Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-2006 Covariance Estimation and Autocorrelation of NORAD Two-Line Element Sets Victor P. Osweiler Follow this and additional works at: https://scholar.afit.edu/etd Part of the Astrodynamics Commons Recommended Citation Osweiler, Victor P., "Covariance Estimation and Autocorrelation of NORAD Two-Line Element Sets" (2006). Theses and Dissertations. 3531. https://scholar.afit.edu/etd/3531 This Thesis is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. COVARIANCE ESTIMATION AND AUTOCORRELATION OF NORAD TWO-LINE ELEMENT SETS THESIS Victor P. Osweiler, Captain, USAF AFIT/GSS/ENY/06-M09 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government. AFIT/GSS/ENY/06-M09 COVARIANCE ESTIMATION AND AUTOCORRELATION OF NORAD TWO-LINE ELEMENT SETS THESIS Presented to the Faculty Department of Aeronautics and Astronautics Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command In Partial Fulfillment of the Requirements for the Degree of Master of Science (Space Systems) Victor P. Osweiler, BS Captain, USAF March 2006 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. AFIT/GSS/ENY/06-M09 COVARIANCE ESTIMATION AND AUTOCORRELATION OF NORAD TWO-LINE ELEMENT SETS Victor P. Osweiler, BS Captain, USAF Approved: ____________________________________ ___________ Lt Col Nathan A. Titus (Chairman) date ____________________________________ ___________ Dr. William E. Wiesel (Member) date ____________________________________ ___________ Lt Col Kerry D. Hicks (Member) date AFIT/GSS/ENY/06-M09 Abstract This thesis investigates NORAD two-line element sets (TLE) containing satellite mean orbital elements for the purpose of estimating a covariance matrix and formulating an autocorrelation relationship. Orbit propagation is performed using Simplified General Perturbations Number 4 (SGP4) analytical model as implemented within Satellite Took Kit. For a given satellite, TLEs from a span of two weeks are used to calculate position and velocity differences of estimated state vectors in order to characterize their variance behavior and compute a covariance matrix for the most recent TLE. Six satellites and eight time spans are investigated, with all state vector differences evaluated in satellite- based coordinate systems. An autocorrelation relationship for each satellite is generated to characterize confidence levels of the orbit predictions. Trends in the deterministic dynamics and errors in the model are observed and discussed. Covariance matrix estimates and associated TLEs are presented. iv AFIT/GSS/ENY/06-M09 To My Mother v Acknowledgments This thesis could not have been accomplished without the support from several incredibly talented and insightful people. First, I would like to express my sincere appreciation to my advisor, Lt Col Nathan Titus, for his support and guidance throughout my entire time at AFIT, especially during the course of this research. Additionally, I would like to extend my gratitude to Dr. William Wiesel for sharing his expertise and recommendations, which shaped this work. I am also indebted to Mr. David Vallado and Dr. T.S. Kelso, who were instrumental in this endeavor. Both generously dedicated their time, knowledge, and assistance for which I will always be grateful. I would like to give thanks to Dr. Michael Gabor for his counsel in improving this body of work. On a more personal level, I want to thank my family and friends who enabled me to persevere through this entire process. Their unconditional support inspired me, making the pursuit of this degree possible. I thank you all! Victor P. Osweiler vi Table of Contents Page Abstract.............................................................................................................................. iv Acknowledgments.............................................................................................................. vi List of Figures.................................................................................................................... ix List of Tables ..................................................................................................................... xi I. Introduction ....................................................................................................................1 Background Information ..............................................................................................1 Problem Statement........................................................................................................3 Research Objectives .....................................................................................................3 II. Literature Review..........................................................................................................4 Space Surveillance........................................................................................................4 Simplified General Perturbations Model......................................................................6 SGP4 History.......................................................................................................... 7 SGP4 Uniqueness. .................................................................................................. 9 Two-Line Element Sets. ........................................................................................ 10 Estimation Theory ......................................................................................................12 Central Limit Theorem. ........................................................................................ 13 Principle of Maximum Likelihood........................................................................ 14 Covariance Matrix................................................................................................ 15 Autocorrelation...........................................................................................................18 Relevant Research ......................................................................................................19 COVGEN Study. ................................................................................................... 19 Space Warfare Center Assessment ....................................................................... 21 INTELSAT Comparisons ...................................................................................... 22 IRIDIUM Comparison.......................................................................................... 23 III. Methodology..............................................................................................................26 Satellite Selection .......................................................................................................26 Satellite-based Coordinate Systems ...........................................................................28 RTC Coordinate System ....................................................................................... 29 VNC Coordinate System....................................................................................... 30 Orbit Propagation .......................................................................................................31 TLE Pair-Wise Differencing................................................................................. 32 vii Page Data Binning. ....................................................................................................... 35 Statistics Formulations ...............................................................................................36 Position Residuals. ............................................................................................... 36 Covariance Matrix Generation. ........................................................................... 37 Autocorrelation Function ..................................................................................... 38 IV. Results and Analysis...................................................................................................44 Position Vector Residuals...........................................................................................44 Covariance Matrix Results .........................................................................................57 Autocorrelation Function Results...............................................................................61 V. Conclusions and Recommendations ............................................................................68 Conclusions ................................................................................................................68 Recommendations for Future Research......................................................................69 Appendix A. TLE Format Full Description......................................................................72 Appendix B: Bins for Expanded Autocorrelation Time Window ....................................74