Development and Implementation of a Trajectory Prediction Methodology by Clay Jackson Robertson A thesis submitted to the Graduate Faculty of Auburn University in partial ful llment of the requirements for the Degree of Master of Science Auburn, Alabama May 7, 2012 Keywords: Time Horizon, Trajectory Prediction, Kalman Filter Copyright 2012 by Clay Jackson Robertson Approved by Dr. Gilbert Crouse, Chair, Associate Professor of Aerospace Engineering Dr. David Cicci, Professor of Aerospace Engineering Dr. John Cochran, Professor and Head of Aerospace Engineering Abstract Operation of unmanned aircraft in the United States? National Airspace System (NAS) is currently severely restricted, primarily due to the need to ensure adequate separation between manned and unmanned aircraft (UA). A particular problem in con ict avoidance algorithm is estimating where con icting tra c is likely to be in the future. While most air tra c spends a large percentage of its time in straight and level ight, maneuvers are still quite common and must be considered. Incorporating uncertainty in tracking algorithms is well established, but current methods primarily only consider uncertainty related to sensor errors and modeling errors. They do not consider the uncertainty of pilot decisions regarding maneuvers. The objective of this research is to quantify the level of uncertainty in aircraft position due to pilot maneuvers and develop methods for incorporating that information into tracking and con ict avoidance algorithms. The uncertainty in position and velocity can arise from di erent sources such as sensor uncertainty, but a signi cant contributor is that the future behavior of non-cooperative air- craft is generally unknown. A pilot may maneuver for quite a number of di erent reasons. While aircraft on a cross-country trip will generally only make small course or altitude ad- justments at various waypoints along their planned track, pilots that are just out "boring holes in the sky" or student pilots practicing various maneuvers may engage in fairly aggres- sive maneuvers unexpectedly. Thus it is helpful to quantify not only where a non-cooperative aircraft would be in the future given that it maintains its current velocity, but also where it could be if the pilot chooses to maneuver. In this study, time histories of aircraft tracks have been used to develop statistical models of aircraft maneuvers. Two sources of aircraft tracks have been used. Auburn University has a small eet of ight training aircraft and ii GPS tracking devices were placed in these aircraft and their movements were tracked over approximately six weeks. Since these aircraft are used almost exclusively for ight training they represent aircraft that are most likely to maneuver. Each GPS unit was packaged with a high capacity NiMH battery pack to allow the unit to operate for up to a week without recharge. The second dataset was obtained from the FAA and includes tracks from aircraft op- erating over the contiguous United States. The Federal Aviation Administration (FAA) database include aircraft that are either operating under Instrument Flight Rules (IFR) or aircraft under Visual Flight Rules (VFR) but using radar ight following. IFR aircraft and VFR aircraft using ight following are typically traveling between two points so these air- craft would not be expected to execute many maneuvers enroute. These two dataset should provide a bound on the frequency of maneuvers. A statistical approach to analyzing the data was used to describe error in the projection due to maneuvers o the projected course. The aircraft tracking data was analyzed to deter- mine how accurately the position of an aircraft could be projected forward in time assuming the aircraft travels at a constant velocity. At each point in time, the aircrafts? position and velocity were estimated using Kalman lter and other straight projection techniques. This position and velocity was projected forward over various time horizons and compared to the aircrafts actual position at that projected time. By accumulating the occurrence of error from the expected projection point, the con dence in projection both along- track and cross-track could be calculated for private, IFR and VFR aircraft. Frequency and extent of deviations for cooperative and non-cooperative air tra c can be used in testing con ict avoidance algorithms for unmanned aircraft. Con dence intervals were developed for com- pliant and non-compliant aircraft in the NAS at various ight levels in terminal and non- terminal environments. iii Acknowledgments The author would like to acknowledge Dr. Gilbert Crouse for his guidance and support throughout this work. The author would also like to thank LTC Trey Kelley and Viva Austin at the US Army UAS Project O ce in Hunstville, Alabama for the opportunity to investigate this problem and the Aerospace Engineering Department at Auburn University for its support and nancial assistance. Finally, the author would like to bestow many thanks to his family and friends for their constant encouragement and unwavering support. This work is dedicated to the author?s parents, Nana J Robertson and Larry D Robertson and sister, Jourdan D Robertson, whose support, dedication, encouragement, humor, advice, inspiration, and love are never without appreciation and held dear. iv Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Estimating Future Position of Air Tra c . . . . . . . . . . . . . . . . . . . . . . 6 3.1 National Airspace System Data . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Auburn University Personal Aircraft GPS Data . . . . . . . . . . . . . . . . 7 4 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.1 Development of Histogram Error . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 Development of Altitude Comparison . . . . . . . . . . . . . . . . . . . . . . 13 4.3 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.4 Other Projection Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.1 National Airspace System Radar Tracks . . . . . . . . . . . . . . . . . . . . 20 5.2 Private Aircraft GPS Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.3 Comparison of Projection Techniques . . . . . . . . . . . . . . . . . . . . . . 48 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 v List of Appendix A Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 List of Appendix A Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A National Airspace System Radar Tracks . . . . . . . . . . . . . . . . . . . . . . 76 B Private Personal Aircraft GPS Tracks . . . . . . . . . . . . . . . . . . . . . . . . 189 C Rhumb Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 vi List of Figures 3.1 NAS data over the United States over a 24 hour period . . . . . . . . . . . . . . 7 3.2 Materials and Aircraft used in GPS track study . . . . . . . . . . . . . . . . . . 8 3.3 All GPS tracks from Auburn, AL . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.1 Progression of histogram error development . . . . . . . . . . . . . . . . . . . . 12 5.1 All tracks, 3 minute projection, di erence in projection to measured histograms 21 5.2 All tracks, 10 minute projection, di erence in projection to measured histograms 22 5.3 Di erence in altitude, all tracks, 3 minute projection . . . . . . . . . . . . . . . 24 5.4 Percent di erence in speed, all tracks, 3 minute projection . . . . . . . . . . . . 25 5.5 Di erence in bearing, all tracks, 5 minute projection . . . . . . . . . . . . . . . 26 5.6 Fort Campbell, 50 km radius, 1 minute projection, di erence in projection to measured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . 27 5.7 Fort Campbell, 50 km radius, 1 minute projection, di erence in altitude . . . . 28 5.8 Fort Campbell, 50 km radius, 1 minute projection, di erence in bearing . . . . . 29 5.9 Las Cruces, 50 km radius, 1 minute projection, di erence in projection to mea- sured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 vii 5.10 Fort Campbell, 50 km radius, 1 minute projection, di erence in projection to measured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.11 Las Cruces, 250 km radius, 1 minute projection, di erence in projection to mea- sured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.12 Fort Campbell, 250 km radius, 1 minute projection, di erence in projection to measured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.13 Las Cruces, 250 km radius, 5 minute projection, di erence in projection to mea- sured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.14 Fort Campbell, 250 km radius, 5 minute projection, di erence in projection to measured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.15 Selected GPS les with and without airports, 1 minute GPS projection using a constant velocity lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.16 Selected GPS les with and without airports, 1 minute GPS projection using a constant velocity lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.17 Selected GPS les with and without airports, 1 minute GPS projection using a constant velocity lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.18 Selected GPS les with and without airports, 1 minute GPS projection using a constant velocity lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.19 Single GPS ight analyzed, breaking the ight into two parts: straight ight and maneuvering ight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 viii 5.20 Cross track error for straight portion of the track using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) . . . . . . . . . . . . . . . . . . . . . 50 5.21 Along track error for straight portion of the track using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) . . . . . . . . . . . . . . . . . . . . . 51 5.22 Cross track error for maneuvering portion of the track using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) . . . . . . . . . . . . . . . 53 5.23 Along track error for maneuvering portion of the track using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) . . . . . . . . . . . . . . . 55 5.24 Cross track error for all GPS tracks using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.25 Along track error for all GPS tracks using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ix List of Tables 5.1 Altitude Con dence, All Tracks, 0-10000 ft Altitudes . . . . . . . . . . . . . . . 37 5.2 Altitude Con dence, All Tracks, 10000-18000 ft Altitudes . . . . . . . . . . . . 37 5.3 Altitude Con dence, All Tracks, Above 18000 ft Altitudes . . . . . . . . . . . . 37 5.4 Altitude Con dence, All Tracks, All Altitudes . . . . . . . . . . . . . . . . . . . 38 5.5 Bearing Con dence, All Tracks, 0-10000 ft Altitudes . . . . . . . . . . . . . . . 39 5.6 Bearing Con dence, All Tracks, 10000-18000 ft Altitudes . . . . . . . . . . . . . 39 5.7 Bearing Con dence, All Tracks, Above 18000 ft Altitudes . . . . . . . . . . . . . 40 5.8 Bearing Con dence, All Tracks, All Altitudes . . . . . . . . . . . . . . . . . . . 40 5.9 Distance Con dence, All Tracks, 0-10000 ft Altitudes . . . . . . . . . . . . . . . 41 5.10 Distance Con dence, All Tracks, 10000-18000 ft Altitudes . . . . . . . . . . . . 41 5.11 Distance Con dence, All Tracks, Above 18000 ft Altitudes . . . . . . . . . . . . 41 5.12 Distance Con dence, All Tracks, All Altitudes . . . . . . . . . . . . . . . . . . . 42 x Chapter 1 Introduction The purpose of this e ort was to assist the Army in their Ground Based Sense and Avoid approach to maintaining separation between unmanned aircraft and other air tra c during free ight. Two separate collections of ight data were analyzed: radar tracks provided by the FAA of the national airspace system (NAS) and global positioning system (GPS) tracks from a ight training aircraft at the Auburn University Airport. The motions of the aircraft in each dataset were analyzed to determine how reliably their position and velocity could be projected forward in time. Sense and avoid e orts typically take current position and velocity estimates of air tra c and project them forward in time to determine if any con icts may arise in the future between the other air tra c and the unmanned aircraft. If the other air tra c maintains its course, speed and altitude this works well, but often air tra c maneuvers for a variety of reasons so additional bu er space must be maintained when projected tracks forward. This task focuses on ascertaining how large a bu er is required for di erent types of air tra c and for di erent operational environments. 1 Chapter 2 Literature Review Several aspects of position prediction using extended time horizons have been investi- gated. Important works exist in the literature related to the concept of aircraft sense and avoidance along with future state prediction. Much of the work in future state prediction and collision avoidance originated from e orts in the robotics community for path planning applications. The methods used by Elnagar is 2001 illustrated that by modeling the move- ment of a free path object with kinematic constraints, a Kalman lter can be used to process the signals from the measurement sensors and predicts the objects? future position [1]. Using a straight projection technique with a known velocity and heading and a short time horizon, collision probability with other moving or stationary objects could be predicted. A generic sampling technique for prediction of an aircraft when given course and heading was develped by Cale in 2001 [2]. The sampling process was developed from the perspective of the air tra c controller such that it anticipated the ight path of the aircraft for several di erent intervals using the current states and ight plan to enhance the projection. The quality of the projection was then determined by comparing the predicted position to the measured at the respective time horizon. This particular method is adopted by many prediction simu- lations with the accuracy increased with the addition of ight plan information, wind eld models, target location and other information about the aircraft?s environment. This method was enhanced by the use of boundary restrictions and multiple sensors. These e orts were extended by Barrios in 2011 to predict the position of a vehicle with multiple models to estimate collision probabilities [3]. Using a separate model for constant position, velocity, acceleration and jerk a comparison between each models prediction, using a Kalman lter, 2 against the measured future position determined the optimum model to use for each portion of the track. Using the robotic community e orts as a platform, the aerospace industry adopted many of the methods of path prediction in the "sense and avoid" movement for unmanned aircraft (UA). The use of multiple models was implemented by Bar-Shalom in a switch lter that helped predict accelerations during maneuvers [4]. Employing a lter to switch between a constant velocity and constant acceleration model enabled the fading memory lter to determine the presence of a maneuver by an accumulation of increased covariance. Realizing the inability to predict maneuvers and the need to address the nonlinear nature of the system, an extended Kalman lter was implemented. The extended Kalman lter uses linearized system equations and introduces an update equation to produce results with the same order of magnitude in accuracy as the Kalman lter. An extended Kalman lter was used in Mao with a stochastic ordinary di erential equation model to solve the ip ambiguity developed by range measurements for two cooperative unmanned aircraft on a third aircraft without GPS signal [5]. Using a priori information about the air vehicle?s course and velocity as well as a speci ed time horizon, the extended Kalman lter provided additional information for the range measurement on where to look for the lost aircraft. The predictability characteristic of the extended Kalman lter was explored by Pr evest [6] over varied time horizons. Using the constant acceleration approach, a motion model was used to project the future position of the aircraft and the quality of the projection was determined by the variance between the true and the prediction. The degradation of the prediction manifested as growing error between the predicted and measured position as the projected time increased, showing that con ict prediction was better with shorter time horizons. This same implementation of the motion model was used by Pr evost again in 2008 with an extension to con ict avoidance [7]. By using the Federal Aviation Administration?s (FAA) regulated separation distance of ve nautical miles in the horizontal plane and 2000 feet in the vertical plane, time horizons were 3 projected to develop ellipsoidal shaped volumes in which the aircraft was projected to be. Given other cooperative aircraft with known position, a design space was developed in which the aircraft could travel safely. This simulation was enhanced by an end-game scenario in which the aircraft must travel through set waypoints as well as avoid projected trajectories from other cooperative aircraft sharing overlapping projected state geometries. Given the need to measure and interpret predictability of an aircraft in free ight, various approaches to developing a metric of prediction probability have been made. A probabilistic approach to the con ict avoidance problem was addressed by Hu who moved to quantify safety in terms of the prescribed separation distance from other aircraft or prohibited airspace [8]. Using an aircraft?s motion based on a stochastic di erential equation model, Markov chains generated paths of greatest safety for aircraft to avoid con ict areas. The look-ahead time horizon, based on a ight plan and an end-game scenario, developed a series of planes that the Markov chains used to develop a path of least resistance for the aircraft. The level of con ict probability was then measured by the ability of the aircraft to avoid the con ict areas as well as make it to the target in a timely fashion. Another method of projection and avoidance used by Richards, termed \mixed-integer linear programming", modeled a design space based on xed characteristics of the aircraft and con ict zones [9]. Designating the maximum turning radius and xing the velocity of the aircraft, possible control inputs to maneuver around con ict zones, including con icting aircraft, were processed through a cost function to minimize ight time while preserving the constrains imposed upon the system. The need for a metric was becoming more apparent such that analytical and numerical solutions to the con ict probability problem were developed. Paielli suggests that the con ict zones between the aircraft and the con ict region, aircraft-to-aircraft or aircraft-to-airspace, could be modeled as an ellipse and sphere, respectively [10]. It is di cult to formulate an analytical solution to the con ict probability based on the geometry of the problem. By changing the coordinate system from spherical to elliptical, the numerical solution could be 4 solved analytically to alleviate computational time and solve for the level of probability, or overlapping of geometries, for each engagement. Based on the ndings of Paielli, Prandini later solved the con ict probability analytically without a coordinate change or complicated numerical solution [11]. This computationally more e cient approach produced a real time probabilistic interpretation to the predictability of the aircraft in free ight. By way of a Monte Carlo simulation, symmetric and streamline encounter situations demonstrated the e ectiveness of the analytical solution by producing histograms of the separation distance. Presenting the level of con ict and predictability in airspace has been the focus in the sense and avoid community. In projecting the position, using methods of con ict detection, the certainty of collision was measured by Al-Basman using a likelihood density approach [12]. The amount of aircraft and coincident trajectories in the environment designated the level of congestion which was ranked through selected levels of intensity. This produced contours of congestions, identifying hazardous airspace through the accumulation of aircraft in the area. The quality of a projection can be broken into its accuracy cross-track and along-track. Gong developed a methodology of determining the e ectiveness of projections in both directions, producing Gaussian distributions of error for planar motion as well as three dimensional aircraft motion [13]. Using ight plans and disregarding ights that deviated from planned waypoints, the quality of a straight projection was presented in histograms, giving shape to the predictability of compliant aircraft in free ight. 5 Chapter 3 Estimating Future Position of Air Tra c Determining whether two aircraft will be in con ict with each other in the future requires estimating where those aircraft will be in the future. There are obvious limitations to how precisely this can be done. The most signi cant limitation is that the future behavior of non-cooperative aircraft is generally unknown. A pilot may maneuver for quite a number of di erent reasons. While aircraft on a cross-country trip will generally only make small course or altitude adjustments at various waypoints along their planned track, pilots that are just out \boring holes in the sky" or student pilots practicing various maneuvers may engage in fairly aggressive maneuvers unexpectedly. Thus it would be helpful to be able to quantify not only where a non-cooperative aircraft would be in the future given that it maintains its current velocity, but also where it could be if the pilot chooses to maneuver. In this study, time histories of aircraft tracks have been used to develop statistical models of aircraft maneuvers. Two sources of aircraft tracks have been used. 3.1 National Airspace System Data Seven days of data were acquired from the FAA through a freedom of information act request to be used for research. Figure 3.1 below shows all the tracks from this data set over a 24 hour period. The FAA database include aircraft that are either operating under Instrument Flight Rules (IFR) or aircraft under Visual Flight Rules (VFR) but using radar ight following. IFR aircraft and VFR aircraft using ight following are typically traveling between two points so these aircraft would not be expected to execute many maneuvers enroute. 6 Figure 3.1: NAS data over the United States over a 24 hour period 3.2 Auburn University Personal Aircraft GPS Data A second dataset was acquired using Auburn University?s small eet of ight training aircraft. GPS tracking devices were placed in these aircraft and their movements were tracked over approximately six weeks. Since these aircraft are used almost exclusively for ight training they represent aircraft that are most likely to maneuver. Each GPS unit was packaged with a high capacity NiMH battery pack to alow the unit to operate for us to a week without recharge as shown in Fig. 3.2a. The aircraft are mostly Cessna 172R models, a single engine high wing aircraft, pictured in Fig. 3.2b. 7 (a) GPS unit and NiMH battery pack (b) Auburn University Cessna 172R Figure 3.2: Materials and Aircraft used in GPS track study Figure 3.3 shows all of the tracks acquired over a six week period. Several long cross- country ights are obvious to Savannah, GA, Huntsville, AL, Rome, GA, Birmingham, AL and the Gulf Coast, but mostly the aircraft are operated within a relatively small area around Auburn, AL. Figure 3.3: All GPS tracks from Auburn, AL 8 The FAA dataset and the AU eet dataset should represent two ends of the maneuver spectrum. Aircraft traveling between two points such as most of those in the FAA dataset should behave very predictably whereas the ight training aircraft such as those in the AU eet should behave much more erratically. 9 Chapter 4 Data Processing To assess the uncertainty associated with pilot maneuvers, the aircraft tracks in both datasets were processed similarly. At each data point in the track, the airplane?s position was projected forward in time assuming the velocity remained constant. The actual position at that future time was then extracted from the track and compared with the projected position. The statistics associated with this \error" between the projected and actual position were accumulated over the entire length of each aircraft?s track and over the entire dataset of di erent aircraft tracks. The FAA dataset provided documented ights of compliant aircraft over the course of a week across the contiguous United States. Geographical latitude and longitude, altitude, and ground speed were given in one minute increments for each aircraft, listed by tail number. Each ight was designated by its provided tail number and used according to the length of its radar track. If the selected time horizon exceeded the recorded ight time of the aircraft, the ight was not used in the error analysis. 4.1 Development of Histogram Error To compare each measured ight with its projected, a system of analyzing the data statistically was developed. The objective was to have a way of organizing the error such that each ight, regardless of the aircraft type, could contribute to the description of the airspace it occupied. A forward projection was used, given an initial position, course, and speed [2]. To overcome the induced error provided by a at Earth assumption, this projection used rhumb 10 lines that accounted for the Earth?s curvature rather than a projection into the XY-plane. A further description of rhumb lines is given in Appendix C. Given the projected location, the relative position and bearing were found between the projected position and the measured position at the projected time. The process of accumulating the histograms is shown below in Fig. 4.1. In the top left of the gure, the rst projection takes place based o prior knowledge of the velocity and bearing. Using the projected position and bearing as the origin and orientation of the grid, respectively, shown in the top right of the gure. Once the location on the grid is recorded, the next projection takes place and the process repeats, as seen in the bottom left of the gure. The error locations were accumulated and saved with each time step, developing occurrences in each two dimensional error bin. Each location on the X-axis of the grid describes the error cross-track of the projected location, while the Y-axis describes the error along-track, providing a two dimensional de- scription of the projected error. Accumulating these points creates a statistical description of the aircraft at various projection lengths. To allow comparisons between aircraft of di ering speeds, each projected error was normalized by the aircraft?s instantaneous speed such that the error of a fast aircraft could be compared to that of a slower one. The result is an error unit expressed in seconds. 11 Measured (X m1 ,Y m1 ) Projected (X p1 ,Y p1 ) Y X Measured (X m1 ,Y m1 ) Projected (X p1 ,Y p1 ) Y X Measured (X m2 , Y m2 ) Projected (X p2 ,Y p2 ) Measured (X m1 ,Y m1 ) Projected (X p1 ,Y p1 ) Y X X Y X Y Figure 4.1: Progression of histogram error development 12 4.2 Development of Altitude Comparison In order to help determine the activity in di erent regions of airspace, the analysis was repeated for di erent altitude ranges. Using the same FAA radar data, the airspace was split into three altitude regions: below 10,000 ft, between 10,000 ft and 18,000 ft, and above 18,000 ft. These altitude ranges correspond to certain operating restrictions associate with the NAS. Below 10,000 ft aircraft are restricted from operating faster than 250 kts IAS. Above 18,000 ft is considered Class A airspace and all aircraft must operate under Instrument Flight Rules. Aircraft tracks that transitioned between the di erent altitude regions were split into segments so that the portions in each altitude range were treated appropriately. 4.3 The Kalman Filter The range and range-rate observations received from satellites by the GPS units arrive with noise in the signal. This noise, developed through the measurement and model receiving the measurements, must be ltered to develop the best measurement derived from the signal. A well-known mathematical approach, developed by R.E. Kalman, for sequentially processing observations of a linear dynamic system can be used to separate noise from the observations in a recursive manor at each discrete time step. The set of recursive equations was developed in 1960 by Kalman and is known as the Kalman lter [14]. The GPS units use a Kalman lter to develop position and rate from the satellite signals. Observations used to develop the statistical model have uncertainties from satellite loss or signal degradation, as well as noise developed from the GPS unit?s Kalman lter. The Kalman lter produces an optimal estimate of a system?s state. The state dynamics must be linear, therefore the current state, x(t), can be expressed as a linear function of the previous state, x(t t). Also, the observation process must be linear such that the observations, z(t), are a linear function of the current state, x(t). Kalman assumed that 13 the state dynamics are perturbed by random excitation, w(t), and the measurements are corrupted by random noise, v(t). Given these assumptions the state dynamics assume the form shown in Eq.(4.1) _x(t) = Fx(t) + w(t) (4.1) where F is a constant state matrix and w(t) is a vector of zero-mean Gaussian random variables. At intervals, the state of the system is observed through Eq.(4.2) z(t) = Hx(t) + v(t) (4.2) where z(t) represents the observations or measurements. H is a matrix which maps the states to the observations and v(t) is a vector of zero-mean Gaussian random variables. The Kalman lter algorithm proceeds using the following steps. First the state and covariance matrix are projected forward in time using Eqs.(4.3) and (4.4). ^x k = k^xk 1 (4.3) P k = kPk 1 Tk + Qk (4.4) Next the Kalman gain, Kk, is calculated in Eq.(4.5) and a corrected estimate of the current state is determined in Eq.(4.6), incorporating the new measurements. The covariance is then updated in Eq.(4.7) using the Kalman gain. Kk = P k HTk HkP k HTk + Rk 1 (4.5) ^xk = ^x k + Kk zk Hk^x k (4.6) Pk = [I KkHk] P k (4.7) 14 Above in Eq.(4.3), is de ned as the transition matrix for the state matrix H. = eH t (4.8) In Eq.(4.3) through (4.7), ^x k is the a priori state estimate at k and the a posteriori state estimate is ^xk. P k and Pk are the a priori and a posteriori estimated error covariance at k, respectively. The time update equations in (4.3) and (4.4) serve as the predictor equations while the measurement update equations in Eqs.(4.5) through (4.7) serve as the corrector equations in this predictor-corrector algorithm. The state space model of the aircraft?s kinematic motion is seen below in Eqs.(4.9) and (4.11). Equation (4.9) presents a constant velocity case for the model, assuming zero change in the acceleration of the aircraft. With this model, the range predictions over the speci ed time horizons were more accurate since the weight on the acceleration from the lter was neglected. Because of the straight projection provided by this lter, maneuvers were modeled as noise and prediction during these maneuvers resulted in a reduction in the quality of the projection. Equation (4.11) represents a constant acceleration model of the aircraft. Because of the higher Kalman gain needed to adjust for unexpected maneuvers, the constant acceleration model su ers in range prediction while providing more accurate projections for maneuvering aircraft. = 2 66 66 66 64 1 0 t 0 0 1 0 t 0 0 1 0 0 0 0 1 3 77 77 77 75 (4.9) where, x = x y vx vy T (4.10) 15 = 2 66 66 66 66 66 66 66 4 1 0 t 0 12 t2 0 0 1 0 t 0 12 t2 0 0 1 0 t 0 0 0 0 1 0 t 0 0 0 0 1 0 0 0 0 0 0 1 3 77 77 77 77 77 77 77 5 (4.11) where, x = x y vx vy ax ay T (4.12) To account for error in the model, noise was added to the system in the form of a process noise to create an arti cial oor for the covariance convergence. Because of the large number of measurements over long tracks, the covariance approaches zero, reducing the Kalman gain such that corrections to the state becomes ine ective. The addition of process noise increases the weight placed on measurements at each time step and reduces the reliance on previous information[15] [16]. For both the constant velocity and constant acceleration models, the process noise was selected to reduce the error between the actual tracks and the projected future positions. Using the distance error between the state at each time horizon and the measured state for all GPS tracks as the a cost function, the diagonals of the process noise matrix were determined using a MATLAB numerical minimization function ga(). This function is a genetic algorithm optimization routine provided in MATLAB used here to optimize the process noise variables. The diagonals of the process noise matrix were optimized for each of the following datasets: all tracks with airports, all tracks without airports, selected tracks with airports, and selected 16 tracks without airports. Q(t) = 2 66 66 66 64 q1 0 0 0 0 q1 0 0 0 0 q2 0 0 0 0 q2 3 77 77 77 75 (4.13) Q(t) = 2 66 66 66 66 66 66 66 4 q1 0 0 0 0 0 0 q1 0 0 0 0 0 0 q2 0 0 0 0 0 0 q2 0 0 0 0 0 0 q3 0 0 0 0 0 0 q3 3 77 77 77 77 77 77 77 5 (4.14) The covariance matrix updates at each time step. The initial position of the aircraft is unknown to the lter, therefore the diagonals of the covariance matrix were initially given high values to allow the lter to rely primarily on the measured position. The cross covariance terms were set to zero because of the initially unknown correlation between each element of the matrix. Eqs. (4.15) and (4.16) present the structure of the covariance matrices for each model of the lter. P = 2 66 66 66 64 2xx xy xvx xvy yx 2yy yvx yvy vxx vxy 2vxvx vxvy vyx vyy vyvx 2vyvy 3 77 77 77 75 (4.15) 17 P = 2 66 66 66 66 66 66 66 4 2xx xy xvx xvy xax xay yx 2yy yvx yvy yax yay vxx vxy 2vxvx vxvy vxax vxay vyx vyy vyvx 2vyvy vyax vyay axx axy axvx axvy 2axax axay ayx ayy ayvx ayvy ayax 2ayay 3 77 77 77 77 77 77 77 5 (4.16) The measurement error matrix is shown in Eq. (4.17). The GPS unit utilizes a di erential global positioning system with an accuracy up to 10 meters. Therefore the standard deviation of the error for the GPS unit is presented in the measurement error matrix as 100 m2. The measurement matrices for the constant velocity and constant acceleration lters are given in Eqs. (4.18) and (4.19), respectively. R = 2 64102 0 0 102 3 75 (4.17) H = 2 641 0 0 0 0 1 0 0 3 75 (4.18) H = 2 641 0 0 0 0 0 0 1 0 0 0 0 3 75 (4.19) 4.4 Other Projection Techniques Other projection techniques were explored to verify the quality of the projection meth- ods. Using the measured signal from the GPS units, the data was projected forward over the time horizon using two other prediction techniques. 18 As described in Appendix C, rhumbline?s can be used to project to a future position given the course and velocity of the vehicle over the time horizon. Similar to the Kalman lter method for a constant velocity model, the rhumbline projection assumes a constant velocity and course over the time horizon in the projection. Rhumbline?s lend themselves to this type of projection because of their course of constant bearing towards the destination. A second constant velocity technique used was to convert the spherical latitude and longitude coordinates into the Cartesian coordinats. This process reduced the complexity of the project, but was only valid for short distances do to the at Earth assumption needed to make the conversion possible. With all ights being centralized around the Auburn University Airport, the at Earth assumption does not cause the projections to diverge as they would for a cross-country track. Lastly, a moving average of the data was taken to smooth the error caused by jumps in sample time. Due to lose of satellite signal the sample rate varies, causing the projection error to increase with longer gaps in sample time. To reduce this e ect on the quality of the projection, a moving average with a window of ve seconds diminished the accumulation of error using a straight projection from the last known velocity and course. 19 Chapter 5 Results 5.1 National Airspace System Radar Tracks Several di erent methods of presenting the results will be used. Figure 5.1 shows results from all aircraft tracks for the three di erent altitude ranges plus a summary of all altitudes when the position is projected three minutes into the future. The two dimensional histogram is presented as a colored contour plot. The X-axis represents cross-track error and the Y-axis represents along-track error. The contour lines represent con dence intervals. The outside of the colored portion is the 99% con dence region. The red star in the image is the location of the aircraft prior to projecting the position forward in time. These graphs demonstrate the expected trends. The graph for above 18,000 ft (lower left of Fig. 5.1) shows that the actual positions are always in front of the previous position. The largest hump in the probability density function is straight ahead at the previous speed. There are some \wings" to the distribution indicating that some aircraft turn left or right, but no aircraft turn completely around. The upper two graphs, representing lower that 10,000 ft on the left and 10,000 ft to 18,000 ft on the right, indicate more maneuvers. The \wings" on the distribution now wrap around the previous spot marked by the star. That indicates some aircraft have turned completely around and are traveling in the opposite direction during the projection interval. 20 X?Error (s) Y?Error (s) 0?10000 Altitudes ?400?300?200?1000 100200300400 ?600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 Altitudes ?400?300?200?1000 100200300400 ?600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 Altitudes ?400?300?200?1000 100200300400 ?600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?400?300?200?1000 100200300400 ?600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 Figure 5.1: All tracks, 3 minute projection, di erence in projection to measured histograms The next gure, Fig. 5.2, shows the same presentation and same altitude ranges, but for a projection 10 minutes into the future. Note that the scales are considerably larger than the previous gure and wrapping around the previous position is evident. As with Fig. 5.1, the predictability of the aircraft increases as the altitude increases. If the projections were perfect, the probability density function would be a spike at the origin. Obviously that is unlikely, but the more of the probability density function that lies ahead of the original position, the more the aircraft could be described as predictable. In the graph for above 21 18,000 ft, even at 10 minutes of projection, the bulk of the error lies in the direction of the vehicle?s path. X?Error (s) Y?Error (s) 0?10000 Altitudes ?1500?1000?500 0 500 10001500 ?2000 ?1500 ?1000 ?500 0 500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 Altitudes ?1500?1000?500 0 500 10001500 ?2000 ?1500 ?1000 ?500 0 500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 Altitudes ?1500?1000?500 0 500 10001500 ?2000 ?1500 ?1000 ?500 0 500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?1500?1000?500 0 500 10001500 ?2000 ?1500 ?1000 ?500 0 500 10 20 30 40 50 60 70 80 90 Figure 5.2: All tracks, 10 minute projection, di erence in projection to measured histograms The next two gures show the motion of the aircraft in the same altitude plane. We can also look at changes in altitude. Figure 5.3 below compares altitude changes with a three minute projection for the four di erent altitude ranges. In the all altitude case (lower right) and the above 18,000 ft case (lower left) both show large peaks centered at zero error. That indicates that aircraft above 18,000 ft rarely change their altitude. Below 18,000 ft however, 22 there are di ering results. There is a large peak at zero error, but there are also very strong peaks away from zero. The 10,000 ft to 18,000 ft range is a transitional range. It is above the altitude most small, single engine aircraft y and below the ight levels (18,000 ft) where transports operate. Consequently, much of the tra c in that region is either climbing to the ight levels or descending from the ight levels. That explains the large peaks at zero error at o to the sides showing climbs and descents. The same holds true for below 10,000 ft where many of the aircraft are involved in terminal operations (takeo , landing and approach). 23 0 1 2 3 4 5 6 7 8x 105 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000Difference in Altitude (m) Occurrences 0 1 2 3 4 5 6x 105 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000Difference in Altitude (m) Occurrences 0 1 2 3 4 5 6 7 8 9 10x 106 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000Difference in Altitude (m) Occurrences 0 2 4 6 8 10 12x 106 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000Difference in Altitude (m) Occurrences Figure 5.3: Di erence in altitude, all tracks, 3 minute projection Figure 5.4 shows changes in speed over the ground over a three minute projection. As with altitude changes, the tracks are most predictable at higher altitude and the variability increases in the lower altitude ranges. The distribution is obviously not Gaussian at less than 10,000 ft and between 10,000 ft to 18,000 ft. The plot showing ights less than 10,000 ft shows a distinct hump for the -25%{10% speed range. This may correspond to aircrafts slowing down for approach and landing. 24 0 1 2 3 4 5 6 7 8 9 10x 105 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 1 2 3 4 5 6 7 8x 105 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 2 4 6 8 10 12x 106 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 2 4 6 8 10 12 14x 106 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure 5.4: Percent di erence in speed, all tracks, 3 minute projection With a ve minute projection, the di erences in bearing at low altitudes are very high as shown in Fig. 5.5. At high altitudes the distribution is a very strong peak and appears to be Gaussian, but at lower altitudes the distribution is much more spread. Below 10,000 ft, the distribution shows clear peaks left and right of zero error. These peaks are also evident for 10,000 ft to 18,000 ft, but de nitely less pronounced. It is not clear what these peaks relate to. 25 0 0.5 1 1.5 2 2.5 3 3.5x 105 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 105 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6 7x 106 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6 7 8x 106 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure 5.5: Di erence in bearing, all tracks, 5 minute projection A slightly di erent way of looking at the probability density functions is through a three dimensional histogram bar chart. These graphs are somewhat more di cult to read quantitatively, but they provide a clear qualitative picture of the statistical distribution. Figure 5.6 below shows the characteristics of air tra c around Fort Campbell in Kentucky. The segments of any air tra c that passed within a 50 km radius around Fort Campbell were included in these plots. As home to the 101st Airborne Division, the area around Fort Campbell sees signi cant military air tra c. These plots show data for a one minute 26 projection in time. As expected, the data show a tighter distribution at higher altitudes than at low altitudes. At low altitudes, a number of peaks are seen around the central peak. This is probably due to the low sample size of this local data. Note that the central peak on the less than 10,000 ft chart is around 40 occurrences compared to a peak of around 800 occurrences above 18,000 ft. ?150?100?50 0 50 100 150 ?200?100 0100 2000 10 20 30 40 50 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?150?100?50 0 50 100 150 ?200?100 0100 2000 20 40 60 80 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?150?100?50 0 50 100 150 ?200?100 0100 2000 200 400 600 800 1000 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?150?100?50 0 50 100 150 ?200?100 0100 2000 200 400 600 800 1000 1200 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure 5.6: Fort Campbell, 50 km radius, 1 minute projection, di erence in projection to measured three dimensional histograms 27 The bar histograms below in Fig. 5.7 show the di erences in altitude for air tra c around Fort Campbell. This is again a one minute projection in time. Here the altitude is predictable as expected above 18,000 ft and also very predictable below 10,000 ft. More variation is seen in the 10,000 to 18,000 ft range, but as seen before, that is most likely due to aircraft transitioning through those altitudes. 0 100 200 300 400 500 600 700 800 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000Difference in Altitude (m) Occurrences 0 100 200 300 400 500 600 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000Difference in Altitude (m) Occurrences 0 2000 4000 6000 8000 10000 12000 14000 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000Difference in Altitude (m) Occurrences 0 5000 10000 15000 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000Difference in Altitude (m) Occurrences Figure 5.7: Fort Campbell, 50 km radius, 1 minute projection, di erence in altitude 28 Figure 5.8 shows the accuracy of the bearing estimates for a one minute projection. There is a strong central peak even at low altitudes, but the distribution is distinctly non Gaussian and it spreads across the large bearing swath. In the ight levels (above 18,000 ft), the distribution returns to looking more Gaussian. 0 50 100 150 200 250 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 50 100 150 200 250 300 350 400 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1000 2000 3000 4000 5000 6000 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1000 2000 3000 4000 5000 6000 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure 5.8: Fort Campbell, 50 km radius, 1 minute projection, di erence in bearing The next two gures compare the tra c around Fort Campbell, Kentucky and Las Cruces, New Mexico. Las Cruces in Fig. 5.9, was selected because while it is lightly populated and hence would not normally see high amounts of air tra c, restricted areas in the area 29 funnel air tra c passing through the region into a narrow section nearby. Consequently, the air tra c should re ect a signi cant amount of aircraft just passing through. This hypothesis is borne out in the data. Fort Campbell in Fig. 5.10, shows that because of its terminal nature, it is less predictable at all altitude ranges than Las Cruces. More error points lie in front of the original position for Las Cruces than Fort Campbell yet both share common error trends at the 10,000 to 18,000 feet ranges. At 10,000 to 18,000 ft altitudes, the Las Cruces region is more concentrated around the zero error point than Fort Campbell, as expected. 30 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 Figure 5.9: Las Cruces, 50 km radius, 1 minute projection, di erence in projection to mea- sured histograms 31 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 Figure 5.10: Fort Campbell, 50 km radius, 1 minute projection, di erence in projection to measured histograms If we expand the radius around Fort Campbell and Las Cruces to 250 km the trends continue approximately the same. Figures 5.11 and 5.12 show a one minute projection for both the areas. As noted before, the density functions for Las Cruces are slightly more compact and are more in front of the original position instead of spreading behind the original position point. 32 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?300?200?100 0 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?300?200?100 0 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?300?200?100 0 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?300?200?100 0 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 Figure 5.11: Las Cruces, 250 km radius, 1 minute projection, di erence in projection to measured histograms 33 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?300?200?100 0 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?300?200?100 0 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?300?200?100 0 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?300?200?100 0 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 Figure 5.12: Fort Campbell, 250 km radius, 1 minute projection, di erence in projection to measured histograms Increasing the projection from one minute to ve minutes changes the plots signi cantly but not the overall character. Both the Las Cruces (Fig. 5.13) and Fort Campbell (Fig. 5.14) data show more predictable ying at above 18,000 ft. Las Cruces tracks are more compact at high altitude than Fort Campbell. 34 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 Figure 5.13: Las Cruces, 250 km radius, 5 minute projection, di erence in projection to measured histograms 35 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 Figure 5.14: Fort Campbell, 250 km radius, 5 minute projection, di erence in projection to measured histograms In order to put these results to use, some metrics have been derived from the data. By sorting the data and nding percentiles within the data, some conclusions can be drawn. The next sets of tables show the con dence intervals for various parameters and altitude ranges. Tables 5.1 through 5.4 show the altitude con dence intervals. Using Table 5.1 for example, one could say that an aircraft will stay within 732 m over one minute 95% of the time, or within 3,139 m over ve minutes 99% of the time. 36 Table 5.1: Altitude Con dence, All Tracks, 0-10000 ft Altitudes Con dence (m) Projection Time 0.75 0.90 0.95 0.99 1 Minute 335 549 732 1067 3 Minute 914 1341 1737 2438 5 Minute 1433 1920 2286 3139 10 Minute 2499 3292 3749 4602 Table 5.2: Altitude Con dence, All Tracks, 10000-18000 ft Altitudes Con dence (m) Projection Time 0.75 0.90 0.95 0.99 1 Minute 549 732 823 1067 3 Minute 1585 2042 2316 2865 5 Minute 2530 3200 3597 4359 10 Minute 4237 5243 5852 6858 Table 5.3: Altitude Con dence, All Tracks, Above 18000 ft Altitudes Con dence (m) Projection Time 0.75 0.90 0.95 0.99 1 Minute 30 427 610 853 3 Minute 274 1280 1676 2347 5 Minute 610 2042 2682 3658 10 Minute 1219 3749 4877 6431 37 Table 5.4: Altitude Con dence, All Tracks, All Altitudes Con dence (m) Projection Time 0.75 0.90 0.95 0.99 1 Minute 274 518 671 945 3 Minute 762 1463 1860 2469 5 Minute 1188 2316 2896 3810 10 Minute 2134 4084 4999 6462 38 Tables 5.5 through 5.8 may be a more reasonable metric, the expected error in bearing (or more precisely heading). This data clearly indicate the aircraft become more predictable at higher altitudes. At 20,000 ft for example, an aircraft will maintain bearing within about 20 degrees for 10 minutes with 95% con dence. Table 5.5: Bearing Con dence, All Tracks, 0-10000 ft Altitudes Con dence (degrees) Projection Time 0.75 0.90 0.95 0.99 1 Minute 22.4 41.9 61.7 96.5 3 Minute 25.1 58.2 88.6 133.2 5 Minute 26.5 58.2 92.2 148.3 10 Minute 21.4 40.7 60.7 141.5 Table 5.6: Bearing Con dence, All Tracks, 10000-18000 ft Altitudes Con dence (degrees) Projection Time 0.75 0.90 0.95 0.99 1 Minute 12.6 21.4 29.8 58.8 3 Minute 12.8 26.3 40.9 94.0 5 Minute 15.1 31.9 52.3 125.2 10 Minute 16.9 34.2 56.8 137.2 39 Table 5.7: Bearing Con dence, All Tracks, Above 18000 ft Altitudes Con dence (degrees) Projection Time 0.75 0.90 0.95 0.99 1 Minute 7.6 10.3 12.7 21.9 3 Minute 5.5 8.6 11.9 26.1 5 Minute 5.5 9.2 13.8 30.5 10 Minute 6.3 11.9 18.2 41.0 Table 5.8: Bearing Con dence, All Tracks, All Altitudes Con dence (degrees) Projection Time 0.75 0.90 0.95 0.99 1 Minute 9.3 16.8 24.9 62.0 3 Minute 7.2 15.4 27.4 82.2 5 Minute 7.3 16.6 28.8 86.8 10 Minute 8.0 17.3 27.5 70.3 40 Table 5.9: Distance Con dence, All Tracks, 0-10000 ft Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 25.6 48.0 64.8 104.6 3 Minute 22.5 42.2 65.4 143.7 5 Minute 29.8 62.9 121.8 305.0 10 Minute 47.2 94.8 166.8 507.5 Table 5.10: Distance Con dence, All Tracks, 10000-18000 ft Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 18.9 35.8 51.4 82.1 3 Minute 15.8 23.7 29.2 47.1 5 Minute 19.4 27.9 34.2 60.3 10 Minute 27.2 38.3 47.6 118.4 Table 5.11: Distance Con dence, All Tracks, Above 18000 ft Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 8.9 13.8 18.0 35.8 3 Minute 5.1 8.6 11.6 19.0 5 Minute 4.7 8.6 11.9 20.5 10 Minute 4.9 10.6 16.4 30.7 41 Table 5.12: Distance Con dence, All Tracks, All Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 11.7 21.7 34.5 71.2 3 Minute 7.3 15.2 23.0 55.6 5 Minute 7.3 17.1 26.9 84.7 10 Minute 7.7 22.8 35.7 107.2 These con dence bounds could be used to develop metrics for a sense and avoid system. A compilation of all the plots made from the FAA radar data have been attached as Appendix A. 5.2 Private Aircraft GPS Tracks Similar processing was done to the data gathered from the Auburn University ight training aircraft. This data wil be presented in histogram form with one minute, three minute, and ve minute projections. In Fig. 5.15 below show the probability density function for a one minute projection in time using a constant velocity Kalman lter. The left gure shows data including the full aircraft track while the gure on the right shows data where portions of the tracks where the aircraft is operating in a tra c pattern reduces the number of maneuvers and the wrap around e ect (where the track is going in the opposite direction of the projected position) is reduced. 42 Figure 5.15: Selected GPS les with and without airports, 1 minute GPS projection using a constant velocity lter Figure 5.16 show results for both a constant velocity Kalman lter (on the left) and a constant acceleration lter (on the right). Both of the gures have had the airport regions removed. The constant acceleration model on the right does a slightly better job of predicting the future positions and has a smaller spread. However, the constant acceleration model shows more along-track variation than the constant velocity lter (note the projection of the blue area forward from the peak and backward behind the peak). 43 Figure 5.16: Selected GPS les with and without airports, 1 minute GPS projection using a constant velocity lter A simple approach to calculating the velocity is to do a backwards di erence between the positions at one point in time and a previous point in time. The left hand gure in 5.17 was prepared using this approach. The right-hand gure was computed using the GPS velocity at each time step. Very little di erence can be noted between the two plots. The lower gure is the set shows results using a moving average of the GPS velocity. These results very closely model the other two methods. 44 Figure 5.17: Selected GPS les with and without airports, 1 minute GPS projection using a constant velocity lter Comparing all the di erent velocity estimation techniques shows that the Kalman lters have a tendency to under predict the wraparound e ect by over predicting the distance but the density of the error is larger while the other projection techniques under predict the 45 change in bearing. The constant acceleration lter better predicts the distance, depicted in the gure by a more consistent radius about the initial point than the constant velocity Kalman lter. Figure 5.18 shows the e ect of di erent projection times on the results. All of the gures have had airport tra c patterns removed. One interesting feature of all of the plots is an apparent tendency to turn to the left more often than the right. That may be a behavioral issue since student pilots may be more comfortable turning toward the side where they are sitting when doing various maneuvers. The gures clearly indicate the increased uncertainty as the projection time increases. 46 Figure 5.18: Selected GPS les with and without airports, 1 minute GPS projection using a constant velocity lter Appendix B contains a compilation of all the GPS data results. 47 5.3 Comparison of Projection Techniques Each of the projection techniques showed to have both strengths and weaknesses during all stages of the ight. As expected by the di erent Kalman lter models, the constant velocity model projected distance better than predicting a maneuver while the constant acceleration model projected changes in bearing better than changes in distance. Looking at the cross-track and along-track error the projection quality of each lter was analyzed to provide information on the lter type needed to predict future position under various ight conditions. Taking a look at one track recorded by the GPS units in Fig. 5.19, the straight ight can be separated from the maneuvering portions to analyze how the lters perform on each segment. Each segment consist of 30 minutes of ight time. The performance of the constant velocity Kalman lter (CVF), the constant acceleration Kalman lter (CAF), GPS projection (GPS), and the moving average projection (MA) were analyzed using bar plots to show the error cross-track and along-track of the projection. 48 Figure 5.19: Single GPS ight analyzed, breaking the ight into two parts: straight ight and maneuvering ight Analyzing the straight portion of the track rst, the ability of each lter in straight ight is characterized by its projection error cross-track (Fig. 5.20) and along-track (Fig. 5.21). 49 Figure 5.20: Cross track error for straight portion of the track using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) The cross-track analysis in Fig. 5.20 shows that the error has a Gaussian distribution. Both the constant velocity and constant acceleration Kalman lter perform equally well in predicting corrections in the straight portion of the ight. The error grows with an extended time horizon but a better prediction in cross-track projection is shown in the 50 constant acceleration case as the projection time grows. The straight GPS projection using course and velocity shows greater error than the moving average. The smoothing of the measurement with the ve second moving average window decreases the e ect of velocity jumps in the measurements and predicts maneuvers better in predominately straight ight. Figure 5.21: Along track error for straight portion of the track using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) 51 The along-track error in Fig. 5.21 is centered around zero for straight ight as expected, with a majority of the error due to underestimating the distance. For the one minute projection, the constant acceleration Kalman lter under predicts the displacement more often then the constant velocity Kalman lter. Both the GPS projection and the moving average tend to over predict the displacement more than the Kalman lters. In the case of the three and ve minute projections, the distribution is Gaussian, giving the straight portion of ight a higher predictability in the along-track projection. 52 Figure 5.22: Cross track error for maneuvering portion of the track using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) In maneuvering ight shown in Fig. 5.22, the range in error grows to several times the error distribution for straight ight in the cross-track case. A majority of the maneuvering error lies to the right of the zero error region, mainly because the turns performed by the aircraft in this situation are left-hand turns. The projection sends the estimated position 53 beyond the turn and under predicts the cross-track position. The constant acceleration Kalman lter holds more error around the zero region than the constant velocity Kalman lter, showing that the acceleration model more accurately predicts maneuvers. The GPS and moving average projections have a Gaussian distribution about the origin of error, more often over predicting the cross-track position, under estimating the magnitude of the maneuvers. 54 Figure 5.23: Along track error for maneuvering portion of the track using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) For the maneuvering portion of the ight along-track shown in Fig. 5.23, the error distributions are Gaussian but not about the origin of error. Each projection technique under predicts the projection but constant acceleration Kalman lter retains more error near 55 the peak of the distribution while the constant velocity Kalman lter has more occurrences away from the origin of the Gaussian distribution. Figure 5.24: Cross track error for all GPS tracks using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) A full ight is presented in Fig. 5.24. Again, the majority of the maneuvers are left- hand turns, developing growing error to the right of the origin because the projection sends 56 the predicted position beyond the left-hand turn. For each time horizon, the distribution is Gaussian and the range of error grows with the projection time. Little can be determined about the quality of projection for each projection method in the cross-track error given that the data has both straight and maneuvering ight patterns. Figure 5.25: Along track error for all GPS tracks using constant velocity Kalman lter (CVF), constant acceleration Kalman lter (CAV), GPS straight projection (GPS), and a moving average lter (MA) 57 Figure 5.25 shows the along-track error distribution for a full ight. With the com- bination of straight and maneuvering segments of ight, the projections under predict the displacement along the track. The along-track error is reduced using a the constant velocity Kalman lter while the cross-track error is reduced using the constant acceleration Kalman lter. The GPS projection has shown to perform as well as the moving average technique yet the moving average controls velocity jumps inherent in the GPS units. In the case of the NAS data with commercial IFR and VFR ights, the tracks pass through predetermined waypoints selected in such a way to reduce aerial con ict and re- duce transit time. Given their straight line nature between their departure and arrival airports, their predictability increases. A more di cult case for predicting future position is the non-compliant aircraft ying without ight plans, tracked only by radar or GPS units. Shown in sections 5.1 and 5.2, the predictability of an commercial aircraft above 18,000 ft is much higher than the predictability of a private pilot training around small airports while commerical aircraft in a terminal environment at lower altitudes develop the same predictability characteristics as non-compliant aircraft. Although highly unpredictably, per- sonal non-compliant aircraft can be tracked with measurable accuracy given the right models and the ability to detect possible maneuvers. Recognizing maneuvers through basic changes in bearing and velocity help determine the model to use and by possibly switching between models, a higher accuracy in displacement and bearing change could be achieved. 58 Chapter 6 Conclusion The ability to predict an aircraft?s future position based on prior knowledge of the states was found to be dependent on the aircraft?s environment. In the case of compliant aircraft sampled in the NAS dataset over a week?s time, the con dence in the quality of the projection varied by altitude levels and whether the aircraft was in a terminal or transit environment. At lower altitudes, aircraft performed many maneuvers either in preparation to land or when leaving the airport, making the performance of the projections less reliable. At altitudes ranging from 10,000 to 18,000 ft, aircraft become more predictable than the lower altitude range. This altitude region served as a transitional environment for commercial aircraft. The change in rate of altitude across the region suggest that aircraft are either beginning their approach or gaining altitude to enter the cruise portion of their ight. At altitudes above 18,000 ft, the projection error reduced and the con dence intervals of the projection produced greater predictability. At each altitude level, a level of certainty in aircraft prediction helped de ne the ight environment. These con dence intervals characterize the aircraft?s motion at various altitudes and can be used to validate simulations for unmanned aircraft. The more challenging task of predicting future position of non-compliant aircraft devel- oped an understanding of the performance of di erent projection techniques under di erent ight conditions. The constant velocity Kalman lter projection more accurately predicted the displacement along the projection as well as reduced the projection error with small maneuvers. For more aggressive maneuvers, a constant acceleration Kalman lter predicted the change in course with less error than the constant velocity Kalman lter yet lacked the along-track performance that the constant velocity Kalman lter displayed. Using the 59 Kalman lters helped mimic the lter used by the GPS units and reduced the error inherent in a straight velocity projection. An improvement in reducing the e ects of satellite loss and jumps in instantaneous velocity was incorporating a moving average. By smoothing the data, the discontinuities in velocity due to inconsistent time steps became less e ective on the projection and reduced the error between the projected and observed position. Looking at the projection success for both compliant and non-compliant aircraft demon- strated the erratic nature of training aircraft at low altitudes. In the hopes of developing models to simulate private pilot activity, combinations of ltering types and projection tech- niques will have to be utilized to adapt to the various ight environments and trajectories seen in a single track. 60 Chapter 7 Future Work With unmanned aircraft entering the NAS, adequate sense and avoid technology is still needed to reduce aerial con icts and open the skies to both manned and unmanned vehicles. Through individual analysis of each projection technique under di erent ight conditions and environments, the need to alternate between projection types and ltering techniques is apparent. In straight and level ight, the constant velocity Kalman lter is very capable yet during maneuvers its performance su ers. By switching between a constant velocity and constant acceleration lter when entering and exiting a maneuver, the prediction performance of the projection improves while keeping the computational cost at a minimum. Bar-Shalom demonstrated the ability of a switch lter when applied to a simulated track consisting of straight and maneuvering portions [4]. Also, much success has been seen in the application of the unscented Kalman lter to the vehicle tracking problem. Due to the highly non-linear nature of the system given the unpredictable changes in course and presence of non-Gaussian noise, the unscented Kalman lter has been used in many tracking applications and path planning models. Another approach to better predicting the future position of aircraft is incorporating more information about the environment into the model. Flight plans, restricted areas, weather conditions, and other information can be applied to the model to notify senors where to look for an aircraft. This approach, used for ground vehicle path planning, divides the airspace into possible regions of interest. The probability of sensing another vehicle or object would then be increased when the sampled environment is reduced. 61 Bibliography [1] Elnagar, A., \Prediction of Moving Objects in Dynamic Environments Using Kalman Filters," Proceedings of 2001 IEEE International Symposium on Computational Intelli- gence in Robotics and Automation, July 29 - August 1, 2001, Ban , Alberta, Canada, Ban , Alberta, Canada, 2001, pp. 414{419. [2] Cale, M., Liu, S., Oaks, R., Paglione, M., Ryan, H., and Summerill, S., \A Generic Sampling Technique for Measuring Aircraft Trajectory Prediction Accuracy," Proceed- ings of 4th USAnEUROPE Air Tra c Management R & D Seminar, December 3rd-7th, 2001 , 2001. [3] Barrios, C. and Motai, Y., \Improving Estimation of Vehicle?s Trajectory Using the Latest Global Positioning System with Kalman Filtering," IEEE Transactions on In- strumentation and Measurement, Vol. 60, No. 12, Dec. 2011, pp. 3747{3755. 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[17] MathWorks, \Great Circles, Rhumb Lines, and Small Circles," http://www. mathworks.com/help/toolbox/map/f5-7173.html#f5-7179, September 2011. 63 Appendices 64 List of Appendix A Figures A.1 All tracks, 1 minute projection, di erence in projection to measured histograms 77 A.2 All tracks, 3 minute projection, di erence in projection to measured histograms 78 A.3 All tracks, 5 minute projection, di erence in projection to measured histograms 79 A.4 All tracks, 10 minute projection, di erence in projection to measured histograms 80 A.5 All tracks, 1 minute projection, di erence in projection to measured three dimen- sional histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A.6 All tracks, 3 minute projection, di erence in projection to measured three dimen- sional histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A.7 All tracks, 5 minute projection, di erence in projection to measured three dimen- sional histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 A.8 All tracks, 10 minute projection, di erence in projection to measured three di- mensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 A.9 All tracks, 0-10000 ft altitudes, di erence in projection to measured histograms 85 A.10 All tracks, 10000-18000 ft altitudes, di erence in projection to measured histograms 86 A.11 All tracks, Above 18000 ft altitudes, di erence in projection to measured histograms 87 A.12 All tracks, All Altitudes, di erence in projection to measured histograms . . . . 88 65 A.13 All tracks, 0-10000 ft altitudes, di erence in projection to measured three dimen- sional histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 A.14 All tracks, 10000-18000 ft altitudes, di erence in projection to measured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 A.15 All tracks, Above 18000 ft altitudes, di erence in projection to measured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 A.16 All tracks, All Altitudes, di erence in projection to measured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 A.17 Di erence in altitude, all tracks, 1 minute projection . . . . . . . . . . . . . . . 93 A.18 Di erence in bearing, all tracks, 1 minute projection . . . . . . . . . . . . . . . 94 A.19 Percent di erence in distance, all tracks, 1 minute projection . . . . . . . . . . . 95 A.20 Percent di erence in speed, all tracks, 1 minute projection . . . . . . . . . . . . 96 A.21 Di erence in altitude, all tracks, 3 minute projection . . . . . . . . . . . . . . . 97 A.22 Di erence in bearing, all tracks, 3 minute projection . . . . . . . . . . . . . . . 98 A.23 Percent di erence in distance, all tracks, 3 minute projection . . . . . . . . . . . 99 A.24 Percent di erence in speed, all tracks, 3 minute projection . . . . . . . . . . . . 100 A.25 Di erence in altitude, all tracks, 5 minute projection . . . . . . . . . . . . . . . 101 A.26 Di erence in bearing, all tracks, 5 minute projection . . . . . . . . . . . . . . . 102 A.27 Percent di erence in distance, all tracks, 5 minute projection . . . . . . . . . . . 103 66 A.28 Percent di erence in speed, all tracks, 5 minute projection . . . . . . . . . . . . 104 A.29 Di erence in altitude, all tracks, 10 minute projection . . . . . . . . . . . . . . . 105 A.30 Di erence in bearing, all tracks, 10 minute projection . . . . . . . . . . . . . . . 106 A.31 Percent di erence in distance, all tracks, 10 minute projection . . . . . . . . . . 107 A.32 Percent di erence in speed, all tracks, 10 minute projection . . . . . . . . . . . 108 A.33 Fort Campbell, 50 km radius, 1 minute projection, di erence in projection to measured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 A.34 Fort Campbell, 50 km radius, 1 minute projection, di erence in projection to measured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . 110 A.35 Fort Campbell, 50 km radius, 1 minute projection, di erence in altitude . . . . 111 A.36 Fort Campbell, 50 km radius, 1 minute projection, di erence in bearing . . . . . 112 A.37 Fort Campbell, 50 km radius, 1 minute projection, percent di erence in distance 113 A.38 Fort Campbell, 50 km radius, 1 minute projection, percent di erence in speed . 114 A.39 Fort Campbell, 50 km radius, 3 minute projection, di erence in projection to measured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 A.40 Fort Campbell, 50 km radius, 3 minute projection, di erence in projection to measured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . 116 A.41 Fort Campbell, 50 km radius, 3 minute projection, di erence in altitude . . . . 117 A.42 Fort Campbell, 50 km radius, 3 minute projection, di erence in bearing . . . . . 118 67 A.43 Fort Campbell, 50 km radius, 3 minute projection, percent di erence in distance 119 A.44 Fort Campbell, 50 km radius, 3 minute projection, percent di erence in speed . 120 A.45 Fort Campbell, 50 km radius, 5 minute projection, di erence in projection to measured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.46 Fort Campbell, 50 km radius, 5 minute projection, di erence in projection to measured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . 122 A.47 Fort Campbell, 50 km radius, 5 minute projection, di erence in altitude . . . . 123 A.48 Fort Campbell, 50 km radius, 5 minute projection, di erence in bearing . . . . . 124 A.49 Fort Campbell, 50 km radius, 5 minute projection, percent di erence in distance 125 A.50 Fort Campbell, 50 km radius, 5 minute projection, percent di erence in speed . 126 A.51 Las Cruces, 50 km radius, 1 minute projection, di erence in projection to mea- sured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 A.52 Las Cruces, 50 km radius, 1 minute projection, di erence in projection to mea- sured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . 128 A.53 Las Cruces, 50 km radius, 1 minute projection, di erence in altitude . . . . . . 129 A.54 Las Cruces, 50 km radius, 1 minute projection, di erence in bearing . . . . . . . 130 A.55 Las Cruces, 50 km radius, 1 minute projection, percent di erence in distance . . 131 A.56 Las Cruces, 50 km radius, 1 minute projection, percent di erence in speed . . . 132 68 A.57 Las Cruces, 50 km radius, 3 minute projection, di erence in projection to mea- sured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.58 Las Cruces, 50 km radius, 3 minute projection, di erence in projection to mea- sured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . 134 A.59 Las Cruces, 50 km radius, 3 minute projection, di erence in altitude . . . . . . 135 A.60 Las Cruces, 50 km radius, 3 minute projection, di erence in bearing . . . . . . . 136 A.61 Las Cruces, 50 km radius, 3 minute projection, percent di erence in distance . . 137 A.62 Las Cruces, 50 km radius, 3 minute projection, percent di erence in speed . . . 138 A.63 Las Cruces, 50 km radius, 5 minute projection, di erence in projection to mea- sured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A.64 Las Cruces, 50 km radius, 5 minute projection, di erence in projection to mea- sured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . 140 A.65 Las Cruces, 50 km radius, 5 minute projection, di erence in altitude . . . . . . 141 A.66 Las Cruces, 50 km radius, 5 minute projection, di erence in bearing . . . . . . . 142 A.67 Las Cruces, 50 km radius, 5 minute projection, percent di erence in distance . . 143 A.68 Las Cruces, 50 km radius, 5 minute projection, percent di erence in speed . . . 144 A.69 Fort Campbell, 250 km radius, 1 minute projection, di erence in projection to measured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 A.70 Fort Campbell, 250 km radius, 1 minute projection, di erence in projection to measured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . 146 69 A.71 Fort Campbell, 250 km radius, 1 minute projection, di erence in altitude . . . . 147 A.72 Fort Campbell, 250 km radius, 1 minute projection, di erence in bearing . . . . 148 A.73 Fort Campbell, 250 km radius, 1 minute projection, percent di erence in distance 149 A.74 Fort Campbell, 250 km radius, 1 minute projection, percent di erence in speed . 150 A.75 Fort Campbell, 250 km radius, 3 minute projection, di erence in projection to measured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 A.76 Fort Campbell, 250 km radius, 3 minute projection, di erence in projection to measured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . 152 A.77 Fort Campbell, 250 km radius, 3 minute projection, di erence in altitude . . . . 153 A.78 Fort Campbell, 250 km radius, 3 minute projection, di erence in bearing . . . . 154 A.79 Fort Campbell, 250 km radius, 3 minute projection, percent di erence in distance 155 A.80 Fort Campbell, 250 km radius, 3 minute projection, percent di erence in speed . 156 A.81 Fort Campbell, 250 km radius, 5 minute projection, di erence in projection to measured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 A.82 Fort Campbell, 250 km radius, 5 minute projection, di erence in projection to measured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . 158 A.83 Fort Campbell, 250 km radius, 5 minute projection, di erence in altitude . . . . 159 A.84 Fort Campbell, 250 km radius, 5 minute projection, di erence in bearing . . . . 160 A.85 Fort Campbell, 250 km radius, 5 minute projection, percent di erence in distance 161 70 A.86 Fort Campbell, 250 km radius, 5 minute projection, percent di erence in speed . 162 A.87 Las Cruces, 250 km radius, 1 minute projection, di erence in projection to mea- sured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 A.88 Las Cruces, 250 km radius, 1 minute projection, di erence in projection to mea- sured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . 164 A.89 Las Cruces, 250 km radius, 1 minute projection, di erence in altitude . . . . . . 165 A.90 Las Cruces, 250 km radius, 1 minute projection, di erence in bearing . . . . . . 166 A.91 Las Cruces, 250 km radius, 1 minute projection, percent di erence in distance . 167 A.92 Las Cruces, 250 km radius, 1 minute projection, percent di erence in speed . . . 168 A.93 Las Cruces, 250 km radius, 3 minute projection, di erence in projection to mea- sured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A.94 Las Cruces, 250 km radius, 3 minute projection, di erence in projection to mea- sured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . 170 A.95 Las Cruces, 250 km radius, 3 minute projection, di erence in altitude . . . . . . 171 A.96 Las Cruces, 250 km radius, 3 minute projection, di erence in bearing . . . . . . 172 A.97 Las Cruces, 250 km radius, 3 minute projection, percent di erence in distance . 173 A.98 Las Cruces, 250 km radius, 3 minute projection, percent di erence in speed . . . 174 A.99 Las Cruces, 250 km radius, 5 minute projection, di erence in projection to mea- sured histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 71 A.100Las Cruces, 250 km radius, 5 minute projection, di erence in projection to mea- sured three dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . 176 A.101Las Cruces, 250 km radius, 5 minute projection, di erence in altitude . . . . . . 177 A.102Las Cruces, 250 km radius, 5 minute projection, di erence in bearing . . . . . . 178 A.103Las Cruces, 250 km radius, 5 minute projection, percent di erence in distance . 179 A.104Las Cruces, 250 km radius, 5 minute projection, percent di erence in speed . . . 180 B.1 Constant velocity and constant acceleration lters, all GPS ights with airports, 1 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 B.2 Straight velocity projection and moving average projection, all GPS ights with airports, 1 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 B.3 Constant velocity and constant acceleration lters, all GPS ights with airports, 3 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 B.4 Straight velocity projection and moving average projection, all GPS ights with airports, 3 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 B.5 Constant velocity and constant acceleration lters, all GPS ights with airports, 5 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 B.6 Straight velocity projection and moving average projection, all GPS ights with airports, 5 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 B.7 Constant velocity and constant acceleration lters, all GPS ights without air- ports, 1 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 72 B.8 Straight velocity projection and moving average projection, all GPS ights with- out airports, 1 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . 197 B.9 Constant velocity and constant acceleration lters, all GPS ights without air- ports, 3 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 B.10 Straight velocity projection and moving average projection, all GPS ights with- out airports, 3 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . 199 B.11 Constant velocity and constant acceleration lters, all GPS ights without air- ports, 5 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 B.12 Straight velocity projection and moving average projection, all GPS ights with- out airports, 5 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . 201 B.13 Constant velocity and constant acceleration lters, selected GPS ights with airports, 1 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 B.14 Straight velocity projection and moving average projection, selected GPS ights with airports, 1 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . 203 B.15 Constant velocity and constant acceleration lters, selected GPS ights with airports, 3 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 B.16 Straight velocity projection and moving average projection, selected GPS ights with airports, 3 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . 205 B.17 Constant velocity and constant acceleration lters, selected GPS ights with airports, 5 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 73 B.18 Straight velocity projection and moving average projection, selected GPS ights with airports, 5 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . 207 B.19 Constant velocity and constant acceleration lters, selected GPS ights without airports, 1 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 B.20 Straight velocity projection and moving average projection, selected GPS ights without airports, 1 minute time horizon . . . . . . . . . . . . . . . . . . . . . . 209 B.21 Constant velocity and constant acceleration lters, selected GPS ights without airports, 3 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 B.22 Straight velocity projection and moving average projection, selected GPS ights without airports, 3 minute time horizon . . . . . . . . . . . . . . . . . . . . . . 211 B.23 Constant velocity and constant acceleration lters, selected GPS ights without airports, 5 minute time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 B.24 Straight velocity projection and moving average projection, selected GPS ights without airports, 5 minute time horizon . . . . . . . . . . . . . . . . . . . . . . 213 74 List of Appendix A Tables A.1 Altitude Con dence, All Tracks, 0-10000 ft Altitudes . . . . . . . . . . . . . . . 181 A.2 Altitude Con dence, All Tracks, 10000-18000 ft Altitudes . . . . . . . . . . . . 181 A.3 Altitude Con dence, All Tracks, Above 18000 ft Altitudes . . . . . . . . . . . . 181 A.4 Altitude Con dence, All Tracks, All Altitudes . . . . . . . . . . . . . . . . . . . 182 A.5 Bearing Con dence, All Tracks, 0-10000 ft Altitudes . . . . . . . . . . . . . . . 183 A.6 Bearing Con dence, All Tracks, 10000-18000 ft Altitudes . . . . . . . . . . . . . 183 A.7 Bearing Con dence, All Tracks, Above 18000 ft Altitudes . . . . . . . . . . . . . 183 A.8 Bearing Con dence, All Tracks, All Altitudes . . . . . . . . . . . . . . . . . . . 184 A.9 Distance Con dence, All Tracks, 0-10000 ft Altitudes . . . . . . . . . . . . . . . 185 A.10 Distance Con dence, All Tracks, 10000-18000 ft Altitudes . . . . . . . . . . . . 185 A.11 Distance Con dence, All Tracks, Above 18000 ft Altitudes . . . . . . . . . . . . 185 A.12 Distance Con dence, All Tracks, All Altitudes . . . . . . . . . . . . . . . . . . . 186 A.13 Speed Con dence, All Tracks, 0-10000 ft Altitudes . . . . . . . . . . . . . . . . 187 A.14 Speed Con dence, All Tracks, 10000-18000 ft Altitudes . . . . . . . . . . . . . . 187 A.15 Speed Con dence, All Tracks, Above 18000 ft Altitudes . . . . . . . . . . . . . . 187 A.16 Speed Con dence, All Tracks, All Altitudes . . . . . . . . . . . . . . . . . . . . 188 75 Appendix A National Airspace System Radar Tracks 76 X?Error (s) Y?Error (s) 0?10000 Altitudes ?400?300?200?1000 100200300400 ?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 Altitudes ?400?300?200?1000 100200300400 ?250 ?200 ?150 ?100 ?50 0 50 100 150 200 250 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 Altitudes ?400?300?200?1000 100200300400 ?250 ?200 ?150 ?100 ?50 0 50 100 150 200 250 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?400?300?200?1000 100200300400 ?250 ?200 ?150 ?100 ?50 0 50 100 150 200 250 10 20 30 40 50 60 70 80 90 Figure A.1: All tracks, 1 minute projection, di erence in projection to measured histograms 77 X?Error (s) Y?Error (s) 0?10000 Altitudes ?400?300?200?1000 100200300400 ?600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 Altitudes ?400?300?200?1000 100200300400 ?600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 Altitudes ?400?300?200?1000 100200300400 ?600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?400?300?200?1000 100200300400 ?600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 Figure A.2: All tracks, 3 minute projection, di erence in projection to measured histograms 78 X?Error (s) Y?Error (s) 0?10000 Altitudes ?4000?3000?2000?10000 1000200030004000?4000 ?3000 ?2000 ?1000 0 1000 2000 3000 4000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 Altitudes ?4000?3000?2000?10000 1000200030004000 ?2500 ?2000 ?1500 ?1000 ?500 0 500 1000 1500 2000 2500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 Altitudes ?4000?3000?2000?10000 1000200030004000 ?1500 ?1000 ?500 0 500 1000 1500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?4000?3000?2000?10000 1000200030004000 ?1500 ?1000 ?500 0 500 1000 1500 10 20 30 40 50 60 70 80 90 Figure A.3: All tracks, 5 minute projection, di erence in projection to measured histograms 79 X?Error (s) Y?Error (s) 0?10000 Altitudes ?1500?1000?5000 50010001500 ?2000 ?1500 ?1000 ?500 0 500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 Altitudes ?1500?1000?5000 50010001500 ?2000 ?1500 ?1000 ?500 0 500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 Altitudes ?1500?1000?5000 50010001500 ?2000 ?1500 ?1000 ?500 0 500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?1500?1000?5000 50010001500 ?2000 ?1500 ?1000 ?500 0 500 10 20 30 40 50 60 70 80 90 Figure A.4: All tracks, 10 minute projection, di erence in projection to measured histograms 80 ?500 0 500 ?500 0 5000 2 4 6 8 x 105 X?Error (s) 0?10000 Altitudes Y?Error (s) Occurrences ?500 0 500 ?400 ?200 0 200 4000 0.5 1 1.5 2 2.5 3 3.5 x 105 X?Error (s) 10000?18000 Altitudes Y?Error (s) Occurrences ?500 0 500 ?400 ?200 0 200 4000 0.5 1 1.5 2 2.5 x 106 X?Error (s) Above 18000 Altitudes Y?Error (s) Occurrences ?500 0 500 ?400 ?200 0 200 4000 0.5 1 1.5 2 2.5 3 3.5 x 106 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.5: All tracks, 1 minute projection, di erence in projection to measured three dimensional histograms 81 ?1000?500 0 500 1000 ?1000 ?500 0 500 10000 0.5 1 1.5 2 2.5 3 3.5 x 105 X?Error (s) 0?10000 Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?1000 ?500 0 500 10000 0.5 1 1.5 2 2.5 x 105 X?Error (s) 10000?18000 Altitudes Y?Error (s) Occurrences ?600?400 ?2000 200400 600 ?1000 ?500 0 500 10000 0.5 1 1.5 2 2.5 x 106 X?Error (s) Above 18000 Altitudes Y?Error (s) Occurrences ?600?400 ?2000 200400 600 ?1000 ?500 0 500 10000 0.5 1 1.5 2 2.5 x 106 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.6: All tracks, 3 minute projection, di erence in projection to measured three dimensional histograms 82 ?5000 0 5000 ?5000 0 50000 1 2 3 4 5 6 7 x 105 X?Error (s) 0?10000 Altitudes Y?Error (s) Occurrences ?5000 0 5000 ?4000 ?2000 0 2000 40000 1 2 3 4 5 x 105 X?Error (s) 10000?18000 Altitudes Y?Error (s) Occurrences ?5000 0 5000 ?2000 ?1000 0 1000 20000 0.5 1 1.5 2 2.5 3 3.5 x 106 X?Error (s) Above 18000 Altitudes Y?Error (s) Occurrences ?5000 0 5000 ?2000 ?1000 0 1000 20000 1 2 3 4 x 106 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.7: All tracks, 5 minute projection, di erence in projection to measured three dimensional histograms 83 ?4000?2000 0 2000 4000 ?4000 ?2000 0 2000 40000 0.5 1 1.5 2 x 105 X?Error (s) 0?10000 Altitudes Y?Error (s) Occurrences ?4000?2000 0 2000 4000 ?4000 ?2000 0 2000 40000 0.5 1 1.5 2 x 105 X?Error (s) 10000?18000 Altitudes Y?Error (s) Occurrences ?4000?2000 0 2000 4000 ?2000 ?1000 0 1000 20000 0.5 1 1.5 2 2.5 x 106 X?Error (s) Above 18000 Altitudes Y?Error (s) Occurrences ?4000?2000 0 2000 4000 ?2000 ?1000 0 1000 20000 0.5 1 1.5 2 2.5 x 106 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.8: All tracks, 10 minute projection, di erence in projection to measured three dimensional histograms 84 X?Error (s) Y?Error (s) 1 Minute Projection ?400?300?200?1000 100200300400 ?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 3 Minute Projection ?500 0 500?800 ?600 ?400 ?200 0 200 400 600 800 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 5 Minute Projection ?3000?2000?10000 100020003000?1500 ?1000 ?500 0 500 1000 1500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10 Minute Projection ?3000?2000?10000 100020003000 ?1500 ?1000 ?500 0 500 1000 1500 10 20 30 40 50 60 70 80 90 Figure A.9: All tracks, 0-10000 ft altitudes, di erence in projection to measured histograms 85 X?Error (s) Y?Error (s) 1 Minute Projection ?400?300?200?1000 100200300400 ?80 ?60 ?40 ?20 0 20 40 60 80 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 3 Minute Projection ?500 0 500 ?600 ?400 ?200 0 200 400 600 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 5 Minute Projection ?3000?2000?10000 100020003000 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10 Minute Projection ?3000?2000?10000 100020003000 ?1500 ?1000 ?500 0 500 1000 1500 10 20 30 40 50 60 70 80 90 Figure A.10: All tracks, 10000-18000 ft altitudes, di erence in projection to measured his- tograms 86 X?Error (s) Y?Error (s) 1 Minute Projection ?400?300?200?1000 100200300400 ?80 ?60 ?40 ?20 0 20 40 60 80 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 3 Minute Projection ?500 0 500 ?600 ?400 ?200 0 200 400 600 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 5 Minute Projection ?3000?2000?10000 100020003000?800 ?600 ?400 ?200 0 200 400 600 800 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10 Minute Projection ?3000?2000?10000 100020003000 ?1000 ?800 ?600 ?400 ?200 0 200 400 600 800 1000 10 20 30 40 50 60 70 80 90 Figure A.11: All tracks, Above 18000 ft altitudes, di erence in projection to measured histograms 87 X?Error (s) Y?Error (s) 1 Minute Projection ?300?200?1000 100200300 ?80 ?60 ?40 ?20 0 20 40 60 80 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 3 Minute Projection ?500 0 500 ?600 ?400 ?200 0 200 400 600 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 5 Minute Projection ?3000?2000?10000 100020003000 ?600 ?400 ?200 0 200 400 600 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10 Minute Projection ?3000?2000?10000 100020003000 ?1000 ?800 ?600 ?400 ?200 0 200 400 600 800 1000 10 20 30 40 50 60 70 80 90 Figure A.12: All tracks, All Altitudes, di erence in projection to measured histograms 88 ?500 0 500 ?200 ?100 0 100 2000 1 2 3 4 x 105 X?Error (s) 1 Minute Projection Y?Error (s) Occurrences ?600?400 ?2000 200400 600 ?1000 ?500 0 500 10000 0.5 1 1.5 2 2.5 3 x 105 X?Error (s) 3 Minute Projection Y?Error (s) Occurrences ?4000?2000 0 2000 4000 ?2000 ?1000 0 1000 20000 1 2 3 4 x 105 X?Error (s) 5 Minute Projection Y?Error (s) Occurrences ?4000?2000 0 2000 4000 ?2000 ?1000 0 1000 20000 5 10 15 x 104 X?Error (s) 10 Minute Projection Y?Error (s) Occurrences Figure A.13: All tracks, 0-10000 ft altitudes, di erence in projection to measured three dimensional histograms 89 ?500 0 500 ?100 ?50 0 50 1000 2 4 6 8 10 12 x 104 X?Error (s) 1 Minute Projection Y?Error (s) Occurrences ?600?400 ?2000 200400 600 ?1000 ?500 0 500 10000 0.5 1 1.5 2 x 105 X?Error (s) 3 Minute Projection Y?Error (s) Occurrences ?4000?2000 0 2000 4000 ?2000 ?1000 0 1000 20000 0.5 1 1.5 2 2.5 3 x 105 X?Error (s) 5 Minute Projection Y?Error (s) Occurrences ?4000?2000 0 2000 4000 ?2000 ?1000 0 1000 20000 2 4 6 8 10 12 x 104 X?Error (s) 10 Minute Projection Y?Error (s) Occurrences Figure A.14: All tracks, 10000-18000 ft altitudes, di erence in projection to measured three dimensional histograms 90 ?500 0 500 ?100 ?50 0 50 1000 2 4 6 8 10 x 105 X?Error (s) 1 Minute Projection Y?Error (s) Occurrences ?600?400 ?2000 200400 600 ?1000 ?500 0 500 10000 0.5 1 1.5 2 x 106 X?Error (s) 3 Minute Projection Y?Error (s) Occurrences ?4000?2000 0 2000 4000 ?1000 ?500 0 500 10000 0.5 1 1.5 2 2.5 3 x 106 X?Error (s) 5 Minute Projection Y?Error (s) Occurrences ?4000?2000 0 2000 4000 ?2000 ?1000 0 1000 20000 0.5 1 1.5 2 x 106 X?Error (s) 10 Minute Projection Y?Error (s) Occurrences Figure A.15: All tracks, Above 18000 ft altitudes, di erence in projection to measured three dimensional histograms 91 ?400?200 0 200 400 ?100 ?50 0 50 1000 2 4 6 8 10 12 x 105 X?Error (s) 1 Minute Projection Y?Error (s) Occurrences ?600?400 ?2000 200400 600 ?1000 ?500 0 500 10000 0.5 1 1.5 2 2.5 x 106 X?Error (s) 3 Minute Projection Y?Error (s) Occurrences ?4000?2000 0 2000 4000 ?1000 ?500 0 500 10000 0.5 1 1.5 2 2.5 3 x 106 X?Error (s) 5 Minute Projection Y?Error (s) Occurrences ?4000?2000 0 2000 4000 ?2000 ?1000 0 1000 20000 0.5 1 1.5 2 x 106 X?Error (s) 10 Minute Projection Y?Error (s) Occurrences Figure A.16: All tracks, All Altitudes, di erence in projection to measured three dimensional histograms 92 0 5 10 15x 105 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1 2 3 4 5 6 7 8 9x 105 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 2 4 6 8 10 12x 106 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 2 4 6 8 10 12 14x 106 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.17: Di erence in altitude, all tracks, 1 minute projection 93 0 1 2 3 4 5 6 7 8 9 10x 105 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6 7 8x 105 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6x 106 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6 7 8x 106 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.18: Di erence in bearing, all tracks, 1 minute projection 94 0 1 2 3 4 5 6 7 8x 105 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1 2 3 4 5 6 7x 105 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1 2 3 4 5 6 7 8x 106 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1 2 3 4 5 6 7 8 9x 106 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.19: Percent di erence in distance, all tracks, 1 minute projection 95 0 0.5 1 1.5 2 2.5x 106 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 2 4 6 8 10 12 14 16 18x 105 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 2 4 6 8 10 12 14x 106 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 2 4 6 8 10 12 14 16 18x 106 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.20: Percent di erence in speed, all tracks, 1 minute projection 96 0 1 2 3 4 5 6 7 8x 105 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1 2 3 4 5 6x 105 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1 2 3 4 5 6 7 8 9 10x 106 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 2 4 6 8 10 12x 106 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.21: Di erence in altitude, all tracks, 3 minute projection 97 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 105 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6x 105 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6 7x 106 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6 7 8x 106 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.22: Di erence in bearing, all tracks, 3 minute projection 98 0 1 2 3 4 5 6 7 8x 105 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1 2 3 4 5 6 7x 105 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 2 4 6 8 10 12x 106 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 2 4 6 8 10 12x 106 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.23: Percent di erence in distance, all tracks, 3 minute projection 99 0 1 2 3 4 5 6 7 8 9 10x 105 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 1 2 3 4 5 6 7 8x 105 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 2 4 6 8 10 12x 106 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 2 4 6 8 10 12 14x 106 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.24: Percent di erence in speed, all tracks, 3 minute projection 100 0 1 2 3 4 5 6 7x 105 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 105 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1 2 3 4 5 6 7 8 9x 106 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1 2 3 4 5 6 7 8 9 10x 106 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.25: Di erence in altitude, all tracks, 5 minute projection 101 0 0.5 1 1.5 2 2.5 3 3.5x 105 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 105 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6 7x 106 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6 7 8x 106 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.26: Di erence in bearing, all tracks, 5 minute projection 102 0 1 2 3 4 5 6x 105 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1 2 3 4 5 6x 105 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1 2 3 4 5 6 7 8 9 10x 106 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 2 4 6 8 10 12x 106 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.27: Percent di erence in distance, all tracks, 5 minute projection 103 0 1 2 3 4 5 6 7x 105 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 1 2 3 4 5 6x 105 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 2 4 6 8 10 12x 106 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 2 4 6 8 10 12x 106 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.28: Percent di erence in speed, all tracks, 5 minute projection 104 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 105 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1 2 3 4 5 6 7x 105 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1 2 3 4 5 6 7 8x 106 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1 2 3 4 5 6 7 8x 106 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.29: Di erence in altitude, all tracks, 10 minute projection 105 0 0.5 1 1.5 2 2.5x 105 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 0.5 1 1.5 2 2.5 3 3.5x 105 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6x 106 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1 2 3 4 5 6x 106 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.30: Di erence in bearing, all tracks, 10 minute projection 106 0 0.5 1 1.5 2 2.5 3 3.5 4x 105 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.5 1 1.5 2 2.5 3 3.5 4x 105 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1 2 3 4 5 6 7 8 9x 106 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1 2 3 4 5 6 7 8 9 10x 106 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.31: Percent di erence in distance, all tracks, 10 minute projection 107 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 105 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 105 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 1 2 3 4 5 6 7 8 9x 106 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 1 2 3 4 5 6 7 8 9x 106 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.32: Percent di erence in speed, all tracks, 10 minute projection 108 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 Figure A.33: Fort Campbell, 50 km radius, 1 minute projection, di erence in projection to measured histograms 109 ?150?100 ?500 50 100 150 ?200 ?100 0 100 2000 10 20 30 40 50 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?150?100 ?500 50 100 150 ?200 ?100 0 100 2000 20 40 60 80 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?150?100 ?500 50 100 150 ?200 ?100 0 100 2000 200 400 600 800 1000 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?150?100 ?500 50 100 150 ?200 ?100 0 100 2000 200 400 600 800 1000 1200 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.34: Fort Campbell, 50 km radius, 1 minute projection, di erence in projection to measured three dimensional histograms 110 0 100 200 300 400 500 600 700 800 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 100 200 300 400 500 600 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 2000 4000 6000 8000 10000 12000 14000 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 5000 10000 15000 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.35: Fort Campbell, 50 km radius, 1 minute projection, di erence in altitude 111 0 50 100 150 200 250 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 50 100 150 200 250 300 350 400 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1000 2000 3000 4000 5000 6000 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1000 2000 3000 4000 5000 6000 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.36: Fort Campbell, 50 km radius, 1 minute projection, di erence in bearing 112 0 50 100 150 200 250 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 50 100 150 200 250 300 350 400 450 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.37: Fort Campbell, 50 km radius, 1 minute projection, percent di erence in distance 113 0 100 200 300 400 500 600 700 800 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 200 400 600 800 1000 1200 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 5000 10000 15000 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.38: Fort Campbell, 50 km radius, 1 minute projection, percent di erence in speed 114 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?400?300?200?1000 100200300400?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?400?300?200?1000 100200300400?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?400?300?200?1000 100200300400?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?400?300?200?1000 100200300400?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 Figure A.39: Fort Campbell, 50 km radius, 3 minute projection, di erence in projection to measured histograms 115 ?500 0 500 ?400?200 0200 4000 20 40 60 80 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?500 0 500 ?400?200 0200 4000 10 20 30 40 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?500 0 500 ?400?200 0200 4000 200 400 600 800 1000 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?500 0 500 ?400?200 0200 4000 200 400 600 800 1000 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.40: Fort Campbell, 50 km radius, 3 minute projection, di erence in projection to measured three dimensional histograms 116 0 100 200 300 400 500 600 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 50 100 150 200 250 300 350 400 450 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.41: Fort Campbell, 50 km radius, 3 minute projection, di erence in altitude 117 0 20 40 60 80 100 120 140 160 180 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 50 100 150 200 250 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.42: Fort Campbell, 50 km radius, 3 minute projection, di erence in bearing 118 0 50 100 150 200 250 300 350 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 50 100 150 200 250 300 350 400 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.43: Fort Campbell, 50 km radius, 3 minute projection, percent di erence in distance 119 0 100 200 300 400 500 600 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 50 100 150 200 250 300 350 400 450 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.44: Fort Campbell, 50 km radius, 3 minute projection, percent di erence in speed 120 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?600?400?2000 200 400 600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 400 500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?600?400?2000 200 400 600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 400 500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?600?400?2000 200 400 600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 400 500 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?600?400?2000 200 400 600 ?500 ?400 ?300 ?200 ?100 0 100 200 300 400 500 10 20 30 40 50 60 70 80 90 Figure A.45: Fort Campbell, 50 km radius, 5 minute projection, di erence in projection to measured histograms 121 ?1000?500 0 500 1000 ?1000?500 0500 10000 10 20 30 40 50 60 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?1000?500 0500 10000 5 10 15 20 25 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?1000?500 0500 10000 100 200 300 400 500 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?1000?500 0500 10000 100 200 300 400 500 600 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.46: Fort Campbell, 50 km radius, 5 minute projection, di erence in projection to measured three dimensional histograms 122 0 50 100 150 200 250 300 350 400 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 50 100 150 200 250 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.47: Fort Campbell, 50 km radius, 5 minute projection, di erence in altitude 123 0 50 100 150 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 50 100 150 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 500 1000 1500 2000 2500 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 500 1000 1500 2000 2500 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.48: Fort Campbell, 50 km radius, 5 minute projection, di erence in bearing 124 0 50 100 150 200 250 300 350 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 20 40 60 80 100 120 140 160 180 200 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.49: Fort Campbell, 50 km radius, 5 minute projection, percent di erence in distance 125 0 50 100 150 200 250 300 350 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 50 100 150 200 250 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.50: Fort Campbell, 50 km radius, 5 minute projection, percent di erence in speed 126 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?100 ?50 0 50 100?150 ?100 ?50 0 50 100 150 10 20 30 40 50 60 70 80 90 Figure A.51: Las Cruces, 50 km radius, 1 minute projection, di erence in projection to measured histograms 127 ?150?100 ?500 50 100 150 ?200 ?100 0 100 2000 10 20 30 40 50 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?150?100 ?500 50 100 150 ?200 ?100 0 100 2000 20 40 60 80 100 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?150?100 ?500 50 100 150 ?200 ?100 0 100 2000 200 400 600 800 1000 1200 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?150?100 ?500 50 100 150 ?200 ?100 0 100 2000 200 400 600 800 1000 1200 1400 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.52: Las Cruces, 50 km radius, 1 minute projection, di erence in projection to measured three dimensional histograms 128 0 100 200 300 400 500 600 700 800 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 100 200 300 400 500 600 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 2000 4000 6000 8000 10000 12000 14000 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 5000 10000 15000 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.53: Las Cruces, 50 km radius, 1 minute projection, di erence in altitude 129 0 50 100 150 200 250 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 50 100 150 200 250 300 350 400 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1000 2000 3000 4000 5000 6000 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1000 2000 3000 4000 5000 6000 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.54: Las Cruces, 50 km radius, 1 minute projection, di erence in bearing 130 0 50 100 150 200 250 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 50 100 150 200 250 300 350 400 450 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.55: Las Cruces, 50 km radius, 1 minute projection, percent di erence in distance 131 0 100 200 300 400 500 600 700 800 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 200 400 600 800 1000 1200 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 5000 10000 15000 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.56: Las Cruces, 50 km radius, 1 minute projection, percent di erence in speed 132 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?400?300?200?1000 100200300400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?400?300?200?1000 100200300400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?400?300?200?1000 100200300400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?400?300?200?1000 100200300400 ?300 ?200 ?100 0 100 200 300 10 20 30 40 50 60 70 80 90 Figure A.57: Las Cruces, 50 km radius, 3 minute projection, di erence in projection to measured histograms 133 ?500 0 500 ?400?200 0200 4000 10 20 30 40 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?500 0 500 ?400?200 0200 4000 20 40 60 80 100 120 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?500 0 500 ?400?200 0200 4000 200 400 600 800 1000 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?500 0 500 ?400?200 0200 4000 200 400 600 800 1000 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.58: Las Cruces, 50 km radius, 3 minute projection, di erence in projection to measured three dimensional histograms 134 0 100 200 300 400 500 600 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 50 100 150 200 250 300 350 400 450 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.59: Las Cruces, 50 km radius, 3 minute projection, di erence in altitude 135 0 20 40 60 80 100 120 140 160 180 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 50 100 150 200 250 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.60: Las Cruces, 50 km radius, 3 minute projection, di erence in bearing 136 0 50 100 150 200 250 300 350 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 50 100 150 200 250 300 350 400 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.61: Las Cruces, 50 km radius, 3 minute projection, percent di erence in distance 137 0 100 200 300 400 500 600 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 50 100 150 200 250 300 350 400 450 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.62: Las Cruces, 50 km radius, 3 minute projection, percent di erence in speed 138 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?600?400?2000 200 400 600 ?600 ?400 ?200 0 200 400 600 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?600?400?2000 200 400 600 ?600 ?400 ?200 0 200 400 600 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?600?400?2000 200 400 600 ?600 ?400 ?200 0 200 400 600 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?600?400?2000 200 400 600 ?600 ?400 ?200 0 200 400 600 10 20 30 40 50 60 70 80 90 Figure A.63: Las Cruces, 50 km radius, 5 minute projection, di erence in projection to measured histograms 139 ?1000?500 0 500 1000 ?1000?500 0500 10000 5 10 15 20 25 30 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?1000?500 0500 10000 20 40 60 80 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?1000?500 0500 10000 100 200 300 400 500 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?1000?500 0500 10000 100 200 300 400 500 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.64: Las Cruces, 50 km radius, 5 minute projection, di erence in projection to measured three dimensional histograms 140 0 50 100 150 200 250 300 350 400 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 50 100 150 200 250 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.65: Las Cruces, 50 km radius, 5 minute projection, di erence in altitude 141 0 50 100 150 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 50 100 150 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 500 1000 1500 2000 2500 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 500 1000 1500 2000 2500 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.66: Las Cruces, 50 km radius, 5 minute projection, di erence in bearing 142 0 50 100 150 200 250 300 350 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 20 40 60 80 100 120 140 160 180 200 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.67: Las Cruces, 50 km radius, 5 minute projection, percent di erence in distance 143 0 50 100 150 200 250 300 350 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 50 100 150 200 250 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.68: Las Cruces, 50 km radius, 5 minute projection, percent di erence in speed 144 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?300?200?1000 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?300?200?1000 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?300?200?1000 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?300?200?1000 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 Figure A.69: Fort Campbell, 250 km radius, 1 minute projection, di erence in projection to measured histograms 145 ?400?200 0 200 400 ?500 0 5000 2000 4000 6000 8000 10000 12000 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?400?200 0 200 400 ?500 0 5000 1000 2000 3000 4000 5000 6000 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?400?200 0 200 400 ?500 0 5000 2 4 6 8 x 104 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?400?200 0 200 400 ?500 0 5000 2 4 6 8 x 104 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.70: Fort Campbell, 250 km radius, 1 minute projection, di erence in projection to measured three dimensional histograms 146 0 0.5 1 1.5 2 2.5 3 3.5 4x 104 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 2000 4000 6000 8000 10000 12000 14000 16000 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5 3x 105 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5 3 3.5x 105 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.71: Fort Campbell, 250 km radius, 1 minute projection, di erence in altitude 147 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 2000 4000 6000 8000 10000 12000 14000 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 2 4 6 8 10 12 14 16x 104 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 105 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.72: Fort Campbell, 250 km radius, 1 minute projection, di erence in bearing 148 0 5000 10000 15000 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 2000 4000 6000 8000 10000 12000 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.5 1 1.5 2 2.5x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.73: Fort Campbell, 250 km radius, 1 minute projection, percent di erence in distance 149 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 104 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3 3.5x 104 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3 3.5 4x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.74: Fort Campbell, 250 km radius, 1 minute projection, percent di erence in speed 150 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?500 0 500 ?800 ?600 ?400 ?200 0 200 400 600 800 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?500 0 500 ?800 ?600 ?400 ?200 0 200 400 600 800 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?500 0 500 ?800 ?600 ?400 ?200 0 200 400 600 800 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?500 0 500 ?800 ?600 ?400 ?200 0 200 400 600 800 10 20 30 40 50 60 70 80 90 Figure A.75: Fort Campbell, 250 km radius, 3 minute projection, di erence in projection to measured histograms 151 ?600?400?200 0 200400 600 ?1000?500 0500 10000 2000 4000 6000 8000 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?600?400?200 0 200400 600 ?1000?500 0500 10000 1000 2000 3000 4000 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?600?400?200 0 200400 600 ?1000?500 0500 10000 1 2 3 4 5 x 104 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?600?400?200 0 200400 600 ?1000?500 0500 10000 1 2 3 4 5 6 x 104 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.76: Fort Campbell, 250 km radius, 3 minute projection, di erence in projection to measured three dimensional histograms 152 0 0.5 1 1.5 2 2.5x 104 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5x 105 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5 3x 105 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.77: Fort Campbell, 250 km radius, 3 minute projection, di erence in altitude 153 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 2 4 6 8 10 12 14 16 18x 104 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 105 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.78: Fort Campbell, 250 km radius, 3 minute projection, di erence in bearing 154 0 2000 4000 6000 8000 10000 12000 14000 16000 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 2000 4000 6000 8000 10000 12000 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.5 1 1.5 2 2.5x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.5 1 1.5 2 2.5 3x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.79: Fort Campbell, 250 km radius, 3 minute projection, percent di erence in distance 155 0 0.5 1 1.5 2 2.5 3x 104 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 5000 10000 15000 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3 3.5x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.80: Fort Campbell, 250 km radius, 3 minute projection, percent di erence in speed 156 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 Figure A.81: Fort Campbell, 250 km radius, 5 minute projection, di erence in projection to measured histograms 157 ?1000?500 0 500 1000 ?2000?1000 01000 20000 1000 2000 3000 4000 5000 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?2000?1000 01000 20000 500 1000 1500 2000 2500 3000 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?2000?1000 01000 20000 1 2 3 4 5 x 104 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?2000?1000 01000 20000 1 2 3 4 5 x 104 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.82: Fort Campbell, 250 km radius, 5 minute projection, di erence in projection to measured three dimensional histograms 158 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5x 105 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5x 105 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.83: Fort Campbell, 250 km radius, 5 minute projection, di erence in altitude 159 0 1000 2000 3000 4000 5000 6000 7000 8000 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 5 10 15x 104 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 2 4 6 8 10 12 14 16 18x 104 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.84: Fort Campbell, 250 km radius, 5 minute projection, di erence in bearing 160 0 5000 10000 15000 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.5 1 1.5 2 2.5x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.5 1 1.5 2 2.5x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.85: Fort Campbell, 250 km radius, 5 minute projection, percent di erence in distance 161 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 104 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.86: Fort Campbell, 250 km radius, 5 minute projection, percent di erence in speed 162 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?300?200?1000 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?300?200?1000 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?300?200?1000 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?300?200?1000 100 200 300?400 ?300 ?200 ?100 0 100 200 300 400 10 20 30 40 50 60 70 80 90 Figure A.87: Las Cruces, 250 km radius, 1 minute projection, di erence in projection to measured histograms 163 ?400?200 0 200 400 ?500 0 5000 200 400 600 800 1000 1200 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?400?200 0 200 400 ?500 0 5000 500 1000 1500 2000 2500 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?400?200 0 200 400 ?500 0 5000 0.5 1 1.5 2 2.5 x 104 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?400?200 0 200 400 ?500 0 5000 0.5 1 1.5 2 2.5 3 x 104 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.88: Las Cruces, 250 km radius, 1 minute projection, di erence in projection to measured three dimensional histograms 164 0 0.5 1 1.5 2 2.5 3 3.5 4x 104 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 2000 4000 6000 8000 10000 12000 14000 16000 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5 3x 105 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5 3 3.5x 105 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.89: Las Cruces, 250 km radius, 1 minute projection, di erence in altitude 165 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 2000 4000 6000 8000 10000 12000 14000 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 2 4 6 8 10 12 14 16x 104 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 105 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.90: Las Cruces, 250 km radius, 1 minute projection, di erence in bearing 166 0 5000 10000 15000 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 2000 4000 6000 8000 10000 12000 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.5 1 1.5 2 2.5x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.91: Las Cruces, 250 km radius, 1 minute projection, percent di erence in distance 167 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 104 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3 3.5x 104 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3 3.5 4x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.92: Las Cruces, 250 km radius, 1 minute projection, percent di erence in speed 168 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?500 0 500 ?800 ?600 ?400 ?200 0 200 400 600 800 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?500 0 500 ?800 ?600 ?400 ?200 0 200 400 600 800 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?500 0 500 ?800 ?600 ?400 ?200 0 200 400 600 800 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?500 0 500 ?800 ?600 ?400 ?200 0 200 400 600 800 10 20 30 40 50 60 70 80 90 Figure A.93: Las Cruces, 250 km radius, 3 minute projection, di erence in projection to measured histograms 169 ?600?400?200 0 200400 600 ?1000?500 0500 10000 200 400 600 800 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?600?400?200 0 200400 600 ?1000?500 0500 10000 500 1000 1500 2000 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?600?400?200 0 200400 600 ?1000?500 0500 10000 0.5 1 1.5 2 x 104 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?600?400?200 0 200400 600 ?1000?500 0500 10000 0.5 1 1.5 2 2.5 x 104 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.94: Las Cruces, 250 km radius, 3 minute projection, di erence in projection to measured three dimensional histograms 170 0 0.5 1 1.5 2 2.5x 104 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5x 105 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5 3x 105 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.95: Las Cruces, 250 km radius, 3 minute projection, di erence in altitude 171 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 2 4 6 8 10 12 14 16 18x 104 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 105 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.96: Las Cruces, 250 km radius, 3 minute projection, di erence in bearing 172 0 2000 4000 6000 8000 10000 12000 14000 16000 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 2000 4000 6000 8000 10000 12000 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.5 1 1.5 2 2.5x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.5 1 1.5 2 2.5 3x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.97: Las Cruces, 250 km radius, 3 minute projection, percent di erence in distance 173 0 0.5 1 1.5 2 2.5 3x 104 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 5000 10000 15000 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3 3.5x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.98: Las Cruces, 250 km radius, 3 minute projection, percent di erence in speed 174 X?Error (s) Y?Error (s) 0?10000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) 10000?18000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) Above 18000 ft Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 X?Error (s) Y?Error (s) All Altitudes ?800?600?400?2000 200400600800 ?1000 ?500 0 500 1000 10 20 30 40 50 60 70 80 90 Figure A.99: Las Cruces, 250 km radius, 5 minute projection, di erence in projection to measured histograms 175 ?1000?500 0 500 1000 ?2000?1000 01000 20000 100 200 300 400 500 X?Error (s) 0?10000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?2000?1000 01000 20000 500 1000 1500 X?Error (s) 10000?18000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?2000?1000 01000 20000 0.5 1 1.5 2 x 104 X?Error (s) Above 18000 ft Altitudes Y?Error (s) Occurrences ?1000?500 0 500 1000 ?2000?1000 01000 20000 0.5 1 1.5 2 x 104 X?Error (s) All Altitudes Y?Error (s) Occurrences Figure A.100: Las Cruces, 250 km radius, 5 minute projection, di erence in projection to measured three dimensional histograms 176 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0?10000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000?18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5x 105 Above 18000 ft Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences 0 0.5 1 1.5 2 2.5x 105 All Altitudes ?5000 ? ?2000?2000 ? ?1000?1000 ? ?500?500 ? ?200?200 ? ?100?100 ? 100100 ? 200200 ? 500500 ? 10001000 ? 20002000 ? 5000 Difference in Altitude (m) Occurrences Figure A.101: Las Cruces, 250 km radius, 5 minute projection, di erence in altitude 177 0 1000 2000 3000 4000 5000 6000 7000 8000 0?10000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000?18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 5 10 15x 104 Above 18000 ft Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences 0 2 4 6 8 10 12 14 16 18x 104 All Altitudes ?180? ? ?90??90? ? ?45??45? ? ?15??15? ? ?10??10? ? ?3??3? ? 3?3? ? 10?10? ? 15?15? ? 45?45? ? 90?90? ? 180?Difference in Bearing (degrees) Occurrences Figure A.102: Las Cruces, 250 km radius, 5 minute projection, di erence in bearing 178 0 5000 10000 15000 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.5 1 1.5 2 2.5x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences 0 0.5 1 1.5 2 2.5x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Distance Occurrences Figure A.103: Las Cruces, 250 km radius, 5 minute projection, percent di erence in distance 179 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 104 0?10000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10000?18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5x 105 Above 18000 ft Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences 0 0.5 1 1.5 2 2.5 3x 105 All Altitudes ?100% ? ?50%?50% ? ?25%?25% ? ?10%?10% ? ?5%?5% ? 5%5% ? 10%10% ? 25%25% ? 50%50% ? 100%Percent Difference in Speed Occurrences Figure A.104: Las Cruces, 250 km radius, 5 minute projection, percent di erence in speed 180 Table A.1: Altitude Con dence, All Tracks, 0-10000 ft Altitudes Con dence (m) Projection Time 0.75 0.90 0.95 0.99 1 Minute 335 549 732 1067 3 Minute 914 1341 1737 2438 5 Minute 1433 1920 2286 3139 10 Minute 2499 3292 3749 4602 Table A.2: Altitude Con dence, All Tracks, 10000-18000 ft Altitudes Con dence (m) Projection Time 0.75 0.90 0.95 0.99 1 Minute 549 732 823 1067 3 Minute 1585 2042 2316 2865 5 Minute 2530 3200 3597 4359 10 Minute 4237 5243 5852 6858 Table A.3: Altitude Con dence, All Tracks, Above 18000 ft Altitudes Con dence (m) Projection Time 0.75 0.90 0.95 0.99 1 Minute 30 427 610 853 3 Minute 274 1280 1676 2347 5 Minute 610 2042 2682 3658 10 Minute 1219 3749 4877 6431 181 Table A.4: Altitude Con dence, All Tracks, All Altitudes Con dence (m) Projection Time 0.75 0.90 0.95 0.99 1 Minute 274 518 671 945 3 Minute 762 1463 1860 2469 5 Minute 1188 2316 2896 3810 10 Minute 2134 4084 4999 6462 182 Table A.5: Bearing Con dence, All Tracks, 0-10000 ft Altitudes Con dence (degrees) Projection Time 0.75 0.90 0.95 0.99 1 Minute 22.4 41.9 61.7 96.5 3 Minute 25.1 58.2 88.6 133.2 5 Minute 26.5 58.2 92.2 148.3 10 Minute 21.4 40.7 60.7 141.5 Table A.6: Bearing Con dence, All Tracks, 10000-18000 ft Altitudes Con dence (degrees) Projection Time 0.75 0.90 0.95 0.99 1 Minute 12.6 21.4 29.8 58.8 3 Minute 12.8 26.3 40.9 94.0 5 Minute 15.1 31.9 52.3 125.2 10 Minute 16.9 34.2 56.8 137.2 Table A.7: Bearing Con dence, All Tracks, Above 18000 ft Altitudes Con dence (degrees) Projection Time 0.75 0.90 0.95 0.99 1 Minute 7.6 10.3 12.7 21.9 3 Minute 5.5 8.6 11.9 26.1 5 Minute 5.5 9.2 13.8 30.5 10 Minute 6.3 11.9 18.2 41.0 183 Table A.8: Bearing Con dence, All Tracks, All Altitudes Con dence (degrees) Projection Time 0.75 0.90 0.95 0.99 1 Minute 9.3 16.8 24.9 62.0 3 Minute 7.2 15.4 27.4 82.2 5 Minute 7.3 16.6 28.8 86.8 10 Minute 8.0 17.3 27.5 70.3 184 Table A.9: Distance Con dence, All Tracks, 0-10000 ft Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 25.6 48.0 64.8 104.6 3 Minute 22.5 42.2 65.4 143.7 5 Minute 29.8 62.9 121.8 305.0 10 Minute 47.2 94.8 166.8 507.5 Table A.10: Distance Con dence, All Tracks, 10000-18000 ft Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 18.9 35.8 51.4 82.1 3 Minute 15.8 23.7 29.2 47.1 5 Minute 19.4 27.9 34.2 60.3 10 Minute 27.2 38.3 47.6 118.4 Table A.11: Distance Con dence, All Tracks, Above 18000 ft Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 8.9 13.8 18.0 35.8 3 Minute 5.1 8.6 11.6 19.0 5 Minute 4.7 8.6 11.9 20.5 10 Minute 4.9 10.6 16.4 30.7 185 Table A.12: Distance Con dence, All Tracks, All Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 11.7 21.7 34.5 71.2 3 Minute 7.3 15.2 23.0 55.6 5 Minute 7.3 17.1 26.9 84.7 10 Minute 7.7 22.8 35.7 107.2 186 Table A.13: Speed Con dence, All Tracks, 0-10000 ft Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 8.6 16.0 21.1 32.3 3 Minute 20.9 31.9 38.8 52.5 5 Minute 28.7 40.5 47.0 60.2 10 Minute 42.4 53.4 58.3 65.8 Table A.14: Speed Con dence, All Tracks, 10000-18000 ft Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 6.4 11.6 15.4 25.3 3 Minute 17.4 27.8 35.3 57.0 5 Minute 26.0 39.3 50.6 81.7 10 Minute 37.7 51.7 67.4 113.3 Table A.15: Speed Con dence, All Tracks, Above 18000 ft Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 1.5 2.9 4.3 8.7 3 Minute 2.8 6.4 9.7 18.2 5 Minute 3.9 9.3 14.3 27.2 10 Minute 6.0 15.8 25.2 54.2 187 Table A.16: Speed Con dence, All Tracks, All Altitudes Con dence (%) Projection Time 0.75 0.90 0.95 0.99 1 Minute 2.6 6.5 11.1 22.4 3 Minute 5.4 14.9 23.4 40.6 5 Minute 7.5 21.2 31.4 51.2 10 Minute 10.8 32.1 44.4 67.6 188 Appendix B Private Personal Aircraft GPS Tracks 189 Figure B.1: Constant velocity and constant acceleration lters, all GPS ights with airports, 1 minute time horizon 190 Figure B.2: Straight velocity projection and moving average projection, all GPS ights with airports, 1 minute time horizon 191 Figure B.3: Constant velocity and constant acceleration lters, all GPS ights with airports, 3 minute time horizon 192 Figure B.4: Straight velocity projection and moving average projection, all GPS ights with airports, 3 minute time horizon 193 Figure B.5: Constant velocity and constant acceleration lters, all GPS ights with airports, 5 minute time horizon 194 Figure B.6: Straight velocity projection and moving average projection, all GPS ights with airports, 5 minute time horizon 195 Figure B.7: Constant velocity and constant acceleration lters, all GPS ights without airports, 1 minute time horizon 196 Figure B.8: Straight velocity projection and moving average projection, all GPS ights without airports, 1 minute time horizon 197 Figure B.9: Constant velocity and constant acceleration lters, all GPS ights without airports, 3 minute time horizon 198 Figure B.10: Straight velocity projection and moving average projection, all GPS ights without airports, 3 minute time horizon 199 Figure B.11: Constant velocity and constant acceleration lters, all GPS ights without airports, 5 minute time horizon 200 Figure B.12: Straight velocity projection and moving average projection, all GPS ights without airports, 5 minute time horizon 201 Figure B.13: Constant velocity and constant acceleration lters, selected GPS ights with airports, 1 minute time horizon 202 Figure B.14: Straight velocity projection and moving average projection, selected GPS ights with airports, 1 minute time horizon 203 Figure B.15: Constant velocity and constant acceleration lters, selected GPS ights with airports, 3 minute time horizon 204 Figure B.16: Straight velocity projection and moving average projection, selected GPS ights with airports, 3 minute time horizon 205 Figure B.17: Constant velocity and constant acceleration lters, selected GPS ights with airports, 5 minute time horizon 206 Figure B.18: Straight velocity projection and moving average projection, selected GPS ights with airports, 5 minute time horizon 207 Figure B.19: Constant velocity and constant acceleration lters, selected GPS ights without airports, 1 minute time horizon 208 Figure B.20: Straight velocity projection and moving average projection, selected GPS ights without airports, 1 minute time horizon 209 Figure B.21: Constant velocity and constant acceleration lters, selected GPS ights without airports, 3 minute time horizon 210 Figure B.22: Straight velocity projection and moving average projection, selected GPS ights without airports, 3 minute time horizon 211 Figure B.23: Constant velocity and constant acceleration lters, selected GPS ights without airports, 5 minute time horizon 212 Figure B.24: Straight velocity projection and moving average projection, selected GPS ights without airports, 5 minute time horizon 213 Appendix C Rhumb Lines A rhumb line, or loxodrome, is a line that crosses all meridians of longitude at the same angle. Although great circles provide the shortest distance between two points on a sphere (for navigation, on the Earth), rhumb lines are easier to because of they keep a constant bearing. Distance Between Two Points In the error calculations, the bearing and velocity between two points in the track for an aircraft are assumed constant [17]. Therefore, rhumb lines can be used to nd the distant and bearing between the two points, as shown below. Given two geodetic points in radians latitude and longitude ( 1; 1) and ( 2; 2), re- spectively. The stretched latitude is de ned as follows. lat = ln tan( 2 2 + 4 ) tan( 12 + 4 ) (C.1) If the stretch latitude, , is zero, i.e. the points share the same parallel, q = cos( 1) (C.2) lon = 2 1 (C.3) otherwise, q = lat (C.4) distance = p 2 +q2 2R (C.5) Where R is the radius of the Earth. The constant bearing, , is de ned as = atan2( lon; ) (C.6) Destination Given Position, Distance and Bearing Given the above equations to nd the distance between two points, the a priori bearing provided by the measurements gives the bearing to project from the current position to the projected position at the given time horizon. The velocity and time horizon give the desired 214 distance to project [17]. = dR (C.7) Where d is the projection distance and R is the radius of the Earth. 2 = 1 + cos( ) (C.8) is the constant bearing between the current point and the projected point. lat = ln tan( 2 2 + 4 ) tan( 12 + 4 ) (C.9) If the constant bearing projects the point on the same parallel q = cos( 1) (C.10) otherwise, q = lat (C.11) = sin( )q (C.12) 2 = mod( 1 + + ;2 ) (C.13) 215