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«UCGE Reports Number 20185 Department of Geomatics Engineering An Analysis on the Optimal Combination of Geoid, Orthometric and Ellipsoidal Height ...»

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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 GNSSlevelling, 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 heightrelated applications.

–  –  –

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.

–  –  –

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.

–  –  –

Approval Page

Abstract

Preface

Acknowledgements

Table of Contents

List of Tables

List of Figures

List of Symbols

List of Abbreviations

1 Introduction

2 Heights, Vertical Datums and GNSS-Levelling





2.1 Geoid heights

2.2 Orthometric heights

2.3 Ellipsoidal heights

2.4 Why combine geoid, orthometric and ellipsoidal height data?

2.4.1 Modernizing regional vertical datums

2.4.2 Global vertical datum

2.4.3 GNSS-levelling

2.4.4 Refining and testing gravimetric geoid models

3 Combined Height Adjustment and Modelling of Systematic Effects

3.1 General combined adjustment scheme

3.2 Role of the parametric model

3.3 Modelling options

3.4 Selecting a parametric model

3.5 Assessing the parametric model performance

3.5.1 Classical empirical approach

3.5.2 Cross-validation

3.5.3 Assessing the goodness of fit

3.5.4 Testing parameter significance

3.6 Summary

vii 4 Results for the Parametric Model Surface Fits

4.1 Results for the Switzerland test network

4.2 Results for the Canadian test network

4.3 Fitting a gravimetric geoid model to the Australian height datum via GPS....... 106

4.4 Remarks on results

5 Overview of Variance Component Estimation

5.1 Introduction

5.2 Review of methods for estimating variance components

5.2.1 Functional models

5.2.2 Stochastic models

5.2.3 Selection of variance component estimation procedures

5.3 The MINQUE method

5.4 Application of MINQUE to the combined height adjustment problem.............. 133

5.5 Remarks on VCE and the combined height adjustment problem

5.6 Iterative minimum norm quadratic unbiased estimation

5.7 Iterative almost unbiased estimation

6 Case Studies: Estimation of Variance Components via a Mixed Adjustment of Ellipsoidal, Geoid and Orthometric Heights

6.1 Description of the Swiss test network data

6.1.1 Initial covariance matrix for the GPS heights

6.1.2 Initial covariance matrix for the geoid heights

6.1.3 Initial covariance matrix for the orthometric heights

6.2 Description of the southern British Columbia/Alberta test network data........... 151 6.2.1 Initial covariance matrix for the GPS heights

6.2.2 Initial covariance matrix for the geoid heights

6.2.3 Initial covariance matrix for the orthometric heights

6.3 Case Study I - Testing a-priori covariance matrices for the height data............. 156 6.3.1 Scaling the covariance matrix for the ellipsoidal heights

6.3.2 Calibration of geoid error models

6.3.3 Assessing the accuracy of orthometric heights

6.4 Case Study II - Non-negative variance components

6.5 Case Study III - Effects of correlations

6.6 Case Study IV - Role of the type of parametric model

6.7 Summary

viii 7 Practical Considerations for Modernizing Vertical Control

7.1 Introduction

7.2 GNSS-levelling: What are the issues?

7.2.1 Computing the accuracy of the levelling data

7.2.2 Computing the accuracy of the GPS data

7.2.3 Computing the accuracy of the geoid height differences

7.2.4 Achievable accuracy of GNSS-levelling for a new baseline.................. 192 7.2.5 Tests in northern Canada

7.3 Combining heterogeneous heights at sea

7.4 Other practical issues

7.5 Summary

8 Conclusions and Recommendations

8.1 Conclusions

8.2 Recommendations for future work

References

–  –  –

2.1: Expected commission and omission errors after GOCE

4.1: Statistics of residuals at points used in the adjustment for various parametric models (Swiss network, units: cm)

4.2: Statistics of cross-validation tests for various parametric models (Swiss network, units: cm)

4.3: Condition numbers for various parametric models in Switzerland

4.4: Summary of selection criteria for Swiss test network

4.5: Statistics of residuals at points used in the adjustment for various parametric models (southern BC/AB network, units: cm)

4.6: Condition numbers for various parametric models in southern BC/AB................ 101 4.7: Statistics of cross-validation tests for various parametric models (southern BC/AB network, units: cm)

4.8: Statistics of residuals at points used in the adjustment for orthonormalized versions of various parametric models (southern BC/AB network, units: cm)..... 104 4.9: Summary of selection criteria for southern BC/AB test network

4.10: Statistics of the residuals before any fit (units: cm)

4.11: Description of Australian regional network geometry

4.12: Statistics of residuals at the points used in the adjustment (upper row) and statistics obtained through cross-validation (bottom row) for Australia (units: cm)

4.13: Condition numbers for various parametric models for the Australian network..... 111 4.14: Final selected models per region and corresponding RMS obtained through cross-validation

5.1: Timeline of key VCE developments in geodetic literature

5.2: Number of GPS-levelling benchmarks in different regions

6.1: Initial CV matrix characteristics for the Swiss network

–  –  –

6.3: Estimated variance components and average standard deviations for local and global geoid models

6.4: Estimated variance components for orthometric height data and average standard deviations for C H

6.5: Estimated variance components using the IMINQUE and IAUE methods........... 173 6.6: Effect of correlations on estimated variance factors (Swiss network)

6.7: Effect of correlations on estimated variance factors (south BC/AB network)....... 176 6.8: Estimated variance components for the Swiss test network using various parametric models

6.9: Estimated variance components for the southern BC/AB test network using various parametric models

7.1: Statistics of various test network configurations (d denotes baseline length in km)

7.2: Standard error of relative height components (in cm) as a function of baseline length

7.3: Results of baselines moving northward from the test network area (units: cm).... 202

–  –  –

1.1: Relationship between ellipsoidal, geoid and orthometric heights

2.1: Computation of regional geoid models using heterogeneous data

2.2: Orthometric height of a point

2.3: Approximation in orthometric heights by neglecting the difference in lengths between the curved plumb line and the ellipsoidal normal

2.4: Reference ellipsoid and geodetic coordinates

2.5: Zenith dependence of station height, receiver clock and tropospheric delay parameters

2.6: Sources of errors for global navigation satellite systems

2.7: Establishment of a reference benchmark height

3.1: Illustrative view of GPS/geoid levelling and the role of the corrector surface........ 62 3.2: First (a), second (b), third (c) and fourth (d) order bivariate polynomial fits.......... 71 3.3: Example of height misclosures before and after parametric model fit

3.4: Classical empirical testing approach

3.5: Cross-validation procedure on a point-by-point basis

3.6: Relationship between R 2 and Rα

3.7: Stepwise procedure for testing parameter significance

3.8: Flowchart of major steps for assessing the performance of parametric corrector surface models

4.1: Swiss test network of GPS-on-benchmarks and original height misclosures.......... 87

4.2a: Low-order corrector surface model fits for the Swiss network

4.2b: High-order corrector surface model fits for the Swiss network

4.3: Statistical measures of goodness of fit for various parametric models (Swiss test network)

–  –  –

4.5: Canadian GPS-levelling benchmark network

4.6: Test network of GPS-on-benchmarks in southern BC/AB and original height misclosures

4.7a: Low-order corrector surface model fits for the southern BC/AB network.............. 98

4.7b: High-order corrector surface model fits for the southern BC/AB network............. 99 4.8: Statistical measures of goodness of fit for various parametric models (southern BC/AB test network)

4.9: Empirical test results for the southern BC/AB region

4.10: 7-parameter differential similarity fit for the southern BC/AB test network......... 106 4.11: GPS-levelling data on the Australian mainland

4.12: Examples of the 4-parameter corrector surface fit (left) and the 10-parameter MRE fit (right) for the Australian mainland

4.13: Selected parametric models for regional fits in Australia (units: m)

5.1: Iterative variance component estimation computational scheme

5.2: Example of estimated variance components at each iteration step

6.1: Swiss test network and error bars for  = h − H − N

6.2: Plot of initial covariance matrix for GPS heights (Swiss network)



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