

  • [49]

    Pelzer H. Zur Analyse geodatischer Deformations-messungen. Verlag der Bayer Akad. d. Wiss, Munchen, Munchen, 410 1971. 164, 1971.

  • [50]

    Paffenholz J-A, Bae K-H. Geo-referencing point clouds with transformational and positional uncertainties. Journal of Applied Geodesy 2012, 6, 1, 33–46.Google Scholar

  • [51]

    Pfeifer N, Dorninger P, Haring A, Fan H. Investigating terrestrial laser scanning intensity data: quality and functional relations. In: 8th Conference on Optical 3-D Measurement Techniques, Zürich, Switzerland, 2007, 328.Google Scholar

  • [52]

    Rao CR, Toutenburg H. Linear Models, Least-Squares and Alternatives. 2nd ed. Springer, New York, 1999.Google Scholar

  • [53]

    Rasmussen CE, Williams C. Gaussian Processes for Machine Learning. The MIT Press, 2006.Google Scholar

  • [54]

    Rueger JM. Electronic distance measurement. Springer-Verlag, Berlin Heidelberg, 1996.Google Scholar

  • [55]

    Satirapod C, Wang J. Comparing the quality indicators of GPS carrier phase observations. Geomatics Research Australasia 2000, 73, 75–92.Google Scholar

  • [56]

    Schön S, Brunner FK. Atmospheric turbulence theory applied to GPS carrier-phase data. J Geodesy 2008, 1, 47–57.Google Scholar

  • [57]

    Soudarissanane S, Lindenbergh R, Menenti M and Teunissen P. Scanning geometry: influencing factor on the quality of terrestrial laser scanning points. ISPRS 2011, 66, 389–399.Google Scholar

  • [58]

    Stein ML. Interpolation of spatial data. Some theory for kriging. Springer, New York, 1999.Google Scholar

  • [59]

    Taylor H, Karlin S. An introduction to stochastic modelling. 3rd edition. Academic Press, https://books.google.de/books?id=UtPgVrVthF8C, 1998.Google Scholar

  • [60]

    Teunissen PJG, Jonkman NF, Tiberius CCJM. Weighting GPS Dual Frequency Observations: Bearing the Cross of Cross Correlation. GPS Sol 1998, 2, 2, 28–37.CrossrefGoogle Scholar

  • [61]

    Tiberius C, Kenselaar F. Variance Component Estimation and Precise GPS Positioning: Case Study. J Surv Eng 2003, 129, 1, 11–18.CrossrefGoogle Scholar

  • [62]

    Vermeer M. The precision of geodetic GPS and one way of improving it. J Geodesy 1997, 71, 4, 240–245.CrossrefGoogle Scholar

  • [63]

    Vosselman G and Maas HG. Airborne and Terrestrial Laser Scanning. Whittles Publishing, Dunbeath, Scotland, UK, 2010.Google Scholar

  • [64]

    Wang J. Towards deformation monitoring with terrestrial laser scanning based on external calibration and feature matching methods. PhD thesis, Wissenschaftliche Arbeiten der Fachrichtung Geodeäsie and Geoinformatik der Leibniz Universität Hannover, No. 308, 2013.Google Scholar

  • [65]

    Wheelon AD. Electromagnetic Scintillation part I Geometrical optics. Cambridge University Press, Cambridge, 2001.Google Scholar

  • [66]

    Wieser A, Brunner FK. An extended weight model for GPS phase observations. Earth Planet Space 2000, 52, 10, 777–782.CrossrefGoogle Scholar

  • [67]

    Wujanz D, Holst C, Neitzel F, Kuhlmann H, Schwieger V, Niemeier W. Survey configuration for terrestrial laser scanning. AVN 2016, 06.Google Scholar

  • [68]

    Wujanz D, Burger M, Mettenleiter M, Neitzel F. An intensity-based stochastic model for terrestrial laser scanners. ISPRS 2017, 125, 146–155.Google Scholar

  • [69]

    Wujanz D, Burger M, Tschirschwitz F, Nietzschmann T, Neitzel F, and Kersten T. Bestimmung von intensitätsbasierten stochastischen Modellen für terrestrische Laserscanner basierend auf 3DPunktwolken. In: Proc. “17. Oldenburger 3D-Tage Optische 3DMesstechnik-Photogrammetrie-Laserscanning”, Oldenburg (Germany), 2018.Google Scholar

  • [70]

    Xu X, Kargoll B, Bureick J, Yang H, Alkhatib H, Neumann. TLS-based profile model analysis of major composite structures with robust B-spline method, Composite Structures 2018, 184, 814–820.CrossrefGoogle Scholar

  • [71]

    Zamecnikova M, Wieser A, Woschitz H, Ressl C. Influence of surface reflectivity on reflectorless electronic distance measurement and terrestrial laser scanning. J of Appl Geodesy 2014, 8, 4, 311–325.Google Scholar

  • [72]

    Zamecnikova M, Neuner N. Untersuchung der Distanzabhängigkeit des gemeinsamen Einflusses des Auftreffwinkels und der Oberflächenrauheit auf die reflektorlose Distanzmessung einer scannenden Totalstation. AVN 2017, 11/12.Google Scholar

  • [73]

    Zhao X, Alkhatib H, Kargoll B, Neumann I. Statistical evaluation of the influence of the uncertainty budget on B-spline curve approximation. J. Appl. Geodesy 2017, 11, 4, 215–230.Google Scholar

  • [74]

    Zhao X, Kargoll B, Omidalizarandi M, Xu X, Alkhatib H. Model selection for parametric surfaces approximating 3D point clouds for deformation analysis, Remote Sensing, 2018, 10, 4, 634, Special Issue: “3D Modelling from Point Clouds: Algorithms and Methods”.CrossrefGoogle Scholar

About the article

Received: 2018-06-01

Accepted: 2019-01-08

Published Online: 2019-01-24

Published in Print: 2019-04-26


Citation Information: Journal of Applied Geodesy, Volume 13, Issue 2, Pages 93–104, ISSN (Online) 1862-9024, ISSN (Print) 1862-9016, DOI: https://doi.org/10.1515/jag-2018-0019.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in