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Earth-Satellite Geometry

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Earth-Satellite Geometry EARTH-REFERENCED AIRCRAFT NAVIGATION AND SURVEILLANCE ANALYSIS Michael Geyer S π/2 - α - θ d h α N U Re Re θ O Michael Geyer Frederick de Wit Navigation Vertical Plane Model Peter Mercator Federico Rostagno Spherical Surface Model Surveillance Project Memorandum — June 2016 DOT-VNTSC-FAA-16-12 Prepared for: Federal Aviation Administration Wake Turbulence Research Office DOT Volpe Center FOREWORD This document addresses a basic function of aircraft (and other vehicle) surveillance and navi- gation systems analyses — quantifying the geometric relationship of two or more locations relative to each other and to the earth. Here, geometry means distances and angles, including their projections in a defined coordinate frame. Applications that fit well with these methods include (a) planning a vehicle’s route; (b) determining the coverage region of a radar or radio installation; and (c) calculating a vehicle’s latitude and longitude from measurements (e.g., of slant- and spherical-ranges or range differences, azimuth and elevation angles, and altitudes). The approach advocated is that the three-dimensional problems inherent in navigation/surveil- lance analyses should, to the extent possible, be re-cast as a sequence of sub-problems: . Vertical-Plane Formulation (two-dimensional (2D) problem illustrated in top right panel on cover) — Considers the vertical plane containing two problem-specific locations and the center of a spherical earth, and utilizes plane trigonometry as the primary analysis method. This formulation provides closed-form solutions. Spherical-Surface Formulation (2D problem illustrated in bottom left panel on cover) — Considers two or three problem-specific locations on the surface of a spherical earth; utilizes spherical trigonometry as the primary analysis method. This formulation provides closed-form solutions. Three-Dimensional Vector Formulation — Utilizes 3D Cartesian vector frame- work; best-suited to situations involving four or more problem-specific points and slant-range or slant-range difference measurements; provides closed-form solutions. Non-Linear Least-Squares (NLLS) Formulation — Employed for the most complex situations, and does not require many of the idealizations necessary for simpler approaches. Provides estimates of the accuracy of its solutions. Drawback is that it requires numerical methods, consequently solution properties are not evident. These techniques are applied, in the context of a spherical earth, to a series of increasing complex situations, starting with two problem-specific points (e.g., a route’s origin and destination) and extending to three or more points (e.g., an aircraft and multiple surveil- lance/navigation stations). Closed-form solutions are presented for measurements involving virtually every combination of ranges, pseudo ranges, azimuth/elevation angles and altitude. The Gauss-Newton NLLS methodology is employed to address the most complex situations. These include circumstances where there are more measurements than unknowns and/or the measurement ‘equations’ cannot be inverted analytically* (including those for an ellipsoidal- shaped earth) and/or are not analytic expressions (e.g., involve empirical data). * The term ‘analytic’ is used for expressions that are described by mathematical symbols, typically from algebra, trigonometry and calculus. ‘Inverting’ refers to solving such expressions for a set of desired unknown variables. i DOT Volpe Center TABLE OF CONTENTS FOREWORD .................................................................................................................... I 1. INTRODUCTION ................................................................................................... 1-1 1.1 Overview of Analysis Methodologies and Their Applications ................................... 1-1 1.1.1 Trigonometric and Vector Analysis Methodologies (Chapters 3 – 5) ...................................... 1-1 1.1.2 Applications of Trigonometric and Vector Methods (Chapters 6 – 7) ..................................... 1-2 1.1.3 Gauss-Newton NLLS Methodology and Applications (Chapter 8) .......................................... 1-3 1.2 Summary of Trigonometric Methodology ................................................................... 1-3 1.2.1 Vertical Plane Formulation ....................................................................................................... 1-3 1.2.2 Spherical Surface Formulation ................................................................................................. 1-4 1.3 Applicability and Limitations of Methodologies ......................................................... 1-5 1.4 Document Outline .......................................................................................................... 1-6 2. MATHEMATICS AND PHYSICS BASICS ............................................................ 2-1 2.1 Exact and Approximate Solutions to Common Equations ........................................ 2-1 2.1.1 Law of Sines for Plane Triangles ............................................................................................. 2-1 2.1.2 Law of Cosines for Plane Triangles ......................................................................................... 2-1 2.1.3 Law of Tangents for Plane Triangles........................................................................................ 2-2 2.1.4 Quadratic Algebraic Equation .................................................................................................. 2-2 2.1.5 Computational Precision........................................................................................................... 2-3 2.1.6 Inverse Trigonometric Functions .............................................................................................. 2-3 2.1.7 Power Series Expansions for arcsin, arccos and arctan ............................................................ 2-5 2.1.8 Single-Variable Numerical Root Finding Methods .................................................................. 2-6 2.2 Shape of the Earth ....................................................................................................... 2-10 2.2.1 WGS-84 Ellipsoid Parameters ................................................................................................ 2-10 2.2.2 Radii of Curvature in the Meridian and the Prime Vertical .................................................... 2-11 2.2.3 Methods for Addressing an Ellipsoidal Earth ......................................................................... 2-12 2.2.4 Surface Area of a Spherical Earth Visible to a Satellite ......................................................... 2-13 3. TWO-POINT / VERTICAL-PLANE FORMULATION ............................................ 3-1 3.1 Mathematical Problem and Solution Taxonomy ........................................................ 3-1 3.1.1 Mathematical Formulation ....................................................................................................... 3-1 3.1.2 Taxonomy of Solution Approaches .......................................................................................... 3-1 3.1.3 Detailed Geometry .................................................................................................................... 3-2 3.2 Accounting for Known User Altitude .......................................................................... 3-3 3.2.1 Need to Account for User Altitude ........................................................................................... 3-3 3.2.2 Method of Accounting for Known User Altitude ..................................................................... 3-4 3.2.3 Conditions for Unblocked Line-of-Sight (LOS) ....................................................................... 3-4 3.3 Computing Geocentric Angle........................................................................................ 3-5 3.3.1 Satellite Altitude and Elevation Angle Known – Basic Method .............................................. 3-5 3.3.2 Satellite Altitude and Elevation Angle Known – Alternative Method ..................................... 3-8 3.3.3 Satellite Altitude and Slant-Range Known ............................................................................... 3-8 3.3.4 Elevation Angle and Slant-Range Known – Basic Method .................................................... 3-10 3.3.5 Elevation Angle and Slant-Range Known – Alternative Method ........................................... 3-11 ii DOT Volpe Center 3.4 Computing Elevation Angle ........................................................................................ 3-12 3.4.1 Satellite Altitude and Geocentric Angle Known – Basic Method .......................................... 3-12 3.4.2 Satellite Altitude and Geocentric Angle Known – Alternative Method ................................. 3-13 3.4.3 Satellite Altitude and Slant-Range Known ............................................................................. 3-13 3.4.4 Geocentric Angle and Slant-Range Known ............................................................................ 3-15 3.5 Computing Slant-Range .............................................................................................. 3-16 3.5.1 Satellite Altitude and Geocentric Angle Known .................................................................... 3-16 3.5.2 Satellite Altitude and Elevation Angle Known ...................................................................... 3-17 3.5.3 Geocentric Angle
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