GPS (eBook)

Preface to the Third Edition 6
References 9
Preface to the Second Edition 10
Preface to the First Edition 13
Contents 17
Abbreviations and Constants 24
Abbreviations 24
1 Introduction 28
1.1 A Key Note on GPS 29
1.1.1 GPS Modernization 31
1.2 A Brief Message About GLONASS 34
1.2.1 The Development of GLONASS 34
1.3 Basic Information on Galileo 36
1.3.1 The Development of Galileo 37
1.4 Introduction of BeiDou 38
1.4.1 The Development of BeiDou 39
1.5 A Combined Global Navigation Satellite System 40
References 41
2 Coordinate and Time Systems 43
2.1 Geocentric Earth-Fixed Coordinate Systems 43
2.2 Coordinate System Transformations 47
2.3 Local Coordinate System 48
2.4 Earth-Centred Inertial Coordinate System 50
2.5 IAU 2000 Framework 54
2.6 Geocentric Ecliptic Inertial Coordinate System 58
2.7 Time Systems 59
References 62
3 Satellite Orbits 63
3.1 Keplerian Motion 63
3.1.1 Satellite Motion in the Orbital Plane 66
3.1.2 Keplerian Equation 70
3.1.3 State Vector of the Satellite 72
3.2 Disturbed Satellite Motion 75
3.3 GPS Broadcast Ephemerides 75
3.4 IGS Precise Ephemerides 77
3.5 GLONASS Ephemerides 78
3.6 Galileo Ephemerides 79
3.7 BDS Ephemerides 79
References 79
4 GPS Observables 80
4.1 Code Pseudoranges 80
4.2 Carrier Phases 82
4.3 Doppler Measurements 84
References 86
5 Physical Influences of GPS Surveying 87
5.1 Ionospheric Effects 87
5.1.1 Code Delay and Phase Advance 87
5.1.2 Elimination of Ionospheric Effects 90
5.1.3 Ionospheric Models 93
5.1.4 Mapping Functions 97
5.1.5 Introduction of Commonly Used Ionospheric Models 100
5.2 Tropospheric Effects 104
5.2.1 Tropospheric Models 105
5.2.2 Mapping Functions and Parameterisation 109
5.2.3 Introduction of Commonly Used Tropospheric Models 112
5.2.4 Tropospheric Model for Airborne Kinematic Positioning 115
5.2.5 Water Vapour Research with Ground-Based GPS Measurement 117
5.3 Relativistic Effects 118
5.3.1 Special Relativity and General Relativity 118
5.3.2 Relativistic Effects on GPS 121
5.4 Earth Tide and Ocean Loading Tide Corrections 123
5.4.1 Earth Tide Displacements of GPS Stations 123
5.4.2 Simplified Model of Earth Tide Displacements 125
5.4.3 Numerical Examples of Earth Tide Effects 127
5.4.4 Ocean Loading Tide Displacement 129
5.4.5 Computation of the Ocean Loading Tide Displacement 132
5.4.6 Numerical Examples of Loading Tide Effects 133
5.5 Clock Errors 134
5.5.1 Introduction of Commonly Used Clock Error Models 136
5.5.2 Impact of Frequency Reference of a GPS Receiver on the Positioning Accuracy 138
5.6 Multipath Effects 139
5.6.1 GPS Altimetry, Signals Reflected from the Earth’s Surface 141
5.6.2 Reflecting Point Positioning 141
5.6.3 Image Point and Reflecting Surface Determination 143
5.6.4 Research Activities in GPS Altimetry 144
5.7 Anti-spoofing and Selective Availability Effects 145
5.8 Antenna Phase Centre Offset and Variation 146
5.9 Instrumental Biases 150
References 151
6 GPS Observation Equations and Equivalence Properties 157
6.1 General Mathematical Models of GPS Observations 157
6.2 Linearisation of the Observation Model 159
6.3 Partial Derivatives of Observation Function 161
6.4 Linear Transformation and Covariance Propagation 165
6.5 Data Combinations 166
6.5.1 Ionosphere-Free Combinations 168
6.5.2 Geometry-Free Combinations 169
6.5.3 Standard Phase–Code Combination 172
6.5.4 Ionospheric Residuals 173
6.5.5 Differential Doppler and Doppler Integration 174
6.6 Data Differentiations 176
6.6.1 Single Differences 177
6.6.2 Double Differences 180
6.6.3 Triple Differences 182
6.7 Equivalence of the Uncombined and Combining Algorithms 184
6.7.1 Uncombined GPS Data Processing Algorithms 185
6.7.2 Combining Algorithms of GPS Data Processing 187
6.7.3 Secondary GPS Data Processing Algorithms 192
6.7.4 Summary 195
6.8 Equivalence of Undifferenced and Differencing Algorithms 196
6.8.1 Introduction 196
6.8.2 Formation of Equivalent Observation Equations 197
6.8.3 Equivalent Equations of Single Differences 199
6.8.4 Equivalent Equations of Double Differences 203
6.8.5 Equivalent Equations of Triple Differences 205
6.8.6 Method of Dealing with the Reference Parameters 206
6.8.7 Summary of the Unified Equivalent Algorithm 207
References 208
7 Adjustment and Filtering Methods 210
7.1 Introduction 210
7.2 Least Squares Adjustment 210
7.2.1 Least Squares Adjustment with Sequential Observation Groups 212
7.3 Sequential Least Squares Adjustment 214
7.4 Conditional Least Squares Adjustment 216
7.4.1 Sequential Application of Conditional Least Squares Adjustment 218
7.5 Block-Wise Least Squares Adjustment 219
7.5.1 Sequential Solution of Block-Wise Least Squares Adjustment 221
7.5.2 Block-Wise Least Squares for Code–Phase Combination 223
7.6 Zhou’s Theory: Equivalently Eliminated Observation Equation System 224
7.6.1 Zhou–Xu’s Theory: Diagonalised Normal Equation and the Equivalent Observation Equation 227
7.7 Kalman Filter 229
7.7.1 Classic Kalman Filter 229
7.7.2 Kalman Filter: A General Form of Sequential Least Squares Adjustment 231
7.7.3 Robust Kalman Filter 232
7.7.4 Yang’s Filter: Adaptively Robust Kalman Filtering 235
7.7.5 Progress in Adaptively Robust Filter Theory and Application 239
7.7.6 A Brief Introduction to the Intelligent Kalman Filter 241
7.8 A Priori Constrained Least Squares Adjustment 241
7.8.1 A Priori Parameter Constraints 242
7.8.2 A Priori Datum 243
7.8.3 Zhou’s Theory: Quasi-Stable Datum 245
7.9 Summary 247
References 249
8 Cycle Slip Detection and Ambiguity Resolution 252
8.1 Cycle Slip Detection 252
8.2 Method of Dealing with Cycle Slips 254
8.3 A General Criterion of Integer Ambiguity Search 254
8.3.1 Introduction 254
8.3.2 Summary of Conditional Least Squares Adjustment 255
8.3.3 Float Solution 257
8.3.4 Integer Ambiguity Search in Ambiguity Domain 258
8.3.5 Integer Ambiguity Search in Coordinate and Ambiguity Domains 259
8.3.6 Properties of Xu’s General Criterion 261
8.3.7 An Equivalent Ambiguity Search Criterion and Its Properties 262
8.3.8 Numerical Examples of the Equivalent Criterion 265
8.3.9 Conclusions and Comments 267
8.4 Ambiguity Resolution Approach Based on the General Criterion 268
8.5 Ambiguity Function 270
8.5.1 Xu’s Conjecture: Maximum Property of Ambiguity Function 271
8.6 Ionosphere-Free Ambiguity Fixing 274
8.6.1 Introduction 274
8.6.2 Concept of Ionospheric Ambiguity Correction 276
8.6.3 Determination of the Ionospheric Ambiguity Correction 279
8.6.4 Integer Ambiguity Fixing Through Ambiguity-Ionospheric Equations 280
8.6.5 Float Ambiguity Fixing 280
8.7 PPP Ambiguity Fixing 280
References 282
9 Parameterisation and Algorithms of GPS Data Processing 285
9.1 Parameterisation of the GPS Observation Model 285
9.1.1 Evidence of the Parameterisation Problem of the Undifferenced Observation Model 286
9.1.2 A Method of Uncorrelated Bias Parameterisation 287
9.1.3 Geometry-Free Illustration 293
9.1.4 Correlation Analysis in the Case of Phase–Code Combinations 294
9.1.5 Conclusions and Comments 295
9.2 Equivalence of the GPS Data Processing Algorithms 296
9.2.1 Equivalence Theorem of GPS Data Processing Algorithms 297
9.2.2 Optimal Baseline Network Forming and Data Condition 299
9.2.3 Algorithms Using Secondary GPS Observables 301
9.2.4 Simplified Equivalent Representation of GPS Observation Equations 302
9.3 Non-equivalent Algorithms 309
9.4 Reference Changing in GPS Difference Algorithm 309
9.4.1 Changing Reference Satellite 309
9.4.2 Changing Reference Station 310
9.5 Standard Algorithms of GPS Data Processing 313
9.5.1 Preparation of GPS Data Processing 313
9.5.2 Single Point Positioning 314
9.5.3 Standard Un-differential GPS Data Processing 319
9.5.4 Equivalent Method of GPS Data Processing 322
9.5.5 Relative Positioning 323
9.5.6 Velocity Determination 324
9.5.7 Kalman Filtering Using Velocity Information 327
9.6 Accuracy of the Observational Geometry 328
9.7 Introduction to the Real-Time Positioning System 330
9.7.1 Network RTK 330
9.7.2 PPP-RTK 333
References 333
10 Applications of GPS Theory and Algorithms 335
10 Applications of GPS Theory and Algorithms 335
10.1.1 Functional Library 335
10.1.2 Data Platform 340
10.1.3 A Data Processing Core 342
10.1.3 A Data Processing Core 343
10.1.3 A Data Processing Core 345
10.3.1 Introduction 346
10.3.2 Concept of Precise Kinematic Positioning 348
10.3.2.1 Combining the Static References with IGS Station 348
10.3.2.2 Earth Tide and Loading Tide Corrections 348
10.3.2.3 Multiple Static References for Kinematic Positioning 349
10.3.2.4 Introducing Height Information as a Condition 351
10.3.2.5 Creation of a Kinematic Tropospheric Model 351
10.3.2.6 Higher-Order Ionospheric Effect Correction 352
10.3.2.7 A General Method of Integer Ambiguity Fixing 352
10.3.3 Concept of Flight-State Monitoring 352
10.3.4 Results, Precision Estimation, and Comparisons 355
10.3.4.1 Multiple Static References for Kinematic Positioning 357
10.3.4.2 Ambiguity of Multiple Static References as a Condition for Kinematic Positioning 357
10.3.4.3 Multiple Kinematic GPS for Flight-State Monitoring and Its Comparison with INS 359
10.3.4.4 Static GPS Data Kinematic Processing 360
10.3.4.5 Doppler Velocity Comparisons 360
10.3.5 Conclusions 360
References 361
11 Perturbed Orbit and Its Determination 363
11.1 Perturbed Equation of Satellite Motion 363
11.1.1 Lagrangian Perturbed Equation of Satellite Motion 364
11.1.2 Gaussian Perturbed Equation of Satellite Motion 367
11.2 Perturbation Forces of Satellite Motion 370
11.2.1 Perturbation of the Earth’s Gravitational Field 370
11.2.1.1 The Earth’s Gravitational Field 370
11.2.1.2 Perturbation Force of the Earth’s Gravitational Field 373
11.2.2 Perturbations of the Sun, the Moon, and the Planets 375
11.2.3 Earth Tide and Ocean Tide Perturbations 376
11.2.4 Solar Radiation Pressure 380
11.2.5 Atmospheric Drag 384
11.2.6 Additional Perturbations 387
11.2.7 Order Estimations of Perturbations 389
11.2.8 Ephemerides of the Moon, the Sun, and Planets 390
11.3 Analysis Solution of the /overline{C}_{20} Perturbed Orbit 394
11.4 Orbit Correction 401
11.5 Principle of GPS Precise Orbit Determination 405
11.5.1 Xu’s Algebraic Solution to the Variation Equation 407
11.6 Numerical Integration and Interpolation Algorithms 409
11.6.1 Runge–Kutta Algorithm 409
11.6.2 Adams Algorithms 413
11.6.3 Cowell Algorithms 416
11.6.4 Mixed Algorithms and Discussions 418
11.6.5 Interpolation Algorithms 419
11.7 Orbit-Related Partial Derivatives 420
References 429
12 Singularity-Free Orbit Theory 431
12.1 A Brief Historical Review of the Singularity Problem 431
12.2 On the Singularity Problem in Orbital Mechanics 434
12.2.1 Basic Lagrangian and Gaussian Equations of Motion 434
12.2.2 Solving Algorithm for the Singularity Problem 439
12.2.3 Xu’s Criteria for Singularity 440
12.2.4 Derivation of Lagrange-Xu Equations of Motion 441
12.2.5 Derivation of Gauss Equations from Lagrange Equations 451
12.2.6 Derivation of Gauss-Xu Equations of Motion 453
12.3 Bridge Between Analytical Theory and Numerical Integration 456
References 457
13 Discussions 460
13.1 Independent Parameterisation and A Priori Information 460
13.2 Equivalence of the GPS Data Processing Algorithms 462
13.3 Other Comments 463
Appendix A: IAU 1980 Theory of Nutation 465
Appendix B: Numerical Examples of the Diagonalisation of the Equations 469
References 475
Index 502