Characterization of Airborne Antenna Group Delay as a Function of Arrival Angle and
its Impact on Accuracy and Integrity of the Global Positioning System
A dissertation presented to
the faculty of
the Russ College of Engineering and Technology of Ohio University
In partial fulfillment
of the requirements for the degree
Doctor of Philosophy
Anurag Raghuvanshi
August 2018
© 2018 Anurag Raghuvanshi. All Rights Reserved. 2 This dissertation titled
Characterization of Airborne Antenna Group Delay as a Function of Arrival Angle and
its Impact on Accuracy and Integrity of the Global Positioning System
by
ANURAG RAGHUVANSHI
has been approved for
the School of Electrical Engineering and Computer Science
and the Russ College of Engineering and Technology by
Frank van Graas
Professor of Electrical Engineering and Computer Science
Dennis Irwin
Dean, Russ College of Engineering and Technology 3 ABSTRACT
RAGHUVANSHI, ANURAG, Ph.D., August 2018, Electrical Engineering
Characterization of Airborne Antenna Group Delay as a Function of Arrival Angle and its Impact on Accuracy and Integrity of the Global Positioning System
Director of Dissertation: Frank van Graas
Antenna group delay variations (AGDV) requirements for single frequency airborne antennas were developed after the completion of ground and airborne specifications for aircraft precision approach operations using the Global Positioning
System (GPS). AGDV were not included in the aircraft position protection level calculations as they were assumed to be part of the allocation for aircraft pseudorange noise and multipath. Unfortunately, the dual-frequency airborne antennas used for the characterization of aircraft pseudorange noise and multipath exhibit smaller group delay variations as a function of signal arrival angle than the allowable errors for single frequency antennas. The Minimum Operational Performance Standards (MOPS) for airborne GPS antennas was updated in 2006, and it was found that single frequency airborne GPS antennas exhibit biases up to several feet due to group delays that are a function of arrival angle into the antenna. Therefore, AGDV must be treated as a new error source for high- accuracy aviation applications such as the Ground Based Augmentation System (GBAS), the Space Based Augmentation System (SBAS) and future Advanced Receiver
Autonomous Integrity Monitoring (ARAIM) concepts. The AGDV requirements were developed using measurements for the worst-case installed antenna but could not be verified during another test campaign. 4 This dissertation provides an approach to characterize and model the AGDV as a function of arrival angle for airborne GPS antennas, and analyze the impact of AGDV on aircraft position and protection levels for single frequency users. The impact of single frequency AGDV on the standard deviation of airborne noise and multipath will be revisited in the dissertation, as the validation of airborne noise and multipath error was performed using a dual frequency antenna. The validation of airborne integrity monitors to protect aircraft position estimation from anomalous ionospheric gradients also employed a dual frequency antenna. The dissertation investigates if the thresholds of the monitors change when a single frequency antenna is used. AGDV as a function of arrival angles will be a significant error for future dual frequency GBAS, SBAS, ARAIM, and other applications involving pseudorange measurements. An initial analysis on the impact of
AGDV on future systems will be performed.
5
To
My parents
Dr. Shyamla Singh & Dr. Mangala Singh
6 ACKNOWLEDGMENTS
I would take this opportunity to thank everyone for their support during the pursue of the doctoral degree. It has been a great journey and wouldn’t have happened without help and support of extraordinary individuals in my life.
Firstly, I would like to thank three most important people who helped me tremendously to achieve this goal, my advisor Dr. Frank van Graas, my mother Dr.
Shyamla Singh and my father Dr. Mangala Singh.
Dr. Frank van Graas has helped me as a mentor and as a parent, I always seek his advice for problems and he always had a correct answer. During my PhD, his unique approach to target a problem and his in depth understanding in the area has inspired me and helped me solve difficult problems with simplistic ease. He has been a great mentor and I have enjoyed being his teaching assistant and loved his undergrad lectures on electromagnetic waves. During my difficult time, he always inspired to persevere. I will be grateful all my life for all the support and help.
My mother, who always said to me that no matter what don’t give up and always believed in me, this dissertation would not have been possible without her support. I know she would be proud and I will try to keep her legacy alive and make her proud. I would like to thank my father who has always supported me, visited me when I needed him and always inspired me with his intelligence and his thoughts. Thanks for the continued support and love.
I would also like to thank Dr. Maarten Uijt de Haag, Dr. Wouter Pelgrum, Dr. Jeff
Dill, Dr. David Juedes, Dr. Harsha Chenji, Dr. Martin Mohlenkamp, Dr. Peter Jung for 7 serving on my dissertation committee and giving feedback on the manuscript and always being prompt on the replies.
I would like to thank Dr. Curtis Cohenour for help with initial experiment setup for my dissertation. I would like to thank Avionics Engineering Center and all its staff for the help they provided during my dissertation. I would like to thank Paul for help with the flight tests used for data collection.
I would like to thank Federal Aviation Administration to support the dissertation in parts.
I would like to thank Dr. Sabrina Ugazio for her support during my PhD, Rakesh,
Akshay, Kevin, Jessica and other Avionics students for easing up the pressure and making
Avionics Engineering Center a fun place to work. I would like to thank Ben, Avinash,
Dheerendra, Nupur, Aisha, Himanshu and Kushal for being there as a friend.
Finally, I would like to thank my cousin sisters and brothers Romila, Shruti, Sanvi,
Akancha, Anand, Ashish and many other to cheer me up. I would like to thank my aunt
Ritu Singh and Saurabh Singh for love, support and taking care of me in USA far from
India and for being my guardian here. I’m very grateful to all the family members for their constant words of encouragement.
8 TABLE OF CONTENTS
Page
Abstract ...... 3 Acknowledgments ...... 6 List of Tables ...... 11 List of Figures ...... 12 List of Acronyms ...... 16 1. Introduction ...... 18 2. Background ...... 22 Antenna Group Delay Variation ...... 22 Literature Review...... 24 2.2.1 Antenna Group Delay Variations for GPS antennas ...... 24 2.2.2 Group delay variation dependency on the feed design for a microstrip antenna ...... 27 2.2.3 Characterization of AGDV for a single frequency airborne antenna ...... 29 2.2.4 Background on effect of AGDV on Vertical Protection Level (VPL) ..... 32 2.2.5 Background on effect of AGDV on Airborne integrity monitors ...... 34 2.2.6 Background on Spherical Harmonics Expansion and Binning Model using Least Squares estimation...... 35 2.2.7 Effects of AGDV on other applications ...... 35 2.2.8 Background on Dual Frequency Smoothing techniques ...... 36 Research questions ...... 37 2.3.1 Contributions and Organization ...... 38 3. Overview of Global Positioning System ...... 42 3.1 Differential Global Positioning System ...... 46 3.1.1 Space Based Augmentation System...... 47 3.1.2 Ground Based Augmentation System ...... 48 3.1.3 Dual Frequency GBAS ...... 52 4. Concept of Accuracy and integrity in GBAS ...... 54 4.1 Overview ...... 54 4.1.1 Error sources for standalone GPS ...... 57 4.1.1.1 Satellite Clock and Orbit Errors ...... 61 9 4.1.1.2 Ionospheric Errors ...... 62 4.1.1.3 Tropospheric Delay ...... 66 4.1.1.4 Multipath and Receiver Noise ...... 66 4.1.1.5 Antenna Group Delay...... 67 4.1.1.6 Phase Wrap up ...... 68 4.1.2 DGPS and Smoothing in GBAS ...... 70 4.1.2.1 Smoothing Algorithm for GBAS ...... 71 4.1.2.2 DGPS in GBAS ...... 72 4.2 Accuracy of the GBAS ...... 75 4.2.1 Model of residual Errors in GBAS ...... 78 4.2.1.1 Airborne Accuracy models...... 78 4.2.1.2 Ground Accuracy Models ...... 80 4.2.1.3 Residual Tropospheric Errors...... 82 4.2.1.4 Residual Ionospheric Errors ...... 83 4.3 Integrity concept for GBAS ...... 84 4.3.1 Protection Level Estimation ...... 85 5. Antenna Group delay variation Characterization ...... 87 5.1 Overview ...... 87 5.2 Methodology to characterize AGDV ...... 87 5.2.1 Code-minus-carrier processing ...... 88 5.2.2 Ground experiments ...... 90 5.2.3 Flight experiments (from [34]) ...... 101 6. Modeling of AGDV ...... 107 6.1 Least Squares Approach ...... 107 6.1.1 Binning model approach to model antenna group delay variation ...... 108 6.1.2 Spherical Harmonics expansion approach to model antenna group delay variation ...... 115 7. Effect of AGDV on position solution and Protection Level ...... 121 7.1 Aircraft Position Errors and Protection Levels ...... 121 7.1.1 Impact of Single Frequency Antenna Group Delay Variations on Position Solution ...... 122 7.1.2 Impact of Single Frequency Antenna Group Delay Variations on Vertical Protection Level ...... 128 10 7.1.2.1 Treat AGDV as a specific risk ...... 128 7.1.2.2 Treat AGDV as average risk ...... 130 7.1.2.3 Show that the protection level doesn’t change significantly in the presence of AGDV ...... 135 7.1.2.4 Reduce the AGDV bound ...... 135 7.1.3 AGDV bounding conclusions ...... 135 8. Effect of AGDV on airborne integrity monitors ...... 137 8.1 Airborne Integrity Monitors ...... 137 8.1.1 Ionospheric threat space ...... 139 8.1.1.1 DSIGMA Range ...... 142 8.1.1.2 Dual-solution vertical bias parameter ...... 145 8.1.1.3 Airborne CCD monitor...... 147 8.2 Flight Experiment ...... 150 8.3 Computer Simulations ...... 152 8.4 Flight Test Results for L1 Antenna ...... 155 8.5 Flight Test Results for Multi-Band Antenna ...... 158 9. Effect oF AGDV on Dual frequency GBAS ...... 163 9.1. Impact of Dual-Frequency AGDV on the ionosphere-free position solution .... 163 9.2 Airborne Integrity Monitoring for Dual Frequency GBAS ...... 168 10. Conclusions and recommendations ...... 172 References ...... 173 Appendix A: Least Squares Estimation ...... 178 Appendix B: Code-Minus-Carrier ...... 181 Appendix C: Spherical Harmonics ...... 184 Appendix D: Dual Frequency GBAS Measurments ...... 188
11 LIST OF TABLES
Page
Table 1. Categories of Precision Approaches with Runway Visual Range and Decision Height ...... 19 Table 2. Pseudorange Error: standalone GPS vs DGPS (From [39]) ...... 47 Table 3. GBAS based landing requirements ...... 56 Table 4. Airborne Accuracy Designators...... 79 Table 5. Ground Accuracy Designators ...... 81 Table 6. Fault-Free Missed Detection Multiplier ...... 86 Table 7. Missed Detection Multiplier ...... 86 Table 8. Model parameters used by Honeywell for simulation (from [24]) ...... 145
12 LIST OF FIGURES
Page
Figure 2.1. Side view of a probe fed microstrip antenna ...... 26 Figure 2.2. From left to right: orthogonal feed, symmetric orthogonal feed, diagonal feed and symmetric diagonal feed ...... 28 Figure. 2.3. Comparison of σpr, gnd, σpr, air, and AGDV bound as a function of elevation angle (from [34]) 31 Figure 3.1. Standalone GPS architecture ...... 45 Figure 3.2. Ground Based Augmentation System architecture...... 48 Figure 3.3. Ground Based Augmentation System across the world. (Courtesy of flygls.net) ...... 51 Figure 4.1. Approach categories for GBAS with Decision Height (DH), vertical and lateral alert limit ...... 55 Figure 4.2. GPS transmitted signal ...... 58 Figure 4.3. Factors affecting the Pseudorange and Carrier Phase measurements ...... 59 Figure 4.4. Delay in the time of transmission due to various factors ...... 60 Figure 4.5. Obliquity Factor...... 64 Figure 4.6. Carrier-smoothed code processing in GBAS ...... 70 Figure 4.7. Position solution for approach service type D (GAST D) ...... 75 Figure 4.8. Airborne Accuracy Designators contribution to overall Pseudorange ...... 80 Figure 4.9. Ground Accuracy Designator contribution to overall Pseudorange ...... 82 Figure 4.10. GBAS integrity allocation ...... 84 Figure 5.1. Ground rotator experiment ...... 91 Figure 5.2. Processing of code minus carrier data to characterize antenna group delay variation for a Dual-Frequency antenna ...... 92 Figure 5.3. Antenna Group Delay Variations for the Dual-Frequency antenna ...... 93 Figure 5.4. Dual-Frequency Antenna Group Delay Variations versus DO-301 Bounds from [2] ...... 93 Figure 5.5. Antenna Group Delay Variations for the Dual-Frequency Antenna for each elevation bin ...... 95 Figure 5.6. Processing of code minus carrier data to characterize antenna group delay variation for a Single-Frequency antenna ...... 96 Figure 5.7. Antenna Group Delay Variation for a Single-Frequency antenna ...... 97 Figure 5.8. Single-Frequency Antenna Group Delay Variations versus DO-301 bounds from [2] ...... 97 13 Figure 5.9. Antenna Group Delay Variation for a Single-Frequency antenna for each elevation bin ...... 98 Figure 5.10. Rounded edge ground plane ...... 99 Figure 5.11. Single frequency antenna group delay variation versus DO-301 bounds for rounded edge ground plane ...... 99 Figure 5.12. Single frequency antenna group delay variation for each elevation bin for rounded edge ground plane ...... 100 Figure 5.13. Multi-Band antenna installed on a Beechcraft Baron (from [34]) ...... 101 Figure 5.14. Aircraft ground track for level orbit (from [34]) ...... 102 Figure 5.15. CMC tracks as a function of time ...... 103 Figure 5.16. CMC tracks as a function of azimuth angle with respect to the antenna ... 104 Figure 5.17. Combined group delay variation and multipath errors versus DO-301 bounds ...... 105 Figure 6.1. CMC as function of elevation and azimuth represented as a grid of azimuth and elevation angles...... 109 Figure 6.2. CMC as a function of azimuth and elevation represented in form of polar grid...... 109 Figure 6.3. Simulated antenna group delay model...... 111 Figure 6.4. Estimated AGDV using simulated AGDV model and binning model approach...... 112 Figure 6.5. Difference between estimated AGDV using simulated AGDV model and binning model approach...... 113 Figure 6.6. Estimated AGDV using real data and Binning Model approach...... 114 Figure 6.7. Estimated AGDV using Binning Model with Least Squares estimation as a function of elevation angle...... 114 Figure 6.8. Estimated AGDV using Binning Model with Least Squares estimation as a function of azimuth angle...... 115 Figure 6.9. Antenna Group Delay estimation using Spherical Harmonics approach. .... 117 Figure 6.10. Difference between AGDV model and estimated AGDV using Spherical Harmonics expansion...... 118 Figure 6.11. Estimated AGDV using Spherical Harmonics expansion approach...... 119 Figure 6.12. Spherical Harmonics expansion model of AGDV as a function of elevation angle compared to DO-301 bound...... 119 Figure 6.13. Spherical Harmonics expansion model of AGDV as a function of azimuth angle...... 120 Figure 7.1. Calculation of nominal and worst-case vertical position errors ...... 123 14 Figure 7.2. Typical and worst case east and north position errors due to single-frequency antenna group delay variations ...... 124 Figure 7.3. Typical and worst case vertical and clock errors due to single-frequency antenna group delay variations ...... 125 Figure 7.4. Poor Geometry typical and worst case east and north position errors due to single-frequency antenna group delay variations ...... 126 Figure 7.5. Poor geometry typical and worst case vertical and clock errors due to single- frequency antenna group delay variations ...... 126 Figure 7.6. Vertical position error compared to the bound for a typical and a poor geometry ...... 127 Figure 7.7. Vertical protection level from equations 4.26 and 7.1, and worst-case vertical position error due to single-frequency AGDV biases ...... 129 Figure 7.8. Poor geometry VPL from equations 4.26 and 7.1, and worst-case vertical position error due to single-frequency AGDV biases ...... 129 Figure 7.9. AGDV model (left), Multipath EM model (middle), Combined model (right) (from [36]) ...... 131 Figure 7.10. Normalized VPL vs VPL (from [36]) ...... 132 Figure 7.11. Code minus carrier as function of elevation angle for a single frequency antenna flight data (from [36]) ...... 133 Figure 7.12. Normalized Position Domain Projection Error for INR data (from [36]) .. 134 Figure 8.1. Nominal ionospheric behavior ...... 138 Figure 8.2. Anomalous ionospheric behavior ...... 139 Figure 8.3. Ionosphere Threat Model (from [37]) ...... 140 Figure 8.4. Dual Solution Ionospheric Gradient Monitoring Algorithm (DSIGMA) Processing Overview ...... 142 Figure 8.5. Geometry for Validation of the Ionospheric Threat Mitigation ...... 145 Figure 8.6. Step Responses of 100-s and 30-s Smoothing Filters and their Difference (from [37])...... 148 Figure 8.7. Piper Saratoga Research Aircraft with Multi-Band and L1-only Antenna (from [37])...... 150 Figure 8.8. Flight Trajectory for DSIGMA and CCD Monitor Evaluation...... 151 Figure 8.9. Satellite Elevation Angles during the Flight (from [37])...... 152 Figure 8.10. Simulation of Difference between 100-s and 30-s Smoothed Pseudoranges due to L1 Antenna Group Delay Variations (from [37])...... 153 Figure 8.11. Simulation of Difference in Vertical Protection Level using 100-s and 30-s Smoothed Pseudoranges due to L1 Antenna Group Delay Variations (from [37]) ...... 154 15 Figure 8.12. Simulation of CCD Monitor Response to L1 Antenna Group Delay Variations (from [37]) ...... 155 Figure 8.13. Flight Test L1 Antenna Difference between 100-s and 30-s Smoothed Pseudoranges (from [37]) ...... 156 Figure 8.14. Flight test L1 antenna difference in Vertical Protection Level using 100-s and 30-s smoothed pseudoranges (from [37]) ...... 157 Figure 8.15. Flight Test L1 Antenna CCD Monitor Response (from [37]) ...... 158 Figure 8.16. Flight Test Multi-Band Antenna Difference between 100-s and 30-s Smoothed Pseudoranges (from [37]) ...... 159 Figure 8.17. Flight Test Multi-Band Antenna Difference in Vertical Protection Level using 100-s and 30-s Smoothed Pseudoranges (from [37])...... 160 Figure 8.18. Flight Test Multi-Band Antenna CCD Monitor Response (from [37]) ...... 161 Figure 9.1. Ionosphere-free pseudorange errors due to L1 and L2 group delay variations as a function of azimuth and elevation angles...... 165 Figure 9.2. Dual-frequency ionosphere-free antenna group delay variations versus DO-301 bounds ...... 166 Figure 9.3. Typical east and north errors due to dual-frequency antenna group delay variations in an ionosphere-free solution ...... 167 Figure 9.4. Typical vertical and clock errors due to dual-frequency antenna group delay variations in an ionosphere-free solution ...... 168 Figure 9.5. Flight test multi-band antenna difference between 100-s and 30-s smoothed ionosphere-free pseudoranges ...... 169 Figure 9.6. Flight test multi-band antenna difference between 100-s and 30-s smoothed divergence-free pseudoranges ...... 170
16 LIST OF ACRONYMS
AD Accumulated Doppler AGDV Antenna Group Delay Variation AMM Airborne Multipath Model BPSK Binary Phase Shift Keying CCD Code Carrier Divergence CMC Code Minus Carrier CORS Continuously Operating Reference Stations CP Carrier Phase DGPS Differential GPS DH Decision Height DSIGMA Dual Solution Ionospheric Gradient Monitor Algorithm EUROCAE European Organization for Civil Aviation Equipment FRPA Fixed reception pattern antennas GAEC GBAS Airborne Equipment Classification GAST D GBAS Approach Service Type D GBAS Ground Based Augmentation System GNSS Global Navigation Satellite System GPS Global Positioning System GS Ground Station ICAO International Civil Aviation Organization IGM Ionospheric Gradient Monitor ILS Instrument Landing System IPP Ionospheric pierce point LAAS Local Area Augmentation System LAL Lateral Alert Limit LTP Landing Threshold Point MASPS Minimum Aviation System Performance Standards MOPS Minimum Operational Performance Standards PR Pseudorange RHCP Right hand circularly polarized 17 RR Reference Receivers RVR Runway Visual Range SC Special Committee VAL Vertical Alert Limit VDB VHF Data Broadcast VHF Very High Frequency WAAS Wide Area Augmentation System WG Working Group
18 1. INTRODUCTION
Navigation is defined in the Oxford Dictionary as “the process or activity of accurately ascertaining one’s position and planning and following a route.” Thus, finding a 3-D position is pertinent to navigate an aircraft from one point to another, and a system is needed which provides a 3-D position for all phases of flight including departure, enroute, landing and airport surface operations. The aviation sector around the world is seeing an exponential growth but the system is still plagued by delays due to inclement weather while landing an aircraft. The motivation behind a landing system that can help aircraft touch down in all-weather conditions originates from such issues, as well as from the desire to increase safety through increasing the availability of landing systems [1].
The Global Navigation Satellite System (GNSS) is an all-weather worldwide position, velocity and timing system. The GNSS is comprised of satellite systems and their augmentations around the world; Global Positioning System (GPS, USA), Galileo
(Europe), GLONASS (Russia), BeiDou (China) and regional satellite systems; QZSS
(Japan) and NavIC (India). There are numerous GNSS applications in the fields of
Agriculture, Environment, Transportation, Surveying and Mapping. Over the years, various navigation aids have been used to navigate, but GNSS being an all-weather, world- wide system is seen as the future of navigation systems. In the aviation sector, GNSS-based systems are already providing the required accuracy, reliability, continuity and availability for all phases of flight, including non-precision approach and Category I precision approach.
The most demanding phase of the flight with respect to the performance requirement aforementioned is the approach and landing phase. Precision approach system 19 performance is divided into three categories with different decision heights as a function of visibility conditions. The decision height is the lowest height above the ground at which the pilot must be able to visually acquire the runway environment. If the pilot is not able to see the runway or runway lights, a missed approach must be initiated. The decision heights and visibility requirements are summarized in Table 1.
Table 1.
Categories of Precision Approaches with Runway Visual Range and Decision Height Category of Precision Approach Runway visual range (RVR) and Decision Height (DH) Category I RVR≥1800 ft, DH≥200ft
Category II RVR≥1200 ft, 100ft≤DH≤200ft
Category III 푅푉푅 ≥1200ft, DH≤100ft
Both the Instrument Landing System (ILS) and the GNSS-based Ground Based
Augmentation System (GBAS) are certified to support Category I precision approaches.
For GBAS, this is also referred to as GBAS Approach Service Type C (GAST C). At the time of this writing, ILS is the only system in the United States certified to support
Category II and III precision approaches. GBAS refers to these approaches as GBAS
Approach Service Type D (GAST D), and its design approval is in process at the time of this writing. Category III landings are important for future Unmanned Aerial Systems
(UAS).
GBAS is seen as an alternative to ILS. The current GBAS is a single frequency system (L1 at 1575.42 MHz) with augmentation from ground reference receivers. The 20 augmentation is needed because the standalone GNSS accuracy doesn’t meet the requirements needed for the precision approach. In GBAS, the airborne receiver’s measurements are augmented by the corrections provided by the ground reference receivers located at surveyed locations on an airport. The corrections improve the quality of the measurements used for the position and velocity solution by the aircraft, and thus the overall performance of the system improves.
GBAS is a complex system and the overall performance depends on various factors.
The mitigation of all risks due to errors in the system is one of the important factors for system approval. Thanks to the efforts of various research groups, major error sources like tropospheric delay, ionospheric delay, ground and airborne multipath have been studied extensively, and its effects on the system have been analyzed. The airborne antenna is an integral part of GBAS, and the performance specifications for the antenna are documented in the Minimum Operational Performance Standards (MOPS) [2]. It also outlines the tests to validate various performance parameters for the airborne antenna.
RTCA, Inc., a standards development organization, updated the MOPS for single frequency airborne antennas in 2006 [2]. The updated MOPS specify the performance requirements and test procedures for an airborne antenna when used to receive GPS signals on an aircraft for a GAST D approach. Requirements for airborne GPS antenna-induced biases known as antenna group delay variation (AGDV) as a function of arrival angle were added to the MOPS. AGDV is an inherent property of antennas. The MOPS put a bound on AGDV as a function of elevation angle, and an antenna must comply with the bound if used for a GAST D precision approach. Although the bounds on AGDV are specified in the updated MOPS, the impact of AGDV on the overall GAST D system performance is 21 not included. Specifically, AGDV has not been included in the error budget, which is used to calculate the position accuracy and integrity of the system. Thus, to determine the impact of AGDV requirements on system performance, a thorough analysis is required.
The motivation of the dissertation is to contribute to the field of GBAS-based precision approaches, by introducing a method to characterize antenna-induced biases as a function of azimuth and elevation angles for a single frequency airborne antenna. Also, to study its impact on the accuracy and reliability of GBAS. The characterization of AGDV in the context of GAST D is necessary for the requirements validation. The dissertation also explores methods to model AGDV, which will be helpful for future simulation and analyses of applications affected by AGDV. The modeling technique is generic and applies to antennas for other GNSS applications as well. The integrity of GBAS is realized using ground and airborne-based monitors. The impact of AGDV on airborne integrity monitors will also be studied. Conclusions on the significance of AGDV as an error source will be provided for current and future GBAS architectures.
22 2. BACKGROUND
Antenna Group Delay Variation
The first component of a GPS receiver is a GPS antenna, which is used to receive the signals transmitted by the GPS satellites. GPS antennas have different requirements based on the application in which they are used. This dissertation concentrates on single frequency airborne GPS antennas, for which [2] contains the Minimum Operational
Performance Standards (MOPS). The main function of the antenna is to receive the signals in the GPS L1 band and reject out-of-band signals.
Antennas can be modeled as a spatial filter. A spatial filter is defined by the amplitude and phase response of an antenna over the bandwidth of the antenna, and over all signal arrival angles. The phase response depends both on the frequency of the signal as well as the elevation and azimuth angles with respect to the antenna of the satellite transmitting the signal. The derivative of the phase response with respect to frequency is known as group delay and thus, is also a function of the frequency of the signal, as well as the elevation and azimuth angles of the signal. The group delay at a certain frequency and arrival angle is the delay experienced by a signal through the antenna.
The group delay as a function of frequency is defined in terms of the phase response of the antenna 휙(휔) for different frequencies 휔 and is calculated as:
푑휙(휔) 휙(휔 + 훥휔) − 휙(휔 − 훥휔) 휏(휔) = ≈ (2.1) 푑휔 2훥휔
It can be interpreted from Equation 2.1 that antenna group delay is zero for antennas with uniform phase response as a function of frequency. The group delay is divided into two components: the difference of delay through the antenna system for two different 23 frequency, and the group delay variation as a function of signal arrival angle, also referred to as the antenna group delay variation (AGDV). The group delay difference between two frequencies according to the antenna MOPS should be less than 25 nanoseconds within the operating bandwidth of ±10.23 MHz [2]. The group delay of a single frequency GPS antenna as a function of azimuth and elevation angles when compared to the zenith direction group delay defines AGDV and is given by Equation 2.2.
( ) ( ) ( ) 훥휏휔푐,퐶/퐴 휃푖, 휑푖 = 휏휔푐,퐶/퐴 휃푖, 휑푖 − 휏휔푐,퐶/퐴 휃 = 90° (2.2) where:
( ) 훥휏휔푐,퐶/퐴 휃푖, 휑푖 is the antenna group delay variation;
휔푐,퐶/퐴 is the carrier frequency (L1 at 1575.42 MHz) modulated with the C/A code;
휃 is the elevation angle of the satellite in the antenna reference frame; and
휑 is the azimuth of the satellite in the antenna reference frame.
AGDV is dependent on the satellite geometry as different satellite elevation and azimuth angles will experience different delays through the antenna. Note that [2] uses the group delay average over all azimuth angles at an elevation angle of 5° as the reference value, instead of the group delay at an elevation angle of 90° as used in Equation (2.2).
There are various reasons to use 90° as the reference. First, the multipath and weaker antenna gain pattern of the antenna at lower elevation angles make the AGDV measurement noisier. Second, the average of group delay at 5° for all azimuth angles can introduce a bias, as the mean can be different from the average of all the azimuth angles for higher elevation angles. The bias will increase the bound of the AGDV at higher elevation angles. Third, at an elevation angle of 90°, AGDV is the same for all azimuth angles, which is enforced by using 90° elevation as the reference. 24 The most recent version of the MOPS includes a new requirement on the AGDV for an airborne single frequency antenna used for GAST D precision approach. The AGDV bound is a function of elevation angle and is defined as [2]:
퐺퐷(휃, 휑) = 2.5 − 0.0462(θ − 5°) ns for 5° ≤ θ ≤ 45°
else, 퐺퐷(휃, 휑) = 0.65 ns (2.3)
Literature Review
The focus of the dissertation is to characterize AGDV for a single frequency airborne antenna planned for use with GAST D precision approach, and to characterize its effect on the accuracy and integrity of the system. This section summarizes previous work on the topic of antenna group delay variations and the methods used to characterize it. The effect of AGDV on other applications is also described.
2.2.1 Antenna Group Delay Variations for GPS antennas
GNSS antenna design depends on the application for which it is used. The gain pattern, polarization axial ratio and multipath mitigation needed for a certain application are some of the factors which influence the antenna design. Also, the size and cost are important and vary from area of usage, for example the GBAS ground reference receiver
(RR) antenna design takes into consideration the mitigation of ground multipath. Thus, it must have a sharp cut-off gain pattern at lower elevation angles. The 2-m tall multipath limiting antenna is a good example of a GBAS ground antenna [3]. The airborne GNSS antenna on the other hand should be small and conformal, and thus the gain pattern is different from the ground reference antenna. Also, if the antenna is used for dual frequency applications it should be wideband [4]. 25 The fixed reception pattern antenna (FRPA) is the most common antenna used in
GNSS systems. It has a nearly omnidirectional pattern in the upper hemisphere and is mostly right hand circularly polarized (RHCP) [4]. The microstrip antenna is a type of
FRPA and is often found in the design of GNSS airborne antennas. The ground antenna and the airborne antennas in GBAS both have AGDV but show different characteristics due to different antenna designs. This dissertation focusses on the characterization of AGDV for a single frequency airborne antenna.
The efforts to mitigate the effects of AGDV for reference station antenna designs for use in Ground-Based and Space-Based Augmentation Systems (GBAS and SBAS) [5,
6] was carried out in the 1990’s. One of the initial reference antennas used for SBAS ground stations had large biases in the pseudorange measurements as a function of azimuth and elevation, which delayed its deployment [7]. AGDV characterization for GBAS ground reference antennas was performed in [5, 8], and it was concluded that calibration of AGDV is not practical due to differences between antennas and configuration control of the antenna installation.
The calibration of an airborne antenna is even more complicated than a ground antenna. The airborne antenna calibration algorithm would need to know the arrival angles for each of the satellites and correct for aircraft roll, pitch and heading angles. Each aircraft might need its own set of calibration parameters for each antenna mounted in a certain orientation, that could change when an antenna is replaced. Finally, the integrity of the calibration process would need to be included in the integrity requirements. The best way forward for an airborne antenna is to understand the design of the antenna and how it influences the group delay variations. The characterization of AGDV can be compared for 26 different antenna designs and the best design can be used which satisfies all other requirements as well. An antenna bound can be defined for the group delay variation of antennas used for a certain application. The bound should be analyzed to predict its implication on the performance parameters of the system. The process assures the antenna is satisfying the performance requirements of the system such as accuracy, integrity, continuity and availability.
The microstrip antennas (or patch antennas) used for airborne applications should meet the Aeronautical Radio Incorporated (ARINC) 743 size requirements. The antennas are restricted to 퐿 × 퐵 × 퐻 = 4.7" × 3" × 0.73". Figure 2.1 shows the side view of a basic microstrip antenna. The antenna consists of a metallic patch on a dielectric substrate and a metallic ground plane.
Figure 2.1. Side view of a probe fed microstrip antenna
Details on patch antenna design can be found in Chapter 14 of [9]. The antenna has an amplitude and a phase response. AGDV is dependent on the phase response as a 27 function of frequency with respect to the elevation and azimuth angles. The phase response and thus the AGDV of the patch antenna depends on the feed points of the antenna, the substrate used, and the design of the antenna. Next, studies conducted on the influence of the feed symmetry on AGDV will be discussed.
2.2.2 Group delay variation dependency on the feed design for a microstrip antenna
In [10], it is pointed out that for high accuracy GPS applications it is important to consider group delay of an antenna as a function of elevation and azimuth angles. A uniform group delay response for all incidence angles is a characteristic needed for high accuracy GPS, else different signals will be delayed by different intervals based on their incidence angles. The ideal phase characteristic is difficult to achieve for practical antennas, but design of the antenna can minimize AGDV for certain applications [10]. The symmetry of an antenna and its feed point(s) also define the group delay response of an antenna as a function of azimuth and elevation angle [10].
The AGDV for antennas with different feed designs was studied in [11]. A computer simulation program from Ansoft was used to theoretically calculate the AGDV.
The different feed configurations are shown in Figure 2.2, which shows the top view of the microstrip antenna with different feed probe configurations.
28
Figure 2.2. From left to right: orthogonal feed, symmetric orthogonal feed, diagonal feed and symmetric diagonal feed
The paper concluded that the symmetric feed configurations has less AGDV, and the orthogonal feed has less AGDV than the diagonal feed [11]. Recently, the German
Aerospace Center (DLR) did a similar study with antennas developed by them, to see the effects of different feed designs on the AGDV estimates [12]. The study showed that symmetric feed configurations had the most uniform response for the group delay as a function of azimuth and elevation angles. Thus, the feed design of the antenna plays an important part in the pseudorange error introduced by the antenna. The other factors that can introduce group delay in an antenna are the pre-select and post-select bandpass filters, the low noise amplifiers due to frequency response of each of these components [4]. Also, the thickness of the dielectric substrate used in the microstrip antenna can influence the
AGDV. Over time, if the substrate is affected by impurities then the AGDV pattern as a function of arrival angle can change. A non-ideal hybrid network to generate right hand circularly polarized radiation for a microstrip antenna can also contribute to the group delay in the antenna [4]. Thus, antenna designs with emphasis on the abovementioned factors can reduce the AGDV to the desired bounds. 29 2.2.3 Characterization of AGDV for a single frequency airborne antenna
The GBAS is based on a single frequency carrier-smoothed code-differential GPS architecture. In basic terms, the RRs have single frequency GNSS receivers connected to antennas installed at surveyed locations, and thus the RRs can measure the errors affecting the system. GBAS broadcasts the errors for each of the visible satellites in the form of pseudorange corrections. An aircraft in the vicinity of the GBAS RRs can then apply these corrections and improve the accuracy of its measurements. The concept of corrections work because the largest errors are spatially correlated, including ionospheric and tropospheric delays, and satellite clock and orbit errors. The RRs and the airborne receiver also apply smoothing techniques to minimize the effects of errors like multipath and receiver noise which are not spatially correlated. Thus, residual multipath error is a major error source for
GBAS. The characterization and modeling of the residual airborne multipath was needed so that a generic model can be used for all aircraft types. A flight test with different aircraft types was used to characterize the standard deviation of the airborne noise and multipath,
휎푝푟,푎푖푟 for Category I approach and positioning services as documented in [13].
A standard error model was derived for airframe multipath and noise and was included in the GBAS MOPS [14]. The validation of the error model was done by a joint effort of FAA, Boeing and Honeywell, and details on how the model was validated can be found in [13]. The paper pointed out that the airborne multipath model is appropriate for
Cat I operations, but it was considered insufficient for more demanding applications like
Cat II and III precision approaches because the flight test was not done with the intention of characterizing the airborne multipath for those operations. 30 Therefore, Boeing started a new program in collaboration with FAA to investigate an improved model for airborne multipath which would also include ground bounce multipath [15]. Extensive flight tests in support of this program were used, and various papers were published based on the analysis [16, 17]. Electromagnetic modeling was also done in conjunction with the flight test to compare the two methods, and it was concluded that the existing Airborne Multipath Model (AMM) would be appropriate for GAST D.
Airborne multipath was also investigated in [18], and it was concluded that the aircraft fuselage is the main contributor to airframe multipath and the contribution of wings and rudders can be ignored.
When 휎푝푟,푎푖푟 was developed, AGDV was assumed to be negligible, but in [13] a common mode error was seen in the code minus carrier measurement and was thought to be due to the difference between group delays at the L1 and L2 frequencies. It is important to note that AGDV and multipath errors are difficult to separate out inflight; both are a function of arrival angle of the signal. Therefore, the assumption was, that if there would be AGDV, it would be included in the airborne multipath model. It is further noted that the development of 휎푝푟,푎푖푟 used a dual frequency antenna to enable the removal of ionospheric delays.
When RTCA SC-159, Working Group 7, started work on a new antenna MOPS for single frequency antennas, it was recognized that AGDV errors are not small, and an error budget was developed to bound the error. The resulting bound was included in the new antenna MOPS [2]. The method used to obtain the bound was published in [19]. A different test campaign [20] with the goal of analyzing the effect of the AGDV bound on the system requirements couldn’t verify the results. 31 A comparison of the AGDV bound with other error sources, such as the standard deviations of the differential corrections, 휎푝푟,푔푛푑, and the standard deviation of the airborne noise and multipath, 휎푝푟,푎푖푟, is illustrated in Figure 2.3.
Figure. 2.3. Comparison of 휎푝푟,푔푛푑, 휎푝푟,푎푖푟, and AGDV bound as a function of elevation angle (from [34])
Although the AGDV bound in Figure 2.3 is larger than the ground and airborne multipath and receiver noise curves, the latter two are standard deviations, while the AGDV is a bias bound. Thus, AGDV is characterized as a bias whereas the other error sources are treated as noise. This is an important observation as bias and noise are treated in a different manner when incorporated into the protection level calculation. In this dissertation, a ground-based method will be introduced to characterize AGDV, and the results will be compared to the bounds in [2]. Next, the AGDV values will be utilized to determine the effects of the error on position accuracy and integrity of the GBAS. 32 A new characterization of the contribution of AGDV to 휎푝푟,푎푖푟 is needed using a flight test, since the previous characterization used a dual frequency antenna to collect airborne data. The ground-based characterization will assist in the understanding of
AGDV, while flight test will aid to comprehend the combined effect of multipath and
AGDV on the system performance, and on the integrity of the system. Future system designs such as dual frequency GBAS and Advanced Receiver Autonomous Integrity
Algorithm (ARAIM) will also be affected by the AGDV so a thorough understanding of the error source is important [21].
2.2.4 Background on effect of AGDV on Vertical Protection Level (VPL)
The integrity of a system is the confidence in the output of the total system and the ability to provide timely warning to the user if the output cannot be trusted. GBAS integrity for GAST D is demanding, and the probability of providing misleading information to the aircraft should be less than 10−9/푎푝푝푟표푎푐ℎ for 15 s exposure time in the vertical direction and 30 s exposure time in the lateral direction [22]. After the system exceeds the protection level, the time to alert the pilot is 2 seconds. The total integrity in GBAS is divided into airborne integrity and Signal-In-Space (SIS) integrity. The SIS integrity is divided into 3 cases or hypotheses which cover the faults that can cause integrity risk. The hypothesis 퐻0 and 퐻1 cover the cases in which the reference receiver is either fault free, or one of the reference receivers is faulty, respectively. The 퐻2 hypothesis covers other faults which can create integrity risk, such as dual, simultaneous reference receiver failures. The faults include failures in the ground subsystem, failures due to atmospheric conditions, failures in GPS satellites, such as excessive code-carrier divergence, excessive acceleration, and orbit errors [22]. The 퐻2 case risks are covered by implementing ground monitoring and 33 sufficient work has been done in this area to cover all the risks. The 퐻0 and 퐻1-associated risks are evaluated using the concept of protection level (PL). The PL is a confidence bound in the position domain and is calculated using satellite geometry and the statistics of the
GBAS errors. The ground reference receivers provide the airborne receivers with integrity information as they are at surveyed locations and can calculate the errors in the system.
The error information is broadcast as a differential correction, and the airborne receiver improves its measurements using the corrections. Using satellite geometry and the measurement error statistics, the PL is calculated and compared to the Alert Limit, which is based on the aircraft operation.
The performance standards documented in [14] and [22] provide the equations to calculate the PL. In the calculation of the PL, it is necessary that the error sources affecting the
GBAS are characterized and modeled. The protection levels are calculated in the vertical and lateral directions. Generally, the Vertical Protection Level is more demanding and has a greater effect on the availability of the GBAS, so it is primarily considered in this dissertation.
The current calculation of VPL was developed without an error budget for
AGDV. The VPL calculation assumes that all error sources are zero-mean and normally distributed, and the biases are covered using an inflated standard deviation. One of the concerns of this method is that if the group delay is bounded using a normal distribution, then the standard deviation that must be used to bound the group delay bias would be several times larger than a bias error bound [23]. The important question is how to incorporate AGDV in the protection level calculations. Four candidate methodologies were identified to incorporate AGDV into the protection level equations: 34 Treat AGDV as a specific risk and add a bias component to the protection level;
Treat AGDV as an average risk and, if necessary, inflate 휎푝푟,푎푖푟;
Show that the protection level doesn’t change significantly in the presence of
AGDV; and
Reduce the AGDV bound.
All four methods will be studied in this dissertation, and a conclusion will be made on which method should be used to incorporate AGDV into the PL, and which method would least affect the service in terms of availability.
2.2.5 Background on effect of AGDV on Airborne integrity monitors
The faults that are not covered in the 퐻0 and 퐻1 integrity risks are mitigated using monitors at the ground reference receivers and airborne receivers. An example of such a risk are anomalous ionospheric gradients. During such conditions, the ground and the airborne receiver encounter different ionospheric delays, and thus the differential corrections are not able to mitigate the error due to ionospheric delay.
A combination of airborne and ground monitors is used to cover the threat space due to ionospheric gradients. The Dual Solution Ionospheric Gradient Monitor Algorithm
(DSIGMA) and Airborne Code Carrier Divergence (CCD) monitor were incorporated into the performance standards to detect ionospheric gradients affecting the measurements of the airborne receivers [14]. The thresholds for the monitors were validated using various flight tests [24]. These flight tests also used a dual frequency antenna.
The DSIGMA and CCD thresholds are dependent on the airborne multipath, and as discussed in the previous section, it is difficult to separate AGDV and multipath, therefore the thresholds might change if a single frequency antenna is used because of the 35 contribution from AGDV. The impact of single frequency airborne AGDV on the airborne monitors will be studied in this dissertation using flight test data and simulations.
2.2.6 Background on Spherical Harmonics Expansion and Binning Model using
Least Squares estimation
Models for the various GBAS error sources are important for system performance analysis for accuracy, integrity and availability, for example the airborne multipath and noise model defines the standard deviation of the residual multipath and noise in GBAS.
The motivation behind the modeling was to avoid testing every aircraft for the multipath characteristics. Similarly, the AGDV of antennas used in specific applications have similarity and thus a model will be helpful to do the analysis involving antennas for certain applications. The method of Least Squares will be used for modeling the AGDV using a binning model and a spherical harmonics expansion. Spherical harmonics expansions have been used previously to model the residuals due to phase variations as a function of azimuth and elevation angles for an Integrated Multipath Limiting Antenna [25], but a binning model approach has not been used before. In this dissertation, the two model approaches are compared.
2.2.7 Effects of AGDV on other applications
AGDV is a generic problem, and even the satellite antennas introduce AGDV as a function of satellite elevation angle as documented in [26]. The Controlled Reception
Pattern Antennas (CRPA) used in the Joint Precision Approach and Landing System
(JPALS) also experiences AGDV and the effects on pseudorange measurements have been studied in [27, 28 ,29]. 36 Various research groups have worked on the characterization of AGDV for several antennas [30, 31]. These contributions highlight the importance of AGDV as an error source by analyzing its impact on various applications, and further validates the effort presented in this dissertation to study AGDV’s impact on the application of precision approach. The approach developed in this dissertation can also be used for other applications affected by AGDV.
2.2.8 Background on Dual Frequency Smoothing techniques
The modernization of GPS added a third civil signal at the L5 (1176.45 MHz) frequency. The signal is in the protected satellite navigation band and thus can be used for navigation purposes. The L1 and L5 frequency measurements can be combined to estimate the ionospheric delay, due to its dispersive properties.
Anomalous ionospheric spatial gradients can be an issue for single frequency
GBAS as pointed out in [24]. Future GBAS will be a dual frequency system and will use the combination of measurement at two frequencies to mitigate the effects of the ionosphere which is a major error source for single frequency GBAS. The impact of AGDV on a combination of L1 and L5 measurements has not been studied, and an initial analysis will be done to evaluate the effects of AGDV on dual frequency GBAS. The observations formed by the combination of L1 and L5 measurements are known as ionosphere free and divergence-free measurements. The multipath remains a major error source, and thus differential GPS using ionosphere-free and divergence-free methods will still employ smoothing techniques as introduced in [32, 33]. The multipath and AGDV are difficult to separate in a flight and thus the dissertation will study the impact of AGDV on the future
GBAS by studying the ionosphere-free and divergence-free observations in detail. 37 Research questions
The following research questions have been formulated based on the literature search.
1. Is the 휎푝푟,푎푖푟 flight test data collected with a dual-frequency antenna representative for
L1-only antennas?
2. Can L1-only antennas meet the DO-301 group delay variation requirements?
3. How should AGDV be measured in the presence of multipath?
4. What is the effect of a ground plane vs. an aircraft installation on antenna group delay
variations?
5. Can a model be defined for AGDV?
6. What is the impact of AGDV on the GBAS position solution?
7. What is the impact of AGDV on the aircraft vertical protection level (VPL)?
8. Should AGDV be treated as a bias or noise?
9. Do AGDV impact the airborne ionosphere integrity monitors?
10. How is the future Dual Frequency GBAS affected by AGDV?
38 2.3.1 Contributions and Organization
The first four research questions focus on the characterization of antenna group delay variation as a function of elevation and azimuth angles. A method to characterize
AGDV will be introduced. The separation of multipath and AGDV for ground experiments is feasible, while flight experiments provide for the characterization of combined multipath and AGDV. These topics are included in this dissertation, based on [34]:
Raghuvanshi, A., Van Graas, F., “Characterization of Airborne Antenna
Group Delay Biases as a Function of Arrival Angle for Aircraft Precision Approach
Operations,” Proceedings of ION GNSS+ 2015, Tampa, FL, September 2015.
Accuracy and integrity are important performance parameters for GBAS. To evaluate accuracy and integrity, each error source needs to be modeled statistically. Thus, the characterization of AGDV and the impact on accuracy performance and position protection levels is provided in this dissertation to answer research questions 6 and 7, based on [35]:
Raghuvanshi, A., Van Graas, F., “Impact of Antenna Group Delay Variations on Protection Levels,” Proceedings of IEEE/ION PLANS 2016, Savannah, GA, April
2016, pp. 857-862.
The original flight tests used to characterize airborne multipath used a dual frequency antenna. Subsequently, it was found in [34] that AGDV for dual frequency antennas can be smaller than those for single frequency antennas. To address the difference, additional simulations were performed, and flight test data were processed by the Boeing
Company and the results were analyzed and documented in [36]. This publication 39 addresses research question 8. The results of this effort are summarized in this dissertation based on [36]:
Harris, M., Miltner, M., Murphy, T., Raghuvanshi, A., Van Graas, F.,
“Bounding GPS L1 Antenna Group Delay Variation for GNSS Landing System
Integrity,” Proceedings of the 2017 International Technical Meeting of The Institute of
Navigation, Monterey, California, January 2017, pp. 591-605.
Airborne integrity monitors are necessary to mitigate the effects of anomalous ionospheric gradients affecting the GBAS operation. The airborne monitors are used in collaboration with ground monitors to mitigate the threat space. The detection thresholds for the monitors were determined using flight tests with a dual frequency antenna. To verify the monitor performance with single frequency antennas, an AGDV model was used in conjunction with flight test data to answer research question 9. The results of this investigation are included in this dissertation based on [37]:
Raghuvanshi, A., Van Graas, F., “The Effect of Aircraft Antenna Group Delay
Variations on Dual Solution Ionospheric Gradient Monitoring,” Proceedings of the
2018 International Technical Meeting of The Institute of Navigation, Reston, Virginia,
January 2018, pp. 267-282.
The last two contributions both use an AGDV model to run a simulation to understand the problem before using flight data. Thus, future analyses for accuracy and integrity would benefit from a generic modeling technique for AGDV. Spherical
Harmonics is a modeling technique that has been used in various applications involving a parameter that is a function of azimuth and elevation angles. It is analogous to Fourier series and thus a signal which is a continuous function in azimuth and elevation angle can 40 be written as a series of spherical harmonics. The AGDV is a function of azimuth and elevation angle, therefore a series of spherical harmonics can used to model the AGDV.
The method is compared to a binning model, and the two models are compared. A journal paper is in preparation on this topic, which is also included in this dissertation to answer research question 5:
Raghuvanshi, A., Van Graas, F., “A Binning Model and a Spherical
Harmonics Expansion to Model Airborne Antenna Group Delay Biases as a Function of Arrival Angle”
A discussion on research question 10 will be provided in this dissertation, with some initial results for question 10 based on flight test and ground-based data.
The dissertation is comprised of 10 chapters. Following this background chapter,
Chapter 3 provides a brief overview of the Global Positioning System. GNSS comprises several satellite systems, but the focus of this dissertation will be on GPS and GBAS from the GPS perspective. Both SBAS and GBAS will be explained briefly. The steps to calculate a position solution in GBAS is explained step-by-step, and the need for various integrity monitors is introduced. The future dual frequency GBAS architecture is explained in a concise manner.
The concepts of integrity and accuracy for GBAS are explained in Chapter 4. The various error sources affecting the single frequency standalone GPS receivers are important to understand the concept of augmentation in GBAS. Spatial and temporal correlation of error sources is also explained, necessary for the understanding of differential GPS (DGPS) and smoothing of code using carrier measurement. Detailed equations explaining the concept of DGPS and measurement smoothing are introduced. The residual errors affecting 41 the accuracy and integrity of GBAS are explained and error models are introduced. The models are used for the calculation of the position solution and corresponding protection levels.
Chapters 5 and 6 explains the methodology used to characterize AGDV and also methods to model AGDV.
Chapters 7, 8 and 9 analyze the impact of AGDV on the position solution and integrity of GBAS. Chapter 9 discusses the dual frequency GBAS architecture and the role of AGDV on the performance parameters for the system. In Chapter 10, conclusions of the dissertation and future research are discussed.
42 3. OVERVIEW OF GLOBAL POSITIONING SYSTEM
In the 1960s the Department of Defense (DoD), the National Aeronautics and Space
Administration, and the Department of Transportation had similar interests in a system that could provide accurate estimates of position, velocity and time, continuous and operable in all weather conditions for worldwide use [38]. In 1964, the Navy’s navigation system
Transit (built by John Hopkins Applied Physics Laboratory) became the first operational satellite navigation system which used Doppler measurements to calculate a position fix, but it could only provide 2-D position accuracy of 25 m (rms) for a stationary user, which could be improved to 5 m 3-D accuracy using data from multiple days and 1 m 3-D when estimated relative to another known point.
During the same period, the Navy, the Air Force and the Army were conducting independent research for a positioning system. The Naval Research Laboratory was testing its own system known as Timation that could provide precise position and timing to a user on the ground. The Air Force was developing its own system known as 621B to provide 3-
D worldwide coverage and positioning. Also, the Army was researching various positioning fix principles such as ranging, angular positioning, etc. Later, the research from all the military units on space-based navigation system was combined to design a single joint-use system by the Office of the Secretary of Defense under the Defense Navigation
Satellite System program. Also, the Navigation Satellite Executive Steering Group was formed to investigate various satellite navigation projects and proposals. The Joint Chiefs of Staff proposed requirements that the final system should satisfy. This led to the development of the first concept of NAVSTAR GPS formulated by the GPS Joint Program
Office. NAVSTAR GPS is now known as GPS [38, 39, 40]. 43 The GPS architecture is divided into three segments: the space segment, the ground control segment and the user segment.
The space segment currently comprises 31 operational satellites. The satellites orbit in 6 planes at an inclination of 55° to the equatorial plane. The user segment takes advantage of ranging signals and data messages transmitted by the satellites. When four satellites are in view, the user can calculate its instantaneous position, velocity and time.
Current information about the space segment can be found in [41]. The ground control segment is comprised of a master control station, monitor stations and ground antennas at various locations on the Earth. It has various functionalities including tracking and monitoring the satellites and maintaining the constellation. It also updates the satellite clock corrections, handles satellite anomalies and checks various parameters that are pertinent while estimating position, velocity and time by a user. Details on the ground segment can be found in various books [39, 40]. The user segment is comprised of the defense user like the Navy, the Army and the Air Force, and the civil user, whose applications include land navigation, civil aircraft navigation, maritime navigation, precise timing, surveying, agriculture and consumer markets.
The positioning provided by the GPS is divided into the precise positioning service
(PPS) primarily intended for military use, and the standard positioning service (SPS) primarily used by the civil sector. The legacy GPS satellites transmit on two different frequencies L1 and L2 at 1575.42 MHz and 1227.60 MHz, respectively. The L1 frequency carries two signals; one intended for civil use which has a code called the coarse acquisition
(C/A code) attached to it, the other code is the precise code or P-code and is intended for
Department of Defense authorized users. The various accuracies achieved by both the 44 services is explained in detail in [39]. Mostly, accuracy around 10 m can be achieved horizontally and around 15 m vertically in both the services without any augmentation. The
GPS modernization program introduced three new civil signals L2C, L5 and L1C. The
GPS IIR(M) satellites started broadcasting L2C in 2005. The L2C signal is used in collaboration with L1 C/A signal for ionospheric corrections using a dual frequency receiver. The L2C signal cannot be used for civil aviation because it is not in a protected navigation band for aviation, but it is used in some aviation applications, such as the Wide
Area Augmentation System (WAAS) reference stations for ionospheric correction estimation over a large area. Non-aviation users, however, use the L2 signal in combination with the L1 signal to measure the ionospheric propagation delay to improve navigation accuracy. The L5 signal at 1176.45 MHz will be used for future civil aviation once the full constellation is achieved because it is in a radio band dedicated for aviation safety services.
The L5 signal was first broadcast with the launch of a GPS IIF satellite in 2010. The L1C will help with interoperability between GPS and other satellite navigation systems and will be broadcast with the launch of GPS III satellites [41].
The user segment uses a combination of two types of measurements to calculate position, velocity and time. The two measurements are the code phase measurement also known as Pseudorange (PR) and the carrier phase (CP) measurement also known as
Accumulated Doppler (AD). The code phase measures the transit time of the signal from the satellite to the user. The carrier phase measurement is the integrated Doppler frequency shift of the received signal. Figure 3.1 shows the basic GPS architecture of a standalone
GPS receiver. There are various errors which are introduced in both the measurements: 45 Errors due to space segment of GPS: satellite clock errors, ephemeris (orbit
prediction) error, relativistic errors;
Atmosphere induced errors: ionospheric and tropospheric delays; and
Error introduced in the GPS receiver due to multipath, antenna group delay and
phase delays, receiver thermal noise and interference.
Figure 3.1. Standalone GPS architecture
The final position, velocity and timing accuracy is dependent on how accurate the measurement are, and how the satellites are distributed in the sky [39, 40]. The satellite clock and ephemeris errors are evaluated by the control stations and are uploaded to the satellites and are rebroadcast to the user to correct for the errors. Standalone GPS applications use models to mitigate the effect due to atmosphere. For example, a single frequency SPS user can achieve horizontal position accuracy better than 10 m. More 46 precise applications like landing an aircraft, road navigation, surveying etc., which require meter level or better accuracy, must incorporate augmentation, which can be differential
GPS (DGPS) [39, 40].
3.1 Differential Global Positioning System
A single frequency user equipped with a single receiver can achieve position accuracy of around 10 m and 15 m horizontally and vertically, respectively. To achieve better performance, DGPS is used. Differential GPS uses the fact that all the errors except for multipath, receiver noise and interference are correlated in time and space. Thus, if we have a GPS receiver at a surveyed location, the GPS receiver can solve for error sources that are spatially correlated, known as differential corrections. The differential corrections can be used by a user and thus the improvement in the user pseudorange and carrier phase measurements is achieved. Table 2 shows the improvement in the pseudorange error introduced using differential GPS techniques. The Differential GPS concept is used in space-based augmentation systems (SBAS) and ground-based augmentation systems
(GBAS).
47 Table 2.
Pseudorange Error: standalone GPS vs DGPS (From [39])
3.1.1 Space Based Augmentation System
The DGPS architecture in which satellites are used to broadcast the differential corrections is known as a space-based augmentation system (SBAS). The service can be provided over a whole continent with 1 to 3 geostationary satellites, and the data are collected through various stations across the coverage area. Corrections are calculated in terms of satellite orbit, ionosphere delay grid points, and range errors. The corrections are then transmitted using a satellite in a form so that individual users can apply them based on their location from an unaugmented GPS. There are various systems in place such as the US Wide Area Augmentation System (WAAS), The European Geostationary
Navigation Overlay Service (EGNOS), the Japanese Multi-functional Satellite 48 Augmentation System (MSAS), and the GPS Aided Geo Augmented
Navigation (GAGAN) system being operated by India.
3.1.2 Ground Based Augmentation System
In a ground-based augmentation system (GBAS), the area covered is small compared to SBAS. The system is supposed to guide aircraft from around 20 NM to the runway threshold. The US version of GBAS is also referred to as the Local Area
Augmentation System (LAAS). Figure 3.2 delineates the basic architecture of the GBAS.
Figure 3.2. Ground Based Augmentation System architecture
The system is divided into the ground segment and the airborne segment. The ground segment consists of 4-6 GPS reference receivers (RR) at surveyed locations and a
Very High Frequency (VHF) Data Broadcast (VDB). The RR generates the differential corrections and broadcast it to airborne users, which apply the corrections and thus achieve 49 better performance due to improvement in the measurements. The step-by-step methodology for GBAS position estimation and calculation of integrity parameters is explained next.
The first part of the process is performed by the ground reference receivers.
→ Step 1: The reference receivers (RRs) at surveyed locations at the airport use the
navigation data transmitted by each of the satellites to calculate the satellite
positions and clock offsets from GPS time. Thus, the distances to the satellites can
be calculated.
→ Step 2: The code measurements (Pseudorange) and carrier phase measurements
(Accumulated Doppler) are obtained at the RRs. The pseudorange is comprised of
the true range and the errors affecting it like ionospheric delay, multipath, etc.
→ Step 3: The pseudorange is smoothed using carrier phase measurements and the
smoothed pseudorange is subtracted from the distance measured in step 1, which
gives the pseudorange errors affecting the system.
→ Step 4: In general, a combination of RR’s is used and differential corrections (DC)
formed by each RR are checked for faults. After fault detection and mitigation, the
DC’s from all the RRs are averaged to form a final DC for each satellite.
Differences between RRs are also calculated for each satellite, which provides the
so-called bias-values (B-values). The corrections are referenced to the centroid of
all RR’s as shown in Figure 3.2.
→ Step 5: The corrections and correction rates are then transmitted to the aircraft along
with the B-values and the standard deviations of the fault-free DCs (needed for
integrity calculation at the aircraft). 50 The other part of the GBAS algorithm is implemented in the aircraft. The aircraft in the vicinity of reference receivers will experience similar errors for spatially correlated error sources.
→ Step 6: The airborne carrier smoothed pseudoranges are formed and the DCs are
applied.
→ Step 7: The standard deviation of the residual errors is calculated using the
information broadcast by the ground RRs, using the airborne accuracy designators
and satellite elevation information.
→ Step 8: The position solution is calculated using the smoothed corrected
pseudorange measurement, satellite geometry and standard deviations from step 7.
→ Step 9: The Protection Level (PL) is calculated using satellite geometry and
standard deviations from step 7 and compared to the Alert Limit (AL). If the
PL>AL, then the GBAS system is flagged as unavailable.
The steps above are the basic steps for the calculation of position of an aircraft during approach using GBAS. The RRs also implement various monitors, for example,
Signal Quality Monitor (SQM), Signal Deformation Monitor (SDM), Measurement
Quality Monitor (MQM), before the corrections are generated and applied to improve accuracy and integrity of the system. The measurements are also passed through the
Multiple Receiver Consistency Check (MRCC), Sigma-Mu monitor and Message Field
Range Check (MFRC) before the data are broadcast to the aircraft using the VDB. The code carrier divergence (CCD) monitor, Ionospheric Gradient Monitor (IGM) and Dual
Solution Ionospheric Gradient Monitor Algorithm (DSIGMA) monitor are also implemented to mitigate the threat space due to anomalous ionospheric gradients. 51 The development of GBAS started in the US in 1994. Figure 3.3 shows GBAS installations around the world in 2017.
Blue /Red: CAT III Prototype/Research Yellow: Ongoing Projects White: Investigations Green: Operational
Figure 3.3. Ground Based Augmentation System across the world. (Courtesy of flygls.net)
Thus, GBAS is a complex system and detection and mitigation of various error sources become important to achieve the accuracy, integrity and availability required for it to be used for precision approach. GBAS requirements for GAST C and D have been harmonized by the International Civil Aviation Organization (ICAO) in the form of
Standards and Recommended Practices (SARPs) [42]. GAST C systems are operational around the world, while GAST D is in the approval process at the time of this writing. 52 3.1.3 Dual Frequency GBAS
The modernization of GPS is underway and a new civil signal at L5 (1176.45 MHz) in the aeronautical radio navigation service band is being transmitted by new generation satellites. The combination of the legacy signal at L1 and L5 can be used for future dual frequency GBAS. The current GBAS is a single frequency system, but research efforts on dual frequency GBAS and multi-frequency, multi-constellation GBAS is being published by various research groups [43, 44].
The single frequency GBAS is based on the concept of differential GPS and smoothing code with less noisy carrier measurements as explained in the previous section.
The major residual errors which affect the accuracy of GBAS include multipath and ionospheric delay, while integrity is complicated due to anomalous ionospheric gradients.
The spatial ionosphere error is dependent on the distance between the ground reference receiver and the airborne receiver. Also, the temporal part of the ionospheric error is dependent on the smoothing filter time constant. Thus, the smoothing of the pseudorange measurements introduces error in the residual ionospheric error. During nominal ionospheric behavior the additional error term doesn’t affect the overall error budget but if there is an anomalous spatial gradient between the ground reference receivers and the aircraft, they will experience dissimilar ionospheric conditions with gradients up to 0.412 m/km.
The advantage of a dual frequency system is that the ionospheric delay can be measured, and thus the ionospheric error contribution can be mitigated. Multipath will still be a major error source such that smoothing is still required, and thus the concept of dual frequency smoothing was introduced in [32, 33]. 53 The measurements used in a dual frequency system are either divergence-free or ionosphere-free. Details of the measurements with equations are explained in Appendix D.
The dual frequency GBAS has certain advantages but a thorough analysis of the error sources affecting the two measurements is necessary. A comparison of two methodologies of dual frequency smoothing is also important.
AGDV is an important error source for a dual frequency GBAS. The reason is that the antenna group delay is a function of frequency as well as a function of azimuth and elevation angles. Thus, for different frequencies it can have different characteristics for each value of azimuth and elevation angles. The divergence and ionosphere-free measurements use a combination of code and carrier measurements for two different frequencies. Thus, the combination of AGDV for the two frequencies needs to be investigated.
54 4. CONCEPT OF ACCURACY AND INTEGRITY IN GBAS
4.1 Overview
RTCA, Inc. developed the Minimum Aviation System Performance Standards
(MASPS) containing the system requirement for the LAAS [22]. LAAS requirements are used for the deployment of the system and its use in the United States (US) National
Airspace System (NAS). The GBAS standards were developed through harmonization between the RTCA, ICAO and European Organization for Civil Aviation Equipment
(EUROCAE) [42]. The antenna group delay variation is defined and bounded similarly in
GBAS and LAAS. The applications which are affected by AGDV have also been defined identically in both the standards. Thus, GBAS will be used throughout the dissertation.
Aircraft landing systems can be classified into Category-I, II and III. The decision heights and visibility requirements were introduced in Chapter 1 and are summarized in
Table 1. Figure 4.1 is a pictorial representation of the approach of an aircraft on a 3° glideslope with different approach categories defined by the DH, VAL and LAL.
55
Figure 4.1. Approach categories for GBAS with Decision Height (DH), vertical and lateral alert limit
In this dissertation, GBAS approach type service D (GAST D) which covers CAT
II and III will be discussed. The MASPS specifies the performance of the GBAS navigation system for each category in terms of accuracy, integrity, availability and continuity [22].
Accuracy for the GBAS is defined in terms of Navigation System Error (NSE) which is the difference between true position and the position output by an airborne GPS receiver.
Integrity is the amount of trust we can put on the output of the system and the ability of the system to provide a timely warning to the user when the output cannot be trusted. In GBAS, if the protection level calculated in the horizontal and vertical domain are larger than the alert limit, the system notifies the user within the time to alert to not use the system.
Availability of the GBAS for approach is defined if the system is available at the initiation of the precision approach. Continuity of the GBAS for approach is defined as the 56 probability that the systems will continue to work throughout the precision approach, given that it was available at the beginning of the approach [22]. Table 3 summarizes the requirements for the navigation system for different approach categories.
Table 3.
GBAS based landing requirements Category Lateral Vertical Integrity Time Lateral Vertical Continuity Accuracy Accuracy Risk to alert Alert Risk (95%) (95%) alert limit limit CAT I 16 m 4 m 2×10−7 6 s 40 m 10 m 8×10−6 in any 150 in any 15 s s
CAT II 5 m 2.9 m 1×10−9 2 s 17 m 10 m 4×10−6 in any 15 s in any 15 s vert, 30 s lateral
CAT III 5 m 2.9 m 1×10−9 2 s 17 m 10 m 2×10−6 in any 15 s in any 15 s vertical & vertical & 30 s lateral 30 s lateral
The next section explains in detail the error sources affecting the accuracy, integrity, continuity and availability of standalone GPS. The errors are similar for other
GNSSs, but in this dissertation GPS will be used as it is fully operational. Differential GPS and smoothing can be used to improve the measurements. Details about DGPS and smoothing will be introduced next, with a focus on GBAS and GAST D. Finally, after applying DGPS and smoothing techniques, the residual errors are bounded and used to 57 calculate the position solution and protection levels. The current standards are reviewed in detail to determine the best approach to add the AGDV requirements.
4.1.1 Error sources for standalone GPS
The accuracy of the position solution is a combination of the quality of the GPS measurements and satellite geometry [39]. The quality of GPS signals is defined in terms of user equivalent range error (UERE) for each satellite. It is a combination of the different error sources affecting the pseudorange. It is approximated as a zero-mean normal distribution with variance defined as the sum of variances of each error source. Thus,
UERE with satellite geometry dictate the final accuracy.
This section introduces the GPS signals and the errors that affect the accuracy of standalone GPS. Spatial and time correlation of the errors will also be discussed and the utilization of correlation in the concept of DGPS and smoothing to achieve better accuracy will be introduced. GBAS is based on the concept of smoothing and differential GPS, thus the explanation of these error sources for a standalone GPS will enhance the understanding of the effects of these error sources on GBAS.
The legacy GPS satellites transmits signal on two frequencies L1 (1575.42 MHz) and L2 (1227.60 MHz). The satellites also transmit pseudorandom codes, the course acquisition code (C/A code) for civil users on the L1 frequency and the encrypted precision code (P(Y)-code) on L1 and L2 for military users. For GBAS, only the C/A code on L1 is used. The C/A code is a random sequence of 1023 bits or chips, which repeats every millisecond. Each satellite transmits a unique C/A code and the receiver has the replica of the codes. The receiver correlates the incoming signal with the replica and thus tracks the satellite. The satellite also transmits navigation data at 50 bits per second. The navigation 58 data has information about satellite orbit, clock offset, and satellite health. The navigation data bits are modulo-2 combined with the C/A code bits. The combined signal is modulated on the carrier signal using binary phase shift keying (BPSK) as shown in Figure 4.2.
Figure 4.2. GPS transmitted signal
The receiver tracks the code and carrier of the GPS signal and outputs two measurements known as code-phase and carrier-phase measurement. The transmitted time is calculated by the correlation of the incoming signal code with the replica code. The difference between time of reception and time of transmission is multiplied by the speed of light to provide the range to the satellite. This measurement is known as Pseudorange
(PR). The receiver also measures the difference in the integrated frequency of the incoming signal and the receiver-generated carrier. This measurement is known as Carrier Phase (CP) or Accumulated Doppler (AD). The total range to the satellite is equal to an integer number of cycles plus a partial cycle. Since the CP is integrated at the start of signal tracking, it 59 contains an unknown constant of integration, but CP can be used to keep track of changes in range to the satellite. The effect of multipath and receiver noise on the carrier tracking loop is less than code tracking loop errors, due to the 19-cm carrier wavelength versus the
293-m code chip length. The errors that affect both the PR and CP measurements are shown in Figure 4.3.
Figure 4.3. Factors affecting the Pseudorange and Carrier Phase measurements
The GPS satellites maintain space vehicle time based on the onboard atomic clocks, and the users maintain their own time. The GPS system maintains a time called GPS time, thus satellite clock and receiver clock offset from GPS time need to be determined. Other sources that delay the C/A code signal by 훿푡푔푝푠 are given by equation 4.1.
훿푡푔푝푠 = 훿푡푎푡푚 + 훿푡푚푢푙 + 훿푡푟푒푐 + 훿푡푔푑 (4.1) where: 60 훿푡푔푝푠 is the total time delay;
훿푡푎푡푚 is the offset due to ionosphere and troposphere delay;
훿푡푟푒푐 & 훿푡푚푢푙is the offset due to receiver noise and multipath; and
훿푡푔푑 is the offset due to delay introduced by antenna group delay.
Figure 4.4 shows the relationship pictorially.
Figure 4.4. Delay in the time of transmission due to various factors
In Figure 4.4, 푇푢& 훿푡푢 are the system time when the signal was received, and receiver offset from GPS time respectively; 푇푠& 훿푡푠 are the time when the signal left the satellite and satellite offset from GPS time respectively; Thus, from Figure 4.4, the true geometric range r, between the satellite and receiver can be written as:
푟 = 푐( 푇푢 − 푇푠) (4.2) 61 The GPS receiver measures Pseudorange 휌 because of offsets due to various factors and is given by
휌 = 푐(푇푢 + 훿푡푢) − 푐(푇푠 + 훿푡푠) + 푐훿푡푔푝푠
= 푐( 푇푢 − 푇푠) + 푐( 훿푡푢 − 훿푡푠 + 훿푡푔푝푠)
휌 = 푟 + 푐( 훿푡푢 − 훿푡푠 + 훿푡푔푝푠) (4.3) where:
휌 is the Pseudorange to a certain satellite.
The navigation data is used to improve the measurement by applying satellite clock corrections. A mathematical model is used to correct for the troposphere delay, while the ionosphere delay is corrected using model parameters from the navigation data [39, 40].
Standalone GPS receivers can use this model to mitigate the ionospheric delay by 50% to
70% [45]. The final accuracy of the standalone GPS is due to the residual errors left after corrections. The measurement after correction is used to calculate receiver position and time offset from the system time. The next section will explain the contribution of each error source, as well as spatial and temporal error correlation characteristics.
4.1.1.1 Satellite Clock and Orbit Errors
Satellite clock time is offset from GPS system time and is corrected using the GPS navigation data that contains the coefficients used for the clock correction. The corrections are estimated by the control segment and broadcast to the satellite to rebroadcast it to the user. The parameters are used to evaluate the clock corrections using a second-order polynomial expression as explained in [39, 40]. The residual error left after applying the clock corrections defines the effect of satellite clock error. The satellite clock error seen by two receivers at different location will be the same i.e. the clock error is spatially correlated 62 so differential techniques completely mitigate residual satellite clock errors. The clock errors change slowly with time and are strongly correlated temporally. Thus, for GBAS, nominal clock errors don’t affect the position solution or accuracy.
Satellite orbits are also estimated by the CS and uploaded to the satellite to broadcast to users via the navigation data. The error in the satellite position estimation can be divided into radial, cross-track and along-track. The radial error is of interest to a standalone user on Earth and has a typical error of 0.18 m, 1 sigma [39, 40]. The cross- track and along-track errors affect GBAS due to different projections of these errors when the user is separated from the GBAS Ground Station. For nominal ephemeris errors, typical projection differences are on the order of 2.5 cm, 1 sigma, for separation distances of 100 km [39]. In addition to a strong spatial correlation, orbit errors also change slowly; therefore, orbit errors are mostly mitigated by GBAS.
4.1.1.2 Ionospheric Errors
The ionosphere is a dispersive medium that delays the code of the GPS signal, and advances the carrier by the same amount. The reason being that the refractive index of the medium is frequency dependent. The refractive index is the ratio of the velocity in vacuum to the velocity in the medium. The relationship between group velocity 푣푔 (C/A code) and phase velocity 푣푝 (carrier signal) for the GPS signal can be approximated as [39]
푑푣푝 푣 = 푣 − 휆 (4.4) 푔 푝 푑휆 where:
휆 is the wavelength of the signal; 63 Thus, the carrier velocity 푣푐 and code 푣퐶/퐴 can be derived using Equation 4.4 and is given by [39, 40]
푐 푣푐 = (4.5) 40.3푛푒 1 − 2 푓푐 푐 푣퐶/퐴 = (4.6) 40.3푛푒 1 + 2 푓푐 where:
푛푒 is the electron density; and
푓푐 is the carrier frequency.
These values are derived using an approximation of the refractive index and neglecting the higher order term in the approximation. The final expression for advance of carrier and delay of code is given by [39, 40]:
−40.3 푇퐸퐶 퐼푐 = 2 (4.7) 푓푐
40.3 푇퐸퐶 퐼퐶/퐴 = 2 (4.8) 푓푐 where:
TEC is the total electron content and represents the total number of electrons in a cylinder with a 1 푚2 cross section extending from the satellite to a user.
The expressions in equations 4.7 and 4.8 are for ionospheric delay in the zenith direction. Generally, satellites are at different elevation angles, and mapping of this vertical error to the desired direction is needed and is achieved by multiplying equations 4.7 and 64 4.8 by an obliquity factor (OF). The geometry to calculate the OF is shown in Figure 4.5,
IPP is the ionosphere pierce point.
Figure 4.5. Obliquity Factor
The Obliquity Factor is defined as:
−1/2 푅 cos 휃 2 푂퐹 = [1 − ( 푒 ) ] (4.9) 푅푒 + ℎ퐼 where:
휃 is the elevation angle of the satellite;
푅푒 is the radius of the Earth; and
ℎ퐼 is the height of maximum electron density.
65 Ionospheric delays are spatially correlated. The equation for the difference in the delays due to elevation angle difference is derived in [39] and a 100 km separation can introduce a delay difference of 3 cm. Typical nominal ionospheric spatial decorrelation is in the range of 0.002-0.005 m/km [48]. However, during non-nominal ionospheric conditions, the spatial variation of TEC can introduce large errors; values of 0.412 m/km have been reported [51]. Thus, anomalous ionosphere gradients can affect the performance of GBAS as the ground reference receivers and airborne receivers could be experiencing different ionospheric delays. The detection and mitigation of such risk is achieved by ground and airborne integrity monitoring. More detail about the monitors will be introduced in the later chapters and the effect of AGDV on the airborne integrity monitors will be analyzed.
Future GBAS systems which may be based on the concept of dual frequencies, such as L1 and L5, have the advantage that they can measure the ionospheric delay, and remove delays both at the ground station and the aircraft. The ionosphere-free measurement uses this concept and forms ionosphere-free pseudoranges using the combination of pseudoranges on the L1 and L5 signals [39, 40]:
휌 − 훾휌 휌 = 퐿5 퐿1 (4.10) 푖표푛표−푓푟푒푒 1 − 훾 where:
2 푓퐿1 훾 ≈ 2. 푓퐿5
The drawback of the measurement is that it is noisier compared to a single frequency measurement. Furthermore, multipath and AGDV from two measurements affect the ionosphere-free pseudorange. It is also possible to create a divergence-free 66 pseudorange by correcting the pseudorange using only the CP from two frequencies, thus removing the change in the ionospheric delay. The latter will only be affected by multipath and AGDV on the L1 frequency. Appendix D derives equations for divergence-free and ionosphere-free measurements, and it will be used as a reference to describe the effect of
AGDV on future GBAS in Chapter 9.
4.1.1.3 Tropospheric Delay
The tropospheric layer is the lowest in the Earth’s atmosphere. It delays the code and carrier by the same amount because the troposphere is non-dispersive for the GPS frequencies. The delay is dependent on the refractive index of the troposphere. The refractive index is dependent on temperature, pressure and relative humidity. The standalone receiver can use models to correct for the tropospheric delay as documented in
[39, 40]. The tropospheric errors are spatially correlated. If the user and the reference station are at the same altitude, then the difference is at the cm-level. The height above the reference station introduces a greater error than that due to the horizontal displacement and should be corrected using a model. An aircraft can experience a rate of change in the tropospheric delay as it descends towards the runway. GBAS requires a tropospheric correction to be made for the error due to the vertical displacement between the GBAS
Ground Station and the aircraft. The remaining error due to lateral separation and non- nominal troposphere is bounded and used in the protection level calculation [14].
4.1.1.4 Multipath and Receiver Noise
Multipath as the name suggests is a scenario in which the GNSS receiver receives the direct signal from the satellite and the reflected and diffracted replicas of the desired signal from multiple paths [39]. Both pseudorange and phase errors are introduced due to 67 signals arriving after the direct signal. The error depends on relative amplitude, delay, relative phase and phase rate with respect to the direct signal. The mitigation of the multipath can be achieved using different methods. The design of antennas that rejects low elevation multipath is one way. The ground station in GBAS employs this concept for the antenna design [3]. Receiver design can be used to mitigate the effects of multipath employing a large precorrelation bandwidth (up to 20 MHz) and correlator designs, such as narrow correlators [39]. The receiver noise also introduces pseudorange and carrier phase measurement errors, but the errors are smaller than multipath errors. Therefore, the multipath error dominates the error budget due to the combined effect of multipath and receiver noise.
In GBAS the ground RRs and airborne receivers encounter different multipath environments and receiver noise, and thus, the differential concept doesn’t mitigate these error sources. The receivers take advantage of the less noisy carrier phase measurements to smooth the code measurement to attenuate the effects of multipath and receiver noise.
The residual multipath error is an important error source and models for ground and airborne multipath and receiver noise have been developed based on statistical data [36].
These models are required to estimate position error and integrity of the position solution.
4.1.1.5 Antenna Group Delay
A signal is delayed by an antenna and the delay is a function of the frequency of the signal and azimuth and elevation angles of the satellite transmitting the signal. It is an inherent property of the receiver antenna and depends on the design of the antenna. The differential technique doesn’t mitigate the effects of AGDV and thus it affects the accuracy 68 and integrity of a differential system. The error can be mitigated using a better antenna design. This error source is detailed in Chapter 2.
4.1.1.6 Phase Wrap up
The Carrier Phase measurement is formed by tracking the phase of the incoming signal and comparing it to the phase of the replica signal generated in the receiver. The rotation of the receiving antenna and the rotation of a satellite around a receiver introduce phase shifts due to the antenna phase pattern. The differential system cancels the error due to the satellite motion, but the aircraft motion will introduce phase wraps that are equal for all satellites. Thus, they primarily affect the clock solution as errors that are common to all measurements do not affect the position solution. When satellites are analyzed on an individual basis, the phase wrap up needs to be corrected.
The GPS antenna needs signals from 4 or more satellites to calculate its position and clock offset from GPS time. The receiver tracks the satellites and forms Pseudorange and Carrier Phase measurements. The pseudorange at time t is defined as:
휌(푡) = 푟(푡)+∈ (푡) + 푐(훿푡푢(푡) − 훿푡푠(푡)) + 퐼(푡) + 푇(푡) + 휂휌(푡) + 휏푔푑(푡) (4.11) where:
푟(푡)is the true geometric range in m;
∈ (푡) is the orbit prediction error in m;
푐(훿푡푢(푡) − 훿푡푠(푡)) is the clock offset from GPS time in m;
퐼(푡) is the ionospheric error in m;
푇(푡) is the tropospheric error in m;
휂휌(푡) is the error due to multipath and receiver noise in m; and
휏푔푑(푡) is the error due to the antenna group delay in m. 69 The Carrier Phase measurements are affected by similar error sources but as discussed, the numerical value for some error sources are different. Also, the carrier phase measurements have an integer ambiguity, a phase wrap up term and the ionosphere advances the phase of the carrier signal. The accumulated Doppler measurement at time t is defined as:
푐푝(푡) = 푟(푡)+∈ (푡) + 푐(훿푡푢(푡) − 훿푡푠(푡)) − 퐼(푡) + 푇(푡) + 휂푐푝(푡) + 휏푝ℎ(푡) + 푁휆 +
휆 퐿1 휑(푡) (4.12) 2휋 where:
휏푝ℎ(푡) is the phase delay in m;
휆 is the wavelength of the signal in m;
휂푐푝(푡) receiver noise and multipath for carrier phase measurement in m;
휑(푡) is the azimuth angle in rad of the satellite being tracked; and
N is the integer ambiguity.
In summary, this section covered all the major error sources affecting the GPS standalone receiver. The spatial and temporal correlations of the errors were also discussed.
The spatial correlation is important for differential GPS. The temporal correlation of the error sources is important for the latency of the system (time gap between the generation of corrections and application). The next section will explain DGPS and measurement smoothing. Both concepts are necessary for the GBAS to achieve the accuracy and integrity required for a precision approach. 70 4.1.2 DGPS and Smoothing in GBAS
GBAS reference receiver positions are surveyed and thus they can calculate the contribution of the range error for each error source mentioned in the previous section. The spatial correlation of most errors, except multipath, receiver noise, and AGDV helps the aircraft to mitigate its measurement error using the differential corrections. As mentioned in the previous section, carrier phase measurement noise and multipath is much smaller than code noise and multipath. Therefore, the carrier measurements are used to smooth the code measurements. The single frequency carrier-smoothed code phase DGPS technique is used for GBAS [46]. Figure 4.6 shows the overview of the GBAS position estimation algorithm.
Figure 4.6. Carrier-smoothed code processing in GBAS
71 The ground reference receiver and aircraft generate raw code and carrier phase measurements. The raw code measurements are smoothed using carrier measurements for both ground and airborne receivers. The smoothed code measurements of the ground receiver are used to calculate the differential corrections, which are broadcast to the aircraft using a VHF Data Broadcast (VDB). The correction rate is also generated and broadcast.
The airborne receiver applies the differential corrections and correction rates to correct for the latency of the measurements. The difference in the height introduces tropospheric errors, and thus, the airborne receiver must compensate for the height difference before it calculates the position solution.
4.1.2.1 Smoothing Algorithm for GBAS
The ground and the airborne receivers used for CAT III compute smoothed pseudorange with 100-s and 30-s smoothing time constants. The smoothing algorithm for
GBAS is documented in [22]. A projected pseudorange is defined as:
휌푝푟표푗 = 휌̃푛−1 + (휑푛 − 휑푛−1) (4.13)
The carrier smoothed pseudorange is obtained from:
휌̃푛 = 훼휌푛 + (1 − 훼)휌푝푟표푗 (4.14) where:
푡 weighting function 훼 is given by, 훼 = 푢푝푑; 푡̃
휌푛 is the current raw pseudorange in m;
휌̃푛 is the current smoothed pseudorange in m;
휌̃푛−1 is the previous smoothed pseudorange in m;
휑푛 is the current raw carrier phase measurement in m; 72 휑푛−1 is the previous raw carrier phase measurement in m;
푡푢푝푑 is the time interval between the samples in s; and
푡̃ is smoothing time constant of the filter in s.
At the start of the filter, 푡̃ can also be the time since the start of filter until 30 s or
100 s have been reached.
4.1.2.2 DGPS in GBAS
The smoothing decreases the effect of the error due to multipath and receiver noise.
Next, the procedure to calculate the differential correction and the application of it to the airborne receiver will be presented.
The Ground Station (GS) knows its surveyed antenna position denoted by
(푥푔, 푦푔, 푧푔). The satellite positions are calculated using the navigation data. Let the satellite position of a single satellite be (푥푠, 푦푠, 푧푠). The range to the satellite can be calculated using:
푠 푠 2 푠 2 푠 2 푟푔푛푑 = √(푥푔 − 푥 ) + (푦푔 − 푦 ) + (푧푔 − 푧 ) (4.16)
The ground receiver also generates the pseudorange measurement to that satellite defined as:
푠 푠 휌푔푛푑 = 푟푔푛푑+∈ +푐(훿푡푔푛푑 − 훿푡푠) + 퐼푔푛푑 + 푇푔푛푑 + 휂푔푛푑,휌 + 휏푔푛푑_푔푑 (4.17)
The parameters are the same as defined in Equation 4.11 but a subscript g is added to denote the measurement is for the ground receiver. The measurement is smoothed using the algorithm explained in the previous section and the smoothed pseudorange is defined by:
푠 푠 ̃ 휌̃푔푛푑 = 푟푔푛푑+∈ +푐(훿푡푔푛푑 − 훿푡푠) + 퐼푔푛푑 + 푇푔푛푑 + 휂̃푔푛푑,휌 + 휏푔푛푑,푔푑 (4.18) 73 Equation 4.18 shows that the ionosphere, the multipath and receiver noise are affected by the smoothing. It is also required to correct for the clock offset of the reference receivers and the satellite clock before sending the corrections. After applying those two corrections and differencing 4.18 from 4.16 we get,
푠 푠 ̃ 푐표푟푟 = 푟푔푛푑 − 휌̃푔푛푑 = −∈ −퐼푔푛푑 − 푇푔푛푑 − 휂̃푔푛푑,휌 − 휏푔푛푑,푔푑 (4.19)
A correction rate 푐표푟푟̇ is also generated using the differential correction from two update intervals and differencing the two corrections divided by the update interval. The correction rate helps to mitigate the effect of correction latency in applying the differential corrections. The aircraft generates a smoothed pseudorange given by:
푠 푠 ̃ 휌̃푎푖푟 = 푟푎푖푟+∈ +푐(훿푡푎푖푟 − 훿푡푠) + 퐼푎푖푟 + 푇푎푖푟 + 휂̃푎푖푟,휌 + 휏푎푖푟,푔푑 (4.20)
The differential corrections are applied to the aircraft smoothed pseudorange as defined in [22]
푠 푠 휌푎,푐표푟푟 = 휌̃푎푖푟 + 푐표푟푟 + 푐표푟푟̇ (푡 − 푡푐표푟푟) + 푇퐶 + 푐훿푡푠 (4.21) where:
푠 휌푎,푐표푟푟 is the corrected pseudorange for the airborne receiver in m;
훿푡푠 is the satellite clock offset in s;
t is the current time in s;
푡푐표푟푟 is the time of generation of the differential correction in s; and
TC is tropospheric correction in m defined in [22] as:
−6 ∆ℎ 10 − ℎ 푇퐶 = 푁푅ℎ표 (1 − 푒 표 ) (4.22) √0.002 + sin 휃2 where:
푁푅 is the refractive index broadcast by the GS; 74 휃 is the elevation angle in rad;
ℎ표 is tropospheric scale height broadcast by the GS in m; and
∆ℎ difference of the height between the aircraft and GS in m.
This correction is important because even at a DH of 200 ft, tropospheric height difference can introduce significant error.
The final corrected pseudorange after applying the tropospheric correction and clock correction is given by:
푠 푠 ̃ ̃ 휌푎,푐표푟푟 = 푟푎푖푟 + 푐(훿푡푎푖푟 − 훿푡푔푛푑) + 퐼푎푖푟 − 퐼푔푛푑 + 푇푎푖푟 − 푇푔푛푑 + 휂̃푎푖푟,휌 − 휂̃푔푛푑,휌 +
휏푎푖푟푔푑 − 휏푔푛푑푔푑 (4.23)
Equation (4.23) can be rewritten as
푠 푠 ̃ 휌푎,푐표푟푟 = 푟푎푖푟 + 푐(훿∆푡) + ∆퐼 + ∆푇 + ∆휂̃ + ∆휏 (4.24) where:
훿∆푡 is the difference of the ground receiver clock offset and airborne receiver clock offset from GPS system time; the majority of the ground receiver clock offset is solved for and removed before sending the differential corrections.
∆퐼̃ is the residual ionospheric error; there is a bound defined in [14] for this error source.
∆푇 is the residual tropospheric error due to the uncertainty in the broadcast refractive index. There is a bound defined in [22] for this error source.
∆휂̃ is the difference between smoothed multipath and receiver noise at the airborne receiver and ground reference receivers. There is a separate bound defined in [14] for this error source; and 75 ∆휏 is the difference between antenna group delay contribution for airborne and reference receiver antennas, respectively.
No bound is defined for this error source in the MOPS [14]. The ground contribution is included in the standard deviation of the differential correction, but the bound identified in the antenna MOPS [2] is not included in the MOPS [14].
4.2 Accuracy of the GBAS
In the previous section, the concepts of DGPS and smoothing were introduced. In addition to the error sources, the accuracy of the GBAS is also dependent on the satellite geometry. Range errors propagate into position errors as a function of satellite geometry.
To detect certain anomalous ionospheric gradients, the aircraft calculates two position solutions for GAST D. Figure 4.7 shows a diagram of the aircraft position calculations.
Figure 4.7. Position solution for approach service type D (GAST D) 76
The GS broadcast corrections based on both 30-s and 100-s smoothed pseudoranges. The airborne receiver also forms 30-s and 100-s smoothed pseudorange measurements and the differential corrections are applied, to generate 30-s and 100-s smoothed position solutions. The vertical and lateral position solutions for 30- and 100-s smoothing are differenced to obtain 퐷푣 and 퐷푙. These differences are added to the vertical and lateral protection levels, respectively.
The position solution model is introduced in [14] and follows a standard procedure for calculation of position using N visible satellites:
∆푧 = 퐻∆푥 + 휉 (4.24) where:
Assuming N no of satellites visible.
∆푧 is the N-dimensional measurement vector and is the difference between the corrected pseudorange and the range of the satellite calculated using the navigation data and assumed user location;
∆푥 is a 4-dimensional vector and is the offset of the user position and clock offset from true position and GPS system time (the four unknowns);
휉 is the N dimensional measurement error vector containing residual error in the corrected pseudorange; and
H is called the geometry matrix. Its dimensions are N×4 and depends on the satellite azimuth and elevation angles. For satellite i, the ith row of the H matrix is given by:
퐻푖 = [− cos 휃푖 cos 휑푖 − cos 휃푖 sin 휑푖 − sin 휃푖 1 ] (4.25) 77 where:
휃 and 휑 are elevation and azimuth angle, respectively.
The weighted least squares approach is used to calculate ∆푥 and is given by:
∆푥 = ((퐻푇푊퐻)−1퐻푇푊)∆푧 (4.26) where:
휎2 0 ⋯ 0 푟,1 2 0 푊−1 = 0 휎푟,2 ⋯ (4.27) ⋮ ⋮ ⋮ ⋱ 2 [ 0 0 …휎푟,푁]
Let 퐺 = ((퐻푇푊퐻)−1퐻푇푊) be the projection matrix that projects the range domain information to the position domain and 휎푟,푖 the standard deviation of all the residual error in a corrected pseudorange for satellite i and is defined as:
2 2 2 2 2 휎푟,푖 = 휎푔푛푑,푖 + 휎푎푖푟,푖 + 휎푖표푛표,푖 + 휎푡푟표푝표,푖 (4.28) where:
휎푔푛푑,푖 is standard deviation of residual noise and multipath for a 30-s smoothed pseudorange for the ground receiver;
휎푎푖푟,푖 is standard deviation of residual noise and multipath for the airborne receiver;
휎푡푟표푝표,푖 is residual tropospheric uncertainty for the airborne receiver; and
휎푖표푛표,푖 is residual ionospheric uncertainty for the airborne receiver. The residual is dependent on the smoothing filter time constant and a 30-s value is used for GAST D.
Next, the model for the residual error sources are explained to understand the contribution of each error separately. 78 4.2.1 Model of residual Errors in GBAS
The LAAS accuracy models were developed by RTCA Working Group-4 Special
Committee-159 and the development of the models was published in [47]. Later, the models were harmonized and are standard models for GBAS as well. The current models for GBAS residual errors are included in [14, 22].
4.2.1.1 Airborne Accuracy models
The airborne contribution is due to receiver noise, interference and multipath experienced by the aircraft GPS receiver. The receiver noise and interference are defined by two airborne accuracy designators (AAD) A and B. The AAD’s are defined in [14, 22] as
AAD A accuracy achievable by using common available receivers (wide- correlators). AAD B accuracy achieved by advanced receiver or future receivers (narrow correlators).
The receiver contribution is represented as a function of elevation angle of the satellite and is given by:
휃 − 푖 휃 휎푛 = 푎표 + 푎1푒 푐 (4.29) where:
휃푖 is the elevation angle to the satellite; and
푎표, 푎1 and 휃푐 depend on the accuracy designator as defined in Table 4.
79 Table 4.
Airborne Accuracy Designators Accuracy Designator 풂풐(m) 풂ퟏ(m) 휽풄(deg)
AAD A 0.15 0.43 6.9
AAD B 0.11 0.13 4.0
The GBAS Airborne Equipment Classification (GAEC) D should be of designator class AAD B.
The airborne multipath is also a contributor to the Pseudorange accuracy. A standard model was developed through the efforts of Boeing, Honeywell and FAA [15-
17]. The reason for a standard model was to avoid testing individual aircraft for multipath contribution. The two designators A and B were also assigned to the multipath model defined in [19] as:
휃 − 푖 휎푚푢푙,퐴퐴퐷 퐴 = 0.13 + 0.53푒 10 (4.30)
휃 − 푖 휎푚푢푙,퐴퐴퐷 퐵 = (0.13 + 0.53푒 10 ) /2 (4.31)
The final airborne contribution as a function of elevation angle is found using
2 2 휎푎푖푟,푖 = √휎푚푢푙 + 휎푛 (4.32)
Figure 4.8 shows the contribution of airborne errors to the pseudorange as a function of elevation angle for different designators. 80
Figure 4.8. Airborne Accuracy Designators contribution to overall Pseudorange
The airborne receiver should satisfy the requirement of AAD B for GAST D.
4.2.1.2 Ground Accuracy Models
The contribution of the ground reference receiver noise, interference and multipath to the overall pseudorange is defined by three accuracy designators A, B and C [22]. The model of the standard deviation of the error on the smoothed pseudorange is defined in
[22] and is given by:
휃 2 − 푖 휃 √(푎표 + 푎1푒 푐 ) 휎 ≤ + 푎 2 (4.33) 푔푛푑,푖 푀 2 where:
M is the number of reference receivers used; 81 휃푖 is the elevation of satellite i; and
푎표, 푎1, θ풄 and 푎2 are defined in Table 5.
Table 5.
Ground Accuracy Designators Accuracy 풂풐(m) 풂ퟏ(m) 휽풄(deg) 휽풊(deg) 풂ퟐ(m)
Designator
GAD A 0.50 1.65 14.3 >5 0.08
GAD B 0.16 1.07 15.5 >5 0.04
0.15 0.84 15.5 >35 0.04
GAD C 0.24 0 -- ≤35 0.04
Figure 4.9 shows the curves for the three ground accuracy designators as a function of elevation angles.
82
Figure 4.9. Ground Accuracy Designator contribution to overall Pseudorange
In GBAS, the standard deviation of the ground differential corrections or fault free measurement error is broadcast to the aircraft [22]. The B-values, which are the error terms calculated by the difference of pseudorange corrections for one reference receiver and the average of pseudorange corrections of all other reference receivers, is also broadcast. These values are important for the calculation of integrity parameters, as they indicate the level of agreement between the reference receivers.
4.2.1.3 Residual Tropospheric Errors
The tropospheric correction is applied to the smoothed airborne pseudorange along with differential corrections using the parameters broadcast to the aircraft by the GS. The
RR’s also broadcast the uncertainty in the refractive index, which is used to calculate the residual tropospheric error as defined in [14] and is given by: 83 −6 ∆ℎ 10 − ℎ 휎푡푟표푝표 = 휎푁푅ℎ표 (1 − 푒 표 ) (4.34) √0.002 + sin 휃2 where:
휎푁푅 is the uncertainty in the refractive index broadcast by the GS;
휃 is the elevation angle;
ℎ표 is tropospheric scale height broadcast by the GS; and
∆ℎ difference of the height between the aircraft and GS;
Additional residual tropospheric errors due to lateral decorrelation and anomalous tropospheric conditions are included as part of the residual ionospheric errors.
4.2.1.4 Residual Ionospheric Errors
The residual ionospheric error is due to the spatial decorrelation of the ionospheric errors due to the distance between the GS and the aircraft, as well as the aircraft velocity.
Therefore, the residual error depends on the speed of the aircraft 푣푎푖푟, the distance between the GS and the aircraft 푥푎푖푟, and the smoothing time constant 휏. It is defined in [14, 22] as:
휎푖표푛표 = 푂퐹. 휎푣푖푔. (푥푎푖푟 + 2. 휏. 푣푎푖푟) (4.34) where:
OF is the obliquity factor defined in Equation (4.9); and
휎푣푖푔 is broadcast by the GS;
The standard deviation of the vertical ionospheric gradient, 휎푣푖푔 is the root-sum- square combination of the nominal ionospheric gradient and the decorrelation due to anomalies tropospheric conditions [14]. The latter was included in the ionospheric broadcast parameters as it was discovered after the message format had been finalized. 84 4.3 Integrity concept for GBAS
For integrity purposes, GBAS is divided into a ground segment, space segment and the airborne segment. Faults from each of these segments can affect the integrity of the overall system. The integrity risk allocation is shown in Figure 4.10 and is defined in [22].
Figure 4.10. GBAS integrity allocation
The 퐻표 case is when the receiver is working normally, and no faults are present.
The 퐻1 case occurs when one of the ground reference receivers experiences a malfunction.
The Protection Level calculation covers the risk due to 퐻표 and 퐻1. The 퐻2 risk covers faults due to multiple RR malfunctions, signal quality degradation, signal deformation, code-carrier divergence and is ensured by the application of various integrity monitors at the ground reference receivers as discussed in the introduction of the dissertation. AGDV potentially affects the Protection Level equations and thus the focus in this dissertation will be on that. Moreover, the Vertical Protection Level (VPL) is more demanding than the
Lateral Protection Level (LPL), and the availability of GAST-D is mostly affected by VPL
[22] and thus, VPL will be primarily studied in this dissertation. 85 4.3.1 Protection Level Estimation
The Protection Levels are computed by the aircraft to cover the integrity risk due to 퐻표 and 퐻1. The first step to calculate the VPL is to use measurements from the same satellite for which differential corrections are available. The aircraft generally tracks more low elevation satellites than the ground reference receivers due to the height of the aircraft.
The VPL for the approach service is defined in [14] and is given by:
푉푃퐿퐴푝푟 = 푚푎푥{푉푃퐿퐴푝푟_퐻표 ,푉푃퐿퐴푝푟_퐻1} (4.35)
The VPL for 퐻표 and 퐻1 are also defined in [22]. The VPL for 퐻표 is given by:
푁 2 2 푉푃퐿퐴푝푟_퐻표 = 퐾푓푓푚푑√∑푖=1 퐺푣푒푟푡,푖휎푟,푖 + 퐷푣 (4.36) where:
° 퐺푣푒푟푡,푖 = 퐺3,푖 + 퐺1,푖 × tan (3 ) and G is the projection matrix defined in Section
푠푡 rd 4.2; 퐺1,푖 and 퐺3,푖 are the 1 and 3 row of the projection matrix;
2 휎푟,푖 is the variance of the user equivalent range error;
2 2 2 2 2 휎푟,푖 = 휎푔푛푑,푖 + 휎푎푖푟,푖 + 휎푖표푛표,푖 + 휎푡푟표푝표,푖 ; the smoothing time for the filter to calculate the ionospheric residual is 100 s;
퐷푣 is the difference between the position solution estimated using 100-s and 30-s smoothed pseudorange; and
퐾푓푓푚푑 is the fault-free missed detection multiplier and is defined in Table 6.
86 Table 6.
Fault-Free Missed Detection Multiplier 퐾푓푓푚푑 M=2 M=3 M=4 5.762 5.810 5.847
The airborne receiver calculates VPL for the 퐻1 case as:
푉푃퐿 = max [푉푃퐿 (푗)] (4.37) 퐴푝푟_퐻1 퐴푝푟퐻1
푉푃퐿 (푗) = ∑푁 |퐺 퐵[푖, 푗]| + 퐾 √∑푁 퐺2 휎2 (4.38) 퐴푝푟퐻1 푖=1 푣푒푟푡,푖 푚푑 푖=1 푣푒푟푡,푖 푟,퐻1,푖
푀[푖] 휎2 = 휎2 + 휎2 + 휎2 + 휎2 (4.39) 푟,퐻1,푖 푈[푖] 푔푛푑,푖 푖표푛표,푖 푡푟표푝표,푖 푎푖푟,푖 where:
퐵[푖, 푗] is the B-value for the 푖푡ℎsatellite and 푗푡ℎ ground reference receiver;
푈[푖] is the number of ground reference receivers whose pseudoranges were used to calculate differential corrections; and
퐾푚푑 is the missed detection multiplier for the faulty ground reference case and is given by Table 7.
Table 7.
Missed Detection Multiplier 퐾푚푑 M=2 M=3 M=4 2.935 2.898 2.878
87 5. ANTENNA GROUP DELAY VARIATION CHARACTERIZATION
5.1 Overview
The characterization of AGDV and the development of the bound was performed by Working Group 7 of RTCA Special Committee 159 using two methods: spherical near field method and far field method using an anechoic chamber [19]. Subsequent efforts reported in [20] used a similar methodology for the characterization but could not re-create the results of [19]. It was pointed out that the difference could be due to the feed mechanism of the patch antenna, and the quality of the antenna range [20]. It was also concluded that
AGDV is expensive and difficult to test. In [20], electromagnetic modeling was used to confirm that it is feasible to design an antenna that meets the antenna MOPS requirements.
The work in this dissertation introduces a method to characterize AGDV using sky measurements, based on a code minus carrier (CMC) analysis. The method was used previously in [8] and achieved similar results to the far-fields methods with experiments performed in an anechoic chamber.
5.2 Methodology to characterize AGDV
The GPS receiver measures pseudorange and accumulated Doppler, also known as code and carrier phase measurements. The errors affecting the measurements were summarized in the previous chapter. The difference of the code and carrier removes similar errors affecting the measurements and the resulting observation has multipath, AGDV and a common bias for a continuous satellite track. In this dissertation the code minus carrier
(CMC) will be used to characterize AGDV. 88 5.2.1 Code-minus-carrier processing
The code minus carrier is defined by Equation 5.1 [34]. Appendix B derives the relationship in detail.
퐶푀퐶푗(휃푖, 휑푖) = ∆휏푔푑,퐿1(휃푖, 휑푖) + 푀(휃푖, 휑푖) + 휓푖 + 퐵푖푎푠푗 (5.1) where:
퐶푀퐶푗(휃푖, 휑푖) is the code-minus-carrier for a satellite j at an elevation 휃푖and azimuth angle 휑푖;
∆휏푔푑,퐿1(휃푖, 휑푖) is the AGDV to be estimated;
푀(휃푖, 휑푖) and 휓푖 are specular and diffuse multipath respectively; and
퐵푖푎푠푗 is the bias for the satellite track and is constant if there is no loss of lock; details of the bias are explained in Appendix B.
The CMC in Equation 5.1 is corrected for the ionospheric delay and the phase wrap up errors. A single frequency antenna will utilize the carrier phase measurements of a nearby antenna to calculate for ionospheric delay, whereas a dual frequency antenna can use its own measurements. The correction is applied using Equation 5.2 [34].
2 푓퐿2 퐼퐿1,푖 = 2(푐푝퐿2,푖 − 푐푝퐿1,푖) ( 2 2 ) (5.2) 푓퐿1 − 푓퐿2 where:
푐푝 is the carrier phase measurement; and
푓퐿1 and 푓퐿2 are the GPS L1 and L2 frequencies, respectively;
The phase wrap-up correction for a dual frequency antenna, is calculated as follows
[34]: 89 0.024 Δ퐴퐷(휑 ) = − ( ) 휑 (5.3) 푖 2휋 푖
When a separate non-rotating dual frequency antenna is used for the phase wrap- up correction, the correction is calculated as follows [34]:
0.1903 Δ퐴퐷(휑 ) = − 휑 (5.4) 푖 2휋 푖
It can be seen from Equation 5.1 that CMC is a combination of multipath and
AGDV. Thus, to utilize CMC for characterizing AGDV, a method to mitigate multipath in the CMC is important.
Meanwhile, it is difficult to separate AGDV and multipath for airborne (CMC) measurements. The standard deviation of airborne multipath and receiver noise also has
AGDV statistics in it. Thus, a combined effect of multipath and AGDV is important to study for airborne applications. The characterization of AGDV when separated from multipath is important to understand the statistics of the error sources in the flight data. In this dissertation, both flight and ground experiments are used to achieve each goal.
There are various techniques to mitigate multipath. In this dissertation, the rotation of the antenna is employed to average the multipath. A 4-ft antenna ground plane as specified by [2] is mounted on an azimuth-only rotator. The antenna is installed in the center of the ground plane and data is collected while the antenna is being rotated. The rotation data can be sorted into elevation and azimuth of the antenna reference frame. This methodology averages out the multipath affecting the data as each bin has data from various multipath scenarios.
Next, CMC data is collected during a flight test and the combined effect of AGDV and multipath is characterized to evaluate 휎푝푟,푎푖푟. This characterization is important 90 because the standard deviation of the airborne multipath and receiver noise is also comprised of AGDV, and this is not part of the current integrity equations. It will be depicted through flight data that specular multipath and AGDV are both are function of azimuth and elevation angles, and thus difficult to separate for airborne receiver data.
The bias of the CMC in Equation 5.1 is removed by taking the mean of each continuous satellite track and subtracting the mean from the CMC. The results are used to characterize AGDV. The next sections describe the ground and flight experiments in detail.
5.2.2 Ground experiments
For the ground rotator experiments, two antennas were used. The first antenna is an active dual-frequency antenna (ANT P/N 42G1215A-XT-1), which is the same type of antenna as used for the Boeing multipath flight tests to characterize 휎푝푟,푎푖푟. The second antenna is an L1-only active antenna (AT 5759W-TNCF000-05-26-NM), which should be representative for a typical aircraft installation [34].
The experimental setup is shown in Figure 5.1. The airborne antenna is installed on a 4-ft ground plane as specified by the MOPS [2]. The ground plane with antenna is installed on a rotator and the installation is done at the airport in an open field to minimize multipath from nearby objects.
91
Figure 5.1. Ground rotator experiment
The rotator is controlled by a computer and changes azimuth by 6° every 10 s and thus completes a full rotation in 10 minutes. Antenna cable wind-up is avoided by successive clockwise and counter clockwise rotations.
A dual frequency GPS receiver (NovAtel OEM-4) was used for the experiment.
The experiment was conducted for a period of 24 hours for the dual frequency antenna, and code and the carrier measurement were collected along with satellite elevation and azimuth angle information.
Code and carrier measurement were used to calculate the CMC for satellite tracks with at least 1 hour of continuous tracking. The ionospheric delay and phase wrap up corrections were applied to the CMC using equations 5.2 and 5.4. The bias was removed by calculating and subtracting the mean for each CMC track. Finally, the data was binned 92 in 5° by 5° azimuth/elevation bins and the median value was calculated for each bin [34].
Figure 5.2 shows the steps used for the process.
Figure 5.2. Processing of code minus carrier data to characterize antenna group delay variation for a Dual-Frequency antenna
Figure 5.3 shows the result for characterization of AGDV for a dual frequency antenna. The variation is between 5 to 10 cm as a function of azimuth and elevation angle.
The AGDV is also plotted as a function of elevation angle and compared to the bound of the AGDV in [2]. It can be seen from Figure 5.4 that the measurements are mostly inside the AGDV bound and the 휎푝푟,푎푖푟 curve, except at high elevation angles, which can be due to a lack of measurement data and multipath due to ground plane diffraction.
93
Figure 5.3. Antenna Group Delay Variations for the Dual-Frequency antenna
Figure 5.4. Dual-Frequency Antenna Group Delay Variations versus DO-301 Bounds from [2] 94 Several conclusions can be drawn from the results in Figure 5.4. First, since the standard deviation of airframe multipath and receiver noise is much larger than the measured AGDV for the dual frequency antenna, it is understandable that the AGDV was not taken into account during previous flight tests that only used dual frequency antennas.
For these tests, it was convenient to use a single antenna to characterize 휎푝푟,푎푖푟 and obtain dual frequency corrections needed for the CMC processing. Second, the multipath environment for the dual frequency antenna mounted on the rotor is low enough to bring the CMC well below the requirements, verifying that the test configuration is sufficient for the purpose of AGDV characterization. Third, a certified antenna for future dual frequency
GBAS will have other performance parameters to consider in addition to AGDV, which may affect AGDV performance.
The plot of AGDV versus azimuth angle also shows the variations of 5 cm to 10 cm as shown in Figure 5.5.
95
Figure 5.5. Antenna Group Delay Variations for the Dual-Frequency Antenna for each elevation bin
Next, the same antenna rotator method is used to characterize AGDV for a single frequency antenna. For the single frequency antenna, the experiment was run for 24 hours.
The processing was equivalent to that of the dual frequency antenna, except that a separate dual-frequency antenna was used for the ionosphere delay variation corrections as shown in Figure 5.6. 96
Figure 5.6. Processing of code minus carrier data to characterize antenna group delay variation for a Single-Frequency antenna
Figure 5.7 shows the result for AGDV for the single frequency airborne antenna.
This antenna is representative of the airborne antenna currently installed for GBAS precision approach. The comparison of the results with dual frequency antenna in Figure
5.4 shows that AGDV for the single frequency antenna can vary up to ±1 m. Figure 5.8 compares the results with the bounds and the antenna exceeds the bound for the lower and highest elevation angles. Multipath is the reason for the breach as residual multipath after averaging is still present in the data. One source of multipath for the higher elevation angles is the diffraction effect due to the ground plane [4]. Figure 5.9 shows the variation of the antenna group delay with azimuth angle for each elevation bin and variations of 0.2 to 1 m can be observed. 97
Figure 5.7. Antenna Group Delay Variation for a Single-Frequency antenna
Figure 5.8. Single-Frequency Antenna Group Delay Variations versus DO-301 bounds from [2]
98
Figure 5.9. Antenna Group Delay Variation for a Single-Frequency antenna for each elevation bin
To investigate the assertion that edge diffraction from the circular ground plane causes multipath, the ground plane was modified with a rounded edge as shown in Figure
5.10. In theory, the rounded edge should mitigate the diffraction error for high-elevation angles but could create potentially worse performance for low-elevation angles due the creeping wave diffraction [4, 19].
The same process was applied to the ground plane with rounded edges for the single frequency antenna and the results are shown in figures 5.11 and 5.12.
99
Figure 5.10. Rounded edge ground plane
Figure 5.11. Single frequency antenna group delay variation versus DO-301 bounds for rounded edge ground plane 100
Figure 5.12. Single frequency antenna group delay variation for each elevation bin for rounded edge ground plane
Improvement in the AGDV measurements is evident by comparing figures 5.11 and 5.12 for elevation angles above 15 degrees to figures 5.8 and 5.9. Note that the 12.5° bin for elevation angles between 10° and 15° is not shown in figures 5.11 and 5.12 due to very high levels of multipath caused by the creeping waves. At the higher elevation angles, the performance improved and almost all measurements in the 87.5° bin are within the bounds. Thus, there is a clear trade-off between the thin edge and the rounded edge ground plane for the ADGV measurements. Further research is recommended for the ground plane design and the height of the ground plane above the surrounding ground. Lowering the ground plane may also reduce the multipath. For the purpose of the research reported in this dissertation, the thin edge ground plane results are acceptable as long as the additional 101 noise due to edge diffraction multipath is recognized. One way to reduce the multipath would be through spatial smoothing, which will be discussed in Section 6.1.1.
5.2.3 Flight experiments (from [34])
A flight experiment was planned to analyze the combined effect of multipath and
AGDV. The statistics of AGDV were known from the ground experiment, and thus a flight test should provide for a characterization of both AGDV and aircraft multipath. A multi- band antenna (Antcom P/N G5Ant-3AT1) installed on the Beechcraft Baron was used for the flight test as shown in Figure 5.13. The antenna is installed forward from the wings away from obstacles to minimize the errors due to other antennas.
Figure 5.13. Multi-Band antenna installed on a Beechcraft Baron (from [34])
The ground track of the flight experiment is shown in Figure 5.14. 102
Figure 5.14. Aircraft ground track for level orbit (from [34])
The portion of the flight data used for the analysis in this paper is shown in red in
Figure 5.14 [34]. A level turn was used to measure the AGDV as a function of azimuth angle and the radius of the orbit was 5 nautical miles, which resulted in an orbit period of approx. 14 minutes. The goal is to find the AGDV as a function of azimuth angle for each tracked satellite at a certain elevation angle. The constant elevation angle was maintained through flat turns with banking of the aircraft restrained to a few degrees.
The code and carrier measurements from the flight data were post-processed. The
CMC were formed using the measurements, and the ionospheric delay was calculated using the dual frequency measurements. The CMC was corrected for ionospheric delay and the resulting data with the mean removed was plotted as a function of time as shown in Figure
5.15.
103
Figure 5.15. CMC tracks as a function of time
The mean satellite elevation angle was used to divide the CMC data into 3 elevation angle groups. The tracks below 30° elevation angle are depicted with red color, tracks between 30° and 60° elevation angle with blue color and tracks above 60° elevation angle by a black color. The CMC as function of time seems random in nature and doesn’t seem to be correlated in time. The tracks have statistics of multipath and AGDV as they cannot be separated during a flight experiment for a fixed pattern antenna, as used in this experiment.
Next, the CMC data is plotted as a function of azimuth angle relative to the antenna.
The aircraft heading information was used to estimate the relative angle by adding satellite azimuth to the aircraft heading. Fortunately, two sets of satellites were at approximately the same elevation angle during the orbit. Specifically, PRN 12, PRN 25 and PRN 31 were 104 at approx. 27° elevation, while PRN 11 and PRN 32 were both at approx. 11° elevation.
The CMC tracks for these satellites are shown in Figure 5.16 [34].
Figure 5.16. CMC tracks as a function of azimuth angle with respect to the antenna
The combination of AGDV and multipath (CMC tracks), are highly correlated as a function of azimuth and elevation angle relative to the antenna frame as shown in Figure
5.16.
The error was also compared to DO-301 bounds for AGDV [2] as shown in Figure
5.17. The CMC of the satellites tracked during the orbit of the aircraft were plotted as a function of elevation angle relative to the antenna. The plot shows that for the multiband antenna used in this dissertation the combined effect of AGDV and multipath exceed the
DO-301 allocation for group delay variations for elevation angles greater than 20° [34].
However, when considering the combined budget for antenna group delay combined with multipath and noise, the antenna could be within the requirements. 105
Figure 5.17. Combined group delay variation and multipath errors versus DO-301 bounds
In summary, for the ground rotator tests, AGDV for the single frequency antenna is larger than that of the dual frequency antenna. The characterization of the airframe multipath model involved several flight tests with a dual frequency antenna. Similar flight tests with single frequency are required to validate the bound for standard derivation of the airframe multipath and receiver noise currently being used. The other important finding is that for the flight test of a multi-band antenna, the combined effect of AGDV and multipath exceeded 휎푝푟,푎푖푟. Since, it is not a guarantee that future dual frequency antenna designs will have smaller AGDV as a function of azimuth and elevation angles, and thus similar testing of these antennas is required. The results presented above address the first four research questions. 106 Is the 휎푝푟,푎푖푟 flight test data collected with a dual-frequency antenna representative for L1-only antennas? No, the antennas have different AGDV, and therefore the data collected with the dual-frequency antenna is not representative for L1-only antennas.
Can L1-only antennas meet the DO-301 group delay variation requirements? Yes, the results show that the antenna tested in the dissertation complies with the DO-301 bounds with slightly exceeding the bounds because of the diffraction effects of the ground plane. The diffraction can be reduced by using a rounded edge ground plane.
How should antenna group delay variations be measured in the presence of multipath? A method of averaging the multipath is used by rotating the antenna, also the antenna should be placed as close to the ground as possible to minimize the effects of diffraction due to creeping waves.
What is the effect of a ground plane vs. an aircraft installation on antenna group delay variations? The ground plane is a representative of the aircraft installation and similar results are expected for AGDV. Multipath and AGDV are difficult to separate in flight as the calibration process is complex and specular multipath has a bias character too.
Thus, a combined model will be the best way forward for mitigation of AGDV with better design of the airborne antenna.
107 6. MODELING OF AGDV
The GBAS error models introduced in Chapter 4 are necessary for position estimation, protection level calculations and to perform availability assessments [47]. The validation of monitors and other applications in GNSS and GBAS includes simulations and real data analysis. The simulations use a generic model of the error source and the model is used to perform initial analysis. In this dissertation a binning model and a spherical harmonics expansion approach are used in combination with Least Squares estimation to model AGDV. The models are compared, and the advantages and limitations of each method are highlighted.
6.1 Least Squares Approach
Start with a measurements vector given by 풚풎×ퟏ where y is a vector having m observations. The objective is to estimate 풙풏×ퟏ, which is a vector of n parameters. Also, the system has many more measurements than parameters to estimate, such that 푚 ≫ 푛.
The relationship between the measurements and the parameters is known and is given by a data-matrix 푯풎×풏. Therefore, the system equation can be written as:
풚 = 푯풙 + 풆 (6.1)
The measurements are corrupted by noise and uncertainties shown as the vector e in Equation 6.1. The least squares method estimates the best solution by minimizing the square of the error ‖흃‖ퟐ = ‖풚 − 푯풙̂‖ퟐ. Appendix A has a detailed explanation of the least squares estimator. The ordinary least squares solution solves for 푥̂, an estimate of 푥, given by equation:
풙̂ = (푯푻푯)−ퟏ푯푻풚 (6.2) 108 6.1.1 Binning model approach to model antenna group delay variation
The characterization of AGDV in the previous section used Code-Minus-Carrier
(CMC) measurements to characterize the AGDV. The relationship between the CMC and
AGDV is given by equation (5.1) introduced in Section 5.2.1.
The ground rotator experiment averages the multipath in the data and thus the equation used to model the antenna group delay is given by:
( ) ( ) 퐶푀퐶푗 휃푖, 휑푖 = 훥휏휔푐,퐶/퐴 휃푖, 휑푖 + 퐵푗 (6.3)
Figures 6.1 and 6.2 show two different depictions of the rotator data. The data in
Figure 6.1 is the track of the CMC as function of elevation and azimuth of the satellite. In
Figure 6.2, the rotator azimuth is also taken into consideration and thus the data is projected into the antenna frame. The point in the middle represents 90° elevation, while each concentric circle defines the next elevation angle, spaced by 10°. The rotation of the antenna maps the data to all the bins of azimuth and elevation angles as seen in Figure 6.2.
Each of the tracks shown in Figure 6.1 have an inherent bias, see Equation 6.3. The goal is to solve for AGDV and the bias for each satellite track.
109
Figure 6.1. CMC as function of elevation and azimuth represented as a grid of azimuth and elevation angles.
Figure 6.2. CMC as a function of azimuth and elevation represented in form of polar grid.
110 There are approximately 80 satellite tracks for 24 hours of data and the rotation of the ground plane projects the data onto all the cells of azimuth and elevation angles with respect to the antenna. Thus, the least squares approach can be used to solve for the group delay in each cell and the bias for each track using Equation 6.3. For example, consider a simple case with two satellite tracks and three cells. In matrix form, Equation 6.3 can be written as
퐶푀퐶1(휃1, 휓1) 1 0 0 1 0 ( ) 푏1 퐶푀퐶1 휃2, 휓2 0 1 0 1 0 푏2 퐶푀퐶1(휃3, 휓3) 0 0 1 1 0 = 푏3 (6.4) 퐶푀퐶2(휃1, 휓1) 1 0 0 0 1 퐵1 퐶푀퐶2(휃2, 휓2) 0 1 0 0 1 ( ) (퐵2) (퐶푀퐶2(휃3, 휓3)) 0 0 1 0 1 where:
퐶푀퐶푖 denote separate satellite tracks;
푏푖 are the values of AGDV; and
퐵푖 are the biases for each satellite track.
Equation 6.4 can be written as:
푪 = 푯풃 (6.5)
The least squares approach is used to solve for AGDV and the track biases:
풃 = (푯푻푯)−ퟏ푯푻푪 (6.6)
A reference cell will be needed to solve the set of linear equations. Following
Equation 2.2, the zenith direction is taken for the reference cell:
퐶푀퐶푗(휃푟푒푓, 휓푟푒푓) = 0 (6.7)
The method explained above will solve for the AGDV for the center of each bin. 111 A simulated AGDV model is used to test the methodology. For the simulation, a known AGDV model is used to generate the input data. The simulation can also be used to evaluate the effect of various factors on AGDV estimation, for example addition of noise to the model and the size of the cells.
The 24-satellite Martinez constellation is used to simulate satellite passes for a 24- hour time period [41]. Athens, Ohio’s latitude and longitude are selected for the location of the antenna. The model for AGDV is similar to the model used in [37], and is given by equations 6.8 and 6.9, where 휃 , 휑 are elevation and azimuth angles, respectively, and the
AGDV model is shown in Figure 6.3:
퐺퐷(휃, 휑) = 훽 sin(2휑) (6.8)
휃 훽 = 0.8 − (0.8 ) (6.9) 90
Figure 6.3. Simulated antenna group delay model.
112 The results for the estimated antenna group delay using the data from the known model are shown in Figure 6.4. Figure 6.5 shows that the estimator residual errors are within ±1 cm.
Figure 6.4. Estimated AGDV using simulated AGDV model and binning model approach.
113
Figure 6.5. Difference between estimated AGDV using simulated AGDV model and binning model approach.
The single frequency antenna data characterized in Section 5.2.2 was used to estimate the AGDV using the binning model with a least squares estimation. Figures 6.6 through 6.8 show the results of the model. 114
Figure 6.6. Estimated AGDV using real data and Binning Model approach.
Figure 6.7. Estimated AGDV using Binning Model with Least Squares estimation as a function of elevation angle. 115
Figure 6.8. Estimated AGDV using Binning Model with Least Squares estimation as a function of azimuth angle.
When compared to results in Figure 5-6-5.8, a smoothing of high frequency multipath has been achieved using Binning Model approach based on least squares.
6.1.2 Spherical Harmonics expansion approach to model antenna group delay
variation
Spherical Harmonics are the solution to the spherical Laplace’s equation. The detailed derivation is shown in Appendix C. The spherical harmonics form a complete orthonormal basis. Therefore, according to the completeness property of spherical harmonics square integral functions of azimuth and elevation angles can be expanded into a uniformly convergent double series of spherical harmonics [56]. The concept is similar to Fourier series as explained in [57]. AGDV is a function of azimuth and elevation angles and can be written as 116 푚 GD(휃, 휑) = ∑푙,푚 푎푙,푚푌푙 (휃, 휑) (6.10) where:
l is the degree;
m is the order of the spherical harmonics;
푚 푚 푌푙 (휃, 휑) = 푁푙,푚(퐴푙,푚 cos(푚휑) + 퐵푙,푚sin (푚휑))푃푙 sin (휃) (6.11)
푁푙,푚 is the normalization coefficient given by:
(2푙+1)(푙−푚)! 푁 = √ (6.12) 푙,푚 4휋(푙+푚)!
푚 and 푃푙 cos (휃) is the Associated Legendre Function.
The combination of equations 6.10 through 6.12 gives the final equation for AGDV as a function of azimuth and elevation angle:
푙푚푎푥 0 푙푚푎푥 푙 GD(휃, 휑) = ∑푙=0 푁푙,0퐴푙,0푃푙 sin(휃) + ∑푙=1 ∑푚=1(푁푙,푚(퐴푙,푚 cos(푚휑) +
푚 퐵푙,푚sin (푚휑))푃푙 sin (휃) ) (6.13)
The goal is to find the coefficients 퐴푙,푚 and 퐵푙,푚. Appendix C has the details of the derivation.
Equation (5.7) shows that the code minus carrier measurement (CMC) corrected for ionosphere and carrier phase wrap consists of multipath and AGDV. The multipath is mitigated using the rotation of the antenna and thus the AGDV as a function of elevation and azimuth angles is left in the data. The Spherical Harmonics expansion method doesn’t solve for the bias of Equation 5.7, and thus the bias is removed by subtracting the mean of the CMC. 117 The validity of the method is verified using a model of AGDV as introduced in
Section 6.1.1 by equations 6.8 and 6.9. The model is used to populate the left-hand side of
Equation 6.13. The goal is to find the coefficients on the right-hand side of Equation 6.13.
Figure 6.9 shows the result for AGDV estimation using the spherical harmonics expansion approach using a maximum value of 30 for the degree of the spherical harmonics, the model used is shown in Figure 6.3.
Figure 6.9. Antenna Group Delay estimation using Spherical Harmonics approach.
The difference between the model of AGDV and the model obtained using Spherical
Harmonics expansion is shown in Figure 6.10.
118
Figure 6.10. Difference between AGDV model and estimated AGDV using Spherical Harmonics expansion.
The residuals are well below 1 mm; thus, the spherical harmonics are a good approach to estimate AGDV. The single frequency antenna CMC is used next to verify if the method works for real data. Figures 6.11 through 6.13 shows the results obtained by using spherical harmonics expansion to model AGDV. The results can be compared to those of the binning model solution, and it can be seen that the results are nearly identical, by comparing figures 6.6 through 6.8 with figures 6.11 through 6.13.
119
Figure 6.11. Estimated AGDV using Spherical Harmonics expansion approach.
Figure 6.12. Spherical Harmonics expansion model of AGDV as a function of elevation angle compared to DO-301 bound.
120
Figure 6.13. Spherical Harmonics expansion model of AGDV as a function of azimuth angle.
This part of the dissertation answers the fifth research question, can we define a model for AGDV? In conclusion, the Binning Model and the Spherical Harmonics
Expansion can be used to model AGDV. The advantage of the Binning Model is that it can be used to solve for the bias in the tracks of the satellites.
121 7. EFFECT OF AGDV ON POSITION SOLUTION AND PROTECTION LEVEL
7.1 Aircraft Position Errors and Protection Levels
The estimation of the position solution and protection level (PL) at the airborne receiver is explained in detail in Chapter 4. The position accuracy and PL are dependent on the variances of the various residual error sources affecting the code and carrier phase measurements and the satellite geometry. The detailed models and position and protection level estimation are specified for an airborne receiver in [14]. However, AGDV is not included in the specification. Chapter 5 shows that AGDV for a single frequency antenna is an important error source for GBAS. The current GBAS standards assume that the errors affecting accuracy and integrity are zero-mean, normally distributed error sources. The errors can be root-sum-squared, and a total variance of the error sources is formed. This variance is used for position and integrity calculations. Errors that cannot be characterized by a Gaussian distribution, such as multipath, are bounded by a Gaussian distribution.
Another method to include the biases is similar to the treatment of the ionospheric bias in
Equation 4.36 and the B-values in Equation 4.38; i.e., a term is added to the VPL to include the worst-possible contribution of the biases bounded by 훽 for each of the measurements:
푁 2 2 푁 푉푃퐿 = 퐾푓푓푚푑√∑푖=1 퐺푣푒푟푡,푖휎푖 + ∑푖=1|퐺푣푒푟푡,푖훽푖| (7.1)
The second term in Equation 7.1 assumes that the biases can combine in the worst- possible way by taking the absolute value of each bias contribution in the vertical position solution. This method is always safe and assumes no knowledge about the biases other than their maximum value 훽. The disadvantage of this method is that the VPL increases, thus reducing availability. 122 The next two sections evaluate the impact of AGDV on the aircraft position accuracy and the VPL. For the position errors, both the actual and worst-case errors are considered. For the VPL, four different methods are investigated to incorporate AGDV.
7.1.1 Impact of Single Frequency Antenna Group Delay Variations on Position
Solution
The measured group delay variation biases of Figure 5.7 were implemented in
Equation 4.26 which resulted in east, north and vertical position errors as well as timing errors. Worst case errors were calculated by taking the DO-301 bound and by combining the biases in the worst-possible way as shown in Equation 7.1. Figure 7.1 shows the overview of the calculations for the nominal and worst-case position errors.
123
Figure 7.1. Calculation of nominal and worst-case vertical position errors
Figure 7.2 shows the typical and worst case east and north errors as a function of time.
124 East Error 5 typical worst case
0 (m)
-5 0 5 10 15 20 25
North Error 5 typical worst case
0 (m)
-5 0 5 10 15 20 25 time(hr)
Figure 7.2. Typical and worst case east and north position errors due to single-frequency antenna group delay variations
From Figure 7.2, the typical east and north position errors are .07 m and .03 m, respectively, while the worst case east and north position errors are 0.62 m and 0.63 m, respectively. Figure 7.3 shows typical and worst case vertical and clock errors as a function of time. Note that the vertical and clock errors are highly correlated.
125 Vertical Error 5 typical worst case
0 (m)
-5 0 5 10 15 20 25
Clock Error 5 typical worst case
0 (m)
-5 0 5 10 15 20 25 time(hr)
Figure 7.3. Typical and worst case vertical and clock errors due to single-frequency antenna group delay variations
Typical vertical and clock errors are .17 m and .15 m, respectively. The worst case vertical and clock errors are 1.33 m and 1.01 m, respectively.
When the geometry is worsened by removing two high-elevation satellites, the east and north errors increase as shown in Figure 7.4.
126 East Error 5 typical worst case
0 (m)
-5 0 5 10 15 20 25
North Error 5 typical worst case
0 (m)
-5 0 5 10 15 20 25 time(hr)
Figure 7.4. Poor Geometry typical and worst case east and north position errors due to single-frequency antenna group delay variations
Similarly, the worst case vertical and clock errors also increase as shown in Figure
7.5.
Vertical Error typical 5 worst case
0 (m)
-5 0 5 10 15 20 25
Clock Error typical 5 worst case
0 (m)
-5 0 5 10 15 20 25 time(hr)
Figure 7.5. Poor geometry typical and worst case vertical and clock errors due to single- frequency antenna group delay variations
127 The position solution for typical and poor geometry show that the errors are close to the bounds at various points in time as shown in Figure 7.6.
Figure 7.6. Vertical position error compared to the bound for a typical and a poor geometry
Based on figures 7.2 through 7.6, it can be concluded that the contribution from
AGDV is generally small at the .31 m (1-sigma) vertical, and less than .77 m under nominal conditions. For poor geometries, the contribution of AGDV increases, but this is also the case for other error sources. It is also noted that the worst-possible error bound can be approached for nominal conditions, while the likelihood of reaching the bound increases for poor geometries.
128 7.1.2 Impact of Single Frequency Antenna Group Delay Variations on Vertical
Protection Level
The AGDV is a bias-like error as shown in Chapter 5, and four methods can be used to incorporate AGDV biases:
1. Treat AGDV as a specific risk and add a bias component to the protection level.
2. Treat AGDV as an average risk and, if necessary, inflate 휎푝푟,푎푖푟.
3. Show that the protection level doesn’t change significantly in the presence of
AGDV.
4. Reduce the AGDV bound.
Each of the four methods are investigated in the following four sections.
7.1.2.1 Treat AGDV as a specific risk
When AGDV is treated as a specific risk, the biases on all satellites are combined in the worst-possible way to ensure that the VPL bounds the error due to AGDV as shown in
Equation 7.1. To evaluate the impact of AGDV on VPL, a 24-hr period is investigated at
Houston, Texas airport using GPS constellation data from 14 January 2016.
Figure 7.7 shows the vertical protection level calculated using only the noise components from Equation 4.26, and the worst case vertical position error due to AGDV.
To calculate the VPL that includes AGDV, the two lower curves in Figure 7.7 should be added, which increases the VPL on average by 1.4 m [35]. When the geometry is worsened by removing two high elevation satellites, the VPL increases and with the addition of
AGDV worst case error reaches the 10 m alert limit at certain points as shown in Figure
7.8. 129 Vertical Protection Level 10 VPLapr__Ho 8 worst case vertical error VPL__bias
6
4
2 VPL VPL Ho Hypothesis(m)
0 0 5 10 15 20 25 time(hr) Figure 7.7. Vertical protection level from equations 4.26 and 7.1, and worst-case vertical position error due to single-frequency AGDV biases
Vertical Protection Level 10 VPLapr__Ho 8 worst case vertical error VPL__bias
6
4
2 VPL VPL Ho Hypothesis(m)
0 0 5 10 15 20 25 time(hr) Figure 7.8. Poor geometry VPL from equations 4.26 and 7.1, and worst-case vertical position error due to single-frequency AGDV biases
On average, the VPL increases by 2 m due to the AGDV biases. Note that the spike that occurs just before 10 hours into the day is not significant as the VPL without the antenna group delay would have exceeded the Vertical Alert Limit (VAL) of 10 m [35]. 130 The benefit of treating AGDV as specific risk is that it is guaranteed safe; i.e., the
VPL always bounds the error with the probability required. The disadvantage is a significant increase in VPL that affects GBAS availability.
7.1.2.2 Treat AGDV as average risk
In average risk, the biases are bounded by normal distributions and root-sum- squared with all other error sources to find the overall standard deviation. The best way for this method would be to make AGDV part of 휎푝푟,푎푖푟, with the justification that AGDV is similar to airborne multipath; both are difficult to predict and can therefore be treated as random variables in the VPL.
The initial analysis was done by using a model for multipath and AGDV to understand the bound of the errors in the range domain and the projection of the errors in the position domain. A multipath electromagnetic model was developed in [17] and the
AGDV model was a derived from the bounds in [2] and is defined as [36]:
퐺퐷(휃푖, 휑푖) = 훽푖 sin(2휑푖) (7.2) where:
휑푖 = azimuth angle from the GPS antenna to satellite i in local level coordinates;
−2.5 − .04625(휃푖 − 5°) 푛푠, 5° ≤ 휃푖 < 45° 훽푖(휃푖) = { ; and 0.65 푛푠, 휃푖 ≥ 45°
휃푖 = elevation angle of satellite i in degrees.
Figure 7.1 shows the models individually and the combined model
131
Figure 7.9. AGDV model (left), Multipath EM model (middle), Combined model (right) (from [36])
It is noted that the Multipath EM model shown in Figure 7.9 is significantly smaller than the multipath contribution to 휎푝푟,푎푖푟 (see Equation 4.31). The combined model pseudorange errors were compared to the bound of the pseudorange error due to multipath by multiplying the standard multipath model with the fault free missed detection multiplier introduced in Table 6. The combined model pseudorange errors were within the bound.
푁 Next, the errors are projected in the vertical position domain by using ∑푖=1(퐺3,푖 × 퐺퐷푀푃푖) where
퐺퐷푀푃 is the combined multipath and AGDV model as depicted in Figure 7.9;
퐺3,푖 is the third row of the projection matrix defined in Section 4.2;
푁 2 2 The result is normalized by the bound 퐾푓푓푚푑 × √∑푖=1(퐺3,푖휎푚푢푙,푖). where
휎푚푢푙 is the standard deviation of airborne multipath as defined in Section 4.2.1.1. 132 The normalized VPL results are shown in Figure 7.10 for up to two satellite failures.
Figure 7.10. Normalized VPL vs VPL (from [36])
It can be concluded from Figure 7.10 that even though the error was bounded in the range domain, the projection in the position domain can exceed the bound due to correlation of the error sources for VPL greater than 7. These results imply that either
휎푝푟,푎푖푟 must be increased, or AGDV bounds must be decreased. Since the AGDV model is based on the RTCA bounds, it is feasible that that an actual aircraft antenna is within those bounds. To investigate the latter, the Boeing Company analyzed single frequency antenna data from a flight test that used both a single frequency antenna and a dual frequency antenna on a Boeing 787. The single frequency antenna was connected to a Honeywell
Integrated Navigation Radio (INR), and a dual frequency antenna was connected to a 133 NovAtel GPS/inertial reference data system as detailed in [36]. The inertial data was used to divide the data into segments. The code measurement was smoothed using a 100-s smoothing filter, and code minus carrier, 퐶푀퐶100,푖, was evaluated for each satellite in view.
The code minus carrier contains multipath, AGDV and receiver noise as explained in
Appendix B. The range domain errors were within the bounds (in blue) for the single frequency antenna for the approach phase of the flight as shown in Figure 7.11.
Figure 7.11. Code minus carrier as function of elevation angle for a single frequency antenna flight data (from [36])
The CMC data from Figure 7.11 was used to project the range domain errors into the position domain using the projection matrix. The normalized position domain projection error (NPDE) was calculated using Equation 7.3: 134 ∑푁 퐺 × {퐶푀퐶 } 푁푃퐷퐸 = 푖=1 3,푖 100,푖 (7.3) 푁 2 2 2 퐾푓푓푚푑 × √∑푖=1{퐺3,푖(휎푚푝,푖 + 휎푛표푖푠푒,푖)} where
휎푚푝 and 휎푛표푖푠푒 are defined in the Section 4.2.
The normalized position domain projection error is shown in Figure 7.12. Figure
7.12 shows that during the approach phase of the flight, the position domain error is well within the bounds.
Figure 7.12. Normalized Position Domain Projection Error for INR data (from [36])
Thus, the current AMM will bound the error due to the combined effect of multipath and AGDV for the single frequency antenna used for the flight tests. 135 7.1.2.3 Show that the protection level doesn’t change significantly in the presence of
AGDV
For both specific and average risk cases, the increase in the vertical position error is significant when DO-301 AGDV bounds are considered as shown in sections 7.1.1, 7.1.2.1 and 7.1.2.2. The specific risk analysis shows an average VPL increase of 1.4 m, which cannot be neglected compared to the limit of 10 m.
7.1.2.4 Reduce the AGDV bound
Reduction of the AGDV bound is necessary, unless 휎푝푟,푎푖푟 is increased. As shown in
Section 7.1.2.2, in order for AGDV to fit within the current 휎푝푟,푎푖푟, an antenna model such as that shown in Equation 7.2 must be used in addition to a decrease in the bounds. If only
Equation 7.2 is used, then the computer simulation shows that the vertical position error bound can be exceeded as shown in Figure 7.10.
7.1.3 AGDV bounding conclusions
Based on the results in sections 7.1.2.1 through 7.1.2.4, AGDV can be treated similar to aircraft multipath using an average risk analysis. This approach was validated for a large commercial aircraft, the Boeing 787, based on the single frequency antenna installed on this aircraft type. Unfortunately, this analysis does not directly apply to other antennas and other aircraft, unless their AGDV are smaller than the antenna installed on the 787 and their multipath error is also smaller than that of a 787. One way to resolve the
AGDV bounding for other aircraft would be to introduce an antenna model shown in
Equation 6.8 or Equation 7.2 with a bound that is smaller than the DO-301 bound shown in Figure 5.4.
This Chapter resolves research questions 6, 7 and 8: 136 What is the impact of antenna group delay variations on the GBAS position solution? Section 7.1 shows the position errors due to AGDV. For nominal geometries, the vertical position error is generally small at .31 m (1-sigma), and less than .77 m.
Corresponding horizontal position errors in the east and north directions are .08 m and .03 m, respectively, while the worst case east and north position errors are 0.62 m and 0.63 m.
What is the impact of AGDV on the aircraft vertical protection level (VPL)? When
AGDV is accommodated as part of 휎푝푟,푎푖푟 as analyzed in Section 7.1.2.2, there is no impact on the aircraft VPL. When AGDV is treated as a specific risk, the average increase in VPL is 1.4 m during nominal geometries.
Should antenna group delay be treated as a bias or noise? How should it be included in the error budget? The recommendation of the research is to treat AGDV similar to aircraft multipath error. Both errors are deterministic and bias-like, but difficult to predict as they require knowledge of the antenna orientation relative to the aircraft and relative to the satellites. In addition, AGDV requires information on the biases as a function of signal arrival angle into the antenna pattern. Therefore, both errors can be treated using an average risk analysis as validated by the results presented in Section 7.1.2.2. In addition, an antenna model should be used as defined in Equation 6.8 or Equation 7.2 with a bound that is smaller than the DO-301 bound shown in Figure 5.4.
137 8. EFFECT OF AGDV ON AIRBORNE INTEGRITY MONITORS
8.1 Airborne Integrity Monitors
For GAST D, a combination of airborne and ground monitors has been specified to detect and mitigate the threat space due to anomalous ionospheric gradients [14]. The airborne monitors include Dual Solution Ionospheric Gradient Monitor Algorithm
(DSIGMA) and Code-Carrier Divergence (CCD) filter. DSIGMA is implemented in the range domain and is the difference of corrected pseudoranges smoothed by carrier measurement for two different time constants. The CCD monitor measures the rate of change of code minus carrier. The thresholds for the monitors were validated through flight tests, but a dual frequency antenna was used for the tests. The monitor thresholds are mostly determined by airborne multipath. The distinction between airborne multipath and AGDV is difficult for airborne data as shown earlier, therefore the threshold will also depend on
AGDV. Since a dual frequency antenna was used for threshold determination, validation of the thresholds for a single frequency antenna is important, as the AGDV for single frequency antennas is larger, and this could affect the thresholds.
GBAS uses a differential GPS and smoothing of code using carrier to achieve better accuracy and integrity than standalone GPS. When the ionosphere is acting nominally, it is spatially correlated, such that the aircraft on a GBAS precision approach and the ground- based reference receivers experience similar ionospheric delays as depicted in Figure 8.1.
Thus, the error can be mitigated using DGPS, except for some residual error due to the distance between the aircraft and the ground reference receivers and smoothing filter effects as explained in Chapter 4. Typical values for nominal ionospheric spatial gradients are in the range of 0.001 to 0.005 m/km. Thus, a spatial separation of 10 km will only 138 introduce range error of 1 cm to 5 cm. Analysis of data in [48-50] shows that rare anomalous ionospheric conditions can occur with gradients up to 0.412 m/km, as illustrated in Figure 8.2. The unusual gradients were derived based on data from Continuously
Operating Reference Stations (CORS) and Wide Area Augmentation System (WAAS) super truth data. Recent analysis using a long-term ionospheric anomaly monitor also verified the values for maximum anomalous gradients [51]. Anomalous gradients can introduce large errors in the GBAS system which can be hazardous for the precision landing [37].
Figure 8.1. Nominal ionospheric behavior
139
Figure 8.2. Anomalous ionospheric behavior
8.1.1 Ionospheric threat space
To analyze the impact of anomalous gradients, an ionospheric threat model, mostly applicable to the mid-latitude region, was developed [52]. In this threat model, the anomaly is considered as a linear semi-infinite wave front, and is defined by the width of the front, the speed of the front, the value of the slope of the gradient, and the pierce point velocities for the airborne and ground antennas as shown in Figure 8.3 [37].
140
Figure 8.3. Ionosphere Threat Model (from [37])
The wedge shape of the ionospheric gradient introduces different ionosphere delays for the aircraft and reference receivers. The rate of change of ionospheric delay introduces error due to smoothing filter employed for mitigation of multipath affects. There is relative motion due to the different velocity of the aircraft 푉푎푖푟, velocity of the ionosphere pierce points for airborne and ground receivers 푉푖푝푝,푎푖푟 푎푛푑 푉푖푝푝,푔푛푑, and velocity of the satellite, 푉푠푎푡 that affect the rate of change of the ionosphere delay and thus additional error is added [53]. The presence of an anomalous ionospheric front could have an impact on the integrity of the GBAS system. Therefore, algorithms and/or monitors are required to mitigate the integrity threat posed by an anomalous ionosphere. Initially, an Ionospheric
Gradient Monitor (IGM) at the GS was proposed. During validation of the IGM it was found that a local tropospheric phenomenon can affect the detection of an ionospheric spatial gradient [24]. A combination of ground and airborne monitors was proposed after extensive analysis and simulation work. The analysis showed that a combination of 141 airborne DSIGMA and CCD monitors should mitigate the threat space due to anomalous ionospheric gradients. Both the GBAS ground station and the airborne receiver employ algorithms and monitors, since, depending on the geometry, the anomaly could impact either the ground, the aircraft, or both simultaneously [37].
For GAST D, the aircraft implements the DSIGMA Range monitor, the CCD monitor, and the addition of dual-solution bias parameters to the horizontal and vertical protection levels. DSIGMA was first introduced by Boeing in [53]. In [14], both the
DSIGMA Range monitor and the dual-solution bias parameters are based on differences between 100-s and 30-s carrier smoothed code measurements. A satellite will not be used for the position solution if the difference between the 100-s and 30-s smoothed corrected pseudoranges exceeds the threshold of 0.976 meters. If the satellite passes the DSIGMA
Range monitor threshold, then it is used in the 30-s and 100-s smoothed position solutions.
The difference between these solutions is added to the protection levels. The airborne CCD monitor is comprised of a cascaded filter which takes the rate of change of code minus carrier as an input and a satellite is excluded from the position solution if the output of the filter exceeds 0.0415 m/s [37]. These thresholds assure that the performance requirement of corrected pseudorange error at the landing threshold point (LTP) exceeding 2.75 m with probability of less than 10−9 per approach is met.
During the validation phase of the DSIGMA Range and CCD monitors, the data used to determine the sigma over bound and the thresholds used a dual frequency antenna.
Since the flight tests were performed during nominal ionospheric conditions, the difference between the 100-s and 30-s smoothed pseudorange measurements is primarily due to noise and multipath error. The rate of change of code minus carrier is also dependent on noise 142 and multipath error. Thus, the DSIGMA Range and the CCD monitor thresholds should account for AGDV as a combined effect of multipath and AGDV is experienced by an airborne receiver.
8.1.1.1 DSIGMA Range
The DSIGMA Range monitor is derived from the DSIGMA monitor first proposed by the Boeing [53]. Figure 8.4 shows an overview of the processing of the 100-s and 30-s smoothed pseudoranges in the range domain and in the position domain.
Figure 8.4. Dual Solution Ionospheric Gradient Monitoring Algorithm (DSIGMA) Processing Overview
The initial DSIGMA monitor was implemented in the position domain. The position solution using 100-s and 30-s smoothed corrected pseudoranges is calculated, and the difference in the vertical and lateral directions is found shown as 푑푖푓푓푣푒푟푡 푎푛푑 푑푖푓푓푙푎푡, 143 respectively [37]. The DSIGMA Range monitor is implemented in the range domain and uses the value of 푃푑푖푓푓 to monitor the ionospheric conditions 푃푑푖푓푓 is defined as:
푃푑푖푓푓 = 휌푐표푟푟,100 − 휌푐표푟푟,30 (8.1) where
휌푐표푟푟,100 and 휌푐표푟푟,30 are the 100-s and 30-s smoothed pseudoranges, respectively, corrected using a ground reference receiver differential correction generated from 100-s and 30-s smoothed pseudoranges [37].
During nominal ionospheric conditions, the corrected pseudoranges derived for
100-s and 30-s smoothed pseudoranges will cancel all error sources except for noise, multipath and antenna group delay variations. Equation 8.1 can be written to depict that for
100-s and 30-s smoothed pseudorange:
휌푐표푟푟,100 − 휌푐표푟푟,30 = ∆퐼̃100−30 + ∆휂̃100−30 + ∆휏100−30 (8.2)
The corrections from the reference antennas are slowly varying; therefore, 푃푑푖푓푓 is dominated by aircraft multipath and noise, and AGDV.
The validation process for the DSIGMA Range monitor is provided in [24] and
[54]. An overbounding sigma of 22 cm was needed to close the validation [55]. A series of flight test conducted by the Federal Aviation Administration (FAA) and Honeywell
(SESAR data) derived the final overbounding sigma of 17.4 cm for a receiver with 0.1 chip correlator spacing and a threshold of 0.976 meters for the airborne DSIGMA Range monitor. The exclusion and re-admittance of a satellite depends on the value of Pdiff in combination with other factors that are provided in detail in [14].
The worst-case analysis approach was used by Honeywell to simulate multiple ionospheric gradient scenarios, and to analyze if using a combination of airborne and 144 ground integrity monitors mitigates the ionospheric threat space. The simulation inputs are ionospheric threat space parameters, a speed profile of an aircraft, and the geographic parameters for aircraft and the reference receiver and integrity parameters as per requirements. Figure 8.5 portrays the geographic input used in the simulation. It assumes that the aircraft always approaches from the north and the runway is aligned in the north- south direction. The angle 훽 controls the direction of the ionospheric gradient, and ranges from -90° to 90°. The velocity of the front also defines the direction, 0° with positive velocity means the front is coming from the North direction and negative velocity means the front is coming from the South. The angle 훼 controls the position of the centroid of the
GBAS reference receivers, which is at a worst-case, constant 5-km distance from the landing threshold point (LTP). The simulation parameters and their range of values are shown in Table 8. It was shown that it is possible to alleviate any hazardous condition due to anomalous ionospheric gradients using the Airborne DSIGMA and CCD monitors, and ground IGM monitors [24]. However, as pointed out, the antenna used to find the threshold for Airborne DSIGMA is a dual frequency antenna and thus the thresholds might change if a single frequency antenna is used.
145
Figure 8.5. Geometry for Validation of the Ionospheric Threat Mitigation
Table 8.
Model parameters used by Honeywell for simulation (from [24]) Gradient Min Max Step size
Mag.(mm/km) 200 500 20
Width (km) 25 200 25
푉푓푟표푛푡(m/s) -750 750 10
훽(deg) -90 90 15
훼(deg) 90 270 15
8.1.1.2 Dual-solution vertical bias parameter
A vertical bias parameter 퐷푣 is also calculated by the airborne equipment during a
GAST D precision approach. The bias is added to the Vertical Protection Level. In GAST 146 D the aircraft receiver estimates two position solutions using the 30-s and 100-s smoothed
Pseudoranges. The position solution is calculated using a linearized weighted least square approach as explained in Chapter 4 [37]:
∆푥 = (퐻푇푊퐻)−1퐻푇푊∆휌 (8.2)
Where ∆푥 is the error in the position solution, ∆휌 is the difference between the measured pseudorange and the pseudorange evaluated from the initial estimate of the user position and clock offset. H is the geometry matrix, W is a weight matrix having only diagonal elements, which are reciprocals of the measurement error variance [14]. The difference of the position solution using a 100-s and 30-s smoothed pseudoranges is given by:
푇 −1 푇 ∆푥 = (퐻 푊퐻) 퐻 푊(∆휌푐표푟푟,100 − ∆휌푐표푟푟,30) (8.3)
or
푇 −1 푇 ∆푥 = (퐻 푊퐻) 퐻 푊(푃푑푖푓푓) (8.4)
The vertical component of ∆푥 is 퐷푣, which is added to the Vertical Protection Level
(VPL) as follows:
푁 푉푃퐿 = 푘 √∑ 푆2 휎2 + 퐷 퐴푝푟퐻0 푓푓푚푑 퐴푝푟푣푒푟푡,푖 푖푥 푣 푖=1
푆 = (퐻푇푊퐻)−1퐻푇푊 where
푘푓푓푚푑 is the fault free missed detection multiplier; and
푆 is the third row of the weighted least square projection matrix S. 퐴푝푟푣푒푟푡 147 Thus, in addition to affecting the airborne DSIGMA Range monitor, the antenna group delay, multipath and receiver noise will also affect the VPL [37].
8.1.1.3 Airborne CCD monitor
The Code Carrier Divergence Filtering implemented in the aircraft is defined in
[14] and is given by:
푍푛 = (1 − 푘) × 푍푛−1 + 푘 × 푑푧푛
퐷푛 = (1 − 푘) × 퐷푛−1 + 푘 × 푍푛
(휌푛 − ∅푛) − (휌푛−1 − ∅푛−1) 푑푧푛 = (8.5) (푡푛 − 푡푛−1)
Where:
휌푛 and ∅푛 are raw code and carrier phase measurements;
k is the time between the consecutive samples divided by 100 s; and
퐷푛 is the output of the filter.
A satellite will be removed from the position solution if |퐷푛| exceeds 0.0415 m/s.
Thus, 푑푧푛, the rate of change of the CMC, is the input to the monitor. Since CMC contains antenna group delay variations, receiver noise and multipath, the effect of antenna group delay variations on |퐷푛| must be taken into consideration [37].
It has been shown in chapter 5 that AGDV for dual frequency is much smaller than for a single frequency antenna. The results shown in chapter 7 manifest that 휎푎푖푟 is still valid in the presence of antenna group delay variations [36]. Note, however, that this is only true for an aircraft antenna that has additional group delay characteristics beyond the bound specified in DO-301. Specifically, it should either be the actual L1 antenna used for 148 the flight tests, or it should have a group delay pattern similar to single frequency AGDV shown in chapter 5.
Monitors that rely on the difference between 100-s and 30-s smoothed pseudoranges are also affected by AGDV in a way that is not covered by 휎푎푖푟. The actual value of the group delay is not as significant as the change in the group delay that will cause a different response for the 100-s and 30-s filters. To illustrate this effect, Figure 8.6 shows the step response of the 100-s and 30-s filters as well as the difference between the filter outputs.
Figure 8.6. Step Responses of 100-s and 30-s Smoothing Filters and their Difference (from [37])
The difference of the responses is the quantity of interest for the DSIGMA Range monitor and also affects the bias parameter Dv. The antenna group delay for a low elevation 149 satellite could change by 1 m if the aircraft executes a turn. This 1-m difference would cause 0.4 m to be added to 푃푑푖푓푓 after approximately 40 s. If the aircraft is not changing attitude or heading, the impact on 푃푑푖푓푓 is negligible. The worst case would be an aircraft heading change of 90 deg for the single-frequency group delays shown in Figure 5.7, which could cause the group delay to change from a maximum positive value to a maximum negative value. According to the DO-301 bound, the change in group delay could reach 5 ns, or 1.5 m for a satellite at a 5° elevation angle. This, in turn, would cause 푃푑푖푓푓 to change by 0.6 m. If the aircraft is on a stable final approach that lasts for at least 300 s, the error reduces to below 0.1 m. For an aircraft on a short final, preceded by a 90-deg turn, the group delay variation would have to be considered. An additional complication is that the
DO-301 bound is only defined for positive elevation angles relative to the aircraft body frame. When the aircraft is turning, satellites that are below the bank angle could also be below the aircraft body horizon, which would be the case for several satellites. In this case, it would not be acceptable to stop tracking the satellites below the aircraft body horizon as it would potentially take hundreds of seconds to re-admit these satellites into the solution.
Furthermore, the aircraft antenna pattern is difficult to design for negative elevation angles since the antenna is mounted on top of the aircraft fuselage and the entire aircraft frame would have to be considered [37].
A flight experiment is required to 1) Address the impact of antenna group delay variations on 푃푑푖푓푓 because previous flight tests used dual-frequency antennas with much smaller antenna group delay variations than those allowed for L1-only antennas, and 2)
Characterize antenna group delay variations for negative elevation angles with respect to the aircraft frame [37]. 150 8.2 Flight Experiment
A flight experiment is used to characterize the effect of the antenna group delay variations on the DSIGMA Range monitor (푃푑푖푓푓), the difference of the vertical component of the solution formed by 100-s and 30-s corrected Pseudoranges (퐷푣), and the CCD monitor. Both a multi-band GNSS antenna (AntCom G5Ant-3AT1) and an L1-only antenna (AntCom AT575-9W) were installed on Ohio University’s Piper Saratoga as shown in Figure 8.7 [37].
Figure 8.7. Piper Saratoga Research Aircraft with Multi-Band and L1-only Antenna (from [37]).
The aircraft flight test trajectory is shown in Figure 8.8. The flight path consists of six approaches to Ohio University’s airport (KUNI) Runway 25 with a 10-nmi straight final starting in the upper right of the Figure 8.8. The duration of the 10-nmi final approach is approx. 500 seconds [37]. 151
Figure 8.8. Flight Trajectory for DSIGMA and CCD Monitor Evaluation.
Satellites with different elevation angles will have different effects on the ionospheric monitors. Therefore, six satellites were selected for a detailed analysis. Each of the satellites were visible for most of the six approaches and their elevation angle profiles as a function of time are shown in Figure 8.9. For example, PRN 29 starts at an elevation angle of approximately 17° and goes down to approximately 5°. PRN 23 rises from approximately 25° to 63°, while PRN 26 starts at 75°, reaches 80° and ends at 53°. Each of the satellites will experience different group delay variations [37]. 152
Figure 8.9. Satellite Elevation Angles during the Flight (from [37]).
8.3 Computer Simulations
Before analyzing the flight test results, the flight path and satellite parameters were first used in a computer simulation to evaluate the impact on the aircraft ionosphere monitors. In order to implement the simulation, an antenna group delay model is added.
The antenna group delay variation model is given in Chapter 6, equations 6.8 and 6.9. Note that it is not possible to use the DO-301 group delay bounds by themselves, as they do not contain information about the shape of the errors as a function of azimuth and elevation angles [37]. Figure 6.3 illustrates the antenna group delay variation model as a function of azimuth and elevation angles. The error in the azimuth direction reflects a double sinusoid, while the error in the elevation direction decreases with increasing elevation angle until it reaches zero at 90°. The model in Figure 6.3 is used to add antenna group delay variations to the measurements from the flight data using elevation and azimuth angles with respect 153 to the antenna frame. The roll, pitch and heading data from the onboard GPS/Inertial
Measurement Unit (GPS/IMU) is used to transform the satellite elevation and azimuth angles from the local horizon frame to the antenna frame. The angles relative to the antenna frame can have elevation angles below the aircraft horizon, because when the aircraft banks, it can receive signals from the satellites which are below the aircraft horizon. Thus, to incorporate such measurements, the antenna group delay model is expanded by mirroring the group delay variation errors with respect to zero elevation angle [37].
Figures 8.10 through 8.12 show the simulated impact of the Group Delay Variation
Model on 푃푑푖푓푓, 퐷푣 and the CCD monitor, respectively. As expected, differences between the 100-s and 30-s smoothed pseudoranges reach up to 0.5 m during aircraft turns but settle down to below 0.1 m as the final approach progresses closer to the runway [37].
Figure 8.10. Simulation of Difference between 100-s and 30-s Smoothed Pseudoranges due to L1 Antenna Group Delay Variations (from [37]). 154 The position differences between the 100-s and 30-s smoothed solutions behave similar to 푃푑푖푓푓, as shown in Figure 8.11. Shortly after the turns, the position difference reaches up to 0.4 m, but quickly goes down below 0.1 m after 100 s [37].
Figure 8.11. Simulation of Difference in Vertical Protection Level using 100-s and 30-s Smoothed Pseudoranges due to L1 Antenna Group Delay Variations (from [37])
The impact on the CCD monitor is much less significant due to the cascaded low- pass filters that reduce the transients introduced by the antenna group delay variations. As shown in Figure 8.12, the maximum impact on the CCD monitor is below 5 mm/s [37]. 155
Figure 8.12. Simulation of CCD Monitor Response to L1 Antenna Group Delay Variations (from [37])
8.4 Flight Test Results for L1 Antenna
The flight test results from the L1 antenna are presented in Figures 8.13 through
8.15. The 푃푑푖푓푓 response is larger than that predicted by the simulation by up to a factor of
2. The larger errors are still within the threshold, especially 100 s after the final approach has been initiated. Several factors mentioned previously contribute to the larger errors: airborne noise and multipath, as well as unknown antenna group delay variations and multipath for satellites that are below the local aircraft horizon during aircraft turns. From
Figure 8.13, it is clear that primarily the low elevation angle satellites, PRN 29 and PRN
14 have large 푃푑푖푓푓 responses. For example, PRN 14’s elevation angle decreases during the flight test, which results in increasing values for 푃푑푖푓푓 [37]. 156
Figure 8.13. Flight Test L1 Antenna Difference between 100-s and 30-s Smoothed Pseudoranges (from [37])
The position differences between the 100-s and 30-s smoothed solutions are also larger than the computer simulation and reach up to 0.7 m. After completion of the turns, the position differences decrease, similar to 푃푑푖푓푓 [37]. 157
Figure 8.14. Flight test L1 antenna difference in Vertical Protection Level using 100-s and 30-s smoothed pseudoranges (from [37])
The impact on the CCD monitor is still below 1 cm/s but shows more variation than that predicted by the computer simulation. 158
Figure 8.15. Flight Test L1 Antenna CCD Monitor Response (from [37])
8.5 Flight Test Results for Multi-Band Antenna
Generally, a multi-band antenna has smaller group delay variations than single frequency antennas. Flight results from the multi-band antenna are presented in figures
8.16 through 8.18. 159
Figure 8.16. Flight Test Multi-Band Antenna Difference between 100-s and 30-s Smoothed Pseudoranges (from [37])
The pseudorange differences between the 100-s and 30-s smoothed solutions shown in Figure 8.16 are smaller than those for the L1 antenna, and reach up to approx. 0.2 m, compared to up to 1.0 m for the L1 antenna. Similarly, the impact on the position difference is also much smaller by a factor of 3, as shown in Figure 8.17 [37]. 160
Figure 8.17. Flight Test Multi-Band Antenna Difference in Vertical Protection Level using 100-s and 30-s Smoothed Pseudoranges (from [37]).
Finally, the impact on the CCD monitor for the multi-band antenna is also well below 5 mm/s, as shown in Figure 8.18.
161
Figure 8.18. Flight Test Multi-Band Antenna CCD Monitor Response (from [37])
In conclusion, the section highlighted the impact of AGDV on the airborne monitors and thus answers the ninth research question: Do antenna group delay variations impact the airborne integrity monitors?
Flight experiment data and simulations were used to analyze the effect of antenna group delay variations on the DSIGMA Range monitor, the CCD monitor, and the dual- solution vertical bias parameter. AGDV affects the difference between 30-s and 100-s smoothed pseudoranges, and thus, affects the DSIGMA Range monitor. The effect on the vertical bias parameter is also observable, which, in turn increases the VPL. Based on the analysis presented in this paper, it was found that the DO-301 [2] group delay variation bound requirement is necessary, but not sufficient for the analysis of 휎푎푖푟 and DSIGMA; 162 specifically, additional information is needed regarding the variation of antenna group delay as a function of azimuth and elevation angles, as well as a characterization of the delay for arrival angles below the antenna horizon frame. The latter can occur during aircraft bank angles that are larger than the satellite elevation angle in the antenna horizon frame. The antenna group delay variation model depicted in Figure 6.3 could be adopted for this purpose [37].
For AGDV errors induced in 푃푑푖푓푓, the difference between the 30-s and 100-s smoothed pseudorange, peak approximately 40 s after an aircraft heading change. During flight tests, the worst-case differences reached up to 0.9 m for a single frequency L1 antenna. Once the aircraft is stabilized on final approach for 100-300 s, the effect of antenna group delay variations are below 0.1 m. Based on the flight test presented in this dissertation, the threshold for DSIGMA Range should be able to accommodate the errors induced by the antenna group delay variations. The CCD monitor is much less sensitive to antenna group delay variations with monitor responses below 5 mm/s. If a short final approach is required using GBAS, then the aircraft might only be in stable flight for 30 s before reaching the runway. In this case, the increase in DSIGMA range should be further investigated, because the aircraft will be banking shortly before the final approach, satellites will be below the aircraft body frame, which is a region that is not specified in terms of antenna performance [37].
163 9. EFFECT OF AGDV ON DUAL FREQUENCY GBAS
Dual frequency GBAS was discussed in Section 3.1.3. Modernized GPS satellites also transmit a civil signal on L5. The combination of signals on two frequencies can help mitigate the effects of ionosphere-related errors. The ionosphere can introduce residual error due to spatial and temporal decorrelation. The spatial errors are due to the difference of delay introduced at the reference station and the aircraft. Nominally, the difference is small and during anomalous ionospheric conditions, monitors mitigate these threats as discussed in Chapter 8. The temporal component is due to the relative motion of the aircraft, ionosphere and the satellites. The error is dependent on the smoothing filter time constant, the velocity of the aircraft and the gradient difference between the ground station and the aircraft.
There are two dual frequency processing approaches: divergence-free and ionosphere-free, which mitigate the effects of error due to spatial and temporal ionospheric effects. Each methodology has its advantages and disadvantages.
9.1. Impact of Dual-Frequency AGDV on the ionosphere-free position solution
Group delay variation results for a dual frequency antenna designed for both L1 and L2 (GPS Link 2 frequency at 1227.6 MHz) was shown in chapter 5 (Future GBAS antenna design will use L1/L5 combination but the concept will be same). Figure 5.4 showed the L1 group delay variations for this antenna in comparison to the single frequency group delay variation bound according to [2].
For dual frequency GBAS, dual frequency antenna L1 and L2 pseudorange measurements are combined to obtain an ionosphere-free pseudorange and uses Equation
9.1 to calculate the ionosphere-free measurement as: 164 푃푅 − 훾푃푅 푃푅 = 퐿2 퐿1 (9.1) 푖표푛표_푓푟푒푒 1 − 훾
2 푓퐿1 where 훾 = 2 ≈ 1.65. Pseudoranges at L1 and L2 are affected by AGDV and when 푓퐿2 pseudorange are combined as in Equation 9.1, the pseudorange errors due to the antenna group delay variations are introduced, the error in the ionosphere-free pseudorange is given by:
Δ푃푅푖표푛표_푓푟푒푒 ≈ 2.5Δ휏퐿1 − 1.5Δ휏퐿2 (9.2)
where Δ휏퐿1 and Δ휏퐿2 are the AGDV at the L1 and L2 frequencies, respectively.
Although the AGDV for the dual-frequency antenna are smaller than those for the single frequency antenna, when the ionosphere-free pseudorange is formed, the error introduced by the group delay variations is a combination of the variations at the L1 and L2 frequencies, which are multiplied by a factor of 2.5 and 1.5, respectively. For example,
Figure 9.1 shows the ionosphere-free measurement error due to the combined group delay variations at the L1 and L2 frequencies.
165
Figure 9.1. Ionosphere-free pseudorange errors due to L1 and L2 group delay variations as a function of azimuth and elevation angles
To further illustrate this point, Figure 9.2 shows the ionosphere-free pseudorange group delay variations relative to the L1 bound [35].
166
Figure 9.2. Dual-frequency ionosphere-free antenna group delay variations versus DO- 301 bounds
The group delay variation biases of Figure 9.1 for an ionosphere-free measurement are implemented in (4.26) which resulted in east, north and vertical position errors as well as timing errors. The typical east and north errors are shown in Figure 9.3.
167 East Error 2 Typical Worst case
0 (m)
-2 0 5 10 15 20 25
North Error 2 Typical Worst case
0 (m)
-2 0 5 10 15 20 25 time(hr)
Figure 9.3. Typical east and north errors due to dual-frequency antenna group delay variations in an ionosphere-free solution
Typical east and north errors are .02 m and .05 m, respectively. The worst case east and north errors are .62 m and .63 m, respectively.
It can be concluded that the combination of L1/L2 AGDV for a dual frequency solution is not as close to the bound as in the case of single frequency AGDV. Figure 9.4 shows the vertical and clock errors, as expected they are correlated but they are also not close to the worst case bound for typical geometries and thus incorporation of AGDV in the VPL for such ionosphere-free measurement can be achieved using an average risk methodology, but further investigation is needed to make a final conclusion, as the dual frequency antenna used for this research is not certified for GAST D.
The corresponding errors in the vertical and clock are shown in Figure 9.4.
168 Vertical Error Typical 2 Worst case
0 (m)
-2 0 5 10 15 20 25
Clock Error Typical 2 Worst case
0 (m)
-2 0 5 10 15 20 25 time(hr)
Figure 9.4. Typical vertical and clock errors due to dual-frequency antenna group delay variations in an ionosphere-free solution
Typical vertical and clock errors are .08 m and .06 m, respectively. The worst case vertical and clock errors are 1.33 m and 1.01 m, respectively.
9.2 Airborne Integrity Monitoring for Dual Frequency GBAS
The multiband antenna data from Chapter 8 is taken and the ionosphere-free and divergence-free measurements are formed. Appendix C explains the measurements in detail. It is noted that for ionosphere-free, the airborne monitor is not needed because the ionosphere delay is removed completely, but divergence-free measurements might still need monitoring.
Figure 9.5 depicts the difference between a 100-s and 30-s smoothed ionosphere- free measurement. The significance of Figure 9.5 is that the ionosphere-free pseudorange removes the effect of delay due to ionosphere but the multipath and AGDV combination have added effects due to combination of measurement on two frequencies. In future dual 169 frequency GBAS the AGDV for L1 and L5 frequency should be characterized and their difference should be analyzed as a major error source for the system.
Figure 9.5. Flight test multi-band antenna difference between 100-s and 30-s smoothed ionosphere-free pseudoranges
Figure 9.6 depicts the difference in the 100-s and 30-s smoothed pseudorange for divergence-free measurement. The effect of the ionosphere is not completely removed when divergence-free measurements are used. The multipath and AGDV for L1 only or L5 only is an important parameter for such measurements. The system using the measurement will have to implement airborne monitors because spatial decorrelation of ionosphere will induce error for such system. The monitor thresholds can be tighter depending on the
AGDV of the antenna used for divergence-free measurements.
170
Figure 9.6. Flight test multi-band antenna difference between 100-s and 30-s smoothed divergence-free pseudoranges
In conclusion, the chapter highlighted the effect of AGDV on future GBAS and answered the tenth research question: How is the future Dual Frequency GBAS affected by antenna group delay variations? Future dual-frequency GBAS performance will be influenced by AGDV. Figures 9.1 and 9.2 show that despite of AGDV for a dual frequency antenna being smaller in magnitude, when used to form ionosphere-free measurement can have significant impact on the performance of future dual frequency GBAS. The typical vertical error due to AGDV for an ionosphere-free measurement is .08 m with standard deviation of 0.2 m (1-sigma) and a maximum value of 0.5 m for typical geometry.
The divergence-free measurement position accuracy will be affected by AGDV for either L1 or L5 frequency and will depend on the future design of the dual frequency antenna. 171 The airborne integrity monitor to mitigate threats due to anomalous ionospheric gradient will not be required for dual frequency based on ionosphere-free measurement but still will be needed for a divergence-free-based system with much tighter bound, depending on the AGDV of the future dual frequency antenna.
172 10. CONCLUSIONS AND RECOMMENDATIONS
A method to characterize AGDV was introduced in this dissertation. The results were similar to the methods used by the RTCA committee [19], which consolidates the fact that AGDV is a major error source that must be included in the overall GBAS error budget.
For GAST C and D precision approach operations, AGDV can be incorporated into the aircraft multipath and noise model given by 휎푝푟,푎푖푟, as long as an antenna model is used as defined in Equation 6.8 or Equation 7.2 with a bound that is smaller than the DO-301 bound shown in Figure 5.4, and the AMM is similar to that for a large commercial aircraft such as a Boeing 787. It was also found that aircraft ionospheric monitors DSIGMA Range and
CCD are not affected by AGDV for final approaches with stable flight for at least 2 minutes.
Future research is recommended to evaluate AGDV for aircraft other than large commercial aircraft, and antennas that do not meet the model in equations 6.8 or 7.2.
Research is also recommended to investigate potential changes in AGDV as a function of time, and antenna malfunctions that affect AGDV.
173 REFERENCES
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178 APPENDIX A: LEAST SQUARES ESTIMATION
Consider the linear system equation given by:
풚 = 푯풙 + 풆 (A.1) where:
풚 is the measurement vector with dimension 풎 × ퟏ
풙 is the unknown vector with dimension 풏 × ퟏ
푯 is the data matrix with dimension 풎 × 풏
풆 is the measurement noise vector with dimension 풎 × ퟏ;
The solution of Equation A.1 can be understood by using a geometric approach, an algebraic approach or using calculus. In this dissertation, the geometric approach is used.
The term 푯풙 can be interpreted as the linear combination of the columns of matrix 퐻. The columns form the column space of 푯. The column space is comprised of the columns and all the linear combination of the columns. In Equation A.1 y is the linear combination of the column of the H matrix. Thus, if y is in column space of 푯 we obtain a solution for x else we can only get an estimate of 풙, written as 풙̂. Since Equation A.1 has measurement noise, y is not known exactly; the best result we can get is an estimate value of x and that is achieved using the concept of least squares. We can write Equation A.1 in terms of estimated values as
풚̂ = 푯풙̂ (A.2)
Then, an error can be defined as
흃 = 풚 − 풚̂ 표푟 흃 = 풚 − 푯풙̂ (A.3)
Thus, the goal is to minimize the error vector so that the best estimate of x can be calculated. 179 To explain the concept, let us assume we have 3 measurement and 2 unknowns. 푯풙 can be written in matrix form as
ℎ11 ℎ12 푥1 푯풙 = (ℎ21 ℎ22) ( ) (A.4) 푥2 ℎ31 ℎ32
which can be written as a combination of the columns of the H matrix.
ℎ11 ℎ21 푯풙 = (ℎ21) 푥1 + (ℎ22) 푥2 (A.5) ℎ31 ℎ32
푯풙 = 퐻1푥1 + 퐻2푥2 (A.6)
Figure A.1 depicts the least squares approach geometrically. If e is zero, y is in the H columns space, and we obtain an exact solution. If e is not zero, the best estimate of x is found by minimizing the distance between y and its projection onto the column space of
H. The minimum distance is the perpendicular distance 흃 which gives the best estimate 풙̂ .
Figure A.1. Geometric Depiction of Least Squares Approach
180 Since the projection 흃 is perpendicular to 푯풙̂, the inner product of the 푯풙̂ and 흃 is equal to zero. Therefore, it can be written in the form of vectors as
(푯풙̂)푇흃 = 0 (A.7)
(푯풙̂)푇(풚 − 푯풙̂) = 0 (A.8)
(풙̂)푇(푯푇풚 − 푯푇푯풙̂) = 0 (A.9)
Which gives
(푯푇풚 − 푯푇푯풙̂) = 0 (A.10)
풙̂ = (푯푇푯)−1푯푇풚 (A.11)
Equation A.11 is the expression for the least squares solution.
181 APPENDIX B: CODE-MINUS-CARRIER
The Pseudorange measurement equation can be written as the sum of true geometric and the error components affecting it:
휌퐿1(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡휌푖) + 퐼퐿1(휃푖, 휑푖) + 푇(휃푖, 휑푖) + 푀(휃푖, 휑푖) + 휓푖 +
휏푔푑,퐿1(휃푖, 휑푖) (B.1) where:
휑 is the azimuth angle of the satellite being tracked;
휃 is the elevation angle of the satellite being tracked;
푟(휃푖, 휑푖) is the true geometric range;
∈푖 is the orbit prediction error;
푐(훿푡휌푖) is the user clock offset from the GPS time for pseudorange;
퐼퐿1(휃푖, 휑푖) is the ionospheric error;
푇(휃푖, 휑푖) is the tropospheric error;
푀(휃푖, 휑푖) is the specular multipath; and
휓푖 is the diffuse multipath.
The carrier phase measurement is written also as a sum of true geometric range and the error sources affecting it as:
푐푝퐿1(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡푐푝,퐿1푖) − 퐼퐿1(휃푖, 휑푖) + 푇(휃푖, 휑푖) + 푚퐿1(휃푖, 휑푖) +
휆 휏 (휃 , 휑 ) + 푁 휆 + 퐿1 휑 (B.2) 푝ℎ,퐿1 푖 푖 퐿1,푖 퐿1 2휋 푖 where:
푚(휃푖, 휑푖) is the specular multipath for carrier phase measurements;
휏푝ℎ(휃푖, 휑푖) is the phase delay; 182 휆퐿1 is the wavelength of the L1 signal;
푁퐿1,푖 is the integer ambiguity for carrier phase measurement at L1 frequency; and
푐(훿푡푐푝푖) is the user clock offset from the GPS time for carrier phase measurements.
The Code-Minus-Carrier (CMC) observation is formed by subtracting B.2 from B.1 and is given by:
퐶푀퐶(휃푖, 휑푖) = 휌퐿1(휃푖, 휑푖) − 푐푝퐿1(휃푖, 휑푖) (B.3)
퐶푀퐶(휃푖, 휑푖) = 푐(훿푡휌푖 − 훿푡푐푝,퐿1푖) + 2 × 퐼퐿1(휃푖, 휑푖) + 푀(휃푖, 휑푖) + 휓푖 − 푚퐿1(휃푖, 휑푖) +
휆 휏 (휃 , 휑 ) − 휏 (휃 , 휑 ) − 푁 휆 − 퐿1 휑 (B.4) 푔푑,퐿1 푖 푖 푝ℎ,퐿1 푖 푖 푖 퐿1 2휋 푖
Dual frequency carrier phase measurements is used to correct for Ionospheric error by using the relationship between ionospheric error at two different frequencies and is given by:
2 푓퐿2 퐼퐿1(휃푖, 휑푖) = 2 2 (퐼퐿2(휃푖, 휑푖) − 퐼퐿1(휃푖, 휑푖)) (B.5) 푓퐿1−푓퐿2
The carrier phase measurement for the L2 frequency is defined by:
푐푝퐿2(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡푐푝,퐿2푖) − 퐼퐿2(휃푖, 휑푖) + 푇(휃푖, 휑푖) + 푚퐿2(휃푖, 휑푖) +
휆 휏 (휃 , 휑 ) + 푁 휆 + 퐿2 휑 (B.6) 푝ℎ,퐿2 푖 푖 퐿2,푖 퐿2 2휋 푖
The difference in the carrier phase measurements is given by
휆 −휆 푐푝 (휃 , 휑 ) − 푐푝 (휃 , 휑 ) = (퐼 (휃 , 휑 ) − 퐼 (휃 , 휑 )) + 퐿1 퐿2 휑 + 푛 + 퐿1 푖 푖 퐿2 푖 푖 퐿2 푖 푖 퐿1 푖 푖 2휋 푖 푗
푚퐿1(휃푖, 휑푖) − 푚퐿2(휃푖, 휑푖) (B.7)
Deriving value of 퐼퐿2(휃푖, 휑푖) − 퐼퐿1(휃푖, 휑푖) from equation (B.7) and inserting in (B.5) gives
휆 −휆 퐼 (휃 , 휑 ) = 훾 × (푐푝 (휃 , 휑 ) − 푐푝 (휃 , 휑 ) − 퐿1 퐿2 휑 − 푛 − 푚 (휃 , 휑 )) (B.8) 퐿1 푖 푖 퐿1 푖 푖 퐿2 푖 푖 2휋 푖 푗 퐿1,퐿2 푖 푖 183 where:
2 푓퐿2 훾 = 2 2 푓퐿1−푓퐿2
The evaluated value of ionospheric delay in equation B.8 is inserted in equation B.4 to form the CMC equation corrected for ionospheric delay and is given by:
퐶푀퐶(휃푖, 휑푖) = 푐(훿푡휌푖 − 훿푡푐푝,퐿1푖) + 2 × 훾 × ( 푐푝퐿1(휃푖, 휑푖) − 푐푝퐿2(휃푖, 휑푖) −
휆 −휆 퐿1 퐿2 휑 − 푛 − 푚 (휃 , 휑 )) + 푀(휃 , 휑 ) + 휓 − 푚 (휃 , 휑 ) + 휏 (휃 , 휑 ) − 2휋 푖 푗 퐿1,퐿2 푖 푖 푖 푖 푖 퐿1 푖 푖 푔푑,퐿1 푖 푖
휆 휏 (휃 , 휑 ) − 푁 휆 − 퐿1 휑 (B.9) 푝ℎ,퐿1 푖 푖 푖 퐿1 2휋 푖
Rearranging (B.9) and putting similar terms together gives
퐶푀퐶(휃푖, 휑푖) = 푐(훿푡휌푖 − 훿푡푐푝,퐿1푖) + 2 × 훾 × ( 푐푝퐿1(휃푖, 휑푖) − 푐푝퐿2(휃푖, 휑푖)) +
푀(휃푖, 휑푖) + 휓푖 − (푚퐿1(휃푖, 휑푖) + 2 × 훾 × 푚퐿1,퐿2(휃푖, 휑푖)) + (휏푔푑,퐿1(휃푖, 휑푖) −
휆 휆 −휆 휏 (휃 , 휑 )) − (푁 휆 + 2 × 훾 × 푛 ) − ( 퐿1 휑 + 2 × 훾 × 퐿1 퐿2 휑 ) (B.10) 푝ℎ,퐿1 푖 푖 퐿1푖 퐿1 푗 2휋 푖 2휋 푖 where:
푛푗 = 푁퐿1,푖휆퐿1 − 푁퐿2,푖휆퐿2;
푁퐿1푖휆퐿1 + 2 × 훾 × 푛푗 is the error due to ambiguity resolution and appears as a bias;
2 × 훾 × ( 푐푝퐿1(휃푖, 휑푖) − 푐푝퐿2(휃푖, 휑푖)) = 0 if it a coherent receiver;
푐(훿푡휌푖 − 훿푡푐푝,퐿1푖) = 0 for a coherent receiver;
푚퐿1(휃푖, 휑푖) + 2 × 훾 × 푚퐿1,퐿2(휃푖, 휑푖) negligible much smaller than code multipath;
and 휏푝ℎ,퐿1(휃푖, 휑푖) negligible.
Must correct for phase wrap up term; the biases will be constant for a satellite track if no loss of lock [34]. Final equation for CMC is given by:
퐶푀퐶푗(휃푖, 휑푖) = 휏푔푑,퐿1(휃푖, 휑푖) + 푀(휃푖, 휑푖) + 휓푖 + 퐵푖푎푠푗 (B.11) 184
APPENDIX C: SPHERICAL HARMONICS
The spherical harmonics is the solution of the Laplace’s equation in spherical coordinates
[56,57]. Spherical coordinates are defined as shown in Figure C.1.
Figure C.1 Spherical Coordinates
A Laplacian of a vector V in spherical coordinates is defined as:
1 휕 휕푉 1 휕 휕푉 1 휕2푉 ∇2푉 = (푟2 ) + (sin 휃 ) + (C. 1) 푟2 휕푟 휕푟 푟2 sin 휃 휕휃 휕휃 푟2푠푖푛2휃 휕휑2
185 When the Laplacian is equal to zero, the separation of variables can be used to find the solution for Equation C.1.
푉(푟, 휃, 휑) = 푅(푟)푃(휃)푄(휑) (C.2)
푟2푠푖푛2휃 Putting 푉 = 푅 × 푃 × 푄 in equation C.1 and dividing the resulting equation by will 푅×푃×푄 result in equation C.3.
1 휕2푄 푠푖푛2휃 휕 휕푅 sin 휃 휕 휕푃 = − (푟2 ) − (sin 휃 ) = −푚2 (C. 3) 푄 휕휑2 푅 휕푟 휕푟 푃 휕휃 휕휃
The differential equation can be solved for Q to find the general solution
푄(휑) = 퐴푚 cos 푚휑 + 퐵푚 sin 푚휑 , 푚 = 0,1,2 …. (C.4)
Now, from equation C.3 we can write
푠푖푛2휃 휕 휕푅 sin 휃 휕 휕푃 (푟2 ) + (sin 휃 ) = 푚2 푅 휕푟 휕푟 푃 휕휃 휕휃
1 휕 휕푅 1 휕 휕푃 푚2 (푟2 ) + (sin 휃 ) = (C. 5) 푅 휕푟 휕푟 푃 sin 휃 휕휃 휕휃 푠푖푛2휃
Rearranging C.5 gives
1 휕 휕푅 푚2 1 휕 휕푃 (푟2 ) = − (sin 휃 ) = 푙(푙 + 1) (C. 6) 푅 휕푟 휕푟 푠푖푛2휃 푃 sin 휃 휕휃 휕휃
1 푑푅 1 푑2푅 (2푟 ) + (푟2 ) = 푙(푙 + 1) (C. 7) 푅 푑푟 푅 푑푟2
The equation C.7 is known as Euler’s equation and the solution is given by
푙 푅(푟) = { 푟 (C.8) 푟−푙−1
Using equation (C.6) and solving for 휃 gives, 186 푚2 1 휕 휕푃 − (sin 휃 ) = 푙(푙 + 1) (C. 9) 푠푖푛2휃 푃 sin 휃 휕휃 휕휃
Rearranging C.9 results in,
1 푑 푑푃 푚2 (sin 휃 ) + [푙(푙 + 1) − ] 푃 = 0 (C. 10) sin 휃 푑휃 푑휃 푠푖푛2휃
Let 푥 = cos 휃 and 푃(휃) = 푍(푥) = 푍(푐표푠휃) and write C.10 in terms on Z and x.
The resulting equation will be
휕2푍 푑푧 푚2 (1 − 푥2) − 2푥 + [푙(푙 + 1) − ] 푍 = 0 (C. 11) 휕푥2 푑푥 (1 − 푥2)
The equation C.11 is called Associated Legendre Differential Equation and the solution is given by
푚 푍(푥) = 푃푙 (푥) 푙 = 0,1,2 … 푚 = 0 … 푙 (C.12)
Where:
푚 푃푙 (푥) are the associated Legendre polynomials;
The final solution can be written as combination of equations C.4, C.8, C.12 as is given by
푙 푉(푟, 휃, 휑) = { 푟 [퐴 푐표푠 푚휑 + 퐵 푠푖푛 푚휑] 푃푚(푐표푠휃) (C.13) 푟−푙−1 푚 푚 푙 where:
l = 0,1,2 … m = 0 … l
We can also write C.13 in terms of spherical harmonics, the real spherical harmonics is given by
(2푙+1)(푙−푚)! 푌푚(휃, 휑) = (−1)푚√ [퐴 푐표푠 푚휑 + 퐵 푠푖푛 푚휑]푃푚(푐표푠휃) (C.14) 푙 4휋 (푙+푚)! 푚 푚 푙 187 Thus, substituting C.14 into C.13 results in
푙푚푎푥 푙 푙 −푙−1 푚 푉(푟, 휃, 휑) = ∑푙=0 ∑푚=0( 퐶푟 + 퐷푟 )푌푙 (휃, 휑) (C.15)
The dissertation is only interested in function called Group Delay as a function of azimuth and elevation angle, thus 푃(휃, 휑) of the solution will be important. Spherical
Harmonics is a complete orthogonal system as sin 푛푥 푎푛푑 cos 푛푥 used for Fourier series.
According to the completeness property of the spherical harmonics any continuous function of θ and φ can be written as a sum of spherical harmonics. The series is a generalized Fourier series and known as Laplace Series.
In terms of real spherical harmonics, it is given by [56]
푙푚푎푥 푙 푚 푚 푚 푚 푃(휃, 휑) = ∑푙=0 ∑푚=0 푁푙 [퐴푙 푐표푠 푚휑 + 퐵푙 푠푖푛 푚휑]푃푙 (푐표푠휃) (퐶. 16) where:
(2푙+1)(푙−푚)! 푁푚 = (−1)푚√ is the normalization factor; 푙 is the degree and 푚 is the 푙 4휋 (푙+푚)! order of the spherical harmonics.
휋 In the dissertation, the elevation angle to the satellite 푒 = − 휃 and the group delay 2 variation of antenna is a function of azimuth and elevation angle of the satellite and thus equation C.16 becomes
푙푚푎푥 푙 푚 푚 푚 푚 푃(푒, 휑) = ∑푙=0 ∑푚=0 푁푙 [퐴푙 푐표푠 푚휑 + 퐵푙 푠푖푛 푚휑]푃푙 (푠푖푛(푒)) (C.17) which can be rewritten using m=0;
푙푚푎푥 0 푙푚푎푥 푙 P(푒, 휑) = ∑푙=0 푁푙,0퐴푙,0푃푙 sin(휃) + ∑푙=1 ∑푚=1(푁푙,푚(퐴푙,푚 cos(푚휑) +
푚 퐵푙,푚sin (푚휑))푃푙 sin (푒) ) (C.18)
Equation C.18 will be used in section 6.2 for modeling of AGDV. 188
APPENDIX D: DUAL FREQUENCY GBAS MEASURMENTS
The dual frequency GBAS architecture takes advantage of dual frequency measurements to mitigate the effects of error due to ionosphere which is a major error source for GBAS.
A combination of code phase and carrier phase measurements will be used to form divergence-free and ionosphere-free pseudoranges which will be used to calculate position and protection levels. The calculation of the position solution and protection levels require the understanding of all the error sources affecting the measurements. The divergence-free and ionosphere-free measurements will be derived in this Appendix and discern if AGDV has any effect on the new measurement or not.
The modernized GPS will have signals on the L5 frequency (1176.45 MHz) in addition to the L1 frequency. A combination of measurements on these two frequencies will be used in future GBAS to mitigate the effect of ionosphere introduced delay, which is a major error source for single frequency GBAS. Next the ionosphere-free and divergence-free measurement formation will be explained using the concept of code and carrier measurement explained in appendix B.
The code phase and carrier phase measurements at the L1 frequency was defined in appendix B by equations B.1 and B.2.
2 2 푓퐿5 1 푓퐿1 Let 훾 = 2 2 and 휅 = − , 1 − 2 푓퐿1−푓퐿5 훾 푓퐿5
The carrier phase measurement for the L5 frequency can be written as the sum of the true geometric range and the errors affecting it and is defined by: 189 푐푝퐿5(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡푐푝,퐿5푖) − 퐼퐿5(휃푖, 휑푖) + 푇(휃푖, 휑푖) + 푚퐿5(휃푖, 휑푖) +
휆 휏 (휃 , 휑 ) + 푁 휆 + 퐿5 휑 (D.1) 푝ℎ,퐿5 푖 푖 퐿5,푖 퐿5 2휋 푖
The difference in the carrier phase measurements equations (B.2) and (D.1) gives
휆 −휆 푐푝 (휃 , 휑 ) − 푐푝 (휃 , 휑 ) = (퐼 (휃 , 휑 ) − 퐼 (휃 , 휑 )) + 퐿1 퐿5 휑 + 푁 휆 − 퐿1 푖 푖 퐿5 푖 푖 퐿5 푖 푖 퐿1 푖 푖 2휋 푖 퐿1,푖 퐿1
푁퐿5,푖휆퐿5 + 푚퐿1(휃푖, 휑푖) − 푚퐿5(휃푖, 휑푖) (D.2)
Dual frequency carrier phase measurements can be used to correct for Ionospheric error by using the relationship between ionospheric error at two different frequencies and is given by
2 푓퐿5 퐼퐿1(휃푖, 휑푖) = 2 2 (퐼퐿1(휃푖, 휑푖) − 퐼퐿5(휃푖, 휑푖)) (D.3) 푓퐿5−푓퐿1
D.1 Divergence-Free Smoothing
Divergence-Free Smoothing utilizes the code phase measurement on L1 as it is, but the carrier phase measurement is changed to
2 푐푝 (휃 , 휑 ) = 푐푝 (휃 , 휑 ) − ( 푐푝 (휃 , 휑 ) − 푐푝 (휃 , 휑 )) (D.4) 퐷퐹 푖 푖 퐿1 푖 푖 휅 퐿1 푖 푖 퐿5 푖 푖 which can be written as:
푐푝퐷퐹(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡푐푝,퐿1푖) − 퐼퐿1(휃푖, 휑푖) + 푇(휃푖, 휑푖) + 푚퐿1(휃푖, 휑푖) +
휆 2 휆 −휆 휏 (휃 , 휑 ) + 푁 휆 + 퐿1 휑 − ((퐼 (휃 , 휑 ) − 퐼 (휃 , 휑 )) + 퐿1 퐿5 휑 + 푛 + 푝ℎ,퐿1 푖 푖 퐿1,푖 퐿1 2휋 푖 휅 퐿5 푖 푖 퐿1 푖 푖 2휋 푖 푗
푚퐿1(휃푖, 휑푖) − 푚퐿5(휃푖, 휑푖) ) (D.5) where:
푛푗 = 푁퐿1,푖휆퐿1 − 푁퐿5,푖휆퐿5
Substituting D.3 into D.5 we get 190 푐푝퐷퐹(휃푖, 휑푖) = (푟(휃푖, 휑푖) +∈푖+ 푐(훿푡푐푝,퐿1푖) − 퐼퐿1(휃푖, 휑푖) + 푇(휃푖, 휑푖) + 푚퐿1(휃푖, 휑푖) +
휆 2 휆 −휆 휏 (휃 , 휑 ) + 푁 휆 + 퐿1 휑 ) − 2(−퐼 (휃 , 휑 )) − ( 퐿1 퐿5 휑 + 푛 + 푝ℎ,퐿1 푖 푖 퐿1,푖 퐿1 2휋 푖 퐿1 푖 푖 휅 2휋 푖 푗
푚퐿1(휃푖, 휑푖) − 푚퐿5(휃푖, 휑푖) )
푐푝퐷퐹(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡푐푝,퐿1푖) + 퐼퐿1(휃푖, 휑푖) + 푇(휃푖, 휑푖) + 푚퐿1(휃푖, 휑푖) +
휆 2 휆 −휆 휏 (휃 , 휑 ) + 푁 휆 + 퐿1 휑 − ( 퐿1 퐿5 휑 + 푛 + 푚 (휃 , 휑 ) − 푚 (휃 , 휑 ) (D.6) 푝ℎ,퐿1 푖 푖 퐿1,푖 퐿1 2휋 푖 휅 2휋 푖 푗 퐿1 푖 푖 퐿5 푖 푖
Code minus carrier is calculated using B.1 and D.6
퐶푀퐶퐷퐹(휃푖, 휑푖) = 푐(훿푡휌,퐿1푖) − 푐(훿푡푐푝,퐿1푖) + 푀퐿1(휃푖, 휑푖) + 휓푖 − 푚퐿1(휃푖, 휑푖) +
2 2 (푚 (휃 , 휑 ) − 푚 (휃 , 휑 ) ) + 휏 (휃 , 휑 ) − 휏 (휃 , 휑 ) − 푁 휆 + (푛 ) − 휅 퐿1 푖 푖 퐿5 푖 푖 푔푑,퐿1 푖 푖 푝ℎ,퐿1 푖 푖 퐿1,푖 퐿1 휅 푗
휆 2 휆 −휆 퐿1 휑 + ( 퐿1 퐿5 휑 ) (D.7) 2휋 푖 휅 2휋 푖
The code minus carrier is passed through the smoothing filter and added to carrier phase measurement for divergence-free measurement in D.6. It can be assumed at this point that the phase wrap-up has been corrected. The pseudorange for divergence free smoothing methodology is given by
휌퐷퐹(휃푖, 휑푖) = 푐푝퐷퐹(휃푖, 휑푖) + 퐶푀퐶푠푚표표푡ℎ,퐷퐹(휃푖, 휑푖)
휌퐷퐹(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡푐푝,퐿1푖) + 퐼퐿1(휃푖, 휑푖) + 푇(휃푖, 휑푖) + 푚퐿1(휃푖, 휑푖) +
휆 2 휆 −휆 휏 (휃 , 휑 ) + 푁 휆 + 퐿1 휑 − ( 퐿1 퐿5 휑 + 푛 + 푚 (휃 , 휑 ) − 푚 (휃 , 휑 ) ) + 푝ℎ,퐿1 푖 푖 퐿1,푖 퐿1 2휋 푖 휅 2휋 푖 푗 퐿1 푖 푖 퐿5 푖 푖
2 푆 × (푐(훿푡 ) − 푐(훿푡 ) + 푀 (휃 , 휑 ) + 휓 − 푚 (휃 , 휑 ) + (푚 (휃 , 휑 ) − 휌,퐿1푖 푐푝,퐿1푖 퐿1 푖 푖 푖 퐿1 푖 푖 휅 퐿1 푖 푖
2 휆 푚 (휃 , 휑 ) ) + 휏 (휃 , 휑 ) − 휏 (휃 , 휑 ) − 푁 휆 + (푛 ) − 퐿1 휑 + 퐿5 푖 푖 푔푑,퐿1 푖 푖 푝ℎ,퐿1 푖 푖 퐿1,푖 퐿1 휅 푗 2휋 푖
2 휆 −휆 ( 퐿1 퐿5 휑 )) (D.8) 휅 2휋 푖
The smoothing will affect multipath term, and the final pseudorange is given by: 191 휌퐷퐹(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡휌,퐿1푖) + 퐼퐿1(휃푖, 휑푖) + 푇(휃푖, 휑푖) + 푚퐿1(휃푖, 휑푖) −
2 (푚 (휃 , 휑 ) − 푚 (휃 , 휑 ) ) + 푆 × (푀 (휃 , 휑 ) + 휓 − 푚 (휃 , 휑 ) + 휅 퐿1 푖 푖 퐿5 푖 푖 퐿1 푖 푖 푖 퐿1 푖 푖
2 (푚 (휃 , 휑 ) − 푚 (휃 , 휑 ) )) + 휏 (휃 , 휑 ) (D.9) 휅 퐿1 푖 푖 퐿5 푖 푖 푔푑,퐿1 푖 푖 where:
S is smoothing factor;
Thus, the AGDV of the dual frequency antenna at L1 frequency is an important factor for divergence-free smoothing and the ionospheric gradient will still introduce error because through divergence-free smoothing technique we have only removed the effect of the smoothing filter on the error introduced in single frequency GBAS.
D.2 Ionosphere-Free Smoothing
Ionosphere-Free Smoothing adjusts both the code and carrier so that effect of ionosphere can be mitigated to the first order. The input to the filter is defined by D.10 and D.11,
1 휌 (휃 , 휑 ) = 휌 (휃 , 휑 ) − ( 휌 (휃 , 휑 ) − 휌 (휃 , 휑 )) (D.10) 퐼퐹 푖 푖 퐿1 푖 푖 휅 퐿1 푖 푖 퐿5 푖 푖
1 푐푝 (휃 , 휑 ) = 푐푝 (휃 , 휑 ) − ( 푐푝 (휃 , 휑 ) − 푐푝 (휃 , 휑 )) (D.11) 퐼퐹 푖 푖 퐿1 푖 푖 휅 퐿1 푖 푖 퐿5 푖 푖 where:
휌퐿5(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡휌,퐿5푖) + 퐼퐿2(휃푖, 휑푖) + 푇(휃푖, 휑푖) + 푀퐿5(휃푖, 휑푖) +
휏푔푑,퐿2(휃푖, 휑푖) (D.12)
휌퐼퐹(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡휌,퐿1푖) + 푇(휃푖, 휑푖) + 푀퐿1(휃푖, 휑푖) + 휓퐿1푖 +
1 휏 (휃 , 휑 ) − ( 푀 (휃 , 휑 ) − 푀 (휃 , 휑 ) + 휓 − 휓 + 휏 (휃 , 휑 ) − 푔푑,퐿1 푖 푖 휅 퐿1 푖 푖 퐿5 푖 푖 퐿1푖 퐿5푖 푔푑,퐿1 푖 푖
휏푔푑,퐿5(휃푖, 휑푖)) (D.13) 192 푐푝퐼퐹(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡푐푝,퐿1푖) + 푇(휃푖, 휑푖) + 푚퐿1(휃푖, 휑푖) + 휏푝ℎ,퐿1(휃푖, 휑푖) +
휆 1 휆 −휆 푁 휆 + 퐿1 휑 − ( 퐿1 퐿5 휑 + 푛 + 푚 (휃 , 휑 ) − 푚 (휃 , 휑 ) ) (D.14) 퐿1,푖 퐿1 2휋 푖 휅 2휋 푖 푗 퐿1 푖 푖 퐿5 푖 푖
퐶푀퐶퐼퐹(휃푖, 휑푖) = 푐(훿푡휌,퐿1푖) − 푐(훿푡푐푝,퐿1푖) + 푀퐿1(휃푖, 휑푖) + 휓퐿1푖 − 푚퐿1(휃푖, 휑푖) +
휆 1 휏 (휃 , 휑 ) − 휏 (휃 , 휑 ) − 푁 휆 − 퐿1 휑 − ( 푀 (휃 , 휑 ) − 푀 (휃 , 휑 ) + 푔푑,퐿1 푖 푖 푝ℎ,퐿1 푖 푖 퐿1,푖 퐿1 2휋 푖 휅 퐿1 푖 푖 퐿5 푖 푖
1 휆 −휆 휓 − 휓 + 휏 (휃 , 휑 ) − 휏 (휃 , 휑 )) + ( 퐿1 퐿5 휑 + 푛 + 푚 (휃 , 휑 ) − 퐿1푖 퐿5푖 푔푑,퐿1 푖 푖 푔푑,퐿5 푖 푖 휅 2휋 푖 푗 퐿1 푖 푖
푚퐿5(휃푖, 휑푖) ) (D.15)
The code minus carrier is passed through the smoothing filter and added to the carrier phase measurement for the divergence-free measurement in D.14. It can be assumed at this point that the phase wrap- up has been corrected. The pseudorange for divergence-free smoothing methodology is given by:
휌퐼퐹(휃푖, 휑푖) = 푐푝퐼퐹(휃푖, 휑푖) + 퐶푀퐶푠푚표표푡ℎ,퐼퐹(휃푖, 휑푖)
휌퐼퐹(휃푖, 휑푖) = 푟(휃푖, 휑푖) +∈푖+ 푐(훿푡푐푝,퐿1푖) + 푇(휃푖, 휑푖) + 푚퐿1(휃푖, 휑푖) + 휏푝ℎ,퐿1(휃푖, 휑푖) +
휆 1 휆 −휆 푁 휆 + 퐿1 휑 − ( 퐿1 퐿5 휑 + 푛 + 푚 (휃 , 휑 ) − 푚 (휃 , 휑 ) ) + 푆 × (푐(훿푡 ) − 퐿1,푖 퐿1 2휋 푖 휅 2휋 푖 푗 퐿1 푖 푖 퐿5 푖 푖 휌,퐿1푖
푐(훿푡푐푝,퐿1푖) + 푀퐿1(휃푖, 휑푖) + 휓퐿1푖 − 푚퐿1(휃푖, 휑푖) + 휏푔푑,퐿1(휃푖, 휑푖) − 휏푝ℎ,퐿1(휃푖, 휑푖) −
휆 1 푁 휆 − 퐿1 휑 − ( 푀 (휃 , 휑 ) − 푀 (휃 , 휑 ) + 휓 − 휓 + 휏 (휃 , 휑 ) − 퐿1,푖 퐿1 2휋 푖 휅 퐿1 푖 푖 퐿5 푖 푖 퐿1푖 퐿5푖 푔푑,퐿1 푖 푖
1 휆 −휆 휏 (휃 , 휑 )) + ( 퐿1 퐿5 휑 + 푛 + 푚 (휃 , 휑 ) − 푚 (휃 , 휑 ) )) (D.16) 푔푑,퐿5 푖 푖 휅 2휋 푖 푗 퐿1 푖 푖 퐿5 푖 푖
The smoothing will affect the multipath term, and the final pseudorange is given by:
1 휌 (휃 , 휑 ) = 푟(휃 , 휑 ) +∈ + 푐(훿푡 ) + 푇(휃 , 휑 ) + 푁 휆 − (푁 휆 − 퐼퐹 푖 푖 푖 푖 푖 푐푝,퐿1푖 푖 푖 퐿1,푖 퐿1 휅 퐿1,푖 퐿1
1 푁 휆 ) + 푆 × (푀 (휃 , 휑 ) + 휓 − (푀 (휃 , 휑 ) + 휓 − 푀 (휃 , 휑 ) − 휓 ) + 퐿2,푖 퐿2 퐿1 푖 푖 퐿1푖 휅 퐿1 푖 푖 퐿1푖 퐿5 푖 푖 퐿5푖
1 휏 (휃 , 휑 ) − (휏 (휃 , 휑 ) − 휏 (휃 , 휑 ))) (D.17) 푔푑,퐿1 푖 푖 휅 푔푑,퐿1 푖 푖 푔푑,퐿2 푖 푖 193 Thus, we see the difference of AGDV at L1 and L2 is important. Thus, the ionospheric affects are completely removed when the ionosphere-free algorithm is used, but the noise on the pseudorange will also increase because of the combination of multipath and
AGDV on L1 and L2 frequencies.
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