7 tesis ppp kongzhe chen
TRANSCRIPT
UCGE Reports Number 20217
Department of Geomatics Engineering
Real-Time Precise Point Positioning, Timing and Atmospheric Sensing
(URL: http://www.geomatics.ucalgary.ca/links/GradTheses.html)
by
Kongzhe Chen
April 2005
THE UNIVERSITY OF CALGARY
Real-Time Precise Point Positioning, Timing and Atmospheric Sensing
by
KONGZHE CHEN
A DISSERTATION
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF GEOMATICS ENGINEERING
CALGARY, ALBERTA
April, 2005
© Kongzhe Chen 2005
iii
ABSTRACT
The availability of precise GPS orbit and clock products has enabled the development of a
novel positioning methodology known as Precise Point Positioning (PPP). Different from
the conventional differential methods, PPP is based on the processing of un-differenced
observations from a single GPS receiver.
The thesis investigates different aspects related to the development of a real-time precise
point positioning system and its application to precise timing and atmospheric sensing. This
includes real-time position determination, receiver clock offset and zenith wet delay
parameter estimation using PPP methodology. The real-time aspect of precise orbit and
clock data distribution has also been analyzed with respect to reliability, timeliness,
convenience and available infrastructure. A comprehensive analysis of the error sources
unique to PPP has been provided along with methods for their modeling and mitigation.
Real-time PPP provides an efficient and flexible method for high precision Precipitable
Water Vapour (PWV) determination. It can significantly improve the timeliness and
temporal resolution of PWV estimates that are essential for severe weather forecast and
operational numerical weather prediction. Methods of real-time PWV estimation using PPP
have been investigated and the major error factors including horizontal gradients, mapping
function selection, antenna phase center variations and elevation cutoff angle have been
analyzed.
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To date, precise point positioning has been implemented using dual-frequency GPS
receivers. Since the majority of GPS receivers in use are single-frequency receivers,
methods of precise point positioning using a single-frequency GPS receiver has been
investigated. A method for the estimation of ionosphere horizontal gradients along with the
zenith delay in single-frequency PPP has been developed. Various comparisons and
analysis have also been conducted using different ionospheric mitigation methods and
under different ionospheric conditions.
Numerical results have indicated that a positioning accuracy of sub-centimetre for static
and sub-decimetre for kinematic positioning is obtainable in real-time using dual-frequency
observations. A sub-nanosecond accuracy has been obtained for the receiver clock offset
estimates. An accuracy of 1 mm for precipitable water vapour has been demonstrated. A
sub-metre to decimetre positioning accuracy has been achieved using PPP with only single-
frequency observations.
v
ACKNOWLEDGEMENTS
I would like to thank my supervisor, Dr. Yang Gao, for his support, encouragement, and
guidance in the past three years.
I am grateful to the examining committee, Drs. Susan Skone, Abraham Fapojuwo, Swavik
Spiewak and Marcelo Santos, for reading the dissertation and giving valuable suggestions.
In the past three years, I have received a lot of support from many researchers. I would like
to express appreciation to Pierre Heroux who has always been ready to discuss with me
about the processing strategies of IGS analysis centers, Ronald Muellerschoen who has
been providing JPL IGDG orbit and clock products in the past two years and answered me
questions about the use of IGDG products, Paul Collins and Ken MacLeod for providing
the NRCan GPS•C real-time code solution orbit and clock products and post-mission phase
solution products via Internet, Dr. Susan Skone and Natalya Nicholson for providing the
radiometer and barometer measurements, Paul Mrstik and Sarka Friedl for providing the
aircraft datasets, Michael Heflin for providing the JPL near real-time precise GPS orbit and
clock products, Dr. Tonie vanDam who provided me atmospheric loading data for some
selected stations and answered me questions about the effects of atmospheric loading in
PPP, Dr. Hans-Georg Scherneck for providing ocean loading coefficients of some selected
stations used in thesis research, Dr. Neil Ashby who answered me questions about
relativistic effects in GPS positioning and timing, Dr. Yong Hu who helped me to conduct
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the Internet data transmission test between University of Calgary and York University, and
Suen Lee for proofreading the thesis.
I also would like to thank my friends for their help and encouragement. They are Xiaobing
Shen, Zhe Liu, Yufeng Zhang, Zhiyu Chen, Wuji Yang, Baichong Chao, Daocang Wu,
Qingyun Hu, Qiaoping Zhang, Hong Li, Shaokui Ge, Chen Xu, Li Sheng, Xiaoji Niu,
Minxue He, Junjie Liu, Chaochao Wang, Wentao Zhang, Changlin Ma, Jau-Hsiung Wang,
Mohamed Abdel-salam, and Adam Wojciechowski.
I would like to thank my parents, and my wife, for their support and encouragement
throughout my studies.
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DEDICATION
To my mother.
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TABLE OF CONTENTS
APPROVAL PAGE ................................................................................................................ ii
ABSTRACT ........................................................................................................................... iii
ACKNOWLEDGEMENTS ................................................................................................... v
DEDICATION....................................................................................................................... vii
TABLE OF CONTENTS .................................................................................................... viii
LIST OF TABLES ................................................................................................................ xii
LIST OF FIGURES ............................................................................................................. xiv
LIST OF SYMBOLS AND ACRONYMS ....................................................................... xviii
CHAPTER 1: INTRODUCTION........................................................................................ 1
1.1 Background ................................................................................................................ 2 1.2 Objectives and Contributions..................................................................................... 7 1.3 Thesis Outline ............................................................................................................ 9
CHAPTER 2: METHODS OF REAL-TIME PRECISE POINT POSITIONING, TIMING AND ATMOSPHERIC SENSING........................................... 11
2.1 Definition of Real-Time........................................................................................... 11 2.2 Real-Time Precise Point Positioning ....................................................................... 13
2.2.1 Concept of Precise Point Positioning........................................................... 15
2.2.2 Advantages of Precise Point Positioning ..................................................... 18
2.2.3 Challenges of Precise Point Positioning ...................................................... 19 2.3 Real-Time GPS Timing ........................................................................................... 21
2.3.1 Time Scales.................................................................................................. 23
2.3.2 GPS Time Transfer Techniques................................................................... 26
2.3.4 Challenges in GPS Time Transfer ............................................................... 32
2.3.5 Real-Time Timing Using IGDG Products ................................................... 35 2.4 Real-Time Atmospheric Sensing ............................................................................. 37
ix
2.4.1 Water Vapour Sensing Techniques.............................................................. 37
2.4.2 GPS Meteorology......................................................................................... 40
2.4.3 Strategies for ZTD Estimation Using GPS Observations ............................ 46
2.4.4 ZTD Estimation Using PPP Methodology................................................... 49
CHAPTER 3: ERROR MITIGATION AND MODELING IN PRECISE POINT POSITIONING .......................................................................................... 53
3.1 Dual-Frequency GPS Observables .......................................................................... 53
3.1.1 Dual-Frequency Code and Carrier Phase Measurements ............................ 54
3.1.2 Ionosphere-free Combinations..................................................................... 61 3.2 Error Mitigation for Dual-Frequency Measurements .............................................. 65
3.2.1 Satellite Orbit and Clock Errors................................................................... 66
3.2.2 Ionospheric Effects ...................................................................................... 66
3.2.3 Tropospheric Delay...................................................................................... 71
3.2.4 Relativistic Effects ....................................................................................... 72
3.2.5 Phase Wind-up ............................................................................................. 75
3.2.6 Hardware Delays.......................................................................................... 78
3.2.7 Initial Phase Offsets ..................................................................................... 81
3.2.8 Multipath and Measurement Noise.............................................................. 83
3.2.9 Antenna Phase Center Offset and Variations............................................... 84
3.2.10 Site Displacement Effects ............................................................................ 87 3.3 Modeling for Dual-Frequency Measurements ......................................................... 97
3.3.1 Functional Model ......................................................................................... 97
3.3.2 Stochastic Model........................................................................................ 101
CHAPTER 4: REAL-TIME PRECISE GPS ORBIT AND CLOCK PRODUCTS AND ANALYSIS...................................................................................... 105
4.1 IGS Precise Orbit and Clock Products................................................................... 106 4.2 NRCan Real-Time Precise GPS Products.............................................................. 110
4.2.1 GPS•C Code Solution Products................................................................. 111
4.2.2 GPS•C Phase Solution Products................................................................ 114 4.3 JPL Real-Time Precise GPS Products ................................................................... 117
x
4.3.1 JPL Near Real-Time Orbit and Clock Products......................................... 118
4.3.2 JPL IGDG Orbit and Clock Products......................................................... 119 4.4 Accuracy Statistics of Real-Time Products ........................................................... 121 4.5 Latency and Age .................................................................................................... 122
4.5.1 Latency and Age of GPS•C Corrections ................................................... 124
4.5.2 Latency and Age of IGDG Corrections ..................................................... 127 4.6 Real-Time Product Distribution Issues.................................................................. 132
4.6.1 Broadcast by Satellite ................................................................................ 134
4.6.2 UDP/IP Multicast....................................................................................... 136 4.7 Real-Time Correction Formats .............................................................................. 139
4.7.1 RTCM-104................................................................................................. 140
4.7.2 RTCA-159.................................................................................................. 141
4.7.3 GPS•C Format ........................................................................................... 141
4.7.4 IGDG Format ............................................................................................. 142
4.7.5 Application of Real-Time Orbit and Clock Products ................................ 143
CHAPTER 5: PRECISE POINT POSITIONING USING SINGLE-FREQUENCY GPS DATA ..................................................................... 145
5.1 Single-Frequency Point Positioning ...................................................................... 145 5.2 Ionospheric Models................................................................................................ 148 5.3 Precise Point Positioning with Ionospheric Delay Estimated................................ 155
5.3.1 Estimating Ionospheric Horizontal Gradients with Un-differenced GPS Measurements ............................................................................................ 157
5.3.2 Ionospheric Mapping Functions ................................................................ 163
5.3.3 Results of Positioning and Ionospheric Delay Estimation......................... 167
CHAPTER 6: NUMERICAL RESULTS AND ANALYSIS ........................................ 178
6.1 Software Development and Parameter Modeling .................................................. 179
6.1.1 P3-RT Software Package ........................................................................... 179
6.1.2 Modeling for Parameters............................................................................ 188 6.2 PPP Using Dual-Frequency Measurements ........................................................... 191
6.2.1 Static PPP Using IGDG Products .............................................................. 191
6.2.2 Kinematic PPP Using IGDG Products....................................................... 194
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6.2.3 PPP Using GPS•C Products ...................................................................... 200
6.2.4 Summary.................................................................................................... 204 6.3 Receiver Clock Offset Estimation Using PPP Methodology................................. 205
6.3.1 Receiver Clock Offset Estimation Using IGDG Products......................... 206
6.3.2 Analysis of Receiver Clock Offset Estimation .......................................... 211 6.4 Atmospheric Sensing Using PPP Technique ......................................................... 212
6.4.1 Comparison with IGS Final Tropospheric Products.................................. 213
6.4.2 Comparison with Radiometer Measurements............................................ 215
6.4.3 Analysis of Real-Time Water Vapour Sensing Results............................. 229 6.5 PPP Using Single-Frequency Measurements......................................................... 233
6.5.1 Positioning at Mid-Latitude Stations ......................................................... 233
6.5.2 Positioning Using Data from Different Ionospheric Regions.................... 238
6.5.3 Positioning Using Kinematic Datasets....................................................... 247
6.5.4 Summary.................................................................................................... 251
CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS................................. 253
7.1 Conclusions............................................................................................................ 253 7.2 Recommendations.................................................................................................. 257
REFERENCES.................................................................................................................... 259
xii
LIST OF TABLES
3.1 Measurements from Cross-Correlation Receivers ....................................................... 55
3.2 Measurements from Non-Cross-Correlation Receivers............................................... 56
4.1 IGS Products of GPS Satellite Orbit and Clock (after IGS Website, 2004) .............. 106
4.2 NRCan Real-Time Precise GPS Products (after Heroux, 2003; Collins, 2004)........ 111
4.3 JPL Real-Time Precise GPS Products (after Muellerschoen, 2003; Heflin, 2004) ... 117
5.1 VTEC of Each Satellite at the Ionospheric Pierce Point............................................ 155
5.2 Ap Indices of GPS Week 1251 .................................................................................. 168
5.3 Positioning Accuracy Using Ionospheric Estimation Model with SLM450 ............. 170
5.4 Positioning Accuracy Using Ionospheric Estimation Model with Different Mapping Functions (Unit: m) .....................................................................................173
5.5 Statistics of VTEC Estimation with Different Mapping Functions (Unit: 0.1 TECU).........................................................................................................................173
5.6 Positioning Accuracy Using Klobuchar Model ......................................................... 176
5.7 Positioning Accuracy Using GIM.............................................................................. 176
6.1 Accuracy Statistics of Static Positioning Results (Unit: cm)..................................... 194
6.2 Accuracy Statistics of Kinematic Positioning Results (Unit: cm) ............................. 198
6.3 Accuracy Statistics of Positioning Using GPS•C Products (Unit: m)....................... 204
6.4 Receiver Clock Offset Estimation Accuracy ............................................................. 210
6.5 ZTD Estimation Statistics .......................................................................................... 214
6.6 Statistics of PWV Comparison .................................................................................. 226
6.7 Accuracy Statistics of Different Strategies (Unit: mm)............................................. 227
6.8 Accuracy Statistics of Single-Frequency Point Positioning at S1 (Unit: m) ............. 237
6.9 Station Coordinates.................................................................................................... 239
6.10 Ap Indices in August 2004 ....................................................................................... 239
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6.11 Accuracy Statistics of Ionospheric Estimation Model for GLPS (Unit: m) ............ 240
6.12 Accuracy Statistics of Ionospheric Estimation Model for S1 (Unit: m).................. 241
6.13 Accuracy Statistics of Ionospheric Estimation Model for FAIR (Unit: m) ............. 241
6.14 Accuracy Statistics of Klobuchar Model for GLPS (Unit: m)................................. 243
6.15 Accuracy Statistics of Klobuchar Model for S1 (Unit: m) ...................................... 243
6.16 Accuracy Statistics of Klobuchar Model for FAIR (Unit: m) ................................. 244
6.17 Accuracy Statistics of GIM for GLPS (Unit: m) ..................................................... 245
6.18 Accuracy Statistics of GIM for S1 (Unit: m)........................................................... 246
6.19 Accuracy Statistics of GIM for FAIR (Unit: m)...................................................... 246
6.20 Single-Frequency Point Positioning with Airborne Dataset (Unit: m)..................... 251
xiv
LIST OF FIGURES
3.1 Solid Earth Tide Effects for S1 from September 2nd to 7th, 2004 ................................ 90
3.2 Ocean Loading Effects for S1 from September 2nd to 7th, 2004.................................. 92
3.3 Pole Tide Effects for S1 from 2000 to 2004 ................................................................ 94
3.4 Atmospheric Loading Effects for S1 from 1995 to 1999 ............................................ 96
4.1 IGS Tracking Network on December 19th, 2004 (from IGS Website, 2004) ............ 107
4.2 AC Solutions with Respect to the IGS Final Orbit Combination (from IGS ACC Website, 2004) ............................................................................................................108
4.3 AC Solutions with Respect to the IGS Final Clock Combination (from IGS ACC Website, 2004)...................................................................................................108
4.4 CACS Network (after ICD-GPS•C, 2001) ................................................................ 112
4.5 CDGPS Radio (after CDGPS Receiver User's Guide, 2003) ................................... 114
4.6 Stations Sharing Real-Time Data with NRCan (after Collins, 2004)........................ 116
4.7 The Global Coverage with 20 Stations (from Collins, 2004) .................................... 116
4.8 IGDG Real-Time Network in 2003 (from Armatys et al., 2003) .............................. 120
4.9 Latency and Age (after Kee, 1996)............................................................................ 123
4.10 Latencies of GPS•C Orbit Corrections ..................................................................... 125
4.11 Latencies of GPS•C Clock Corrections.................................................................... 125
4.12 Ages of GPS•C Orbit Corrections ............................................................................ 126
4.13 Ages of GPS•C Clock Corrections ........................................................................... 127
4.14 Latencies of IGDG Corrections ................................................................................ 129
4.15 Ages of IGDG Orbits ................................................................................................ 129
4.16 Ages of IGDG Clocks ............................................................................................... 130
4.17 Difference between Orbits Aged 28 s and 0 s........................................................... 131
4.18 Difference between Clocks Aged 5 s and 0 s ........................................................... 132
xv
4.19 Transmission Time over the Internet ........................................................................ 136
4.20 Packet Loss Rate Using UDP Transport Protocol .................................................... 139
5.1 The Equatorward Increase of TEC at Local Time 16:00........................................... 159
5.2 The West to East Increase of TEC at Local Time 6:00 ............................................. 159
5.3 The East to West Increase of TEC at Local Time 18:00 ........................................... 160
5.4 The VTEC at Ionospheric Pierce Point for Satellites at 30° Elevation Angle against Azimuth at Different Local Time ...................................................................160
5.5 Mapping Functions .................................................................................................... 166
5.6 Mapping Function Difference.................................................................................... 166
5.7 Positioning Errors Using Ionospheric Estimation Model for GPS Week 1251......... 170
5.8 Zenith Ionospheric Delay Estimates for GPS Week 1251......................................... 171
5.9 Ionospheric Gradients in the East Direction .............................................................. 171
5.10 Ionospheric Gradients in the North Direction........................................................... 172
5.11 Vertical TEC Comparison......................................................................................... 172
6.1 P3-RT Interface – Setup............................................................................................. 181
6.2 P3-RT Interface – Kinematic Processing................................................................... 182
6.3 P3-RT Interface – Static Processing .......................................................................... 183
6.4 P3-RT Interface – Trajectory ..................................................................................... 184
6.5 P3-RT Interface – Position Errors.............................................................................. 184
6.6 Real-Time PPP........................................................................................................... 186
6.7 Post-Mission PPP....................................................................................................... 187
6.8 Real-Time Static Positioning Using IGDG Products................................................. 192
6.9 Static Positioning Using IGS Dataset ........................................................................ 193
6.10 Positioning Errors with Vehicle Dataset................................................................... 195
6.11 Vehicle Trajectory on September 30th, 2003 ............................................................ 196
6.12 Positioning with Aircraft Dataset.............................................................................. 197
xvi
6.13 Aircraft Trajectory on August 28th, 2004.................................................................. 197
6.14 Satellite Geometry for Vehicle Dataset on September 30th, 2003 ............................ 199
6.15 Satellite Geometry for Aircraft Dataset on August 28th, 2004 ................................. 200
6.16 Kinematic Positioning Using GPC•C Code Solution Products................................ 201
6.17 Satellite Geometry of S2 on December 2nd, 2003..................................................... 202
6.18 Kinematic Positioning Using GPC•C Phase Solution Products ............................... 203
6.19 Receiver Clock Offset and ZTD Estimates on June 12th, 2004 ................................ 208
6.20 Receiver Clock Offset and ZTD Estimates on June 13th, 2004 ................................ 209
6.21 Receiver Clock Offset and ZTD Estimates on June 14th, 2004 ................................ 210
6.22 ZTD Estimates Compared with IGS Tropospheric Products.................................... 214
6.23 Radiometer, GPS Antenna and MET3A Instruments ............................................... 217
6.24 Niell Mapping Functions .......................................................................................... 218
6.25 Difference between the Wet and Hydrostatic Mapping Functions........................... 219
6.26 Difference between the Calculated and Measured Pressures ................................... 221
6.27 PWV from GPS and WVR on September 2nd, 2004................................................. 223
6.28 PWV from GPS and WVR on September 3rd, 2004 ................................................. 223
6.29 PWV from GPS and WVR on September 4th, 2004 ................................................. 223
6.30 PWV from GPS and WVR on September 5th, 2004 ................................................. 224
6.31 PWV from GPS and WVR on September 6th, 2004 ................................................. 224
6.32 PWV from GPS and WVR on September 7th, 2004 ................................................. 224
6.33 PWV from GPS and WVR on September 8th, 2004 ................................................. 225
6.34 PWV Comparison between GPS and WVR from September 2nd to 8th, 2004.......... 225
6.35 Positioning Errors Using Ionospheric Estimation Model on December 3rd, 2003 ... 234
6.36 Zenith Ionospheric Delay Estimates on December 3rd, 2003 ................................... 234
6.37 Positioning Errors Using Ionospheric Estimation Model on July 27th, 2004 ........... 236
xvii
6.38 Zenith Ionospheric Delay Estimates on July 27th, 2004 ........................................... 236
6.39 Satellite Geometry of S1 on July 27th, 2004 ............................................................. 237
6.40 Positioning Using Ionospheric Estimation Model .................................................... 248
6.41 Positioning Using Klobuchar Model with Code Measurements .............................. 249
6.42 Positioning Using Klobuchar Model with Code and Phase Measurements ............. 249
6.43 Positioning Using GIM with Code Measurements ................................................... 250
6.44 Positioning Using GIM with Code and Phase Measurements .................................. 250
xviii
LIST OF SYMBOLS AND ACRONYMS
Symbols
mT weighted mean temperature of the atmosphere
sT surface temperature
NG tropospheric horizontal gradient in north direction
EG tropospheric horizontal gradient in east direction
gm tropospheric gradient mapping function
a azimuth angle
e elevation angle
hzD tropospheric zenith hydrostatic delay
wzD tropospheric zenith wet delay
hm hydrostatic mapping function for troposphere
wm wet mapping function for troposphere
nG ionospheric horizontal gradient in north direction
eG ionospheric horizontal gradient in east direction
1P P-Code pseudorange measurement on 1L
1C C/A-Code pseudorange measurement
xix
2P P-Code pseudorange measurement on 2L reported by non-cross-correlation
receivers
'P2 P-Code pseudorange measurement on 2L reported by cross-correlation
receivers
iΦ carrier phase measurement on iL
srρ true geometric range
c speed of light
sdt satellite clock error
rdt receiver clock error
orbd satellite orbit error
tropd tropospheric delay
Li/iond ionospheric delay on iL
gdT group delay differential of satellite
21 P/PDCB differential code bias between 1P and 2P of receiver
11 C/PDCB differential code bias between 1P and 1C
reld relativistic effects
iwδ phase windup on iL
iλ wavelength on iL
iN integer phase ambiguity on iL
xx
)t(ri 0φ initial phase offset of the receiver on iL
)t(si 0φ initial phase offset of the satellite on iL
Pidm P-code multipath on iL
1Cdm C/A code multipath
imδ carrier phase multipath on iL
sol,rrδ solid earth tides
pol,rrδ pole tide
ocn,rrδ ocean loading
atm,rrδ atmospheric loading
ant,rrδ antenna phase center offset and variations of receiver antenna
Acronyms
AC IGS Analysis Center
ACC IGS Analysis Center Coordinator
ARP Antenna Reference Point
BIPM Bureau International des Poids et Mesures
C/A Coarse Acquisition
CACS Canadian Active Control System
CDGPS Canadian-Wide DGPS Service
CDPD Cellular Digital Packet Data
xxi
CODE Center for Orbit Determination in Europe
CS Control Segment
CSRS Canadian Spatial Reference System
DCB Differential Code Bias
DGPS Differential GPS
DOP Dilution of Precision
EFEC Earth-Fixed Earth-Centered
ERP Earth Rotation Parameters
GFZ GeoForschungsZentrum
GGN Global GPS Network
GIM Global Ionospheric Model
GPS Global Positioning System
GPS•C GPS Correction Service
GPST GPS Time scale
GRAPHIC Group And Phase Ionospheric Correction
GSD Geodetic Survey Division
GSM Global System for Mobile Communication
ICD Interface Control Document
IFB Inter-Frequency Bias
IERS International Earth Rotation and Reference Systems Service
IGDG Internet-based Global Differential GPS System
IGP Ionospheric Grid Point
xxii
IGF IGS Final products
IGR IGS Rapid products
IGS International GPS Service
IGST IGS Time scale
IGU IGS Ultra-Rapid products
IMS Integrity Monitor Station
IODE Issue Of Data Ephemeris
IONEX IONosphere map EXchange format
IPP Ionospheric Pierce Point
ITRF International Terrestrial Reference Frame
IWV Integrated Water Vapour
JPL Jet Propulsion Laboratory
MCS Master Control Station
MS Master Station
MSAT Mobile Satellite system
MSLM Modified Single Layer Model
NASA National Aeronautics and Space Administration
NGS National Geodetic Survey
NRCan Natural Resources Canada
USNO United States Naval Observatory
OTF On-The-FLY
P-Code Precise Code
xxiii
PPP Precise Point Positioning
PWV Precipitable Water Vapour
PRN Pseudo Random Noise
QoS Quality of Service
RCP Right Circularly Polarized
RINEX Receiver Independent Exchange Format
RMS Root Mean Square
RS Reference Station
RTACP Real-Time Active Control Point
RTCA Radio Technical Commission for Aviation
RTCM Radio Technical Commission for Marine
RTG Real-Time GIPSY
RTK Real-Time Kinematic
RTMACS Real-Time Master Active Control Station
RTNT Real-Time Net Transfer
SA Selective Availability
SBL Special Bureau on Loading
SISRE Signal-In-Space Range Error
SLM Single Layer Model
SNR Signal-to-Noise Ratio
SP3 Standard Product # 3
SPP Single Point Positioning
xxiv
SPS Standard Positioning Service
STD Standard Deviation
SWD Slant Wet Delay
SWV Slant Water Vapour
TAI International Atomic Time
TCP Transmission Control Protocol
TEC Total Electron Content
TWSTT Two-Way Satellite Time-Transfer
UDP User Datagram Protocol
UT Universal Time
UTC Coordinated Universal Time
VACP Virtual Active Control Point
VLBI Very Long Baseline Interferometry
VPN Virtual Private Network
VTEC Vertical Total Electron Content
WAAS Wide Area Augmentation System
WADGPS Wide Area Differential GPS
WVR Water Vapour Radiometer
ZHD Zenith Hydrostatic Delay
ZTD Zenith Tropospheric Delay
ZWD Zenith Wet Delay
1
CHAPTER 1
INTRODUCTION
Ever since Zumberge et al. (1995; 1997) introduced the concept of carrier phase-based
Precise Point Positioning (PPP), which can achieve centimetre to decimetre level accuracy
using dual-frequency code and phase measurements and precise satellite orbit and clock
products, PPP, whose position determination is based on the processing of un-differenced
GPS observations from a single receiver, has now been widely recognized as a new high-
precision positioning technology using GPS. This is because in addition to the
computationally efficient characteristic, the original reason for which it was introduced
(Zumberge et al., 1995; 1997), people have identified a number of unique features of PPP
including simplified operation, cost-effectiveness, no base stations required and positioning
accuracy comparable to the double-difference approach (Kouba and Heroux, 2001a).
However, PPP so far are still used in post or near real-time applications because of the lack
of real-time precise GPS orbit and clock products and the methodology and software to
support real-time data processing.
On the other hand, point positioning using single-frequency observations has not been
considered capable of providing accurate position solutions at centimetre to decimetre level
similar to PPP with dual-frequency measurements, because of ionospheric effects.
2
Investigation of models for mitigating ionospheric effects to support precise point positioning
using single-frequency GPS data is highly demanded by numerous applications.
In addition to positioning, PPP is capable of supporting precise timing and atmospheric
sensing (Senior and Ray, 2001b; Chen, 2004). Due to its un-differenced nature in data
processing, the method can offer several unique advantages compared to the use of
differential techniques. Investigating its real-time capability is of great interest to many
applications.
1.1 Background
The fundamental navigation technique for the Global Positioning System (GPS) is to use
one-way ranging information from the GPS satellites that are also broadcasting their
estimated positions (Parkinson, 1996). The initial design goal of the system was to provide a
Single Point Positioning (SPP) service for position determination using a single GPS
receiver. SPP is however subject to the effects of all GPS error sources including satellite
orbit and clock errors in the GPS navigation messages, atmospheric effects, and receiver
related errors, etc. After Selective Availability (SA) was turned off in May 2000, the
dominant errors are the errors in the broadcast satellite orbit and clock and ionospheric
effects (Clynch, 2000). Even with a dual-frequency GPS receiver whereby the ionospheric
effects could be effectively eliminated, SPP can only provide position solutions at the
accuracy of a couple of metres.
3
To obtain positional accuracy at the decimetre to centimetre level, the Real-Time Kinematic
(RTK) positioning technique has been developed and is now widely used in practice. RTK
positioning requires the combination of simultaneous observations from a minimum of two
GPS receivers, with at least one serving as the base receiver with precisely known
coordinates and the rest as rover receivers whose positions are to be determined. Double
differencing between satellites and receivers can remove common errors among GPS
satellites and receivers or reduce errors that are spatially correlated among satellites and
receivers. If equipped with integer carrier phase ambiguity resolution techniques, RTK is
able to provide centimetre accurate positions. There are several drawbacks, however, that are
associated with the current RTK positioning techniques. One drawback is that base receivers
are required to set up at stations with precisely known coordinates, which is not always
feasible in practice. Another drawback related to the current RTK positioning systems is that
the rover receivers must be within the vicinity of the base receivers, typically less than 20 km
(Kouba and Heroux, 2001a).
With the advent of precise orbit and clock products currently available from a number of
organizations including the International GPS Service (IGS), Natural Resources Canada
(NRCan) and the Jet Propulsion Laboratory (JPL), the two major error sources, namely
satellite orbit errors and clock errors, can be effectively mitigated by the use of precise
satellite orbit and clock data. Point positioning using code measurements and the precise GPS
orbit and clock data is able to provide position solutions at metre level accuracy (Lachapelle
et al. 1994a; Heroux and Kouba, 1995). The phase-based point positioning was first
4
introduced by Zumberge et al. (1995, 1997), who recognized that by processing dual-
frequency code and phase measurements, point positioning could provide comparable
accuracy to that of double differencing method with a greatly reduced computational burden.
This is known as Precise Point Positioning (PPP) and it differs from the conventional SPP
method in several aspects: the use of both dual-frequency code and phase measurements, the
use of precise GPS orbit and clock products and much higher positioning accuracy at
centimetre to decimetre level. PPP is able to overcome the shortcomings associated with the
current RTK methods in that no base stations are required and it has the potential to provide
positioning accuracy similar to the conventional double difference RTK positioning systems
(Gao and Shen, 2002).
PPP has received increasing interest within the GPS positioning and navigation community
in the past decade. Automated GPS data analysis services using precise GPS products have
been set up by Zumberge (1999). Testing by email and ftp, static positioning accuracy of a
couple of centimetres has been reported (Witchayangkoon and Segantine, 1999).
Witchayangkoon (2000) has also developed a PPP system for post processing, which has
demonstrated kinematic position solutions at sub-metre level accuracy and static position
solutions at sub-decimetre level accuracy. After SA was switched off, it was possible to
interpolate the IGS satellite clocks, currently sampled at 5 minute intervals (Kouba and
Springer, 2001b), to any data rate with interpolation error less than 4 cm (Zumberge and
Gendt, 2000). Most of the processing, so far, has been conducted in post-mission since the
time delay for the necessary precise data can be up to several days. For example, the IGS
5
final products, which can provide GPS satellite orbit and clock accurate to 5 cm and 0.1 ns
respectively, have a latency of 13 days. Even after real-time precise GPS orbit and clock
products became available from some agencies, such as JPL and NRCan (Muellerschoen et
al., 2000; Collins et al., 2001), the tests with real-time orbit and clock products were mostly
done in a simulated real-time mode, which is to “simulate the processing as if in real-time”
(Muellerschoen et al., 2001). Broadcast over the geostationary satellites (Armatys et al.,
2003), the real-time precise GPS orbit and clock products from JPL have also been reported
to improve real-time positioning accuracy using some commercial receivers, such as
receivers from NavCom (Bisnath et al., 2003). However, the reported accuracy of a couple of
decimetres is much worse than the accuracy of sub-decimetre, which has been obtained in
post-mission or simulated real-time mode (Zumberge et al., 1998; Muellerschoen et al., 2001,
Armatys et al., 2003). This is because several error sources, such as solid earth tides and
ocean loading, have not been corrected in these commercial receivers. The potential of PPP
technique would not be fully exploited if these error sources, which are normally reduced in
differential techniques, have not been corrected (Kouba, 2003). There is a significant need
for a systematic investigation of different aspects related to the development of a real-time
precise point positioning system due to its great promise in positioning accuracy and
operational flexibility, and its great potential in enabling numerous new applications.
The use of ground-based GPS observations to estimate the amount of Precipitable Water
Vapour (PWV) in the troposphere has been investigated extensively since 1992 (Bevis et al.,
1992). Due to the importance of high spatial and temporal resolution, all-weather capability
6
for severe weather forecast and operational numerical weather prediction, it is essential for
ground-based GPS meteorology to obtain high-precision Zenith Wet Delay (ZWD) estimates,
which are nearly proportional to the PWVs overlying the GPS receiver (Bevis et al., 1992),
from GPS observations with latency as short as possible. The estimation so far, however, is
feasible only in post-mission or near real-time (30 minutes to one hour latency) because of
the time required for collecting and compiling GPS raw data, the latency of the precise GPS
products and the complexity of the data processing (Gendt et al., 2001; Pacione et al., 2002;
Reigber et al., 2002; Ware et al., 2004). The real-time PPP provides a new way to enable
real-time ZWD estimation. Unlike traditional double-difference approach, its time latency
and temporal resolution are only limited by the data rate of GPS receivers, so it can support
real-time GPS meteorology applications.
GPS has long been proven to be a primary tool for precise time dissemination and transfer
(Allan and Weiss, 1980). The potential of GPS timing would not be completely exploited if
only code measurements are used in processing (Schildknecht, et al., 1990). Carrier phase
time transfer has been widely investigated by many researchers (Larson and Levine, 1999;
Schildknecht and Dudle, 2000). The sub-nanosecond accuracy so far was only obtained in
post-mission using IGS precise orbit and/or clock products (Larson et al., 2000). A real-time
PPP system has the potential to realize carrier phase time transfer in real-time at sub-
nanosecond accuracy.
Though most GPS receivers in use are single-frequency receivers, point positioning using
single-frequency measurements has not been considered able to provide positioning accuracy
7
at centimetre to decimetre level, because of ionospheric effects. Different models have been
developed to mitigate ionospheric effects to provide position solutions at the accuracies of a
couple of metres (Klobuchar, 1996; Ovstedal, 2002; Montenbruck, 2003; Beran et al., 2003).
Real-time sub-metre or even decimetre level point positioning accuracy is highly demanded
by single-frequency GPS users.
1.2 Objectives and Contributions
The major objective of this thesis is to develop methodology for real-time precise point
positioning, timing and atmospheric sensing using real-time precise satellite orbit and clock
products. Specific research tasks are described in the following.
1) To identify and investigate the characteristics of various error sources critical to PPP
and further to develop methods to mitigate them in real-time.
2) To investigate appropriate approaches for real-time precise orbit and clock data
transmission with respect to the latency and age.
3) To study different models to mitigate ionospheric effects for PPP using single-
frequency data.
4) To investigate methods for real-time atmospheric sensing using PPP methodology.
8
5) To investigate the potential of PPP technique for real-time precise timing by studying
the achievable accuracy of receiver clock offset estimates using real-time GPS orbit
and clock products.
6) To develop an operational real-time PPP software system for real-time positioning,
receiver clock offset and zenith tropospheric delay estimation.
The major contributions of this research are summarized in the following.
1) A real-time precise point positioning software system has been developed capable of
real-time positioning, receiver clock offset and zenith tropospheric delay estimation
using real-time precise orbit and clock products.
2) Techniques for real-time distribution of precise orbit and clock products have been
investigated with respect to latency, reliability, coverage, and cost.
3) Error sources relevant to precise point positioning using dual-frequency or single-
frequency observations have been investigated along with the approaches to mitigate
them in real-time.
4) Issues related to real-time precise water vapour estimation, including mapping
function selection, zenith hydrostatic delay modeling, elevation cut-off angle, and
horizontal gradient estimation, have been investigated. The test results have indicated
that PWV accuracy better than 1 mm is obtainable in real-time.
9
5) A new timing method has been proposed with the capability of recovering
UTC(USNO) at an accuracy of a few nanoseconds in real-time using a single GPS
receiver.
6) Sub-nanosecond receiver clock offset estimates have been obtained by PPP method
with real-time precise orbit and clock products. The tests have demonstrated the
potential of real-time timing using PPP method.
7) A new model has been proposed to mitigate ionospheric effects for PPP with single-
frequency measurements. Test results show that the model can provide sub-metre or
even decimetre level accuracy in real-time.
8) Real-time tests have been conducted to assess the obtainable accuracy of real-time
PPP methodology using two different real-time precise orbit and clock products
which are useful for industrial applications and product development.
1.3 Thesis Outline
Chapter 2 of this thesis gives a brief overview of precise point positioning, GPS timing, and
GPS atmospheric sensing. The background, challenges, methodologies, and real-time aspects
of these applications are discussed.
10
Dual-frequency GPS observable equations are presented in Chapter 3. In this Chapter, error
sources that are relevant to PPP data processing are described. The characteristic of these
error sources is analyzed along with the methods to mitigate them.
Chapter 4 presents several types of precise GPS orbit and clock products, including those
from IGS, NRCan, and JPL. The accuracy, latency, and sample interval of these products are
also analyzed in this chapter. Appropriate approaches for real-time precise orbit and clock
data transmission are investigated.
In Chapter 5, different ionospheric models are studied for PPP using single-frequency data
with regard to the obtainable accuracy and timeliness. A new model is proposed with great
promise to perform real-time positioning with an accuracy that is only obtainable by other
models in post-mission with a latency of approximately 11 days.
Chapter 6 analyzes the performance of precise point positioning using dual-frequency and
single-frequency data, receiver clock offset estimation, zenith tropospheric delay and
precipitable water vapour estimation, using real-time precise GPS orbit and clock products.
The real-time software package P3-RT, which has been developed to conduct all data
processing tasks in this thesis, is also described in this chapter.
Finally in Chapter 7, the conclusions and recommendations of this thesis are presented.
11
CHAPTER 2
METHODS OF REAL-TIME PRECISE POINT POSITIONING,
TIMING AND ATMOSPHERIC SENSING
This chapter will describe the methods of precise point positioning, timing and atmospheric
sensing. The definition of real-time will be described first. The concept of precise point
positioning will be given next along with its advantages and challenges. Timing with carrier
phase measurements will also be investigated. A new timing method will be proposed with
the capability of recovering UTC(USNO) at an accuracy of a few nanoseconds in real-time
using a single GPS receiver. GPS meteorology using precise point positioning methodology
and real-time precise GPS orbit and clock products will be finally presented. Challenges and
methods for real-time precise point positioning, timing and atmospheric sensing are
emphasized.
2.1 Definition of Real-Time
This research is focused on real-time applications of PPP methodology, including
positioning, timing and atmospheric sensing. As defined by Hofmann-Wellenhof et al.,
(2000), a position solution is not real-time, unless it is reported in the field immediately or
12
while still on the station. This requirement is too strict especially for high dynamic
positioning, because it takes time to process but there is also a less stringent definition. If the
processing time is negligible, the position solutions can still be treated as real-time though
they are obtained with a very short delay (Hofmann-Wellenhof et al., 2000). In this research,
real-time data processing is conducted in a computer with real-time precise orbit and clock
products received over the Internet or from a serial port. As tested with Javad GPS receivers,
which output raw observations at 20 Hz, the solution was reported before the next epoch data
was received. Therefore, the processing time for a single epoch data should be less than 50
ms using PPP technique.
In some applications, such as meteorology, real-time is not a strict terminology. It depends
on both the need and expectation (Gutman and Benjamin, 2001). In their position paper for
the real-time applications and products session, Bar-Sever and Dow (2002) have given the
definition of real-time and near real-time:
“We think that the term ‘real-time’ (rt) is too restrictive, and prefer the more flexible ‘near
real-time’ (nrt) to describe latencies ranging from 0 to 6 hours. The boundary between rt and
nrt may be drawn at the latency below which batch processing and data handling is no
longer practical. The IGS is currently producing hourly RINEX files from a large sub-set of
the network, which feed a number of batch-type processes, including the ultra-rapid orbit
determination. We define, therefore, Real Time processes and application as those that
require sub-hourly latency.”
13
Sub-hourly latency is a general definition. For different applications the timeliness
requirement varies significantly. Real-time positioning, navigation and timing allow latency
of seconds (Bar-Sever and Dow, 2002). In numerical weather forecasting applications, during
stable weather conditions, Precipitable Water Vapour (PWV) estimates can still be
considered as real-time with latency up to 1 hour. During active weather conditions,
however, data for weather forecast should be updated within a few minutes. If the data is
only updated every half an hour, the information can only be considered near real-time
(Gutman and Benjamin, 2001). In GPS meteorology, the sooner the PWV estimates are
provided, the more useful they are for weather forecast.
In this research, data is processed with the computationally efficient PPP method and real-
time precise GPS orbit and clock products; the results, including the coordinates, the receiver
clock offset and the zenith tropospheric delay estimates, can be provided with very short
latency, i.e. less than one second.
2.2 Real-Time Precise Point Positioning
The reason GPS was developed in the first place is to determine coordinates of a position
with a single GPS receiver. This positioning method is often referred to as absolute
positioning or point positioning (Goad, 1998). Point positioning is however subject to the
effects of all GPS error sources including satellite orbit and clock errors in the GPS
navigation messages, atmospheric effects and receiver related errors, etc. Currently, the
14
dominant errors are the broadcast satellite orbit and clock errors and ionospheric effects
(Clynch, 2000). Therefore, point positioning can only provide position solutions at the
accuracy of a couple of metres using broadcast ephemeris. To obtain better results,
differential positioning approaches are usually used, which require a minimum of two GPS
receivers, with at least one set up at a station with known coordinates (Kouba and Heroux,
2001a). Differencing procedure between satellites and receivers can remove common errors
among GPS satellites and receivers or reduce errors that are spatially correlated among
satellites and receivers. However, the requirement that base receiver(s) be set up at station(s)
with precisely known coordinates is not always feasible in practice. Another drawback
related to differential positioning systems is that the rover receivers must be within the
vicinity of the base receivers (Kouba and Heroux, 2001a). This is because the cancellation of
spatially correlated errors among the GPS satellites and receivers becomes less effective as
the separation or baseline length between the base and rover stations increases. For large-
scale applications such as aerial survey and mapping, current RTK techniques often become
problematic in practice due to the increased operational cost and complexity.
The availability of precise GPS satellite orbit and clock products has enabled the
development of a novel positioning methodology known as precise point positioning (PPP).
PPP processes un-differenced GPS code and/or carrier phase measurements using precise
orbit and clock products. Because of its high precision and operational flexibility, PPP began
to receive increased attention within the GPS positioning, timing, and atmospheric sensing
communities.
15
In this section, the concept of precise point positioning is described along with the
advantages and challenges of this novel technique. Methods for a real-time precise point
positioning system are introduced. Details of error mitigation and modeling for real-time
precise point positioning methodology will be investigated in Chapter 3.
2.2.1 Concept of Precise Point Positioning
Although the concept of precise point positioning was introduced by Anderle (1976) to
denote point positioning using precise ephemeris and Doppler satellite observations,
Zumberge et al. (1995) were the first to describe PPP method for centimetre accuracy
applications using GPS carrier phase as the principle observable. There are four milestones in
the development of GPS precise point positioning, each resulting in great improvement in
accuracy or practicability.
Code-based PPP (1980s~1995)
The International GPS Service (IGS) has been providing precise GPS orbit and clock
products since 1994 (Kouba and Heroux, 2001a). Some government agencies even started the
work from 1980s (Lachapelle et al., 1996). However, before 1995, only code observations
were used to conduct point positioning with precise GPS ephemeris. Carrier phase
measurements, if even involved, were only used to smooth code measurements (Hatch,
1982). Point positioning using precise GPS ephemeris and code measurements can improve
the accuracy by an order of 2 with respect to the standard point positioning using broadcast
16
ephemeris, to a couple of metres from about 100 m with SA on (Heroux et al., 1993;
Lachapelle et al. 1994a; Heroux and Kouba, 1995). However, because only the noisy code
measurements were used, the metre level accuracy cannot satisfy the requirements of
centimetre accurate positioning applications such as geodetic surveys where carrier phase
based double differencing methods with ambiguity resolution must be used.
Phase-based PPP (1995~)
It did not take long before PPP method offered centimetre level positioning accuracy by
processing both code and phase observations. Zumberge at al. (1995, 1997) presented daily
precision of a few millimetres in the horizontal and centimetres in the vertical for static
positioning. Sub-decimetre level accuracy was also obtained using receivers set up on a
vehicle (Zumberge at al., 1998). After introducing carrier phase measurements as primary
observables, the accuracy has been improved by an order from metre level to decimetre level.
PPP with SA off (2000~)
The demise of SA greatly benefited civil GPS users who conduct point positioning using
broadcast GPS ephemeris. It is also significant to precise point positioning users, because the
precise clocks become more predictable (Neilan et al., 2000; Zumberge and Gendt, 2000).
With SA on, PPP can only provide centimetre level position solutions with data that
coincides with times when precise satellite clocks have been estimated (Zumberge at al.,
1998). Most users could only access the 15-minute clock solutions from IGS before 2000,
though higher rate clocks were available from several analysis centers (Neilan et al., 2000).
17
As investigated by Zumberge at al. (1998), using precise clocks with a sample interval of 30
s, kinematic positioning can achieve approximate 7 cm 3D accuracy with GPS observations
at the same data rate. The accuracy would degrade by a factor of three for higher rate data.
After SA was switched off, the 30 s clocks can be interpolated to process data at any rate
with just about 4 mm degradation in the interpolated clocks, compared to about 8 cm (or 20
times larger) degradation in the SA era (Zumberge and Gendt, 2000). Even using the 5-
minute clocks from IGS (Kouba and Springer, 2001b), the errors introduced by interpolation
are less than 4 cm with SA off. Therefore, kinematic PPP is applicable to any data rate after
May 2, 2000, using IGS products.
Real-Time PPP (2000~)
The latest IGS precise GPS orbit and clock products, which are accurate enough to perform
PPP with centimetre level accuracy, are the observed half of the Ultra-Rapid (IGU) products.
But even the IGU products (observed half) still have a latency of about 3-hour. The latency is
even longer for the more accurate and reliable products, such as IGS Rapid and Final orbits
and clocks. So IGS precise products can only support centimetre level accuracy PPP in post-
mission currently.
The advent of real-time precise GPS orbit and clock products from some agencies has made
real-time PPP practicable, for example, the real-time products from JPL and NRCan
(Muellerschoen et al., 2000; 2001; Collins et al., 2001). However, even using the real-time
products, most data processing was still conducted in post-mission to simulate real-time
processing because of the lack of real-time PPP software (Muellerschoen et al. 2001). In this
18
research, a software package has been developed to perform real-time PPP processing using
orbit and clock products from both JPL and NRCan.
2.2.2 Advantages of Precise Point Positioning
As discussed above, PPP can provide sub-decimetre level kinematic and centimetre level
static positioning accuracy, which is comparable to double-difference technique with On-
The-FLY (OTF) ambiguity resolution. Compared with double-difference methods, PPP has
the following advantages.
Cost Saving and Simplified Operation
PPP brings not only great flexibility to field operations, but also reduces labour and
equipment cost and simplifies operational logistics by eliminating the need for base stations.
On the other hand, errors will be introduced to the position solutions of the rover stations in
differential techniques if the coordinates of the base stations are not precisely determined
(Heroux et al., 2004).
Globally Consistent Solutions
PPP can provide globally consistent position solutions. PPP solutions are consistent with the
precise GPS orbit and clock products, which are usually based on a globally consistent
reference frame. For example, IGS products are currently based on ITRF2000 which is also
the reference frame used by JPL to estimate real-time orbit and clock products.
19
Computational Efficiency
PPP is computationally efficient for the analysis of GPS data from large networks (Zumberge
et al., 1997). Even today, with advanced computational technology, this property of PPP is
still very useful for real-time or near real-time applications. For example, in GPS
meteorology, PWV estimates should be provided as soon as possible. Data from a GPS
network can be processed in parallel on multiple computers using the PPP method.
Therefore, the processing time of PPP can be much shorter than that of the traditional double
differencing methods employed to process the same datasets from a network. In this research,
real-time atmospheric sensing will be investigated using the computationally efficient PPP
methodology.
Other Useful Parameters
Absolute receiver clock offset and Zenith Tropospheric Delay (ZTD), which are treated as
nuisances in double differencing techniques, can be estimated in PPP and used for timing and
meteorological applications. In this research, the absolute receiver clock offset will be
estimated in PPP processing using real-time orbit and clock products to investigate the
potential of real-time timing using PPP method.
2.2.3 Challenges of Precise Point Positioning
Currently, PPP also has several challenges which are described in this section.
20
Availability of Precise GPS Orbit and Clock Products
After SA was turned off, high precision GPS orbits and clocks from IGS or other agencies
can be interpolated to any rate for post-mission PPP processing. But in real-time applications,
users are still seeking real-time precise orbit and clock products. Several types of orbit and
clock products are available in real-time currently, such as products from JPL and NRCan,
but most of them are still at developing or testing stages. Receiving and decoding devices for
real-time products are also under development. In Chapter 4, approaches for real-time
distribution of orbit and clock products will be investigated. The latencies of products
received over the Internet and their impact to positioning will also be discussed. Real-time
software systems are also required to conduct real-time data processing using the real-time
products. A real-time software system has been developed and will be presented in Chapter
6.
Convergence Time
Because of receiver- and satellite-specific phase offsets, namely initial phase offsets, which
are generally float values and unknown, the ambiguities in the un-differenced carrier phase
measurements will no longer be integers (Zumberge et al., 1997). It takes time for the float
ambiguities in PPP to converge. Achievements have been made by some researchers to
pseudo-fix or even fix the float ambiguities and a convergence time of several minutes would
be highly desired (Gao and Shen, 2002).
21
Error Mitigation
Some error sources, which are cancelled out or partially mitigated in differential processing,
have to be properly modeled or estimated in PPP processing. These sources will be studied in
Chapter 3 to fully exploit the potential of PPP methodology. In order to achieve sub-
decimetre level accuracy, dual-frequency receivers are required to mitigate ionospheric
effects. How to obtain comparable accuracy using single-frequency receivers is a big
challenge for GPS researchers but is significant for GPS community because most GPS
receivers currently in use are single-frequency receivers. Point positioning using single-
frequency observations and precise orbit and clock products has been investigated by
researchers with an accuracy of a couple of metres in post-mission (Ovstedal, 2002;
Montenbruck, 2003). In this research, a new ionospheric model will be proposed, which has
great promise to provide sub-metre level accuracy in real-time using single-frequency
observations and precise GPS orbit and clock products. The details of PPP with single-
frequency measurements are discussed in Chapter 5.
2.3 Real-Time GPS Timing
The clocks operating on the GPS satellites need to be precisely synchronized so that GPS
signals can be used for positioning. To synchronize the clocks, one Master Control Station
(MCS) was set up at Falcon Air Force Base near Colorado Springs, Colorado, which collects
the GPS satellite tracking data from five monitor stations around the world. The time error,
22
frequency error, and frequency drift for each clock operating on the GPS satellite are thus
estimated. The clock errors are then uploaded to each satellite and broadcast to users in real-
time. The process allows clocks across the constellation to be synchronized to within a few
nanoseconds (Francisco, 1996). The synchronization of the clocks at a level of a few
nanoseconds makes possible for metre level accurate position determination, and it also
brings great promise for timing at a few nanoseconds accuracy. GPS has quickly become a
primary tool for high precise time dissemination and transfer ever since it became operational
(Klepczynski, 1996).
Time scale is the foundation of timing and time transfer. There are several different time
scales relevant to GPS time transfer. Before discussing the time transfer techniques, these
time scales will be described first. Currently, only the noisy GPS code observables are used
for time dissemination and transfer in real-time. The more precise carrier phase
measurements, however, are only employed in post-mission because of the latency of precise
GPS orbit and clock products.
As real-time PPP can determine receiver position up to centimetre level accuracy, the
potential of real-time receiver clock offset estimation at the sub-nanosecond level is evident.
PPP has become a preferred choice for precise timing and time transfer. In this research, the
potential of real-time timing using precise point positioning method will be investigated. A
new timing method is proposed with the capability of recovering UTC(USNO) at an
accuracy of a few nanoseconds in real-time using a single GPS receiver. Receiver clock
23
offset estimates at sub-nanosecond accuracy using real-time precise orbit and clock products
will be presented in Chapter 6.
2.3.1 Time Scales
There are several time scales related to GPS time transfer, including Universal Time,
Coordinated Universal Time, GPS Time and IGS Time.
Universal Time
Universal time (UT) is a time scale based on the Earth rotation on its axis. UT is not uniform
since the angle velocity of the Earth’s rotation is not constant. The fluctuations are partly
caused by the changes in the polar moment of inertia exerted by tidal deformation and other
mass transports. Other factors include the oscillations of the Earth’s rotational axis itself. The
universal time, after corrected for the polar variations, is denoted by UT1 (Hofmann-
Wellenhof et al., 2000). UT1 is still not uniform because of small changes caused by both
regular seasonal variations and irregular and unpredictable changes in the Earth’s rotation
period (Spilker, 1996b).
Coordinated Universal Time
More precise and uniform time scales have been introduced with the availability of high-
precision atomic clocks. The International Atomic Time (TAI) is an atomic time scale
coordinated by the Bureau International des Poids et Mesures (BIPM). TAI is a uniform and
24
continuous time scale. Because the universal time scale is still widely used, to keep the TAI
close to the universal time scale UT1, a compromise time scale, Coordinated Universal Time
(UTC), was introduced. The lengths of the seconds used in the generation of TAI and UTC
are the same. UTC differs from TAI by an integral number of seconds. The UTC scale is
adjusted by introducing positive or negative leap seconds to ensure approximate agreement
with UT1 within 0.9 s (Spilker, 1996b). At 0 hour on January 1, 1958, TAI and UT1 were
coincident. By the end of 2004, 32 positive leap seconds were inserted into UTC, therefore,
TAI-UTC=32 s.
UTC is not a directly available clock in real-time. UTC(BIPM) time is normally generated
with about 1 month latency. In order to obtain a real-time estimate of UTC, 50 timing centers
around the world generate their own current estimates of UTC, namely UTC(k) (Allan et al.,
1997). For example, UTC(USNO) is a real-time estimate of UTC kept by the U.S. Naval
Observatory (USNO). All of the timing centers are keeping their UTC(k)s within 100 ns of
UTC. In 1996, most UTC(k)s were kept within 10 ns of UTC(BIPM) (Allan et al., 1997).
Currently, the differences between UTC(k)s and UTC(BIPM) have been further reduced
(Lewandowski and Tisserand, 2004)
GPS Time
The GPS time (GPST) is maintained by a set of atomic clocks operating at GPS control
segment (CS) and space. Thus, the GPS time is also based on atomic time but without the
leap seconds of UTC. The introduction of leap seconds in GPS time would make the P-code
25
receiver out of lock. Other than the leap second effect, however, GPS time is kept by the GPS
control segment to within 1 µs of UTC(USNO) time (modulo 1 s) (Spilker, 1996b).
GPS time was set to agree with UTC at 0 hour on January 6, 1980. Therefore, if ignoring the
fractional part less than 1 µs, GPST-TAI=-19 s and by the end of 2004, GPST-UTC=13 s.
GPS Time is also traceable back to UTC(USNO) with several nanoseconds uncertainty,
which will be discussed in the following.
IGS Time Scale
Since GPS week 1087 (November 5th, 2000), the IGS has provided two sets of clock
estimates, a rapid combination with a latency of 17 hours, and a final set with a latency of 13
days (Kouba and Springer, 2001b). IGS hopes to keep all the IGS clock information,
including those of satellites and receivers, referenced to a common, consistent timescale. At
the beginning of the clock products, the IGS has used a simple linear alignment of the clocks
to the broadcast GPS time for each separate day (Ray and Senior, 2003). However, the
instability of GPS time, which is only kept to within 1 µs of UTC(USNO) time (modulo 1 s),
is so large that this procedure introduces large day-to-day discontinuities in the time and
frequency of IGS clock products. The discontinuities do not affect position solution using the
clock products, but they definitely degrade the performance of time and frequency
dissemination (Ray and Senior, 2003).
To exploit the geodetic receiver for precise time transfer using IGS precise products, a
workshop, namely “IGS/BIPM Pilot Project to Study Accurate Time and Frequency
26
Comparisons using GPS Phase and Code Measurements” was organized in December 1997
(Ray and Petit, 1999). The project has not only helped refine the IGS clock products and their
link to UTC, it has also provided new opportunity for the BIPM to improve the time transfer
results. Recently, a final timescale, namely IGST, has been generated with regard to the final
IGS combined clock products (Ray and Senior, 2003). These new time scale can greatly
benefit the users who employ IGS products for time transfer.
2.3.2 GPS Time Transfer Techniques
GPS can be used for precise time transfer in several ways, namely, one-way, common-view,
melting-pot and carrier phase.
One-Way
The simplest GPS time transfer technique is the one-way GPS time transfer, which is also
known as “direct-access” or “passive” GPS time transfer. In one-way GPS time transfer, a
GPS receiver, which is set up on a fixed site whose coordinates have been precisely
determined, can calculate the difference between the local clock driving the receiver and the
GPS time using the pseudorange measurements, and then recover UTC(GPS) using broadcast
GPS navigation messages (ICD-GPS-200C, 2000). The broadcast GPS navigation messages
provide the parameters required to relate the GPS time to UTC(GPS), which is used to denote
the time scale of GPS delivered prediction of UTC(USNO) (Hutsell, et al., 2002). The
coefficients are determined by monitoring the clocks on GPS satellites at the USNO
27
(Klepczynski, 1996). Therefore, user can even link the clock under test to UTC using a single
GPS receiver with C/A code observations.
This method is easy to implement and it can directly recover UTC(USNO) in real-time.
Moreover, it does not need to communicate with other systems. On the other hand, the
accuracy of the recovered UTC(USNO) is limited, e.g., 20 ns after SA switched off
(Lombardi et al., 2001). The error sources include the satellite orbit and clock errors,
ionospheric and tropospheric errors, multipath effects and measurement noise in code
observations. The performance of this technique is also limited by the uncertainty in the
broadcast inter-frequency bias (Tgd). As defined in ICD-GPS-200C, the group delay
differential between 1L and 2L signals is specified as consisting of random and bias
components. The absolute value of the bias part should not exceed 15 ns and the random
variations should not exceed 3 ns (2σ) (ICD-GPS-200C, 2000). The uncertainty between
UTC(USNO) and UTC(GPS) also affects the accuracy of recovered UTC(USNO).
After SA was turned off, the performance of this technique has been improved about a factor
of 2. Moreover, by using multi-channel GPS receivers, and precise ionospheric products, a
10 ns accuracy is achievable. The use of precise GPS orbits and clocks can improve the
precision of receiver clock estimates, but not all time references of the precise orbit and clock
data are traceable to UTC. For example, the traceability of the time reference of IGS Final
precise orbits and clocks, IGS Time scale, is still under development. The traceability of one-
way GPS time transfer will be discussed in Section 2.3.4.
28
Common-View
Common-view (CV) provides the direct comparison of two clocks at remote locations. In this
method, by differencing GPS observations at two sites to the same satellite at the same
instant, the difference between the two clocks driving GPS receivers at two sites can be
obtained (Allan and Weiss, 1980). By differencing, some common errors can be removed
completely, such as satellite clock errors, uncertainties of broadcast inter-frequency biases.
Some spatially correlated errors can also be reduced, including satellite orbit errors,
ionospheric and tropospheric errors. Therefore, the accuracy of common-view time
comparison is much better than the accuracy of recovered UTC(USNO) using the one-way
method. A 10 ns accuracy can be obtained using single channel GPS receivers. By using
multi-channel receivers, the accuracy can be improved to better than 5 ns (Lombardi et al.,
2001). The accuracy can be further improved by averaging measurements over a long period.
Moreover, if one of the clocks is directly linking with UTC, then the other clock under test
can recover UTC more accurately than the one-way method.
For example, to maintain the TAI and UTC, the BIPM uses the common view technique
based on GPS C/A code observations from time receivers installed in the time laboratories.
At first, only single-channel receivers were used. Multi-channel receivers were later
introduced to the procedure (Defraigne and Petit, 2003).
Common-view requires communication between two receivers. It is mainly used in post-
mission. The performance of common-view is also affected by satellite orbit errors,
ionospheric and tropospheric errors, especially when conducting intercontinental time
29
transfer. Precise ionospheric products and precise GPS orbits can be used to improve
accuracy in post-mission.
Melting-Pot
Common-view method has been widely accepted. However, it requires exactly simultaneous
observations at both locations, a requirement which is not easy to fulfill except at
measurement laboratories (Klepczynski, 1996). Actually, if precise orbit and clock products
are used, the measurement uncertainty of common-view technique might be just slightly
smaller than the one-way technique when the receivers are widely separated. Moreover, the
satellites, which are commonly viewed by two receivers, are limited in number and in
tracking time, when the receivers are widely separated. In this case, a more robust technique,
melting-pot, can be used (Klepczynski, 1996).
In melting-pot method, a local clock is also compared with a remote clock. It differs from the
common-view, which only uses satellites tracked by two receivers simultaneously, while for
melting-pot, each receiver tracks satellites all-in-view and calculates offset of the local clock
with respect to a common reference time, such as GPS time or IGS time (depending on the
orbit and clock data). The difference of the offsets obtained from two receivers at the same
instant will be the difference of two clocks under test. This method is more robust than the
common-view method because it utilizes satellites all-in-view at each site and sequentially
there are no gaps in data (Klepczynski, 1996). The melting-pot method is probably slightly
less accurate than the common-view method, but it is more robust and derives the offset from
30
all observations. In melting-pot time transfer, if one of the clocks is traceable to UTC, then
the other clock can recover UTC as well.
Carrier Phase
Common-view time transfer using single-frequency C/A-code GPS receivers, as now widely
used, is limited mainly by the code multipath effects, which are not zero-mean and cannot be
removed by averaging, and over long distances, by error introduced by ionospheric delays,
which can not be mitigated effectively using single-frequency measurements. Though GPS
satellite orbit errors can be greatly minimized by precise GPS orbit products, IGS ionospheric
products, which are used in post-mission time-transfer, can only provide about 2 TECU
accuracy of vertical total electron content at grid points (Ovstedal, 2002). These factors limit
the accuracy of GPS C/A-code time transfer to a couple of nanoseconds. Even after
multichannel P-code dual-frequency GPS receivers are introduced, the accuracy of traditional
common-view time transfer is still limited by the noise and multipath effects in the code
measurement. The potential of GPS time transfer would not be fully exploited unless both the
phase and code measurements from all satellites in view are involved (Schildknecht and
Springer, 1998). Using geodetic GPS receivers, ionospheric effects can be mitigated
effectively by dual-frequency measurements. Carrier phase multipath and noise is also much
smaller than those of C/A code in magnitude. Moreover, the carrier phase multipath is zero-
mean and thus can be further minimized by averaging (Ray, 2000). Therefore, the accuracy
of time transfer using carrier phase measurements is promising. Different from code
31
measurement, GPS carrier phase measurement contains an unknown ambiguity, which must
be resolved before it is used for precise time transfer.
High accurate position solutions (millimetre to centimetre) achieved by GPS carrier-phase
observations with ambiguities resolved also indicate the potential of GPS carrier-phase for
time-transfer. Since receiver clock errors are closely related to position errors, the use of
geodetic receivers for precise frequency and time transfer has been proposed a long time ago
(Schildknecht, et al., 1990, Schildknecht and Springer, 1998). The technique using geodetic
receivers for time transfer is often called carrier phase time transfer or geodetic receiver time
transfer. In the past years, this technique has been widely investigated (Larson and Levine,
1999; Schildknecht and Dudle, 2000; Ray et al., 2001; Dach et al., 2002; Defraigne and Petit,
2003). Larson et al. (2000) presented an agreement of 1 ns using IGS final orbits and a few
nanoseconds using IGS predicted orbits with the Two-Way Satellite Time-Transfer
(TWSTT) measurements apart from a constant offset duo to the unknown hardware delays in
both ends.
It is important to note that the accuracy of carrier phase time transfer is also affected by
hardware delay instabilities. Therefore, in order to take full advantage of this promising
technology, the receivers should be calibrated carefully. Also, precise GPS satellite orbit
and/or clock products should be used to exploit the precise carrier phase measurements.
Carrier phase time transfer is often referenced as a novel technique different from one-way
and common-view techniques, but it can be implemented in one-way or common-view mode
(Lombardi et al., 2001).
32
If carrier phase measurements are involved in one-way time transfer, a PPP time transfer is
applied. As PPP can provide position solutions at centimetre level accuracy, the potential of
PPP timing is evident (Senior and Ray, 2001b). In Section 2.3.5, a PPP timing method will
be proposed with the capability of recovering UTC(USNO) at an accuracy of a few
nanoseconds in real-time using a single GPS receiver.
2.3.4 Challenges in GPS Time Transfer
GPS is now widely accepted as a primary tool for time dissemination and transfer. Several
challenges for this application are described in the following.
Traceability of One-Way Time Transfer
Traceability is defined as an unbroken chain of comparison with stated uncertainties
(Lombardi et al., 2001). In GPS timing, traceability is the capability of linking the clock
under test to UTC. In common-view and melting-pot time transfer, if one of the clocks is
precisely linked to UTC, then the other clocks are also traceable to UTC. In one-way GPS
time transfer, the traceability of the clock under test depends on the time reference of the
orbit and clock data. If broadcast orbit and clock data is used, the clock estimates are then
referenced to GPS time. GPS navigation messages contain the parameters needed to relate
GPS time to a time scale of GPS delivered prediction of UTC(USNO), named UTC(GPS), as
already discussed. The uncertainty between UTC(USNO) and UTC(GPS) is still significant.
33
As investigated by Hutsell et al. (2002), the RMS of daily UTC(GPS) - UTC(USNO) is 5.84
ns for the period from October 2000 to September 2002. It is 6.32 ns for the period from
January 1999 to September 2000 (Gifford et al., 2000). The RMS difference is 7.84 ns and
6.94 ns in 1997 and 1998, respectively (Rivers and Osborne, 1999). Though the performance
of recovering UTC(USNO) using the parameters in the navigation messages has been
improved year by year, the uncertainty is still significant, especially considering that carrier
phase measurements can be used to estimate the clock offset with an accuracy of sub-
nanosecond.
A new method will be proposed in Section 2.3.5. The method estimates receiver clock in the
same way as one-way time transfer, but it uses dual-frequency code and phase
measurements, and the IGDG precise orbit and clock products from JPL, which make real-
time sub-nanosecond receiver clock estimates possible. Moreover, the use of JPL IGDG
products allows the users to recover UTC(USNO) with a much higher accuracy than using
the broadcast messages because JPL IGDG products are referenced to AMC2, one the
alternate master clocks of UNSO.
Calibration of GPS Receiver
The receiver clock offset estimate from GPS data processing is a virtual internal clock
estimate. How to relate the internal clock estimates to the external hardware clock, which is
used to drive the GPS receiver, is a receiver calibration issue (Schildknecht and Dudle,
2000). The external hardware clock is usually the clock to be tested by time transfer
techniques. The problem is that there are very few geodetic receiver systems which have
34
been calibrated to accurately relate their internal clocks to the external clock standards (Ray
and Petit, 1999a).
Unlike geodetic position determination, in which any effect that is bias-like will be absorbed
into the clock estimates, all those biases should be separated from the clock estimates for
GPS time transfer (Ray and Senior, 2003). No matter which technique is used, the GPS time
transfer performance is affected by hardware delays, including delay in antennas, cable and
receivers. Frequency transfer may not be as sensitive to these delays as time transfer. The
hardware delay in GPS satellite has been calibrated to ns level in the factory before the
satellite launch (ICD-GPS-200C). In order to use GPS receiver for timing, all instrumental
delays from antenna to receiver should also be calibrated (Petit et al., 2001). As investigated
by many researchers, the uncertainty of the instrumental delays is still at 2 ns level even after
careful calibrations (Petit et al., 2001; Landis and White, 2002). The uncertainty in receiver
calibration is a limiting factor for high precision time transfer using geodetic GPS receivers,
as carrier phase can provide sub-nanosecond level clock offset estimates (Ray and Senior,
2003).
Real-Time Carrier Phase Time Transfer
One-way GPS C/A code time transfer is very easy to implement in real-time. Common-view
GPS time transfer needs communication between receivers. Currently, the common-view
GPS C/A code time transfer is also practicable in near real-time or even real-time with
advanced communication technology. Carrier phase time transfer requires precise GPS orbit
and/or clock products. The tests so far were mostly carried out in post-mission using IGS
35
final orbit and/or clock products. Predicted GPS orbits have also been used with degraded
performance (Larson et al., 2000). The demand for real-time time transfer is significant from
the industry, such as electric power companies. In this thesis, the research results of precise
receiver clock offset estimation will be presented in Chapter 6 using precise point positioning
method and JPL real-time orbit and clock products. The details of JPL real-time products
from the Internet-based Global Differential GPS System (IGDG) will be described in Chapter
4. PPP using IGDG products also provides a new way to recover UTC(USNO), which will be
described in the next section.
2.3.5 Real-Time Timing Using IGDG Products
In the IGS Analysis Centers (ACs) or other agencies that provide precise GPS orbit and clock
products, one receiver clock equipped with a hydrogen maser external frequency is usually
fixed and used as a time reference. For example, JPL IGDG orbits and clocks are referenced
to the clock of IGS station AMC2 (Muellerschoen, 2003). Because of the importance of
AMC2 station in both GPS and timing communities, estimating receiver clock offset using
PPP method and the IGDG real-time orbit and clock products provides a new way to recover
UTC(USNO) in real-time using a single GPS receiver with a much better performance than
the traditional one-way time transfer.
AMC2 station, which is known as one of the “Alternate Master Clocks (AMCs)” because its
clock is selected as a back-up realization of UTC(USNO), is located at the USNO’s Alternate
Master Clock at Schriever APB in Colorado (Matsakis et al., 1999). Hourly Two-Way
36
Satellite Time-Transfer (TWSTT) is performed by USNO between the Alternate Master
Clock and the Master Clock which serves as a real-time estimate of UTC, namely
UTC(USNO) as already discussed, to steer the clock of AMC2 to the UTC(USNO) (Senior et
al., 1999). This particular relationship between the AMC2 clock and UTC(USNO) provides a
new method for precise real-time recovery of UTC(USNO) using a single GPS receiver. As
described in Powers (2002), since July 3rd, 2002, all timing calibration biases have been
removed from the pseudorange measurements of AMC2 during the data collection procedure.
Therefore, any clock solution referenced to this station would recover UTC(USNO) with an
accuracy of a few nanoseconds (Powers, 2002).
Because AMC2 clock is fixed by JPL during data processing to estimate IGDG real-time
precise orbit and clock products, the receiver clock offset estimates in precise point
positioning using JPL IGDG products are also referenced to AMC2 clock. As presented in
Section 6.3, receiver clock offset estimates at 100 ps accuracy are obtainable in real-time.
Therefore, precise point positioning using IGDG products have the potential of recovering
UTC(USNO) with an accuracy of a few nanoseconds. Timing using PPP methodology and
IGDG products, which keeps the flexibility of the one-way time transfer, can offer a much
better performance to recover UTC(USNO) in real-time than the traditional one-way time
transfer method, in which the accuracy is limited to a level of about 20 ns (Lombardi et al.,
2001). The capability of this method to recover UTC(USNO) is comparable to the common-
view method which however requires one clock precisely linked to UTC(USNO).
37
2.4 Real-Time Atmospheric Sensing
The use of GPS observations to sense the water vapour in atmosphere has been extensively
investigated since 1992 (Bevis et al., 1992). Due to importance of high spatial and temporal
resolution, all-weather capabilities of the GPS estimated Precipitable Water Vapour (PWV)
for severe weather forecasting and operational numerical weather prediction, the key issue of
ground-based GPS meteorology is how to get high precision Zenith Wet Delay (ZWD)
estimate, which is nearly proportional to the vertically integrated water vapour overlying the
GPS receiver, from GPS observations with as short latency as possible. The estimation so far,
however, is feasible only in post-mission or near real-time (about 30 minutes to one hour
latency) because of the time spent in collecting and compiling GPS raw data, the latency of
the precise GPS products and the complexity of the data processing (Gendt et al., 2001;
Pacione et al., 2002; Reigber et al., 2002; Ware et al., 2004). In this research, a methodology,
which is capable of providing high precision PWV in real-time using a single GPS receiver,
is described. Key factors that would affect tropospheric delay estimation will also be
addressed including horizontal gradients, mapping function selection, antenna phase center
variations, and elevation cut-off angle.
2.4.1 Water Vapour Sensing Techniques
Water vapour in the atmosphere is quite variable and it has significant impact on the global
climate change and small-scale weather development. Therefore, precisely sensing
38
atmospheric water vapour plays a vital role in meteorological and climatologic research,
including numerical weather prediction and global climate change study (Dodson et al.,
2001). A variety of techniques have been developed to measure the spatial distribution of
water vapour in the atmosphere, including radiosonde, water vapour radiometer (WVR),
Very Long Baseline Interferometry (VLBI) and GPS. Each approach has its own advantages
and disadvantages.
Radiosondes have been widely used all over the world as a reliable tool for determining the
distribution of water vapour in the atmosphere. Installed on balloons, radiosondes can
provide pressure, temperature and relative humidity measurements through a profile of the
atmosphere to a height of about 30 km (Dodson et al., 2001). The advantages of radiosondes
are that they can provide water vapour profile information. But the operational cost restricts
that they are only launched by most meteorological agencies twice per day at limited stations.
For example, in Alberta, radiosondes or rawinsondes are launched twice per day at Stong
Plain (Skone, 2005). Therefore, the temporal and spatial resolution of radiosonde
measurements is not adequate enough, though they can provide good vertical resolution
(Bevis et al., 1992). Another limitation of radiosondes is that the accuracy of humidity
measurements degrades at cold region and at high altitude. Humidity sensors of radiosondes
do not work normally at temperature below –40°C (Dodson et al., 2001). Despite these
limitations, radiosondes are still widely used by meteorological agencies of many countries.
Another water vapour sensing instrument, the water vapour radiometer can be used in two
ways: ground-based and space-based. Ground-based WVRs measure the background
39
radiation emitted by atmospheric water vapour and then provide integrated water vapour
(IWV) estimates (Bevis et al., 1992). The advantage of ground-based WVRs is that they can
provide good temporal resolution of water vapour distribution in atmosphere, by providing
IWV estimates with high rate and in real-time. But WVRs are expensive devices, so they
cannot be installed densely enough to provide high spatial resolution on the ground. On the
other hand, the space-based WVRs possess some complementary characteristics of the
ground-based ones. These downward-linking WVRs can provide water vapour measurements
with good spatial resolution. But the temporal resolution of their measurements is limited
because the satellites only cycle the Earth few times per day. The recovery of IWV from
space-based WVRs is very complicated over land because the temperature of the hot
background is extremely difficult to sense (Bevis et al., 1992). It is obvious that
measurements of ground-based and space-based WVRs are complementary. Ground-based
WVRs provide high temporal resolution measurements over land, while space-based WVRs
provide high spatial resolution measurements over ocean. A common limitation of ground-
based and space-based WVRs is that their accuracy is degraded when it is raining or heavily
cloudy (Bevis et al., 1992).
VBLI can also be used for atmospheric water vapour sensing. The idea of VBLI for water
vapour measuring is similar to that of ground-based GPS in that it also provides tropospheric
delay estimates. But it is not practical for meteorological applications because of economic
reasons (Niell et al., 2001).
40
GPS has long been proven to be a powerful tool for atmospheric sensing (Bevis et al., 1992).
It shows great advantages over traditional techniques, which will be discussed in the
following.
2.4.2 GPS Meteorology
The use of GPS observations to sense the water vapour in atmosphere has been extensively
investigated since 1992 (Bevis et al., 1992). Compared with other atmosphere sensing tools,
such as radiosonde and ground-based and space-based WVRs, the advantages of GPS include
the high temporal and spatial resolution of PWV estimates, low cost, and all-weather
capability.
Like WVRs, GPS can be used in two ways to sense atmospheric water vapour. The first
technique utilizes the ground-based, stationary GPS receivers, which have been largely
deployed in continuously operating networks being constructed around the world in a
regional or global scale (Bevis et al., 1992; Businger et al., 1996). The second technique is
the GPS occultation, which uses GPS receivers operating on low earth orbit satellites (Yuan
et al., 1993; Hajj et al., 2002). Water vapour estimated by GPS can play an important role in
the study of climate changes (Yuan et al., 1993), and is of crucial importance for severe
weather forecast and operational numerical weather prediction (Kuo et al., 1993). Therefore,
GPS water vapour sensing can be used in both meteorological and climatologic applications.
The requirements for these applications are different. The key requirements for meteorology
are accuracy and timeliness. Climate research and climatology require even higher accuracy
41
than meteorology but they are less strict on timeliness. This means that water vapour
estimates for climatology can be achieved in post-mission using the most accurate GPS orbit
and clock products. In this research, ground-based GPS for meteorological applications will
be investigated.
The fundamental of GPS atmospheric sensing is the tropospheric propagation delays. The
tropospheric delays, which are noise in geodetic research, have become signals in
meteorology (Bevis et al., 1992). The tropospheric delay is caused by the larger refractive
index n (n>1) of atmospheric gases than that of free space (n=1), which slows down the
speed of signal to below its speed in vacuum. The spatially varying refractive index also
curves the signal travelling path. The tropospheric delay can be expressed as (Spilker,
1996c):
( ) gpathactualpathgeopathactualtrop Ndsdsds)s(nd ∆+=−= −
−
−−
610 (2.1)
where g∆ is the difference between the curved and geometric straight paths, and N is the
refractivity and can be expressed as (Spilker, 1996c):
( ) 6101 ×−= nN (2.2)
The refractivity N is a function of pressure and temperature of the atmosphere as provided
by Thayer (1974):
123
12
11
−−− ++= vvvvdd Z)T/P(kZ)T/P(kZ)T/P(kN (2.3)
42
where 1k =(77.604±0.014) K/mbar, 2k =(64.79±0.08) K/mbar, 3k =(3.776±0.004)×105
K2/mbar, T is the atmosphere temperature (in degrees Kelvin), dP and vP are the partial
pressure of dry air and water vapour (in mbars). 1−dZ and 1−
vZ are the inverse compressibility
factors for dry air and water vapour. As investigated by Davis et al. (1985), the constants 1k ,
2k and 3k can be calculated to a relative accuracy of about 0.02%.
The total tropospheric delay tropd is normally decomposed into two components, which are
the “hydrostatic delay” and the “wet delay”. Hydrostatic delay depends only on the surface
pressure and contributes to about 90% of the total delay. Wet delay is a function of water
vapour distribution (Saastamoinen, 1972; Davis et al., 1985). Mapping functions can be used
to relate the slant hydrostatic delay at any elevation angle to a Zenith Hydrostatic Delay
(ZHD), and the slant wet delay to a Zenith Wet Delay (ZWD) (Niell, 1996). Zenith
Tropospheric Delay (ZTD) is the sum of ZHD and ZWD. Therefore, if ignoring the
horizontal gradients, which will be discussed in Section 2.4.4, the tropd and ZTD can be
expressed as:
wzhztz
wzwhzhwshstrop
DDd
D)e(mD)e(mDDd
+=
+=+= (2.4)
where hsD and wsD are the slant hydrostatic delay and wet delay, respectively, )e(mh and
)e(mw are the hydrostatic and wet mapping functions, respectively, e is the elevation angle,
tzD , hzD and wzD are the ZTD, ZHD and ZWD, respectively.
43
Several mapping functions have been developed with small difference at low elevation
angles (Davis et al., 1985; Niell, 1996). The ZHD, which is approximately 2.3 m at sea level,
can be calculated to better than 1 mm given the surface pressure measurement accurate to 0.3
mbar or better, using the following formula (Saastamoinen, 1972):
H.cos.P.
Dhz 0002802002660100227680 0
−−=
φ (2.5)
where 0P is the pressure in mbars; φ is the latitude and H is the height above the geoid (km).
The ZWD can be converted to Precipitable Water Vapour (PWV) overlying a GPS receiver,
using the following formula (Bevis et al., 1994):
ZWDPWV ×Π= (2.6)
where the ZWD is given in the unit of length, the dimensionless constant of proportionality
Π can be roughly given as 0.15 (Bevis et al., 1994).
Similarly, the Slant Wet Delay (SWD) can also be related to the integrated Slant Water
Vapour (SWV) as follows:
SWDSWV ×Π= (2.7)
The SWV can then be used for water vapour tomographic modeling (Skone and Shrestha,
2003).
44
The value of Π can vary as much as 20% different regions and seasons, but it is possible to
predict it with an RMS relative error of less than 2% given only surface temperature
observations at the site. The following expression can be used to calculate the value of Π
(Bevis et al., 1994):
( )[ ]123
610mkkT/kR mv −+
=Πρ
(2.8)
where ρ is the density of liquid water and vR is the specific gas constant for water vapour;
1k , 2k and 3k are the physical constants from the atmospheric refractivity equation (2.3); m
is the ratio of the molar masses of water vapour and dry air; and mT is the weighted mean
temperature of the atmosphere. mT is defined by Davis et al. (1985) as
( )( )dzT/P
dzT/PT
v
v
m
=
2 (2.9)
As investigated by Bevis et al. (1994), the relative error in Π is close to the relative error in
mT . Different regressions have been developed to relate the value of mT with the surface
temperature sT (Schuler et al., 2001; Mendes et al., 2000b). The relationship between mT and
sT varies with location, altitude, weather and season. In this research, mT is calculated from
surface temperature using the linear relation given by Bevis et al. (1994) as follows:
sm T..T 720270 += (2.10)
45
Both mT and sT are given in degrees Kelvin. This regression was determined by investigating
data in a 2-year period from 13 stations in the United States. These stations spread from
Fairbanks, Alaska, to West Palm Beach, Florida. It was considered suitable for the stations in
Calgary and with a similar accuracy (Gao et al., 2004). The linear relation was demonstrated
with an RMS relative error of less than 2% for the investigated stations (Bevis et al., 1994).
As shown in Equation 2.4, after the removal of the Zenith Hydrostatic Delay (ZHD), which
can be determined with an accuracy of better than 1 mm with precise surface pressure, the
Zenith Wet Delay (ZWD) is obtained from the GPS derived Zenith Tropospheric Delay
(ZTD). Since ZWD is nearly proportional to the quantity of PWV integrated along the zenith
direction, the total PWV can be extracted from ZWD to a relative accuracy of a few percent,
given only surface temperature measurements at the site. Because the high accuracy and
timeliness requirement of water vapour measurements for numerical weather prediction
(Gutman and Benjamin, 2001), the key issue for GPS meteorology is how to obtain high-
precision ZTD estimates from GPS observations with latency as short as possible. Due to the
time spent in collecting and compiling GPS raw data, the latency of the precise GPS products
and the complexity of the data processing, the estimation so far has been conducted only in a
post-mission or near real-time processing mode. In the following, strategies for ZTD
estimation will be discussed.
46
2.4.3 Strategies for ZTD Estimation Using GPS Observations
An important factor that allows precise ZTD estimation is the availability of GPS satellite
orbits with sufficient accuracy (Gutman and Benjamin, 2001). Before the advent of precise
orbit products available from organizations such International GPS Service (IGS), ZTD had
to be estimated along with satellite orbits and other station parameters. The requirement of
estimating the satellite orbits makes the approach too complex to be feasible for real-time or
near real-time ZTD estimation (Collins et al., 2002). This method could only be used in post-
mission for meteorological research or climatology. For example, currently the IGS uses this
method to estimate ZTD for IGS stations (Heroux, 2003). But as proposed by Bar-Server
(2004), this method will be replaced by PPP methodology for IGS tropospheric products.
As more and more precise GPS orbit products have become available, the processing
procedure can be greatly simplified. In particular, the availability of ultra-rapid products
(predicted half) from IGS has pushed the investigation of real-time GPS meteorology (Kruse
et al., 1999; Fang et al., 2001; Douša, 2001). Because the satellite clock errors in IGS ultra-
rapid products are still considerable (5 ns in the predicted half), normally the double
differencing approach with integer ambiguity resolution capability has to be used to remove
the satellite clock errors in order to derive more precise tropospheric delay estimates (Rocken
et al., 1997), which however can only provide the relative estimates of tropospheric delays or
Slant Water Vapours (SWVs) for short baselines (Bevis et al, 1992). In order to obtain
absolute tropospheric delays or SWVs, either 1) long baselines must be included to reduce
the correlation of the tropospheric delays between stations (Duan et al., 1996; Alber et al.,
47
2000), or 2) absolute SWVs from a Water Vapour Radiometer (WVR) collocated at one GPS
station should be available to “lever” the relative SWVs to absolute ones (Rocken et al.,
1995). The latter is known as “WVR-levering”. Though both methods can provide PWV
estimates at an accuracy of 1~2 mm (Tregoning et al., 1998), they have several shortcomings
that will be addressed below, besides the time spent in collecting and compiling GPS raw
data from a regional network.
As to the first method, Rocken et al. (1993) have simulated that the absolute PWV cannot be
recovered to better than 1.5 mm even using perfect orbits and dry delays for baselines up to
1000 km. Tregoning et al. (1998) also demonstrated that PWV solutions, which did not
include baselines longer than 2000 km, had a larger bias and a higher scatter with respect to
the radiosonde measurements. To process data from a network including extremely long
baselines, special efforts also have to be taken to reduce the satellite orbit error effects for
near real-time PWV estimation (Ge et al., 2000). As a result, the complexity and the required
processing time are significantly increased. Since the valid time period of PWV estimates is
relatively short for weather forecasts, a forecaster for instance may look for updated
estimates every few minutes during active weather conditions (Gutman and Benjamin, 2001).
Efficient communication and processing techniques to minimize the latency thus have
become a major concern for meteorological applications using this method (Collins et al.,
2002). A discussion on the required accuracy and timeliness of GPS meteorological data can
be found in Gutman and Benjamin (2001).
48
As to the second method, the “levering” process requires the use of other devices such as
radiometers. Since radiometers are not “all-weather” instruments, they may not generate
useful data under heavy rain conditions and their accuracy may also be degraded when it is
heavily cloudy or lightly raining (Bevis et al., 1992). This will break the all-weather property
of GPS meteorology.
As already discussed, the availability of precise GPS satellite orbit and clock products has
enabled the development of a novel positioning methodology known as precise point
positioning (Zumberge et al., 1997). Two advantages of PPP are that it is computationally
efficient and some absolute information is retained, such as receiver clock offset and zenith
tropospheric delay. PPP provides a new way to perform real-time absolute ZTD estimation
using real-time precise GPS orbit and clock products. PPP does not require spending time in
collecting and compiling GPS raw data from a regional or global GPS network. GPS raw
data can be processed in parallel in local computers. Only ZTD estimates are transferred to a
processing center. Differing from traditional double-difference approach, the timeliness and
temporal resolution of PPP ZTD estimates are only limited by the data rate of GPS receivers,
so it can support real-time GPS meteorology applications.
Haase et al. (2003) have applied PPP approach in post-mission to estimate meteorological
parameters using precise orbits and clocks with a latency of several hours or days. As
investigated by researchers (Rocken et al., 2003; Wang and Dare, 2004), comparable
accuracy of zenith tropospheric delay estimates has been obtained using both PPP processing
and double-difference processing in post-mission. PPP method was also found in some near
49
real-time applications to save processing time (Reigber et al., 2002; Gendt et al., 2003). In
these applications, high-quality GPS orbits and clocks were first estimated from a global
network and ZTDs were then estimated using parallel PPP processing based on the precise
orbits and clocks from the first step. This idea is the same as what Zumberge et al. (1995;
1997) applied to save processing time. As real-time precise orbit and clock products have
become available from organizations, such as JPL, with sub-decimetre accuracy for orbits
and sub-nanosecond accuracy for clocks (Muellerschoen et al., 2000), the process can be
further simplified. Given the real-time precise orbit and clock products, precise real-time
zenith tropospheric delay estimation becomes feasible using PPP methodology. Its high
timeliness and temporal resolution are of great importance to severe weather forecasting
(Chen, 2004; Gao et al., 2004).
2.4.4 ZTD Estimation Using PPP Methodology
The observation models used in PPP processing will be described in Chapter 3. Several error
sources, which can be completely eliminated or partially mitigated in double difference
method, must be modeled in un-differenced GPS data processing. These error sources will
also be discussed in detail in Chapter 3. Several issues have to be considered to get precise
ZTD estimates using PPP methodology, including tropospheric horizontal gradients,
mapping function selection, antenna phase center variation, elevation cut-off angle, etc.
The estimation of troposphere horizontal gradients was shown to be beneficial for both GPS
positioning and meteorology in Bar-Sever et al. (1998). Bar-Sever (2004) even proposed to
50
adopt the PPP method to estimate ZTD along with horizontal gradients for IGS tropospheric
combinations. The following equation has been used to model tropospheric effects in this
research (McCarthy and Petit, 2004):
)]asin(G)acos(G)[e(m
D)e(mD)e(md
ENg
wzwhzhtrop
+
++= (2.11)
where
hzD , wzD , )e(mh , )e(mw are the same variables as in Equation 2.4;
NG , EG are the tropospheric horizontal gradients in north and east directions;
)e(mg is the gradient mapping function and
a , e are the azimuth and elevation angles.
The Saastamoinen model in Equation 2.5 has been applied to determine the zenith
hydrostatic delay.
The following gradient mapping function has been used (Chen and Herring, 1997):
003201
.)etan()esin()e(mg +
= (2.12)
The mapping functions, elevation cut-off angle and antenna phase center variation are also
important issues in tropospheric delay estimation (Fang et al., 1998). Niell Mapping
Functions (NMFs) are used for hydrostatic and wet mapping functions in this research (Niell,
1996), which are among the most accurate mapping functions when observations at
51
elevations angles below 15° are to be included (Mendes and Langley, 2000a). Fang et al.
(1998) have demonstrated that systematic errors have been reduced in ZTD estimation when
using NMFs. Since NMFs are independent of surface meteorology, the users can benefit
when meteorological data is unavailable or unreliable during processing (Niell, 1996). They
are also useful in tropospheric gradient estimation, in which GPS data at low elevation angles
would be included to get precise ZWD estimates. NMFs are valid at elevations as low as 3°
(Niell, 1996).
In some researches, ZWD and ZHD were estimated as a total ZTD using a single mapping
function, which can be either the hydrostatic mapping function or wet mapping function
(Duan et al., 1996). However, because of the difference in mapping functions, especially at
low elevations (Niell, 1996), estimating total delay using a single mapping function will lead
to a bias in the zenith tropospheric delay estimates, which will directly affect the accuracy of
zenith wet delay estimates. In this research, precise pressure measurements (if available) or
pressure models would be used to model zenith hydrostatic delays which are thus mapped
using the hydrostatic mapping function in PPP processing. Therefore, ZWD, possibly
including un-modeled ZHD, is estimated with wet mapping function. The modeled ZHD and
the estimated ZWD will be output as a total ZTD, which will be subtracted by the ZHD
calculated from precise pressure measurements and result in the true ZWD. The details about
why two mapping functions should be used instead of using a single mapping function will
be discussed in Section 6.4.
52
Bar-Sever and Kroger (1996) found that an elevation cut-off angle of 15° led to a significant
bias between GPS-based estimates of ZWD and the WVR-based estimates. The bias was
reduced dramatically when a cut-off angle of 7° was used. GPS data at low elevation angles
should also be included to separate gradient components from the azimuthally homogeneous
components (Bar-Sever et al., 1998). As investigated by Rothacher et al. (1997), including
low elevation data can increase the number of observations and obtain a better decorrelation
between the estimated height and tropospheric delay parameters. However, the measurement
noise and multipath effects increase at lower elevation angles. In this research, an elevation
cut-off angle of 7° is therefore used.
Both satellite and receiver antenna phase center variations were found to affect tropospheric
delay estimation (Fang et al. 1998; Schmid and Rothacher, 2002). In this research, the IGS
elevation-dependent phase center models have been used.
53
CHAPTER 3
ERROR MITIGATION AND MODELING IN
PRECISE POINT POSITIONING
This chapter describes the error mitigation and modeling for precise point positioning using
dual-frequency GPS data. The error sources, which are only relevant to precise point
positioning using single-frequency GPS data, will be investigated later on in Chapter 5. Dual-
frequency GPS measurements will be first described. General error sources, either modeled
or estimated to support precise point positioning, will be investigated. Function models and
stochastic models for precise point positioning will also be presented.
3.1 Dual-Frequency GPS Observables
There are two advantages to use dual-frequency GPS measurements for precise point
positioning. The first advantage is that the first-order ionospheric effects can be removed
completely. The second is that most GPS orbit and clock products, including those in the
broadcast and precise ephemeris, are consistent with the dual-frequency ionosphere-free
combinations. In this section, dual-frequency GPS measurements and ionosphere-free
combinations will be discussed.
54
3.1.1 Dual-Frequency Code and Carrier Phase Measurements
Currently, for positioning purpose, GPS satellites transmit signals on two frequencies,
namely, Link 1 or 1L , and Link 2 or 2L , which are centered at 1575.42 MHz and 1227.6
MHz, respectively (Spilker, 1996a). The 1L and 2L signals are modulated by pseudorandom
noise (PRN) codes so that GPS receivers can track the signals for positioning. 1L is
modulated by both Coarse/Acquisition code (C/A code) and Precision code (P code). 2L is
only modulated by P code. When Anti-Spoofing (AS) is activated, a secure Y code is used
instead of the P code to modulate 1L and 2L signals. The chip rate of Y code is the same as
that of P code, but the Y code is only provided to authorized U.S. government users (Spilker,
1996a). The AS mode was activated on January 31, 1994 and is still operational on Block II
satellites (Langley, 1998a).
For civil GPS receivers, several techniques have been developed to track the encrypted P
code (Y code) (Langley, 1998b). In the civil GPS community, dual-frequency GPS receivers
are usually categorized into two groups based on the tracking techniques used. The first
group are the cross-correlation receivers such as AOA Rogue/TurboRogue and Trimble 4000
serial receivers. The second group are the non-cross-correlation receivers such as Ashtech Z-
XII and AOA Benchmark/ACT receivers (Ray, 1999b). The measurements provided by these
two types of receivers are different and are described in the following.
The cross-correlation style receivers can output at least four types of code and carrier phase
measurements, which are described in Table 3.1.
55
Table 3.1 Measurements from Cross-Correlation Receivers
Observables Description
1C C/A code at 1L frequency 'P2 Codeless pseudorange at 2L frequency
1L ( 1C ) C/A-based phase at 1L frequency 'L2 cross-correlated phase at 2L frequency
The reason that they are called cross-correlation receivers is that the 'P2 and 'L2
measurements are formed by cross-correlation technique which measures the cross-correlated
( 2P - 1P ) pseudorange and the ( 2L - 1L ) phase differences. Hence, 'P2 and 'L2 can be
expressed as (Ray, 1999b):
'P2 = 1C + ( 2P - 1P ) (3.1)
'L2 = 1L ( 1C ) + ( 2L ( 2P )- 1L ( 1P )) (3.2)
Therefore, the observables ( 1C and 'P2 or 1L ( 1C ) and 'L2 ) are no longer independent of each
other, which will affect the stochastic modelling when using the P1-P2-CP ionosphere-free
combinations to be discussed in Section 3.1.2.
The non-cross-correlation receivers can provide direct measurements of 1P and 2P without
the use of the Y-code by using methods such as squaring technique (Hofmann-Wellenhof et
al., 2000). They can provide six (or more) range measurements as shown in Table 3.2.
56
Table 3.2 Measurements from Non-Cross-Correlation Receivers
Observables Description
1C C/A code at 1L frequency
1P Y1-codeless pseudorange at 1L frequency
2P Y2-codeless pseudorange at 2L frequency
1L ( 1C ) C/A-based phase at 1L frequency
1L Y1-codeless phase at 1L frequency
2L Y2-codeless phase at 2L frequency
The C/A based measurements 1C and 1L ( 1C ) are the same as the corresponding
measurements from the cross-correlation receivers, but the measurements on 2L are different
between those two types of receivers. Moreover, the non-cross-correlation receivers can
provide two carrier phase ( 1L ( 1C ) and 1L ) and two code ( 1C and 1P ) measurements on 1L .
Normally, 1L ( 1C ) with a higher signal-to-noise ratio (SNR) is reported in RINEX files (Ray,
1999b). The differences between 1C and 1P are not zero-means and are called ( 1P - 1C )
biases. ( 1P - 1C ) biases are relatively stable in time but their magnitude can be up to ~ 2 ns
(Jefferson et al., 2001, Gao et al., 2001). CODE has been estimating ( 1P - 1C ) biases since
GPS week 1057 (Schaer, 2000).
There are four types of pseudorange measurements that could be reported in a RINEX file.
Which pair of measurements ( 1C or 1P , 'P2 or 2P ) should be used to form the ionosphere-
free combinations in PPP data processing depends on the clock products. Before GPS week
1056 (April 02, 2000), IGS precise clock products are consistent with the 1C / 'P2 code
57
measurements. Starting from GPS week 1056 (April 2nd, 2000), precise clocks of IGS
products is fully consistent with the 1P / 2P code measurements (Kouba, 2003). The clock
errors in the broadcast ephemeris are also consistent with 1P / 2P code measurements (ICD-
GPS-200C, 2000). The same convention ( 1P / 2P ) is also used by JPL and NRCan to estimate
their real-time orbit and clock products (Heroux, 2003; Muellerschoen, 2003). Therefore,
users who choose 1C and/or 'P2 pseudorange measurements should apply the ( 1P - 1C ) code
biases when they use clock information which is consistent with 1P / 2P .
According to specifications, the phase bias ( 1L ( 1C )- 1L ( 1P )) is a constant (Ray, 1999b).
Therefore, difference between 'L2 and 2L is also a constant. As phase observables are
inherently ambiguous, the difference between 'L2 and 2L would be absorbed into the float
ambiguity in precise point positioning.
Some non-cross-correlation receivers, such as NovAtel OEM4 serials, TRIMBLE 5700,
LEICA CRS1000, etc, only report the higher SNR C/A based measurements. In the RINEX
files, these receivers normally report four types of code and phase measurements, 1C , 2P ,
1L ( 1C ) and 2L . Although they are non-cross-correlation receivers, ( 1P - 1C ) code biases
should be applied to their 1C measurements before they are used to form ionosphere-free
combinations.
As discussed above, there are six (or more) types of measurements which may be reported by
dual-frequency GPS receivers in RINEX files. Based on the current conventions, in which
58
clock information is consistent with 1P / 2P measurements, the following observation equation
models can be formed:
)P(dm)DCBT(c
dddd)dtdt(cP
PP/Pgd
relL/iontroporbs
rsr
1121
11
ερ
++−+
++++−+= (3.3)
)C(dm)DCBDCBT(c
dddd)dtdt(cC
CC/PP/Pgd
relL/iontroporbs
rsr
111121
11
ερ
++−−+
++++−+= (3.4)
)P(dm)DCBT(c
dddd)dtdt(cP
PP/Pgd
relL/iontroporbs
rsr
2221
22
εγρ
++−+
++++−+= (3.5)
)P(dmDCBc)DCBT(c
dddd)dtdt(c
)PP(CP
'PC/PP/Pgd
relL/iontroporbs
rsr
'
' 21121
2
1212
2εγ
ρ
++⋅−−+
++++−+=
−+=
(3.6)
)( mw)N)t()t((
dddd)dtdt(c
iiiis
ir
ii
relLi/iontroporbs
rsri
Φ++++−+
+−++−+=Φ
εδδφφλ
ρ
00
(3.7)
where,
1P is the P-Code pseudorange measurement on 1L (m);
1C is the C/A-Code pseudorange measurement (m);
2P is the P-Code pseudorange measurement on 2L reported by non-cross-correlation
receivers (m);
59
'P2 is the P-Code pseudorange measurement on 2L reported by cross-correlation
receivers (m);
iΦ is the carrier phase measurement on iL (m);
srρ is the true geometric range (m);
c is the speed of light (m/s);
sdt is the satellite clock error (s);
rdt is the receiver clock error (s);
orbd is the satellite orbit error (m);
tropd is the tropospheric delay (m);
Li/iond is the ionospheric group delay on iL (m);
gdT is the group delay differential of satellite (s);
21 P/PDCB is the differential code bias between 1P and 2P of receiver (s);
11 C/PDCB is the differential code bias between 1P and 1C (s);
reld is the relativistic effects (m);
iwδ is the phase windup on iL (m);
iλ is the wavelength on iL (m/cycle);
iN is the integer phase ambiguity on iL (cycle);
)t(ri 0φ is the initial phase offset of the receiver on iL (cycle);
)t(si 0φ is the initial phase offset of the satellite on iL (cycle);
60
Pidm is the P-code multipath on iL (m);
1Cdm is the C/A code multipath (m);
imδ is the carrier phase multipath on iL (m);
)(∗ε is the measurement noise (m);
and
22
2
1
6077
=
=
ffγ (3.8)
To calculate the geometric range srρ , the satellite and receiver positions should refer to the
same reference frame. The satellite and receiver positions defined in the Earth-fixed frame
can be expressed by (Rothacher and Beutler, 2002b):
)t(r)t(r)t(r ssant
ss,e
sse δ+= 0 (3.9)
ant,ratm,rocn,rpol,rsol,rr,e,rre,r rrrrr)t(r)t(r δδδδδ +++++= 0 (3.10)
where
)t(r ss,e 0 is phase center of satellite antenna at emission time st ;
)t(r ssantδ is antenna phase center offset and variations of satellite antenna;
)t(r r,e,r 0 is tide-free position of receiver at reception time rt ;
sol,rrδ is solid earth tides;
61
pol,rrδ is pole tide;
ocn,rrδ is ocean loading;
atm,rrδ is atmospheric loading and
ant,rrδ is the antenna phase center offset and variations of receiver antenna.
To fully exploit the potential of precise point positioning methodology, the error sources and
tide effects shown in Equations 3.3 to 3.10 should be considered and they will be discussed
in Section 3.2.
3.1.2 Ionosphere-free Combinations
After precise GPS orbit and clock products are applied, the biggest error source remaining in
un-differenced GPS measurements is from the ionosphere. Although the range errors due to
the troposphere may be as big as the ionosphere effects, they are more stable spatially and
temporarily. Dual-frequency users can remove the first-order ionospheric effects by forming
ionosphere-free combinations. If ignoring higher-order ionospheric effects, which will be
investigated in Section 3.2, the first-order ionospheric group delay or carrier phase advance
can be expressed in units of distance as follows (Klobuchar, 1996):
= Ndlf
.d
iLi/ion 2
340 (3.11)
62
where Ndl is the Total Electron Content (TEC), in unit of el/m2, integrated along the path
from an observer to a GPS satellite, and N is electron density.
In precise point positioning, two types of ionosphere-free combinations can be formed. In the
following, only measurements from non-cross-correlation receivers are discussed unless
otherwise specified. Measurements from cross-correlation receivers, after applying the
differential code biases, should be equivalent to the corresponding measurements from non-
cross-correlation receivers except for the correlation between measurements. The first type of
ionosphere-free combination, as presented in Equations 3.12 to 3.14, is the traditional one
based on dual-frequency measurements using the frequency dependent characteristic of
ionospheric refraction.
)P(dmddd)dtdt(c
ffPfPf
P
IFIFreltroporbs
rsr
IF
ερ +++++−+=
−⋅−⋅= 2
22
1
22
212
1
(3.12)
)( mwFddd)dtdt(c
ffff
IFIFIFIFreltroporbs
rsr
IF
Φ+++++++−+=
−Φ⋅−Φ⋅
=Φ
εδδρ
22
21
22
212
1
(3.13)
where
22
21
202022101011
ff)N)t()t((cf)N)t()t((cf
Fsrsr
IF −+−−+−
=φφφφ
(3.14)
63
The float ambiguity IFF in this ionosphere-free combination includes the initial phase offsets
of the receiver and satellite on both 1L and 2L . If we assume that the noises in 1P and 2P are
uncorrelated, the noise level of the code combination can be determined by applying random
error propagation law as follows:
( ) ( ) ( ) ( )1
2
2
2
1 31
11
PPPPIF εεγ
εγ
γε ≈
−+
−= (3.15)
The noise of the phase combination can be determined in a similar way.
The first-order ionospheric effects on code and carrier phase are equal in magnitude with
opposite sign. Therefore, another type of ionosphere-free combination can be formed by
averaging code and carrier phase measurements on the same frequency. This combination
has been first introduced to mitigate ionospheric effects for processing single-frequency
measurements, known as GRAPHIC (Group And Phase Ionospheric Correction) (Yunck,
1996a). It has been introduced into precise point positioning using dual-frequency
measurements by Gao and Shen (2002) as:
222
22
2
11111
10101121
111
)()P(
mdmw
)N)t()t(()DCBT(c
ddd)dtdt(c
PCP
P
srP/Pgd
reltroporbs
rsr
Φ+++++
+−+−
+
+++−+=
Φ+=
εεδδ
φφλ
ρ (3.16)
64
222
22
2
22222
20202221
222
)()P(
mdmw
)N)t()t(()DCBT(c
ddd)dtdt(c
PCP
P
srP/Pgd
reltroporbs
rsr
Φ+++++
+−+−
+
+++−+=
Φ+=
εεδδ
φφλγρ
(3.17)
Plus the traditional ionosphere-free carrier phase combination given in Equation 3.14, there
will be three ionosphere-free observables for each satellite in each epoch. The above P1-P2-
CP (Gao and Shen, 2002) code-phase combinations will reduce the noise level by half
compared to the original code observations or to about one sixth to the traditional
ionosphere-free code combination, since the phase measurement noise is about two orders
smaller than that of code measurements. Precise point positioning using P1-P2-CP code-
phase combination allows estimating the ambiguities associated with 1L and 2L separately.
However, though the combinations keep the integer nature of the ambiguities, the presence of
hardware delays and initial phase offsets only allow float ambiguities to be estimated as:
2210101121
1
)N)t()t(()DCBT(cF
srP/Pgd
CP
+−+
−=
φφλ (3.18)
2220202221
2
)N)t()t(()DCBT(cF
srP/Pgd
CP
+−+
−=
φφλγ (3.19)
The relationship between the float ambiguities 1CPF , 2CPF and IFF can be expressed as:
22
21
22
212
12ff
)FfFf(F CPCP
IF −⋅−⋅
= (3.20)
65
If hardware delays and initial phase offsets can be resolved, integer ambiguities are
achievable using the code-phase combinations. Progress has been made by researchers to
estimate the initial phase offsets (Gabor and Neren, 2002). gdT has also been provided in
navigation messages with an accuracy of nanosecond level (ICD-GPS-200C, 2000). On the
other hand, Gao and Shen (2002) have proposed a method known as pseudo-fixing to fix the
float ambiguities (including hardware delays and initial phase offsets) to float values. The
method has been demonstrated to reduce the ambiguity convergence time. Fixing or pseudo-
fixing the ambiguities in PPP is beyond the scope of this thesis. In this thesis, all processing
using dual-frequency measurements is based on the traditional ionosphere-free combinations.
3.2 Error Mitigation for Dual-Frequency Measurements
Precise point positioning, which is aimed at providing centimetre level positioning accuracy,
is subject to the influence of all possible error sources in the un-differenced GPS
observations. Most of those error sources can be eliminated completely or mitigated to a
negligible level through single or double differencing. Some of these error sources can also
be safely neglected in traditional point positioning using only code measurements for less
accurate positioning applications. In the following, major error sources that should be taken
into account in PPP are described and their characteristics and methods to mitigate them are
investigated.
66
3.2.1 Satellite Orbit and Clock Errors
In double differencing techniques, satellite clock errors can be removed completely. Satellite
orbit errors are also reduced by differencing between receivers. In precise point positioning,
precise orbit and clock products are used, which can reduce satellite orbit and clock errors to
5 cm and 0.1 ns in post-mission or sub-decimetre and sub-nanosecond level in real-time.
Precise orbit and clock products used in this research will be discussed in Chapter 4.
3.2.2 Ionospheric Effects
If precise orbit and clock products are used, the ionospheric effects become the biggest error
source in un-differenced measurements. A comparable magnitude error source, tropospheric
effects, is much easier to model, which will be discussed in the next section.
Ionosphere is a dispersive medium for frequencies greater than 100 MHz. The ionospheric
refraction can be approximated by a series expansion in the reciprocal of frequency 1−f .
Considering that GPS signals are right-hand circularly polarized and omitting the higher
orders, the refractive index of the code (group) and phase can be approximated as follows
(Bassiri and Hajj, 1993):
( )
+++++= BBgr cosYXXcosXYXn θθ 22 121
43
21
1 (3.21)
67
( )
++−−−= BBph cosYXXcosXYXn θθ 22 121
41
21
21
1 (3.22)
where Bθ is the angle of ray with respect to the Earth’s magnetic field, ( )2f/fX p= and
f/fY g= . pf , gf and f are the plasma, gyro and carrier phase frequencies, respectively.
The gyro frequency gf is typically 1.5 MHz, and the plasma frequency pf is rarely exceeds
20 MHz (Klobuchar, 1996). The measured range can then be expressed as:
= ndsmρ (3.23)
The geometric range between the satellite and receiver srρ can be obtained by setting the
refractive index n = 1. Here, by omitting other error sources, the GPS observables discussed
in Section 3.1 can be rewritten as follows (Bassiri and Hajj, 1993):
41
31
21
1 fr
fs
fq
P sr +++= ρ (3.24)
42
32
22
2 fr
fs
fq
P sr +++= ρ (3.25)
41
31
21
111 32 fr
fs
fq
Nsr −−−+=Φ λρ (3.26)
42
32
22
222 32 fr
fs
fq
Nsr −−−+=Φ λρ (3.27)
68
where
TEC.dLfq p ×== 34021 2 (3.28)
dLcosNBcdLcosffs BBpg θθ == 02 7527 (3.29)
( )dLcosNB.NdLfr Bg θ20
22 1107442437 +×+= (3.30)
where N is the density of electrons, TEC is the total electron content along the GPS signal
propagation path and 0B is the magnitude of the Earth’s magnetic field. q , s and r
represent the first-, second- and third-orders ionospheric effects, respectively.
As discussed in Section 3.1, for dual-frequency users, the following ionosphere-free linear
combinations can be formed to remove the first-order ionospheric effects, and mitigate part
of the second- and third-orders ionospheric effects as follows (Bassiri and Hajj, 1993):
22
212121
222
21
22
122
21
21
ffr
)ff(ffs
Pff
fP
fff
P
sr
IF
−+
−=
−−
−=
ρ (3.31)
22
212121
22
21
2211
222
21
22
122
21
21
32 ffr
)ff(ffs
ffNcfNcf
fff
fff
sr
IF
++
+−−
+=
Φ−
−Φ−
=Φ
ρ (3.32)
69
As investigated by Bassiri and Hajj (1993), the second-order ionospheric group delay can be
calculated by the following equation:
TECEsincosAcosEcossinHR
R.
fs
m'mmm
'm
E
Ei
i
×−×
+×= − θθλ 210612
3
183 (3.33)
where ER is the average Earth radius, H is the ionospheric shell height and was set to 300
km by Bassiri and Hajj (1993), 'mθ is the magnetic colatitude of the sub-ionospheric point,
mA and mE are the azimuth and elevation of satellite in a local magnetic East-North-Vertical
coordinates. mE is the same as the elevation angle in local geodetic coordinates E . The
relationship between mA and the azimuth in local geodetic coordinates A has been given by
Bassiri and Hajj (1993).
The magnitude of the second-order ionospheric group delay, as calculated from Equation
3.33, is of the order of 0.16 and 0.33 mm per TEC Unit (TECU) for 1P and 2P respectively
(Bassiri and Hajj, 1993). The residual range error caused by the second-order ionospheric
group delay in the ionosphere-free code combination is then -0.11 mm per TECU. From
Equation 3.24 to 3.27, we can see that the second-order ionospheric phase advances in 1Φ
and 2Φ are half of the group delays in 1P and 2P in magnitude. The residual range error in
the ionosphere-free phase combination is also half of that in the code combination. The
maximum vertical ionospheric electron content is typically about 100 TECU at a solar
maximum near the geomagnetic equator (Langley, 1997). Therefore, the residual range error
in ionosphere-free code combination can be up to –1.1 cm for a satellite at the zenith. The
70
value can be about 3 times bigger for a satellite at 5° elevation angle. The precision of PPP
solutions is dominated by the un-modeled errors in the phase observations after the
ambiguity convergence. In the mid-latitude regions during a solar maximum, a typical
ionospheric content is about 40 TECU for a daytime average (Langley, 1997), which
corresponds a 2.2 mm residual range error caused by the second-order ionospheric phase
advances in the phase combination for a satellite at the zenith. Therefore, for the sub-
decimetre level position determination, the second-order ionosphere residuals are not critical,
but they will definitely affect the sub-centimetre level zenith tropospheric delay estimation,
which will be discussed in Chapter 6.
The third-order ionospheric group delay is about one order less than that of the second-order
ionospheric group delay. As investigated by Bassiri and Hajj (1993), for TEC=100 TECU,
the third-order ionospheric group delay is about 0.86 mm and 2.4 mm for 1P and 2P
respectively. The residual range error of the third-order ionospheric group delay in the
ionosphere-free code combination is –0.66 mm. The third-order ionospheric phase advances
in 1Φ and 2Φ are one third of the group delays in 1P and 2P . The residual range error in the
ionosphere-free phase combination is also one third of that in the code combination.
Therefore, the third-order ionospheric effects can be safely neglected in precise point
positioning, time transfer and meteorological applications.
It is obvious that the residual range errors of the higher-order ionospheric effects in the P1-
P2-CP ionosphere-free combinations are even smaller than those in the traditional
combinations. With GPS modernization, the measurements on three or more frequencies can
71
be used to mitigate the ionospheric effects more effectively than the dual-frequency
combinations. For example, triple-frequency combinations can eliminate the ionospheric
effects up to the second-order (Xu, 2003).
For precise point positioning using single-frequency observations, the broadcast Klobuchar
model or the more accurate Global Ionospheric Model (GIM) can be used. The broadcast
Klobuchar model can mitigate about 50% of the ionospheric effects using coefficients in
navigation messages (Klobuchar, 1996). GIM has the potential to support sub-metre level
positioning using post-mission ionospheric products. Ionospheric effects mitigation for
precise point positioning using single-frequency GPS data will be investigated in Chapter 5.
3.2.3 Tropospheric Delay
The troposphere is the lower part of the Earth’s atmosphere. It extends to a height of less than
9 km over the poles and more than 16 km over the equator (Langley, 1998a). In the
troposphere, the refractive index n (n>1) is larger than that in a free space (n=1), which
causes the speed of radio signals to decrease below its speed in vacuum c. The spatially
varying refractive index also causes the signal path to have a slight curvature with respect to
the geometric straight path. Both of them constitute the tropospheric delay (Bevis et al.,
1992). The tropospheric delays in GPS measurements may be even bigger than the
ionospheric effects during ionospheric quiet periods, especially for satellites at low
elevations. But the tropospheric delays are much more stable and easier to model than the
ionospheric effects (Klobuchar, 1996).
72
Usually, the total troposphere delay tropd is decomposed into two components, which are the
“hydrostatic delay” and the “wet delay”. The hydrostatic delay depends only on the surface
pressure and contributes to about 90% of the total delay. The wet delay is a function of the
water vapour distribution (Saastamoinen, 1972; Davis et al., 1985), which makes GPS a
promising tool for water vapour sensing. Mapping functions can be used to relate the slant
hydrostatic delay at any elevation to a Zenith Hydrostatic Delay (ZHD), and the slant wet
delay to a Zenith Wet Delay (ZWD). The ZHD, which is approximate 2.3 m at sea level, can
be calculated to better than 1 mm given the surface pressure measurement accurate to 0.3
mbar or better (Saastamoinen, 1972).
However, water vapour is a highly variable component in atmosphere. Because of the spatial
and temporal variability of water vapour, the ZWD can hardly be modeled to better than 1~2
cm even using precise meteorological measurements (Bevis et al., 1992). As discussed in
Section 2.4, ZWD can be estimated to centimetre level accuracy for meteorological
applications. Therefore, in this research, ZHD will be modeled while ZWD will be estimated
when processing dual-frequency GPS measurements. Issues related to ZWD estimation using
un-differenced GPS data have been discussed in Section 2.4.
3.2.4 Relativistic Effects
Relativity affects GPS positioning in several ways. Relativistic effects are relevant for
satellite orbit, satellite signal propagation and satellite and receiver clocks (Hofmann-
73
Wellenhof et al., 2000). For point positioning users, regardless of whether broadcast or
precise ephemeris data is used, the following relativistic effects should be taken into account.
Relativistic Effects on Satellite Clock
GPS satellite clocks are affected by both special (relative velocity of the satellite) and general
(gravity field of the earth) relativistic effects. Therefore, the fundamental frequency of the
satellite clocks, which was selected as 10.23 MHz, has been slightly tuned by a constant
offset of –0.00457 Hz before satellite launch, based on the assumption of a circular orbit
(Ashby and Spilker, 1996).
Because of the eccentricity of the GPS orbits, satellite clocks show periodic variations. The
GPS community has already developed two conventions to calculate the periodic special
relativity variations (Ashby and Spilker, 1996):
cVX
Esineac
ssclk,rel
22 == µδρ (3.34)
where µ is the earth’s gravitational constant, a is the semi-major axis, e is the eccentricity,
E is the eccentric anomaly, sX and sV are the satellite position and velocity vectors, and c
is the speed of light. Users may select the first convention when using broadcast ephemeris.
On the other hand, the second one is often used for SP3 format ephemeris.
74
Relativistic Effects on Signal Propagation
In addition to these special relativistic effects on the clock, there is still a small delay of the
GPS signal as it passes through the Earth’s gravity field. The corresponding correction is
given by (McCarthy and Petit, 2004):
−+++
=srsr
srsr
gra,rel RRRR
lnc ρ
ρµδρ2
2 (3.35)
where rR and sR are the geocentric distances of receiver and satellite. This term ranges
between about 12.7 mm and 18.7 mm. The effects can be mostly removed in differential GPS
data processing. But the effects should be taken into account in point positioning using un-
differenced data.
Sagnac Effects
Sagnac effects are caused by the Earth’s rotation during the time of the satellite signal
transmitting to the receivers. It can be expressed in the following form (Ashby and Spilker,
1996).
22 rse
sag,rel
rrc
×⋅
Ω+=δρ (3.36)
where eΩ is the Earth angular rotation rate (WGS-84), 7.2921151467×10-5 rad/s, sr and rr
are the position vectors of satellite and receiver at the instant a signal is transmitted,
respectively, and × denotes a vector cross product.
75
This term can be tens of metres in value and should be taken into account all the time,
regardless of whether the broadcast or precise ephemeris data is used. Equation 3.36 is
widely applied when using SP3 format ephemeris. When broadcast ephemeris are used to
calculate the satellite coordinates, the Sagnac effects can be simply removed during the
calculation of the corrected longitude of ascending node of satellite as follows:
)tt()tt)(( eeee
.
∆+Ω−−Ω−Ω+Ω=Ω 0 (3.37)
where et is the reference time for ephemeris, 0Ω is the longitude of ascending node of
satellite at et , .
Ω is the rate of right ascension, t∆ is the signal propagation time.
3.2.5 Phase Wind-up
GPS satellites transmit right circularly polarized (RCP) radio waves. The electromagnetic
wave from the GPS satellite can be described as a rotating electric field propagating from the
satellite antenna to the receiver antenna. If both antennas keep static, the measured carrier-
phase would be the geometric angle between the instantaneous electric field at the receiver
antenna and a reference direction on the antenna. If the orientation of the receiver antenna or
satellite antenna changes, the reference direction or the direction of the electric field will also
change and thus the measured phase. Therefore, a carrier-phase measurement depends on the
orientation of the antennas of the satellite and the receiver as well as the direction of the line
of sight between the satellite and the receiver (Wu et al., 1993). The effect introduced to the
76
phase measurements because of the rotation of antennas is called “phase wind-up” or “phase
wrap-up” (Wu et al., 1993; Tetewsky and Mullen, 1997).
The phase wind-up because of the rotation of receiver antenna can be removed completely by
double differencing or absorbed to the receiver clock offset in point positioning. Thus, the
receiver antenna rotation affects PPP timing results (Tetewsky and Mullen, 1997). In timing
applications, the receiver antennas are usually static. The receiver antenna rotation effects are
not considered in this research.
However, satellite antennas rotate slowly because satellites orient their solar panels towards
the Sun (Kouba, 2003). Unlike phase wind-up due to receiver antenna rotation, the effects of
satellite antenna rotation cannot be removed by double differencing because they depend on
the direction of the line of sight between the satellite and the receiver. As investigated by Wu
et al. (1993), the residual phase shift after double differencing could be as large as half a
cycle depending on the baseline length. Neither can they be absorbed to receiver clock offset
in PPP. In PPP, if they are not corrected, the constant part of the effects would be absorbed
into the float ambiguities, the variations will affect the ambiguity convergence and
subsequently affect the results. Wu et al. (1993) have provided two expressions to mitigate
the effects, which give the same results. One of them is more suitable for computer
programming and presented in the following.
As studied by Wu et al. (1993), the phase wind-up correction is determined by the angle
between the two effective dipoles of the satellite and the receiver, and its past history as
follows:
77
φπ ∆+=∆Φ N2 (3.38)
where φ∆ is a fractional part of a cycle and can be calculated as follows (Wu et al., 1993):
)D'D/D'D(cos)(sign
⋅=∆ −1ζφ (3.39)
'D
and D
are the effective dipole vectors of the satellite and receiver computed from the
current satellite body coordinate unit vectors ( ',',' zyx ) and the local receiver unit vectors
( zyx ,, ), respectively. They are expressed as follows (Wu et al., 1993):
')'('' ykxkkxD ×−⋅−= (3.40)
ykxkkxD ×+⋅−= )( (3.41)
)'( DDk
×⋅=ζ (3.42)
where k
is the unit vector pointing from the satellite to the receiver.
The integer N is given as follows (Wu et al., 1993):
]/)int[(nN prev πφ 2∆−∆Φ= (3.43)
where intn denotes the nearest integer function and prev∆Φ is the previous value of phase
wind-up correction. Equation 3.43 is given based on the assumption that the data rate is high
enough so that the change in the correction is less than half a cycle between successive
computations (Wu et al., 1993). The value of N can be arbitrarily set to any integer,
78
normally zero, at the beginning of a phase tracking session. It is obvious that the equations
above just provide the information of the change of phase wind-up from the beginning of a
phase tracking session. The initial phase wind-up angles would be combined into the
ambiguities (Leick, 2004).
The correction from Equation 3.38 is given in phase angle. The corrections in range are then
expressed as ∆Φ= 11 λω , ∆Φ= 22 λω , ∆Φ=−−
22
21
2211
ff
ffIF
λλω for 1Φ , 2Φ and IFΦ ,
respectively. From these expressions, it is obvious that an applicable method to remove phase
wind-up effects is to form the widelane phase combination.
Differing from carrier-phase measurements, the rotation of satellite or receiver antenna does
not affect the code measurements. Therefore, no wind-up corrections are required for the
code measurements (Leick, 2004).
3.2.6 Hardware Delays
The points where satellite signals are generated and transmitted are different. Also the phase
center of the receiver antenna is located at a different point from the signal correlation in the
receiver. Therefore, the signal is delayed by both satellite and receiver hardware. This type of
signal delays are called hardware delays or group delays. The hardware delays are different
for 1P and 2P , and the difference between them is called hardware bias, or group delay
differential ( gdT ) for satellites, and Inter-Frequency Bias (IFB) or Differential Code Bias
79
(DCB) for receivers, respectively. In addition to the DCB between 1P and 2P , there is also
systematic difference between 1P and 1C , which is satellite dependent, and has been
discussed in Section 3.1.
In the case of double differencing between satellites and receivers, the hardware delays of
both satellites and receivers can be completely removed. In the case of point positioning,
where the hardware delay of the receiver is the same to all satellite measurements, it would
be completely absorbed to the receiver clock offset estimation when single-frequency
observations are used. For point positioning using dual-frequency observations, the receiver
hardware delay would be removed or absorbed to the float ambiguities depending on which
type of ionosphere-free combinations are used.
The group delay differentials ( gdT ) of satellites are provided in the navigation messages. The
corrections would benefit single-frequency users. They are also useful for some applications
in addition to positioning, such as ionospheric modeling. In navigation messages, the satellite
clock information is based on the dual-frequency ionosphere-free code combination (ICD-
GPS-200C, 2000). Therefore, the traditional ionosphere-free combinations are immune from
the hardware delays if the broadcast ephemeris is used.
As described by Kouba (2003), no group delay calibration corrections are applied for the
receiver and satellite ( 2L - 1L ) biases in IGS analyses. Thus, no calibrations are to be applied
when the IGS clock products are used in precise point positioning using the traditional
ionosphere-free combinations.
80
According to ICD-GPS-200C (2000), users who only use the measurements on 1L frequency
shall modify the satellite clock correction with the following equation:
gdss Tdtdt −=1 (3.44)
where sdt is the satellite clock correction calculated from the clock correction coefficients in
the navigation messages. The user who only uses the measurements on 2L frequency shall
modify the code phase with the equation below:
gdss Tdtdt ⋅−= γ2 (3.45)
The value of gdT can be expressed as follows (ICD-GPS-200C, 2000):
( )2111
ttTgd −
−=
γ (3.46)
where it is the GPS time corresponding to the ith frequency signal transmitted from the
satellite.
Precise point positioning using single-frequency measurements will be discussed Chapter 5.
The accuracy of the broadcast gdT values directly affects the single-frequency point
positioning performance, especially after SA turned off. The ionosphere community has been
estimating the group delay differentials since 1993. In 1998, a world wide cooperative
analysis was initiated to determine new gdT values. The first set of new gdT values based on
JPL’s estimates was uploaded to satellite in April 1999. Currently JPL provides updated
81
estimates of gdT for uplink every quarter or as needed and also monitors the values daily to
identify any abrupt changes in the gdT values due to configuration changes of the satellites
(Wilson et al., 1999).
Hardware delays of both satellite and receiver also exist in carrier phase observations. The
hardware delays in phase measurements would be removed completely by double
differencing. Because hardware delays are different between the code and phase
measurements (Xu, 2003), in precise point positioning, only parts of the hardware delays in
phase measurements are absorbed to the receiver clock estimate. Because phase observables
are inherently ambiguous, the residuals (not absorbed to clock estimates) of hardware delays
in phase measurements would be absorbed by the ambiguity parameters in point positioning
(Leick, 2004). The residuals, along with the initial phase offsets, destroy the integer nature of
ambiguities in point positioning. The integer parts of the residuals are combined into the
integer ambiguities, while the fractional parts would be combined into the initial phase
offsets which are discussed in the following.
3.2.7 Initial Phase Offsets
As shown in Equation 3.7, the un-differenced phase observable includes two non-zero initial
fractional phase terms. The first term, )t(ri 0φ , is the un-calibrated component of phase delay
in the receiver and would be common to all channels of the receiver. The second term,
)t(si 0φ , exists in the satellite transmitter and is different from satellite to satellite (Blewitt,
82
1989). They are usually called initial (or fractional) phase offsets of the receiver and satellite
(Teunissen and Kleusberg, 1998; Han et al., 2001). The initial phase offsets are part of the
receiver- and satellite-generated signals. Carrier-phase measurements at a specific receiver
on a specific frequency should contain the same receiver-based initial phase offset. Similarly,
carrier-phase measurements to a specific satellite on a specific frequency should contain the
same satellite-based initial phase offset (Gabor and Nerem, 2002).
These initial phase offsets can be completely removed by double differencing between
satellites and receivers. They can also be treated as constants for each receiver or satellite so
that they can be removed by time differencing between phase measurements to the same
satellite at two consecutive epochs (Han et al., 2001). But initial phase offsets are absorbed
into the ambiguity in un-differenced phase measurement, thus they destroy the integer nature
of the ambiguity. Solving these items can help the ambiguity resolution in PPP thus improve
the accuracy. Currently, there are still no values of initial phase offsets available for satellites
and receivers. Gabor and Nerem (2002) have tried to estimate these items to reconstruct the
integer nature of the ambiguity, which has improved the mean baseline length repeatability
about 40~50 percent using the PPP method.
Because the hardware delays in phase measurements are also satellite and receiver
dependent, the integer parts of the hardware delays are absorbed to the integer parts of the
ambiguities, while the fractional parts are absorbed to the initial phase offsets.
83
3.2.8 Multipath and Measurement Noise
Multipath is mainly caused by reflecting surfaces near the receiver antenna, such as metallic
objects, ground and water surfaces, or introduced by reflections at the satellite during signal
transmission (Hofmann-Wellenhof et al, 2000). The reflected signals may interfere with the
direct signals, and lead to noisier measurements. They may also confuse the signal tracking
in GPS receivers and lead to biased measurements (Langley, 1997).
The theoretical maximum multipath error that can occur in code measurements is
approximately half the code chip length or 150 m for C/A code ranges and 15 m for P(Y)
code ranges. Typical errors are much lower (generally less than 10 m). The carrier phase
multipath error can be up to about one-quarter of the wavelength, which corresponds to
approximately 4~5 cm for phase measurement on 1L or 2L (Braasch, 1996). A common
method for reducing code multipath effects is to smooth code measurements with carrier
phase measurements (Hatch, 1982). This method also reduces the noises of code
measurements. In PPP processing, multipath effects not only degrade the accuracy but also
delay the ambiguity convergence. A Choke Ring antenna would be necessary in the presence
of multipath.
Measurement noises exist in both code and carrier phase observations. The noise magnitude
in code and phase measurements is significantly different. Code noise, which is generally
less than 1% of the chipping rate (<3 m for C/A code and < 0.3 m for P(Y) code), is much
bigger than the phase noise, which is approximately 2 mm or equivalent to 1% of the
84
wavelength (Langley, 1998b). Currently, because of the advance in signal processing
technology, the noise level of some receivers is much smaller than the theoretical one. As
given by Hofmann-Wellenhof et al. (2000), the typical noise is 10~300 cm, 10~30 cm and
0.2~5 mm for C/A code, P(Y) code and carrier phase, respectively. The noise level of the
ionosphere-free combinations has been discussed in Section 3.1.
Both multipath error and measurement noise are related to the elevation of the observed
satellite. Normally, they increase while the satellite elevation decreases. The residuals of the
ionospheric and tropospheric effects also show similar characteristics. The characteristics of
these error sources provide an approach for stochastic modeling, which will be discussed in
Section 3.3.
3.2.9 Antenna Phase Center Offset and Variations
Phase center offset and variations are relevant to both GPS satellite antenna and receiver
antenna. Antenna phase center is the point where a signal is transmitted or received.
However, it is not a well-defined physical point and varies with the changing direction of the
incoming or outgoing signals. The antenna phase center can be usually given in the form of a
mean offset from a physically defined point on the antenna, normally, antenna reference
point (ARP) is selected, plus a variation. Rothacher and Beutler (2002b) have described the
offset and variation using the following two terms:
85
• A mean offset, 0r , which is a vector from the ARP to the location of the mean
phase center (average over a certain elevation range).
• An elevation ( e ) and azimuth (α ) dependent phase center variation ( )α,e∆Φ with
respect to the mean phase center 0r .
Both 0r and ( )α,z∆Φ are frequency dependent and different for 1L and 2L . The total
correction for the observation between the station r and the satellite s due to the receiver
antenna phase center offset and variations can be calculated using the following equation
(Rothacher and Beutler, 2002b):
( )ααδρ ,eer),e( srant ∆Φ+⋅= 0 (3.47)
where sre is the unit vector between the station r and the satellite s.
Receiver antenna phase center offset and variations may introduce up to 10 cm of vertical
positioning error, and sub-centimetre horizontal positioning error, if they are not accounted
for (Rothacher and Beutler, 2002b). The effects introduced by the phase center offset of
satellite antenna may be much bigger than those from receiver antenna because satellite
orbits can be referred to a point far from the phase center. The phase center offset of GPS
satellite antenna can be more than one metre for post-mission precise GPS orbit and clock
products. Unlike orbit and clock information provided by the broadcast ephemeris, which
refer to the phase center of the satellite antenna, precise GPS orbit and clock products could
refer to any point chosen by the product providers (Zhu et al., 2003). For example, the force
86
models used by the IGS community for satellite orbit modeling refer to the center of mass,
their precise GPS satellite coordinates also refer to the same point in IGS products (Kouba,
2003). Normally, the offsets between the phase center and mass center are calibrated before
the satellite launch. For Block IIR satellites, the phase center and mass center locate at the
same point. The offsets for Block II/IIA satellites are normally provided in the precise
products. Real-time precise orbit and clock products are normally based on the broadcast
ephemeris and also refer to the satellite phase center.
IGS has started to standardize the receiver antenna names since 1992 (Rothacher and Beutler,
2002b). Currently, phase offset and variations information of both GPS satellite antennas and
a subset of GPS receiver antennas have been provided in the IGS antenna phase center offset
tables (Rothacher and Mader, 2002a; Kouba, 2003). The variations have been given in two
formats, elevation-dependent, and elevation- and azimuth-dependent (Rothacher and Beutler,
2002b).
The National Geodetic Survey (NGS) of USA has also calibrated a subset of GPS antennas
(Mader, 1999). The format is similar to IGS elevation-dependent format. NGS calibrates
antennas in both relative and absolute methods, while IGS provides only relative calibrated
values. In both NGS and IGS relative calibration, all antenna phase center offsets and
variations are calibrated with respect to the AOAD/M_T antenna, which is a choke ring
antenna and is the most common antenna type within the global network of the International
GPS Service (IGS).
87
3.2.10 Site Displacement Effects
The Earth is not a rigid body, and therefore is deformed by the gravitational potential. The
gravitational potential in the vicinity of the Earth is a combination of the tidal gravitational
potential of external bodies, including the Moon, the Sun, and other planets, and the Earth's
own potential. Both the external potential and the Earth's own potential contain time
independent part, which is permanent, and time dependent part, which is periodic. The tidal
potentials cause solid Earth deformations, and thus lead to site displacements, which also
include permanent and periodic parts. Correspondingly, two crusts, namely “mean tide” crust
and “tide free” crust, are introduced by IERS (International Earth Rotation and Reference
Systems Service). Site positions are on the “mean tide” crust if the time dependent part of the
tidal contributions is removed. It is obvious that the permanent deformation is still present in
the “mean tide” crust. By removing the permanent deformation from the “mean tide” crust, a
“tide free” crust is obtained. However, a truly “tide free” crust is unobtainable because the
gravitational potential is present all the time. Instead, the “conventional tide free” crust is
usually used (McCarthy and Petit, 2004). In the following, some models, which are used by
IERS to get the “conventional tide free” terrestrial reference frame, will be presented. The
coordinates after applying the corrections from the models are also called “conventional tide
free”, or simply “tide free”, coordinates (McCarthy and Petit, 2004).
Actually, these tidal corrections are not necessarily applied in PPP data processing all the
time. When they should be applied in data processing depends on where the datasets are
collected. For example, when airborne dataset, which is free from the tidal effects of the earth
88
(Xu, 2003), is processed using PPP method, no tidal corrections are required to be applied to
get the tide-free coordinates. However, sometimes users would find big discrepancies
between the coordinates obtained from PPP and coordinates from differential techniques in
which the base stations are set on the Earth. This is because the differential software has not
considered the differences in the tidal effects between the base on the Earth and rover in the
air, or the software has not been set to process baseline between stations on the Earth and in
the air.
Solid Earth Tides
Deformations due to solid earth tides are mainly caused by the gravitational attractions acting
on the elastic body of the Earth by the Moon and the Sun. The effects of solid earth tides can
reach about 30 cm in the radial direction and 5 cm in the horizontal direction (Kouba and
Heroux, 2001a). Solid earth tides are represented by the Love number hnm and the Shida
number lnm, which weakly depend on station latitude and tidal frequency (McCarthy and
Petit, 2004).
Models have been provided by IERS to calculate the displacement due to the second-degree
tides and displacement due to the third-degree tides. For a 5 mm precision in height
component, only the second-degree tides are necessary (Kouba and Heroux, 2001a).
The following equation can be used to calculate the second-degree tides (McCarthy and Petit,
2004):
89
( ) ( ) ( )[ ]
⋅−⋅+
−⋅=∆ =
rrRRrRlrRrhRGM
RGMr jjjj
j je
ej2
2
2
3
23
4
321
23
(3.48)
where,
jGM is the gravitational parameter for the Moon (j = 2) or the Sun (j = 3),
eGM is the gravitational parameter for the Earth,
jR , jR is unit vector from the geocenter to Moon or Sun and the magnitude of that vector,
eR is the Earth's equatorial radius,
r is the unit vector from the geocenter to the station,
2h nominal degree 2 Love number,
2l nominal degree 2 Shida number.
Figure 3.1 shows solid earth tide effects for station S1 on roof of the Engineering Building at
the University of Calgary from September 2nd, 2004 to September 7th, 2004. As shown in the
figure, the effects can be up to 15 cm in height, and 5 cm in north and east.
90
Figure 3.1 Solid Earth Tide Effects for S1 from September 2nd to 7th, 2004
Ocean Loading
Site displacement due to ocean loading is caused by ocean tides. Ocean tides change the
ocean mass distribution and the water load on the crust, thus cause time-varying
deformations of the Earth (McCarthy and Petit, 2004). The deformations are more significant
for coastal sites than sites far from the sea. The displacement can be several centimetres for
coastal sites in the height component. The following simplified models have been provided
by IERS (McCarthy and Petit, 2004) to calculate the corrections in height, east and north
components for a particular site at time t:
91
( )hjjjjj
jhj tcosfAh Φ−++=∆ =
µχω11
1
(3.49a)
( )wjjjjj
jwj tcosfAe Φ−++−=∆ =
µχω11
1
(3.49b)
( )sjjjjj
jsj tcosfAn Φ−++−=∆ =
µχω11
1
(3.49c)
The summation over j represents the eleven partial tides including the semidiurnal waves M2,
S2, N2, K2; the diurnal waves K1, O1, P1, Q1; and the long-period waves Mf, Mm, and Ssa. The
nodal parameters, jf and jµ , depend on the longitude of the lunar node and can be
calculated from equations provided by IERS. The astronomical argument jχ at t = 0h can be
computed using subroutine from IERS also. The angular velocities jω at t = 0h are provided
by IERS as constants. hjA , wjA and sjA are the amplitudes in height, west and south
components. hjΦ , wjΦ and sjΦ are the phases in height, west and south components. Both
the amplitudes and phases for particular site can be calculated from some ocean loading
service agencies. The coefficients are applicable for stations within 10 km from the particular
site (McCarthy and Petit, 2004). In kinematic positioning, the coefficients can be interpolated
from precomputed sites to the receiver position at each epoch. In this research, coefficients
provided by Scherneck (2003) are used.
Figure 3.2 shows ocean loading effects for the station S1 from September 2nd to 7th, 2004.
Ocean loading effects are about one order smaller than those of solid earth tides for S1.
92
Figure 3.2 Ocean Loading Effects for S1 from September 2nd to 7th, 2004
Pole Tide
The pole tide is caused by the centrifugal effect of polar motion. The variation of station
coordinates caused by the pole tide can be up to a couple of centimetres (McCarthy and Petit,
2004). The displacements in the height, east and north components can be calculated by the
following equations (McCarthy and Petit, 2004):
( )λλφ sinmcosmsinh 21232 +−=∆ (3.50a)
( )λλφ cosmsinmcose 2129 −=∆ (3.50b)
93
( )λλφ sinmcosmcosn 2129 +=∆ (3.50c)
where the corrections are given in millimetres. φ and λ are the latitude and longitude of the
station. 1m and 2m represent the variations of the pole from the mean pole, given in seconds
of arc which can be expressed as follows (McCarthy and Petit, 2004):
pp xxm −=1 (3.51a)
( )pp yym −−=2 (3.51b)
where ( px , py ) are the pole position coordinates in seconds of arc, which are provided by
IGS or some other agencies in real-time or post-mission. ( px , py ) are the mean pole position
coordinates in seconds of arc, which are estimated by IERS in a yearly basic. The mean pole
position coordinates can also be calculated as follows (McCarthy and Petit, 2004):
)t(x)tt()t(x)t(x ppp 000−+= (3.52a)
)t(y)tt()t(y)t(y ppp 000−+= (3.52b)
where )t(x p 0 =0.054, )t(y p 0 = 0.357. px and py are given in seconds of arc.
)t(x p 0 =0.00083 and )t(y p 0
=0.00395, in seconds of arc per year. t is given in year and 0t
is 2000.
94
Because 1m and 2m vary at most 0.8 second of arc, the maximum height and horizontal
displacements are approximately 25 mm and 7 mm, respectively (McCarthy and Petit, 2004).
Figure 3.3 shows pole tide effects for station S1 from 2000 to 2004. As shown in the figure,
pole tide changes very slowly. In a short period, the effects vary only a small amount. In
position solutions, not considering the pole tide would introduce biases. In timing and
meteorological applications, if station coordinates are estimated along with receiver clock
offset and tropospheric delay, pole tide effects can be absorbed into coordinate estimation
and safely neglected.
Figure 3.3 Pole Tide Effects for S1 from 2000 to 2004
95
Atmospheric Loading
Atmospheric loading is caused by the variations of atmospheric pressure. The temporal
variations in the distribution of atmospheric mass can cause site displacement up to a couple
of centimetres (McCarthy and Petit, 2004). The site displacement due to atmospheric loading
has been widely investigated (vanDam and Wahr, 1987; vanDam et al., 1994; Sun et al,
1995). Pressure loading effects are larger at high-latitude sites because of more intensive
weather there. Effects are smaller for sites near the sea due to the inverted barometer
response of the sae surface, which is characterized by that changes in sea surface height is
inversely proportional to changes in the surface pressure. The horizontal deformations are
about one-third the amplitude of the vertical deformations (McCarthy and Petit, 2004).
Several methods for computing atmospheric loading corrections have been developed so far.
The geophysical model approach was adopted by the IERS at the 2002 IERS Meeting in
Munich, Germany. Regression coefficients determined from a geophysical model could be
used along with local pressure measurements to calculate the vertical deformations, while the
much smaller horizontal deformations are usually neglected in this model. The regression
coefficients can be determined by fitting local pressure to the vertical deformation predicted
by the geophysical model (McCarthy and Petit, 2004).
The regression coefficients for a subset of IGS stations have been estimated by the Special
Bureau on Loading (SBL), which was established by the IERS in February 2002. The
regression coefficients for a given site are applicable for stations within 10 km from the
given site (vanDam, 2003). In kinematic positioning, the regression coefficients can be
96
interpolated from precomputed sites to the receiver position at each epoch. In this research,
regression coefficients provided by vanDam (2003) are used. In addition to regression
coefficients, atmospheric pressure measurements are also required to calculate the
deformation.
Figure 3.4 shows atmospheric loading effects for station S1 from 1995 to 1999. The data was
provided by vanDam (2003). The pressure variations were calculated by removing the mean
pressure over the period from the observed pressures. From the figure, we can see
atmospheric pressure can change the station height up to 1 cm with millimetre level changes
in north and east. The changes in station height show strong negative correlation with the
changes in pressure. Regression coefficients are estimated based on the relationship between
changes in station height and atmospheric pressure on the station.
Figure 3.4 Atmospheric Loading Effects for S1 from 1995 to 1999
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3.3 Modeling for Dual-Frequency Measurements
In this research, the Kalman filter was used to perform optimal estimation. The principle of
the Kalman filter was first introduced by Kalman in 1960 and has been extensively described
in literature (Gelb, 1974; Brown and Hwang, 1992). In the Kalman filter, the relationship
between the measurements and the parameters is described by the functional (or
mathematical) model, while the noise characteristics of the measurements are described by
the stochastic model.
3.3.1 Functional Model
Functional models should be specified to describe the relationship between observables, i.e.
the ionosphere-free code and phase combinations in this research, and the unknown
parameters including receiver clock offset, tropospheric delay parameters, ambiguities,
and/or receiver coordinates. In the following, the observables and unknowns will be
described.
Observables
Dual-frequency GPS measurements and ionosphere-free combinations have been discussed
in Section 3.1. As already discussed, there are two types of ionosphere-free combinations that
could be used for precise point positioning. Although the C1-P1-CP combinations can be
used to facilitate the fixing of integer ambiguities, the integer nature of the ambiguity is
98
currently corrupted by the existence of non-zero initial phase offset and hardware delay in
un-differenced phase measurements. In this research, the more widely used traditional
ionosphere-free combinations are used for position determination, receiver clock and
tropospheric delay estimation. Equation 3.12 and 3.13 are rewritten in the following as the
primary observables in the Kalman filter.
)P(dmddd)dtdt(c
ffPfPf
P
IFIFreltroporbs
rsr
IF
ερ +++++−+=
−⋅−⋅= 2
22
1
22
212
1
(3.53)
)( mwFddd)dtdt(c
ffff
IFIFIFIFreltroporbs
rsr
IF
Φ+++++++−+=
−Φ⋅−Φ⋅=Φ
εδδρ
22
21
22
212
1
(3.54)
where
22
21
202022101011
ff)N)t()t((cf)N)t()t((cf
Fsrsr
IF −+−−+−
=φφφφ
(3.55)
The ionosphere-free code combination can be smoothed by the ionosphere-free phase
combination to reduce its noise and multipath (Hatch, 1982):
n)n(
nn
P
))(P(n
nn
PP
nIF
n
i
iIF
n
i
iIF
nIF
nIF
nSM
nIFn
SM
Φ−+Φ
−=
Φ−Φ+−+=
−
==
−−
1
1
1
11
11
(3.56)
99
where nSMP and 1−n
SMP are the smoothed code combinations at the nth and the (n-1)th smoothing
epochs, respectively, iIFP and i
IFΦ are the ionosphere-free combinations of code and phase at
the ith smoothing epoch. Phase smoothing is also helpful to shorten the ambiguity
convergence time.
Unknowns
The number of unknown parameters varies for different applications. The receiver clock
offset and ambiguities, which are usually considered as nuisance parameters, should be
estimated all the time. In this research, to mitigate tropospheric delay, the zenith wet delay
will be estimated, instead of estimating the total zenith tropospheric delay that has done in
some researches.
As discussed in Section 2.4, estimating troposphere horizontal gradients can improve both
zenith wet delay estimation and the repeatability of the static site position, but will also
introduce two additional parameters. To estimate horizontal gradients, GPS observations at
low elevation angles must be included to separate gradient components from the azimuthally
homogeneous components. In static positioning and meteorological research, when antenna
is normally set up in good observing condition and multipath-friendly environment, satellites
at low elevation, i.e. down to 7°, can be tracked and used for data processing. In this
research, tropospheric gradients will be estimated in static positioning and meteorological
processings. Estimating tropospheric gradients is also applicable in timing application, as
timing receivers are normally set up in good observing condition. Improved zenith wet delay
100
estimates can also improve the receiver clock offset estimation due to strong correlation
between them (Hackman and Levine, 2003). In kinematic positioning, when observing
condition varies, the satellites tracked may be less than the number required to obtain a
solution for gradients. Multipath effects may also be strong for satellites at low elevations
under kinematic environments. Therefore, only zenith wet delay will be estimated in
kinematic positioning.
In timing and meteorological applications, a GPS antenna is usually set up at a fixed site. In
post-mission, the antenna coordinate parameters can be resolved first, and then be removed
from the unknowns. But in this research, they will be estimated in real-time or simulated
real-time data processing.
In summary, the state vector for static positioning, timing and atmospheric sensing
applications is:
( )TnENwzr F,,F,F,G,G,D,dt,h,, 21λϕ (3.57)
where ϕ , λ , and h are the latitude, longitude and height of the station. rdt is the receiver
clock offset. wzD , NG and EG are the zenith wet delay, horizontal gradients in north and east
directions. n is the number of satellites included in processing. iF is the 1L and 2L combined
float ambiguity in the ionosphere-free phase combination, described in Equation 3.55.
The state vector for kinematic positioning is:
101
( )Tnwzr F,,F,F,D,dt,h,, 21λϕ (3.58)
where there are no tropospheric gradient parameters.
3.3.2 Stochastic Model
The stochastic model describes the statistical properties of the observables, including both
the precision of observables and the correlation between the observables. The stochastic
model is generally given in the form of the variance-covariance (vc-) matrix, in which the
diagonal terms are the variances and the off-diagonal terms are the covariances (or
correlations) between the observables. To get optimal estimates, the inverse of the vc-matrix
of the observables serves as the weight matrix in the Kalman filter (Tiberius et al., 1999).
Errors in the stochastic model will not only affect the estimation accuracy but also the
ambiguity convergence.
In GPS positioning, different weighting schemes have been applied for satellites at different
elevation angles. The simplest one is equal weight for all observations. This scheme is not
adequate to describe the actual stochastic relationship among the observables, especially for
observables at low elevation (they are included to estimate the gradient parameters in
meteorological applications) because the noise and multipath effects increase when elevation
angle decreases. Another weighting scheme is based on the signal-to-noise ratio (SNR)
information of observables (Talbot, 1998). The SNR can partly reflect the precision of
observables. The problem of this scheme is that not all receivers will output such
102
information. Although there are fields in RINEX file reserved for SNR information, it is not
uncommon for there to be no data in those fields in the RINEX files or for the data to be
provided in the GPS receiver manufacturer’s own proprietary format. Given the above, this
scheme is not applicable or would lead to incorrect modeling in some cases, especially in
real-time.
Currently, a widely used scheme is the elevation-dependent weighting (Rothacher et al.,
1997). Measurement noise and multipath error increase significantly for observables at low
elevation. The elevation-dependent measurement noise is induced mainly by the receiver
antenna’s gain pattern. Atmospheric attenuation to GPS signal also contributes to the
elevation-dependent measurement noise (Tiberius et al., 1999). The increased multipath
errors in observations acquired from low elevation satellites are due to reflection property of
the circularly polarized GPS signals (Braasch, 1996). The troposphere and ionosphere
residuals are also elevation-dependent because of the elevation-dependent mapping
functions. In this research, the following weighting function is used:
( ) ( )esinew 2= (3.59)
where e is the elevation angle.
The variance and covariance of the code and phase measurement should be given properly to
fully exploit the precise phase measurements. The measurement noises in code and phase
measurements have been discussed in Section 3.2. The noise levels of 1 m and 1 cm have
been used for code and phase measurements by Zumberge et al. (1998). In this research, the
103
noise levels of 0.3 m and 2 mm are used for code and phase measurements. For satellite at
elevation e, the variance of ionosphere-free combinations will be determined using the
following equations:
( ) ( ) ( )esinesine PP
PIF
IF 2
2
2
22 1
3σσσ ≈= (3.60)
( ) ( ) ( )esinesine IF
IF 2
2
2
22 1
3 ΦΦ
Φ≈=
σσσ (3.61)
where, 21Pσ and 2
1Φσ are the variances of the code and phase measurements for satellite at
zenith, equal to 9×10-2 m2 and 4×10-6 m2, respectively.
Phase smoothing is used in this research for the reduction of the noise level and multipath
effects in the code measurements. The smoothed code and phase observables become
correlated after the smoothing process. Following the smoothing scheme given in Equation
3.56 and assuming that the code and phase measurements are independent before smoothing,
the variance and covariance for the smoothed ionosphere-free code and the ionosphere-free
phase observables will be determined as follows:
104
( )
( ) ( )
( )
−
−−++
=
−
−
−
−−−=Φ
ΦΦ
ΦΦ
=
−
=Φ
Φ
Φ
2
2
2
2
2
1
1
1
22
2
2
2
2
1
11
11
01
01
01
01
10000
11111
1
1
nIF
nIF
nIF
nIF
iIF
iIF
nIF
IF
nIF
IF
n
nn
n
n
n
nn
n
n
n
n
nn
nnnn,PCov
n
i
n
iP
P
P
nIF
nSM
σσ
σσσσ
σ
σσ
σ
(3.62)
where 2
iIFP
σ and 2
iIFΦ
σ are the variance of the ionosphere-free combinations of code and
phase at the ith smoothing epoch.
To avoid overweighting the code measurement, an alternative smoothing scheme proposed
by Lachapelle et al. (1986) can be used. They proposed a sliding weight scheme and the
weights of code and phase in the smoothed code measurement would be held fixed after a
selected amount of smoothing epochs.
105
CHAPTER 4
REAL-TIME PRECISE GPS ORBIT AND CLOCK
PRODUCTS AND ANALYSIS
An important element of precise point positioning is precise GPS orbit and clock products.
Today, the advent of several types of real-time precise GPS orbits and clocks has made real-
time PPP possible. In post-mission, the performance of PPP is only affected by the accuracy
and sample interval of precise GPS products. In real-time, however, the precise GPS products
will cause additional degradations to the accuracy of PPP through their latency and age. The
latency and age of the precise GPS products depend on the data processing strategy,
communication issues, and data format of the products.
The purpose of this chapter is to present the development of real-time precise orbit and clock
products, followed by the latency and age analysis of two types of real-time precise GPS
products. The communication issues and formats will also be discussed as how to reduce the
latency and age. For a comparison purpose, some IGS post-mission products will be
discussed at first.
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4.1 IGS Precise Orbit and Clock Products
Since 1994 the International GPS Service (IGS) has been providing precise GPS products to
the scientific community with increased accuracy and timeliness (Kouba and Heroux,
2001a). Today, IGS products with different latencies and accuracies are available to GPS
users. Table 4.1 demonstrates the characteristics of different IGS precise satellite orbit and
clock products with respect to different accuracies and timeliness. Compromises are
necessary between “Accuracy” and “Timeliness” when choosing IGS products. From IGS
Ultra-Rapid (predicted half) to IGS Final, the corresponding accuracy of products increases
as well as the latency.
Table 4.1 IGS Products of GPS Satellite Orbit and Clock (after IGS Website, 2004)
Satellites Orbits / Clocks Accuracy Latency Updates Sample Interval
Orbits ~200 cm 2 hours Broadcast
Clocks ~7 ns real-time daily
2 hours
Orbits ~10 cm 15 minutes IGS Ultra-Rapid (predicted half) Clocks ~5 ns
real-time four times daily 15 minutes
Orbits <5 cm 15 minutes IGS Ultra-Rapid (observed half) Clocks ~0.2 ns
3 hours four times daily 15 minutes
Orbits <5 cm 15 minutes IGS Rapid
Clocks 0.1 ns 17 hours daily
5 minutes
Orbits <5 cm 15 minutes IGS Final
Clocks <0.1 ns ~13 days weekly
5 minutes
IGS orbit and clock products are based on a global tracking network and several contributing
Analysis Centers. The IGS global tracking network consists of more than 300 permanent,
107
continuously operating GPS stations. This number is increasing yearly as more stations are
added to the network. As of December 19th, 2004, the total number is 381 (IGS website,
2004). Figure 4.1 shows the IGS global tracking network.
Figure 4.1 IGS Tracking Network on December 19th, 2004 (from IGS Website, 2004)
Currently, eight IGS Analysis Centers (AC) contribute daily ultra-rapid, rapid and final GPS
orbit and clock solutions to the IGS combinations (Kouba, 2003). As investigated by Beutler
et al. (1995), the combination procedure can average out the random-like noises produced by
different analysis centers using different approaches and modeling, and typically result in
more robust and precise solutions. Figure 4.2 shows the orbit RMS (mm) of the individual
AC solutions with respect to the IGS Final orbit combination. Figure 4.3 shows the clock
RMS (mm) of the individual AC solutions with respect to the IGS Final clock combination.
For display purposes the RMS values of the summaries in Figures 4.2 and 4.3 are weekly
averaged. Both figures were obtained from the website of IGS Analysis Center Coordinator
(ACC) at GFZ Potsdam.
108
Figure 4.2 AC Solutions with Respect to the IGS Final Orbit Combination (from IGS ACC
Website, 2004)
Figure 4.3 AC Solutions with Respect to the IGS Final Clock Combination (from IGS ACC
Website, 2004)
109
From Figures 4.2 and 4.3, we can see that IGS orbit and clock products have been improving
in the past years. This is because more and more stations have been added to the network and
the data processing strategy has been improved over years.
Precise products with different accuracies and latencies can be applied to different
applications. For example, the IGS Final and Rapid products, whose 5-minute clock
sampling allows an interpolation of SA-free satellite clocks well below the decimetre level
(Zumberge and Gendt, 2000), can be used to support sub-centimetre to sub-decimetre level
static and kinematic positioning in post-mission to achieve the highest accuracy (Kouba and
Heroux, 2001a), or used to support climate and timing transfer researches. The IGS Ultra-
Rapid (observed half) products with just slightly degraded accuracy and a latency of 3 hours
can still satisfy near real-time PPP applications. The IGS Ultra-Rapid (predicted half)
products are available in real-time. Although Ultra-Rapid (predicted half) clocks are not
accurate enough for carrier phase-based precise point positioning, the Ultra-Rapid orbits,
which is much more accurate than the broadcast orbits, can be used to improve results of
differential data processing in real-time. For example, they have been widely used in ground-
based GPS meteorology (Fang et al., 2001; Douša, 2001). As for real-time PPP application,
real-time orbit and clock products from organizations such as JPL and NRCan, which will be
discussed in the following, are much better than IGS Ultra-Rapid (predicted half), because of
their much more accurate clocks and shorter sample intervals. To save the bandwidth for data
transfer, the real-time orbit and clock products are usually broadcast in the format of
corrections to the broadcast orbits and clocks. IGS also plans to generate real-time products
110
based on a real-time tracking network. The Real Time Working Group (RTWG) was
established in 2002 to develop real-time infrastructure and processes for IGS real-time
products. Its current focus is on the development of a prototype architecture which makes use
of data from a number of agencies’ real-time subnetworks (IGS RTWG website, 2004).
Currently, there are about 20 stations which can share real-time GPS data to IGS Analysis
Centers, with additional stations scheduled to join the real-time network (MacLeod, 2004).
The IGS orbit/clock products are consistent with the IGS global reference frame which
conforms to the International Terrestrial Reference Frame (ITRF) (Kouba, 2003). Therefore,
PPP using IGS products offers a simple and direct access to the IGS realization of ITRF.
Since its beginning in 1994, IGS has used six different official realizations of ITRF (ITRF92,
ITRF93, ITRF94, ITRF96, ITRF97 and ITRF2000). IGS products have adopted the IGS
ITRF2000 realization (IGS00) since December 2, 2001 (GPS Week 1143) (Kouba, 2003).
Kouba (2002) has also given the transformation parameters between frames and a program to
do transformation.
4.2 NRCan Real-Time Precise GPS Products
The Geodetic Survey Division (GSD) of Natural Resources Canada (NRCan), an Analysis
Center (AC) of the IGS, has provided two types of precise GPS orbit and clock products,
which are real-time or will be real-time in the future. One is based on code processing using
data from a wide area GPS tracking network in Canada. The second type is based on phase
111
processing using data from a global GPS tracking network. The parameters of these two
products are described in Table 4.2. Details of the tracking network and the accuracy will be
discussed in the following.
Table 4.2 NRCan Real-Time Precise GPS Products (after Heroux, 2003; Collins, 2004)
Satellite Orbit / Clock Products Accuracy Latency Updates Sample Interval
Orbits ~20 cm 20 s GPS•C (code solution)
Clocks ~2 ns ~5 s 2 s
2 s
Orbits ~10 cm 20 s GPS•C (phase solution)
Clocks ~1 ns ~8 hours 2 s
2 s
4.2.1 GPS••••C Code Solution Products
In Canada, the Canadian Active Control System (CACS) was established to improve the
performance of GPS positioning and to provide a direct access to the Canadian Spatial
Reference System (CSRS). Based on CACS, precise GPS satellite orbits and clocks have
been estimated daily since 1992. In 1996, the accuracies of the precise orbits and clocks are
about 10 cm and 1 ns, respectively. Since 1996, the Geodetic Survey Division (GSD) of
Natural Resources Canada (NRCan) has developed a prototype of real-time GPS Correction
Service (GPS•C) based on the CACS stations (Caissy et al., 1996). Figure 4.4 shows the
CACS network.
112
Figure 4.4 CACS Network (after ICD-GPS••••C, 2001)
The GPS•C system consists of four components, which are the Real-Time Active Control
Points (RTACPs), Real-Time Master Active Control Station (RTMACS), Virtual Active
Control Points (VACPs) and Integrity Monitoring Stations (IMSs) (ICD-GPS•C, 2001).
The RTMACS is used to control and acquire GPS and meteorological data from all RTACPs,
which are set up for data collection, validation and communication. The data is processed at
the RTMACS to generate corrections to the broadcast orbits and clocks, and ionospheric
vertical delay grids. These corrections are then sent to the VACP and IMS stations via
multicast network. VACPs are the primary distribution points of GPS•C corrections. IMSs
113
are set up to monitor the differences between the local GPS corrections and the GPS•C
derived corrections (ICD-GPS•C, 2001).
After several years of tests, the GPS•C service was launched on October 14, 2003 (Kassam,
2003). Currently, real-time GPS•C products are broadcast via the MSAT (Mobile Satellite
system). CDGPS radios required for GPS users to receive the GPS•C products over satellite
downlink have also been developed. GPS•C products are encoded in a modified RTCA-159
(MRTCA) format which includes a subset of RTCA-159 messages and GPS•C proprietary
messages. The CDGPS radios can localize the corrections to the standard RTCM-104 format,
which enables single-frequency pseudorange users to enhance their positioning precision
(Kassam, 2003). For the users equipped with dual-frequency receivers, CDGPS radios will
also relay the GPS•C orbit and clock products in the modified RTCA-159 format, allowing
the most demanding GPS•C users to achieve the highest possible accuracy. Before GPS•C
products were broadcast by MSAT geostationary satellites, they have been broadcasted to
users over the Internet using multicast technology (Chen et al., 2002). NRCan plans to keep
this service in the future. When broadcast over the Internet, GPS•C products are
encapsulated in the GPS•C IP Multicast format for transmission (ICD-GPS•C, 2001). Figure
4.5 shows the CDGPS Radio.
114
Figure 4.5 CDGPS Radio (after CDGPS Receiver User's Guide, 2003)
The GPS•C code solution products comprise corrections to broadcast satellite orbits and
clocks, and ionospheric parameters. When broadcast over the Internet, the maximum update
interval is 2 s for clock corrections, 120 s for orbit corrections, and 300 s for ionospheric
parameters. When broadcast over satellites, the update interval of clock corrections is 2 s,
and 20 s for orbit corrections. The GPS•C corrections are based on the NAD83 reference
frame. In Chapter 6, some results obtained using real-time GPS•C code solution orbit and
clock products will be presented.
4.2.2 GPS••••C Phase Solution Products
The GPS•C products were primarily based on processing code observations from CACS
network. Therefore, the accuracy of the products is limited and only practicable for code
processing. Since 2001, NRCan has been enhancing its real-time GPS•C system with carrier
phase processing. This development is aimed at improving the quality of the clock
115
corrections to facilitate carrier phase-based precise point positioning (Collins, et al., 2001).
However, the performance of the corrections based on carrier phase processing using data
from CACS network is not satisfactory, especially for users at the boundary of CACS
tracking network (Collins, 2004). This is because it takes time for the ambiguities in phase
solution to converge. Each satellite is tracked by CACS network twice daily. The tracking arc
only consists of several hours of data for each pass. The geometry of each satellite is poor
because each satellite is tracked by stations in a limited region. Therefore, users at the
boundary of CACS tracking network can just use the degraded products from processing the
phase observations with un-converged ambiguities, which are only as accurate as those of
code solution. Therefore, to generate phase solution products, a network with global
coverage is preferred.
In 2004, NRCan tried to acquire real-time data from some stations operated by different
agencies outside Canada, some of which are IGS real-time stations. In the middle of 2004,
data from 20 stations can be acquired in near real-time by NRCan. As investigated by Collins
(2004), these 20 stations can fulfill a global coverage with satellites that can be tracked by a
minimum of 4 stations most of the time. Figure 4.6 shows the 20 stations sharing near real-
time data with NRCan. The stations are plotted in different colours indicating different
agencies operating the stations. Figure 4.7 shows the global coverage using the 20 stations.
The colours indicate the number of stations observing each satellite sub-point.
116
Figure 4.6 Stations Sharing Real-Time Data with NRCan (after Collins, 2004)
Figure 4.7 The Global Coverage with 20 Stations (from Collins, 2004)
It takes several hours for NRCan to accumulate data from the 20 stations. Therefore, GPS•C
phase solution products based on global network have a latency of about 8 hours currently.
NRCan plans to real-time these phase solution products in the future. NRCan GPS•C phase
117
solution products keep the same format and the same reference frame as the real-time code
solution products. In Chapter 6, some results obtained using NRCan GPS•C phase solution
orbit and clock products will be presented.
Before the development of GPS•C phase solution products, GPS•C products only consisted
of the real-time GPS•C code solution products. In the remainder of this thesis, the term
GPS•C products are meant to refer to the real-time GPS•C code solution products only
unless otherwise specified.
4.3 JPL Real-Time Precise GPS Products
JPL, another Analysis Center of IGS, has generated two types of real-time or near real-time
products based on the NASA Global GPS Network (GGN), which is operated and maintained
by JPL. Table 4.3 shows details of the products. The accuracy statistics of the products are
provided by Heflin (2004) and Muellerschoen (2003), respectively.
Table 4.3 JPL Real-Time Precise GPS Products (after Muellerschoen, 2003; Heflin, 2004)
Satellite Orbit / Clock Products Accuracy Latency Updates Sample Interval
Orbits ~22 cm Near Real-Time (NRT)
Clocks ~0.7 ns ~3
minutes 15
minutes 5 minutes
Orbits ~18 cm ~28 s Internet-based Global Differential GPS System (IGDG) Clocks ~1 ns
~4 s 1 s 1 s
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4.3.1 JPL Near Real-Time Orbit and Clock Products
JPL near real-time orbit and clock products can be downloaded from JPL website (Heflin,
2000), and denoted as JPL NRT products in this thesis. When it was introduced, the latency
of JPL NRT is 15 minutes, which has now been shortened to about 3 minutes (Heflin, 2004).
Therefore, JPL NRT should also be considered a type of real-time products according to the
definitions given by Bar-Sever and Dow (2002). JPL NRT orbits are provided in the format
of SP3, while clocks are formatted in RINEX. The reference frame of the products is ITRF
2000 (Heflin, 2004).
JPL NRT products are very useful for GPS meteorological applications, in which IGU
(predicted half) products are widely used currently. As shown in Tables 4.1 and 4.3, JPL
NRT orbits are slightly worse than IGU (predicted half) orbits, but JPL NRT clocks are much
more accurate than those of IGS Ultra-Rapid (predicted half) products. JPL NRT orbits and
clocks are accurate enough for zenith tropospheric delay estimation using PPP methodology.
To access JPL NRT products, users do not need to maintain continuous communication with
the server. Just one access per 15 minutes is enough, which is more robust than using JPL
IGDG products, which needs continuous communication with the server or geostationary
satellite. In Chapter 6, zenith tropospheric delay estimates using JPL NRT products will be
presented.
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4.3.2 JPL IGDG Orbit and Clock Products
Other types of real-time products of JPL’s are the real-time Internet-based Global
Differential GPS products (Muellerschoen et al., 2000), denoted as JPL IGDG products.
Decimetre level kinematic positioning accuracy has already been obtained in some
experiments using the real-time products in a simulated real-time mode (Muellerschoen et al.,
2000; 2001). The reference frame of the products is ITRF 2000 (Muellerschoen, 2003).
The prototype of JPL real-time global differential GPS system has been investigated since
the middle of 1990’s (Yunck et al., 1996b). A software package, Real-Time Net Transfer
(RTNT), was then developed for returning data from the NASA Global GPS Network
(GGN), which is operated and maintained by JPL with some stations ready for Internet
connections. From these stations, 1 Hz data was returned over the Internet. In 2000, 15 of
these stations could return better than 95% of the data in less than 3 s (Muellerschoen et al.,
2000). The GPS orbit and clock products are then estimated using the Real-Time GIPSY
(RTG) software. Positioning accuracy of ~10 cm RMS in the horizontal and < 20 cm RMS in
the vertical has been obtained by Muellerschoen et al. (2000) using the orbit and clock
products in a simulated real-time mode. In 2001, the number of real-time stations was
increased to 25 (Muellerschoen et al., 2001). As shown in Figure 4.8, the number of real-time
stations was increased to more than 60 in 2003 (Armatys et al., 2003). Consequently, the
accuracy and reliability of the orbit and clock products were also increased through
redundancy. Sub-decimetre in horizontal and decimetre in vertical kinematic positioning
accuracy was obtained in a simulated real-time processing (Armatys et al., 2003). But
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expanding stations will also add to the complexity of processing which as a result, would
delay the orbit and clock products.
Figure 4.8 IGDG Real-Time Network in 2003 (from Armatys et al., 2003)
In 2001, most of the data from the GGN real-time network arrived at the central processing
center within 2 s of its GPS time-tag. Data with latencies less than 2 s was used in computing
the GPS satellite clocks. Older data, up to 6 s in latency, was used to compute the GPS
satellite orbits since they are more predictable than clocks (Muellerschoen et al., 2001). In
2001, with only 25 real-time stations, the latency of the correction message with an Internet
connection was around 2.5 s, with about 2 s spent to accumulate the global data over the
Internet at the central processing center, and 0.5 s to process the clock solutions. Currently,
with more than 60 stations, the latency with an Internet connection has increased to about 4 s.
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JPL IGDG orbit and clock products are also broadcast by the geostationary Inmarsat
satellites (Armatys et al., 2003). The latency through Inmarsat is around 1.5 s longer than the
Internet latency. It takes about 1.5 s to uplink the global correction messages to the
geostationary satellite and downlink to the user (Muellerschoen et al., 2001).
4.4 Accuracy Statistics of Real-Time Products
There are no published data about the accuracies of GPS•C code or phase solution products,
JPL NRT and IGDG products. The data in the Tables 4.2 and 4.3, which was provided by
Heroux (2003), Collins (2004), Heflin (2004) and Muellerschoen (2003), was obtained by
comparisons with other more precise products such as the IGS Rapid products
(Muellerschoen, 2003). The orbit and clock comparisons were performed separately.
Sometimes, it is difficult to separate the errors in orbits and clocks, as they can be correlated
with each other, especially in the case of real-time. As investigated by Parkinson (1996), the
orbit error and clock error of each satellite were shown to be negatively correlated, which
means that they would cancel each other partially. In the real-time case, because orbits
update less frequently than clocks, errors in the orbits, especially errors in the radial
component, are “filtered out” by the clock solution (Muellerschoen, 2003). Different
components of orbit errors affect the range measurements in different ways. Signal-In-Space
Range Error (SISRE) is usually used to calculate the effects of GPS orbit and clock errors to
the range measurements. SISRE can be computed by the following equation (Malys et al.,
1997):
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)CA(CLK)-(RSISRE 2 22
491 ++= (4.1)
where R is the radial orbit error, A is the along-track orbit error, C is the cross-track orbit
error and CLK is the clock error.
According to equation (4.1), the accuracy statistics of the orbit and clock products discussed
above does not necessarily present their effects on measurements, especially in the real-time
case when radial orbit errors are almost cancelled out by the fast updated clock corrections.
The good performance of these real-time products, which will be shown in Chapter 6, has
proven this point.
4.5 Latency and Age
In real-time applications, two key factors affect the validity of real-time orbit and clock
corrections. They are the latency and age of the corrections.
Latency is the time taken to collect raw data from reference stations and estimate the
correction parameters plus the time spent for the correction data to reach the users (Kee,
1996). Age is the total delay of the correction data including the above correction data
latency and the interval from the time when the correction data is received to the time when
the correction data is actually applied by the users. Since users have to apply one set of
corrections till a new set arrives, corrections could maintain a constant latency for a period of
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time while the age is actually increasing. Figure 4.9 shows the relationship between the
latency and the age.
Figure 4.9 Latency and Age (after Kee, 1996)
Theoretically, the best results would be obtained if the orbit and clock corrections were
applied to the measurements whose GPS time-tag coincides with that of the measurements
used to estimate the corrections. However, in real-time application, non-zero latency and age
are unavoidable because of the time taken to estimate the correction parameters and for
transmission. People usually try to shorten the latency and age of the corrections to minimize
accuracy degradation. After SA was turned off in 2000, the age of the corrections is not as
vital as before regarding its effect on the achievable positioning accuracy. Before SA was
turned off, satellite clock errors changed rapidly and randomly. Normally clock corrections
aged up to one minute may not improve the accuracy. After SA was turned off, errors in the
satellite clocks do not change as quickly as the random shifts generated by SA. Clock
corrections can still be helpful in improving the solution even when aged up to one minute.
But minimizing the age values is still of importance as a method to further improve the
positioning accuracy. Satellite orbit error is more predictable than that of satellite clock. By
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using correction-rates along with orbit corrections, the orbit corrections can remain valid for
several minutes.
The latency and age of NRCan GPS•C corrections and JPL IGDG corrections will be
presented in the following as received over the Internet. For satellite access, the latency and
age would be about 1.5 s longer because of a longer correction transmission time.
4.5.1 Latency and Age of GPS••••C Corrections
In this section, latency and age of GPS•C code solution orbit and clock corrections are
presented. In 2002, for test purpose, NRCan GPS•C products were sent to a selected port and
IP address of a computer set up in the University of Calgary over an open Internet connection
using UDP (User Datagram Protocol) multicast. The latency and age presented in the
following are obtained from the test conducted on July 8, 2002.
GPS•C orbit and clock products are encoded in a modified RTCA-159 format. When
broadcast over the Internet, the messages are encapsulated in the GPS•C IP Multicast format
for transmission (ICD-GPS•C, 2001). Orbit corrections and clock corrections are encoded
into different messages with different update intervals. Therefore, the latencies and ages of
orbit corrections and clock corrections are different.
Given in Figures 4.10 and 4.11 are the latencies of orbit corrections and clock corrections,
respectively. Though encoded into different message, the latencies for orbit corrections and
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clock corrections are similar, ranged between 5 and 6 s. Most of the latencies are
approximately 5.55 s.
Figure 4.10 Latencies of GPS••••C Orbit Corrections
Figure 4.11 Latencies of GPS••••C Clock Corrections
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Figures 4.12 and 4.13 show the ages of orbit corrections and clock corrections for satellite
PRN 21. During the period when the test was conducted, GPS•C clock corrections updated
every 2 s. The update interval for orbit corrections was not constant. Only an upper limit of
120 s is defined. The ages of orbit corrections range between 6 and 60 s. The spiked age of
about 150 s in Figure 4.12 is because one or two orbit messages were lost, or were not sent
out from the server. Because correction-rates were also broadcast, the corrections were still
valid even aged up to 150 s (Chen et al., 2002). Another spiked age of about 30 s occurred
because the orbit corrections were updated within 25 s at that time. As shown in Figure 4.13,
the ages of clock corrections range between 6 and 7 s. The spikes exist because clock
messages were lost or not sent out.
Figure 4.12 Ages of GPS••••C Orbit Corrections
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Figure 4.13 Ages of GPS••••C Clock Corrections
As shown in Figures 4.12 and 4.13, there may be some packets lost using UDP multicast
over open Internet. UDP is considered connectionless and it sends and receives datagrams on
a “best effort” basis. The spikes in Figure 4.12 and 4.13 may also be caused by messages
which were expected but were not sent out in time. As discussed below, no message loss was
found when testing with JPL corrections using UDP transport protocol. Section 4.6 will
discuss the correction distribution issues.
4.5.2 Latency and Age of IGDG Corrections
Since the middle of 2003, for research purpose, JPL IGDG orbit and clock products have
been sent to a selected port number and IP address of a computer set up in the University of
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Calgary over an open Internet connection. The UDP transport protocol is also used to
distribute the corrections. The latency and age presented in the following were obtained from
the test conducted on June 28, 2004.
IGDG orbit and clock corrections are encoded in a JPL proprietary message format. Orbit
corrections and clock corrections are encoded into the same message, which updates every
second. Each 44 bytes message contains updated centimetre-level clock corrections (fast
clock corrections) for all satellites, orbit corrections (including correction-rates) and metre-
level clock corrections (slow clock corrections) for one satellite (Armatys, 2002). Thus,
completely sending the sequence of all satellites takes about 28 seconds, depending on the
number of available satellites. Therefore, the update interval is 1 s for clock corrections and
about 28 s for orbit corrections. The latencies of orbit corrections and clock corrections are
the same, but the ages are different for orbit corrections and clock corrections.
Figure 4.14 shows the latencies of IGDG corrections. The latencies of IGDG corrections are
very consistent. All of the latencies range between 3.5 and 4.5 s.
Figures 4.15 and 4.16 show the ages of orbit corrections and clock corrections for satellite
PRN 1. For a latencies of 3.5 to 4.5 s, the ages of orbit corrections range between 4 and 32 s
with an update interval of about 28 s. As shown in Figure 4.16, the ages of clock corrections
range between 4 and 5 s. The ages of both orbit and clock corrections were obtained by
testing with a GPS data sample interval of 1 s. No clocks aged over 5 s or orbits aged over 32
s are found in the plots, this is because there was no message lost during the test.
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Figure 4.14 Latencies of IGDG Corrections
Figure 4.15 Ages of IGDG Orbits
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Figure 4.16 Ages of IGDG Clocks
Figure 4.17 shows the difference between applying an orbit correction aged 28 s along with
orbit correction-rate and a zero-latency orbit correction. From the plot, we can see, most of
the time, the difference is less than 5 mm in each coordinate component. The RMS
differences during the test are 3.897 mm in x component, 3.678 mm in y component and
3.678 mm in z component. Using orbit corrections aged 28 s add less than 1 cm range error
to the user’s measurements. Therefore, with orbit correction-rates also broadcast, an update
interval of 28 s has little impact to the validity of the orbit corrections.
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Figure 4.17 Difference between Orbits Aged 28 s and 0 s
Figure 4.18 shows the difference between applying a clock correction aged 5 s and a zero-
latency clock correction. From the plot, we can see, most of the time, that the difference is
less than 10 mm. The RMS difference during the test is 5.482 mm. Therefore, a latency of 5 s
in clock corrections adds less than 1 cm range error to the user’s measurement, which is not
significant considering the sub-nanosecond accuracy of the corrections. The differences are
mainly due to the resolution of the clock correction data, which is 1/128 m or about 7.8125
mm. From the plot, we can see that all differences are multiples of 7.8125 mm. As the
accuracy of IGDG clock corrections is sub-nanosecond level, a resolution of 1/128 m is good
enough. However, if the accuracy of these clock corrections can be improved to the accuracy
of the IGS Final clock products (0.1 ns), the resolution should be also improved as well.
Currently, the effective round-off error of the IGDG clock corrections is 1/2 of the resolution,
or about +/- 3.9 mm.
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Figure 4.18 Difference between Clocks Aged 5 s and 0 s
4.6 Real-Time Product Distribution Issues
Several factors affect the correction latency and age, including the time taken to accumulate
GPS measurements from the tracking network over the Internet or satellites, the time taken to
estimate the corrections, and the time taken to distribute the corrections.
The time taken to accumulate GPS measurements varies significantly depending on the
coverage of the network and the communication techniques. As for GPS•C code solution
corrections, only data from 12 stations in Canada is used. The maximum communication
delays are less than 0.5 for land communication link and about 1.5 s for satellite
communication link (Lahaye et al., 1998). Therefore, it takes about 1.5 s for GPS•C to
accumulate GPS measurements. But it takes a little bit longer time for IGDG to accumulate
measurements, which acquires data from over 60 stations in a global network. Plots of certain
stations regarding data latency and number of satellites observed in the last hour can be
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found at JPL IGDG website (2004). Because of redundancy in JPL real-time tracking
network, processing can be started before data from all stations is returned. For example,
early-arrived data can be used in computing the GPS clocks. Older data can be used to
compute the GPS orbits since they are more predictable than clocks due to their orbital
dynamics (Muellerschoen, 2001). Some communication techniques can be used to further
shorten the accumulation time. For example, NRCan adopted Virtual Private Network (VPN)
to transmit data between some RTACPs and RTMACS (ICD-GPS•C, 2001). VPN is a
mature technology that can be introduced to shorten transmission time between tracking
stations and processing center. VPN offers a cost-effective, scalable, and manageable way to
create a private network over a public infrastructure such as the Internet or over a service
provider's Frame Relay, ATM (Asynchronous Transfer Mode), wireless network, or even
satellite connections. VPN creates tunnels across an IP network and provides users the same
policies as a private network, including security, quality of service (QoS), manageability, and
reliability.
The time taken to estimate the corrections varies depending on the number of measurements
used. For example, in 2000, it takes 0.5 s to process the clock solution using measurements
from 25 real-time stations (Muellerschoen et al., 2001). It was extended to about 1 s for over
60 stations (Muellerschoen, 2003).
The time taken to distribute the corrections varies depending on the techniques used. Usually,
the corrections can be broadcast by geostationary satellites or distributed over the Internet.
Some WADGPS systems, such as OmniSTAR and NavCom, have adopted the former, while
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JPL and NRCan plan to adopt both of them to distribute the IGDG and GPS•C products. The
advantages and limitations of satellite- and Internet-based transmission will be discussed in
the following.
4.6.1 Broadcast by Satellite
Some WADGPS systems broadcast their precise data over geostationary satellites. The main
advantage of satellite-based broadcasting is that the receiving device can be integrated within
a GPS receiver because the corrections can be broadcasted using the same band (L-Band) as
GPS signals. At the same time, the bandwidth is not so serious a problem in satellite-based
broadcasting when compared with Internet-based distributing.
Still, compared with the Internet, satellite-based broadcasting also has several disadvantages.
Firstly, special receivers are required. It takes time and effort to develop a special receiver to
receive and decode corrections broadcasted by satellites. Internet, on the other hand, is a
public infrastructure, for which users have many accessible devices. This is especially
relevant for meteorological and timing applications, in which processing is conducted indoor
where Internet connections are normally available. Secondly, the cost of satellite-based
broadcasting is much more expensive than that of Internet-based distributing. Thirdly, it
takes a longer time to uplink the correction messages to satellite and downlink to the users.
For example, as tested by JPL, it takes 1.5 s to uplink the global correction messages to the
geostationary satellite and broadcast to the users (Muellerschoen et al., 2001). Similar
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transmission time has also been demonstrated by Lahaye et al. (1998). A much shorter
transmission time is required for Internet distributing. Figure 4.19 shows the transmission
time over the Internet, which was obtained by sending 106-byte length packets from a
computer at the University of Calgary to a computer at the University of York and back to
the sender. In this one-day’s test, the packets were sent out at a rate of 1 Hz using UDP
transport protocol. The transmission time was calculated by subtracting the time of sending
from the time of returning. From the figure, we can see, most of time, the packet was
returned within 60 ms after over 7000km traveling. Less than 1% packets were returned over
100 ms. Fourthly, in satellite-based broadcasting, users can rarely reply to the server. In this
case, the two-way communication and extensible data transmission rate of Internet-based
distributing are attractive to some special users. Finally, the satellite-based broadcasting
approach cannot provide global coverage. Geostationary satellites can only provide
corrections to users at latitudes from 76°N to 76°S because of the limitation of their orbits
(Armatys et al., 2003). For example, currently, real-time GPS•C corrections are broadcast via
the MSAT geostationary satellites (Kassam, 2003). But they cannot provide stable and
reliable communication links to users in all parts of Canada. Since geostationary satellites
such as MSATs are seen at low elevation in high latitudes, they are vulnerable to signal
blockage.
Considering the above, the satellite-based broadcasting and Internet-based distributing
methods are complementary to each other. Users will benefit from the coverage and
reliability if the orbit and clock products are broadcast through both approaches. Therefore,
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some organizations plan to adopt both of them to distribute their corrections, such as JPL and
NRCan. As for users, they can choose either approach based on the reliability, timeliness,
convenience, and available devices.
Figure 4.19 Transmission Time over the Internet
4.6.2 UDP/IP Multicast
In this research, JPL real-time orbit and clock products are received over the Internet. Several
techniques can be used for distributing products efficiently over the Internet.
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Multicast
Unlike satellite-based broadcasting, Internet-based distributing will encounter the problem of
bandwidth exhaustion. There are two distinct methods with which packets can be sent over
the Internet: unicast and multicast. Unicast is point-to-point communication whereas
multicast is point-to-multipoint or multipoint-to-multipoint communication. Unicasting
creates a stream for each receiver. This can be bandwidth intensive when multiple streams
are requested and cause quite a burden on the internal network. In the multicast case, the
server puts together a packet with a group address and gives it to the network. The routers
recognize the addresses included in the group address and subsequently duplicate the packet
for each receiver. Multicast technology eases the burden on the broadcast server as well as
the internal network. Multicast technique is also suitable for returning GPS data from real-
time tracking network to processing centers, because real-time stations can return real-time
data stream to multiple processing centers. For example, currently, JPL shares real-time data
stream from some stations with NRCan.
UDP Transport Protocol
There are two types of transport-layer protocols widely used in the Internet: Transmission
Control Protocol (TCP) and User Datagram Protocol (UDP) (Fapojuwo, 2003). TCP is a
connection-oriented protocol, meaning that TCP will set up, maintain, and tear down a
connection. The sender will wait for a reply from the receiver, and resend the packets if the
receiver has not received the packet correctly. TCP is designed for reliable transmission, but
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not for timely transmission. On the other hand, UDP is based on connectionless technology,
which means that the datagram is sent without first setting up a connection. The UDP is
designed to transfer packets in a timely manner, and does not re-send the lost packets. UDP is
also a much simpler protocol than TCP and has fewer overheads. Another advantage of UDP
is that it is much easier to implement multicast using UDP than TCP. Because UDP is a one-
to many and many-to-one protocol, while TCP is a one-to-one protocol. The drawback of
UDP is that it does not guarantee that the packet will arrive at its final destination.
When timeliness is a more important factor than reliability, UDP is usually used. Otherwise,
TCP is used. Therefore, UDP is more suitable for real-time services than TCP, such as real-
time distribution of GPS observations and products. Real-time orbit and clock products
update at a high rate, 1~2 s for clock corrections, 20~50 s for orbit corrections. Low packet
loss rate is compensated by the high update rate. Currently, GPS data stream and GPS•C
corrections are transmitted over NRCan's network using UDP multicast (ICD-GPS•C, 2001).
UDP is also adopted by JPL and GFZ for data transmission. For example, IGDG products are
distributed using UDP transport protocol to users. UDP transport protocol is also
recommended by IGS Real-Time Working Group for real-time distribution of data and
products (Muellerschoen and Caissy, 2004).
When directly receiving orbit and clock products from servers using UDP transport protocol,
such as the servers of GPS•C and IGDG, users do not know whether the missing packets are
unsent or lost. Therefore, here, the test done to show the transmission time in Section 4.6.1 is
used to show the packet loss rate of distributing products using UDP transport protocol. As
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shown in Figure 4.20, the packet loss rate is zero most of time. The packet loss rate was
calculated on a one-minute basis. Only in two minutes did the packet loss rate reach 5%,
which means 3 out of 60 packets were lost. Only 69 packets were lost in the entire test and
the total packet loss rate is 69/86400=0.08%. Less than one packet was lost out of 1000
packets on average.
Figure 4.20 Packet Loss Rate Using UDP Transport Protocol
4.7 Real-Time Correction Formats
The formats of the real-time orbit and clock products will determine the update interval and
bandwidth required to distribute the products. Several industry standard formats have been
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implemented for differential services, such as RTCM-104 and RTCA-159. They are widely
used in real-time kinematic or wide area differential GPS systems. However, they are not
suitable to encode the high precision orbit and clock products for global coverage. Two
formats have been proposed by NRCan and JPL, which are currently used to carry their real-
time orbit and clock products. These formats will be discussed in the following.
4.7.1 RTCM-104
RTCM-104 was first recommended by the Radio Technical Commission for Maritime
Services Special Committee 104 as standards for DGPS correction messages in November
1983. Types of messages have been added ever since then to the subsequent versions
(Muellerschoen and Caissy, 2004).
RTCM-104 has been widely accepted as a standard for local area DGPS services. Many GPS
receivers are now integrated with RTCM capability. But the scalar corrections in the RTCM
message are only applicable to measurements whose errors are correlated with those in the
measurements used to calculate the corrections. The spatial decorrelation of these scalar
corrections in this format makes them unsuitable for carrying the globally effective
corrections.
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4.7.2 RTCA-159
RTCA-159 is a standard format developed by the Radio Technical Commission for Aviation
Special Committee 159 for aircraft navigation using GPS. It was first proposed as the
standard to serve the Wide Area Augmentation System (WAAS) (Muellerschoen and Caissy,
2004). Subsequently, it was also adopted by some other wide area differential systems. In
this standard, corrections are encoded in a vector format including separate corrections for
satellite clocks, orbits and ionospheric effects. In this point, it can be chosen to carry real-
time corrections which are globally effective. RTCA-159 has also been widely used in GPS
community as a standard and is integrated into many GPS receivers. But the resolutions of
this format are limited. For example, the resolution for clock corrections is 0.125 m, and the
same resolution is applied to orbit corrections in each positioning component (ICD-GPS•C,
2001). Considering that the corrections of this format are originally proposed for code
processing, the resolutions were adequate. On the other hand, some corrections discussed in
this chapter are obtained by processing using both code and phase measurements, and the
accuracy of the corrections has been improved to decimetre level. A modified RTCA-159
format has been proposed by NRCan for the high precision corrections.
4.7.3 GPS••••C Format
GPS•C uses a subset of the RTCA-159 message types and has implemented several non-
standard RTCA-159 messages. Therefore, GPS•C products are encoded in a modified
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RTCA-159 (MRTCA) format. For example, messages of types 32, 33, 34, 35 and 45, which
were reconstructed from the standard RTCA-159 message 2 and 25, respectively, carry fast
and slow corrections with a higher resolution. In these types of messages, the resolution of
the corrections was increased to 4 mm for orbits and clocks. The bandwidth required to
distribute GPS•C messages is 106 byte/s for update intervals of 2 s and 20 s for clocks and
orbits respectively (ICD-GPS•C, 2001).
4.7.4 IGDG Format
GPS•C adopted a modified RTCA-159 format, which keeps some aspects of the original
RTCA-159. For example, the active PRN list (PRN mask) is encoded in a separate message
(Message Type 1). The full time of week and GPS week number are put in another separate
message (Message Type 12). The preambles of the messages carrying orbit and clock
corrections only provide the GPS time modulo 256 s (ICD-GPS•C, 2001). Message Type 1
and 12 are not sent out as frequently as the orbit and clock correction messages. If one of
them is lost, which is likely to happen when distributed using UDP, the integrity of the
correction data will be broken and a long sequence of corrections will be useless. The
separate message format also adds difficulties for decoding. For example, to decode the clock
corrections of a single satellite, at least information in four messages should be used,
including messages for PRN mask, full time tag, slow clock corrections, and the fast clock
corrections.
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A simplified format was proposed by JPL and used to encode the IGDG orbit and clock
products. The format has been described in details by Armatys (2002). The IGDG 44-byte
messages are generated at 1 Hz. There are no separate PRN mask and time tag messages. The
44-byte message contains orbit corrections and metre-level clock corrections for one satellite,
and centimetre-level clock corrections for all satellites. Therefore, it is very convenient to
decode the corrections, and one message loss only affects the availability of one satellite. The
bandwidth required to distribute IGDG messages is only 44 byte/s with update interval of 1 s
for clock corrections. The resolutions of IGDG clock corrections is 1/128 m=7.8125 mm,
which may be not high enough for corrections with improved accuracy, as discussed in
section 4.5.2.
4.7.5 Application of Real-Time Orbit and Clock Products
Currently, many wireless technologies are Internet accessible, such as GSM (Global System
for Mobile Communication) and CDPD (Cellular Digital Packet Data). Furthermore, the
third generation wireless network’s devices can provide millions of bits per second in terms
of bandwidth (Fapojuwo, 2003). Therefore, users can adapt wireless devices, which are very
important in kinematic application, to receive orbit and clock products distributed over the
Internet. For example, even CDPD, an older first-generation technology with a bandwidth of
14.4 kbps (Fapojuwo, 2003), is enough to receive GPS•C and IGDG orbit and clock
products.
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The real-time orbit and clock corrections are based on the broadcast ephemeris, which
normally update every two hours. If corrections based on one set of broadcast ephemeris are
applied to another set of broadcast ephemeris, the positioning accuracy will be degraded. The
Issue Of Data Ephemeris (IODE), which exists in broadcast ephemeris with a numerical
range of 0 to 255, is also introduced into the real-time corrections. The IODE items in the
corrections indicate which set of broadcast ephemeris has been used in generating the
corrections. Normally, corrections based on old ephemeris will continue for several minutes
for user accumulating new ephemeris between the ephemeris updates.
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CHAPTER 5
PRECISE POINT POSITIONING USING
SINGLE-FREQUENCY GPS DATA
This chapter investigates precise point positioning method using only single-frequency GPS
data. A new ionospheric estimation model has been proposed which can estimate vertical
TEC accurate to 1~2 TECU and has shown great promise for real-time single-frequency
point positioning. The chapter begins with a brief description of single-frequency point
positioning methods, followed by studying several models used for ionospheric effects
mitigation. The new ionospheric estimation model is then proposed which estimates
horizontal gradients of ionospheric delay using a single GPS receiver. The chapter concludes
with some numerical results to investigate the accuracy of zenith ionospheric delay
estimation and positioning using the new model. Some positioning results using other
ionospheric mitigation models are also presented as a comparison.
5.1 Single-Frequency Point Positioning
Most GPS receivers in use today are single-frequency receivers. Point positioning using
single-frequency observations and precise GPS orbit and clock products has been
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investigated by researchers for a long time (Heroux and Kouba, 1995). However, the precise
GPS orbit and clock products, which have been proven to be significant for point positioning
using dual-frequency measurements, can only offer limited improvement for point
positioning using single-frequency data.
Compared with point positioning using broadcast orbits and clocks, the accuracy of dual-
frequency point positioning has improved from tens of metres with SA on or a couple of
metres with SA off to sub-decimetre by applying precise GPS orbit and clock products. On
the other hand, precise GPS orbits and clocks can just improve the accuracy of single-
frequency point positioning to a couple of metres from about 10 m when broadcast orbits and
clocks are used after SA was turned off (Kouba, 2003). Ionospheric effects are the main
factors for single-frequency users to further improve positioning accuracy using un-
differenced data. Lachapelle et al. (1994b; 1996) presented kinematic point positioning
accuracy of a couple of metres using shipborne and airborne datasets with precise orbit and
clock products, and found ionospheric effects were the dominant error sources. Therefore,
the key issue for single-frequency point positioning is how to mitigate ionospheric effects.
In the past decade, ionospheric products, new mapping functions and analysis models have
been developed to improve the accuracy of single-frequency point positioning. IGS has been
providing the total electron content of ionosphere on a global scale since 1998 (Feltens and
Schaer, 1998). Using the products, Ovstedal (2002) has demonstrated an accuracy of 1~2
metres with IGS precise orbits and clocks. But only code measurements have been used. In
this research, improved accuracy will be presented by using both code and phase
147
observations. Ionosphere-free combination can also be formed on single-frequency GPS
observations, known as GRAPHIC (Group And Phase Ionospheric Correction) (Yunck,
1996a). Montenbruck (2003) has obtained 1.5 m 3D positioning accuracy in LEO (Low Earth
Orbit) satellite orbit determination using this model. In some research, ionospheric delay
parameters have also been estimated as unknowns along with coordinates. Beran et al. (2003)
have presented a couple of metres accuracy using single-frequency observations from a static
geodetic receiver with one or two (a bias and a drift) zenith ionospheric parameters
estimated.
In these researches, because the ionospheric effects, which became the dominant error
sources after precise orbit and clock products were used, could not be mitigated effectively
using un-differenced single-frequency data to fully exploit the precise carrier phase
observables, only accuracies of one metre to a couple of metres were obtained. Therefore, the
PPP methodology, which is promised to provide centimetre level accuracy, is only applicable
to dual-frequency GPS users at present.
Consequently, single-frequency point positioning has not been considered to be able to
provide high precision positioning results at accuracy such as decimetre level. To achieve
decimetre or even centimetre level accuracy using single-frequency observations, differential
methods were widely used, such as Differential GPS (DGPS) and Real-Time Kinematic
(RTK) systems. Differential methods, however, need the supports of base stations with
precisely known coordinates, which not only increase the operational cost of equipment and
labour but it is also not easy to establish in remote survey areas.
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In this research, precise point positioning method using single-frequency GPS data will be
investigated. It aims to derive decimetre to sub-metre accurate position solutions by
processing un-differenced single-frequency code and carrier phase observations with precise
GPS orbit and clock products. To mitigate ionospheric effects, a new ionospheric estimation
model has been proposed and will be described. The new model, which estimates horizontal
gradients along with the zenith delay of ionosphere, has demonstrated its potential to provide
position solutions at decimetre level accuracy using single-frequency data. On the other hand,
several models, such as the Klobuchar model and the Global Ionospheric Model (GIM), will
also be examined by processing both code and carrier phase measurements to investigate the
obtainable accuracy of these models.
5.2 Ionospheric Models
In single-frequency data processing, the observation equation models can be described as
follows:
)()(
)(
112/1
1
PdmDCBTc
dddddtdtcP
PPPgd
reliontroporbs
rsr
ερ
++−+
++++−+= (5.1)
)C(dm)DCBDCBT(c
dddd)dtdt(cC
CC/PP/Pgd
reliontroporbs
rsr
111121
1
ερ
++−−+
++++−+= (5.2)
149
)(w m)N)t()t((
dddd)dtdt(csr
reliontroporbs
rsr
111101011
1
Φ++++−+
+−++−+=Φ
εδδφφλ
ρ (5.3)
where,
1P is the P-Code pseudorange measurement on 1L (m);
1C is the C/A-Code pseudorange measurement (m);
1Φ is the carrier phase measurement on 1L (m);
srρ is the true geometric range (m);
c is the speed of light (m/s);
sdt is the satellite clock error (s);
rdt is the receiver clock error (s);
orbd is the satellite orbit error (m);
tropd is the tropospheric delay (m);
iond is the ionospheric delay on 1L (m);
gdT is the group delay differential of satellite (s);
21 P/PDCB is the differential code bias between 1P and 2P of receiver (s);
11 C/PDCB is the differential code bias between 1P and 1C (s);
reld is the relativistic effects (m);
1wδ is the phase windup on 1L (m);
1λ is the wavelength on 1L (m/cycle);
150
1N is the integer phase ambiguity on 1L (cycle);
)t(r01φ is the initial phase offset of the receiver on 1L (cycle);
)t(s01φ is the initial phase offset of the satellite on 1L (cycle);
1Pdm is the P-code multipath on 1L (m);
1Cdm is the C/A code multipath (m);
1mδ is the carrier phase multipath on 1L (m).
In both GPS receivers and satellites, GPS signals are delayed by hardware. The delays in 1P
and 2P pseudorange measurements are different. For satellites the differences is normally
called group delay differential gdT . For receivers, the term Differential Code Bias ( 21 P/PDCB )
is often used. With double differencing between satellites and receivers, these biases are
removed completely. As discussed in Chapter 3, users do not need to consider these terms
when applying the traditional dual-frequency ionosphere-free combinations, regardless if
they use broadcast ephemeris or IGS precise orbits and clocks. However, for single-
frequency point positioning, they have to be taken into account (Kouba, 2003). Users can
obtain the gdT information from the broadcast navigation messages (Wilson et al., 1999),
while the 21 P/PDCB can be absorbed into receiver clock offset estimation in single-frequency
point positioning. There are also biases between 1P and 1C measurements, namely 11 C/PDCB ,
which are satellites related (Jefferson et al. 2001). As the clock information in broadcast
ephemeris or in IGS precise clock products is fully consistent with P1/P2 code
measurements, for users who use 1C measurements, 11 C/PDCB should also be applied. The
151
values of 11 C/PDCB for all GPS satellites can be downloaded from CODE DCB website
(2004).
Satellite orbit error orbd and clock error sdt can be reduced to centimetre level by precise
orbits and clocks. Zenith tropospheric delay can be modeled to decimetre or even centimetre
level accuracy using simple pressure, humidity and temperature models or meteorological
measurements. It can also be estimated, but the estimation of additional parameters may
degrade the solution if errors in observables have not been mitigated effectively. Relativistic
effects reld and phase windup 1wδ can be modeled to centimetre level accuracy. The initial
phase offsets of both satellite and receiver are absorbed by the float ambiguity estimation.
The ionospheric delay iond , which can be almost removed using dual-frequency
observations, can be mitigated by models discussed in the following when only single-
frequency observations are available. As discussed in Chapter 3, antenna phase center offset
and variations and site displacement effects such as solid earth tides and ocean loading
should be taken into account in precise point positioning using single-frequency data.
Several methods have been developed to deal with ionospheric effects in single-frequency
point positioning, including:
I. Using broadcast Klobuchar model (Klobuchar, 1996);
II. Using Global Ionospheric Model (GIM) provided by IGS and other organizations
(Schaer et al., 1998);
152
III. Using single-frequency ionosphere-free combination (Yunck, 1996a; Montenbruck,
2003);
IV. Estimating ionospheric delay parameters (Beran et al., 2003).
The first and likely the most popular method, is using the Klobuchar model with broadcast
ionospheric coefficients (Klobuchar, 1996). This method can be implemented in real-time or
post-mission using broadcast ephemeris or precise orbit and clock products. The drawback to
the Klobuchar model with broadcast ionospheric coefficients is that it can only mitigate
50~60% of total ionospheric errors (Klobuchar, 1996). Even using precise orbit and clock
products, the end results can only be accurate to a couple of metres (Ovstedal, 2002). Post-fit
ionospheric coefficients have been developed that can help improve the performance of the
Klobuchar model. Since the middle of July 2000, CODE has been providing Klobuchar-style
ionospheric coefficients on a regular basis that best fit its IONEX data. The post-fit
coefficients have demonstrated much better performance than the coefficients originally
broadcast by the GPS for the single-frequency user (Ovstedal, 2002). Currently, CODE post-
fit coefficients have several days’ latency, so they can only be used in post-mission. CODE is
also computing predicted Klobuchar-style coefficients, but the improvement is not as
significant as the post-fit ones.
The second method is to use the Global Ionospheric Model (GIM) provided by IGS and other
organizations (Schaer et al., 1998). Since 1998, IGS has provided ionospheric TEC grid
parameters with a latency of about 11 days. Currently, the IGS ionospheric products can
provide an accuracy of 2 TECU (1 TECU corresponds to 0.163 m range error in 1P ) at grid
153
points (Ovstedal, 2002). But the accuracy does degrade for interpolated points as their
temporal resolution is 2 hours and spatial resolution is 5 degrees (longitude) × 2.5 degrees
(latitude) (IGS website, 2004). As investigated by Ovstedal (2002), this method could
provide better results than Klobuchar model using the same GPS dataset and ephemeris. But
as only code measurements were processed, the achievable accuracy is limited at a couple of
meters. The use of the model is also limited by the 11 days’ latency of the IGS ionospheric
products despite that the precise GPS orbits and clocks have been available with much
shorter latency or even in real-time as discussed in Chapter 4. In the following, this model
will also be used along with both code and phase measurements to further exploit the
potential of the IGS ionospheric products.
The third method is based on the use of a single-frequency ionosphere-free combination
(Montenbruck, 2003). The single-frequency ionosphere-free combination, which averages
the code and carrier-phase measurements on the same frequency, has been introduced as
GRAPHIC (Group And Phase Ionospheric Correction) and can be expressed as follows
(Yunck, 1996a):
222
222
1111101011
21111
)()P(mdm)N)t()t((
)DCBT(cwddd)dtdt(c
P
Psr
P/Pgdreltroporb
sr
Φ++
++
+−+
−+++++−+=
Φ+
εεδφφλ
δρ (5.4)
Though the first-order ionospheric error can be completely removed with the combination,
the phase ambiguity is introduced and the noise of this combination is dominated by the code
noise (about half of the magnitude of the code noise). It is also impossible to solve coordinate
154
parameters and ambiguities using observations from only a single epoch. A global
adjustment using cumulative measurements has to be carried out, and a long convergence
time is required for the float ambiguities, i.e. 2~4 hours (Heroux et al., 2004). This method is
suitable for post-mission with long data tracking sessions. Montenbruck has demonstrated
1.5 m 3D positioning accuracy in determining orbits of LEO satellites. Because this research
is focused on real-time applications, this model will not be used.
The last method is to estimate zenith ionospheric delay using code and/or phase observations
(Beran et al., 2003). Mapping functions are used to map the zenith ionospheric delay to slant
delays. Using post-mission precise GPS orbits and clocks, they obtained an accuracy of a
couple of metres with metre level biases using single-frequency observations from a static
geodetic receiver with one or two (a bias and a drift) estimated ionospheric parameters. This
method is also applicable in real-time navigation using real-time precise GPS orbit and clock
products. However, zenith ionospheric delays for different satellites at ionospheric pierce
points vary significantly, it is not adequate to model ionospheric delays for all satellites using
only one zenith delay and mapping functions. The limitations of using one zenith delay and
mapping functions have been investigated in Klobuchar et al. (1993) who described that
applying mapping function to regions with large horizontal electron density gradients would
lead to errors of several TECU. Therefore, overall this model does not show any
improvement over the simple Klobuchar model since it could only achieve comparable
accuracy by using IGS precise orbit and clock products and both code and phase
155
measurements (Beran et al., 2003). In the next section, estimating ionospheric horizontal
gradients using single-frequency observables from a single GPS receiver will be discussed.
Given in Table 5.1 are the vertical TEC (VTEC) of each satellite at the ionospheric pierce
point at a specific epoch obtained via interpolation from IGS final ionospheric products to the
S1 pillar on the roof of Engineering Building at the University of Calgary. IGS Final
ionospheric products were interpolated to the ionospheric pierce point of each satellite using
the recommended interpolation procedure (Schaer et al., 1998). The epoch is from December
3, 2003, which is an ionospheric quiet day with Ap index of 4. From the table, we can see
that even with a quiet ionospheric condition and at mid-latitude station, the differences in the
VTECs at the ionospheric pierce points of different satellites can be up to 11 TECU.
Table 5.1 VTEC of Each Satellite at the Ionospheric Pierce Point
PRN 1 2 3 8 10 13 27 28 29 30
Azimuth (deg) 153 61 60 286 290 166 34 229 318 111
Elevation (deg) 11 19 32 54 25 39 87 23 12 45
VTEC at ionospheric pierce point (TECU) 25.5 22.9 22.6 20.8 18.3 23.3 22.0 21.1 14.3 23.4
5.3 Precise Point Positioning with Ionospheric Delay Estimated
Beran et al. (2003) could only obtain positioning accuracy of a couple of metres by
estimating zenith ionospheric delay parameters along with coordinates. They have not
considered the horizontal gradients of electron density, which may account for the metre
156
level biases. Giffard (1999; 2000) has also tried to estimate vertical TEC using observables
from a single GPS receiver. Without estimating the horizontal gradients, he only obtained an
accuracy of 3 to 10 TECU of vertical TEC estimates when compared with IGS products.
Horizontal electron density gradients have long been investigated by researchers. As
described in Leitinger (1993), the most typical gradients are:
the general equatorward increase of total electron content (TEC) in mid-latitudes
during daytime;
the west to east increase of TEC in the morning for all seasons;
and the east to west increase in the afternoon in winter.
Though some models, such as the Klobuchar model, have considered these gradients, the
actual gradients can differ considerably from the average values because of the day-to-day
variability of the ionization (Leitinger, 1993). Normally, people use TECU/km to denote
electron content changes versus distance or TECU/deg to denote electron content changes
versus latitude or longitude. Vo and Foster (2001) have shown TEC gradients are correlated
with the background TEC. High gradients values occurred in the sunlit sector with TEC
gradients up to 10 TECU/deg found in the post-noon ionosphere. Hernández-Pajares et al.
(1998) suggested a 2 TECU/deg gradient with low solar activity for tomographic modeling.
Horizontal electron density gradients have been described as a common phenomenon in
middle-latitude region (Gail et. al., 1993), but they have also been investigated in other
regions (Huang, 1997; Ohta and Hayakawa, 2000). Schaer et al. (1999b) and Bock et al.
157
(2000) attempted to introduce ionospheric gradient parameters in GPS network processing,
and they found these parameters might absorb part of the satellite- and epoch-specific biases.
Dai et al. (2001) have also made similar attempts to estimate ionospheric gradient parameters
for ambiguity resolution in the hopes that the ionospheric gradient parameters could absorb a
significant amount of the spatially correlated ionospheric biases. No work has been reported
on estimating ionospheric gradient parameters using un-differenced single-frequency GPS
data.
5.3.1 Estimating Ionospheric Horizontal Gradients with Un-differenced GPS
Measurements
IGS Final ionospheric TEC grids, which are accurate up to 2 TECU or even better at grid
points (Ovstedal, 2002), can be used to demonstrate the gradients numerically. The IGS Final
ionospheric TEC grids for December 31st, 2003 were obtained for this purpose. The Ap index
of that day is 19, which means a typical ionospheric condition. A mid-latitude IGS station,
AMC2 (38.8° N, 104.5° W), was selected to test the ionospheric gradients. The ionospheric
TEC grids were interpolated to the ionospheric pierce points for satellites observed at AMC2
with different azimuth and elevation angles.
Figures 5.1 to 5.4 show VTEC at ionospheric pierce points for satellites observed by AMC2
at different azimuth and elevation angles. The VTEC shown in these figures is consistent
with the three types of gradients discussed above. In Figure 5.1, VTEC was interpolated to
158
the ionospheric pierce points for satellites in the north (0° azimuth) and south (180° azimuth)
with elevation angles from 0° to 90° at 16:00 local time. It presents the equatorward increase
of TEC in the mid-latitudes during daytime. In Figure 5.2, VTEC was interpolated to the
ionospheric pierce points for satellites in the east (90° azimuth) and west (270° azimuth) with
elevation angles from 0° to 90° at 6:00 local time. It highlights the west to east increase of
TEC in the morning. In Figure 5.3, VTEC was interpolated to the ionospheric pierce points
for satellites in the east (90° azimuth) and west (270° azimuth) with elevation angles from 0°
to 90° at 18:00 local time. It illustrates the east to west increase of TEC in the afternoon. In
Figure 5.4, VTEC was interpolated to the ionospheric pierce points for satellites at an
elevation angle of 30° with azimuth angles from 0° to 360° at different local time. It shows
the general TEC changes against the azimuth angles at different local time periods.
If no horizontal gradients existed, the vertical TEC should be the same at the ionospheric
pierce points for satellites with different elevation angles and different azimuth angles. The
tangent-like curves in Figures 5.1, 5.2 and 5.3 and the cosine-like curves in Figure 5.4
indicate that ionospheric horizontal gradients are present at different times and directions.
159
Figure 5.1 The Equatorward Increase of TEC at Local Time 16:00
Figure 5.2 The West to East Increase of TEC at Local Time 6:00
160
Figure 5.3 The East to West Increase of TEC at Local Time 18:00
Figure 5.4 The VTEC at Ionospheric Pierce Point for Satellites at 30°°°° Elevation Angle against
Azimuth at Different Local Time
161
The cosine-like behaviour of the VTEC gradients in Figure 5.4 and the tangent-like
behaviour of the VTEC gradients from Figures 5.1 to 5.3 suggest that mapping functions can
be derived by tilting the zenith.
Following Klobuchar (1996) and Davis et al. (1993), the ionospheric group delay in the
measurement to a satellite whose signal penetrates the ionosphere at altitude z and
horizontal position vector x
as measured from the site, can be expressed as:
( ) ( )
( ) ( )
( ) ( )α
α
,eded
dlxzNf
.dlzN
f.
dllNf
.,ed
g
g
ion
+=
⋅+=
=
0
202
2
340340
340
(5.5)
where e is the elevation angle, α is the azimuth angle, 0N is the electron density with
absence of gradients and gN
is the horizontal gradient of electron density at x
=0, and
( ) ( ) ( )eFddlzNf
.ed z 0020
340 == (5.6)
( ) ( ) ⋅= dlxzNf
.,ed gg
2
340α (5.7)
( )ed 0 , zd and ( )eF0 are the slant group delay, zenith delay and mapping function with the
absence of horizontal gradients of electron density, when the group delay is independent of
azimuth. ( )α,ed g is the delay caused by the horizontal gradients of electron density.
162
( )α,ex
, ( )zN g
, and dl can be expressed as (Davis et al., 1993):
( ) ( )ααα sinecosnecotz,ex +≈ (5.8)
( ) ( ) ( )ezNnzNzN eng +=
(5.9)
( )dzeFdl 0≈ (5.10)
where n and e are unit vectors in north and east directions, nN and eN are the horizontal
gradients of electron density in north and east directions, respectively.
Therefore, the gradient delay can be calculated as:
( ) ( )
( ) ( )
( ) [ ]αα
αα
α
sinGcosGecoteF
dzzzNf
.sindzzzN
f.
cos
ecoteF,ed
en
en
g
+=
+×
≈
0
22
0
340340 (5.11)
where nG and eG are the horizontal gradients in north and east directions respectively, and
they can be expressed as:
( )
( )
=
=
dzzzNf
.G
dzzzNf
.G
ee
nn
2
2
340
340
(5.12)
Therefore, the total ionospheric group delay for satellite with elevation e and azimuth α is:
163
( ) ( ) ( ) [ ]ααα sinGcosGecoteFeFd,ed enzion ++= 00 (5.13)
If a single layer model mapping function is used, ( )eF0 can be expressed as (Manucci et al.,
1993):
( )( )
20
1
1
+−
=
ecosHR
ReF
E
E
(5.14)
where ER is the mean earth radius (6371 km) and H is the ionospheric shell height, which
can be roughly selected between 300 km and 500 km (Schaer, 1999a). For example, the value
of 450 km has been used by IGS ionospheric products. Additional details about mapping
functions will be discussed in Section 5.3.2.
The new model, as shown in Equation 5.13, can be used to estimate zenith ionospheric delay
along with the horizontal gradients using single-frequency code and carrier phase
measurements. It is denoted as “ionospheric estimation model” in the thesis. In Section 5.3.3
and Section 6.5, the performance of this model will be compared with the Klobuchar model
and the Global Ionospheric Model.
5.3.2 Ionospheric Mapping Functions
Except for the ionosphere-free model, all the previously discussed models need to use
ionospheric mapping functions. Several mapping functions are used with slight differences at
164
low elevation. These mapping functions include the broadcast model, Single Layer Model
(SLM) and Modified Single Layer Model (MSLM) mapping functions.
The broadcast model mapping function is given as (Klobuchar, 1987):
−⋅+=90
9621
e)e(F (5.15)
where e is the elevation angle.
The single-layer model (SLM) mapping function was given in Equation 5.14. The SLM
assumes that ionospheric electron density is concentrated on a thin shell of height H above
the mean Earth radius ER (Manucci et al., 1993). In practice, the shell height can be selected
from 300 to 500 km. 350, 400 and 450 km are most commonly used, and are denoted as
SLM350, SLM400 and SLM450 in this thesis.
The modified single-layer model (MSLM) mapping function can be expressed as (Schaer,
1999a):
( )( )( )
2
901
1
−
+−
=
esinHR
ReF
E
E β (5.16)
where β is a correction factor which is close to unity. In this research, the coefficients,
which best fit the JPL extended slab model mapping function, are adopted with H = 506.7
165
km and β = 0.9782 (CODE Ionosphere Map website, 2004). It is denoted as MSLM506 in
the thesis.
The mapping functions discussed above differ primarily at low elevation angles, e.g., lower
than 15°. At high elevation angles, they are very close. This behaviour can be seen in Figure
5.5 which shows the mapping functions. To highlight the differences more clearly, using the
IGS selected SLM450 as a reference, the values from which other mapping functions differ
are plotted in Figure 5.6 using a large scale. At elevation greater than 15°, the differences of
all mapping functions to SLM450 are less than 0.2. Even at elevation 5°, the differences are
still less than 0.3, which is less than 10% of the value of the mapping functions at this
elevation. It is interesting to see that at all elevations, MSLM506 is smaller than SLM450,
SLM350 and SLM400 are bigger than SLM450. On the other hand, the broadcast model
mapping function is smaller than SLM450 at elevations less than 55°, but bigger than
SLM450 at elevation greater than 55°. Because of the strong correlation existing between the
mapping functions and the zenith ionospheric delay estimates, these mapping functions may
provide similar position solutions but different zenith ionospheric delay estimates. The
numerical results will further confirm this.
166
Figure 5.5 Mapping Functions
Figure 5.6 Mapping Function Difference
167
In this research, the broadcast model mapping function will be used along with Klobuchar
model. The Global Ionospheric Model will adopt SLM450 to keep consistent with IGS
ionospheric products. The same mapping function will also be used in the proposed
ionospheric estimation model, because the use of SLM450 provides better position solutions
than other mapping functions as tested in Section 5.3.3.
5.3.3 Results of Positioning and Ionospheric Delay Estimation
To test the performance of the ionospheric estimation model in positioning and ionospheric
delay estimation, a dataset collected in GPS week 1251 from IGS station AMC2 (38.8° N,
104.5° W, 1911.4 m) was processed. The necessary GPS observation files and IGS Final
orbit and clock products were obtained from the IGS website. A dual-frequency receiver was
used at AMC2 station but only single-frequency observations, 1P and 1Φ , were processed.
The P3-RT software package, which will be discussed in Chapter 6, was used to conduct the
processing. The ionospheric activity was typical throughout the selected week. The Ap
indices are shown in Table 5.2.
The index Kp and its related indices ap and Ap have been widely used to measure the world
wide geomagnetic activity. The three-hour-range Kp is obtained as the mean value from the
standardized K index (Ks) of 13 magnetic observatories. The three-hour-range ap index is
derived from the Kp index by a linear scale. The Kp index can be one of the 28 values ranged
from 0o to 9o. The corresponding ap index is ranged from 0 to 400. To measure the daily
168
world wide geomagnetic activity, the daily Ap index is usually used, which is the average of
the 8 three-hour-range ap indices in one day. In this thesis, the Ap index is obtained from the
International Service of Geomagnetic Indices (ISGI) website (2004).
Table 5.2 Ap Indices of GPS Week 1251
Day 28 29 30 31 1 2 3
Ap index 13 5 6 19 24 15 22
The zenith ionospheric delay, zd , was modeled as a random walk process with a spectral
density of h/m2 . The gradient parameters, nG and eG , were also modeled as random
walk processes with a spectral density of h/dm2 . An elevation cut-off angle of 7° was
used. The receiver’s clock was modeled as a white noise process. The float ambiguity
parameters were estimated as constants. Station coordinates were estimated as random walk
processes with a spectral density of h/km60 . The sample interval of GPS measurements is
30 s.
The positioning errors are shown in Figure 5.7 while Figures 5.8, 5.9 and 5.10 show the
estimated zenith ionospheric delay and horizontal gradients in the east and north respectively.
As shown in Table 5.3, about 30 cm accuracy in the horizontal components and 50 cm
accuracy in the vertical component were obtained at this mid-latitude station for one week. In
Figure 5.7, the correlation between ionospheric conditions and positioning errors was also
indicated. The Ap indices are bigger in December 31st, January 1st and January 3rd than in
169
other days. As a result, the positioning errors are bigger in these days as compared to other
days.
Figures 5.8 to 5.10 have clearly shown the daily ionospheric activity at AMC2. The half
cosine patterns in Figure 5.8 show the diurnal variations of TEC, with some disturbances.
Figures 5.9 and 5.10 show decimetre level gradients in the east and north directions, and they
also indicate three typical ionospheric gradients found at mid-latitude stations. Figure 5.9
indicates that TEC increases from west to east in the morning and from east to west in the
afternoon. A positive gradient in the morning (about 9:00 local time) and a negative gradient
in the afternoon (about 18:00 local time) were obtained in the east direction daily. The
negative gradients in the north direction shown in Figure 5.10 indicate TEC increases
towards equator at mid-latitude regions.
The zenith ionospheric delay estimates were also transferred to VTEC and compared with
IGS Final ionospheric products. Estimates obtained at every 2 hours were used for the
comparison as that matched the temporal resolution of IGS ionospheric TEC grids. The first
estimate at GPS time (1251, 0) was also omitted because of the un-converged ambiguity. The
comparison was shown in Figure 5.11. The unit is in 0.1 TECU to be consistent with IGS
products. The RMS difference is about 1.2 TECU with a bias of about -0.6 TECU.
170
Figure 5.7 Positioning Errors Using Ionospheric Estimation Model for GPS Week 1251
Table 5.3 Positioning Accuracy Using Ionospheric Estimation Model with SLM450
RMS (m) BIAS (m) STD (m) Latitude 0.260 -0.097 0.241
Longitude 0.248 -0.084 0.234
Height 0.486 -0.050 0.483
171
Figure 5.8 Zenith Ionospheric Delay Estimates for GPS Week 1251
Figure 5.9 Ionospheric Gradients in the East Direction
172
Figure 5.10 Ionospheric Gradients in the North Direction
Figure 5.11 Vertical TEC Comparison
To compare the performance of different mapping functions for positioning and ionospheric
delay estimation, the same dataset was also processed using the broadcast model mapping
173
function, SLM350, SLM400 and MSLM506. The positioning accuracy statistics are shown in
Table 5.4. The VTEC estimation statistics, as compared with IGS ionospheric products, are
presented in Table 5.5.
Table 5.4 Positioning Accuracy Using Ionospheric Estimation Model with Different Mapping Functions (Unit: m)
Broadcast SLM350 SLM400 SLM450 MSLM507
RMS BIAS RMS BIAS RMS BIAS RMS BIAS RMS BIAS
Latitude 0.264 -0.095 0.265 -0.092 0.260 -0.094 0.260 -0.097 0.268 -0.103
Longitude 0.254 -0.097 0.260 -0.098 0.253 -0.091 0.248 -0.084 0.243 -0.067
Height 0.490 -0.066 0.504 -0.128 0.489 -0.086 0.486 -0.050 0.502 0.009
Table 5.5 Statistics of VTEC Estimation with Different Mapping Functions (Unit: 0.1 TECU)
Broadcast SLM350 SLM400 SLM450 MSLM507
RMS 20.60 21.69 17.53 12.28 11.25
BIAS -18.19 -19.39 -14.37 -5.89 -0.05
As shown in Tables 5.4 and 5.5, different mapping functions provide similar positioning
accuracy with small differences in the height component, even though their VTEC estimates
vary significantly. The ionospheric estimation model provides the best positioning results
when using SLM450, but its VTEC estimates are not the best among these mapping
functions when compared with the IGS ionospheric products. On the other hand, the VTEC
estimates using MSLM507 are most consistent with the IGS ionospheric products, which use
the SLM450 mapping function. Using the same mapping function, SLM450, a -0.6 TECU
bias between the ionospheric estimation model and IGS ionospheric products exists. The
accuracy of IGS ionospheric grids, which were selected as references in this research, is
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about 2 TECU at the grid points at a sample interval of 2-hour. The accuracy degrades after
interpolation to the ionospheric pierce point at a selected epoch. The biases may also be due
to the errors in the IGS ionospheric grids. The accuracy of vertical TEC estimation will be
further investigated if more accurate ionospheric products are available to be used as
references.
The broadcast model, SLM300 and SLM400 seem to have underestimated the vertical TEC
with negative biases of 1 to 2 TECU when compared to the IGS ionospheric products. The
mapping function comparison in Figure 5.6 can be used to explain the biases. As shown in
Figure 5.6, SLM300 and SLM400 are bigger than SLM450, and the broadcast model is also
bigger than SLM450 at elevations less than 55°. When mapped to the slant TEC, the
underestimated vertical TEC is counteracted by the overestimated mapping functions. As a
result, these mapping functions provided similar slant ionospheric group delays or phase
advances for observations, and offered similar positioning results.
To compare the positioning accuracy with other models, the dataset was also processed using
the Klobuchar model and GIM. An elevation cut-off angle of 10° was used instead of the 7°
used for the ionospheric estimation model to estimate the horizontal gradients. The settings
for coordinate and receiver clock parameter estimation are the same as used in the
ionospheric estimation model. The zenith tropospheric delay, if estimated, is estimated as a
random walk process with a spectral density of h/cm1 . The float ambiguity parameters, if
estimated, were estimated as constants. The results are shown in Tables 5.6 and 5.7.
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When using the Klobuchar model, two processing strategies have been tested. The first
processing strategy uses only code observables, while the second one uses both code and
phase observables. As shown in Table 5.6, the phase observables can slightly improve the
positioning accuracy for this mid-latitude station. The improvements are mainly in the
vertical component. As will be seen in Chapter 6, the phase observables would not
necessarily improve the position solutions all the time when using Klobuchar model.
Another processing strategy has been tested using GIM in addition to the code processing,
and the processing using both code and carrier phase. Using both code and phase
observables, it estimates zenith tropospheric delay (ZTD) instead of modeling it as done in
the first two processing strategies. As shown in Table 5.7, the phase observables can improve
the positioning accuracy but estimating zenith tropospheric delay does not improve the
accuracy. When processing using the Klobuchar model, because of the big ionospheric
residuals, ZTD estimation is not recommended. Because there are 7 parameters (3
coordinates, 1 receiver clock offset and 3 ionospheric parameters) to be estimated in the
ionospheric estimation model in addition to the float ambiguity parameters, estimating ZTD
will degrade the solution if errors in measurements have not been mitigated effectively. With
this in mind, the zenith tropospheric delay will not be estimated in the following processing,
it will be modeled using Saastamoinen model and meteorological settings.
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Table 5.6 Positioning Accuracy Using Klobuchar Model
Code (m) Code & Phase (m) Latitude 0.872 0.613
Longitude 0.304 0.978
Height 1.743 1.056
Table 5.7 Positioning Accuracy Using GIM
Code (m) Code & Phase (m) Code & Phase with ZTD estimated (m)
Latitude 0.501 0.319 0.319
Longitude 0.221 0.352 0.354
Height 0.756 0.417 0.436
From Tables 5.3, 5.5, 5.6 and 5.7, several conclusions can be made. VTEC (or zenith
ionospheric delay) can be estimated at 1~2 TECU accuracy using un-differenced single-
frequency GPS observations at mid-latitude region, which has great promise for single-
frequency point positioning. Decimetre level accuracy is obtainable using single-frequency
measurements. Phase observables can improve positioning accuracy at mid-latitude stations
using the GIM, although estimating ZTD cannot improve the positioning accuracy. The
improvement of using phase observables to the Klobuchar model is not obvious. The
ionospheric estimation model and GIM perform better than the Klobuchar model. The
ionospheric estimation model can provide comparable accuracy to GIM, which uses the IGS
ionospheric products with a latency of about 11 days. Because several types of real-time
precise GPS orbit and clock products have become available with comparable accuracy to the
IGS post-mission orbits and clocks, the accuracy obtained using the ionospheric estimation
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model and Klobuchar model is obtainable in real-time. In Section 6.5, single-frequency
precise point positioning results will be presented using JPL real-time GPS products.
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CHAPTER 6
NUMERICAL RESULTS AND ANALYSIS
This chapter will present numerical results for precise point positioning, receiver clock offset
estimation and atmospheric sensing using real-time orbit and clock products. Section 6.2 will
show position determination results using dual-frequency measurements. Datasets were
collected from receivers at a fixed site as well as on a vehicle and aircraft. These datasets are
processed to show the performance of positioning under different dynamic environments.
The results of receiver clock offset estimation are shown in Section 6.3. Section 6.4 shows
zenith tropospheric delay estimation and water vapour sensing results. The results of precise
point positioning using single-frequency measurements are presented in Section 6.5. Some of
the processing was conducted in real-time using real-time orbit and clock products received
over the Internet or a serial port, while other processing was done in a simulated real-time
mode with products generated for real-time applications. All processing tasks were
accomplished by the P3-RT software package, which will be described in the next section.
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6.1 Software Development and Parameter Modeling
A software package, P3-RT, has been developed to perform all processing tasks required in
this thesis. Details of the software and the modeling for parameters are described in the
following.
6.1.1 P3-RT Software Package
Interface
P3-RT was developed using C++ on the Microsoft Windows Operating System and features
a user-friendly interface. Figure 6.1 shows the interface of P3-RT for setting up new
processing tasks.
In the Settings dialog, user can set basic settings such as elevation cut-off angle and
measurement noise. If a standardized antenna name is selected, antenna phase center offset
and variations would be applied in the software according to the parameters in the
standardized IGS antenna table.
Initial coordinates can also be set for the processing of static positioning, timing or
atmospheric sensing. In static positioning, if the precisely known coordinates are given, the
coordinates can be used to calculate accuracy statistics. In timing and atmospheric sensing,
precisely determined coordinates can be used to fix the position parameters, which thus can
be removed from the unknowns in post-mission PPP processing. In this research however,
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coordinates are still estimated in timing and atmospheric sensing data processing in real-time
or simulated real-time mode.
Estimating ZWD and modeling ZHD are recommended in this thesis when dual-frequency
data is used. Meteorological settings can be used to calculate an apriori ZWD. User can also
choose to model both ZHD and ZWD based on precise meteorological measurements which
can be input from a source file. But as discussed in Section 3.2, the spatial and temporal
variability of water vapour makes modeling ZWD difficult and cannot be modeled any better
than 1~2 cm even when using precise meteorological measurements.
Finally, users can also exclude “unhealthy” satellites from the processing manually in post
processing.
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Figure 6.1 P3-RT Interface – Setup
Figures 6.2 and 6.3 show the interface of P3-RT during kinematic and static processing,
respectively.
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Figure 6.2 P3-RT Interface – Kinematic Processing
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Figure 6.3 P3-RT Interface – Static Processing
Figures 6.4 and 6.5 show the interface of P3-RT for results analysis. Figure 6.4 displays the
trajectory in horizontal and vertical components separately for kinematic positioning. If
ground-truth coordinates, such as position solutions from double differencing techniques, are
provided, the position errors can also be displayed as shown in Figure 6.5.
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Figure 6.4 P3-RT Interface – Trajectory
Figure 6.5 P3-RT Interface – Position Errors
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Real-Time and Post-Mission Capability
The primary purpose of P3-RT is for real-time processing. Currently, it can also perform
post-mission processing using precise products (SP3 format orbits and RINEX format
clocks) from IGS or other agencies, or simulated real-time processing using products
generated for real-time applications. All of these modes have been used in this research.
In real-time mode, GPS raw data can be accessed through a computer’s serial port or over the
Internet. Acquiring GPS raw data over the Internet is useful for atmospheric sensing, in
which data from a GPS network is normally processed. Currently, the data stream from
several types of receivers can be processed directly in P3-RT. These include receivers
manufactured by Javad, Leica, Trimble, etc. More data stream formats will be supported in
the future. Precise GPS orbit and clock products can also be received through the serial ports
or over the Internet. Two formats of real-time orbit and clock products are currently
supported. These formats are the GPS•C format and the IGDG format discussed in Chapter
4. Figure 6.6 shows the real-time PPP settings.
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Figure 6.6 Real-Time PPP
In post-mission mode, GPS raw data can be input in the standard RINEX format. More
precise GPS products are also supported in post-mission. Orbits in the SP3 format and clocks
in RINEX format from IGS or other agencies at any sample intervals are supported. Products
with a low sample rate for orbits and clocks would be automatically interpolated to the time
tag of GPS measurements. Though designed for real-time applications, the high sample rate
orbit and clock products in the GPS•C and IGDG formats can also be used in post
processing. Figure 6.7 shows the PPP settings for post-mission.
In simulated real-time mode, only previously logged real-time orbit and clock products in the
GPS•C and IGDG formats are supported. The software intentionally delays the logged real-
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time orbit and clock products for a period. This delay is used to simulate the real-time latency
of the products that would be found in a normal real-time process. The simulated real-time
processing has also been used by other researchers when a pure real-time processing is
impractical (Muellerschoen et al., 2001). In this research, the test for receiver clock offset
estimation was conducted in this simulated real-time mode because the GPS raw data of
station AMC2 was not available in real-time.
Figure 6.7 Post-Mission PPP
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Positioning, Timing and Atmospheric Sensing with P3-RT
P3-RT is a realization of PPP methodology. It can be used in positioning, timing and
atmospheric sensing applications in real-time and post-mission.
For positioning applications, P3-RT can offer position solutions using dual-frequency or
single-frequency measurements. As will be demonstrated later in this chapter, it can provide
sub-centimetre level static position solutions or sub-decimetre level kinematic position
solutions in real-time or post-mission using dual-frequency receivers. The models discussed
in Chapter 5 have been implemented into P3-RT for single-frequency data processing. Using
single-frequency measurements, sub-metre level kinematic positioning accuracy has also
been obtained.
Sub-nanosecond level receiver clock offset estimates that will be presented in Section 6.3
show the promise of P3-RT in timing applications.
Sub-centimetre level ZTD estimates will be shown in Section 6.4. Provided precise pressure
and temperature measurements are available in real-time, P3-RT is capable of outputting sub-
millimetre level PWV estimates in real-time.
6.1.2 Modeling for Parameters
In P3-RT, invariable parameters, such as the receiver coordinates in static mode and the
ambiguity parameters, are estimated as constants. The receiver clock offset is modeled as a
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white noise process. Other variable parameters, such as the coordinates in kinematic mode,
the tropospheric parameters and/or the ionospheric parameters, can be modeled as random
walk processes or first-order Gauss-Markov processes. Typical modelling settings for the
processing in this research are discussed in the following.
In this research, the receiver coordinate parameters are estimated as constants in the static
mode. For kinematic positioning, they are modeled as random walk processes with spectral
densities that will be defined according to the actual dynamic environments. For example, for
land vehicles, spectral densities of h/km60 and h/km6 can be used for horizontal and
vertical components respectively. While for airborne datasets, spectral densities of
h/km100 ~ h/km1000 and h/km20 can be used for horizontal and vertical
components respectively. For shipborne datasets, spectral densities of h/km60 and
h/m60 can be used for horizontal and vertical components respectively. When static
datasets are processed in an epoch-by-epoch mode, a spectral density of h/km60 is used
for both horizontal and vertical components.
The tropospheric zenith wet delay, if estimated, is modeled as a random walk with a spectral
density of h/mm6 to h/cm2 depending the dynamic environments and meteorological
conditions. The tropospheric horizontal gradient parameters, if estimated, are modeled as
random walk processes with a spectral density of h/mm.30 .
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In single-frequency point positioning using the ionospheric estimation model, the zenith
ionospheric delay is modeled as a random walk with a spectral density of h/m2 . The
ionospheric horizontal gradient parameters are modeled as random walk processes with a
spectral density of h/dm2 .
The receiver clock is modeled as a white noise process. The float ambiguity parameters are
estimated as constants.
The apriori uncertainties for parameters would be set based on the initial values of the
parameters. For example, the uncertainties for coordinate parameters can be safely set as 100
km if only approximate coordinates are provided.
The initial values of the float ambiguities are normally calculated from the code
measurements. An uncertainty of 10 m is used in this research.
The initial value of the tropospheric zenith wet delay is calculated from meteorological
settings. Because the zenith wet delay is normally at the decimetre level, plus the residual of
the zenith hydrostatic delay, which should be less than one decimetre using a pressure model
that will be discussed in Section 6.4, the uncertainty of the zenith wet delay can be set as
0.1~1 m. For the tropospheric horizontal gradients, which are normally at the centimetre
level, an uncertainty of 0.01 m is used.
In single-frequency point positioning using the ionospheric estimation model, the initial
value of the zenith ionospheric delay is calculated from the broadcast Klobuchar model.
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Considering that the model performance and typical zenith ionospheric delay, the uncertainty
of the zenith ionospheric delay can be set as 3 m. For the ionospheric horizontal gradients,
which are normally at the decimetre level, an uncertainty of 0.1 m is used.
Normally, a large uncertainty should used for receiver clock offset, i.e., 3×105 m in range or
1 ms in transmission time. This is due to the fact that most receivers try to synchronize their
internal clocks with GPS time within 1 ms.
6.2 PPP Using Dual-Frequency Measurements
In this section, several datasets collected using dual-frequency GPS receivers are processed
using real-time orbit and clock products, including JPL IGDG products, NRCan GPS•C code
solution products and GPS•C phase solution products. Some datasets are processed in
simulated real-time or post-mission mode because GPS measurements or orbit and clock
products are not available in real-time.
6.2.1 Static PPP Using IGDG Products
A real-time static test was carried out on December 3rd, 2003. One Javad Legacy dual-
frequency receiver was set up on the S2 pillar on the roof of the Engineering Building at the
University of Calgary. JPL IGDG orbit and clock products were received over the Internet.
The sample interval was set to 10 s. The coordinates of S2, which were determined in ITRF-
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1993 with centimetre level accuracy, were transformed to ITRF-2000 using coefficients
provided by McCarthy and Petit (2004) and used as true coordinates. The positioning errors
are shown in Figure 6.8. The accuracy statistics of the position solutions after convergence is
presented in Table 6.1.
It took approximately 20 minutes for the positioning errors to converge to the decimetre
level. But it took more than 1 hour to converge to the centimetre level. The long convergence
time may be caused by multipath effects on the roof, which have been investigated in Ray
(2000). The antenna used in this test is a JPSLEGANT antenna with a small ground plane,
which can partially mitigate the multipath effects. After convergence, the RMS of the
positioning errors is about 2 cm in each positioning component. The centimetre level biases
would be partially caused by the reference coordinates. The coordinates of S2 were
determined several years ago with an accuracy of centimetre level.
Figure 6.8 Real-Time Static Positioning Using IGDG Products
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To further test the obtainable accuracy of PPP in static mode using JPL IGDG products,
accurate coordinates for the test stations in ITRF-2000 are required to serve as the ground-
truth. The coordinates of IGS stations are estimated on a daily basis in the ITRF-2000 frame
currently. IGS stations are also normally set up in multipath-friendly environment with a
Choke Ring antenna. Raw data and coordinates of these stations can be downloaded from
IGS website. In this test, GPS data acquired for the day of August 4, 2004 at IGS station
ALGO was processed in a simulated real-time mode using the JPL real-time orbit and clock
products acquired from JPL server. The positioning results are shown in Figure 6.9 and the
accuracy statistics for results after convergence are given in Table 6.1. It can be seen that the
coordinate estimates could converge to the centimetre level within 30 minutes. This
convergence time is much shorter than the convergence time with data affected by multipath.
After the convergence, all position coordinate components are accurate at the sub-centimetre
level. The results in Table 6.1 indicate that PPP is capable of providing real-time sub-
centimetre level accuracy for static control survey.
Figure 6.9 Static Positioning Using IGS Dataset
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Table 6.1 Accuracy Statistics of Static Positioning Results (Unit: cm)
Real-time dataset IGS dataset
Latitude 2.3 0.9
Longitude 1.4 1.0 RMS
Height 2.6 0.7
Latitude -1.7 0.8
Longitude -1.2 0.3 BIAS
Height 2.4 0.0
Latitude 1.6 0.3
Longitude 0.8 0.9 STD
Height 0.8 0.7
6.2.2 Kinematic PPP Using IGDG Products
Two kinematic datasets were processed using JPL IGDG orbit and clock products. The first
dataset was collected on a vehicle and processed in real-time. Another dataset was collected
by an aircraft and processed in a simulated real-time mode.
Land Vehicle Kinematic Positioning
A kinematic positioning test using a land vehicle was conducted on September 30th, 2003.
The vehicle was driven along the highway at a speed of 80 km/h near Springbank, Alberta. In
order to establish a reference trajectory for the vehicle, a reference receiver was set up at one
control point of the Springbank Baseline Network so double difference data processing could
be performed to establish a reference for accuracy assessment. Both the control point and
vehicle used Javad Legacy dual-frequency receivers with the JPSLEGANT antennas. A
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CDPD radio was used to receive JPL IGDG real-time precise orbit and clock products via the
Internet. The sample rate of the two GPS receivers was set to 1 Hz. The PPP solutions are
obtained using P3-RT software while the double difference solutions are obtained using the
commercial software package GrafNav from the Waypoint Consulting Inc. With a relatively
short baseline length (7 km on average and maximum of 12 km), the ambiguity-fixed
position results from the GrafNav can serve as the ground-truth to assess the positioning
accuracy of PPP solutions.
The positioning differences between PPP and double difference solutions are shown in
Figure 6.10, the trajectory of vehicle is presented in Figure 6.11, and the positioning accuracy
statistics after convergence is given in Table 6.2. They indicate that sub-decimetre accurate
positioning results have been obtained in real-time using the precise point positioning method
with a convergence time of about half an hour.
Figure 6.10 Positioning Errors with Vehicle Dataset
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Figure 6.11 Vehicle Trajectory on September 30th, 2003
Airborne Kinematic Positioning
The airborne dataset, which was provided by the Mosaic Mapping System Inc. (2004), was
collected on August 28th, 2004 at 40 km north of Halifax, Nova Scotia. A NovAtel Black
Diamond GPS receiver and Model 512 antenna were set up on a helicopter. The sample rate
of the two GPS receivers was 1 Hz. The helicopter was flying at an altitude of approximately
250 m above the ground at 50 knots. The distance between the rover and base is less than 10
km. During the test, a NovAtel DL-4 receiver and an antenna with a ground plane were used
as the base station. The double-difference ambiguity-fixed position solutions, provided by
Mosaic Mapping System Inc. (2004), have been used as the ground-truth.
The positioning errors, trajectory and accuracy statistics after position convergence are
shown in Figure 6.12, Figure 6.13, and Table 6.2.
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Figure 6.12 Positioning with Aircraft Dataset
Figure 6.13 Aircraft Trajectory on August 28th, 2004
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Table 6.2 Accuracy Statistics of Kinematic Positioning Results (Unit: cm)
Vehicle dataset Aircraft dataset
Latitude 7.8 2.8
Longitude 5.9 6.8 RMS
Height 7.9 4.9
Latitude 3.6 -0.2
Longitude -3.5 -1.5 BIAS
Height -0.7 -1.5
Latitude 6.9 2.8
Longitude 4.8 6.7 STD
Height 7.9 4.6
As shown in Figures 6.10 and 6.12, about 20 to 30 minutes was taken for the positioning
errors converge to the decimetre level in kinematic mode. After convergence, sub-decimetre
level accuracy was obtained in each positioning component. These tests confirmed that PPP
is a very promising technique for high precision real-time kinematic positioning.
The results obtained from processing the aircraft dataset are even better than those obtained
from the vehicle dataset. It took only 20 minutes for the positioning errors to converge to the
decimetre level. Satellite geometry, observation environment, and improvement in the IGDG
orbit and clock products are possible reasons for the better results.
Shown in Figures 6.14 and 6.15 are the satellite geometry for the aircraft and vehicle
datasets. It is obvious that the observation environment in the air is much better than that on a
vehicle. Though at least 5 satellites were tracked when driving along the high way, many
loss-of-locks were found in the vehicle dataset because signals were blocked by hills or
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buildings. The loss-of-locks definitely degraded the results because it takes time for the float
ambiguities of the new tracked satellites to converge.
The aircraft dataset was collected about 1 year later than the real-time vehicle testing. More
tracking stations have been added to the JPL IGDG real-time tracking network during this
period and new processing strategies were adopted by JPL in April 2004 (Muellerschoen,
2004). All these led to improvements of JPL IGDG products.
Figure 6.14 Satellite Geometry for Vehicle Dataset on September 30th, 2003
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Figure 6.15 Satellite Geometry for Aircraft Dataset on August 28th, 2004
6.2.3 PPP Using GPS••••C Products
In this section, the two types of NRCan precise orbit and clock products were tested in real-
time and post-mission for precise point positioning.
Positioning with GPS••••C Code Solution Products
GPS•C code solution products are broadcast over satellite and the Internet in real-time
(Kassam, 2003; Chen et al., 2002). Using orbit and clock products received over the Internet,
half-metre accuracy has been obtained using data from IGS stations. The accuracy was
degraded to the metre level when processing a dataset collected on the roof of Engineering
Building because of multipath effects in code measurements (Chen et al., 2002).
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In this test, a CPGPS radio was used to receive GPS•C code solution orbit and clock
products. The real-time test was carried out on December 2nd, 2003. The settings are the same
as the real-time static test on December 3rd, 2003 that has been described in Section 6.2.1.
The phase smoothed, ionosphere-free code observables were processed in real-time in an
epoch-by-epoch mode. The coordinates from PPP processing, which were based on NAD83,
were transformed into ITRF93 using coefficients provided by Kouba (2002) and compared
with the true coordinates. The positioning errors and satellite geometry are shown in Figures
6.16 and 6.17. The accuracy statistics are presented in Table 6.3.
The positioning errors are correlated with the satellite geometry. Sub-metre level accuracy
was achieved in real-time. The positioning errors were dominated by the measurement noises
and multipath effects in the ionosphere-free code combinations, and errors in the GPS•C
code solution orbit and clock products.
Figure 6.16 Kinematic Positioning Using GPC••••C Code Solution Products
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Figure 6.17 Satellite Geometry of S2 on December 2nd, 2003
Positioning with GPS••••C Phase Solution Products
GPS•C phase solution orbit and clock products are still at the testing stage. Their latency is
still several hours, and they are not continuously available because of occasionally missing
real-time raw data from several tracking stations (Collins, 2004). For this test, several days
worth of orbit and clock products were provided by NRCan. One dataset collected on S1
pillar on the roof of Engineering Building on June 10th, 2004 using a Javad receiver was
processed using the phase solution products with a sample interval of 10 s. Dual-frequency
code and phase measurements were used for position determination in an epoch-by-epoch
mode. The positioning errors are presented in Figure 6.18 and Table 6.4.
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The positioning accuracy and convergence time are comparable with those obtained using the
kinematic dataset with the IGDG orbit and clock products. But the GPS•C products are still
not as consistent as the IGDG products. The large errors during the time period from GPS
time 403200 s to 417600 s were due to the degraded orbit and clock products (Collins, 2004).
Currently only 20 stations are used by NRCan to estimate the phase solution orbit and clock
products as compared to over 60 stations used by JPL for IGDG products computation.
Figure 6.18 Kinematic Positioning Using GPC••••C Phase Solution Products
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Table 6.3 Accuracy Statistics of Positioning Using GPS••••C Products (Unit: m)
Code Solution
(December 2, 2003) Phase Solution (June 10, 2004)
Latitude 0.426 0.048
Longitude 0.232 0.065 RMS
Height 0.686 0.065
Latitude 0.122 0.010
Longitude -0.070 0.007 BIAS
Height -0.117 0.011
Latitude 0.408 0.047
Longitude 0.221 0.065 STD
Height 0.676 0.064
6.2.4 Summary
From the results presented above, the following conclusions can be made.
Sub-centimetre to sub-metre level accuracy was obtained by processing dual-frequency
measurements. But the accuracy is related to the performance of precise orbit and clock
products. Using the IGDG real-time orbit and clock products, sub-centimetre level static and
sub-decimetre kinematic accuracy is obtainable in real-time. GPS•C code solution products,
which are only suitable for code processing, can provide sub-metre level kinematic accuracy
in real-time. GPS•C phase solution products, which are not yet available in real-time, have
the potential to offer comparable accuracy to that of the IGDG products but they are still not
as consistent. The tests using datasets collected on the roof of the Engineering Building have
indicated the qualities of these products. Though the datasets were collected at different
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periods and/or different sites, the datasets are one full day in length. Therefore, the satellite
geometry should be similar for these tests on the roof. The observation conditions on the roof
are similar. The ionospheric effects might be different significantly for different seasons, but
dual-frequency ionosphere-free combinations were used in these test.
When processing phase measurements, the convergence time is affected by a variety of
factors, including satellite geometry, multipath effects, performance of precise orbit and
clock products, etc. As demonstrated by the discussed tests, the typical convergence time is
about 20~30 minutes for the positioning errors to converge to the decimetre level. In static
mode, 30 minutes to 1 hour is required for the positioning results to converge to centimetre
level. To shorten the convergence time, fixing or pseudo-fixing the ambiguity may be
required in real-time positioning (Gao and Shen, 2002). In post-mission processing,
backward processing can be used to get consistent accuracy for the entire data period (Gao et
al., 2005).
6.3 Receiver Clock Offset Estimation Using PPP Methodology
In addition to position determination, PPP can also output receiver clock offset solutions
which have the potential to support precise timing applications. Unlike the positioning
accuracy assessment, in which the ground-truth can be established using either double
differencing solutions or comparing with coordinates from precisely surveyed stations, the
receiver clock offset estimates are time reference dependent and normally no ground-truth is
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available for the clock offset value of a specific receiver. To estimate precise orbit and clock
products, usually one receiver clock, equipped with a hydrogen maser external frequency, is
fixed and used as a time reference. For example, the JPL IGDG orbit and clock products are
generated using the clock of IGS station AMC2 as the reference clock (Muellerschoen,
2003). One can assess the accuracy of receiver clock offset estimation from PPP by
processing the GPS data from the AMC2 station equipped with the reference clock. The
resultant receiver clock estimates from PPP solutions for station AMC2 should theoretically
equal zero using the precise orbit and clock products that are referenced to the clock, and the
variations in the solutions should then directly reflect the quality of the clock solutions using
PPP method. As described in Section 2.3, PPP using the IGDG products also provides a new
method to recover UTC(USNO) with an accuracy of a few nanoseconds in real-time using a
single GPS receiver.
6.3.1 Receiver Clock Offset Estimation Using IGDG Products
In this test, the receiver clock offset was estimated as a white noise process using GPS data
from ACM2 station acquired from June 12th, 2004 to June 14th, 2004. The IGDG orbit and
clock products were used to do processing in a simulated real-time mode. The station
coordinates were estimated as constants. ZWD was estimated along with horizontal gradients
as random walk processes, while ZHD was modeled using Saastamoinen model. Figures
6.19, 6.20 and 6.21 present the receiver clock offset estimates for ACM2 station. The clock
offset estimates are also considered as errors in clock offset estimation as the theoretical
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value of the clock offset is zero. For the purpose of correlation analysis, which will be
discussed in Section 6.3.2, ZTD estimates were also displayed in the figures. Table 6.4
provides the statistics of the estimation accuracy.
It took about half an hour for the float ambiguities to converge for processing the data each
day. The receiver clock offset estimates remain less than 0.2 ns after the ambiguity
convergence. The RMSs of receiver clock offset estimates after the ambiguity convergence
in the three test days are 0.077 ns, 0.106 ns and 0.091 ns.
The accuracy of about 0.1 ns indicates that PPP is capable of providing real-time receiver
clock offset estimates at sub-nanosecond accuracy, making it a promising tool for time
transfer.
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Figure 6.19 Receiver Clock Offset and ZTD Estimates on June 12th, 2004
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Figure 6.20 Receiver Clock Offset and ZTD Estimates on June 13th, 2004
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Figure 6.21 Receiver Clock Offset and ZTD Estimates on June 14th, 2004
Table 6.4 Receiver Clock Offset Estimation Accuracy
Day RMS (ns) BIAS (ns) STD (ns)
June 12th, 2004 0.077 0.018 0.075
June 13th, 2004 0.106 -0.003 0.106
June 14th, 2004 0.091 0.037 0.083
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6.3.2 Analysis of Receiver Clock Offset Estimation
In the test above, the float ambiguities and the station coordinates were estimated as
constants. On the other hand, ZHD was modeled and ZWD was estimated along with its
horizontal gradients. The ZTD presented in Figures 6.19, 6.20 and 6.21 is the sum of the
modeled ZHD and the estimated ZWD. A negative correlation between errors in ZWD
estimates and errors in clock offset estimates has been investigated by various researchers.
Hackman and Levine (2003) illustrated the following relationship between them:
wetclk d.d 43−= (6.1)
where clkd and wetd are the errors in clock offset estimate and ZWD estimate, respectively.
The errors in receiver clock offset estimates and the errors in ZWD estimates in the test
above should follow the similar relationship. However, in the test, no precise ZWD
measurements were available for stations AMC2 to serve as references. Therefore, the errors
in ZWD estimates could not be calculated to show the correlation between errors in clock
offset estimates and ZWD estimates. IGS tropospheric products, which has been claimed to
be accurate up to 4 mm for IGS stations (IGS website, 2003), only provide ZTD for IGS
stations. The sample interval of the products is 2 hours, which is too long to demonstrate the
correlations. Instead, in Figures 6.19, 6.20 and 6.21, the ZTD estimates were presented. The
negative correlation between the ZTD and the receiver clock offset estimates is still obvious,
especially before the ambiguity convergence.
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Due to the correlation between the ZWD and the receiver clock offset estimation, the error
sources affecting the ZWD estimation will also corrupt the clock solution. The error sources
include errors in orbits, clocks, multipath and the higher-order ionospheric effects. They will
be analyzed in Section 6.4 and they would introduce about triple errors in the receiver clock
offset estimates than in the ZWD estimates.
Still, in order to use the estimates for timing, all instrumental biases should be calibrated to
relate the internal clock to the external hardware clock driving the receiver (Petit et al.,
2001). Special cables that are less temperature sensitive may also be required (Larson et al.,
2000).
6.4 Atmospheric Sensing Using PPP Technique
Atmospheric sensing with GPS, as discussed in Section 2.4, is equivalent to ZWD or ZTD
estimation, given that precise pressure measurements accurate up to the sub-mbar level are
available. To assess the accuracy of atmospheric sensing with PPP method, accurate values
of ZTD, ZWD or PWV are required for using as the ground-truth. IGS has provided high
accuracy ZTD combinations for IGS stations since 1998 (Gendt, 1998). The current accuracy
of this combination is given to be at the 4 mm level (IGS website, 2004), which is
sufficiently good enough for the products to be used as the ground-truth.
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Radiometers, which can be calibrated to provide ZWD at about 1.8 mm accuracy or PWV at
0.3 mm accuracy (Rocken et al., 1993), are usually used to provide reference measurements
to assess the accuracy of ZWD or PWV estimates from GPS.
6.4.1 Comparison with IGS Final Tropospheric Products
In this test, three types of precise orbit and clock products with different accuracies and
latencies were used to assess the obtainable accuracy of ZTD estimates. They are the IGS
Final, JPL NRT and JPL IGDG products. JPL IGDG orbit and clock products were tested
with simulated real-time processing, while the other two were used in post-mission tests.
Raw data from GPS week 1251 for IGS station AMC2 and the IGS Final products were
downloaded from the IGS website. The JPL NRT products for the same period were
downloaded from the JPL website. IGDG real-time orbit and clock products were received
over the Internet. P3-RT was used to process the GPS raw data with different precise
products. The real-time IGDG products were used in a simulated real-time processing.
Though the IGS Final products are not available in real-time, the products were used to
process the data from AMC2 for comparison. Only observations at 5-minute interval from
AMC2 were processed to avoid clock interpolation. The sample interval of IGS Final
tropospheric delay combinations is 2 hours, so only the ZTD estimates at those epochs were
used for comparison. There were a total of 84 such epochs and the results are shown in
Figure 6.22. A summary of the results obtained using different products is provided in Table
6.5.
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Figure 6.22 ZTD Estimates Compared with IGS Tropospheric Products
Table 6.5 ZTD Estimation Statistics
Products RMS (mm) BIAS (mm) STD (mm)
IGS Final 4.1 -2.0 3.6
JPL IGDG 5.2 -2.2 4.7
JPL NRT 5.9 0.1 5.9
The statistics shows that PPP method has the potential to provide ZTD estimates at an
accuracy of about 5~6 mm in real-time. As shown in Figure 6.22, almost all the differences
when using IGS Final products are less than 10 mm while the differences can be more than
15 mm when using JPL IGDG products or NRT products. The processing results using the
IGS Final products provided the best results, but the IGS Final products are not available in
real-time, as it has a latency of 13 days. The -2 mm biases when using IGS Final and JPL
IGDG may be caused by the use of different processing strategies in the ZTD estimation. The
ZTDs of IGS stations are estimated as a total delay using a single hydrostatic mapping
215
function in some analysis centers (Heroux, 2003). In this test, the ZHD was modeled and
mapped with the hydrostatic mapping function and pressure model, and ZWD, including the
un-modeled ZHD, is estimated with the wet mapping function. The output ZTD is the sum of
the modeled ZHD and the estimated ZWD. The effects of different mapping functions for the
zenith delay estimation will be investigated in Section 6.4.2. The spikes in the results of JPL
IGDG products and NRT products may be caused by the errors in orbit and clock products.
The statistics in Table 6.5 show that the quality of the satellite orbit and clock products
directly affects the obtainable accuracy of ZTD estimates.
Because ZHD can be determined to better than 1 mm given accurate surface pressure (Bevis
et al., 1992), it is possible for PPP method to produce real-time ZWD at an accuracy of 6~7
mm if precise pressure measurements can be obtained in real-time. The vertical integrated
water vapour overlaying a receiver, in terms of Precipitable Water Vapour (PWV), can be
related to the ZWD at the receiver using Equation 2.6. The dimensionless constant of the
transfer factor Π is approximately equal to 0.15. Therefore, PPP has the potential to
determine real-time PWV to an accuracy of 1 mm to satisfy the required accuracy for GPS
meteorological applications (Gutman and Benjamin 2001).
6.4.2 Comparison with Radiometer Measurements
From July to September in 2004, a real-time water vapour sensing test was conducted in the
University of Calgary. One week’s results, from September 2 to 8 will be discussed in this
section.
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A Radiometrics 1100 WVR, which has been set up on the roof of the Engineering
Building, was used to derive the “true” PWV values in this test to assess the PWV estimates
derived from GPS observations using PPP methodology. The radiometer was set up to make
direct measurements of line-of-sight slant water vapour to all GPS satellites during the test
period. The WVR tracks each satellite for approximately 40 s. Consequently, it takes about 6
minutes to track all satellites in view in a given cycle (Gao et al., 2004). In order to obtain
optimal accuracy for the hydrostatic delay values, precise pressure sensors are required at the
observation site. A Paroscientific MET3A sensor was set up beside the radiometer. The
MET3A collected meteorological data continuously during the test period – measuring
pressure, temperature and humidity at 30 s interval. The pressure measurements were
interpolated to the sample interval (10 s) of GPS measurements to calculate the zenith
hydrostatic delay. The accuracy of the pressure observations is better than 0.1 mbar for this
instrument (Nicholson, 2004), so that the corresponding errors in the hydrostatic delay values
are considered negligible. The temperature measurements of MET3A, which were used to
calculate the transfer factor Π , are accurate up to 0.1 degree (Nicholson, 2004).
A Javad Legacy dual-frequency receiver was used during the test period to output GPS
measurements at an interval of 10 s. A Javad JPSLEGANT antenna was set up on S1 pillar
(51°04’ N, 114°07’ W, 1116.82 m) on the roof beside the radiometer (about 1.5 m away).
JPSLEGANT is an antenna with a flat ground plane so it can partially mitigate the multipath
effects. The MET3A is located about 2 m above the GPS antenna, which was set up at almost
the same height as the radiometer. Assuming a pressure scale height of 8 km, a 2 m height
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difference will lead to about 0.5 mm ZHD difference. The pressure measurements of the
MET3A were corrected to the height of GPS antenna in post-mission to isolate the ZHD. The
setup of the radiometer, JPSLEGANT antenna and MET3A sensor is shown in Figure 6.23.
Figure 6.23 Radiometer, GPS Antenna and MET3A Instruments
In some researches, ZWD and ZHD were estimated as a total ZTD using a single mapping
function, which can be either the hydrostatic mapping function or wet mapping function
(Duan et al., 1996). This method is practical when no gradient components were estimated
and the elevation cut-off angle is as big as 15°, since the difference between the hydrostatic
and wet mapping functions at 15° is ~0.03 (Niell, 1996). In this research, GPS data at low
elevation angles was included to separate gradient components from the azimuthally
homogeneous components. The difference between the hydrostatic and wet mapping
Radiometer
GPS Antenna
MET3A
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functions can be up to 0.26 at 7°, which is the elevation cut-off angle used in this research.
Estimating only a total delay using a single mapping function will therefore lead to a bias in
zenith tropospheric delay estimates. Figure 6.24 shows the Niell Mapping Functions for the
testing site during the testing period, where the hydrostatic and wet mapping functions are
denoted as NMFh and NMFw, respectively. The difference between the hydrostatic and wet
mapping functions is shown in Figure 6.25.
Figure 6.24 Niell Mapping Functions
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Figure 6.25 Difference between the Wet and Hydrostatic Mapping Functions
Therefore, in this test, ZHD was modeled and mapped using the hydrostatic mapping
function, while ZWD was estimated using the wet mapping function. However, precise
pressure measurements were only available with a latency of several hours. So in real-time
processing a pressure model, the model based on Equation 6.2, has been used to calculate
pressure values. The un-modeled ZHD was absorbed to ZWD estimates. The ZWD plus the
un-modeled ZHD was then estimated using wet mapping functions. The modeled ZHD and
estimated ZWD (including the un-modeled ZHD) were output as a total ZTD from the real-
time processing. The precise pressure measurements were then used to isolate the hydrostatic
delays from the ZTD in post-mission before they were transferred to PWV and compared
with the measurements from WVR.
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During the test, the RMS difference between the calculated pressure value and the precise
pressure measurements from MET3A is just 5.92 mbar, which corresponds to about 13 mm
ZHD. Estimating such a small amount of un-modeled ZHD with the wet mapping function
did not greatly affect the total ZTD. The contribution of the small un-modeled ZHD to the
errors in the PWV estimates will be discussed in Section 6.4.3. The difference between the
calculated pressure value and the precise pressure measurements from MET3A is shown in
Figure 6.26, where the pressure value is calculated using the following formula by assuming
a pressure scale height of 8 km.
( )80000
/hePP −×= (6.2)
where 0P is the pressure at sea level and 1013.25 mbar was selected in this research, h is the
site height in metre and 1118.62 m, which is the height of MET3A, was used to calculate the
pressure values in Figure 6.26 and compare with MET3A measurements, and 1116.82 m,
which is the height of the GPS antenna, was used in the real-time processing.
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Figure 6.26 Difference between the Calculated and Measured Pressures
During September 2-8, 2004, JPL IGDG real-time precise orbit and clock products were
acquired over the Internet from a JPL server at a rate of 1 Hz. Real-time GPS observations
were output at interval of 10 s from the Javad receiver. The PPP numerical computation was
conducted using the software package P3-RT installed in a computer at the office ENF405
with an Internet connection and a serial port connected to the GPS receiver. ZTDs were
output in real-time.
After removed the zenith hydrostatic delays, which were calculated using the Saastamoinen
model and precise pressure measurements from MET3A that have been converted to the
height of radiometer and GPS antenna, the remaining ZWD were converted to PWV and
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compared with water vapour measurements from the WVR. Equation 2.8 and 2.10 given by
Bevis et al. (1994) were used to calculate the transfer factor Π . The surface temperatures
required in these equations were obtained from the MET3A measurements.
The slant water vapour measurements to each satellite from the WVR were mapped to zenith
water vapour using the wet Niell Mapping Function. The zenith water vapour measurements
for each cycle of observations were averaged and then compared with the average value of
GPS-derived zenith water vapour estimates over the same time period, about 6 minutes per
cycle. The averaged PWV measurements from the radiometer, the averaged GPS-derived
PWV, and the differences between them are shown in Figures 6.27 to 6.33. To demonstrate
the correlation between days, the differences between GPS-derived and radiometer-measured
PWV are also presented in Figure 6.34, from top to bottom in the order of September 2, 3, 4,
5, 6, 7, and 8. The results are given in local time series and are offset 5 mm between days.
For each day, the first one-hour ZWD estimates were used only for the ambiguity
convergence. Because of an Internet connection problem during the test period, a 4-hour
period of orbit and clock products were lost on September 2, 2004. The statistics of the
results are shown in Table 6.6.
It can be observed that the PWV difference between the WVR measurements and GPS
estimates is about 1 mm, with very small (less than 0.3 mm) random biases over the seven
days. The results indicate the potential to determine PWV to an accuracy of 1 mm in real-
time using precise orbit and clock products and PPP methodology.
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Figure 6.27 PWV from GPS and WVR on September 2nd, 2004
Figure 6.28 PWV from GPS and WVR on September 3rd, 2004
Figure 6.29 PWV from GPS and WVR on September 4th, 2004
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Figure 6.30 PWV from GPS and WVR on September 5th, 2004
Figure 6.31 PWV from GPS and WVR on September 6th, 2004
Figure 6.32 PWV from GPS and WVR on September 7th, 2004
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Figure 6.33 PWV from GPS and WVR on September 8th, 2004
Figure 6.34 PWV Comparison between GPS and WVR from September 2nd to 8th, 2004
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Table 6.6 Statistics of PWV Comparison
Date RMS (mm) BIAS (mm) STD (mm)
Sept. 2nd, 2004 0.93 0.04 0.92
Sept. 3rd, 2004 0.99 0.08 0.99
Sept. 4th, 2004 1.09 0.30 1.05
Sept. 5th, 2004 0.77 -0.28 0.72
Sept. 6th, 2004 0.65 0.01 0.65
Sept. 7th, 2004 0.64 0.12 0.62
Sept. 8th, 2004 0.94 -0.21 0.91
To demonstrate the importance of mapping functions in zenith tropospheric delay estimation,
the GPS data was also processed in post-mission with different estimation strategies. The
first method is to model ZHD with a hydrostatic mapping function and the precise pressure
measurements from MET3A, and to estimate ZWD with a wet mapping function. Only ZWD
was output and converted to PWV and compared with the radiometer measurements. Two
other methods are to estimate ZTD with a single mapping function, wet or hydrostatic
mapping function. Still, ZHD calculated from the precise pressure measurements was
removed from the estimated ZTD. The remaining ZWD was thus converted to PWV. Table
6.7 shows the accuracy statistics of different methods when compared with radiometer
measurements.
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Table 6.7 Accuracy Statistics of Different Strategies (Unit: mm)
Estimate ZWD Estimate ZTD with NMFh Estimate ZTD with NMFw Day
RMS BIAS STD RMS BIAS STD RMS BIAS STD
2 0.90 0.04 0.90 1.10 0.64 0.89 14.90 -14.76 2.10
3 1.02 0.13 1.01 1.36 0.80 1.10 14.37 -14.23 1.97
4 1.02 0.16 1.01 1.37 0.94 0.99 14.88 -14.79 1.71
5 0.70 -0.11 0.69 0.74 0.18 0.71 14.31 -14.23 1.51
6 0.70 0.22 0.66 1.02 0.69 0.76 13.65 -13.58 1.42
7 0.63 0.20 0.60 0.96 0.67 0.69 13.16 -13.05 1.69
8 0.87 -0.22 0.84 1.04 0.38 0.96 14.61 -14.49 1.83
As shown in Table 6.7, estimating ZWD with a wet mapping function and modeling ZHD
with a hydrostatic mapping function and pressure measurements provides the best results.
Though they are slightly worse than the real-time results in September 6 and 8, the results
overall are better than the real-time ones. Estimating ZTD with a hydrostatic mapping
function also provides millimetre level accuracy, but the biases are positive in all days,
compared with the random biases when estimating only ZWD in real-time or post-mission
processing. The biases are strongly correlated with the PWV values. If we use the results
from the first method as the reference, the biases will be about 0.5~0.6 mm in September 2,
3, 4, 7 and 8, and about 0.3~0.4 mm in September 5, 6. The PWVs range between 10 and 18
mm in September 2, 3, 4, 7 and 8, and between 7 and 13 mm in September 5 and 6.
Therefore, mapping ZWD with a hydrostatic mapping function produces a bias of about 3%
to 4% of the ZWD value in magnitude. The bias is consistent with the difference between the
hydrostatic and wet mapping functions at the elevation cut-off angle, which is 7° in this test.
The difference between the hydrostatic and wet mapping functions is about 0.26 at the
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elevation angle of 7°, which is about 3.3% of the wet mapping function at this elevation
(~7.92).
On the other hand, estimating ZTD with a wet mapping function would not provide useful
information to PWV estimation because the ZTD is dominated by ZHD. The bias introduced
by mapping ZHD with a wet mapping function is also about 3% to 4% of the ZHD value in
magnitude (about 2 m ZHD during the test). The difference between the hydrostatic and wet
mapping functions is about 3.5% of the hydrostatic mapping function at the elevation angle
of 7° (~7.65).
Therefore, the relative bias introduced by mapping ZWD using a hydrostatic mapping
function or mapping ZHD using a wet mapping function is determined by the relative
difference between the mapping functions at the elevation cut-off angle.
Based on the analysis above, the best way for PPP meteorology is to model and map ZHD
with precise pressure measurements and the hydrostatic mapping function, and estimating
ZWD with the wet mapping function. If pressure measurements are not available during the
data processing, a pressure model can be used to calculate an approximate pressure value as
was adopted by the real-time processing in this research. If even a pressure model is not
practicable, ZTD should be estimated with a hydrostatic mapping function. Estimating ZTD
with a wet mapping function should not be used in any case.
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6.4.3 Analysis of Real-Time Water Vapour Sensing Results
The 1 mm level RMS of GPS-WVR precipitable water vapour difference can be mainly
attributed to errors in GPS zenith wet delay estimates, the transfer factor Π (as calculated
from surface temperature measurements) and the WVR retrieval coefficients and/or
calibration errors. Details of these potential error sources are described in the following.
Errors in GPS Orbits and Clocks
Errors in the orbit and clock products will directly affect the zenith wet delay estimation. As
shown in Section 6.4.1, the difference between ZTD estimates using the IGS Final products
and the IGDG products can be up to more than 1 mm (4.1 for the IGS Final and 5.2 for the
IGDG). If one assumes that the errors in orbits and clocks are not correlated with other
errors, the ZTD errors introduced by the IGDG orbit and clock products would be about 3
mm, which corresponds to 0.45 mm PWV.
The contribution of errors in orbits and clocks can also be analyzed in another way. As
investigated in Section 6.2, IGDG real-time orbit and clock products can provide sub-
centimetre level positioning accuracy in static mode. The zenith tropospheric error caused by
the errors in orbits and clocks should be less than 3.3 mm considering that the positioning
errors in height component are typically 3 times as large as the zenith tropospheric errors
(Rocken et al., 1993). Therefore, their contribution to PWV errors should be less than 0.5
mm.
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In summary, the contribution of errors in orbits and clocks should be in the range of 0.4 to
0.5 mm. They are the main error sources in the PWV estimation.
Errors in Zenith Hydrostatic Delay
In this research, the pressure measurements from the MET3A barometer were used along
with Saastamoinen model to calculate and remove the ZHD from the ZTD output from GPS
data processing. Since the accuracy of the pressure observations is better than 0.1 mbar in
this research, even after interpolation, the corresponding errors in hydrostatic delay values
therefore are considered negligible (Gao et al., 2004). In real-time processing, a pressure
model was used to calculate pressure value, which led to approximately 13 mm ZHD being
combined into the ZWD and mapped with a wet mapping function. As discussed above,
when an elevation cut-off angle of 7° is used, mapping ZWD using a hydrostatic mapping
function or mapping ZHD with a wet mapping will introduce an error of about 3.5% of the
mapped parameter value in magnitude. In this case, the error would be approximate 0.46 mm
in ZWD, equivalent to 0.07 mm in PWV. The total contribution to PWV errors should be less
than 0.1 mm.
Multipath Effects
The test site was located at an environment with significant multipath effects as previously
discussed. Even though an antenna with a small ground plane was used, the residual
multipath effects seem evident. Similar diurnal variations in PWV errors can be found in
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Figure 6.34, especially during 12:00 to 14:00 local time. The big negative differences (with
respect to WVR truth) during this period could be caused by multipath effects.
Higher-order Ionospheric Effects
In this research, dual-frequency observations were used to form ionosphere-free
combinations which have neglected the higher-order ionospheric effects. The residual
second-order ionospheric effect is about 0.11 mm/TECU for code combinations and half of
this effect for phase combination (Bassiri and Hajj, 1993). During the test period, the vertical
TEC was less than 30 TECU, ranging between 5 to 15 TECU most of times. Considering that
ionospheric mapping function is smaller than tropospheric mapping function as shown in
Figure 5.5 and Figure 6.24, the higher-order ionospheric effect on the PWV estimation
should be at a 0.1 mm level. In post-processing, corrections for the second-order ionospheric
effects have been calculated using the equations discussed in Section 3.2. But they have not
shown any improvement over the ionosphere-free combinations, because the IGDG orbit and
clock products are estimated using ionosphere-free combinations without considering the
higher-order ionospheric effects (Muellerschoen, 2003).
Mapping Functions
Both wet and hydrostatic NMFs were not 100% perfect at very low elevations. When
compared with the radiosonde profiles, NMFh and NMFw have shown biases of –0.0011 and
–0.0179 respectively at 5° elevation (Niell, 1996). Considering big mapping functions at this
elevation (NMFh=10.1428, NMFw=10.7441), the relative errors in the mapping functions
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are approximately 0.01% and 0.17%. The errors are even smaller at 7° elevation. Therefore,
the contribution should be less than 0.1 mm for an elevation cut-off angle of 7°.
Errors in Transfer Factor Π
The relative error in Π is approximately equal to the relative error in mT (Bevis et al., 1994).
In this research, mT is calculated from surface temperature using the linear relation estimated
by Bevis et al. (1994) with an RMS relative error of less than 2%. This regression was
determined by investigating data in a 2-year period from 13 stations in the United States,
from Fairbanks, Alaska, to West Palm Beach, Florida. The linear relation was considered
suitable for the test site in Calgary, with a similar accuracy (Gao et al., 2004). From the
figures, it is observed that the PWV in Calgary during the test period is about 15 mm.
Therefore, the uncertainty in mT contributes to the PWV difference should be less than 0.3
mm.
WVR Retrieval Coefficients and Calibration
Biases may be introduced to the PWV measurements if the radiometer is not calibrated
carefully. Rocken et al. (1993) claimed that a radiometer could be calibrated to provide ZWD
at 1.8 mm accuracy or PWV at 0.3 mm accuracy. Because the retrieval coefficients for the
radiometer used in this research were derived from radiosonde data local to Boulder,
Colorado, an uncertainty of about 1 mm level for the radiometer PWV observations can be
assumed (Gao et al., 2004).
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6.5 PPP Using Single-Frequency Measurements
In this section, precise point positioning results using single-frequency measurements are
presented. All data was processed using the IGDG orbit and clock products either in real-
time or simulated real-time mode. As discussed in Chapter 5, the key issue for precise point
positioning using single-frequency measurements is how to mitigate the ionospheric effects.
Therefore the performance of single-frequency precise point positioning is correlated with
the ionospheric conditions, which is also regionally dependent (Skone, 1999; Komjathy,
1997). Data collected at different ionospheric conditions and from different ionospheric
regions was therefore processed in this research. Different ionospheric mitigation models
were used to investigate the performances of these models at different ionospheric
conditions.
6.5.1 Positioning at Mid-Latitude Stations
A real-time static test was carried on December 3rd, 2003. One Javad Legacy dual-frequency
receiver was set up on the S1 pillar on the roof of the Engineering Building at the University
of Calgary. Only measurements on 1L were used in processing using the ionospheric
estimation model proposed in Chapter 5. The IGDG orbit and clock products were received
over the Internet. The sample interval was set to 10 s. The coordinates of S1, which were
determined in ITRF-1993, were transferred to ITRF-2000 using coefficients provided by
McCarthy and Petit (2004) and used as the true coordinates. The positioning errors and
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zenith ionospheric delay estimates are shown in Figure 6.35 and 6.36. The accuracy statistics
is presented in Table 6.8.
Figure 6.35 Positioning Errors Using Ionospheric Estimation Model on December 3rd, 2003
Figure 6.36 Zenith Ionospheric Delay Estimates on December 3rd, 2003
235
The date for the real-time test was selected randomly, but it turned out to be an ionospheric
quiet period. The Ap index is 4 on December 3rd, 2003. The half cosine shape of the zenith
ionospheric delay also indicates a quiet ionospheric condition. At the beginning of the
processing, an apriori value, which was calculated using the initial coordinates and the
Klobuchar model, was set as the initial zenith ionospheric delay. As shown in Figure 6.36,
the initial zenith ionospheric delay did not deviate from the true value noticeably. About 30-
minute was required for the zenith ionospheric delay and ambiguities to converge. A
positioning accuracy of about half metre in 3D was obtained in the real-time test for that day.
To test the performance of the ionospheric estimation model under ionospheric disturbed
conditions, a dataset of 24 hours was processed in a simulated real-time mode using the
saved IGDG orbit and clock products. The Ap index on July 27th, 2004 is 186, indicating a
day with an extremely disturbed ionosphere. The GPS data was collected on the S1 pillar
with a sample interval of 10 s to match the first test. The positioning errors, zenith
ionospheric delay estimates, and satellite geometry are shown in Figures 6.37, 6.38 and 6.39.
The accuracy statistics is presented in Table 6.8.
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Figure 6.37 Positioning Errors Using Ionospheric Estimation Model on July 27th, 2004
Figure 6.38 Zenith Ionospheric Delay Estimates on July 27th, 2004
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Figure 6.39 Satellite Geometry of S1 on July 27th, 2004
Table 6.8 Accuracy Statistics of Single-Frequency Point Positioning at S1 (Unit: m)
December 3rd, 2003 July 27th, 2004
Latitude 0.277 0.490
Longitude 0.223 0.434 RMS
Height 0.371 0.691
Latitude 0.027 -0.228
Longitude -0.180 -0.017 BIAS
Height 0.085 -0.070
Latitude 0.268 0.433
Longitude 0.195 0.434 STD
Height 0.361 0.688
As shown in Figure 6.39, many satellites lost lock during testing because of the disturbed
ionosphere, which could even lead to missing solutions in some epochs when the number of
satellites is less than 7. The positioning errors are not as smooth as those in the ionospheric
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quiet day. The estimates for the zenith ionospheric delay also do not have a half cosine-like
trend and are very noisy. Sub-metre level accuracy was obtained for each positioning
component. The accuracy is degraded by a factor of 2 when compared with the results
obtained on December 3rd, 2003 with an ionospheric quiet condition.
Therefore, in the middle latitude regions, sub-metre level positioning accuracy is obtainable
in real-time, using the ionospheric estimation model and real-time precise GPS orbit and
clock products.
6.5.2 Positioning Using Data from Different Ionospheric Regions
In this section, datasets collected at different ionospheric conditions from three stations
located at different ionospheric regions were processed using different models, including the
ionospheric estimation model, the Klobuchar model, and the Global Ionospheric Model
(GIM). The purpose of the tests is to compare the performance of these models in different
regions and under different ionospheric conditions. All datasets were processed in a
simulated real-time mode using the IGDG orbit and clock products acquired from the real-
time water vapour sensing test in August 2004. The IGDG orbit and clock products were
available continuously in real-time during that month except for several hours which were
lost because of an internal network problem on August 25th. The data that has been processed
includes five days with the most disturbed ionospheric conditions, five days with the quietest
ionospheric conditions, and one day with typical ionospheric conditions in August 2004. The
approximate coordinates of the three stations and the Ap indices of these days are shown in
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Table 6.9 and 6.10. The three stations are GLPS located at the equatorial region, the S1
located at the mid-latitude region, and FAIR located at the high-latitude region. The data
interval is 30 s for GLPS and FAIR and 10 s for S1. For the purpose of consistency, all
datasets were processed independently epoch by epoch at an interval of 30 s. August 19th was
selected instead August 25th as an ionospheric quiet day because part of the orbit and clock
products was lost in August 25th. August 14th was selected as a day with typical ionospheric
conditions. The ionospheric products required for processing using GIM were downloaded
from IGS website.
Table 6.9 Station Coordinates
GLPS S1 FAIR
Latitude -00° 44’ 35” 51° 04’ 46” 64° 58’ 41”
Longitude -90° 18’ 13” -114° 07’ 58” -147° 29’ 57” Height 1.8 m 1116.8 m 319.0 m
Table 6.10 Ap Indices in August 2004
Date 4 8 24 3 19 14 20 10 21 31 30
Ap index 3 3 3 3 4 7 15 16 16 28 42
Given in Tables 6.11, 6.12 and 6.13 are the accuracy statistics (RMS) using the ionospheric
estimation model for stations GLPS, S1 and FAIR in these 11 days. The tables are sorted in
ascending Ap order. From the tables, we can see the ionospheric estimation model can
provide metre level accuracy at all stations and for all testing days. In ionospheric quiet days
at mid- and high- latitude stations, half metre level accuracy was obtained.
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The performance of the ionospheric estimation model is strongly correlated with ionospheric
conditions. At each station, the accuracy is higher in ionospheric quiet days than in disturbed
days. Accuracy is best at mid-latitude, worst at equatorial regions and high-latitude is in
between, which is consistent with the ionospheric conditions in these regions. In the
equatorial region, the values of the peak electron density are the highest among the three
regions while the mid-latitude ionosphere is the least variable and undisturbed (Komjathy,
1997). The ionospheric activity is more complicated in high-latitude region than mid-latitude
region due to factors such as auroral activity and ionospheric trough (Skone, 1999;
Komjathy, 1997).
Table 6.11 Accuracy Statistics of Ionospheric Estimation Model for GLPS (Unit: m)
Day Ap Latitude Longitude Height
4 3 0.167 0.396 0.451
8 3 0.270 0.286 0.532
24 3 0.289 0.472 0.740
3 3 0.111 0.359 0.485
19 4 0.189 0.695 0.740
14 7 0.421 0.820 0.734
20 15 0.190 0.352 0.532
10 16 0.262 0.348 0.497
21 16 0.282 0.853 1.006
31 28 0.286 0.556 0.735
30 42 0.369 0.772 0.917
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Table 6.12 Accuracy Statistics of Ionospheric Estimation Model for S1 (Unit: m)
Day Ap Latitude Longitude Height
4 3 0.259 0.295 0.364
8 3 0.237 0.281 0.406
24 3 0.281 0.247 0.342
3 3 0.314 0.322 0.398
19 4 0.319 0.276 0.407
14 7 0.306 0.308 0.392
20 15 0.325 0.260 0.621
10 16 0.363 0.293 0.520
21 16 0.260 0.399 0.434
31 28 0.205 0.253 0.547
30 42 0.449 0.270 0.636
Table 6.13 Accuracy Statistics of Ionospheric Estimation Model for FAIR (Unit: m)
Day Ap Latitude Longitude Height
4 3 0.176 0.191 0.415
8 3 0.231 0.207 0.686
24 3 0.233 0.222 0.605
3 3 0.231 0.258 0.456
19 4 0.246 0.302 0.631
14 7 0.388 0.410 0.702
20 15 0.457 0.563 0.741
10 16 0.483 0.339 0.628
21 16 0.279 0.240 0.749
31 28 0.300 0.248 0.652
30 42 0.494 0.319 1.057
Shown in Tables 6.14 to 6.16 are the accuracy statistics (RMS) for the three stations using
the Klobuchar model over the 11 days. The results from the processing using only code
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observations and the results from the processing using both code and phase observations are
shown in order to demonstrate the improvement using phase observations.
Compared to the ionospheric estimation model, the accuracy from the Klobuchar model is
much worse; only 1 to 3 metres accuracy was obtained. But the Klobuchar model is not as
sensitive to ionospheric conditions as the ionospheric estimation model. Unlike the
ionospheric estimation model, the Klobuchar model provides the best results at the high-
latitude station FAIR, though it still provides better results at the mid-latitude station S1 than
the equatorial station GLPS. This may be due to the fact that the Klobuchar model just uses 8
coefficients to fit the ionospheric activity on a global scale. The coefficients perform better at
some regions, which may not be characterized with quieter ionospheric conditions, than other
areas.
Processing using both code and carrier phase observations does not guarantee better results
than the code only processing when using the Klobuchar model. This is especially true at the
equatorial station GLPS where the accuracy even degraded when processing using code and
phase observations as shown in Table 6.14. The Klobuchar model is not accurate enough to
exploit the higher accuracy of phase observations at equatorial stations. As shown in Tables
6.15 and 6.16, only slight improvements are found at mid- and high- latitude stations for
phase processing solutions.
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Table 6.14 Accuracy Statistics of Klobuchar Model for GLPS (Unit: m)
Code Code & Phase Day Ap
Latitude Longitude Height Latitude Longitude Height
4 3 0.511 0.653 1.566 0.477 1.551 1.927
8 3 0.607 0.373 1.257 0.701 0.894 2.836
24 3 1.317 0.500 1.496 1.483 1.084 1.818
3 3 0.514 0.436 1.587 0.757 0.712 1.154
19 4 1.172 0.613 1.749 0.906 1.519 2.069
14 7 1.171 0.761 1.223 1.058 1.186 2.254
20 15 0.834 0.606 1.669 0.756 1.093 1.890
10 16 0.689 0.328 1.215 0.989 1.183 2.338
21 16 1.212 0.627 1.184 1.022 1.141 1.772
31 28 0.882 0.775 2.578 1.158 0.980 2.187
30 42 0.869 0.541 2.936 0.745 1.290 3.225
Table 6.15 Accuracy Statistics of Klobuchar Model for S1 (Unit: m)
Code Code & Phase Day Ap
Latitude Longitude Height Latitude Longitude Height
4 3 0.465 0.288 1.257 0.703 0.516 1.095
8 3 0.480 0.308 1.311 0.764 0.761 1.609
24 3 0.530 0.322 2.066 0.523 0.534 1.352
3 3 0.460 0.291 0.976 0.335 0.477 0.669
19 4 0.518 0.333 2.367 0.527 0.588 2.097
14 7 0.440 0.326 1.923 0.536 0.627 0.874
20 15 0.488 0.330 2.603 0.734 0.470 1.287
10 16 0.693 0.353 1.380 0.412 0.572 0.838
21 16 0.584 0.323 2.361 0.607 0.574 1.071
31 28 0.578 0.348 1.180 0.422 0.411 1.009
30 42 0.536 0.368 1.069 0.498 0.400 0.927
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Table 6.16 Accuracy Statistics of Klobuchar Model for FAIR (Unit: m)
Code Code & Phase Day Ap
Latitude Longitude Height Latitude Longitude Height
4 3 0.409 0.238 0.844 0.253 0.187 0.865
8 3 0.693 0.452 0.818 0.388 0.236 0.885
24 3 0.788 0.478 1.653 0.499 0.465 0.987
3 3 0.358 0.298 0.850 0.357 0.290 0.820
19 4 0.891 0.552 1.803 0.462 0.362 1.147
14 7 1.131 0.659 1.432 0.647 0.362 1.313
20 15 0.790 0.499 2.047 0.482 0.280 0.768
10 16 0.643 0.558 0.927 0.432 0.367 0.740
21 16 0.841 0.538 1.549 0.388 0.295 1.074
31 28 0.317 0.256 0.929 0.358 0.307 0.824
30 42 0.530 0.445 0.891 0.652 0.548 1.335
Given in Tables 6.17 to 6.19 are the accuracy statistics (RMS) for the three stations using the
GIM over the 11 days. The results from the processing using only code observations and the
results from the processing using both code and phase observations are presented.
The GIM provides much better results at all stations than the Klobuchar model. It can even
provide slightly better results than the ionospheric estimation model at the high latitude
station, though the latter is much better at the equatorial station. The accuracy of the GIM is
metre or even sub-metre level at mid- and high-latitude stations but it is about 1.5 metres at
equatorial stations.
Like the ionospheric estimation model, the GIM provides the best results at mid-latitude
stations and performs the worst at equatorial stations. This is consistent with the ionospheric
245
conditions and the performance of the IGS ionospheric products in these regions. The IGS
tracking network, which is used to create the ionospheric products, is unevenly distributed
(Fedrizzi et al., 2002). It is much denser in the mid-latitude region than in the equatorial
region (Komjathy, 1997). Also, the resolution of the IGS final ionospheric TEC grids, which
is 5 deg (longitude) x 2.5 deg (latitude) at a 2-hour’ interval, is not high enough to recover
TEC at any given location and time in the equatorial region. This is because the vertical TEC
can change up to 20-TECU within one hour or several degrees in this region.
Unlike the Klobuchar mode, the carrier phase observations can almost always improve the
ionosphere estimation accuracy when using the GIM because the IGS ionospheric products
can model ionospheric delay much better than the broadcast Klobuchar coefficients.
Table 6.17 Accuracy Statistics of GIM for GLPS (Unit: m)
Code Code & Phase Day Ap
Latitude Longitude Height Latitude Longitude Height
4 3 0.338 0.486 1.578 0.364 0.309 0.980
8 3 0.425 0.318 1.920 0.432 0.790 1.026
24 3 0.643 0.406 1.771 0.636 0.644 1.001
3 3 0.348 0.378 1.670 0.260 0.323 1.341
19 4 0.420 0.483 1.907 0.594 0.645 1.456
14 7 0.578 0.640 1.978 0.598 0.960 1.406
20 15 0.496 0.471 2.162 0.360 0.336 1.185
10 16 0.376 0.352 1.999 0.431 0.517 1.035
21 16 0.586 0.540 1.974 0.363 0.522 1.349
31 28 0.655 0.612 1.649 0.663 0.625 1.252
30 42 0.515 0.361 1.584 0.346 0.603 0.979
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Table 6.18 Accuracy Statistics of GIM for S1 (Unit: m)
Code Code & Phase Day Ap
Latitude Longitude Height Latitude Longitude Height
4 3 0.374 0.221 0.615 0.188 0.215 0.355
8 3 0.391 0.215 0.573 0.245 0.219 0.474
24 3 0.418 0.217 0.631 0.238 0.281 0.494
3 3 0.410 0.219 0.565 0.243 0.290 0.270
19 4 0.363 0.231 0.629 0.256 0.263 0.466
14 7 0.387 0.233 0.613 0.279 0.363 0.403
20 15 0.375 0.230 0.599 0.324 0.244 0.540
10 16 0.413 0.226 0.638 0.241 0.228 0.442
21 16 0.462 0.235 0.564 0.469 0.289 0.654
31 28 0.391 0.300 0.730 0.322 0.236 0.671
30 42 0.406 0.260 0.804 0.308 0.199 0.523
Table 6.19 Accuracy Statistics of GIM for FAIR (Unit: m)
Code Code & Phase Day Ap
Latitude Longitude Height Latitude Longitude Height
4 3 0.249 0.146 0.517 0.145 0.165 0.371
8 3 0.277 0.255 0.570 0.199 0.205 0.621
24 3 0.278 0.205 0.620 0.248 0.228 0.551
3 3 0.281 0.198 0.573 0.200 0.171 0.407
19 4 0.264 0.214 0.580 0.186 0.211 0.505
14 7 0.352 0.316 0.691 0.333 0.278 0.824
20 15 0.377 0.221 0.679 0.355 0.218 0.574
10 16 0.285 0.281 0.711 0.338 0.278 0.658
21 16 0.283 0.246 0.701 0.234 0.186 0.724
31 28 0.325 0.222 0.617 0.285 0.230 0.656
30 42 0.437 0.312 0.921 0.481 0.327 0.880
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6.5.3 Positioning Using Kinematic Datasets
To test the performance of these ionospheric models in pure kinematic positioning, the
airborne dataset, used for dual-frequency positioning in Section 6.2, was processed with the
IGDG orbit and clock products in a simulated real-time mode again. This time however, only
measurements on 1L were used. The double differencing ambiguity-fixed position solutions
served as the ground-truth. The Ap index of that day is 7.
Figures 6.40 to 6.44 show the positioning errors in each positioning component using the
three models. The accuracy statistics for all processing is shown in Table 6.20.
Figures 6.40, 6.42 and 6.44 present results from the processing using both code and phase
observations, while Figures 6.41 and 6.43 show the much noisier results from the code only
processing. It took approximately 10 to 20 minutes for the positioning errors to converge to a
sub-metre level with phase processing. When using the ionospheric estimation model and
GIM, the positioning errors remained within the sub-metre level after ambiguity
convergence. However the ambiguities may diverge when using the Klobuchar model. This
may be caused by the change over time of large ionosphere residuals, which are evident by
the varying bias in the height component in Figure 6.41 when processing using only code
measurements. When processing phase measurements using the Klobuchar model, the float
ambiguities can absorb part of the ionospheric residuals for short periods, but the varying
residuals would make the ambiguities diverge, leading to worse solutions. This can explain
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why processing phase measurements using the Klobuchar model does not assure better
results than processing code measurements, as indicated in Section 6.5.2.
As shown in Table 6.20, promising results were obtained using different models. About 20
cm accuracy was obtained in each positioning component when using the ionospheric
estimation model and GIM by phase processing. The accuracy is though much worse when
using the Klobuchar model, but a metre level accuracy was still obtained through phase
processing. When processing using only code measurements, the GIM and Klobuchar model
can only provide accuracies of about 0.7 m and 2.5 m respectively. The accuracy obtained
with the kinematic dataset is even better than the accuracy obtained with static datasets. This
is because the accuracy of single-frequency precise point positioning is correlated with not
only the ionospheric conditions but also the satellite geometry. The performance of these
models in kinematic positioning confirms that sub-meter level accuracy is obtainable using
single-frequency measurements.
Figure 6.40 Positioning Using Ionospheric Estimation Model
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Figure 6.41 Positioning Using Klobuchar Model with Code Measurements
Figure 6.42 Positioning Using Klobuchar Model with Code and Phase Measurements
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Figure 6.43 Positioning Using GIM with Code Measurements
Figure 6.44 Positioning Using GIM with Code and Phase Measurements
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Table 6.20 Single-Frequency Point Positioning with Airborne Dataset (Unit: m)
Klobuchar GIM Ionospheric
Estimation Code Code & Phase Code Code & Phase
Latitude 0.154 0.419 0.418 0.482 0.131
Longitude 0.271 0.322 0.347 0.222 0.243
Height 0.217 2.317 0.840 0.448 0.262
6.5.4 Summary
The following conclusions can be drawn from the analysis of the single-frequency precise
point positioning results:
The performance of all ionosphere models is correlated with the ionospheric conditions. For
each model, the positioning accuracy is higher on ionospheric quiet days than on disturbed
days, and performance is better at mid- and high-latitude stations than at equatorial stations.
At mid- and high-latitude stations, almost all models can provide the metre level accuracy on
ionospheric quiet days. At equatorial stations, even the best model can only provide about
one metre level accuracy during ionospheric disturbed periods.
Based on the same dataset, the ionospheric estimation model and GIM offer better
performance than the Klobuchar model. The ionospheric estimation model and GIM provide
comparable accuracy at mid-latitude stations. But the GIM is slightly more accurate at high-
latitude stations, while the ionospheric estimation model is much better at equatorial stations.
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Carrier phase observations can always improve positioning accuracy when using the GIM
and precise GPS orbit and clock products, but they can only slightly improve the positioning
accuracy at mid- and high-latitude stations when using the Klobuchar model.
The demonstrated positioning accuracy using the Klobuchar model and the ionospheric
estimation model is obtainable in real-time using IGDG real-time orbit and clock products.
The same positioning accuracy is only obtainable with latency of about 11 days when using
the GIM and IGS ionospheric products.
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CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
Real-time position determination, receiver clock offset estimation and atmospheric sensing
using PPP methodology have been investigated in this thesis. From the research results
presented in this thesis, some conclusions and recommendations are summarized in the
following.
7.1 Conclusions
A number of error sources, which can be removed completely or mitigated partially by
double differencing techniques, should be taken into account in PPP to exploit this novel
technique for different applications.
Several types of real-time precise GPS orbit and clock products are now available from
different agencies, such as JPL and NRCan. Internet multicast and satellite broadcast are two
complementary approaches for real-time distribution of precise GPS orbit and clock
products. Users can choose either approach based on the reliability, timeliness, convenience,
and available devices that best suit their field of application.
254
JPL IGDG real-time orbit and clock products can be received over the Internet with a latency
of about 4 s, which only adds an extrapolation error of less than 1 cm in real-time processing.
A real-time software package, P3-RT, has been developed for real-time precise point
positioning, timing and atmospheric sensing. The software can output position solutions,
receiver clock offset and ZTD estimates within 50 ms provided the precise GPS orbit and
clock products are accessible in real-time. The computational efficiency and implementation
flexibility of PPP make it a promising tool for real-time positioning, timing and atmospheric
sensing.
Sub-centimetre level static and sub-decimetre level kinematic positioning accuracy is
obtainable in real-time using dual-frequency code and phase measurements, and real-time
precise GPS orbit and clock products.
Among several types of real-time orbit and clock products tested in this research, IGDG
products from JPL provide the most consistent and accurate results in real-time; GPS•C
phase solution products, which are still at the development stage and have a latency of
several hours, provide comparable accuracy but with less consistency as the IGDG products.
On the other hand, GPS•C code solution products can provide sub-metre level positioning
accuracy in real-time with dual-frequency code measurements.
The convergence time of the float ambiguity in un-differenced phase measurement is affected
by a variety of factors, including satellite geometry, multipath effects, performance of precise
orbit and clock products, etc. Typical convergence time is about 20~30 minutes for the
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positioning errors to converge to the decimetre level. In a static mode, a longer time,
generally in the range of 30 minutes to 1 hour, is required for the positioning results to
converge to the centimetre level.
Using real-time precise GPS orbit and clock products and PPP methodology, precipitable
water vapour overlying the GPS receiver can be estimated in real-time with an accuracy of
about 1 mm, if precise pressure and temperature measurements are available in real-time.
Issues related to real-time atmospheric sensing have been investigated. Besides troposphere
horizontal gradients, elevation cut-off angle, antenna phase center offset and variations,
mapping functions also play an important role in the ZWD or PWV estimation. The best
strategy is to model and map the ZHD with precise meteorological measurements and
hydrostatic mapping function and estimate the ZWD with a wet mapping function. If precise
meteorological measurements are not available in real-time, a pressure model can be used
instead. Estimating the total ZTD using a single mapping function will lead to big bias in the
ZWD or PWV estimates, which is related to the relative difference between the hydrostatic
and wet mapping functions at the elevation cut-off angle.
Errors in real-time orbit and clock products contribute most to the errors in real-time
atmospheric sensing using PPP methodology. Other main error contributions come from the
multipath, the higher-order ionospheric effects and the transfer factor. The multipath effects
can be reduced by site and antenna selection. A more accurate transfer factor can be obtained
by an analysis on the local data. However, the higher-order ionospheric effects cannot be
256
mitigated by users unless they can be taken into account in the real-time orbit and clock
products.
Receiver clock offset can be estimated in real-time to an accuracy of about 100-picosecond
with respect to the reference clock of the precise GPS orbit and clock products, therefore
holding great promise for real-time time transfer applications.
JPL IGDG real-time products provide a new method to recover UTC(USNO) at an accuracy
of a few nanoseconds in real-time using a single GPS receiver. Timing using PPP
methodology and IGDG products, which keeps the flexibility of the one-way time transfer,
can offer much better performance for recovering UTC(USNO) in real-time than the
traditional one-way time transfer method. The capability of this method to recover
UTC(USNO) is comparable to the common-view method which requires that one clock be
precisely linked to UTC(USNO).
Ionosphere horizontal gradients have been investigated at a mid-latitude station. An
ionospheric estimation model has been proposed to estimate the ionosphere horizontal
gradients along with the zenith delay. The new model can provide VTEC estimates with an
accuracy of 2 TECU when compared with the IGS Final ionospheric products.
The performance of single-frequency precise point positioning is correlated with the
ionospheric activity. Several models have been investigated which can mitigate ionospheric
effects at different levels. The ionospheric estimation model can offer sub-metre level
position solutions with single-frequency data in real-time. The performance of the
257
ionospheric estimation model is comparable to the global ionospheric model using the IGS
ionospheric products which have a latency of 11 days.
The real-time tests which have been conducted with the P3-RT software for positioning using
single or dual-frequency GPS measurements and for atmospheric sensing have indicated the
potential of PPP for real-time positioning and meteorology applications.
7.2 Recommendations
Real-time precise GPS orbit and clock products are essential for the implementation of PPP
processing for real-time positioning, timing and meteorological applications. Currently, only
several types of precise GPS products can be provided to users in real-time. This is because a
real-time global tracking network is required to estimate high precision GPS orbits and
clocks. The denser the network, the more consistent and accurate the products can be.
Therefore, more agencies should be involved to provide real-time GPS data to processing
centers.
The 20 to 30 minutes convergence time of the float ambiguities will affect PPP for real-time
applications. Progresses have been made by researchers to improve the ambiguity
convergence. A convergence time of several minutes would be highly desired.
Higher-order ionospheric effects, which can be up to several centimetres in magnitude at
zenith during times of high TEC, are one of the major error sources for meteorological
parameter determination using GPS. Although algorithms have been developed to mitigate
258
the higher-order ionospheric effects to the millimetre level, they would not benefit PPP users
unless the higher-order ionospheric effects have been taken into account in the data
processing for the orbit and clock products. Some agencies have plans to mitigate the higher-
order ionospheric effects in their products. For example, the IGS has considered mitigating
the second-order ionospheric effects during data processing (Dow, 2004).
The IGS ionospheric products, which are currently accurate up to 2-TECU at grid points, can
be used to produce sub-metre level positioning solutions for single-frequency GPS users.
However, the accuracy degrades under ionospheric disturbed conditions and in equatorial
regions. In addition, the products are only available with a delay of 11 days. As precise GPS
orbit and clock products are expected to be widely available in real-time in the near future,
precise ionospheric products with higher resolutions and a shorter latency will be in high
demand by single-frequency users.
The absolute zenith and slant tropospheric delays estimated from PPP data processing can be
used for troposphere tomographic modeling. Also the absolute zenith and slant ionospheric
delays obtained from PPP using un-differenced data can be used for ionosphere tomographic
modeling.
Real-time PPP processing using precise GPS orbit and clock products, precise pressure and
temperature measurements for real-time precipitable water vapour estimation is significant
and should be investigated in the future.
259
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