Radio Occultations Using Earth Satellites: A Wave Theory Treatment (Deep-Space Communications and Navigation Series 6)

1.1 Introduction......Page 23
1.1.1 History of the Occultation Technique......Page 24
1.1.2 Occultations from Earth Satellites......Page 25
1.1.3 The Global Positioning System......Page 30
1.1.4 Timing......Page 33
1.1.5 Ephemerides......Page 34
1.2.1 Connected Phase......Page 35
1.2.3 Inverting Radio Occultation Data......Page 36
1.2.4 Assimilating Radio Occultation Data......Page 39
1.2.5 Rays and Stationarity......Page 40
1.2.6 Excess Doppler......Page 43
1.3 Scientific Applications of GPS Occultation Observations......Page 45
1.3.1 Weather......Page 46
1.3.2 Climate......Page 49
1.4 Problems from Multipath and Some Remedies......Page 50
1.4.1 Spectral/Holographic Techniques......Page 54
1.4.2 Back Propagation......Page 56
1.4.4 The Impact Parameter Space Curve......Page 63
1.4.5 A Full-Spectrum Wave Theory......Page 65
1.5.1 Chapter 2......Page 69
1.5.2 Chapter 3......Page 70
1.5.3 Chapter 4......Page 71
1.5.4 Chapter 5......Page 72
1.5.5 Chapter 6......Page 74
1.5.6 Appendices......Page 82
1.6.2 Placing the Occulted GPS Satellite at Infinity......Page 83
1.6.5 Coplanarity......Page 85
1.6.7 Treating the GPS Signal as a Harmonic Wave......Page 86
1.7 Recommendations for the Next Chapters......Page 87
References......Page 88
Acronyms......Page 0
2.1 Introduction......Page 98
2.1.1 Multipath Scenarios......Page 100
2.1.2 An Overview of Chapter 2......Page 103
2.2 Geometric Optics in a Spherical Medium......Page 104
2.2.1 Multipath, Shadow Zones, and Caustics According to Geometric Optics......Page 107
2.2.2 Thin-Atmosphere Conditions......Page 111
2.3 Thin Phase Screen Models......Page 113
2.3.1 The Helmholtz–Kirchoff Integral Theorem......Page 114
2.3.2 The Space Curve for Impact Parameter......Page 115
2.3.3 The Fresnel Phase Screen......Page 121
2.3.4 Suitability of the Thin-Screen Model for Diffraction Analysis......Page 122
2.3.5 A Phase Profile for the Thin Screen......Page 123
2.3.6 Bending-Angle Perturbations......Page 124
2.3.7 Case A: A Discontinuity in Refractivity......Page 126
2.3.8 Case B: A Discontinuity in Scale Height......Page 128
2.4 Multipath Using a Thin Phase Screen Model......Page 130
2.5 Scalar Diffraction: The Rayleigh–Sommerfeld Integral......Page 135
2.6 The Stationary-Phase Technique......Page 137
2.6.1 Necessary Conditions for Validity of the Geometric Optics Approach......Page 142
2.7 Numerical Results Using Thin-Screen/Scalar Diffraction......Page 143
2.7.1 Fresnel Response to a Discontinuity in Lapse Rate......Page 146
2.7.2 Fresnel Response to a Discontinuity in Refractivity......Page 149
2.7.3 A Boundary Layer......Page 152
2.8 Sensing a Boundary in the Ionosphere......Page 154
2.8.1 Fresnel Effects at a Boundary......Page 157
2.8.2 Amplitude Effects at a Boundary......Page 160
2.8.3 Ray Splitting......Page 161
2.8.4 Doppler Information at a Boundary......Page 167
2.8.5 Fresnel Effects Using an Improved Electron Density Model......Page 169
2.9 The Error in the Recovered Refractivity Resulting from Fresnel Phase Perturbations......Page 173
2.10 Fresnel Transform Techniques......Page 177
2.10.1 Adjoining the Constraint A(u) equivalent 1......Page 179
2.10.3 A Numerical Example......Page 180
2.10.4 Fresnel Aliasing......Page 182
2.10.5 Numerical Results......Page 183
References......Page 188
3.1 Introduction......Page 191
3.2 Scalar Potentials......Page 194
3.2.1 Series Expansions for Scalar Potentials......Page 196
3.3 Multiple Internal Reflections......Page 199
3.4 Fresnel Formulas for Reflection and Transmission Amplitudes......Page 201
3.4.1 Conservation Principle......Page 204
3.4.2 Scattering Angles and Intensities......Page 205
3.4.3 Caustics......Page 209
3.5.1 Transmission and Reflection Coefficients for Incident External Waves......Page 212
3.5.2 Transmission and Reflection Coefficients for Incident Interior Waves......Page 216
3.5.3 Aggregate Scattering......Page 218
3.6 The Problem of Slow Convergence......Page 220
3.7 The Sommerfeld–Watson Transformation......Page 221
3.8 Evaluating Scattering Coefficients with Asymptotic Expansions......Page 223
3.9 Expressing Scattering Coefficients in Terms of Phasors......Page 229
3.10 Asymptotic Forms for the Hankel and Legendre Functions Evaluated at the LEO......Page 234
3.11 Geometric Optics Interpretation of Mie Scattering Theory......Page 236
3.12 Evaluating Mie Scattering by Integration of the Scattering Phasor......Page 238
3.13.1 Stationary Phase......Page 240
3.13.2 Comparison with Diffraction from a Knife-Edge......Page 243
3.13.3 Stationary-Phase Points for the Scattering Integrals......Page 244
3.13.4 Vestigial Rainbow Effects......Page 249
3.13.5 Asymptotic Solutions for a Transparent Refracting Sphere......Page 252
3.13.6 Correspondence Between the Stationary-Phase Value for Spectral Number and Snell’s Law......Page 256
3.13.7 Stationary Value for the Scattering Phase......Page 257
3.13.8 Evaluating the Scattering Integrals Using the Stationary-Phase Technique......Page 260
3.15.1 Numerical Integrations......Page 263
3.15.2 Comparison with Thin-Screen/Scalar Diffraction Results......Page 266
3.16.1 Rainbow Effects on the Refracting Sphere......Page 269
3.16.2 Rainbow Effects at LEO Distances......Page 271
3.16.3 Assessing Rainbow Effects Using the Third-Order Stationary-Phase Technique......Page 273
3.17.1 The Perfectly Reflecting Sphere......Page 277
3.17.2 Geometric Optics Interpretation......Page 283
3.17.3 Limiting Cases: The Strongly Absorbing Sphere......Page 285
References......Page 289
4.1 Introduction......Page 291
4.2.1 Cartesian Stratification......Page 292
4.3 The Characteristic Matrix......Page 296
4.4 The Stratified Medium as a Stack of Discrete Layers......Page 297
4.4.1 The Characteristic Matrix when n (x) = constant......Page 298
4.4.2 A Stack of Homogeneous Layers when n (x) is Piecewise Constant......Page 299
4.4.3 Range of Validity......Page 302
4.4.4 The TM Case......Page 303
4.5 The Characteristic Matrix for an Airy Layer......Page 304
4.6 Incoming and Outgoing Waves and Their Turning Points......Page 308
4.6.1 Eikonal and Cophasal Normal Paths......Page 313
4.6.2 Defocusing......Page 316
4.7 Concatenated Airy Layers......Page 318
4.8 Osculating Parameters......Page 323
4.8.1 At a Turning Point......Page 329
4.9 Airy Functions as Basis Functions......Page 330
4.10 Wave Propagation in a Cylindrical Stratified Medium......Page 331
4.11 Wave Propagation in a Spherical Stratified Medium......Page 338
4.12 Correspondence between Characteristic Matrices for Cartesian and Spherical Stratified Airy Layers......Page 344
References......Page 347
5.1 Introduction......Page 348
5.2 Maxwell’s Equations in a Stratified Linear Medium......Page 353
5.2.1 Scalar Potential Functions......Page 354
5.3 Modified Spherical Bessel Functions......Page 356
5.4 Asymptotic Forms......Page 360
5.5 Modified Mie Scattering in a Spherical Stratified Medium......Page 362
5.5.1 Incoming Waves......Page 366
5.5.2 Evaluating the Spectral Coefficients in a Stratified Medium......Page 371
5.5.3 Outgoing Waves......Page 376
5.5.4 Correspondence between Cartesian and Spherical-Stratified Phase Quantities......Page 377
5.5.5 Absorption......Page 378
5.6 More Geometric Optics: Cumulative Bending Angle, Bouguer’s Law, and Defocusing......Page 379
5.6.1 Defocusing......Page 382
5.7 More Asymptotic Forms......Page 385
5.7.1 Equating dG[rho^ † (nu),nu]/dnu to tilde alpha(nu,nu) and d^2G[rho^ †(nu),nu]/dnu^2 to d tilde alpha(nu,nu)/d nu......Page 392
5.7.2 Fixing the Form for g (y \Z)  when y \Z> 0 by Asymptotic Matching......Page 393
5.7.3 Behavior of the WKB Solution at a Turning Point......Page 396
5.8 Spectral Representation of an Electromagnetic Wave in a Spherical Stratified Medium......Page 398
5.8.1 Behavior of  partial G /partial nu......Page 400
5.8.2 Accuracy of the Osculating Parameter Technique......Page 404
5.8.3 Numerical Comparisons......Page 408
5.8.4 Comparison of Phase Delays in an Airy Layer from Wave Theory and Geometric Optics......Page 417
5.8.5 Asymptotic Matching the Spectral Coefficients for Incoming and Outgoing Waves......Page 418
5.9.1 Geometric Interpretation of the Phasors......Page 422
5.9.2 Stationary Phase Conditions......Page 424
5.9.3 Stationarity of Psi ( ±,± )......Page 425
5.9.4 Plane Waves......Page 426
5.9.5 The Electric Field for an Incoming Wave......Page 428
5.9.6 Evaluating the Electric Field Vector Using the Stationary Phase Technique......Page 431
5.9.7 An Outgoing Electric Field......Page 435
5.10 Comparison of Geometric Optics and Wave Theory......Page 437
5.10.1 Comparison of Wave Theory with Geometric Optics......Page 438
5.10.2 Duality between Systems......Page 442
5.10.3 Amendments to Account for Wave-Front Curvature from the Finite Distance of the Emitting GPS Satellite......Page 444
5.11 The Electric Field at a Turning Point......Page 445
5.11.1 Fourier Transform of the Airy Function......Page 448
5.11.2 Fresnel Transform of the Airy Function......Page 450
5.12 Caustics and Multipath......Page 451
5.12.1 A Numerical Example of Multipath and Caustics......Page 452
5.12.2 A Geometric Optics Interpretation of Multipath......Page 459
5.12.4 Third-Order Theory......Page 462
5.12.5 Reduction of Multipath Spectral Width by Defocusing......Page 471
5.12.6 Combined Water Vapor and Dry Air Refractivity Model......Page 472
5.13 Spectral Coefficients in a Spherical Refracting Medium with an Embedded Discontinuity......Page 477
5.13.1 A Medium with a Discontinuity in its Refractive Gradient......Page 478
5.13.2 A Transparent Sphere Embedded in a Refracting Medium......Page 487
5.14 The Scattered Field from a Perfectly Reflecting Sphere Embedded in a Refracting Medium......Page 488
5.14.1 Stationary Phase Analysis......Page 494
5.14.2 Results from Wave Theory......Page 502
5.14.3 Law of Reflection......Page 505
References......Page 507
6.1 Introduction......Page 509
6.2 GPS Receiver Operations......Page 510
6.2.1 Adverse Signal Conditions......Page 514
6.2.2 Flywheeling......Page 520
6.2.3 Refractivity Error from a Single-Ray Paradigm......Page 522
6.3 Spectral Representation of the Field at the LEO......Page 524
6.3.2 The Obliquity Factor......Page 527
6.3.3 Doppler Variability......Page 529
6.4 Refractivity Recovery......Page 532
6.4.1 Super-Refractivity......Page 539
6.4.2 Improving the Accuracy of G†(v)......Page 544
6.4.3 Resolution Issues......Page 546
6.5 Summary......Page 547
References......Page 549
A.1 Bending Angle and Bouguer’s Law, Eq. (2.2-2)......Page 551
A.2 Defocusing......Page 553
A.3 Excess Doppler......Page 554
A.4 Scalar Diffraction Theory......Page 560
A.5 The First Fresnel Zone......Page 562
A.6 The Abel Transform......Page 563
A.8 Perturbation of the Bending-Angle Profile Near a Boundary......Page 564
A.9 Bending-Angle Perturbation......Page 568
A.10 Bending-Angle Perturbation by a Discontinuous Refractivity, Case A......Page 569
A.11 The Fresnel Transform Pair......Page 570
A.12 Ray Path Phase Delay......Page 571
References......Page 573
Appendix B Caustic Surfaces......Page 574
References......Page 578
C.1 Case C: A Discontinuous Lapse Rate......Page 579
C.2 Case B: A Discontinuous Scale Height......Page 581
C.3 Case A: A Discontinuous Refractivity......Page 582
D.1 Maximum Flaring......Page 586
D.2 Minimum Signal Amplitude in a Shadow Zone......Page 589
D.3 Accuracy of the Stationary Phase Technique......Page 594
References......Page 595
Appendix E Bending by a Gaussian Electron Density Distribution......Page 596
Appendix F The Effect of Cycle Slips on Recovered Refractivity......Page 598
G.1 Introduction......Page 602
G.2 Application to a Perfectly Reflecting Sphere......Page 606
G.3 Application to a Refracting Sphere......Page 608
G.4 Aggregate Scattering......Page 611
Reference......Page 613
Appendix H Characteristic Matrix in a Stack of Airy Layers......Page 614
Appendix I Field Equations in a Stratified Medium......Page 620
Appendix J Conditions for Near-Equivalence between dG^†(nu) / d nu and tilde alpha (nu, nu), and between d^2G ^†(nu)/ •d nu^2 and d tilde alpha(nu, nu)/d nu......Page 622
Glossary of Terms......Page 626
Acronyms......Page 634