December, 2000. Advisory Committee: ... December, 2000 ... I would like to express my sincere gratitude towards my thesis advisor Dr. Alfred Leick for his.
ELEMENTS OF GPS PRECISE POINT POSITIONING By Boonsap Witchayangkoon B.Eng. (Honors) King Mongkut’s University of Technology Thonburi, Bangkok, 1992 M.S. University of Maine, 1997
A THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy (in Spatial Information Science and Engineering)
The Graduate School The University of Maine December, 2000
Advisory Committee: Dr. Alfred Leick, Professor of Spatial Information Science and Engineering, Advisor Dr. Neil Comins, Professor of Physics and Astronomy Dr. Ray Hintz, Associate Professor of Spatial Information Science and Engineering Dr. Richard B. Langley, Professor of Geodesy and Geomatics Engineering, University of New Brunswick Dr. Charles Mundo, Adjunct Professor of Spatial Information Science and Engineering
Copyright 2000 Boonsap Witchayangkoon
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LIBRARY RIGHTS STATEMENT
In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of Maine, I agree that the Library shall make it freely available for inspection. I further agree that permission for "fair use" copying of this thesis for scholarly purposes may be granted by the Librarian. It is understood that any copying or publication of this thesis for financial gain shall not be allowed without my written permission.
Signature: Boon Witchayangkoon Date: August 25, 2000
ELEMENTS OF GPS PRECISE POINT POSITIONING By Boonsap Witchayangkoon Thesis Advisor: Dr. Alfred Leick
An Abstract of Thesis Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy (in Spatial Information Science and Engineering) December, 2000
The International GPS Service (IGS) now regularly makes accurate GPS ephemeris and satellite clock information available over the Internet. The satellite coordinates are given in the International Terrestrial Reference Frame (ITRF). This thesis investigates Precise Point Positioning (PPP) using dual and singlefrequency pseudorange and carrier phase observations. Both the static and kinematic modes are investigated. The static PPP solution examples use six-hour data sets from four stations. The observations were made while Selective Availability (SA) was active and after it had been discontinued. The static solutions agree to within 10 cm with published coordinates or with solutions obtained from the Jet Propulsion Laboratory (JPL) PPP Internet service.
The
kinematic solutions show a discrepancy of less than one meter, mostly around half a meter. For observations with low multipath, the research shows that single-frequency ionosphere-free PPP solutions are equivalent to the dual-frequency solutions. In case of single-frequency observations the pseudorange dominates the solution.
Using a priori tropospheric information does not seem to improve dual-frequency PPP solutions as compared to the case when the vertical tropospheric delay is estimated as part of the Kalman filter solution. However, a priori tropospheric information seems to provide benefits to single-frequency kinematic PPP.
The Saastamoinen model is used when
computing the zenith tropospheric delay. In all cases, the Neill's mapping function is applied. The studies show high correlation between receiver clock and the up coordinate. The troposphere has a high negative correlation with receiver clock and the up coordinate. However, the troposphere is more correlated with the receiver clock than the up component. All solutions incorporate corrections for solid earth tides, relativity, and satellite antenna phase center offsets. Corrections have not been applied for the phase wind-up angle. The widelane and ionospheric functions are used to detect and fix cycle slips in a semigraphical manner. Since even a single cycle slip significantly falsifies PPP solutions, it is suggested that between-satellite carrier phases be used as another way of detecting slips (now since SA has been discontinued). The software consists mostly of highly modular Mathcad functions that form an excellent base for continued research of PPP.
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude towards my thesis advisor Dr. Alfred Leick for his support and guidance, and the members of my thesis advisory committee: Dr. Richard Langley, Dr. Ray Hintz, Dr. Chales Mundo, and Dr. Neil Comins, for their cheerful guidance and cooperation during this thesis. My thanks also go to numerous of people whom I may not name all, for their grateful and joyful support, encouragement and helpfulness, especially Dr. Paulo Cesar Lima Segantine and his family, Dr. Lee YoungChan and his family, Dr. James Zumberge, Dr. Peter Kuntu Mensah and his family, Dr. Kate Beard, Dr. Ed Ferguson, Dr. Ramesh Gupta, Dr. Roop Goyal and his family, Dr. Kridayuth Choompooming, Dr. Carol Bult, Dr. Hans-Georg Scherneck, Zuheir Altamimi, Dennis Manning, Karen Kidder, Troy Jordan, Haci Mustafa Palancioglu, Sharron J. Macklin, Dilnora Azimova, Jeanne Timmons, Karen and Robert Liimakka, Mike Pearson, Teresa Cail, Puttipol Dumrongchai, Nakarin Satthamnuwong, Balkaran Samaroo, Edward P. Wells, Brian J. Naberezny, Stephanie Sturtevant, Tom Noonan, Carolyn Leick, Haci Mustafa Palancioglu, Samantha and Kurt Wurm, Pratya Levan, Wararat Sophanowong, Ramaswamy Hariharan, Sawat Pararach, Taweesak Kijkanjanarat, Cheng Tee Tang, Cynthia Henny, Jake Bogar, Tracey Nightingale, Young Su Kim, Saharat Buddhawanna, Shinsuke Sasanawin, Auay Wanasen, Piriya Panwichai, Angsana Tokitkla, Sunisa and Wattanachai Smittakorn, and Premwadee Furodchanakul. I thank the Fogler Library for an excellent source of references for this research. I also thank the Mathcad technical support for programming debug helps.
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Very special thanks are due to my parents for providing me excellent upbringing, education, and guidance, which helped me reaching at this point in my life. Heartfelt thanks go to my uncles, my aunts, my cousins, my sister, and my brothers, for their unconditional support and understanding that always encouraged me to follow the path I have chosen. Finally I would like to thank the Royal Thai Government and the Thammasat University for financial assistance.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS ............................................................................................................III
LIST OF TABLES .........................................................................................................................XII
LIST OF FIGURES ..................................................................................................................... XIV
1
2
INTRODUCTION .................................................................................................................... 1 1.1
RESEARCH GOALS ............................................................................................................... 1
1.2
MOTIVATION....................................................................................................................... 3
1.3
PREVIOUS RELEVANT WORKS .............................................................................................. 4
1.4
APPROACH .......................................................................................................................... 9
1.5
THESIS ORGANIZATION........................................................................................................ 9
BACKGROUND..................................................................................................................... 11 2.1
THE GPS SYSTEM ............................................................................................................. 12
2.1.1
General Information ................................................................................................... 12
2.1.2
The Undifferenced Observation Equations .................................................................. 14
2.2
THE GLONASS SYSTEM ................................................................................................... 16
2.3
COMPONENTS OF PPP........................................................................................................ 18
2.3.1
Geophysical Models ................................................................................................... 19
2.3.2
Atmosphere................................................................................................................ 20
2.3.3
Reference Frames....................................................................................................... 23
2.3.4
IGS ............................................................................................................................ 23
2.3.5
Phase Wind-up Error .................................................................................................. 24
2.3.6
Receiver Antenna Phase Center Offset........................................................................ 27
2.3.7
Satellite Antenna Phase Center Offset and Satellite Orientation................................... 29
2.3.8
Satellite Clocks .......................................................................................................... 31
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3
4
5
2.3.9
Group Delay Differential ............................................................................................ 32
2.3.10
Relativity ................................................................................................................... 34
GEOPHYSICAL MODELS ................................................................................................... 36 3.1
DEFORMABLE E ARTH ........................................................................................................ 36
3.2
SOLID E ARTH TIDES .......................................................................................................... 37
3.2.1
Brief History .............................................................................................................. 37
3.2.2
The Tidal Potential ..................................................................................................... 39
3.2.3
Solid Earth Tide Displacements .................................................................................. 42
3.3
OCEAN LOADING ............................................................................................................... 43
3.4
PLATE TECTONIC MOTION ................................................................................................. 46
3.5
ATMOSPHERIC TIDES ......................................................................................................... 49
INTERNATIONAL TERRESTRIAL REFERENCE FRAME (ITRF)................................. 51 4.1
GENERAL STATEMENTS ON REFERENCE FRAMES ................................................................ 51
4.2
THE ITRF ......................................................................................................................... 53
4.3
TRANSFORMATION BETWEEN ITRFS .................................................................................. 53
4.4
ORIENTATION AND ORIGIN OF THE ITRF ............................................................................ 55
4.4.1
Orientation ................................................................................................................. 55
4.4.2
Origin ........................................................................................................................ 56
4.5
THE DRAFT ITRF-2000 REFERENCE FRAME ....................................................................... 56
4.6
GPS WGS-84 ................................................................................................................... 57
4.7
AGREEMENT BETWEEN WGS-84 AND ITRF ....................................................................... 58
TROPOSPHERE AND IONOSPHERE................................................................................. 59 5.1
STANDARD ATMOSPHERE .................................................................................................. 60
5.2
TROPOSPHERE ................................................................................................................... 64
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5.2.1
Tropospheric Models.................................................................................................. 66
5.2.1.1 Hopfield Model ..................................................................................................... 67 5.2.1.2 Saastamoinen Model.............................................................................................. 69 5.2.2
Mapping Functions..................................................................................................... 72
5.2.2.1 Marini Mapping Function ...................................................................................... 73 5.2.2.2 Marini & Murray Mapping Function ...................................................................... 73 5.2.2.3 Chao Mapping Function......................................................................................... 74 5.2.2.4 Lanyi Mapping Function (Lanyi) ........................................................................... 75 5.2.2.5 Davis Mapping Function (CfA-2.2)........................................................................ 76 5.2.2.6 Herring Mapping Function (MTT) ......................................................................... 78 5.2.2.7 Niell Mapping Function (NMF) ............................................................................. 78 5.3
IONOSPHERE...................................................................................................................... 81
5.3.1
Spatial and Temporal Variations................................................................................. 81
5.3.2
Ionospheric Range Delay............................................................................................ 85
5.3.3
Ionosphere Models ..................................................................................................... 87
5.3.4
Functions of Observables............................................................................................ 87
5.3.4.1 Dual-Frequency Ionosphere-free ............................................................................ 88 5.3.4.2 Dual-Frequency Ionosphere ................................................................................... 88 5.3.4.3 Single-Frequency Ionosphere-free Code and Phase................................................. 89 6
PRECISE IGS ORBIT AND SATELLITE CLOCK............................................................. 91 6.1
IGS ORBITAL ANALYSIS AND ITS PRODUCTS ...................................................................... 91
6.1.1
IGS Structure and Operation....................................................................................... 91
6.1.2
Products ..................................................................................................................... 92
6.2
THE SP3 EPHEMERIS ......................................................................................................... 98
6.2.1
The SP3 GPS Orbital Format and Data Accuracy ........................................................ 98
6.2.2
Precise Satellite Clock Information............................................................................. 99
6.3
LAGRANGE INTERPOLATION .............................................................................................. 99
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7
MATHEMATICAL IMPLEMENTATIONS....................................................................... 101 7.1
DILUTION OF PRECISION .................................................................................................. 101
7.2
CYCLE SLIP DETECTION AND REMOVAL ........................................................................... 102
7.2.1
Multipath ................................................................................................................. 103
7.2.2
Widelane.................................................................................................................. 104
7.2.3
Cycle Slips ............................................................................................................... 105
7.2.3.1 Between Satellite Differences .............................................................................. 105 7.2.3.2 Undifferenced Observation Cycle Slip Detection and Fixing ................................ 110 7.2.3.1.1 Cycle Slip Detection in the Widelane Combination........................................ 110 7.2.3.1.2 Cycle Slip Detection in the Ionospheric Combination .................................... 111 7.2.3.1.3 Fixing Cycle Slip of Undifferenced Observation............................................ 112 7.3
KALMAN FILTER ............................................................................................................. 113
7.3.1
Extended Kalman Filter............................................................................................ 116
7.3.2
Discrete Gauss-Markov Process................................................................................ 118
7.3.3
PPP Implementation ................................................................................................. 119
7.3.3.1 Partial Derivatives ............................................................................................... 119 7.3.3.2 Receiver Clock Estimation................................................................................... 120 7.3.3.3 Ambiguity Estimation.......................................................................................... 120 7.3.3.4 Observation Weighting Schemes.......................................................................... 121 7.3.4 8
Computational Flow and Software Components........................................................ 124
NUMERICAL STUDY AND RESULTS.............................................................................. 127 8.1
DATA SETS ..................................................................................................................... 127
8.2
A PRIORI KALMAN FILTER SETTINGS ............................................................................... 129
8.3
ANALYSIS E XAMPLE........................................................................................................ 130
8.3.1
Widelane.................................................................................................................. 131
8.3.2
Ionospheric Carrier Phase......................................................................................... 132
8.3.3
Multipath ................................................................................................................. 132
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8.3.4
OMC for P1 ............................................................................................................. 133
8.3.5
OMC for P2 ............................................................................................................. 134
8.3.6
OMC Pseudorange Ionosphere-free .......................................................................... 135
8.3.7
Pseudorange OMC Difference Between Satellites ..................................................... 136
8.3.8
Pseudorange Ionospheric Variations ......................................................................... 137
8.3.9
Elevation Angle and Azimuth................................................................................... 138
8.3.10
Sky Plot ................................................................................................................... 139
8.3.11
North East & Up ...................................................................................................... 141
8.3.12
Receiver Clock Error Estimate.................................................................................. 142
8.3.13
Ionosphere-free Pseudorange OMC (Innovation) ...................................................... 142
8.3.14
Ionosphere-free Carrier Phase OMC (Innovation) ..................................................... 143
8.3.15
Reparameterized Ambiguity Estimates...................................................................... 144
8.3.16
Variance of Estimated Ambiguity ............................................................................. 145
8.3.17
Number of Satellites Used in Kalman Filter and DOPs.............................................. 146
8.3.18
Relativistic Correction.............................................................................................. 147
8.3.19
Approximated Tropospheric Effect Using Saastamoinen Model ................................ 148
8.3.20
Estimated Tropospheric Zenith Delay ....................................................................... 149
8.3.21
Correlation of the Estimated Parameters ................................................................... 149
8.4
EXPERIMENTS ................................................................................................................. 152
8.4.1
Discrepancies With Respect to Published Coordinates............................................... 152
8.4.2
Verification of Solid Earth Tides .............................................................................. 155
8.4.3
Estimated Troposphere vs. Saastamoinen Model ....................................................... 156
8.4.4
Impact of Satellite Antenna Offset ............................................................................ 157
8.4.5
Impact of Relativity.................................................................................................. 158
8.4.6
Impact of Single Cycle Slip ...................................................................................... 159
8.4.7
Using A Priori Tropospheric Information.................................................................. 162
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8.4.8
Kinematic Positioning .............................................................................................. 164
8.4.8.1 Dual-Frequency Kinematic Positioning ................................................................ 164 8.4.8.2 Single-Frequency Kinematic Positioning.............................................................. 169 9
CONCLUSIONS AND RECOMMENDATIONS................................................................ 172 9.1
CONCLUSIONS ................................................................................................................. 172
9.1.1
Dual-Frequency Solutions ........................................................................................ 173
9.1.2
Impact of IGS Products and Service.......................................................................... 173
9.1.3
Single-Frequency PPP .............................................................................................. 174
9.1.4
Correlation Between the Up Coordinate, Zenith Tropospheric Delay, Receiver Clock, and Ambiguities ............................................................................................ 174
9.1.5
A Priori Tropospheric Information............................................................................ 175
9.1.6
Kinematic PPP ......................................................................................................... 175
9.2
RECOMMENDATIONS ....................................................................................................... 176
9.2.1
Cycle Slip ................................................................................................................ 176
9.2.2
Phase Wind-up Error ................................................................................................ 176
9.2.3
Receiver Antenna Phase Center Offset...................................................................... 177
9.2.4
A Priori Tropospheric Information............................................................................ 177
9.2.5
GLONASS............................................................................................................... 177
REFERENCES ............................................................................................................................. 179
APPENDIX A: SP3 EPHEMERIS FORMAT............................................................................. 191
APPENDIX B: RELEVANT MATHCAD FUNCTIONS........................................................... 197 B.1 CYCLE SLIP DETECTION AND FIXING .................................................................................... 197 B.2 OMC COMPUTATIONS ......................................................................................................... 217 B.3 GRAPHING AND DATA EDITING ............................................................................................ 238 B.4 KALMAN ESTIMATION.......................................................................................................... 245
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APPENDIX C: PSEUDORANGE PERFORMANCE USING PRECISE EPHEMERIS........... 263
APPENDIX D: HOURLY JUMPS IN RINEX FILES................................................................ 264
BIOGRAPHY OF THE AUTHOR............................................................................................... 265
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LIST OF TABLES
Table 2. 1 GPS satellite antenna phase center offset adopted by IGS (Kouba and Springer, 1998) ..............................................................................29 Table 2. 2 Typical sample of TGD (extracted from the GPS broadcast navigation message DOY2(2000))......................................................................................33 Table 3. 1 Relative contributions to tidal potential from various celestial bodies...................41 Table 3. 2 Sample of ocean loading file ................................................................................44 Table 3. 3 Cartesian rotation vector for each plate using the NNR-NUVEL1A kinematic plate model (IERS Conventions (1996, p. 14). The units are radians per million years.).................................................................................49 Table 4. 1 Transformation parameters from ITRF-94 to other ITRFs....................................55 Table 5. 1 Sea level standard values.....................................................................................60 Table 5. 2 The fundamental seven layers of the U.S. Standard Atmosphere 1976 .................62 Table 5. 3 Frequently used refractivity constants..................................................................66 Table 5. 4 Coefficients of the hydrostatic NMF mapping function (Niell, 1996) ...................79 Table 5. 5 Coefficients of the wet NMF mapping function (Niell, 1996) ..............................80 Table 6. 1. The current IGS structure/components................................................................93 Table 6. 2 Approximate Availability and Accuracy of the IGS Products...............................94 Table 6. 3 Comparisons of IGS Rapid and IGS Final combined EOP with the IERS Bulletin A for 1997 (units: mas – milli-arc-sec.; ms - millisec.) (Kouba et al., 1998) ..........................................................................................................97 Table 6. 4 Pre-computed denominators of l i for the standardized fixed nodes....................100
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Table 8. 1 Station hardware ...............................................................................................128 Table 8. 2 Published coordinates of ARP ...........................................................................128 Table 8. 3 RINEX data sets................................................................................................129 Table 8. 4 Correlation coefficients of the estimated parameters [WES2, DOY2(2000)] .................................................................................................150 Table 8. 5 The difference between the PPP solutions and the published coordinates for station WES2 (cm) ....................................................................................153 Table 8. 6 The difference between the PPP solutions and the published coordinates for station NJIT (cm).......................................................................................153 Table 8. 7 The difference between the PPP solutions and the published coordinates for station USNO (cm) ....................................................................................154 Table 8. 8 The difference between the PPP solutions and the published coordinates for station TSEA (cm).....................................................................................154 Table 8. 9 Averaged discrepancies before and after SA-off (cm) for dual-frequency solutions .........................................................................................................154 Table 8. 10 The impact of satellite antenna offset on the station solutions (3D errors) ........158 Table 8. 11 Station coordinates offset due to ignoring relativistic effect .............................158
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LIST OF FIGURES Figure 2. 1 Instantaneous positioning error before and after SA (available from the US Space Command (IGEB, 2000) ).......................................................................14 Figure 2. 2 Troposphere and other atmospheric layers...........................................................20 Figure 2. 3 Ionosphere regions..............................................................................................22 Figure 2. 4 Geometric effect on phase (Wu et al., 1993).......................................................25 Figure 2. 5 GPS satellite clock correction for PRN5 prior to and after SA discontinuation..................................................................................................32 Figure 2. 5 Relativity corrections .........................................................................................35 Figure 2. 6 Plate tectonics....................................................................................................46 Figure 3. 1 Schematic of lunar tidal force (Vaní¹ek and Krakiwsky, 1982)...........................40 Figure 3. 2 Tidal Potential ...................................................................................................41 Figure 3. 3 Graphic representation of the M2 loading effect in vertical displacement (Courtesy of Hans-Georg Scherneck, Onsala Space Observatory, Chalmers University of Technology).................................................................45 Figure 3. 4 Major tectonic plates of the world .......................................................................48 Figure 5. 1 Hopfield single-layer polytropic model atmosphere............................................67 Figure 5. 2 Schematic of Saastamoinen tropospheric and stratospheric spherical layered dry atmosphere .....................................................................................70 Figure 5. 4 Sunspot count 1700-1800 (top), 1800-1900 (middle), and 1900-2000 (bottom) (NOAA, 2000)....................................................................................82 Figure 5. 5 Sunspot number prediction for cycle 23 (NASA, 2000b) ....................................83 Figure 5. 6 Monthly mean sunspot numbers (NOAA, 2000).................................................84 Figure 5.7 shows GPS ionospheric range errors as functions of TECU and frequency. .........86
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Figure 5. 7 GPS Ionospheric range errors as functions of TECU and frequency....................86 Figure 7. 1 Multipath on P1 and P2 for PRN29 [NJIT, DOY137(2000)].............................104 Figure 7. 2 Phase OMC between satellites without satellite clock correction applied. L1 (top), L2 (middle), and ionosphere-free (bottom). The base satellite is PRN25. [WES2, DOY138(2000)] ..................................................................107 Figure 7.3 Phase OMC between satellites with satellite clock correction applied. L1 (top), L2 (middle), and ionosphere-free (bottom). The base satellite is PRN25. [WES2, DOY138(2000)] ...................................................................108 Figure 7. 4 Phase OMC between satellite ionosphere-free PRN21-PRN25 without (top) and with (middle) satellite clock correction applied, and satellite clock correction difference (bottom). [WES2, DOY138(2000)] ......................109 Figure 7. 5 The Kalman filter computation recursive scheme .............................................115 Figure 7. 6 The EKF computation recursive scheme...........................................................117 Figure 7. 7 Potential weighting functions, comparison between cosecant and cosecant-squared.............................................................................................122 Figure 7. 8 Potential step function weightings, comparison between equations 7. 33 (line) and 7. 34 (dotted line) ............................................................................124 Figure 7. 9 PPP Algorithm.................................................................................................126 Figure 8. 1 Widelane before and after cycle slip detection (top) and the difference(bottom) for PRN5 [WES2, DOY2(2000)].......................................131 Figure 8. 2 Ionospheric carrier phase for PRN5 [WES2, DOY2(2000)] ..............................132 Figure 8. 3 Multipath on P1 and P2, for PRN5 [WES2, DOY2( 2000)] ..............................133 Figure 8. 4 OMC for P1 for PRN5 (top) and all satellites (bottom) [WES2, DOY2(2000)] .................................................................................................134
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Figure 8. 5 OMC for P2 for PRN5 (top) and all satellites (bottom) [WES2, DOY2(2000)] .................................................................................................135 Figure 8. 6 OMC for pseudorange ionosphere-free for PRN5 (top) and all satellites (bottom) [WES2, DOY2(2000)] ......................................................................136 Figure 8. 7 OMC difference between satellites for P1 (top) and pseudorange ionosphere-free (bottom). The base satellite is PRN5 [WES2, DOY2(2000)] .................................................................................................137 Figure 8. 8 Pseudorange ionospheric variations PRN5 [WES2, DOY2(2000)]....................138 Figure 8. 9 Elevation and azimuth for PRN5 [WES2, DOY2(2000)] ..................................139 Figure 8. 10 Sky plot for PRN5 (top) and all satellites (bottom) [WES2, DOY2(2000)] .................................................................................................140 Figure 8. 11 North (top), east (middle) and up (bottom) [WES2, DOY2(2000)]..................141 Figure 8. 12 Receiver clock estimate(top) and its variance (bottom) [WES2, DOY2(2000)] .................................................................................................142 Figure 8. 13 Ionosphere-free pseudorange OMC (innovation) for PRN5 (top) and all satellites (bottom) [WES2, DOY2(2000)]........................................................143 Figure 8. 14 Ionosphere-free carrier phase OMC (innovation) for PRN5 (top) and all satellites (bottom) [WES2, DOY2(2000)]........................................................144 Figure 8. 15 Reparameterized ambiguity estimates PRN5 (top) and all satellites (bottom) [WES2, DOY2(2000)] ......................................................................145 Figure 8. 16 Variance of estimated ambiguity PRN5 (top) and all satellites (bottom) [WES2, DOY2(2000)] ....................................................................................146 Figure 8. 17 Number of SV used in the computation (top) and the respective DOPs (bottom) [WES2, DOY2(2000)] ......................................................................147 Figure 8. 18 Relativistic correction for PRN5 [WES2, DOY2(2000)].................................148
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Figure 8. 19 Approximated tropospheric error PRN5 (top) [WES2, DOY2(2000)] .............148 Figure 8. 20 Estimated zenith tropospheric error [WES2, DOY2(2000)] ............................149 Figure 8. 21 Correlation coefficients between the estimated parameters: clock-uptroposphere (top), north-east-up(second from top), ambiguities (second from bottom), ambiguity-north-east-up-troposphere (bottom) [WES2, DOY2(2000)] .................................................................................................151 Figure 8. 22 Station solution (top) compared with the solid earth tides corrections (bottom) [WES2, DOY2(2000)] ......................................................................155 Figure 8. 23 Comparison of tropospheric zenith delay between the estimated value and the Saastamoinen model [WES2, DOY136(2000)]....................................156 Figure 8. 24 Adding a slip of one cycle to PRN29 at epoch 300 [USNO, DOY138(2000)]..............................................................................................160 Figure 8. 25 Subtracting a slip of one cycle to PRN29 at epoch 300 [USNO, DOY138(2000)]..............................................................................................161 Figure 8. 26 Solutions comparison between zenith troposphere estimated and approximated [WES2, DOY136(top), 137(middle), and 138(bottom) (2000)]............................................................................................................163 Figure 8. 27 Dual-frequency kinematic solutions comparison between troposphere estimated and approximated (top) with the associated DOPs (middle) and SV used (bottom) [WES2, DOY136(2000)] ....................................................165 Figure 8. 28 Horizontal offset comparison of kinematic solutions between zenith troposphere estimated (top) and approximated (bottom) [WES2, DOY136(2000)]..............................................................................................166
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Figure 8. 29 Dual-frequency kinematic solutions comparison between troposphere estimated and approximated (top) with the associated DOPs (middle) and SV used (bottom) [USNO, DOY137(2000)] ....................................................167 Figure 8. 30 Horizontal offset comparison of kinematic solutions between zenith troposphere estimated (top) and approximated (bottom) [USNO, DOY137(2000)]..............................................................................................168 Figure 8. 31 Single-frequency kinematic solutions comparison between troposphere estimated (top) and approximated (second from top) with SV used (second from bottom) and DOPs (bottom) [WES2, DOY2(2000)] ...................170 Figure 8. 32 Horizontal offset comparison of kinematic solutions between zenith troposphere estimated (top) and approximated (bottom) [WES2, DOY2(2000)] .................................................................................................171 Figure C. 1 Pseudorange performance [WES2, DOY136 (top) and 137 (bottom) (2000)]............................................................................................................263 Figure D. 1 Hourly jumps in pseudorange P1+P2 (top), but not carrier phase L1 + L2 (bottom) for PRN4 [GAIT, DOY2(2000)] .......................................................264
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1 Introduction
1.1 Research Goals The most popular GPS technique currently in use for accurate positioning is double differencing. This technique is popular because common mode errors cancel for short baselines or their impact is drastically reduced in the case of long baselines.
This is
particularly true for satellite orbital errors and receiver clock errors. Another distinguishing characteristic of double differencing is the relative ease of constraining double difference ambiguities to integer values. Several ambiguity fixing strategies have become available that typically rely on statistical tests and search strategies.
The least squares ambiguity
decorrelation adjustment (LAMBDA) developed at the Delft University of Technology (Teunissen, 1994) seems to enjoy an ever increasing popularity. Single differencing has recently found a new value in connection with the processing of GLONASS observations (Leick et al., 1998a and 1998b). Since GLONASS satellites transmit generally at different frequencies, the receiver clock errors do not cancel in the double difference carrier phase equations. Appropriate scaling prior to forming the double differences results in cancellation of the receiver clock errors, but introduces a non-integer double difference ambiguity.
These difficulties can be avoided when processing single
differences and fixing double difference ambiguities implicitly.
1
Little emphasis has thus far been given to undifferenced processing of carrier phases, i.e. using observation from one receiver only. The reason being is that common mode errors do not cancel in this case. However, this situation is likely to change as highly accurate orbital positions AND satellite clock data become available from the IGS (International GPS Service). See Neilan et al. (1997) and Kouba et al. (1998) for a brief description of these services. The availability of very accurate satellite clock data is of fundamental importance for the undifferenced processing technique (the GPS satellite clock stability is intentionally degraded when Selective Availability is active). The maturing of the IGS products makes it opportune to subject precise point positioning (PPP) to a rigorous scientific study and analysis. Per definition, PPP uses accurate orbital data AND accurate satellite clock data as provided e.g. by the IGS, and DUAL-frequency pseudorange AND carrier phase observations collected by the user, and uses the ionosphere-free function of the observables. PPP implies single receiver positioning (as far as the user is concerned). The second receiver, needed in the double difference approach, is essentially replaced by the collection of continuously operating receivers around the world.
The achievable accuracy of PPP approaches that for the
underlying network of receivers, i.e. we expect centimeter position accuracy. The research addresses the following: • Investigate the performance of PPP as a function of the accuracy of IGS orbits and IGS satellite clock data. This research is expected to provide recommendations for the design of future IGS products and services. • Investigate the correlations between vertical coordinate, tropospheric zenith delay, receiver time, and ambiguities. The research will shed light on the minimum requirements to make the vertical parameter, the tropospheric zenith delay, the receiver clock and the ambiguities independent (separable). The research will focus on the accuracy of the vertical coordinate and on limitations resulting from biases that 2
might be unavoidable. The latter aspect is of particular interest for vertical deformation applications. • The performance of PPP will be investigated as a function of the length of observations ranging from single epoch solutions (kinematic mode) to six hours. • Investigate the possible contribution of connecting an atomic clock to the receiver. Such a combination could be particularly useful for kinematic application. • Clarify and investigate limitations and constraints of the PPP technique that may still be caused by lingering Selective Availability. • Investigate the utility of L1-only receivers for PPP (single-frequency PPP). In this case the ionospheric correction requires special considerations. • Study and document existing "geo" models that must be incorporated to make centimeter positioning meaningful. Examples include solid earth tides, ocean loading, plate tectonic motion, and atmospheric models. • Develop Mathcad software to test components of PPP and support future research.
1.2 Motivation Algorithmic developments as well as hardware developments have steadily improved since about 1982 when civil applications of GPS positioning began. Highlights in these developments include the introduction of dual-frequency receivers, narrow correlation receivers, the antenna swap method, and OTF (ambiguity fixing On-the-Fly). The IGS has quietly but effectively contributed to bringing GPS applications to a higher level of accuracy. By virtue of having observation stations distributed worldwide and having organized an efficient data management and analysis operation, the IGS now provides GPS orbital accuracy at the 5 cm level (1σ). Equally important, it makes satellite clock information
3
available with correspondingly high accuracy. This is a major accomplishment that potentially makes single receiver positioning competitive with relative positioning between two receivers. Accomplishing one's high-accuracy "positioning mission" with just one receiver is probably as revolutionary as the introduction of GPS itself. The major advantage of course is that the user has to operate only one receiver at a time. It is a characteristic of GPS that applications are too numerous to be listed. Many applications are still emerging. For example, PPP is useful to researchers for studying diurnal tidal motions of the solid earth center (Scherneck and Webb, 1998). Tregoning et al. (1998) attempt to measure isostatic rebound in Antarctica using continuous remote GPS observations. No doubt, these and other applications can benefit from PPP, primarily because singlereceiver operation and expected simplicity in processing. Since positioning with certain accuracy implies time transfer capability with comparable accuracy, PPP is becoming a preferred candidate for accurate time transfer. Already a major international campaign is in preparation by the BIPM (Bureau International des Poids et Measures) and IGS (Ray, 1998). Whereas research groups and research institutes are expected to pay increasing attention to PPP and respective scientific applications, the proposed research specifically keeps the needs of practicing surveyors in mind. Surveyors constitute a very large GPS user group with specific needs. Not only will they appreciate operating only a single receiver, but also the decreasing reliance on the CORS (Continuous Operating Reference Stations) and other high accuracy reference networks.
1.3 Previous Relevant Works Relevant work on PPP has been carried out primarily at JPL (Jet Propulsion Laboratory). In fact the very term "Precise Point Positioning" seems to have its origin at JPL. There are two 4
publications from JPL researchers that are particularly relevant to PPP, indeed these papers sparked the interested in the proposed research. The theoretical foundation of PPP is documented in Zumberge et al. (1997a). They set out to develop an efficient approach to make the "accuracy achieved at IGS for global solutions" available to users. They recognized that various networks of stations have been established around the world to satisfy the need for high accuracy positioning. An example is the CORS network consisting of some 80 plus stations operated by the NGS (National Geodetic Survey). These networks typically serve on the premise of relative positioning, i.e. the user will position himself relative to the nearest CORS station. Sophisticated users might reference to several CORS station. The JPL researchers state: "To keep the computational burden associated with the analysis of such data economically feasible, one approach is to first determine the precise GPS satellite positions and clock corrections from a globally distributed network of GPS receivers. Then, data from the local network are analyzed by estimating receiver-specific parameters with receiver-specific data: satellite parameters are held fixed at their values determined in the global solution. This 'precise point positioning' allows analysis of data from hundreds or thousands of sites every day with 40 Mflop computers, with results comparable in quality to the simultaneous analysis of all data." In order for users of PPP to achieve the highest possible accuracy, it is important that the solution be "consistent." One not only must use their precise ephemeris and THEIR clock data but also use the same "geo" models like earth tides, etc. to avoid a degradation of achievable accuracy. Zumberge et al. (1997a) document the validity of their approach by analyzing daily sets of carrier phase data achieving millimeter repeatability in the horizontal components and centimeter precision in the vertical. 5
Even more astounding results are reported in Zumberge et al. (1997b).
They
computed orbits and clock information solutions from the Flinn global network (this is a subset of the IGS network). Many of the Flinn stations are equipped with a hydrogen maser or a good quality rubidium or cesium clock. Thus a very stable time reference is available at the receiver site.
This is crucial when estimating high-rate satellite clock corrections (in
particular when SA is active). All Flinn network receivers record data at a 30 second interval. GPS high-rate clocks are then estimated at the same respective 30-second epoch based on a free-network solution for Flinn. They utilized such high rate clock information to analyze carrier phase data from a single receiver for both static and kinematic mode. For the static mode with 5 minutes of data, the 3-D positional accuracy was 0.44 cm, but with a daily repeatability of 1.86 cm. The kinematic mode provided a 3-D positional accuracy of 3.4 cm. This result is remarkable! It seems clear that PPP constitutes a major step forward in the development of high accuracy positioning, and that it is a complex technique. Zumberge and his colleagues are the only researchers that have thus far reported PPP results of such a high accuracy. However, there is no analysis of PPP reported in the literature and no explanation can be found why other researchers seem to have failed to achieve comparable accuracy. The results reported above were achieved with the software GIPSY/OASIS-II developed at JPL. Additional information on this software is provided below. Currently, the Bernese software, but not GAMIT, has PPP capability and the Bernese group is still experimenting with certain components of PPP (Hugentobler, private communication). However, there is no evidence in the literature. Other researchers have previously reported results using PPP. However as we will see, their results are at an order of magnitude worse than what has been achieved at JPL.
6
Héroux et al. (1995) reported the accuracy of submeter for single point positioning using pseudoranges in conjunction with the use of JPL's GIPSY/OASIS-II software and 30 second interval GPS orbits and clock corrections from the Geodetic Survey Division (GDS), Natural Resources Canada (NRCan). Lachapelle et al. (1996) applied kinematic single point positioning to the aircraft in the post-processed mode.
Single-frequency pseudoranges were used in the analysis and
precise ephemerides and clock corrections were also obtained from the NRCan.
Using
GIPSY/OASIS-II software, the analytical results were then compared to DGPS and showed consistent accuracy at 1 m (rms) in latitude and longitude and 2 m (rms) in height. They concluded that the ionosphere degraded the accuracy, particularly in the height component because a single frequency receiver was used. Henriken et al. (1996) tested stand-alone positioning with single- and dual- frequency pseudoranges.
Using precise ephemerides and clock corrections from NRCan, the post-
processed analysis was conducted based on epoch-to-epoch solutions.
The
low passed
filtered Fast Fourier Transform (FFT) was then applied to remove high frequency receiver noise. The results are accurate to 0.5-1.5 m horizontally and 1.5-3 m vertically depending upon how one accounts for ionospheric corrections. It must be noted that for the experiments listed above, the standard software was JPL's GIPSY/OASIS-II. It should be noted that the Canadian experiments relied on pseudoranges only. Pseudoranges are not as accurate as carrier phases; however carrier phases add complexity because of additional ambiguity parameters and the possibility of cycle slips. The Canadians apparently did not use a free-network solution, i.e. their results may have been negatively affected by inaccuracies in the reference network. As IGS continuously improves the solutions, the need for using a free-network solution as reference is expected to diminish.
7
In any case, PPP has been demonstrated by JPL at the couple of centimeter level and by Canadian researchers at the sub-meter level. It is a technique to be reckoned with. JPL's GIPSY/OASIS-II software has been developed typically in piecemeal fashion over many years and by many people.
Its primary purpose is satellite orbit and earth
orientation determination. It is the "working horse" for JPL's researchers. PPP capability has been added apparently around 1994/95. The source code is Fortran that runs on UNIX. JPL makes executables of GIPSY/OASIS-II available upon request. However, because GIPSY's "shape and form" is that of an "internal research software", transporting it to other computers and actually using it is no easy task. JPL has found it necessary to contract with Raytheon to provide installation and consulting services to users (Zumberge, private communication). In exceptional cases, the source code can be made available to researchers at universities or government agencies. A formal contract between the University of California and the home institute is required.
The process can take up to several months (Zumberge, private
communication). It appears that GIPSY/OASIS-II and the Bernese software are the only software available that can deal with PPP.
Even if the considerable effort is made to install
GIPSY/OASIS-II software package at the University of Maine, its value in terms of supporting dissertation research is highly questionable. Dissertation research requires that the source code is totally understood and modifications/additions to the program can be made easily by the student. This does not seem to be the case with GIPSY/OASIS-II. It is, therefore, unavoidable that a new, trimmed-down, dedicated PPP research software be developed as part of this dissertation.
8
1.4 Approach This research is involved with a study and utilization of existing geo-models to accomplish centimeter positioning globally with GPS using IGS products. This study will further deal with the current and planned IGS products to accomplish this goal. Respective software will be developed and tested with real data. In order to facilitate the study, software components developed by other researchers will be used. Examples are Neill’s tropospheric corrections and the solid Earth tides software.
1.5 Thesis Organization Chapter 2 provides background on the navigation satellite systems and the equations for the basic undifferenced observables. Components of PPP are also summarized. The geophysical models will be dealt with in detail in chapter 3. The treatment begins with the concept of deformable earth followed by details on solid Earth tides, ocean loading, and plate tectonic motion. Chapter 4 deals with the International Terrestrial Reference Frame (ITRF). This encompasses the history of the development of geodetic reference frame, precise definition of the current frame, and the effects of temporal variations. Chapter 5 discusses the tropospheric and ionospheric effects on the GPS signals. The global mapping functions for tropospheric corrections are given and the ionosphere-free functions of the observables are listed. Chapter 6 focuses on the precise IGS ephemeris and satellite clock information for GPS and gives the Lagrange interpolation formulation. Chapter 7 addresses the mathematical implementation of PPP. Two major sections deal with cycle slip detection for undifferenced observations and the actual implementation of the Kalman filter. The numerical results are given in Chapter 8. This is followed by conclusions and recommendations in Chapter 9. There are four appendices: the
9
SP3 ephemeris format, a collection of Mathcad functions developed for this study, pseudorange solutions, and hourly jumps in RINEX files.
10
2 Background
The development of satellite-aided positioning or navigation is far from completed. While the beginning of satellite positioning can be traced back to the days of the Sputnik satellite, the Global Positioning System (GPS) has made satellite positioning available to a world-wide community of users since the early eighties and is currently undergoing a modernization phase.
In the nineties the Russian GLObal NAvigation Satellite System (GLONASS)
commanded some attention. Combining the signals of both systems attracted the curiosity of scientists to alleviate shortcomings of the individual systems. Since early 1998, the European community has been planning to launch a similar satellite navigation system called Galileo. This system is currently in the design phase (Hein, 2000). In addition to paying attention to developing the space component of satellite systems and refining positioning algorithms, a complete positioning infrastructure has been developed consisting of world-wide and/or national reference networks, the IGS (International GPS Service), and so on. This chapter provides background on GPS and GLONASS and the various components that are typical and essential for Precise Point Positioning.
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2.1 The GPS System
2.1.1
General Information
GPS is a weather-independent 24-hour position and navigation system that is maintained and operated by the U.S. Department of Defense (DoD). The first satellites of this system were launched in 1978. The system achieved initial operability capability (IOC) in 1993 when the orbital constellation reached 24 space vehicles orbiting at an altitude of about 20180 km and 55 degrees inclined orbital planes (Block I satellite at 64 degrees). The U.S. Air Force Space Command (AFSC) announced full operational capability (FOC) in 1995 when the constellation consisted of only Block II satellite.(USNO, 2000a). UTC(USNO) is the reference for GPS time (not adjusted for leap seconds). The broadcast ephemeris refers to the WGS-84 geodetic reference frame. The U.S. Coast Guard Navigation Center maintains a GPS home page (Navcen, 2000) that is the best source for current information. This web address contains many downloadable papers and reports, among others the ICD-GPS-200 which contains system specifications. Since the first satellite was launched in 1978, there have been three generations of GPS satellites, so-called Block I, Block II/IIA, and Block IIR. Block I and II/IIA were manufactured by the Rockwell company, whereas Block IIR by Lockheed Martin. The next GPS generation is called Block IIF, which the U.S. Air Force awarded a contract to Rockwell in 1996. The Block IIF satellite, the latest generation of GPS managed by the NAVSTAR GPS Joint Program Office at the Space and Missile System Center (SMC), has improvements over previous blocks of GPS satellites including a design life of 12.7 years, a dramatic increase in the growth space for additional payloads and missions, and provision for a new high accuracy civilian signal. The contract calls for 33 satellites and is valued at about $1.3 billion. The first delivery of the Block IIF satellites is expected in 2005 at the earliest. 12
Some components of the GPS signal structure will be changed and supplemented as part of on-going modernization efforts which aim to make the system perform even better and more reliable for the two major, and yet distinct user communities, i.e. military and civilians. Currently GPS transmits two carriers, L1 = 10.23 x 154 = 1575.42 MHz (wavelength
λ1 ≈19.0 cm) and L2 = 10.23 x 120 = 1227.6 MHz (wavelength λ2 ≈24.4 cm). The carriers are modulated with a precision (P) code (L1 & L2), and a coarse acquisition code C/A (L1). The P(Y)-code has been encrypted (Anti Spoofing AS) and is henceforth referred to as Ycode. The chipping rates for the P-code and C/A-code are 10.23 MHz and 1.023 MHz respectively. The C/A-code is normally available on L1 only, but could be activated on L2 by the ground control. Upon completion of the modernization phase, the GPS satellites are expected to transmit the C/A-code on L2 and have a new (third) civil signal, called L5 at 1176.45 MHz. Of course, L1 also carries the navigation message modulated at 50 bps. The GPS system has two levels of service, Standard Positioning Service (SPS) and Precise Positioning Service (PPS). SPS is a positioning and timing service continuously available to (civilian) users worldwide with no charge. SPS refers to L1 C/A-code positioning whereas PPS refers to P-code positioning. Until the recent discontinuation of selective availability (SA), the broadcast ephemeris (epsilon) and the satellite clock (dither) were intentionally falsified. As a result stated SPS positioning accuracy was 100 m horizontally and 156 m vertically, and the time transfer accuracy was 340 ns (95% probability) (USNO, 2000b). The White House decision resulted in the discontinuation (setting to zero) of SA at 4.00 UTC on May 2, 2000. Consequently, civilian C/A-code users have immediate access to accuracy better than 20 meters (95%). Figure 2.1 shows the impact of discontinuing SA.
13
Figure 2. 1 Instantaneous positioning error before and after SA (available from the US Space Command (IGEB, 2000) )
PPS, which is highly accurate and based on cryptographic changes of P(Y)-code, is available to only authorized (military) users. By having a military P(Y)-code capable receiver, PPS yields positioning accuracy (95% probability) of 22 m horizontally, 27.7 m vertically, and 200 ns for the time transfer accuracy to UTC (USNO, 2000b). Anti-spoofing (AS) has been implemented to guard against unauthorized transmissions of satellite data that mimic the actual satellite signals via the encryption of the P-code to form the Y-code.
2.1.2
The Undifferenced Observation Equations
Both the carrier phase and pseudorange observables are important to PPP. The relevant expressions are in units of distance (e.g., Leick, 1995; Hofmann-Wellenhof, 1997):
Φ ip,k = ρ kp − cdt k + cdt p + + d k ,i ,Φ +
d kp,i ,Φ
+
dip,Φ
c p N + I kp,i ,Φ + Tkp f i i ,k +
ε kp,i ,Φ 14
2. 1
Pi ,pk = ρ kp − cdt k + cdt p + I kp,i , P + Tkp + d k ,i ,P + d kp,i , P + dip, P + ε kp,i ,P where i = subscript identifying L1 or L2 fi = frequency
k= receiver station identifier p = satellite identifier
Φ k = measured carrier phase scaled to distance (meters) p
Pkp = measured pseudorange ρkp = geometric topocentric distance p
N k = integer ambiguity dtk = receiver clock error dt p = satellite clock error I kk ,i ,Φ , I kk ,i,P = ionosphere for phase and pseudorange and frequency fi Tkp = troposphere
dk ,i ,Φ , dk ,i,P = receiver hardware delay for phase or range respectively dkp,i ,Φ , dkp,i,P = multipath for phase or range respectively p dip,Φ , di,P = satellite hardware delay for phase or range respectively
ε kp,i ,Φ ,ε kp,i ,P = random measurement noise for phase or range respectively
The tropospheric and ionospheric effects are discussed in a later chapter.
15
2. 2
2.2 The GLONASS System GLONASS is a Russia-based positioning system managed and maintained by the Russian Space Forces. GLONASS is very similar to GPS. The full GLONASS constellation also calls for 24 satellites. While FOC has briefly been achieved, there are currently only 10 GLONASS healthy satellites (SFCSIC, 2000). The satellites are located in three orbital planes with 64.8 degrees inclination. At 19100 km orbital altitude their period is 11 hours and 15 minutes. General information about GLONASS can be obtained from the Coordinated Scientific Information Center, the Russian Space Forces web site (SFCSIC, 2000), and the ICDGLONASS (1998). The GLONASS carriers L1 and L2 are also modulated with P-codes and C/A-codes. In contrast to GPS, each L-band frequency is different for each GLONASS satellite as given below (some GLONASS satellites on opposite sides of the orbit use the same frequency). For satellite p being the GLONASS almanac number, and frequency channel number n, frequency allocation for L1 and L2, respectively (ICD-GLONASS, 1998):
f 1 p = ( 2848 + n) 0.5625 MHz ,
2. 3
f 2p = ( 2848 + n) 0.4375 MHz .
2. 4
During 1998-2005, GLONASS applies frequency channels n = 0...12 without any restrictions, but n = 0 and n = 13 are intended for technical purposes. Beyond 2005, GLONASS will use n = (-7… +6), but n = +5 and +6 will be for technical purposes. The GLONASS observation equations are similar to those of GPS, except that proper identification of the carrier frequency now requires a subscript to identify the carrier and a superscript to identify the satellite. The reference frame for the broadcast ephemeris is PZ-90. Over the last 10 years much effort has been made to relate the PZ-90 and WGS-84 coordinate systems. Typically 16
seven parameters were estimated to locate the origin (∆u, ∆v, ∆w), determine the orientation (ε, β, ω) and scale (s). For example, Misra et. al. (1996) used
1 u ∆u v = ∆v + (1 + s ) − ω β w ∆w
− β x ε y 1 z
ω 1 − ε
2. 5
and obtained ∆u = 0, ∆v = 2.5 m., ∆w = 0, s = 0, ε = 0, β = 0, and ω = -1.9x10-6 radians. GLONASS time is UTC(SU) and accounts for leap seconds according to the IERS notification. Users are usually notified in advance of the planned corrections. During the leap second correction, GLONASS time is also corrected by changing enumeration of second pulses of all onboard cesium clocks. Difference between GLONASS time and UTC(SU) is expected to be within 1 millisecond. Like GPS, GLONASS satellite clock correction is included in the navigation message.
Recent efforts to increase the GLONASS time
synchronization is reported in Mikhail (2001). A systematic effort has been made through the International GLONASS EXperiment (IGEX) campaign in 1998 to study and test the GLONASS system in general, derive the best transformation parameters between PZ-90 and WGS-84, and to relate GLONASS time and GPS time (Slater et al., 2000). Integrity monitoring of IGEX-98 data is reported by Jonkman and Jong (2000a, 2000b, and 2000c). After ending the experiment the participants agreed on a continuation. In late May, 2000, the campaign evolved into the International GLONASS Service Pilot Project (IGLOSPP), sponsored by the International GPS Service (IGS) (Slater, 2000). The pilot service will operate for a period of up to four years, from 2000-2003. The combined GLONASS orbits (IGEX-solution) as well as weekly summaries for the entire year of 1999 have been uploaded
17
to the CDDIS which is available on anonymous logon (CDDIS, 2000a). More information about IGLOS-PP can be found at IGLOS-PP (2000). The different frequencies of GLONASS cause extra complication when attempting to fix double difference ambiguities (Leick and Mundo, 1997; Wang et al., 2000).
These
difficulties are not relevant to PPP because one never attempts to fix undifferenced ambiguities. The combined GPS/GLONASS PPP solution, however, would require separate receiver clock parameters for GPS and GLONASS (even though the same receiver observes the signals from both satellite systems). The research described in this thesis did not use GLONASS observations.
2.3 Components of PPP Unlike in relative positioning, common mode errors do not cancel in PPP. Station movements that result from geophysical phenomena such as tectonic plate motion, Earth tides and ocean loading enter the PPP solution in full, as do observation errors resulting from the troposphere and ionosphere. Relevant satellite specific errors are satellite clocks, satellite antenna phase center offset, group delay differential, relativity and satellite antenna phase wind-up error. Receiver specific errors are receiver antenna phase center offset and receiver antenna phase wind-up. This section contains a brief description of these error sources and of other components that are parts of PPP. Critically important components will be amplified in the next chapters.
18
2.3.1 Geophysical Models Tidal deformations occur in the solid Earth, in the oceans, and in the atmosphere, and all three interact in different ways with the Earth’s rotation. In addition to causing the Earth's rotation axis to precess and nutate in space, the gravitational lunar and solar attractions, in combination with inertial force (centrifugal force) resulting from the orbital movement of the Earth around the common gravity center of the Earth-Moon or Earth-Sun system, cause the tides (Lambeck, 1980). The Earth deforms because it has a certain degree of elasticity. This tidal component, called solid Earth tides, is accurately computable. The Earth tides are similar to ocean tides, however the latter are strongly affected by coastal geography and ocean topography. There are four measurable tidal constituents large enough for geodetic consideration. These are the lunar diurnal, the lunar semidiurnal, the solar diurnal, and the solar semidiurnal tides. Diurnal tides have a period of about 1-day (24 hours and 50 minutes) whereas semidiurnal tides are about half a day (12 hours and 25 minutes). Ocean loading, i.e. primarily vertical variation of the crust in primarily coastal areas, is caused by sea level fluctuations due to the tides. In order to achieve truly centimeter-level global geodesy, ocean loading must be included in the site positional analysis. Tectonic plate motions have been the subject of intense geodetic research for several decades using Satellite Laser Ranging (SLR), Very Long Baseline Interferometry (VLBI), GPS, and other space geodetic systems. Normally, tectonic plate motion is detected by using regional or global tectonic networks. Data tracking of observations over a long period of time from one station to another yield vectors of plate tectonic displacements. Currently, there are hundreds of IGS sites around the world recording daily data for analyzing plate motion. In addition, there are 64 GPS sites in California which are being applied to study earthquakes in the Los Angeles area. For each site, data is analyzed into time 19
series to see movement trends in terms of latitude, longitude and height. This shows the evolution of positions and best-fitting velocities as can be examined over the Internet (JPL, 2000b).
2.3.2 Atmosphere The tropospheric and ionospheric regions of the atmosphere affect the propagation of GPS signals. The troposphere is the bottom portion of the Earth’s atmosphere, which is the layer of weather on the Earth. The thickness varies between 8 - 16 km from pole to equator. The tropospheric temperature normally varies inversely with height, approximately − 6.5°C per km (NOAA/NASA/USAF, 1976).
Figure 2. 2 Troposphere and other atmospheric layers
The troposphere normally has about 75% of the atmosphere’s mass and most of the water vapor in the atmosphere. High water vapor concentration ranges from 4% in the tropical regions (humidity about 60-80% or more), but diminishes to just trace amounts in the polar areas. The average atmospheric pressure is 1.03 kg/cm2 holding nitrogen 78%, oxygen 21%, and other gases 1% (e.g., argon, hydrogen, ozone, and methane). There is a small 20
amount of carbon dioxide, but the concentration has doubled since 1900. Water cycling takes place in the troposphere as the exchange and movement of water between the Earth's surface and atmosphere. Solar energy causes water to evaporate, and wind circulates the moisture. Air rises, then expands and cools down condensing water vapor and thus developing clouds. Various types of precipitation happen depending upon size and temperature of water particles. Consequently, the troposphere is changing according to temporal and seasonal variations. Stratosphere layer and tropopause are above the troposphere. The stratosphere, tropopause, and troposphere are considered the electrically neutral atmosphere, which is a non-dispersive medium for radio waves at frequency less than about 20 MHz. The influence of troposphere refraction on both the carrier phases and code modulation is identical. However, a part of the signals’ energy is absorbed by non-ionized gases (e.g., carbon dioxide) and water molecules. Consequently, these matters delay the signal up to 2.5 meters in the zenith direction and 30 meters close to a horizon angle causing a longer signal travel time from satellite to receiver as compared to vacuum.
These delays vary with temperature,
pressure, and humidity as well as spatial and physical location of the receiver. Tropospheric refraction cannot be eliminated with dual-frequency observation. The ionosphere is a layer or layers of ionized air surrounding the Earth extending from almost 80 km above the Earth’s surface to altitudes of 1000 km or more. The air is extremely thin at these altitudes. When the atmospheric particles are ionized by radiation (e.g., ultraviolet radiation and X-rays from the Sun), they tend to remain ionized due to few collisions between free negatively charged electrons and positively charged atoms and molecules called ions. These ions characterize the ionosphere. The free electrons affect the propagation of radio waves, thus the GPS signals. Unlike the troposphere, the ionosphere is a dispersive medium for radio waves, which means that the modulations on the carrier and carrier phases are affected differently and this effect is a function of carrier frequency. The 21
impact decreases with the increased frequency.
Normally, the radiated energy from a
transmitter gets through the ionosphere, in part absorbed by the ionized air, and in part refracted or bent downward again towards the Earth’s surface. Further, carrier frequencies below about 30 MHz are reflected by the ionosphere, thus only higher frequencies, such as GPS signals, television and frequency-modulation (FM) radio, can normally penetrate the ionosphere.
Figure 2. 3 Ionosphere regions There are two distinct regions in the nomenclature of ionization as shown in Figure 2.3 . The magnetosphere refers to the outermost region where the particle motion is controlled by the geomagnetic field. The ionosphere can be divided into two main layers called the E layer (sometimes called the Heaviside layer or Kennelly-Heaviside layer, from about 80 to 113 km.) and the F layer (sometimes called the Appleton layer, which is above the E layer). The E layer reflects low frequency radio waves while the F layer reflects higher-frequency radio signals.
The F layer is composed of two layers: the F1 and F2 layers, which start
approximately at 180 and 300 km above the Earth’s surface, respectively. The thickness of the F layer changes at night, thus altering its reflecting characteristics (Jursa, 1985).
22
The dispersive characteristics of the ionosphere closely follows the 11-year cycle of sunspots (see e.g. Knight et al., 1996; Klobuchar and Doherty, 1998; Kunches and Klobuchar, 1998).
2.3.3 Reference Frames Many reference systems and frames have been introduced and made available to users. Examples are WGS-84 and PZ-90 used by GPS and GLONASS respectively for the broadcast ephemerides. The International Terrestrial Reference Frame (ITRF) has been established by the International Earth Rotation Service (IERS).
The ITRF frame is frequently updated
according to the new data obtained from various geodetic observation systems, thus producing a time series of reference frames. The transformation from one reference frame to another is generally accomplished with a seven-parameter transformation. This will be discussed in Chapter 4. The IGS precise ephemeris is referenced to the ITRF.
2.3.4 IGS The International GPS Service (IGS) was formally established by the International Association of Geodesy (IAG) in 1993 and officially started its operations on January 1, 1994 after a successful pilot phase of more than one year (IGS 1997; Neilan et al., 1997). IGS is composed of more than 200 globally distributed permanent GPS tracking sites, three Global Data Centers, five Operational or Regional Data Centers, seven Analysis Centers, an Analysis Center Coordinator, and a Central Bureau (IGS, 1998). Each IGS site of the global network operates a dual-frequency GPS receiver that records measurements at 30-second intervals (Zumberge et al., 1997b; and Neilan et al., 1997). The Jet Propulsion Laboratory (JPL) serves as the Central Bureau and, since 1999, the Center of Orbit Determination in Europe (CODE)
23
serves as the Analysis Center Coordinator (Kouba et al., 1998). IGS is a member of the Federation of Astronomical and Geophysical Data Analysis Services (FAGS) and it operates in close cooperation with the International Earth Rotation Service (IERS). The IGS service and its products are described in Chapter 6.
2.3.5 Phase Wind-up Error Phase wind-up problem is associated with the antenna orientation, both at the satellite and at the receiver. This is due to the electromagnetic nature of circularly polarized waves intrinsic in the GPS signals. Ideally, at the receiver the measured angle of carrier phase equals the geometric angle between the instantaneous electric field and a reference direction at the receiving antenna.
Thus, when the antenna orientation changes, so does the reference
direction, and subsequently the measured phase. Likewise, the change of satellite antenna orientation changes the direction of the electric field at the transmitting antenna and, as a result, the change at the receiving antenna, thus the measured phase. Wu et al. (1993) derived the phase wind-up correction for a crossed dipole antenna, but is applicable to more general cases. A crossed dipole antenna consists of two equal-gain dipole elements perpendicular to each other. Let xˆ and yˆ be the unit vectors in the directions of the two dipole elements at the receiver antenna (horizontal plane), see Figure 2.4. Similarly, let xˆ′and yˆ′be the unit vectors in the directions of the two dipole elements at the transmitting antenna.
Symbol φ is a azimuth angle from the receiver antenna x-dipole
direction to the satellite and θ is a satellite zenith angle. Angles φ′and θ ′are at the satellite, measured similar to that at the receiver.
24
At the receiving antenna, let the phase signal from the x-dipole be received 90°earlier relative to that from the y-dipole element. The signals from both dipoles are added to form the antenna output.
Figure 2. 4 Geometric effect on phase (Wu et al., 1993)
25
Undifferenced observation: The phase wind-up correction is given as
∆φ = φ + φ′ +π
2. 6
π refers to the ground receiver pointing upward away from the center of the Earth.
Phase wind-up for single difference: Let the receiver apparatuses be set such that the x-axes of the two antennas, k and m, point horizontally in the azimuthal direction along the baseline, observing satellite p which is on the left of m when looking at k. The phase wide-up for single difference is the difference of the phase wide-up of the two receivers:
∆φSD = ∆φmp − ∆φkp
2. 7
Apply the associated phase wind-up corrections:
∆φSD = φmp − φkp + (φ' mp − φ' mp ) = β1 − (π − α 1 ) + γ
2. 8
= Ω1 in which α1, β1, and γare the three inner spherical triangle angles formed by projecting k, m, and p onto a unit sphere concentric with the earth. This result indicates that phase wind-up correction for a single difference is equal to the inner area Ω 1 of spherical triangle on a unit sphere. If the satellite is on the other side of the baseline, the phase wind-up will have the same magnitude, but opposite sign.
Phase wind-up for double difference: Similar to the single difference case, by adding satellite q, the double difference phase-wind-up correction is
∆φDD = ±Ω
2. 9
Ω = α + β + γ+ δ− 2π
26
where α, β, δ , and γare the inner angles of the spherical quadrilateral formed by the projection of the two satellites and the two receivers onto the unit sphere. Effect of carrier phase wrap-up induced by rotating GPS antennas has been studied by Tetewsky and Mullen (1997). 2.3.6 Receiver Antenna Phase Center Offset A recent study about GPS antenna phase center offset has been conducted at the National Geodetic Survey (NGS) by Mader (1999). The effect occurs because a GPS range observation is measured from a satellite transmitted signal to the electrical phase center of the receiving antenna. The electrical phase center variation (PCV) is a function of a particular antenna's phase pattern (Aloi, 1999). GPS antenna phase center is neither a single well-defined physical point nor stable spot, but rather varies with the changing direction of the incoming satellite signal. However, practically, users assume that the received signal point stays constant over the observation period, which is often referred to as the phase center of the antenna. Mader (1999) experimented in a series of tests using baselines to study relative antenna phase center position with respect to the reference antenna. But, absolute antenna calibrations have not been clearly demonstrated. For very short baselines using identical antennas at the opposite ends, the phase center variations should cancel out and no effect is seen. On the other hand, when different antenna types are used and these variations are disregarded, the baseline solution will be the weighted average of the individual phase centers of the two antennas.
The antenna phase center offset may reflect significant vertical
positioning accuracy of up to 10 cm and sub-centimeter in the horizontal.
27
Normally, PCV is a function of both elevation and azimuth (Wubbena et al., 1997; Aloi, 1999). However, it is not easy to model PCV variations due to high temporal correlation with signal reflection multipath and specific antenna. As a matter of simplicity by assuming azimuthal symmetry, one simple model is rather to assume that the phase center varies as a function of satellite elevation angle only. If absolute antenna calibrations were known, it would be possible to include this information with reference code and phase observations to position a physical point such as an external antenna reference point (ARP):
ϕ kp,i (t ) =
fi p f ρ k (t ) + N kp,i + i Θ i ( E ) sin E + εϕ c c
2. 10a
Pkp,i (t ) = ρ kp (t ) + Θ i ( E ) sin E + ε P
2. 10b
where Θ i (E ) is the calibrated vertical distance between phase center and ARP for Li (i = 1, 2). The symbol E denotes the satellite elevation angle.
In equation (2.10) the clock,
tropospheric, ionospheric, and multipath terms are ignored for simplicity. The antenna phase variations will effect the ionosphere-free observables as:
ϕ kp,IF (t ) = =
f12 f 12
−
f 22
ϕ kp,1 (t ) −
f1 f 2 f 12
−
f 22
ϕ kp,2 (t )
f1 p f2 f f ρ k (t ) + 2 1 2 N kp,1 − 2 1 2 2 N kp,2 c f1 − f 2 f1 − f 2
[
+ f13Θ 1 ( E ) − f 1 f 22 Θ 2 ( E )
]c ( fsin− Ef
28
2 1
2 2 )
2. 11
Pkp, IF (t ) =
f 12 f12
−
f 22
Pkp,1 (t ) −
[
f 22 f12
−
f 22
Pkp,2 (t )
= ρkp (t ) + f12 Θ 1 ( E ) − f 22 Θ 2 ( E )
]f
2. 12
sin E 2 1
− f 22
2.3.7 Satellite Antenna Phase Center Offset and Satellite Orientation Satellite antenna phase center offsets do not cancel for PPP and must be dealt with accordingly. In double differencing, these offsets cancel. These offsets are given in the same satellite-fixed coordinate system that is also used to express solar radiation pressure (Leick, 1995, p. 57). The origin of the coordinate system is at the satellite's center of mass, the k-axis points toward the Earth center, the j-axis points along the solar panel axis, the i-axis completes the right-handed coordinate system and lies in the Sun-satellite-Earth plane. This definition breaks down when the Sun, the satellite and the Earth are colinear. In this case and when the satellite is in the earth's shadow the satellite attitude becomes unstable and complicated to model (Bar-Sever, 1996). Starting on 1998-Nov-29 (GPS Week 986, day 0) the IGS products incorporated the antenna phase center offsets given in Table 2.1. Table 2. 1 GPS satellite antenna phase center offset adopted by IGS (Kouba and Springer, 1998)
Block II/IIA: (i, j, k) = (0.279m, 0.000m, 1.023m) Block IIR
: (i, j, k) = (0.000m, 0.000m, 0.000m)
The satellite ephemeris refers to the center of mass of the satellite. It can readily be envisioned that the k-offset will be absorbed by the receiver clock estimate if not corrected 29
properly. GPS 43 (PRN 13) was the only Block IIR satellite available at the time this research was conducted. Let XSat and XSun be the GPS satellite and the Sun coordinates in the ECEF system. XSat is obtained from the SP3 satellite ephemeris and XSun (or Xmoon) is calculated from the planetary ephemeris. Unit vector e at the satellite and pointing towards the Sun is,
− X Sat v X e = Sun X Sun − X Sat
2. 13
Unit vector at the satellite center of mass and pointing to the Earth's center is,
v − X Sat k = X Sat
2. 14
The unit vector along the solar panel axis is,
v v v j = k ×e
2. 15
The direction that completes the satellite-fixed right handed coordinates system is,
v v v i = j ×k
2. 16
If O denotes the antenna phase offset expressed in the satellite fixed (i, j, k) coordinate system and given in Table 2.1 then,
∆X Sat = R − 1 O
2. 17
is the offset expressed in the ECEF coordinates system where the rotation matrix R is:
v i T v R = j T v k T
2. 18
30
The satellite phase center antenna in ECEF is
X SV = X Sat + ∆X Sat
2. 19
2.3.8 Satellite Clocks Taking the speed of light as approximately 3 x 108 m/s, a satellite clock error of 1 µs causes an error in the computed topocentric distance of 300 m. Accurate knowledge of the satellite clock errors is of central importance to PPP. Without knowing the satellite clock error there would be no PPP technique because the respective solution would be in the same "class" as the standard navigation solution (which only corrects the satellite clock errors as provided in the broadcast message). Of course, one major advantage of single and double differencing is the elimination of the satellite clock error. Each of the Block II/IIA satellites carries two cesium (CS) and two rubidium (RB) atomic clocks. For Block II/IIA, CS clocks are considered the best satellite clocks (USNO, 2000a). One of the atomic clocks defines space vehicle time (others operate as spare). The satellite clock errors are estimated by the IGS in connection with their satellite ephemeris production using a global data set. The SP3 ephemeris files contain a respective column for the satellite clock. Figure 2.5 shows a typical example of a satellite clock variation computed by JPL under the influence of SA and after discontinuation of SA.
31
Figure 2. 5 GPS satellite clock correction for PRN5 prior to and after SA discontinuation
2.3.9 Group Delay Differential Group delay differential (TGD) is the L1-L2 instrumental bias that differs from satellite to satellite. The L1-L2 correction is given by bits 17 through 24 of word seven of the navigation message. TGD is pre-calculated by the Control Center based on measurements made by the SV contractor during factory testing (ICD-GPS-200C). The value of TGD equals 1 /(1 − γ) times the delay differential, i.e.:
TGD =
t Pp1 − t Pp2
2. 20
1− γ
32
where γ= ( f1 f 2 ) 2 and t p is the time when the signal for each frequency is transmitted. Single frequency users must correct the satellite clock (as computed from the polynomial coefficient given in broadcast message) as follows: (ICD-GPS-200C):
dt Lp1 = dt p − TGD
2. 21a
dt Lp2 = dt p − γTGD
2. 21b
Single frequency users that process pseudoranges and carrier phases for PPP do not have to correct for TGD because they are absorbed by the estimated ambiguities. In case of dualfrequency observation substituting (2.21) into the dual-frequency ionosphere-free function (5.40) cancels TGD . Table 2. 2 Typical sample of TGD (extracted from the GPS broadcast navigation message DOY2(2000))
PRN
TGD (ns)
1 2 4 5 6 7 8 9 10 13 16 17 18 24 26 27 30
-3.259629011154 -1.396983861923 -6.053596735001 -4.190951585770 -5.122274160385 -1.862645149231 -4.190951585770 -5.587935447693 -1.862645149231 -1.210719347000 -9.313225746155 -1.862645149231 -5.122274160385 -9.313225746155 -6.519258022308 -4.190951585770 -7.916241884232
Currently JPL provides updated estimates of TGD to the US Air Force Second Space Operations Squadron (2SOPS) every quarter and also monitors the values daily to identify any
33
abrupt changes in the TGD values due to configuration changes on the satellites. The first complete set of biases were uploaded on 29 April 1999 (Wilson, 1999).
Example of TGD
values given in the broadcast ephemeris for DOY2, 2000, is shown in Table 2.2. Having the correct TGD values provides three important benefits (Wilson, 1999): 1) single-frequency users who are not subject to SA (which has been discontinued) gain higher positioning accuracy because the satellite clock error can be computed more accurately. 2) For WADGPS, it provides more consistent use of fast clock corrections. 3) For the ionospheric community, when the L1-L2 bias for the receiver is known or estimated, it increases the capability to extract the absolute TEC from dual-frequency observations.
2.3.10 Relativity Because of GPS orbital eccentricity, it is necessary to take into account the small relativistic clock correction as suggested in the ICD-GPS-200C:
dt rel =
2 c2
X ⋅X&
2. 22
where X and X& are position and velocity of a GPS satellite. Relativistic correction changes from satellite to satellite and from epoch to epoch are seen in Figure 2.5. Unlike in differential GPS, the relativistic corrections must be applied in PPP.
34
Figure 2. 5 Relativity corrections
35
3 Geophysical Models
3.1 Deformable Earth The Earth is primarily composed of three basic components: solid (e.g., rock), liquid (e.g., ocean), and atmosphere.
These components make the Earth far from absolutely rigid.
Therefore, in order to study the Earth’s deformation, the more realistic Earth’s model should be somewhere in between being rigid (all considered solid) and being liquid. The Earth must be seen as a deformable body over a wide range of time scales in response to changing surface loads in the atmosphere, ocean, and hydrosphere (Lambeck, 1989). In addition, we have the knowledge about the Earth’s motion that the Earth revolves around the Sun, together with other planets, and at the same time the Earth also rotates or spins around its instantaneous axis of rotation. The Earth’s motion, combined with solar and lunar attraction forces, causes Earth tides, which are of interest regarding their effects on geocentric coordinates. In general, the Earth’s temporal deformations occur locally as well as globally. Tidal deformations take place in the solid Earth, in the oceans, and in the atmosphere. There are two potentials relevant for each point of the Earth’s surface. First, the gravity potential results from the Newtonian attraction from the whole mass of the Earth. Second, the Earth's rotation causes the centrifugal potential. The forces corresponding to the
36
difference in the potential cause the tides. The oceanic tides are characterized by the fact that the sea surface steadily adjusts itself to the potential surface. The Earth’s physical properties are rather quite complex (elasticity, viscosity, and plasticity) and cause the Earth to react to forces in a complex manner. In terms of temporal variations the shape of the Earth and consequently the positions of the points on it can be classified into three categories according to Vaní¹ek and Krakiwsky (1982): • Secular – linear, slow, creeping • Periodic – with period ranging from fractions of a second up to tens of years • Episodic – suddenly accelerating and decelerating In the following sections, we are concerned with Earth tides, ocean loading, and plate tectonic motion.
3.2 Solid Earth Tides
3.2.1
Brief History
The following succinct history about the solid Earth tides is extracted from the first chapter of Melchor (1978 and 1983). "In 1824, the mathematician Abel was the first who pointed out that the direction of the vertical does not stay constant but changes according to the influence of the attraction forces from the Sun and the Moon. In 1844, C.A. Peters published the first calculation of this effect. Around 1876, Lord Kelvin drew attention to the deformable effects of the Earth itself indicating that it was no longer acceptable to consider the Earth as being completely rigid. 37
Kelvin then demonstrated that the amplitudes observed at the Earth’s surface, for each phenomenon derived from the tidal potential (i.e., oceanic tides, deviations of the vertical, variations of the gravity force) would be affected by the deformation of the surface on which our measurements are made. In the early nineteenth century, the thought of a not entirely rigid Earth, but deformable as a result of the tides, had therefore begun to be accepted. In the meantime, some astronomers doubted the periodic variations in the latitudes and studied the oscillation of the direction of the vertical and the local deformations of the Earth crust. The horizontal pendulum, invented in 1832 by Hengler and Zöllner, was the first basic instrument in the study of Earth tides and Seismology. Due to imperfections of the suspension wire available at that time, von Rebeur Paschwitz conceived a suspension on metallic points. This became the first instrument to record deviations of the vertical caused by the Earth tides.
In 1890, research institutes of Potsdam, Strasbourg, and
Teneriffe equipped with Paschwitz’s instruments demonstrated that there is an existence of periodic oscillations of the vertical. Concurrently, Küstner and Marcuse were experimentally showing the actual existence of the periodic displacements of the instantaneous rotation axis of the Earth. Surprisingly, the two phenomena of the variations of latitude and Earth tides governed by the same theory were discovered at the same epoch in the same country, Germany. The simplest method that provides a clear demonstration that bodily tides do exist is rested on a very simple logic. When observing ocean tides relative to the marks fixed on the crust accompanied with the tide gauge, these marks would be perfectly fixed if the globe was perfectly rigid, and the observed 38
amplitude of the ocean tide would then be equal to that calculated. On the other hand, if the solid part is deformed, then the measured amplitude will be equal to the difference between oceanic and Earth tides.
G. Darwin
implemented this procedure in observing long period oceanic tides (monthly and semi-monthly lunar tides)."
3.2.2
The Tidal Potential
The Earth tides occur according to the variations of the gravitational force, or the so-called tidal force, exerted by celestial bodies such as the Moon. For any point on or within the Earth’s surface, the gravitational force exerted by a celestial body is a sum of two components, see Figure 3.1. The first component is the force that governs the Earth’s motion as a whole. This equals the gravitational force acting at the Earth’s center of gravity.
The second
component is the tidal force that equals the remainder of the force acting at the considered point. Interestingly, the tidal force at a far point (e.g., point D) acts in the outward direction to the celestial body. The reason is that the Earth is accelerating towards the attracting body at the same rate as its center of mass M, but the near side (e.g., point A) is accelerating more and, on the contrary, the far side (e.g. point D) is accelerating less than the center of mass. Viewing all as a whole, the tidal force attempts to deform the equipotential surface of the Earth’s gravity field causing its shape to prolate in the celestial direction. In other words, the shapes are likely to elongate in the direction of the resultant force exerted by the configuration of the celestial bodies.
39
Figure 3. 1 Schematic of lunar tidal force (Vaní¹ek and Krakiwsky, 1982)
Beginning with Newton's law of gravitation, the tidal potential can readily be derived. Vaní¹ek and Krakiwsky (1982) give the expression:
Wt ( P ) = m
G m m dM
r ∑ dPm n= 2 M ∞
n
P (cos Z P ) n
3. 1
The symbols have the following meaning: m
the mass of the Moon, 7.38 x 1022 kg
G
the universal constant of gravitation, 6.673 x 10-11 m3/(kg s2).
d Mm
average distance between center of the Earth and center of the Moon
ZP
the zenith angle of the Moon at P
Pn
associated with a series of the Lagendre's functions
rP
the distance from center of mass of the Earth to point P 40
Figure 3. 2 Tidal Potential
A similar equation can be obtained for the solar tidal potential by simply replacing the notation of Moon (m) with the Sun (s). Typically, the solar potential is about 46% of the lunar potential. Other celestial bodies contribute much less as shown in Table 3.1.
Table 3. 1 Relative contributions to tidal potential from various celestial bodies (Vaní¹ek and Krakiwsky, 1982)
Celestial Bodies
Tidal Potential
Moon Sun Venus Jupiter Mars
1.0 0.4618 0.000054 0.0000059 0.0000010
As it should be expected at any point in and on the Earth, the luni-solar potential varies temporally. This is primarily due to the temporal changes in geocentric distances
dMm , dMs and zenith distances Z m , Z s . The largest amplitudes of these periodic variations are semidiurnal and diurnal.
Diurnal period band is caused by lunar and solar motion.
41
Whether the celestial body is overhead or under the observer, the semidiurnal results in an identical tidal potential. Lunar semidiurnal, which has the period of half a lunar day, is the major contribution to the tidal potential.
3.2.3
Solid Earth Tide Displacements
Taking the mathematical gradient of the tidal potential (3.1), the tidal force components in the geocentric coordinate system are obtained. These force components are relatively easy to compute because only the well known expressions for the celestial motion of the sun and the moon are needed.
To convert the force components to actual displacements requires
knowledge of the so-called Love and Shida numbers. These numbers are "conversion factors" that reflect the non-rigidy of the Earth, or reflect the yield of the actual earth to the tidal forces. Because of the complexity of the Earth's deformation property these numbers have been determined experimentally and are continuously getting refined (IERS Conventions, 1996). Observed positions on the surface of the solid Earth must be corrected for solid Earth tide displacement in order to obtain coordinates in the time-invariant ITRS reference frame. For example,
x 0 x (t ) y = y (t ) − 0 z 0 z (t )
∆x (t ) ∆y (t ) , ∆z (t )
3. 2
where x (t ), y (t ), and z (t ) are coordinates of an observed position at time t. The solid Earth tides corrections ∆x (t ), ∆y (t ), and ∆z (t ) are obtained by using JPL's Development Ephemeris DE403 (planetary) and Lunar Ephemeris LE403 (JPL, 2000a), and a Fortran program downloaded from IERS96 (2000). 42
3.3 Ocean Loading The Ocean loading tide is the deformation of the sea floor and adjacent land responding to the redistribution of seawater which takes place during the ocean tide (Zlotnicki, 1996). The pure ocean tide can primarily be measured by using tide gauges, whereas altimeters measure the sum of ocean, loading and Earth body tides (Zlotnicki, 1996).
The pure ocean tide can be
directly observed at the beach from rising and falling with respect to a benchmark. The tide gauges installed at the coastlines measure and record these transitions. One can also put a pressure gauge at the ocean floor to detect the dynamics of ocean tide. The sum of pure ocean tide, loading tide, and Earth tide is called geocentric tide, which can be sensed from space using an altimeter.
Elastic ocean tide is the sum of the ocean and ocean loading tide
(Zlotnicki, 1996). A site displacement component ∆c (radial, west, south) at a particular site at time t can be written as
∆c = ∑ f j Acj cos(ω j t + χ j + u j − Φ cj ) ,
3. 3
j
where ψ
colatitude
χj
mean longitude of Sun, Moon, lunar perigee
fj , uj Acj , Φ cj
functions of longitude lunar node site-specific elements that reflect the coastal geography, the elastic and density structure of an earth model.
43
The subscript j runs over the major lunar and solar tides. They are: M2 (principle semidiurnal), S2, N2, K2 (semidiurnal), K1, O1, P1, Q1, (diurnal), and
Mf, Mm, and Ssa (long-
period). Table 3.2 shows a sample of ocean loading file for station WES2, Westford, Massachusetts (OSO, 2000). Figure 3.3 shows a snapshot of ocean loading along the East coast. Table 3. 2 Sample of ocean loading file
$$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$
COLUMN ORDER: M2 "PTM" = COMBINED From RRAY: From SCHW: ROW ORDER: AMPLITUDES (m) RADIAL TANGENTL EW TANGENTL NS PHASES (deg) RADIAL TANGENTL EW TANGENTL NS
S2 N2 K2 K1 O1 P1 Q1 MF MM SSA SOLUTION: M2 S2 N2 K2 K1 O1 P1 Q1 MF MM SSA
WES2 WES2 $$ GOT99.2_R.Ray_CC_PP_PTME ID: Feb 3, 2000 16:12 PTM $$ Computed by H.G.Scherneck on gere.oso.chalmers.se, 2000 $$ 40440S020 P WES2, IGS, GPS RADI TANG lon/lat: 288.5062 42.6129 .00716 .00193 .00191 .00067 .00422 .00281 .00139 .00048 .00043 .00015 .00060 .00353 .00065 .00089 .00019 .00047 .00022 .00015 .00003 .00011 .00004 .00020 .00188 .00041 .00039 .00015 .00033 .00034 .00011 .00011 .00003 .00003 .00013 -171.3 -163.1 173.1 -162.9 -9.4 -3.2 -8.9 3.6 12.8 61.4 -86.2 -129.2 -127.3 -156.2 -134.0 -21.4 15.7 -19.6 66.7 -3.2 170.7 -95.2 -22.5 9.0 -37.1 4.3 173.3 -155.3 176.5 -171.4 -55.7 -93.0 26.9
Ocean tide loading is the largest perturbation in the solid Earth tide predictions. Both amplitude and phase of ocean loading effects are heavily station and frequency dependent, normally having magnitude of centimeters, and where the vertical displacement is approximately three times larger than the horizontal components. The conventional IERS models to compute ocean loading displacements (IERS Conventions, 1996) do not include the motion of the origin of the coordinate system (motions
44
of the center of mass), but contain only the displacements due to deformation with respect to the center of gravity of the solid earth (Scherneck, 1998a).
Figure 3. 3 Graphic representation of the M2 loading effect in vertical displacement (Courtesy of Hans-Georg Scherneck, Onsala Space Observatory, Chalmers University of Technology)
45
3.4 Plate Tectonic Motion The Earth’s surface layer thickness ranging from approximately 40 to 90 kilometers is assumed to be composed of a set of large and small plates all together consisted the rigid lithosphere (the outer part of solid Earth) having average density 2.67 x 103 kg/m3. The lithosphere lies on and slides over an underlying, weaker layer of partially molten rock resulting from heat and pressure, called the asthenosphere, with density 3.27 x 103 kg/m3. The lithosphere plate movements across the surface layer of the Earth are driven by stress forces and interact along the plate boundaries producing divergence, convergence, or slippage of plate boundaries.
Figure 2. 6 Plate tectonics
Historically speaking, around the eighteenth century, Leonhard Euler, Swiss mathematician, described Earth’s surface plate movement by using the spherical geometry theorem which describes such movement as a rotation around the pole. About 1908 to 1917, the German geologist Alfred Wegener proposed the theories of continental drift. He along with others also recognized that continental plates rupture, drift apart, and eventually collide
46
with each other. Plate tectonic has had a pervasive impact on Earth sciences since around 1967-68 when geodetic space techniques became available. Plate tectonic theory is associated with lithosphere which is divided into a small number of plates that float on or move independently over the Earth’s mantle. The sizable and sudden plate’s movement causes Earthquake due to its stresses and/or volcanic activity. The nature of plate tectonic activity during most of the Earth history is still ambiguous. The plate tectonics hypothesis has developed to synthesize Earth’s dynamic behavior, thus simplifying plate tectonics concepts. Geodesy has provided an important role for plate tectonics study with high temporal resolution of the plate movements, particular from space state-of-the-art technologies such as GPS and VLBI. Lambeck (1989) has given axioms of the plate tectonics hypothesis as follows: § The plate tectonic motions are uniform on time scale of a million years or longer, but this may be an artifact of the resolution of the geological observations. This agrees with many recent geodetic observations that the present-day plate motions are very similar to average motions for the past few million years. § All inter-plate motion occurs on the plate boundaries. Geodetic technologies are used to observe how the motions between adjacent plates are absorbed, which relates stress and strain fields across the plate boundaries. § Considering the points away from their boundaries, the plates function essentially as rigid bodies, moving relative to each other without experiencing distortion. This implies that either the deformations are small compared with the motions at the plate boundaries, or these internal distortions are small when averaged over periods of millions of years.
47
These key hypothesis are important in understanding plate tectonics process involved in geodesy. Other assumptions can also be made to facilitate the plate tectonics study. The modern geodetic observations are able to answer to what extent these hypothesis are valid.
Plate Motion Model: The IERS96 recommends the NNR-NUVEL1A model for the plate motions given by DeMets et al. (1994).
Figure 3. 4 Major tectonic plates of the world (Courtesy of the Hawaii Natural History Association) (HNHA, 2000)
Figure 3.4 shows map of the tectonic plates.
Cartesian rotation vector for each plate of the
NNR-NUVEL1A kinematic plate model is given in Table 3.3. The actual transformation of Cartesian coordinates (X0, Y0, Z0) and (X, Y, Z) of the epochs t0 and t is given by
X = X 0 + 10 − 6 [ Ω Y Z 0 − Ω Z Y0 ](t − t 0 )
Y = Y0 + 10 − 6 [ Ω Z X 0 − Ω X Z 0 ](t − t 0 ) . Z = Z 0 + 10 − 6 [ Ω X Y0 − Ω Y X 0 ](t − t 0 ) 48
3. 4
Table 3. 3 Cartesian rotation vector for each plate using the NNR-NUVEL1A kinematic plate model (IERS Conventions (1996, p. 14). The units are radians per million years.)
Plate Name Pacific Africa Antarctica Arabia Australia Caribbean Cocos Eurasia India Nazca North America South America Juan de Fuca Philippine Rivera Scotia
ΩX
ΩY
ΩZ
-0.001510 0.000891 -0.000821 0.006685 0.007839 -0.000178 -0.010425 -0.000981 0.006670 -0.001532 0.000258 -0.001038 0.005200 0.010090 -0.009390 -0.000410
0.004840 -0.003099 -0.001701 -0.000521 0.005124 -0.003385 -0.021605 -0.002395 0.000040 -0.008577 -0.003599 -0.001515 0.008610 -0.007160 -0.030960 -0.002660
-0.009970 0.003922 0.003706 0.006760 0.006282 0.001581 0.010925 0.003153 0.006790 0.009609 -0.000153 -0.000870 -0.005820 -0.009670 -0.012050 -0.001270
3.5 Atmospheric Tides The gravitational pull of the sun and the moon affects the solid Earth, the oceans, and the atmosphere in different ways because of the properties of the material involved. Atmosphere tides fundamentally affect the ocean and Earth tides in an indirect way. Sea level is affected as a result of atmospheric pressure variations. Spatial and temporal variations of atmospheric mass deform the Earth's surface. Many studies demonstrate possible vertical displacements of up to 25 mm, but about one-third of this amount for horizontal displacements.
A simplified form in computing vertical displacement (mm) is (IERS
Conventions, 1996),
∆r = − 0.35 p − 0.55 p
3. 5
49
where p is the local pressure anomaly with respect to the standard pressure (101.3 kPa) andp is the averaged pressure anomaly within the 2000 km radius surrounding the site. Both quantities have units of mbar (or 0.1 kPa).
50
4 International Terrestrial Reference Frame (ITRF)
Many reference systems and reference frames have been introduced and made available to the public. Examples are the World Geodetic System (WGS-84), PZ-90, and the highly accurate International Terrestrial Reference Frame (ITRF).
4.1 General Statements on Reference Frames The Earth, as with all celestial bodies, is not static in nature. The Earth moves, rotates and undergoes deformation. Since motion and position are not absolute concepts, they can be mathematically described only with respect to some reference of coordinates (Kovalevsky and Mueller, 1989) called a reference frame. According to Kovalevsky and Mueller (1989), the purpose of a reference frame is to provide the means to materialize a reference system so that it can be used for the quantitative description of positions and motions on the Earth (terrestrial frames) or of celestial bodies including the Earth in space (celestial frames). In constructing the reference frame, a set of parameters must be chosen. Thus the term ‘convention’ has been used to characterize this choice.
After defining the model in detail employed in the
relationship between its configuration of the basic structure and its coordinates, the coordinates are thus thoroughly defined, but not necessarily accessible, hence the term a conventional reference system. The term ‘system’ refers to the inclusion of the description of the physical environment as well as the theories utilized in the coordinate definition. In 51
making a reference system available to users, it is normally materialized through a number of points, objects, or coordinates, and a set of parameters. These define a conventional reference frame. The reference frame must be accessible and clearly defined without ambiguity in writing equations of motion of a body whose coordinates are referred to in the frame. In 1988, the International Earth Rotation Service (IERS) was established at a site in Paris under the cooperation of the International Astronomical Union (IAU) and the International Union of Geodesy and Geophysics (IUGG). IERS assumed the responsibility of the Bureau International de l’Heure (BIH). The goal of IERS is to provide to the worldwide scientific and technical community the reference values for Earth orientation parameters and reference realizations of internationally accepted celestial and terrestrial reference systems (LAREG, 2000a). The International Terrestrial Reference Frame (ITRF) was established and is maintained by the Terrestrial Reference Frame Section of the Central Bureau (CB) of the IERS. Currently, there are three products generated by the IERS CB including the ITRF, the realization of the International Celestial Reference System (ICRS) (a space-fixed system that refers to the positions of quasars and other celestial objects), and the determination of Earth orientation parameters (EOP) (i.e., Universal Time, nutation corrections, and polar motion coordinates) which relate the ITRS and the ICRS. The ITRF implementation was originally based on the combination of Sets of Station Coordinates (SSC) and velocities derived from observations of space-geodetic techniques such as Very Long Baseline Interferometry (VLBI), Lunar Laser Ranging (LLR), and Satellite Laser Ranging (SLR). IERS augmented the methodology to include GPS in 1991 and the Doppler Orbitography and Radio-positioning Integrated by Satellite (DORIS) in 1994 (Boucher et. al., 1996). IERS regularly performs annual ITRF solutions, which are published
52
in the IERS Annual Reports and Technical Notes. Since 1988, IERS has evolved many ITRF solutions, namely ITRF-97, 96, and 94 to 88. The on-going ITRF effort is called the ITRF2000 (LAREG, 2000d) which includes not only active space geodetic instruments, but also useful markers. In addition, the IGEX-98 GLONASS stations are expected to be part of the ITRF-2000 solution.
4.2 The ITRF The ITRF represents the International Terrestrial Reference System (ITRS). ITRS has an origin at the mass center of the whole Earth that takes the oceans and the atmosphere into account (Boucher and Altamimi, 1996; LAREG, 2000a). The ITRS is realized by estimates of the coordinates and velocities of a set of observing stations of the IERS. The ITRS uses International Standard (SI) meter for its length unit defined in a local Earth frame in the meaning of a relativistic theory of gravitation. According to the resolutions by the IAU and the IUGG, the orientation of the ITRS axes is consistent with that of the BIH System at 1984.0 within ± 3 milli-arc-second (mas) and the time evolution in orientation of ITRS has no residual rotation relative to Earth’s crust (Boucher and Altamimi, 1996).
4.3 Transformation Between ITRFs Presently, in the ITRF computation process, the following basic procedure has been implemented (Boucher and Altamimi, 1996): ?
Reduction of each individual SSC’s at a common reference epoch t0 using their respective station velocity models (fixed geophysical plate motion models or estimated velocity fields),
53
?
At a reference epoch t0, the least-squares estimation yields ITRF station coordinates, in addition to seven transformation parameters, for each SSC with respect to the ITRF. The combination procedure uses the standard model based on Euclidean similarity involving the seven parameters with the general transformation form between ITRF coordinates (X, Y, Z) and individual solution coordinates (Xs, Ys, Zs) having three respective translations: T1, T2, and, T3; three respective rotations: R1, R2, and R3; and the scale factor: D.
Xs X Ys = Y + Zs Z
T 1 T 2 + T 3
− R3 R 2 X D R3 − R1 D Y S − R 2 R1 Z
4. 1
?
Appropriate variance is assigned for local ties between co-located stations
?
The ITRF velocities have been estimated either by 1. Combination, similar to the procedure used in the combination of station coordinates; therefore, the method of combining velocities is equivalent to and consistent with the method for combining station coordinates and can make use of derivatives from the model, or 2. Differentiating combined coordinates at two different epochs.
Table 4.1 provides parameters from ITRF-94 to previous ITRF series, published in previous IERS technical notes. From equation (4.1), (X, Y, Z) are the coordinates in ITRF-94 and (Xs, Ys, Zs) are the coordinates in the other frames. Rates must be applied for ITRF-93. By construction, the transformation parameters between ITRF-94, ITRF-96 and ITRF-97 are zero (Altamimi, 2000). The time epoch is used to indicate the position in a time series of ITRF.
54
Table 4. 1 Transformation parameters from ITRF-94 to other ITRFs (McCarthy, 1996; Altamimi, 2000)
4.4 Orientation and Origin of the ITRF
4.4.1
Orientation
From versions ITRF-88 through ITRF-92, the orientation was defined such that no rotation existed between these frames. However, the orientation of ITRF-93 was constrained to be consistent with the IERS series of Earth Orientation Parameters at epoch 88.0. The ITRF-94 orientation is again constrained to be consistent with the ITRF-92 at epoch 1988.0. For ITRF96 and ITRF-97, the reference frame definition (origin, scale, orientation, and time evolution) of the combination is achieved in such a way that ITRF-96 and ITRF-97 are in the same system as the ITRF-94 (LAREG, 2000b; LAREG, 2000c).
55
4.4.2
Origin
The ITRS origin is located at the center of mass of the whole Earth, including the oceans and the atmosphere.
The origins from series ITRF-88 through ITRF-92 were fixed to the
respective ICRS SLR solutions included in each ITRF calculation.
4.5 The Draft ITRF-2000 Reference Frame ITRF-2000 is an adoption of the newest ITRS realization. ITRF-2000 consists of not only a set of positions and velocities of global network tracking stations, but also related useful markers recognized by a wide application community such as geodesy, cartography, etc. ITRF-2000 includes the previous ITRF stations and expands to cover other types of points: • PRARE stations, • IGEX-98 GLONASS stations, • points located at tide gauges, following the GLOSS (Global Sea Level Observing System) and related programs such as EOSS (European Sea-level Observing System) or EUVN (European Vertical Reference Network) • points linking high accuracy gravity sensors or time/frequency laboratories • calibration sites for satellite altimetry, and • markers useful for national surveying agencies.
The quality of each individual point is specified according to the ITRF-2000 quality criteria guidelines.
In particular, the stations fulfilling the International Space Geodetic
Network (ISGN) criteria are identified in the publication. In addition, a validation process has been established, which had not been applied in all previous ITRF realization processes.
56
4.6 GPS WGS-84 The military reference frame World Geodetic System 1984 (WGS-84) is applied to the GPS system. The WGS-84 Coordinate System is a Conventional Terrestrial Reference System (CTRS) utilizing a right-handed Earth-fixed orthogonal coordinate system. Its z-axis is in the direction of the IERS Reference Pole (IRP) that corresponds to the direction of the BIH Conventional Terrestrial Pole (CTP) at epoch 1984 with an uncertainty of 0.005” (IERS Conventions, 1996, page 11; NIMA, 1997, page 2-2). Its x-axis is the intersection of the IERS Reference Meridian (IRM) with the plane passing through the origin and normal to the z-axis (NIMA, 1997, page 2-2). The IRM is coincident with the BIH Zero Meridian) at epoch 1984 with an uncertainty of 0.005” (IERS Conventions, 1996, page 11). Its y-axis completes a right-handed ECEF orthogonal coordinate system. The latest realization of WGS-84 frame is at epoch 1997 and the name has been given as WGS-84(G873).
This realization is
implemented by the National Imagery and Mapping Agency (NIMA). The letter ‘G’ indicates that the observation coordinates were obtained through GPS techniques and that Doppler data were not included in the observations materializing the frame. The number ‘873’ is the GPS week number at epoch 0h UTC on 29 September 1996 of a first date when coordinate frame was made available through NIMA GPS ephemerides.
The WGS-84(G873) represents
NIMA’s latest geodetic and geophysical modeling of the Earth from a geometric, geocentric, and gravitational standpoint based on data, techniques, and technology available through 1996 (NIMA, 1997, page 1-1). This is the third edition of WGS-84. The previous versions of WGS reference frames are WGS-84 and WGS-84(G730), which were implemented in the NIMA GPS precise ephemeris estimation process ranging from 1 January 1987 to 1 January 94 and 2 January 94 to 28 September 1996, respectively. The station coordinates which compose the operational WGS-84 reference frame are those of the permanent DoD GPS monitor stations. WGS-84(G873) was implemented in the GPS Operational Control Segment (OCS) and
57
incorporated into the Kepler elements of the broadcast message on 29 January 1997. The WGS-84 origin serves as the geometric center of the WGS-84 Ellipsoid, and the z-axis serves as the rotational axis of this ellipsoid of revolution (NIMA 1997, page 3-1).
4.7 Agreement Between WGS-84 and ITRF A comparison of coordinates between two reference frame systems can be made after the adjustment of a best fitting seven-parameter transformation. Malys and Slater (1994) reported an agreement between WGS-84(G730) and ITRF-92 at the 0.1 m level. Daily comparisons of WGS-84(G873) and ITRF-94 through their respective precise orbits reveal systematic differences no larger than 2 cm. The day-to-day dispersion on these parameters indicates that these differences are statistically insignificant (NIMA, 1997, page 2-5).
58
5 Troposphere and Ionosphere
Satellite signals travel through the atmosphere which affects the state of the signals. These are divided into two effects, tropospheric and ionospheric. Each effect influences the satellite signals differently. Since the troposphere is a non-dispersive medium, tropospheric refraction causes an identical effect on both code and phase modulation. The troposphere causes a signal delay of up to 30 meters for a horizontal path. Therefore, the effect from the troposphere is considered one of the major sources of errors imposed on the satellite signals. On the other hand, the ionosphere is a dispersive medium of the ionized atmosphere layer(s). Thus, the ionosphere affects the signal code and phase modulation in an opposing way. Moreover, the ionospheric effect is a function of carrier frequency. Fortunately, the ionospheric effect can be eliminated via dual frequency observation.
The environments of the troposphere and
ionosphere are described in Chapter 1 (see section 2.4.2 ).
In this chapter, the emphasis will be given to the recent tropospheric model or so-called global mapping function. The global mapping function must be utilized for the global positioning analysis.
For the ionospheric effect, the popular ionosphere-free linear combination
expression will be given.
59
5.1 Standard Atmosphere In the 1920s, the first modern standard atmosphere definition was developed in the U.S. by the National Advisory Committee for Aeronautics (NACA) and in Europe by the International Commission for Aerial Navigation (ICAN) to fulfill a need to standardize aircraft instruments and improve flight performance. Theoretical aspects of the physics of the air were utilized to build the standard atmosphere. In 1952, the discrepancy between the two independently developed NACA and ICAN standards was eliminated through the adoption of a new standard atmosphere definition by the International Civil Aviation Organization (ICAO) with computed tables ranging from 5 km below to 20 km above mean sea level. The U.S. Committee on Extension to Standard Atmosphere (COESA) was formed in 1953, and in 1961 a working group was convened to define a new standard atmosphere up to an altitude of 700 km. The work of COESA led to the new versions of the U.S. Standard Atmosphere with slight modifications from those previously adopted. These models were published jointly by the National Oceanic and Atmospheric Administration (NOAA), the U.S. Air Force, and NASA. The U.S. Standard Atmosphere 1976 has been used until now. The standard atmosphere is essentially defined in terms of an ideal air obeying the perfect gas law and by assumption that the atmosphere is static with respect to the earth (Laurila, 1976). It is based on the standard values for air density, temperature, and pressure at sea level as given in the Table 5.1. Table 5. 1 Sea level standard values Temperature, TSea
288.15 K
Pressure, Psea
101325 Pa (or N/m2) (1013.25 mb)
Gas constant, R
8.31432 x 103 Nm/(kmol K)
Density, ρsea
1.225 kg/m3
Gravity acceleration, gsea
9.80665 m/s2
60
In addition to perfect gas theory, rocket and satellite data atmospheric pressure, density, and temperature were used to represent the Earth's atmosphere from sea level to 1000 km. Single profiles representing the idealized, steady-state atmosphere for moderate solar activity are applied for the U.S. Standard Atmospheres 1958, 1962, and 1976 (NOAA/NASA/USAF, 1976; NASA, 2000a). Below 32 km the U.S. Standard Atmosphere agrees with the ICAO standard atmosphere for all practical purposes. However, the U.S. Standard Atmosphere does not necessarily represent an average of the vast amount of atmospheric data today from observations within that height region, particularly for heights below 20 km. Parameters listed include temperature, pressure, density, acceleration caused by gravity, pressure scale height, number density, mean particle speed, mean collision frequency, mean free path, mean molecular weight, sound speed, dynamic viscosity, kinematic viscosity, thermal conductivity, and geopotential altitude. The altitude resolution varies from 0.05 km at low altitudes to 5 km at high altitudes. Units in all tables are given in English (foot) as well as metric (meter) units. The U.S. Standard Atmosphere Supplements (1966) includes tables of temperature, pressure, density, sound speed, viscosity, and thermal conductivity for five northern latitudes (15, 30, 45, 60, and 75 degrees), for summer and winter conditions (NASA, 2000a), which departs from the U.S. Standard Atmosphere. The U.S. Standard Atmosphere utilized the linearly segmented temperature high profile, and the assumption of hydrostatic equilibrium, in which the air is treated as a homogenous mixture of the several constituent gases. The fundamental seven layers of the Standard Atmosphere (1976) from sea level to 86 km (Table 5.2) includes (geopotential) height and temperature gradient by altitude (temperature lapse rate).
61
Table 5. 2 The fundamental seven layers of the U.S. Standard Atmosphere 1976
Layer
H1 (km) From
H2 (km) To
α = dT / dh (°K/km)
1 2 3 4 5 6 7
0 11 20 32 47 51 71
11 20 32 47 51 71 84.852
− 6.5 0.0 1.0 2.8 0.0 − 2.8 − 2.0
Based on the standard sea level values given in Table 5.1, the atmospheric parameters can then be computed using information given in Table 5.2.
The temperature variation can be
expressed as a sum of a series of lower layers, linear altitude. The temperature T at height h (in km) falling in layer n is written as:
T = TSea + α n ( h − H1, n ) +
n− 1
∑ α i ( H 2, i − i =1
H 1, i )
5. 1
where TSea is the temperature at sea level. In the range of each atmosphere layer where temperature varies linearly as a function of altitude, the pressure can be calculated from the following expression: g M − 0 αnR
α n ( h − H 1,n ) + T P = Pn ⋅ T
, for α ≠ 0
5. 2a
g M P = Pn exp 0 ( h − H 1,n ) RT
, for α = 0
5. 2b
62
where Pn is the pressure at the initiative point of each altitude range, gravity
g 0 = 9.80665 m/s2, and universal gas constant R is given Table 5.1. Having pressure and temperature, the density at a specific altitude can be calculated from
ρ=
PM RT
5. 3
in which M is a constant mean molecular weight of the gas. At greater heights (i.e. between 86 and 1000 km), the definitions governing the Standard are far more complex due to dissociation and diffusion processes producing significant departure from homogeneity. Therefore, the temperature height profile cannot be expressed as a series of linear functions like those employed at lower attitudes. Relevant expressions and standard tables are given in NOAA/NASA/USAF (1976). The variations of gravity acceleration g defined by U.S. Standard Atmosphere through manipulating the inverse-square law of gravitation, with sufficient accuracy for most model atmosphere computations is 2
r0 g = g0 r + Z 0
5. 4
where r0 = 6,356,766 m is the effective radius of the earth at sea-level in which the centrifugal acceleration is taken into account, Z is the geometric height at a specific latitude, and g0 = 9.80665 m/s2. For the U.S. Standard Atmosphere, this equation is valid from sea level to the geometric altitude of 1000 km.
63
5.2 Troposphere The neutral atmosphere, which is the non-ionized part of atmosphere, can normally be divided into two components, the hydrostatic (dry) and wet portions of the troposphere.
The
hydrostatic component consists of mostly dry gases (normally referred to the dry part) , whereas the wet component is a result of water vapor. The troposphere causes radio signal delay.
The hydrostatic fraction contributes approximately 90% of the total tropospheric
refraction (Leick, 1995, page 308). For high accuracy positioning, correcting the delay of radio signals as they traverse the neutral atmosphere is necessary as it is one of the dominant error sources. Tropospheric effect is frequency-independent and cannot be eliminated via dual-frequency observations. The tropospheric path delay can be defined as (e.g., Janes et al., 1991; Mendes, 1994):
Tkp =
∫cscθ ( r ) dr − − + [ n ( r ) 1 ] csc θ ( r ) dr ∫ path path
csc ε ( r ) dr ∫ path
5. 5
where r is geocentric radius, n is the refractive index, and θ and ε , respectively, refer to refracted (apparent) and non-refracted (geometric or true) satellite elevation angle; n relates to the tropospheric refractivity N Trop as given below:
N Trop = (n − 1) ⋅10 6
5. 6
Note that equation (5.5) holds for a spherically symmetric atmosphere, and n is allowed to vary along the signal path as a function of geocentric radius. The first term characterizes the deviation of electromagnetic path s from geometric length of the refracted transmission path. The bracketed term is the geometric delay accounting for path curvature (ray bending), which is the difference in the geometric lengths of the electromagnetic and rectilinear paths from the satellite to the observing station (Janes et al., 1991). Such curvature 64
effect is essentially significant for satellite elevation angles of 10-20 degrees; therefore, in practice, the bracketed term is often omitted. For satellite signals, we obtain tropospheric delay:
Tkp = 10 − 6
∫N
Trop
ds
5. 7
path
Allowing for the hydrostatic and the wet components, tropospheric delay can be rewritten as:
Tk p = 10 − 6
∫( N
Trop d
+ N wTrop ) ds
path
= 10
−6
Trop −6 ∫N d ds + 10
path
Trop ∫N w ds
5. 8
path
where N dTrop and N wTrop correspond to tropospheric refractivity of the hydrostatic and the wet components. In 1974, Thayer expressed the refractivity NT in term of absolute temperature and partial pressure of the dry gases (Pd) and of water vapor ( e0 ) in millibars, viz. (e.g., Mendes et al., 1994):
N T = K1
Pd − 1 Zd + T0
e0 e + K 3 02 Z w− 1 K 2 T0 T0
5. 9
where the constant coefficients K1, K2, and K3 are empirically determined. T0 is absolute temperature in Kelvins at the tracking station. Zd and Zw are corresponding compressibility factors for dry air and water vapor, which account for the departure of the air behavior from that of the ideal gas and rest on the partial pressure due to dry gases and temperature. The first term on the right side of equation (5.9) refers to N dTrop , whereas the bracketed terms refer to
N wTrop . The frequently used sets of refractivity constants are given in Table 5.3.
65
Table 5. 3 Frequently used refractivity constants (e.g., Bean et al., 1966; Mendes et al., 1994; Langley, 1996)
Refractivity coefficients
Smith and Weintraub [1953]
Thayer [1974]
K1 (K/mb) K2 (K/mb) K3 (K2/mb)
77.61±0.01 72 ± 9 (3.75 ± 0.03) 105
77.604 ± 0.014 64.79 ± 0.08 (3.776 ± 0.004) 105
Normally the vertical wet and dry refractions are related to the refraction of a particular elevation angle by the mapping function.
The wet component depends on water vapor
content. Much research has been focused on modeling water vapor content. The water vapor pressure e0 can be calculated from a priori knowledge of environmental information such as relative humidity and temperature at the tracking station. Water vapor pressure in millibars recommended in the IERS Conventions (1996) is
e0 = 0.0611 RH 10
7.5(T0 − 273.15) 237.3+ T0 − 273.15
5. 10
where RH is the relative humidity at the observing station in percent.
5.2.1
Tropospheric Models
Due to the significance of tropospheric effects on radio signal propagation, many studies to formulate tropospheric correction have been performed. Various tropospheric models exist. In addition, different mapping functions which illustrate signal delay as a function of elevation angle are also given. The following provides descriptions of frequently used tropospheric models and mapping functions.
66
5.2.1.1 Hopfield Model Hopfield empirically developed a tropospheric model in 1969 using worldwide data. The Hopfield model applies a single layer polytropic model atmosphere ranging from the Earth's surface to altitudes of about 11 km and 40 km for the wet and dry layers, respectively (Hopfield, 1969; Janes et al. 1991; Hofmann-Wellenhof et al., 1997), see Figure 5.1.
Figure 5. 1 Hopfield single-layer polytropic model atmosphere The Hopfield model shows dry and wet refractivity components as a function of tracking station height h above the Earth's surface and is given in the following forms:
N dTrop
N wTrop
=
H d N dTrop ,0
=
H w N wTrop ,0
µ
− h Hd
5. 11a
µ
− h Hw
5. 11b
where
67
µ=4
empirically determined power of the height ratio,
Hd = 40136 + 148.72(T − 273.16)
a polytropic thickness for the dry part (m),
Hw = 11000
a polytropic thickness for the wet part (m),
N dTrop ,0 = K1
P0 T0
dry tropospheric refractivity for the station at the
Earth's surface as a function of pressure (millibars) and temperature (Kelvin),
N dTrop ,0 = K 2
e0 e + K 3 02 T0 T0
wet tropospheric refractivity for the station at the
Earth's surface as a function of water vapor, pressure, and temperature.
Inserting equation (5.11) into equation (5.8), and integrating each element with the respective integration ranges along the vertical direction (i.e. from h = 0 to h = Hd and from h = 0 to h = Hw for the dry and wet components), we then obtain tropospheric zenith delay in units of meters:
[
]
10 − 6 Trop T = N d ,0 H d + N wTrop ,0 H w . 5 Z k
5. 12
The values of dry and wet polytropic thickness, Hd and Hw, are typically in the range of 40-45 km and 10-13 km, respectively (Hofmann-Wellenhof et al., 1997).
Hopfield's zenith
tropospheric delay equation (5.12) can be employed together with a mapping function to obtain tropospheric delay at a specific satellite elevation angle.
68
5.2.1.2 Saastamoinen Model Saastamoinen (1971) applied the law of Gladstone and Dale; that the height integral
∫( n − 1) dr
of the atmospheric refractivity for radio microwaves taken from ground level to
the top of the stratosphere is, in a dry atmosphere, directly proportional to the ground pressure. Derivation of the Saastamoinen model involves thinking of the atmosphere as a mixture of two ideal gases, dry air and water vapor. Gas laws are then applied to derive refractivity. The temperature in the troposphere from sea level to about ten kilometers decreases with height at a fairly uniform rate which varies slightly with latitude and season, although in the polar regions there is a permanent inversion in the lower troposphere where the actual temperatures initially increase with height. Saastamoinen divided the dry atmosphere into two layers: a polytropic troposphere extending from the surface to an altitude of approximately 11-12 km, and an isothermal stratosphere continuing from the troposphere to approximately 50 km as shown in Figure 5.2 . Atmospheric water vapor is confined to the troposphere only. For the normal mid-latitude conditions, the Saastamoinen model is given in units of meters as:
1255 2 Tk p = 0.002277 sec z P0 + + 0 . 05 e − Ω tan z T 0 0
5. 13
where z is the true zenith distance, P0 is the pressure at the observed station in millibars, and the coefficient
R P0 T0 − ( Rβ / g ) p 0T 0 Ω= . rg 1 − Rβ / g
5. 14
69
Figure 5. 2 Schematic of Saastamoinen tropospheric and stratospheric spherical layered dry atmosphere R is the gas constant, r is the earth’s radius, T0 is temperature at the tracking station, and p0 and T0 are pressure and temperature at the bottom of the stratosphere. β is the vertical gradient of temperature ( dT / dh ). Figure 5.3 shows a plot of the coefficient Ω varied with station height above sea level as given by Saastamoinen (1971). For a station at sea level, Ω is approximately 1.16. For the signal coming in the zenith direction (z = 0), the term involving coefficient Ω is zero, and equation 5.13 can be rewritten for tropospheric zenith delay as:
1255 TkZ = 0.002277 P0 + + 0 . 05 T e0 0
70
5. 15
Coefficient for Saastamoinen Model 1.2 1.1
coefficent
1 Ω
0.9 0.8 0.7 0.6 0.5 0.4
0
1
2
3
4
5
6
h Station height km above sea level
Figure 5. 3 Coefficient Ω for Saastamoinen model versus height The first term is a function of surface pressure and refers to the hydrostatic component, while the rest is a function of water vapor pressure and corresponds to the wet component. A correction to standard gravity at the observing station has been incorporated into the Saastamoinen model (Janes and Langley, 1989; Janes et al., 1991):
TkZ =
0.002277 1255 P0 + + 0.05 e0 g′ T
5. 16
where
g′ = 1 − 0.0026 cos 2ϕ − 0.00028 h0
5. 17
has units m/s2, and ϕ and h0 are station latitude and orthometric height (km) (Saasamoinen, 1971). Saastamoinen estimates the accuracy of the hydrostatic and the wet components as 2-3 mm and 3-5 mm rms, respectively. Mendes (1998) found that the hydrostatic component can be predicted with sub-millimeter accuracy from the Saasamoinen model if accurate measurements of surface pressure are available. For the wet component he found that it is 71
much more difficult to use the surface meteorological data to predict with the best models show rms of a few centimeters .
5.2.2
Mapping Functions
The mapping function describes the elevation angle dependence of the delay of the signals that travel through the neutral atmosphere (Niell, 1996). Each neutral atmospheric component, the zenith delay and a mapping function are used to model the line of sight delay. Azimuthal symmetry is usually assumed. The mapping function parameters are normally involved with temperature, pressure, and relative humidity. Moreover, some mapping function models, e.g. MTT by Herring (1992) and NMF by Niell (1996), take latitude and height above sea level into account.
An extensive comparison between mapping functions and ray traces of
radiosonde profiles is given in Mendes (1998). Generally, the tropospheric delay correction is defined in terms of the contributions of the hydrostatic and wet components. Therefore, the total tropospheric delay can be described by:
τ a ( E ) = τ hz mh ( E ) + τ wz m w ( E )
5. 18
where the total tropospheric delay τa for unrefracted observation elevation angle E is being considered.
The symbols τ hz ,τ wz , mh , and m w are the zenith delays, and the mapping
functions for the hydrostatic and wet atmospheric components, respectively. The nominal zenith delays for a site at sea level are 2300 mm for the hydrostatic and 100 mm for the wet component. Wet zenith delay extreme values are 300 and 400 mm for midlatitude and tropical regions, respectively (Niell, 1996). However, the more specific values of zenith delay may be obtained from the tropospheric models.
72
Because of the atmospheric curvature, the mapping function does not just change as the cosecant of the elevation angle E, which would be expected for a plane parallel refractive medium. The hydrostatic mapping function will change according to changes in the ratio of the atmospheric thickness to the Earth’s radius, as a result of the changes in temperature (Niell, 1996).
Therefore, the temporal change of mapping function is associated with
variability of temperature at various atmospheric heights, which can be obtained from radiosonde profiles.
5.2.2.1 Marini Mapping Function In 1972, J. W. Marini developed a tropospheric correction, which shows that the elevation angle E dependence of any horizontally stratified atmosphere can be approximated by expanding in a continued fraction in term of 1 / sin E . The general form of the Marini mapping function is written as (Marini, 1972):
m( E ) =
1 sin E +
5. 19
a b
sin E + sin E +
c sin E + K
E is the geometric (unrefracted) satellite elevation angle and a, b, c,... are profile dependent coefficients. The Marini mapping function does not explicitly separate the hydrostatic and wet components of tropospheric delays.
5.2.2.2 Marini & Murray Mapping Function The Marini & Murray mapping function maps the total delay based on Saastamoinen zenith delay viz.(Marini and Murray, 1973): 73
m( E ) =
1+ ζ
ζ 1+ ζ sin E + sin E + 0.015
5. 20
,
5. 21
wherein
ζ =
G=
G TkZ
0.002644 [− 0.14372 h0 ] e g′
5. 22
Surface meteorology information as well as station altitude and latitude are incorporated into the Marini & Murray mapping function. TkZ and g ' are previously given in equations (5.16) and (5.17). The Marini & Murray formula is specified to be valid for an elevation angle greater than 10°. In addition, the comparison of the Marini & Murray formula with ray traces of radiosonde data revealed that the standard deviation of the range correction increases from 20 mm at the zenith to nearly 200 mm at 10°.
5.2.2.3 Chao Mapping Function In 1974, C. C. Chao derived a tropospheric mapping function to be used for radio tracking corrections of the 1971 Mariner Mars spacecraft. Chao treats the wet and dry components separately through empirical fitting to an average refractivity profile derived from two years of radiosonde data (Chao, 1974).
74
mh ( E ) =
mw ( E ) =
1
5. 23a
ah sin E + tan E + bh 1
5. 23b
aw sin E + tan E + bw
where a h = 0.00143, bh = 0.0445,
a w = 0.00035, and bw = 0.0170 .
It should be
noted that the term tan(E ) was used to ensure that both dry and wet mapping functions are one at the zenith. The accuracy of the Choa's dry mapping function is 1% down to 1° with respect to the ray trace of the average annual refractivity profiles. The error at this level is too large for accurate geodetic VLBI application (Niell, 1996). Due to vast spatial and temporal variability, Choa's wet mapping function was sufficiently accurate for space geodetic measurements until the introduction of the Herring mapping function (Niell, 1996).
5.2.2.4 Lanyi Mapping Function (Lanyi) Lanyi (1984) constructed a total mapping function that has both the dry and wet components. The Lanyi mapping function was expected to be applicable to a 6°minimum elevation angle. The parameterization of the Lanyi mapping function includes surface temperature, height of isothermal surface layer, and temperature lapse rate.
The greater parameterization in
temperature profiling of the Lanyi mapping function allows inclusion of an isothermal layer of variable height beginning at the surface. The IERS Conventions (1996) prefers the Lanyi mapping function if information about the vertical temperature distribution in the atmosphere is available.
75
5.2.2.5 Davis Mapping Function (CfA-2.2) Both the Marini & Murray (1973) and Chao (1974) mapping functions are in the generalized forms, and thus are exposed to the influence of variability in the refractivity profile and lateral gradients. To increase accuracy at low elevation angles and to better accommodate local and seasonal variations, further modification was performed by Davis et al. (1985) by adding the fraction of the sine term to Chao's mapping function. The Davis mapping function was dubbed "CfA-2.2":
mh (ε ) =
1 sin ε +
5. 24
a tan ε +
b sin ε + c
in which through the least-square fit the ray trace yields coefficients as linear functions of the surface weather conditions, i.e. pressure, temperature, and relative humidity.
a = 0.001185 [ 1 + 6.071 ⋅10 − 5 ( P0 − 1000) − 1.471 ⋅10 − 4 e 0 + 3.072 ⋅10 − 3 (T0 − 20) + 0.01965(α + 6.5) − 0.005645( H t − 11.231)] 5. 25
b = 0.001144 [ 1 + 1.164 ⋅10 − 5 ( P0 − 1000) − 2.795 ⋅10 − 4 e0 + 3.109 ⋅10 − 3 (T0 − 20) + 0.03038(α + 6.5) − 0.001217( H t − 11.231)] 5. 26
c = − 0.0090
5. 27
From above, H t is the height of the tropopause (km). α is the tropospheric temperature lapse rate value. Even though Davis et al. (1985) evaluated parameters a and b by least-squares fit
76
to ray traces of idealized pressure, temperature, and humidity profiles of spherical symmetric layered atmosphere, they remarked that errors of 1 to 2 mm are present for elevation angles from 20° to 60°. This is due to the incorrectness of the tangent term since tan E does not approach sin E quickly enough. However, the advantage of CfA-2.2 is its simplicity, both in calculating the mapping function itself and in calculating partial derivatives of the mapping function with respect to the parameters to be estimated (Davis et al., 1985). The tropospheric temperature lapse rate (α) normally has values ranging from − 6 to − 7°K/km, but − 6.5°K/km is the standard value according the U.S. Standard Atmosphere (NOAA/NASA/USAF, 1976). However, Mendes and Langley (1998) reported a mean global value of 6.17 ±0.82°/ km . Mendes and Langley (1998) also observed a correlation between α and surface temperature as expressed in the linear model as a function of temperature:
α (°/ km ) = − 5.930 − 0.0359(T0 − 273.16)
5. 28
The rms agreement of this mapping function (equation 5.24) compared with ray tracing is less than 5 mm for all elevations above 5°. Mendes and Langley (1998) averaged the global tropopause height data and reported H t = 11.3 ± 2.6 km, while Davis et al. (1985) suggested a value of 11.231 km. The large standard deviation of Mendes and Langley (1998) reflects the large latitudinal and seasonal variations, and therefore cannot be accounted for through a single nominal number. The tropopause is highest at the equatorial regions with very small seasonal variation and attenuates as it approaches polar areas where large seasonal variations are detected from middle to high latitudes. Mendes and Langley (1998) found that tropopause height is highly correlated with surface temperature according to the expression, in km
H t = 7.508 +
T0 − 273.16 2.421 e 22.90
5. 29
77
5.2.2.6 Herring Mapping Function (MTT) In 1992, T. A. Herring applied the Marini mapping function, but this correction has been normalized to unity at zenith as given below (Herring, 1992):
1+ m( E ) =
sin E +
a 1+
b 1+ c a
sin E +
5. 30
b sin E + c
The coefficients of the physical quantities a, b, and c can be estimated from the leastsquares fits of m(E) to ray traces of idealized temperature and humidity profiles for different values of pressure, temperature, etc.
These coefficients depend linearly on surface
temperature, the cosine of the station latitude and the height of station above the geoid (ranging from 0-1600 m).
5.2.2.7 Niell Mapping Function (NMF) Recently, Niell (1996) proposed the new mapping function (NMF) based on temporal changes and geographic location rather than on surface meteorological parameters. He argued that all previously available mapping functions have been limited in their accuracy by the dependence on surface temperature, which causes three dilemmas. All of these are because there is more variability in temperature in the atmospheric boundary layer, from the Earth's surface up to 2000 m. First, diurnal alterations in surface temperature cause much smaller variations than those calculated from the mapping functions.
Second, seasonal changes in surface
temperature are normally larger than upper atmosphere changes (but the computed mapping
78
function yields artificially large seasonal variations). Third, the computed mapping function for cold summer days may not significantly differ from warm winter days. For example, actual mapping functions are quite different than computed values because of the difference in lapse rates and heights of the troposphere. The new mapping functions have been derived from temperature and relative humidity profiles, which are in some sense averages over broadly varying geographical regions. Niell (1996) compared NMF and ray traces calculated from radiosonde data spanning about one year or more covering a wide range of latitude and various heights above sea level. Such comparison was to ascertain the validity and applicability of the mapping function NMF. Through the least-square fit of four different latitude data sets, Niell (1996) showed that the temporal variation of the hydrostatic mapping function is sinusoidal within the scatter of the data. Table 5. 4 Coefficients of the hydrostatic NMF mapping function (Niell, 1996)
Coefficients
Latitude ϕ i 15°
30°
45°
60°
75°
1.2196049e-3 2.9022565e-3 63.824265e-3
1.2045996e-3 2.9024912e-3 64.258455e-3
3.4000452e-5 7.2562722e-5 84.795348e-5
4.1202191e-5 11.723375e-5 170.37206e-5
Average a b c
1.2769934e-3 2.9153695e-3 62.610505e-3
1.2683230e-3 2.9152299e-3 62.837393e-3
a b c
0.0 0.0 0.0
1.2709626e-5 2.1414979e-5 9.0128400e-5
1.2465397e-3 2.9288445e-3 63.721774e-3
Amplitude 2.6523662e-5 3.0160779e-5 4.3497037e-5
Height Correction aht bht cht
2.53e-5 5.49e-3 1.14e-3
79
For the hydrostatic NMF mapping function, the parameter a at tabular latitude ϕ i at time t from January 0.0 (in UT days) is given as:
t − T0 a(ϕ i , t ) = aavg (ϕ i ) + aamp (ϕ i ) cos 2π 365.25
5. 31
where T0 is the adopted phase, DOY28. The linear interpolation between the nearest a(ϕ i , t ) is used to obtain the value of a(ϕ , t ) . For parameters b and c, a similar procedure was followed.
Table 5. 5 Coefficients of the wet NMF mapping function (Niell, 1996)
Latitude ϕ
Coefficients aw bw cw
15°
30°
45°
60°
75°
5.8021897e-4
5.6794847e-4
5.8118019e-4
5.9727542e-4
6.1641693e-4
1.4275268e-3
1.5138625e-3
1.4572752e-3
1.5007428e-3
1.7599082e-3
4.3472961e-2
4.6729510e-2
4.3908931e-2
4.4626982e-2
5.4736038e-2
The coefficients for the wet NMF mapping function are shown in Table 5.5. No temporal dependence is included in the wet NMF mapping function. Therefore, only an interpolation in latitude for each parameter is required. Height correction associated with the NMF is given as:
∆m( E ) =
dm( E ) H dh
5. 32
dm( E ) 1 = − f ( E , a ht , bht , c ht ) dh sin( E )
80
5. 33
where f ( E , a ht , bht , c ht ) is a three-term continued fraction (equation 5.30). The parameters aht, bht, cht as given in Table 5.4 were determined by a least-squares fit to the height correction at nine elevation angles, and H is the station height above sea level. Mendes (1998) analyzed the large number of mapping functions by comparing against radiosonde profiles from 50 stations distributed worldwide (32,467 benchmark values). The models that meet the high standards of modern space geodetic data analysis are Ifadis, Lanyi, MTT, and NMF. He found that for elevation angle above 15 degrees, the models Lanyi, MTT, and NMF yield identical mean biases and the best total error performance. At lower elevation angles, Ifadis and NMF are superior.
5.3 Ionosphere Ionospheric effect is a result of electromagnetic waves of GPS signals travelling through a dispersive atmosphere to the antenna. The effect inversely varies with the square of frequency f of the signals. Having dual-frequency observation, ionospheric range errors can be removed from observation data. Major ionospheric effects correspond to rise and fall of the number of sunspots − the solar cycle.
5.3.1
Spatial and Temporal Variations
Normally, at mid-latitudes the ionospheric effect on GPS signals can be negligible. On the other hand, the ionospheric scintillation activity is becoming more significant at lower latitudes, especially in the hours immediately after sunset (Knight and Finn, 1996).
In
addition, the ionospheric effects rise and fall according to number of sunspots. The sunspot cycle, which has a vast effect, was discovered in 1843 by Samuel Heinrich Schwabe. Around
81
1848, Johann Rudolph Wolf, Swiss astronomer and astronomical historian, confirmed Schwabe's discovery of a cycle in sunspot activity through the use of previous records that defined the cycle's length more accurately, to be an average of 11.1 years. Figure 5.4 shows annual sunspot numbers since 1700.
Figure 5. 4 Sunspot count 1700-1800 (top), 1800-1900 (middle), and 1900-2000 (bottom) (NOAA, 2000) 82
Wolf discovered a daily technique to measure solar activity by simply counting the number of individual spots and groups of spots on the sun's surface. He introduced the term "the Zurich relative sunspot number" (or Wolf's sunspot number), a value equal to the sum of the spots plus 10 times the number of groups. This method is still used today and daily observations of sunspots are averaged to find annual values. Wolf sunspot counts rise and fall roughly every 11 years, and its cycle is asymmetrical with an average 4.3 years to rise from a minimum to the maximum and another 6.6 years to drop to a minimum once again.
Figure 5. 5 Sunspot number prediction for cycle 23 (NASA, 2000b)
From the record, the largest annual mean number (190.2) occurred during 1957-58. The peak in the current sunspot cycle (number 23) is approaching around the middle of 2000 (see Figure 5.5). Sunspot numbers can be obtained from the archive at NOAA (2000). Figure 83
5.6 depicts monthly variation of sunspot counts. Wolf also discovered that the sunspot cycle coincided with disturbances in the Earth's magnetic field.
140
120
Counts
100
80
60
40
Monthly Sunspot Numbers
Ja n98 M ar -9 M 8 ay -9 8 Ju l-9 8 Se p98 No v-9 8 Ja n99 M ar -9 M 9 ay -9 9 Ju l-9 9 Se p99 No v-9 9 Ja n00 M ar -0 M 0 ay -0 0
20
Figure 5. 6 Monthly mean sunspot numbers (NOAA, 2000)
Some of the phase variations are transformed to amplitude and phase via diffraction resulting in an irregular but rapid variation in amplitude and phase, called scintillation (Leick, 1995, page 297).
In other words, ionospheric scintillation is caused by small-scale
irregularities in the ionospheric electron density, and can disturb the amplitude and phase of traversing radio signals. Such effects can cause severe fades in the signal or rapid phase gradients that exceed a receiver ability to hold “lock” on the signal; several cases of loss of lock under such conditions have been reported (Nordwall, 1996).
84
Among others, solar ultraviolet (UV) activity is a major cause of ionospheric turbulence.
For GPS, the major ionospheric effects pertinent to solar UV activity are
ionospheric range delays and amplitude fading and phase scintillation effects.
Direct
measurements of solar UV radiation cannot be made from the Earth's surface due to atmospheric absorption, instead data from solar cycles collected over a 300 year period are used. Another surrogate measure of this UV radiation is the solar radio flux at wavelength 10.70 cm; however, this method is less subjective than measurements of sunspot numbers (Klobuchar and Doherty, 1998).
5.3.2
Ionospheric Range Delay
Ionospheric range delays are directly proportional to the total electron content (TEC), which varies along the transmission path and can be defined as
TEC =
∫N
( s ) ds
e
5. 34
path
where Ne is the local electron density (electrons/m3). The TEC represents the total number of free electrons contained in a column with cross-sectional area of 1-square meter along the path of signal between satellite and receiver. The TEC is in units of (el/m2). The Total Electron Content Unit (TECU) is defined as TECU = 1 ⋅1016 el / m 2 . Transforming the time delay of a code sequence or the phase advancement to the corresponding distance (in meters) is:
I kp, f ,P = =
40.28 f
2
TEC =
40.28 c f
2
40.28 f
2
∫N e ds
path
5. 35
∫N e dt
path
85
The above equation is the ionospheric range delay or advance between receiver k and satellite p for the carrier frequency f and c is the speed of light. The corresponding time delay or advance follows as
vf =
I kp, f ,P c
=
40.3 TEC cf 2
5. 36
Figure 5.7 shows GPS ionospheric range errors as functions of TECU and frequency.
30
I(f1)
I(f2)
Ionosphere (m)
25 20 15 10 5 0 0
25
50
75
100 90 80 70 60 50 40 30 20 10 0 100
Time delay (ns)
GPS Ionospheric Range Error
TECU
Figure 5. 7 GPS Ionospheric range errors as functions of TECU and frequency
For GPS signals, it is necessary to identify the delays of the P1 and P2-codes and the advances of the L1 and L2 carrier phases. Normally, the ionospheric code delay has a unit in meters while carrier phases have a unit in cycles, except when the carrier phases have been scaled to distance.
I kp,1, P = − I kp,1, Φ = − I kp,2, P
=−
I kp,2, Φ
c p I f1 k ,1,ϕ
5. 37
c p =− I f 2 k ,2,ϕ
86
Ionospheric relations between two frequencies can be formed, for code and phase:
I kp,1,P I kp,2,P
I kp,1,ϕ I kp,2,ϕ
5.3.3
=
=
f 22
5. 38
f12
f2 f1
5. 39
Ionosphere Models
There are a few ionospheric models available to estimate ionospheric effect. Examples are the ionospheric plate model, daily cosine model, and ionospheric point model.
Ionospheric
coefficient for the cosine model included in the navigation message compensates approximately 50% of the actual group delay. Details of these models are summarized in Leick (1995, pages 299-302).
5.3.4
Functions of Observables
The effects from the ionosphere on GPS analysis can vary depending on many factors. Such factors include geomagnetic variations, spatial locations, upper atmospheric chemical composition and temperature, wind circulation, duration of the sunspot cycle, season, time of the day, and line of sight. Since ionospheric effect is a function of a signal's frequency, having dual frequency data can eliminate almost all of the ionospheric effects.
In addition, an
extremely precise measurement of relative TEC can be formed from the linear combination of the two carrier phases of two signals (Musman et al., 1998).
Even the most dominant
ionospheric correction is the lowest order term ( 1 / f 2 ), the higher order terms 1 / f 3 and
87
1/ f
4
might be important by affecting the observation accuracy at a few centimeters and
millimeters respectively (Bassiri and Hajj, 1992).
5.3.4.1 Dual-Frequency Ionosphere-free Exploiting the ionospheric frequency relation equations (5.38) and (5.39), a combination to obtain ionosphere-free functions for codes and phases can be formed (Leick, 1995; Langley, 1996):
Pkp, IF (t ) =
ϕ kp, IF (t ) =
f12 f12
−
f 22
f12 f12
−
f 22
Pkp,1 (t ) −
ϕ kp,1 (t ) −
f 22 f12
−
f 22
f1 f 2 f12
−
f 22
Pkp,2 (t )
5. 40
ϕ kp,2 (t )
5. 41
In equation (5.41), the ambiguity term is not an integer due to scaling factors employed to eliminate the ionospheric term.
5.3.4.2 Dual-Frequency Ionosphere Pseudorange linear combinations for the ionospheric solution can be established directly:
PI p, k = P1,pk − P2p, k p = (1 − α ) I Pp, k + c (1 − α ) TGD
5. 42
+ d1, P , k + d1p, P , k + d1p, P − d 2, P , k − d 2p, P , k − d 2p, P where α = ( f 1 / f 2 ) 2 . The receiver clock cancels in this combination. The TGD term is a constant over a period of time, as given in the broadcast navigation message. Thus, this
88
ionospheric combination contains the difference of hardware delays and multipath that normally are small magnitudes compared to the ionospheric effect, see Figure 8.8. The dual-frequency phase ionospheric function can be written as:
f1 p ϕ (t ) f 2 k ,2
ϕ kp,I (t ) = ϕ kp,1 (t ) − =
f1 p f N k ,2 (t ) − 1 (1 − α )I kp,1, P (t ) f2 c
N kp,1 (t ) −
5. 43
The scale factor on the L2 carrier phase is used to scale it to have the same frequency as in L1. This destroys the integer ambiguity nature of phase on L2.
The phase ionospheric
combination can be used to detect cycle slips. Figure 8.2 shows a phase ionospheric plot.
5.3.4.3 Single-Frequency Ionospheric-Free Code and Phase Since the ionospheric effect disrupts the code and phase differently (thus the term "group delay and phase advance"), it is possible to use this knowledge to eliminate common ionospheric error:
Ξkp, i (t )
=
Pkp, i (t ) + Φ kp, i (t ) 2
=
ρ kp (t ) −
+
d k ,i ,P (t )
+
c dt k + c dt p + Tkp (t ) + N kp, i
2 d k ,i ,Φ (t ) 2
+ +
d kp,i ,P (t ) 2 d kp,i ,Φ
(t )
2
+ +
d ip,P (t ) 2 dip,Φ 2
(t )
+ +
ε i ,P
5. 44
2 ε i ,Φ 2
with the reparameterized ambiguity
N kp, i =
TGD, i c N kp, i + β c 2 fi 2
5. 45
89
This equation can be applied to both L1 (β = 1) and L2 (β = α). However, clock errors do not cancel out, but multipath and noises contribute at half of the pseudorange and carrier phase p
values. The analysis process must be able to estimate the reparameterized ambiguity, N k .
90
6
Precise IGS Orbit and Satellite Clock
The International GPS Service (IGS) generates precise ephemerides for the satellites together with by-products such as Earth orientation parameters (EOP) and GPS clock corrections. The IGS service is built upon a global network of permanent tracking stations and provides information and data products from computational centers to all GPS users through data archive and exchange centers. In this chapter, the IGS analysis is introduced. The GPS orbital information is normally given in the standard SP3 format which will be discussed in detail in the following section.
Format and description of satellite clock corrections will also be
covered.
6.1 IGS Orbital Analysis and Its Products
6.1.1
IGS Structure and Operation
Since its establishment, the IGS accomplishes its mission through the following components: • Networks of tracking stations • Global and Regional Data Centers • Analysis and Associate Analysis Centers • Analysis Coordinator 91
• Central Bureau • International Governing Board The tracking data are available at various Data Centers, the individual orbits determined by the Analysis Centers at the Global Data Centers, and the official IGS orbits are combined at the Central Bureau and the Global Data Centers (IGS, 1998). Table 6.1 provides the current IGS components/structure, beginning with the IGS Operation Centers and the IGS station network that rigorously apply IGS standards for station monument/hardware, data quality, submission formats, and delivery delays.
6.1.2
Products Since the IGS test operations started in June 1992, a continuous set of highly accurate
daily GPS orbital data and EOP have been available from individual processing centers. Since November 1992, the IGS Analysis Center Coordinator has regularly compared the orbits of the individual processing centers (Goad, 1993). The IGS product accuracy has improved from approximately 1m (orbits) and 1mas (EOP) (Beutler, 1994) to about 5 cm (orbits) and about 0.1 to 0.2 mas (EOP) (Neilan et al., 1997). This improvement indicates that the IGS orbit is becoming more accurate, stable, and reliable. Table 6.2 shows approximate availability and accuracy of IGS products. The EOP are combined with those determined by means of satellite and lunar laser observations as well as VLBI observations by the IERS (Beutler, 1994). After the discontinuation of SA, the final satellite clock correction has accuracy 0.1 ns (Kouba, 2000).
92
Table 6. 1. The current IGS structure/components
Operation Centers/ IGS Station Network: Global Data Centers: CDDIS - Crustal Dynamics Data Information System, GSFC, NASA, USA IGN - Institut Geographique National, France SIO - Scripps Institute of Oceanography, Univ. of Cal., USA Regional Data Centers: AUSLIG - Australian Land Information Group, Australia BKG - Bundesamt für Kartographie und Geodäsie, Germany JPL - Jet Propulsion Laboratory, Caltech, USA NGS - National Geodetic Survey, NOAA, USA NRCan - Geodetic Survey, NRCan, Canada Analysis Centers: CODE - Center For Orbit Determination in Europe, Univ. Bern, Switz. NRCan - Geodetic Survey, NRCan, Canada ESA - European Space Operation Center, ESA, Germany GFZ - GeoForschungsZentrum, Potsdam, Germany JPL - Jet Propulsion Laboratory, Caltech, USA NGS - National Geodetic Survey, NOAA, USA SIO - Scripps Institute of Oceanography, Univ. of Cal., USA Associated Analysis Centers (AAC): USNO - AAC for Rapid Service, U.S. Naval Observatory, USA JPL - Global Network AAC (GNAAC), Jet Propulsion Lab., USA MIT - GNAAC, Massachusetts Institute of Technology, USA NCL - GNAAC, University of New castle upon Tyne, UK RNAACs – 15 Regional Network AACs in Europe, N.A., Asia and Australia Analysis Center Coordinator : (T. Springer/CODE) Central Bureau (CB): hosted by JPL, includes IGS CB Information System (Director: R. Neilan) International Governing Board: 15 members
Normally, the high-quality GPS data is online within one day and data products are online within two weeks of observations. The IGS global network of permanent tracking stations, each equipped with a GPS receiver, generates raw orbit and tracking data. The Operational Data Centers, which directly contact the tracking sites, collect the raw receiver data in Receiver INdependent EXchange format (RINEX) (Gurtner, 1997) and then forward these data to the Regional or Global Data Centers. For efficiency and to reduce electronic network traffic, the Regional Data Centers collect data from several Operational Data Centers before transmitting them to the Global Data Centers. Data not used for global analyses are archived and available online at the Regional Data Centers. The Global Data Centers archive 93
and provide online access to tracking data and data products which normally must be available to users for at least 60 days (Kouba et al., 1998). The online data are employed by the Analysis Centers to create a range of products which are then transmitted to the Global Data Centers for public use.
The IGS Central Bureau Information System, accessible on the
Internet, provides both IGS member organizations and the public with a gateway to all the IGS global data and data product holdings along with other valuable information. Table 6. 2 Approximate Availability and Accuracy of the IGS Products Units: mas – milli-arc-second; ms - millisecond (Kouba et al., 1998) Note that the predicted clock accuracy refers to the case of SA being active
IGS Products
GPS Satellites
IGS Station Positions
Availability
Interval
Accuracy
Ephemerides Predicted Rapid Final Clocks Predicted Rapid Final
Real Time 1-2 days 10-12 days
15 min 15 min 15 min
50 cm 10 cm 5 cm
Real Time 1-2 days 10-12 days
15 min 15 min 15 min
150 ns. 0.5 ns. 0.3 ns.
Weekly Solutions
< 4 weeks
7 days
3-5 mm
1-2 days 10-12 days
1 day 1 day
0.2mas 0.1mas
1-2 days 10-12 days
1 day 1 day
0.4 mas/day 0.2 mas/day
1-2 days 10-12 days
1 day 1 day
0.20ms 0.05ms
1-2 days 10-12 days < 4 weeks
1 day 1 day 2 hours
0.06ms/day 0.03ms/day 0.4cm
Pole Rapid Final Pole Rates Rapid Final Earth Orientation UT1− UTC Rapid Final Length of Day Rapid Final Tropospheric Zenith Delay
Since January 1994, contributions from the seven current IGS Analysis Centers make IGS official orbits possible and available to a user community. Other than supporting a variety of government and commercial interests, the IGS also develops international GPS data
94
standards and specifications. With a multi-national affiliation, the IGS collects, archives, and distributes GPS observation data sets with sufficient accuracy to satisfy the objectives of a wide range of applications and experiments.
IGS uses these data sets to generate data
products as follows: § High-quality orbits for all GPS satellites (estimated accuracy better than 5 cm (one sigma) for the final ephemerides) § Earth Rotation Parameters § Contributions to determine the tracking site coordinates and velocities in International Terrestrial Reference Frame (ITRF), in close cooperation with the International Earth Rotation Service (IERS) § Phase and pseudorange observations in daily RINEX files for each IGS tracking site § GPS satellite and tracking station clock information § Ionospheric information § Tropospheric information § Other data products in support of geodetic and geophysical research activities. Highly accurate and reliable data and data products supplied by the IGS that meet the demands of a wide range of applications and experimentation are available within two weeks of observation. These data can be accessed through the Internet via the Information System managed and maintained by the IGS Central Bureau (JPL), which is sponsored by the National Aeronautics and Space Administration (NASA). IGS near real time high-quality GPS data and data products provided by the IGS global system of satellite tracking stations, Data Centers, and Analysis Centers meet the objectives of a wide range of scientific and engineering applications, and research. Even 95
though improvements are frequently made, the current accuracy of various IGS products is sufficient to support, improve, and extend current scientific objectives including: • Realization of global accessibility to ITRF • Improvement of ITRF • Monitoring deformations of the solid Earth • Monitoring Earth rotation • Monitoring variations in the liquid Earth (sea level, ice-sheets, etc.) • Scientific satellite orbit determinations • Ionospheric monitoring • Climatological research, eventually weather prediction
Precise GPS Satellite Ephemerides and Clock Information: To obtain the final orbits, the combination of ephemerides can be accomplished via many means. The official IGS orbits are produced by forming a weighted combination of the ephemerides submitted by each individual analysis center (IGS, 1997). Each week the Analysis Center Coordinator provides the precise orbits and clocks and a summary report file documenting the combination process. Based upon IGS orbit comparison and combination, final GPS satellite ephemerides precision is better than 5cm (one sigma) in each coordinate.
Normally, a rapid solution can be
computed from 15 to 20 worldwide distributed stations after the end of the day using data available at the time. The rapid solution is made available usually within 21 hours following the observations with its estimated accuracy better than 50cm (one sigma) in each coordinate component. Generally, this degradation over the final solution has minor impact on the positioning accuracy for most GPS users. Approximate availability and accuracy of the IGS products are listed in Table 6.2 (Kouba et al., 1998).
96
IGS has taken two actions to improve the consistency between the combined IGS orbits and the combined IGS clocks. First is an improved clock weighting scheme using the clock estimates from one AC as reference instead of the satellites without SA. Second is to correct the AC clock, before the combination, based on the difference in the radial component between the AC orbit and the IGS combined orbit. It is expected that all high quality IGS products should be more reliable and at least as accurate as, if not more than, the solutions obtained from each individual analysis center (Springer et al., 1998). More attention has been given to the improved precision of the IGS combined orbit prediction (IGP), LOD/UT combination and satellite clock combination. Comparing with the IGS rapid solution (IGR), the IGP outlier detection can be performed (Kouba et al., 1998). This considerably enhances the IGP reliability and consistency. The resulting ephemerides and clocks are output to daily files in the SP3 format and these product files can be obtained from the CBIS or any of the Global Data Centers. Table 6.3 shows the IGS and IGR comparisons with the IERS Bulletin A for 1997. Table 6. 3 Comparisons of IGS Rapid and IGS Final combined EOP with the IERS Bulletin A for 1997 (units: mas – milli-arc-sec.; ms - millisec.) (Kouba et al., 1998)
IGS Final PMy LOD (mas) (ms)
Comparison
PMx (mas)
Mean
.28
.15
Standard Deviation
.07
.07
IGS Rapid PMy LOD (mas) (ms)
UT (ms)
UT (ms)
PMx (mas)
.001
.015
.40
.26
− .004
.043
.026
.044
.24
.27
.034
.203
It became clear that the IGS orbits and clocks were inconsistent at the 200 mm level and improvement has been carried out by improving the clock weighting scheme and correcting the AC clocks before the combination (Springer et al., 1998). The update and
97
combination strategies including orbit combination and evaluation statistics and remarks are given in the IGS final summary file. The current IGS products are based on ITRF-97.
6.2 The SP3 Ephemeris The currently used GPS satellite orbital format distributed by the IGS is the format defined by the National Geodetic Survey (NGS), called SP3 format.
SP3 is an ASCII
representation that includes the satellites’ position and clock corrections. ECF3 and EF18 are binary counterparts to SP3, which are regarded as the second generation of orbital formats. All formats have been carefully designed by taking many factors into consideration including their use for GLONASS and geostationary satellites (Remondi, 1993).
While the previous
generation could handle only 35 satellites, the second generation can accommodate up to 85 satellites (NGS, 2000; Remondi, 1993).
Additionally, the header section of the second
generation has been developed to allow changes and insert new information, e.g., orbital accuracy information for each satellite.
6.2.1
The SP3 GPS Orbital Format and Data Accuracy
Following the NGS study, it appeared that for all application purposes the velocity data does not need to be distributed, since it can be calculated to an accuracy about 0.004 mm/s from the positional data (Remondi, 1993). However, the velocity data can optionally be included in the SP3 format. NGS provides programs for users to recover velocity information as well as translate one format to another. The SP3 format is given to 1 mm and 1 ps (pico-second). For each information line for a given satellite, the flag notation “P” for SP1, SP2, and SP3 refers to position-only, and “V” for SP3 indicates velocity. Both position and velocity are required for the velocity flag. The velocity data has units of decimeters/s with an accuracy of 10− 4 98
mm/s. The rate of change of clock correction at the last column of the velocity line has units of 10− 4 µs/s with precision of 1 ps/s. See appendix for the SP3 format.
6.2.2
Precise Satellite Clock Information
The satellite clock information is given in the SP3 data file in units of microseconds. It is given at the same epochs for which the satellite positions are given. Its precision information is discussed above: 1 ps for the clock and 1 ps/s for the rate of change of clock correction.
6.3 Lagrange Interpolation GPS ephemeris and clock is given at a nominal epoch. The interpolation is needed to obtain the satellite position and clock correction at the transmission epoch. Lagrange is the Newton form of interpolating polynomials. Lagrange is probably the most convenient and efficient method and has several advantages (Cheney and Kincaid, 1994, page 134). Lagrange can be applied to the GPS orbit interpolation as utilized by Remondi (1989). We can also apply Lagrange to the satellite clock interpolation as well. Having a set of fixed nodes x1 , x 2 ,K , x n , the system (called cardinal function in interpolation theory) will have polynomials of degree ( n − 1) . Such system has n special polynomials denoted by l1 , l 2 ,K , l n with the following property
0 li ( x j ) = 1
if i ≠ j if i = j
6. 1
The Lagrange formula, a linear combination of polynomial l i , can then be used to interpolate any function f:
99
n
p( x ) = ∑ l i ( x ) f i ( xi )
6. 2
i =1
p(x ) is the interpolating polynomial of degree ( n − 1) for the function f.
p(x ) yields
f ( x j ) at x j . The formula for l i , the product of ( n − 1) linear factors, is given as follows: n x− x j li ( x) = ∏ j ≠ i xi − x j
1 ≤i ≤ n .
6. 3
j =1
l i is therefore a polynomial of degree ( n − 1) and having the required property given in equation 6.1. Note that the denominators of l i are just numbers. For equal epoch intervals ephemeris and for a given degree of polynomial one can pre-compute the denominators by standardizing the fixed nodes, e.g., 1, 2, … , n. This will save computation time and resources. Table 6.4 provides pre-computed denominators of l i for the standardized fixed nodes.
Table 6. 4 Pre-computed denominators of l i for the standardized fixed nodes Number of standardized fixed nodes (n): 1, 2, … , n
Denominators of l i i = 1, 2, … , n
3
2 -1 2
5
24 -6 4 -6 24
7
720 -120 48 -36 48 -120 720
9
40320 -5040 1440 -720 576 -720 1440 -5040 40320
11
3628800 -362880 80640 -30240 17280 -14400 17280 -30240 80640 -362880 3628800
100
7 Mathematical Implementations
This chapter provides mathematical techniques and implementations utilized in PPP analysis. These include cycle slip detection and removal and mathematical consideration associated with Kalman filter implementation for PPP. In addition, the computation flow and software components are also given.
Information from previous chapters is used as fundamental
principles of the software development.
7.1 Dilution of Precision The term dilution of precision (DOP) has been accepted to characterize the effect of the geometric satellite distribution on the accuracy of the navigation solution. The DOP factors include vertical dilution of precision (VDOP), horizontal dilution of precision (HDOP), positional dilution of precision (PDOP), time dilution of precision (TDOP), and geometric dilution of precision (GDOP). The DOP expressions are given below (Langely, 1999):
101
VDOP = HDOP = PDOP = TDOP = GDOP =
σh σ σ n2 + σ e2 σ σ n2 + σ e2 + σ h2 σt σ
7. 1
σ
σ n2 + σ e2 + σ h2 + σ t2 c 2 σ
where σ is usually taken to be equal to the total user equivalent range error (UERE) and
σ n2 , σ e2 , σ h2 , and σ t2 are the corresponding variances of position in Northing, Easting, Height (Up), and Time. The DOPs vary from epoch to epoch according to the change of satellite geometry.
7.2 Cycle Slip Detection and Removal The carrier phase measurements can be continued only when the receiver has the ability to maintain lock on the incoming GPS signal, which is directly related with the integer number of wavelengths (called ambiguity) referenced to the first phase measurement when the receiver starts to lock on the signal. But in some situations, the receiver loses lock of the phase lock loop which is due to the poor reception, very fast acceleration changes, or shadowing of satellite signals by obstacles in their path. Such situations cause discontinuities in the phase measurements, called "cycle slips," triggering a sudden jump of the phase.
102
7.2.1
Multipath
Multipath varies greatly depending upon a variety of factors, for example, the receiversatellite-reflector geometry and the strength and the delay of the reflected signal compared to the line-of-sight signal.
Therefore, multipath is considered one of the major limitations
imposed on the accuracy of the observables. Multipath distorts the C/A-code and P-code modulations as well as the carrier phase observation. Multipath affects pseudoranges much more than phases, therefore phases provide more precise solutions. The multipath expressions applied to GPS/GLONASS are given in equations 7.2 and 7.3 (Li, 1995),
Pk p,1 + (2 β − 1)Φ kp,1 − 2 βΦ kp,2 = (2 β − 1)λ1p N kp,1 − 2 βλ2p N kp,2
+ {d1,P }+ (2 β − 1){d1,Φ }− 2 β {d 2,Φ }
Pk p,2 + 2αβΦ kp,1 − (1 + 2αβ )Φ kp,2 = 2αβλ1p N kp,1 − (1 + 2αβ )λ2p N kp, 2
+ {d 2, P }+ 2αβ {d1,Φ }− (1 + 2αβ ){d 2,Φ }
7. 2
7. 3
where
f1 p α ≡ p f 2 β ≡
2
1 1− α
Φ k = measured carrier phase scaled to distance (meters) p
{d }= d i,Φ
{d }= d i, P
k , i, Φ
k ,i , P
(t ) + d kp, i , Φ (t ) + d ip, Φ (t ) + ε k , i , Φ
(t ) + d kp, i , P (t ) + dip, P (t ) + ε k , i , P
103
for carrier phase, i = 1, 2 for pseudorange, i = 1, 2.
Assuming there are no cycle slips and a specific satellite is being considered, the first two terms on the right side of equations (7.2) and (7.3) should provide a constant straight line. The rest of the terms can be considered noises from multipath and hardware delays. Accordingly, a plot using the terms on the left side of both equations (i.e., the observations) can be used to identify such noises. More importantly, cycle slips may be detectable to a certain extent from such plots.
Figure 7. 1 Multipath on P1 and P2 for PRN29 [NJIT, DOY137(2000)]
Multipath plots on P1 (equation (7.2)) and P2 (7.3) are illustrated in Figure 7.1, together with the elevation angle. Multipath appears to be greater at low elevation angles. The plots of equations (7.2) and (7.3) can be used in the preprocessing step to visualize quality of the data as well as to detect cycle slips.
7.2.2
Widelane
Widelane is a linear combination of carrier phase that increases the effective GPS signal wave length.
Widelane ambiguity can be computed from pseudorange and carrier phase
observation.
104
N wp = ϕ wp −
f1 P1p + f 2 P2p
7. 4
( f1 + f 2 ) λw
where
ϕ wp = ϕ 1p − ϕ 2p
7. 5
and
λw =
c ≅ 86.2 cm fw
7. 6
f w = f1 − f 2
7. 7
Widelane is a useful linear combination for the purpose of ambiguity estimation in differential GPS (Teunissen, 1997). Theoretically, widelane ambiguity should be constant. Due to spatial changes of the satellite signal's path causing multipath, the widelane ambiguity function has a small variation particularly at low satellite elevation angles.
7.2.3
Cycle Slips
Attempts can be made to detect and correct the change in phase ambiguity. If a cycle-slip is found and fixed, no further action is needed. But in the case of loss of lock the respective ambiguity must be re-estimated.
7.2.3.1 Between Satellite Differences A cycle slip causes a sudden jump in the phase observation by an integer number of cycles and all observations obtained after the cycle slip are shifted by the same integer amount. One way to find a cycle slip is to plot phase observations versus time which will show a step function or 105
individual outliers. Cycle slips can be any size ranging from one to millions of cycles. The plot should have the capability to be viewed at different scales (or different aspect ratios). Typically, steep slopes are seen from the plots of undifferenced observations as a result of receiver clock error. In such cases, small slips may not be visible. For single stations, a useful function for slip detection is the between-satellite singledifference (one receiver and two satellites) of carrier phases as presented in equation 7.8.
ϕ k (t ) ≡ ϕ k (t ) − ϕ k (t ) . pq
p
q
7. 8
Notice that equation 7.8 is free of receiver clock error. Slips are removed by adding integer values to restore continuity of the carrier function. However, visual cycle slip fixing is not easy because (7.8 ) is strongly dependent on time. Carrier phase OMC (Observed Minus Computed) between satellites shows less of a time dependency. Receiver clock error is eliminated. When satellite clock correction is not applied, a slope might still be visible due to high clock drift. See Figure 7.2.
The plots use
observations after SA had been turned off. A high variation would be seen with SA-on observations. The satellite clock correction removes most of the slope in the OMC plot as seen in Figure 7.3. Note that slips do not necessarily produce integer steps in the ionospherefree function (Leick, 1995, page 356). Single-frequency users are limited, of course, to L1 OMCs only. To detect cycle slips, the Kalman filter (Mertikas and Rizos, 1997) may be applied to the OMC.
106
Figure 7. 2 Phase OMC between satellites without satellite clock correction applied. L1 (top), L2 (middle), and ionosphere-free (bottom). The base satellite is PRN25. [WES2, DOY138(2000)]
107
Figure 7.3 Phase OMC between satellites with satellite clock correction applied. L1 (top), L2 (middle), and ionosphere-free (bottom). The base satellite is PRN25. [WES2, DOY138(2000)]
108
Figure 7. 4 Phase OMC between satellite ionosphere-free PRN21-PRN25 without (top) and with (middle) satellite clock correction applied, and satellite clock correction difference (bottom). [WES2, DOY138(2000)]
Figure 7.4 shows the ionosphere-free phase OMC without and with satellite clock correction applied and satellite clock correction for the difference PRN21-PRN25.
The
IGS(JPL) produced satellite clock correction has the same but mirrored variation as seen in the OMC when satellite clock correction is not applied (compare top and bottom figures). The
109
reversal seen after epoch 550 in the middle plot reveals possible errors in the satellite clock correction. Prior to the reversal the clock corrections are not available for several epochs.
7.2.3.2 Undifferenced Observation Cycle Slip Detection and Fixing Undifferenced cycle slip detection and fixing as discussed in Blewitt (1990) has been implemented in this study. Since cycle slips can occur concurrently and differently on L1 and L2, the slip on each signal must be independently detectable. The undifferenced cycle slip detection involves linear combinations of the observations: the widelane combination and the ionospheric combination. Let N1 and N2 be ambiguities for L1 and L2, respectively.
7.2.3.2.1
Cycle Slip Detection in the Widelane Combination
The widelane ambiguity is written as:
Nw = ϕw −
f 1 P1 + f 2 P2 . ( f 1 + f 2 ) λw
7. 9
For every observation epoch, the widelane ambiguity is evaluated. Setting a priori rms of half widelane cycles, the sequential recursive algorithm for the widelane ambiguity and its variance is given as
Nw
i
= Nw
σ i2 = σ i2− 1 +
i− 1
{(
+
[
]
7. 10
)2 − σ i2− 1 }
7. 11
1 N w,i − N w i
1 N w,i − N w i
i− 1
i− 1
where N w is the mean value of Nw, σi is the standard deviation of Nw , and i refers to the current data epoch being evaluated. The subsequent epoch is required such that N w,i + 1 is
110
within 4σi of the running mean N w i . Otherwise it is assumed that a cycle slip has occurred and the respective number of widelane cycle slips is recorded. However, consecutive slips with equal widelane slips but negative sign are later treated as outliners in the cycle slip fixing stage.
7.2.3.2.2
Cycle Slip Detection in the Ionospheric Combination
The idea of ionospheric slip detection is used in the very unlikely case that the slip in L1 equals the slip in L2, which the widelane slip detection is unable to detect. The ionospheric combination for carrier phase observation is
ϕ 1 ϕ 2 Φ iono = c f − f 1 2 = δI + λ1 N1 − λ2 N 2
7. 12
= δI + λ1 ( N1 − N 2 ) + (λ1 − λ2 ) N 2 = δI + λ1 N w − λI N 2
f12 I = where δ 2 − 1I P1 and IP1 is the ionospheric delay on P1-code. λI = (λ1 − λ2 ) is the f2
ionospheric wavelength. The pseudorange ionospheric combination is
Piono = P2 − P1 = δI
.
7. 13
Since the pseudorange does not have integer cycle discontinuity, it is possible to construct a polynomial fit Q to Piono and subtract it from Φ iono ( i.e. Φ iono − Q ), then look for discontinuities. Blewitt (1990) suggested an empirical formula to compute the degree of polynomial fit as
111
N m = min + 1, 6 100
7. 14
where N is the number of observations in the data. It should be noted that this simple fit is only used for discontinuity detection, not for the value of cycle slip. The algorithm is given as
( LI i − Qi ) − ( LI i − 1 − Qi − 1 ) > k cycles
7. 15
( LI i + 1 − Qi + 1 ) − ( LI i − Qi ) < 1 cycles
7. 16
where epoch i is the first good data point after the occurrence of a cycle slip if both of the above conditions are met. The default value of k is set to 6 ionospheric cycles (6 x 5.4 cm = 32.4 cm), but can be set to a more appropriate value depending on the ionospheric conditions. The reasons for using high tolerance k value are because (1) receivers at high latitudes often have large phase variations due to ionospheric activity. This should not be confused with cycle slips. (2) there are only slim chances of having equal cycle slips in L1 and L2 of less than 6 ionospheric cycles. Equal cycle slips are detected by the ionospheric combination, the respective widelane cycle slips are therefore set to zero.
7.2.3.2.3
Fixing Cycle Slip of Undifferenced Observation
Let ∆N1 and ∆N2 be the number of cycle slips for L1 and L2, respectively. The number of widelane slips is: ∆Nw = ∆N1 − ∆N2 . The cycle slip can be proceeded by a polynomial fit to
Φ iono just before the slip occurrence and then extrapolated to the slip epoch or after the slip epoch if in case of a data gap. This is denoted as Φ iono, estimate . The value of real Φ iono at
112
the
slip
epoch
is
also
calculated,
Φ iono, at _ slip .
The
difference
is
∆Φ iono = Φ iono, at _ slip − Φ iono, estimate . We can apply the knowledge from Φ iono :
∆Φ iono = λ1∆N w − λI ∆N 2 .
7. 17
∆Nw is obtained from the widelane slip detection, or zero in case of equal slip detected by the ionospheric algorithm. ∆N2 can therefore be computed from the above equation. Since ∆Nw = ∆N1 − ∆N2 , now the number of cycle slips ∆N1 in L1 is very obvious. Having ∆N1 and ∆N2 values, cycle slips in the phase data can be fixed by adding (or subtracting) the integer cycles to all the respective subsequent data points.
7.3 Kalman Filter The Kalman filter is an optimal estimation technique that minimizes the estimation error in a well defined statistical sense (Gelb, 1974). As a linear filter using a recursive algorithm which processes measurement information sequentially in time, the Kalman filter involves two main steps: filtering and prediction. Filtering is the estimation of the state vector at the current epoch based on all previous measurement information. Prediction involves the estimation of the state vector x at a future time. The Kalman filter system state vector (dynamic model) which evolves with time can be written as
x k + 1 = Φ k x k + wk
7. 18
corresponding to the measurement vector (measurement model)
zk = H k xk + vk
7. 19
113
where w k ~ N k (0, Qk ) and v k ~ N k (0, Rk ) are the system and measurement noises which are mutually uncorrelated vectors. Subscript k refers to the epoch of time. Φ is the transition matrix. H is the measurement connection matrix. Elements of H are the partial derivatives of the predicted measurements with respect to each stage and must be computed for every epoch. The respective Kalman filtering algorithm involves Kalman Gain (K), covariance update ( Pk ) and prediction ( Pk−+ 1 ), in the time update and measurement update steps, as shown in Figure 7.5.
114
Initial Estimates of xˆk− and Pk−
Measurement Update (Filter or correct) (a) Calculate Kalman Gain
K = Pk− H kT ( H k Pk− H kT + Rk ) − 1 (b) Using measurement to update estimate
xˆk = xˆk− + K ( z k − H k xˆk− ) (c) Update the error covariance
Pk = (I − K k H k )Pk−
Time Update (Predict) (a) Project the state ahead
x k− + 1 = Φ k xˆk (b) Project the covariance ahead
Pk−+ 1 = Φ k Pk Φ Tk + Qk
Figure 7. 5 The Kalman filter computation recursive scheme
115
7.3.1
Extended Kalman Filter
The extended Kalman filter (EKF) is an extension of the standard Kalman filter (SKF). EKF is used when the process to be estimated and/or the measurement relationship to the process is non-linear (Welch and Bishop, 1997).
EKF simply applies SKF through the linearization
(through approximation of Taylor series) around the previous state estimate. The EKF provides a simple but efficient algorithm to process a nonlinear system. In this work, we deal with a nonlinear measurement system and a linear dynamic model. The state vector system which evolves with time can be written as
x k + 1 = Φ k x k + wk
7. 20
corresponding to the measurement vector
z k = h( x k ) + v k
7. 21
The respective governing equations that linearize an estimate about 7.20 and 7. 21 are
x k + 1 = Φ k xˆk + wk
7. 22
zk = ~ z k + H ( xk − ~ xk ) + vk
7. 23
H is a partial derivative of measurement equation h(o) with respect to the estimated vector x, around the approximated state vector. The EKF computation recursive scheme is given in Figure 7.6.
116
Initial Estimates of xˆk− and Pk−
Measurement Update (Filter or correct) a) Calculate Kalman Gain
K = Pk− H kT ( H k Pk− H kT + Rk ) − 1 b) Using measurement to update estimate
xˆk = xˆk− + K ( z k − h( xˆk− )) c) Update the error covariance
Pk = ( I − K k H k )Pk−
Time Update (Predict) a) Project the state ahead
x k− + 1 = Φ k xˆk b) Project the covariance ahead
Pk−+ 1 = Φ k Pk Φ Tk + Qk
Figure 7. 6 The EKF computation recursive scheme
117
7.3.2
Discrete Gauss-Markov Process
A Markov process is a random process that allows users to link the process with simple filters. A fundamental model is first-order Markov if the probability distribution for the process ( x k ) depends only on the value at one point immediately in the past ( x k − 1 ) (Gelb, 1974, page 42). The differential equation for continuous first-order Markov process x(t) is given as
x&(t ) = −
1 x (t ) + w (t ) τ
7. 24
where τ is the correlation time and w is white noise. If the condition is added that the probability density function of w and therefore x also are Gaussian, the given process is a Gauss-Markov process. A discrete version (or a first-order difference equation) of a GaussMarkov process can be written as
xk + 1 = κ xk + wk
7. 25
where
κ=
T eτ
7. 26
T is the data interval. Let β = (1 / τ ) , β is called the dampening coefficient. A larger β value yields a shorter correlation length. This permits a large variable change from one epoch to the next. On the other hand, a small β value describes high correlation in the following epochs, and thus allows only a small variation (Mertikas and Rizos, 1997). The associated covariance in discrete time for the Gaussian white noise w is
cov( w) =
q [1 − e − 2Tβ ] . 2β
7. 27
118
q is the variance of the process noise. If τ equals zero, the model becomes a pure white noise model without correlation time which implies κ = 0 .
But if the correlation time τ
approaches infinity [ κ = 1 ], the process is called pure random walk (Gelb, 1974 pages 43 and 79). Station coordinates are usually modeled as pure random walk, so are reparameterized ambiguities. Troposphere and receiver clock can be modeled as either white noise or random walk (Tralli et al., 1990).
7.3.3
PPP Implementation
Mathematical issues for PPP implementation are given. These include partial derivatives for EKF, receiver clock estimation, ambiguity estimation, and observation weighting scheme.
7.3.3.1 Partial Derivatives The submatrix H for satellite p at epoch i for the state vector of position x, y, z, ambiguity, receiver clock, and tropospheric effect can be written as.
x i− − X ip H i (Φ ) = p ρi
y i− − Yi p
z i− − Z ip
ρip
ρip
x i− − X ip H i (P) = p ρi
y i− − Yi p
z i− − Z ip
ρip
ρip
p
p
1 1 TMF 0 1 TMF
7. 28
ρip = ( x i− − X ip ) 2 + ( y i− − Yi p ) 2 + ( z i− − Z ip ) 2 p
p
p
where ( X i , Yi , Z i ) are the coordinates of satellite p at the transmission time and ( x i− , y i− , z i− ) are the estimated or projected receiver coordinates.
For the carrier phase
observation Φ (scale ϕ to distance), the partial derivative for the reparameterized ambiguity of 119
the respective satellite is 1. The partial derivative for zenith troposphere is the tropospheric mapping function (TMP) (e.g., NMF). Rows of H for dual-frequency analysis are twice that for single-frequency.
7.3.3.2 Receiver Clock Estimation Receiver clock can be estimated in EKF using a Gauss-Markov process. Zumberge et al. (1997b) suggested that the receiver clock can be modeled as white noise. However, it is better to model it as random walk. Since we have pseudorange observations which include the clock term, an approximate ratio of the change in the clock from the current epoch to the next in the EKF dynamic model can be directly computed.
Ri = Pi p − ( ρip − cdt ip )
7. 29
Ri + 1 = Pi +p1 − ( ρip+ 1 − cdt ip+ 1 )
Φ clk =
Ri + 1 Ri
7. 30
Because many satellites are observed at the same time, it is additionally better to average out the random error by averaging the ratio of all available satellites. The PPP solution obtained from the random walk process with a priori information of the ratio of receiver clock shows a slight solution improvement than that using the white noise process.
7.3.3.3 Ambiguity Estimation The ambiguity should be constant as long as the receiver is locked to the signal. ambiguity number changes when the receiver loses lock.
The
Unlike the double differenced
carrier phase, the undifferenced PPP analysis estimates only real numbers of ambiguities, or
120
"floated solutions."
In fact, PPP ambiguities are reparameterized ambiguities which are
collections of ambiguities of L1 and L2, with carrier phase ionosphere-free scaling factors.
)
The reparameterized ambiguities N are being estimated every epoch, as a pure random walk process. The dynamic model is in the form
) ) N ip+ 1 = N ip + w
7. 31
where w is Gaussian noise.
7.3.3.4 Observation Weighting Schemes Recent scientific GPS research has started to pay more attention to observation weighting methods (Vermeer 1997; Teunissen, 1998; Collins and Langley 1999; Hartinger and Brunner 1999). Because different weight methods yield different solutions particularly in the height component, the observation must be given appropriate weight. As alternatives to constant weighting, normally used in GPS analysis, other possible weighting schemes may be used. For example: a) Exponential weighting schemes that weight corresponding observations observed from near horizon satellites is lowered (Euler and Goad, 1991) b) Weighting that reflects receiver generated signal-to-noise ratio (SNR) values (Collins and Langley, 1999; Hartinger and Brunner, 1999). Normally, SNR represents the carrier-tonoise-power-density ratio (C/N0) which varies with the elevation of the arriving signal. Langley (1997) derived phase variance (m) using C/N0 (dB-Hz) values as follows 2
λ 2 σ Li = B i 10 − 0.1 C / N0 , 2π
i = 1, 2
121
7. 32
where B is the carrier tracking loop bandwidth (Hz). It should be noted that some receiver manufacturers do not provide either SNR or C/N0, and some provided are in an arbitrary format. c) Weighting as a cosecant function of the satellite elevation angle (Vermeer, 1997; Collins and Langley, 1999). This is because the amount of signal noise increases towards the horizon, similar to the tropospheric error which has a cosecant shape, according to various models of the tropospheric mapping function (e.g., Marini, Chao, Davis, and Herring mapping functions). d) Weighting as square of a cosecant function of the satellite elevation angle (Vermeer, 1997; Hartinger and Brunner, 1999). This is from the fact that GPS residuals reveal a more swiftly increasing noise level for low elevation angles. e) Combination weighting using C/N0 information together with satellite elevation knowledge.
Figure 7. 7 Potential weighting functions, comparison between cosecant and cosecantsquared
122
Hartinger and Brunner (1999) used the SIGMA-ε model where the phase variances are computed using C/N0 values and thus observation weight directly echoes signal quality. Their experimental results show that baseline rms of the SIGMA-ε model is much less than that of equal-weighting, especially at a low elevation cutoff angle. Collins and Langley (1999) reported that, in the presence of multipath, the cosecant and SNR weighting schemes yield a great improvement over the equal-weighting scheme. Moreover, according to the scaling effect of the a posteriori variance factor, the cosecant and SNR schemes are almost numerically equivalent. The amount of observation noise increases and exhibits presence of multipath which mostly occurs in signals from low satellite elevation angles. It should be more appropriate to apply a step function using a combination of uniform weight for high elevation angle observation and lower weight at low elevation angle.
Deweighting observations at high
elevation angles will lose valuable information. The step function variance may be given as
σ 2 σ (E) = 2 2 σ cosec ( E )
E>α E α E
