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An Analysis on the Optimal Combination of Geoid, Orthometric and Ellipsoidal Height Data (URL UCGE Reports Number 20185 Department of Geomatics Engineering An Analysis on the Optimal Combination of Geoid, Orthometric and Ellipsoidal Height Data (URL: http://www.geomatics.ucalgary.ca/links/GradTheses.html) by Georgia Fotopoulos December 2003 UNIVERSITY OF CALGARY An analysis on the optimal combination of geoid, orthometric and ellipsoidal height data by Georgia Fotopoulos A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF GEOMATICS ENGINEERING CALGARY, ALBERTA DECEMBER, 2003 © Georgia Fotopoulos 2003 ABSTRACT The main objective of this research is to present a detailed analysis of the optimal combination of heterogeneous height data, with particular emphasis on (i) modelling systematic errors and datum inconsistencies, (ii) separation of random errors and estimation of variance components for each height type, and (iii) practical considerations for modernizing vertical control systems. Specifically, vertical control networks consisting of ellipsoidal, orthometric and geoid height data are investigated. Although the theoretical relationship between these height types is simple in nature, its practical implementation has proven to be quite challenging due to numerous factors that cause discrepancies among the combined height data. To address these challenges a general procedure involving empirical and statistical tests for assessing the performance of selected parametric models is developed. In addition, variance component estimation is applied to the common adjustment of the heterogeneous heights. This leads to an in-depth analysis of the effects of correlation among heights of the same type, provisions for computing non-negative variance factors, and the intrinsic connection between the proper modelling of systematic errors and datum inconsistencies with the estimated variance components. Additional numerical studies include the calibration of geoid error models (both regional and global), scaling the GPS-derived ellipsoidal height covariance matrix, and evaluating the accuracy of orthometric heights obtained from national/regional adjustments of levelling data. Ultimately, one of the main motivations for this work is embedded in the eminent need to introduce modern tools and techniques, such as GNSS- levelling, in establishing vertical control. Therefore, part of this research is aimed at bringing to the forefront some of the key issues that affect the achievable accuracy level of GNSS-levelling. Overall, the analysis of the optimal combination of the heterogeneous height data conducted herein provides valuable insight to be used for a variety of height- related applications. iii PREFACE This is an unaltered version of the author's Doctor of Philosophy thesis of the same title. This thesis was accepted by the Faculty of Graduate Studies in December, 2003. The faculty supervisors of this work were Dr. M.G. Sideris and Dr. N. El-Sheimy, and the other members of the examining committee were Dr. J.A.R. Blais, Dr. Y. Gao, Dr. L. Lines, and Dr. A.H.W. Kearsley. iv ACKNOWLEDGEMENTS I would like to express my appreciation and thank my supervisors, Dr. M.G. Sideris and Dr. N. El-Sheimy, for their support and guidance throughout my graduate studies. Their continuous encouragement and advice were greatly appreciated. The other member of my supervisory committee, Dr. J.A.R. Blais, is also acknowledged for answering various questions throughout my studies at the University of Calgary. I would also like to thank Dr. A.H.W. Kearsley and Dr. W.E. Featherstone for hosting my visit as a research fellow in Australia during my studies. It was an excellent opportunity and an inspiring place to work. My friend and colleague, Christopher Kotsakis, is also thanked for his thorough proofreading of the initial draft of this thesis. Funding for my PhD studies was provided from numerous sources, including NSERC, iCORE, GEOIDE, Alberta Heritage Foundation, Killam Scholarship Fund, the University of Calgary, and the Department of Geomatics Engineering. Urs Marti from the Swiss Mapping Authority is gratefully acknowledged for making the Swiss data available. Also, Marc Véronneau, Mike Craymer, and André Mainville from the Geodetic Survey Division at NRCAN are thanked for providing the Canadian data and answering all of my questions. Christian Malmquist is also thanked for his work on the covariance matrix for EGM96. Finally, I would like to thank my brothers Costa, Alex and Chris for keeping me grounded during the writing of this manuscript. Last, but not least, I thank my parents for their endless support and encouragement. I could not have done it without them. v Στους γονείς µου, Αθανάσιο και Ειρήνη. vi TABLE OF CONTENTS Approval Page ...................................................................................................................ii Abstract.............................................................................................................................iii Preface............................................................................................................................... iv Acknowledgements............................................................................................................ v Table of Contents ............................................................................................................vii List of Tables...................................................................................................................... x List of Figures..................................................................................................................xii List of Symbols ............................................................................................................... xvi List of Abbreviations...................................................................................................... xix 1 Introduction ................................................................................................................. 1 1.1 Background ............................................................................................................. 1 1.2 Thesis Objectives .................................................................................................... 5 1.3 Thesis Outline ......................................................................................................... 7 2 Heights, Vertical Datums and GNSS-Levelling........................................................ 9 2.1 Geoid heights........................................................................................................... 9 2.2 Orthometric heights............................................................................................... 19 2.3 Ellipsoidal heights................................................................................................. 26 2.4 Why combine geoid, orthometric and ellipsoidal height data?............................. 35 2.4.1 Modernizing regional vertical datums....................................................... 36 2.4.2 Global vertical datum................................................................................ 41 2.4.3 GNSS-levelling ......................................................................................... 46 2.4.4 Refining and testing gravimetric geoid models......................................... 48 3 Combined Height Adjustment and Modelling of Systematic Effects ................... 51 3.1 General combined adjustment scheme.................................................................. 51 3.2 Role of the parametric model................................................................................ 59 3.3 Modelling options ................................................................................................. 63 3.4 Selecting a parametric model ................................................................................ 69 3.5 Assessing the parametric model performance....................................................... 72 3.5.1 Classical empirical approach..................................................................... 73 3.5.2 Cross-validation ........................................................................................ 75 3.5.3 Assessing the goodness of fit .................................................................... 76 3.5.4 Testing parameter significance.................................................................. 80 3.6 Summary ............................................................................................................... 84 vii 4 Results for the Parametric Model Surface Fits ...................................................... 86 4.1 Results for the Switzerland test network............................................................... 87 4.2 Results for the Canadian test network................................................................... 95 4.3 Fitting a gravimetric geoid model to the Australian height datum via GPS ....... 106 4.4 Remarks on results .............................................................................................. 113 5 Overview of Variance Component Estimation..................................................... 115 5.1 Introduction ......................................................................................................... 115 5.2 Review of methods for estimating variance components.................................... 119 5.2.1 Functional models..................................................................................
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