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UCGE Reports Ionosphere Weighted Global Positioning System Carrier UCGE Reports Number 20155 Ionosphere Weighted Global Positioning System Carrier Phase Ambiguity Resolution (URL: http://www.geomatics.ucalgary.ca/links/GradTheses.html) Department of Geomatics Engineering By George Chia Liu December 2001 Calgary, Alberta, Canada THE UNIVERSITY OF CALGARY IONOSPHERE WEIGHTED GLOBAL POSITIONING SYSTEM CARRIER PHASE AMBIGUITY RESOLUTION by GEORGE CHIA LIU A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF GEOMATICS ENGINEERING CALGARY, ALBERTA OCTOBER, 2001 c GEORGE CHIA LIU 2001 Abstract Integer Ambiguity constraint is essential in precise GPS positioning. The perfor- mance and reliability of the ambiguity resolution process are being hampered by the current culmination (Y2000) of the eleven-year solar cycle. The traditional approach to mitigate the high ionospheric effect has been either to reduce the inter-station separation or to form ionosphere-free observables. Neither is satisfactory: the first restricts the operating range, and the second no longer possesses the ”integerness” of the ambiguities. A third generalized approach is introduced herein, whereby the zero ionosphere weight constraint, or pseudo-observables, with an appropriate weight is added to the Kalman Filter algorithm. The weight can be tightly fixed yielding the model equivalence of an independent L1/L2 dual-band model. At the other extreme, an infinite floated weight gives the equivalence of an ionosphere-free model, yet perserves the ambiguity integerness. A stochastically tuned, or weighted, model provides a compromise between the two ex- tremes. The reliability of ambiguity estimates relies on many factors, including an accurate functional model, a realistic stochastic model, and a subsequent efficient integer search algorithm. These are examined closely in this research. Two days of selected Swedish GPS Network data sets from ionospherically active (up to 15 ppm) and moderate (up to 4 ppm) days, forming a maximum baseline length of 400 km, have been analyzed. All three ionosphere weight models yielded a 90% range of correct widelane ambiguities within three minutes, regardless of inter- iii station distance. Of the three ionospheric weighting schemes analyzed, the weighted model appears to be the optimal choice, with smaller instantaneous estimate errors and possessing the convergence characteristic. Upon parameter removal of the resolved widelane ambiguities, the precision of L1 ambiguity estimates improved between a few precent to 59 percent for baselines over one hundred kilometres. However, the improvement is not sufficient for resolving the L1 ambiguities. iv Acknowledgements First and foremost, I wish to thank many individuals at the Geomatics Department at the University of Calgary. Without the continuous encouragement and support from my supervisor, Professor G. Lachapelle, this thesis would not have been pos- sible. Further gratitude extents to my colleague, Luiz-Paulo Fortes, and Dr. Susan Skone for many constructive discussions. Second, I would like to thank Dr. Stefan Schaer at the Astronomical Institute at the University of Berne, for responding to many of my email questions related to ionosphere modelling. His prompt and unequivocal replies are much appreciated. I would also like to thank Professor Peter Teunissen at the Delft University of Technol- ogy for his valuable advise related to the integer ambiguity resolution and providing the Matlab version of the LAMBDA code. Lastly, further acknowledgements extend to the Swedish National Geodetic Sur- vey for providing the GPS data sets, the Onsala Astronomical Institute for the water vapour radiometer and meteorological data, the Scripps Institute of Oceanography for the orbit data, and the Astronomical Institute of the University of Berne for the Global Ionosphere Models. v Table of Contents Approval Page ii Abstract iii Acknowledgements v Table of Contents vi Notations xiii 1 Introduction 1 1.1 Background . 1 1.2 Motivation . 4 1.3 Objectives . 7 1.4 Outline . 9 2 GPS Signals, Observables, and Errors 12 2.1 Signal Structure . 12 2.2 Signal Processing . 16 2.2.1 Antenna . 17 2.2.2 RF Section . 19 2.3 GPS Observation Models . 22 2.3.1 Fundamental GPS Observation Equations . 22 2.3.2 Forming Differences . 24 2.3.3 Linearly Combined Observations . 26 3 Atmospheric Effects And Modelling 32 3.1 Ionosphere . 33 3.1.1 Ionization Processes . 33 3.1.2 Ionospheric Effects on GPS Signals . 37 3.1.3 TEC Modelling And Parameterization . 40 3.1.4 Application of Ionosphere Models . 49 3.2 Troposphere . 50 3.2.1 Troposphere Modelling . 53 3.2.2 Inverse GPS Method . 55 vi 4 Implementation 57 4.1 Overview of the Kalman Filter . 57 4.2 Non-Linear Measurement Model . 60 4.3 Random Processes . 61 4.4 Functional Model . 63 4.5 Stochastic Model . 65 4.5.1 System Noise . 65 4.5.2 Measurement Noise . 66 4.5.3 Measurement (Co)Variance Matrix, R . 69 4.6 Ambiguity Resolution: The LAMBDA Method . 72 4.6.1 Decorrelation Transformation . 72 4.6.2 Integer Search . 73 4.6.3 Admissible Integer Estimators . 74 4.6.4 Validation . 76 4.6.5 Ambiguity Fixing . 77 4.7 Summary . 78 5 Stochastic Analyses 81 5.1 Data Set Characterization . 83 5.2 Measurements . 84 5.2.1 Elevation Dependence . 84 5.2.2 Measurement Cross-Correlation . 86 5.2.3 Measurement Time Correlation . 88 5.3 Distance Dependence . 90 5.3.1 Geometrical Errors . 90 5.3.2 Ionospheric Errors . 94 5.3.3 Ionospheric Time Correlation . 102 5.4 Summary . 104 6 Results 106 6.1 Sensitivity of Ionosphere Weighting . 107 6.2 Unconstrained Ambiguity Solution . 114 6.3 Widelane Ambiguity Resolution Performance . 120 6.3.1 Widelane Constrained Position and L1 Ambiguity Estimates . 122 6.4 L1 Integer Ambiguity Resolution Performance . 124 6.5 Ionosphere-Free Solution . 124 6.6 Summary . 129 vii 7 Conclusions and Recommendations 131 7.1 Conclusions . 131 7.2 Recommendations . 133 References 135 A Double Difference Zero-Baseline Measurement Noise 147 B Autocorrelation Functions of Double Difference Ionosphere 150 C Sensitivity of Ionosphere Weighting 155 viii List of Tables 2.1 Summary of LC and Their Characteristics . 31 3.1 SLM Function Vs Broadcast Mapping Function at z=80◦ . 43 5.1 Standard Deviation of Double Difference Measurements . 86 5.2 Standard Deviation of Zero Difference Measurements . 86 5.3 Summary of Double Difference Noise Level . 88 5.4 Summary of Correlation Coefficients . 88 5.5 Summary of Baselines Evaluated . 90 5.6 Double Difference Geometrical Errors, April 7 (24Hr) . 93 5.7 Double Difference Geometrical Errors, June 21 (24Hr) . 93 5.8 Double Difference Ionosphere Error, April 7 (24Hr) . 97 5.9 Double Difference Ionosphere Error, June 21 (24Hr) . 97 5.10 Summary of Ionospheric Error . 100 5.11 Means and Standard Deviations Of Double Difference Ionospheric De- lays, April 7 (24Hr) . 101 5.12 Means and Standard Deviations Of Double Difference Ionospheric De- lays, June 21 (24Hr) . 101 5.13 First-Order Correlation Time in Seconds, 1/e Point . 104 6.1 Ionospheric Estimate Errors at the Start and the End of Filtering . 111 6.2 24 Hr Model Versus 3 Hr Model . 112 6.3 Summary of Estimate Errors At 90-Percentile † . 119 † 6.4 Summary of Error Estimates At 90-Percentile With Constrained NWL 123 ix List of Figures 1.1 Current Sunspot Cycle Number 23 (NASA, 2001) . 4 1.2 Swedish SWEPOS Network (SWEPOS, 2001) . 8 1.3 Kp Indices, April 7 and June 21 2000 (NOAA, 2001) . 11 2.1 GPS Signal Structure . 15 2.2 SA Effect on Position and Range . 16 2.3 Double Difference . 25 3.1 Atmosphere Layers . 33 3.2 Spatial and Temporal TEC Variations . 38 3.3 Single Layer Model (SLM) . 42 3.4 Mapping Functions At Various Shell Heights . 43 3.5 CODE Global Ionosphere Map (GIM), April 7, 2000 . 47 3.6 Broadcast Ionosphere Model (BIM), April 7, 2000 . 47 3.7 CODE Global Ionosphere Map (GIM), June 21, 2000 . 48 3.8 Broadcast Ionosphere Model (BIM), June 21, 2000 . 48 3.9 The Impact of Ionosphere Models on Double Difference L1 Measure- ments Over 145 km Baseline, ONSA-VANE . 51 3.10 The Impact of Ionosphere Models on Double Difference L1 Measure- ments Over 35 km Baseline, ONSA-GOTE . 52 3.11 Empirical, WVR, GPS TZD Estimations, Sept 16, 1998 at Station Onsala . 56 4.1 Standard Kalman Filter Algorithms (Brown and Hwang 1997) . 59 4.2 Ionosphere Weighted GPS Carrier Phase Ambiguity Resolution Flowchart 80 5.1 GDOP and Number of Satellites Tracked Above 15◦ at Station Onsala On April 7 and June 21 . 83 5.2 Double Difference Time Correlations . 89 5.3 Double Difference Geometrical Error, April 7 . 92 5.4 Double Difference Geometrical Error, June 21 . 92 5.5 Double Difference Ionospheric Error, ONSA-VANE (145 km) . 95 5.6 Double Difference Ionospheric Error, April 7 . 96 5.7 Double Difference Ionospheric Error, June 21 . 96 5.8 Active Ionospheric Error, April 7(00-02Hr) . 99 5.9 Distributions of Double Difference Ionospheric Delay . 102 5.10 Elevations of PRN05, 09, and 30 . 103 x 5.11 Comparison of Ionospheric Autocorrelation between Two Satellite Pairs over JONH-LEKS (335 Km) . 104 6.1 Comparative Pseudo-Observable Weightings, GOTE-VANE (110 km), April 7 . 108 6.2 Sensitivity of Ionosphere Weighting . 109 6.3 Comparative Performance Between 2 Hr and 24 Hr Ionospheric Mod- els, April 7 . 113 6.4 Horizontal Accuracy at 90-Percentile . 115 6.5 Vertical Accuracy at 90-Percentile . 116 6.6 Unconstrained NWL Ambiguity Accuracy at 90-Percentile . 117 6.7 Unconstrained NL1 Ambiguity Accuracy at 90-Percentile . 118 6.8 Characteristics of Ionosphere Fixed, Weighted, and Floated Pseudo- Observables . 119 6.9 Widelane Ambiguity Resolution Performance, April 7 ..
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