TRENDS IN COMMUNICATIONS SATELLITES
INTRODUCTION HISTORY OF THE OCCULTATION TECHNIQUE OCCULTATIONS FROM EARTH SATELLITES THE GLOBAL POSITIONING SYSTEM TIMING EPHEMERIDES INFORMATION CONTENT IN GPS OCCULTATION OBSERVATIONS CONNECTED PHASE SAMPLE RATE VERSUS VERTICAL RESOLUTION INVERTING RADIO OCCULTATION DATA ASSIMILATING RADIO OCCULTATION DATA RAYS AND STATIONARITY EXCESS DOPPLER SCIENTIFIC APPLICATIONS OF GPS OCCULTATION OBSERVATIONS WEATHER CLIMATE PROBLEMS FROM MULTIPATH AND SOME REMEDIES SPECTRAL/HOLOGRAPHIC TECHNIQUES BACK PROPAGATION THE CANONICAL TRANSFORM TECHNIQUE THE IMPACT PARAMETER SPACE CURVE A FULL-SPECTRUM WAVE THEORY OVERVIEW CHAPTER 2 CHAPTER 3 CHAPTER 4 CHAPTER 5 CHAPTER 6 APPENDICES LIMITATIONS AND SIMPLIFICATIONS IONOSPHERE PLACING THE OCCULTED GPS SATELLITE AT INFINITY TIME SPHERICAL SYMMETRY COPLANARITY CIRCULARITY TREATING THE GPS SIGNAL AS A HARMONIC WAVE RECOMMENDATIONS FOR THE NEXT CHAPTERS REFERENCES CHAPTER 2: SCATTERING OF ELECTROMAGNETIC WAVES FROM A SPHERICAL BOUNDARY USING A THIN PHASE SCREEN MODEL AND SCALAR DIFFRACTION THEORY MULTIPATH SCENARIOS AN OVERVIEW OF CHAPTER 2 GEOMETRIC OPTICS IN A SPHERICAL MEDIUM MULTIPATH, SHADOW ZONES, AND CAUSTICS ACCORDING TO GEOMETRIC OPTICS THIN-ATMOSPHERE CONDITIONS THIN PHASE SCREEN MODELS THE HELMHOLTZ–KIRCHOFF INTEGRAL THEOREM THE SPACE CURVE FOR IMPACT PARAMETER THE FRESNEL PHASE SCREEN SUITABILITY OF THE THIN-SCREEN MODEL FOR DIFFRACTION ANALYSIS A PHASE PROFILE FOR THE THIN SCREEN BENDING-ANGLE PERTURBATIONS CASE A: A DISCONTINUITY IN REFRACTIVITY CASE B: A DISCONTINUITY IN SCALE HEIGHT CASE C: A DISCONTINUITY IN LAPSE RATE MULTIPATH USING A THIN PHASE SCREEN MODEL SCALAR DIFFRACTION: THE RAYLEIGH–SOMMERFELD INTEGRAL THE STATIONARY-PHASE TECHNIQUE NECESSARY CONDITIONS FOR VALIDITY OF THE GEOMETRIC OPTICS APPROACH NUMERICAL RESULTS USING THIN-SCREEN/SCALAR DIFFRACTION FRESNEL RESPONSE TO A DISCONTINUITY IN LAPSE RATE FRESNEL RESPONSE TO A DISCONTINUITY IN REFRACTIVITY A BOUNDARY LAYER SENSING A BOUNDARY IN THE IONOSPHERE FRESNEL EFFECTS AT A BOUNDARY AMPLITUDE EFFECTS AT A BOUNDARY RAY SPLITTING DOPPLER INFORMATION AT A BOUNDARY FRESNEL EFFECTS USING AN IMPROVED ELECTRON DENSITY MODEL THE ERROR IN THE RECOVERED REFRACTIVITY RESULTING FROM FRESNEL PHASE PERTURBATIONS FRESNEL TRANSFORM TECHNIQUES 2 10 1 ADJOINING THE CONSTRAINT A U ( ) 2 10 2 DIFFRACTION INTEGRAL WITH MULTIPATH 2 10 3 A NUMERICAL EXAMPLE 2 10 4 FRESNEL ALIASING 2 10 5 NUMERICAL RESULTS CHAPTER 3: SCATTERING FROM A LARGE TRANSPARENT SPHERE BASED ON MAXWELL’S EQUATIONS: MIE SCATTERING THEORY SCALAR POTENTIALS SERIES EXPANSIONS FOR SCALAR POTENTIALS MULTIPLE INTERNAL REFLECTIONS FRESNEL FORMULAS FOR REFLECTION AND TRANSMISSION AMPLITUDES CONSERVATION PRINCIPLE SCATTERING ANGLES AND INTENSITIES CAUSTICS MIE SCATTERING THEORY: OBTAINING THE SCATTERING COEFFICIENTS AT A BOUNDARY TRANSMISSION AND REFLECTION COEFFICIENTS FOR INCIDENT EXTERNAL WAVES TRANSMISSION AND REFLECTION COEFFICIENTS FOR INCIDENT INTERIOR WAVES AGGREGATE SCATTERING THE PROBLEM OF SLOW CONVERGENCE THE SOMMERFELD–WATSON TRANSFORMATION EVALUATING SCATTERING COEFFICIENTS WITH ASYMPTOTIC EXPANSIONS EXPRESSING SCATTERING COEFFICIENTS IN TERMS OF PHASORS ASYMPTOTIC FORMS FOR THE HANKEL AND LEGENDRE FUNCTIONS EVALUATED AT THE LEO GEOMETRIC OPTICS INTERPRETATION OF MIE SCATTERING THEORY EVALUATING MIE SCATTERING BY INTEGRATION OF THE SCATTERING PHASOR INTERPRETING SCATTERING USING THE STATIONARY- PHASE TECHNIQUE 3 13 1 STATIONARY PHASE 3 13 2 COMPARISON WITH DIFFRACTION FROM A KNIFE-EDGE 3 13 3 STATIONARY-PHASE POINTS FOR THE SCATTERING INTEGRALS 3 13 4 VESTIGIAL RAINBOW EFFECTS 3 13 5 ASYMPTOTIC SOLUTIONS FOR A TRANSPARENT REFRACTING SPHERE 3 13 6 CORRESPONDENCE BETWEEN THE STATIONARY-PHASE VALUE FOR SPECTRAL NUMBER AND SNELL’S LAW 3 13 7 STATIONARY VALUE FOR THE SCATTERING PHASE 3 13 8 EVALUATING THE SCATTERING INTEGRALS USING THE STATIONARY-PHASE TECHNIQUE DUALITY BETWEEN STATIONARY-PHASE CONCEPTS IN ELECTRODYNAMICS AND IN GEOMETRIC OPTICS DIFFRACTION FROM A LARGE, TRANSPARENT, REFRACTING SPHERE USING MIE SCATTERING THEORY 3 15 1 NUMERICAL INTEGRATIONS 3 15 2 COMPARISON WITH THIN-SCREEN/SCALAR DIFFRACTION RESULTS LOOKING FOR RAINBOWS 3 16 1 RAINBOW EFFECTS ON THE REFRACTING SPHERE 3 16 2 RAINBOW EFFECTS AT LEO DISTANCES 3 16 3 ASSESSING RAINBOW EFFECTS USING THE THIRD-ORDER STATIONARY-PHASE TECHNIQUE LIMITING CASES 3 17 1 THE PERFECTLY REFLECTING SPHERE 3 17 2 GEOMETRIC OPTICS INTERPRETATION 3 17 3 LIMITING CASES: THE STRONGLY ABSORBING SPHERE 3 17 4 LIMITING CASES: REFLECTIONS FROM A SPHERE OF INFINITE RADIUS CHAPTER 4: WAVE PROPAGATION IN A STRATIFIED MEDIUM: THE THIN-FILM APPROACH THIN-FILM CONCEPTS CARTESIAN STRATIFICATION THE CHARACTERISTIC MATRIX THE STRATIFIED MEDIUM AS A STACK OF DISCRETE LAYERS THE CHARACTERISTIC MATRIX WHEN N (X) = CONSTANT A STACK OF HOMOGENEOUS LAYERS WHEN N (X) IS PIECEWISE CONSTANT RANGE OF VALIDITY THE TM CASE THE CHARACTERISTIC MATRIX FOR AN AIRY LAYER INCOMING AND OUTGOING WAVES AND THEIR TURNING POINTS EIKONAL AND COPHASAL NORMAL PATHS DEFOCUSING CONCATENATED AIRY LAYERS OSCULATING PARAMETERS AT A TURNING POINT AIRY FUNCTIONS AS BASIS FUNCTIONS WAVE PROPAGATION IN A CYLINDRICAL STRATIFIED MEDIUM WAVE PROPAGATION IN A SPHERICAL STRATIFIED MEDIUM CORRESPONDENCE BETWEEN CHARACTERISTIC MATRICES FOR CARTESIAN AND SPHERICAL STRATIFIED AIRY LAYERS CHAPTER 5: PROPAGATION AND SCATTERING IN A SPHERICAL- STRATIFIED REFRACTING MEDIUM MAXWELL’S EQUATIONS IN A STRATIFIED LINEAR MEDIUM SCALAR POTENTIAL FUNCTIONS MODIFIED SPHERICAL BESSEL FUNCTIONS ASYMPTOTIC FORMS MODIFIED MIE SCATTERING IN A SPHERICAL STRATIFIED MEDIUM INCOMING WAVES EVALUATING THE SPECTRAL COEFFICIENTS IN A STRATIFIED MEDIUM OUTGOING WAVES CORRESPONDENCE BETWEEN CARTESIAN AND SPHERICAL-STRATIFIED PHASE QUANTITIES ABSORPTION MORE GEOMETRIC OPTICS: CUMULATIVE BENDING ANGLE, BOUGUER’S LAW, AND DEFOCUSING MORE ASYMPTOTIC FORMS EQUATING DG D ? ? ? ? † , ( ) [ ]/ TO ˜? ? ? ( , ) AND D G D ? ? ? ? † , ( ) [ ]/ TO D D ˜? ? ? ? ( ) / , FIXING THE FORM FOR G Y ( )ˆ WHEN ˆY > BY ASYMPTOTIC MATCHING BEHAVIOR OF THE WKB SOLUTION AT A TURNING POINT SETTING G[ , ] ? ? FOR ? ? ? ? †( ) SPECTRAL REPRESENTATION OF AN ELECTROMAGNETIC WAVE IN A SPHERICAL STRATIFIED MEDIUM BEHAVIOR OF ? ?? G / ACCURACY OF THE OSCULATING PARAMETER TECHNIQUE NUMERICAL COMPARISONS COMPARISON OF PHASE DELAYS IN AN AIRY LAYER FROM WAVE THEORY AND GEOMETRIC OPTICS ASYMPTOTIC MATCHING THE SPECTRAL COEFFICIENTS FOR INCOMING AND OUTGOING WAVES INTERPRETING WAVE THEORY IN A REFRACTING MEDIUM USING THE STATIONARY PHASE TECHNIQUE GEOMETRIC INTERPRETATION OF THE PHASORS STATIONARY PHASE CONDITIONS STATIONARITY OF ?( , ) ± ± PLANE WAVES THE ELECTRIC FIELD FOR AN INCOMING WAVE EVALUATING THE ELECTRIC FIELD VECTOR USING THE STATIONARY PHASE TECHNIQUE AN OUTGOING ELECTRIC FIELD THE ELECTRIC FIELD AT THE LEO COMPARISON OF GEOMETRIC OPTICS AND WAVE THEORY 5 10 1 COMPARISON OF WAVE THEORY WITH GEOMETRIC OPTICS 5 10 2 DUALITY BETWEEN SYSTEMS 5 10 3 AMENDMENTS TO ACCOUNT FOR WAVE-FRONT CURVATURE FROM THE FINITE DISTANCE OF THE EMITTING GPS SATELLITE THE ELECTRIC FIELD AT A TURNING POINT 5 11 1 FOURIER TRANSFORM OF THE AIRY FUNCTION 5 11 2 FRESNEL TRANSFORM OF THE AIRY FUNCTION CAUSTICS AND MULTIPATH 5 12 1 A NUMERICAL EXAMPLE OF MULTIPATH AND CAUSTICS 5 12 2 A GEOMETRIC OPTICS INTERPRETATION OF MULTIPATH 5 12 3 AMPLITUDE VARIABILITY FROM GEOMETRIC OPTICS 5 12 4 THIRD-ORDER THEORY 5 12 5 REDUCTION OF MULTIPATH SPECTRAL WIDTH BY DEFOCUSING 5 12 6 COMBINED WATER VAPOR AND DRY AIR REFRACTIVITY MODEL SPECTRAL COEFFICIENTS IN A SPHERICAL REFRACTING MEDIUM WITH AN EMBEDDED DISCONTINUITY 5 13 1 A MEDIUM WITH A DISCONTINUITY IN ITS REFRACTIVE GRADIENT 5 13 2 A TRANSPARENT SPHERE EMBEDDED IN A REFRACTING MEDIUM THE SCATTERED FIELD FROM A PERFECTLY REFLECTING SPHERE EMBEDDED IN A REFRACTING MEDIUM 5 14 1 STATIONARY PHASE ANALYSIS 5 14 2 RESULTS FROM WAVE THEORY 5 14 3 LAW OF REFLECTION CHAPTER 6: THE INVERSE PROBLEM: USING SPECTRAL THEORY TO RECOVER THE ATMOSPHERIC REFRACTIVITY PROFILE GPS RECEIVER OPERATIONS ADVERSE SIGNAL CONDITIONS FLYWHEELING REFRACTIVITY ERROR FROM A SINGLE-RAY PARADIGM SPECTRAL REPRESENTATION OF THE FIELD AT THE LEO STOPPED FIELD AT THE LEO THE OBLIQUITY FACTOR DOPPLER VARIABILITY REFRACTIVITY RECOVERY SUPER-REFRACTIVITY IMPROVING THE ACCURACY OF G†( ) ? RESOLUTION ISSUES SUMMARY