GEODYN Systems Description Volume 1 updated by: Jennifer W. Beall Prepared By: John J. McCarthy, Shelley Rowton, Denise Moore, Despina E. Pavlis, Scott B. Luthcke, Lucia S. Tsaoussi Prepared For: Space Geodesy Branch, Code 926 NASA GSFC, Greenbelt, MD February 23, 2015 from the original December 13, 1993 document 1 Contents 1 THE GEODYN II PROGRAM 6 2 THE ORBIT AND GEODETIC PARAMETER ESTIMATION PROB- LEM 6 2.1 THEORBITPREDICTIONPROBLEM . 7 2.2 THE PARAMETER ESTIMATION PROBLEM . 8 3 THE MOTION OF THE EARTH AND RELATED COORDINATE SYS- TEMS 11 3.1 THE TRUE OF DATE COORDINATE SYSTEM . 12 3.2 THE INERTIAL COORDINATE SYSTEM . 12 3.3 THE EARTH-FIXED COORDINATE SYSTEM . 13 3.4 TRANSFORMATION BETWEEN EARTH-FIXED AND TRUE OF DATE COORDINATES................................. 13 3.5 COMPUTATION OF THE GREENWICH HOUR ANGLE . 14 3.6 PRECESSION AND NUTATION . 15 3.6.1 Precession . 17 3.6.2 Nutation . 19 3.7 THE TRANSFORMATION TO THE EARTH CENTERED FIXED ADOPTED POLE COORDINATE SYSTEM . 20 4 LUNI-SOLAR-PLANETARY EPHEMERIS 22 4.1 EPHEMERIS USED IN GEODYN VERSIONS 7612.0 AND LATER . 22 4.1.1 Chebyshev Interpolation . 23 4.1.2 Coordinate Transformation . 24 5 THE OBSERVER 27 5.1 GEODETICCOORDINATES ......................... 27 5.1.1 Station Constraints . 33 5.1.2 Baseline Sigmas . 38 5.1.3 Station Velocities . 39 5.2 TOPOCENTRIC COORDINATE SYSTEMS . 39 5.3 TIME REFERENCE SYSTEMS . 40 5.3.1 Partial Derivatives . 41 5.4 POLAR MOTION . 43 5.4.1 E↵ect on the Position of a Station . 43 5.4.2 Partial Derivatives . 44 5.5 SOLID EARTH TIDES . 49 5.5.1 Station Displacement Due to Luni-Solar Attraction . 49 2 5.5.2 Station Displacement Due to Pole Tide . 50 5.5.3 Partial Derivatives for h2 and l2 . 51 5.6 THE UPLIFT DUE TO OCEAN TIDES . 53 5.6.1 The Hendershott Tidal Model . 53 5.6.2 The NOAA Tide Model . 55 5.7 TECTONIC PLATE MOTION . 57 5.8 OCEAN LOADING . 58 6 MEASUREMENT MODELING AND RELATED DERIVATIVES 59 6.1 THE GEOMETRIC RELATIONSHIPS . 62 6.2 THE GEOMETRIC PARTIAL DERIVATIVES . 73 6.3 THE TIME DERIVATIVES . 76 6.4 MATHEMATICAL MODEL FOR SST MEASUREMENTS . 80 6.4.1 Definition of SST Terms . 80 6.4.2 SST Range Measurements and Partials . 81 6.4.3 SST Range Rate Measurements . 82 6.4.4 Range Rate Partials . 89 6.5 PCE MEASUREMENT TYPES . 90 6.6 AVERAGE RANGE RATE MEASUREMENT TYPES . 91 6.7 THE 8906 OCEAN TIDE ALTIMETRIC MEASUREMENT CORRECTION 94 6.8 PLANETARY EPHEMERIS CORRECTION . 95 6.9 TABLE COMPRESSION VIA 2D FOURIER TRANSFORMS . 99 6.9.1 Introduction . 99 6.9.2 Theoretical Basis . 99 6.9.3 Application of tile Range Correction Algorithms . 100 7 DATA PREPROCESSING 104 7.1 TIMEPREPROCESSING............................ 104 7.2 REFERENCE SYSTEM CONVERSION TO TRUE OF DATE . 105 7.3 TRANSPONDER DELAY AND GATING EFFECTS . 105 7.4 ABERRATION . 105 7.5 REFRACTIONCORRECTIONS. 106 7.6 TRANET DOPPLER OBSERVATIONS . 113 7.7 SATELLITE-SATELLITE TRACKING DATA PREPROCESSING . 113 7.8 CORRECTIONS TO TIME MEASUREMENTS DUE TO THE EFFECTS OF GENERAL RELATIVITY . 114 7.9 SCALING TO A NEW VALUE OF SPEED OF LIGHT . 115 7.10 RELATIVISTIC SPACECRAFT CLOCK DRIFT CORRECTIONS . 116 8 FORCE MODEL AND VARIATIONAL EQUATIONS 119 8.1 EQUATIONS OF MOTION . 119 3 8.2 THE VARIATIONAL EQUATIONS . 120 8.3 THE EARTH’S POTENTIAL . 123 8.3.1 Spherical Harmonic Expansion . 123 8.3.2 Temporal Variations . 132 8.4 SOLAR, LUNAR, AND PLANETARY GRAVITATIONAL PERTURBA- TIONS....................................... 132 8.5 SOLAR RADIATION PRESSURE . 133 8.5.1 Radiation Pressure Models for GPS Satellites . 134 8.6 ATMOSPHERIC DRAG . 136 8.7 ATMOSPHERIC DENSITY . 138 8.7.1 Jacchia 1971 Density Model . 138 8.7.2 Thermospheric Drag Model (DTM) . 153 8.8 TIDAL POTENTIALS . 158 8.8.1 The Love Model . 158 8.8.2 The Expanded Tide Model . 159 8.8.3 The 8906 Tide Model . 160 8.9 GENERAL ACCELERATIONS . 163 8.10 DYNAMIC POLAR MOTION . 165 8.11 EARTH RADIATION . 166 8.12 THERMAL DRAG EFFECT ON LAGEOS . 168 8.13 RELATIVISTIC CORIOLIS FORCE . 169 8.14 LENSE-THIRRlNG . 170 8.15 PLANETARY MOON POINT MASS PERTURBATION . 171 8.16 ORIENTATION INITIAL CONDITIONS - VARIATIONAL EQUATIONS: T. Sabaka . 172 8.16.1 Introduction . 172 8.16.2 Application to computing angular velocity and reference frame through time.................................... 173 8.16.3 Application to computing partials of angular velocity and reference frame with respect to initial conditions through time . 175 8.16.4 Application to IAU reference frames . 182 8.16.5 Application to computing partials of angular velocity and reference frame with respect to moments of inertia through time . 188 9 INTEGRATION AND INTERPOLATION 192 9.1 INTEGRATION . 192 9.2 THE INTEGRATOR STARTING SCHEME . 194 9.3 INTERPOLATION . 195 10 THE STATISTICAL ESTIMATION SCHEME 195 10.1 BAYESIAN LEAST SQUARES ESTIMATION . 195 4 10.2 THE PARTITIONED SOLUTION . 199 10.3 DATA EDITING . 204 10.4 ELECTRONIC BIAS . 205 10.5 CONDITION NUMBERS . 207 11 GENERAL INPUT I OUTPUT DISCUSSION 207 11.1 INPUT . 207 11.2 OUTPUT . 209 11.3 COMPUTATIONS FOR RESIDUAL SUMMARY . 213 11.4 KEPLER ELEMENTS AND NON-SINGULAR KEPLER ELEMENTS . 215 11.4.1 Partial Derivatives of the Elements . 221 11.4.2 Node Rate and Perigee Rate . 226 11.4.3 Period Decrement and Drag Rate . 227 12 REFERENCES 229 5 1THEGEODYNIIPROGRAM The GEODYN II program, which evolved from GEODYN, is a totally redesigned software system. It has been written for the NASA/Goddard Space Flight Center by EG&G Wash- ington Analytical Services Center, Inc., and has been operational since November 1985. The main user group is NASA/GSFC, where GEODYN II first operated on an IBM 3081 Computer and on a CDC Cyber 205 vector processing computer. The highly efficient op- timized operation on the Cyber has been the main reason GEODYN has been redesigned. These computers are not available any more at GSFC, and GEODYN now operates on a Cray Y-MP. In the mid-1990.s, GEODYN was optimized to take advantage of the Cray parallel processors. With the evolution of small workstations, GEODYN II has mainly been used on HP 900 series computers and during the last few years on SUN workstations. Now in the year 2002, the maintenance of the GEODYN code and executables is done entirely on a SUN workstation. Currently, the maintenance and development of GEODYN II is performed by RAYTHEON ITSS under contract. Like GEODYN, GEODYN II is used extensively for satellite orbit determination, geodetic parameter estimation, tracking instrument calibration, satellite orbit prediction, as well as for many other items relating to applied research in satellite geodesy using virtually all types of satellite tracking data. The GEODYN II documentation is web-based and features an Operations Manual, a File Description chapter, and maintenance history. 2 THE ORBIT AND GEODETIC PARAMETER ESTIMA- TION PROBLEM The purpose of this section is to provide an understanding of the relationship between the various elements in the solution of the orbit and geodetic parameter estimation problem. As such, it is a general statement of the problem and serves to coordinate the detailed solutions to each element in the problem presented in the sections that follow. The problem is divided into two parts: the orbit prediction problem, and • the parameter estimation problem. • 6 The solution to the first of these problems corresponds to GEODYN’s orbit generation mode. The solution to the latter corresponds to GEODYN’s data reduction mode and of course is based on the solution to the former. The reader should note that there are two key choices that dramatically a↵ect the GEO- DYN solution structure: Cowell’s method of integrating the orbit, and • a Bayesian least squares statistical estimation procedure for the parameter estimation • problem. 2.1 THE ORBIT PREDICTION PROBLEM There are a number of approaches to orbit prediction. The GEODYN approach is to use Cowell’s method, which is the direct numerical integration of the satellite equations of motion in rectangular coordinates. The initial conditions for these di↵erential equations are the epoch position and velocity; the accelerations of the satellite must be evaluated. The acceleration-producing forces that are currently modeled in GEODYN are the e↵ects of: the geopotential, • the luni-solar potentials, • planetary potentials, • radiation pressure, • solid Earth and ocean tidal potential, • atmospheric drag, and • general acceleration. • Perhaps the most outstanding common feature of these forces is that they are functions of the position of the satellite relative to the Earth, Sun, Moon, or Planets and of the Sun and Moon relative to the Earth. Only atmospheric drag is a function of an ad- ditional quantity,[1] specifically, the relative velocity of the satellite with respect to the atmosphere. The accurate evaluation of the acceleration of a satellite therefore involves the solution in two concomitant problems: The accurate modeling of each force on the satellite - Earth - Sun - Moon - Planet • relationship, and 7 The precise modeling of the motions of the Earth, Sun, Moon, and Planets. • The specific details for each model in these solutions are given elsewhere in Sections 3, 4, and 8. The question of how these models fit together is, in e↵ect, the question of appropriate coordinate systems. The key factor in the selection of coordinate systems for the satellite orbit prediction problem is the motion of the Earth. For the purposes of GEODYN, this motion consists of: 1. precession and nutation, and 2. rotation. We are considering here the motion of the solid body of the Earth, as versus the slippage in the Earth’s crust (polar motion) which just a↵ects the position of the observer.