PERFORMANCE IMPROVEMENT METHODS FOR TERRAIN DATABASE

INTEGRITY MONITORS AND TERRAIN REFERENCED NAVIGATION

A thesis presented to

the Faculty of the

Fritz J. and Dolores H. Russ

College of Engineering and Technology

of

Ohio University

In partial fulfillment

of the requirements for the degree

Master of Science

Ananth Kalyan Vadlamani

March 2004 This thesis entitled

PERFORMANCE IMPROVEMENT METHODS FOR TERRAIN DATABASE INTEGRITY

MONITORS AND TERRAIN REFERENCED NAVIGATION

BY

ANANTH KALYAN VADLAMANI

has been approved for

the School of Electrical Engineering and Computer Science

and the Russ College of Engineering and Technology by

Maarten Uijt de Haag

Assistant Professor of Electrical Engineering and Computer Science

R. Dennis Irwin

Dean, Russ College of Engineering and Technology VADLAMANI, ANANTH K. M.S. March 2004. Electrical Engineering and Computer Science

Performance Improvement Methods for Terrain Database Integrity Monitors and Terrain Referenced Navigation (115pp.)

Director of Thesis: Maarten Uijt de Haag

Terrain database integrity monitors and terrain-referenced navigation systems are based on performing a comparison between stored terrain elevations with data from airborne sensors like radar altimeters, inertial measurement units, GPS receivers etc. This thesis introduces the concept of a spatial terrain database integrity monitor and discusses methods to improve its performance. Furthermore, this thesis discusses an improvement of the terrain-referenced aircraft position estimation for aircraft navigation using only the information from downward-looking sensors and terrain databases, and not the information from the inertial measurement unit. Vertical and horizontal failures of the terrain database are characterized. Time and frequency domain techniques such as the Kalman filter, the autocorrelation function and spectral estimation are designed to evaluate the performance of the proposed integrity monitor and position estimator performance using flight test data from Eagle/Vail, CO, Juneau, AK, Asheville, NC and Albany, OH.

Approved: Maarten Uijt de Haag Assistant Professor of Electrical Engineering and Computer Science To my parents

ACKNOWLEDGEMENTS

During my study at Ohio University, I’ve worked on many projects, as part of the coursework under the guidance of my teachers, but none was as challenging or as extensive as this one: the Masters’ thesis. Challenging, because it led me to explore and understand concepts as an engineer and I wish to thank everyone who have helped me during the course of this research.

I express my sincere thanks to my advisor, Dr. Maarten Uijt de Haag, who got me involved in research and encouraged me at every stage. Thank you Maarten, for your patience, attention and the faith that motivated me to go on. Also, for the numerous reviews and inputs, in the past for various conference papers and now for this thesis, which have helped to make this work meaningful. We have brought many a project to fruition and I’m sure we will continue to do so in future.

I am grateful to my thesis committee members Dr. Michael Braasch, Dr. Frank van Graas and Dr. William (Gene) Kaufman for their time and effort in reviewing my thesis and their useful comments. I thank Dr. Braasch for the great learning experience during all the courses I took with him these last two and a half years that have contributed immensely to my understanding and my research. I thank Dr. van Graas for introducing and laying a solid foundation to the concepts that will remain with me throughout my career.

I thank Jacob Campbell for a lot of things: patiently explaining his thesis to me, providing me with the data and some initial routines to work on, for the useful discussions that helped me burst through a plateau phase in my research, for his support and for simply being there, so I knew I could run up to him in case of problems.

I thank Steve Young for his useful inputs during conferences and Dr. Robert Gray for his initial research that got it all started. I am thankful to the NASA B757 ARIES flight crew for their support and expertise during the EGE flight trials. I am deeply grateful to the Ohio University King-Air C90 pilots Brian Branham and Jamie Edwards for the flight tests at JNU, also to Dr. Richard McFarland for the flight tests at AVL and KUNI conducted on the Ohio University DC3 and the chief of airborne laboratories, Jay Clark, for his help and support during the said flight- testing. Support from the terrain database providers: NIMA, NGS, and Jeppesen is greatly appreciated. The research presented in this thesis was supported and funded through NASA under Cooperative Agreement NCC-1-3511.

I thank the faculty, staff and all my colleagues at the Avionics Engineering Center who have all been part of this learning experience. In fact, I believe with all our interactions, I have learned as much in the break-room, hallways and student offices at AEC, as I have in formal coursework. It’s a great place with great people. I thank all my friends who have supported, tolerated, and motivated me, who have heard me out, who have set examples in all aspects of life and from whom I learn constantly.

And most of all, I thank my parents, Ramanand and Kanaka Durga Vadlamani, for their love and encouragement and for making me the person I am. Although words cannot express my gratitude, I owe everything to them, my brother, Ravi, who has been there to share my joys and sorrows and my family who have constantly supported me. It is wonderful to be amongst you.

1 Opinions, interpretations, conclusions and recommendations are those of the author and are not necessarily endorsed by NASA 7

TABLE OF CONTENTS

Abstract...... 3 Acknowledgements ...... 5 List of Tables ...... 9 List of Figures...... 10 List of Acronyms...... 13 1 Introduction...... 15 1.1 Contributions ...... 16 1.2 Outline of the Thesis ...... 17 2 Synthetic Vision Systems ...... 18 2.1 SVS Component Description ...... 21 2.1.1 Sensors...... 22 2.1.2 Terrain Databases ...... 22 2.1.3 Displays ...... 23 2.2 SVS Predecessors ...... 24 2.2.1 Ground Proximity Warning Systems (GPWS) ...... 24 2.2.2 Terrain Awareness and Warning System (TAWS) ...... 24 3 Terrain Database Integrity Monitor ...... 26 3.1 Statistical Method...... 30 3.1.1 Formulation of Hypotheses...... 31 3.1.2 Test Statistic for Decision Making...... 33 3.1.3 Pseudo-Random Noise Analysis ...... 36 3.2 Vertical Domain Integrity Monitor...... 37 8 3.3 Horizontal Domain Integrity Monitor ...... 38 3.4 Spatial Integrity Monitor ...... 41 4 Terrain Referenced Navigation...... 43 5 Theoretical Background ...... 52 5.1 The Kalman Filter...... 55 5.1.1 Test Statistic Using Kalman Estimates ...... 58 5.1.2 Pseudo-Random Noise Analysis Revisited...... 60 5.2 Autocorrelation Function Estimation ...... 62 5.3 Modern Spectral Estimation ...... 65 5.3.1 Blackman – Tukey Spectral Estimation ...... 66 5.3.2 Maximum Entropy Spectral Estimation...... 66 6 Spatial Failure Detection and Estimation ...... 70 6.1 EGE Flight Test and System Description...... 71 6.1.1 NASA ARIES Triple Redundant Radar Altimeter ...... 72 6.1.2 Slant Range Errors ...... 72 6.1.3 Altitude Dependent Errors...... 74 6.1.4 Mid-Value Select...... 74 6.2 Terrain Database Integrity Monitor...... 75 6.2.1 Vertical Domain Integrity Monitor...... 76 6.2.2 Horizontal Domain Integrity Monitor ...... 78 6.2.3 Spatial Integrity Monitor ...... 81 6.3 Application to Terrain Referenced Navigation...... 82 6.3.1 Lateral Position Estimation...... 83 6.3.2 Vertical Bias Estimation ...... 88 6.3.3 Spatial Position Estimation ...... 89 7 Additional Flight Test Results...... 92 7.1 Flight Test Environment...... 93 7.1.1 JNU Flight Test...... 93 7.1.2 AVL Flight Test...... 93 7.1.3 KUNI Flight Test ...... 94 7.2 Terrain Database Integrity Monitor...... 94 7.2.1 Vertical Domain Integrity Monitor...... 95 7.2.2 Horizontal Domain Integrity Monitor ...... 95 7.2.3 Spatial Integrity Monitor ...... 95 7.3 Applications to Terrain Referenced Navigation...... 99 7.3.1 Lateral Position Estimation...... 99 7.3.2 Vertical Bias Estimation ...... 101 7.3.3 Spatial Position Estimation ...... 102 8 Summary and Conclusions...... 104 9 The Road Ahead...... 107 References...... 109 Appendix...... 113 A.1 Terrain Database Specifications ...... 113 A.2 Kalman Filter Design Parameters...... 114 9

LIST OF TABLES

Table 3.1 Decision Making in Hypothesis Testing ...... 32 Table 3.2 Over-Bounded Absolute Disparity Distributions for Sensors and Data...... 32 Table 4.1 A comparison of Several TRN Systems...... 51 Table 6.1 Mean E-N Coordinates of Horizontal Position Estimates...... 84 Table 6.2 Mean E-N Coordinates of Horizontal Position Estimates of Figure 6.13 ...... 85 Table 6.3 Mean E-N Coordinates of Horizontal Position Estimates of Figure 6.14 ...... 85 Table 6.4 Mean E-N Coordinates of Horizontal Position Estimates of Figure 6.15 ...... 87 Table 7.1 Mean E-N Coordinates of Horizontal Position Estimates of Figure 7.6 ...... 99 Table 7.2 Mean E-N Coordinates of Horizontal Position Estimates of Figure 7.7 ...... 100 Table 7.3 Mean E-N Coordinates of Horizontal Position Estimates of Figure 7.8 ...... 101 Table 7.4 Mean E-N Coordinates of Horizontal Position Estimates of Figure 7.10 ...... 103 Table A.1 Terrain Database Specifications...... 113

10

LIST OF FIGURES

Figure 2.1 – SVS Concept Architecture Overview...... 21 Figure 2.2 – Depiction of (a) Linear Error and (b) Circular Error in the DEM ...... 23 Figure 3.1 – Sample Synthetic Vision System Fault Tree ...... 29 Figure 3.2 – Measurement of Synthesized Terrain Profile ...... 30

Figure 3.3 – Distribution of T for H0 and H1...... 34 Figure 3.4 – Threshold value for chi-square statistic...... 35 Figure 3.5 – Non-Centrality parameter look-up plot ...... 35 Figure 3.6 – Operating Characteristics (OC) Curve ...... 36 Figure 3.7 – T Statistic values for Approaches to R/W 25 at EGE...... 37 Figure 3.8 – T value for 0, 25m and 35m biases...... 38 Figure 3.9 – Region of Missed Detection within –20 to 20 Resolution Grid in Latitude and Longitude ...... 40 Figure 3.10 – Space Envelope of Missed Detection ...... 41 Figure 3.11 – Sample Space Envelope Plot at EGE ...... 42 Figure 4.1 – The TERCOM System ...... 44 Figure 4.2 – Measurements and Terrain Correlation in the TERCOM System...... 44 Figure 4.3 – The SITAN System ...... 46 Figure 4.4 – Frequency Domain Terrain Correlation Scheme of [37]...... 48 11

Figure 5.1 – P and Q as a function of µB ...... 59

Figure 5.2 – Autocorrelation Sequence of Kalman Estimates for µB of 34.2m, 25m and 15m...... 60

Figure 5.3 – Correlation Coefficient (ρ) of Kalman Estimates for µB of 34.2m, 25m and 15m ....61

Figure 5.4 – OC Curve of Kalman estimates for µB of 34.2m, 25m and 15m ...... 61 Figure 5.5 – Autocorrelation Function of a WSS Random Process...... 62 Figure 5.6 – Autocorrelation of finite length constant signal ...... 63 Figure 5.7a, 5.7b – Autocorrelation of 50 sample length Gaussian noise having mean 15 and σ = 18.9...... 63

Figure 5.8 – Rxx(0) Estimation Using a Straight Line Fit ...... 64 Figure 5.9a, 5.9b – Mean and Standard Deviation of Bias Estimate ...... 65 Figure 5.10 – Spectral Estimates Using Blackman-Tukey and Maximum Entropy Methods ...... 67 Figure 5.11a – Zero Frequency Power Estimates for Various Vertical Biases...... 68 Figure 5.11b – Higher Frequency Power Estimates for Various Vertical Biases...... 68 Figure 6.1 – Runway 07 and 25 Approaches and Departures at EGE ...... 71 Figure 6.2 – Illustration of Plumb-Bob Height and Slant Range...... 73 Figure 6.3 – Illustration of Altitude Dependent Errors...... 74 Figure 6.4 – Mid-Value Select Scheme ...... 75 Figure 6.5 – Different Test Statistics During Approach to R/W 25 at EGE for Biases of 0, 25m and 35m...... 77 Figure 6.6 – Augmentation Scheme for DEM Integrity Monitor ...... 78 Figure 6.7 – Improvement in RoMD within –20 to 20 Resolution Grid Points in Latitude and Longitude ...... 79 Figure 6.8 – Improvement in RoMD at Locations shown in Figure 3.9...... 80 Figure 6.9 – RoMD during Entire Flight Path at EGE...... 81 Figure 6.10 – Improvement/Reduction in the Volume of Space Envelopes...... 82 Figure 6.11 – Terrain Navigation using Various Statistics and their E-N Error Comparison ...... 83 Figure 6.12 – Horizontal Position Fixes on the DEM...... 84 Figure 6.13 – Horizontal Position Fixes on the DEM with Horizontal Bias ...... 85 Figure 6.14 – Horizontal Position Fixes on the DEM with Horizontal and Vertical Biases ...... 85 Figure 6.15 – Horizontal Position Fixes on the DEM with Horizontal and Vertical Biases using Equations (6.1) and (6.2)...... 86 Figure 6.16 – Terrain Information Metric Variation...... 87 Figure 6.17 – East and North Direction Error Variation with ‘I’...... 88 12 Figure 6.18a – Vertical Bias Estimates using Mean and ACF Estimators...... 89 Figure 6.18b – Pitch and Roll Angles...... 89 Figure 6.19 – Illustration of Spatial Position Estimation...... 90 Figure 6.20 – Horizontal Position Fixes using Spatial Position Estimation ...... 91 Figure 6.21 – Spatial Position Estimation of East, North and Vertical Biases ...... 91 Figure 7.1(a) – Ohio University King Air C90 Flying laboratory, ...... 93 Figure 7.1(b) – DEM Integrity Monitor Experiment (DIME) equipment ...... 93 Figure 7.2 – Ohio University DC-3 Flying laboratory...... 94

Figure 7.3 – T and TKF values during flight segments at JNU, AVL and KUNI for biases of 0, 25m and 35m...... 96 Figure 7.4 – Improvement in RoMD at EGE, JNU, AVL and KUNI...... 97 Figure 7.5 – Reduction in Space Envelope using the Kalman Filter method ...... 98 Figure 7.6 – Horizontal Position fixes on the DEM at JNU ...... 99 Figure 7.7a,b and c – Horizontal Position fixes on the DEM at AVL without biases ...... 100 Figure 7.7d,e and f – In the presence of both Horizontal and Vertical biases ...... 100 Figure 7.8a,b and c – Horizontal Position fixes on the DEM at KUNI without biases ...... 101 Figure 7.8d,e and f – In the presence of both Horizontal and Vertical biases ...... 101 Figure 7.9 – Vertical Bias Estimates using Mean and ACF Estimators at JNU, AVL and KUNI ...... 102 Figure 7.10 – Horizontal Position Fixes on the DEM at JNU, AVL and KUNI ...... 102 Figure 7.11 – Spatial Estimation of Vertical Biases at JNU, AVL and KUNI ...... 103 Figure A.1 – Illustration of a shortest interval and its associated probability (area) under a normal curve...... 114 Figure A.2 – Steady state Kalman filter parameter values...... 115

13

LIST OF ACRONYMS

ADS-B Automatic Dependant Surveillance - Broadcast ACF Autocorrelation Function AD Absolute Disparity AGL Above Ground Level AHRS Attitude Heading Reference Set ATC Air Traffic Controller BT Blackman - Tukey CaB Commercial and Business CDTI Cockpit Display of Terrain Information CEP Circular Error Probability CFIT Controlled Flight Into Terrain COTS Commercial Off The Shelf DEM Digital Elevation Model DGPS Differential GPS DoD Department of Defense DTED Digital Terrain Elevation Data DWL Downward-Looking ECEF Earth Centered Earth Fixed EGPWS Enhanced Ground Proximity Warning Systems EKF Extended Kalman Filter EVS Enhanced Vision Systems FAA Federal Aviation Administration FHA Functional Hazard Assessment FLIR Forward Looking Infrared FLTA Forward Looking Terrain Avoidance 14 FTA Fault Tree Analysis FWL Forward-Looking GA General Aviation GNSS Global Navigation Satellite System GPS Global Positioning System GPWS Ground Proximity Warning System HDD Head Down Display HMD Head Mounted Display HUD Head Up Display ILS Instrument Landing System IMC Instrument Meteorological Conditions INS/IRS/IMU Inertial Navigation/Reference/Measurement System/Unit LAAS Local Area Augmentation System LEP Linear Error Probability LiDAR Light Detection And Ranging MDB Minimum Detectable Bias MESE Maximum Entropy Spectral Estimation MMWR Millimeter Wave Radar MPDLIM Multiple Path Downward Looking Integrity Monitor MSE Mean Squared Error MSL Mean Sea Level MTI Misleading Terrain Information NASA National Aeronautics and Space Agency NGA National Geospatial-Intelligence Agency NIMA National Imagery and Mapping Agency PDF Probability Density Function

PFFD Probability of Fault-Free Detection PMD Probability of Missed Detection PSD Power Spectral Density RA Radar Altimeter RADAR Radio Detection and Ranging RoMD Region of Missed Detection RTCA Radio Technical Commission for Aeronautics SA Situation Awareness SBAS Space Based Augmentation System SRTM Shuttle Radar Topography Mission SVS Synthetic Vision System TAWS Terrain Awareness and Warning System TIS-B Traffic Information Systems - Broadcast TRN Terrain Referenced Navigation TSO Technical Standard Order USGS United States Geological Survey WAAS Wide Area Augmentation System WGS84 World Geodetic System 1984 WxRadar Weather Radar 15

1

INTRODUCTION

Most aircraft are equipped with instruments to aid the pilot during flight in Instrument Meteorological Conditions (IMC). IMC are characterized by poor or no visibility due to bad weather conditions and darkness. This lack of visibility reduces the number of visual cues available to the pilot and may result in loss of spatial orientation. In order to address the problem of low visibility during a flight and to improve aircraft and airport operations, in general, research on Synthetic Vision Systems (SVS) is being conducted by the National Aeronautics and Space Agency (NASA) in collaboration with various industry and academic institutions, Ohio University being one of them. In this context, this thesis addresses two issues: firstly, development of methods to improve terrain database integrity monitors for use in SVS and secondly, extension of the integrity monitor concepts for application to terrain referenced navigation (TRN).

The basic idea of SVS displays is to provide computer-generated moving images to the pilot that must be a true representation of the outside world. Out of all the information that the computer has to collect in order to generate this ‘outside world’ picture, information about the terrain features is derived on-board from terrain elevation databases. As computers are incorporated in all aspects of flight, the greatest challenge facing avionics researchers is to ensure the accuracy, availability, integrity and continuity of operation of the modern systems. These required 16 navigation performance parameters have to be considered during the design phase and depend mainly on the inherent technologies used. Integrity stems from the component level and depends heavily on the reliabilities of the individual components, which can be studied rigorously using, among others, fault-tree analysis. Terrain database integrity refers to the capability to detect errors in the terrain database that could cause or contribute to the failure of a system function. This thesis deals with improving the integrity of the information derived from terrain databases using sensor information in real-time and estimation techniques such as Kalman filtering and spectral estimation. The methods thus developed will find applications in a Terrain Database Integrity Monitor and for Terrain Referenced Navigation.

1.1 Contributions

The work presented in this thesis elaborates on some of the fundamental work done previously by: Gray, R. A., in his doctoral dissertation titled “Inflight Detection of Errors for Enhanced Aircraft Flight Safety and Vertical Accuracy Improvement using Digital Terrain Elevation Data with an Inertial Navigation System, Global Positioning System and Radar Altimeter”, at Ohio University, Athens, Ohio, in June 1999. [1] Campbell, J., in his masters’ thesis titled “Characteristics of a Real-Time Digital Terrain Database Integrity Monitor for a Synthetic Vision System”, at Ohio University, Athens, Ohio, in November 2001. [2]

The sections of the aforementioned work relevant to this thesis are described in Chapter 3. Some of the material presented in this thesis has previously been published as the conference papers:

Vadlamani, A. and Uijt de Haag, M., “Application of Spectral Estimation Techniques to Terrain Database Integrity Monitors and Terrain Navigation Systems”, in the proceedings of the ION GPS/GNSS 2003, pp. 2389-2398, September 2003. Vadlamani, A. and Uijt de Haag, M., “Improving the Detection Capability of Spatial Failure Modes Using Downward-Looking Sensors in Terrain Database Integrity Monitors”, in the proceedings of the 22nd Digital Avionics Systems Conference (DASC), session 9C5, October 2003. 17 Vadlamani, A. and Uijt de Haag, M., “Improved Downward-Looking Terrain Database Integrity Monitor and Terrain Navigation”, in the proceedings of the IEEE Aerospace Conference, March 2004.

1.2 Outline of the Thesis

This thesis consists of three parts in addition to the two introductory chapters and the summary. Previously published works on terrain database integrity monitors and TRN systems are summarized and their relevance explained in Chapters 3 and 4, respectively. Chapter 5 gives some theoretical background on the estimation techniques used to improve the terrain database integrity monitor performance and perform TRN, namely: Kalman filtering and spectral estimation, tailored towards this thesis’ perspective. The actual research and results are presented in Chapter 6; specifically, results using the Kalman filter estimator, Autocorrelation function estimator and the Maximum Entropy Spectral Estimator. The data used to illustrate these concepts was collected over Eagle/Vail County, CO airport (EGE) and a description of the flight test and data collection system is also provided. Results showing the integrity-monitoring scheme using the various estimation techniques are provided. Also treated are extensions to the integrity monitor concepts that are evaluated for applications to TRN. Additional flight test results at locations other than EGE are discussed in Chapter 7. These locations include Asheville, NC (AVL), Albany, OH (KUNI), and Juneau, AK (JNU). The flight test setup and system description are given in brief at the beginning of Chapter 7. At this point, it is regarded essential to explain that the purpose of this thesis is to build upon the previous work on terrain database integrity monitoring and to improve or extend the original ideas. 18

2

SYNTHETIC VISION SYSTEMS

The aviation industry constantly tries to solve the problems that arise from limited visibility. These problems affect the safe operation of all phases of flight (take-off, en route and landing), the efficient management of aircraft traffic around the terminal area as well as on the ground, airport delays and aircraft runway incursions [3]. Poor or no visibility construe Instrument Meteorological Conditions (IMC) due to bad weather conditions like clouds, rain, fog, snow or darkness. Though most modern aircraft are equipped with instruments to aid the pilot during IMC flight, they only help to alleviate the problem to some degree. The lack of visibility reduces the number of visual cues available to the pilot and may result in loss of spatial orientation. Examples of deteriorated visual cues are a blurred horizon, the extent and contrast of the ground versus sky. Furthermore, under poor visibility conditions, pilots with limited experience might continue to fly into worsening weather that could result in reduced Situation Awareness (SA) and lead the aircraft to unexpectedly fly into mountains, ground, water, obstacles or other air traffic, while the pilots are in complete control of the vehicle. Such an accident is referred to as a Controlled Flight Into Terrain (CFIT) accident. [3]

In commercial aviation, over 30% of all fatal accidents worldwide are characterized as CFIT [4]. Several early attempts to mitigate the poor visibility problem have used advanced sensors such as active or passive radar (RAdio Detection And Ranging) and forward-looking infrared (FLIR) 19 sensors that can penetrate darkness, clouds or fog and enhance some of the significant terrain or obstacle features [4]. The terrain and obstacle data thus obtained is then depicted onto the pilot displays that can give some indication of the real world. Such systems are known as “Enhanced Vision Systems” (EVS). While enhanced vision systems have been used previously in military systems, they have not been found feasible in commercial aircraft due to cost and technical complexity until recently [5]. Some of the government organizations that are investigating possible solutions for CFIT and for the limited visibility problem, in general, are:

The National Geospatial-Intelligence Agency’s (NGA, formerly known as National Imagery and Mapping Agency (NIMA)) Ron Brown Airfield Initiative program [6] that is collecting geospatial information around airfields to support Global Navigation Satellite System (GNSS) instrument approach procedures. The Federal Aviation Administration’s (FAA) partnership with industry in the Safe Flight 21 (SF-21) program [7] that is exploring the use of Automatic Dependent Surveillance Broadcast (ADS-B) to provide real-time traffic information to both air traffic controllers and flight crews. NASA’s Aviation Safety Program (AvSP) [8], a partnership with FAA, the Department of Defense (DoD), the aviation industry and academia, is aimed at developing technologies that would contribute to a reduction in the aviation fatal accident rate by a factor of 5 by year 2007. The goal of the AvSP is to increase aviation safety for both Commercial and Business (CaB) and general aviation (GA) aircraft.

Additional information about these programs and others are summarized in [2] or can be obtained in detail from their respective WebPages.

SVS are cockpit display systems being developed by various companies and government agencies such as NASA. An SVS provides the pilots with either a Heads Up Display (HUD) or a Heads Down Display (HDD) containing information about aircraft state, guidance, navigation, traffic and a virtual depiction of the terrain as viewed “from the cockpit”. With the help of highly accurate terrain and obstacle databases, and knowledge of the aircraft’s position, attitude and heading, it is possible to generate a ‘virtual’ picture of the outside world from the flight deck’s perspective. Taking into account the aircraft’s motion, a new virtual picture is generated based on the most recent position/attitude/heading. By depicting these pictures on the display in quick succession, it is possible to re-create three-dimensional moving scenes of the outside world, in 20 real-time. The Global Positioning System (GPS) and horizontal and vertical guidance from the Wide Area Augmentation System (WAAS), GALILEO (the European Satellite Navigation System, under development) or other forms of differential GPS can be used to determine the aircraft position. WAAS was commissioned by the FAA at 12:01 AM on July 10, 2003. The accuracy of WAAS is within 1-2 meters in the horizontal and 2-3 meters in the vertical direction throughout most of Continental U.S. and Alaska [9]. Even though the position/attitude/heading information is derived from sensors in real-time, the terrain and obstacle information depicted on the displays is obtained from on-board databases. Guaranteeing the integrity of these databases becomes an important issue. The databases might have inaccuracies due to errors in data- collection as well as a change in terrain features and obstacle locations over the years.

Some of the focus areas in SVS research in recent years include [4]: EVS sensors research [10] Millimeter Wave Radar (MMWR) Forward-Looking Infrared (FLIR) flight research HUD development ([11]) Head-Mounted Display (HMD) development Datalinks: Controller-Pilot Datalink Communications (CPDLC), Traffic Information Systems – Broadcast (TIS-B), Automatic Dependent Surveillance – Broadcast (ADS-B) Cockpit Display of Traffic Information (CDTI) Increasingly Accurate Terrain Database: Shuttle Radar Topography Mission (SRTM), Light Detection and Ranging (LIDAR) maps, Improvement in Commercial Off-The-Shelf (COTS) graphics processors improvement driven by video gaming/entertainment industry. Developments in GPS, Inertial Navigation Systems (INS), and GPS/INS integration.

This list is by no means complete. Various technological advancements at the sub-system level may enable SVS to be certified for critical flight operations. Any new technology brings challenges of its own. As the data for SVS displays becomes increasingly accurate, it is possible to significantly reduce the risk of air accidents like CFIT. An experiment conducted by the 21 NASA Langley Research Center corroborates the belief that an SVS will improve a pilot’s ability to detect and avoid a potential CFIT compared to conventional flight instrumentation [12]. A resource of documents and publications pertaining to SVS can be found at [13].

2.1 SVS Component Description

An SVS consists of three major components: sensors, databases, and pilot (the inputs), SVS main processing unit, and the graphics display (the output). Figure 2.1 presents a conceptual overview of the SVS architecture. Only the components deemed relevant for this thesis are described hereunder. Detailed description of individual blocks can be found in [14], from which some of the blocks in Figure 2.1 have been reproduced.

Sensors/Databases

Aircraft Hazards Terrain State Data Weather Information FLIR MMWR Database, (INS, AHRS, Radar (Datalink, DGPS Radar TAWS) Altimeter)

SVS Computer

Perspective Image Object Integrity Transformation Detection Monitoring

Symbology Interface and Data Fusion Generation Communication

Figure 2.1 – SVS Concept Architecture Overview Display: NASA Langley SVS Depiction of Approach to Asheville, NC (AVL), courtesy [3] 22 2.1.1 Sensors In the absence of visual cues, a pilot primarily relies on the aircraft’s external sensors for situational awareness. Differential GPS (DGPS), Inertial Reference System (IRS) and Attitude/Heading Reference Set (AHRS) provide spatial position and orientation information. GPS receivers are typically capable of providing position information with reference to various coordinate frames such as the Earth-Centered-Earth-Fixed or ECEF coordinate frame and the World Geodetic System 1984 or WGS ’84 reference frame. The height information could be either with respect to the WGS84 global reference ellipsoid model for the earth, or with respect to WGS84 reference geoid (this is equivalent to the height above mean sea level (MSL) [15]). Many countries are researching Space Based Augmentation Systems (SBAS) for improving GPS positioning accuracy. U.S. SBAS is called WAAS. In addition, Local Area Augmentation Systems (LAAS) are undergoing research by industry and universities, some prototype LAAS are in the testing phase.

A radar altimeter is the primary sensor for providing height above ground level (AGL). For the purpose of terrain database integrity monitoring, sensors are classified as downward-looking (DWL) or forward-looking (FWL). Radar or even laser altimeters are typically DWL sensors. A laser altimeter that uses a scanning beam laser can be used as both DWL and FWL sensor. FLIR, MMWR and Weather Radar (WxRadar) are examples of FWL sensors since they illuminate terrain and objects ahead of the aircraft. Terrain database integrity monitors that use DWL sensor inputs are conceptually easier to design compared to the same using FWL sensors due to the complexity involved in perspective transformation, object detection and data fusion.

2.1.2 Terrain Databases The source of terrain information depicted on the SVS display is a terrain elevation database or Digital Elevation Model (DEM); a digital look-up table or database containing terrain heights and their corresponding position coordinates. The position coordinates are expressed in predefined horizontal and vertical datums. DEM examples are NIMA’s Digital Terrain Elevation Database (DTED) levels 0, 1 and 2 and Jeppesen® terrain databases. The DEMs are specified by a number of parameters such as their post spacing (resolution), the horizontal and vertical reference datums, and linear and circular errors [16]. Linear error specifies the accuracy in the vertical direction whereas the circular error specifies the accuracy in the horizontal direction. A depiction of Linear and Circular errors is given in Figure 2.2 [17]. More information about the DEMs used for this research can be found in the Appendix. 23

Circular Error

Post Spacing e tud Linear i

Error Lat

Top View Longitude Figure 2.2 – Depiction of (a) Linear Error and (b) Circular Error in the DEM

Ideally, the DEM represents the height or elevation of the terrain at a set of given coordinates (e.g. latitude and longitude) expressed in a predefined vertical datum. However, the given elevations deviate from the true elevations due to the presence of systematic faults and randomly distributed errors, primarily due to the way in which they are generated from different sensor technologies such as photogrammetry, remote sensing, etc., coordinate transformation mismatches, discrepancies in vertical and horizontal datums used to express the heights as well as the manual post-processing process. These errors manifest in the form of:

Bias in the vertical and horizontal domains Ramps in the vertical and horizontal domains Random distributed errors in the vertical domain and circularly distributed random errors in the horizontal domain.

This thesis mainly concentrates on the bias errors in both the vertical and horizontal domains.

2.1.3 Displays Displays are an important component of SVS that are used to depict the terrain features and other navigation and guidance information. SVS prototypes, both in simulation and testing, have experimented with HUDs, head down Primary Flight Displays (PFD) or a dedicated page in a Navigation Display (ND). The industry and NASA have developed a few display formats, color schemes, and symbology. The primary considerations for selecting an appropriate display are the size and type of aircraft cockpit, interface with other aircraft systems and general consensus among pilots. HUDs have been found to be most suitable for better situational awareness during landing and touchdown phases of flight and SVS mission accomplishment in general. [4][14] 24 2.2 SVS Predecessors

In the early 1970’s, a number of studies investigating the occurrence of CFIT, as evidenced from early flight recorder data, Air Traffic Control (ATC) records and experiences of pilots in CFIT incidents, have identified common conditions that precede such accidents [4]. It was concluded that the CFIT accidents could have been avoided, had there been a system on-board having forward-looking capability that could provide the flight crew with situational awareness and alert them of imminent danger, such as approaching terrain, excessive climb or descent rates etc. Since then, Ground Proximity Warning Systems (GPWS) have entailed a long history of research.

2.2.1 Ground Proximity Warning Systems (GPWS) The aircraft state is determined using various sensor inputs that are part of the standard aircraft avionics suite. By a dynamic projection of the aircraft state into the near future, the GPWS compares projected state to the hazardous conditions preceding a CFIT accident and generates an alert whenever the conditions exceed the limits for safe operation. Radio Technical Commission for Aeronautics (RTCA) DO-161A specifies the ‘Minimum Performance Standards for Airborne Ground Proximity Warning Equipment’ [18]. In the classical GPWS operation, the primary sensor providing aircraft state with respect to the terrain is the radar altimeter that measures the height AGL. Aircraft descent is detected by decreasing height AGL values. Over steeply rising terrain, however, this method may not allow sufficient time to perform a corrective action.

Advances in terrain mapping technology permitted the development of a new type of GPWS that provides greater situational awareness for flight crews. The FAA has approved certain installations of this type of equipment, known as the Enhanced Ground Proximity Warning System (EGPWS). EGPWS incorporates terrain, obstacle and airport databases. The projected aircraft state vector is compared to a three-dimensional digital map of the real world and potential hazardous situations can be identified much earlier than the classical GPWS. In recent years, the distinction between EGPWS and Terrain Avoidance and Warning Systems (TAWS) has dwindled and they are used synonymously with each other, although TAWS is much more clearly defined in literature.

2.2.2 Terrain Awareness and Warning System (TAWS) In March 2000, the FAA issued two rules that require all turbine-powered U.S.-registered airplanes type certified to have six or more passenger seats (excluding pilot and copilot seats) and 25 manufactured after March 29, 2002, to be equipped with an FAA approved TAWS. Those airplanes manufactured on or before March 29, 2002 are exempt until March 29, 2005, after which all airplanes are required to comply with the rule [19]. The basic TAWS functions for all Technical Standard Order (TSO) approved systems include a Forward Looking Terrain Avoidance (FLTA). The FLTA looks along, below and sideways of the airplane’s flight path and provides the pilot with suitable aural and visual alerts if a potential CFIT threat exists or if the airplane is hazardously below the 3 degree glideslope during final approach to the nearest runway. The TAWS improves upon existing GPWS systems by providing the flight crew with much earlier aural and visual warning of impending terrain, forward looking capability, and continued operation in the landing configuration. These improvements provide more time for the flight crew to make smoother and gradual corrective action.

The FAA issued a final version of TSO-C151b [20], which prescribes the minimum operational performance standards that TAWS equipment must meet to be identified with the TSO-C15b Class A, B or C marking. Class A and B TAWS equipment requirements are covered by Title 14 of the Code of Federal regulations (14CFR) parts 91, 135 and 121. Class A TAWS equipment must present terrain information on a display system and must provide indications of imminent contact with the ground for excessive rates of descent; excessive closure rate to terrain; altitude loss after take-off; flight into terrain when not in a landing configuration; excessive downward deviation from an Instrument Landing System (ILS) glideslope; and a voice callout when the airplane descends to 500 feet above the terrain or nearest runway elevation. EGPWS falls under the TAWS Class A category. Class B TAWS equipment does not require a display and must provide indications of imminent contact with the ground only for excessive rates of descent; altitude loss after take-off and include a voice callout when the airplane descends to 500 feet above the terrain or nearest runway elevation. Class C TAWS equipment is intended for voluntary installations on aircraft not covered by the requirements in 14CFR parts 91, 135 and 121 for Class A and B TAWS.

TAWS have not been certified for critical flight operations but only as advisory or warning systems. They provide alerts or advisories but it is at the discretion of the flight crew whether to take corrective action or not. This is due to the high failure rate of the equipment for false alarms and missed detections. To be certified for critical flight operations, such failures must be “highly improbable”. 26

3

TERRAIN DATABASE INTEGRITY MONITOR

SVS displays provide a clear, daylight view of the outside world at all times, even in poor visibility conditions. Due to the compelling nature of the displays, it is quite possible for the pilots to use the displays for functions other than the function they are designed for. This is referred to as the “unintended use” of SVS displays. Unintended use can lead to situations in which safety of life is threatened. This is especially the case for an SVS that is only approved as an advisory function. For example, the pilot might lose touch with reality for a few minutes and rely for his tactical navigation decisions solely on the SVS display. It is under these circumstances that the terrain database or DEM must be guaranteed to a high level of integrity. In order to be certified for a level of reliability more stringent than advisory, SVS as a whole must meet or exceed the required reliability ratings based on the derived Fault Tree Analyses (FTAs) and other safety assessment tools. The SVS includes the terrain database server that could provide misleading terrain information (MTI) to the pilots. To ensure that the terrain elevation data used for the generation of SVS display imagery meets a required failure rate, an integrity monitor function may be added to the terrain database server. Both the defined terrain database failure rate and the additional integrity monitor will then guarantee the specified probability of fault-free detection (false alarm) and probability of missed detection. This terrain database integrity monitor performs a consistency check between the terrain database and the sensor measurements of the real world. 27

The DEM integrity monitor performance is specified by probabilities of Missed Detection (PMD),

Fault–Free Detection (PFFD), Minimum Detectable Bias (MDB) and a Time-to-Alarm.

Inaccuracies in a DEM larger than a nominal level are considered DEM failures. PMD is the probability that the presence of such a failure goes undetected. PFFD is the probability of failure detection, when a failure condition does not exist. Such an alert is also known as false alarm or nuisance alert. The time it takes for the system/algorithm to detect a failure condition and generate an alarm is called the Time-to-Alarm. It is the sum of the time to accumulate the inputs and the time the system takes to process them and generate an alarm. In the present scenario [1][2], the inputs are the height measurements over time. If the algorithm operates on ‘N’ inputs

(x1 , x2 , x3 ,..., xN ) , measured at a rate of ‘s’ samples/second, and it takes ‘τ’ seconds to process the inputs and generate an alert, the Time-to-Alert, intuitively, would be (N*s) + τ seconds. Thus, considering the sampling rate, the number of measurements that are used as the input to the system must be chosen judiciously. This involves a tradeoff between speed and efficiency of the integrity monitor that is justified by the characteristics of a DEM failure. As an aircraft is traversing the terrain, a bias in the DEM does not show up suddenly from a particular measurement, but its detection is a gradual process. ‘N’ must not be chosen too small so as to compromise on efficiency, nor must it be chosen too large as to exceed the necessary Time-to- Alert.

Depending on the impact of a system failure on the aircraft and its passengers, the FAA has classified avionics system failures for the purpose of certification. Failure is a loss of function, or a malfunction, of a system or a part thereof. A failure condition is defined as the effects of the failure on the airplane and its occupants, both direct and consequential, caused or contributed to by one or more failures, considering relevant adverse operational or environmental conditions. Failure conditions may be classified according to their severities as follows, a thorough explanation of which can be found in [21]:

Minor: Failure conditions which would not significantly reduce airplane safety, and which involve some inconvenience to occupants. 28 Major: Failure conditions which would reduce the capability of the airplane or the crew to cope with adverse operating conditions involving a large reduction in safety margins or functional capabilities or some discomfort to occupants. Catastrophic: Failure conditions that would prevent continued safe flight and landing.

Systems whose failure can be classified as minor are referred to as non-essential system, systems whose failure can be classified as major are referred to as essential systems, and finally systems whose failure can be classified as catastrophic are referred to as critical systems. Essential systems should have a probability of failure between 10-9 and 10-5. Critical systems must have a probability of failure less than 10-9 per flight hour. Non-essential systems are all remaining systems allowing a probability of failure greater than 10-5 per flight hour.

Three SVS application categories were identified in [22]: SVS advisory applications, SVS strategic applications, and SVS tactical applications. Based on the applications, these SVS are classified as non-essential, essential, and critical, respectively. Strategic and tactical applications include the take-off and landing operations that involve a certain CFIT risk due to possible negative climb and excessive descent rates. Deployment of SVS during these phases of flight is safe only if the terrain databases meet a high level of integrity. Some work is in progress to create High-Quality Databases (HQ-DB) for terrain, obstacles and airports and the databases for specific SVS applications to be derived from them in such a way that would allow certification [23]. The future availability of a HQ-DB is interesting as it could significantly simplify the Functional Hazard Assessment (FHA) for an SVS failure, but for the databases currently in use, an independent integrity monitor is required to ensure that the presented information for critical flight applications conforms to a certain level of integrity.

A conceptual and simplified SVS fault tree is depicted in Figure 3.1 [17]. This tree shows the contribution of a terrain database and integrity monitor failure to the overall SVS failure. An undetected failure in any one of the SVS subsystems can lead to an SVS failure. This is indicated by the ‘OR’ operation. If considered as just another component, the inclusion of a DEM integrity monitor appears to increase the probability of a DEM failure. But if the integrity monitor is designed to a failure rate much smaller than that of the DEM, it could alert the flight crew of a 29

PSVS

P Display P P Etc. Computer PTerrain Obstacle

DEM Integrity SVS Obstacle SVS Display SVS Computer Monitor Database

SVS Terrain Database

Figure 3.1 – Sample Synthetic Vision System Fault Tree

DEM failure, and cause them to take corrective action, thus reducing the effect of a DEM failure on the above fault tree. Had the integrity monitor not been included, the aircraft would have continued on its course, oblivious of a DEM failure, with possible hazardous consequences. The inclusion of a DEM integrity monitor thus reduces the probability of a terrain database failure to the failure rate of the integrity monitor.

Much work has been done on terrain database integrity monitors using both downward-looking and forward-looking sensors. Forward-looking sensors like X-band weather radars have been used along with GPS inputs to implement terrain feature detection algorithms to monitor DEM integrity [24]. The work on downward-looking sensors has been summarized in this chapter to lay a foundation for the improvements presented in Chapter 6. Due to the inherent presence of sensor measurement noise and random errors in the DEM, it is not possible to detect vertical biases and horizontal translations in the DEMs in an absolute sense, but only in a statistical manner. The statistical method is described as a baseline method for reliability conformance in Section 3.1. Any improvements or tradeoffs must contrive from this method. The integrity monitor function in the vertical and horizontal domain is presented in Sections 3.2 and 3.3. Section 3.4 combines the vertical and horizontal domains into a spatial integrity monitor. 30 3.1 Statistical Method

The DEM integrity monitor concept was originally proposed in [1] and a description of its various details and extension can be found in [2]. The statistical method compares the terrain database elevation profile with an independent terrain profile synthesized from a downward- looking external sensor in real-time. The method uses Radar Altimeters (RA) to measure the height AGL and GPS WAAS information to derive the aircrafts height above MSL. Since GPS and RA inputs are necessary for this method, failure of either one of them would result in a failure of the integrity monitoring capability. Figure 3.2 illustrates the sensor measurements.

GPS Antenna

lar RA Antenna

hDGPS AGL

Terrain h DEM MSL

Figure 3.2 – Measurement of Synthesized Terrain Profile (Aircraft Image courtesy of Boeing)

The difference between both sensor measurements is, ideally, equal to the height of the ground or terrain above MSL, i.e. the quantity that is stored in the terrain database. The synthesized height measurements are formed as:

hsynt (ti ) = hDGPS (ti )− hRA (ti )− lar (3.1)

where, hDGPS is the height above MSL as derived from DGPS measurements, hRA is the height

AGL obtained from RA measurements and lar is the distance offset between the GPS and the RA antennas, typically the distance between the roof and the belly of the aircraft. All measurements

are referenced to time epoch ti . In order to compare the database elevations and the elevations synthesized from the sensor measurements a metric is chosen which is sensitive to the errors that 31 must be detected i.e. the bias errors. This so-called consistency metric is referred to as the Absolute Disparity (AD or p) and given by:

p( ti ) = hsynt ( ti )− hDEM ( ti ) (3.2)

Some other metrics than can be used (although not explored in this thesis) to express the consistency or degree of agreement between the terrain database profile and the synthesized terrain profile are:

Successive Disparity: s(ti ) = p(ti ) − p(ti−1 ) (3.3) This quantity is formed by subtracting two consecutive absolute disparities, thus getting rid of constant biases in the sensor measurements. Successive disparities are sensitive to ramp errors. Cross Correlation: This quantity is the magnitude of the correlation function or the correlation coefficient as a measure of agreement between the DTED and synthesized terrain profiles.

For an aircraft flying at a height of at least 290 meters and a speed of no less than 60 meters/second, the absolute disparities, obtained by subtracting the Digital Terrain Elevation Database (DTED 1, 3” x 3” spatial resolution) terrain profile from the synthesized terrain profile, can be approximated as an independent Gaussian random variable (RV) with mean zero (assuming no biases on the DTED) and a standard deviation of 18.9 meters [1]. Since the

2 absolute disparities are normally distributed, their squares p ( ti ) are chi-squared distributed. The sum-of-squares of N consecutive absolute disparities will be used as the test statistic in the Hypothesis testing section of this chapter. The statistical derivations in the next section refer to the integrity monitor application but also apply to the TRN application as presented in Chapter 6.

3.1.1 Formulation of Hypotheses The integrity monitor fault detection capability is based on the following hypothesis testing steps [25]:

1. Formulate a Null Hypothesis (H0) and an appropriate Alternate Hypothesis (H1) that is accepted when the Null Hypothesis must be rejected. 32

2. Specify the probability of a Type I error for H0; if necessary, also specify the probability

of a Type II error for H1. 3. Construct a criterion for testing the null hypothesis against the given alternative hypothesis based on the sampling distribution of an appropriate statistic. 4. Calculate, from the data, the value of the statistic on which the decision is to be based. 5. Decide whether to reject the null hypothesis, accept it or to reserve judgment.

Rejection of the Null hypothesis H0 when it is true is referred to as either a Type I error, a fault-

free detection or a false alarm. Acceptance of hypothesis H0 when it is false is referred to as Type II error or a missed detection.

Table 3.1 Decision Making in Hypothesis Testing

Accept H0 Reject H0

Type I error H is true Correct Decision 0 (Fault-free Detection)

Type II error H is false Correct Decision 0 (Missed Detection)

Under the fault-free condition (no bias present), the over-bound of the probability density of the errors on the sensors and the terrain database yields a normal Probability Density Function (PDF), which leads to the following null hypothesis or fault-free hypothesis:

2 Η 0 : p0 ~ N(0,σ p ) (3.4)

2 2 2 where N(0,σ p ) is a normal distribution with mean zero and variance σ p . The variance σ p is derived from the convolved variances of the individual sensor error Probability Density Functions (PDFs), errors due to ground cover and the specified error characteristics of the DEM [1].

Table 3.2 Over-Bounded Absolute Disparity Distributions for Sensors and Data Vertical Error Source Absolute Disparity Distribution Radar Altimeter N(0, (1.8) 2) Kinematic GPS Height N(0, (0.22) 2) 2 DTED Vertical Error N(0, (18.2) ) refer Appendix Ground Cover N(0, (4.6) 2) N(0, (18.9) 2) Convolved Distribution (σ P ) 33 When a failure mode exists in the form of a bias on either the synthesized height or the terrain database elevations or both, the bias shows up in the PDF of the absolute disparities in the PDF mean value, giving rise to the following alternate or faulted hypothesis:

2 Η 1 : p1 ~ N(µ,σ p ) (3.5) where µ is the failure in the form of a bias. [2] furthermore defines the minimum detectable bias

(MDB), µ B as the smallest bias that can be detected with a probability of 1- PMD.

3.1.2 Test Statistic for Decision Making The chi-square distribution is a special case of the gamma distribution:

1 − x ⎧ xα −1e β ,x>0,α >0,β >0 ⎪ β α Γ ( α ) f ( x ) = ⎨ (3.6) 0 ,otherwise ⎩⎪

ν with α = , β = 2 , and the parameter ν which is referred to as the ‘degrees of freedom’ of the 2 distribution [25]. ν plays a vital role in statistical inference and hypotheses testing. If

x1 , x2 ,..., xN is a random sample of size N taken from a normal population having a mean µ and

the variance σ 2 , then

N 2 ∑()xi − µ χ 2 = i (3.7) σ 2

is a random variable with a chi-square distribution with ν = N degrees of freedom. If the sample mean X is unknown, then

N 2 ∑()xi − X χ 2 = i (3.8) σ 2

is a random variable having the chi-square distribution with ν = N −1 degrees of freedom [25]. 34 Forming the chi-square statistic as in equation (3.7) by scaling the sum of squares of the ADs by the variance of the noise on the ADs under nominal conditions provides the integrity monitor Test Statistic, T:

1 N T = p2 t (3.9) 2 ∑ (i ) σ p i=1

Under H0, T is chi-square distributed with N degrees of freedom and under H1, T is non-central chi-square distributed with N degrees of freedom and non-centrality parameter λ [26]. Figure 3.3

(not to scale) shows a representation of the distributions of the test statistic, T, under H0 and H1 hypotheses [2][26].

y t H0

H1 Probability Densi

PMD PFFD

TD Test Statistic (T) N + λ

Figure 3.3 – Distribution of T for H0 and H1

-4 For the purpose of this thesis, PFFD (Type I error) is chosen to be 10 and PMD (Type II error) is specified as 10-7. The degrees of freedom ν or N is chosen as 50 successive measurements. For

the given PFFD and N = 50 successive measurements, the threshold value of the chi-square statistic ‘T’, on which the decision has to be based can be found in chi-square distribution tables or calculated indirectly from equation (3.6). Figure 3.4 shows a look-up plot for the threshold

values for various degrees of freedom and various PFFD, the values having been generated using MATLAB™, since look-up tables are not easily available for higher number of significant digits. 35

300 -5 PFFD=10 -4 PFFD=10 -3 PFFD=10 -2 250 PFFD=10 →

eshold 200 Thr istic

t 150 uare Sta

q 96 Chi-S 50

25 50 75 100 125 150 175 200 Number of Samples, N (Degrees of Freedom) Figure 3.4 – Threshold value for chi-square statistic

-4 For PFFD = 10 and N = 50, the test statistic threshold (TD) was found to be 96. Figure 3.5 shows a look-up graph used to find the non-centrality parameter λ, given a value for PMD and the -7 threshold value of the test statistic. For PMD = 10 , TD = 96 and N = 50, λ is found to be 164.17.

-9 PMD=10 -8 PMD=10 300 -7 PMD=10 -6 PMD=10 →

) 250 λ (

200

164.17 150 rality Parameter t n e 100 Non-C 50

20 40 60 80 96 120 140 160 180 Chi-Square Statistic Threshold→ Figure 3.5 – Non-Centrality parameter look-up plot 36

Assuming a constant minimum detectable bias µ B on the disparities, the non-centrality parameter λ is related to the bias error [26][27] as follows:

λ µ = σ (3.10) B p N

2 2 Using equation (3.10) with σ p = (18.9) , µB is found to be equal to 34.2 meters, i.e. H0 is

rejected with a probability larger than 1-PMD if a bias of 34.2 meters or more is present. This is shown to be true by a Monte Carlo analysis in the next section.

3.1.3 Pseudo-Random Noise Analysis A Monte Carlo analysis with 10000 simulations was performed using MATLAB™, generating normally distributed pseudo-random numbers with a mean varying from 0 to 50 m and a standard

deviation equal to σ P = 18.9m . The T value was calculated for each run and compared to

threshold, TD. The results are shown in Figure 3.6 as the probability of accepting H0 under the presence of a bias, µ.

1

0.8 0 H 0.6 of t

cep 0.4 ac P 0.2

0 0 5 10 15 20 25 30 34.2 40 45 50 Bias (meters) Figure 3.6 – Operating Characteristics (OC) Curve

The Operating Characteristics (OC) curve shows the probability of accepting the null hypothesis under the presence of a bias fault. This probability decreases as the bias increases and at a bias of 34.2m and above, the null hypothesis is accepted only with a probability of less than 10-7. The plot was not generated rigorously with 107 simulations due to practical reasons, but it is just included to convey the general idea of the integrity monitor detection capability. 37 3.2 Vertical Domain Integrity Monitor

Ideally, p( ti ) equals zero but due to noise on the DEM and sensors, the variance of p( ti ) can be over-bound by (18.9)2; this condition is referred to as the nominal operating condition. In case of the presence of biases on the sensors and DEM, the absolute disparities are given by:

p = noisesensors+DEM + biassensors+DEM (3.11)

Figure 3.7 shows the computed values of the T statistic using GPS WAAS measurements of aircraft position and height MSL [28] for various approaches to runway 25 during NASA’s flight tests at Eagle-Vail (EGE), CO using their Boeing 757 ARIES test aircraft in 2001.

T Statistic Value Using WAAS Measurements 200

180

160

140

120 tic s 100

T Stati 80

60

40

20

0 -200 -150 -100 -50 0 50 100 150 200 Time From Threshold Crossing Figure 3.7 – T Statistic values for Approaches to R/W 25 at EGE

The T statistic values are much higher during the banking/turning phase of the aircraft (between 100 and 150). Such a characteristic is caused by a systematic error introduced in the radar altimeter measurements, as it measures a slant height instead of the ‘plumb-bob’ height [2]. Compensation techniques such as the ‘spot-beam algorithm’ [2] have not been implemented in this work, since one of the motivating factors behind this thesis is the presumption of horizontal biases in the terrain database. 38 Using various deliberately introduced biases, the T values during approaches to runway 25 at EGE have been plotted in Figure 3.8. The first plot is for zero bias (same as Figure 3.7 but scaled differently) and the next two are for biases of 25m and 35m, respectively.

Bias = 0 250 200 c i t

s 150 i t a t 100 T S 50

0 -200 -150 -100 -50 0 50 100 150 200 Bias = 25m 250 200 c i t

s 150 i t a t 100 T S 50

0 -200 -150 -100 -50 0 50 100 150 200 Bias = 35m 250 200 c i t

s 150 i t a t 100 T S 50

0 -200 -150 -100 -50 0 50 100 150 200 Time From Threshold Crossing Figure 3.8 – T value for 0, 25m and 35m biases

The plots clearly show that for a bias of 25 meters (less than MDB), not all points exceed the threshold of 96, but for a bias of 35 meters all points exceed the threshold of 96, thus forcing us to reject the null hypothesis H0 and provide the pilots with an aural or visual alert.

3.3 Horizontal Domain Integrity Monitor

In this thesis we introduce a metric to express the sensitivity of the DEM integrity monitor to horizontal displacements: the Region of Missed Detection or RoMD. The RoMD is defined as the region within which the probability of a missed detection is larger than or equal to the required probability of missed detection. The RoMD concept was introduced in [1] but will be used more extensively in this thesis. A smaller RoMD is preferred, thus, any improvements to the performance of the integrity monitor must reduce the region of missed detection. The concept of using downward-looking sensors for detection of horizontal failure modes was introduced in [2] and referred to as the Multiple Path Downward Looking Integrity Monitor (MPDLIM). In 39 MPDLIM the T value is computed for multiple flight paths horizontally offset from the nominal flight path (given by GPS positions). The horizontal offsets form a rectangular search grid around the nominal position. The T value at each grid point represents the measure of similarity between the terrain database profile below the offset flight path and the synthesized terrain profile. In essence, the vertical domain integrity monitor has been extended so that multiple terrain profiles from the terrain database would be compared to a single synthesized terrain profile, and all the grid points whose associated T value does not exceed the integrity monitor threshold are probable undetected terrain database offsets (horizontal translations). These grid points form the RoMD. The expression for the test statistic T, can symbolically be written as:

1 N T m,n = h t − h x,y 2 (3.12) () 2 ∑()SYNT ()i DEM ( ) σ i=1 where

x = lat()t + lat _offset for m = −P to P i m

y = lon()ti + lon _offsetn for n = −P to P

Terrain elevation plots and RoMDs are shown in Figure 3.9 for an approach to runway 25 at EGE. The dark areas in Figure 3.9c, 3.9d, 3.9e and 3.9f represent the regions of missed detection. The units on the x-axis and y-axis are in terms of terrain database grid points or 3 arc-seconds, or 90m approximately.

The performance of the proposed horizontal direction integrity monitor is a function of the terrain signature (roughness and non-periodicity). The difference in the horizontal error detection capability at all four flight segments in Figure 3.9 can be directly related to the terrain signature underneath the flight path at the respective locations (see equation (3.13)). The performance relies greatly on the deviation of the DEM terrain profile from the synthesized profile over the different points in the search grid. For terrain with a large spatial similarity such as flat or periodic terrain, the computed T values are similar over the entire search grid, resulting in a decreased MPDLIM performance. However, while traversing such terrain, the CFIT risk is in general, much smaller. Rough, quickly varying and non-periodic terrain offers improved performance of the horizontal integrity monitor as well as increased risk of CFIT. 40

c d b

a

"T" Value "T" Value "T" Value "T" Value 20 20 20 20

15 15 15 15

10 10 10 10

5 5 5 5

0 0 0 0

-5 -5 -5 -5

-10 -10 -10 -10

-15 -15 -15 -15

-20 -20 -20 -20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 a b c d Figure 3.9 – Region of Missed Detection within –20 to 20 Resolution Grid in Latitude and Longitude

A measure of the terrain roughness can be obtained from the average size of the terrain gradient, called the information content or roughness energy of the terrain [29], or:

N 1 2 I = ∑(∆hsynth ) (3.13) N i=2 41 3.4 Spatial Integrity Monitor

Having discussed both vertical and horizontal domain terrain database integrity monitors, it is possible to combine the two ideas in order to form a spatial failure mode detector/integrity monitor. The concept of a ‘Spatial Integrity Monitor’ is introduced here. The idea is to intentionally introduce positive and negative vertical biases on the synthesized elevations and use the horizontal integrity monitor concept to form RoMDs for each of the intentionally introduced biases. Stacking the RoMDs on top of each other results in a volume or ‘space envelope’ of missed detection. The concept of space envelopes is illustrated in Figure 3.10. The position of the aircraft indicates GPS position.

Generate Alert

Space Envelope of Missed Detection Approaching Rough Terrain

Figure 3.10 – Space Envelope of Missed Detection

Only the lower half of the ‘space envelope’ is shown in Figure 3.10 and all results that follow since the lower half is significant for generating a CFIT alert. A similar upper half of the ‘space envelope’ exists, having decreasing RoMDs with increasing magnitude of vertical bias. The space envelope of missed detection signifies the volume within which the probability of a missed detection is larger than or equal to the required probability of missed detection, or, in physical terms, an aircraft present anywhere within that volume fails to generate a vertical domain terrain 42 database bias/integrity alert. By thus forming the ‘space envelope’, an integrity alert can be generated if any part of the space envelope intersects the actual terrain.

Space Envelope ( T )

-5

-10

-15 ters me -20

-25

300 200 200 100 100 0 0 -100 -100 -200 Latitude (meters) Longitude (meters) Figure 3.11 – Sample Space Envelope Plot at EGE

Figure 3.11 shows a sample space envelope plot at a particular instant during flight at EGE. The star ‘*’ in Figure 3.11 represents the GPS aircraft position and the space envelope has been plotted as a contour map. Note how the space envelope tapers off with increasing magnitude of vertical biases as more points exceed the T statistic decision threshold. 43

4

TERRAIN REFERENCED NAVIGATION

Navigation is the determination of the position and velocity of a moving vehicle [30]. In terrain- referenced navigation (TRN), external sensors are integrated with terrain databases to obtain position and/or velocity estimates that will enable user navigation. Often, filtering techniques are used to reduce the sensor measurement noise and implement the sensor integration/fusion process. In this chapter, some existing TRN techniques will be reviewed and their salient features highlighted. A comparison of the various TRN techniques with respect to their principle of operation and the sensors used is provided for reference, in contrast to the navigation scheme presented in Chapter 6.

Any geographical location along with its surroundings has unique features with respect to its topography i.e., the distribution of terrain elevations. If a sufficiently large number of terrain heights are compared to an existing database of terrain elevations, it may be possible to determine a user’s location on the earth. Pioneer work on terrain-referenced navigation began in the early 1960s with the development and testing of a guidance technique known as Terrain Contour Matching (TERCOM). TERCOM can be used to either periodically update a dead-reckoning system such as an inertial measurement unit (IMU) or to provide continuous guidance on its own in combination with an IMU. Similar to fingerprinting, TERCOM operates on the principle of 44 matching a previously stored signature with one acquired during flight, the signature being that of the terrain.

BAROMETRIC RADAR ALTIMETER ALTIMETER

ALTITUDE MEASURED ELEVATION CORRELATION NAVIGATOR PROCESSING

DIGITAL MAP-INDICATED MAP ELEVATION

POSITION CORRECTION Figure 4.1 – The TERCOM System

A description of TERCOM can be found in [31][32][33]. A block diagram of the TERCOM system is shown in Figure 4.1 [33]. TERCOM uses a radar altimeter and barometric altimeter to measure the vertical terrain profile, an IMU to provide crude positions and an on-board database of terrain elevations. The radar altimeter measures the ground clearance or height above ground level (AGL) and the barometric altimeter provides a reference height, such as above MSL, for synthesized height calculation (similar to equation (3.1)). An illustration of the height measurements and a sample terrain correlation plot is shown in Figure 4.2.

Maximum Correlation RA Antenna AGL titude Al ight - e o r H Ba

Bias

hsynt hDTED Baro-Altimeter Reference Height MSL

Figure 4.2 – Measurements and Terrain Correlation in the TERCOM System

45 A barometric altimeter or baro-altimeter provides height measurements by sensing the air pressure and can be calibrated to provide heights above MSL. A baro-altimeter, however, suffers from large bias errors and a sluggish behavior. Fortunately, this highly correlated bias error is common among all measurements and will therefore only increase or decrease the magnitude of the complete correlation function, the maximum value being the one of interest. Some TERCOM systems integrate the barometric sensor with a vertical accelerometer to overcome the sluggish behavior and provide a stable height reference from which the radar altimeter data is subtracted to obtain terrain contour [31]. The required radar altimeter accuracy is not large, as the system needs to determine the vertical profile only and not the absolute height of the vehicle AGL. The vertical terrain profile thus obtained is correlated with the sequence of heights retrieved from the terrain database and several parallel flight paths that are offset from the predicted flight path by increments in the cross and along track directions. The predicted flights path coordinates are provided by the IMU or any Doppler-type navigation systems. The point within the terrain database search grid for which the magnitude of the correlation reaches a maximum is considered to be the position of the navigator. A sequence of such position fixes is input to a Kalman filter to update the position, velocity and other modeled states in an Inertial Navigation System (INS). The accuracy of the system is a function of the drift rates of accelerometers in the IMU; greater the drift rates, greater will be the magnitudes of cross-track and along-track corrections.

TERCOM and many more techniques that followed, until the mid 1990s, invariably used an IMU to derive position estimates in a dead-reckoning way, because GPS was still on the drawing board in the late 1960s and 70s and was not declared to full operational capability until April 27, 1995. Inertial navigation systems drift over time but are stable and GPS derived positions are more accurate but exhibit ‘noise’. Integration of GPS and INS can give accurate and stable position estimates. GPS can also replace the use of barometric altimeters as GPS positions do not suffer form biases and can provide an accurate height measurement above MSL. The accuracy of TRN techniques is limited by the resolution of the terrain database, but GPS/INS integration can provide meter-level accuracy. Hence, the use of GPS integrated with INS gained popularity over TRN.

Although a Kalman filter is eventually utilized, TERCOM and its variations are essentially correlation-based guidance schemes; the fundamental quantity used as one of the Kalman filter inputs is a position fix derived from the correlation between the sensed and stored terrain- 46 clearance data. A significantly different approach treats each radar altimeter terrain-clearance profile as an input to the Kalman filter. Such a concept has been pursued at Sandia National Laboratories since 1974 and is known as SITAN (Sandia Inertial Terrain Aided Navigation) [32][33][34]. SITAN has the ability to model significant INS and measurement errors into the Kalman filter and, hence, reduces the sensitivity of the navigation solution to these errors. The system is said to operate “continuously” as it does not involve complex matching algorithms to account for vehicle maneuvers. The covariance analysis inherent to Kalman filtering offers the ability to predict system performance. Due to the undulating nature of the terrain, the radar altimeter’s terrain-clearance measurement is a nonlinear function of vehicle position and since the user position with respect to the terrain is constantly changing, the nonlinearities are functions of time. These nonlinearities must be properly accounted for to achieve satisfactory performance. A block diagram of the SITAN system is shown in Figure 4.3 [33].

BAROMETRIC ALTIMETER ALTITUDE CORRECTION

ALTITUDE MODIFIED KALMAN NAVIGATOR POSITION FILTER

DIGITAL TERRAIN MAP PARAMETERS RADAR ALTIMETER

POSITION CORRECTION

Figure 4.3 – The SITAN System

SITAN takes a linearization approach for treating the terrain nonlinearities by applying an Extended Kalman Filter (EKF) and aiding this filter by terrain slope information. The standard EKF approach to linearization is to expand a nonlinear measurement function in a Taylor series about the current state estimates, retain the linear terms and neglect the higher order terms. A 27m circular error probability (CEP) was achieved using this approach, which is a significant improvement over a 120 m CEP using an unaided inertial system [32]. A drawback of SITAN is a high probability of filter divergence due to the highly nonlinear nature of the terrain, especially, 47 when the linearization error is comparable in magnitude to the measurement error. A stochastic linearization approach for obtaining approximate nonlinear filters is another variation of the system. Stochastic linearization applies a least squares plane fit to the terrain map in order to compute the slopes, which are then input to the EKF [32]. Parallel SITAN [35] uses a bank of three or five state EKFs that process identical altimeter measurements but are centered at different latitudes and longitudes within a circular uncertainty region around the true position. After some updates, the filter with the minimum average weighted residual squared (AWRS) value is identified as having the correct position estimate.

The relationship between Bayes’ theorem and the Kalman filter is explored in [36], the application of Bayes’ theorem to Terrain Contour Navigation being the subject of the work. A statistical approach to TRN using the Bayes’ theorem is also presented in [29]. The TRN method described in [36] is based on the concept of the “likelihood function”. This “likelihood function” is considered Gaussian for a sufficient length of the vertical terrain profile measurements (called ‘a batch of Terrain Contour Navigation data’). The probability density of a set of vertical terrain profile data is a function of the aircraft position. If an aircraft measures a set of vertical terrain

profile data YM, which is very similar (‘similar’ due to the presence of noise) to a vertical terrain profile YD from a terrain database corresponding to a position XD, then there is a high probability

that the aircraft is located at that position XD. This is the concept that is stated formally and expressed mathematically by the “likelihood function”.

A correlation-based TRN scheme that operates in the frequency domain using a Discrete Cosine Transform (DCT) is presented in [37]. TERCOM correlation algorithms require long terrain profile data lengths of about 5-10 km and do not have good accuracy in high noise environments. SITAN systems diverge when the uncertainty region is larger than several hundred meters. Some of these deficiencies in conventional terrain correlation approaches are overcome using the frequency domain technique. In principle, time domain or frequency domain processing are equivalent in terms of performance, however, frequency domain processing techniques often allow for reduced computational complexity and increased observability. In the frequency domain correlation system (implemented separately from the navigation system), measured terrain profile data is correlated in the frequency domain with reference elevations stored in map format to determine a two-dimensional position error estimate. The result of this correlation is 48 input to the independently operating navigation system for a better estimate of the vehicle position.

ALTIMETER (RADAR, BARO)

DISCRETE COSINE TRANSFORM

DEAD RECKONING NAVIGATION FREQUENCY DIGITAL MAP DOMAIN KALMAN GENERATOR CORRELATION SENSORS FILTER PROCESSOR

Figure 4.4 – Frequency Domain Terrain Correlation Scheme of [37]

Variations of terrain correlation techniques that aid in vehicle navigation and guidance are seemingly endless as evidenced by the numerous U.S. patents issued in this area. However, it is interesting to study some of them conceptually, to comprehend the different ways of generating the state estimates and the filtering techniques used to arrive at the optimum.

In [38], a method of estimating the vehicle position using a Kalman filter whose input position estimates were arrived at in two ways is described. Correlating ‘point-by-point’ data samples of measured terrain profile with a ‘point-by-point’ map terrain profile (the points/positions on the map provided by the navigation system) at equally spaced time intervals provides the first three- dimensional position estimate. A similar computation at the next time instant provides the second position estimate. A velocity vector is computed using the two position estimates. The third position estimate is computed in two ways: one, similar to the first two estimates, and the other by projecting the velocity vector forward in time. These two position estimates are then input to a Kalman filter to arrive at a better estimate. In [39], an autonomous navigational system is disclosed that updates a vehicle’s position and heading coordinates at various checkpoints along a predefined flight path for a pilot-less airborne vehicle. It uses radar altimeter sensed terrain profile information to compute the vehicle position at each of the various checkpoints and update the navigation system periodically. 49 Another U.S. patent that is noteworthy is [40] that uses aircraft weather radar to obtain terrain reflectivity data. The terrain reflectivity data is then compared to terrain data stored in a terrain database using a test statistic. The test statistic indicates the degree to which the radar terrain reflectivity data and the terrain database data agree. Note that the method uses GPS derived vehicle positions to access the terrain database contrary to the formerly mentioned methods that have historically used an INS as the primary navigation system. One of the limitations of the method in [40] that will be addressed in the terrain referenced navigation section of Chapter 6 in this thesis is when the ground has little or no variation within the radar map area. In such cases the test statistic is insensitive to errors in latitude and longitude (horizontal errors) but is still sensitive to altitude errors (vertical errors). Over moderately hilly terrain that offers sufficient variation, it is not only possible to reject the current aircraft position as highly unlikely (as evidenced by the test statistic), but also to search the digital terrain database for the most probable aircraft position.

A recent U.S. patent [41] describes a day/night, all weather terrain aided navigation method that uses a narrow beam, low power, frequency hopping, spread-spectrum emission radar altimeter as its primary sensor. The radar altimeter consists of one transmitter and multiple receivers. This method differs from other TRN methods in a way that it utilizes the radar sensed terrain features to aid a primary navigation solution (state estimate) rather than for position recognition only by comparing with an on-board terrain database. The primary navigation solution is computed using any or all of the following: satellite based, inertial, ground based, Doppler and dead reckoning navigation. Correlation of the synthesized terrain profile and database profile yields the expected position/orientation of the sensed terrain features with respect to the primary navigation solution. The difference between the expected and the primary positions is input to a processor that updates the primary navigation solution to minimize the navigation error.

A significantly different TRN approach is the Continuous Visual Navigation (CVN) system described in [42]. Instead of terrain contour or terrain feature matching schemes, [42] uses a scheme for matching linear features that are extracted from terrain images. A DWL video/infra- red camera is used to take a picture of the terrain features and man-made objects like houses, roads, a line or trees, etc. Linear features are extracted from the picture, using image-processing techniques. Based on an INS position estimate, expected linear features are predicted from on- 50 board databases and compared to observed linear features. The offsets between the predicted and observed features are used to refine the INS error components.

Developments in laser technology have enabled the use of airborne laser scanners for TRN applications. Such a concept has been explored in [43] that uses a laser range scanner to sense terrain elevations a high-resolution terrain database (2m post-spacing). Data from the laser range scanner and IMU is integrated to the terrain database to estimate the aircraft position using a test statistic approach. Data from the laser range scanner and GPS is integrated to estimate the aircraft attitude.

A system that is currently being used on a variety of aircraft, especially military aircraft, is BAE Systems’ (merger of British Aerospace plc and General Electric Company’s (GEC) Marconi Electronic Systems) TERPROM®, a terrain profile matching system. TERPROM® uses stored digital terrain elevation data, and inputs from the aircraft’s navigation system and radar altimeter to produce a highly accurate terrain-referenced navigation solution. Its applications include terrain-referenced navigation, predictive ground collision avoidance, missile guidance and passive ranging. Many enhancements to the algorithm during the 1990s have helped in the development of TERPROM® II which can achieve an accuracy of better than 20m. More information on TERPROM® can be found at the BAE Systems website [44]. A comparative table of the aforementioned methods is provided in Table 4.1. 51

Table 4.1 A comparison of Several TRN Systems TRN Method Sensors Integration Principle Ref. Radar, baro-altimeter, Correlation, Kalman [31][32][33] TERCOM IMU, Terrain database Filtering of position estimates SITAN Radar, baro-altimeter, Extended Kalman Filter [32][33][34] IMU, Terrain database Parallel SITAN Radar, baro-altimeter, Bank of 3-5 state [35] IMU, Terrain database Kalman Filters Bayesian Approach to Radar, baro-altimeter, “Likelihood Function” [36] Terrain Contour Navigation IMU, Terrain database System for Correlation and Radar, baro-altimeter, Discrete Cosine [37] Recognition of Terrain IMU, Terrain database Transform, Kalman Elevation Filter Radar, baro-altimeter, Kalman Filter to [38] Vehicle Guidance System IMU, Terrain database combine two position estimates Autonomous Check-Pointing Radar altimeter, IMU, Terrain feature [39] Navigational System for an Terrain feature matching Airborne Vehicle database Aircraft Position Validation Weather radar, Terrain Comparison using test [40] Using Radar and Digital database statistic Terrain Elevation Database Method and System for Low power, frequency Aiding a primary [41] Terrain Aided Navigation hopping, spread navigation solution spectrum radar altimeter Future Terrain Referenced DWL video/IR camera, Linear feature matching [42] Navigation Techniques linear feature database Exploiting Sensor Synergy Light Detection and Laser range scanner, Comparison using test [43] Ranging-Based Terrain high-resolution terrain statistic, Kalman filter Navigation – A Concept database, GPS, IMU Exploration TERPROM® Radar altimeter, Terrain profile matching [44] aircraft’s navigation solution, terrain database

52

5

THEORETICAL BACKGROUND

The use of sensors for the measurement of a physical quantity never yields its exact value. There always exists an error, some deviation from the actual value. If the measurements deviate from the actual value by a constant amount, then there is a bias present. If the deviation of measurements increases or decreases linearly as a function of time, the errors are said to exhibit a ramp, or alternately, a non-linear exponential that generally tends to increase over time. Errors that can be classified as a higher order polynomial are rare in the practical world, and if present, they are usually the result of an incorrect system model. Errors whose magnitudes do not exhibit a regular trend, in other words, which are random but seem to be clustered together are called noise. A broader classification of noise is any unwanted signal, may it be as interference or as a simple deviation in the measurement from an exact value. We restrict ourselves to random errors whose occurrence has a regular trend or distribution.

Wherever noise is present, methods exist to reduce the noise level, provided, of course, that its nature is understood. A historical summary of the approaches for noise mitigation and improvement of system performance follows:

a) Low-pass, high-pass, and band-pass filters are used to suppress certain frequency components in the analog and digital domain. Examples of digital filters are the FIR 53 (Finite Impulse Response) and IIR (Infinite Impulse Response). Whether analog or digital, the filter operation can be observed in the frequency domain, i.e., an insight on the frequencies of the signals and noise was necessary for proper design of the filters. The Fourier Transform and its inverse are useful tools to go back and forth between time and frequency domains. The Fourier transform (and its inverse) for discrete sequences of finite length, N, called the Discrete Fourier Transform (DFT, IDFT) is given by:

⎛ n−1 ⎞ N − j2π ( k −1 )⎜ ⎟ X( k ) = F{}x( n ) = ∑ x( n )e ⎝ N ⎠ 1 ≤ k ≤ N n=1 (5.1) ⎛ n−1 ⎞ 1 N j2π ( k −1 )⎜ ⎟ x( n ) = F −1{}X( k ) = ∑ X( k )e ⎝ N ⎠ 1 ≤ n ≤ N N k =1

where, x( n ) is the discrete time sequence and X( k ) is its spectrum. b) The development of State Space Models allowed the visualization of system processes in the time domain. The classical transfer function methods of control theory were essentially frequency domain techniques. State space models are especially useful for multi-input multi-output (MIMO) continuous or discrete time systems. The state space models helped the formulation of time domain filter algorithms like the Weiner filter and the Kalman filter. c) Use of Fourier transform methods for direct estimation of the Power Spectral Density (PSD) has the drawback of having poor frequency resolution. Many alternative techniques that promise better spectral (frequency) resolution have been developed, known as the modern methods of spectral estimation.

This thesis principally investigates the Kalman filter, the Autocorrelation Function estimator and the Maximum Entropy Spectral Estimator for applications with terrain database integrity monitors and TRN.

Discrete time processes may arise in two ways, either from events that take place naturally in discrete steps or from sampling a continuous process at discrete times. The sampling interval for discretization may be under the control of the designers or may be forced upon them due to measurement constraints. Such is the case for choosing a measurement interval of 1 second for 54 the RA in the integrity-monitoring concept [1]. Considering the speed of the aircraft and the RA beamwidth, as well as the RA operating principle (continuous wave or pulsed), the measurement interval must be sufficiently long so as to ensure that the RA measurements are independent, i.e. the RA beam illuminates a different cell of the terrain database for each measurement. For this reason, the absolute disparities are considered to be representative of an independent Gaussian noise sequence. A one second measurement interval is ideal because all measurements can be synchronized to GPS time (GPST) using the pulse-per-second (PPS) signal output from most commercial GPS receivers.

But whatever the reason for discretization to arise, it is desirable to fit all processes to the following discrete time model based on [45]:

xk +1 = φk xk + wk (5.2)

yk = Bk xk + vk

where, xk = vector state of the process/system at time tk

φk = matrix that relates xk to xk +1 in the absence of a forcing function (in the sampled version of a continuous process, this is the state transition matrix)

wk ,vk = vectors whose elements are white sequences

Bk = linear connection matrix between output yk and state xk

Estimation of a signal in the presence of noise can be implemented through a whole series of recursive algorithms that estimate the desired signal parameters based on some prior knowledge of the system. Though these algorithms are referred to as ‘filters’, as their function is similar to classical analog filters, there is nothing physically tangible about them. They are a set of mathematical equations to be solved recursively and they provide an expression for the estimates. Since a priori knowledge is essential for the equations, the better the system model, the better will be the estimates. A Kalman filter is one such filter whose performance depends upon the system model and the noise characteristics.

This chapter is intended to introduce the concepts used in this thesis. The implementation results will be discussed in the next two chapters. In section 5.1, the Kalman filter is discussed within a 55 scope having direct bearing to this work. The filter equations are listed followed by a discussion of the tradeoffs that arise from the usage of the filter. Section 5.2 presents a simple estimation scheme by exploring the properties of a finite length autocorrelation function of the absolute disparities. The theory and mathematical framework pertaining to the Blackman-Tukey (BT) spectral estimation and the Maximum Entropy Spectral Estimation (MESE) are the main focus of Section 5.3.

5.1 The Kalman Filter

In 1960 R. E. Kalman published his paper that describes the linear filtering problem of discrete measurement data [46]. More than a decade prior to the appearance of Kalman’s paper, pioneer work in the field of stationary statistics and control theory led to the formulation of the Wiener filter. Methods for designing the Wiener filter suffer from many limitations as described in [46] that restrict their practical usefulness. An important class of theoretical and practical problems in communication and control theory are of a statistical nature. These problems include: (i) prediction of random signals; (ii) separation of random signals from random noise; and (iii) detection of signals of known form (pulses, sinusoids) in the presence of random noise. The Kalman filter provides a set of linear recursive equations that can be implemented practically to solve these problems. For a more explicit understanding of the types of problems, consider the following situation. Let the signal be x( t ) and the noise be η( t ). Only the sum

y( t ) = x( t ) +η( t ) can be measured. Suppose that the exact values y( t0 ),..., y( t ) have been

observed. From these observations, the value of the signal at t = t1 must be inferred. t1 may be

less than, equal to or greater than t . If t1 < t , the process of inference is referred to as data

smoothing (interpolation). If t1 = t , this process is referred to as filtering. And if t1 > t , the process is referred to as prediction. Since a Kalman filter is sufficiently generic to include all of these and similar problems, the filter operation is called estimation [46]. Because of its potential benefits, the Kalman filter is used extensively for optimal estimation and sensor fusion in navigation and tracking problems.

Various types of Kalman filters can be found in literature including the linearized, extended and complementary Kalman filters. A Kalman filter consists of a set of linear equations. State space models of non-linear systems must first be linearized to use the Kalman filter equations. When 56 linearization is done about a pre-determined nominal trajectory, it is called a Linearized Kalman Filter. Here, the term trajectory is used in a broad sense to denote the changing values of the state variables over time, though it is easier to visualize as the progress of a vehicle’s position coordinates over time. When the state space models are linearized about the filter’s predicted trajectory, it is called an Extended Kalman Filter (EKF). An EKF estimates the incremental state variables that are added to the predictions to obtain an estimate of the total state variables. A Complementary Kalman Filter uses independent measurements made at the same time from two different instruments. One is considered a reference/nominal and other is linearized about the former. The measurements from the different sensors may have different accuracy and noise characteristics and the filter endeavors to obtain the best of both worlds, like the integration of GPS and INS. Many Kalman filter variations are described in literature [45] but there is only one Kalman Filter. All Kalman filter variations mainly differ in the mathematical formulation of the system model; irrespective of how the linearization is carried out, if one is consistent in using the state variables, the basic Kalman filter equations are the same. The basic Kalman filter equations are reproduced here from [45]. Many alternative forms for the equations have been formulated that are available in most textbooks, which aim to reduce the computational complexity. This is not considered an issue for this thesis whose objective is to evaluate the performance of an integrity monitoring/navigation method that uses a Kalman filter, and not to evaluate the performance of the filter itself.

The random process to be estimated is modeled in the form

xk +1 = φk xk + wk (5.3)

The observation (measurement) of the process is assumed to occur at discrete points in time according to the linear relationship

zk = Η k xk + vk (5.4) where,

xk = (n ×1) process state vector at time tk . n is the dimension of the vector equal to the number of state variables.

φk = (n × n) matrix relating xk to xk +1 in the absence of a forcing function. 57

wk = (n ×1) process noise vector – assumed to be a white sequence with known covariance structure.

zk = (m ×1) measurement vector at time tk .

Η k = (m × n) matrix giving the ideal (noiseless) connection between the measurement and the

state vector at time tk . This matrix is also referred to informally as the domain transformation matrix.

vk = (m ×1) measurement error vector – assumed to be a white sequence with known

covariance structure and having zero cross-correlation with the wk sequence.

The covariance matrices for the wk and vk vectors are given by

Τ ⎧Qk , i = k E[]wk wi = ⎨ (5.5) ⎩ 0, i ≠ k

Τ ⎧Rk , i = k E[]vkvi = ⎨ (5.6) ⎩ 0, i ≠ k

Τ E[wkvi ]= 0 for all k and i (5.7)

where E[]⋅ denotes the expectation operator and ‘ T ’ denotes the transpose.

A Kalman filter is used on the absolute disparities described in Chapter 6 to reduce the nominal noise (on sensors + terrain database) and estimate a potential bias error (on sensors + terrain database). If the nominal errors are Gaussian, the Kalman filter is an optimal estimator for the bias error in the Minimum Mean Squared Error (MMSE) sense. The system model for the absolute disparities is given by a bias component plus a noise component or z = µ +η , where η is zero mean Gaussian noise with a standard deviation of 18.9 meters for the absolute disparity model described in Chapter 3. The Kalman filter equations to be implemented for recursive estimation are given by the following sequence:

− − 1. Specify initial prediction xˆ0 and the initial prediction error variance P0 . 2. Compute the Kalman filter Gain:

− Τ − Τ −1 Kk = Pk Η k (Η k Pk Η k + Rk ) (5.8) 58 3. Update Estimate with Measurement:

− − ˆxk = ˆxk + Kk (zk − Η k ˆxk ) (5.9) 4. Compute estimation error Variance:

− Pk = (Ι − Kk Η k )Pk (5.10) 5. Predict ahead: ˆx− = φ ˆx k +1 k k (5.11) − Τ Pk +1 = φk Pkφk + Qk 6. Repeat from Step 2.

where the ‘^’ (hat) denotes an estimate and the ‘ – ’ (super minus) denotes an estimate formed without using the current measurement, in other words, a prediction, and,

− ˆx0 is zero as per the assumed system model,

− 2 2 P0 can be a value, roughly between (15) and (20) (Refer Appendix),

φk the state transition matrix is unity,

zk are the measurements at time tk ; absolute disparities,

Η k is the domain transformation matrix; unity,

2 Rk is the measurement error variance; the variance on the absolute disparities (18.9) , constant,

Qk is the system noise variance, also called the tuning parameter of the filter, also a constant.

5.1.1 Test Statistic Using Kalman Estimates Chapter 3 described the T statistic (T-value) as a ratio of the sample variance to the population variance, scaled by the number of samples (equation (3.7), (3.9)). Consider a situation in which an aircraft flies over the entire area described in the terrain database and uses its sensors to measure each terrain database elevation. When the filter is run for all the absolute disparities, the statistical variance of the estimates, given by the P matrix in the Kalman recursion, would be the variance of the complete population of filtered estimates over the terrain database area. Note that value of the P matrix is only a theoretical value computed by the filter and only represents the true value for correctly modeled linear systems. Thus, the T-value using the filtered estimates can be computed as follows: 59 1 N ˆ 2 TKF = ∑ x (tk ) (5.12) P k=1

Based on the choice for the tuning parameter Q, the variance of the estimates, P, attains a constant steady-state value over time. Q can be varied in such a way that P attains a desired value.

Then, the value σ p in equation (3.10) can be substituted by P and a new MDB, µB, for the

Kalman filtered case can be computed. Or, given a required value of µB, the desired estimator variance P and tuning parameter Q can be computed. Figure 5.1 shows a plot of both P and Q as a function of µB.

1000 Variance of Estimates, P 900 Tuning parameter, Q

800

700

600 2

ter 500 me 400

300

200

100

0 0 5 10 15 20 25 30 Minimum Detectable Bias (meters)

Figure 5.1 – P and Q as a function of µB

The Cramer-Rao lower bound on the variance of a constant bias value in the presence of white

2 Gaussian noise is the variance of the ordinary least squares filter [47], i.e. σ P / N = 7.14 and a lower bound on the µB is found to equal 4.84 meters by substituting the Cramer-Rao lower bound on the variance into equation (3.10).

However, the application of a Kalman filter (or any filter) to filter the absolute disparities invalidates the basic assumption that the underlying random variables, xi, are independent; 60 equations (3.7) and (3.9) are only valid for independent samples taken from a normal distribution. The filtered estimates are no longer independent but are highly correlated and equation (5.12) no longer applies in a strict sense. The amount of correlation of the estimator outputs, in time, depends on the filter’s tuning parameter, Q. Considering independent measurements, a large value of Q assumes a noisy process (system) and puts more weight on the measurements. A small value of Q assumes a linear system and puts more weight on the filter’s predictions of the state variables, rather than on the measurements resulting in highly correlated estimates. Thus,

for large values of Q (and thus µB), equation (5.12) still applies without violating the independency assumption. This is further explored in the next section.

5.1.2 Pseudo-Random Noise Analysis Revisited A Monte Carlo analysis with 10000 simulations was run to approximate the autocorrelation functions of unfiltered Gaussian noise and Kalman filtered Gaussian noise. Values for Q were

derived from two values for the minimum detectable bias, µB; a value of µB equal to 25 meters

and a value of µB equal to 15 meters. Figure 5.2 shows that decreasing µB and therefore the tuning parameter Q results in an increase of the sample de-correlation time. The same notion is conveyed by the correlation coefficients as a function of time difference between samples in

Figure 5.3. The plot labeled ‘Unfiltered Noise’ corresponds to a µB of 34.2 m.

200 Unfiltered Noise µ = 25m B µ = 15m B 150

100

50

0

-50 -50 0 50 2N-1 length Autocorrelation Sequence

Figure 5.2 – Autocorrelation Sequence of Kalman Estimates for µB of 34.2m, 25m and 15m 61 1.2 Unfiltered Noise µ = 25m B 1 µ = 15m B )

ρ 0.8 t ( n

0.6

0.4 elation Coefficie r 0.2 Cor

0

-0.2 0 10 20 30 40 50 Time

Figure 5.3 – Correlation Coefficient (ρ) of Kalman Estimates for µB of 34.2m, 25m and 15m

To obtain near-independent samples with minimum time-to-alarm when using a Kalman filter, an estimate must be picked every 5 seconds. This would reduce the degrees of freedom in equation (3.7) by a factor of 5. In order to maintain the same number of degrees of freedom, the number of

1 Unfiltered Noise = 25m µB 0.9 = 15m µB

0.8

0.7

0.6 0 H of t 0.5 accep P 0.4

0.3

0.2

0.1

0 0 5 10 15 20 25 30 35 Bias (meters)

Figure 5.4 – OC Curve of Kalman estimates for µB of 34.2m, 25m and 15m 62 original samples can be increased five-fold. It must be noted that such an increase in samples would result in a time-to-alert which is 5 times as long as the original time-to-alert.

The operating characteristic curves for the three cases discussed in Figure 5.2 and Figure 5.3 are

shown in Figure 5.4. The OC curve for µB of 25 meters looks encouraging, with the probability of accepting H0 becoming very small.

5.2 Autocorrelation Function Estimation

Consider a Wide Sense Stationary (WSS) random process, x , having an infinite number of samples. The mean and variance of process x are X and σ 2 , respectively. The autocorrelation function (ACF) of this process would look similar to Figure 5.5 [48]. As the time delay τ approaches infinity, the ACF tends to settle to a constant value equal to the square of the mean

2 2 (X ). The difference between the value of the ACF at τ = 0 (denoted by Rxx (0)) and (X ) is

the variance of the process σ 2 .

RXX

σ 2

X 2

τ Figure 5.5 – Autocorrelation Function of a WSS Random Process

Now, consider a constant process or signal with a finite number of samples:

x1 = x2 = ... = xN = X = c . The autocorrelation function of this signal is a (2N-1) length sequence given by the formula: 63 1 N Rxx(τ ) = ∑ x( k )x( k +τ ) (5.13) N k =1

for τ = −( N −1) to ( N − 1)

The autocorrelation function of this constant process is plotted in Figure 5.6.

250

200

150

100

50

0 -50 -40 -30 -20 -10 0 10 20 30 40 50 ← τ →

Figure 5.6 – Autocorrelation of finite length constant signal

Shown in figure 5.6 is the autocorrelation function of a 50-sample length constant signal with magnitude A=15. From Figure 5.6 it can be observed that Rxx (0) =15 = A.

600

500

400

300

200

100

0 -50 -40 -30 -20 -10 0 10 20 30 40 50 ← τ →

250

200

150

100

50

0 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 ← τ → Figure 5.7a, 5.7b – Autocorrelation of 50 sample length Gaussian noise having mean 15 and σ = 18.9 64 Next, zero-mean Gaussian random noise with a standard deviation of σ = 18.9 is added to the signal. The average autocorrelation function was derived from 1000 Monte-Carlo simulations and is shown in Figure 5.7a. Figure 5.7b shows a magnified view of the first 49 autocorrelation

samples. Rxx(0) can be found by an extrapolation of the previous 49 samples in the sequence. Since the samples form a straight line, a first-order least-squares polynomial fit is used to estimate

Rxx(0), the square of the bias. An example is shown in Figure 5.8.

250

209.28

150

100

50

0 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 ← τ →

Figure 5.8 – Rxx(0) Estimation Using a Straight Line Fit

In this example, Rxx (0) = 209.28 = 14.467 . The mean of 1000 Monte-Carlo simulations for a range of biases from 0m to 50m (Figure 5.9a.) shows poor estimates for lower biases of up to 10m and more accurate estimates for larger biases. A similar idea is conveyed by their standard deviations in Figure 5.9b that are less than 3m.

The ACF method is useful for estimating biases in the vertical direction. For horizontal translations/biases in the DEM, the correlation coefficient for the synthesized terrain profile and the DEM terrain profile is computed at zero lag.

1 N ⎡(h ( t )− h ) (h ( t ) − h )⎤ ρ = ⎢ SYNT i SYNT DEM i DEM ⎥ (5.14) N ∑ σ σ i=1 ⎣⎢ hSYNT hDEM ⎦⎥

The correlation coefficient is computed at each point of the search grid described in the previous section and the point having the maximum correlation coefficient is chosen as the most probable horizontal aircraft position with reference to the DEM coordinate frame. 65 Mean of Bias Estimates 50

45

40

35

30

25

20

15

10

5

0 0 5 10 15 20 25 30 35 40 45 50

Standard Deviation of Bias Estimates 3.5

3

2.5

2

1.5 0 5 10 15 20 25 30 35 40 45 50 Bias (meters) →

Figure 5.9a, 5.9b – Mean and Standard Deviation of Bias Estimate

5.3 Modern Spectral Estimation

Improved performance in estimating the power spectral density function for short data segments is the major advantage of modern spectral estimation techniques over traditional methods. A good overview of both classical and modern spectral estimation techniques can be found in [49]. The power spectral density (PSD) is a measure of the power/strength of a signal at various frequencies. If the terrain database has a vertical bias, the absolute disparities tend to exhibit a bias that corresponds to a PSD component at zero frequency. If the terrain database has a horizontal bias, in the presence of significant terrain undulations, the absolute disparities exhibit a high noise component that corresponds to power at higher frequencies of the PSD. Estimation of the PSD of the absolute disparities and analysis of the power contained at various frequencies can 66 provide insights about the agreement of the terrain features. The equations of two methods of spectral estimation are provided here followed by a comparison of their spectral resolutions; Blackman-Tukey (BT) spectral estimation and Maximum Entropy spectrum estimation (MESE).

5.3.1 Blackman – Tukey Spectral Estimation One class of spectral estimation techniques referred to as the classical method estimates the PSD based on known ACF samples. An example of the classical methods is the Blackman-Tukey (BT) Spectral Estimator:

( n−1 ) ˆ − j2πfk PBT ( f ) = ∑w[ k ]rxx [ k ]e (5.15) k =−( N −1 )

where the ‘^’ represents an estimate, rxx are known ACF values and w[k] is a Bartlett lag window given by:

⎧ k ⎪1 − k ≤ N − 1 ⎪ N − 1 (5.16) w[]k = ⎨ ⎪ 0 k ≥ N − 1 ⎪ ⎩

One disadvantage of the BT spectral estimator is that it can only resolve detail to 1 (N −1) ≅ 0.02 cycles/sample, for N = 50 successive measurements, resulting in ‘spreading’ or ‘smoothing’ of the spectrum. Another disadvantage is the appearance of side-lobes due to the windowing function [49].

5.3.2 Maximum Entropy Spectral Estimation A modern spectral estimation technique that results in a higher resolution PSD is the Maximum Entropy Spectral Estimator (MESE). The MESE is equivalent to an Auto-Regressive (AR) spectral estimator for Gaussian random processes and known ACF samples. MESE has been selected as a spectral estimation technique because it involves little complexity in contrast to other AR spectral estimators that involve an unwieldy computation of the AR parameters. 67 The power at a certain frequency is given by:

σ 2 Pxx( f ) = (5.17) N −1 2 1+ ∑a[ k ]e− j2πfk k =1

where { a[1],a[ 2],...,a[ N − 1],σ 2 } are found by solving the Yule-Walker equations using known ACF samples expressed as:

2 ⎡ rxx [0 ] rxx [ −1] L rxx [ −( N − 1)]⎤⎡ 1 ⎤ ⎡σ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ r [1] r [0 ] r [ −( N − 2 )] a[1] 0 ⎢ xx xx L xx ⎥⎢ ⎥ = ⎢ ⎥ (5.18) ⎢ ⎥ ⎢ M M O M ⎥⎢ M ⎥ M ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣rxx [ N − 1] rxx [ N − 2] L rxx [0 ] ⎦⎣a[ N − 1]⎦ ⎣⎢ 0 ⎦⎥

The difference in spectral resolution between the classical and modern methods described in this section is illustrated in Figure 5.10 using pseudo-random Gaussian noise sequences generated in MATLAB™.

80 BT MESE MESE

70

] 60 B y [d t i s n e D

l 50 a ectr p S r e w

o 40 P

30

20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Frequency (cycles/sample) Figure 5.10 – Spectral Estimates Using Blackman-Tukey and Maximum Entropy Methods

68 The BT spectral estimate of the PSD appears flat due to its limited resolution; the MESE on the other hand shows a less ‘smooth’ graph. The dotted line is the MESE estimate of the PSD at double the resolution. The power spectrum component at zero frequency can be attributed to the bias component of the signal, whereas the power at higher frequencies can be attributed to the noise component. Henceforth, the lower resolution of MESE is used to compute the PSD, which is still better than the BT spectral estimate. For the MESE to be applicable in DEM integrity monitoring, a threshold must be chosen above which to reject the Null hypothesis and accept the Alternative hypothesis. The plots that follow would aid such a decision.

Figure 5.11a and Figure 5.11b show three different Monte-Carlo runs of 1000 simulations for biases ranging from 0m to 40m. The mean value of the zero frequency PSD estimates computed for the 1000 simulations for each bias as well as the minimum value at each bias is plotted versus the inserted bias in Figure 5.11a. Figure 5.11b is a plot of the mean and the minimum value of the sum of the higher frequency PSD estimates at each bias over 1000 simulations versus the inserted bias.

Power at Different Biases at Zero Frequency Power at Different Biases at Higher Frequency 120 55 Mean Mean Minimum Minimum

100 50

80

45 ] ] B B d d [ [ r 60 r e e w w Po Po 40

40

35 20

0 30 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Bias (meters) Bias (meters) Figure 5.11a – Zero Frequency Power Estimates for Various Vertical Biases Figure 5.11b – Higher Frequency Power Estimates for Various Vertical Biases

The zero-frequency power spectrum component shows an increase in magnitude as a function of in the inserted bias. The mean value, however, is very noisy. A reasonable value to which to set the threshold can be found from the Monte-Carlo graph and equals 20dB. The noise power spectrum components at higher frequencies show a decreasing trend with increasing biases, but it is not a correct assumption to expect the ADs with lower inserted biases to exhibit high noise 69 power since our assumption of Gaussian noise characteristics is an over-bound value and the actual noise might be much lower. 35dB is taken as a reasonable threshold value for noise power. Different frequency resolutions require different thresholds to be selected for the noise power. As seen in Figure 5.11a and Figure 5.11b, the spectral estimation methods are noisy in character and for this reason; they are unlikely to provide the best results. It should be noted that the discussed spectral estimation method is highly qualitative and that statistical evaluation of the methods is still required in order for them to satisfy reliability requirements. Hence this method, as well as the ACF method must augment the statistical method in order to achieve improved performance.

Both the ACF and MESE methods do not have a strong statistical foundation as opposed to the statistical integrity monitoring method described in Chapter 3. The threshold values that are selected for the integrity monitoring purpose are based on a cautious conjecture, inspired by simulation results, instead of a theoretical derivation. Similar in approach to the statistical method, the Kalman filtering method goes a step further by using the filter estimates of the absolute disparities, but the ACF and MESE are drastically different approaches to the problem. For this reason, an augmentation scheme for the statistical method using the ACF and/or the MESE methods is described in the next chapter. 70

6

SPATIAL FAILURE DETECTION AND ESTIMATION

It is important to differentiate between ‘detection’ and ‘estimation’, as applicable to the present context. Broadly, the research presented in this thesis covers the concepts of Integrity Monitoring and Terrain Referenced Navigation (TRN). Assuming the occurrence of a failure, Integrity Monitoring involves the detection of the failure. Estimation and consequent correction of the failure constitutes TRN. The theoretical concepts presented in the previous chapter are all essentially estimation schemes, but their applicability has been explored for the dual purpose of Integrity Monitoring as well as Navigation. At first glance, this might seem a misappropriation, but a careful examination of the methods, their input data and their results would dispel such doubts. The process of detection is carried out by forming a test statistic and comparing it against a predetermined threshold. All four detection schemes, discussed in the previous chapter and whose results are provided in this chapter, follow this basic approach. Estimation schemes attempt to arrive at a solution that is as close to the truth as possible. The estimation schemes in this chapter follow the test statistic approach by selecting the minimum or the maximum value of the test statistic (as the case may be depending upon what the test statistic signifies) and the corresponding solution is considered an estimate of the truth.

The Kalman filter is used on the absolute disparities to reduce the measurement noise and estimate a potential bias. Hypothesis testing based on a test statistic is then used for detecting a 71 failure. The ACF estimator is useful for estimating biases in the vertical direction and the correlation coefficient is used as a tenable statistic for the detection of horizontal biases. Spectral estimation methods, like the MESE, are estimation schemes for the PSD, based on known autocorrelation samples, but are used here for detection of horizontal biases by selecting reasonable thresholds for power values. The following sections provide a description of the flight test, system components and subsequent results of applying the techniques introduced in Chapter 5 to the Integrity Monitoring and the Terrain Referenced Navigation problem.

6.1 EGE Flight Test and System Description

To determine the safety benefits of SVS displays for terrain awareness, NASA conducted research flight tests at the Eagle County, CO regional airport (FAA airport identifier: EGE) from August 21 to September 7, 2001. Six evaluation pilots, representing three airlines, FAA and Boeing flew eighty-seven approaches and departures to Runway 07 and 25 near Eagle/Vail’s mountainous terrain using NASA’s Airborne Research Integrated Experiments System (ARIES) [50]. ARIES is a modified Boeing 757-200 passenger jet. Two approaches to EGE were considered: an approach to Runway 25 and an approach to runway 07. Although runway 25 is the most common approach, tailwind restrictions might require a pilot to make the circle-to-land approach at runway 07. A photograph of the two approaches and a MATLAB™ generated view of the surrounding terrain using Jeppesen terrain database is shown in Figure 6.1a [50] and 6.1b.

Figure 6.1 – Runway 07 and 25 Approaches and Departures at EGE 72 The data used from approaches to both runways is the radar altimeter and GPS position information. The terrain database used is a Jeppesen terrain database with a 3 arc second post spacing data derived from the USGS’s National Elevation Dataset (NED) product. The ARIES is equipped with a prototype Rockwell-Collins WAAS receiver and triple redundant radar altimeters. Some issues related to these radar altimeters are discussed in the following subsection.

6.1.1 NASA ARIES Triple Redundant Radar Altimeter Three sets of transmitters and receivers constitute the triple redundant radar altimeter on the ARIES B-757-200 test aircraft. A triple redundant architecture is a common feature on most commercial aircraft to increase a system’s fault tolerance. In addition, each RA has two processors to calculate the height that provides integrity to the data. The RA antennas are located just in front of the wings on the belly of the aircraft. They are named right, center and left or 1, 2 and 3. The radar altimeters use a Frequency-Modulated Continuous Wave (FM-CW) modulation having a 3 dB beamwidth of 90º (± 45º) on each antenna. A study of the altimeter characteristics and their relative performance can be found in [2].

6.1.2 Slant Range Errors A radar altimeter measures terrain clearance or height AGL. This is absolutely true when the aircraft is in level flight over flat terrain wherein the radar altimeter measures the perpendicular height above the ground or “plumb-bob” height. Plumb-bob height is defined as the distance between the aircraft and the terrain along a vector originating from the center of the aircraft and pointing towards the center of the earth [2]. Whereas in most cases, when the aircraft is flying over mountainous terrain, the radar altimeter measures the distance to a point within its beam having the shortest reflection time or the strongest reflected signal power. The transmitted radar signal bounces off the sides of raised terrain (mountains) and reaches the receiver antennas faster and probably with higher signal power due to less free-space loss. The distance thus recorded by the radar altimeter is different from the plumb-bob height and referred to as the “Slant Range” as illustrated in Figure 6.2. Slant ranges are also measured over flat terrain when the aircraft makes a pitching or a rolling maneuver, when the radar antenna boresight no longer points vertically downwards. The FM-CW radar altimeter on the B757 ARIES measures an average or composite range to the terrain within its beam. Due to the large beamwidth of the radar altimeter, even level flight over flat terrain does not provide a ‘plumb-bob’ height measurement and a slant range 73 component is invariably present. Banking maneuvers of the aircraft has a similar effect of introducing slant range measurements. The terrain database height corresponding to a set of position coordinates always represents the plumb-bob height. This discrepancy in the two heights that are not truly symbolic of each other might be the primary cause of false alerts generated by the integrity monitor. In fact, results of terrain correlation show very poor agreement between the synthesized and database profiles when the radar altimeters measure the slant range.

hSlant _ Range Bob − umb Pl h

Figure 6.2 – Illustration of Plumb-Bob Height and Slant Range

One method that attempts to compensate for the slant range errors is the “spot-beam algorithm” [2] that uses the geometry of angles to predict the point where the radar antenna boresight intersects the terrain and chooses the database height, within the radar beam or “spot”, that gives the shortest range to the aircraft. Such an algorithm must consider the circularly distributed horizontal errors, a non-trivial issue. The scenario becomes even more complicated with the presence of horizontal biases on the terrain databases. Pitch and roll maneuvers violate the assumption of Gaussian noise on the absolute disparities and also result in mis-modeling of the Kalman filter. For this reason, data obtained during the banking phase of flight should not be used for the purpose of integrity monitoring. The results in this chapter were generated without any compensation for the slant range errors due to which, some phases of flight with high pitch and roll angles exceed the T statistic threshold and show very low terrain correlation. 74 6.1.3 Altitude Dependent Errors Radar altimeters are subject to altitude dependent bias errors. The altitude accuracy of the NASA 757-200 ARIES radar altimeters is specified as 1.0 ft or 2% of range; whichever is greater. The altitude error compensation has been included in the results after altitude dependent variable biases were observed in the initial results during different segments of flight in the EGE data. Other altitude dependent errors are: (i) The banking of aircraft at higher altitudes over terrain having significant undulations and (ii) Illumination of larger areas by the radar beam [28]. Suppose an aircraft flying over a mountain performs a banking maneuver and its radar altimeter beam skirts the mountain and illuminates a valley or the side of an adjoining mountain. The range calculated by the altimeter based on the radar returns will be much greater than the true plumb-bob height. Consider another scenario where an aircraft is flying through a valley. Its radar beam illuminates a certain area of the terrain underneath. Now, if the aircraft flies level along the same ground track but at a higher altitude, its radar beam illuminates a much larger area of the terrain underneath. The larger area might include the slopes of nearby mountains that provide faster and high-powered radar returns resulting in slant range measurements as a function of aircraft altitude. An illustration of these effects is shown in Figure 6.3 below.

Figure 6.3 – Illustration of Altitude Dependent Errors

6.1.4 Mid-Value Select A technique that is being used on many triple redundant systems, especially in Boeing aircraft, for increased reliability and fault tolerance is called Mid-Value Select. It means selecting the median, in magnitude, of the three redundant measurements. In case of a single failure, Mid- Value select will not select the faulty measurement. An example of the mid-value select scheme is shown in Figure 6.4a for sine waves and in Figure 6.4b using the three ARIES radar altimeter 75

1 300

0.8

0.6 290

0.4 280 0.2

0 270 -0.2

-0.4 260 S RA -0.6 1 1 S RA 2 2 S 250 RA -0.8 3 3 S RA mid mid -1 0 0.5 1 1.5 2 400 405 410 415 420 Figure 6.4 – Mid-Value Select Scheme measurements. The mid-value select scheme helps to alleviate the slant range measurement problem. An important consideration while using this scheme is the modified standard deviation of the measurements, that follows from order statistics. Monte-Carlo simulation results have shown that the standard deviation of noise on the measurements decreases by a factor of 0.67. Hence, the standard deviation of noise on the radar altimeter from Table 3.2 reduces to 0.67 ×1.8 = 1.206m and the convolved standard deviation of the noise on the absolute disparities is 18.8 m, not very different from the original 18.9 m. As it can be observed, the major contributor to the error budget is the terrain database.

6.2 Terrain Database Integrity Monitor

The integrity monitor reviewed in Chapter 3 calculates the absolute disparity between the synthesized height profile and the terrain database height profile and forms a chi-squared test statistic for hypothesis testing and decision-making. In this section the concepts presented in Chapter 5 (Kalman filtering, ACF, BT, and MESE) will be applied to the DEM integrity monitor and resulting performance improvements will be presented. The integrity monitoring operation in both the vertical domain and the horizontal domain will be discussed separately in section 6.2.1 and 6.2.2 and their integration into a Spatial Integrity Monitor is the focus of section 6.2.3.

The terrain database integrity monitor uses the absolute disparities, p( ti ), as its primary input. These absolute disparities consist of a noise component and, in the presence of a failure, a bias component. The MDB is one of the key parameters that determines the capabilities of the 76

integrity monitor and depends not only on PFFD, PMD and N, but also on the nominal noise variance of the absolute disparities. The lower the noise variance, the lower the value of the MDB that the integrity monitor is able to detect with a preset probability. Lowering the noise variance is exactly what the Kalman filter method endeavors to achieve. Considering the frequency spectrum of the absolute disparities, a bias component is a constant value and shows up at the zero frequency or very low frequency in the spectrum, whereas the noise component occupies all frequencies. By observing the power at various frequencies, it might be possible to isolate, or at least determine qualitatively, the contribution of each component. In the frequency domain version of the horizontal domain integrity monitor, the absolute disparities computed over a search grid of flat or slowly varying terrain like plains and fields tend to have biases on them, whereas those computed over a search grid of rough, mountainous terrain tend to have a large noise component. The rationale behind selecting the frequency domain is to achieve separation between the noise and bias components and improvement of the performance of the statistical integrity monitor method.

6.2.1 Vertical Domain Integrity Monitor

The values of the test statistic ‘T’, the test statistic based on the Kalman filter estimates, ‘TKF’, the correlation coefficient ‘ρ’, the bias power ‘PB’ and noise power ‘PN’ in dB along with their respective thresholds are shown in Figure 6.5 for three different biases: 0, 25m and 35m. Figure 6.5a is identical to Figure 3.8 with different scales. Comparison of the T-values in Figure 6.5a with the TKF-values Figure 6.5b shows an improved sensitivity to vertical biases and triggering of the integrity threshold for smaller vertical biases. In order to allow the Kalman filter to settle down to a steady state, 60 seconds of absolute disparities are run through the Kalman filter and

only the last 50 seconds of estimates used to compute the TKF statistic. Figure 6.5c shows the variation of the correlation coefficient ρ along the flight path during approaches to runway 25 at EGE on September 1, 2001. The values show good agreement between the synthesized and terrain database height profiles except during the flight segment just before the runway threshold. This poor correlation can be attributed to the slant range measurements of the radar altimeter during its banking maneuver as discussed in sections 6.1.2 and 6.1.3. The effects of the same can be seen in the other plots too. An interesting observation in Figure 6.5c is that the ρ values are almost the same for all three bias values. The reason for this lies in equation (5.14) wherein the mean of the heights is subtracted out in the formula. For this reason, the correlation coefficient, by itself, is not useful for the integrity monitoring application. In the next sections we will see 77 that the ρ value is applicable for TRN. Figure 6.5d and Figure 6.5e show an increase in the power values for larger biases. Although this is a desirable trend, the high amount of jitter and spikes in the power values limit the usefulness of the magnitude of power as a measure for integrity monitoring.

Bias = 0 Bias = 0

200 200 c i t s s i e t i t a i t 100

ar 100 T S sp i D

e

0 t 0 u

-200 -150 -100 -50 0 50 100 150 200 l -200 -150 -100 -50 0 50 100 150 200 Bias = 25m Bias = 25m so b A 200 200 ed c i t er s t i l t i a F t 100 100 an T S m l a

0 K 0 -200 -150 -100 -50 0 50 100 150 200 g -200 -150 -100 -50 0 50 100 150 200 n

Bias = 35m i Bias = 35m s c U i 200 t 200 s c i t ati s t i t S a t 100 100 KF T T S

0 0 -200 -150 -100 -50 0 50 100 150 200 -200 -150 -100 -50 0 50 100 150 200 Time From Threshold Crossing Time From Threshold Crossing a Bias = 0 b

1

0.5

0

) -200 -150 -100 -50 0 50 100 150 200

ρ Bias = 25m t (

en 1 ci i eff o

C 0.5 n o i at el

r 0 r

o -200 -150 -100 -50 0 50 100 150 200

C Bias = 35m

1

0.5

0 -200 -150 -100 -50 0 50 100 150 200 Time From Threshold Crossing c Bias = 0 Bias = 0

40 40 ) ) 20 N 20 B P P ( (

D 0 0 S PSD P f

of -20 -20 -200 -150 -100 -50 0 50 100 150 200 -200 -150 -100 -50 0 50 100 150 200 E Bias = 25m Bias = 25m S ESE o E M M

40 m 40 o r om f r

] f

] 20 20 B B d d [ r r [ 0 e 0 e w w o Po

P -20 -20

y y -200 -150 -100 -50 0 50 100 150 200 c -200 -150 -100 -50 0 50 100 150 200

Bias = 35m n Bias = 35m nc e u q que 40 e 40 r F

r Fre e o

r 20 20 h g i Ze 0 H 0

-20 -20 -200 -150 -100 -50 0 50 100 150 200 -200 -150 -100 -50 0 50 100 150 200 Time From Threshold Crossing Time From Threshold Crossing d e Figure 6.5 – Different Test Statistics During Approach to R/W 25 at EGE for Biases of 0, 25m and 35m 78 6.2.2 Horizontal Domain Integrity Monitor The MPDLIM scheme is used with the various test statistics to evaluate the performance of the integrity monitor in the horizontal/lateral domain. Each of the ACF and Spectral Estimation methods are used in conjunction with the statistical method by superimposing their regions of missed detections and choosing the common regions. A block diagram is provided to illustrate this scheme.

RoMD Correlation RoMD Bias RoMD Noise RoMD 'T' Statistic RoMD 'T' Statistic Coefficient Power Power

RoMD Correlation RoMD Spectral Augmentation Augmentation

Figure 6.6 – Augmentation Scheme for DEM Integrity Monitor

The improvement in the horizontal detection capability of the integrity monitor is illustrated by the RoMDs as shown in Figure 6.7. The ‘+’ sign at the grid coordinates (0,0) indicates the GPS position which is considered the aircraft’s ‘true’ position. The dark (blue) areas are the terrain database offset or probable aircraft positions (according to the DEM) for which the integrity monitor fails to generate an integrity alert. Figure 6.7a is the RoMD using the original statistical method and Figure 6.7b, d and g are the RoMD formed using the Kalman filter method, the Correlation Augmentation and the Spectral Augmentation to the statistical method, respectively. Clearly, Figure b, d and g show an improved RoMD performance of the integrity monitor

compared to the Statistical Method. Figure 6.7a and b use a threshold of 96 for the ‘T’ and ‘TKF’ statistics. Figure 6.7c, e and f use thresholds of 0.98, 20dB and 35dB for the correlation coefficient, power at zero frequency and power at higher frequency, respectively. A note on the

choice of thresholds for ρ, PB and PN: During level flight, when the RA measures plumb-bob

height, a higher threshold for ρ and lower thresholds for PB and PN can be chosen, in fact, the choices for the PB threshold (20dB) and the PN threshold (35 dB) are based on Monte Carlo simulations instead of theoretical analyses. During banking phases of flight, when the 50 seconds 79

of absolute disparities might exhibit a ramp, a lower threshold for ρ and higher thresholds for PB and PN must be chosen.

"T " Value "T" Value KF 20 20

15 15

10 10

5 5

0 0

-5 -5

-10 -10

-15 -15

-20 -20 -20 -10 0 10 20 -20 -10 0 10 20 a b

"T" Value Terrain Correlation (ρ) T + ρ 20 20 20

15 15 15

10 10 10

5 5 5

0 0 0

-5 -5 -5

-10 -10 -10

-15 -15 -15

-20 -20 -20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 a c d

Bias Power Spectrum (P ) Noise Power Spectrum (P ) T + P + P "T" Value B N B N 20 20 20 20

15 15 15 15

10 10 10 10

5 5 5 5

0 0 0 0

-5 -5 -5 -5

-10 -10 -10 -10

-15 -15 -15 -15

-20 -20 -20 -20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 a e f g Figure 6.7 – Improvement in RoMD within –20 to 20 Resolution Grid Points in Latitude and Longitude

A reduction in RoMD at all four points along an approach to runway 25 at EGE as shown in the horizontal domain integrity monitor scheme of Figure 3.9 in Chapter 3, is shown in Figure 6.8 on the next page. 80

"T" Value "T" Value "T" Value "T" Value 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20 "T " Value KF "TKF" Value "TKF" Value "TKF" Value 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20

T + ρ T + ρ T + ρ T + ρ 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20

T + P + P B N T + P B + PN T + P B + PN T + P B + PN 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20

Figure 6.8 – Improvement in RoMD at Locations shown in Figure 3.9

Figure 6.9a, b, c and d show the areas of missed detection computed at every 5-second interval during an approach to runway 25 at EGE over the entire flight path. A very wide region of missed detection is observed during the banking/turning phase of the aircraft due to mis-modeling of the radar altimeter. For the same reason, the correlation augmentation plot does not have a RoMD as all points exceed the threshold. The effects of the radar altimeter modeling error becomes prominent in the Kalman filtered case after some delay because of the filter’s characteristic of storing past history. A performance improvement of the integrity monitor can be seen in Figure 6.9b, c and d compared to 6.9a during level flight. 81

Region of Missed Detection (T) Region of Missed Detection (TKF) 39.7 39.7

39.65 39.65 ] ] s e 39.6 es 39.6 e re r g g e e d d [ [ e

39.55 d 39.55 ude t tu i t ti a La L

39.5 39.5

39.45 39.45

-107.1 -107.05 -107 -106.95 -106.9 -106.85 -106.8 -106.75 -106.7 -106.65 -107.1 -107.05 -107 -106.95 -106.9 -106.85 -106.8 -106.75 -106.7 -106.65 Longitude [degrees] Longitude [degrees] Region of Missed Detection (T + P + P ) Region of Missed Detection (T + ρ) B N 39.7 39.7

39.65 39.65 ] ]

es 39.6 es 39.6 e e r r g g e e d d [ [ e e

d 39.55 d 39.55 tu tu ti ti a a L L

39.5 39.5

39.45 39.45

-107.1 -107.05 -107 -106.95 -106.9 -106.85 -106.8 -106.75 -106.7 -106.65 -107.1 -107.05 -107 -106.95 -106.9 -106.85 -106.8 -106.75 -106.7 -106.65 Longitude [degrees] Longitude [degrees] Figure 6.9 – RoMD during Entire Flight Path at EGE

6.2.3 Spatial Integrity Monitor The vertical and horizontal domains of the integrity monitor schemes have been integrated to form a spatial integrity monitor, similar to the ‘Space Envelope’ shown in Figure 3.10 and Figure 3.11 by introducing an intentional vertical bias of increasing magnitude on the synthesized heights and applying the MPDLIM scheme to form RoMDs corresponding to each vertical bias.

The integrity monitor improvement methods using the TKF, (T+ρ) and (T+PB+PN) statistics are applied to the spatial integrity monitor and a reduction in the extent of the ‘space envelope’ is shown in Figure 6.10. The vertical biases in the ‘space envelope’ represent only the intentionally introduced biases in the DEM as the radar altimeter’s altitude dependent biases have been compensated for. The star ‘*’ in Figure 6.10 represent the GPS derived position of the aircraft,

considered to be the truth reference. The space envelopes using TKF, (T+ρ) and (T+ PB+PN) statistics have a reduced extent compared to the one formed using just the T statistic. It is interesting to note that the space envelopes for T and TKF do not extend beyond 35m and 25m, respectively in the vertical direction, which are the MDB for each case. 82 Space Envelope ( T ) Space Envelope ( T ) KF

-5 -5

-10 -10 s s -15 r -15 te meter me -20 -20

-25 -25

300 200 300 200 200 100 200 100 100 0 100 0 0 -100 0 -100 -100 -200 -100 -200 Latitude (meters) Longitude (meters) Latitude (meters) Longitude (meters) Space Envelope ( T + P + P ) Space Envelope ( T + ρ ) B N

-5 -5

-10 -10 s s -15 -15 meter meter -20 -20

-25 -25

300 200 300 200 200 100 200 100 100 0 100 0 0 -100 0 -100 -100 -200 -100 -200 Latitude (meters) Longitude (meters) Latitude (meters) Longitude (meters) Figure 6.10 – Improvement/Reduction in the Volume of Space Envelopes

6.3 Application to Terrain Referenced Navigation

The T, TKF, ρ and PB values computed by the methods discussed previously were also used for TRN, also known as Map-aided Navigation. Note that the form of terrain navigation addressed in this section does not include an INS. The T (and TKF) value is a measure of the agreement between the synthesized terrain profile and the terrain database profile; a minimum T implies a maximum agreement between the two terrain profiles and therefore a high probability that the aircraft exists at that location. The same is true for ρ except that a maximum value of ρ implies a maximum correlation between the synthesized and database elevations. Navigation using minimum PB values seems crude in comparison with the other statistics, but has been plotted alongside to put it in perspective with the other methods. 83 6.3.1 Lateral Position Estimation

Minimum T Position Estimates Minimum T Position Estimates KF

39.74 39.74

39.72 39.72

39.7 39.7 ] ] s s

e 39.68 e 39.68 e e r r g g

e 39.66 e 39.66 [d [d e e

d 39.64 d 39.64 tu tu ti ti a a

L 39.62 L 39.62

39.6 39.6

39.58 39.58

39.56 39.56 -107 -106.95 -106.9 -106.85 -106.8 -106.75 -106.7 -106.65 -107 -106.95 -106.9 -106.85 -106.8 -106.75 -106.7 -106.65 Longitude [degrees] a Longitude [degrees] b East Error East Error 500 500

200 200

s 100 100 r rs e e t t 0 0 -100 -100 me me -200 -200

-500 -500 -400 -300 -200 -100 0 100 200 -400 -300 -200 -100 0 100 200

North Error North Error 500 500

200 200 100

100 rs rs e e 0 0 -100 -100 met met -200 -200

-500 -500 -400 -300 -200 -100 0 100 200 -400 -300 -200 -100 0 100 200 Time From Threshold Crossing Time From Threshold Crossing Minimum P Position Estimates Maximum ρ Position Estimates B

39.74 39.74

39.72 39.72

39.7 39.7 ] ] 39.68 39.68 ees ees r r g g

e 39.66 e 39.66 [d [d e e

d 39.64 d 39.64 tu tu ti ti a a

L 39.62 L 39.62

39.6 39.6

39.58 39.58

39.56 39.56 -107 -106.95 -106.9 -106.85 -106.8 -106.75 -106.7 -106.65 -107 -106.95 -106.9 -106.85 -106.8 -106.75 -106.7 -106.65 Longitude [degrees] c Longitude [degrees] d East Error East Error 500 500

200 200 100 100 rs rs e e

t 0 t 0 -100 -100 me -200 me -200

-500 -500 -400 -300 -200 -100 0 100 200 -400 -300 -200 -100 0 100 200

North Error North Error 500 500

200 200 100 100 rs rs

e 0 e 0 -100 -100 met -200 met -200

-500 -500 -400 -300 -200 -100 0 100 200 -400 -300 -200 -100 0 100 200 Time From Threshold Crossing Time From Threshold Crossing Figure 6.11 – Terrain Navigation using Various Statistics and their E-N Error Comparison 84

Figure 6.11 depicts the minimum T, TKF, maximum ρ and the minimum PB horizontal position estimates and their respective East and North position errors computed at 5-second intervals during an approach to runway 07 at EGE. The trajectories of the minimum T, TKF and maximum ρ positions closely follow the ‘true’ GPS positions except during the time period directly after the aircraft performs a sharp turn (refer section 6.1.2-6.1.3). Referring to the East and North direction errors, one finds that during level flight, the magnitude of error is within ±200 m, i.e. two resolution grid point spacing. Accumulating the position coordinates as offsets from the ‘true’ GPS positions over time can give useful insights about the presence of horizontal biases on the DEM.

Two-dimensional plots of the deviations of the minimum T, TKF, maximum ρ and the minimum

PB positions from the ‘true’ GPS positions are shown in Figure 6.12. The aircraft position is estimated by computing the test statistics at parallel flight paths that are offset in latitude and longitude, from the nominal path represented by GPS position coordinates.

T P T KF ρ B 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20 a b c d

Figure 6.12 – Horizontal Position Fixes on the DEM

The units on both axes represent 3” (arc-second) increments in latitude and longitude as offsets

from the GPS position. All minimum T, TKF and maximum ρ positions cluster around the GPS

truth reference position (grid coordinates of (0,0)), showing no horizontal bias. The minimum PB positions are more spread out. The coordinates of the mean of the position fixes over time for the four figures are given in Table 6.1.

Table 6.1 Mean E-N Coordinates of Horizontal Position Estimates

T TKF ρ PB Grid Point Spacing (-0.61, 0.09) (-0.91, 0.11) (-0.38, -0.76) (-0.14, -0.84) Meters (-43.97, 8.85) (-65.03, 10.46) (-27.25, 70.05) (-9.91, -78.10)

85 Since the resolution of the terrain database is a limiting factor, rounding the resolution grid coordinates to the nearest integer gives exactly the same horizontal bias estimate. Figure 6.13 shows a plot similar to Figure 6.12 but with a horizontal bias of 30"×30" introduced into the DEM. All the position estimates cluster around the grid coordinates of (10, 10) indicating the presence of a horizontal bias.

T P T KF ρ B 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20 a b c d

Figure 6.13 – Horizontal Position Fixes on the DEM with Horizontal Bias

Table 6.2 Mean E-N Coordinates of Horizontal Position Estimates of Figure 6.13

T TKF ρ PB Grid Point Spacing (9.38, 10.09) (9.08, 10.11) (9.61, 9.24) (9.16, 8.73) Meters (668.34, 934.85) (647.28, 936.46) (685.06, 855.94) (652.85, 809.2)

Now, in addition to a horizontal bias of 30"×30", a vertical bias of 25 m is introduced into the DEM and the corresponding horizontal position fixes are shown below.

T P T KF ρ B 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20 a b c d

Figure 6.14 – Horizontal Position Fixes on the DEM with Horizontal and Vertical Biases

Table 6.3 Mean E-N Coordinates of Horizontal Position Estimates of Figure 6.14

T TKF ρ PB Grid Point Spacing (4.45, 10.87) (2.39, 10.34) (9.61, 9.24) (3.52, 9.7) Meters (317.13, 1007.3) (170.33, 958.20) (685.06, 855.94) (250.86, 901.0)

86 Figure 6.14 shows that the introduction of a vertical bias in the terrain database deteriorates the

performance of the horizontal position estimators based on the T, TKF and PB test statistics. The correlation coefficient ρ, on the other hand, is completely insensitive to the new introduced error. The reason for this, as mentioned in section 6.2.1, is that in the mathematical expression for ρ, the mean of the heights is removed. Although ρ is shown to be useless for reliable integrity monitoring, it is extremely useful for terrain navigation, as can be seen in the literature and TRN systems in use.

1 N ⎡(h ( t )− h ) (h ( t ) − h )⎤ ρ = ⎢ SYNT i SYNT DEM i DEM ⎥ Copy of (5.14) N ∑ σ σ i=1 ⎣⎢ hSYNT hDEM ⎦⎥

Based on the expression for ρ, the T and TKF statistics (of equations (3.9) and (5.12)) are modified by subtracting the mean of the absolute disparities and Kalman filter estimates (of absolute disparities) respectively as follows:

1 N T = p t − p 2 (6.1) 2 ∑[()i ] σ p i=1

N 1 2 ˆ ˆ TKF = ∑[x()tk − x] (6.2) P k =1

Even though these modified equations decrease the degrees of freedom of the system from 50 to 49, it is immaterial for terrain navigation, as it involves no decision-making based on a threshold value. The T statistic methods of position estimation perform well in the presence of horizontal biases as long as the search grid includes the true aircraft position. Figure 6.15 shows scatter plots of the horizontal position estimates computed based on equations (6.1) and (6.2).

T P T KF ρ B 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20 a b c d

Figure 6.15 – Horizontal Position Fixes on the DEM with Horizontal and Vertical Biases using Equations (6.1) and (6.2) 87 Table 6.4 Mean E-N Coordinates of Horizontal Position Estimates of Figure 6.15

T TKF ρ PB Grid Point Spacing (9.41, 9.26) (9.37, 9.14) (9.61, 9.24) (8.16, 8.24) Meters (670.82, 857.55) (667.72, 847.08) (685.06, 855.94) (581.62, 763.3)

The performance of the modified minimum T and TKF statistic position estimates are much better and comparable to the maximum ρ position estimates: all estimates cluster closely around the true aircraft position (10, 10) with respect to DEM coordinates.

Figure 6.16 shows an example distribution of the terrain information content metric (equation (3.13)) at a particular point along the flight segment at EGE as a function of horizontal offset from the flight path. The terrain information content is computed using the terrain database heights instead of the synthesized heights. This plot can be used to predict the horizontal bias uncertainties while flying a straight path. Figure 6.17 can aid predictions of the magnitude of the East and North direction bias uncertainties based on ‘I’.

Information Content Contours Information Content Variation 20 20

15 15

10 10

5 5

0 0

-5 -5

-10 -10

-15 -15

-20 -20 -20 -10 0 10 20 -20 -10 0 10 20 Figure 6.16 – Terrain Information Metric Variation

The mean of the deviation from the true east and north positions are plotted against the terrain information metric, I (equation (3.13)) in Figure 6.17 for multiple approaches to runways 07 and 25 at EGE. For all the ‘I’ bins except the ones with high standard deviations, the magnitude of mean error is between 50m and 100m in both east and north directions, which are approximately, equal to one grid point spacing of the terrain database. 88

Mean of East Direction Error Mean of North Direction Error 50 0

0 -50 s s r r e t e

t -50 me me -100 -100

-150 -150 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Standard Deviation of East Direction Error Standard Deviation of North Direction Error 250 300

250 200 200

150 s s r r e e t

t 150 me me 100 100

50 50

0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Terrain Information Metric Terrain Information Metric Figure 6.17 – East and North Direction Error Variation with ‘I’

6.3.2 Vertical Bias Estimation Vertical biases have been estimated by three methods: (i) By the taking the mean of the 50 absolute disparities at each point along the flight path, (ii) by taking the mean of the 50 absolute disparity estimates of the Kalman filter and (iii) by the ACF method described in Section 5.2. The mean and the ACF estimators are applied at 1-second interval over the entire flight path for an approach to runway 07 at EGE (the same flight segment of Figure 6.11). A vertical bias of 25m is deliberately introduced into the DEM and the vertical bias estimates using all three methods are shown in Figure 6.18. Radar altimeter bias errors and altitude dependant errors have been compensated for, so that the performance of these methods in the presence of terrain database vertical bias can be studied. The vertical bias estimated by all three estimators is very close within a meter. For short data lengths, the ACF method can give better estimates in the presence of a few anomalous data points as they are disregarded in the straight line fit of the autocorrelation function. It must be noted that the ACF method performs well to estimate the magnitude of vertical bias but cannot distinguish the direction of bias; the limitation stems directly from the properties of an autocorrelation function. 89

Mean AD Mean AD 30 KF ACF s r

e 25 t Me

20

-350 -300 -250 -200 -150 -100 -50 0 50 100 150

20 Pitch Roll

10 ees r 0 g e D -10

-20 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 Time from Threshold Crossing

Figure 6.18a – Vertical Bias Estimates using Mean and ACF Estimators Figure 6.18b – Pitch and Roll Angles

6.3.3 Spatial Position Estimation Figure 6.19 illustrates the scheme for a DEM-referenced spatial position estimation by which a horizontal translation (east and north or longitude and latitude) can be estimated together with a vertical bias offset. The algorithm identifies the position that corresponds to the minimum ‘T’ value within the ‘space envelope’ of Section 6.2.3 and computes the East, North and Up offsets from the true GPS position.

The ‘T’ values are computed using the equation (3.9). An explanation of the scheme and a few definitions are in order. The star ‘*’ denotes the electrical phase center of the GPS antenna, i.e., the GPS position of the aircraft. Consider an aircraft navigating with respect to a biased DEM that has a profile below the actual terrain by a distance ‘B’. If the aircraft intends to maintain a constant altitude ‘H’ above the terrain and relies on the DEM for situational awareness, due to the bias in the DEM, the aircraft actually has a terrain clearance of only h = Η − B . This can lead to an increased risk of CFIT around challenging terrain as illustrated in Figure 6.19. The spatial 90

Minimum 'T' Position * Rise to Safety B *

h Η

Actual Terrain B hsynt

Biased DEM

hDEM MSL

Figure 6.19 – Illustration of Spatial Position Estimation

position estimation scheme identifies the most probable aircraft position (with reference to the true terrain), by introducing a set of positive and negative vertical biases with increasing magnitudes on the synthesized elevations and computing the ‘T’ value over a horizontal search grid corresponding to each vertical bias. In this way, a ‘space envelope’ or volume of ‘T’ values are computed and the position that gives the minimum value is identified as the most probable aircraft position. The difference between the spatial integrity monitor and the spatial position estimation schemes is that the former compares all the ‘T’ values to a threshold while the latter identifies the minimum.

For the purpose of this scheme, a positive bias is defined as a potentially dangerous bias that could result in increased risk of CFIT. A DEM is said to have a positive bias when the DEM profile is lower than the actual terrain profile. An aircraft relying on the DEM assumes it is away from terrain but actually is much closer. In the presence of such a positive bias, the aircraft must rise higher to safety. A negative bias is defined as a bias that could result in a so-called loss of terrain. A negative bias exists when the DEM profile is higher than the actual terrain profile. Although a negative bias does not pose a threat of CFIT, it could be potentially dangerous for operations that require the aircraft to be in proximity to the ground, such as landing on the runway. 91 The spatial position estimation scheme is applied to the same flight data used in Sections 6.3.1 and 6.3.2 and the superimposed lateral position estimates are shown in Figure 6.20 (coordinates of their mean are ((9.42, 9.25) in resolution grid coordinates or (671.1m, 857.36m)). The East, North and Vertical bias estimates at each measurement time are shown in Figure 6.21. The vertical bias step size is taken as 0.5m.

T 20

15

10

5

0

-5

-10

-15

-20 -20 -10 0 10 20 Figure 6.20 – Horizontal Position Fixes using Spatial Position Estimation

Eas t Bias Es t im at e

800 s

r 700 e Met 600

500 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 Nor th Bias Es tim ate 1000

800 s r e t 600 Me

400

-350 -300 -250 -200 -150 -100 -50 0 50 100 150 Vertical Bias Estimate 40

35

30 s r e t 25 Me 20

15 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 Time from Threshold Crossing

Figure 6.21 – Spatial Position Estimation of East, North and Vertical Biases 92

7

ADDITIONAL FLIGHT TEST RESULTS

Besides EGE, flight test data was collected by Ohio University aircraft on three other locations: (a) Juneau (JNU), Alaska, (b) Asheville (AVL), North Carolina, (c) and Albany (KUNI), Ohio. In this chapter this flight data is analyzed and the results of the proposed integrity monitoring and terrain navigation methods are presented. These three flight test locations have drastically different terrain features as described in the following sections. One of the reasons for showing the flight test results at these locations is to illustrate and study the effects of different geography on the performance of the integrity monitor and TRN system. The integrity monitor has a solid foundation in probability theory, which gives an idea of the integrity monitor’s failure rate, but provides little information regarding the magnitude of failures, which are in this case, the vertical and horizontal biases represented by the region of missed detection. The results illustrated in Section 7.2 raise user awareness to the integrity monitor’s limitations with respect to its terrain- dependent horizontal fault detection capability. Performance of the proposed position estimation schemes (navigation) at the three locations has been illustrated in the results provided in Section 7.3 for comparison with the results shown in Section 6.3. 93 7.1 Flight Test Environment

7.1.1 JNU Flight Test A flight test using Ohio University’s King Air C-90 (see Figure 7.1a) was performed in the vicinity of Juneau (JNU), Alaska from September 10 through September 17, 2002. The plots in this chapter are for a segment of flight over the Gastineau Channel during approaches flown to runway 08 and runway 26 at JNU. During this phase, the aircraft was flying over water with terrain on either side; the aircraft was actually flying below the level of the terrain. The flight test instrumentation consisted of the DEM Integrity Monitor Equipment (DIME) package. The DIME package consists of a positioning processor, an SVS graphics computer, and the real-time DEM integrity monitor. The positioning computer consists of a Novatel OEM4 receiver and a Condor Engineering ARINC 429 interface card providing accurate positioning measurements. The positioning processor was capable of using corrections provided by either WAAS or a Local-Area Augmentation System (LAAS) prototype ground station via a data radio [24]. The terrain database was a 3-arcsec DEM obtained from the USGS and formatted for use in the test equipment. The terrain information content metric (equation (3.13)) computed using synthesized terrain heights while flying along the Gastineau channel had a range between 0.5 and 3.6.

Figure 7.1(a) – Ohio University King Air C90 Flying laboratory, Figure 7.1(b) – DEM Integrity Monitor Experiment (DIME) equipment

7.1.2 AVL Flight Test Flight tests at AVL were performed using Ohio University’s DC-3 flying laboratory (see Figure 7.2) on September 26 and 27, 2000. Post processed positioning data from Kinematic GPS (KGPS) and height AGL measurements from a Honeywell radar altimeter is used to evaluate the 94 performance of the proposed methods of integrity monitoring and navigation. The terrain database used is the DTED® Level 1 described in the Appendix. AVL presents a rough terrain ranging from a height of 620m to 1129m [2], but not as mountainous and challenging as EGE. The terrain information metric variation was found to be between 1.4 and 11.8 for one particular flight segment.

Figure 7.2 – Ohio University DC-3 Flying laboratory

7.1.3 KUNI Flight Test Tests at the Ohio University airport at Albany, OH were also conducted using the DC-3 aircraft on August 28 and 31, 2000. The sensors used were a Honeywell radar altimeter and a prototype LAAS GPS system developed under a Cooperative Agreement with the FAA. The database used for the results is the DTED® Level 1. KUNI is located is southeast Ohio at the foothills of the Appalachian ranges. The terrain has a relatively large spatial similarity as compared to EGE and AVL with the terrain heights ranging from 190m to 305m above MSL [2] and the terrain information metric, I, ranging from 5 and 10 during one segment of flight.

7.2 Terrain Database Integrity Monitor EGE flight test results were discussed in detail in Chapter 6. At the three flight test locations in this chapter only the results are shown of those methods best suitable for the specific applications. 95 7.2.1 Vertical Domain Integrity Monitor

The test statistic ‘T’, computed using equation (3.9) and ‘TKF’, computed using equation (5.12) have been plotted in Figure 7.3 for flight segments at the three locations of JNU, AVL and KUNI.

Improved performance of the integrity monitor is observed in the vertical domain, where the TKF values have greater magnitudes and exceed the threshold in the presence of biases. The ‘T’ and

‘TKF’ plots for JNU are smooth, whereas those for AVL and KUNI appear jittery. This jitter can be attributed to the characteristics of the measurement sensors on board the aircrafts; the JNU flight test was conducted on the King Air C-90 using a COTS FM-CW radar altimeter with a 45- degree beam width, and the DC-3 aircraft was used for the tests at AVL and KUNI with a pulsed radar altimeter with a 15-degree beam width.

7.2.2 Horizontal Domain Integrity Monitor Integrity monitoring schemes over a horizontal search grid are shown in Figure 7.4 for EGE, JNU, AVL and KUNI, respectively. EGE shows the smallest RoMD whereas JNU has the largest. The difference in the horizontal bias detection capability at all four geographical locations can be directly related to the terrain signature underneath and around the flight path at the respective locations (given by equation (3.13)). Flying through the Gastineau channel near JNU shows a poor along-track detection performance due to the lack of terrain undulations along the aircraft’s track (over water). However, the cross-track detection performance is better due to the presence of mountains on either side of the channel. Note that the along-track CFIT risk is also much smaller than the cross-track CFIT risk. The high terrain information content at EGE, on the other hand, provides a smaller RoMD. KUNI, which presents a rough, hilly terrain, has a smaller RoMD compared to the terrain for the flight segment shown at AVL that has a large spatial similarity. The correlation augmentation seems to perform better, but it involves an empirical selection of the decision-making threshold for the correlation coefficient. Furthermore, while flying over water and the terrain database contains zero MSL heights, the correlation coefficient produces indeterminate values.

7.2.3 Spatial Integrity Monitor The concept of space envelopes of missed detection, as described in Section 3.4, has been applied to the flight segments at JNU, AVL and KUNI. Figure 7.5 shows the space envelopes of missed detection as contour plots at a particular time instant during the aircraft’s flight. The units on all three axes are in meters although the vertical axis is scaled differently for easy observation. The 96 space envelope of missed detection is a collection of all possible aircraft positions within a spatial search grid that do not exceed the ‘T’ statistic threshold of 96, pertaining to the vertical domain integrity monitor. A reduction in the extent of the space envelopes of missed detection at all three locations is observed by the use of a Kalman Filter.

JNU, Bias = 0 JNU, Bias = 0

100 100 c i t is t a

t 50 50 s e i T S it r a p s 0 i 0 D 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 e t

Bias = 25m lu Bias = 25m o s

400 b 400 A d

e 300 300 r c i e t ilt is t F 200 a 200 n t a T S lm 100 100 a g K

0 in 0 0 50 100 150 200 250 300 350 400 450 500 s 0 50 100 150 200 250 300 350 400 450 500 U

c

Bias = 35m i Bias = 35m t is t a t 600 600 S KF c i T t is

t 400 400 a t

T S 200 200

0 0 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Measurement Time Measurement Time

AVL, Bias = 0 AVL, Bias = 0

100 100 c i t s i t a t 50 50 s e i T S it r a p s 0 i 0 0 500 1000 1500 2000 2500 3000 3500 4000 D 0 500 1000 1500 2000 2500 3000 3500 4000 e t

Bias = 25m lu Bias = 25m o

500 s 500 b A

400 d 400 e r c i e t 300 300 s ilt i t F a n t 200 200 a T S lm

100 a 100 g K

0 in 0 0 500 1000 1500 2000 2500 3000 3500 4000 s 0 500 1000 1500 2000 2500 3000 3500 4000 U

c

Bias = 35m i Bias = 35m t is t a

600 t 600 S c KF i t 400 T 400 is t a t

T S 200 200

0 0 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Measurement Time Measurement Time

KUNI, Bias = 0 KUNI, Bias = 0

100 100 c i t is t a t

50 s 50 ie T S it r a p s 0 i 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 D 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 e t

Bias = 25m lu Bias = 25m o

300 s 300 b A d e r c

i 200 200 e t ilt is t F a n t 100 a 100 T S lm a K

0 ing 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 s 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 U

c

Bias = 35m i Bias = 35m t

500 is 500 t a t

400 S 400

c KF i t 300 T 300 is t a t 200 200 T S 100 100

0 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Measurement Time Measurement Time

Figure 7.3 – T and TKF values during flight segments at JNU, AVL and KUNI for biases of 0, 25m and 35m 97

EGE, "T" Value JNU, "T" Value AVL, "T" Value KUNI, "T" Value 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20

"TKF" Value "TKF" Value "TKF" Value "TKF" Value 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20

T + ρ T + ρ T + ρ T + ρ 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20

T + P B + PN T + P B + PN T + P B + PN T + P B + PN 20 20 20 20

10 10 10 10

0 0 0 0

-10 -10 -10 -10

-20 -20 -20 -20 -20 0 20 -20 0 20 -20 0 20 -20 0 20

Figure 7.4 – Improvement in RoMD at EGE, JNU, AVL and KUNI 98 Space Envelope ( T ) at JNU Space Envelope ( T ) at JNU KF

0 0

-5 -5

-10 -10

-15 -15 s r e ters t -20 -20 me me

-25 -25

-30 -30

-35 -35

500 1000 500 1000 0 0 0 0 -500 -1000 -500 -1000 Longitude (meters) Latitude (meters) | Longitude (meters) Latitude (meters) Space Envelope ( T ) at AVL Space Envelope ( T ) at AVL KF

0 0

-5 -5

-10 -10

-15 -15 s s r r e e t t -20 -20 me me

-25 -25

-30 -30

-35 -35

500 1000 500 1000 0 0 0 0 -500 -1000 -500 -1000 Longitude (meters) Latitude (meters) | Longitude (meters) Latitude (meters) Space Envelope ( T ) at KUNI Space Envelope ( T ) at KUNI KF

0 0

-5 -5

-10 -10

-15 -15 s s r r e e t t -20 -20 me me

-25 -25

-30 -30

-35 -35

500 1000 500 1000 0 0 0 0 -500 -1000 -500 -1000 Longitude (meters) Latitude (meters) | Longitude (meters) Latitude (meters) Figure 7.5 – Reduction in Space Envelope using the Kalman Filter method

99 7.3 Applications to Terrain Referenced Navigation

7.3.1 Lateral Position Estimation

Minimum values of the ‘T’, ‘TKF’ and the maximum value of the correlation coefficient ‘ρ’ are used to estimate the aircraft position within the horizontal search grid. Based on the findings of

Section 6.3, ‘T’ and ‘TKF’ are computed using equations (6.1) and (6.2) respectively and ‘ρ’ using equation (5.14). A superimposed search grid containing the estimated aircraft positions over a straight segment of flight at JNU (flying along the Gastineau channel) is shown in Figure 7.6.

T T KF ρ 20 20 20

10 10 10

0 0 0

-10 -10 -10

-20 -20 -20 -20 0 20 -20 0 20 -20 0 20 a b c

Figure 7.6 – Horizontal Position fixes on the DEM at JNU

Table 7.1 Mean E-N Coordinates of Horizontal Position Estimates of Figure 7.6

T TKF ρ Grid Point Spacing (1.42, 3.3) (2.75, 2.12) (-0.48, 6.43) Meters (69.06, 305.58) (133.96, 196.44) (-23.75, 595.66)

The horizontal position fixes are observed to lie all along the RoMD at JNU (refer Figure 7.4). Due to the lack of significant terrain undulations in the RoMD, the algorithm simply identifies the position with the minimum T or TKF, which, in this case, are not good estimates of the true aircraft position. The correlation coefficient algorithm has another limitation over water: it produces indeterminate values for ρ over the RoMD. Figure 7.6c shows the mean position fixes of the indeterminate values of ρ.

Horizontal position fixes during a straight flight segment at AVL are shown in Figure 7.7a, b, and c. In Figure 7.7d, e and f both an intentional bias of 30 arc seconds in each of the East and North directions and a vertical bias of 25 meters were introduced in the DEM. The position estimators 100 produce terrific results as the closely clustered dots represent the convergence of almost two hundred points (position estimates).

T T KF ρ 20 20 20

10 10 10

0 0 0

-10 -10 -10

-20 -20 -20 -20 0 20 -20 0 20 -20 0 20 a b c

T T KF ρ 20 20 20

10 10 10

0 0 0

-10 -10 -10

-20 -20 -20 -20 0 20 -20 0 20 -20 0 20 d e f Figure 7.7a,b and c – Horizontal Position fixes on the DEM at AVL without biases Figure 7.7d,e and f – In the presence of both Horizontal and Vertical biases

Table 7.2 Mean E-N Coordinates of Horizontal Position Estimates of Figure 7.7

No Intentional Bias T TKF ρ Grid Point Spacing (-0.92, 0.01) (-0.92, 0.01) (-0.33, 0) Meters (-69.81, 1.14) (-69.81, 1.14) (-25.13, 0) With Bias T TKF ρ Grid Point Spacing (9.07, 10.01) (9.07, 9.98) (9.66, 10) Meters (684.14, 927.14) (684.14, 924.85) (728.82, 926)

Similar plots providing horizontal position fixes over a straight flight path at KUNI are shown in Figure 7.8. 101 T T KF ρ 20 20 20

10 10 10

0 0 0

-10 -10 -10

-20 -20 -20 -20 0 20 -20 0 20 -20 0 20 a b c

T T KF ρ 20 20 20

10 10 10

0 0 0

-10 -10 -10

-20 -20 -20 -20 0 20 -20 0 20 -20 0 20 d e f Figure 7.8a,b and c – Horizontal Position fixes on the DEM at KUNI without biases Figure 7.8d,e and f – In the presence of both Horizontal and Vertical biases

Table 7.3 Mean E-N Coordinates of Horizontal Position Estimates of Figure 7.8

No Intentional Bias T TKF ρ Grid Point Spacing (-0.84, 0.30) (-1.06, 0.53) (-0.74, 0.21) Meters (-60.75, 28.70) (-76.26, 49.27) (-53.21, 20.03) With Bias T TKF ρ Grid Point Spacing (8.09, 9.33) (8.15, 9.77) (7.97, 8.94) Meters (580.35, 864.80) (584.54, 905.42) (571.55, 828.52)

7.3.2 Vertical Bias Estimation The mean and the ACF estimators, as detailed in Section 6.3.2, are used to estimate an intentionally introduced vertical bias of 25 meters on the DEM, during straight flight paths at JNU, AVL and KUNI. The measurements are not compensated for any kind of errors, viz. radar altimeter bias errors or altitude dependent errors. The manifestation of these errors can be observed in Figure 7.9. 102 JNU 25

20 s r e t Mean AD Me 15 Mean AD KF ACF

20 40 60 80 100 120 140 AVL 35

s 30 r e t Me 25

10 20 30 40 50 60 70 80 KUNI

35 s r e 30 Met

25 50 100 150 200 250 300 Measurement Times Figure 7.9 – Vertical Bias Estimates using Mean and ACF Estimators at JNU, AVL and KUNI

7.3.3 Spatial Position Estimation Similar to the results shown in Section 6.3.3 for EGE data, the spatial position estimation scheme is applied to data at JNU, AVL and KUNI. For an intentionally introduced bias of 30"×30" (10"×10"in grid post spacing) in the East and North directions and a 25m vertical bias, the lateral position fixes are shown in Figure 7.10.

T at JNU T at AVL T at KUNI 20 20 20

10 10 10

0 0 0

-10 -10 -10

-20 -20 -20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 a b c Figure 7.10 – Horizontal Position Fixes on the DEM at JNU, AVL and KUNI 103 Table 7.4 Mean E-N Coordinates of Horizontal Position Estimates of Figure 7.10 ‘T’ Values JNU AVL KUNI Grid Point Spacing (13.04, 10.08) (9.07, 10.01) (8.19, 9.35) Meters (633.71, 933.93) (684.14, 927.14) (586.89, 866.25)

The vertical bias estimates at each location are shown in Figure 7.11. Comparison with Figure 7.9 indicates similar trends although Figure 7.11 appears more discrete due to the discrete step size (0.5m) of the vertical bias search bin.

V e r tical Bias Es tim ate (JNU) 26

25 s r e t 24 Me

23

20 40 60 80 100 120 140 Vertical Bias Estimate (AVL) 26

25

24 s r e t 23 Me 22

21 10 20 30 40 50 60 70 80 V e r tical Bias Es tim ate (KUNI) 35 s r

e 30 Met

25 50 100 150 200 250 300 Measurement Times

Figure 7.11 – Spatial Estimation of Vertical Biases at JNU, AVL and KUNI

104

8

SUMMARY AND CONCLUSIONS

Terrain database integrity monitors may be a necessary component of an SVS to support intended functions that require FAA system certification at a level other than advisory. The digital terrain database is an important component of the system that provides terrain information for the SVS display. If the integrity of the terrain database cannot be guaranteed by the terrain database providers at the time of generation, it is necessary to include an integrity monitor at times of operational usage. This thesis explores the concept of a spatial integrity monitor for terrain databases and a few methods to improve its performance. Chapter 3 provides the concepts developed in previous works [1][2] on the integrity monitor in the vertical and horizontal domains. The spatial integrity monitor using downward-looking sensors like the radar altimeter is based on these concepts. Statistical characterization of the vertical domain integrity monitor with regard to its minimum detectable bias is straightforward, but statistical characterization of the horizontal detection capability is more challenging. Data was used to study the performance of the horizontal domain integrity monitor from four flight tests locations with a different terrain characterization; one at Eagle/Vail, CO in the vicinity of the Rocky Mountains (Chapter 6), others at Juneau, AK, Asheville, NC and at Albany, OH around Ohio University (Chapter 7). The results confirm the theoretical understanding that the horizontal fault detection capability is dependent upon the information content in the terrain. 105 The terrain database is a major contributor to the overall error budget, having an error standard deviation of 18.2 meters compared to the overall 18.9 meters. The different standard deviations of the individual sensors in all the different aircrafts used do not significantly change the overall error standard deviation. For this reason, a standard deviation of 18.9 meters was retained for the tests at all locations, also considering the slight variations for the different terrain databases used. Due to the presence of noise on the sensors and terrain databases that affects the performance of the integrity monitor, Kalman filtering and transform domain techniques have been explored to achieve a performance improvement. The theoretical background of these techniques has been covered in Chapter 5.

The Kalman filter, like any other filter, alters the statistical properties of the system model. An analysis of the effects of filtering has been carried out in Section 5.1 from a reliability point of view and the involved tradeoffs between strict theory and practical applicability have been discussed. It is found in Section 6.2 that the application of the Kalman filter to the integrity monitor improves its statistical detection capability for both vertical biases and horizontal translations. Although the Kalman filter is used for the estimation of a constant bias on the absolute disparities, this scheme provides a tunable filter that can be designed to be sensitive to any µB by just changing one parameter; Q. The concept of using the Autocorrelation function method and the Spectral Estimation method to augment the statistical baseline method for the purpose of terrain database integrity monitoring was introduced in this thesis. The ACF and MESE methods are highly qualitative and do not satisfy reliability requirements independently and so must be used in conjunction with the statistical method (Section 6.2.2). The horizontal bias detection capability of all four methods described herein depends greatly on the terrain signature, but to different extents. Since they are based on different techniques, one in conjunction to the other gives improved performance.

Integrity monitor concepts based on the test statistic approach have been extended to the application of terrain-referenced navigation. Chapter 4 reviews some well known terrain referenced navigation systems for comparison with the methods of this work. The minimum T- value point on the search grid gives the most likely horizontal position of the aircraft. The same goes for values of TKF, ρ and PB, although their performance (closeness to true position) varies.

From Section 6.3, the alternate formulations of T and TKF perform as good as the correlation

coefficient ρ even in the presence of vertical and horizontal biases. The PB statistic does not 106 perform as good as the others for navigation. Position estimates formed using TKF are much closer to the ‘true’ GPS positions during banking phases of flight than those formed using ‘T’. The ACF estimator is shown to estimate vertical biases within a meter range of the mean estimators. The spatial position estimation method (Section 6.3.3) is a true conceptual extension of the spatial integrity monitor for application to terrain navigation. It can be imagined as a shrinking of the ‘space envelope’ by reducing the T statistic threshold continuously, until the space envelope condenses to a single point (corresponding to the minimum T value), in three- dimensional space. This point is the aircraft position estimate. The positions thus obtained could be used for en-route navigation with an acceptable horizontal translation error that depends upon the terrain signature. More precise terrain database position estimates require closer spacing of the search grid points which in turn requires a higher resolution terrain database. Integration with other sensors such as an INS can also be considered.

Analysis of the horizontal fault detection capability during terrain navigation (Section 6.3.1) revealed that predictions of horizontal biases based on the terrain information metric must be done in real time depending upon the flight trajectory. The metric must be computed using terrain database heights and its variation over the multiple parallel path search grid can give useful insight about the magnitude of horizontal biases. 107

9

THE ROAD AHEAD

Listed below are a few recommendations for research that could improve the terrain database integrity monitor and take SVS further to being certifiable for safety critical operations:

Development of robust algorithms for the treatment of radar altimeter slant range and altitude dependent errors in the presence of vertical and horizontal biases on the terrain database. Combination of the sensors with an inertial navigation system could be considered for aircraft attitude determination. The aircraft attitude and orientation with respect to known terrain features could then be used to develop a compensation algorithm.

The results presented in Chapters 6 and 7 emphasize the dependence of the horizontal bias detection capability on the terrain signature. What the results signify, is that at each

point within the RoMD, the vertical bias is less than the minimum detectable bias µB,

with a probability of (1-PMD). But the methods/results do not include the statistical characterization of the horizontal biases. A look at the DTED specifications (appendix) shows values for a linear error probability (LEP) and a circular error probability (CEP). Further research is needed on statistics and probability in order to relate the CEP with the 108 horizontal fault detection capability and place a probabilistic bound on the presence of horizontal biases and increase our confidence in the integrity monitor.

Addition of another state to the Kalman filter to include the correlation of estimates in time could further improve the performance of the Kalman filter method.

The terrain database error is the major contributor to the overall error budget. Improved performance of the integrity monitor (in vertical + horizontal domains), indicated by a

low value of µB, requires a highly accurate terrain database. Improvements for the horizontal domain require a database with higher spatial resolution. More precise terrain database position estimates for navigation have a similar requirement. Although very difficult at present, the generation of high-resolution, highly accurate terrain databases that conform to specified probabilities of fault-free and missed detection, could eventually do away with an integrity monitor. But changes in the terrain features, either due to obstacles or natural phenomenon like volcanic eruptions and earthquakes make an integrity monitor a better choice for SVS rather than frequent mapping of the terrain, which can be quite unfeasible.

By projecting the flight path into the future, Section 6.3.1 provides a method for prediction of horizontal uncertainties based on a terrain information metric computed using database elevations. Downward-looking sensors do not provide sufficient information to validate the predictions but forward-looking sensors certainly can. Development of integrity monitoring schemes using forward-looking sensors is strongly encouraged for increased situational awareness. At the time of writing this thesis, there have been ongoing research projects using forward-looking weather radar [24] and a scanning beam laser altimeter (LiDAR) [51] for the purpose of monitoring terrain database integrity. It is speculated that by using LiDAR generated terrain databases and real-time integrity monitoring schemes, the along-track and cross-track information provided by a LiDAR can greatly improve the performance of the integrity monitor, as well as terrain-referenced navigation. 109

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APPENDIX

A.1 Terrain Database Specifications

Various terrain databases having different resolutions and coverage areas are available, some of which have been used in this thesis. These include the Digital Terrain Elevation Database (DTED levels 0, 1 and 2) [16], the United States Geological Survey (USGS) DEM and a high resolution National Geodetic Survey (NGS5) DEM created solely for the AVL airport area. Table A.1 gives an overview of the characteristic parameter values of some of these databases. [2][17]

Table A.1 Terrain Database Specifications DEM Post Absolute Vertical Absolute Horizontal Vertical Horizontal Spacing Accuracy Accuracy Datum Datum DTED 0 30 arc-sec <30m, 90% L.E.P <50m, 90% C.E.P MSL WGS84 DTED 1 3 arc-sec <30m, 90% L.E.P <50m, 90% C.E.P MSL WGS84 DTED 2 1 arc-se <18m, 90% L.E.P <23m, 90% C.E.P MSL WGS84 USGS 3 arc-sec N/A N/A NGVD27 WGS84 NGS5 5m 1m, 90% 1m, 90% MSL WGS84 L.E.P = Linear Error Probability and C.E.P = Circular Error Probability 114 Assuming normally distributed errors on the DTED level 1 DEM heights, 90% L.E.P corresponds to 1.645σ, which equals 30m. Hence, σ = 30m/1.645 = 18.2m. An illustration to this effect is shown in Figure A.1.

0.3989

0.3

0.2 1.645σ 90% 0.1

0 -5 -4 -3 -1.645 -1 0 1 1.645 3 4 5 ← X.σ → Figure A.1 – Illustration of a shortest interval and its associated probability (area) under a normal curve

A.2 Kalman Filter Design Parameters

− − In Section 5.1, the Kalman filter design parameters, namely, xˆ0 , P0 ,φk , H k , Rk ,Qk were

− specified. The prediction error variance P0 can be taken as the square of the bias that has a relatively higher chance of going undetected, based on the OC curve of Figure 3.6. An analysis of the estimation error variance, Pk , and the Kalman filter gain, K k , shows that the choice of

− P0 does not have a great effect on the steady state performance of the filter. Figure A.2 shows

− the plots of Pk and K k over 60 successive measurement times for various values of P0 . The

− 2 2 values of P0 range from (10m) to (25m) . As can be seen, the various curves for Pk settle to

the same constant value at steady state. Similar the case with K k . When using the last 50

− estimates for the computation of the TKF statistic, the choice of P0 has a little but not a

significant influence on the value of TKF. One downside of running 60 successive absolute disparities through the Kalman filter and using only the last 50 successive estimates for the

computation of TKF is the increased time-to-alarm of 60 seconds. Letting the Kalman filter to run for the entire flight and using 50 successive estimates at a time is definitely a better option, but one must be careful to take the aircraft dynamics into account. As the history of measurements has an impact on the current filter estimates, the filter must be reset whenever there are bad 115 measurements due to the pitching and rolling maneuvers of the aircraft, unless they have been compensated.

Estimation Error Variance 250

200

150 Use Last 50 Estimates

100

50 0 10 20 30 40 50 60

Kalman Filter Gain 0.7

0.6

0.5

0.4

0.3

0.2 0 10 20 30 40 50 60 Figure A.2 – Steady state Kalman filter parameter values