 | Hoque M.M. Higher order ionospheric propagation effects and their corrections in precise GNSS positioning / Deutsches Zentrum für Luft- und Raumfahrt Institut für Kommunikation und Navigation, Neustrelitz. - Köln: DLR, Bibliotheks- und Informationswesen, 2009. - xii, 235 p.: ill. - (Forschungsbericht; 09-09). - ISSN 1434-8454
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1 INTRODUCTION ................................................. 1
1.1 Motivation .............................................. 1
1.2 Structure of this thesis ................................ 2
1.3 Concept of satellite positioning ........................ 3
1.4 GPS overview ............................................ 4
1.4.1 GPS constellation ................................ 4
1.4.2 GPS signal ....................................... 5
1.4.3 Calculating position ............................. 7
1.4.4 Error sources and accuracy ....................... 7
1.4.5 GPS modernizations ............................... 9
1.5 The European program for global navigation services,
Galileo ................................................ 10
1.5.1 Galileo signal .................................. 11
1.5.2 Availability .................................... 12
2 RADIO WAVE PROPAGATION IN THE IONOSPHERE .................... 13
2.1 Propagation in a dispersive medium ..................... 13
2.2 Propagation in an anisotropic medium ................... 14
2.3 Wave polarization ...................................... 16
2.4 Plasma and gyro frequencies ............................ 16
2.5 Ray path equation by Bouguer's law ..................... 17
2.6 Geometric distance or true range ....................... 18
2.7 Ionospheric refractive index ........................... 19
2.8 Ionospheric phase and group delay ...................... 21
2.9 Dual-frequency observables ............................. 22
2.10 Comparison of higher order ionospheric terms ........... 25
3 ELECTRON DENSITY AND GEOMAGNETIC FIELD MODEL ................ 27
3.1 Characteristics of the ionosphere ...................... 27
3.2 Chapman layer .......................................... 29
3.3 Simulation bed: ionosphere model with a broader
variety of profile shapes .............................. 30
3.3.1 Multi-layered ionosphere ........................ 30
3.3.2 Comparison with CHAMP reconstructed profiles .... 33
3.4 Integration of Chapman layer to calculate TEC .......... 34
3.5 Parabolic and quasi parabolic (QP) layer ............... 37
3.5.1 Parameter determination of a QP layer ........... 39
3.6 Multi quasi parabolic (MQP) fitting .................... 39
3.7 CHAMP vertical profile ................................. 41
3.8 Geomagnetic field model: ICRF .......................... 42
4 RAY TRACING THROUGH THE IONOSPHERE .......................... 45
4.1 Introduction ........................................... 45
4.2 Ray tracing algorithm .................................. 45
4.3 Ray direction in the ionosphere ........................ 51
4.4 Homing-in method ....................................... 53
4.4.1 Nelder-Mead simplex algorithm ................... 53
4.5 Validation of ray tracing results ...................... 54
4.5.1 Step size and computational error ............... 54
4.5.2 Comparison with Bmnner and Gu results ........... 55
5 SECOND ORDER IONOSPHERIC TERM AND ITS CORRECTION ............ 58
5.1 Introduction ........................................... 58
5.2 Magneto-ionic effect or second order ionospheric
delay .................................................. 58
5.3 Dual-frequency second order residual error Δs2 ......... 59
5.4 Second order delay and its correction for GNSS users
in Germany ............................................. 59
5.4.1 Second order term correction .................... 63
5.4.2 Validation of the correction formula ............ 65
5.4.3 Δs2 map over Germany ............................ 66
5.5 Global Δs2 variation at zenith ......................... 66
5.6 Δs2 variation at geomagnetic conjugate points .......... 67
5.7 Second order term and its correction for GNSS users
in Europe .............................................. 69
5.7.1 Δs2 correction: computing average magnetic
field component B cos Θ ......................... 69
5.7.2 Two-circle-approximation and parameter
determination ................................... 71
5.7.3 Model formulation ............................... 73
5.7.4 Model errors .................................... 75
5.8 Second order delay and its correction for GNSS users
in south-east Asia ..................................... 78
5.8.1 The correction formula .......................... 80
5.8.2 Validation ...................................... 80
5.9 Error in case of computing B cos Θ at ionospheric
pierce point ........................................... 82
6 THIRD ORDER IONOSPHERIC TERM AND ITS CORRECTION ............. 88
6.1 Third order phase delay ................................ 88
6.2 Dual-frequency third order residual error Δs3 .......... 89
6.3 Third order delay correction ........................... 91
6.3.1 Dual-frequency residual error correction ........ 95
7 EXCESS PATH LENGTH AND ITS CORRECTION ....................... 97
7.1 Excess path length (arc-to-chord) ...................... 97
7.2 The correction formula ................................ 100
7.3 Alternative correction formula ........................ 102
7.4 Dual-frequency residual error due to excess path
Δsb(length) ............................................. 105
7.5 Ray path deviation from the vacuum path ............... 106
8 RANGE ERROR DUE TO TEC DIFFERENCE AT TWO FREQUENCIES AND
ITS CORRECTION ............................................. 110
8.1 Introduction .......................................... 110
8.2 TEC difference between L1 and L2 signal paths ......... 110
8.3 Reasons of high TEC difference between L1 and L2
paths ................................................. 112
8.4 Range error due to TEC difference between L1 and L2
paths ................................................. 114
8.5 Analytical solution of TEC difference ................. 115
8.6 Δsb(TEC) correction using the analytical solution ...... 119
8.7 Empirical formula for TEC difference .................. 120
8.8 Validation of empirical formula ....................... 121
9 EFFECTS OF HORIZONTAL GRADIENTS ............................ 124
9.1 Introduction .......................................... 124
9.2 Ionospheric model with horizontal gradients ........... 124
9.3 Effects of horizontal gradients on Δs2 ............... 125
9.4 Effects of horizontal gradients on Δs3 ................ 127
9.5 Effects of horizontal gradients on Δsb(length) .......... 128
9.6 Effects of horizontal gradients on Δsb(TEC) ............ 129
10 HIGHER ORDER IONOSPHERIC EFFECTS ON USER POSITION .......... 131
10.1 Range computation using carrier-phase measurements .... 131
10.2 Position error computation using point positioning
technique ............................................. 132
10.3 Positioning improvements of higher order
corrections ........................................... 133
10.3.1 User position in northern hemisphere ........... 133
10.3.2 User position in southern hemisphere ........... 135
10.4 Performance of correction formulas .................... 137
11 REFRACTION OF GPS SIGNALS DURING RADIO OCCULTATION ......... 140
11.1 Introduction .......................................... 140
11.2 Occultation geometry and ray path deviation ........... 140
11.3 Refractions during CHAMP-CPS occultation .............. 142
11.4 Refraction effects using CHAMP vertical profile ....... 144
11.4.1 Case study I ................................... 145
11.4.2 Case study II .................................. 149
11.5 Excess path length formula ............................ 152
11.6 Refraction effects on signals received on LEO
satellites at 200 km height ........................... 158
12 FARADAY ROTATION CORRECTION IN SAR INTERFEROMETRY .......... 160
12.1 SAR principle ......................................... 160
12.2 Ionospheric Faraday rotation .......................... 161
12.3 Correction formula .................................... 162
12.4 Validation of correction formula ...................... 163
13 MODERNIZED GPS AND FUTURE GALILEO SYSTEMS .................. 167
13.1 Triple-frequency combination for range estimation ..... 167
13.2 Triple-frequency residual error ....................... 169
13.3 Triple-frequency error correction ..................... 170
13.3.1 Residual error (Δsb(TEC))tr correction ........... 170
13.3.2 Residual error (Δs3)tr correction .............. 171
13.3.3 Excess path length (Δsb(length))tr correction .... 171
13.4 Validation of correction formulas ..................... 172
13.5 Quadruple-frequency combination for range
estimation ............................................ 173
13.6 Quadruple-frequency residual error .................... 175
13.7 Residual error for new dual-frequency combinations .... 176
14 CONCLUSIONS ................................................ 180
14.1 Concluding remarks .................................... 180
14.1.1 Second order ionospheric term .................. 180
14.1.2 Third order ionospheric term ................... 181
14.1.3 Excess path length ............................. 181
14.1.4 Range error due to TEC difference between L1
and L2 signal paths ............................ 181
14.1.5 Higher order effects considering horizontal
gradients ...................................... 181
14.1.6 Higher order ionospheric effects on user
position ....................................... 182
14.1.7 Refraction effects during GPS occultation ...... 182
14.1.8 Higher order effects in triple- and
quadruple-frequency measurements ............... 182
14.2 Future work ........................................... 183
REFERENCES .................................................... 185
APPENDIX A: CONVERSION OF EARTH'S MAGNETIC FIELD COMPONENTS:
GEODETIC TO 3D CARTESIAN COORDINATE SYSTEM ................. 194
APPENDIX B: PROJECTION OF VECTORS ON A PLANE .................. 198
APPENDIX C: DERIVATION OF TEC DIFFERENCE BETWEEN TWO GNSS
SIGNAL PATHS ............................................... 199
APPENDIX D: DERIVATION OF TEC EXPRESSION AT ZENITH AND OTHER
ELEVATION ANGLES ........................................... 206
APPENDIX E: DERIVATION OF ∫n2edl AT ZENITH AND OTHER
ELEVATION ANGLES ........................................... 213
APPENDIX F: TEC DERIVATIVE WITH RESPECT TO RECEIVER HEIGHT .... 216
APPENDIX G: POINT POSITIONING TECHNIQUE: LINEARISED MODEL ..... 221
LIST OF ACRONYMS .............................................. 223
LIST OF SYMBOLS ............................................... 225
LIST OF FIGURES ............................................... 228
LIST OF TABLES ................................................ 234
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