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A Bell & Howell Information Com pany 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA 313/761-4700 800/521-0600 INVESTIGATION OF STOCHASTIC MODELS TO IMPROVE
THE GLOBAL POSITIONING SYSTEM SATELLITE ORBITS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
in the Graduate School of The Ohio State University
By
C. David Chadwell, B.S.C.E., B.S.S., M.S.
*****
The Ohio State University
1995
Dissertation Committee Approved by
Clyde C. Goad
Christopher Jekeli Adviser Richard H. Rapp Department of Geodetic Science and Surveying UMI Number: 9526006
UMI Microform 9526006 Copyright 1995, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized copying under Title 17, United States Code.
UMI 300 North Zeeb Road Ann Arbor, MI 48103 DEDICATION
To Susan ACKNOWLEDGMENTS
This research work has been the possible through the help of many people. First,
I would like to thank my adviser, Professor Clyde C. Goad for his guidance, availability,
enthusiasm and flexibility. I thank the reading committee: Professor Christopher Jekeli
and Professor Richard R. Rapp for their valuable comments that improved this work. I would like to express my sincere gratitude to Professor Ivan I. Mueller for his comments
and for the use of his well-organized library. Also, thanks to Dr. Erricos C. Pavlis and
Mr. David Rowlands of the Space Geodesy branch, Goddard Spaceflight Center, who answered my many questions about the GEODYNII software.
During my stay at Ohio State, I met many people with whom discussions of things, geodetic and other, have made my time there quite enjoyable. Most recently,
I want to thank Ms. Dorota Grejner-Brzezinska, Mr. Ming Yang, Mr. Jeong Hee Kim,
Dr. Yan Ming Wang, and Mr. Jarir Saleh, and from earlier times Mr. Kevin Stacy and
Mr. Rob Tudhope. Thanks also to Dr. Lonnie G. Thompson for two years of GRA support at the Byrd Polar Research Center and hiring an undergraduate surveyor many years ago. A special thanks to my wife Susan for her typing, and more importantly, her constant enthusiasm during the many years of my graduate studies. Thanks to NASA/GSFC for the support to work on an interesting project involving their GEODYNII orbit determination software on Grant NAG 5-2041 Many thanks to the Department of Geodetic Science and Surveying for support as a Graduate
Teaching Associate and a Graduate Research Associate. I’d also like to thank the Ohio
Supercomputer Center for providing computer support for this study.
iv TABLE OF CONTENTS
DEDICATION ...... ii
ACKNOWLEDGMENTS...... iii
LIST OF TABLES ...... viii
LIST OF FIGURES ...... x
CHAPTER PAGE
I INTRODUCTION ...... 1
II ORBITAL DYNAMICS AND SITE DISPLACEMENT...... 7 2.1 Reference F ram e s ...... 11 2.1.1 Celestial Reference Fram e ...... 12 2.1.2 Terrestrial Reference Fram e ...... 12 2.1.3 Transformations ...... 13 2.1.4 Relativistic Effects ...... 15 2.2 Earth’s Geopotential ...... 18 2.2.1 Admission o f C2i ,S 2l Coefficients ...... 21 2.2.2 Pole T ide ...... 22 2.3 Third Body Acceleration and the Indirect Oblation E ffect ...... 23 2.3.1 Indirect Oblation Perturbation ...... 24 2.4 Solar Radiation Pressure ...... 25 2.4.1 ROCK4 and ROCK42 M o d e ls ...... 27 2.4.2 Y-Bias Acceleration ...... 31 2.4.3 Earth Radiation (Albedo) ...... 33 2.4.4 GPS Eclipse Effects on Acceleration M odeling ...... 33 2.4.5 New GPS Attitude M odel ...... 36 2.4.6 Resonant Acceleration M odel ...... 37 2.5 Ocean Tide M odel ...... 38 2.6 Solid Earth Tide M odel ...... 40 2.6.1 Treatment of Permanent T id e ...... 42
v 2.7 Station Displacement Due to Pole T id e ...... 43 2.8 Station Displacement Due to Solid Earth ...... 45 2.9 Ocean Loading ...... 47
III OBSERVATION MODELING ...... 49 3 .1 The GPS Observables...... 50 3.2 Selective Availability (SA) and Anti-Spoofing (A S ) ...... 52 3.3 Satellite and Receiver C locks ...... 53 3.4 Propagation M edium ...... 55 3 .4.1 Ionospheric Refraction ...... 56 3.4.2 Troposphere ...... 60 3.4.3 Relativistic Propagation ...... 65 3.5 M ultipath...... 66 3 .6 Antenna Offset and Orientation ...... 66
IV. ESTIM A TO R S...... 67 4.1 Filtering and Sm oothing ...... 70 4.2 Square Root Information Filter and Smoother (SRIF/SRIS) 75 4.3 Epoch State SRIF/SRIS...... 82 4.4 Time-Varying Stochastic Parameter M odels ...... 96 4.5 The Dynamic Stochastic Parameter Transition M atrix ...... 98 4.6 Observable Decorrelation and Normalization ...... 100
V ORBIT DETERMINATION USING A BATCH LEAST-SQUARES ESTIMATOR AND DETERMINISTIC M ODELS ...... 102 5.1 Experiments A, B, and C, Parameter Selection ...... 103 5 .2 The International GPS Geodynamics Service (IGS) Tracking Data and O rb its ...... 109 5.3 Experiments A, B, and C, Deterministic Dynamic Model Fits to the IGS O rb its ...... 114 5 .4 Ground Station Tracking Data Preprocessing ...... 116 5.5 Experiment A, B, and C, Batch Least-Squares GPS Orbit Determination Using IGS Tracking Stations ...... 124
VI ORBIT DETERMINATION USING A SQUARE ROOT INFORMATION FILTER/SMOOTHER AND STOCHASTIC MODELS 127 6.1 Empirical Autocorrelation Functions ...... 128 6.2 Stochastic Modeling of the ‘Wet’ Tropospheric Refraction Zenith D elay...... 133 6.3 Experiments E and F, Parameter Selection ...... 142 6 .4 Stochastic Model Results of the Solar Radiation Pressure Scale Factor and Y-Bias Acceleration ...... 146 6.5 Stochastic Modeling Results of the Radial, GPS X-Axis and GPS Y-Axis Accelerations ...... 155
vi VII CONCLUSIONS AND RECOMMENDATIONS 164
APPENDICES A OSUORBFS and Control Card Descriptions ...... 170 B. OSUORBFS Subroutine Descriptions ...... 185 C. Householder Transformation Subroutine ...... 190 D. Comments on GEODYNII and Its Role As a Trajectory Preprocessor for OSUORBFS ...... 193
LIST OF REFERENCES...... 198
vii LIST OF TABLES
TABLE PAGE
1. Acceleration of GPS Satellite ...... 8
2. Systematic Errors Affecting GPS Observables ...... 50
3. OSUORBFS Parameter Types...... 84
4. Experiment A, Batch Least-Squares Estimated Parameters on Day 016, 1994 ...... 106
5. Experiment B, Batch Least-Squares Estimated Parameters on Day 016, 1994 ...... 107
6 Experiment C, Batch Least-Squares Estimated Parameters on Day 016, 1994 ...... 108
7. The IGS Global Tracking Stations Used in This Study ...... I l l
8 Transformation Parameters Between IGS Analysis Center GPS Orbits on Jan. 16, 1994 ...... 113
9. Experiments A, B, and C, GEODYNII RMS2 of Fit (cm) to the IGS O rbit...... 115
10. Transformation Parameters Between GEODYNII GPS Satellite Orbits and IGS GPS Satellite Orbits for Experiments A, B, and C on Jan 16, 1994 Using IGS GPS Satellite Orbits as D a ta ...... 116
11. Baseline Combinations on Jan. 16, 1994 Used in the GPS Satellite Orbits Solution ...... 122
12. Experiments A, B, and C, Batch Least-Squares GPS Satellite Orbit RMS2 of Fit (m) Using IGS Tracking Station D a ta ...... 124
viii 13 Transformation Parameters Between GEODYNII GPS Satellite Orbits and IGS GPS Satellite Orbits for Experiments A, B, and C on Jan 16, 1994 Using Ion-Free Double-Differenced Phase Ranges 125
14. Autocorrelation Functions of Selected Stochastic M odels ...... 132
15. Experiment D, Square Root Information Filter/Smoother Estimated Parameters on Day 016, 1994 ...... 137
16. Experiment D, OSUORBFS RMS3 of Fit (cm) Using First-Order Gauss-Markov and Random Walk Models for Tropospheric Refraction‘Wet’ Zenith D elay ...... 138
17. Transformation Parameters from the IGS GPS Satellite Orbit to the OSUORBFS Orbits for Experiment D on Jan. 16, 1994 Using First- Order Gauss-Markov and Random Walk Models for Tropospheric Refraction‘Wet’ Zenith D elay ...... 139
18. Experiment D, Square Root Information Filter/Smoother Estimated Parameters on Day 016, 1994 ...... 144
19. Experiment F, Square Root Information Filter/Smoother Estimated Parameters on Day 016, 1994 ...... 145
20. Experiment E, OSUORBFS (Smoothed) RMS3 of Fit (cm) Using First-Order Gauss-Markov and Random Walk Models for the Solar Radiation Pressure Scale and Y-Bias Acceleration and Ion-Free Double-Differenced Phase R anges ...... 148
21. Transformation Parameters Between IGS Orbit and OSUORBFS Orbit for Experiment E on Jan. 16, 1994 ...... 148
22. Experiment F, OSUORBFS RMS3 of Fit (cm) Using First-Order Gauss-Markov and Random Walk Models for Radial, GPS X-Axis, and GPS Y-Axis Accelerations and Ion-Free Double-Differenced Phase Ranges ...... 155
23 Transformation Parameters from the IGS GPS Satellite Orbit to the OSUORBFS Estimated Satellite Orbits for Experiment F on Jan 16, 1994 Using Ion-Free Double-Differenced Phase R anges ...... 157
24. OSUORBFS Time A ccounting ...... 175
ix LIST OF FIGURES
FIGURE PAGE
1. GPS Spacecraft Coordinate System ...... 29
2. Flowchart Showing the Operations Involved in the Described P ro ced u re ...... 69
3 IGS International GPS Tracking Network ...... 110
4. Data Preprocessing Procedure ...... 117
5. GPS Satellite Positions at Tracking Receiver Collection Time . . . 118
6. Receiver Clock Offset at Quincy on Jan. 16, 1994 Determined by Static Pseudorange Solution ...... 121
7. Receiver Clock Offset at Fortaleza on Jan. 16, 1994 Determined by Static Pseudorange Solution ...... 121
8. Example of Ion-Free Double-Differenced Residuals on Jan. 16,1994, Orbit Solution Using Solar Radiation Pressure Scale and Y-Bias Accelerations Model, Experiment A ...... 129
9. Autocorrelation Function of Ion-Free Double-Differenced Residuals on Jan. 16, 1994, for Baseline WETT to GRAZ and Orbit Solution Using Solar Radiation Pressure Scale and Y-Bias Accelerations Model, Experiment A ...... 130
10. Random Number Time Series Autocorrelation Function ...... 133
11. Random Walk Variation in the Tropospheric Refraction Zenith Delay at Wettzell on January 16, 1994 with y]q con = 1 cm / Vhr . . 140
X 12. Autocorrelation Function of Ion-Free Double-Differenced Residuals on Jan. 16, 1994, OSUORBFS Solution Using a First-Order Gauss-Markov Model for the Troposphere ‘Wet’ Zenith Delay, Experiment D ...... 141
13. Experiment E, Solar Radiation Pressure Scale Factor Correction for PRN02 Using a First-Order Gauss-Markov (FOGM) Model with yjqcm = 0.05 and r = 12 H ours ...... 150
14. Experiment E, Solar Radiation Pressure Scale Factor Correction Sigma Using a Random Walk (RW) Model with -Jqcon ~ 0.05 .... 151
15. Experiment E, Comparison of Solar Radiation Pressure Scale Factor Correction Sigma for PRN02 Using a First-Order Gauss-Markov (FOGM) Model with ^jq con = 0.05 and r = 12 Hours and Random Walk (RW) Model with q mn = 0 .0 5 ...... 152
16. Experiment E, Comparison of Y-Bias Acceleration for PRN02 Using a First-Order Gauss-Markov Model (FOGM) with y]qcon = 1 x 10“9 m / s2 and r = 12 Hours and a Random Walk (RW) Model with -yjqcon = 1 x 10“9 m / s2...... 153
17. Experiment E, Comparison of Y-Bias Acceleration Sigma for PRN02 Using a First-Order Gauss-Markov Model (FOGM) with yjqcon - 1 x 10 9 m / s2 and r = 12 Hours and a Random Walk (RW) Model with y jq ^ = 1 x 10'9 m / s2 ...... 154
18. Experiment F, Comparison of Radial Accelerations Using a First- Order Gauss-Markov (FOGM) Model with yjqco„ = 1 x 10 9 m / s2 and r = 12 Hours and Using a Random Walk (RW) Model with = lxl0-9m /s2...... 158
19. Experiment F, Comparison of Radial Acceleration Sigmas Using a First-Order Gauss-Markov (FOGM) Model with yJqcon = 1 x 10"9 m / s2 and r = 12 Hours and Using a Random Walk (RW) Model with -Jqcon = 1 x 10'9 m / s2...... 159
xi 20. Experiment F, Comparison of GPS X-Axis Accelerations Using a First-Order Gauss-Markov (FOGM) Model with yjqcon = 1 x 10“9 m / s2 and r = 12 Hours and Using a Random Walk (RW) Model with yjqcon = 1 x 10 9 m / s2...... 160
21. Experiment F, Comparison of GPS X-Axis Acceleration Sigmas Using a First-Order Gauss-Markov (FOGM) Model with yjqcm = 1 x 10~9 m / s2and r = 12 Hours and Using a Random Walk (RW) Model with y]qcon = 1 x 10“9 m / s2...... 160
22. Experiment F, Comparison of GPS Y-Axis Accelerations Using a First-Order Gauss-Markov (FOGM) Model with yJqco„ = 1 x 10 9 m / s2 and r = 12 Hours and Using a Random Walk (RW) Model with ylqcon = 1 x 10“9 m / s2...... 161
23. Experiment F, Comparison of GPS Y-Axis Acceleration Sigmas Using a First-Order Gauss-Markov (FOGM) Model with ^ q co„ = 1 x 10~9 m / s2and r = 12 Hours and Using a Random Walk (RW) Model with y [q ~ = 1 x 10"9 m / s2...... 162
xii CHAPTER I
INTRODUCTION
In the early 1980’s the Global Positioning System (GPS) was first used for geodetic positioning (Goad and Remondi, 1984). The accuracy of relative baseline determination and absolute positioning were limited to some extent by the GPS orbit accuracy. The dynamic models used to determine the orbits included a specific solar radiation pressure model, ROCK4, developed at Rockwell by Porter (1976).
Investigators soon learned that the Porter model underestimated the acceleration experienced by the satellite by about 20% (Fliegel et al., 1992). By introducing a solar radiation scale factor parameter that was estimated along with the satellite initial elements 10 m orbit accuracy was achieved. Investigators also reported an additional acceleration along the GPS Y-axis (Fliegel et al., 1985). To compensate, an additional
GPS Y-axis acceleration parameter was also estimated. This is sometimes called the y- bias. The standard model became the estimation of the solar radiation pressure scale and the y-bias.
Orbit accuracy continued to improve with the introduction of the phase observable, refinement to the tropospheric modeling, and the densification and expansion of the tracking station network. In the late 1980’s, the GPS satellite position predicted by the Standard Positioning Service (SPS) broadcast ephemeris was typically within 5 m
l 2
of the position estimated using actual tracking data in a post-fit, precise orbit
determination (Remondi and Hofmann-Wellenhof, 1990). Post-fit orbits were routinely
available with 1-2 m accuracy and specialized efforts using fiducial stations could push this below one meter (Lichten and Bertiger, 1989). Even with the introduction of
Selective Availability, which degraded the SPS broadcast elements and clock offsets and dithered the satellite frequency standard to the extent that the position difference from
precise post-fit orbits increased to 50 m or more, the precise post-fitted orbits and the corresponding satellite clock models were still determined at non-SA levels. The non-
SA levels were possible since post processed double differenced observables are insensitive to SA (Davis, et al., 1990). During the early 1990’s the establishment of an extensive global tracking network helped drive the routinely achieved accuracy below
lm. In 1994, with the operational status of the International GPS Geodynamics Service
(IGS), orbits are now determined on a daily basis by seven analysis center at the 20 cm level (Kouba et al., 1993).
For GPS arcs of a few days or more, the standard approach cannot model the accelerations to the meter level. The acceleration model must be expanded to include more terms. The source or sources of this unmodeled acceleration may result from deficiencies in the deterministic model or possibly the accelerations are random processes. Deficiencies in the deterministic model may arise from the solar radiation force and the acceleration it imparts to the satellite. Several possible sources deserve mentioning. The physical properties of absorption and reflectance of the actual satellite are not completely understood. The GPS Z-axis may be pointing slightly away from the 3
negative geocentric radial direction. The solar panel axis is most likely not perpendicular
to the GPS X-Z axis plane. Finally, the solar radiation force model itself may be slightly
incorrect. All these impart residual accelerations that can be resolved into components
of acceleration along the GPS X, Y, Z body axes.
Other possible sources of residual accelerations include departures from the true gravitational, and third-body point-mass accelerations, and tidal forces, from their
adopted models. These unmodeled accelerations should be smaller than that from the unmodeled solar radiation pressure. At an altitude of 20,000 km, a GPS satellite is primarily affected by the first 8 or 12 degree and order terms of the gravity field. By tracking much lower satellites, these have been determined to a precision higher than that needed for GPS. However, because of their high altitude, the GPS satellites are affected significantly by the third-body attractions of the sun and moon. The sun and moon positions and masses have been determined independently to a relatively higher precision and can be modeled satisfactorily. The tidal accelerations are again small, relative to the solar radiation acceleration, and a few percent error in the tidal accelerations is an even smaller effect.
Researchers have modeled the residual acceleration as deterministic parameters and as stochastic parameters. Colombo (1986) suggests these residual accelerations are filtered by the orbital motion, which acts as a type of bandpass filter, into effects that resonate at a zero frequency (i.e. secular) and the nominal orbit frequency (once per revolution). He suggests using a three-term Fourier expansion about the nominal GPS orbital frequency along each spacecraft body axis. Using this technique Colombo (1989) obtained proportional baseline repeatability of lx l 0 7 using six day arcs with the rather limited constellation available in 1985. The advantage of Colombo’s approach is that it can be implemented in a conventional least-squares correction algorithm. The primary disadvantage is that it substantially increases the number of dynamic parameters that must be estimated and for which variational equations must be integrated.
The Jet Propulsion Lab (JPL) group adopted stochastic parameter models for the acceleration along the GPS Z-axis, GPS X-axis, and GPS Y-axis (Lichten and Bertiger,
1989). They suggest these stochastic processes be modeled as first-order Gauss-Markov processes with process noise covariance of 2xl(T2Skm2 / sec4 and correlation time of
12 hours. Using this approach they achieve submeter accuracy over a one-week arc.
The advantage is that only three additional acceleration terms are estimated. The stochastic model is less constrained than the deterministic model. It can respond to erratic variations more readily than the deterministic model. The disadvantage is that a sequential filter/smoother estimator is required.
The question arises as to how best to model these residual accelerations.
• Are additions to the deterministic models appropriate or should stochastic models be introduced?
• If so, which stochastic models are best?
• What are the appropriate process noise covariance and if required, correlation times for the stochastic model?
• Can stochastic models improve GPS satellite orbits? 5
To begin to answer these questions an extensive review of the acceleration effecting the satellite is undertaken. Over 99% of the force can be modeled with the dynamic models developed in Chapter 2 and the measurement models developed in
Chapter 3. The implementation of a stochastic model requires a sequential filter/smoother. To fill this role, a square root information filter/smoother (SRIF/SRIS) algorithm is developed and then interfaced with the batch algorithm in Chapter 4. Two mistakes in Bierman’s (1977, pp. 158-160) SRIF time propagation algorithm are shown.
Also, the discrete forms of two stochastic models are reviewed.
In Chapter 5, the following three deterministic dynamic model experiments are implemented in the batch least-squares estimator:
(A) solar radiation pressure scale and y-bias acceleration model in a batch least-squares estimate,
(B) solar radiation pressure scale and the nine parameter acceleration model of Colombo in a batch least-squares estimator,
(C) solar radiation pressure scale and bias accelerations in the radial, GPS X-axis, and GPS Y-axis in a batch least-squares estimator.
The dynamic models are tested by fitting to a precisely determined International GPS
Geodynamics Service (IGS) orbit on January 16, 1994. Then, the GPS satellite orbits of
January 16, 1994 are independently determined based entirely on ground tracking data for each of the deterministic dynamic modeling experiments.
In Chapter 6, empirical autocorrelation functions are constructed from the residuals of a GPS orbit fit using deterministic models. These are compared with the 6 characteristic autocorrelation functions of common stochastic models. Based on these comparisons appropriate stochastic models are selected The IERS standards (McCarthy,
1992) recommend that the ‘wet’ tropospheric refraction zenith delay be modeled as a random walk. This procedure is reexamined to explore other possible stochastic models and the appropriate defining parameters.
(D) ‘wet’ tropospheric refraction zenith delay modeled as a deterministic bias and a stochastic process in a sequential estimator.
A series of experiments are conducted to assess the capabilities of stochastic models of the solar radiation pressure and/or residual acceleration, to improve the GPS orbit. The following modeling scenarios are investigated:
(E) solar radiation pressure scale and y-bias acceleration as stochastic models in a sequential estimator,
(F) solar radiation pressure scale as a deterministic bias and stochastic models for the radial, GPS X-axis, and GPS Y-axis in a sequential estimator.
In Chapter 7, the findings of this study are summarized, conclusions drawn and recommendations given. Appendix A contains a description of the program
OSUORBFS and control cards. Appendix B contains descriptions of the OSUORBFS subroutines. Appendix C comments on the use of GEODYNII as a trajectory pre processor for OSUORBFS. CHAPTER II
ORBITAL DYNAMICS AND SITE DISPLACEMENT
The forces that act on the GPS satellites result primarily from the gravitational attraction of the masses of earth, moon, and sun, and the solar radiation pressure which imparts a force to the satellite. The mass structure of the earth is not uniform, but is composed of many non-spherical components and is dominated by the equatorial bulge.
This mass structure also is not static, but is redistributed by the rotation of the earth about its axis along with the attraction of the moon and sun. The net acceleration of the satellite caused by these forces can be represented as the superposition of the individual accelerations in the form of the equations of motion,
X = \s7 ,' + X wsl.: + X A/ + X , + X w + X w + X lo + X /;T + X,.7< + X „ + X OT (2-1) where the accelerations are due to:
X Sh. the mass of the spherical uniform earth,
X A,A7, the nonspherical mass distribution of the earth,
X A/ the point mass of the moon,
X s. the point mass of the sun,
X w the solar radiation pressure,
X,„ the GPS Y-axis bias, X
X lo the indirect oblation perturbation,
X,,T the attraction of the solid earth masses displaced by the tidal force
attraction of the sun and moon,
the earth radiation (Albedo),
Xss the Schwarzschild relativistic effect,
Xor the attraction of the ocean masses displaced by the tidal force attractions of
the sun and moon, plus the attraction of the crust displaced by the loading
of the ocean masses.
The approximate magnitude of these accelerations are given in Table 1.
Table 1. Acceleration of GPS Satellite (Reigber, 1989; Zhu and Groten, 1988).
Source of Designation in Magnitude of Acceleration (m / ,v2) Acceleration Equation (2-1) a = 26559 km, A/M - 2.0x1 O'2 m2 / kg Kepler Term 0.6
V C20, harmonics NSE 5.0xl0 -\ 3.0x 10‘7 Third body X „ X A, 5.0x10 6 Solar Radiation 1.0x10 7
Indirect Oblation X;0 2.0x10 9 Y bias 0.8x10 9
Earth tides XCT 1.0 x 10“9
Relativistic Effect X* 0.3x10 9 Earth Radiation X ^ 0.4x10 9 Ocean tides Xo, 1.0x10 ,0 9
The Schwarzschild relativistic effect is not a true force, but results from the departure of the geocentric reference frame from an inertial frame This topic is covered in section 2.1 4.
By adopting a priori values for the initial position and velocity of the satellite
(state vector) and the other parameters in the equations of motion, these equations can be integrated numerically to obtain the satellite’s state vector at a later time. The a priori values of the model parameters can be improved by estimating corrections through a differential orbit correction process
XmJ =x;j + AX™J (2-2) where X mJ is the updated (a posteriori) force or measurement modeled parameters,
X™'1 is the a priori force or measurement modeled parameters,
AX”',; is the correction to the a priori force or measurement modeled parameters.
This process requires observing a measurement (O) that can be related through function
( f ) to the modeled parameters (X),
O = f(X) (2-3)
This is the observation equation.
For a set of observations, equation (2-3) is linearized about the a priori modeled parameters
di dX{ O C = A X 0m + AXJ0+e (2-4) KdX{ j \OX{j where O is the observation, with covariance E 10
C = f(X 0) is the computed observation based on the a priori values of the
modeled parameters,
d f —^ is the partial derivative of the observation equation with respect to the
epoch model parameters at time t,
dX — 1 is the state translation matrix, or variational partials, which relates the state o X Q
of force model parameters at time /, to initial values at time t0,
AX{ are corrections to the initial force model parameters, e.g. position, velocity,
solar radiation pressure coefficient,
AX” are the corrections to the measurement model parameters, e.g. tropospheric
refraction scale coefficient, receiver clock bias,
£ is the observation noise.
The corrections to the apriori model parameters are traditionally estimated using a least squares approach
AX0 = (Ar A) ' Ar L^(O-C) (2-5)
d o dX where A = \ d X j \ ud X n
The differential equations, to compute variational partial derivatives, have the form
4 t (Y) = DY + e 4(Y) + F (2-6) dt dt
dX(t) where Y are the variational partial derivatives = x(0’ a \( t) D = ^ x (/) are the partial derivatives of the equations of motion with respect to
position,
E = ^ are the partial derivatives of the equations of motion with respect to
velocity,
respect to modeled parameters.
GEODYNII uses a Cowell type integrator with a fixed step size to integrate the equations of motion and the variational equation (2-6).
The computational models for the accelerations in equation (2-1) and the necessary partial derivatives D, E, and F in equation (2-6) are developed in this chapter.
The observation equations are discussed in Chapter 3. An alternative to the least- squares estimator is given in Chapter 4.
2.1 Reference Frames
Near-earth satellite orbits about the earth and are tracked by stations on the earth’s surface. The integration of the equations of motion of the satellite can avoid complicated rotation terms by adopting a ‘quasi-inertial’ reference frame. The definition of ‘quasi inertial’ is addressed in 2.1.4. A terrestrial frame, rotating with the earth, is the only convenient way to describe tracking stations on the earth. But, this terrestrial frame 12 is not a ‘quasi inertial’ frame Thus, it is necessary to have a celestial and a terrestrial reference frame, and a mathematical transformation between the two.
2.1.1 Celestial Reference Frame
The celestial frame used by the International GPS Geodynamics Service (IGS) is the one adopted by the IERS (McCarthy, 1992):
“The IERS Celestial Reference Frame (ICRF) is based on the extragalactic objects determined by the VLBI and is a realization of a system of directions which are consistent with those of the FK5. The origin of the right ascensions of the frame was implicitly defined in the initial realization by the adoption of the right ascensions of 23 radio sources catalogs obtained by the Goddard Space Flight Center, the Jet Propulsion Laboratory, and the National Geodetic Survey.”
“The ICRF polar axis points in the direction of the mean pole at J2000.0 as defined by the IAU conventional models of precession and nutation.”
The origin is defined as the center of mass of the solar system (i.e barycentric).
2.1.2 Terrestrial Reference Frame
The terrestrial frame is defined by the IERS Terrestrial Reference Frame (ITRF) and is defined to meet the following criteria:
“a) It is geocentric, the center of mass being defined for the whole earth, including oceans and atmosphere. b) Its scale is that of a local Earth Frame, in the meaning of a relativistic theory of gravitation. c) Its orientation is given by the BIH orientation at 1984.0. d) Its time evolution in orientation will create no residual global rotation with regards to the crust.”
Its realization is maintained by the IERS, the latest being the International
Terrestrial Reference Frame 1992 (ITRF92) At any time t, the coordinates in the
ITRF92 system are defined as 13
x(') = X„(/„) + V„(/ - 1J + 1 AX,(l,) (2-7) I
where X0( f J are the coordinates at an initial epoch ta (1988),
v„ = V plale + Vr, a velocity term,
Vpiare *s based upon the NNR-NUVEL1 (Argus and Gordon, 1991) plate
model,
Vr a parameter to be estimated,
ZAX,(/) are corrections to station position (i.e. solid earth tide, ocean loading
pole tide).
The IERS standards recommend that permanent earth tide be treated as a
correction to the station coordinates (McCarthy, 1992). However, the ITRF92 system
has not applied the permanent tide correction (zero frequency), so these coordinates do
contain the permanent earth tide displacement (Boucher et al., 1993).
2.1.3 Transformations
The combined motion of the ITRF in the ICRF is due to the combined effect of the response of the earth to ‘free motion’ and the lunisolar attraction or ‘forced motion’.
The secular portion of this combined motion is defined as precession. The remaining
motion is modeled as nutation which describes the periodic motion of an adopted axis in
the ICRF, and polar motion which describes the motion of the earth’s crust with respect to the adopted axis. A convenient formulation for the IAU 1976 model for precession is
given by Lieske et al (1977). The nutation series was developed by Wahr (1981) for an 14 elastic earth with restrictions. Additional corrections to the nutation series have been determined using VLBI (Herring, 1987). These corrections are thought to be due primarily to the difference between the real earth and Wahr’s restricted model
The ‘free’ nutation and polar motion cannot yet be predicted very accurately because of the limited knowledge of the earth’s interior structure. Thus, the ‘free’ portion is observed. The IERS determines the polar motion of the ITRP with respect to the CEP defined by the adopted theory of nutation and precession. Additionally, IERS determines the rotation of the earth (UTC-UT1). Final values of x p, y p and UTC-UT1 are given in the monthly IERS Series Bulletin B. Preliminary values are given in the weekly IERS Series Bulletin A. Both of these series contain predicted values (McCarthy and Luzum, 1991).
The transformation from the inertial frame to the terrestrial frame is given as follows
X; iw.=SNPX/£HF (2-8) where S is the earth orientation parameter (EOP) matrix at t[;
N is the nutation matrix at /,,
P is the precession matrix from time /„ to /,;
See Mueller (1969) or Seidelmann (1992) for the standard definition of these matrices.
The true of reference date (TORD) frame is used in GEODYNII to integrate the equations in motion. However, GEODYNII evaluates most accelerations in another system, the true of date (TOD) frame. Retaining only the terms involving X T0D and 15 neglecting N0, P0, and N(, at any time 1t, the transformation from the TOD to the
TORD system is
V - N P N Y TORD ~~ 1 ’ 0 1 O.iJ1 i TODTORD (2-9) where N0 is the Nutation matrix at time tTORD,
P0 i is the Precession matrix from tt to tTOHD,
Nj is the Nutation matrix at time 1t.
At any time t., the positions in the ITRF system are transformed to the TORD system by
TORD (2- 10) where Sj is the EOP matrix at time /,.
For a near-earth satellite Zhu et al., (1987) have shown that a geocentric system is sufficient where the sun and moon can be included as point masses with their tidal effects, and with relativistic corrections appended.
2.1.4 Relativistic Effects
Traditionally, the equations of motion are developed using Newtonian physics assuming an inertial frame. In classical Newtonian mechanics an inertial frame is one that is not rotating and moves with a constant translational velocity, i.e. the origin experiences no acceleration. Also, in classical mechanics, a ‘quasi inertial’ frame has no rotation, but the origin experiences a small change in the translational velocity, i.e. the origin is accelerated Thus, a geocentric coordinate system with an orientation fixed in 16 space that experiences a small acceleration as the earth orbits the sun has been viewed historically as a ‘quasi inertial’ frame in which the Newtonian formulation is valid
(Moritz and Mueller, 1987)
The classical interpretation of the non-rotating geocentric frame as a ‘quasi inertial’ frame, with the application of the Newtonian mechanics, is not valid for satellite geodesy with centimeter-level measurement for satellite laser ranging and GPS phase observations. The non-rotating geocentric frame, a ‘quasi inertial’ frame, must be reformulated in terms of general relativity theory.
In the theory of general relativity, space is viewed as a four dimensional space time continuum that is curved due to a gravitational field. Fortunately, the consequent departures from Newtonian theory are small and at the current modeling accuracy of l.Ox 10 ,3w/.y2 they can be handled, to a first-order approximation, as appended corrections to the Newtonian equations of motion.
The dominant term is due to the presence of the gravitational attraction of the earth. Its effect on the acceleration of the satellite is given as the Schwarzschild (1916) solution by Huang et al. (1990)
(2- 11) where GMU is the gravitational constant multiplied by the mass of the earth,
c is the vacuum speed o f light,
R, R are the geocentric satellite position and velocity vectors, respectively,
R= | r | , r = |r |. 17
Here only the point mass effect is included. The point mass has an acceleration of approximately 0.3 x 10 ~9ml s2 for GPS orbits (Zhu and Groten, 1988) The quadrapole
(J2) effect is on the order of 10 ~nm/ s2 and is excluded from GPS orbits, but may be included for LAGEOS orbits (Soffel et al., 1988). The point mass also causes a gravitational time delay which will be considered in the next chapter.
The next effect, in order of magnitude, is the Lense-Thirring acceleration due to the gravito-magnetic field generated by the earth’s rotation about its axis (Huang et al.
1990)
(2- 12) where J is the earth’s angular momentum vector per unit mass, |./| = 9.8 x 10'/w2 / sec,
and all other parameters defined as above.
For GPS satellites it has an acceleration of approximately 1.0 x 10 ~l2m/ sz (Soffel, 1989).
There are additional smaller effects due to the earth rotation that are not considered at this time (Huang et al., 1990).
Finally, the rotation of the earth about the sun results in the geodesic precession
(de Sitter, 1916) effect (Huang et al., 1990)
X g /)= 2 [G x R] (2-13) where Q = |[ v ff-V s]
V,., Vs. are the barycentric velocity vectors of the earth and sun, GMS is the gravitational parameter of the sun,
R l;s is the barycentric vector from the earth to the sun, Rhs = | R FS | .
Its magnitude is approximately 2.0x ]0~um /s 2 for GPS orbits. It is also called the relativistic Coriolis effect.
The IERS standards recommend that the correction for the point mass effect be included in the equations of motion. The effects of the quadrapole, Lense-Thirring, and geodesic precession are not included for GPS satellites. GEODYNII applies corrections for the point mass effect, the Lense-Thirring effect, and the geodesic precession The quadrapole is not included. With GEODYNII, it is possible either to apply all three corrections or not apply any of the corrections. To accommodate best the IERS standards all three are applied. Since the accelerations due to the Lense-Thirring and geodesic precession are on the order of 10 12/n/.v2, this should not affect the comparisons of one-day GPS orbits at the decimeter level.
2.2 Earth’s Geopotential
The earth’s geopotential is represented as a point mass effect and an expansion of spherical harmonics to represent the nonspherical effects of the earth’s mass (Torge, a 'P e U NSli ~ GM il t Pnm(sin
r is the geocentric satellite distance,
,7m« *s upper limit for the summation (highest degree),
ac is the Earth’s mean equatorial radius,
A is the satellite east longitude,
/^(sin^) indicate the associated Legendre functions, and
Cnm and Snm are the unnormalized gravitational harmonic coefficients.
The body frame acceleration in the spherical coordinate system is given as the gradient
(V) applied to the potential
d d d V = — u r + — u , + — (2-15) d, “r + d
The spherical coordinates (
Cartesian coordinates (x,y,z) as 20
r-y jx 2 + y 2 + z2 (2-16c)
The transformation from the spherical coordinate system to the Cartesian coordinate system is
a (f> ax ar Ax ax ax axA^ a
Thus, acceleration is transformed into body frame Cartesian coordinates x, y, z by the transformation
-xz y X r2p ~ p 2 r -yz X y V(7 (2-18) ‘ SE.NSE r2 p 7 r SE.NSE z 0
1 r . where p = -Jx2 +y2 ,
r = i x2 +y2 +z2 .
Now the &NSE is the acceleration in body frame Cartesian coordinates due to the geopotential. The acceleration must be transformed into the inertial system
X^=(SNP)ra^A,5E (2-19)
The associated partials for the variational equations are given as
s e .NSE 1 SE.NSE ax ax = (SNP)r a Vi SNP, ax~° ( 2 -2 0 ) 21
GEODYNII uses as equivalent, but alternate form
SE.N SE . . —t-. — A 3 U (/2r=C,rf/2C1 + I — C2t (2-21) 2- 1 2 1 dek where ek ranges over the elements and,
d 2U U2 is the matrix whose /, / element is given by
d e, C, is the matrix whose i,j'h element is given by ,
d 2ek C2k is the matrix whose /, / element is given by dr,tdrt ’
r - SNPx.
The IERS standard gravity field for GPS satellites is the Goddard Earth Model
T3 (GEMT3) up to and including degree and order 8. (McCarthy, 1992; Lerch et al.,
1992). The GM(GEMT3) = 398600.436 AtwV and the value of a (GEMT3) =
6378137.0 m should be used with equation (2-14b). The more recent value of
GM(IERS) = 398600.4415 krtr' / s2 contains oceans, atmosphere and should be used with the two-body term or the Kepler term in equation (2-14a). GEODYNII does not allow the specification of two separate GM terms. The GM(IERS) value is specified.
2.2.1 Admission of C2I, ,V21 Coefficients
The values of C21 and S2] in the GEMT3 solution were constrained to zero, thus, forcing the principal axis of inertia or mean figure axis to coincide with the mean rotation or z axis of the adopted body frame coordinate system Any difference would be due to 22 long period fluid motion of the earth and this effect has not yet been independently verified (Wahr, 1987, 1990). The adopted IERS reference pole is not the same pole used for the GEMT3 geopotential coefficients. Thus, admitting the terms C21, .S’2I accounts for the offset of the IERS pole from the GEMT3 pole. This is not the same as admitting
C2I, S2} in the GEMT3 model solution. The coefficients were computed by Lambeck,
(1971)
C„ = xC;„V3 (2-22a)
Sn = - y C ,J 3 (2-22b) where x , y are the coordinates, in radians, of the mean figure axis with respect to the
adopted mean rotation axis,
C2 o the normalized geopotential coefficient,
V3 is needed because C20 is normalized.
With x = 2.0362 x 10 7(0.*042) and y = 1.421 x 10'6 (0."293), the values are
C21 (IERS) = -0.17 x 1 O'9 (2-23a)
S21(IERS) = 1.19 x 10 9 (2-23b)
2.2.2 Pole Tide
The principal rotational deformation of the potential has the frequency of one
Chandler period. The change in geopotential is proportional to the tidal potential
SV = kv (2 -2 4 ) where k - 0.3 is the Love Number. 23
And, combining equation (2-22) with equation (2-24) gives
AC,, = *W30!() (2-25a)
AA„=-iJ!V3C„. (2-25b)
The corrections are given in the IERS standards as
AC21 = -1.3x 10 (2-26a)
A.V21 = -1.3 x 10 9( - ^ ;,) (2-26b) where xp=mx- y ,
yP=m1 -y,
mx, m2, are the instantaneous polar motion values,
x , y are the 6-year mean values of the polar motion.
Thus, C21, .V21 will have a mean of zero and no permanent station shift is introduced.
2.3 Third Body Acceleration and the Indirect Oblation Effect
The direct attraction of the moon and planets on the near-earth satellite perturbs the orbit. The point mass effects of these third bodies are modeled in the inertial reference frame
(r ,- r ) R, \ S =GMS (2-27) |r .v _ r | Ir J
(Rai -R) r M X u = OMu (2 -2 8 ) Rm-R| R 24 where GMS, GMhl are the gravitational parameters of the sun and moon, respectively,
R s> Ra,, R are the inertial true of reference date geocentric position vectors of
sun, moon, and satellite, respectively.
The associated partial derivatives for the variational equations are
(2-29)
(2-30)
The planetary and lunar ephemerides recommended for the IERS standards are the JPL
Development Ephemeris DE200 and the Lunar Ephemeris LE200. These provide the barycentric positions of the sun, moon, and the earth-moon baracenter which are then transformed to geocentric positions (Newhall, 1989).
2.3.1 Indirect Oblation Perturbation
In addition to the point mass effects, two additional accelerations arise. First, the attraction of the equatorial mass of the earth (C20) on the point mass of the moon and thus its equal and opposite acceleration on the earth. Second, the similar attraction of the equatorial mass of the moon (c20) on the point mass of the earth The total indirect oblation acceleration is
(2 -3 1 ) where X a,(A) is the acceleration on the point mass moon due to the oblate earth,
Xe(M) is the acceleration on the point mass earth due to the oblate moon
Now X M (E) is
-s in
ac is the equatorial radius of the earth,
rm is the lunar position vector in true of date coordinates, rm = |rm |,
(j>m, X m are the geocentric latitude and longitude of the moon, respectively.
The magnitude of this acceleration is approximately 2.0 x 10~9 m is1. The acceleration term X h\M ) has a similar form, but the magnitude of the acceleration is approximately
1.0 x 10 13 m is1. Thus, this contribution is not included in GEODYNII. These effects are not included in the variational partial derivatives. Neither contribution is addressed in the IERS standards.
2.4 Solar Radiation Pressure
The solar radiation acceleration is 26 where v is an eclipse factor and is the ratio of sunlight not eclipsed by the earth or
moon,
v = 0 if satellite is in shadow (umbra),
v = 1 if satellite is in sunlight,
0 < v < 1 if satellite is in penumbra,
S is the mean solar flux of at one astronomical unit,
c is the speed of light,
Rs is the semi major axis of the earth’s orbit, i.e. one astronomical unit,
CR is the reflection coefficient for the satellite,
A is the cross sectional area o f satellite,
M is the mass of satellite,
R is the position vector of the sun in the inertial frame,
R s is the position vector of the sun in the inertial frame.
The associated partial derivative of equation to be used in the variational equations is given as (Cappellari et al., 1976)
and the force partial with respect to the velocity is
(2-35)
Now the norm squared of equation (2-33) is 27
(2-36)
Thus, equation (2-34) can be represented as
(2-37)
i 3[r-rJr-rJ where A = R-R,
2.4.1 ROCK4 and ROCK42 Models
For GPS satellites the value of Cn ■ A is quite complicated due to the presence of the solar panels and many other hardware features that protrude from the satellite body, and the different reflective and absorptive properties of the construction materials. This is in contrast to spherical satellites, such as LAGEOS, for which CR- A is nearly a constant. These complex interactions have been modeled for GPS satellites and given in a convenient approximation known as the Rockwell models, ROCK4 and ROCK42
(Porter, 1976; Fliegel et al., 1992). These models account for the accelerations along the
X aps and Z GPS satellite body axes. An additional Y-bias acceleration along the YGKV axis must be estimated (Fliegel et al., 1985).
The GPS spacecraft body frame is defined as
(2 -3 8 a ) 28
Z GPS x 1 GPS (2-38b) ^ G P S X ^ SAT. SUN
YfiPX X Ziaps v - GPS ~ (2-38c) Ygps X'ZJGPs\
where RSAT.SUN is the sun position vector from the satellite, R SUN R< S4T
R ^ is the inertial geocentric position vector of the satellite,
Rsun is the inertial geocentric position vector of the sun.
The sun-satellite-earth angle (B) is
^ SAT.SUN ' ^ B = ARCCOS (2-39) R SAT.SUN
The Z GPS axis (see Figure 1) is intersection of the sun-satellite-earth plane and the satellite orbit plane, and is positive towards the earth. The \ aps axis is in the sun- satellite-earth plane and is perpendicular to Z aps . The Yaps axis is perpendicular to the sun-earth-satellite plane and completes a right-hand system. Also, twice a year the sun
lies in the orbit plane. Then the Z aps and R s a t .s u n are parallel when
B = 0° and #=180°. At these positions the cross product Z craxR S4mv is zero.
Here, the XGPS and Yaps axes lie in the plane perpendicular to the Zaps axis and
R S.4T,SUN vector, but their directions in the plane are not defined. Thus, the entire acceleration occurs along the Z aps axis 29
GPS
o Sun
GPS 7 GPS
Figure 1. GPS Spacecraft Coordinate System
The original ROCK models were developed by Porter at Rockwell and are referred to as the ‘standard’ model (Porter, 1976). Upon implementation of these models during the 1980’s, practitioners found that the model underestimated the force by about 20% along the \ GPS and Z GPS axes. About 5% of this was due to thermal radiation, the standard model included only visible light effects, radiation and neglected infrared radiation. About 9% percent was due to transient effects that frequently occur with recently launched satellites. These effects are now thought to be outgassing of water vapor from the Multi-Layer Insulation (MLI) (Fliegel et al., 1992). The effects decay and reach equilibrium in 18 to 22 weeks. The remaining 6% was due to a plume shield that was added to the satellite after release of the ‘final’ drawings upon which the
Porter model was based. To address the deficiencies of the standard model, Fliegel, Gallini and Swift developed a thermal model for BlockI and Blockll satellites that includes the plume shield (Fliegel et al., 1992) The transient effects due to outgassing still occur, but reach equilibrium after 18-22 weeks. There remain several known error sources in the thermal model as outlined by Fliegel et al. (1992). However, the accuracy of their thermal model is now 2-3% of the total observed solar radiation pressure effect. The remaining error is primarily manifested as a scale change, and not a change in the shape of the approximation functions. Thus, these effects are adequately modeled by estimating a scale factor for the ROCK model accelerations. Additional problems arise during the eclipse of GPS satellites. These are addressed later in Section 2.4.4.
For Block I satellites, the thermal model (T10) force in 10“5 Newton are (Fliegel et al., 1992)
X GPS = -4.34 sin ( B) + 0.1 sin (2 B +1.1) - 0.05 cos (4 B) + 0.06 (2-40a)
Z o ps = “ 4.34 cos (B) + 0.17 sin (IB - 0.4) - 0.05 sin (4 B) - 0.06 (2-40b)
For Block II satellites, the thermal model (T20) force in 10 ’ Newton, (Fliegel et al.
1992) are
XG/,, = -8.96 sin ( B) + 0.16 sin (3 B) + 0.10 sin (5 B)- 0.07 sin (IB) (2-4la)
Z ow = -8.43 cos (B) (2-4 lb) where B is the is the sun-satellite-earth angle. The IERS standards recommend the T10 and T20 solar radiation pressure model for GPS orbit modeling. 31
2.4.2 Y-Bias Acceleration
In addition to the deficiencies described above, practitioners observed an acceleration of about 10x 10 9n?/.v2 along the \ ars axis This Y-bias acceleration has the form (Feltens, 1989)
Yops=v2 Cr Pj ;Q (2-42) where v is the eclipse factor defined before,
CR = reflectivity constant of the solar panels,
P = solar radiation pressure,
A = area of the solar panels,
M = the spacecraft’s mass,
Q = sum of the angles dx,d 1, and dr
d, Q - d i + — + c/3,
dx - misalignment angle of the solar sensor,
d2 = angle of one solar panel with respect to the other,
di = yaw altitude control bias.
Solar panel alignment errors of Q= 0.5° to 1.0° can cause such accelerations (Fliegel et al., 1985). Since Q cannot be determined prior to launch, the Y-bias is typically estimated as an additional acceleration.
The transformation of accelerations from the GPS body axis system to the TOD inertial frame is 32
V V ^ ron X GPS (0 YGrs(0 ^ GPS (0 GPS v Y 1 TOD = ^ors (./) YcrsU) ^G /,.f(-/) GPS (2-43) 7 7 ^ TOD * g p s (*) Y„rs(k) Zarsi*)- L ^ GPS J where X GPS(i) etc. are the coefficients of the TOD inertial frame unit vectors computed in equation (2-38).
Some additional concerns must be addressed when implementing the ROCK models. First, it is assumed that the ROCK model uses the solar pressure at 1
Astronomical Unit (AU). Since the satellite-sun distance varies from 1AU these forces should be scaled by The dominant variation is the geocentric position R-R c vector o f the sun, Rs. The earth’s orbit about the sun can vary by -1.7% of 1AU over
2 R one year The position of the satellite has a fractional variation of only —-, less than
0.04%, over the 12-hour sidereal orbit period, which is negligible. The appropriate scale
. K factor is . Additionally, there is a dependence on the 11-year solar cycle. If the k solar flux variation is obtained from other observations, GEODYNII can accept such a new solar constant value more appropriate for the time period.
Finally, the associated partial derivative for the variational equations of ROCK4 model may be approximated. For the spherical satellite model in equation (2-33) the associated partial derivatives of the acceleration with respect to position in the inertial frame were computed by post-multiplying the norm of the inertial acceleration in equation (2-36) by a linear operator in equation (2-37). The acceleration in the inertial 3.1
frame for the cannon ball model is relatively easy to compute since the cross sectional
area of the satellite does not depend on the position vectors. For computing the
acceleration on GPS satellites, the cross sectional area does depend upon the position
vectors. The partial variational derivatives should include this dependence. An
approximation for the ROCK4 variational partial derivatives can be computed neglecting
the dependence of the cross sectional area on the position vectors and including only the
more direct dependence on the position vectors. This is obtained by computing the
ROCK accelerations in the body frame, rotating to the inertial frame, taking the norm
and post-multiplying by the equation (2-37).
2.4.3 Earth Radiation (Albedo)
Rizos and Stolz (1985) estimated that the earth albedo effect causes a 1-2 meter
effect on GPS orbits after two days. This is in agreement with Landau (1988) who states that the acceleration is about 4.0x 10~l0m /s 2. Fliegel et al. (1985) show that it is at
most 2% of the force of the direct solar radiation pressure. The effect is about the
magnitude of the ocean tides, but the IERS standards currently recommend that it be
ignored for GPS.
2.4.4 GPS Eclipse Effects on Acceleration Modeling
The eclipse season effects are the focus of active research as they affect GPS
orbits at the centimeter level Eclipse seasons occur twice a year each lasting about 30 to 40 days The eclipse period occurs twice a day, once per revolution, and each of 34 these periods can last up to one hour. The primary concerns are the discontinuity introduced into the integration, thermal radiation during the eclipse, solar panel misalignment following eclipse, earth sensor misalignment prior to the eclipse, and thermally shocked outgassing.
An umbra and penumbra model is used to determine the amount of sunlight that may be eclipsed by the earth or moon. During an eclipse, the satellite passes from sunlight to shadow or vice versa in a continuous manner. The sunlight-shadow boundary is not a discontinuity. But, due to be the presence of the atmosphere the sunlight is diffused, smoothing the transition.
Typically, GPS satellite orbits are integrated using a 6 minute step size. The maximum time in shadow for GPS satellites is approximately one hour and is often less than this. Although any transition is continuous, it can occur between the discrete integrator steps. So, it is very easy to ‘step’ over the transition. Thus, to the integrator, this appears as a discontinuity relative to smoother variations that occur for most gravitational forces. These discontinuities can be handled more effectively by specifying a smaller integration step size. Of course, the disadvantage is that all the other accelerations are now integrated at the same small step size at an increased computational cost. At the extreme, the small step size may even cause the accumulation of round-off errors causing the integrator to diverge. Rowlands et al
(1994) have reformulated the equations of motion to integrate the solar radiation pressure separately ffom the other equations A small step size can be used for the solar 35 radiation pressure, while maintaining a larger step size for the other accelerations. This technique may be incorporated into later versions of GEODYNII (Rowlands, 1994, private communication)
Solar panel misalignment occurs during an eclipse. The panels are at rest during the eclipse and then ‘catch up’ with the sun in about 20 minutes after the eclipse.
According to Fliegel et al. (1985) about 1/4 of the solar force on the SV is acting along track during these 20 minutes, causing a velocity increment of approximately
2 x 10 ’m /s .
When the satellite enters shadow then the direct solar pressure force is attenuated. However, the satellite continues to emit thermal energy for some time and thus the satellite is accelerated by this thermal force. Relative to steady-state conditions, this produces a small impulse away from the sun upon entering the eclipse and toward the sun on leaving the eclipse. This results in 1.0x ]0 9m /s 2 acceleration acting for the
20 minutes it takes the panel to reach thermal equilibrium. However, Gallini computed that the cumulative effect over 7 days is only about 0.4 m (Fliegel et al., 1992)
These effects are too small to account for the effects seen at the eclipse period
Fliegel et al. (1985) proposes two other possible sources for this. The first effect occurs just prior to the eclipse when the sun sensor mistakes the earth’s rim as the sun and incorrectly points the earth sensor. This Y bias acceleration was observed by Swift
(1987). The second effect occurs with a new satellite when the outgassing can be amplified by the thermal shock induced by the eclipse 2.4.5 New GPS Attitude Model
The eclipse season anomalies may finally be satisfactorily resolved. According to
Bar-Sever (1994), the problems arise from residual electronic noise in the sun sensor.
Normally the sun sensor circuitry initiates a yaw driving mechanism to rotate the solar panels to align normal to the sun. On entering an eclipse, the sun sensor circuitry is not set to exactly zero, because as an analog system it contains some residual noise This electronic noise behaves randomly sometimes being positive and sometime negative and for each case the noise initiates the yaw driving mechanism to rotate the solar panels toward the sun in a different direction. This residual noise can drive the yaw at a full rate of 0.12 deg/sec and can rotate over 360 degrees during the 50 minutes of an eclipse. On leaving the eclipse, the satellite can be in yaw error by 180 degrees and take about one hour to correct. Since neither the rate nor direction is known, it is not easily modeled.
To remedy the problem, on June 6, the Air Force began to basis the yaw +0.5 degrees on all satellites except SVN 11, 14, and 20. This bias overrides the noise and forces the direction of the yaw driving and the rate is estimated as part of the solution. This approach requires the changing of orbit generation software, since the yaw bias changes the dynamics, the measurement modeling software, since the antenna phase bias is changed; and the estimation software, since new parameters must be estimated. The necessary modifications have not yet been made to the GEODYNII software. 2.4.6 Resonant Acceleration Model
An alternative approach to accommodating the small mismodelled or unmodelled accelerations has been developed by Colombo (1989). He recognized that although the forces that affect the GPS satellite are complex, they occur periodically and resonate with the orbit. This arises from the constant pointing of the solar panels to the sun and the antennae towards the earth as the satellite orbits the earth. Since only the departures from the adopted force model are considered, a uniform spherical earth and circular GPS orbits are acceptable approximations. This leads to a set of linearized differential equations for the accelerations in the across track, along track and radial (x, y, z) directions known as Hill’s equations
A x = -n\ Ax + — A fx (2-44a) tn
Ay = -2nl Az + — A f (2-44b) m '
1 Az = -3/;0 Az + 2n0 Ay + — A f. (2-44c) m where n0 is the mean orbital frequency,
m is the GPS satellite mass,
A f x, A f y, A f. are the small mismodeled forces in x, y, z directions, respectively,
Ax, Ay, Az, and Az are the departures in coordinates and velocities from the
nominal values. 38
The solution to equation (2-44) is given by Colombo (1989, 1986) and Kaplan
(1976). Colombo then constructs the Fourier transforms of the acceleration error and the corresponding orbit error and identifies the resonant frequency for the x axis at n0 and at they' and z axis at 0 and //„. Thus, the accelerations can be represented as the sum of the Fourier components as
= AXi cos n0t + Bx sin / y + Cx> (2-45) where / = 1, 2, 3 for x, y, z,
Ax<, Bx , and Cx are parameters to be estimated in the adjustment.
This model is available in GEODYNII.
2.5 Ocean Tide Model
The modeling of the ocean tides of the earth is a difficult problem due to the complex hydrodynamical response to the tidal forces. Using a combination of a numerical solution of the Laplace’s Tidal Equations and tidal gauge data, Schwiderski
(1983) developed the ocean tides due to the 11 main tidal constitutes (Ssa, Mm, Me, Ql,
01, PI, Kl, N2, M2, S2, K2). His model gives the amplitude and phase on a 1" xl" grid over the surface of the earth. At any point on the ocean, the tide can be computed as the summation over all eleven tidal frequencies of the amplitude and phase as a function of the Doodson astronomical argument. The potential due to this water mass is computed by treating the tidal layer as a thin film of varying height over the spherical earth. This differential mass is integrated over the sphere to give the time varying potential due to 39 the ocean tides. The details of this approach are given by Goad (1977). Following a similar approach, Christodoulidis and Smith (1988) developed an expansion for the ocean tide potential that is used in GEODYNII
^ 1 "4” A" * ^ ^ ^ ^ * 1 U OCEAN ~ I I 4nG Rp0 — - 4 C " J - J PIq(s\n
R is the mean earth radius,
p 0 is the average density of sea water,
k\ is the load deformation coefficient,
Cjqj is the surface spherical harmonic of the tidal amplitude expansion,
<}),X,r are the geocentric latitude, east longitude and geocentric radius of the
satellite position,
a'Ut f is the ocean angular argument, given in Christodoulidis and Smith (1988).
For GPS altitudes of ~3 earth radii, only the dominant tide constituents are considered. The side bands typically have amplitudes of about 10% of the dominant tide.
Thus, they can be excluded when only GPS satellites are modeled. The ocean tide amplitude and phase up to degree 6, order 2 for the 11 main tidal constitutes are modeled. If low orbiting satellites are tracked with GPS, then additional lines, side bands and higher degree and order terms should be considered for the lower satellites.
Using equation (2-46) the ocean tide acceleration on the satellite can be computed using equations (2-15, 18, and 19). The associated partial derivatives for the 40 variational equations are not computed. They can be approximated if the Eanes et al.
(1983) implementation is used, as discussed at the end of the solid earth tide section.
2.6 Solid Earth Tide Model
The tidal attraction of the sun and moon on the elastic earth causes the masses to respond periodically by deforming and thus changing the earth’s geopotential and displacing the station position. The solid earth tide model depends upon the adopted lunisolar ephemeris and the (elastic) earth model. Historically, the mean elements from
Brown’s lunar theory were used in Doodson’s, and Cartwright and Tayler’s expansion of the tidal potential (Doodson, 1921; Cartwright and Tayler, 1971, Cartwright and Edden,
1973). More recently, osculating elements from a planetary ephemeris such as JPL
DE/LE 200 have been used in the development of Christodoulidis and Smith (1988).
Wahr (1981) used an elastic earth model 1066A in the development of his nutation series. For an elastic earth model, the deformation and the resulting change in geopotential are proportional to the frequency of the Doodson constituents. This frequency dependency complicates the evaluation of the geopotential change. The change is dominated by the response using a nominal frequency independent Love number k2,
(2-47) where k2 is the frequency independent (nominal) Love number,
Re is the mean earth radius, 41
r is the geocentric position vector of the satellite, r = |r|,
is the geocentric position vector of the disturbing body (j=2) moon, (j=3) sun,
R.j = R }
GMj is the gravitational parameter of the disturbing body.
This contains the full spectrum of Doodson constitutes for (l,m=2,0;2,l;2,2) tidal potential terms. A second step uses the frequency dependent Love numbers to compute corrections to the geopotential change computed in the first step.
Christodoulidis and Smith (1988) developed the solid earth tide potential as
- (3 -m Y R V Us, = L fk tA,Gj — L /'2m(sin0)cosaf (2-48) / V r where As is the equivalent of a Doodson coefficient,
Gd is the equivalent of a Doodson constant,
a j E is the equivalent of the Doodson astronomical argument.
To reduce the number of Doodson tidal constituents to correct, a rationale was adopted to correct only those terms which contribute significantly to the geopotential.
In other words, the Doodson frequencies whose amplitude were below a certain threshold could be neglected without an appreciable change in the geopotential.
The eight Doodson tidal frequencies exceeding the cutoff are 145.555, 163 .555,
165.545, 165 555, 165.565, 166.554, 255.555, and 273.555 (McCarthy, 1992) For step one, GEODYNII uses equation (2-47) and the planetary ephemeris. For step two, 42
GEODYNII uses Christodoulidis’s (1988) formulation given in equation (2-48) The solid earth tide accelerations are computed by applying equations (2-15, 18, and 19)
GEODYNII includes associated partial derivatives with respect to tidal coefficients and Love numbers, but not with respect to position. Because of the small magnitude of the acceleration, this is appropriate, since when these tidal constituents and
Love numbers are estimated they contribute more significantly to the variational partial derivatives.
2.6.1 Treatment of Permanent Tide
The IERS standards adopt the implementation of Eanes that represent the geopotential change as a periodic correction to the harmonic coefficients to the geopotential model. The changes in the normalized second degree geopotential coefficient is derived by Sanchez (1974) and given by Eanes et al. (1983) and McCarthy
(1992)
1 7(3 3 CM. I v
A(- = v s k'G ir e h ~ i f 7- ( sin^ ) ( 2 - 4 9 ) where k2 is again the frequency independent or nominal second degree Love number,
Re is the equatorial radius of the earth,
GMe is the gravitational parameter for the earth,
GMj is the gravitational parameter for the moon (j=2) and sun (j=3),
r is the distance from geocenter to moon or sun, 43
Greenwich) of the sun or moon, respectively.
The mean of equation (2-49) is not necessarily zero This bias is called the permanent tide or zero frequency tide. The nominal value k2 is not appropriate to apply to the permanent tide so the permanent tide contribution should be included in C20 and not be a part of the periodic correction AC20. The GEMT3 model C20 does not contain the permanent tide. The IERS standards recommend that the permanent tide be included in the definition of C20(GEMT3) McCarthy (1992)
C20(Gm T3) = -484.16499 x 10 6 - (1.39119 x 10 8) x 0.30
= -484.169164x10 6 (2-50)
Equation (2-48), used in GEODYNII, computes the periodic effect and the permanent tide together, i.e. the mean of equation (2-48) is nonzero For GEODYNII the permanent tide should not be included in C20(GEMT3). Since GPS is not used currently to estimate k2, then it is permissible to exclude the permanent tide from C’20(GEMT3)
2.7 Station Displacement Due to Pole Tide
If the rotation axis and the ITRF pole always coincide, then the rotational potential at a site on the earth’s crust would remain constant. However, this is not the case. The rotation axis describes a quasi-circular path of approximately 10 m radius and with a frequency of approximately 430 days, the Chandler frequency Thus, a site experiences a change in rotational potential as where co2 is the rotation rate of the earth,
r is the radial distance to the station,
6 is the co-latitude of the station,
A is the east longitude,
mx,m2 are the polar motion components.
The displacement of the site on an elastic earth is
A V h2co2r 2 i \ Sr = h2 = - — sin20[XpcosA-YpsinA) (2-52a)
l2 S A V (o2r 2 / \ S„ = — _ = -l2 coslOyXp cosX-Yp sin A j (2-52b)
L 1 S A V L(o2r 2 , \ S, = r ~ r c . = ------cos# A\, sinA-T.cos/l (2-52c) g sin 6 8 X g v 1 p ’ where h2 is the Nominal second degree Love number,
/2 is the Nominal second degree Shida number,
X p =mx- X (2-53a)
Yp =-m2- Y (2-53b)
The 6-year average position ofthe rotation axis may not coincide with the ITRF reference pole. Thus, using m, and m2 in equation (2-53) above would give corrections that have a nonzero mean This would bias the station coordinates. With X and Y as the mean position of the rotation axis, the average of this periodic correction is zero
Any permanent effect is retained in the site coordinates. 45
An alternative use of equation (2-53) arises when a Terrestrial Frame other than the ITRF is used. In this case, X and Y describe the position of the alternate frame relative to the ITRF. The IERS polar motion values relate the ITRF reference pole and the Celestial Ephemeris Pole (CEP). To use the IERS values of w, and m2, the values of
X,Y must be subtracted to make the X Yp values consistent with the alternative terrestrial frame.
2.8 Station Displacement Due to Solid Earth
The displacement of the station due to the solid earth tide is handled in the same two step process as was used to compute the solid earth tide change in the geopotential.
First, the station displacement is computed using frequency (Doodson) independent second degree Love (/f2) and Shida (/2) numbers:
3 GM r 4 (h V h i ) J f / \1 ( h , 1 / \2 h . I 3/, R.-r R + 3 — - / , R , - r - — | (2-54) I 2' 1 /J j 7--2 [ g m e r ) J . 1 2 2 A J ' 2 . where GM} = gravitational parameter for the Moon (j=2) or the Sun (j=3),
GMe = gravitational parameter for the Earth,
R, = unit vector from the geocenter to Moon or Sun, R, = R 7 ,
r = unit vector from the geocenter to the station,
h2 = 0.609 is the frequency independent (nominal) second degree Love number,
/2 = 0.0852 is the frequency independent (nominal) second degree Shida number. 46
For the second step, the frequency dependent Love and Shida numbers are used to correct equation (2-54). Only frequencies for which the correction exceeds 0.005 m are considered Thus, just the kx Frequency Love number correction to the radial station direction must be computed
(2-55) where 8 hk< = hk (Wahr) - h2 (nominal) = -0.0887,
Hki = 0.36878m is amplitude of the term (165.555),
(j) = geocentric latitude of station,
X = east longitude of station,
6ki = tide argument = r + ,v = 6 g+ n
6 g = Greenwich hour angle.
The correction is a maximum at
The correction computed using equation (2-55) has a nonzero average over time.
Thus, the correction contains the permanent tide effect and applying the correction removes the permanent tide from the site coordinates. The Chapter 7 IERS standards
(McCarthy, 1992) recommend removing the permanent tide from the correction, i.e. leave it in the site coordinates. However, in the Introduction, the IERS standards recommend treating the permanent tide as a correction to the site coordinate, and override the recommendation given in Chapter 7. 47
2.9 Ocean Loading
The displacement of the oceans due to the tidal attractions of the moon and sun cause a periodic loading and relaxing of the crust. The tracking stations fixed to the crust experience a periodic radial and tangential displacement relative to the ITRF frame.
The range of this motion is about 5 cm. The local station displacements, radial, east- west and north-south for a number of sites were computed by Schemeck (1991). He used the Schwiderski tide model, Green’s functions for an elastic earth on sites whose
300 km cap is probably on ocean lithosphere, and Green’s functions for a visco-elastic structure for stations on continental shelves. The coast lines were taken from the
ETOPO05 data set of the National Geophysical Data Center. The harmonic expansion for the radial and tangential displacements is given for each o f the 11 main tidal constitutes by Schemeck (1991) as
AC = £ / ; Ag cos fa t + Xj +»7 ~ % ) (2-56) where AC are the displacement components, radial, positive up; tangential east-west,
positive west; tangential north-south, positive south,
Ag ,OtJ are the amplitude and phase for each tidal constitute at the specified
location as computed by Scherneck and given in IERS Technical
Note 13, Table 9.1 of McCarthy (1992),
fj,u, are slowly varying functions of the lunar node and perigee as given in
Table 26 of Doodson (1928),
o)j is the phase rate of change, frequency, for each tidal constitute, 48
X} is the astronomical argument at beginning of day t=0,
1 is the time from beginning of day.
In GEODYNII, the tangential north-south direction is positive north and the tangential east-west direction is positive east. Also, equation (2-56) is decomposed as
AC= Za,cos( bJ=fJAg sin(-M,+J. CHAPTER III OBSERVATION MODELING In this chapter, the magnitudes of effects that influence the GPS observables, and in turn the orbit determination, are examined. The various instrumental and enivronmental systematic effects on the observable can be expressed as ( h o r r = Oraw + ^ AS + ^ . V 4 + ^ S.4T ^ STA ^ tropo ^ 1 0 no ^ ^ REL. ^ A f P ^ AS’T where Ocorr is the corrected GPS observable, Orm. is the observed GPS observable that contains the systematic errors, A^. is the effect of anti-spoofing (AS), A&l is the effect of selective availability (SA). A7x.(7. is the satellite clock error, A i s the receiver clock error, AIWpo *s the tropospheric refraction, Awno is the ionospheric delay/advance, ARm is the effect of the relativistic propagation, A klp is the effect of satellite and receiver multipath, A _, V7. is the effect of antenna phase bias In Table 2, the expected magnitudes of these systematic effects and the modeling approach to mitigate them are given 49 50 Table 2. Systematic Errors Affecting GPS Observables Source Effect Modeling Technique ^ AS no Pcode Pcode not implemented for orbit determination ^ SA 400 m double difference, estimate orbit W SAT 1 x 10”12'"13 double difference, estimate satellite clock ATSTA 1x10" double difference ^ tropo -2 .7 m modified Hopfield model, estimate zenith delay ^ wno -20 m form ionospheric free combination 18 mm double difference to reduce effect ^ MT <1 cm good site selection, robust antenna ^ ANT 0.03-1 m correct for offsets, phase centers, orientation 3.1 The GPS Observables The launch of SVN36/PRN6 on March 10, 1994, brings the number of BLOCKII satellites to 24. This is a full constellation of GPS satellites as planned by the U.S. Air Force. This allows the Operational Control Segement (OCS) to issue a full operational capability (Gibbons, 1994). The satellites have 12 sidereal hour orbits at a height of approximately 20,000 km distributed in six orbit planes spaced 60“ apart in right ascension and inclined at approximately 55° from the equator (Green et al., 1989). The constellation is divided into 21 active satellites and three in orbit spares. Each Blockll satellite carries two rubidium frequency standards and two cesium frequency standards with an accuracy of 1 x 10 12 to 1 x 10“u (Van Melle, 1990). Each satellite broadcasts navigation data on two L band frequencies derived from a base frequency of 10.23 MHz f Ll = 1575.42MHz = 154x10.23MHz (3-2a) f LI = 1227.6MHz = 120 x 10.23MHz (3-2b) 51 Additional signals are modulated on these frequencies. First, the course acquistion code (C/A) with a period of ~1 millisecond is modulated on LI Second, a precise code (P) with a period of about 267 days is modulated on both LI and L2 (Spilker, 1978). Additionally, the satellite ephemerides, GPS time, satellite clock characteristics, and transmitter status information are modulated on the navigation signal at a rate of 50 bits per second for 30 seconds tranferring 1500 bits of data in a frame (Van Dierendonck et al., 1978). An additional Y-code is modulated on the P-code to deny access of P-code to all but authorized users of the Precise Positioning Service And the accuracy of the Standard Positioning Service is intentionally degraded to deny 30 m absolute positioning to the standard users. Intentional system degradation will be discussed in section 3.2. The pseudorange observables available are on LI C/A code (Cl), LI P-code (PI), and L2 P-code (P2). The pseudorange is represented as c[tR - 1,]= p + cAt-cAt^. + £ (3-3) where tH, At are the receiver clock time and its correction to GPS time, respectively, t,, Atn. are the satellite clock time and its correction to GPS time, respectively, p is the geometric range, s is the noise. Additonally, the phase observable is available on LI and L2. The phase observable is represented as = t S('«)"-J -tM + K+e (3-4) 5 2 where X is the LI or L2 wavelength, N r is the unknown number of integer cycles. To measure the time of Cl, PI and/or P2 arrival, the receiver replicates the code and by cross correlation of the two data streams finds the peak Equations (3-3) and (3-4) define observables between one satellite and one receiver. The double difference (DD) observable is a linear combination between two satellites and two receivers DD =[(S1 -)• Tl) - (S2 -> Tl)] -[(SI -> T2) - (S2 -► T2)J (3-5) where (SI —> Tl) etc. are the satellite-station range or phase observations (e.g., satellite 1 to station 1). 3.2 Selective Availability (SA) and Anti-Spoofing (AS) The Department of Defense (DoD) implemented SA to reduce the accuracy of single point positioning using the Standard Positioning Service (SPS) from 30 to 40 m to approximately 100 m (Georgiadou and Doucet, 1990). Specifically, during 95% of the time the user can acheive ±100 m real time point positioning. The errors are no more than 300 m for 99.9% of this remaining 5% of the time. This is accomplished by dithering the frequency standard, the so called S process and by the addition of bias terms to the broadcast ephemeris elements, the so-called £ process These directly effect the the accuracy of real-time point positioning as is DoD’s intention. The effect is 5 3 diminished for static relative baseline determination by using differenced data and precise post-fit orbits (Rocken and Meertens, 1991). When post-processing double differences for orbit determination the dithering is mostly cancelled and a precise orbit without the effects of SA can be determined (Bertiger and Yunck, 1990). There is an additional encryption of a Y-code onto the P-code signal. This effectively denies the P-code pseudorange to unauthorized users. It is a problem for kinematic applications that rely on a combination of code and phase. But most orbit determinations use the phase range double difference and do not rely directly on the code observable. 3.3 Satellite and Receiver Clocks The satellite clock has a frequency based on rubidium and cesium standards These are typcally stable to about 1 x 10 12 and 1 x 10 l3, respectively. However, these clocks continue to drift and are reset periodically by the OCS The corrections are broadcast to the user in the navigation message (Spilker, 1978). The receiver clock is typically a quartz clock that has a stablility of 1 x 10 11. The receiver clock offset is estimated using a static pseudorange solution that solves for the clock offset at each epoch and the static position of the receiver. The GPS satellite and receiver are moving at different velocities in a quasi-inertial frame and are experiencing different gravitational potentials. Thus, the two clocks are running at different rates. From general and special relativity, the two clocks can be related by 54 T r GM 1 w , v 3 GM^ Ts = T+~1 , +^n2(4a+yL)-i~-\-zr{*s # v J (3-6) V' ^ l a ) c where ts is the satellite clock time, t is the earth bound station clock time, c is the speed of light, GM is the gravitational parameter of the earth, is the magnitude of the position vector of the earthbound station in geocentric coordinates, O is the rotation rate of the earth, x^.A is the x component of the earthbound station in geocentric coordinates, is the y component of the earthbound station in geocentric coordinates, a is the semi-major axis of the satellite orbit, xA. is the position vector of the satellite in geocentric coordinates, v 4 is the velocity vector of the satellite in geocentric coordinates. The second term in equation (3-6) causes a secular offset that can be corrected by setting the satellite clock frequency lower by 4.4x1 O'10 MHz from the nominal frequency of 10.23 MHz (Spilker, 1978). The third term causes a periodic variation that has a maximum magnitude of 14 m for single point positioning, but cancels out with between-station differences. (Zhu and Groten, 1988). 3.4 Propagation Medium Any medium that is not a vacuum contains free electrons and ions that interact with electromagnetic energy that passes through it. If the wavelengths of the transmitted energy are very different from the orbits of the electrons of the ions, the only effect is to slow the propagation because of the finite time delay caused by the shifting of the electron configurations through the medium. This is a non-dispersive medium. If the wavelengths of the propagated waves are close to the wavelength of the partial medium, then near resonance exists which affects the propagation differently. This is a dispersive medium. A constant wavelength phase is affected by the phase velocity (Vp), and a wave packet or group formed by waves with close but different wavelengths is affected by the group velocity (Vg). These are related by (Wells, 1974) dV W-*!* ( 3 ' 7 ) where A is the wavelength considered. And, to a first-order approximation, the refractive indices are related as dn <3 - 8 > where ng, np are the refractive indices of the group and phase velocities, respectively. For microwaves, the ionosphere is a dispersive medium, and the troposphere is non-dispersive. For any non-dispersive medium, V = V , i.e. the troposphere for microwaves. The phase velocity (LI and L2) can exceed the speed of light in a vacuum; 56 the group velocity (C1,P1, and P2) cannot. For GPS, the code is affected by the group velocity, the phase by the phase velocity. 3.4.1 Ionospheric Refraction The ionosphere is the region of the atmosphere extending from about 50 km above the earth outward to nearly 1,000 km. It contains free electrons and ions that have been released from ionized atmospheric gases by the sun’s ultraviolent radiation. These electrons and ions are also influenced by the geomagnetic field (Wells, 1974). The ionized content in the atmosphere varies according to the solar radiation fluctuations, amount of ionized gases in the atmosphere and the activity of the geomagnetic field of the earth (Bassiri and Hajj, 1993). For GPS signals of ~1 GHz the ionosphere is a dispersive medium. To a first-order approximation the refractive index for the phase velocity is (Hartmann and Lettinger, 1984) C-n. / 2 where (’ - 40.3 is a constant, ne is the electron density, f is the signal frequency, and from equation (3-8) the refractive index for the group velocity is C-n. r The effect on GPS code signals can be up to 100 meters (Klobuchar, 1991). Double differencing the phase data can reduce the effect somewhat, but differential ionospheric effects over, for example, a baseline of 40 km can result in baseline errors of up to 10 cm (Kleusburg, 1986). The GPS broadcast ephemeris contains a model of the ionosphere that accounts for 50-75% for the ionospheric effect at any one time (Klobuchar, 1987). However, these are not satisfactory for geodetic positioning at 1x10 7 or better. The first-order effects of the ionosphere can be removed by using the frequency dependence of the refraction of the GPS LI and L2 signals in the dispersive ionosphere. The double differenced P-code pseudorange is DD(P\)= p*+ J^T + £ x (3-11) J i 79/J>(P2)= + e 2 (3-12) . / 2 where p * is the geometric range difference, A —j is the ionospheric delay on PI and P2 respectively, J t e , is the measurement noise on P 1 and P2 respectively. Now a linear combination that is free of the first-order ionospheric effects is obtained by applying the conditions DIJIF(P) = a ,IJD(P 1) + a 2DD(P2) (3-13) where a , and a 2 satisfy the conditions, 58 and a ,+ a 2= 1 (3-15) Solving for a , and a 2 one gets ■ 2 a II f7 = 2.546 (3-16) / , 2 - n f 2 a 2= -y-;—? =-1.546 (3-17) ./I •/ 2 For the phase observable the LI and L2 double difference combinations are DD(L\) = Z^ - + - ^ + N i + e i (3-18) /.i .//i 0/7(L2) = - ^ 1- + - 7 - + W2 + f 2 (3-19) ^ / 2 A; where —— and —— are the phase advance effect of the ionosphere on the LI and L2 J 21 .2 22 frequencies, respectively, ,N 2 are the double difference ambiguities on LI and L2, respectively, c , is measurement noise on LI and L2 frequencies, respectively. Here the conditions applied are DIJIF(L) = p ,DD(LI) + p 2DIJ(L2) (3-20) where P , and p 2 satisfy the conditions I I + 0 = 0 (3-21) yfn J v/ , 2 ) The second condition assigns the wavelength of LI as the wavelength of the ion-free combination. The higher order ionospheric effects remain after forming the linear combinations above and can result in errors of the distance from receiver to satellite of 2-3 cm (Gu and Brunner, 1990). Brunner and Gu (1991) report that the second and third order effects result from the interaction of the geomagnetic field on the ionospheric field and the LI and L2 signal propagation along slightly different bent ray paths, and neither propagate along a straight line as is assumed to remove the first-order correction. Brunner and Gu give an expanded form of the refractive index that depends on the geomagnetic field and terms that account for the difference in ray paths. They claim that their model is accurate to a few millimeters. Bassiri and Hajj (1993) give an approximation to Gu and Brunner’s model that accounts for the second order effect, but neglects the third order effect and the difference in ray paths by assuming a straight line path. The Bassiri and Hajj approximation is less complicated to evaluate than Gu and Brunner’s model. Bassiri and Hajj claim their model accounts for 90% of the effect given by the more exact formula of Gu and Brunner. The IERS standards make no direct recommendations. But, the IGS centers routinely use the ion-free combination given in equation (3-20) (Zumberge and Goad, 60 1993). The solar cycle is in downward cycle until about the year 1996, then it turns upward. Brunner and Gu (1991) predict a maximum second and third order effect of nearly 10 cm even at elevations angles of 30 degrees during extreme conditions that could occur during a solar maximum. It is likely that in the near future ionosphere modeling of the second and third order effects will be necessary to remove the solar cycle signature which could corrupt the long term repeatability of baseline coordinates. The models given above assume the ionosphere to be stable during the transit of the signal from a satellite to the receiver. In the equatorial and polar regions the ionosphere fluctuates rapidly causing scintillations. Scintillations can be refractive causing changes in the direction and propagation, or diffractive corrupting the amplitude and phase of the signal. These result in receiver losses of lock (Tseng et al., 1989) The amplitude scintillations can be monitored by plotting the receiver outputs of the signal to noise ratio (S/N). The phase scintillations can be monitored by plotting the gradient of the ionospheric refraction over two epochs at about 1 minute intervals (Wanninger, 1993) 3.4.2 Troposphere Up to a height of 50 km above the earth’s surface and below the ionized portion of the atmosphere is the neutral atmosphere. It is electrically neutral and non-dispersive for frequencies below 30 GHz. The troposphere extends from the earth’s surface to a height of approximately 40 km. It contains approximatley 80% of the mass of the neutral atmosphere. Temperatures decrease with height in the troposphere. Above the troposphere at a height of approximately 40 km is the tropopause. Here the air 61 temperature is nearly constant. Above the tropopause is the stratosphere which extends up to a height of approximately 50 km. Temperature increases with height in the stratosphere. Since the bulk of the atmospheric mass lies in the troposphere, the refraction is called often the troposphere refraction, though it refers to the entire neutral atmospheric refraction. The neutral atmosphere consists of dry air and water vapor. The dry air (dry) portion at a height of 40-44 km exhibits no significant variation with latitude and height (Smith and Weintraub, 1953). The water vapor (wet) at a height of 10-14 km exhibits wide variations in latitude and height. Assuming negligible signal bending, the refraction of the electromagnetic wave along the geometric path (gp) can be written as A r = 10~6 J N(s)ds (3-23) gp where N (s) = (n -1 ) x 106 is the reffactivity, n is the index of refraction. The delay is a function of the distance traveled through the atmosphere. The shortest distance is in the zenith direction. Due to the earth’s curvature the distance through the atmosphere increases with increasing zenith angle or decreasing elevation angle. The delay along the line at elevation angle ( E) can be written simply as the zenith delay mulitiplied by a mapping function that maps the zenith delay to the delay along the line of sight at elevation angle ( E) A r(E) = m(E)A r( 90“) (3-24) where m(E) is the mapping function. 62 At a height of approximately 10 km the wet content is zero, but the dry content is still about half of its surface value. Thus, it is convenient to separate the delay caused by the dry and wet content in equation (3-23) as A r = 10 6 J N d(s)cfs+ 10'6 J NJs)ds (3-25) gdp gwp where J and J are the integrals over the geometric dry path and the geometric wet gdp * » ■ / ) path, respectively, N d is the dry refractivity, is the wet refractivity. Likewise, equation (3-24) can be written in terms of zenith delays and mapping functions Ar(E) = md(E)*rd(90-) + mw(E)Arw(90'-) (3-26) where A^(90) and ArM (90°) are the zenith path delays (Elevation angle = 90 ) for the dry and wet effects, respectively, md(E) and mw(E) are the mapping functions for the zenith path delays of the dry and the wet effects, respectively. Thayer (1974) assumes an ideal gas, and gives the refractivity in terms of a hydrostatic component that is mostly predictable and the water vapor that remains highly unpredictable. Saastamoinen (1972, 1973) derives expressions for the hydrostatic zenith delay and water vapor zenith delay from the equation given by Thayer (ibid.). Hopfield (1969, 1971, 1977) adopted a quartic model to represent the refractivity of the dry and wet portions from which the ‘dry’ and ‘wet’ zenith delays are computed. Various mapping functions have been derived for the Hopfield (ibid.) model (Yionoulis, 1970) 6 3 and the Saastomoinen (ibid.) models. One widely used formulation is that of Goad and Goodman (1974). They represent the height, in terms of the range to the top of the ‘dry’ or ‘wet’ component and the elevation angle, and expand this as a Taylor series including terms up to second order The equations for the ‘dry’ and ‘wet’ zenith delays from the Saastamoinen and Hopfield models are equated, and after a substitution into the mapping funcion, the results use the Saastamoinen values for the zenith delays. The modified Hopfield model is used in GEODYNII. The LERS recommends the mapping equations of Lanyi (1984). This mapping function was developed for use with VLBI that required more accurate mapping down to elevation angles of 5 degrees It has the form of A W = *”(£)1 sin E (3_27) where is a complicated mapping function that includes the mappings of the ‘dry’ and ‘wet’ zenith delays and ray bending. The original expression for m(E) can be found in Lanyi (1984). An updated expression is given by Sovers and Border (1987) and Sovers (1991) which corrects a sign error. In this study, the GPS data are only used when the satellite elevation angle is above 20 degrees. Kaniuth et al. (1989) report that the Lanyi mapping function argees at the mm level with a mapping function developed by Davis et al. (1985). They repeated the estimation of baselines and orbits using first the Davis and then the modified Hopfield models. They report the average difference in heights of five stations to be 7 mm using a 20 degree elevation cutoff. Janes et al. (1989) compared the Davis, and Goad and Goodman mapping functions with ray-tracing. The two mapping functions have an 64 average difference of 8 mm. They also show that ray bending is a 2-3 mm effect. Thus, by adopting elevation angle cutoff of 20 degrees, the difference between the Lanyi and modified Hopfield models should not execeed 1 cm The IERS recommends estimating the ‘wet’ component using a random walk model. This capability has been added to the filter to be discussed in the next chapter. However, GEODYNII estimates a tropospheric scale factor correction that is the combined effect of the ‘dry’ and ‘wet’. The ‘dry’ effect is stable, typically varying only about 2 cm over 12 hours, the ‘wet’ is much more variable. So, the combined estimate used in GEODYNII is dominated by the variability of the ‘wet’ delay. Current tropospheric refraction models compute a zenith delay and through a mapping function project this zenith delay along the line of site from receiver to satellite. This mapping function assumes a radial symmetry, i.e. the mapping is a function of the elevation angle only and not the azimuth. The departure of zenith delay from the model can be estimated as a parameter in a least-squares fit over some time interval, or as a time varying parameter in a Kalman filter. The question arises: is there any azimuthal dependency on the line-of-sight delay? To access this dependency, Ware et al. (1993) pointed a water vapor radiometer (WVR) along the line-of-sight to the satellite from each end of the 50 km baseline. They also modeled the line-of-sight with a tropospheric model which estimates the zenith variation as a random walk and uses a mapping function to project the zenith variation along the line-of-sight. He reports a discrepancy of up to 70% between the vertical RMS repeatability of the baseline between the WVR and the random walk model. He concludes that at some sites, assuming a radially 65 symmetric mapping can degrade the vertical RMS significantly. This suggests that the azimuthal dependence should be investigated and possibly the mapping functions modified to accommodate azimuthal dependence. 3.4.3 Relativistic Propagation The difference in the gravitational potential experienced by the satellite and the receiver, which is at a lower altitude and usually on, or near the earth, results in the wave velocity increasing as it approaches the earth. This is the so-called relativistic propagation. The effect on a phase measurement is 2GM r, +r, +rv St = In (3-28) r.+ rj-rv where rtJ = H',". rj ,r, are the geocentric positions of the satellite at transmission time /, and the receiver at reception time /,, respectively. The effect for the range is given as c8l and for a single phase measurement the maximum value is 19 mm (Zhu and Groten, 1988). The effect can be reduced, but not cancelled using station and satellite differences. 6 6 3.5 Multipath Multipath is the arrival of reflected and diffracted signals in addition to the direct signal. The effect can be as large as several tens of meters for the code measurement For the carrier phase, the effect can be up to 5 cm for highly reflective environments. With the intelligent configuration of antennas (e.g. choke rings) and the site selection to minimize the interactions from local reflective structures, the effect is reduced to approximately 2 mm. Additionally, for stationary sites, it is possible to extract multipath signals from the satellite repeat path and correct for the multipath. 3.6 Antenna Offset and Orientation The satellite phase antenna offsets from the spacecraft center of mass are given in IERS Technical Report No. 13 (McCarthy, 1992). The LI and L2 phase centers at the antenna receivers vary from manufacturer. These corrections are available from the manufacturer. The observed carrier phase depends on the orientation of the antennas of the receiver and the satellite. Wu et al. (1993) have shown that time effect does not cancel for double differenced phase observables. They report a 0.8 cm change in the length of a 4200 km baseline and a 5 cm change in the GPS orbit. The correction to the double differenced phase is A ESTIMATORS A conventional batch least-squares differential corrector algorithm includes deterministic models of the physical environment. Conventional algorithms have been used to process differenced phase and pseudorange data to determine daily GPS orbits with 20 cm consistency (Kouba, 1993). The conventional batch least-squares algorithm cannot accommodate stochastic models; only a stochastic estimation algorithm is suitable, such as a sequential filter/smoother. For this study, a conventional least-square corrector algorithm and a sequential filter/smoother are required. The National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC) orbit determination program GEODYNII was selected as the conventional algorithm. A Square Root Information Filter/Square Root Information Smoother (SRIF/SRIS) called the Ohio State University Orbit Filter/Smoother (OSUORBFS) was developed as the sequential filter/smoother. GEODYNII is the orbit determination software used by the NASA GSFC for processing satellite tracking data for the determination of the low degree and order coefficients of the earth’s gravitational field (Lerch et al., 1992). GEODYNII has been used to analyze data sets containing optical, microwave and laser ranging data to satellites covering more than thirty years. More recently GEODYNII is being used to 6 7 6 8 determine TOPEX orbits TOPEX data, with its reliance on GPS, has required refinement of GPS processing capabilities of the GEODYNII software. The SRIF/SRIS was selected because the information-related algorithms are best suited for problems involving large amounts of data that may involve almost no a priori knowledge of the model parameters. Covariance-related algorithms (UD factorization), on the other hand, are best suited for problems involving frequent solutions for the parameter estimates, scalar measurements, and nearly perfect a priori knowledge of the model parameters. The information-related algorithms seems best suited for the filtering process of the orbit determination problem. The SRIF/SRIS formulation was chosen over the UD formulation (covariance) since for GPS geodetic orbit estimation, which requires smoothing, the SRIF/SRIS is as much as 50% faster than the UD filter/smoother (Lichten and Bertiger, 1989). GEODYNII consists of over 160,000 lines of FORTRAN code. From a development point of view, it was advantageous to develop a filter/smoother software separate from GEODYNII that required only the standard GEODYNII output products. These include the measurement partial derivatives, and residuals, the variational partial derivatives, and nominal satellite trajectory. The alternative approach was to modify the GEODYNII code to incorporate the filter/smoother. This was deemed unrealistic under the limits of the project budget and manpower resources Also, GEODYNII currently cannot model the correlation among data values Differenced pseudorange and especially differenced phase are precise data types that improve the GPS orbit precision (Counselman and Abbot, 1989) To overcome this 69 limitation a correlated double difference (measurement type 87) data processing capability was added to OSUORBFS modeled after Goad and Mueller (1988) The development approach has been to interface the standard GEODYNII Variational (V-Matrix, FORT.80) and measurement partial (FORT.90) files with software modules containing the stochastic estimator, the stochastic models for random walk and first-order Gauss-Markov models, and the correlated differenced phase range processing routine. Thus, no modifications to the original GEODYNII software have been required. A schematic of the implementation is shown in Figure 2. The observational data are edited in the preprocessor and the data are passed to GEODYNII as one of its standard data types. A reference orbit is determined using GEODYNII as a batch least- squares processor and the GEODYNII output files FORT.80, FORT.90 are generated. Daily IGS Data Planetary Ephemeris j GEODYNII EOP Gravitational Reference Orbit, Residuals, Model Measurement Partial Derivatives, Variational Partial Derivatives SRIF/SRIS Refined daily orbit Figure 2 Flowchart Showing the Operations Involved in the Described Procedure. 70 These are passed to OSUORBFS to estimate an improved trajectory, station coordinates, etc. Here parameters can be treated as having stochastic characteristics. The common foundation of the information and covariance filters and the efficiency of the Householder transformation to compute an equivalent square upper triangular matrix from a larger rectangular matrix are shown in section 4.1. The SRIF and SRIS algorithms are outlined in section 4.2. The SRIF and SRIS algorithms are expanded to include three parameter types, stochastic, pseudoepoch, and bias in section 4.3. The first-order Gauss-Markov and random walk models are shown in section 4.4. A general expression for the dynamic variational matrix for the dynamic stochastic parameters is developed in section 4.5. A technique to decorrelate and normalize the correlated observations is given in section 4.6. 4.1 Filtering and Smoothing The following discussion assumes some familiarity with filtering and smoothing theory as developed, for example in Brown (1983) and Gelb (1974). Additional familiarity is assumed with the square root information filtering and smoothing algorithm as developed by Bierman (1977) and implemented by Swift (1987). The discrete form of the stochastic state equations is (4-la) (4-lb) where x , is the state at time / 7 1 x' is a ‘deterministic’ state estimate that does not involve white noise process, state lJ+l, w j is the vector of white noise process terms with a covariance matrix Q . with dimw < dimx, G maps the source white noise process into the state with dimx. The discrete form of the linear measurement model is (4-2) where %j is the vector of measurements at time , A j is the matrix of partial derivatives of the measurement model with relation to the state at t \j is the vector of measurement noise with the covariance Pc. Without a loss of generality, a set of observations is constructed that has a diagonal covariance matrix with each diagonal element equal to one. The procedure is given in section 4.6. A solution to this problem was first proposed by Kalman in early 1960's (Kalman, 1960; Kalman and Bucy, 1961). A solution for the state and its covariance can be derived by applying Bayes' rule. This derivation can be found in Maybeck (1979, p. 118). The results are repeated here with a slight change in notation (4 -3 ) 7 2 *7 = P/[p;'^+A>J <4-4) The symbol refers to the propagated (predicted) estimate of the state and covariance at t}. The ,AI symbol refers to the estimate of the state or covariance after incorporating the measurement at ty The state is propagated from to time tyX using equation (4-lb). The covariance P; is propagated to tJtX by + (4-5) Notice that the inverse of P , is required in equations (4-3) and (4-4). To avoid the inversion of P , at each step, a direct propagation of P7'J, is desired. This can be developed by applying the following lemma to equation (4-5). (A + X ty )~' = A-1 - A~'XT(l +YA~'Xt ) ' y A"1 (4-6) where A = P^ Oj, x t = g ,Q„ Y = GJ, and defining 1VI/+1 (4-7) = M ,.,-C,G ;M ;., =[ i -C j GJ] m j, 1 (4-8) where the gain C, is c , = m,.1c ,[c x ,1g ;+q ;'] ' (4 -9) 7 3 To propagate and update the state equations in terms of P*1 and PJ1, the state estimates are replaced by (4-10) The equations for the state update and propagation are (4-12) (4-13) The state estimates can be found at any time by solving equations (4-10) and/or (4-11) for Xj and/or x;. Equations (4-8) to (4-13) are an algorithm to solve the problem defined by equations (4-1) and (4-2). Here, the inverse covariance is propagated. This algorithm is one of the class o f Bayes' filters. The inverse covariance is also called an information matrix leading to the name information filter. This algorithm requires computing the inverse of an n x n matrix where n is the number of states. This algorithm is most efficient when the number of measurements, m, is relatively larger than the number of states, //, and when the solutions for the state and covariance are needed infrequently. The usual Kalman filter can be derived from equations (4-3) and (4-4) by applying the matrix lemma [P 1 + A r A] ' = P - PA r [A PA r ] ' AP (4-14) These optimal filters, either the Bayes' or Kalman, exhibit numerical instabilities that cause the state estimates to diverge (Bierman and Thornton, 1977). More numerically 7 4 stable gain matrix expressions have been derived for both the covariance and information matrix forms (Maybeck, 1979). However, these require a significant number of additional matrix computations and thus are not completely satisfactory. A more comprehensive approach was to reformulate the filter algorithm in terms of square roots of the covariance or the information matrix. The square root filter maintains numerical accuracy to approximately the same number of digits with half the computer word-length required by a conventional non-square root algorithm The square root (or more correctly the Cholesky factorization) of an n x n matrix N is defined as N = SSr (4-15) where S is a lower triangular matrix. The square roots are not unique. Any orthogonal transformation (T) of a square root matrix S is also a square root of N. The useful properties of the orthogonal trans formation can be shown by factoring PJ1 into the product of its Cholesky factors p; 1 = R jR ,, (4-16) A 1 and thus, P' becomes p ; = r ; r , = r ; r > + a ; a , (4-17) R; where R (4-18) l \ R R.rJ N ow j = I (4-19) _ 0 . k J 75 The transformation T is an orthogonal transformation. Its columns form an orthonormal basis for R; of n vectors since has rank of n. The first n vectors span the range space of R, and the vectors /;•*-1 to n m are orthogonal to this spanning set Thus, the last m rows of R, are zero. Also, the basis vectors were chosen in a manner that R ; is upper triangular. A Householder transformation T is used to compute R^ (Bierman, 1977). This allows the square root matrices of dimension ( n+m ) by n to be transformed to an equivalent form of an upper triangular square matrix of dimension //. 4.2 Square Root Information Filter and Smoother (SRIF/SRIS) The application of the Householder transformation to an augmented matrix containing equations describing the state and measurements is the essence of the Square Root Information Filter (SRIF) and Square Root Information Smoother (SRIS). The equations are most easily constructed using the 'data equation' point of view (Bierman, 1977, pp. 17-18). A ‘data equation’ is just a convenient form in which to define the linear system. More common alternative forms are the normal equations in a least- squares adjustment algorithm, and the state and covariance matrices in a Kalman filter algorithm. First, concerning the notation, quantities referring to the time t} will be denoted by ‘(j)’ or [ ] The state status by for propagated, ‘A’ for the measurement updated, or for smoothed, and the parameter type associated with a matrix is indicated by subscripts. 7 6 The state 'data equation' at is defined as *xO') = Rx(7)*(y') + v Jt(y) (4-20) where zx(j) is the right side of a linear equation of the form Ax = b, R x(j) is a nonsingular square matrix, \(j) is the state to be estimated, vx(j) is zero mean noise with unit covariance. Next, the initialization of the system and computation the state estimate and its covariance are addressed. First, to initialize the system, the relationship (4-21) shows that the matrix R x(y) is the inverse of the square root of the covariance matrix of the a priori information. Also, upon initialization, the vector zx(j) is zero. The solution at tj for the state (x(y)) is x O ) = [ r ;' zJ ^ (4-22) In fact, equations (4-21, 4-22) are a general set of relationships that are used at any time after a measurement update or state propagation or smoothing, procedures to be discussed in the following paragraphs, to compute the state estimate and its covariance. The next step is to incorporate measurements into the system defined by equation (4-20). A ‘data equation’ is defined for the measurement that is analogous to the so called ‘observation equation’ in least-squares adjustment. The measurement ‘data equation’ is equation (4-2). The matrices A (j) and z(j) are the measurement partial 7 7 derivatives and O - C values to be added to the a priori information given in equation (4-20). This is accomplished by applying a Householder transformation (T,) to the combined a priori information from equation (4-20) and the measurements from equation (4-2) in an augmented form R, . A z _ . 0 e_ J where Rx(j) is the nonsingular square matrix of the state incorporating the measurements at t}, ix(j) is the right side o f the linear system, e(y) is the residuals of fit for the measurements z(j). The left side of equation (4-23) is equivalent to the right side as discussed near the end of section 4.1 and shown in equation (4-19). The top row of the right side of (4-23) is the state ‘data equation’ incorporating the measurements at ty *xO') = RxO,)iU) + vIC/) (4-24) where all terms are analogous to those in equation (4-20), except they now contain the contribution of the measurements at ty The solution for the state \(j) and its covariance P x(j) are obtained by applying equations (4-21, 4-22) with the ‘A’ matrices replacing matrices. Bierman (1977, pp. 69-71) shows the application of the Householder transformation in equation (4-23) is equivalent to adding the current measurement partials and the O - C values to the normal equations in the case of a step-wise least- 78 squares solution, and as the measurement update step in a Kalman filter application See also Swift (1987, Appendix H). The propagation of the state and covariance were given in equations (4-lb) and (4-5). The propagation of the state and covariance can also be incorporated into the SRIF algorithm. First, Bierman (1977, p. 116) defines the state ‘data equation’ in equation (4-20) in terms of x(j + 1) by solving equation (4-la) for x(j) and substituting into (4-20) giving KU) = + W)[W) - G W(./)] (4-25) Second, Bierman (1977, p 116) defines an a priori process noise ‘data equation’ at t} as O') = Ra> O) w(y) + va(j) (4-26) where (j) is the right side of a linear equation of the form Ax = b, R« 0 ) is the inverse of the square-root of the process noise covariance Q. i.e. Q = R i'iR L ')1 w (j) is the white noise, v a, (j) *s the zero-mean noise with unit covariance. Equations (4-25) and (4-26) involve terms w (j) and x(j + 1). To propagate the system from t} to /7+1, the system defined by equations (4-25) and (4-26) must be transformed into a system involving only x(j +1). This is accomplished by applying a Householder transformation to the augment system of the state ‘data equation’ from equation (4-25) and the process noise ‘data equation’ from equation (4-26) where R„,(y +1)is a matrix relating w(/) to the propagated process noise ‘data equation’ at /,,,, R ^ (y + 1) is a matrix relating x(y +1) to the propagated process noise ‘data equation’ at / z j j +1) is a vector, the right side of the propagated process noise ‘data equation’ at /,,,, R x(y' + 1) is a matrix relating x(y +1) to the state ‘data equation’ at tJ4], zx(j +1) is a vector, the right side of the state ‘data equation’ at /;+1. This removes w (y) terms from the bottom row of the right side of equation (4-27). The ‘data equation’ containing the state from to f7+1 is XU +1) = +1) * 0 +1) + v x0 + 0 (4-28) where all terms are analogous to those in equations (4-20) except they now refer to tJtX. Using relationships from equations (4-21) and (4-22), the state (x(y + l)) and its covariance (P (j + 1)) can be computed. The top row of the right side of equation (4-27) is no longer needed in the filtering algorithm, but this propagated process noise ‘data equation’ links w (j) and x(y + l) and the matrices ^ ^ (y + 1), R ^O + l), and z^y + l) are saved off-line to be used in the smoothing algorithm to be discussed in the following Bierman (1977, pp. 122-124) and Swift (1987, Appendix H) show that the ‘data 8 0 equation’ propagation in equation (4-27) is equivalent to the state and covariance propagation in a Kalman filter. The data update in equation (4-23) and the state propagation in equation (4-27) can be successively applied to measurements at subsequent steps up to the end of the data interval (ts ). At any step tj < tN the state estimate and its covariance can be computed from equations (4-21, 4-22). The Square Root Information Filter (SRIF) algorithm is defined by equations (4-21, 4-22, 4-23, 4-27). The orbit determination problem uses measurements collected during an interval from /„ to iN. The filter estimate at 1}, where t0 For the orbit determination problem, the fixed interval smoother is appropriate because the measurements are available in the fixed interval t0 to tN. For Kalman filtering, the Rauch-Tung-Striebel (RTS) smoothing algorithm is a widely used implementation of a fixed-interval smoother (Brown, 1983). A general formulation for inverse covariance smoothing is given by Maybeck (1979). A Square Root Information Smoother (SRIS) has been developed by Bierman (1977, pp. 214-219). At tN, the filtered estimate and covariance are contained in the state ‘data equation’ as ix(N) = Rx(N)x(N) + \x(N) (4-29) At tN, the smoothed (*) estimate and covariance are contained in the state ‘data equation’ as 8 1 z\.(N) = R‘x(N)x(N) + v'x(N) (4-30) Bierman (1977, p. 215) recognized that at the end of the fixed-internal tK, the filtered and smoothed estimates coincide, i.e. zx(N) = zx(N) (4-3 la) RX(N)=R*X(N) (4-3 lb) vx(N) = v'x(N) (4-31 c) And using equation (4-22) to compute the solutions for the states, x(N) = \(N) (4-3 Id) Now, return to the augmented form in equation (4-27) that was used to propagate ffom /A,_, to tN. The top row of the right side of equation (4-27) is now R .(tf )w(tf -1) + Rm{N)x(N) = %.{N)- vm(N) (4-32) At tN, the bottom row of the equation (4-27) is the propagated (~) ‘data equation’ at (N. Then, a measurement update is made using equation (4-23) giving equation (4-29). Because the filtered and smoothed estimates coincide at tN, equation (4-30) can replace equation (4-29) which gives Rl(N)x{N) = z'(N)-v'x(N) (4-33) Next, use the relationship in equation (4-la) at tN, relating the state at tN , and tN, and substitute this expression for x(N) in equations (4-32) and (4-33) to give [RJN) + Rm( N)G\w(N - 1) + R ^iN W N , N - 1 )x(N - 1) = zJN) - vJN) (4-34) R ’(A^)Gw(A - 1) + R'x(N) Combine equations (4-34 and 4-35) into an augmented matrix and apply a Householder transformation to remove w( N - 1) from the bottom row resulting in Ram + R mm G R0JN)®(N,N-\) zJN) R'X(N)G R'X(N)Q*(N,N - 1) z'x(N). K W - \ ) R H N -\) zx( N - 1) (4-36) 0 R*(//~ 1) z\{N -1). The top row of the right side is discarded, but the bottom row contains the smoothed estimate at (N_} in the form of a ‘data equation’ zx(N - 1) = R *X(N- \)x(N - 1) + \'X(N +1) (4-37) The smoothed estimate and covariance are computed by applying the relationships in equation (4-21) and (4-22). The smoothing procedure can be continued from tN_x to /„ using the successive top row of equation (4-27) that was saved off-line during the propagation step. This is Bierman’s Square Root Information Smoother (SRIS). Swift (1977, Appendix I) has shown its equivalence to the Rauch-Tung-Striebel smoother. 4.3 Epoch State SRIF/SRIS The equations in section 4.2 defining the SRIF and SRIS are perfectly valid for the orbit determination (OD) problem. These equations would use a single variable type x to represent all unknowns in the OD problem. These variables would be updated at each time step t}. In practice, the OD problem variable types are divided into three main classes One, the stochastic parameter state variables (Ap) that directly involve process noise and include random walk or first-order Gauss-Markov models. Two, pseudoepoch state variables (Ax) that indirectly involve the process noise. These are the satellite initial elements (e g position and velocity at some initial epoch). The reason to formulate the SRIF/SRIS in terms of the initial elements is that the batch least-squares algorithm provides measurement partial derivatives and variational partial derivatives for the satellite initial elements. These can be used directly in the SRIF/SRIF algorithm with pseudoepoch state variables. Otherwise, a substantial amount of recomputing would be required to reformulate the measurement partial derivatives and variational partial derivatives in terms of current state satellite elements. The word pseudoepoch (Bierman, 1977, pp. 152-153, Swift, 1987, Appendix A) is used instead of epoch because the states at any time 1} are the corrections to the state going from t} to mapped back to the epoch /„ using the variational partial derivatives. Third, bias parameter (Ay) types are parameters that involve no process noise parameters. The stochastic and bias parameters are further divided into orbit-related, if they involve dynamic models and thus the variational partial derivatives, and measurement-related, if they involve only measurement models. Examples of these different parameter types are given in Table 3. In section 4.2, x was a generic representation of all states in the SRIF/SRIS algorithm. In the remainder of the text, Ax has a specific definition as pseudoepoch state variables and it should not be confused with the x used in section 4.2. In this section, the equations from section 4.2 for the state propagation (4-la), state ‘data equation’ (4-20), state and covariance estimate (4-21,22), measurement ‘data equation’ (4-2), measurement update (4-23), state and covariance propagation (4-27), and smoothing (4-36) are redefined by partitioning the generic variable x into Ap, Ax, 8 4 A y . In this way, the operational equations implemented in the SRIF/SRIS algorithm for the OD problem are shown Table 3. OSUORBFS Parameter Types Type # Description Example 1 orbit-related stochastic solar radiation pressure, y-bias acceleration general acceleration 2 measurement-related tropospheric refraction correction stochastic 3 pseudoepoch state satellite initial elements 4 measurement-related double difference bias, tropospheric constant refraction correction, station coordinates, station and satellite clock models 5 orbit-related constant solar radiation pressure coefficient, y-bias, general acceleration Partitioning the propagation equation (4-la) into stochastic states, pseudoepoch states and bias states equation (4-la) becomes Ap M 0 0 Ap W 7 Ax - I 0 Ax + 0 -Ay. . 0 0 I. -Ay. . 0 . 7 * 1 J where V, = T0) is the inverse of the state transition matrix interpolated from the GEODYNII V-matrix file (FTN80), 8 5 \ p(fj+i,tj) is the transition matrix of the time-varying parameters from to (see section 4.5), M is the stochastic parameter transition matrix (see section 4.4), 7’ is the initial epoch time. The parameter in the state ‘data equation’ in equation (4-20) is partition into the three parameter types which gives K K K Ap - R, R„ Ax + V, (4-39) Lv 0 0 R,_ M. .V,. This is analogous to equation (4-20) and the definition of the terms in equation (4-39) follow ffom the definition in equation (4-20). The R matrices are subscripted with either one or two subscripts the first letter corresponds to the ‘data equation’ and the second to the parameter type. For example, R IV relates the Ay parameters to the Ax ‘data equation’. A single subscript indicates the ‘data equation’ and parameter type are the same. For example R x relates the Ax parameter to the Ax ‘data equation’. The solution for the state estimates and covariances in equations (4-21,22) are partitioned to include the three parameter types. First, the solution for the state is Ap R* R p x M Ax 0 R w (4-40) LAyJ R 'z For the covariance, the R \j) matrix analogous to the R^(y') in equation (4-20) is 8 6 R R R R r P P * px r : R_,(y) = 0 R x 0 R R. (4-41) 0 r .' The Ap and Ax parameters involve process noise and their covariance matrix can be separated from the covariance matrix of the Ay parameters, which do not involve process noise. The separated equations are R n R ’ i P p x RnP R. P * P „( / > o a H \ - 0 R . t 1 R R R R P px p y R, R,» py r :' r ; (4-42) 0 R. R. 0 R . R Pv (./) - [R v' R ' ^ (4-43) Equations (4-40) and (4-43) are written without any superscript notation The reader should by now recognize that equations for the estimate and its covariance, like equation (4-21) and (4-22), represent a general set of relationships that can be applied to the propagated (~), or measurement update (A), or smoothed (*) ‘data equations’. The measurement ‘data equation’ analogous to equation (4.2) is z(j) = Ap(j)Ap(j) + Ax(y)Ax(y) + Av(y)Ay(y) +v(y) (4-44) where A p ,Ax, and A v are the measurement partial derivatives with respect to Ap, Ax, and Ay, respectively. The number of stochastic, pseudoepoch, and bias parameters are np , nx, and n>, respectively. The measurement update transformation in equation (4-23) is written as a 87 two-step transformation that saves storing a block of //,. x ( np + nx) zeroes that would be present if equation (4-23) was used in its original form (Bierman, 1977, pp. 136-138) giving R„ N> R, R,* P> X R,R„ K •'a = 0 (4-45) p x K R, K X R. K X La „ A, A,. z 0 0 A z R, X Ry X (4-46) -Ay z 0 e _ The state ‘data equation’ incorporating the measurements at and analogous to equation (4-24) is R/« K Ap = I z* 0 R, R* Ax + V, (4-47) Ix 0 0 R v- M. .V.v. The state estimates and covariance can be computed by applying equations (4-40, 4-42, 4-43). Along with each stochastic parameter a corresponding bias parameter type is estimated (Swift, 1987, p. 25). This scheme estimates the mean as a bias and the stochastic process is then modeled as a zero-mean process with zm = 0 Partitioning the propagation in equation (4-26) with three parameter types, equation (4-26) becomes • -R„M 0 0 0 R, K r ; r ; 0 - 0 R;u (4-48) R y “ RpxVy KK i p R; k x> 0 N> 0 ,R*/| “ R*-Vy R, * Rx K Z*_ K J K The top row of the right side of equation (4-47) contains the matrices that are no longer needed for filtering, but are saved for the smoothing. Following the adopted notation of Bierman (1977, pp 142) and Swift (1987, pp. 45-46), these matrices are given the smoothed symbol to indicate they are now part of the smoothing algorithm. The R* matrix corresponds to the R ffl matrix in equation (4-26). The R^,, R ^, and R^ matrices correspond to the R K matrix partition into three parameter types. And, z* corresponds to z^. The remaining R matrices correspond to the Rx matrix partition into the three parameter types. The Ay parameter types are not time varying and thus are not included in the propagation step. The propagated state ‘data equation’ analogous to (4-28) is K Ap = R*,, K Ax + v , 0 0 K. -Ay. -Vy- Again, the state estimates and covariances can be computed by applying equations (4-40, 4-42, 4-43). The user is cautioned regarding the use of the subroutine given on pages 158-160 in Bierman (1977). Bierman (ibid.) developed his algorithm for the time propagation step storing only a subblock (S) of the entire matrix on the left side of equation (4-48). The S matrix is 89 The FORTRAN code given by Bierman (1977, pp. 158-160) implementing his algorithm is listed below with line numbers for reference. 1 DO J1 = 1, NP 2 IF (Jl < Nd) then 3 DO 1=1, Npx 4 DO K=l, Nx 5 S(I. 1) = S(I, 1) - S(I, Np + K) * Vp (K, Jl) 6 ENDDO 7 ENDDO 8 ENDIF 9 a = -Rw (Jl) * DM (Jl) 10 a = a * * 2 11 DO 1=1, Npx 12 V(I) = S(I,1) 13 cr=cr + v(I)**2 14 ENDDO 15 a = SQRT (a ) 16 a = a - cr 17 SIG (Jl) = o 18 o-= l.D0/(o- * a) 19 DO J2 = 2, Ntot 20 8 = 0.D0 21 DO 1=1, Npx 22 8 = 8 + S (I, J2) * v (I) 23 ENDDO 24 8 = 8 * a 25 L = J2 - 1 26 IF (J2 > Np) L = J2 27 S(Npx + Jl, L) = 8 * a 28 DO 1=1, Npx 29 S (I, L) = S (I, J2) + 8 * v (I) 30 ENDDO 31 ENDDO 32 8 = a * Rw (Jl) * o 33 S (Npx + Jl, Np) = Rw (Jl) + 8 * a 34 DO 1=1, npx 35 S (I, np) = 8 * v (I) 36 ENDDO 37 ENDDO 90 The general formulation in equation (4-48) is correct. However, Bierman assumes the off-diagonal elements of the matrix R* are all zero, i.e. that R* is diagonal This is incorrect. Also, in Bierman’s algorithm, the application of the Householder transformation to column j, where Jl < j < nd and Jl is the current column to orthogonalize and nd is the number of orbit-related stochastic parameters, neglects the products involving V^. These inadequacies can be shown by comparing the matrix products between Bierman’s algorithm and the application of the Householder transformation in equation (4-48). The following notation is assumed: [/] indicates the /th column of the matrix, (/) indicates the /th element of a vector, (/,/) indicates the element at /th row and yth column of matrix. Also, in the following, y will be used to represent the entire matrix on the left side of equation (4-48). The Householder transformation algorithm is given Bierman (1977, Chapter 4) and the reader is directed to this reference for details and background. The Householder transformation (7’) applied to the first column of the matrix on left side of equation (4-28) is = yti] - ^ »[!] = o' (4‘51) o-u(l) where .v/tf//(y(l))(y[l]ry[l])' \ u[l] = y[l]+o-, u(l) = y(l) + Q-. 91 Before proceeding an additional notational convention must be defined. The A A ^ expression (R f - R^V^} is the matrix that results from subtracting the product from Rf And {Rp - R^V^} (ij) is the element in the /th row andyth column of this matrix. The Householder transformation in (4-51) is computed correctly in Bierman’s algorithm which can be shown by computing a from the Householder transformation applied to equation (4-48). From equation (4-48) the dot product (y[l] r y[l]) is up f . . . y[iry[i]=l{R„M}(/,i){R„M}(/,i) + l[|R p - R„V,)(,,l) *(r , - R„Vp)(/,l)] (4-52) + S [(r „ - r , v „}(u ) ♦ (r „ -R«V,j(,,l)] l-lL J Since R ^M is diagonal l(R„M)(/,l) {R„Mj(,,l)=[R„M(U)]! (4 53) t=l This is computed correctly in lines 9 and 10 of Bierman’s algorithm. The products (Rp-R^V,)[l]and |R„-R„ Vpj[l] are computed in lines 3 to 7. Notice that only the first column of this matrix is computed. This key point is why the algorithm fails when the Householder transformation is applied to the second column. The components of the dot product on lines 2 and 3 of equation (4-52) are correctly computed on lines 11 to 14 o f Bierman’s algorithm. The square root is computed on line 15. Finally, the value of (J is assigned to SIG(l), i.e. R*,(l), on line 17. 92 Now, when the Householder transformation ( Tu) is applied to the second column (y[2]) the algorithm fails. The Householder transformation applied to the second column of the matrix on the left side of equation (4-48) has the form 7;y[2] = y[2]-y[21 “[1]u[l] (4-54) CTU(l) where y[2] is the second column, and all other terms are those defined in equation (4- 51). The problem arises in the computation of y[2]r u[l]. To illustrate, the correct dot product is computed from equation (4-48) as y[2]r u|l] = l[{R„M }(/,2) ■ {R„ m)(/,d] + (R„mJ(1,2) * [{ r.m |(1 ,1 ) + a] + i[(r , - R „ VP}(/,!) * ( r , - R „ Vp )(/,2)] (4-55) + l ||R w - R 1V,j(/,l)*jR„-R,V„M] Since and M are diagonal matrices the first row in equation (4-53) is zero, i.e. 2 [{ R „ m )(/,2). {r „ m )(/,1)] = 0 (4-56) |R„M)(1,2) * [{R„ M)(l,l) + ct] - 0 (4-57) However, in general, the following subblocks are non-zero matrices (4-58) (4-59) Thus, the terms on the last two lines of equation (4-55) are non-zero, i.e. g ( R p -R„V,)(;,l) *{r , - R„V„}(/,2)] * 0 (4-60) 93 £ [ |r „ - R,V„}(/,1) *( r ,, - R,V„)(/,2)] * 0 (4-61) Using these results, the two problems in Bierman’s algorithm are shown. First, from equation (4-54) the dot product y[2]r u[l] * 0, as shown in equations (4-60, 4-61) y[2]r u[l] and thus, ------77^*0. Since cr* 0 [see equation (4-52) and the discussion that ctu (1) follows it], then u(l)*0, and ^ ^ ^ u[l] * 0. Now, since is diagonal, cru(l) y(l,2)=0, thus the first row of 7^y[2] * 0. Therefore, the matrix R^,, in general, contains non-zero off-diagonal elements. Now, to show the second problem, first recognize that the second column (y[2]) of the matrix on the left side of equation (4-48) is |R „M )[2] y[2]=j jR,-R^V,j[2l (4-62) L(R„-R,V„j[2l_ Next, follow Bierman’s approach to computing the terms on lines 2 and 3 of equation (4-55). As mention earlier, in lines 3 to 7 only the products R^V^l] and R^V^p] are computed, i.e. only for the first column. The loop from lines 19 to 31 applies the Householder transformation, used to orthogonalize the Jl column (here Jl = l), to the remaining columns (2 Jr 1 - | R J ^ ®'erman s a^8or'thm fails to compute the products R ^ Vp [2] and R* Vp [2] prior to computing the dot product y[2]r u[l]. Thus the product (£ ) formed on line 22 is S = 2 ( r „ - R„ VJ ( u ) * (r ^(2,/)} (4-63) + £(Rw - R xV,)(l,i)*(Rx(2,i)) (4-64) 1 = 1 A comparison of equations (4-63, 4-64) with lines 2 and 3 of equation (4-55) shows the y[2]r u[l] products involving V are missing. Thus, the factor ------——, needed in equation (4- p cru(l) 54), is incorrect in Bierman’s algorithm. In line 29, u[l] is scaled by an incorrect scalar before being subtracted from y[2]. In addition, y[2] itself is incorrect since it lacks the products involving \ p. These deficiencies were evident in a numerical comparison of Bierman’s algorithms and equation (4-48). In fact, the problem was discovered in this way. One might propose to modify Bierman’s algorithm to correct these mistakes, and retain using the subblock S matrix. This could be done, but the correction would involve computing A A A A and the storing the product R p - R pv \ p and R x;, - R x\ p matrices. The original idea behind Bierman’s algorithm was to avoid storing precisely these matrices. Also, the neglected upper-triangular portion of R *, must be stored. Thus, this is not a satisfactory resolution, since the desired reduction in computer storage would not be realized. The only savings of storage would be the elements below the diagonal of R The formulation in equation (4-48) was used in OSUORBFS. The equation for the implementation of Bierman's (1977, p. 216) SRIS pseudoepoch formulation is obtained by partitioning the state variable in equation (4-35) into the three parameter types. See also Swift (1987, pp. 45-47) The SRIS equation is The terms correspond to the **’ terms when propagating from t} to /,,, in equation (4-48). Admittedly, this is a somewhat awkward notation, but this maintains consistency with the earlier references (Bierman, 1977, p. 216; Swift, 1987, p. 47), The M and V;, matrices are those from equation (4-38). The correspondence to the matrices in equation (4-36) is made by inspection. For example, R ’ in equation (4-36) corresponds to the matrix subblock containing R^., R^,, R*, R^„ in equation (4-65). The top row of the right side of equation (4-65) is discarded. The other terms form the smooth state ‘data equation’ at or 96 * K r ;* r ; Ap vr Ax + r :„ r ; R Iv V* * 0 0 R v_ M. -V.v- J j j The solution for the state and covariance can be computed by applying equations (4-40, 4-42, 4-43). 4.4 Time-Varying Stochastic Parameter Models Some physical processes are not easily expressed as deterministic models. This is prevented by the random nature of the process, or a lack of understanding of the underlying physics. For the former, a time varying stochastic model is appropriate. For the latter, all that may be known or postulated about the process is its stochastic characteristics. This a priori information is used by specifying a time varying stochastic model. The models are defined by stochastic differential equations involving white noise The first-order Gauss-Markov process has an exponential correlation function. The random walk is a white noise process integrated in time. Many physical processes can be modeled by a first-order Gauss-Markov model or a random walk model. The discrete mathematical expression for these models can be found in most textbooks on the subject, for example, Maybeck (1979) or Gelb (1974). The first-order Gauss-Markov process describes the physical process where the state at /,,, depends only on the previous state at tj. The discrete form of the state update is 97 Ap(y + 1) = M(y + lJ)A p(y) + w(j) (4-67) to where M(> +1,j) = e T , At = lJtl - w(/) is the white noise. The discrete process noise variance (qaJ is (Id, (4-68) where qcon is the continuous process noise variance, r is the correlation time. The random walk model is a special case of the first-order Gauss-Markov where t —> oc. The discrete state update becomes ApO' + l) = ApO') + w(y) (4-69) The discrete process noise variance is To implement the first-order Gauss-Markov process, the correlation time (r) and the continuous process noise variance ( qcon) are specified. The matrix M is computed in equation (4-67) and qAs is computed in equation (4-68). The random walk model is specified by defining the continuous process noise variance ( qcon), here M = I, and qAi is computed using equation (4-70). 98 4.5 The Dynamic Stochastic Parameter Transition Matrix The orbit-related stochastic parameters require the matrix V (/7<1, l}) that maps the effects of the stochastic parameters Ap(/'), on the pseudoepoch state parameters Ax(y + 1). The matrix has dimension nx (number of pseudoepoch state parameters) by nd (number of orbit-related stochastic parameters) and has the general form of O' 0 0 0 \ (4-71) 0 0 o d/"a' where the /th satellite contribution (t x,t ), a 6 by nd(i) matrix. The nd(i) is the number of orbit-related stochastic parameters for the /th satellite. A general expression for dynamic stochastic parameter transition matrix can be developed using the variational partial derivative matrices generated by GEODYNII. From equation (4-38), the state update for Ax is separated, as Ax(y +1) = V„(/,+1,/, )Ap(» + Ax(y) (4-72) This is the difference equation. Equation (4-72) contains variational partial derivative matrix relating the state at and tJ. Only the variational partial derivative matrix relating the state at /^, to the initial epoch state T0 is available from the batch least - squares algorithm Equation (4-72) is expanded to show that the desired variational 99 partial derivative matrix Vp(/7+I,^) can be computed from V (/ ,,7j). Now, equation (4-72) can be expanded as AXj+i = Vp(tj^,tj)Apj + ZV/,(/,,f,_I)Apj+Ax0 (4-73) t = 1 If a< b< c, the relationship J = J - J holds. Based on this relationship, the b a a expression for the transition matrix can be further expanded as Ax,.1=[V,(/J„,7;)-V,(l,,7'0)]4p, + £[v,(/„7;)-V ,(«M,7;)]4p,.1+Ax0 (4-74) where V ,(/,+1,T0) = d r ’ (/>+1, T0)O , (tJ+l ,T0), V/1(^ ,7 ;) = ;’ O '1 (tJ+l, T0) and d>“‘ (t}, T0) are the inverses of the 6 x 6 state transition matrices from T0 to tJ+l and t} , respectively, (ljU, T0) and (tJ, 70) are the variational partial derivatives of the dynamic parameters from T0 to /J+1 and tJ , respectively. For selected times the matrices 1 r dXj d x , ax,ax,ax,ax,ax, ax, axndYn dZ {, at. az{, atat dY} dY} at at aY, at at at dX, 0Y9 K<',.7o>] = a x , d X , ax, ax, ax, ax,(4-75) , L d Z ( d X0 a t a t . at l where are the variational partial derivatives relating the state at /, to the initial state parameters at 7’. The initial state parameters include the satellite initial epoch elements ( X 0, Y0, Z0, X0,Y0,Z0) and constant parameters (I]), e.g. solar radiation pressure scale factor, y-bias acceleration, etc. Values at times between 1, and t,^ are interpolated using a ninth-order interpolator. 4.6 Observable Decorrelation and Normalization As the reader may have noticed, the SRIF/SRIS algorithms are constructed assuming the observations are uncorrelated and normalized. In general, observations are correlated and unnormalized. However, this poses no problem as long as the covariance matrix is regular because a set of observations can be decorrelated and normalized using the procedure given below. This procedure is available in OSUORBFS for double differenced observations. 101 The general form of the observation equations as defined in equation (4-2) The observation error v is a zero mean, E(\) = 0, but is correlated, £(w T) = Pv. A set of uncorrelated observations with unit covariance can be constructed from the lower triangular square root of Pv Pv = K W (4-76) Here, Lv can be computed by a lower Cholesky factorization of Pv. The desired independent set of observations is obtained by multiplying equation (4-2) by L"' giving Lv'z = L“'Ax + Lv‘v (4-77) CHAPTER V ORBIT DETERMINATION USING A BATCH LEAST-SQUARES ESTIMATOR AND DETERMINISTIC MODELS In Chapter 2, the models accounting for over 99% of the GPS satellite accelerations were investigated. In Chapter 3, the systematic errors affecting the GPS observable were given along with models and/or measurement techniques to reduce the error. In Chapter 4, the SRIF/SRIS algorithms were developed and discrete forms of the first-order Gauss-Markov and random walk models given. In this chapter, the three deterministic modeling scenarios introduced in Chapter 1 are implemented in a batch least-squares estimator. The stochastic modeling tests are performed in Chapter 6. In section 5.1, the specific model parameters for each of the three deterministic dynamic modeling scenarios, i.e., Experiments A, B, and C, are given. In section 5.2, the collection of ground station GPS tracking data and the estimation of post-fit orbits by the International GPS Geodynamics Service (IGS) are discussed. In section 5.3, to verify that the dynamic models are correctly implemented, a fit is made to a precisely determined International GPS Geodynamics Service (IGS) orbit of January 16, 1994, by rotating X, Y, and Z satellite positions from ECEF into TOD coordinates and introducing these as direct observations of the satellite position. This fit is performed for each of the three deterministic dynamic modeling scenarios, i.e. Experiments A, B, and 102 C. The compatibility of the dynamic models used in the batch least-squares processor and IGS analysis centers is demonstrated. In section 5 4, the preprocessing of the ground station GPS tracking data is discussed The formulation of the ion-free double- differenced observables ffom the tracking data, collected at 33 IGS ground stations on January 16, 1994, is presented. Finally, in section 5.5, the batch least-squares estimator is augmented with appropriate models for the troposphere, earth tide and ocean loading effects on the tracking stations. Also, additional parameters are introduced which include station positions, double-differenced phaserange biases, polar motion and UT1 variation. Then, the GPS satellite orbits on January 16, 1994, are independently determined based entirely on ground tracking data for each of the three deterministic dynamic modeling experiments. For each deterministic dynamic modeling approach the estimated GPS satellite orbits are compared to the Final IGS GPS satellite orbits. 5.1 Experiments A, B, and C, Parameter Selection The reasons for choosing the particular combinations in Experiments A, B, and C are now given. As discussed in Chapter 1, the standard model estimates the solar radiation pressure scale factor and a y-bias acceleration term. This is Experiment A. In Chapter 6, these two parameters will be estimated as stochastic processes. Conceptually, a more general approach is to estimate the solar radiation pressure scale factor, a radial, a GPS X-axis, and a GPS Y-axis acceleration term. This model separates the solar radiation pressure scale factor ffom the residual accelerations that are absorbed by the three acceleration terms along each spacecraft body axis. This approach is Experiment 104 B. In Chapter 6, the three acceleration terms will be modeled as stochastic processes An attempt to model the time variation in a deterministic way has been outlined by Colombo (1989) See section 2 4 6 He introduces three three-term Fourier expansions about the nominal GPS orbital frequency (with a period of approximately 12 hours) along each GPS body axis. This is the approach in experiment C. Since this model already accommodates a time variation in the coefficients of the sine and cosine terms of the expansion, it is not appropriate to model further these terms as time-varying stochastic processes. The following experiments considered in Chapter 5 are: (A) solar radiation pressure scale and y-bias acceleration model in a batch least-squares estimate, (B) solar radiation pressure scale and bias accelerations in the radial, GPS X-axis, and GPS Y-axis in a batch least-squares estimator, (C) solar radiation pressure scale and the nine parameter acceleration model of Colombo in a batch least-squares estimator. The selection of model parameters estimated in the batch least-squares estimation for Experiment A is given in Table 4. Twenty-six satellites were modeled with a total of 156 satellite initial elements to be estimated. The a priori sigma values for the position and velocity are set to large values on the order of 106 m and 106 m/s, respectively. For each of the 26 satellites, a solar radiation pressure scale and y-bias acceleration coefficient are estimated. The a priori sigma for the solar radiation pressure scale is 10 and the y-bias acceleration is 1 x 10 7 m / s2 For each parameter, the a priori sigmas are given much larger values than the realistic uncertainty in these parameters. This frees the 105 batch least-squares algorithm to determine the parameters without constraining any parameter. These model parameters are used in the GEODYN fits to the Final IGS orbit in section 5 3 Additional model parameters must be specified for the GEODYN GPS orbit determination in section 5.5 which uses the IGS ground station GPS tracking data. The formulation of the double-differenced phaserange observations is discussed in section 5.4. A total of 895 double-difference phaserange biases are estimated. One bias is included for each unique double-differenced combination. Initial phaserange ambiguity values are estimated ffom the pseudorange. The biases are typically known to be better than 10 m. Again the a priori uncertainty is set unrealistically large to free the estimation process ffom any incorrect a priori parameter values. For each station, a tropospheric reffaction scale bias is estimated for each 12-hour segment. From Table 2, the total tropospheric reff action zenith delay is approximately 2.7 m, the ‘wet’ portion variability can be as much as 30 cm. Thus, the scale biases are typically on the order of 0.1. For each station, a measurement time bias is estimated with a priori uncertainty of 1 x 10 6 sec. Initial station clock estimates at each epoch are determined ffom the pseudo range solution. This is discussed in section 5.4. The polar motion and UT1 variation are estimated as constants over a 12-hour span. The a priori values are available ffom the IERS Bulletin A. The a priori uncertainties are 0.1 mas and 1 x lO^6 sec, respectively The thirteen IGS core tracking stations (see Table 7) are constrained to 0.001 m and the other tracking station uncertainties are 0.1 m. 106 Table 4. Experiment A, Batch Least-Squares Estimated Parameters on Day 016,1994 PARAMETER DESCRIPTIONS NO A PRIORI SIGMA satellite epoch 156 3.2 x 106 m elements 3.2x 106 m /s solar radiation pressure scale 26 10 Y-bias acceleration coefficient 26 1 x 10“7 m / s2 double-difference range biases 895 3 x108 m tropospheric refraction scale bias 68 1.0 station clock bias 34 1 x 10 6 s polar motion 4 0.1 mas and UT1 2 1x10 6 s station 36 0.001 m positions 60 0.100 m Total 1307 The selection of model parameters in the batch least-squares estimator for Experiment B is given in Table 5. For this experiment, the solar radiation pressure scale factor and y-bias combination is supplemented with a radial and GPS X-axis acceleration terms. Following the same reasoning used in Experiment A, the satellite initial elements, solar radiation pressure scale factor, and acceleration biases are estimated with large uncertainties. The other parameters estimated and their a priori uncertainties remain the same as in Experiment A. 107 Table 5. Experiment B, Batch Least-Squares Estimated Parameters on Day 016,1994 PARAMETER DESCRIPTIONS NO A PRIORI SIGMA satellite epoch 156 3.2 x 106 m elements 3.2x 106 m /s solar radiation pressure scale 26 10 radial acceleration bias 26 0.1 m/s7 GPS X-axis acceleration bias 26 0.1 m/s2 GPS Y-axis acceleration bias 26 0.1 m / s2 double-difference biases 895 3 x 108 m tropospheric refraction scale bias 68 1.0 station clock bias 34 lxl0~6s polar motion 4 0.1 mas and UT1 2 1 x 10~6 s station 36 0.001 m positions 60 0.100 m Total 1359 The selection of modeled parameters for the batch least-squares estimation for Experiment C is given in Table 6. For Experiment C, the solar radiation pressure scale factor is estimated, but now the Colombo nine parameter acceleration model supplements the radial, GPS X-axis and GPS Y-axis bias parameters with a sine and a cosine coefficient with a frequency of one nominal GPS orbital frequency (with a period 108 of approximately 12 hours). Again, the satellite initial elements and solar radiation pressure scale are given large a priori uncertainties. The a priori sigma values for the acceleration terms are those used by GSFC (Pavlis, 1994, private communication) When processing TOPEX GPS data, GSFC achieved their most consistent results compared to University of Texas TOPEX GPS orbits by assigning large uncertainties to the GPS X-axis and GPS Y-axis acceleration biases and small uncertainties to radial acceleration bias, sine, and cosine coefficients. The other parameters estimated and their a priori uncertainties remain the same as in Experiment A and B. Table 6. Experiment C, Batch Least-Squares Estimated Parameters on Day 016, 1994 PARAMETER DESCRIPTIONS NO. A PRIORI SIGMA satellite epoch 156 3.2 x 106 m elements 3.2x 106 m /s solar radiation pressure 26 10 radial acceleration bias 26 1 x 10"30 m/s2 sine coefficient 26 1 x 10~31 m/s2 cosine coefficient 26 1 x 10'32 m / s2 GPS X-axis acceleration bias 26 0.1 m/s2 sine coefficient 26 1 x 10~50 m/s2 cosine coefficient 26 1 xlO~M m/s2 GPS Y-axis acceleration bias 26 0.1 m / s2 sine coefficient 26 1 x 10 60 m/s2 cosine coefficient 26 lxlO'6' m/s2 double-difference biases 895 3 x 10* m tropospheric refraction scale bias 68 1.0 10‘) Table 6 (continued) station clock bias 34 1 x 10"6 s polar motion 4 0.1 mas and UT1 2 1x10 6s station 36 0.001 m positions 60 0.100 m Total 1515 5.2 The International GPS Geodynamics Service (IGS) Tracking Data and Orbits For this study, data from the International GPS Geodynamics Service (IGS) are used from a set of 33 tracking stations in the IGS global network. The geographic locations are shown in Figure 3. The names, ID codes, locations and receiver types for some of the IGS stations is given in Table 7. To be an IGS station, the data must be collected at a 30-second interval without interruption and must be accessible over some electronic network at least once every 24 hours (Mueller, 1992). Data are then transferred to regional data centers and the complete global tracking set is maintained at three global data centers. There are currently seven analysis centers whose task is to compute the GPS orbits on a daily basis. Typically, the orbits are available as products, accessible to users in approximately one week. A weighted average of the satellite positions, X, Y, and Z, from of all seven analysis center orbits, called the IGS rapid orbit, is available within two weeks. And after about a month, when the final Bulletin IERS earth rotation parameters (erp’s) have been determined, the final IGS orbit is made available. 110 Figure 3 IGS International GPS Tracking Network. Ill Table 7. The IGS Global Tracking Stations Used in This Study Station Name Four Longitude Latitude ID Number Rec Character (") ('*) Type ID (1) Tromso TROM * E 18 56 N 69 40 10302003 RG Madrid MADR* W 04 15 N 40 25 13407012 RG Kootwijk KOSG * E 05 48 N 52 10 13504003 RG Wettzell WETT * E 12 52 N 49 08 14201009 RG Hartebeesthoek HART * E 27 42 S 25 53 30302002 RG Algonquin ALGO * W 78 04 N 45 57 40104002 RG Yellowknife YELL * W 114 28 N 62 28 40127003 MR Goldstone GOLD * W 116 47 M 35 14 40405031 RG Fairbanks FAIR * W 147 29 N 64 58 40408001 RG Kokee Park KOKB * E 200 20 N 22 10 40424004 RG Santiago SANT * W 70 40 S 33 09 41705003 RG Tidbinbilla TIDB * E 148 58 S 35 23 50103108 RG Yarragadee YAR1 * E 115 20 S 29 02 50107004 RG Onsala ONSA E 11 55 N 57 23 10402004 TB Metsahovi METS E 24 23 N 60 13 10503011 MR Graz GRAZ E 15 29 N 47 04 11001002 MR Matera MATE E 16 42 N 40 38 12734008 MR Herstmonceux HERS E 00 20 N 50 52 13212007 MR Usuda USUD E 138 22 N 36 08 21729007 RG Taipei TAIW E 131 32 N 25 01 23601001 RG Maspalomas MASP W 15 38 N 27 46 31303001 MR Saint John’s STJO W 52 41 N 47 36 40101001 MR Penticton DRAO W 11937 N 49 19 40105002 RG Quincy QUIN W 120 56 N 39 58 40433004 TR Pie Town PIE1 W 108 07 N 34 18 40456001 TR McMurdo MCMU E 166 40 S 77 51 66001001 RG Parnate PAMA W 151 02 S 16 44 92201003 RG Kourou KOUR W 52 37 N 05 08 97301210 TR Bermuda BRMU W 64 39 N 32 21 42501004 TR Fortaleza FORT W 38 35 S 03 45 41602001 TR McDonald MDOl W 104 01 N 30 40 40442012 TR Richmond RCM5 W 80 23 N 25 36 40499018 TR Westford WES2 W 71 29 N 42 37 40440020 TR (1) TR=Turbo Rogue, MR=Mini Rouge, RG= Rogue * IGS Fiducial Station 112 Table 8 shows the results for the intercomparison of the IGS orbits for January 16, 1994. This comparison is made by solving for the seven parameter transformations between any two orbits using software developed by Gerhard Beutler of the University of Bern (Goad, 1993). This allows the isolation of a consistent difference in the ECEF reference frame definition used by the individual centers. For example, a bias in the x origin between two analysis centers would appear as an orbit difference if the orbits were compared directly. This effect is correctly estimated as a translation in the x direction, and its effect on the orbit comparison is correctly modeled by using the seven parameter transformation. Inconsistencies of the reference frame is much less of a problem now that the analysis centers, for the most part, hold the same IGS stations fixed in their adjustment. Table 8 shows that, during this period, the IGS analysis center orbits were consistent at the 20 cm RMS level. The RMS of the transformation from a Final Analysis Center or Final IGS orbit to another Final Analysis Center or GEODYNII or OSUORBFS determined orbit is RMS,(77MMS) = iJ(TX(X0J - Xm )2 In (5-1) where Xofls is the satellite position vector from a Final Analysis Center or GEODYN or OSUORBFS estimated orbit, X FIN is the Final IGS orbit, TX is a seven-parameter transformation from the Final Analysis Center or GEODYN or OSUORBFS orbit to the Final IGS orbit, 113 n is the number of position coordinate comparisons, i.e. three times the number of common epochs times the number of satellites. Table 8. Transformation Parameters Between IGS Analysis Center GPS Orbits on Jan. 16,1994 DX DY DZ RX RY RZ SCALE RMS] FROM TO (m) (m) (m) (mas) (mas) (mas) (ppb) (m) -0.027 -0.013 -0.018 -1.4 -0.9 -0.3 0.1 0.11 igs0732 cod0732 -0.005 0.016 0.011 -2.9 -1.3 0.1 0.3 0.12 igs0732 emri)732 -0.015 -0.011 -0.020 -1.8 -1.0 -0.2 0.2 0.18 igs0732 esa0732 0.056 -0.001 -0.019 -1.8 -2.1 0.6 0.5 0.12 igs0732 gfz0732 0.011 -0.048 0.005 -1.3 -0.8 0.4 -0.3 0.12 igs0732 jpl0732 -0.017 0.076 0.052 -1.2 0.0 -0.2 -1.2 0.29 igs0732 ngs0732 -0.013 0.054 0.070 -1.9 -1.1 -0.8 -0.5 0.19 igs0732 sio()732 0.022 0.029 0.029 -1.6 -0.5 0.4 0.2 0.18 cod0732 emr0732 0.012 0.003 -0.002 -0.4 -0.1 0.1 0.1 0.19 cod0732 esa0732 0.083 0.013 -0.001 -0.4 -1.3 0.9 0.4 0.18 cod0732 gfz.0732 0.038 -0.035 0.024 0.0 0.0 0.7 -0.4 0.17 cod0732 jpl0732 0.010 0.090 0.070 0.1 0.8 0.1 -1.3 0.33 cod0732 ngs0732 0.014 0.070 0.088 -1.2 -0.2 -0.5 -0.6 0.24 cod0732 sio0732 -0.010 -0.027 -0.030 1.1 0.4 -0.3 -0.1 0.24 emr0732 esa0732 0.061 -0.017 -0.030 1.1 -0.8 0.5 0.1 0.17 emr0732 gfz0732 0.016 -0.064 -0.005 1.6 0.5 0.3 -0.6 0.18 emr0732 jpl0732 -0.012 0.060 0.042 1.7 1.3 -0.3 -1.5 0.32 cmr0732 ngs0732 -0.008 0.038 0.059 0.4 0.3 -0.9 -0.8 0.24 emr0732 sio0732 0.071 0.010 0.001 0.0 -1.1 0.7 0.3 0.23 csa0732 gfz0732 0.026 -0.037 0.025 0.5 0.2 0.6 -0.5 0.24 esa0732 jpl0732 -0.002 0.087 0.072 0.6 0.9 0.0 -1.4 0.37 esa0732 ngs0732 0.001 0.067 0.089 -0.7 -0.1 -0.7 -0.7 0.29 esa0732 sio()732 -0.045 -0.047 0.024 0.5 1.3 -0.2 -0.8 0.18 gfz0732 jpl0732 -0.073 0.077 0.071 0.6 2.1 -0.8 -1.6 0.32 gfz0732 ngs0732 -0.070 0.055 0.088 -0.7 1.1 -1.4 -1.0 0.25 gfz0732 sio()732 -0.029 0.124 0.047 0.1 0.8 -0.6 -0.9 0.34 jpl0732 ngs0732 -0.023 0.103 0.065 -1.2 -0.2 -1.2 -0.2 0.25 jpl0732 sio0732 0.004 -0.021 0.018 -1.3 -1.0 -0.6 0.6 0.30 ngs0732 siol)732 114 5.3 Experiments A, B, and C, Deterministic Dynamic Model Fits to the IGS Orbits For this data analysis, the final IGS GPS satellite positions are rotated from the ITRF92 ECEF frame to the TOD inertial frame using the rotation matrix given in equation (2-8). The X, Y, Z satellite positions, available at 15-minute intervals, are then introduced as observations of the satellite positions and GEODYNII solves for a new orbit. The analysis center orbits are determined using ion-free double-differenced observations that are affected by the forces given in Table 1 and the propagation medium as given in Table 2. Any mismodeling of these propagation effects (e.g. troposphere, ionosphere, geometric effects) affects the orbit positions. The GEODYNII orbit-fit models only the dynamic parameters given in Table 1. Thus, the orbit-fit residuals contain not only unmodeled dynamic effects in GEODYNII, but also the unmodeled dynamic and propagation effects used by the IGS analysis center software. For each experiment, the solution was computed with the batch least-squares estimator, GEODYNII. The RMS of fit of each solution is given in Table 9. In general, the RMS values for GEODYNII and OSUORBFS are computed as KMS2(GEODYN) = V(0-C)-(0-C)//n (5-2) RMS}(OSUOKBFS) = ^[(O-C)-AXf - [ ( 0 - 0 - AX]/m (5-3) where O - C is the observed minus computed from GEODYNII, A is the matrix of measurement partial derivatives from GEODYNII, X is the state update from OSUORBFS, m is the number of measurements. 115 The RMS values in Table 9 decrease by approximately 0.5 mm as the acceleration models increase in complexity. This is not necessarily a significant result, but it does follow an expected trend. Table 9. Experiments A, B, and C, GEODYNII RMS2 of Fit (cm) to the IGS Orbit Experiment GEODYNII RMS2 of Fit (cm) A 4.75 B 4.72 C 4.66 The transformation parameters between the IGS GFS satellite orbits that were input to GEODYNII as observations (X, Y, Z positions) and the GEODYNII estimated orbits were determined. The transformation parameters are given in Table 10. All three experiments have RMS, of fits of approximately 4.3 cm. This is approximately 4 mm smaller than the RMS, values in Table 9 indicating that 4 mm of the difference is absorbed in the transformation parameters. The RMS, values can be interpreted as revealing the consistency of the dynamic models used in the experiment and those used by the IGS analysis centers. Interestingly, the RMS, value for each experiment is approximately 4.3 cm. This suggests that for one-day orbits, even the two parameter models used in experiment A is sufficient for 20 cm GPS orbit determination. The more complex models for the accelerations used in experiments B and C are unnecessary for 116 20 cm level GPS orbit determination Also, introducing the radial acceleration bias appears to absorb any scale differences because for experiment B and C the scale factor correction was estimated to be zero. Table 10. Transformation Parameters Between GEODYNII GPS Satellite Orbits and IGS GPS Satellite Orbits for Experiments A, B, and C on Jan. 16,1994 Using IGS Satellite Orbits as Data Experiment DX DY DZ RX RY RZ S RMS, (m) (m) (m) (mas) (mas) (mas) (PPb) (m) A 0.000 0.012 0.003 0.0 0.2 0.1 -0.3 0.043 B -0.002 0.011 0.003 0.0 0.2 0.1 0.0 0.043 C -0.002 0.011 0.003 0.0 0.2 0.1 0.0 0.042 5.4 Ground Station Tracking Data Preprocessing The tracking data must be processed to remove systematic errors and construct the appropriate observations for orbit determination. Likewise, the broadcast ephemeris elements can be used to compute a set of satellite initial elements from which to begin the orbit determination These procedures, using the IGS tracking data, and GPS broadcast ephemeris elements, are shown in Figure 4 and described in the following. 117 ITRF92.STA DAILY DATA FILES OPTBASE RINEX RINEX OBS FILE NAV FILE PSTATIC COMBNAV COMPOSITE RINEX NAV FILE FORMGA BEEPCE FTN40 FTN05 SAT. INITIAL DDIF DATA MBIAS CARDS ELEMENTS Figure 4. Data Preprocessing Procedure. The receiver clock sets itself to GPS time during some initialization by sequencing with the transmitted signal from the satellite. The data collection at a measurement time is triggered by the receiver clock. Since the receiver clock is typically a quartz clock, the receiver clock can be offset and drift from the GPS time. The double-difference observation is typically conceptualized assuming the measurements are collected at the receivers at the same GPS time epoch, as shown in Figure 5(a). In this case, the double-differenced observable is A2 t (1Q) are the one-way time delays ffom satellite i to station j, 118 N is the integer ambiguity, e is the measurement noise. The time delay is computed from the range as where pv is the geometric range between satellite i and station j, c is the speed of light 2 (b) 2 Figure 5. GPS Satellite Positions at Tracking Receiver Collection Time. If one receiver clock is offset from GPS time, then its observable actually refers to a slightly different time as shown in Figure 5(b). Since GPS receivers are driven by independent oscillators and even though they attempt to ‘set’ themselves to GPS time, the situation in Figure 5(b) occurs in general. In this case, the double-difference observable is 119 A20 = /o(ri 2 ( 0 - r2I('i)-rn('i)+ r 22('0)) + N + £ (5-6) The time delays at /, can be written in terms of time delays at /0 and time delay rates as = / o ( r , 2 ( ' o ) - ( r 2]('o ) + r 21(/0)A/) -(rM(/0) + r„(/0)A/) + r22(/0)) + N + e (5-7) where A /= If /, is in error, i.e., the clocks at the two receivers are offset by an unknown amount, the computed (C) double-difference observable can be in error. This occurs because the rate of change of the ranges are not equal, i.e. pn * pu, and in general they do not cancel. The effect on the double-difference is given in the following. First, assume that two receiver clocks are offset, i.e., Figure 5(b), equation (5-7), but that in modeling the double-difference the assumption is that the clocks are synchronized, i.e., Figure 5(a), equation (5-6). The error, in cycles, in computing (C) under this assumption is simply the terms remaining after differencing equations (5-6) and (5-7), i.e. A(AV) (cycles) = / 0(-r'2lA/ -f„A/) (5-8a) c Now, multiply equation (5-8a) by the wavelength X = — to get the error in meters and write the time delays in terms of the range rates to give 120 A(A2^) (meters) = cj-Af - A/ (5-8b) A(A2^) (meters) ~ (-p 21 - PU)&1 (5-8c) A nominal range rate difference - ( pn - />,,) is 100 m/s. A clock offset of 0.01 secs, if neglected, results in approximately one meter error in the computed double differenced phaserange observable. The receiver clock offset must be applied for GPS satellite orbit solutions at the 20 cm level. For static positioning, the point positioning solution using pseudoranges is implemented to estimate, over the entire data span, one set of station positions and the receiver clock offset at each epoch. This is performed by routine PSTATIC in Figure 4. In this study, as a matter of practice, the estimated receiver clock offsets (A/Kirc)are added to the receiver’s version of GPS time ( tREC) to determine the correct GPS Time (tGPS) of the phase measurement at the receiver, i.e. (5-9) Two examples of the resulting receiver clock offset from GPS time are given in Figures 5 and 6. In Figure 5 the receiver clock offset from GPS Time at station Quincy is shown. The magnitude o f 0.01 secs is relatively large. 121 QUIN Clock Offset -1.0E-2 ° ® -1.2E-2 -1.4E-2 -1.6E-2 0 20000 40000 60000 60000 Seconds of Day Figure 6. Receiver Clock Offset at Quincy on Jan. 16, 1994 Determined by Static Pseudorange Solution. In Figure 7, a more stable receiver clock is shown at station Fortaleza. The magnitude of 6 x 10 7 secs has no appreciable effect on the computed double-differenced observable and does not effect the orbit determination solution. 8.0E-7 E o FORT Clock Offset S'S" 7.0E-7 a> E oota. 6.0E-7 > o a> 5.0E-7 20000 40000 60000 80000 Seconds of Day Figure 7. Receiver Clock Offset at Fortaleza on Jan. 16, 1994 Determined by Static Pseudorange Solution. 122 Beginning with a set of r simultaneously operating tracking stations, there are r(r -1) / 2 possible baselines of which only r - 1 are independent. Of all the possible r - 1 baseline sets, the set composed of the shortest baseline distances is a logical choice since this allows the maximum amount of common satellite view from any two stations and thus, the maximum number of DD observables are available. The technique of Goad and Mueller (1988) was used to construct an independent set of r - 1 baselines. The selection is made by first forming all possible r{r -1) / 2 baselines, ordering by distance, and beginning with the shortest baseline, adding each additional baseline and identifying dependent baselines by using Cholesky and Gram matrices. This task is performed in a routine OPTBASE in Figure 4. The baselines used in this study are given in Table 11. Table 11. Baseline Combinations on Jan. 16,1994 Used in the GPS Satellite Orbits Solution STATION FROM STATION TO BASELINE (m) WETTZELL GRAZ 302.046 KOOTWIJK HERSTMONCEUX 406.737 PIETOWN MCDONALD 556.708 KOOTWIJK WETTZELL 602.529 GOLDSTONE PEQ QUINCY 618.448 ALGONQUIN WESTFORD 642.554 KOOTWIJK ONSALA 700.521 GRAZ MATERA 719.368 ONSALA METSAHOVI 784.249 GOLDSTONE PEQ PIETOWN 810.969 TIDBINBILLA HOBART 832.192 PENTICTON QUINCY 1042.983 TROMSO METSAHOVI 1081.879 MADRID HERSTMONCEUX 1211.605 YELLOWKNIFE PENTICTON 1495.415 ST. JOHNS WESTFORD 1572.091 YELLOWKNIFE FAIRBANKS 1630.820 MADRID MASPALOMAS 1745.467 KOUROU FORTALEZA 1884.078 USUDA CURREN TAIPEI 2016.770 123 Table 11 (continued) RICHMOND WESTFORD 2044.528 MCDONALD RICHMOND 2369.748 TIDBINBILLA YARRAGADEE 3197.086 KOUROU RICHMOND 3651.024 HERSTMONCEUX ST. JOHNS 3739.120 KOKEE PARK QUINCY 4075.402 KOKEE PARK PAMATAI 4431.379 SANTIAGO KOUROU 4546.786 FAIRBANKS USUDA CURREN 5510.427 YARRAGADEE TAIPEI 5795.876 HARTEBEESTHO FORTALEZA 7027.188 Finally, the double-differenced ion-ffee phase ranges are constructed between the satellites and the stations over the chosen r -1 baselines. An elevation cutoff of 20 degrees is used. A few cycle slips still remain in the data collected above the elevation angle of 20 degrees. These cycle slips are handled in one of two ways; either they are edited from the data or a new phase range bias is introduced. The corrections for the receiver antenna phase difference of the LI and L2 are applied to the data (see section 3 .6). For the ROGUE type antennas the maximum effect at 90 degrees elevation angle of 0.018 m. The data are formatted for input into GEODYNII. The task is completed entirely by the routine FORMGA. Additionally, the various option cards used to set up the GEODYNII program run must be formulated for the dataset. This is partially completed by this routine as shown in Figure 4. If no a priori satellite initial elements are available, they can be constructed from the GPS Broadcast Ephemeris. The Broadcast Elements are given at one-hour intervals, the routine COMBNAV sorts the many RINEX NAV files to provide an ordered, 124 concatenated file of GPS Broadcast Ephemeris The broadcast elements are transformed from the ECEF to the TOD as performed in BEEPCE also shown in Figure 4. 5.5 Experiment A, B, and C, Batch Least-Squares GPS Orbit Determination Using IGS Tracking Stations For each of the three deterministic modeling approaches, the GPS orbits are determined using 19,352 ion-free double-differenced observations. The observations are spaced at 5 minute intervals. The estimated parameters and their a priori uncertainties are given in Tables 4, 5, and 6 for experiments A, B, and C, respectively. The RMS2 of the solutions are given in Table 12. Table 12. Experiments A, B, and C, Batch Least-Squares GPS Satellite Orbit RMS2 of Fit (m) Using IGS Tracking Station Data Experiment RMS2 of Fit (m) A 0.012 B 0.011 C 0.011 The RMS2 of fit shows no significant variation among the three modeling approaches. The two-parameter model in experiment A, which includes only the solar radiation pressure scale factor and the y-bias acceleration terms, is as appropriate as the other more complex models for GPS satellite orbits determination. The batch least- 125 squares determined orbits from this study were compared to the Final IGS GPS satellite orbit by solving for the 7-parameter transformation between the Final IGS GPS satellite orbits and the batch least-squares GPS satellite orbits These transformation parameters are given in Table 13. Comparing the transformation parameter values of any of the three least-squares determined orbits with the transformation parameter values of the IGS orbits among the IGS analysis centers given in Table 8 shows no significant systematic difference between the orbits solved for in this study and those of the IGS analysis centers. Table 13. Transformation Parameters Between GEODYNII GPS Satellite Orbits and IGS GPS Satellite Orbits for Experiments A, B, and C on Jan. 16,1994 Using Ion-Free Double-Differenced Phase Ranges Experiment DX DY DZ RX RY RZ S RMS, (m) (m) (m) (mas) (mas) (mas) (ppb) (m) A -0.016 -0.021 -0.026 0.4 1.2 -0.1 0.6 0.26 B -0.015 -0.020 -0.024 0.3 1.1 -0.1 0.4 0.25 C -0.016 -0.022 -0.025 0.3 1.1 0.0 0.4 0.25 Also, the RMS, values of 0.26 m, 0.25 m, and 0.25 m are comparable with the RMS differences among the IGS analysis center orbits. The level of RMS, can be interpreted to show that all the three deterministic dynamic modeling experiments are compatible at the 25 cm level for one-day arcs. Again the two-parameter approach of experiment A is sufficient for 26 cm level orbit determination. A small improvement is evident by replacing the two-parameter model with the more complex model of experiment B, which includes 4 parameters: the solar radiation pressure scale factor, the radial, GPS X-axis, GPS Y-axis acceleration terms. For one-day arcs at the 25 cm level, there does not appear to be any gain by including the more complex 10-parameter model. CHAPTER VI ORBIT DETERMINATION USING A SQUARE ROOT INFORMATION FILTER/SMOOTHER AND STOCHASTIC MODELS In Chapter 5, the deterministic dynamic models were tested in the batch least- squares estimator by fitting to IGS orbits. Then, the batch least-squares estimator solved for the GPS satellite orbits using only ion-free double-differenced tracking data from IGS tracking stations. Three different deterministic models, Experiments A, B, and C, were implemented. In this chapter, experiments A and B are further investigated to ascertain if the deterministic dynamic models and/or the measurement models can or should be modeled as stochastic time-varying processes. The ion-free double-differenced phaserange residuals from the Experiment A solution are used to construct an empirical autocorrelation function. This empirical autocorrelation is compared to the known autocorrelation functions of several stochastic process models. Based on these comparisons and general characteristics of the physical phenomena, stochastic models are selected. Based on the nominal magnitudes of the residuals and the characteristics of the empirical autocorrelation function, a range of values is selected for the continuous process noise standard deviation and the correlation time. The chapter includes an examination of the stochastic modeling of the tropospheric refraction zenith delay. Also, experiments are conducted to access the 127 128 effectiveness of the model and the most appropriate process noise standard deviation and correlation time for modeling the solar radiation pressure scale factor and the ‘residual’ acceleration. 6.1 Empirical Autocorrelation Functions In this section, the stochastic processes evident, if any, in the data are investigated. The approach used is the standard technique of computing the autocorrelation from the data (Davis, 1986 and Gelb, 1974). The empirical autocorrelation function is compared to the autocorrelation functions of several common stochastic processes. In this case, the data are actually the ion-free double-differenced phaserange residuals from the deterministic dynamic model fit when solar radiation pressure scale factor, and y-bias acceleration are modeled as constant parameters. If the deterministic model is inadequate, the process may be evident in the residuals Experiment A is the simplest dynamic model implemented in this study. This is the standard model, and as explained in Chapter 1, it is widely used. Thus, it is appropriate as the basis for the consideration of additional modeling. A sample period of about 3000 seconds of the ion-free double-differenced phaserange residuals from Experiment A are shown in Figure 8. The double-differenced observable involves two stations and two satellites. With n satellites visible, n-\ independent double-differenced observables, involving the two stations, are possible. To include the total effect at a station, the residuals from all observations involving the station are averaged at each epoch and the empirical autocorrelation function is computed and shown in Figure 9 Residuals (m) Figure 8. Example of Ion-Free Double-Differenced Residuals on on Residuals Double-Differenced Ion-Free of Example 8. Figure -0.04 -0.05 -0.03 - - 0.02 0.03 0.04 0.02 0.00 0.01 0.01 0.01 0.01 10 20 3000 2000 1000 0 Experiment A. Model, Experiment Accelerations Y-Bias and Scale Radiation Solar Pressure Using Solution Orbit 1994, 16, Jan. eod o Day ofSeconds 129 Time (minutes) Figure 9. Autocorrelation Function of Ion-Free Double-Differenced Residuals on Jan. 16, 1994, for Baseline WETT to GRAZ and Orbit Solution Using Solar Radiation Pressure Scale Y-Bias Accelerations Model, Experiment A. There is a correlation evident in Figure 9. Several questions immediately arise. First, is this process stationary and ergodic? To attempt an answer, the autocorrelation function is recomputed after resetting the ‘0’ lag location to different times, approximately four hours apart. This was repeated for 6 different initializations. Basically, the same correlation appeared in all realizations of the autocorrelation function. The variances ranged from 1.55 to 1.76 cm2. The conclusion is that the decaying correlation is real. Since the process does not exhibit a dependence on the time, but only on the time difference it is assumed to be a stationary process. Ergodicity cannot be established from the data available, it is assumed the process is ergodic. 131 The next question to ask is what type of process is it. In Table 14 are the autocorrelations for several simple stochastic processes. In Figure 10 is the autocorrelation function for a random number time series, i e. a discrete version of white noise that has no correlation after lag 0. Clearly the empirical autocorrelation function in Figure 9 does not resemble that of the random constant, random walk, or random ramp. The autocorrelation function in Figure 9 has the general shape of the autocorrelation function of an exponentially correlated random variable. However, the empirical autocorrelation crosses the abscissa whereas the exponential autocorrelation function does not. In this sense, the autocorrelation function in Figure 9 somewhat resembles the random number time series autocorrelation, but clearly the empirical autocorrelation function is correlated after lag 0, whereas the random number time series is not. A least- squares fit of the first-order Gauss-Markov autocorrelation function yields a correlation time of 72 minutes and a process noise standard deviation of 2.3 cm. But, can anything be said about the underlying physical processes? Is it caused by mismodeled measurements or unmodeled accelerations? The correlation time is approximately the time variation reported by researchers modeling the tropospheric refraction zenith delay variation (Tralli et al., 1988). In Experiment A, the tropospheric refraction zenith delay variation has been estimated as a constant over two 12-hour periods. This may be too ‘coarse’ a model to absorb sufficiently the tropospheric refraction zenith delay variation. In the next section, the tropospheric refraction scale bias is modeled as a stochastic process Table 14. Autocorrelation Functions of Selected Stochastic Models (Gelb, 1974) NAME AUTOCORREl A! ION FuNCtlOr^ < f(Y ) RANDOM CONSTANT t . r ) RANDOM qt I WALK / i •t e [.’( oi ] RANDOM RAMP ' -"■ t " ' ■ i ! i EXPONENTIALLY CORRELATED RANDOM VARIABLE v>( t ) i PERIODIC I if (0 )•' ^ Tco» I w | r I) | RANDOM VARIABLE \ — r 133 I i l l z -1 J i— r O 5 10 15 20 25 Time (minutes) Figure 10 Random Number Time Series Autocorrelation Function (Davis, 1986). 6.2 Stochastic Modeling of the ‘Wet' Tropospheric Refraction Zenith Delay The physical relationship defining the tropospheric model was developed in section 3 .3 .2. The a priori dry component of the zenith troposphere is nominally 2.200 m. The tropospheric refraction scale bias is the scale bias of that value, i.e. tropospheric refraction scale bias of 0.1 is 0.220 m. This is just the convention of the software used for this study. The variation is dominated by the ‘wet’ tropospheric zenith delay. Herring et al. (1990) adopted a random walk model after their simulation study indicate it is a sufficient model for the ‘wet’ tropospheric zenith delay in VLBI solutions. 134 They did not explicitly test a first-order Gauss-Markov model. Lichten and Bertiger (1989, p. 174) state that a random walk model of the tropospheric zenith delay gave the best baseline repeatability and agreement with VLB I They did not state what other process noise models, if any, were tested. A year later, Tralli and Lichten (1990) used a random walk and first-order Gauss Markov model. They conclude that either model is sufficient. The IERS recommends modeling the variation in the ‘wet’ tropospheric zenith variation as a random walk (McCarthy, 1992). Clearly, from the empirical autocorrelation functions in Figure 9, the shape is that of an exponentially correlated process. The first experiment will model tropospheric refraction scale bias (essentially the ‘wet’ tropospheric refraction zenith variation) as a random walk, because this is a widely adopted model and as a first-order Gauss-Markov process, because this is more evident from the empirical autocorrelation function. The deterministic dynamic model includes the solar radiation pressure scale factor, and the y- bias acceleration term. The parameter selection and a priori uncertainties are given in Table 15. For the random walk, typical values found in the literature for the continuous process noise standard deviation is 1 cm / Vhr to 2 cm / Vhr (Herring et al., 1990). The correlation time and process noise standard deviation values in Table 16 are those taken to bracket the values determined by the least-squares fit. As discussed in section 4.6, the ion-free double-differenced phaserange residuals must be decorrelated and normalized to be processed in the SRIF/SRIS algorithm. The full covariance matrix for the double-difference range data must be constructed since GEODYNII does not accept or provide measurement correlations At any particular 135 epoch, the m = (#stations-1) x (^satellites— 1) linear independent double difference range data types can be formed These m observations are independent in the sense of linear algebra, but are statistically correlated Each of the m observations has the form given in equation (3-5). For small regional networks, a single satellite-station pair is selected as the base satellite-station and the m observations are constructed by differencing the remaining satellite-stations with the base pair. For a global network, the distance between stations may prevent using a single base pair to construct all observations at that epoch. Thus, no consistent numerical structure exists that would permit a symbolic construction of the decorrelated measurement set. Pv and Lv' must be computed (GEODYNII does not provide Pv or L'.1) numerically at each epoch. Pv is computed using conventional error propagation P(, = ct,2 GGr (6-2) where o) is the standard deviation of the single one-way range measurement, G is the matrix of partial derivatives of the observation equation with respect to the one-way range. The GGr matrix contains only the elements -1,0,1 which form the linear combination of one-way ranges that define the double difference. The decorrelated residuals set with unit (normalized) covariance is obtain from equations (4-76) and (4-77). After the batch least-squares estimator (GEODYNII) solution for Experiment A has converged, the measurement partials, (FORT.90) and variational partial derivatives (FORT. 80) are output With OSUORBFS, the square root information 136 filtered/smoothed solution is computed as discussed in Chapter 4 and Appendix A The RMS,’s of fit of the OSUORBFS solutions, using the various combinations of stochastic model and defining parameters, are given in Table 16 With the first-order Gauss-Markov model, for a given yjqcnn, as the r increase the RMS3 increase. This is consistent with equation (4-68) that shows that for a given cIcon A/, as r increases, q Ai decreases, and the transition M from equation (4-67) approaches one. Thus, the model is more tightly constrained. In this case, the model ‘relies’ more on the ‘state’ than on the ‘measurements’ and the RMS, increases. With both the random walk and the first-order Gauss-Markov models, for a given r , as the iJqcon decrease, the RMS3 increase. For the first-order Gauss-Markov model, this is consistent with equation (4-68) that shows that for a given r and A/, as qmn decreases, qds decreases, and M is unchanged. Thus, the model is more tightly constrained. For the random walk, this is consistent with equation (4-70) that shows that for a given A/, as q con decrease, q As decrease, and the model is more tightly constrained. In this case, for both models, the model ‘relies’ more on the ‘state’ than on the ‘measurements’ and the RMS3 increases. In Table 17 are the transformation parameters from the Final IGS GPS satellite orbits to the OSUORBFS estimated GPS satellite orbits using the ion-free double- differenced observable and the stochastic model combinations given in Table 16. In Table 17, in general, there is an improvement of approximately 2-3 cm in RMS, over the purely deterministic model when the tropospheric refraction zenith variation was 137 modeled as a constant over two 12-hour spans There is little variation in results among the two different stochastic models. The random walk models and first-order Gauss- Markov models both show approximately the same level o f improvement in the RMS, From Table 17, it is not immediately clear which is better, the random walk or first-order Gauss-Markov models. This is consistent with the findings of Tralli and Lichten (1990). But, based on the fact the empirical autocorrelation function most closely resembles a first-order Gauss-Markov model, the first-order Gauss-Markov model with 1-hour correlation time and 2 cm process noise standard deviation is recommended. This is different than the current standard of practice of using a random walk model (McCarthy, 1992). However, ffom a purely pragmatic point of view, it does not really appear to matter which stochastic model is used. Table 15. Experiment D, Square Root Information Filter/Smoother Estimated Parameters on Day 016,1994 PARAMETER OSUORBFS DESCRIPTIONS No. Type A priori sigma V^con r satellite epoch 156 3 .35 m elements 3.7 xlO"5 m /s2 solar radiation 26 5 10 pressure scale Y-bias acceleration 26 5 1x10 7 m / s2 coefficient 1 3 8 Table 15 (continued) double difference 895 4 100 m biases tropospheric refraction 68 4 0.1 (0.22 m) (1) (1) scale bias 68 5 0.1 (0.22 m) station clock bias 34 4 1x10 6 s polar motion 4 5 0.1 mas and UT1 2 5 1xlO6s station 36 4 0.001 m positions 60 4 0.100 m Total 1375 (1) See Table 16 for yjqcon and rvalues. Table 16. Experiment D, OSUORBFS RMS3 of Fit (cm) Using First-Order Gauss-Markov and Random Walk Models for Tropospheric Refraction ’Wet’ Zenith Delay con First-Order Gauss-Markov Random Walk - r = l hr t -2 hrs r - 4 hrs - 0.03 m 2.49 2.62 2.74 2.21 0.02 m 2.81 3.00 3.20 2.45 0.01 m 3.24 3.44 3.72 2.94 139 Table 17. Transformation Parameters from the IGS GPS Satellite Orbit to the OSUORBFS Orbits for Experiment D on Jan. 16,1994 Using First-Order Gauss-Markov and Random Walk Models for Tropospheric Refraction ’Wet’ Zenith Delay Model r DX DY DZ RX RY RZ S RMS, co n hr m m m mas mas mas PPb m FOGM 0.03 m 1 0.006 -0.005 0.007 -0.2 -0.1 0.1 -0.3 0.23 FOGM 0.02 m 1 0.022 -0.008 0.007 -0.1 0.0 -0.4 -0.3 0.23 FOGM 0.01 m 1 0.016 0.008 -0.008 0.0 0.1 -0.1 -0.3 0.23 FOGM 0.03 m 2 0.000 -0.020 0.013 -0.2 0.2 -0.1 -0.4 0.23 FOGM 0.02 m 2 0.003 -0.002 0.006 0.0 0.0 -0.3 -0.3 0.23 FOGM 0.01 m 2 0.003 0.010 0.001 0.0 -0.1 -0.1 -0.3 0.24 FOGM 0.03 m 4 0.009 -0.021 0.017 -0.2 0.1 -0.1 -0.4 0.23 FOGM 0.02 ni 4 0.001 -0.009 0.013 -0.1 0.1 0.2 -0.3 0.23 FOGM 0.01 m 4 -0,002 0.008 0.002 -0.2 -0.1 -0.1 -0.4 0.24 RW 0.03 m - 0.004 -0.021 0.023 0.1 -0.1 4). 2 0.1 0.24 RW 0.02 m - -0.006 -0.017 0.014 -0.1 0.2 -0.1 -0.1 0.24 RW 0.01 m - 0.006 0.021 0.006 -0.1 -0.1 -0.2 -0.4 0.23 Note: FOGM = First-Order Gauss-Markov model and RW = Random Walk model. In Figure 11 is an example of the variation in the tropospheric refraction zenith delay modeled as a random walk at Wettzell on January 16, 1994. •“ -0.02 ------* r------"i------,------1----- 0 20000 40000 60000 80000 Seconds of Day Figure 11. Random Walk Variation in the Tropospheric Refraction Zenith Delay at Wettzell on January 16, 1994 with yjq con = 1 cm / Vhr. The deterministic model of the tropospheric refraction zenith variation was revisited. One might think that the orbit could be improved simply by decreasing the time span over which the batch least-squares estimator estimates a constant for the tropospheric refraction zenith delay. Thus, an additional solution was performed estimating the tropospheric refraction zenith delay modeled as a constant over consecutive four-hour time spans. The RMS, of fit of the transformation from the Final IGS GPS satellite orbits to the batch least-squares estimated GPS satellite orbits is reduced from 26 cm to 25 cm. This is an improvement over the 12-hour fits, but not as good as using a stochastic model. 141 After implementing the stochastic models for the tropospheric refraction zenith variation, it is reasonable to look again at the empirical autocorrelation function. In Figure 12 is the empirical autocorrelation function of the ion-free double difference residuals after including a first-order Gauss-Markov model for the tropospheric zenith variation, Experiment D. This autocorrelation function closely resembles that of a random number time series given in Figure 10. Thus, no process remains evident in the residuals. 1.1 0.8 —t> 0.7 o o 0.5 0.3 E 0.1 o V/' r - 0.1 “T‘ "T ~ ” T~ 10 20 30 40 50 60 70 Lag Figure 12. Autocorrelation Function of Ion-Free Double-Differenced Residuals on Jan. 16, 1994, OSUORBFS Solution Using a First-Order Gauss-Markov Model for the Troposphere ‘Wet’ Zenith Delay, Experiment D. 142 6.3 Experiments E and F, Parameter Selection There is no satisfactory approach to generate an empirical autocorrelation correlation function for the accelerations from the range residuals. Lichten and Border (1989), for example, justify the selection of first-order Gauss-Markov models for modeling the residual satellite accelerations based on the fact that they improve the baseline repeatability and agreement with VLBI estimates of the baseline. It is not clear which, if any, other stochastic models were tested. In this experiment the first-order Gauss-Markov model is compared with a random walk model, to verify that one-day orbits benefit from stochastic modeling of the acceleration, and to investigate which of these two models is better for modeling the ‘residual’ acceleration. The analysis will proceed by implementing the first-order Gauss-Markov model and the random walk model for the deterministic dynamic models used in Experiment A and B. Values of the process noise standard deviation are selected as 0.1%, 1%, and 10% of the nominal acceleration magnitude. As noted in Chapter 2, over 99% of the accelerations are modeled deterministically. The above combination should encompass a realistic range of possible process noise standard deviations. Lichten et al. (1989) estimated week-long orbits using correlation times of 0.5 days to several days. They achieved sub-meter GPS orbits over a week-long GPS orbit. In this study, one-day GPS orbits are estimated with approximately 25 cm accuracy. Since this level of accuracy is achievable with bias terms (e.g. Experiment A, B, and C) stochastic processes that may remain may have correlation times less than or equal to one day. This is supported by Colombo’s (1986, 1989) derivations since he has shown dynamic effects are ‘filtered’ 143 into effects with a period corresponding to the nominal orbit period Thus, the correlation times are chosen as one-half, one, and two times the nominal orbit period of the GPS satellite Both first-order Gauss-Markov and random walk models are implemented The combinations tested are for correlation times of 6 hours, 12 hours, and 24 hours, and yfq ^ values of 0.1, 0.05, and 0.01 for the solar radiation pressure scale, 1 x 10~8 m / s2, 1 x 10 9 m / sz, and 1 x 10~'° m / s2 for the accelerations. The estimated parameters and a priori sigmas, initializing the square root information filter/smoother in Experiment E, are given in Table 18. The correlation times and continuous process noise values given in Table 20. The a priori sigmas of the satellite epoch element position and velocity are the a posteriori sigmas from the GEODYNII solution of 0.35 m and 3.7 x 10-5 m/s2, respectively. For the solar radiation pressure scale and y-bias acceleration, each parameter is estimated as a bias and as a stochastic parameter. The stochastic processes should be zero mean, and this is accomplished by simultaneously estimating a bias term (Tralli and Lichten, 1990, p. 131). The solar radiation pressure scale and y-bias values are lightly constrained. The stochastic process, is controlled more by the process noise standard deviation and the correlation time from Table 20. The various combinations of process noise standard deviation and correlation time that were tested are given in Table 20. 144 Table 18. Experiment D, Square Root Information Filter/Smoother Estimated Parameters on Day 016,1994 PARAMETER OSUORBFS DESCRIPTIONS No. Type A priori sigma con r satellite epoch 156 3 .35 m elements 3.7 x 10 "J m/s2 solar radiation 26 5 10 pressure scale 26 1 10 (1) (1) Y-bias acceleration 26 5 1 x 10 7 m / s2 coefficient 26 1 1 x 10 7 m/s2 (1) (1) double difference biases 895 4 100 m tropospheric refraction 68 4 0.1 scale bias 68 5 0.1 0.02 m 70 min station clock bias 34 4 1x10 6s polar motion 4 5 0.1 mas and UT1 2 5 lxlO^s station 36 4 0.001 m positions 60 4 0.100 m Total 1427 (1) See Table 20 for and r values. The estimated parameters and a priori sigmas initializing the square root information filter/smoother in Experiment F are given in Table 19. The a priori sigmas for the satellite initial elements are the a posteriori sigmas from the GEODYNI1 solution The solar radiation pressure scale is estimated only as a bias with large uncertainty. The radial, GPS X-axis and GPS Y-axis accelerations are estimated as biases and as 145 stochastic parameters with large a priori uncertainties. The process noise standard deviation and correlation times are given in Table 22. The solar radiation pressure scale should not realistically vary that much over 1 day Thus, it seems appropriate to model it just as a bias. Conceptually, the small residual accelerations that the GPS satellite experiences are appropriately modeled as time-varying acceleration resolved along each of the three GPS spacecraft axes. Therefore, only the radial, GPS X-axis, and GPS Y- axis biases are estimated as stochastic processes. Table 19. Experiment F, Square Root Information Filter/Smoother Estimated Parameters on Day 016,1994 PARAMETER OSUORBFS DESCRIPTIONS No. Type A priori sigma •\Al con r satellite epoch 156 3 .35 m _ elements 3.7x10’ m/s2 solar radiation pressure 26 5 10 - - scale radial acceleration 26 5 0.1 m/s2 bias 26 1 0.1 m/s2 (1) (1) GPS X-axis 26 5 0.1 m / s2 acceleration bias 26 1 0.1m/ s2 (1) (1) GPS Y-axis 26 5 0.1 m/s2 acceleration bias 26 1 0.1 m / s2 (1) (1) double difference biases 895 4 100 m tropospheric refraction 68 4 0.1 scale bias 60 5 0.1 0.02 m 70 min 146 Table 19 (continued) station clock bias 34 4 1 x 10"6 s polar motion 4 5 0.1 mas and UT1 2 5 lxlO'6S station 36 4 0.001 m positions 60 4 0.100 m Total 1505 (1) See Table 22 for >/qcon and r values. 6.4 Stochastic Model Results of the Solar Radiation Pressure Scale Factor and Y-Bias Acceleration After the batch least-squares estimator (GEODYNII) solution for Experiment A has converged, the measurement partial derivatives, (FORT.90) and variational partial derivatives (FORT.80) are output. The square root information filtering/smoothing (OSUORBFS) solution is computed as discussed in Chapter 4 and Appendix A. The tropospheric refraction zenith delay is modeled as a first-order Gauss-Markov process as discussed in section 6.2. As shown in Table 18, nine first-order Gauss-Markov and three random walk models of the solar radiation pressure scale and y-bias acceleration have been implemented. The RMS3 of fit of the OSUORBFS solution for each of twelve solutions is given in Table 20. The OSUORBFS RMS3 of fit ranges from 2.25 cm for the random walk model with ^ q con =0. 1, 1 x 10 ® m / s2 for the solar radiation pressure 147 scale and y-bias acceleration, respectively, to 3.22 cm for the first-order Gauss-Markov model with - J q ^ =0.01, 1 x 10 ‘ m / s2, and r = 24 hours. With the first-order Gauss-Markov model, for a given yjqc0n, as the r increase the RMS, increase. This is consistent with equation (4-68) that shows that for a given qcon and At, as x increases, qAs decreases, transition M from equation (4-67) approaches one, and the model is more tightly constrained. In this case, the model ‘relies’ more on the ‘state’ and does not ‘relax’ to fit the ‘measurements’ and the RMS, increases. With both the random walk and the first-order Gauss-Markov models, for a given x , as the -Jqcon decrease, the RMS, increase. For the first-order Gauss-Markov model, this is consistent with equation (4-68) that shows that for a given x and At, as qcon decreases, qds decreases, M is unchanged, and the model is more tightly constrained. For the random walk, this is consistent with equation (4-70) that shows that for a given At, as qcon decrease, q As decrease, and the model is more tightly constrained. For both models in this case, the model relies more on the ‘state’ and does not ‘relax’ to fit the ‘measurements’ and the RMS, increases. On the basis of the RMS, fit alone, the smaller values would suggest using a random walk, but this judgment should be delayed until after the solutions are compared with the IGS orbits, in the following. The transformation parameters from the Final IGS GPS satellite orbits to the Experiment E OSUORBFS GPS satellite orbits using ion-free double-differenced phase ranges were determined. These transformation parameters are given in Table 21. The 148 RMS, of fit varies from 22.5 cm to 25.4 cm. The RMS, values indicate that the fit to the Final IGS GPS satellite orbits is improved by introducing the stochastic models The random walk models have a slightly larger RMS, of fit. Table 20. Experiment E, OSUORBFS (Smoothed) RMS3 of Fit (cm) Using First-Order Gauss-Markov and Random Walk Models for the Solar Radiation Pressure Scale and Y-Bias Acceleration and Ion-Free Double-Differenced Phase Ranges ■\Alcon First-Order Gauss-Markov Random Walk s.r.p.s. r =6 hrs r =12 hrs r =24 hrs - y-bias 0.1 2.38 2.45 2.55 2.25 1 x 10“* m / s2 0.05 2.67 2.89 2.92 2.38 1 x 10“9 m / s2 0.01 3.01 3.10 3.22 2.53 1 x KT10 m / s2 Table 21. Transformation Parameters Between IGS Orbit and OSUORBFS Orbit for Experiment E on Jan. 16,1994 Model DX DY DZ RX RY RZ S RMS, 4 FOGM 0.1 6 0.013 -0.008 0.016 -1.2 0.9 -0.3 -0.4 0.229 1 x 10-8 m / s2 FOGM 0.05 6 0.004 -0.001 0.017 -0.1 1.1 -0.4 -0.3 0.234 1 x 10-9 m / s2 149 Table 21 (continued) FOGM 0.01 6 0.016 0.002 0.003 0.8 0.7 -0.2 -0.3 0.236 lx 10~10m /s 2 FOGM 0.1 12 0.009 -0.021 0.017 -1.2 0.4 -0.6 -0.4 0.225 1 x 10~8m / s2 FOGM 0.05 12 0.003 -0.008 0.010 0.9 -0.8 -0.3 -0.3 0.232 lx 10~9m /s2 FOGM 0.01 12 0.017 0.006 0.006 0.8 -1.1 -0.1 -0.2 0.229 1 x H f 10m /s 2 FOGM 0.1 24 0.006 -0.021 0.013 -0.2 0.6 -0.8 -0.4 0.231 1 x 10 8 m / s2 FOGM 0.05 24 0.001 0.005 0.004 1.0 0.1 -0.6 -0.3 0.225 lx 10~9m /s2 FOGM 0.01 24 -0.006 0.011 0.004 0.9 1.1 -0.1 -0.3 0.224 . lf>—10 , 2 1 x 10 m / s RW 0.1 - -0.009 -0.010 0.017 0.5 0.1 -0.2 0.0 0.254 1 x10~8 m/s2 RW 0.05 - -0.017 -0.022 0.014 -0.6 1.1 -0.1 -0.1 0.246 1 x 10 9 ml s2 RW 0.01 - -0.008 -0.020 0.013 -0.6 0.2 -1.2 -0.3 0.244 1 x 10 10ml s2 Note: FOGM = First-Order Gauss-Markov model and RW = Random Walk model. The first-order Gauss-Markov model ( ^ q con - 0.05 and r = 12) hours for the solar radiation pressure scale factor correction for PRN02 is shown in Figure 13 . The variation in the correction factor is about 0.4% of the factor’s nominal value of 1.0 This order of magnitude appears reasonable per the discussion in section 2.3. 1 5 0 0.004 £ om u> 0.003 2 «= 3 O M 0 .0 0 2 M O o © FOGM cO 0.001 O • 2 0.000 1 - -0.001 o CO -0 .0 0 2 0 10000 20000 30000 40000 50000 60000 70000 Seconds of Day Figure 13. Experiment E, Solar Radiation Pressure Scale Factor Correction for PRN02 Using a First-Order Gauss-Markov (FOGM) Model yj^con = 0-05 and r = 12 Flours. The random walk model (-Jqcon = 0.05) of the same correction is shown in Figure 14. Typically, the correction factor is -0.002, but it has a large departures from +0.08 and to -0.06. This range of values is not realistic as discussed in section 2.3. The values should vary about 4% over one year. Here the range is about 14% over one day. Admittedly, these departures occur at just a few points, around 20000 seconds of day, and near end point, at approximately 70000 seconds of day. However, the first-order Gauss-Markov model appears less inclined to these large variations. v i - 0.10 — j i r i r r r 0 10000 20000 30000 40000 50000 80000 70000 Seconds of Day Figure 14. Experiment E, Solar Radiation Pressure Scale Factor Correction Sigma Using a Random Walk (RW) Model with ^ q can = 0.05. The sigma values of the solar radiation pressure scale factor correction from the first-order Gauss-Markov model and the random walk models are compared in Figure 15. The random walk has large variations of 0.02 to 1.0 which appears excessive. The first-order Gauss-Markov ranges from 0.006 to 0.02 which are more reasonable, based on the anticipated magnitudes of the solar radiation pressure scale factor correction as discussed in section 2.3. 152 10 FOGM CO RW M C w .001 0 10000 20000 30000 40000 50000 60000 70000 Seconds of Day Figure 15. Experiment E, Comparison of Solar Radiation Pressure Scale Factor Correction Sigma for PRN02 Using a First-Order Gauss-Markov (FOGM) Model with yjqmn =0.05 and r = 12 Hours and Random Walk (RW) Model with J q ^ n = 0.05. The y-bias acceleration model estimate and sigmas using the first-order Gauss- Markov ( ^ q con = 1 x 10 9 m / s2 and r = 12 hours) and random walk models ( ^ q con = 1 x 10~9 m / s2 ) are compared in Figure 16 and 17, respectively. Again, in general, the random walk has more variation, especially near the end points, and changes more rapidly than the first-order Gauss-Markov model. However both models give physically meaningful results over the entire data span. Figure 16. Experiment E, Comparison of Y-Bias Acceleration for PRN02 for Acceleration Y-Bias of Comparison E, Experiment 16. Figure Y-Blas Acceleration (m/s**2) 10 10 10 -10 '* -8 ih 1 910x = with with (FOGM) Model Gauss-Markov First-Order a Using 1 x 10 9 m / s2 and 1s2 9 and / m x10 10000 20000 30000 40000 50000 80000 70000 RW FOGM eod o Day ofSeconds - r m/s2. 12 Hours and a Random Walk (RW) Model Model (RW) Walk Random a and 12Hours \ I i / 1 Jcn - yJqcon 153 154 t\T ID'7 FOGM «* E -8 RW CD 10 CO 10-e ® o WtYlYY^YVWWVYwVt^YiWrt o < m co ■10 m >- 10000 20000 30000 40000 50000 60000 70000 Seconds of Day Figure 17. Experiment E, Comparison of Y-Bias Acceleration Sigma for PRN02 Using a First-Order Gauss-Markov Model (FOGM) with yjqc„„ = 1 x 10 9 m / s2 and r = 12 Hours and a Random Walk (RW) Model with J q Z , = 1 x 10"9 m/s2. The random walk model varies more rapidly with larger variations than the first- order Gauss-Markov model as shown in Figures 13 to 17. The random walk models have a smaller RMS3 of fit than the first-order Gauss-Markov model. However, when compared to the Final IGS orbit the first-order Gauss-Markov have RMS, of fit approximately 1 cm smaller. Thus, the first-order Gauss-Markov models appears to model better the solar radiation pressure scale factor correction and y-bias acceleration. It is not clear that any of the tested correlation time and process noise standard deviation gives significantly better results than any other. Thus, the first-order Gauss-Markov model with any of the tested defining values is an acceptable model. 155 6.5 Stochastic Modeling Results of the Radial, GPS X-Axis and GPS Y-Axis Accelerations This is Experiment B combined with the tropospheric refraction zenith variation modeled as a first-order Gauss-Markov process as discussed in section 6.2. The process noise standard deviation for the accelerations used in the first-order Gauss-Markov and random walk models and the correlation times for the first-order Gauss-Markov model are given in Table 22. The solar radiation pressure scale is not estimated as a stochastic parameter. The OSUORBFS smoothed solution RMS3 of fit for the radial, GPS X-axis, and GPS Y-axis accelerations as first-order Gauss-Markov and random walk models are given in Table 22. Table 22. Experiment F, OSUORBFS RMS3 of Fit (cm) Using First-Order Gauss-Markov and Random Walk Models for Radial, GPS X-Axis, and GPS Y-Axis Accelerations and Ion-Free Double-Differenced Phase Ranges V q con First-Order Gauss-Markov Random Walk Acceleration r=6 hrs r= 1 2 hrs r =24 hrs - 1 x 10"8 m / s2 1.42 1.45 1.58 1.38 1 x 10“9 m / s2 2.58 2.67 2.82 3.22 1 x 10"'° m / s2 2.64 2.72 2.92 1.80 1 5 6 The RMS, of fit ranges from 1.38 cm for a random walk model with yjqcon = 1 x 10 '° m / s2 to 3.22 cm for a random walk model with yjq^ = 1 x 10'9 m / s2. In general, with the first-order Gauss-Markov model, the RMS, increase for a given t and decreasing y]qcn„ or for a given yjqcm and increasing correlation time. This follows from the same reasoning developed in sections 6.2 and 6.4. However, the random walk estimates exhibit very curious behavior as the -Jqcon is decreased. It changes by a factor of 2 between the -yjq ^ - 1 x 10“9 m / s2 and 1 x 10 10 m / s2. This erratic behavior makes the random walk suspect as a model for the radial, GPS X and Y axes accelerations. This will become more clear when the solution is compared to the IGS orbits. The transformation parameters from the Final IGS GPS satellite orbits to the OSUORBFS estimated GPS satellite orbits for Experiment F are given in Table 23. First, the random walk models at -yjqcon = 1 x 10~‘ m / s2 and = 1 x 10“9 m / s2 have large RMS, values of 1.052 m and 0.553 m. The random walk model with *jqcnn = 1 x 10~10 m / s2 is more consistent with the RMS, values of the first-order Gauss-Markov model. But, all random walk models have relatively large scale factor corrections. Apparently, the random walk allows the radial acceleration to vary undesirably. This is explored in more detail, later in this section. For the first-order Gauss-Markov models, the RMS, has no discernible trend with correlation time and process noise standard deviation. The smallest RMS, (0.226 m) occurs for correlation time of 12 hours, and continuous process noise standard deviation of 1 x 10 9 m / s2. However, there is really no significant difference between the correlation times of 6, 12, and 24 hours. This 1 5 7 result is consistent with Lichten et al.(1989, p. 175) who found the results were insensitive to the correlation time. Table 23. Transformation Parameters from the IGS GPS Satellite Orbit to the OSUORBFS Estimated Satellite Orbits for Experiment F on Jan. 16,1994 Using Ion-Free Double-Differenced Phase Ranges Model DX DY DZ RX RY RZ S RMS, •J^con r hr m m m mas mas mas PPb acceleration m - 8 , 2 -0.9 -0.4 0.231 FOGM 1 X m / s 6 0.076 -0.009 0.040 -0.5 -1.2 - 9 , 2 FOGM 1 x 0 m / s 6 0.016 0.028 -0.028 0.9 -0.4 1.3 0.9 0.228 FOGM 1 x 0_1°m / s2 6 -0.017 0.026 -0.018 1.4 -0.1 0.5 0.8 0.227 FOGM 1 x 0~8 m/ s2 12 0.039 -0.024 -0.012 0.1 -1.0 -0.5 -1.1 0.228 -9 . 2 0.5 0.7 0.226 FOGM 1 X m / s 12 0.028 0.029 -0.029 -1.4 -0.2 -1 0 , 2 1.5 0.233 FOGM 1 X m / s 12 0.028 0.016 -0.024 0.4 0.5 -0.3 -1.4 -0.5 -0.2 0.231 FOGM 1 X 8m/s2 24 0.029 -0.028 -0.014 0.2 0.228 FOGM 1 X 0 9 m / s2 24 0.018 0.025 -0.018 -0.6 0.2 0.4 -0.2 -1 0 . 2 0.9 0.233 FOGM 1 X m /s 24 0.021 0.020 -0.023 0.4 1.3 0.5 RW 1 x 0 ~8 m m // sr 2 - -0.063 0.221 -0.098 -1.4 -5.2 -4.8 -8.0 1.052 -2.0 -5.3 -7.2 0.552 RW 1 X 0 9 m / s2 - -0.028 0.068 0.025 -0.5 RW 1 x - 10m/s2 - 0.086 -0.024 0.006 0.2 -4.9 -2.2 -2.2 0.301 Note: FOGM = First-Order Gauss-Markov model and RW = Random Walk model. The first-order Gauss-Markov model (y]qcon - 1 x 10“9 m/s2 and r = 12 ) and random walk model (yjqc0n = 1 x 10 9 m / s2) for the radial acceleration is given in Figure 18. The random walk estimate of approximately 1 x 10"3 m / s2 is unrealistic. Apparently, even with y]qcon = 1 xlO 9 m/s2 the random walk allows large variation in the radial accelerations. On the other hand, the first-order Gauss-Markov show magnitudes consistent with the discussion in section 2.3. The random walk sigma shown in Figure 19 exhibits the same unrealistically large value The first-order Gauss-Markov sigma is again consistent with the physical understanding of the magnitude expected for the ‘residual’ accelerations. 10'2 w 10'3 « *M -4 4 10 10-5 A FOGM * ID’6 f- RW ® .7 ® 10 ' o < 10 -8 10 '" ca -10 cc 10 10000 20000 30000 40000 50000 60000 70000 Seconds of Day Figure 18. Experiment F, Comparison of Radial Accelerations Using a First-Order Gauss-Markov (FOGM) Model with yjqcon = 1 x 10 9 m / s2 and r = 12 Hours and Using a Random Walk (RW) Model with yJqcon= 1 x 10 9 m / s2. 159 — 10-1 CM 10 ' 0-3 :E . at 10 -4 (O FOGM 10* RW to 10' o 4> O 10 ' O < 10' j i i i ~ 5? \ ' . YV.. . . V. . . V . . .VvV.V.V. .V ..V Y y ..Y .i.Y l.iV .Y . Seconds oi Day Figure 19. Experiment F, Comparison of Radial Acceleration Sigmas Using a First-Order Gauss-Markov (FOGM) Model with y]qcon = 1 x 10'9 m / s2 and r = 12 Hours and Using a Random Walk (RW) Model with yjqcon = 1 x 10'9 m / s2. The first-order Gauss-Markov model ( ^ q co„ = 1x10 9 m / s2 and r = 12 ) and random walk model {yjqcon = 1 x 10'9 m/s2) for the GPS X-axis acceleration is given in Figure 20. The first-order Gauss-Markov estimate again is consistent with the anticipated magnitude. The random walk estimate’s magnitude is relatively large, but within the limits o f possibility. The corresponding sigmas are compared in Figure 21. The random walk estimate shows wider variation than the first-order Gauss-Markov estimate. Figure 21. Experiment F, Comparison of GPS X-Axis Acceleration Sigmas Using a First- a First- Using Sigmas Acceleration X-Axis GPS of Comparison F, Experiment 21. Figure X-Axis Acceleration Sigma (m/s**2) ^ X-Axis Acceleration (m/s**2) re 20. Experiment F, Comparison of GPS X-Axis Accelerations Using a First-Order First-Order a Using Accelerations X-Axis GPS of Comparison F, Experiment 20. re -9E-9 -7E-9 -5E-9 -3E-9 -1E-8 -IE-9 10 10 10 -7 ‘ -8 Hours and Using a Random Walk (RW) Model with with Model (RW) Walk Random a Using and Hours Order Gauss-Markov (FOGM) Model with with Model (FOGM) Gauss-Markov Order Hours and Using a Random Walk (RW) Model with with Model (RW) Walk Random a Using and Hours with Model (FOGM) Gauss-Markov 10000 20000 30000 40000 50000 60000 70000 0 10000 20000 30000 40000 50000 60000 70000 0 V**1 eod o Day ofSeconds eod o Day ofSeconds RW FOGM ...... FOGM y]qcon 1x1- 2 n = 12 = r 1 and s2 x = 10-9m/ con q ^ ‘v/ = 1 x 10 9 m / s2 and 1s2 and 9 m / x 10 = yJqcon con q ^ 11x = s2. O'9m / = 1 x 10'9 m / s2. 1/ m 10'9 x = r =12 =12 160 161 The first-order Gauss-Markov model ( ^ q co„ = 1 x 10 9 m/s2 and r = 12 ) and random walk model (yjqcnn = 1 x 10‘9 m / s2) for the GPS Y-Axis acceleration is given in Figure 22. The first-order Gauss-Markov estimate again is consistent with the anticipated magnitude. The random walk estimate’s magnitude is relatively large, but again within the limits of possibility. The corresponding sigmas are compared in Figure 23. The random walk estimate again shows wider variation than the first order Gauss-Markov model. 5E-9 CM « CO 4E-9 3E-9 OS 2E-0 a> \ ! o® o 1E-9 < £ 0E+0 < - -IE-9 0 10000 20000 30000 40000 50000 60000 70000 Seconds of Day Figure 22. Experiment F, Comparison of GPS Y-Axis Accelerations Using a First-Order Gauss-Markov (FOGM) Model with yJqcon = 1 x 10 9 m / s2 and x = 12 Hours and Using a Random Walk (RW) Model with y]qcon = 1 x 10"9 m / s2. 162 « CO FOGM *£ « o E 10 B JEP co oc -9 S 10 o o < c■= io-10 0 10000 20000 30000 40000 50000 80000 70000 Seconds of Day Figure 23. Experiment F, Comparison of GPS Y-Axis Acceleration Sigmas Using a First-Order Gauss-Markov (FOGM) Model with -J(jcnn - 1 x 10 9 m / s2 and t = 12 Hours and Using a Random Walk (RW) Model with v t = 1*10 9m/s2. Based on Experiments E and F for the solar radiation pressure scale, and residual accelerations, the first-order Gauss-Markov model is the better choice. It does not exhibit unexpected erratic variations, but changes more gradually than the random walk variation. Basically, for one-day orbits, any of the models used in Experiments A, B, C, D, E, or F are sufficient to maintain 25 cm RMS consistency. As the length of the arc becomes longer, or the accuracy requirements more demanding, then the model from Experiment F, solar radiation scale bias, radial, GPS X-axis, and GPS Y-axis accelerations modeled stochastically is recommended. Based on this study, the proper choice of the defining parameters is a first-order Gauss-Markov model with = 163 1 x 10 9 m / s2 and r = 12 hours. The stochastic modeling of the solar radiation pressure scale factor and the y-bias term gives nearly the same improvement However, physically, the solar radiation pressure scale does not vary much over just one day When this parameter is modeled stochastically, it is absorbing the accelerations that are more correctly modeled as accelerations along the radial and the GPS X and Y axes. Summarizing this section, the random walk model estimates of the radial acceleration greatly exceed (see Figures 18-23) the values expected from the magnitude possible from a physical understanding of the problem as given in Table 1. The RMS, values are significantly greater than the RMS, value of the first-order Gauss-Markov model. The first-order Gauss-Markov model estimates of the accelerations are more reasonable considering anticipated range of the residual accelerations. Based on this study, first-order Gauss-Markov model with yjqco„ = 1x10 9m/s2 and t = 12 hours has the lowest RMS, value. Though, all combinations of correlation time and process noise standard deviation give similar RMS,. CHAPTER VII CONCLUSIONS AND RECOMMENDATIONS The force models required for GPS satellite modeling have been discussed in Chapter 2. The model of forces ffom gravitational harmonics, third body, earth tides, and ocean tides have been reviewed. These force models agree with those recommended by the IGS at the centimeter level. From this review, the following three items are noteworthy: • In sections 2.5 and 2.6, the correspondence of Eanes’ formulation of the ocean tide model and solid earth tide model with that of Christodoulidis’ is shown. Christodoulidis neglects any contribution of ocean tide model and solid earth tide model to the variational partial derivatives. While this may be appropriate, it is worth mentioning that Eanes’ implementation allows an approximation to this contribution. • The indirect oblation perturbation is not addressed in the IERS standards, even though as shown in section 2.3.1 the acceleration is approximately 10 9 m is1, i.e. at the same order of magnitude as the earth tides or y-bias. • As discussed in section 3 .4.1, the second and third order ionosphere effects are a few centimeters in magnitude and are not currently modeled nor are they addressed in the IGS/IERS standards. These may be necessary to maintain long-term repeatability of baseline coordinates. 164 165 The capacity of a batch least-squares differential corrector algorithm to provide the necessary products for a filter/smoother was investigated and verified in Chapter 4. The square root information filter/smoother was derived and interfaced to batch least- squares estimator. • The algorithm to compute equation (4-48) coded in the subroutine given on pages 158-160 in Bierman (1977) is incorrect. In section 4.3, this algorithm was shown to neglect the upper triangular elements with R^,. It was also shown that the incorporation of products involving the V matrices is incorrect. Bierman’s subroutine should not be used. Instead, the more general procedure given in equation (4-48) is recommended. A subroutine is given in Appendix C. • In section 4.6, a capability to construct a correctly correlated data set was developed and implemented. Its cost in computation time was less than 1% and as expected, slightly raised the RMS3 of fit over the incorrect approach of neglecting the data correlation. • In section 4.5, an efficient technique for computing the transition matrix relating the stochastic orbit related parameter effects and the pseudoepoch state parameters was developed using the variational partial derivatives provided by the batch least-squares algorithm. This approach does not require derivation of unique approximations for each new stochastic orbit-related parameter. Instead, it is general and can accommodate any parameter for which the batch least-squares algorithm provides variational partial derivatives. 166 In Chapter 5, the deterministic dynamic models in the batch least-squares estimator are verified by fitting to Final IGS GPS satellite orbits. Then, the GPS satellite orbits were independently determined based entirely on ion-free double-differenced phase ranges. Three different deterministic models were implemented in the batch least- squares estimator: (A) solar radiation pressure scale and y-bias acceleration model in a batch least-squares estimate, (B) solar radiation pressure scale and bias accelerations in the radial, GPS X-axis, and GPS Y-axis in a batch least-squares estimator, (C) solar radiation pressure scale and the nine parameter acceleration model of Colombo in a batch least-squares estimator, • All three modeling approaches are sufficient for one-day arcs with approximately 25 cm RMS,. Even the two-parameter model, solar radiation pressure scale factor, and y- bias acceleration is sufficient for one-day arcs at the 26 cm RMS, level. In Chapter 6, the empirical autocorrelation function of the ion-ffee double differenced phase range observables was shown to have a process that could be modeled as a first-order Gauss-Markov process with correlation time of 72 minutes and a process noise standard deviation of 2.3 cm. The magnitude and correlation time resembled that of the tropospheric zenith delay. Stochastic modeling of the troposphere was examined in the following experiment: (D) ‘wet’ tropospheric refraction zenith delay modeled as a deterministic bias and a stochastic process in a sequential estimator. 167 It was found that either the first-order Gauss-Markov model or the random walk model improved the RMS, of the orbit by approximately 2 cm. Stochastic modeling of the acceleration on the GPS satellite was also investigated; two experiments were conducted: (E) solar radiation pressure scale and y-bias acceleration as stochastic models in a sequential estimator, (F) solar radiation pressure scale as a deterministic bias and stochastic models for the radial, GPS X-axis, and GPS Y-axis in a sequential estimator. • The random walk models proved unsatisfactory for modeling the radial accelerations. The GPS satellite orbits were improved by approximately 1 cm when the residual accelerations were modeled as first-order Gauss-Markov processes. • The first-order Gauss-Markov stochastic process model with continuous process noise standard deviation (ijq c0n ) for the acceleration of 1x10 9 m/s2 and the correlation time of r =12 hours is recommended for the acceleration terms; and a continuous process noise standard deviation for the solar radiation pressure scale of 0.01 (unitless scale factor) and r = 12 hours is recommended for the solar radiation pressure scale factor. The results were insensitive to the correlation times of 6, 12, or 24 hours • The solar radiation pressure scale factor is used to absorb some of the residual acceleration. Actually, the solar radiation pressure scale changes slowly over the 11 -year solar cycle. Thus, for longer arcs, the solar radiation pressure scale should be estimated as a bias and the radial, GPS X-axis, and GPS Y-axis acceleration biases (F) as 168 stochastic processes. This model separates the solar radiation pressure scale from the residual accelerations that are absorbed by the three time varying acceleration terms along each spacecraft body axis • The OSUORBFS runtime on a CRAY Y-MP8/864 is approximately 20 minutes for a one-day global GPS dataset consisting of 26 satellites and approximately 40 stations. • The coupling of a filter/smoother estimator (OSUORBFS) with a batch least-squares estimator (GEODYNII) is logical. Batch least-squares estimator is extremely robust and provides clean dataset to the filter/smoother, while the filter/smoother extends the modeling capabilities available to the batch least-squares estimator data processing. In conclusion, the questions posed in Chapter 1 are now addressed. • Are additions to the deterministic models appropriate or should stochastic models be introduced? For one-day orbit solutions and comparison of deterministic and stochastic models for the residual acceleration, deterministic models are sufficient for GPS orbit determination with 25 cm consistency. The stochastic modeling of the tropospheric refraction zenith variation improved the orbit by 2 cm. Also, stochastic models of acceleration on the GPS satellite further improved the orbit by 1 cm. • If so, which stochastic models are best? From the numerical results, either the random walk or the first-order Gauss-Markov are appropriate models for the tropospheric refraction. However, since the empirical autocorrelation function of the ion-free double-differenced phaserange residuals is a correlated process, the first-order Gauss-Markov model is the recommended choice for 169 the stochastic model. For the solar radiation pressure factor and the y-bias acceleration, the first-order Gauss-Markov model is recommended. For radial, GPS X-Axis and GPS Y-Axis accelerations, the first-order Gauss-Markov model is also recommended. The random walk model was not satisfactory. • What are the appropriate process noise covariance and if required, correlation times for the stochastic model? For the tropospheric zenith delay, a random walk with yjqcon = 1 cm / Vhr or a first-order Gauss-Markov model with yjq con = 0.023 m, r =72 min are appropriate. For the acceleration, the first-order Gauss-Markov model with yjq c00 = 1 x 10“9 m / s2 and r = 6, 12, and 24 hours is recommended for the acceleration terms; and yjq con = 0.01 (unitless scale factor) and r =6, 12, or 24 hours is recommended for the solar radiation pressure scale factor. • Can stochastic models improve GPS satellite orbits? For one day GPS orbit solutions, the stochastic models of the troposphere and of the residual accelerations can improve the orbits by 3 cm compared to the Final IGS satellite orbit. APPENDIX A OSUORBFS AND CONTROL CARD DESCRIPTIONS 170 171 OSUORBFS The program OSUORBFS is designed to filter and smooth the GEODYNII batch solutions OSUORBFS requires from GEODYNII the measurement partials file (FTN90) and the variationals V-matrix file (FTN80) The user must supply a user input file (FSN05) and a file of satellite identification numbers and masses (SATMAS.TAB). The GEODYNII processing proceeds in the usual way with TDF, G2S, and G2E program executions. On the last iteration an output of the setup deck (FTN05) which contains the current parameter estimates is requested using PUNCH to output the new setup deck in file FTN07. Then FTN07 is modified to include global cards PARFIL and EMATRX to output files FTN90 and FTN80. The maximum iteration numbers for the global (outer) and arc (inner) are set to one on the ENDGLB and REFSYS statements. Now program executables TDF, G2S, and G2E are executed and FTN06, FTN80, and FTN90 files are output. An alternative approach avoiding the restart of GEODYNII would be to permit the output of the FTN90 and FTN80 files on the last inner of the last global iteration. This option is not available in the current version of GEODYNII. This option should be added. It would simplify the implementation of OSUORBFS The user must construct the control file, FSN05, for OSUORBFS. This file contains eight mandatory control statements (REFSYS, EPOCHS, DECORR, FILSMT, UPDTRJ, CONPRT, SATMAS, PRESID) to control the configuration of the filter/smoother solutions. 1 7 2 FSN05 must also contain the parameter labels from FTN06. The measurement partials and variationals form GEODYNII are identified by internal parameter labels as described in the GEODYNII manual volume 5 For OSUORBFS to recognize these partial derivatives, the parameter labels as they appear in FTN80 and FTN90 must be specified in FSN05 These labels can be accessed by printing the EMATRIX header record in FTN06 during the last iteration of the GEODYN run. They must be manually edited and placed in FSN05. FSN05 must also contain the parameter types as defined in Table 3. For each parameter type the a priori standard deviation must be specified. Additionally, for the first-order Gauss-Markov model, the continuous process noise standard deviation (^jqcon) and the correlation ( r ) time must be specified. For the random walk model, the continuous process noise standard deviation (yjqco„) and a negative correlation time (—r, which acts as a flag) must be specified. These are read in a free format. The order of the parameter types in FSN05 is arbitrary; the file is sorted and the time-varying stochastic parameters are moved to the top of the file to accommodate the implementation of the Vp as an nx x nd matrix to save space. The stochastic parameters (types 1, 2) are assumed to be zero mean processes. Typically, the physical process modeled is not zero mean. The non-zero mean is estimated as a constant (types 4, 5). Thus, the constant (types 4, 5) and the stochastic parameters (types 1, 2) are estimated together. The constant can be 173 estimated without a stochastic parameter, but a stochastic parameter must be estimated with a constant unless of course the process is known to be zero mean Since GEODYNTI does not have time-varying models, a type of time variation can be implemented as physical processes that are modeled as constants over consecutive segments of time. For example, the tropospheric scale correction in GEODYNII may be modeled over a 24 hour period by estimating a constant over the first 12 hours and another constant over the second 12 hours. In OSUORBFS, the one constant and a time-varying stochastic parameter would be estimated over the entire span of 24 hours. This requires the measurement partial derivatives from the two consecutive constant estimates in GEODYNII to be concatenated This is controlled by the CONPRT statement. OSUORBFS can be implemented as a conventional least-squares sequential estimator by specifying all parameters as pseudo epoch state (Type 3) and constant parameters (Types 4, 5). The partial file should not be concatenated. The full covariance matrix may be generated. TIMING STUDY The OSUORBFS software was developed on a Personal Computer (PC) while maintaining portability to the CRAY. The original reductions of the GIG91 dataset using OSUORBFS was carried out on the PC, specifically, an Intel 60 MHz Pentium. (Pentium is a registered trademark of the Intel Corporation ) The timing for the main processes of OSUORBFS on the Pentium is given in Table 24 for GIG91 (Day 044), GIG92 (Day 280), and GIG92 (Day 360). For the GIG91 solution, the time of 3 hours and 15 minutes is not prohibitive and in an operational mode these computations could be done on a PC, for example, overnight. However, for the development of the software, the CRAY proved invaluable. As is shown in Table 24, only 3 minutes and 18 seconds were required to complete the solution on the CRAY. This results primarily from the fact that the two computationally intensive routines, filtering and smoothing, have at their core the Householder transformations. The subroutines that perform these transformations can be vectorized, thus the significant reduction in computation time. For datasets of 25-30 stations and 26 satellites, the runtime for OSUORBFS is approximately 25 minutes. The option to construct the full covariance matrix and then decorrelate the data was initially thought to be time consuming but as is evident in Table 27, implementing the decorrelation option only increases the CRAY runtime by 1-2% of the total execution time. Table 24. OSUORBFS Time Accounting EXECUTION TIME (EIH:MM:SS) GIG91 (Day 044 GIG92 (Day 280) GIG92 (Day 360) PENTIUM CRAY CRAY CRAY CRAY CRAY CRAY w DECORR w/o DECORR wDECORR w/o DECORR w DECORR w/o DECORR w DECORR PREPROCESSING 0:02:47 0:00:04 0:00:04 0:00:04 0:00:05 0:00:06 0:00:06 FILTER 1:46:35 0:01:36 0:01:40 0:04:02 0:04:19 0:03:15 0:03:27 SMOOTHER 1:39:16 0:01:32 0:01:32 0:03:04 0:03:04 0:02:52 0:02:52 SMOOTHER RESIDUAL 0:00:59 0:00:01 0:00:01 0:00:01 0:00:01 0:00:01 0:00:01 UPDATE TRAJECTORY 0:00:36 0:00:02 0:00:02 0:00:02 0:00:02 0:00:03 0:00:03 TOTAL 3:30:15 0:03:18 0:03:21 0:07:17 0:07:34 0:06:19 0:06:31 PENTIUM = Intel 60 MHz Pentium (Pentium is a registered trademark of the Intel Corporation.) CRAY = Y-MP8/864 with UNICOS 8.0.2.2 1 7 6 REFSYS REFSYS 910208000000.0000000 COLUMNS FORMAT DESCRIPTION UNITS 1-6 A6 REFSYS - Specifies the True of Reference Date (TORD) reference system used by GEODYNII. The time MUST be the same as the time used on the REFSYS statement in FTN05 (G2S). Only TORD is valid. Mean of Date (MOD) J2000 is not currently implemented 7 blank 21-26 16 Year,month,day of reference date (YYMMDD) 27-30 14 Hour, minute of reference date (HHMM) 31-40 D10.8 Seconds of reference date (SS .sssssss) 1 7 7 EPOCHS EPOCHS 910208000000.0000000 360.0 COLUMNS FORMAT DESCRIPTION UNITS 1-6 A6 EPOCHS - Specifies the UTC time of epoch of UTC the satellite elements on the EPOCH card used by GEODYNII. The program assumes all satellite epochs are the same. This time MUST be the same as the time used on all the EPOCH statements in FTN05 (G2S). Also, this card specifies the ‘mini batch’ interval size. If no interval is specified, the state is updated at each nominal measurement epoch. 7 blank 21-26 16 Year,month,day of epoch date (YYMMDD) 27-30 14 Hour, minute of epoch date (HHMM) 31-40 D10.8 Seconds of epoch date (S S. sssssss) 46-60 D15.9 ‘Mini Batch’ interval. secs. = 0.0, State updates are computed at each nominal measurement epoch. 178 DECORR DECORR 1 o.io COLUMNS FORMAT DESCRIPTION UNITS 1-6 A6 DECORR - controls the computation of the full covariance matrix and decorrelation for double difference ranges 7 blank 8 II = 0, measurements assumed uncorrelated, measurements whitened by dividing by the GEODYNII supplied weight. = 1, full covariance matrix computed for double differenced ranges, then decorrelated and whitened. 9-10 blank 11-20 D10.5 standard deviation for a one-way range meters measurement 179 FILSMT FILSMT 1111111111 COLUMNS FORMAT DESCRIPTION UNITS 1-6 A6 FILSMT - controls the filter/smoother operations, controls the computation of the estimates/covariances of the stochastic and pseudo epoch parameters (px) and the constants (y), and controls the output of estimates/covariances to output files FSN06, FSN16, and FSN17. 7 blank 8 II = 0, do not filter data = 1, filter data 9 II = 0, do not compute the estimate px at each filter step and output to FSN06 = 1, compute the estimate px at each filter step and output to FSN06 10 II = 0, do not compute the estimate px at each filter step and output to FSN16 = 1, compute the estimate px at each filter step and output to FSN16 11 II - ' do not compute the covariance px at each filter step and output to FSN16 = 1, compute the covariance px at each filter step and output to FSN16 12 II = 0, do not compute the estimate y at each filter step and output to FSN06 = 1, compute the estimate y at each filter step and output to FSN06 180 13 II = 0, do not compute the estimate y at each filter step and output to FSN17 = 1, compute the estimate y at each filter step and output to FSN17 14 11 = 0, do not compute the covariance y at each filter step and output to FSN17 = 1, compute the covariance y at each filter step and output to FSN17 15 II = 0, do not smooth data = 1, smooth data 16 II = 0, do not compute the estimate px at each smoother step and output to FSN06 = 1, compute the estimate px at each smoother step and output to FSN06 17 II = 0, do not compute the estimate px at each smoother step and output to FSN16 = 1, compute the estimate px at each smoother step and output to FSN 16 18 11 = 0, do not compute the covariance px at each smoother step and output to FSN 16 = 1, compute the covariance px at each smoother step and output to FSN 16 Note: The option to compute and output the covariance at each filter and/or smoother step (columns 11, 14, 18) requires the estimates to be computed at each filter step. For column 11, either 9 and/or 10 must be enabled (=1). For column 14, either 12 and/or 13 must be enabled. For column 18, either 16 and/or 17 must be enabled. Note: If the filter is enabled and estimates are not computed at each epoch (i.e., columns 9 through 14=0), the filter estimate PX and Y are output to FSN06 on the final epoch. Also, if the PX covariance is output at each epoch, when smoothing, but not when filtering, the Y covariance at the final epoch will be output to FSN 17. 181 UPDTRJ U P D T R J 1 COLUMNS FORMAT DESCRIPTION UNITS 1-6 A6 UPDTRJ - controls the satellite trajectory computation and output in a TORD system to file 7 blank 8 II = 0, do not update trajectory = 1, update trajectory 182 CONPRT CONPRT 1 COLUMNS FORMAT DESCRIPTION UNITS 1-6 A6 CONPRT - controls the concatenation of the piece-wise measurement partials to the first partial location 7 blank 8 II = 0, do not concatenate = 1, concatenate Note: This option must be enabled (=1) when a bias parameter and a stochastic parameter share the measurement partials. 1 8 3 SATMAS SATMAS 0 COLUMNS FORMAT DESCRIPTION UNITS 1-6 A6 SATMAS - controls which satellite number system to use 7 blank 8 II = 0, GEODYN international satellite id = 1, OSU modified international satellite id 184 PRESID PRESID 1111 COLUMNS FORMAT DESCRIPTION UNITS 1-6 A6 PRESID - controls the output of filter/smoother residuals to files FSN06 and FSN15. 7 blank 8 11 = 0, do not output filter residuals to FSN06 = 1, output filter residuals to FSN06 9 11 = 0, do not output filter residuals to FSN 15 = 1, output filter residuals to FSN15 10 11 ~ o, do not output smoother residuals to FSN06 = 1, output smoother residuals to FSN06 11 11 = 0, do not output smoother residuals to FSN 15 = 1, output smoother residuals to FSN 15 APPENDIX B OSUORBFS Subroutine Descriptions 185 186 SUBROUTINE DESCRIPTION bsmoth Builds the smoother matrix. ckdiml Checks if maxnp, maxnx, maxnd, maxny, and/or maxndy values are exceeded. Outputs summary to fsn06. ckdim2 Checks if maxobs, maxntr, and maxntc are values are exceeded cholow Computes the lower Cholesky factorization of a two dimensional positive definite array p(maxobs,maxobs). P has physical dimensions maxobs by maxobs and logical dimensions of m by m. The lower triangular portion of p is replaced with the lower triangualar factorization L of P=LLt. chtmdf Computes time difference returns character string. cl8005 Compares the parameter lists in fsn05 and fort.80. Warns user of unused labels and stops program if user requested label is not found. C19005 Compares the parameter lists in fsn05 and fort.90. Warns user of unused labels and stops program if user requested label is not found. cmpbin Generate the time bins for the epoch. coefD9 Computes the coefficients for the 9-th order polynomial interpolator. The equations for computing the odd and even coefficients are the same. covinf Computes the covariance matrix for the information formulation. The inverses were computed and stored in the S and Ry matrices in the subroutine SOLINF. This routinue returns the covariance matricies as upper triangular matrices in the Ry and in the upper triangular matrix occupying the npx by npx sub-block of S. The last column (ntot) of s and the last column of Ry are destroyed. decorr Forms the full observation covariance matrix for measurement type 87, then decorrelates and normalizes the obsersvation residuals and design matrix elements. 187 SUBROUTINE DESCRIPTION dotg Computes dot product of two N-size vectors. fsinit Initialize the filter array for processing. hsttpx Apply a sequence of elementary Householder transformations to partially triangularize the a priori information and new data array, nulling terms below the diagonal of the first Np + Nx columns hstty Apply a sequence of elementary Householder transformations to combine the Y a priori with the modified y observation. htppxin Apply a sequence of elementary Householder transformations to partially triangularize the square root information array, nulling terms below the diagonal of the first 2*Np + Nx columns. This propagates forward from t to t+dt. The smoother coefficient are computed and output. indexx Heapsort algorithm to chronologicaly order and index the measurement partial array. into09 A 9-th order polynomial interpolator where the 9 equations in 9 unknowns have been solved symobolically rather than by least squares. Once the polynomial is fit to the provided 9 points, the position is computed. Similarly the velocity is computed from the differentiated polynomial. In practice the polynomial coefficients rarely change and the file I/O program is responsible to inform this function through the samecf logical. invlwt Computes the inverse of a lower triangular matrix stored in the two dimensional array P. P has physical dimensions of maxobs by maxobs and logical dimensions of m by m. The orginal lower triangular matrix is replaced by its inverse. invuda Computes the inverse of the npx by npx submatrix, an upper triangular matrix, that is stored as a double array. The npx by npx submatrix is replaced by its inverse. 188 SUBROUTINE DESCRIPTION invuvc Computes the inverse of an upper triangular matrix that is stored columnwise as a vector. The original vector is replaced by its inverse. lodvar Loads the variational partials from direct access file into buffer for interpolation or direct read. mjd Change gregorian day to modified julian day and vice versa. rd90hl Read the header record determine the total number of estimated parameters and the number of parameter groups. rd80h2 Read the second set of header information. rd90h2 Read the second set of header information. read80 This subroutine reads the G2E binary force model partial (variational) file (V-Matrix) partial file FTN80. The force model partials are given in the true o f reference date. read90 Read the group identifiers, number of parameters, parameter labels, current parameter values, and a priori parameter values from the measurement partials file (FTN90). Store the parameter labels and current values for later use resout Output FILTER and/or SMOOTHER residuals to files FSN06 and/or FSN15. rffs05 Read the control file (FFS05) control statements, parameter sequence numbers, parameter labels, parameter types, apriori standard deviations, process noise standard deviations, correlation times. Store in direct access binary file. Determine the values of np,nd, nx,ny,ndy. secymd Converts from MJDS (from 2430000.5) to YYMMDDHHMMSS sss solinf Solves information formulation for the parameters. solout Write solution to file FSN06. 189 SUBROUTINE DESCRIPTION stor05 Resort parameter list and move all dynamic parameters (stochastic orbit-related) to the beginning of the array NOTE: Currently assumes uncorrelated parameter apriori covariance svdcmp Single Value Decomposition (SVD) algorithm. tcomb Combine two numbers stored as the (real) integer part and fractional part (check comments in subroutine TNORM) example: ta( 1) is a (real) integer part ta(2) is the fractional part, ta = tb + factor * tc, normally factor will be either +1 dO or -1 dO. util90 Gather time and number of observations. varvp Constructs the matrix Vp( t(j+l), t(j)) using the variational equations ymdmjd Converts from YYMMDDHHMMSS.sss to MJDS (from 2430000.5). APPENDIX C Householder Transformation Subroutine 190 191 subroutine htppxm(s,vm,np,nx,ntr,ntc,maxntr,maxntc,ntot) C======5======c Apply a sequence of elementary Householder transformations to c partially triangularize the square root information array, nulling c terms below the diagonal of the first Np columns. This propagates c forward from t to t+dt. The smoother coefficient are computed c and output. Adapted from ’Factorization Methods for Discrete Sequential Estimation* by Gerald J. Bierman, p.219. Version Comments Pgmr. 9402.1 D. Chadwell c Variable I/O Type Description c c np i i + 4 Number of stochastic parameters c nx i i*4 Number of dynamic parameters c ny i i + 4 Number of bias parameters c vm(2np+nx) 1 r*8 Local work vector c Sm(ntr,ntc) i r*8 SRIF array at t, with c ntr = np+nx+max( np,maxnme ) c ntc = 2*np+nx+ny+l c c SRIF array at t+1, lower part c c c Sm on input (all elements refer to time j) : c c Np Np Nx Ny 1 c c 1 1 1 1 1 c | -RwM | Rw 1 0 1 o 1 o 1 Np c 1I 1| 1 1 1I c Tppx# | ARp-ARpxVp | 0 1 A Rpx 1 ARpy 1 Azp 1 Np c 1I Il 1 i 1 c 1 -A RxVp | 0 1 A Rx I ARxy 1 AZ x I Nx c 1 1 1 1 1 c c c c Sm on output (all elements refer to time j+1) . c c Np Np Nx Ny 1 c c 1 1 1 1 c I * Rp | * Rpp 1 *Rpx 1 *Rpy 1 *zp Np c | | 1l 1 c 1 0 1 ~Rp 1 ~Rpx 1 ~Rpy 1 -Zp Np c 1 |i 1 1 c | 0 | ~Rxp 1 ~Rx I ~Rxy 1 ~Zx Nx c 1 1 1 1 c c c- implicit none 192 integer*4 i,j,k,np,nx,ntr,ntc,nct,nntr integer* 4 maxntr,maxntc,ntot double precision vm(maxntr) !! vm(2*maxnp+maxnx) double precision s(maxntr,maxntc) double precision z,sigma,alpha integer*4 icnt nct= np ! Number of columns to triangularize nntr=2*np+nx z=0.dO do j=l,nct Triangulize columns 1 to net sigma=z do i=j,nntr v m (i)=s (i,j; s(i,j)=z Forces jth column to be zero ! Yj t*Yjsigma = sigma +vm(i)*vm(i) ! Yj t*Yjsigma enddo if(sigma.le.z)goto 4 sigma = dsqrt(sigma) ! (Yjt*Yj)l/2 if(vm(j).gt.z)sigma = -sigma !SIGMAj ! =sgn(Yj(j))*(Yj*tYj)1/2 s (j , j)=sigma v m (j)=vm(j)-sigma ! Uj (1)=Yj(1)+SIGMAj sigma=l.dO/(sigma*vm(j)) ! =l/SIGMAj*Uj(j) do k=j+l,ntc ! Cycle thru columns j+1 to Ntot alpha=z do i=j,nntr ! Rows above j are not changed alpha = alpha +s(i,k)*vm(i) ! = Ykt*Uj (k>j) enddo alpha = alpha * sigma ! = Ykt*Uj/SIGMAj*Uj(j) do i =j,nntr s(i,k)=s(i,k)+alpha*vm(i) Tuj Yk=Yk- (Ykt*Uj'/SIGMAj *Uj (j ) ) *Uj enddo enddo continue enddo return endC APPENDIX D COMMENTS ON GEODYNII AND ITS ROLE AS A TRAJECTORY PREPROCESSOR FOR OSUORBFS 193 194 The GEODYNII software has been used for several years to determine the gravitational coefficients of the earth from satellite tracking data Laser ranging data has been the primary data source for this procedure More recently, GPS has been added as a geodetic system. In response to this, GEODYNII software has been frequently updated to include GPS observables and models. GEODYNII’s capacity to process GPS ground station data is still evolving and should soon reach completion. Comparison of the GEODYNII determined GPS orbits with IGS GPS orbits revealed significant departures of up to a 1 meter RMS. The primary purpose of this study was to investigate, using a filter/smooiher, to improve modelling of GPS orbits. This led to a necessary review of the GEODYNII dynamic models and measurement models. The review of the dynamic models is given in Chapter 2, and the review of the measurement models is given in Chapter 3. Significant discoveries were: (1) incorrect ROCK4 model implementation, (2) earth reradiation model is not correct for GPS satellites. These effects were not likely identified earlier because Colombo’s resonant acceleration model, which is routinely implemented for TOPEX solutions, absorbs these mismodelled accelerations. Rather than rely soley on empherical acceleration terms, an attempt has been made to implement dynamic models correctly as the current level of understanding permits. This has led to some recommendations to refine the dynamic models These comments are given in the following: The comparisons are made with respect to the IGS (IERS) recommendations for standards given in IERS Technical Note 13 (McCarthy, 1992) These standards are recommended for IGS analysis centers for 195 the operational production of orbits for the geodynamic community, and provide a realistic frame work in which to discuss the GPS orbit modeling. In some cases, GEODYNlI’s capability exceeds that of the IGS; in other cases, GEODYNII is deficient These discrepancies are outlined in the following discussion. • The earth radiation model in GEODYNII, while correct for spherical satellites, is not correct for GPS satellites. The error can be as great as 30-40%, and varies with the cross sectional area of the satellite that changes as the satellite rotates once per orbit. • GEODYNII does not allow a specification of two separate GM terms. The IERS standards recommend using the GM that was determined with geopotential model with the nonspherical geopotential coefficients and a second GM (IERS) that includes oceans and atmospheres with the two-body term. To conform to this standard, one could use GM (IERS) and scale coefficients of the gravitational model by a factor GM(model)/GM(IERS) However, this will affect the computation of the C2I and .V21 and pole tide corrections, since they involve the C20 term. • An approximation to the solar radiation pressure variational partials was implemented to add to the variational equations of GEODYNII. While a small contribution to the variationals (10~15), it is theoretically correct and more consistent to add this contribution. • The solar radiation pressure in GEODYNII now includes a normalization of a vector product and the angular argument for all of the ROCK4 terms The ROCK4 model 1% accelerations are computed at 1 AU A time varying scaling factor should be added to account for the variation of the sun-earth distance of ±1.7% from 1AU over one year The scale factor for the 12-hour orbit period is negligible ( « 1%) The solar flux constant (at 1AU) which has an 11-year cycle should be updated periodically to agree more closely with the ambient solar radiation pressure Realistic satellite mass values should be used. • GEODYNII includes associated partials with respect to tidal coefficients and Love numbers, but not with respect to position • GEODYNII computes the periodic effect and the permanent tide together, in effect, the mean of equation (2-48) is nonzero. Thus, for GEODYNII, the permanent tide should not be included in C-20 GEMT3. Since GPS is not used currently to estimate K 2, this is appropriate. • The station displacement due to solid earth tide corrections are computed using equation 2-55 has a nonzero average over time. Thus, the correction contains the permanent tide effect and applying the correction removes the permanent tide from the site coordinates. The Chapter 7 IERS standards (McCarthy, 1992) recommend removing the permanent tide from the correction, i.e., leave it in the site coordinates. However, in the introduction, the IERS standards recommend treating the permanent tide as a correction to the site coordinate, and override the recommendation given in Chapter 7. The ITRF92 coordinates contain the permanent tide effect. 197 • GEODYNII requires a double difference bias to be estimated for each unique satellite station pair even when the bias is a linear combination of previously defined biases This leads to the introduction of more than a minimal set of biases If a minimal set is formed, then the information extracted from the data is maximized If more than a minimal set is used, the information content in the data is distributed among the dependent set. For example, the phase biases are integers but are initially estimated as real numbers. The solution can be improved by resolving and fixing the biases to their correct integer value If more than a minimal set is used, then the integer nature of the bias is not as easily resolved. 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