# AN ENVIRONMENTAL MODEL FOR CALCULATING THE ANTENNA TEMPERATURE OF EARTH-BASED MICROWAVE ANTENNAS

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Dissertation contains pages with print at a slant, filmed as received__________ 16. Other__________________________________________ ______________________________ . Text follows. University Microfilms International Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AN ENVIRONMENTAL MODEL FOR CALCULATING THE ANTENNA TEMPERATURE OF EARTH BASED MICROWAVE ANTENNAS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University by Kevin M. Lambert * * * * * The Ohio State University 1987 Dissertation Committee: Approved By: Prof. Leon Peters, Jr. Prof. Curt A. Levis Prof. Roger C. Rudduck ^ Adviser Dept, of Electrical Engineering Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To my family ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENT With sincere gratitude, I wish to acknowledge the people who have assisted me in the research and production of this dissertation. My adivsor, Professor Roger C. Rudduck, provided guidance and many helpful suggestions for the practical implementation of the various theoretical models. Professor Curt A. Levis was instrumental by introducing me to the concept of antenna temperature and to the theory of radiative transfer. Our discussions and his careful review of the document were of great value in achieving the final result. Professor Leon Peters, Jr. provided his perspective and helpful comments during his review of the study. Professor Robert K. Crane of Dartmouth College assisted in the development of the model by providing the computer code of the Global Model. The production of this dissertation was done by the editorial and drafting staff of the ElectroScience Laboratory. I wish to thank Becky Thornton for her perseverance through many difficulties in generating the text. She was assisted by Julie Riegler and Amy Henderlong. Thanks also to Bob Davis and Jim Gibson for their preparation of the figures. I also wish to thank my wife, Patricia, for her encouragement and patience during the research and the writing of this dissertation. Finally, I would like to acknowledge and express my appreciation to my parents, James R. Lambert and Marilyn A. Lambert. Their lifelong efforts ultimately provided me with the opportunity to achieve this goal. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VITA November 21, 1959 ....... Born, Hamilton, Ohio June 1982 ............... B.S., Electrical and Computer Engineering, University of Cincinnati, Cincinnati, Ohio January 1979-September 1982 Cooperative Education Student, Communications Satellite Corp., Earth Station Antenna Department, Clarksburg, MD March 1984 .............. M.Sc., The Ohio State University Columbus, Ohio October 1982-present .... Graduate Research Associate, The Ohio State University ElectroScience Laboratory, Department of Electrical Engineering, Columbus, Ohio FIELDS OF STUDY Major Field: Electrical Engineering Studies in Antennas Professor Roger C. Rudduck Studies in Electromagnetics Professor Robert G. Kouyoumjian Studies in Astronomy Professor Geoffrey Keller Studies in Communications Professor Aharon A. Ksienski Studies in Mathematics Professor John T. Scheick iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PUBLICATIONS "Far Field Measurements of 8-Foot Reflector Antennas in the Compact Range at the Ohio State University," (co-authors R.C. Rudduck and T.H. Lee) to be presented at the 1987 International IEEE/AP-S Symposium, Virginia Tech., Blacksburg, VA, June 15-19, 1987. "Microwave Antenna Technology, Final Report, Volume I: Antenna Designs, Calculations and Selected Measurements," (co-authors, R.C. Rudduck and T.H. Lee), The Ohio State University ElectroScience Laboratory, Report 717822-1, May 1987. "Microwave Antenna Technology, Final Report, Volume II: Complete Set of Antenna Pattern Measurements," (co-authors, T.H. Lee and Roger C. Rudduck), The Ohio State University ElectroScience Laboratory, Report 717822-2, May 1987. "Microwave Antenna Technology, Final Report, Volume IV: An Environmental Model for Calculating the Antenna Temperature of Earth Based Microwave Antennas," The Ohio State University ElectroScience Laboratory, Report 717822-4, May 1987. "Microwave Antenna Technology, Final Report, Volume V: Antenna Temperature Code — User's Manual," The Ohio State University ElectroScience Laboratory, Report 717822-5, May 1987. "An Analysis of Backscatter from Plates with Known Surface Errors," (co-author, Roger C. Rudduck) The Ohio State University ElectroScience Laboratory, Report 718141-3, June 1986. "Analysis of Deterministic Phase Errors in Aperture Antennas," The Ohio State University ElectroScience Laboratory, Report 715559-4, April 1985. "Wide Angle Sidelobe Reduction of a Horizontally Polarized Antenna and Symmetric, Three-Layer, Planar Radome System," The Ohio State University ElectroScience Laboratory, Report 713712-3, October 1984. "Preliminary Designs for a Dual Reflector Torus Antenna," Internal Report, Communications Satellite Corp., Clarksburg, MD, September 1982. "Theoretical and Experimental Gain of the Waveline, Model 2999, Standard Gain Horn," Internal Report, Communications Satellite Corp., Clarksburg, MD, August 1982. "Scientific Atlanta 7.7m Antenna Sidelobes," Internal Report, Communications Satellite Corp., Clarksburg, MD, July 1982. "Synthesis of Shaped Beam Antenna Pattern for Improved Global Coverage from a Geostationary Satellite," Internal Report, Communications Satellite Corp., Clarksburg, MD, April 1980*. v Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ACKNOWLEDGMENT........................................... iii VITA..................................................... LIST OF TABLES............ iv viii LIST OF FIGURES......................................... ix GLOSSARY OF SYMBOLS...................................... xvii CHAPTER PAGE I INTRODUCTION ....................................... 1 II THE DEFINITION AND EVALUATION OF ANTENNA TEMPERATURE ... 5 A. B. C. D. III IV INTRODUCTION .................................. THE DEFINITION OF ANTENNA TEMPERATURE ........... EVALUATION OF ANTENNA TEMPERATURE ............... SUMMARY....................................... 5 5 11 20 RADIATIVE TRANSFER ................................. 22 A. B. C. D. INTRODUCTION .................................. THE EQUATION OF RADIATIVE TRANSFER .............. SOLUTIONS TO THE EQUATION OF RADIATIVE TRANSFER -SUMMARY....................................... 22 23 28 33 THE ATMOSPHERE WITHOUT RAIN ........................ 36 A. B. C. D. E. V INTRODUCTION ................................... 36 DESCRIPTION OF THE ATMOSPHERE .................... 37 ABSORPTION COEFFICIENTS FOR WATER VAPOR AND OXYGEN .. 39 A MODEL OF THE GASEOUS ATMOSPHERE ................ 57 EVALUATION OF THE EQUATION OF RADIATIVE TRANSFER -- 67 THE ATMOSPHERE WITH RAIN ........................... 114 A. B. C. D. 114 115 117 E. INTRODUCTION .................................. A BRIEF INTRODUCTION TO RAIN AND RAIN MODELS ..... THE GLOBAL MODEL OF RAIN ATTENUATION ............ USE OF THE GLOBAL MODEL FOR BRIGHTNESS TEMPERATURE CALCULATION.................................... SUMMARY....................................... vi with permission of the copyright owner. Further reproduction prohibited without permission. 133 136 VI NON-ATMOSPHERIC NOISE SOURCES ....................... 141 A. B. C. D. 141 142 144 156 INTRODUCTION ................................... EXO-ATMOSPHERIC SOURCES ......................... THE GROUND MODEL................................ SUMMARY ........................................ VII COORDINATE AND VECTOR TRANSFORMATION ........ A. B. C. D. INTRODUCTION ................................... COORDINATE TRANSFORMATION ...................... VECTOR TRANSFORMATION ........................... SUMMARY ........................................ 158 158 159 168 171 VIII ANTENNA NETWORK LOSSES .............................. 173 A. B. C. D. E. F. G. H. IX INTRODUCTION .................................... NETWORK CONTRIBUTION ........................... THE PERTURBATION THEORY ....................... SMOOTH WALL CIRCULAR CYLINDRICAL WAVEGUIDE ........ SMOOTH WALL CIRCULAR CONICAL WAVEGUIDE ............ CORRUGATED CIRCULAR CYLINDRICAL WAVEGUIDES ........ CORRUGATED CIRCULAR CONICAL WAVEGUIDE ............ SUMMARY ........................................ 173 173 175 179 180 190 200 203 SOME EXAMPLES OF ANTENNA TEMPERATURE CALCULATIONS USING THE ENVIRONMENTAL MODEL ....................... 207 A. INTRODUCTION ................................... B. THE ANTENNA USED IN THE ANTENNA TEMPERATURE EXAMPLES ...................................... C. ANTENNA TEMPERATURE FOR CLEAR SKY CONDITIONS ...... D. ANTENNA TEMPERATURE FOR A CLOUDY CONDITION........ E. ANTENNA TEMPERATURE DURING RAIN .................. F. SUMMARY ........................................ 207 210 227 236 241 SUMMARY AND CONCLUSIONS ............................. 246 REFERENCES ............................................. 252 APPENDICES ............................................. 257 X A 207 EVALUATION OF THE PHI INTEGRAL IN THE ANTENNA TEMPERATURE CALCULATION ............................ 257 B DERIVATION OF SLANT PATH LENGTHS IN THE ATMOSPHERE .... 262 C HORIZONTAL PROJECTIONS OF SLANT PATHS FOR THEGLOBAL MODEL ............................................. 266 vii Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES TABLE 1 PARAMETERS FOR WATER VAPOR ABSORPTION COEFFICIENT CALCULATION ...................... 45 TABLE 2 RESONANT FREQUENCIES OF ATMOSPHERIC OXYGEN ...... 51 TABLE 3 METHODS USED TO CALCULATE THE BRIGHTNESS TEMPERATURE OF THE ATMOSPHERE WITHOUT RAIN ..... 107 VALUES FOR THE GLOBAL MODEL POINT RAIN RATE DISTRIBUTIONS ............................. 123 SPECIFIC ATTENUATION PARAMETERS FOR A 20°C DROP TEMPERATURE AND A LAWS AND PARSON DROP SIZE DISTRIBUTION .............. 128 TABLE 4 TABLE 5 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I LIST OF FIGURES Figure 2.1. Equivalent circuit for the definition of antenna temperature................................. 6 Figure 2.2. Arbitrary antenna and spherical coordinate system..................................... 7 Figure 3.1. An elemental volume of the atmosphere.......... 24 Figure 3.2. Propagation in a nonrefractive atmosphere...... 26 Figure 3.3. Origin of the cosecant law.................... 31 Figure 4.1. WVABSORB generated water vapor absorption for p=7.5 gm/m3, P=1013 mbar and T=300 K........... 46 Figure 4.2. Water vapor absorption as reported by Waters [13]....................................... 47 Figure 4.3. WVABSORB generated water vapor absorption for p=10 gm/m3, P=1013 mbar and T=318 K ........... 48 Figure 4.4. WVABSORB generated water vapor absorption for p=7.5 gm/m3, P=1013 mbar and T=300 K for 100-220 GHz........................................ 48 Figure 4.5. Oxygen absorption for v=58.82 GHz and T=295 K. . 53 Figure 4.6. Oxygen absorption as calculated by Rosenkranz [18] for 58.82 GHz and 295 K ................. 54 Figure 4.7. Oxygen absorption coefficient as calculated by Smith [17] for P=1013 mbar and T=293 K........ 55 Figure 4.8. Oxygen absorption for P=1013 mbar and T=293 K. . 56 Figure 4.9. Temperature and dry air pressure profiles of the U.S. Standard Atmosphere, 1976, as calculated by ATMOD...................................... 60 Figure 4.10. Water vapor density and water vapor partial pressure profiles in ATMOD for 7.5 gm/m3 surface water vapor density and 2 km scale height..... 64 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.11. Temperature and dry air pressure profiles for h =0.4 km, TSURF=300 K, PS1_ F=1023 mbar and 7.5 gS/m3 surface water vapor density with 2 km scale height................................ 68 Figure 4.12. Water vapor density and water vapor partial pressure profiles for h = 0.4 km, TgURF=300 K, PgUFF=1023 mbar and 7.5°gm/m3 surface water vapor density with 2 km scaleheight........... 69 Figure 4.13. Absorption coefficient profile of water vapor at 11 GHz. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 water vapor density at MSL with 2 km scale height included........................ 71 Figure 4.14. Absorption coefficient profile of water vapor at 22.235 GHz. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 water vapor density at MSL with 2 km scale height included..................... 72 Figure 4.15. Absorption coefficient profile of oxygen at 11 GHz. U.S. Standard Atmosphere, 1976.......... 74 Figure 4.16. Absorption coefficient profile of oxygen at 40.0 GHz. U.S. Standard Atmosphere, 1976...... 75 Figure 4.17. Absorption coefficient profile oxygen at 60.3061 GHz. U.S. Standard Atmosphere,1976. .. 76 Figure 4.18. Local earth geometry for a zenith line integral.................................... 77 Figure 4.19. Attenuation profile of oxygen at 11 GHz, U.S. Standard Atmosphere, 1976.................... 79 Figure 4.20. Attenuation profile of oxygen and water vapor at 11 GHz. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 surface water vapor density with 2 km scale height................................ 81 Figure 4.21. Attenuation profile at 22.235 GHz of oxygen and water vapor. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 surface water vapor with 2 km scale height...................................... 82 Figure 4.22. Zenith attenuation at MSL as a function of frequency. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 surface water vapor density with 2 km scale height included........................ 85 x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.23. Variation of zenith attenuation with station elevation. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 water vapor density at MSL with 2 km scale height included........................ 86 Figure 4.24. Schematic of the atmosphere for evaluation of the equation of radiative transfer............ 87 Figure 4.25. Variation of zenith brightness temperaturewith station elevation. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 water vapor density at MSL with 2 km scale height included............... 92 Figure 4.26. Slant lengths in the earth's atmosphere...... 93 Figure 4.27. Horizon observation path. 94 .............. Figure 4.28. Incremental ray path length due to earth curvature. Horizon direction for R =8500 km and A=10 m..........................?........... 97 Figure 4.29. Variation of horizon attenuation with station elevation. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 water vapor density at MSL with 2 km scale height included........................ 98 Figure 4.30. Variation of horizon bx. with station elevation. Atmosphere, 1976 and 7.5 density at MSL with 2 km 99 " temperature U.S. standard gm/m3 water vapor scale height.......... Figure 4.31. Brightness temperatureprofile of the atmosphere from MSL. Data obtained by integration along the indicated observation paths. U.S. Standard Atmosphere, 1976 with 7.5 gm/m3 surface water vapor density and 2 km scale height..................................... 101 Figure 4.32. Brightness temperatureprofile of the atmosphere from MSL. U.S. Standard Atmosphere, 1976 with 7.5 gm/m3 surface water vapor density and 2 km scale height....................... 105 Figure 4.33. Sky noise temperature, or brightness temperature as calculated by Smith [17]. U.S. Standard Atmosphere, 1976 with 7.5 gm/m3 surface water vapor density with 2 km scale height included................................... 106 xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.34. Typical range of absorption due to clouds. Cumulus cloud with M=1.00 gm/m3 and moderate fog with M=0.02 gm/m3 at T=280 K .................. Figure 4.35. Brightness temperature profile of the atmosphere from MSL. U.S. Standard Atmosphere, 1976 with 7.5 gm/m3 surface water vapor density and 2 km scale height. Cloud layer with M=1.00 gm/m3 present from 0.66 km to 2.7 km. .. Figure 5.1. 110 112 Global Model rain rate climate regions for the world. From Crane [44]...................... 119 Global Model rain rate climate regions for the continental United States and southern Canada. From Crane [45]............................. 120 Global Model rain rate regions for Europe. From Crane [45].................................. 121 Global Model point rain rate distributions as a function of the percent of year the rain rate is exceeded. From Crane [44]................... 124 Global Model effective 0°C isotherm height. From Crane [44]............................. 125 Global Model path averaging factors. From Crane [44]........................................ 127 Horizontal projection of slant paths for the Global Model................................ 130 Figure 5.8. Example calculations of the Global Model....... 132 Figure 5.9. Brightness temperature profile at 11 GHz for 1 mm/hr. rain................................. 137 Figure 5.10. Brightness temperature profile at 11 GHz for 12.7 mm/hr. rain............................ 138 Figure 5.11. Brightness temperature profile at 11 GHz for 25.4 mm/hr. rain............................ 139 Figure 5.2. Figure 5.3. Figure 5.4. Figure 5.5. Figure 5.6. Figure 5.7. Figure 6.1. Ground model for antenna temperature calculations................................. 146 Figure 6.2. Power reflection coefficients of a flat earth with er=10.0................................. 149 xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6.3. Clear sky example of the brightness temperature profile at 11 GHz, horizontal polarization 150 Figure 6.4. Clear sky example of the brightness temperature profile at 11 GHz, vertical polarization........ 152 Figure 6.5. Brightness temperature profile at 11 GHz for 12.70 mm/hr rain, ground model included. Horizontal polarization....................... 154 Figure 6.6. Brightness temperature profile at 11 GHz for 12.70 mm/hr rain, ground model included. Vertical polarization........................ 155 Figure 7.1. The station coordinate system.................. 159 Figure 7.2. The antenna coordinate system.................. 160 Figure 7.3. Coordinate systems coincident................. 161 Figure 7.4. Positioning in azimuth....................... 163 Figure 7.5. Positioning in elevation...................... 163 Figure 7.6. Polarization alignment....................... Figure 8.1. Antenna network.............................. 174 Figure 8.2. Smooth wall circular cylindricalwaveguide Figure 8.3. Smooth wall circular conical waveguide......... Figure 8.4. Truncated conical waveguide................... 183 Figure 8.5. Example for T E ^ mode attenuation in a section of conical waveguide......................... Figure 8.6. Figure 8.7. Figure 8.8. Figure 8.9. 164 179 TE^j mode attenuation coefficient for the waveguide of Figure 8.5. 11 GHz frequency Example for T M ^ mode attenuation in a section of conical waveguide......................... mode attenuation coefficient for the waveguide of Figure 8.7. 11 GHz frequency Examples of microwave horn antennas........... Figure 8.10. Corrugated circular cylindrical waveguide geometry................................... xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 185 186 187 188 189 191 Figure 8.11. Attenuation coefficient of an H E ^ mode in corrugated waveguide, with r.,=30 mm, ^ r./r =0.737, b=10 mm, p=ll mm, and <r=1.57 x 10 S/m.° From [57]............................. 199 Figure 8.12. HE-, mode attenuation coefficient as calculated by CI2RC_SUB. Corrugated waveguide geometry, r.=30 mm, r_/r =0.737, b=10 mm, p=ll mm and o±1.57 x 10 S?m............................ 201 Figure 8.13. A section of corrugated circular conical waveguide................................... 202 Figure 8.14. Corrugated horn with 15° flare angle........... 205 Figure 8.15. Corrugated horn with 24° flare angle........... 206 Figure 9.1. H-plane pattern of a prime focus fed parabolic reflector. Diameter=8 feet, focal point to diameter ratio=0.5, frequency = 11 GHz, corrugated horn feed......................... 209 Figure 9.2. Clear sky brightness temperature profile. Vertical polarization........................ 211 Figure 9.3. Clear sky example of antenna temperature. Vertical polarization. Network effects excluded.................................... 212 Figure 9.4. Clear sky example of antenna temperature. Vertical polarization. Network effects included.................................... 214 Figure 9.5. Clear sky example of antenna temperature. Vertical polarization. Total antenna temperature with sky and ground contributions. . 215 Figure 9.6. Clear sky example of antenna temperature. Vertical polarization. Total antenna temperature with main beam contribution.........218 Figure 9.7. G/T ratio for the clear sky example of antenna temperature. Vertical polarization........... 219 Figure 9.8. Boresight G/T ratio for the clear sky example of antenna temperature. Vertical polarization and 36,000 km range. Terrestrial source for 0 >90°. 221 z Figure 9.9. Clear sky brightness temperature profile. Horizontal polarization...................... 222 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 9.10. Clear sky example of antenna temperature. Horizontal polarization. Total antenna temperature with sky and ground contributions. . 223 Figure 9.11. Clear sky example of antenna temperature. Horizontal polarization. Total antenna temperature with main beam contribution......... 224 Figure 9.12. G/T ratio for the clear sky example of antenna temperature. Horizontal polarization.......... 225 Figure 9.13. Boresight G/T ratio for the clear sky example of antenna temperature. Horizontal polarization and 36,000 km range. Terrestrial source for 9 >90°...................................... 226 z Figure 9.14. Brightness temperature profile for a cloudy condition. Vertical polarization.............. 228 Figure 9.15. Example of antenna temperature for a cloudy condition. Vertical polarization. Total antenna temperature with main beam contribution. 229 Figure 9.16. G/T ratio for the cloudy sky example of antenna temperature. Vertical polarization........... 230 Figure 9.17. Boresight G/T ratio for the cloudy sky example of antenna temperature. Vertical polarization and 36,000 km range. Terrestrial source for 0 >90°...................................... 231 z Figure 9.18. Brightness temperature profile for a cloudy condition. Horizontal polarization........... 232 Figure 9.19. Example of antenna temperature for a cloudy condition. Horizontal polarization. Total antenna temperature with main beam contribution. 233 Figure 9.20. G/T ratio for the cloudy sky example of antenna temperature. Horizontal polarization......... 234 Figure 9.21. Boresight G/T ratio for the cloudy sky example of antenna temperature. Horizontal polarization and 36,000 km range. Terrestrial source for 0 >90°...................................... 235 z Figure 9.22. Brightness temperature profile for the rain example. Vertical polarization............... 237 xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 9.23. Antenna temperature for the rain example. Vertical polarization. Total antenna temperature with main beam contribution.........238 Figure 9.24. G/T ratio for the rain example of antenna temperature. Vertical polarization............ 239 Figure 9.25. Boresight G/T ratio for the rain example of antenna temperature. Vertical polarization and 36,000 km range. Terrestrial source for 6 >90.0°.................................... 240 z Figure 9.26. Brightness temperature profile for the rain example. Horizontal polarization............. 242 Figure 9.27. Antenna temperature for the rain example. Horizontal polarization. Total antenna temperature with main beam contribution.........243 Figure 9.28. G/T ratio for the rain example of antenna temperature. Horizontal polarization.......... 244 Figure 9.29. Boresight G/T ratio for the rain example of antenna temperature. Horizontal polarization and 36,000 km range. Terrestrial source for 0 >90.0°.................................... 245 z Figure B.l. Geometry for the calculation of slant path length...................................... 262 Figure C.l. Geometry for the calculation of the horizontal projection of the slant path L ................ 266 xvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. GLOSSARY OF SOME SYMBOLS USED ENGLISH waveguide radius PQ attenuation between point P and point Q slant path attenuation beamwidth BV(T) radiation intensity of a blackbody radiator velocity of light amplitudes of oxygen absorption lines D horizontal projection of a slant path 0(9,4-) antenna directive gain in direction (0,40 E energy E1 Em E(9,4») Eco(9’*> Ecross<e’^ f (v»V *1 molecular energy of a rotational state molecular energy of a rotational state antenna electric field in direction (0, 4*) co-polarized electric field cross-polarized electric field line shape function statistical weight of a molecular state h Planck's constant h max maximum height of the atmosphere station altitude H altitude of the 0°C isotherm H station altitude XVI1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. antenna magnetic field in direction (9,4)) i co 1cross unit vector in the co-polarized direction unit vector in the cross-polarized direction radiation intensity of incident radiation volume emission coefficient k free space wavenumber k Boltzmann's constant k volume absorption coefficient v k' specific attenuation due to rain k c,nm cutoff wavenumber ^cloud v absorption coefficient of clouds or fog absorption coefficient of water vapor absorption coefficient of oxygen L LI loss in the antenna network slant length in the atmosphere MSL mean sea level N maximum number of antenna pattern regions p rnm zeroes of the Bessel functions P' nm zeroes of the derivatives of the Bessel functions P pressure P(z) power in an antenna network PA PDM PD fSURF ph 2o antenna contribution to the total system noise power dry air pressure from the atmospheric model dry air pressure at the surface water vapor pressure xvm Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. H20,SURF LOSS water vapor pressure at the surface power lost in an antenna network. thermal noise power available from a resistor RAD SURF total power radiated by an antenna total pressure at the surface total pressure r effective path average factor R path averaged rain rate R. antenna resistance effective earth radius resistive part of the surface impedance exhibited by waveguide walls P point rain rate RH relative humidity T physical temperature T(s) physical temperature along path s T(0,*) brightness temperature without an explicit polarization dependence antenna temperature including antenna network effects constant physical temperature Tco(e’« brightness temperature resolved into the co-polarized direction Tcross'(0,4>) ’T/ brightness temperature resolved into the cross polarized direction antenna temperature excluding antenna network effects Tg(0,<f>) brightness temperature in direction (0,$) atmospheric temperature from the atmospheric model xix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. mean atmospheric temperature PQ TSKY t surf atmosr’ieric emission between points P and Q atmospheric emission incident upon the surface surface temperature U(9,+) antenna radiation intensity uave average antenna radiation intensity U antenna co-polarized radiation intensity U antenna cross-polarized radiation intensity co cross V wb t 4 WN ,WN diagonal elements of the oxygen transition rate matrix off diagonal elements of the oxygen transition rate matrix antenna reactance N interference coefficients of the oxygen absorption coefficient Z partition function Z free space impedance antenna impedance xx Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. GREER a attenuation coefficient in an antenna network a rain specific attenuation parameter a absolute value of a negative elevation angle 0 rain specific attenuation parameter r(0) Global Model path averaging factor T. in input reflection coefficient 8(D) Global Model path averaging factor 8 s skin depth incremental distance in the atmosphere energy difference between molecular states Akg q ^ AT empirical correction to the water vapor absorption coefficient correction to the atmospheric temperature water vapor linewidth parameter water vapor linewidth parameter Av^Cl^O) water vapor linewidth parameter er relative dielectric constant of the surface (0p,<f>Q> angles defining the antenna orientation v frequency \>^m resonant frequency of a molecule vj£ resonant absorption frequencies of oxygen p water vapor density pjj q water vapor density PH 0 M water vapor density from the water vapor model xxi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pa n chop bUKr water vapor density at the surface a conductivity aR reflection coefficient Fresnel reflection coefficient for horizontal polarization o^j Fresnel reflection coefficient for vertical polarization Tp angle defining the orientation of the aperture electric field xnn rlt atmospheric opacity between points P and Q opacity transition matrix element 4>p upper bound of an antenna pattern region <J>q lower bound of an antenna pattern region fractional population Q spherical surface of space xxii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I INTRODUCTION The role of the receiving antenna in a communication system is to provide a detectable signal to the receiver from the electromagnetic energy that is incident upon the antenna. In this sense, the ideal receiving antenna would be an antenna which would deliver only the desired sigral energy to the receiver input. are not that selective. unwanted. Real antennas, however, They respond to all incident energy, wanted and The unwanted energy enters the system and contributes to the overall system noise. The antenna temperature is a measure of the noise contributed by the antenna. The total system noise includes the noise from the receiver and the transmission lines between antenna and receiver, in addition to the noise contributed by the antenna. Early in the development of microwave communication systems, receiver noise dominated the total system noise and the antenna and transmission line noise contributions were negligible. Later development of low noise receivers and amplifiers has reduced the noise contributions of the receiving system to the same order of magnitude as the antenna noise. Thus antenna noise temperature has become of greater importance in the design of communication systems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Information on antenna noise temperature would be of great value during the design stage of a microwave communication system using a lownoise receiver. With this information, the system designer would be able to work with a model which is more representative of the operational state of the system. This would facilitate the specification of system components and provide increased confidence that the system will perform as designed. Antenna noise temperature is a function of the particular antenna under consideration and the environment in which the antenna operates. Therefore, in order to obtain antenna temperature information, the system designer must be provided with the ability to design and analyze the antenna and analyze the environment as it appears to the antenna. The capability for the design and analysis of some of the most commonly used microwave antennas is presently available in the OSU — Numerical Electromagnetic Code — Reflector Antenna Code [1]. This code has been used successfully to design horn antennas as well as prime focus and Cassegrain reflector systems [2]. The designs were fabricated and measured to verify the accuracy of the code [3]. There is very good agreement between the calculated patterns and the measured patterns as can be witnessed by the references. The code can model additional antenna types and contains a wide range of design and analysis capabilities for other applications. However, the code does not contain an environmental model. The objective of this study is to develop, as a supplement to the Reflector Antenna Code, an environmental model for earth based microwave 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. antennas operating in the 1-40 GHz frequency range. The designer can use the Reflector Antenna Code to determine hov the antenna responds to incident power. The environmental model can then be used to calculate what the incident noise power density is. Together they can provide the antenna noise temperature information needed by the designer. Alternatively, they can be used to analyze an already existing communication link. The antenna is sensitive to the environment in which it is placed because the environment contains numerous sources of microwave radiation. These sources can be of man-made or natural origin. For an earth based antenna, examples of such sources include: the sun and other celestial objects, the atmosphere, the earth and nearby structures. The antenna receives a portion of the power radiated by these sources. Since the desired information is not carried by this power, the overall system noise is increased. This noise appears in the system because of the properties of the antenna. For this reason, this component of the total system noise is considered purely an antenna parameter. The antenna temperature is an effective temperature representing this noise power. Accounting for all radiators in a model for general applicability is not practical. Each location of an antenna would have to be described in detail before a calculation could be made. The wide range of possible station locations on the surface of the earth would require essentially a new model for each location considered. Additionally, the cost of performing the calculation with such a detailed model could be 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. prohibitive. However, if the resources are available, this type of detailed calculation could be made for a given location. The complexity of the problem is in supplying the model with the details of the specific station environment. This study will not address this problem but will use simple models instead. The model to be presented here includes the significant radiation mechanisms that are common to all station locations. Since the positions of the sun and other celestial objects, nearby structures and other communication links are dependent on station location, they will not be included in the model. The model does simulate the effects of atmospheric oxygen and water vapor, clouds, fog, rain and the surface of the earth. This model simplifies the treatment of the local environment by assuming a flat earth for the model of the surface. Additionally, no azimuthal variation of the environment is included in the model. More complex modeling, beyond the assumptions made in this study, are beyond the scope of the present research and are reserved for future implementation. This study begins with a presentation of the theoretical background behind the calculation of antenna temperature. A brief discussion on the theory of radiative transfer, which describes the transfer of energy by radiation through a medium, is also given. From this foundation, the various components of the model are presented along with an explanation on how they are implemented within the model. to demonstrate the use of the model. Several examples follow Finally, recommendations are given for the future development of the model. 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER II THE DEFINITION AND EVALUATION OF ANTENNA TEMPERATURE A. INTRODUCTION The concept of antenna temperature is well established. Many authors have presented excellent discussions on the topic [4-7]. Derivations of the expression for antenna temperature may be found there. Since the orientation of this study is toward the evaluation of the expression, a derivation is not given. Instead, emphasis is placed on broad descriptions intended to reinforce the concept. Development of the implementation of the expression begins with the standard definition of antenna temperature. significance of each term is given. A discussion on the Then the basic equation is manipulated into a form suitable for calculation. A method for reducing the number of computations needed to evaluate the antenna temperature is presented. B. THE DEFINITION OF ANTENNA TEMPERATURE Antenna temperature is an effective temperature and not the physical temperature of the antenna structure. Recall that the thermal noise power P^ which is available from a resistor at temperature T is given by [8], 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P = k T B n watts -23 where k = 1.38 x 10 bandwidth in hertz. (2.1) joule/kelvin is Boltzmann's constant and B is the Now consider an antenna with impedance Z^, connected to a receiver as shown in Figure 2.1(a). RECEIVER Z R. + i (a) RECEIVER ra + * x a (b) Figure 2.1. Equivalent circuit for the definition of antenna temperature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The antenna contributes a noise power power. to the total system noise Imagine the antenna replaced by a circuit element as shown in Figure 2.1(b). the antenna. The circuit element has the same value of impedance as The antenna temperature is defined as the temperature that the resistor must have in order to produce the same noise power, PA , as the antenna. Consider the antenna shown in Figure 2.2. The radiation pattern of an antenna describes the sensitivity of the antenna to incident radiation. The lossless antenna receiving the radiation that results from thermal emission in the environment, will have an antenna temperature, at a given frequency, defined by, ta = h. i^ (2 .2 ) D(0,<|,) TB(0»+) dS ’ A X A y Figure 2.2. Arbitrary antenna and spherical coordinate system. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where D(0, <f) is the directive gain of the antenna. Tfi(0, $) is the brightness temperature, a function representing the environmental radiation, incident from direction (0,«#>). The integration is over all of space. The brightness temperature refers to the equivalent blackbody temperature of the incoming radiation. A blackbody is a fictitious object in thermodynamic equilibrium that absorbs electromagnetic energy perfectly. In order to maintain thermodynamic equilibrium, the blackbody must also be a perfect radiator. The intensity of radiation emitted by a blackbody at temperature T and frequency v is given by Planck's radiation law, Jv ■ V T> ■ ^ c hv/kT e - 1 <2-3> where I -2 = intensity in watts m Hz -1 -2 rad -34 h = Planck's constant, 6.63 x 10 joule sec v = frequency in Hz c = velocity of light in m sec-* -23 -1 k = Boltzmann's constant, 1.38 x 10 joule kelvin T = temperature in kelvin. In the microwave region of the spectrum, hv « kT, so the exponential term can be approximated by the first two terms in the power series 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. expansion. Applying this to Equation (2.3) and converting to wavelengths results in the Rayleigh-Jeans radiation law, (2.4) This relationship allows a temperature value to be assigned to all radiation incident on the antenna even though the radiation is not coming from a blackbody. The temperature that is given to the radiation is the temperature that a blackbody would need to have in order to radiate with the same intensity. Note that Equation (2.4) is valid for use in the antenna temperature calculation only when a single frequency is being considered. It is unlikely that the incident radiation behaves like a blackbody in the frequency domain. Therefore for each frequency considered, a different equivalent blackbody temperature would have to be assigned to the radiation. The variation of the equivalent blackbody temperature, as a function of frequency, could be made explicit by writing Equation (2.4) as Iv = Bv(T(v)) - 2kT(v) X2 (2.5) In this equation T(v) is the equivalent blackbody temperature at frequency v. The frequency dependence of the antenna temperature enters into the problem only when it is desired to convert the antenna temperature into 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the noise power received by the antenna. This conversion is provided through the generalization of Equation (2.1), U P = k n (2 .6) T(v) dv where and B is the bandwidth under consideration. The function T(v) represents the frequency dependence of the equivalent blackbody temperature shown in Equation (2.5) in addition to the frequency dependence of the antenna pattern. It is obvious that for T(v) = T, Equation (2.6) reduces to Equation (2.1). The operation indicated by Equation (2.2) is now apparent. The integrand is the product of the directive gain of the antenna and a term related to radiation intensity. Therefore, the integrand represents the power collected by the antenna from an element of space dQ. The integration and division by 4n averages over all of space the power collected from each element. The result is an effective temperature representing the power collected by the antenna. With the exception of the desired transmission, this power collected by the antenna is detrimental to the operation of the system. Therefore T^ represents the unwanted or noise power that the antenna contributes to the system. 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The environment enters into the calculation of antenna temperature through the brightness temperature. In order to obtain the antenna temperature, the brightness temperature profile must be known for all of space. This requires knowledge of what is radiating in the environment, where the radiator is located and how much radiation is being produced. The function of the environmental model is to provide answers to these questions. The other factor of the integrand, the directive gain, depends on the radiation pattern. For this study pattern information is provided by the Reflector Antenna Code. independent of the pattern. However, the incoming radiation is Thus the environmental model does not depend on the source of the pattern. Pattern information could be provided by any technique. Structurally, the pattern is input information to the calculation of antenna temperature, as is the environmental information. They must be combined and integrated to produce the antenna temperature. integral must be performed numerically. This The implementation of this integration is the topic of the next section. C. EVALUATION OF ANTENNA TEMPERATURE The evaluation of antenna temperature requires the complete radiation pattern of the antenna, in addition to, the brightness temperature distribution in the environment of the antenna. This section assumes these two pieces of information are known and studies the mechanics involved in evaluating the integral in Equation (2.2). 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Recall that the directive gain is related to the radiation pattern through the radiation intensity, DO, <>) = , (2.7) ave where U(0,40 is the radiation intensity. Radiation intensity is the power radiated in a given direction per unit solid angle. uave is the average value of the radiation intensity, where the average is taken over all of space, 1_ Uave = = . 4-= ji (2 .8 ) U(9,40 dS Note that since U(0,4O is a power density, the integral in Equation (2.8) represents the total power radiated by the antenna. Therefore U may also be written as, ave J ’ u _ ave " (Z 9) An ' Using Equation (2.7) in Equation (2.2) results in the following expression for antenna temperature, TA “ 4it U ave JQ U(0, ♦) T_(0, 40 dQ . D Substitution of Equation (2.8) into Equation (2.10) gives, 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.10) JQ U(0,*) TB(0,+)dS T . = ---- =-----------JQU(0,*) dS (2.11) This expression demonstrates that two integrals actually have to be evaluated to calculate the antenna temperature. The denominator, as shown before, is the radiated power of the antenna. The numerator is similar, but requires environmental information. The form of Equation (2.11) and the similarity between the integrals have two advantages that can be exploited during numerical evaluation. similar. The first is that the code for the integrations will be Evaluation of the numerator just requires an extra call to an environment subroutine to get the brightness temperature. The second advantage is that since the denominator is the radiated power, Equation (2.7) and Equation (2.8) can be used to get the directivity of the antenna. Since the Reflector Antenna Code also produces the directivity by integrating the primary feed pattern rather than the secondary reflector pattern, this calculation can serve as a check on the antenna temperature calculation. Another interesting property is revealed by Equation (2.11). If the radiation incident on the antenna is independent of 0 and <t>, for example, (2 .12) then the integrations cancel and the antenna temperature is given by, 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.13) An investigation of radiation intensity is required before Equation (2.11) can be put into a form which can be evaluated. The radiation intensity is defined from the radiated electric and magnetic fields, U(9,4>) = Re fe(9,<fr) x H*(9,$)] • r2r , (2.14) where r is the radius of the sphere over which the fields are defined and r is the radial unit vector. When the far field approximations are applicable, the plane wave relationship, HO,*) = i- r x E(9,-f>) , o (2.15) with Zq being the free space impedance, may be introduced into Equation (2.14). This action results in the equation, U(9,4>) = Relj3(9,*) x (r x E*(9,4>))] • r2r . (2.16) o Employing a vector identity, U(9,4>) = Re [|e (9, <*>)|2 r - (e (9,*) • r) E*(9, *)]-r2r , o 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.17) but since there is no radial component of the electric field in the far field, U(6,40 = |e <0, 4.) I2 r2 (2.18) o In light of the far field approximations, the electric field is a vector field which can be defined by any two orthogonal components which are both mutually orthogonal with the radial vector. Typically, in theoretical work, the electric field is resolved into 0 and $ components in the standard spherical coordinate system, centered on the antenna. However, the field components usually measured are those described in Definition 3 of Ludwig [9]. Since the measurement is a simulation of the operational mode of the antenna, these are the components that will be used in this development. Let the two components of the field in this system be called the co-polarized component and the cross-polarized component. Then a general expression for the electric field can be written as, i -jkr E(0,*) = [eco(0,4>) ico + Ecross (9, «*>') icrossj1 ---r where ECO and ECrOSS are in o general r -complex. (2.19) Substituting o this field expression into Equation (2.18) results in, u<e,« - i - [IEco<e,« 2 * 2 e cross ]• 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2 .20) Equivalently, u<e,4>) = uco(e,4>) + ucross < e ,« (2 .21 ) where, uco<e,«0 = Eco<e>*>|2 Z o (2 .22 ) and (2.23) Written in the manner of Equation (2.21), the total radiation intensity appears to have originated from two antennas. The first antenna radiates the co-polarized pattern while the second antenna radiates the cross-polarized pattern. This simulation of the antenna is necessary in order to calculate the noise power collected in the co polarized pattern as well as what is received in the cross-polarized pattern. Additionally, this consideration will allow the environmental model to include noise sources which have a polarization dependence. With the two antenna simulation, the antenna temperature becomes, T Jb T c o <9’W Bc .<9-*>M < - f A 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.24) Written in terms of the field quantities, the antenna temperature is, T A (2.25) Note that there are two types of integrals to solve in Equation (2.25). The first appears in each term of the numerator and has the form, (2.26) The other type, appearing in each term of the denominator, has the form, (2.27) Because of the similarity in these two forms, the approach to their numerical solution is the same. The result for I^ can be obtained from the result for 1^ by using T(0,<J>)=1. Further analysis of these integrals requires some statements pertaining to the radiation pattern of the antenna. Imagine an antenna centered in a spherical coordinate system with the z axis normal to the aperture of the antenna as shown in Figure 2.2. Antenna patterns are usually measured and calculated as a series of individual phi patterns. A phi pattern is the resonse of the antenna as a function of theta for a 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fixed value of phi. This is done because the pattern varies a great deal as a function of theta but is slowly varying as a function of phi. Therefore higher resolution is required in theta. The number of phi patterns taken during a measurement, or calculated, gives the resolution in phi. The number of patterns required depends on the desired resolution and the characteristics of the antenna. However, in most cases, only a few phi patterns are needed to describe the complete twodimensional antenna pattern. Finally, it is most convenient to measure the pattern by not strictly adhering to the conventions of the standard spherical coordinate systems. An antenna is usually measured for the range of values, -n<0<n, 0<<f><n instead of O<0<n, 0<4><2n. This is done to simplify the measurement and it demonstrates the symmetry of the antenna pattern. Patterns are often calculated in this manner as well. Therefore, consider Ij for patterns specified as -lt<0Ol, 0<<)><Jl, I T(0,4>) 1 E(0,4>) 2 sin 10 1 d0d<|> . (2.28) Assuming N+l phi patterns are available, the phi integration can be divided into N regions and Equation (2.28) is written as, (2.29) where ^ ,$p are the boundaries of each region and 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The summation is over all regions. The advantage of writing the integral in this manner is that the phi integration can be eliminated. As mentioned previously, antenna patterns are typically slowly varying functions of phi. The brightness temperature is also a relatively smooth angular function. This will be demonstrated when the brightness temperature profiles are presented in Chapter VI. Exploiting the characteristics of these two factors, linear interpolation in phi may be used on the terms in the integrand and the phi integration solved analytically. In this manner, the evaluation of Ij requires a single numerical integration instead of a double integration. Using linear interpolation, the number of terms N needed to evaluate 1^ in phi is far fewer than the number required by a numerical integration. The details of the analytic evaluation of the phi integral are tedious and are therefore removed to Appendix A. The results are all that are required here. Evaluation of 1^ results in, (2.33) where 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I 0.(9) = E(0,*p ) 2T(0,* 1 i + 3 ) + E(9f+n ) 2T(0,*n ) + E(0,* ) 2T(0,*n ) + E(0,* )|2T(0,+p ) I i I i ' i • *i (t (0,*p ) + T(0,* )][e V , ^ )E(0,*Q )+E(0,^) e \0,* q )] (2.34) Although formidable, this equation is readily implemented on the computer. Note, the term involving conjugate values of the field is real as shown in Appendix A. By setting all the temperature functions to unity in Equations (2.33) and (2.34) a similar result is obtained for I2, rfi- V - V Q2(0) sin|0| d0 -It i=l (2.35) where Q„(0) = E(0,<> ) ' i E(0,$Q ) | + + | [e *(0,*p )E(0,^) + E(0, ^p^)E*(0, )j (2.36) Applying Equations (2.33) and (2.35) to Equation (2.25), the antenna temperature is evaluated. The integral over theta is evaluated numerically by using the trapezoidal rule. The sampling intervals used to determine the number of phi regions and the number of theta values must be defined by the particular situation. *The desired accuracy, the 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. structure of the antenna pattern and the environment must be taken into consideration. D. SUMMARY The concept of antenna noise temperature has been introduced. The basic formula was given and transformed into an expression that accounts for power received by the antenna in both polarizations. However, the expression contains three terms, each of which requires an integration over all of space. In order to reduce computations, the characteristics of the antenna pattern were exploited in a method that uses linear interpolation to allow simple evaluation of certain integrals. This chapter has treated the antenna noise temperature by assuming that all of the noise sources in the environment are located in the far field region of the antenna. This allows the far field pattern to be used in the calculation of antenna temperature. This model will not consider the effects of sources which are actually in the near field region of the antenna. Only the far field pattern will be used which implies that all objects in the environment, regardless of their proximity to the antenna, will be in the far field region. With the mechanics of the antenna temperature calculation in place, this study can proceed to investigate the inputs to the calculation. The antenna pattern will be produced from some outside source, namely the Reflector Antenna Code. The development of the brightness temperature profile of the environment begins with the next chapter. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER III RADIATIVE TRANSFER A. INTRODUCTION The calculation of antenna temperature requires knowledge of the radiation that is incident on the antenna from all directions. To acquire this knowledge, all sources of radiation, which can be sensed by the antenna, must be known. In general, any object with physical temperature above absolute zero will produce thermal radiation. However, the energy radiated by the source is not necessarily the energy that impinges on the antenna. For an earth based antenna, the energy must propagate through the atmosphere before reaching the antenna. The atmosphere is composed of constituents which attenuate and scatter the radiation as the radiation propagates between source and antenna. The constituents, having temperatures above absolute zero, result in additional emission of electromagnetic energy along the propagation path. These properties make the atmosphere an important consideration in the calculation of antenna temperature. In fact, because all incident radiation passes through the atmosphere, the description of the atmosphere serves as the foundation for the entire environmental model. The objective of this chapter is to present the mathematical basis for a model of the atmosphere to be used within the 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. environmental model. This analysis of atmospheric propagation, based on the conservation of energy, is called the theory of radiative transfer. B. THE EQUATION OF RADIATIVE TRANSFER The equation of radiative transfer is the mathematical description of the absorption, emission and scattering which occurs as electromagnetic energy is transported through the atmosphere. The classical work on the subject is considered to be that of Chandrasekhar [10], although the development can be found in several texts dealing with the transport of electromagnetic energy through media [11,12]. aim here is not to present the development of the theory. The Instead a general description of the factors which affect energy transport will be given. The intention of this description is to acquaint those not familiar with the theory and to serve as a reference for the assumptions made in the atmospheric model. Consider the elemental volume of the atmosphere shown in Figure 3.1. Being composed of atmospheric materials, the volume may contain fundamental units of matter such as free electrons, ions and atoms, mixed with more complex units such as gas molecules and suspended liquids and solids. The intent of the radiative transfer theory is to obtain the change in electromagnetic intensity, dl^, along path s, after passage through the volume. The volume interferes with the intensity by three mechanisms: absorption, emission and scattering. The level of interference by these mechanisms is a function of the properties of the atmospheric 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I A dV Figure 3.1. An elemental volume of the atmosphere. constituents. The amount of absorption and scattering caused by the atmospheric constituents is proportional to the strength of the incident intensity. Also, all effects are functions of frequency. In the microwave portion of the spectrum, wavelengths are large enough to allow the effects of smaller particles such as ions, electrons and atoms to be neglected. However, power is absorbed and emitted by the gas molecules and the liquid and solid particles. Additionally, the suspended liquid and solid particles may be of sufficient size to cause scattering effects. For example, a portion of the radiation along s may be 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. scattered into the path along u or radiation from another direction, * A perhaps t, can be scattered into the path s. Scattering effects are quite complicated to calculate. The effect of a single particle can be found but extending this to a region of space, containing many particles, can require extensive calculation. Fortunately, scattering by atmospheric constituents in the 1-40 GHz frequency range of concern here is small. The one constituent that can cause significant scattering, under certain conditions, is rain. For this reason and because rain is a time varying component of the atmosphere, two separate atmospheric conditions will be defined and considered. These are the atmosphere with and without rain. More will be said about rain in Chapter V. A clear sky atmospheric condition is when the atmosphere contains only gases. Consider the observation path through a slab of atmosphere shown in Figure 3.2, and assume that atmosphere is nonscattering and nonrefractive. Specifying thermodynamic equilibrium in the atmosphere allows absorption to be related to emission and the equation of radiative transfer to be written as [13], Iv(s) = lv(0)e T',(0’S) + ® Bv(T(s'))e ^ ’S>kv(S')ds' . (3.1) In Equation (3.1), Iv(s) is the intensity at position s, which arrived along s at frequency v. Bv(T(s')) is Rayleigh-Jeans approximation to Planck's law which was given in Equation (2.4). The term k^(s') is the 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.2. Propagation in a nonrefractive atmosphere. volume absorption coefficient of the atmospheric material. The absorption coefficient has units of reciprocal length and integration of k (s') along the path produces what is known as the optical depth, or opacity of the atmosphere. Tv(s',s) = k (s")ds" (3.2) s' In terms more familiar in engineering, is twice the attenuation in nepers. 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since Tv(s',s) is the attenuation between s' and s, then the first term in Equation (3.1) is the contribution to lv(s) from the intensity that was incident on the slab of atmosphere. suffers an attenuation of t ^(0,s ) This incident intensity during propagation through the atmosphere. The integral term represents the emission in the atmosphere. Under the thermodynamic equilibrium condition, the emission coefficient j is related to the absorption coefficient by the Kirchhoff-Planck law, (3.3) rv-V T>• Thus in Equation (3.1) the term B^(T(s'))k (s'), in the integrand, is the emission from the segment ds'. This emitted intensity experiences an attenuation t (s',s) before reaching the point s, and the integral sums the contributions from all elements along the path. By using the Rayleigh-Jeans approximation to Planck's law (Equation (2.4)), the intensities in Equation (3.1) can be represented by equivalent blackbody temperatures, T_(s) = T_(0)e B B -t (0,s) fs -T (s',s) v + T(s’)e v k(s')ds' . jQ v (3.4) Thus, Tg(s) is the brightness temperature of the atmosphere, at frequency v and location s, which has arrived along direction s. Tg(0) is the equivalent blackbody temperature of the radiation incident on the 27 with permission of the copyright owner. Further reproduction prohibited without permission. atmosphere, at frequency v, and T(s') is the physical temperature of the atmosphere at s'. In order to produce the antenna temperature of an earth-based antenna, Equation (3.4) must be solved for observation paths taken through the atmosphere of theearth. for equation (3.4) However, a closedform solution has not beenfound for theatmosphere.Therefore, the solution will have to be found numerically. A solution, made possible by making certain assumptions, is a useful exercise for the study of transport theory. C. SOLUTIONS TO THE EQUATION OF RADIATIVE TRANSFER The structure of the atmosphere of the earth prevents the evaluation of the integral in Equation (3.4) in closed form. However, by assuming that the temperature profile in the atmosphere is constant, the integral can be evaluated. Let T(s) = T (3.5) and note from Equation (3.2) that dxv = -kvds (3.6) where the negative sign originates because x and s increase in opposite directions. With this simplification, Equation (3.4) becomes, 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This form of the radiative transfer equation can be used to demonstrate some significant characteristics of the atmosphere. If attenuation by the atmosphere is small, tv (0,s ) = 0 (3.8) and Tjj(s ) = Tb (0) . (3.9) Thus the atmosphere is transparent and the temperature at s is essentially the temperature that was incident on the atmosphere. Alternatively, if the atmospheric attenuation is large, Tv(0,s) » 1 then Tg(s) = T . In this case, the atmosphere is opaque, hence the origin of the term opacity. The incident radiation is completely attenuated and the brightness temperature is the physical temperature of the surrounding medium. Finally, consider a case where, 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T = Tb (0) then Tfi(s) = Tfi(0) In this situation, the emitted radiation is equal to the absorbed radiation, resulting in a brightness temperature that is equal to the incident temperature. Many researchers have used Equation (3.7) to measure the attenuation in the atmosphere. For example, an earth based receiver can be used to measure the radiation arriving from directly overhead, the zenith direction. In this situation, T„(0) is the radiation incident on the atmosphere from extraterrestrial sources. Provided that the sun or some other bright celestial object is not in the zenith path, T_(0) is D negligible at microwave frequencies. TB(s) = T(1 - e Then Equation (3.7) becomes ) (3.8) Equation (3.8) is called the radiometric formula because a radiometer is used to measure Tfi(s). The attenuation is calculated from, (3.9) where, in this sense, T represents some mean physical temperature of the atmosphere. This mean temperature must be provided by other technique. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Empirical data, atmospheric models and an analysis based on path parameters have all been used to determine the mean temperature [14,15]. The zenith attenuation is convenient to work with because use of a simple relationship provides the attenuation for some other directions. Let z be the path along the zenith direction and let the s path be defined by the elevation angle 0 as shown in Figure 3.3. Assuming the earth is locally flat, the atmosphere can be divided into units of dz as shown in the figure. For angles 0>1O°, refraction is negligible and the ray path s can be considered straight. The ray path increment ds is related to the zenith path increment dz, by (3.10) ZENITH ATMOSPHERE T7>n 11 n )nn/ EARTH Figure 3.3. Origin of the cosecant law. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since absorption coefficients usually depend only on altitude and not on the incremental path length, use of Equation (3.10) in Equation (3.2) shows that, Tv(s',s) 0>1OC kv(z") csc(0) dz" (3.11) Jz' Since the cosecant term is not a function of z, the integral is simply the zenith opacity of the atmosphere. Therefore, Equation (3.11) can be written as, Tv(s',s) = tv(z',z) csc(0) 0>1O° . (3.12) This is known as the cosecant law and it is widely used to approximate slant path attenuation from zenith path attenuation. At lower elevation angles, refraction in the atmosphere and the curvature of the earth are significant to the attenuation. Refraction in the atmosphere bends the path taken by the radiation, thereby increasing the total path length. Because the earth curves, the atmosphere curves with it, making a horizon path remain longer in the lower atmosphere than the zenith path. Specific attenuation is more significant in the lower atmosphere and thus the earth curvature tends to increase total attenuation for 0<1O°. Horizon paths are treated in greater detail in the next chapter. 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Although slant path attenuation for large elevation angles is simply related to zenith attenuation, the same is not true for brightness temperature. For the sake of demonstration, assume T_(0)=0 and that the cosecant law is valid. The brightness temperature calculation must be taken through the entire atmosphere which can be taken as having infinite extent. Then Equation (3.4) becomes for an observation point at near sea level, -X (z',“ )csc(0) TB<e> = T(z')e V k (z')csc(0)dz' v 0 where 0 is now used to indicate the observation path. (3.13) Examination of this equation shows that the cosecant relationship does not hold for brightness temperature because of the argument of the exponential function. Therefore separate calculations of brightness temperature are required for each elevation angle. This is significant to the antenna temperature calculation because the brightness temperature must be known for all elevation angles. D. SUMMARY In this chapter, the transport of energy by electromagnetic waves propagating through the atmosphere has been described. The form of the equation of radiative transfer for a nonscattering, nonrefracting atmosphere in thermodynamic equilibrium was given. This equation is valid at microwave frequencies for an atmosphere containing only gases. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The equation requires a numerical solution because the equation cannot be solved otherwise. To demonstrate some properties of the atmosphere which are contained in the radiative transfer equation, a constant atmospheric temperature assumption was made. The equation can be solved in this case and the effects of a transparent and an opaque atmosphere were demonstrated. These characteristics are important to the discussion of antenna temperature. The opacity of the atmosphere determines what the antenna 'sees' in the environment. The form of the radiative transfer equation known as the radiometric formula was given. The cosecant law was presented to demonstrate the elevation angle dependence of attenuation. The cosecant law is used later to reduce the number of calculations needed to obtain attenuation in the atmosphere. As was pointed out in the chapter, the brightness temperature does not adhere to the cosecant law. However, in the next chapter, a method is presented which uses the cosecant law in conjunction with the radiometric formula to produce the brightness temperature profile without excessive calculation. The equation of radiative transfer models the role of the atmosphere in the antenna temperature calculation. However this equation only provides the technique by which brightness temperature can be calculated. In order to use the equation, more information is needed on the characteristics of the atmosphere, specifically, those constituents of the atmosphere which affect microwave propagation. The constituents whose principal effect is absorption are treated in the 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. next chapter. Rain, which absorbs and scatters microwave radiation, discussed separately in the subsequent chapter. Sources of radiation which are incident on the atmosphere are covered in Chapter VI. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER IV THE ATMOSPHERE VITHOUT RAIN A. INTRODUCTION The previous chapters have attempted to develop some background on the subject of antenna temperature. Chapter II defined antenna temperature and showed that it depends on the antenna pattern and the brightness temperature profile of the environment. Chapter III shows that the brightness temperature for a given direction in space can be found by using the equation of radiative transfer. The equation of radiative transfer demonstrates the role of the atmosphere in the propagation of energy from a noise source to the antenna. Thus the equations for the calculation of antenna temperature are in place. What is still needed are the models of the noise sources and the atmospheric constituents which affect propagation. Together, these individual models will form the environmental model that is needed to perform the antenna temperature calculation. The presentation of the individual models will begin with this chapter. The bases of the individual models which will be presented are not new. They are drawn from the works that the various authors have made available in the literature. The purpose of this work is to assemble these models and implement them in such a way that the brightness temperature profile of the environment may be calculated. For this 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. reason, the focus of the discussion in the following chapters will be on the implementation of the individual models and on how they are used in conjunction with the overall environmental model. Details on the derivation and development of the individual models will not be given but may be found in the references cited. This chapter contains a general description of the atmosphere. The atmospheric constituents which have significant effects on microwave propagation are pointed out. The remainder of the chapter deals with modeling the atmosphere when rain is not present. This requires models of gaseous absorption as well as cloud and fog absorption. The absorption coefficients of these constituents depend on the distribution of meteorological parameters within the atmosphere. This leads to a need for an atmospheric model which is also discussed. B. DESCRIPTION OF THE ATMOSPHERE The greatest atmospheric effects on microwave propagation occur below an altitude of 80 km. In this region the atmosphere is composed principally of gases, although some liquid and solid matter are present. In the absence of liquids, solids and water vapor, the atmosphere is, by volume, 78.084% nitrogen, 20.946% oxygen, and 0.934% argon. These are the most abundant gases and the remainder are trace gases which can be found listed in Table IV, Chapter 2.3 of [13]. The amount of these gases are stable and the dynamics of the atmosphere keep them well mixed so that no pockets of an individual gas are created. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Vater vapor, ozone, and carbon dioxide are all gaseous constituents of the atmosphere but their abundance is variable. Vater vapor originates from the evaporation of surface water. The atmosphere distributes this water vapor in such a manner that the water vapor concentration varies with time at a single location and it varies between locations. the atmosphere. Ozone is created and destroyed by reactions within The rate of these reactions are not constant resulting in varying levels of ozone. The amount of carbon dioxide is believedto be increasing due to industrial activity [16]. The liquid and solid matter are chiefly forms of condensed water vapor. The condensed water vapor may appear as rain, clouds, fog, hail, ice or snow. Suspended surface material and particulates from pollution sources may also be present in the lower atmosphere. The properties of the atmospheric constituents which are important to an antenna temperature calculation are absorption, emission and scattering. Assuming the atmosphere is in thermodynamic equilibrium allows emission to be simply related to absorption. Thus models must be chosen which describe the absorptive and scattering effects of the constituents. At the microwave frequencies, gas molecules are too small to cause scattering, but they can have absorptive effects. The liquid water drops in rain, clouds, and fog, also absorb radiation and, depending on the size of the drop, can scatter significantly. form, is a relatively poor absorber. Solid water, in any However, snow and hail often 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. contain significant amounts of liquid vater which causes these constituents to have appreciable absorption. The specific frequency range being addressed in this study is 1-40 GHz. In this region, the atmospheric gases which significantly affect microwave propagation are oxygen and water vapor. gases is discussed in the next section. considered in this model. Absorption by these No other gases will be In this frequency range the size of the vater drops in clouds and fog is small enough to allow scattering to be ignored. Therefore absorption by these constituents is undertaken in section D. Rain drops scatter in this range so discussion of the rain model is reserved for Chapter V. The effects of snow and hail, which are complicated to model, and suspended surface material or pollutants which are highly variable, will not be included. C. ABSORPTION COEFFICIENTS FOR WATER VAPOR AND OXYGEN According to Quantum Theory, the energy of a single photon of electromagnetic radiation is given by, E = hv (4.1) where h is Planck's constant and v is the frequency of the radiation. The rotational energy of a gas molecule may also be quantized. rotational state is associated with a different energy. Each The molecule can attain a higher energy state from a lower energy state if it is supplied with an energy given by, (4.2) 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Em is the energy of the high state and lower state. is the energy of the If the gas molecule is in an electromagnetic field, the energy to change states is available from the photons. A photon with E = AElm (4.3) can impart energy to the molecule putting the molecule in the higher rotational state. This process annihilates the photon and the gas molecule is said to have absorbed it. Similarly the molecule may also create a photon by falling from a higher state to a lower state. The difference in energy between the states is radiated with a frequency, E -E. m l lm ' T - ,, ,. ' <4 -4> Absorption and emission can only occur at frequencies corresponding to the energies between states. These energies are an intrinsic property of the type of gas molecule. Thus the frequencies represented by Equation (4.4) are the resonant frequencies of the molecule. The molecular absorption causes the gas to produce an absorption line, in a continuum of radiation, at the resonant frequencies. The discrete frequencies predicted by Equation (4.4) are for a single gas molecule. In a gas composed of numerous molecules, collisions between molecules perturb the energies of the rotational states. The result is that the absorption lines of the molecule are not 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sharp but are broadened into a small range of frequencies about the theoretical resonant frequency. This is called collisional broadening and it is the dominant type of line broadening mechanism in the atmosphere below an altitude of 70 km. There are other types of line broadening mechanisms but they can be neglected in this analysis. The atmospheric gases which have significant absorption lines in the microwave spectrum are water vapor and oxygen. resonates at 22.235 GHz, 183.31 GHz and 325.152 GHz. Water vapor Oxygen has a series of resonances near 60 GHz and a single resonance at 118.75 GHz. Ozone also has appreciable absorption lines in the microwave region. However the lowest frequency at which a significant absorption occurs is 67.356 GHz. This line does not affect the spectrum between 1-40 GHz and therefore ozone will be neglected. Other gases with microwave spectra occur in such trace amounts that they can be neglected also. A procedure for calculating the absorption coefficients of water vapor and oxygen, for use in a radiative transfer model of the gaseous atmosphere, has been reported by Smith [17]. Smith uses the expressions for the absorption coefficients that are given in Waters [13], except that the Rosenkranz [18] expression is used for the oxygen absorption coefficient. He also suggests that the Gaut and Reifenstein [19] empirical adjustment for water vapor absorption be included in the calculation. This procedure is straightforward and is outlined in detail in the references. The references also provide results of the absorption coefficient calculations which compare favorably with 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. measurements. Therefore this absorption coefficient calculation was chosen for the environmental model. A Fortran program WVABSORB has been written to calculate the absorption coefficient of water vapor from the equations and data given by Waters [13]. Using the notation of Waters, the absorption coefficient is given by, -3/2 pVr kH20 = 1'U L r -E,/kT -E /kTl 1 m e -e gll*lm l ►+ all transitions + Ak, (4.5) h 2o where the chemical formula of water has been used to identify the absorber. In this equation p is the water vapor density in grams per meter cubed, v is the frequency in GHz, T is the temperature in kelvin, Em and E^ are the internal energies of the upper and lower molecular states, g^ is the statistical weight of the lower state, <J>^m is the transition matrix element and f is a function describing the shape of the line. The function f has units of GHz-1 and the summation is over all water vapor transitions. 1 f(v’vlm> n Waters uses the kinetic line shape, 4v v, Av. lm lm ,2 2,2 . 2.2 lm -v > * 4v lm with the linewidth parameter A\>^m given by, 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.6) lm = » P r p U T r X fi ,r ,rt-3 PTfAvlm(H20> ,ll ,, lmUOI 3J V.300J 1 + A’6 X 10 P .0 “ 1 <4,7) Avl« )] where P is total pressure is millibars and additional linewidth parameters. an(* ^vlm are Note, the units of millibars for pressure was the choice of the authors in the references used in this study. Therefore, millibars will be used in this study even though it is not a standard unit. Waters states that sufficient accuracy is obtained for frequencies less than 300 GHz if the summation is only taken over the lowest ten frequency transitions. of the parameters, g^, He provides the values Em > E^> ^vlm’ ^vlm^H2 ^ an(* x reproduced here in Table 1. -. which are The additive term, Aky q is the Gaut and 2 Reifenstein empirical correction factor, Akjj Q = 1.08 x 10 (4.8) The units of E, and E in Table 1 are cm 1 m . These numbers must be multiplied by Planck's constant and the velocity of light in order to be used in Equation (4.5). Finally, the units of the absorption coefficient as given by these equations are cm *. Conversion to attenuation is provided by 1 cm ^ = 4.34 x 10^ dB/km . (4.9) 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.1 displays the water vapor absorption coefficient as calculated by WVABSORB for a case where p=7.5 gm/m3, P=1013 mbar and T=300 K. This data was calculated for comparison with results published in Waters [13], which are reproduced in Figure 4.2. Comparisons should be made with the solid curve of Waters, whichisfor the model described above. curve, generatedbyWVABSORB, for Figure 4.3 shows the GHz inset in Figure 4.2. the 15-40 Figure 4.4 containsthe WVABSORB calculations for the 100-220 GHz inset of Figure 4.2. Comparison of the data shows that WVABSORB calculates the amplitude and the shape of the absorption lines properly. However, the WVABSORB curve runs slightly lower than Waters' curve in the continuum region between lines. No explanation has been found for this behavior, but in the 1-40 GHz region the difference is only 0.01 dB/km. Thus no significant effects to the environmental model are expected. If the environmental model were to be extended past 40 GHz, this item might have to be investigated in greater detail. The absorption coefficient of atmospheric oxygen, as developed by Rosenkranz, is given by, k = C P [v/T]2F (4.10) 2 which is in the notation of the original work, except for kn . In this 2 equation, P is pressure in millibars, v is frequency in GHz, T is temperature in kelvin and 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE 1 PARAMETERS FOR WATER VAPOR ABSORPTION COEFFICIENT CALCULATION E m Resonant Frequency* (GHz) X 447.30 446.56 2.85 13.68 .626 .1015 142.27 136.16 2.68 14.49 .649 3 .0870 1293.80 1283.02 2.30 12.04 .420 325.152919 1 .0891 326.62 315.78 3.03 15.21 .619 380.197322 3 .1224 224.84 212.16 3.19 15.84 .630 390.18 1 .0680 1538.31 1525.31 2.11 11.42 .330 437.34667 1 .0820 1059.63 1045.03 1.50 7.94 .290 439.150812 3 .0987 756.76 742.11 1.94 10.44 .360 443.018295 3 .0820 1059.90 1045.11 1.51 8.13 .332 448.001075 3 .1316 300.37 285.42 2.47 14.24 .510 22.235080 3 .0549 183.310091 1 321.225644 (cm H2°> (GHz) l+lmI2 ) AV ) *1 (cm .o Av, lm E1 * More accurate frequency data was available in [20]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ion of the copyright owner. Further reproduction prohibited without permission. O z: CD □ Z o h— Q_ OC O CD QQ cr CM 0. 50. 100 . 150. FRFQIIFNTY Figure 4.1. 200. 250. 300 . fr,H7l WVABSORB generated water vapor absorption for p=7.5 gm/m3 P=1013 mbar and T=300 K. (d B / k m ) COEFFICIENT H z0 A B S O R P T I O N 100 140 180 220 I02 10,-4 0.1 E o e 10 Ll I Io'5 20 30 40 CJ u. LibJ o O 10 Q_ o: o o' 10 CO CD < 0 CM 1 50 100 150 FREQUENCY Figure 4.2. 200 250 (GHz) Water vapor absorption as reported by Waters [13]. Comparisons should be made with the solid curves. 10-300 GHz plot: T=300 K, P=1013 mbar and p=7.5 gm/m3; 20-40 GHz inset: T=318 K, P=1013 mbar and p=7.5 gm/m3; 100-220 GHz inset: T=300 K, P=1013 mbar and p=7.5 gm/m3. 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15. 23. 25. 30. 35. F n ii Q U L N C T Figure 4.3. (GH Z i WVABSORB generated water vapor absorption for p=10 gm/m3, P=1013 mbar and T=318 K. FREQUENCY Figure 4.4. HO. (GHZ) WVABSORB generated water vapor absorption for p=7.5 gm/m3, P=1013 mbar and T=300 K for 100-220 GHz. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ' 0.333 for kn in nepers/km 2 c = « , 1.434 for k_ (4.11) in dB/km 2 The function F is written by Rosenkranz as f 39 0.70 wfa F = P * £»<~v) + £»(v> + £«('v)] YL N=1 N,0DD ~2 2 v +(Pwb) (4.12) where f±,v) * The wn = M (dg)2" (4.13) )2 ♦ K )2 are interference coefficients given by, ftt+ “ T N+2WN - dN lVN"VN+2 + Aj+ 'J' w. dN-2WN - --VN“VN-2 (4.14) VN V60 The parameters w^ and wb are the diagonal elements of the transition rate matrix and are evaluated from w.T N = 1.16 x 10 0.85 -3 (4.15) ffl and 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vb = 0.48 x 10-3 (300)^' (4.16) The off diagonal elements of the matrix are generated by recursion from $ wm 9 = WN N-2 N *N_2 (4.17) WN = W WN ( 4 -18) and t with the starting point, w^g = 0 and works down to N=l. 4* is not used to calculate w-1 which is set at zero. Equation (4.18) *K N7 is the fractional population of the state, calculated by, *n = (2N±1) e~2.0685N(N+1)/T (4>W) where Z is the partition function, Z = 0.725T . (4.20) The v* are the resonant absorption frequencies of oxygen and they are listed in Table 2. The amplitude of the v* frequency line is, 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE 2 RESONANT FREQUENCIES OF ATMOSPHERIC OXYGEN N 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 VN 56.2648 58.4466 59.5910 60.4348 61.1506 61.8002 62.4112 62.9980 63.5685 64.1278 64.6789 65.2241 65.7647 66.3020 66.8367 67.3694 67.9007 68.4308 68.9601 69.4887 118.7503 62.4863 60.3061 59.1642 58.3239 57.6125 56.9682 56.3634 55.7838 55.2214 54.6711 54.1300 53.5957 53.0668 52.5422 52.0212 51.5030 50.9873 50.4736 49.9618 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. r N(2N+3) 1 (4.21) L(N+1)(2N+1)J and that of the line is, r(N+l)(2N-l)1 (4.22) = L N(2N+1) J These equations have been implemented in the Fortran program 02ABS0RB. Figure 4.5 is a curve of oxygen absorption which was generated by the program for comparison with data published by Rosenkranz. This figure shows the variation in the absorption coefficient with pressure at a frequency of 58.82 GHz and 295 K temperature. The curve presented by Rosenkranz is shown in Figure 4.6. Comparison shows that the curves are identical. In his paper, Smith gives the oxygen absorption coefficient variation with frequency for 1013 mbar pressure and 293 K temperature. Figure 4.7. This data is reproduced in Figure 4.8 shows this case as generated by 02ABS0RB. Again, the curve is identical to the published curve. Therefore 02ABS0RB appears to work as intended. Oxygen and water vapor are the only gaseous constituents which are included in the environmental model. Examination of the equations for the absorption coefficients of these gases shows that they depend on the atmospheric parameters of water vapor density, temperature and total pressure. The equation of radiative transfer, in which these coefficients are used, requires an integration through the atmosphere. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O ABSORPTION (DB/KM) LO m CM O 100.0 200.0 PRESSURE Figure 4.5. 300.0 (MB) Oxygen absorption for v=58.82 GHz and T=295 K. 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 ABSORPTION ( dB/km) 4 3 2 0 0 100 200 300 400 P R E S S U R E ( mb) Figure 4.6. Oxygen absorption as calculated by Rosenkranz [18] for 58.82 GHz and 295 K. The solid line is the calculation of Rosenkranz. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I ■o o u_ O o O 'J- A E 0.06 o 0.04 CO CO * * 0.02 0.01 0.006 0.004 0.002 0.001 10 20 50 100 200 350 FREQUENCY, GHz Figure A.7. Oxygen absorption coefficient as calculated by Smith [17] for P=1013 mbar and T=293 K. Comparisons should be made with the dotted 02 curve only. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I •fa o ABSORPTION C D B/ KM ) "o o o O' FREQUENCY Figure 4.8. (GHZ) Oxygen absorption for P=1013 mbar and T=293 K. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since the atmospheric parameters are not constant, their variation with altitude must be known. The variation in the parameters is contained in the atmospheric model which is the subject of the next section. D. A MODEL OF THE GASEOUS ATMOSPHERE The calculation of the absorption coefficients of oxygen and water vapor, through the atmosphere, requires knowledge of the structure of the atmosphere. Specifically, the variation of temperature, pressure and density with altitude must be known. provided by a number of techniques. This information may be Actual measurement of the vertical structure of the atmosphere could be taken during the conditions for which the absorption coefficient profile is desired. Also, a sophisticated model atmosphere, based on the ideal gas law and equation of hydrostatic equilibrium, could be used to simulate the conditions in the atmosphere. Alternatively, the model atmosphere could be used to generate a general description of the atmosphere for average conditions as well. This final method is the most practical approach for a system application such as this one. A profile of temperature and dry air pressure for average atmospheric conditions around 45°N latitude is available from the U.S. Standard Atmosphere, 1976 [21]. The standard atmosphere provides the value of a number of parameters over an altitude range from -5 km to 1000 km. Alternately, there are supplemental atmospheres available which take into account latitudinal differences and seasonal changes not considered in the standard atmosphere. Valley [22] lists supplemental 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. atmospheres for tropical (15°N), subtropical (30°N), mid-latitude (45°N), sub-arctic (60°N) and arctic (75°) climates. Atmospheres for January and July are given for all latitudes except for 15°N which is not sensitive to seasonal variation. These sources provide a data base from which typical atmospheric conditions at all geographic locations can be simulated. The U.S. Standard Atmosphere, 1976 will be used for the remainder of this study. It represents the conditions of the atmosphere which are typical of Columbus, Ohio (40°N). Additionally, most atmospheric radiative transfer data, available in the literature, use it. The supplemental atmospheres could be implemented simply by programming the data into the environmental model being discussed here. To date, this has not been done and user selection of the atmospheric model is a development which can be addressed in the future. The Fortran program ATMOD has been written to generate the temperature and dry air pressure variations from the standard atmosphere data. The standard atmosphere data are stored in subroutine STANDARD_ATMOS within ATMOD. to 100 km have been stored. above 100 km are negligible. Only the data from the altitude range 0 km Atmospheric effects on microwave radiation The data are stored in 1 km increments. Finer resolution is available in the standard atmosphere but it is not necessary for this application. Note, supplemental atmospheres would be implemented through STANDARD_ATMOS. The temperature data in the standard atmosphere is a piecewise linear approximation to the temperature variation with altitude. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore, linear interpolation is used to obtain temperatures between sample points. Mathematically, (h-h )T(h ) + (h -h)T(h ) T(h) = --- £■--- \ .. U----- * — Q nP (4.23) where T is the temperature, h is the altitude and hp, h^ are successive sampled altitudes such that hp < h < hQ . (4.24) The dry air pressure profile in the standard atmosphere has an approximately exponential nature with altitude. Therefore, linear interpolation is used on the logarithm of the pressure to form a continuous function. This is written as la [P(h)J rrrhn = -----------<h-hP>1" F (hQ>] V h)1" F (hP>] In r— *— (r-----------Q (4.25) p where P is the pressure, In is the natural logarithm and Equation (4.24) holds. The pressure is obtained from Equation (4.25) by P(h) = elnIp(h>l . (4.26) The temperature profile and dry air pressure profile of the U.S. Standard Atmosphere, 1976, as calculated by ATMOD, are shown in Figure 4.9. The altitude shown on the vertical axis of these plots is 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ll» .’ 190. 200 . 210 . 220 . 230 . 290 . 2S0 . 260 . 270. 290 . 200. TEMPERATURE (KELVIN) - j j i l i f " B .| | ::. I j p ^ j | J J H 3 i I I 4li4----- 4t»*rr —t n ! 11'11 cr''';;Vo'-j V m l Itf - - •) Inn | llji ~f~“j " ' H o-" ‘S c F 'H & ' S o’ ORT RIR PRESSURE (MB) Figure 4.9. Temperature and dry air pressure profiles of the U. Standard Atmosphere, 1976, as calculated by ATMOD. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. referenced to mean sea level (MSL). The values of the parameters at MSL are 288.15 K for temperature and 1013.25 mbar of dry air pressure. No information is available from the standard atmosphere on water vapor because the water vapor content of the atmosphere is highly variable. Therefore, an additional model must be provided to generate the water vapor profile. Measured data indicates that water vapor density exhibits an exponential dependence on altitude in the lower atmosphere. Data above 7 km are scarce and models for this region are subjective 122]. Smith and Waters (23] have used an exponential model of water vapor, with scale height 2 km, up to an altitude of 12 km in their radiative transfer model. Based on their experience, this water vapor model will be adapted for use in this environmental model. The water vapor content of the atmosphere is typically measured in terms of the relative humidity. The input variable to the model is relative humidity, which is related to water vapor density by, , ,„39 KB(222)‘ 1010-9.834<300/T) (4 2?) where T is kelvin, RH is the relative humidity in percent, and pu n is 2 the water vapor density in gm/m3. Using the exponential variation of water vapor density with a scale height of 2 km, the water vapor profile is given by, 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I -h/2 h<12 km ( 4 .2 8 ) h>12 km where h is the altitude in km. Since the water vapor models for the higher altitudes are subjective, this profile as stated in Equation (4.28). reportwill use the water vapor Using this profile in brightness temperature calculations produces results which compare favorably with the calculations of Smith and Waters. later. These results will be presented Smith and Waters include water vapor data above 12 km, and since comparison of the calculations are favorable, the water vapor above 12 km should not have a large effect on antenna temperature. The presence of water vapor in the atmosphere contributes to the total pressure. For this model the total pressure, P^., is given by Pt = PDM + PH20 (4*29) where P^y is the dry air pressure given by the U.S. Standard Atmosphere, 1976 and Py q is the partial pressure of water vapor. The partial pressure of water vapor is related to water vapor density by, V where p f T ■ »h 20 1.216.57 (4.3C) in gm/m3 and T in kelvin give Py ^ in millibars. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This water vapor density profile has been included in ATMOD. Figure 4.10 shows the water vapor density profile and corresponding water vapor partial pressure as generated by ATMOD for p„ n(0) = 7.5 2 gm/m3. The U.S. Standard Atmosphere, 1976 temperature profile was used in the pressure calculation. The temperature and water vapor density at h=0 correspond to a relative humidity of 58.64%. Care must be taken to be sure that the relative humidity does not exceed 100% at any point in the atmosphere. Clearly, if the absorption coefficients depend on altitude, then the result of the radiative transfer calculation will depend on where in the atmosphere the integration begins. In terms of the antenna temperature calculation, this corresponds to the location of the antenna within the vertical structure of the atmosphere. In this study, the elevation or altitude of the station above MSL will be used to locate the antenna in the atmosphere. Also since the meteorological parameters of temperature, total pressure and relative humidity can be measured at the station, some correction should be made to the model, as outlined thus far, to compensate for these parameters. The model being presented in this study will use a simplescheme for correcting for local meteorological parameters. Such a correction is based on the assumption that the structure of the atmosphere does not change. For the temperature and dry air pressure profiles of the standard atmosphere, this means that the envelope of the curve will just shift in order that the temperature and dry pressure of the model agree with the measurement of those parameters. The same is true of the water 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. _J_i_t-t H -H -J W i 4 — J .i.liiJ £ 4~i - l- ff!4 ' m i n j ; ir ^ l# i|1 i^ N l llE m iS E E E ^ a iu ^ 4zT 1 : <; ,, ; ^ | -i .) t , :ijiu --B..;■ ® 10:*~i $ * $ & H s \ q '1 i rTTETSFftf 5 i <i$ WATER VAPOR DENSITY (CM/H*»<3) x se UJ o r> * _j a WATER VAPOR PARTIAL PRESSURE (MB) Figure 4.10. Water vapor density and water vapor partial pressure profiles in ATMOD for 7.5 gm/m3 surface water vapor density and 2 km scale height. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vapor profile. Note that this assumption is overly simplified for the atmosphere and future study should investigate methods which more adequately represent local variations. Imagine a station at altitude hQ where the parameters Tg^p, Pg^p and RH (relative humidity) are recorded. Since the temperature in the standard atmosphere varies linearly, a correction given by, AT - w where - v v <4-31> is from the standard atmosphere, is needed to shift the temperature profile. Therefore, for these station conditions, ATMOD calculates T(h) = TM(h) + AT (4.32) for the temperature profile. Before a correction is made to the dry air pressure profile, the dry air pressure at the surface must be extracted from the total pressure Pg^p* This requires converting the relative humidity to water vapor density by Equation (4.27), and then using Equation (4.30) to get the partial pressure of the water vapor. Once this is known, then the dry air pressure at the surface is found from, PD,SURF " PSURF PH20,SURF * (4.33) By the exponential nature of the pressure, the correction to the dry air pressure is done on the logarithmic scale, 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^ (4.34) “ ln P d ,SURf ] * ln[PDM(ho)] Then the shifted pressure profile is, l n p D(h>] . ln[PD„(h)] „ or equivalently, which is calculated by ATMOD. The water vapor density profile is exponential, so a correction to it is related to that for dry air pressure. Rewriting Equation (4.35) in terms of water vapor density gives (4.36) (hQ) pH20,SURF where the subscript M represents the model. For this density, the model was a mathematical one given by Equation (4.28). Using Equations (4.28) in Equation (4.36) results in the corrected water vapor density profile, (ho-h)/2 pH20,SURFe h<12 km h>12 km 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.37) As a demonstration of the compensation for station altitude, surface temperature, surface total pressure and relative humidity, ATMOD was used to calculated atmospheric profiles for h =0.4 km, Tei__=300 K, O ^SURF=^ ^ oUKr mbar and 29.4% relative humidity (7.5 gm/m3 of water vapor). The results of this calculation are shown in Figures 4.11 and 4.12. This atmospheric model can now be applied to generate the absorption coefficient profile of oxygen and water vapor through the atmosphere. Using these profiles in a radiative transfer calculation will produce the brightness temperature profile of the atmosphere which is a major step toward the goal of an antenna temperature model. Implementation of the equation of radiative transfer, with absorption coefficient and brightness temperature profiles, are covered in the next section. E. EVALUATION OF THE EQUATION OF RADIATIVE TRANSFER The equation of radiative transfer must be evaluated numerically when applied to a complex atmosphere such as that of the earth. Thus it is instructive to examine the behavior of each individual term in the equation in order to develop some limits on the calculation. The limits will serve to keep the calculation practical without the loss of good engineering information. The equation of radiative transfer is repeated here for convenience, 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f^ ,i,a\ ^ l,lu\ 0L^ ' b L'r "\(T1B,Y(r ,ra,iif',n"\&' ""'\cf DRY R1R PRESSURE (MB) Figure 4.11. Temperature and dry air pressure profiles for hQ=0.4 km, TSURp=300 K, Psurf=1023 mbar and 7.5 gm/m3 surface water vapor density with 2 km scale height. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UJ a <i46>B9| icf rjTTwio1 HATER VRPOR DENSITT (GM/H«><3) HfiTER VRPOR PRRTIRL PRESSURE (MB) Figure 4.12. Water vapor density and water vapor partial pressure profiles for hQ= 0.4 km, TSURp=300 K, PSURp=1023 mbar and 7.5 gm/m3 surface water vapor density with 2 km scale height. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I 's Tb (s ) = Tfi(0)e T(0’s) T(s')e T(s',s)k(s')ds' + (4.38) JO where the opacity is ’s t (s ', s ) = Js' k(s")ds" (4.39) and the frequency dependence is implicit. T(s) is the physical temperature profile of the atmosphere. This profile results from the atmospheric model and was covered in the previous section. The atmospheric model provides the distribution of meteorological parameters as a function of altitude. By using this model in conjunction with WVABSORB and 02ABS0RB, the profiles of the prominent absorption coefficients may be generated and studied. First consider the absorption coefficient profile of water vapor. The profile for a water vapor density of 7.5 gm/m3 at MSL is shown in Figure 4.13 for v=ll GHz. A frequency of 11 GHz is chosen to display data because future antenna temperature measurements are anticipated at this frequency. An interesting point to note from this curve is that at 12 km the coefficient is three orders of magnitude below the coefficient at the surface. This demonstrates the limited effect that water vapor has on microwave radiation at high altitudes. Also, this provides some justification of truncating the water vapor model at 12 km. Since 11 GHz is away from the water vapor line at 22.235 GHz, Figure 4.14 is provided to show the behavior of the coefficient at the line frequency. Here, the coefficient at 12 km is two orders of magnitude down from the value at the surface. Thus, truncating the model here should provide sufficient engineering accuracy. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CM O) RLTITUDE (KM) ■A a . A t— 10' r TTTl r-l 1 O'5 1 — I I I'tlTT X 1 A 1 — I I'1 n I'l 10 H20 ABSORPTION COEFFICIENT Figure 4.13. a 1— rr 109 u 10 ,-2 (DB/KM) Absorption coefficient profile of water vapor at 11 GHz. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 water vapor density at MSL with 2 km scale height included. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I CM 0) ALT ITUDE (KM) CO CO CO CM H2G ABSORPTION COEFFICIENT Figure 4.14. (DB/KM) Absorption coefficient profile of water vapor at 22.235 GHz. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 water vapor density at MSL with 2 km scale height included. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The oxygen absorption coefficient profile at 11 GHz is shown in Figure 4.15, up to an altitude of 100 km in the standard atmosphere. Note here that the absorption coefficient at 100 km is 13 orders of magnitude different from the surface value. This explains why oxygen has little effect on microwave propagation at very high altitudes, provided that the frequency is not near an oxygen absorption line. The oxygen profile for a frequency of 40 GHz is shown in Figure 4.16. This curve is shown because it is at the frequency, within the frequency range under consideration, that is closest to the oxygen line region around 60 GHz. This curve also shows that most oxygen absorption, at 40 GHz, occurs in the lower atmosphere. The oxygen profile at the absorption line of 60.3061 GHz is shown in Figure 4.17, although this is outside of the frequency range of interest in this study. line. This data shows the strength of the oxygen Also this data demonstrates that at the line frequencies, oxygen absorption in the upper atmosphere is significant. Integrating the absorption coefficient profile, along the trajectory taken by an electromagnetic wave, results in the attenuation suffered by the wave as it propagates. This is shown by Equation (4.39) and is also known as the opacity or optical depth of the atmosphere between the points, s and s'. For a zenith calculation of atmospheric opacity, the observation path would be as shown in Figure 4.18. observation path. Two variables are shown for the The s system, introduced when radiative transfer was introduced, increases in the direction of ray travel. The top of the 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 90. 80. 70. (KM) 60. 50. >10. ALTITUDE 30. 20. v,dj,”ioj|fiioii!\ m B"*! o,mi o-n 02 ABSORPTION COEFFICIENT Figure 4.15. o(DB/KM) Absorption coefficient profile of oxygen at 11 GHz. Standard Atmosphere, 1976. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U.S. 100 90. 80. 70. (KM) 60. 50. HO. ALT ITUDE 30. 20. itmantnij'WiPUino ABSORPTION COEFFICIENT Figure 4.16. (DB/KMI Absorption coefficient profile of oxygen at 40.0 GHz. U.S. Standard Atmosphere, 1976. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100. 90. 80. 70. (KM) 60. 50. 140. ALTITUDE 30. 20. b'-'K’ rtthm m Inbnif\d-'HiMoninino* 02 A B S O R P T I O N COEFFICIENT Figure 4.17. (OB/KM) Absorption coefficient profile oxygen at 60.3061 GHz. U.S. Standard Atmosphere, 1976. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O B S E R V A T IO N P A TH z = z, s =o s = s. t>ii/1) )))))n )n*r)i))777)i)n )}>>))> z = 0 Figure 4.18. Local earth geometry for a zenith line integral. atmosphere is s=0 and the bottom is s=smax* The zenith system, represented by z, is more familiar to earth-bound observers. The z origin is at the surface and z increases to the top of the atmosphere, which is z=z . The two systems are related by max z - V..-S • The differential of this equation dz = -ds (4.41) 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shows the opposite directions of increase. Substituting Equation (4.40) and Equation (4.41) into Equation (4.30) produces the attenuation as a function of altitude, ’z' t(z',0) = k(z")dz" . (4.42) 0 This equation describes the amount of attenuation between point z' and the surface. The variable z' can be placed anywhere along the zenith path, but it is most instructive to allow it to vary from the surface out to the top of the atmosphere. This has been done for oxygen at 11 GHz in Figure 4.19. The way this graph is interpreted is that the vertical scale is the z' in Equation (4.42). The horizontal scale is the atmospheric attenuation between z' and the surface. Note that most of the attenuation is contributed by the lower atmosphere. This is due to the rate of decrease in the absorption coefficient shown in Figure 4.15. Above 10 km very little is being added to the total attenuation and above 20 km nothing significant is being contributed. Past this altitude, the attenuation has converged to a specific value. This value is the opacity of the atmosphere at 11 GHz, due to oxygen. In reality, the atmosphere will contain some water vapor and that will contribute to the attenuation profile. When both oxygen and water vapor are present, the total absorption coefficient is given by, 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o co ir> CM o CM LU Q t— _J CE o O TTTT 10' ATTENUATION Figure 4.19. (DB) Attenuation profile of oxygen at 11 GHz, U.S. Standard Atmosphere, 1976. 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.20 shows the same oxygen profile as Figure 4.19 but with 7.5 gm/m3 surface water vapor density added. As should be expected from the water vapor absorption coefficient profile, Figure 4.13, the main contributions to the opacity are in the lower atmosphere. The attenuation does not change significantly above approximately 15 km. The inclusion of water vapor has increased the opacity of the atmosphere by about .02 dB. The attenuation profile data starts to indicate that calculating attenuation above 15-20 km is unnecessary for the application being contemplated for this study. The greatest opacity in the 1-40 GHz range is at the 22.235 GHz line of water. Figure 4.21 displays the attenuation profile of the atmosphere for that frequency and 7.5 gm/m3 surface water vapor density. Although the opacity at this frequency is ten times larger than at 11 GHz, it is all formed in the lower atmosphere. Recall that the water vapor model shuts off at 12 km, but the attenuation profile shows very little change above 10 km. A great deal of water vapor would have to be present above 12 km to contribute to the opacity and this is physically unlikely. The rapid convergence of the opacity is due to the rapid decrease in the absorption coefficient profiles. Simultaneous observation of the attenuation profile and the absorption coefficient profiles seems to indicate that convergence of the opacity occurs when the absorption 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. &n CM CM RLTITUDE (KM) o O o R T T E N U f l T I O N (DB) Figure 4.20. Attenuation profile of oxygen and water vapor at 11 GHz. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 surface water vapor density with 2 km scale height. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ATTENUATION Figure 4.21. (DB) Attenuation profile at 22.235 GHz of oxygen and water vapor. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 surface water vapor with 2 km scale height. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. coefficients have changed by a few magnitudes. the application of the model. This is significant to The altitude at which the absorption coefficients have dropped a couple of magnitudes depends on where the reference is. In other words, convergence of the opacity depends on the elevation of the station with respect to MSL. This application is concerned with earth-based microwave antennas. The surface of the earth has finite elevation and thereby so do possible station elevations. This study will concentrate on stations with elevations of 5 km or less. This should encompass most operational stations. With these considerations and knowledge of the convergence of the atmospheric opacity, a limit on the attenuation integral will be set at 20 km. This limit is not necessary for this development and is only mentioned to indicate the practical aspects of implementing this theory. This limit should give enough margin for stations with elevations in the 0-5 km range. A user with an application that is outside of this range must increase the upper limit on the integration to acquire adequate results. Judging from the opacity calculations, a good rule of thumb would be to terminate the integration at an altitude that is 20 km greater than the station elevation. The accuracy of this limit with respect to low noise or high altitude stations, should be established in the future. Using the 20 km upper limit, the data in Figure 4.22 were calculated. This figure shows zenith attenuation at MSL over the frequency range of interest. The 40-60 GHz region has been included to show that the upper end of the 1-40 GHz range is influenced by the tail 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the 60 GHz oxygen line. Oxygen only, and oxygen with 7.5 gm/m3 surface water vapor are shown together to demonstrate the effects of water vapor. The zenith opacity is significantly lower for stations above MSL than for those at MSL. This is because the stations at the higher elevations get above the heavy absorption in the lower atmosphere. The zenith opacity for these stations can be written as, rhmax T (h ,h ' max' o ) k(z')dz' = (4.43) h o where h is the elevation of the station and h is the maximum extent o max of the atmosphere, which is taken to be 20 km here. Figure 4.23 shows the variation of zenith attenuation for the range of station elevations being considered in this study. The elements needed to calculate the zenith brightness temperature, by using the equation of radiative transfer, have now been presented. Chapter VI will treat the cosmic background radiation and the surface of the earth. These are the environmental constituents which will comprise the Tg(0) for this study. chapter, assume Tfi(0) is zero. For now and the remainder of the From Equation (4.38), the equation of radiative transfer for the zenith direction can be written as max w = T(z') e h k(z')dz' o 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.44) ZENITH RTTENURTION (DB) •b o 0& ONLY o o 0. Figure A.22. 10 . 20. 30. FREQUENCY (10. 50. 60. (GHZ) Zenith attenuation at MSL as a function of frequency. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 surface water vapor density with 2 km scale height included. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. STATION ELEVAT ON M 03 O z D <r ZD Z Ll ) I— t— d z UJ M I' 10. 15. 20. FREQUENCY Figure 4.23. 25. 30. 35. (GHZ) Variation of zenith attenuation with station elevation. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 water vapor density at MSL with 2 km scale height included. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where again h is the station elevation and h is the maximum height ° o max 6 of the atmosphere. This equation has been implemented in the Fortran program RADTRAN. RADTRAN uses the information provided by ATMOD, WVABSORB and 02ABS0RB to get the attenuation profile, t(z,ho), as well as brightness temperature. The manner in which the integration is done by RADTRAN can be explained with the aid of Figure 4.24. Finite sampling of the atmospheric model produces an atmosphere composed of spherically symmetric layers. The atmospheric parameters have to be assumed to be constant within a layer. For the zenith path, these layers appear flat as shown in the figure. Assume for this demonstration that the brightness temperature is desired at MSL, z=0. The atmosphere is divided into layers such that the boundary between layers is given by z=IA, where I is an integer. The atmospheric parameters within each layer are defined by sampling the t z = 3A V k3 z = 2A T 2 » Kg z =A A /2 T T, , k, 7 7 7 7 T 7 T 7 T r 7 7 ? > > / > / 77 7 7 7 / ?/J Figure 4.24. z =0 Schematic of the atmosphere for evaluation of the equation of radiative transfer. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. atmospheric model at every IA+A/2 and assuming the parameters are constant in the layer. is the physical temperature of the layer and the absorption coefficient routines are used to find kj for the layer. Under these conditions, the attenuation coefficient can be written as, J-l kjdz' kjdz' + t(z,0) L JI-1 1=1 (4.45) M-l where J is the layer containing z. Since the function k as implemented in Equation (4.45) is not a continuous function, the integrals can be evaluated resulting in, J-l t(z,0) = A kj + (z—Zj^1) kj . (4.46) 1=1 This is the formula used by RADTRAN to calculate zenith attenuation. Also, these conditions enable the following form of the radiative transfer equation to be written, M Tje T<z'’0) kj dz' = Y ~ *i t b (°) = V ~ fcl M Zj .ZI-1 (4.47) fcr where M is the highest layer of the atmosphere. Note that since the zenith path is being used, Tg(0) represents the brightness temperature 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where M is the highest layer of the atmosphere. Note that since the zenith path is being used, Tg(0) represents the brightness temperature at the surface and not the temperature of a source outside the atmosphere as it did when the s path was being used. The constant parameters allow the integral in Equation (4.47) to be evaluated. By substituting Equation (4.46) for x(z,0) in Equation (4.47), the integral appears as, T ' 1-1 -A .Z. k .- (z'- z_ -)k7 e kjdz' (4.48) I e 1-1 Taking the constant terms outside of the integral, #1 . Tjkj 1-1 -A .E, k.+ zT ,kT 1=1 l 1 -1 I . JI -z'k. dz' ' (4.49) 1-1 Evaluating the integral, *! = 1-1 -A .I- k. -AkT e 1-1 1 Tj (1 - e X) (4.50) By relating this equation back to the radiometric formula, Equation (3.8), the physical significance of this equation may be seen. Layer I -Ak is acting as a radiation source of strength, Tj(l-e ). Tj represents the amount of thermal power in the layer that is available for 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. radiation. The term U-e ) represents the absorption coefficient and f ) is by thermodynamic equilibrium and the Kirchoff-Planck law, Tj. U-e representative of the emission coefficient of the layer. The other term in Equation (4.50), 1-1 -A .Z. k. i=l 1 e , is the attenuation due to the layers between the source layer and the ground. Substituting Equation (4.50) into Equation (4.47) has the result, M r— Tg(O) = 2 _ I“1 -A .I. k. -Ak_ e 1 ^(1-e X) . (4.51) 1=1 Thus the total brightness temperature at the ground is the result of summing the contributions of each layer with the attenuation of the intervening layers included. This is the same interpretation given to the integral form of the radiative transfer equation. The difference is that here the source regions are layers and in the integral form the source regions are points. This interpretation of radiative transfer has been used in RADTRAN. Figure 4.25 shows the result of applying RADTRAN to calculate the zenith brightness temperature for various station elevations. The sampling 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the antenna temperature calculation to be performed, the brightness temperature must be known for all directions. directions that are near the horizon. This includes As was mentioned previously, refraction and the curvature of the earth affect microwave propagation along these directions. Therefore these effects must be addressed before a brightness temperature profile of the atmosphere can be generated. One concept that has been used to account for ray bending due to atmospheric refraction is the concept of the effective earth radius [24]. This concept assumes the ray path remains straight but the radius of the earth is changed to compensate for bending. Although only strictly valid when the refractive index varies linealry with altitude, the effective earth radius shall beapplied in this model to simulate refraction at low elevation angles. Consider the view of the earth that is shovn in Figure 4.26. In this figure, Re is the effective radius of the earth and z^ is the altitude of an atmospheric layer boundary as before. Radiation, incident from a direction defined by the elevation angle 0, must travel a distance Lj to reach the station. According to Ippolito [25], the effective radius of the earth at microwave frequencies is usually taken to be 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. STATION ELEVATION OKM cc => (X cc tn cn cc m M IS . 20. 25. FREQUENCY Figure 4.25. 30. (GHZ) Variation of zenith brightness temperature with station elevation. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 water vapor density at MSL with 2 km scale height included. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.26. Slant lengths in the earth's atmosphere. (4.52) R = 8500 km e and the path length is given by LT = R I e -sin(0) + 2 2zt fZ T l sin2(0) + ^— + R" e . ej which is derived in Appendix B. For 0 in the range, 1O°<0<9O°, and Zj« Rg, this reduces to 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.53) ^1 (4.54) sin(0) Equation (4.54) is the cosecant law that was discussed earlier. The manner in which Equation (4.54) is obtained from Equation (4.53) is also shown in Appendix B. The role that the curvature of the earth plays in affecting propagation at low elevation angles can be seen in Figure 4.27. The horizon observation path is drawn because it presents the worst case for a perfectly spherical earth. Note that the distance along the path in a given layer is no longer constant but a function of the altitude of the layer. The Aj for each layer can be found by applying Equation (4.53) in, Aj = Lj. - Lj _2 (4.55) ZENITH HORIZON Figure 4.27. Horizon observation path. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the horizon path, the length L^. can be obtained from Equation (4.53) by setting 0=0°. This gives L. ’I 0= 0 ° (4.56) Substituting Equation (4.56) into Equation (4.55) results in (4.57) For the spherical layers of uniform A spacing, Equation (4.57) can be manipulated into / 2 R 2 R e (I-DA + 1 (4.58) This form shows why the incremental path lengths for slant paths are longer in the lower atmosphere than for the zenith path. In the upper atmosphere, I-*» and Equation (4.58) shows that Aj.-frA, so the incremental path lengths for the horizon path would be smaller here than in the lower atmosphere. However, it is the lower atmosphere which significantly affects microwave radiation for the frequencies under consideration, and opacity is larger along the horizon path. For perspective, a plot of Aj up to 1=2000, is included in Figure 4.28 for R =8500 km and A=10 m. e 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The horizon opacity for the spherically symmetric atmosphere can be written as, M T. horizon Y (4.59) ki4i I=J where J is the layer the station resides in and M is the maximum layer. RADTRAN uses Equation (4.59) to calculate horizon attenuation. Figure 4.29 is an example of the calculation for various station elevations. Note that the horizon attenuation is 2 orders of magnitude larger in dB than the zenith attenuation shown in Figure 4.23. Similarly, A^ may be substituted for A in the expression for zenith brightness temperature (Equation (4.51)) in order to obtain the horizon brightness temperature. This results in 1-1 (4.60) I=J which is also implemented by RADTRAN. layer that the station resides in. Again, in this equation, J is the Figure 4.30 shows the horizon brightness temperature as calculated by RADTRAN, for station elevations in the range of interest. Knowing ^ as a function of elevation angle indicates a method for obtaining the brightness temperature for all directions. A_f as a 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. | ' “*T ■ 1 ■ 1 i l 1 LD- D i %; INCREMENTAL i 1 PRTH LEN G TH (KM) J'- •. . D 1 I i i i l i • I -L 1* ■■ 1 1 D - nP Figure 4.28. ‘ IM I 11 i n? i i 11 i n3 i i Incremental ray path length due to earth curvature. Horizon direction for R =8500 km and A=10 m. e 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i iu 10* I •b OV CO- STATION ELEVATION ST- (DB) co- OKM cm - IKM HORIZON R T T E N U R T 1 ON 2KM 3KM 4KM 5KM co- O ioS'CO- 0 Figure 4.29. 5. 10 . 15 20. 25. FREQUENCY (GHZ) 30. 35. 40. Variation of horizon attenuation with station elevation. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 water vapor density at MSL with 2 km scale height included. 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (K) I TEMPERATURE STATION ELEVATION OKM HORIZON BRIGHTNESS \4KM 15. 20. 25. 30. F R E Q U E N C Y (GHZ) Figure 4.30. Variation of horizon brightness temperature with station elevation. U.S. Standard Atmosphere, 1976 and 7.5 gm/m3 water vapor density at MSL with 2 km scale height. 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. function of elevation angle, can be substituted into the summation of Equation (4.60) and the calculation performed for each desired observation path. Figure 4.31. This has been done for a range of elevation angles in A disadvantage with obtaining the brightness temperature profile in this manner is the amount of computational time required to perform the integration. Calculation of the antenna temperature of an antenna with a complicated or finely structured pattern, would require the brightness temperature to be calculated for a large number of observation paths. Thus, the total time required to generate the brightness temperature profile could be prohibitive, even though the time to calculate a single path is insignificant. Therefore, a method was sought which would allow rapid calculation of the brightness temperature profile. A method has been developed which calculates the brightness temperature, for almost all observation paths by applying the mean temperature of the atmosphere at zenith to the other angles. RADTRAN is used to calculate the zenith opacity, t (90°), and the zenith brightness temperature, Tg(90°). These two known quantities are then used in the radiometric formula to find the unknown, which is the zenith mean temperature, T (90°) TM = l e -T(90°) ' (4.61) With Tm at zenith known, the brightness temperature for other angles can be found by the radiometric formula and by assuming that the mean 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I P A R A M E T E R - E L E V A T IO N ANGLE IT) TEMPERATURE (K) CD- OJ- 00- f— oLn- 01' BRIGHTNESS OJ O co coCO LO 20 30 CO 60' o 0. 5. 10 . 15. 20 . FREQUENCY Figure 4.31. 25. 30. 35. (GHZ) Brightness temperature profile of the atmosphere from MSL. Data obtained by integration along the indicated observation paths. U.S. Standard Atmosphere, 1976 with 7.5 gm/m3 surface water vapor density and 2 km scale height. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. temperature for the other observation paths does not differ greatly from the zenith mean temperature. This will be a good approximation, if the mean temperature of the atmosphere is not strongly dependent on the elevation angle, because x(0) = t t (90°) c s c (0) obeys the cosecant law, . 0>1O° rs»» Therefore, using the zenith mean temperature, TM, for angles where the cosecant law is valid, TB<0) = TM \} ~ e_T(90°)csc(e)] 0>1O° or e-T(9O°)csc(0) Tb (0) = Tb (90°) 1 _ e-*(90°) 0>1O° (4.62) This approximate technique gives good results as will be seen later. The advantage of using it is that integration along each observation path has been replaced by an algebraic equation. This algebraic equation will allow for a more rapid calculation of the brightness temperature profile. For the region where the cosecant law breaks down, an interpolation scheme is used. Numerical experimentation showed that the calculation shown in Equation (4.62) held up well at 5° when compared with Smith's data. However, Equation (4.62) failed to reproduce the curve of Smith 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. at 0°. Therefore, an interpolation scheme is used between the horizon and 5° to obtain the brightness temperature. RADTRAN is used to calculate the brightness temperature at the horizon and Equation (4.62) is used to get Tg(5°). The brightness temperature for the points in between are found from, - e/ 5 c In t b (0°) o°<e<5° in[rB (e)] = In [rB <0°)] (4.63) In Tb (5°) The approach of extending the zenith mean temperature to other observation angles in Equation (4.62), and the interpolation of Equation (4.63), are implemented in the Fortran program ENVIR. With the capabilities of ENVIR coupled with RADTRAN, it is possible to produce, without excessive calculation, the brightness temperature profile of the atmosphere that has been sought. Figure 4.32 is an example of a brightness temperature distribution for the 1-40 GHz frequency range. For comparison, the data of Smith is reproduced in Figure 4.33. The agreement between the curves of Figure 4.32 and Figure 4.33 is very good. The shape of the curves is the same between the two figures and the amplitude of the water vapor line is reproduced. The level of the tails of the absorption line, away from the center frequency, are slightly lower in Figure 4.32 when compared to Smith. This difference is probably due to the slight differences in the water vapor absorption coefficient that were noted in Section C of this chapter. Comparison of Figure 4.32 with the numerically integrated data of Figure 4.31 is also very good. The curves are identical for a 0° 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. elevation and for all elevations starting with 10° and greater. error can be seen in the interpolated data between 0° and 5°. Some This error may have arisen because the 5° curve is in error, due to extending the cosecant law, and that data is used as a seed in the interpolation. Hence, the error gets carried through all of the data. However, because of the difficulty in describing horizon effects in general, this error is relatively insignificant for the application being considered for the model. This scheme for generating the brightness temperature profile has been adopted for the environmental model. Table 3 is presented to summarize the methods used to obtain the brightness temperature profile of the gaseous atmosphere. The last major constituent that contributes to the brightness temperature of the atmosphere without rain, is condensed water in the form of clouds and fog. P. A model for their effects is discussed in the next section. CLOUD AND FOG ABSORPTION Clouds and fog occur in the atmosphere of the earth with enough frequency that they should be included in an antenna temperature model. Clouds and fog are composed of drops of liquid water that usually have diameters which are less than 0.1 mm. Therefore the scattering effects are negligible in the 1-40 GHz frequency range of concern here. Absorption is the major effect. Measurements of cloud attenuation have 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PA R A M E T E R - ELEVATION ANGLE o GO in TEMPERATURE (K) cn CM- O ). 00 CO in CD BRIGHTNESS cu O cn- 00 CO 30 CD 60 CM 90' 0. Figure 4.32. 5. 10 . 15. . 20 FREQUENCY 25. 30. 35. 40. (GHZ) Brightness temperature profile of the atmosphere from MSL. U.S. Standard Atmosphere, 1976 with 7.5 gm/m3 surface water vapor density and 2 km scale height. 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY C H l Figure 4.33. Sky noise temperature, or brightness temperature as calculated by Smith [17]. U.S. Standard Atmosphere, 1976 with 7.5 gm/m3 surface water vapor density with 2 km scale height included. TABLE 3 METHODS USED TO CALCULATE THE BRIGHTNESS TEMPERATURE OF THE ATMOSPHERE WITHOUT RAIN ELEVATION ANGLE METHOD 0=0 RADIATIVE TRANSFER O<0<5 INTERPOLATION 5<0<9O 0=90 EXTENSION OF THE ZENITH MEAN TEMPERATURE RADIATIVE TRANSFER Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shown that the noise temperature of the clouds can dominate the brightness temperature profile [14]. Therefore the designer should be provided with the means to calculate antenna temperature under conditions of clouds or fog. The difficulty in modeling clouds for this application is in the large number of cloud types which can occur. Additionally, more than one type may be present or several layers of clouds may exist in the structure of the atmosphere. Also the clouds may be stratus clouds, meaning continuous in an atmospheric layer, or they may be individual clouds, randomly located. For these reasons the cloud model chosen implemented here is general with respect to cloud type and location. An absorption coefficient for clouds has been used by Slobin [26] for generating statistics on cloud attenuation. His expression is u 1rtO.Ol22(29l-T)-l , k - M X 10 X 1'16 cloud " J (L W (4‘63) where M is the cloud water particle density in gm/m3, T is the cloud temperature in kelvin, X is in cm and kcloU(j is in NP/km. The cloud water particle density is what differentiates between the types of clouds. Slobin provides a table of typical fog and cloud models for mid-latitude conditions. The table lists 14 cloud types ranging from moderate fog with M=0.02 gm/m3 to cumulus clouds with M=1.00 gm/m3. The absorption coefficients for these two types are plotted in Figure 4.34. Slobin also recommends that since clouds are highly variable, other 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sources of cloud information should be used in calculating absorption from clouds. Upon this recommendation, the cloud model used here does not contain cloud parameters for typical situations. Instead, it is structured to allow the user to describe the cloud conditions. The user supplies the model with the cloud particle density, the altitude of the base of the cloud layer and the altitude of the ceiling of the cloud layer. With this information the Fortran program, CL0UD_ABS0RB calculates the absorption coefficient of the cloud according to Equation (4.63). The temperature is assumed to be the same temperature of the atmospheric layer obtained from ATMOD. When a cloud is present in an atmospheric layer, the total absorption coefficient is k = kH20 + k02 + kCLOUD • (4.64) Thus, when the environmental model is incrementing through the atmosphere to generate the absorption coefficient profile, it tests to see if a cloud is present. included. If it is, the absorption of the cloud is If no cloud is present, ^clo UD^" subsequent absorption coefficient profile is submitted to the radiative transfer program and the brightness temperature profile is calculated as described in the previous section. The number of cloud layers that can be included is not limited, but this model can only handle stratus clouds consisting of layers of 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (DB/KM) <w R B S Q R P T I ON o o o 0. s. 10 . 15. 20 . FREQUENCY Figure 4.34. 25. 30. 35. (GHZ) Typical range of absorption due to clouds. Cumulus cloud with M=1.00 gm/m3 and moderate fog with M=0.02 gm/m3 at T=280 K. 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. continuous clouds having the same water content. Simulation of widely separated single clouds is not possible at this time. As a worst case example of the effects of a single cloud layer on the brightness temperature, consider Figure 4.35. This figure is for the same conditions as Figure 4.32 except that a cloud layer with M=1.00 gm/m3 has been included from 0.66 km to 2.7 km. The differences between Figure 4.32 and Figure 4.35 demonstrate the effect of clouds on the brightness temperature. This concludes the description of the brightness temperature profile of the atmosphere when rain is not present. The treatment of rain is the next major topic to be addressed. G. SUMMARY This chapter has described the various components of the environmental model that are used to generated the brightness temperature profile of the atmosphere in the absence of rain. The constituents of the atmosphere, which significantly affect microwave propagation, were identified and the implementation of their effects into the model was described. The program ATMOD was written to provide the distribution of the atmospheric parameters with altitude. The data base used by ATMOD is the U.S. Standard Atmosphere, 1976 [21] which gives typical values for mid-latitude regions. Other atmospheric models are available in the literature [22] and can be implemented by changing the data base in 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P A R A M ETER - ELEVATION ANGLE O) CO' (O' (O' BRIGHTNESS TEMPERRTURE (K) (O' cu CD (O' CD (O O eo r~ (O (O' (M' 0. Figure 4.35. 5. 10. 20. 15. FREQUENCY 25. 30. 35. U0. (GHZ) Brightness temperature profile of the atmosphere from MSL. U.S. Standard Atmosphere, 1976 with 7.5 gm/m3 surface water vapor density and 2 km scale height. Cloud layer with M=1.00 gm/m3 present from 0.66 km to 2.7 km. 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ATMOD. Vater vapor density is included in ATMOD by assuming an exponential variation of the density with altitude. The user specifies the surface density and the scale height. The program WABSORB was written to generate the absorption coefficient of vater vapor. Vaters [13]. The expressions that are used are those of Good agreement between the results calculated by WVABSORB and those published by Waters were achieved. The program 02ABS0RB was written to calculate the absorption coefficient of oxygen. Rosenkranz [18]. This program uses the equations given by There was excellent agreement with data published by Rosenkranz and by Smi th [17]. The brightness temperature profile was obtained by using radiative transfer in the program RADTRAN and by methods contained in the program ENVIR. The brightness temperature profile which was used as an example is nearly identical to the radiative transfer data of Smith [17]. Finally, CLOUD_ABSORB was written to account for the possible presence of clouds and fog in the atmosphere. coefficient used is that of Slobin [26]. The absorption The program cannot model individual clouds, but instead simulates stratus clouds. These programs give the user flexibility in describing the atmospheric environment and the agreement with previously published results validates the numerical models. Therefore the programs discussed in this chapter should contribute to a reasonable and versatile model for the calculation of antenna temperature. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER V THE ATMOSPHERE WITH RAIN A. INTRODUCTION The effects of rain on microwave propagation have received a great deal of attention since the advent of satellite communications [27-29]. Early systems made use of the frequencies in the atmospheric transmission window where the effects of the clear atmosphere were minimal. However, during a severe rain event, the satellite link, would suffer significant degradation and an increase in system noise. Therefore systems had to be designed with enough margin to allow operation of the link to continue during these events. Research into the effects or rain was motivated by the desire to anticipate the appropriate margin for the link. The result of years of this research is that numerous rain models are available. For this study, these models were reviewed and judged on their applicability to the type of antenna temperature model that was desired. A choice was made and that model was adapted to fit within the framework of the antenna temperature calculation. In this chapter a brief description of the effects of rain on microwave propagation and how they are modeled is given. The reasons behind the choice of a model are given and the implementation of that model is described. 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B. A BRIEF INTRODUCTION TO RAIN AND RAIN MODELS Rain is a characteristic of the lower atmosphere. It begins as atmospheric conditions allow water vapor to condense on some particle that is present in the atmosphere. The conditions allow condensation to continue and a raindrop is formed. As the drop achieves the mass necessary to allow gravitational forces on the drop to overcome the convectional forces that keep it suspended, the drop will fall. Since the conditions that produced the drop are likely to be present over an appreciable volume of the atmosphere, many such drops will form and fall to the earth as rain. Experimental data has shown that the size of raindrops can be related to the rate at which rain falls. the drop size distribution. This relationship is known as There are three distributions that are commonly used in rain models, the Laws and Parsons [30], Marshall-Palmer [31] and Joss thunderstorm and drizzle [32]. There are many distributions because rain is a complex meteorological condition and the drop size distributions can vary for the same rain rate and rain type at different locations. The rain rate can also be used to describe the spatial distribution of rain intensity. Personal experience of rain and scientific measurements of rain rate show that the intensity of rainfall associated with a rain event can be nearly constant over large geographical areas or vary rapidly between two nearby points. Again this is a result of the complex meteorological processes that cause rain. On the other hand, measurement of the vertical profile of rain shows that the rain 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rate remains constant to a certain height and then falls off rapidly [34]. This height is the 0°C isotherm which is about 3.5 km for the continental U.S. although it has a seasonal variation. The classical approach to the study of the effects of rain on radiovave propagation is to solve the problem of a single rain drop in the presence of a dynamic electromagnetic field [35-37]. Vater is a lossy dielectric and the absorption and scattering cross sections of the drop are desired. Solutions to the problem may be obtained by approximating the shape of the drop as a sphere or as an oblate spheroid, and applying Mie scattering theory [38-41]. The result of the single drop is then applied to a unit volume of rain by assuming a drop size distribution such as one of the ones mentioned earlier. This produces the specific attenuation for that volume of rain. Specific attenuation has the units of attenuation per unit length. Thus if the total effects of rain are to be known, then the distribution of the specific attenuation must be known over the path taken by the radiation. A rain model may take a deterministic approach to this problem by assuming the specific attenuation distribution is known. This would be accomplished through specification of the rate rate along the path. However, given the complex nature of rain, this type of model is of limited use for practical engineering applications [33]. An alternate approach to a theoretical distribution of rain rate is a statistical distribution based on empirical data. Typically these begin with a local rain rate chosen on the basis of a cumulative probability distribution, so that the end result is the likelihood that 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a given attenuation will be exceeded. The horizontal variation in rain rate may be accounted for by using statistical data for these variations. Because of the practicality of such a model for system design, a similar approach will be used for the antenna temperature model. Use of a statistical model for antenna temperature will produce probability distributions that will guide the system designer. Determination of the probable increase in the system noise due to rain will enable the designer to choose the correct margin for the communications link. The disadvantage to using a statistical rain model for an antenna temperature calculation, is the difficulty in including scattering by rain. Brightness temperature calculations, which include scattering, have been done by assuming a specific, deterministic storm model [42]. However, models which extract brightness temperature information from the statistics of rain are apparently unavailable. Therefore, in this study, scattering will be neglected. This will allow a statistical rain model to be used to generate statistics on brightness temperature. For this application, neglecting scattering is seen as a less serious assumption than assuming a deterministic model of a storm. Ippolito et al. [43] present a summary of six statistical models. These models were evaluated and the Global Model of Crane was chosen for the antenna temperature calculation. A discussion of the Global Model and the adaptation of it for use in the antenna temperature calculation are the subjects of the next section. 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C. THE GLOBAL MODEL OF RAIN ATTENUATION The model used in this study to predict the attenuation due to rain is the Global Model of Crane [44]. This model estimates the annual probability distribution of attenuation for a specified propagation path. By using the model for the range of elevation angles from horizon to zenith, the expected attenuation can be found for all paths from the antenna. No azimuth variation in rain will be included. The brightness temperature profile may be obtained from the attenuation information using a method to be discussed later. In this manner the antenna temperature probability distribution can be found. The Global Model depends on extensive empirical data from around the world to estimate the probability distribution of surface point rain rate. Similarities in climates around the world allowed the world to be divided into eight broad climate regions based on rain rate distributions. In defining the regions, Crane took into account expected variations in terrain, storm type, storm motion and atmospheric circulation which may affect each local climate. Figure 5.1 is a world map showing the correspondence between rain rate climate regions and geographical location. Since the Global Model was first introduced, more data has become available for North America and Europe. This has allowed these continents to be further subdivided into more rain climate regions. Figure 5.2 shows the more detailed map of North America and Europe is shown in Figure 5.3. 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. RAIN RATE CLIMATE REGIONS POLAR: 0A 0 0 TEMPERATE: Tundia (Dfy) 0C Telga (Modetale) 0 0 SUB TROPICAL H e Maritime Conllnenlal Q p TROPICAL wei B lo Arid B ll Modefale Wei HI? 3 ZT CD CD T 3 o Q. O 3 T-5D vO « o CD Q. 3 " O c ■o CD LONGITUDE (Deo) ( /> V) o' 3 Figure 5.1. Global Model rain rate climate regions for the world. Crane [44]. From O Q Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.2. Global Model rain rate climate regions for the continental United States and southern Canada. From Crane [45]. Figure 5.3. Global Model rain rate regions for Europe. 145]. From Crane 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The point rain rate probability distributions for each region as published by Crane are available in tabular and graphical format. Table 4 contains the numerical values of the distributions and Figure 5.4 contains plots of the distributions. The Global Model accounts for variations of rain intensity along the path by finding an effective path average factor r, defined by where Rp is the point rain rate and R is the path averaged rain rate. The path average factor must account for vertical variations as well as horizontal variations along the path. The vertical variations along the path are due to latitudinal and seasonal changes in the 0°C isotherm. Also, the intense rain cells characteristic of heavy rains can carry rain drops up to the -5°C isotherm. The Global Model assumes that the rain rate remains constant to the 0°C isotherm but uses an algorithm to determine the effective height of the isotherm for a given probability of occurance. Figure 5.5 displays the 0°C isotherm height calculated in this manner for several probabilities. The horizontal variation of rain rate in the Global Model uses a power law expression, r = y(D)Rp6(D) (5.2) 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE 4 VALUES FOR THE GLOBAL MODEL POINT RAIN RATE DISTRIBUTIONS FROM IPPOLITO ET AL. [43] RAIN CLIMATE REGION Percent o f Year A B1 ' B B2 C °1 d>d2 °3 E F G H Minutes Hours per per Year Year 0.001 28.5 45 57.5 70 78 90 108 126 165 66 185 253 5.26 0.09 0.002 21 34 44 54 62 72 89 106 144 51 157 220.5 10.5 0.18 O.OOS 13.5 22 28.5 35 41 50 64.5 80.5 118 34 120.5 178 26.3 0.44 0.01 10.0 15.5 19.5 23.5 28 35.5 49 63 98 23 94 147 52.6 0.88 0.02 7.0 11.0 13.5 16 18 24 35 48 78 15 72 119 105 1.75 0.05 4 .0 6.4 8 .0 9 .5 11 14.5 22 32 52 B.3 47 8 6 .5 263 4.38 0.1 2.5 4 .2 5 .2 6 .1 7 .2 9 .8 14.5 22 35 5.2 32 64 526 8.77 0 .2 1.5 2 .8 3 .4 4 .0 4 .8 6 .4 9 .5 14.5 21 3.1 21.8 4 3 .5 1052 17.5 0 .5 0.7 1.5 1 .9 2 .3 2 .7 3 .6 5.2 7.8 10.6 1.4 12.2 2 2 .5 2630 43.8 1.0 0.4 1.0 1 .3 1 .5 1 .8 2 .2 3 .0 4.7 6 .0 0.7 8 .0 12 .0 5260 87.7 2 .0 0.1 0.5 0 .7 0 .8 1.1 1.2 1.5 1.9 2 .9 0.2 5 .0 5 .2 10520 175 5 .0 0.0 0.2 0 .3 0 .3 0 .5 0 .0 0 .0 0 .0 0 .5 0.0 1.8 1.2 26298 438 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I a) A LL REGIONS BAIN RATE (mmfh) 150 100 50 -A 0 PERCENT OF YEAR RAIN RATE VALUE EXCEEDED b) SUBREGIONS OF THE U S A RAIN RATE (mm/h) REGION 100 50 0 0.001 0.01 0.1 1.0 10.0 PERCENT OF YEAR RAIN RATE EXCEEDED Figure 5.4. Global Model point rain rate distributions as a function of the percent of year the rain rate is exceeded. From Crane [44]. 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 5 4 PRO BABILITY OF .OCCURRENCE UJ 3 0 .001% LU 2 0 .01% 1 0 .1% 0 10 20 30 50 40 60 70 L A TIT U D E (DEG) Figure 5.5. Global Model effective 0°C isotherm height. [44]. From 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where y(D) and 5(D) are parameters that best fit r to the empirical data. Figure 5.6 shows the variation of these parameters with D. The Global Model uses a theoretical model to determine the specific attenuation from the point rain rate and then finds the total attenuation along the path using the path average factor. The rain rate is related to the attenuation by k' = a RpP (5.3) where k' is in dB/km and Rp is in mm/hr. The parameters a and B are found by assuming a Laws and Parson drop size distribution in the solution of the raindrop scattering problem. Table 5 lists these parameters as a function of frequency for horizontal and vertical polarization and 20°C drop temperature [41]. With this statistical and theoretical information, the attenuation is found from, A(Rp ,D) = a Rp uPd e -1 uB bPecPd cB bPecpD cB d<D<22.5 km (5.4) or A(Rp,D) = a RpK [euPD-l‘ . uB . 0<D<d 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.5) 5 o o l ___________ 0 5 L 10 16 20 D - BASAL PATH LENGTH (km) 0 .4 Z Ui Z 8 02 UI 0.1 0 5 10 IB 20 D - BASAL PATH LENGTH (km) Figure 5.6. Global Model path averaging factors. From Crane [44]. 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE 5 SPECIFIC ATTENUATION PARAMETERS FOR A 20°C DROP TEMPERATURE AND A LAVS AND PARSON DROP SIZE DISTRIBUTION (FROM [41]) a Frequency Horizontal (GHz) Polarization 1.0 2.0 4.0 6.0 8.0 10.0 12.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0 120.0 150.0 200.0 300.0 400.0 0.0000387 0.000154 0.000650 0.00175 0.00454 0.0101 0.0188 0.0367 0.0751 0.124 0.187 0.263 0.350 0.442 0.536 0.707 0.851 0.975 1.06 1.12 1.18 1.31 1.45 1.36 1.32 0 Vertical Polarization 0.000035 0.000138 0.000591 0.00155 0.00395 0.00887 0.0168 0.0347 0.0691 0.113 0.167 0.233 0.310 0.393 0.479 0.642 0.784 0.906 0.999 1.06 1.13 1.27 1.42 1.35 1.31 Horizontal Polarization 0.912 0.963 1.120 1.31 1.33 1.28 1.22 1.1 1.10 1.06 1.02 0.979 0.939 0.903 0.873 0.826 0.793 0.769 0.753 0.743 0.731 0.710 0.689 0.688 0.683 Vertical Polarizat: 0.88 0.923 1.07 1.27 1.31 1.26 1.20 1.13 1.07 1.03 1.00 0.963 0.929 0.897 0.868 0.824 0.793 0.769 0.754 0.744 0.732 0.711 0.690 0.689 0.684 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where A is in dB. u = The coefficients, ln[becd] b = 2.3 R, (5.6) -0.17 (5.7) c = 0.026 - 0.03 In d = 3.8 - 0.6 In (5.8) Pp] (5.9) Pp] are empirical constants used by the Global Model. The distance D, is the horizontal projection of the path distance between the station and the 0°C isotherm as shown in Figure 5.7. Mathematically, D is given by 9>10c f(H-Ho)/tan(0) D = (5.10) Re«j>, \p in radians 6<10e where H is the 0°C isotherm height, Hq is the station height, 0 is the elevation angle and • -1 ip = sin COS0 (H+Re) |^Ho+Rg}2 sin2(0)+2Ro(H-H^)+H2-H? - (H^+R^)sin(0) e o' o e (5.11) 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. H 0 ° ISOTHERM f t -S U R F A C E (a) e>10° Local Flat Earth Approximation H * (b) e<io° Earth Curvature Included Figure 5.7. Horizontal projection of slant paths for the Global Model. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where R =8500 km is the effective radius of the earth. e The derivation of t|/ is shown in Appendix C. Finally, the model calculates the total attenuation along a slant path from the attenuation of the projected horizontal path by As = <5,12) where D/cos(0) 0>1O° (5.13) L = .J(Re+ Ho)2+(Re+H)2-2(Re+Ho)(Re+H)cos(i)0 0<1O°. which is also shown in Appendix C. Upon request, Prof. Crane generously provided a computer code listing of the Global Model to this study [46]. A Fortran version of the model, GRAM, was written from the original listing which was in PASCAL. Figure 5.8 shows two sample calculations from GRAM. The Global Model allows choice of climate zone, slant or terrestrial path, frequency, latitude or path length, elevation angle, station height and polarization. The output consists of the attenuation (ATTEN) in dB, rain rate (RAINR) mm/hr, 0°C isotherm height (HEIGHT) km, horizontal projection (DISTANCE) km, and the probability (PTOTAL) that the rain rate will be exceeded in percent. The choice of the Global Model for the antenna temperature calculation was made for several reasons. First, it is a statistical 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CLIMATE ZONE - D1 SLANT PATH FREQUENCY - 14.500 GHZ LATITUDE - 43.500 DEGREES ELEVATION ANGLE - 15.000 DEGREES STATION HEIGHT 0.400 KM LINEAR POLARIZATION 63.000 DEGREES FROM HORIZONTAL « ATTEN RA1NR HEIGHT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.062 0.178 0.373 0.661 1.061 1.595 2.287 3.168 4.275 5.656 7.371 9.492 12.078 15.146 18.650 22.487 26.554 30.768 35.078 39.459 0.329 0.694 1.185 1.829 2.656 3.706 5.028 6.690 8.780 11.421 14.776 19.048 24.435 31.053 38.856 47.660 57.244 67.430 78.1C6 89.233 0.961 1.130 1.299 1.468 1.637 1.806 1.975 2.144 2.313 2.482 2.651 2.820 2.989 3.158 3.327 3.496 3.665 3.834 4.003 4.172 DISTANCE PTOTAL 3.585 4.215 4.846 5.477 6.108 6.739 7.369 8.000 8.631 9.262 9.893 10.523 11.154 11.785 12.416 13.047 13.678 14.308 14.939 15.570 4.95303 3.14873 2.00171 1.27252 0.80896 0.51427 0.32693 0.20784 0.13213 0.08399 0.05340 0.03395 0.02158 0.01372 0.00872 0.00554 0.00352 0.00224 0.00142 0.00091 DISTANCE PTOTAL 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 22.500 4.95303 3.14873 2.00171 1.27252 0.80896 0.51427 0.32693 0.20784 0.13213 0.06399 0.05340 0.03395 0.02158 0.01372 0.00872 0.00554 0.00352 0.00224 0.00142 0.00091 CLIMATE ZONE - D1 TERRESTRIAL PATH FREQUENCY - 14.500 GHZ PATH LENGTH - 23 .000 KM CIRCULAR POLARIZATION * ATTEN RAINR HEIGHT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1.475 2.093 2.738 3.444 4.231 5.119 6.128 7.285 8.625 10.193 12.048 14.252 16.853 19.851 23.182 26.743 30.441 34.210 38.019 41.861 0.344 0.715 1.213 1.864 2.701 3.763 5.101 6.781 8.894 11.565 14.961 19.282 24.727 31.405 39.261 48.109 57.726 67.937 78.637 89.785 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Figure 5.8. Example calculations of the Global Model. 132 with permission of the copyright owner. Further reproduction prohibited without permission. model which is necessary because of the temporal and spatial variation in the characteristics of rain. These characteristics are such that they cannot be modeled adequately using a deterministic analysis. Second, the world wide applicability of the Global Model complements the atmosphere without rain model developed in the last chapter. The various atmospheric models that can be defined for different climates in that model are similar to having the rain climate regions in the Global Model. For these reasons the Global Model should provide adequate design information on attenuation and antenna temperature during rain. The modifications needed to enable the program GRAM to produce the brightness temperature profile of the atmosphere with rain, are covered in the next section. D. USE OF THE GLOBAL MODEL FOR BRIGHTNESS TEMPERATURE CALCULATION Only a few modifications of the Global Model code, GRAM, were required to produce a brightness temperature profile. made, were made simply for convenience. Those that were The first was a modification that allows the attenuation to be calculated for a range of elevation angles instead of a single angle. This modification allows an attenuation profile to be generated. The second modification allows the user to choose either the rain rate or the probability. The quantity that is not specified is then calculated by the code. The environmental model approximates the brightness temperature due to rain by using the Global Model generated attenuation in the radiometric formula, 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t b <0> = tm (5.14) [1-10 A(e)/10)] where A(0) is the attenuation in dB and rain. is the temperature of the Since the specific attenuation parameters, a and P, being used are for a rain temperature of 20°C, TM is set equal to 293 K. This choise of TH is likely to produce higher temperature than would be experienced on the path. The choice of Tu as 293 K in this environmental model has been done to allow the model to predict high levels of brightness temperature to compensate for the approximations that were made. This method for obtaining the brightness temperature during rain i likely to be a coarse first approximation to the true value. more sophisticated methods are reserved for future study. However, A more rigorous, but also more difficult, approach has been suggested [47]. This approach would attempt to combine the precise definition of Tu for M the non-scattering case [14,15], with the horizontal rain statistics of Crane. For the current model, the coefficients a and P could be calculated for other rain temperatures and used in the Global Model if particular situation warranted their use. Again it must be noted that calculation of brightness temperature by Equation (5.14) ignores scattering into the path [42]. However, significant scattering occurs only at the top end of the 1-40 GHz range of concern here or at very heavy, low probability, rains in the lower end. From a system design point of view, this is a less serious approximation than using a deterministic rain model to get scattering 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. into the path. Rain is too complex a meteorological condition to allow a deterministic model to provide adequate design information on a world wide basis. This brightness temperature calculation also ignores the contributions of clouds and the atmosphere above the rain. In most cases rain will dominate the noise temperature and the contributions of the other constituents will be insignificant. For light rains, the contributions from clouds could be on the same order so some error may be incurred under those conditions [25]. This problem could be resolved through coupling the radiation transfer model with the Global Model. The proposed approach mentioned earlier would address this problem [47]. This however has not been done and is reserved for future implementation. The Fortran programs GRAM11 and GLMC contain the modifications just described. These programs were used to generate the brightness temperature profiles at 11 GHz shown in Figures 5.9, 5.10 and 5.11. profiles are for the parameters listed above the curves. The attenuation calculated by the Global Model is polarization sensitive because oblate spheroidal rain drops are assumed. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The E. SUMMARY This chapter has discussed the subject of modeling the effects of rain on microwave propagation, with the goal of finding a model that is suitable to be used in an antenna temperature calculation. The theoretical determination of the specific attenuation due to rain was accepted, but models based on storm profile assumptions were rejected as not being practical for this application. Instead models which use statistical analysis to evaluate path profiles were adopted. Of the statistical models available, the Global Model of Crane was chosen as the one most compatible with the goals of this study. The statistical modeling and global applicability of the model make it attractive for use in the environmental model for the antenna temperature calculation. The Global Model is a model for predicting the attenuation due to rain. Since the brightness temperature is required for the antenna temperature calculation, the brightness temperature must be obtained from the attenuation data when using the Global Model. the radiometric formula has been used. In this study, This formula is likely to be a coarse approximation to the brightness temperature because of the implicit assumption on the mean temperature along the path and because it ignores scattering into the path. This method for producing the brightness temperature is likely to predict a higher temperature than would actually be incurred in all but light rains. to the choice of the mean temperature. This is due mainly However, for a system designer, a method which predicts higher temperatures is more acceptable than a 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X) "O CD o Q. C o CD Q. -o CD Frequency = 11.00 GHz Station Height = 0.238 km Rain Rate = 1.00 mm/hr. Climate zone = D2 Latitude = 40°N Rain temperature = 20°C Probability = 2.8810% </> C/5 o' 3 a> o o ■a >< cq' Vertical Polarization Horizontal Polarization 3 CD > 7 3X ^ Ui X CD Ui Ui CD -o O Q. O 3 "O o Ui "sj <n <n in ui CD Q. CD CD ■o CD 0 .0 1 5 . 0 3 0 . 0 6 0 . 0 ELEVATION ANGLE 10EGREES) 7 5 . 0 9 0 . 0 0.0 1 5 . 0 3 0 . 0 6 0 . 0 7 5 . 0 ELEVATION ANGLE (0EGREES) Cfl o' 3 Figure 5.9. Brightness temperature profile at 11 GHz for 1 mm/hr. rain. 9 0 . 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission Climate zone = D2 Latitude = 40°N Rain temperature = 20°C Probability = 0.1357% Frequency = 11.00 GHz Station Height = 0.238 km Rain Rate = 12.7 mm/hr. Horizontal Polarization Vertical Polarization z > -J UJ x UJ (U u> 03 < =c 3 a tc b j a. z UJ UJ ► — <n UJ z X o t c ff i 0 .0 15.0 3 0 . 0 6 0 . 0 ELEVATION ANGLE (DEGREES) Figure 5.10. 7 5 . 0 9 0 . 0 o.o 1 5 . 0 3 0 . 0 6 5 . 0 _ _ _ ELEVATION ANCLE (DEGREES! Brightness temperature profile at 11 GHz for 12.7 mm/hr. rain. 7 5 . 0 9 0 . 0 7J "O CD — 5 o Q. C o CD Q. ■o CD Frequency = 11.00 GHz Station Height = 0.238 km Rain Rate = 25.4 mm/hr. Climate zone = D2 Latitude = 40°N Rain temperature = 20°C Probability = 0.0388% CO C/5 o' 13 of the copyright owner. Further reproduction prohibited without permission. Horizontal Polarization Vertical Polarization UJ UI UI VO Q_ CL UJ to UJ X tn 03 0.0 1 5 . 0 3 0 . 0 15.0 6 0 . 0 ELEVATION ANGLE (DEGREES) Figure 5.11. 75.0 9 0 . 0 o.o 15.0 3 0 . 0 15.0 9 0 . 0 ELEVATION ANGLE (DEGREES) Brightness temperature profile at 11 GHz for 25.4 mm/hr. rain. 7 5 . 0 9 0 . 0 method which predicts lower temperatures. A more sophisiticated method of generating the brightness temperature due to rain can be addressed in future work. In this version of the environmental model, the radiative transfer model and the rain model are kept separate. This means that the clouds and atmosphere above a rain event and gaseous absorption within the rain are not included in the antenna temperature calculation during rain. The effect of this omission is not expected to be significant except perhaps under light rain conditions. This problem can be addressed in future work. The computer code as supplied by Crane was slightly modified to allow brightness temperature profiles to be generated. were given of brightness temperature profiles at 11 GHz. Three examples These examples demonstrate the severe effect rain can have on system noise temperature. 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER VI NON-ATMOSPHERIC NOISE SOURCES A. INTRODUCTION The previous two chapters have developed the models for the atmosphere, the medium through which all radiation must travel to get to the antenna. The atmosphere is important to an antenna temperature 1 calculation because it is both an absorber and emitter of radiation. Knowing the absorbing properties of the atmosphere allows the amount of radiation that reaches the antenna from a source outside the atmosphere to be determined. The purpose of this chapter is to discuss sources which are outside the atmosphere and develop models for them. The structure of the atmosphere provides a natural classification of the outside sources. The atmosphere may be imagined as a hollow shell dividing the universe into two regions, a region outside the shell and one inside the shell. occupied by the earth. terrestrial sources. The region inside the shell is of course The sources here can be classified as Outside the shell is the rest of the universe and the sources here are extra-terrestrial or exo-atmospheric in origin. This chapter will discuss these sources according to these classifications. Additionally, the models used to represent them in the antenna temperature calculation will be presented. 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B. EXO-ATMOSPHERIC SOURCES The calculation of radiation incident on the antenna from exo- atmospheric sources is straightforward. The temperature of that radiation is simply given by Tb (9) = Ti(0)e"T(0) (6.1) where T^(0) is the temperature of the radiation incident on the atmosphere from the 0 direction and t(0) is the opacity of the atmosphere in that direction. In the 1-40 GHz range, the exo- atmospheric sources are generally discrete sources. Therefore the difficulty in this calculation is describing the position of the sources with respect to the antenna. The dynamics of the bodies within the universe make the position of the sources a function of time. Including them in an antenna temperature calculation would require a radio map of the sky for the time at which the temperature was wanted. This is not practical from a systems point of view. In most cases the sources are not of sufficient strength to significantly affect the antenna temperature. A weak source can only be seen by the main beam or one of the first few sidelobes of a highly directional antenna. For broad beam antennas or if the source is in a minor lobe region, the source will not be detectable. A source must be near the boresight of the high gain antenna to have any influence at all, and even then the influence is small. Therefore, including a radio 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. map of the sky for an engineering calculation of antenna temperature is not justified. One discrete source that could dominate antenna temperature is the sun. At microwave frequencies the equivalent blackbody temperature of 4 5 the radiation emitted by the sun is of the order of 10 -10 K for quiet sun conditions. When solar activity creates disturbances within the sun, the radiation can be 10-100 times higher. Thus the sun can determine the antenna temperature especially if it is near boresight. However, the daily and yearly variation of the position of the sun make including the sun in a general environmental model difficult. reason the sun is excluded from the model. For that The user may include the sun in the model without too much difficulty if it is known that the sun will be near the boresight. One source of exo-atmospheric radiation that is not discrete is the background radiation that is thought to be left over from the creation of the universe. This radiation has an equivalent blackbody temperature that is usually taken as 3 K for this frequency range. This temperature has a significant effect only at frequencies where the atmosphere is nearly transparent. The radiative transfer model and the rain model both have provisions to include exo-atmospheric temperature. The exo-atmospheric temperature is included as one of the inputs to the environmental model. This temperature is assumed to be incident on the atmosphere from all directions and thus it is included mainly to account for the background radiation. Modeling other sources, such as the sun, requires adding a 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. directional dependence to the temperature. This could be done for specific environmental conditions but it has been excluded from this version of the model. In the broadest sense, exo-atmospheric temperatures are only significant for antenna elevation angles near zenith. As the elevation approaches the horizon, atmospheric attenuation increases, making these sources less visible. Also, sources on the surface of the earth start to dominate antenna temperatures at near horizon angles. These sources are the topic of the next section. C. THE GROUND MODEL The region interior to the atmosphere is occupied by the earth. All objects on the surface of the earth have a temperature above absolute zero and therefore they emit thermal radiation. warm body so it also radiates. The earth is a Man made sources of emission, such as other communication links, are also present. All of these sources contribute to the antenna noise temperature. Including them in the environmental model would require a complete description of the man made and natural features of the antenna station. in a general model, they will be excluded. Since this is not possible The one terrestrial source that is common to all stations, the surface of the earth, will be included in the model. The radiation properties of the surface depend on the physical temperature of the surface, the roughness of the surface and the dielectric and conductive properties of the surface. The physical 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. temperature controls the thermal activity of the electrons in the surface material. The roughness and dielectric properties determine how well the surface radiates. The roughness also causes the surface to scatter incident energy and the dielectric properties give the surface a reflection coefficient. well. Thus the surface reflects incident energy as Note the conductivity can be accounted for by using a complex dielectric constant. Again, the problem of adequately describing local features of station environment appear. the Personal experience shows that terrain changes rapidly at some geographical locations. Generally, the roughness and dielectric properties are not constant within the line of sight of the antenna. Furthermore, the elevation of the local terrain can change rapidly away from the station location. For example, a station might be located in a valley or on a mountain overlooking a valley. Thus it is not possible to develop a general surface model. Assuming maximum ignorance of the local terrain of a station, a general model can be developed. study. This is the approach taken in this The surface of the earth will be assumed to be flat, smooth and homogeneous. The flatness and homogeneity assumptions should be good for antennas which are reasonably close to the ground. depends on the wavelength and the local terrain. The smoothness With a smooth surface, scattering is not present and the only effects are emission and reflection [48]. 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PO Figure 6.1. SKY Ground model for antenna temperature calculations. Under these assumptions the local terrain of the station would look like the drawing shown in Figure 6.1. From this figure it is desired to find the brightness temperature profile at a point, Q, which is a distance h above the ground. The angle 6 defines the elevation angle, so negative elevation angles involve the ground. a = 10| Let a be defined by 9<0° . (6.2) The brightness temperature at Q in the direction a is given by [48], 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tb(«) - Tpq<«) * (Tsky(«) |,R(«)|2 ♦ Tsurf(l-|aR(«)|2))e' P0 . (6.3) In this equation, Tpg is the temperature of the emission from the atmosphere between the two points and Tpg is the atmospheric attenuation between the two. is the radiation incident on point P and op is the reflection coefficient of the surface. Since the earth is assumed to be flat, the reflection coefficient is given by the Fresnel coefficients, cos(P) - [e -sin2 (P)l °r h ---------- p — :— is <6-4> cos(p) + |er-sin (P)J and ercos(P) - |^er-sin2(P)j °r v - --------- r — r - i (6-5) e r c o s ( e ) + [er - s i n 2(p)J where the H and V subscripts denote horizontal and vertical polarization. A plot of the magnitude squared of these coefficients as a function of elevation angle, for er=10 is given in Figure 6.2. The atmospheric models developed in the previous two chapters can be used to provide the atmospheric information needed to perform Equation (6.3). For the clear sky, radiative transfer is used to find Tsky’ TPQ am* XPQ‘ Tsky *s slmP1y t*ie brightness temperature profile at 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the surface, so the technique described in Chapter IV is used to find it. TpQ and Tpg are found fromdoing a radiative transfer calculation (Equations (3.2) and (3.4)) for the case whena=90°. The Tpg for other angles is found from the cosecant law and the Tp^ for other angles uses the mean temperature approach. For the clear sky calculation, Equation (6.3) is performed in the program ENVIR. As an example of a brightness temperature profile for a clear atmosphere with the ground model included, consider Figure 6.3. This curve is for horizontal polarization and for the environmental conditions listed above the curve. The different terms contributing to the brightness temperature for negative elevation angles are plotted separately. It is evident in the curve that the reflected contribution mirrors the sky profile for 0>O, except that the reflected term is weighted by the reflection coefficient. The antenna height of two meters means that atmospheric emission and attenuation between antenna and ground is small until the elevation angle approaches the horizon. At that point the path length through the atmosphere is approaching infinity so the atmospheric emission dominates and the atmospheric absorption attenuates the other contributors. spike in the curve at 0=0. This is the source of the The spike is a direct result of using the flat earth model and it always approaches the ambient temperature of the atmosphere. As the elevation angle approaches -90°, the atmospheric attenuation between antenna and ground is negligible and ground emission dominates the profile. 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ---------------------- H O R IZO N T A L P O L A R IZ A T IO N ----------------------- V E R T IC A L P O LA R IZ A T IO N ■ ■ ■ " i---...■ — . .. .. ..j._ ....'. ----- ...4 ------i_ _ _ ....... .. ......... _ _ _ _i .......1-.....! / | .-i_ _ _ _ :.. _ ...... *. - ...:------........ i ------1 -------:-------— J- I - \ ,----- - i....... ...— ; • ;L . . . . . _ _ _ i _ _ _ _ ....r ... - " ------1 -------j.------------:— /.:—J --------j------? -----.......f ~ . .......1 ~'i------------ - f - ■* I . •_ _ _ __i_ ; ... ------j------;-----------U - / ■ : ---------4 ----^ a z z iz ** .. "t - — ! 'L -...... . . |— 4 -. . . !„ : .-z ; ------------ 7 -------f-----.......:...........-j «_ _ _ _ _ _ _ .......;. . . . . . . . s±. —fc--i —H i -....-7 -...- ■ z in r _ .... 1 -----........ . .. . .. .. ■ z r z / . .:: H Z ..r j ' .....!------L -----.-- • -1 -1 ------:-------i------i----■i.....-4- j .....H ~-~: ... 1 —i. . . . . . . -; i:....w .....;..j ../ - / --------r— J i ■ ...‘ 4 ------1 : ri_ ------! ! -------j------__ _ _i. ... . . . i....... • • •i — i....- i f : 1 -. ...; -X. .. : ....-i- .....*-------. • ----------1 * — ... -----i—i - .... / ----4.. 4- - - 1 -----T? ; • -------i-------;------ ------f------ j ..... r ........... ------;----■ ....4— - j. . :.... . _ _ _ . .... _ _i_...... ■ .....— ■ ......... • J j -... ..._..„4- - - .- - - ... ....! _ j:_ .. ZZ] Z.ZZZ ;.... I . : 1 REFLECTION COEFFICIENT r-1 .. --------J -----A .. .. .. ... •_ _ __ _ _ _ :------ .------- .... --r...... •■~ ...— ;------ -....| POWER -. ; { 4- . _... . _ “ —■1 ..J.. _ _ i_ _ _ -.....-•....................- I.........J. . J . . . i. . . . . . . .--- - --- 1 . . - - - i .... ‘— . . .:_ _ _ - Z...-j-~ - -.........r...... -------i— ..... T ... : ,- - - ! -----1- - - .... .— .. . . .. _ i....... ------\ ...- ......i.......i...... ..--- —.. - -.-p— , ------ . „ 1 | ---1-- 1-- 1 -75.0 -60.0 -«5.0 ELEVRTION RNGLE Figure 6.2. .. / - .. i. - » .......■ ■ ■ ■/ : - 7 -i.......i f . .... ---- ...i.... r i -30.0 . ..■7 ,x ----.; .... y . 1 -15.0 1 0.0 (DEGREES) Power reflection coefficients of a flat earth with £^=10. 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY IGH2) - 11.00 H0R120NTAL P0LAR12RTI0N STATION HEIGHT (KM) - 0.23B ANTENNA HEIGHT. IM) - 2.000 EXO-ATMOS. TEMPERATURE (K) - 3.00 SURFACE TEMPERATURE !K) - 2B8.15 SURFACE PERMITTIVITY - 10.000.00 SURFACE H20 DENSITY IDM/M««3) - 7.50 •b BRIGHTNESS TEMPERATURE (KELVIN) o>00- *b cnCD- i * J i i i i i i j i i i 1 1 ; i i ! i ! ~CD en00r— ; ! S | I 1 i • i | i! j • i ii i ;! i ! i j i ■90.0 ! i • ! | : -60.0 / ' i/ : /--■ i / : 7 i i! wX 1^ i ; \ \ ! V \ / !'t / j \ j -j i TOTAL ' ‘ I REFLECTED I SURFACE EMISSION ATMOSPHERIC EM ISSIO N " i i ! ! ! ! <5 i i\ • i i !• NV 1. i il_J— :— ---- y N ^Ji h--------------- ■ *b ■ — —— —■ ----- ■ i | -30. 0 j • ! \ i U ’ | i i : !i I I 1 ! i i i 1 ! ; i | | n i ) L ( ---------- | i i x . : j i | . 1 1 i i ! i 0. 0 ELEVATION ANGLE Figure 6.3. I 1 j ! ii ii i i i I i i i ! i I i 30.0 i ! i? ! ii ! : i I , i J ! i i i } \ | j -------- | j j I j j i | 60.0 ; 1 | i ; j i i ! !i — 90. (DEGREES) Clear sky example of the brightness temperature profile at 11 GHz, horizontal polarization. 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6.4 shows the brightness temperature profile for the same conditions but for vertical polarization. Here, the temperature reaches a maximum equal to the surface temperature because the reflection coefficient goes to zero at the Brewster angle. This causes a hump in the brightness temperature profile of the ground not seen in the horizontal case. Note that both polarizations converge to the same temperature for 0=-9O° because the reflection coefficients converge on the same value. Also no difference is seen between the sky profiles of the two polarizations because the clear sky has no polarization dependence. The spike again appears on the horizon as a consequence of the flat earth model. The horizon spike appears as a slope discontinuity in the profile and the question is raised of whether nature actually behaves like this. It certainly appears that the profile could be smoothed in the horizon region to provide a continuous curve. The problem of doing this is in the difficulty of describing the actual physical processes which occur for a horizon path. Earth curvature certainly plays a part so the flat earth is only an approximation. If an antenna could actually see the horizon, a radiative transfer calculation would be needed for small negative elevation angles. This is because the antenna would be looking down through the atmosphere to the horizon and then back out through atmosphere to space. In any case, there is a discontinuity between atmosphere and surface at some point so perhaps a discontinuity in the temperature profile is justified. unobstructed views of the horizon. Alternately, antennas rarely have The objects obstructing the view of 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY (GHZ) - 11.00 VERT1CRL POLARIZATION STATION HEIGHT (KM) - 0.238 RNTENNR HEIGHT. (M) - 2.000 EXO-RTMOS. TEMPERATUR E (K) « 3.00 SURFRCE TEMPERATURE IK) - 288.15 SURFACE PERMITTIVITY - 10.00 0.00 SURFRCE H20 DENSITY (GM/M*«3) - 7.50 •b ov TOTAL REFLECTED SURFACE EMISSION ATMOSPHERIC EMISSION co co (KELVIN) in cn BRIGHTNESS TEMPERATURE CD- CO re co in- o» rj O incn- 90.0 -60.0 30.0 0.0 ELEVATION ANGLE Figure 6.4. 30.0 60.0 90.0 (DEGREES) Clear sky example of the brightness temperature profile at 11 GHz, vertical polarization. 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the antenna are likely to have blackbody temperatures approaching the ambient temperature. In this case, the discontinuity of the profile may be more like the physical situation. Because of these unknown factors, the ground model strictly adheres to the flat earth assumption. Modification in the horizon direction requires analysis of empirical data which is unavailable at the present time. Under conditions of rain, the atmospheric brightness temperature and attenuation are provided by the rain model. is again the brightness temperature profile as seen at the surface, so the calculation of it is the same as described in Chapter V. temperature Tpg and attenuation The are found from the Global Model by obtaining the terrestrial path attenuation A(D), along the horizontal path D given by, D = t i ^ * <6*6> The attenuation along the slant path is found as A - A <D > 7) PQ " cos(a) • (5,/' The radiometric formula is then used to get the temperature. This processing as well as the ground model of Equation (6.3) are contained within the rain model program, GRAM11. 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CLIMATE ZONE - 02 RAIN RATE (MM/HR) - 12.70 FREQUENCY (GHZ) - 11.00 PROBABILITY - 0.1357 1ATITUDE (DEG) - *JO.JOO STATION HEIGHT (KM) - 0.238 ANTENNA HEIGHT. (M) - 2.000 HORIZONTAL POLARIZATION SURFACE TEMPERATURE (K) - 288.15 SURFACE PERMITTIVITY - 10.00 0.00 RAIN TEMPERATURE IK) - 293.18 •b r 0500- - | 1 ; < \ ! ( (KELVIN) i i ; ; ; 1 ■ 1 1 — i TEMPERATURE : R E F L E C T E D I S U R F A C E | i : 1 ; t j i i 4 k 00- i y ! t y / ! : ^ • " -------- i i ! / l ; . 1 / * * I S ' , i i I 1 i i I j | * i i j j j ? ; : j 0 5 -] \ . ! i I * ! ; i l ; J j l i i ! | i | • | . i ; S j i ! : ' j 1 ; i i j ; !! i i ; | » ! i 1 ------------------- ■------- ^ J : i j ! i 1 t 1 i 1 I I: <i ■ • • » ! • ‘ • * i ; ; i j j ; j i i | | j j i1i j •! i ; I j i • 1 I | i i i1 1 -60.0 i ! ; T - i j i \ i * b -30. 0 0. 0 ELEVATION ANGLE Figure 6.5. ! i \ / -90. 0 j -------------- - _ _ _ _ ;_ _ _ _ _ _ •b E M IS S I O N A T M O S P H E R I C M — | E M I S S I O N --------- 05- BRIGHTNESS — — —- Ojj l b - 1— - T O T A L 30. 0 • i « > ! 1 | 60.0 90. (DEGREES) Brightness temperature profile at 11 GHz for 12.70 mm/hr rain, ground model included. Horizontal polarization. 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CLIMATE ZONE - 132 RAIN RATE (MM/HR) - 12.70 FREQUENCY IGHZ3 - 11.00 PROBABILITY - 0.1357 LATITUDE 1DEG) - VO.00 STATION HEIGHT IKM) - 0.238 RNTENNR HEIGHT. CM) - 2.000 VERTICAL POLARIZATION SURFACE TEMPERATURE (K) - 288.15 SURFRCE PERMITTIVITY - 10.00 0.00 RAIN TEMPERATURE (K) - 293.18 •b cn T O T A L : REFLECTED SURFACE EMISSION ATMOSPHERIC EMISSION CD (KELVIN) CD in - coCM BRIGHTNESS TEMPERATURE cn CO' CO' in CVJ' cn CD' cn- s -90.0 -60.0 -30.0 0.0 ELEVATION ANGLE Figure 6 .6. 30.0 60.0 90.0 (DEGREES) Brightness temperature profile at 11 GHz for 12.70 mm/hr rain, ground model included. Vertical polarization. 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figures 6.5 and 6.6 are examples of the brightness temperature profile during rain, with the ground model included. These profiles use the same environmental conditions as the profiles shown in Figure 5.9. The same trends that were seen in Figures 6.3 and 6.4 are evident in the ground profiles of these figures. the same structure. contributions. This is because the ground model has The only change has been in the atmospheric The rain rate chosen for the example makes the atmosphere more opaque than in the clear sky example. This leads to higher temperatures across the range of elevation angles. The peak shows up along the horizon and is once again due to the flat earth model. D. SUMMARY This chapter has considered the terrestrial and extra-terrestrial sources of noise that affect antenna temperature. It was pointed out that the atmospheric models provide the means to include these effects in the antenna temperature calculation but there is difficulty in programming the location of these sources. Since the environmental model is intended to be a model of general applicability, only two sources which are common to all stations are included. The first source is the cosmic background radiation of 3 K that is incident on the atmosphere from all directions. the surface of the earth. and homogeneous. The second source is The surface was assumed to be flat, smooth For this model, the contribution to the brightness 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. temperature was from reflection and emission. The Fresnel reflection coefficients are used. Other sources not included in this model can be put in through further programming. The foundation for treating these sources exist within the atmospheric models. By using those models, an environmental model can be created for a specific location. Also, unusual circumstances which may occur at the station and increase system noise can be studied. Further, later development of this model may allow for some terrain description or inclusion of exo-atmospheric and man made sources. At this point the model to produce the brightness temperature profile of the environment, which is needed for an antenna temperature calculation, has been presented. However, the rain and ground model have introduced a polarization dependence into the profile. Therefore, before an antenna temperature calculation can be made, the conversion from a horizontal and vertical polarization system to a co-polarized and cross-polarized system must be investigated. This is the subject of the next chapter. 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER VII COORDINATE AND VECTOR TRANSFORMATION A. INTRODUCTION The calculation of antenna temperature requires the integration of the product of the antenna pattern with the brightness temperature profile of the environment. The previous chapters have demonstrated how the calculation may be implemented and they have provided a model to produce the brightness temperature profile. The environmental model defines the profile in the coordinate system of the station with the z axis in the zenith direction. The antenna pattern is defined in the coordinate system of the antenna, where the z axis is in the direction normal to the aperture. Therefore, the coordinate transformation between the two systems must be found in order to calculate the antenna temperature. The objective of this chapter is to produce the coordinate transformation. The environmental model also calculates the noise temperature in terms of horizontal and vertical polarizations. The antenna pattern is defined using a co-polarized and a cross-polarized system. Therefore the conversion between these two systems is also discussed. 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B. COORDINATE TRANSFORMATION The station coordinate system is the local coordinate system of the environment surrounding the antenna. The environmental model generates a brightness temperature profile that is defined in this coordinate system. The basic definition of this system is that the z direction be the zenith direction as shown in Figure 7.1. the station coordinate system. in the direction of north. The s subscript denotes The xg axis is arbitrarily chosen to be This forces the yg axis to be in the west direction in order to have a right handed coordinate system. The brightness temperature profile was defined using the elevation angle which is the angle between the rg axis and the xs_yg plane. Note this is not the horizon system of coordinates used in astronomy. In that Zs (ZENITH) Figure 7.1. The station coordinate system. 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. system, points in space are located by altitude (elevation angle) and azimuth (-$ ). 5 The antenna coordinate system is the system in which the antenna pattern is defined. The basic premise in this system is that the z direction is normal to the aperture of the antenna, as shown in Figure 7.2. The A subscript represents the antenna coordinate system. The direction of the electric field in the aperture of the antenna is shown by the arrow and it defines the yA direction. The xA direction is then defined to produce a right handed coordinate system. A y, Figure 7.2. The antenna coordinate system. 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The integration resulting in the antenna temperature is done in the antenna coordinate system. Thus the brightness temperature profile must be transformed into the antenna coordinate system. The transformation matrix can be developed by imagining the two systems are coincident as shown in Figure 7.3. The antenna is mounted on a pedestal which allows for azimuth, elevation and polarization movement. antenna be moved in the x -y plane. s s M « pointing in the The azimuth lets the The elevation adjustment permits * plane and thus the y^ axis is the elevation axis. Polarization movement allows the antenna polarization to be changed. The pedestal allows the boresight of the antenna to be directed at a point (0p,$p) in the station coordinate system, with a polarization angle of Tp. polarized. Note, this study assumes that the antenna is linearly The extension to circular polarization should be addressed in the future. A A x„ ,x Figure 7.3 Coordinate systems coincident. 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Assume the antenna is positioned in azimuth first. coordinate system will be as shown in Figure 7.4. The the This transformation is given by, = T, (7.1) where T^ is the 3 x 3 transformation matrix, cos Op) sin(<f>p) 0 -sin(<f>p) cos(«f«p) 0 (7.2) 0 Now positioning the antenna in elevation produces the situation shown in Figure 7.5. This change is accounted for by the transformation matrix, cos(0p) T 2 = 0 -sin(6p) (7.3) 1 sin(Gp) 0 cos (ep) The total transformation is given by, 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.4. Positioning in azimuth. A Figure 7.5. Positioning in elevation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Finally, the polarization angle is to be set to Tp. If Tp is defined from the x axis, then the antenna would have to be rotated from '*t xA to xA as shown in Figure 7.6. This transformation is given by, A = T„ (7.5) *A Zk A Figure 7.6. Polarization alignment. 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where T3 cos(Tp) sin(Tp) 0 -sin(Tp) COS(Tp) 0 (7.6) 1 Using this matrix, the total transformation is given by, (7.7) T3T2T1 or more compactly, (7.8) XA = T 1X A s where r 1- . (7.9) and the inverse is used for notational simplicity later on. Therefore T is the transformation matrix needed to go between coordinate systems. In the procedure of the antenna temperature calculation, an antenna pattern point (G^*^) will be known in the antenna system and the same point will have to be found in the station system (8 s ,<|> ). s To do this a 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fortran program CONVERT has been written. rectangular coordinates of CONVERT initially finds the *n the antenna system as, xA = cos<f>A sin0A (7.10) yA = sin+A sinOA 2A - C0SeA • Then the rectangular coordinates in the station system are found by using the transformation matrix, X = T X. s A (7.11) Finally the spherical coordinates, (©g,4*s) are found from, 6_ = cos -1 (7.12) 2 2 2 x + y + z •'s s J s and -1 ♦g = tan x s 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Also, the antenna pattern is generated in non-standard spherical coordinates vhere, -180° < 9. < 180° ~ A and 0° < ♦ < 180° . Therefore CONVERT must generate the standard spherical coordinates in order to use the above procedure. f4>A-180° e.co A A e.>o A- fl0 A i e.co A A e.>o A- This is done by (7.13) ♦a = and (7.14) This procedure allows the brightness temperature at the point (0A ,<j>A ) to be found using the models of the previous chapters. With a brightness temperature known in the coordinate system of the pattern, the antenna temperature integration can be performed. The horizontal and vertical components of the brightness temperature will be resolved into the co-polarized and cross-polarized components of the antenna in the next section. 167 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C. VECTOR TRANSFORMATION The vertical and horizontal polarization directions in the station coordinate system are also the 9 and $ directions in that system. s s The 0k 0k co-polarized and cross-polarized directions are related to the 9^ and <(>A directions of the antenna system by Definition 3 of Ludwig [9], i CO sin(*A ) cos(<|>A) i . cross. - c o s (<J>a ) sin(<|>A ) (7.15) Thus the vector transformation from vertical and horizontal to copolarized and cross-polarized is a problem of finding the transformation between two spherical coordinate systems. The transformation can use the matrix T of the last section if the spherical coordinate vectors are converted to rectangular coordinate vectors. Operating on the spherical coordinate vectors in the station system, X„ = S R s s s (7.16) ' where sin(9g)cos(<j>s) Ss = sin(9g)sin(Ag) cos(9g) cos(9 )cos(A ) -sin(Ag) s s cos(9 )sin(A ) cos(Ag) s s -sin(9g) and 168 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (7.17) X = s ( 7 .1 8 ) ' RS = sj sj Transformation between rectangular systems is accomplished by modifying Equation (7.11), X. = T A 1 X s (7.19) or X. = T'1 S R A s s (7.20) where (7.21) XA = Conversion from rectangular to spherical in the antenna system is done by, (7.22) R a = S. X. A A A where 169 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sin(0A)cos(<|>A) sin(0A)sin((J)A) cos(0A)cos(<f>A) cos(0A)sin(<f>A) -sin(0A) -sin(«|>A) cos(0A) cos(0A) (7.23) and 0. RA (7.24) Using Equation (7.22) in Equation (7.20) gives the transformation between spherical systems, (7.25) R. = S. T *S R A A s s Let B be the matrix, (7.26) B = S.T *S A s with elements b... ij 0A b22 23 b32 33J II i Then, 0 s (7.27) Using this equation in Equation (7.15) results in the vector transformation, 170 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I • « ■ i CO = C i . cross. 9 s (7 .2 8 ) .♦s. where sin(4> A) cos(«f»A ) rb22 b23 C = (7.29) -cos(*A) sin(«f«A) b32 b33 The transformation matrix C is used to transform fields between the two systems. Since noise temperature is a power quantity, the elements of the C matrix must be squared to transform power. T CO T . cross. <CU >2 (C21)2 (C12)21 <c22)2 Thus, horizontal (7.30) vertical is the transformation that allows the antenna temperature calculation to be performed in the co-polarized, cross-polarized system. This transformation is also implemented in the Fortran program CONVERT. D. SUMMARY This chapter presented the coordinate and vector transformations that allow the antenna temperature integration to be performed in the coordinate system of the antenna pattern. The brightness temperature profile is most conveniently described and calculated in a coordinate system that is referenced to the local environment of the antenna. 171 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The antenna pattern is calculated in a system that has reference to the antenna aperture. The transformations developed in this chapter were needed to relate the two systems to one another. The environmental model is now intact and can be used to calculate the antenna temperature. an ideal antenna. However, this antenna temperature would be for A more realistic calculation would have to include the losses and noise contributions of the antenna structure. These topics are covered in the next chapter. 172 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER VIII ANTENNA NETWORK LOSSES A. INTRODUCTION The antenna feed network is the transmission line and associated devices that bring the power collected by the antenna to the receiver. For microwave antennas this usually consists of a horn feed and a length of waveguide. The feed network affects the antenna temperature because 2 it is a lossy system. The I R or ohmic loss of the network attenuates the power received by the antenna. 2 Additionally, the I R loss allows the network to introduce noise in the system, effectively increasing the noise temperature. The voltage standing wave ratio (VSWR) of the network influences the power available to the receiver as well. This antenna network can be included in the broad definition of the antenna system. For this reason the factors of the network which affect the antenna temperature should be included in the antenna temperature calculation. This chapter will present the models used to account for thse factors. B. NETWORK CONTRIBUTION A schematic representation of the antenna network is shown in Figure 8.1. The antenna network is defined as the feed and the transmission line network that is used to couple the feed to the receiver. Thus, the terminals of the network are taken at the receiver- 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o- T, A o Figure 8.1. Antenna network. antenna network junction. In Figure 8.1, is the noise temperature that the antenna collects from the environment. T is the physical o temperature of the antenna structure and I\n is the input reflection coefficient of the network. The antenna temperature, T , which includes cl the effects of the network is given by [7], T. + (L-l) T 1 (8 .1) 2 where L is the I R loss of the network. The reflection coefficient of the network is very difficult to calculate because it depends on numerous factors. these factors is not available. Accurate modeling of However, measurement of the reflection coefficient of the network can be done rather easily. Therefore the 1.74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VSWR of the network is left as an input to the antenna temperature calculation which is specified by the user. Note the VSWR measurement must be done of the feed and the coupling network as a whole unit, for that is how the antenna network was defined. Equation (8.1) is only valid for the entire network when taken as a single unit. 2 Techniques do exist for the calculation of the I R loss. The techniques used in the antenna temperature model are the subject of the remainder of the chapter. C. THE PERTURBATION THEORY 2 I R loss, or attenuation, occurs in a waveguide system because of the finite conductivity of the metal which is used to form the walls of the waveguide. A commonly used method used in engineering practice to evaluate this loss is based upon the perturbation theory. The electric and magnetic field inside a particular waveguide with finite conducting walls are different from the fields that would exist if the walls were infinitely conducting. If it were possible to find the fields in the guide with finite conductivity, evaluation of the attenuation would be straightforward. Generally, these fields cannot be found, and when they can be, the solution can be quite complex. The perturbation theory [49-51] assumes that the fields in the waveguide do not change as the conductivity becomes finite. This assumption is valid provided the conductivity is very high, which it is for the metals commonly used in waveguide. Since the fields are the 175 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. same, the currents flowing on the guide walls are the same. This permits evaluation of the losses from the known current distribution. If the waveguide is sufficiently far away from cutoff, and the skin depth is considerably greater than the surface irregularities, a simple first order approximation to the perturbation theory, known as the power loss method, may be used [50,52]. in an arbitrary waveguide. Let z be the direction of propagation According to the power loss method, the power at any point z is related to the power at z=0 by P(z) = P(0)e 2az, (8.2) where a is the attenuation coefficient. The rate of power loss down the guide is equal to the rate of decrease of power propagated. ?loss ■ - 2<*(°>e-2“z * Hence, • (8-3) By Equation (8.2), the attenuation coefficient can be written as a ~ ,n .. (8.4) Ploss 2P where P is the total power in the waveguide. This power is given by the Poynting vector, P - | Re (e x H*) • ds (8.5) 176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where E and H are the expressions for the fields in the waveguide and s is the cross-sectional surface of the waveguide, orthogonal to the direction of propagation. Since P^ogs is the power loss associated with the currents flowing on the walls, P - ^ loss 2 where .j guide walls J -J * dl s s (8 .6) is the resistive part of the surface impedance exhibited by the walls. This resistance is given by rm ■ T T - <8' 7 > S with a as the conductivity and 6g as the skin depth. The surface current density, J~ is equal to the tangential magnetic field. P Therefore, Equation (8 .6) may be written as, loss - ^ (n x H)•(n x H*) dl 2 (8 .8 ) guide walls where n is the inward directed normal to the guide walls. Noting that, (n x H)*(n x H*) = n*[H x (n x H*)] 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = n-[(H-H*)n - (H-n)H*] = H-H (8.9) since H-n=0 by the boundary conditions. Using Equation (8.9) in Equation (8 .8) results in, |H|2 dl (8 .10) . guide walls With Equations (8.5) and (8.10) used in Equation (8.4), the expression for the attenuation coefficient becomes, Rm ( a Jwalls p 2Rg (H-H*) dl (8 .11) (E x H*)-ds Examination of Equation (8.11) shows that evaluation of the attenuation coefficient, within the limits of the power loss method, only requires the determination of fields inside the perfect waveguide. Commonly used waveguide and feed systems in microwave antennas employ sections of cylindrical and conical waveguide of circular crosssection. Additionally, the inside walls of the feed or waveguide may be smooth or corrugated. The antenna temperature model contains provisions to calculate the loss of these four types of waveguide. The manner in 178 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which the perturbation theory is used to evaluate these losses is covered in the following sections. D. SMOOTH WALL CIRCULAR CYLINDRICAL WAVEGUIDE A longitudinal and cross-sectional view of a smooth wall circular cylindrical waveguide is shown in Figure 8.2. The field inside the waveguide can be decomposed into the familiar series of orthogonal modes. The expressions for the modes are well known and widely published. Therefore, the implementation of Equation (8.11) is straightforward and only the results, as given in [52], will be presented here. TE RM a = —— nm aZ o ^ ^\\v For the TE modes, nm 1 k c,nm + 2 n \v\\\\ TE pm TM Figure 8.2. Smooth wall circular cylindrical waveguide 179 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (8 .12 ) where a is the guide radius, Z q is the impedance of free space, and k =p' /a. c,nm nm The p' are the zeros of the derivatives of the Bessel nm function of order n. TM nm kQ=2n/X For the TM modes nm 1 rm aZ~ 0 T = = = ~ I A 1 (8-13) - k2 /k2 c,nm o with k =p /a where the p are the zeros of the Bessel Function of c,nm nm rnm order n. A Fortran program I2R has been written to calculate the coefficients. to n=2 and m=3. I2R will calculate the attenuation of TE and TM modes up Recall that the above expressions are valid only if the frequency is sufficiently above cutoff. Therefore the code issues a warning when the frequency of operation is less than 10% above cutoff. If the frequency is below cutoff, the program will issue a warning and not calculate any attenuation. At very high frequencies the coefficients given here predict lower attenuation than would be experienced in practice. This is because this method fails to account for surface roughness. E. SMOOTH WALL CIRCULAR CONICAL WAVEGUIDE The geometry of the conical waveguide is given in Figure 8.3. angle 3 is the half angle of the cone. The The field inside the cone may be found by solving Maxwell's Equations is spherical coordinates and applying the appropriate boundary conditions [53,54]. In order to 180 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. obtain an exact expression for the fields, a transcendental equation must be solved to produce the separation constants in the solution of the Helmholtz equation. Although approximate techniques for reducing the complexity of this equation exist [55] a lot of rigor and computational time will need to be used to apply the power loss method to these fields. Alternatively, for cones with small flare angles, the fields inside the cone should not be very different from those in a smooth wall circular cylindrical waveguide. This idea has been used for a long time to obtain the fields radiated by a conical horn, with good engineering results. The same idea should be able to produce good attenuation information in a conical waveguide of small angle. Z q Figure 8.3. ( z ) Smooth wall circular conical waveguide. 181 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. With the field specified, the attenuation can be calculated. By treating each circular cross-section of radius a(z) of the cone as a section of a smooth wall circular cylindrical waveguide of radius a, the results of the previous section may be used. The radius is a function of z and is given by (8.14) a(z) = z tan(P) . The attenuation coefficients may be written by substituting Equation (8.14) for a in Equations (8.12) and (8.13). TE. . _ °nm ' RM " Zqz This gives, 1_________ ta n (g ) ?rm'koz P' nm K. z tan(3) o / + n tan(B>J 2 (8.15) and (8.16) Z z tan(3) pnm/koz tan(S)) A conical horn or flared section of a horn would not use an infinite cone as is shown in Figure 8.3. Instead the cone would be truncated as shown in Figure 8.4. 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a (z2 ) a(z.) ZJ& Figure 8.4. z Truncated conical waveguide. In order to apply Equations (8.15) and (8.16) to this waveguide, the assumption that the truncation of the cone does not affect the field structure must be made. Also, the mode for which the attenuation is being calculated must be above cutoff at a(z). The total attenuation of the truncated conical waveguide is given by. (8.17) This integral can be evaluated in closed form for both TE and TM modes. For TE modes, the attenuation is 183 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. z2/r J 2 I z2 - l M ^TE nm , ' 2 2 (P v*nm ) ~n Z z tan(0) o ' In z+'Jz -] nepers z^/y' (8.18) where y' = nm kQtan(P) For TM modes, the attenuation is ^TM nm with y = z2/y M Z tan(g) o In |z+iz2-l Z]/Y nepers (8.19) nm kQtanO) * As an example of the calculation of attenuation using this method, consider a T E ^ mode propagating in the section of copper waveguide shown in Figure 8.5. The attenuation coefficient in dB at 11 GHz is plotted as a function of the radius in Figure 8 .6 . Integration of the curve or evaluation of Equation (8.18) results in a total attenuation of A_ = 6.02 x 10 3 dB. 11 184 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <x = 5.8 X I0 7 S/m 7.5 1.91 c m 10.34cm Figure 8.5. Example for T E ^ mode attenuation in a section of conical waveguide. Note that if the diameter had been held to 1.91 cm the attenuation would _2 be 4.37 x 10 dB and if the diameter had been held to 10.34 cm the _3 attenuation would be 1.75 x 10 dB.- Hence the conical waveguide attenuation falls between the two extremes as would be expected. Similarly, the attenuation coefficient of a T M ^ mode propagating in the section of copper waveguide of Figure 8.7 is shown in Figure The frequency is 11 GHz. The total attenuation of the mode in the waveguide is = 4.50 x 10-3 dB . 185 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 .8 . *b 05' CD RTTENURTIQN COEFFICIENT (DB/M) (O' (O' OJ' O 05 CO (O in (O' Cd o CD in on CM- O 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 R (CM) Figure 8 .6 . mode attenuation coefficient for the waveguide of Figure 8.5. 11 GHz frequency. 186 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a * 7.5 3.84 c m Figure 8.7. 5.8 x I07 S / m 7.62cm Example for waveguide. mode attenuation in a section of conical A mode in a circular cylindrical waveguide with a 3.84 cm diameter _3 would be 9.81 x 10 dB. The same mode in a circular cylindrical _3 waveguide with 7.62 cm diameter would suffer 2.66 x 10 dB of attenuation. Fortran programs have been written for the antenna temperature model which allow the attenuation of microwave horns to be calculated. Often, horns are a combination of circular cylindrical sections and flared sections. The programs use the method of the last section for the circular waveguide and the method of this section for the flared sections. For example, the conical horn of Figure 8.9a is calculated to _2 have 2.50 x 10 dB of attenuation at 11 GHz assuming that only the dominant TE ^ mode is propagating. 8.9b. A dual mode horn is shown in Figure In this horn the TE^^ mode propagates through the entire structure and a T M ^ mode is also present in the large diameter guide. The total power lost in the guide is the sum of the power loss of each individual mode. The total attenuation for this horn is calculated to 187 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o ay (DB/M) CD' (O' RTTENUflTION COEFFICIENT in oo cu- <M O 0.0 1.0 3.0 2.0 4.0 A (CM) Figure 8 .8 . mode attenuation coefficient for the waveguide of Figure 8.7. 11 GHz frequency. 188 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.0 I 5.08 c m 32.05 c m 10.34 c m 7.5 cr= 1.557X I O 7 S/m (a) conical horn 2.90 c m 10.06cm 53.84cm .91cm cr= 1.557 X I07 S / m (b) dual mode horn Figure 8.9. Examples of microwave horn antennas. 189 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.99cm be 0.16 dB. The perturbation theory will next be applied to waveguides with corrugated walls. F. CORRUGATED CIRCULAR CYLINDRICAL WAVEGUIDES The earlier discussion of the perturbation theory demonstrated how attenuation may be evaluated from knowledge of the field structure. Thus the problem of evaluating the attenuation, in any type of waveguide, is essentially one of determining the fields in the perfectlyconducting guide. For the corrugated waveguide, this is a complicated problem. The geometry of a corrugated circular cylindrical waveguide is shown in Figure 8.10. The geometry is described by the slot depth s, the slot width b, the ridge width t and the period p. interior of the waveguide is divided into two regions. Note that the The inner region, defined by r<r^, and the outer region, defined by r^<r<rQ. The standard procedure for finding the fields in this waveguide is to postulate a field expansion for each region, which satisfies Maxwell's equations in that region. The field expansions are then matched at the common boundary (r=r^) to determine the unknowns. Detailed analysis on how this is done is included in [56], from which most of this work on corrugated waveguides is drawn. Only a brief description of the analysis will be presented here. 190 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I s r,o Figure 8.10. Corrugated circular cylindrical waveguide geometry. In the inner region of the waveguide, the electromagnetic energy sees a periodic structure. By Floquet's theorem, the fields may be written as a series of terms which are harmonic in space. Thus the longitudinal components of the field have the form, 0 0 Aj^yK^e _jPNZ N cos(n0) (8 .20) N=-® CD "jPNz yoBNJn(KNr)e cos(n0) N=-“ for r<r^. The separation constant is given by 191 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (8 .21) and the propagation coefficient is (8.23) and is the propagation coefficient of the fundamental space harmonic. The free space wave number, k, is (8.24) and the admittance of free space is y . The remaining field components may be derived from the longitudinal components. For r>rj, the waveguide resembles a cavity so solutions to Maxwell's equations take the form of standing waves. The slots will support a TM mode (E ,E ,E.) and an infinite set of TM and TE no v z’ r’ <f> nm nm standing waves. The longitudinal components are written as, 00 m=0 0 0 m=0 192 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where, SD = J (r r)Y (T r ) - J (T r )Y (T r) n n' m ' nN m o' nv m o' nv m ' SD ' =J (r'r)Y (r'r ) - J (r'r )Y (r'r) n nv m ' nv m o' nv m o' nv m ' R° = J (T r)Y'(T r ) - j'(T r )Y (T r) n nv m ' nv m o' nv m o' nv m ' R°' = J (r'r)Y'(T r ) - j ' d ' r )Y (r'r) n nv m ' nv m o' n' m o' nv m ' ' 2 (ry = 2 2 - (h)z and ' = n(2m-l) in b The unknowns in these equations are the mode coefficients, t BN , t cm » cm> dm, dm and the propagation constants, and K^. These unknowns are found by equating the z and <f> directed components of the total field at the interface r=r1< -3 By using the orthogonality of the e and sin(»^z), cosCr^z) functions, a system of algebraic equations for the 193 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. mode coefficients in the inner region can be developed. This system may be written in matrix form as, (8.27) in which U,V,X,Y are elements which depend on the propagation factor {3^. In order for non-trivial solutions of A,B to exist, the determinant, (8.28) must be exactly zero. The solution to Equation (8.28) gives the proper value of Equation (8.28) is called the characteristic equation for this reason. With known, the matrix equation (8.27) may be solved to determine A^,B^. These coefficients are used with the boundary conditions to find 9 9 c , c , d and d , the coefficients in the outer region. m m m m e With this done, the field in the corrugated waveguide is specified. This procedure is known as the space-harmonic model of corrugated waveguide fields. In practice, the number of field terms in Equations (8.20), (8.21), (8.25) and (8.26) must be truncated to a finite number in order to permit calculation. Thus for r<r^, K N=-K E z'N (8.29) 194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. K H z ■ _V_ ■ l ~ „ ZN N=-K and for r>r^, L> E = ) z / —a m=0 E z m (8.30) L H = ) H z I— , zm m=l The actual value of K and L is very important in the practical application of this model. K and L determinethe amount of operations necessary to solve the characteristic equation. Since the characteristic equation is transcendental, it must be solved numerically and computational accuracy and number of operations become important parameters. In Clarricoats et al. [57], the authors state that although computational accuracy is retained for integer values of K up to 15, it is found sufficient to take K=L=1 for accurate computation of the propagation coefficient. The special case of the space-harmonic model, for K=L=1 implies just one pair of space harmonics in the inner region in addition to the fundamental. For the outer region, one higher order TE-TM standing wave combination is included with the lowest order TM field. Clarricoats et al. also state that, except in cases where the frequency is close to the 195 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. low frequency or high frequency cutoff of a particular mode, the K=L=0 case produces sufficient results. In this case, the characteristic equation takes the much more manageable form [58], (8.31) where S = Kr n and The K=L=0 model assumes only the fundamental mode is present in the inner region and the TMnQ mode is present in the slot. The special case of the space-harmonic model with K=L=0 is very similar to the more familiar surface impedance model of corrugated waveguide [59,60]. The difference between the models is that the surface impedance model neglects the thickness of the ridges and assumes a very large number of slots per wavelength. Mathematically, 196 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f = 1.0 and b -» 0 such that sin(P ~~7 1 * 1 3 2 This reduces the characteristic equation to © s Sn n2a2 g = 0 . V Krl> - TWT) nv 1' (8.32) If the surface impedance model is used, and the conditions for using it are not met, the resulting inaccuracy in the calculation of f5 by Equation (8.32) will result in serious errors in the calculation of attenuation. This will be demonstrated later. Assuming a particular model for representing the fields has been chosen, the attenuation coefficient is calculated from the expression, “ - IFF <8-33> where P. is the power lost in one period and P is the total power. L O This is the same basic form used previously except that the period appears in the denominator. The period must be included because the loss mechanisms within one entire period must now be included and the 197 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. period p in the denominator is necessary for normalization. Alternatively, a can be written as, L L 5 pL.m + 5 p ^— x ^— pr L„m a = K y N=-K + -— „ pL-N m=0 1 m=0 2 3 ------------ £--------------2P £ x (8.34) P°N where Pqn = power flow in the N 1*1 space harmonic in region 1, P.L^m = power loss on the side walls of the slots, P. = power loss on the base of the slot, 2m N = Power ^oss on the top of a ridge at r=r^. For each special case of the space-harmonic model, the series in Equation (8.34) are truncated to the proper value of K,L and for the surface impedance model, K=L=0, and PT is removed. 3 The differences in the calculation of attenuation by each model is demonstrated in Figure 8.11 which is reproduced from [57]. undesirability of the surface impedance model. Note the Not only is it inaccurate in calculating the attenuation, but more importantly, it 198 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. misses the frequency of minimum attenuation. This difference is solely due to the different values of P as calculated by the respective characteristic equations. Also note the relatively minor improvement between K=L=1 and K=L=5. For the antenna temperature model, the space-harmonic model with K=L=0 is used. This decision is based on the reasoning that, although this model is slightly more complex than the surface impedance model, it has a small price to pay for the improved accuracy. However, the feed types and accuracy required for this study do not warrant the complexity and increased accuracy of a more sophisticated space-harmonic model. A Fortran program CI2RC_SUB has been written to calculate the attenuation of the dominant HEjj mode in corrugated waveguide. This program is based on the attenuation program, written in Basic, that has 1 0 -1 C ■o 2 o 1 0 *2 o - - - - Surface Impedance Model 3 c V Q Space-Harmonic Model Parameter, K=L 10*3 10 14 fre q u e n c y .G H z Figure 8.11. Attenuation coefficient of an HEjj mode in corrugated waveguide, with r.j=30 mm, r^r^O.737, b=10 mm, p=ll mm, and a=1.57 x 10^ S/m. From [571. 199 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. been published in Clarricoats and Olver [56]. The major difference between the program is that the program in Clarricoats uses the surface impedance characteristic equation to find form. 8. CI2RC_SUB uses the K=L=0 Other differences are due to typographical errors or minor oversights in the published version. Figure 8.12 is the attenuation coefficient for the waveguide of Figure 8.11, as calculated by CI2RC_SUB. Agreement with Figure 8.11 is good. Also results calculated by CI2RC_SUB agree favorably with results shown in other publications. As with the smooth wall version of the attenuation calculation, the program CI2RC_SUB issues a warning when an attempt is made to calculate attenuation when the HEjj mode is cut off. This attenuation calculation may be applied to corrugated circular conical waveguide as shown in the next section. G. CORRUGATED CIRCULAR CONICAL WAVEGUIDE As with smooth wall conical waveguide, published procedures for attenuation calculation are apparently not available for the corrugated circular conical waveguide. However, since the flare angles to be considered will be relatively small, the same method used in the smooth wall case should be applicable here. There it was assumed that each dz of the conical section was a section of circular cylindrical guide and the a(z) was calculated for each section. the integral of a(z)dz. The total attenuation is for For corrugated conical waveguide the approach will be the same except that the a(z) will be of corrugated cylindrical 200 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.0 8.0 9.0 10.0 FREQUENCY Figure 8.12. 11.0 12.0 13.0 1U.0 (GHZ) H E ^ mode attenuation coefficient as calculated by CI2RC SUB. Corrugated waveguide geometry, r.=30 mm, i7 r./r =0.737, b=10 mm, p=ll mm and <y=1.57 x 10 S/m. 1 o 201 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. waveguide. Thus for the waveguide shown in Figure 8.13, the total attenuation is, a(z)dz A = nepers (8.35) Applying this approach to corrugated horns ignores the fact that the slot depths may vary as a function of z. the case. This does not have to be The attenuation for each slot could be calculated, accounting for the geometry of the slot, and then the total obtained as the sum over all slots. However, this method would not be desirable from the system design point of view. Typically, the designer, when using the program, will want information on the Z Figure 8.13. 2 A section of corrugated circular conical waveguide. 202 Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. performance of a corrugated horn but the details of the horn dimensions will not be known at that point. Thus the approach as described will be used. Additionally, given the low loss of the H E ^ mode in corrugated guide, and the short length of the horns, accuracy should not suffer by the manner in which the horn is modeled. The Fortran program CI2RC_SUB has been given the capability to calculate the attenuation in corrugated conical waveguide. Using this program and the smooth wall program I2R_SUB, the total attenuation for a corrugated horn may be calculated. in Figure 8.14. sections. For example, consider the horn shown This horn is composed of four different waveguide It has two smooth wall circular cylindrical sections, a smooth wall circular conical and a corrugated circular conical section. Using the methods presented in this chapter, this horn is expected to _2 have 1.88 x 10 GHz. dB of attenuation at an operational frequency of 11 An example of a larger corrugated horn is shown in Figure 8.15. This horn is also composed of four waveguide sections and the calculated _2 attenuation is 6.64 x 10 H. dB. SUMMARY This chapter has presented two practical aspects that affect the antenna temperature. These effects are the voltage standing wave ratio of the antenna and the inherent ohmic losses associated with the antenna structure. VSWR is a quantity that cannot be reliably calculated. 203 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore the VSWR of the feed network is a left as an input to the antenna temperature calculation. Practical methods exist for the calculation of ohmic losses and this chapter discussed the models used in the antenna temperature calculation. Four types of waveguide that are commonly used in microwave antennas were modeled. These types are: smooth wall circular cylindrical waveguide, smooth wall circular conical waveguide, corrugated circular cylindrical waveguide and corrugated circular conical waveguide. could be modeled. Any antenna which uses sections of these waveguides The antenna temperature calculation is set up to specifically handle dual mode horns, conical horns and corrugated horns. Examples of the attenuation calculation of these horn types were given. The discussion given in this chapter is the last discussion on an individual component of the antenna temperature model. In the following chapter several examples of antenna temperature calculations will be given. 204 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to to 0) CM 0) 205 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 8.14. Corrugated horn with 15° flare angle. fO I Corrugated horn with 24° flare angle. in co Figure 8.15. in Cl 1 1 E E ro m cr> 206 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER IX SOME EXAMPLES OF ANTENNA TEMPERATURE CALCULATIONS USING THE ENVIRONMENTAL MODEL A. INTRODUCTION The previous chapters have dealt with the presentation and explanation of the individual models which compose the antenna temperature model. Numerous examples have been given to demonstrate how each particular model works. In this chapter, those previous examples will be used together to demonstrate the antenna temperature model as a whole unit. An antenna pattern that has been generated by the Reflector Antenna Code [1] will be used to simulate the operation of an antenna in three different environments. These environments will correspond to a typical Columbus, Ohio location under clear, cloudy and rainy conditions. Discussions of the resulting antenna temperatures and comparisons between the various environmental conditions will be given. B. THE ANTENNA USEDIN THE ANTENNA TEMPERATURE EXAMPLES The antenna that has been chosen for use in the antenna temperature calculations is an 8 foot (2.44 m) diameter, prime focus reflector. The reflector has a focal point to diameter ratio of 0.5 and it is fed by 207 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the corrugated horn shown in Figure 8.14. The far field pattern of this antenna was calculated at 11 GHz using the Reflector Antenna Code. The pattern was calculated for 0° < <f> < 180° 180° < 6 < 180° with sampling spacings of A$ = 18° A9 = 0.2° . The calculated H-plane, or <f>=0° cut, pattern is shown in Figure 9.1. The main features of this pattern are the main beam at 8=0°, the direct feed spillover regions near G=±120° and the backlobe regions near 9=±180°. Other cuts of the pattern are not shown because the corrugated horn produces a circularly symmetric pattern. Thus the pattern of the reflector is symmetric in the absence of blockage and scattering mechanisms of the feed mounting structure, which were not included in this calculation. The antenna temperature of this antenna will be calculated for three different environments which were discussed earlier. 208 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 -20 Db -30 -40 -50 -60 -70 -80 -180 -150 -120 -90 -60 -30 Theta Figure 9.1. 0 30 60 90 120 150 180 (Degrees) H-plane pattern of a prime focus fed parabolic reflector. Diameter=8 feet, focal point to diameter ratio=0.5, frequency = 11 GHz, corrugated horn feed. 209 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C. ANTENNA TEMPERATURE FOR CLEAR SKY CONDITIONS The first example of the antenna temperature calculation will be for the clear sky environmental conditions first given in Figure 6.4. Figure 9.2 shows the vertically polarized brightness temperature profile, for those same conditions, plotted as a function of zenith angle. The zenith angle will be represented by 0z in the text. The surface water vapor density has been converted to relative humidity for the label on the plot. The antenna temperature, for the antenna vertically polarized, is shown in Figure 9.3. In this calculation, the VSWR and ohmic loss of the feed network are ignored. The profile of the antenna temperature is a near replica of the brightness temperature profile. This result is expected for a narrow beam antenna since the brightness temperature profile may be viewed as the antenna temperature profile of an antenna with an impulsive pattern. The finite beamwidth of the antenna pattern can be seen at the horizon where the beam tends to smooth out the discontinuity in the brightness temperature profile. The feed spillover and backlobe of the antenna are evident at small zenith angles. At these angles, the antenna temperature is nearly double that of the brightness temperature. This is because the spillover and backlobes are directed at the surface of the earth for those angles. Note that the small dip in antenna temperature that occurs near the 40° zenith angle is due to the fact that the spillover lobes in one hemisphere of the pattern enter a cold region of sky. 210 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY (GHZ) = 11.00 STATION HEIGHT IKM) * 0.230 ANTENNA HEIGHT. IM) * 2.000 RELATIVE HUMIOITY AT SURFACE 17.) = 58.61 WATER VAPOR SCALE HEIGHT (KM) « 2.00 SURFACE PRESSURE (MBAR) = 1023.0 NUMBER OF CLOUD LAYERS =0. ' EXO-ATMOSPHERIC TEMPERATURE (K) = 3.00 SURFACE TEMPERATURE IK) = 288.15 SURFACE PERMITTIVITY = 10.00 -J 0.00 VERTICAL POLARIZATION o CD' p .- CO' (KELVIN) w CO' BRIGHTNESS TEMPERATURE CO' CD' fU' o CO 00' pCD ID' CO' O 0.0 30.0 60.0 90.0 ZENITH ANGLE Figure 9.2. 120.0 150.0 180.0 (DEGREES) Clear sky brightness temperature profile. polarization. Vertical 211 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY (GHZ) - 11.00 STATION HEIGHT (KM) - 0.238 ANTENNA HEIGHT. (M) - 2.000 RELATIVE HUMIDITY AT SURFACE 17.) = 58.64 WATER VAPOR SCALE HEIGHT (KM) - 2.00 SURFACE PRESSURE (MBAR) * 1023.0 NUMBER OF CLOUD LAYERS *0. EXO-ATMOSPHERIC TEMPERATURE (K) = 3.00 SURFACE TEMPERATURE (K) - 288.15 SURFACE PERMITTIVITY = 10.00 -J 0.00 GAIN (DBI) * 48.30 TAUP (DEC.) « 90.00 cv CD (KELVIN) CO in cry C\J TEMPERATURE CD CD CO' in m CM' ANTENNA O in co CM 0.0 30.0 60.0 90.0 ZENITH ANGLE Figure 9.3. 120.0 150.0 (DEGREES) Clear sky example of antenna temperature. polarization. Network effects excluded. Vertical 212 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180.0 The gain that is printed on the plot is the free space gain which results from the numerical integration of the antenna pattern. Figure 9.4 shows the same antenna temperature profile with the effects of the antenna network included. assumed for the feed. A VSWR of 1.065 has been This corresponds to a 30 dB return loss. loss in the feed is seen to slightly reduce the gain. The Additionally, noise from the feed losses raises the antenna temperature slightly in the range from 0° to about 80°. Past that point ground emission into the main beam dominates the antenna temperature and emission due to the feed is insignificant. Since this is a small horn, a larger horn in the same situation, or the same horn with a longer waveguide run, is expected to show noticeably increased temperature for small zenith angles. The individual contributions to the antenna temperature of the sky and ground can be determined by breaking up the integral over all of space into a separate integral over each hemisphere. for the current example and is shown in Figure 9.5. This has been done This curve demonstrates the drop off in ground contribution as the spillover region begins to point to the cold sky. The ground contribution continues to decrease until the main beam begins to see the ground. This causes a sharp increase in the ground contribution until it totally dominates when the full main beam illuminates the ground. When the main beam is illuminated by the sky noise, Figure 9.5 shows that the sky contribution is most significant. The exception to this is below 30° where all the spillover lobes see the ground. This breakdown of the contributions 213 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY tGHZ) - 11.00 STATION HEIGHT (KM) - 0.238 RNTENNfl HEIGHT. (M) - 2.000 RELATIVE HUMIDITY AT SURFACE 17.) - 58.64 WATER VAPOR SCALE HEIGHT (KM) «= 2.00 SURFACE PRESSURE (MBAR) - 1023.0 NUMBER OF CLOUD LAYERS *0. EXO-ATMOSPHERIC TEMPERATURE* (K) - 3.00 SURFACE TEMPERATURE (K) = 288.15 SURFACE PERMITTIVITY = 10.00 -J 0.00 * b GAIN ID8I) =■ 48.28 TAUP IDEG.) * 90.00 (KELVIN) co (O' CM TEMPERATURE CO cnCM ANTENNA O CP' CDCO CO- CM- 0.0 30.0 60.0 ZENITH Figure 9.4. 90.0 ANGLE 120. 0 150.0 180.0 (DEGREES) Clear sky example of antenna temperature. polarization. Network effects included. Vertical 214 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY (GHZ) « 11.00 STATION HEIGHT (KM) - 0.23B ANTENNA HEIGHT. IM) - 2.000 RELATIVE HUMIDITY AT SURFACE (*/.) - 58.64 HATER VAPOR SCALE HEIGHT (KM) - 2.00 SURFACE PRESSURE IMSRR) - 1023.0 NUMBER OF CLOUD LAYERS -0. EXO-ATMOSPHERIC TEMPERATURE (K) - 3.00 SURFACE TEMPERATURE (K) - 288.15 SURFACE PERMITTIVITT - 10.00 -J 0.00 *b GAIN (DBI) - 48.28 TAUP (DEG.) - 90.00 O' CO' TOTAL ID' O' GROUND (KELVIN) SKY CO CM' TEMPERATURE O' CD CO in to- CM- ANTENNA "o CD' oin CD' CM 0.0 30.0 60.0 ZENITH Figure 9.5. 90.0 ANGLE 150.0 180.0 (DEGREES) Clear sky example of antenna temperature. Vertical polarization. Total antenna temperature with sky and ground contributions. 215 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. from each hemisphere demonstrates the effect of spillover for stations pointed near zenith. Spillover more than doubles the antenna temperature in that region for this example. Breakdown of contributions such as this would be a useful analysis tool in the design of low noise antennas. Another possible way to break down contributions to the total antenna temperature is on the basis of the pattern. the contribution due to the main beam. Figure 9.6 shows This curve also shows the balance between spillover contributions and the main beam contributions, or from another point of view, the importance of the earth in determining antenna temperature under transparent sky conditions. Since the Antenna Temperature Code calculates the gain of the antenna and the antenna temperature, the gain over temperature ratio, (G/T), of the antenna may be calculated as, G/T = G - 101og10(Ta) (dBK-1) where G is the antenna gain in dB and T cl kelvin. (9.1) is the antenna temperature in The G/T of the antenna, under the environmental conditions being described, is shown in Figure 9.7. Note that this G/T does not represent the figure of merit for the system. Calculation of the figure of merit requires knowledge of the noise figure of the receiver. The environmental model contains within it the equations and parameters that would enable a calculation of attenuation, to a specified point in the environment, to be performed. If the source of 216 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the desired transmission is located at the specified point, then the environmental model can be used to calculate the atmospheric attenuation between the receiving antenna and the source. Although not common in practice, some systems designers prefer to include the effect of this atmospheric attenuation, along with the G/T ratio, in a different parameter sometimes known as the boresight G/T. boresight G/T = G - 101og1()(Ta) - A where G is the antenna gain in dB, T di Mathematically, (dBK-1) (9.2) is the antenna temperature in kelvin and A is the atmospheric attenuation in dB. Thus, for this study, the boresight G/T is defined as the G/T of the antenna reduced by the atmospheric loss to the source. Since atmospheric attenuation depends on the path length in the atmosphere, boresight gain depends on the distance between the antenna and the source of the desired radiation. The antenna temperature model assumes that the source is in the boresight of the antenna and allows the user to specify the range to the source. Hence the source can be terrestrial, airborne or outside the atmosphere, such as a satellite. For an atmosphere without rain, the Antenna Temperature Code calculates the attenuation to the source by integrating the absorption coefficient profile along the path to the target. Figure 9.8 shows the boresight gain over temperature ratio for the clear sky example and a source range of 36,000 km. Note for this case, this ratio is the ratio 217 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY IGH2) - 11.00 STATION HEIGHT IKM) - 0.238 ANTENNA HEIGHT. IM) - 2.000 RELATIVE HUHIDITY AT SURFACE 17.1 « 58.6*1 WATER VAPOR SCALE HEIGHT (KM) - 2.00 SURFACE PRESSURE 1MBAR) - 1023.0 NUMBER OF CLOUD .LAYERS -0. EXO-ATMOSPHERIC TEMPERATURE (K) - 3.00 SURFACE TEMPERATURE IK) - 288.15 SURFACE PERMITTIVITY - 10.00 -J 0.00 * b GAIN (DBI) - *18.28 TAUP (DEG.) - 90.00 OV CD' TOTAL (O' (KELVIN) in MAIN BEAM (O' ru ff) TEMPERRTURE CD' COin try OJ RNTENNR O CO' (O in rr (O' ru 0.0 30.0 60.0 ZENITH Figure 9.6. 90.0 ANGLE 120.0 (DEGREES) 150.0 180.0 Clear sky example of antenna temperature. Vertical polarization. Total antenna temperature with main beam contribution. 218 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY (GHZ) - 11.00 STATION HEIGHT (KM) - 0.230 RNTENNR HEIGHT. (M) - 2.000 RELATIVE HUMIDITY RT SURFACE (’/) = 58.614 WATER VRPOR SCALE HEIGHT (KM) = 2.00 SURFACE PRESSURE (MBAR) = 1023.0 NUMBER OF CLOUD LAYERS *0. EXG-flTMQSPHERIC TEMPERATURE (K) ■= 3.00 SURFACE TEMPERRTURE IK) - 280.15 SURFACE PERMITTIVITY - 10.00 -J 0.00 GRIN (DBI) - 48.28 TAUP (DEG.) = 90.00 m .. r o O G/T (dBK~ CD . o - o. -! . 30.0 60.0 ZENITH Figure 9.7. 90.0 ANGLE 120.0 150.0 180.0 (DEGREES) G/T ratio for the clear sky example of antenna temperature. Vertical polarization. 219 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for any space source since it was assumed that the atmosphere only extends to 20 km. Hence, this might be the boresight G/T of an earth station for satellite communications. The boresight G/T calculation is allowed to continue past the horizon, (0z>90°), by assuming the source is on the surface of the earth. Thus at these angles, the specified range has no meaning, because the range is determined by the slant length to the surface. This is advantageous because it allows the boresight G/T to be calculated for antennas which may be on a tower, looking down to a source on the surface of the earth. A similar set of curves have been prepared for when the antenna is horizontally polarized. Figure 9.9 shows the brightness temperature profile of the environment as a function of zenith angle, for horizontal polarization. The resulting antenna temperature and sky and ground contributions are shown in Figure 9.10. main beam is shown in Figure 9.11. The contribution due to the TheG/T is shown in Figure 9.12 and the boresight G/T for a source range of 36,000 km is shown in Figure 9.13. The same characteristics that were evident in the vertically polarized case are seen in these curves. The next example will demonstrate the effects of clouds on the antenna temperature by inserting a cloud layer in the atmosphere. 220 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY (GHZ) « 11.00 STATION HEIGHT (KM) - 0.238 ANTENNA HEIGHT. (M) - 2.000 RELATIVE HUMIDITY AT SURFACE I'/) = 58.64 HATER VAPOR SCALE HEIGHT (KM) = 2.00 SURFACE PRESSURE (MBRR) = 1023.0 NUMBER OF CLOUD LAYERS =0. EXO-ATMOSPHERIC TEMPERATURE (K) »= 3.00 SURFACE TEMPERATURE (K) = 288.15 SURFACE PERMITTIVITY = 10.00 -J 0.00 GAIN (DBI) = 48.28 RANGE (KM! = 36000.00 TAUP (OEG.) = 90.00 o in i ; i .... ......... ! 3 i 3 i : i o f\J i i __ ! !’ i \ 1 .. !.._ : ! i i l ...... ........L . _ i : i i ___ I ......L..... i ...... : i i j ! _ i i T .. ; i ... i......" r i ....r “ 7 ■ : i 30.0 | ! • 60.0 ZENITH j i r• *, i • 1 j i 1 i . . . !... _. i i i i ......i.... ' ] ...... ...... ; .. —T 1 ..... I...... I.......... i i i i ; : : I i i ! ! r ] ..._! ....._ 1 ! ...... i ...... i..._ • Figure 9.8. I i i ! I _ j._ ... • '0.0 1 .. . j j ...... . . ; i i i ^.. f 1 v i : .....j .T ... ... . ] ...... i i i I ..." • .. ... ! I * i I ! ~ : ! .... I _ J___ . ! 1 : "" N i : o o i j.. i ■ ; ^ i j t i 5 ! i ! ! I i ■“ _"i.... i ! ; 1i ......|...... i ‘ i 1 o .. . ! 1 ! ro G/T (dBK~ I 1 : ! ___J ___....... BORESIGHT I j i 90.0 ANGLE | ! 120.0 150.0 ..... 1.. ... ... 1.... 1 ! j 160.0 (DEGREES) Boresight G/T ratio for the clear sky example of antenna temperature. Vertical polarization and 36,000 km range. Terrestrial source for 0 >90°. z 221 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY (GHZ) = 11.00 STATION HEIGHT (KM) - 0.238 RNTENNR HEIGHT. (M) = 2.000 RELATIVE HUMIDITY RT SURFACE 17.) = 58.611 WATER VflPQR SCALE HEIGHT (KM) - 2.00 SURFACE PRESSURE IMBRR) = 1023.0 NUMBER OF CLOUD LAYERS =0. EXO-ATMOSPHERIC TEMPERATURE (K) = 3.00 SURFACE TEMPERATURE IK) * 288.15 SURFACE PERMITTIVITY * 10.00 -J 0.00 HORIZONTAL POLARIZATION 05CD' P- BRIGHTNESS TEMPERATURE (KELVIN) CO' in- O' CO' co in- O 05' CO CO in- CD- OJ 0.0 30.0 60.0 ZENITH Figure 9.9. 90.0 ANGLE 12 0. 0 150.0 180.0 (DEGREES) Clear sky brightness temperature profile. polarization. Horizontal 222 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY (GHZ) - 11.00 STRTION HEIGHT IKM) - 0.238 ANTENNA HEIGHT. (M) - 2.000 R E L ATIVE HUMIDITY RT SURFACE 17.) -= 58.64 WATER VAPOR SCALE HEIGHT (KM) - 2.00 SURFACE PRESSURE CMBAR) - 1023.0 NUMBER OF CLOUD LAYERS -0. EXO-ATMOSPHERIC TEMPERATURE (K) - 3.00 SURFACE TEMPERATURE (K) « 288.15 SURFACE PERMITTIVITY « 10.00 -J 0.00 * b GAIN IDBI) - 48.28 TAUP IDEG.) - 0.00 WSO' (O' (KELVIN) in TOTAL GROUND CO' SKY C\J ov TEMPERATURE CD- co in co- ANTENNA "o in co 0.0 30.0 60.0 ZENITH Figure 9.10. 90.0 ANGLE 120.0 150.0 180.0 (DEGREES) Clear sky example of antenna temperature. Horizontal polarization. Total antenna temperature with sky and ground contributions. 223 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY (GHZ) - 11.00 STATION HEIGHT (KM) - 0.238 ANTENNA HEIGHT. IM) *> 2.000 RELATIVE HUMIDITY AT SURFACE (*/) « 58.64 WATER VAPOR SCRLE HEIGHT (KM) « 2.00 SURFACE PRESSURE (MBAR) - 1023.0 NUMBER OF CLOUD LAYERS -0. EXO-ATMOSPHERIC TEMPERATURE (K) - 3.00 SURFACE TEMPERATURE IK) - 286.15 SURFACE FERMITTIVITY - 10.00 -J 0.00 * b GAIN (DBI) - 48.28 TAUP (DEG.) - 0.00 CD- f^CD XT)' ANTENNA TEMPERATURE (KELVIN) TOTAL co — MAIN B E A M - c\j CD' in to- CM1 O cn OD' cn ru- 0.0 30.0 60.0 ZENITH Figure 9.11. 90.0 ANGLE 120.0 (DEGREES) 150.0 180.0 Clear sky example of antenna temperature. Horizontal polarization. Total antenna temperature with main beam contribution. 224 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. GRIN (DBI) - 48.28 TAUP (OEG.) = 0.00 30.0 20.0 G/T (dBK ” ) '10.0 50.0 FREQUENCY (GHZ) - 11.00 STATION HEIGHT (KM) - 0.238 RNTENNfl HEIGHT. (M) = 2.000 RELATIVE HUMIDITT RT SURFACE 17.) = 58.S4 WRTER VAPOR SCALE HEIGHT (KM) = 2.00 SURFACE PRESSURE (MBAR) - 1023.0 NUMBER OF CLOUD LAYERS -0. EXO-ATMOSPHERIC TEMPERATURE (K) = 3.00 SURFACE TEMPERATURE IK) = 288.15 SURFACE PERMITTIVITY = 10.00 -J 0.00 I - 10.0 1 -.. ! . © 30.0 60.0 ZENITH Figure 9.12. 90.0 ANGLE 120. 0 150.0 180.0 (DEGREES) G/T ratio for the clear sky example of antenna temperature. Horizontal polarization. 225 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. GAIN (OBI) = 48.28 RANGE (KM) = 36000.00 TAUP (DEG.) = 0.00 30.0 20.0 to.o - t . 0.0 B0RESIGHT G/T (dBK" ) 40.0 50.0 FREQUENCY (GHZ) - 11.00 STRTION HEIGHT (KM) - 0.238 ANTENNA HEIGHT. (M) - 2.000 RELATIVE HUMIDITY RT SURFACE (/.) = 58.64 WATER VAPOR SCRLE HEIGHT (KM) = 2.00 SURFACE PRESSURE (MBAR) «= 1023.0 NUMBER OF CLOUD LAYERS =0. EXO-ATMOSPHERIC TEMPERATURE (K) = 3.00 SURFACE TEMPERRTURE (K) = 288.15 SURFRCE PERMITTIVITY «= 10.00 -J 0.00 30.0 60.0 ZENITH Figure 9.13. 90.0 ANGLE 120.0 150.0 180.0 (DEGREES) Boresight G/T ratio for the clear sky example of antenna temperature. Horizontal polarization and 36,000 km range. Terrestrial source for 9 >90°. z 226 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D. ANTENNA TEMPERATURE FOR A CLOUDY CONDITION This example consists of using the same atmosphere shown in the last section, but with a cloud layer included. The cloud layer extends from an altitude above MSL of 0.66 km to 2.7 km and has a liquid water density of 1.0 gm/m3. This is the same atmosphere for which a brightness temperature profile was given in Figure 4.29. Figure9.14 shows the brightness temperature profile against zenith angle for vertical polarization. The antenna temperature profile for the vertically polarized antenna is given in Figure 9.15. comparing this curve to the clear sky example of Figure 9.6 shows the main beam contribution is more significant under cloudy conditions. This is because the atmosphere has become more opaque while the ground contribution has remained essentially constant. This example also demonstrates the significant effect that clouds can have on antenna temperature. The G/T ratio is shown in Figure 9.16 and the boresight G/T ratio for a source at 36,000 km is shown in Figure 9.17. The profiles for the case of horizontal polarization are given in Figures 9.18 and 9.19. The G/T ratio is shown in Figure 9.20 and the boresight G/T ratio for a source at 36,000 km is shown in Figure 9.21. The next example will be for antenna temperature during a rain event. 227 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCY IGHZ) ■- 11.00 STATION HEIGHT (KM) - 0.238 ANTENNA HEIGHT. (M) - 2.000 RELATIVE HUMIDITY AT SURFACE 17.) = 58.64 HATER VAPOR SCALE HEIGHT (KM) - 2.00 SURFACE PRESSURE (MBAR) - 1023.0 NUMBER OF CLOUD LAYERS -1. LAYER 1. BASE (KM) - 0.660 CEILING (KM)- 2.700 EXO-ATMOSPHERIC TEMPERATURE (K) - 3.00 SURFACE TEMPERRTURE (K) - 2e8.15 SURFACE PERMITTIVITY - 10.00 -J 0.00 VERTICAL POLARIZATION DENSITY (GM/M«*31- 1.0 (KELVIN) toin- m BRIGHTNESS TEMPERRTURE OV 03' (O' in- p> ruCO mCO' (O' in(TV ruo 0.0 30.0 60.0 90.0 12 0.0 150.0 180.0 Z E N I T H R N G L E (DEGREES) Figure 9.14. Brightness temperature profile for a cloudy condition. Vertical polarization. 228 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FRE3UENCY (GHZ) - 11.00 STATION HEIGHT IKMJ - 0.230 ANTENNA HEIGHT. (H) - 2.000 RELATIVE HUMIDITY RT SURFACE (7.) - 58.64 HATER VRPOR SCALE HEIGHT tKM) - 2.00 SURFACE PRESSURE IMBAR) - 1023.0 NUMBER OF CLOUD LAYERS -1. LRYER 1. BASE (KM)- 0.660 CEILING (KM)- 2.700 EXO-ATMOSPHERIC TEMPERATURE (K) - 3.00 SURFRCE TEMPERATURE IK) - 28B.15 SURFACE PERMITTIVITY - 10.00 -J 0.00 *b 1_ o- GAIN (051) - 48.28 i RNTENNfl TEMPERATURE (KELVIN) ! i I | S * i ! ! i i ; i i I 1 : i ; i i 1 ; { i 1 ! I; i • : i ; ! 1 > j : t ! I S * i 1 * ^ 1 1/ // . /, 1/ : 1 : i 1 ! i ) i ! / | t i i i i i ! i i ; ! i i i ; i 1 ! | i ! i I / -- MMIIN DC.A M i 1 ! 1 i ! ! i i t | | 1 ! i • i ! 1 1 1 i i 1 i | I ; 30.0 60. 0 i i i i i i I i ! 1 ! i ! i ! : ! i i ! ! ! i ! i ! ! ! ! i ! 90.0 ZENITH ANGLE Figure 9.15. i l i i ! i I I 1 0. 0 ! 1U IAL y.' i ■b < ! i ! j i 1 » ~o o~ TAUP (DEG.) - 90.00 i O- DENSITY (GM/M— 3J- 1.0 i 1 ; ! J 1 i i > ! i 120. 0 1 1 i i ! ‘ 1 i i ! ! 1 i i ; i : i ; ! ! i ! 1 150.0 j i i ! i i 180. (DEGREES) Example of antenna temperature for a cloudy condition. Vertical polarization. Total antenna temperature with main beam contribution. 229 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I FREQUENCY (GHZ) - 11.00 STATION HEIGHT (KM) - 0.238 ANTENNA HEIGHT. IM) - 2.000 RELATIVE HUMIDITY AT SURFACE IZ) - 58.64 HATER VAPOR SCALE HEIGHT (KM) - 2.00 SURFACE PRESSURE (MBAR) - 1023.0 NUMBER OF CLOUD LAYERS -1. LAYER 1. BASE (KM)- 0.660 CEILING (KM)« 2.700 EXO-RTMQSPHERIC TEMPERATURE (K) - 3.00 SURFACE TEMPERATURE (K) - 288.15 SURFACE PERMITTIVITY - 10.00 -J 0.00 o o GAIN (OBI) - 48.28 DENSITY (GM/M**3) - 1.0 TAUP (DEG.) * 90.00 o (dBK I o G/T o < M 30.0 60.0 90.0 ZENITH RNGLE Figure 9.16. 12 0. 0 150.0 180.0 (DEGREES) G/T ratio for the cloudy sky example of antenna temperature. Vertical polarization. 230 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCT (GHZ) - 11.00 STATION HEIGHT (KM) - 0.238 RNTENNR HEIGHT. (M) - 2.000 RELATIVE HUMIDITY RT SURFACE (Z) - 58.64 HATER VAPOR SCALE HEIGHT (KM) • 2.00 SURFACE PRESSURE (M8AR) - 1023.0 NUMBER OF CLOUD LAYERS -1. LAYER 1. BASE (KM) ■ 0.660 CEILING (KM) - 2.700 EXO-ATMOSPHERIC TEMPERATURE (K) - 3.00 SURFACE TEMPERATURE (K) « 288.15 SURFACE PERMITTIVITT - 10.00 -J 0.00 DENSITY (GM/M*«3) - 1.0 GRIN (DBI) = 48.28 RANGE (KM) =■ 36000.00 TRUP (OEG.) - 90.00 o o BORESIGHT G/T (dBK m t\j - o o 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Z E N I T H ANGLE (DEGREES) Figure 9.17. Boresight G/T ratio for the cloudy sky example of antenna temperature. Vertical polarization and 36,000 km range. Terrestrial source for 0 >90°. z 231 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCT IGHZ) - 11.00 STATION HEIGHT 1KM) - 0.238 ANTENNA HEIGHT. (H) - 2.000 RELATIVE HUMIDITY AT SURFACE m - 58.64 HATER VAPOR SCALE HEIGHT (KM) - 2.00 SURFACE PRESSURE (MBRR) - 1023.0 NUMBER OF CLOUD LRYERS «1. LATER 1. BASE (KM)- 0.660 CEILING (KM)* 2.7G0 EXO-ATMOSPHERIC TEMPERATURE (K) - 3.00 SURFACE TEMPERATURE (K) - 268.15 SURFACE PERMITTIVITY - 10.00 -J 0.00 HORIZONTAL POLARIZATION DENSITT (GM/M**3)« 1.0 BRIGHTNESS TEMPERRTURE (KELVIN) co- CVJ- o- co- to- CD O JCDco inCO- 0.0 30.0 60.0 90.0 1 2 0.0 150.0 180.0 Z E N I T H R N G L E (DEGREES) Figure 9.18. Brightness temperature profile for a cloudy condition. Horizontal polarization. 232 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCT (GHZ) - 11.00 STATION HEIGHT (KM) - 0.238 ANTENNA HEIGHT. (M) - 2.000 RELATIVE HUMIDITT AT SURFACE (X) - 58.64 HATER VAPOR SCALE HEIGHT (KH) - 2.00 SURFACE PRESSURE IHBAR) - 1023.0 NUMBER OF CLOUD LATERS -1. LATER 1. BASE IKM)» 0.660 'CEILING (KM)- 2.700 EXO-ATMOSPHERIC TEMPERATURE (K) - 3.00 SURFACE TEMPERATURE (K) - 288.15 SURFACE PERMITTIVITY - 10.00 -J 0.00 *b CHIN (DBI) - 46.28 DENSITT IGH/M««3)- 1.0 TAUP IDEG.) - 0.00 CDid - (KELVIN) total o MAIN BEAM CM* O RNTENNA TEMPERATURE CO COID- CO- O cnco- co incn- 0.0 30.0 60.0 90.0 12 0.0 150.0 Z E N I T H A N G L E (DEGREES) Figure 9.19. Example of antenna temperature for a cloudy condition. Horizontal polarization. Total antenna temperature wi main beam contribution. 233 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I GAIN IOBI) - 48.28 DENSITT IGM/M--3) - 1.0 TAUP IDEG.) - 0.00 30.0 20.0 10.0 G/T (dB K ” ) 40.0 50.0 FREQUENCT (GHZ) - 11.00 STRTION HEIGHT (KM) - 0.238 RNTENNR HEIGHT. (M) - 2.000 RELRTIVE HUHIDITT RT SURFACE 17.) - 58.64 HATER VAPOR SCALE HEIGHT (KM) - 2.00 SURFACE PRESSURE (MBflR) - 1023.0 NUMBER OF CLOUD LATERS =1. LATER 1. BASE (KM)- 0.660 CEILING (KM) - 2.700 EXO-ATMOSPHEP.IC TEMPERATURE (K) - 3.00 SURFACE TEMPERATURE IK) - 288.15 SURFACE PERMITTIVITT - 10.00 -J 0.00 o. 0.0 30.0 60.0 120.0 150.0 180.0 Z E N I T H A N G L E (DEGREES) Figure 9.20. G/T ratio for the cloudy sky example of antenna temperature. Horizontal polarization. 234 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREQUENCT (GHZ) - 11.00 STATION HEIGHT (KM) - 0.230 ANTENNA HEIGHT. (M) - 2.00G RELATIVE HUM ID ITT AT SURFACE 17.) - 56.64 HATER VAPOR SCALE HEIGHT (KM) - 2.00 SURFACE PRESSURE IMBAAl - 1023.0 NUMBER OF CLOUD LAYERS «1. LATER 1. BASE (KM)- 0.660 CEILING (KM) - 2.700 EXO-ATMOSPHERIC TEMPERATURE IK) - 3.00 SURFACE TEMPERATURE (K) - 2 0 8 . 1 5 SURFACE PERMITTIVITY - 10.00 -J 0.00 o DENSITY (GM/M — 3) - 1.0 GAIN IDBI) - 48.20 RANGE (KM) - 36000.00 TAUP IDEG.) = 0.00 o T3 t> . If) U J O CO o o 30.0 60.0 90.0 120.0 150.0 180.0 Z E N I T H R N G L E (DEGREES) Figure 9.21. Boresight G/T ratio for the cloudy sky example of antenna temperature. Horizontal polarization and 36,000 km range. Terrestrial source for 9z>90°. 235 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E. ANTENNA TEMPERATURE DURING RAIN The brightness temperature profile for a condition of rain that was presented in Figures 6.5 and 6.6 calculate antenna temperature. will be used in this example to Figure 9.22 shows the brightness temperature profile as a function of zenith angle for the case of vertical polarization. Figure 9.23 is the corresponding antenna temperature for vertical polarization. This example clearly demonstrates the serious effect that rain has on antenna temperature. The main beam is seen to collect almost all of the noise temperature contribution. For the rain curves, the probability stated is the probability that the rain rate will exceed the percentage probability in one year. Thus, when applied to antenna temperature, the probability is the probability that the antenna temperature will exceed the value given, because of rain. The G/T ratio of the antenna during this rain event is plotted in Figure 9.24. Comparison of this curve to one of the earlier G/T curves shows the degradation in system performance that rain causes. This is further demonstrated by the boresight G/T curve shown in Figure 9.25. The range in this example is again 36,000 km. As in the earlier examples, once the zenith angle is greater than 90°, the source is assumed to be on the surface of the earth. To generate the boresight G/T for a rain condition, the rain model must be used to find the attenuation to the source. For the sources outside the rain or earth-based sources, this requires the same application of the Global Model as discussed in Chapters V and VI. 236 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For CLIMATE ZONE - 02 LATITUDE IDEG) = 40.00 FREQUENCT IGHZ1 - 11.00 STATION HEIGHT IKM) «= 0.238 ANTENNA HEIGHT. (M) - 2.000 RRIN RATE IMM/HR) - 12.70 PROBABILITY = 0.1357 EXO-ATMOSPHERIC TEMPERATURE (K) = 3.00 SURFACE TEMPERATURE IK] * 288.15 SURFACE PERMITTIVITY = 10.00 -J 0.00 VERTICAL POLARIZATION O' CO' (KELVIN) ID in ro od BRIGHTNESS TEMPERATURE o CD ID in oj O O' CD ID in rn O 0.0 30.0 60.0 ZENITH Figure 9.22. 90.0 ANGLE 120.0 150.0 180.0 (DEGREES) Brightness temperature profile for the rain example. Vertical polarization. 237 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CLIMATE ZONE - D2 LATITUDE IDEG) - 90.00 FREQUENCY (GHZ) « 11.00 STATION HEIGHT (KM) - 0.238 A N TENNA HEIGHT. (Ml - 2.000 RAIN RATE (MM/HR) « 12.70 PROBABILITY - 0.1357 E X O - ATMOSP HERIC TEMPERATURE (K) - 3.00 SURFACE TEMPERRTURE IK) - 288.15 SURFACE PERMITTIVITY - 10.00 -J 0.00 * b GAIN (OBI) - *48.28 TAUP (DEG.) - 90.00 O' r*CD' total TEMPERRTURE (KELVIN) id- : MAIN BEAM “ o ev en (O' CDCO- cv RNTENNfl "o CD CD CDcntv- 0.0 30.0 60.0 ZENITH Figure 9.23. 90.0 RNGLE 120.0 (DEGREES) 150.0 180.0 Antenna temperature for the rain example. Vertical polarization. Total antenna temperature with main beam contribution. 238 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CL IMR TE ZONE - 02 LATITUDE (DEG) - *10.00 FREQUENCT (GHZ) « 11.00 STATION HEIGHT (KM) = 0.238 ANTENNA HEIGHT. 1M) * 2.000 RAIN RATE (MM/HR) = 12.70 PROBABILITY = 0.1357 EXO-RTM0SPHERIC TEMPERATURE (K) - 3.00 SURFACE TEMPERATURE (K) «= 288. 15 SURFACE PERMITTIVITY = 10.00 -J 0.00 o GAIN (OBI) = U8.28 TAUP (DEG.) *= 90.00 in o o © “ 1 i !. PU O o 30.0 60.0 ZENITH Figure 9.24. 90.0 ANGLE 12 0. 0 150.0 180.0 (DEGREES) G/T ratio for the rain example of antenna temperature. Vertical polarization. 239 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CLIMRTE ZONE - D2 LATITUDE (DEG) - NO.00 FREQUENCT (GHZ) - 11.00 STATION HEIGHT (KM) = 0.238 ANTENNA HEIGHT. (M) = 2.000 RAIN RATE (MM/HR) - 12.70 PROBABILITY = 0.1357 EXO-ATMOSPHERIC TEMPERATURE (K) * 3.00 SURFACE TEMPERATURE (K) * 288.15 SURFACE PERMITTIVITY *= 10.00 -J 0.00 GAIN (DBI) = 48.28 RANGE 'KM) = 36000.00 TAUP (DEG.) = 90.00 o in ! i .....r ....r .... . .. : .... ] ... 1 ! i 1 ! i i i • i j ; 1 ! i i ... J........ o i 1 i i ......i___j...... ! r i [ I ..... L _ J _ i m i i i ! ! : jz 0 < f= V> CO IxJ i CO o : V ; i • .....i I : : • i i ; I i I ! • ! ..... :....... J . ...... ___ 4 ... .... ' I ......... : 5 i : 30.0 60.0 ... 4 ^ ... ; i J .. i i i i ; ? ......!__ ; r! ’i— ; ! ! i —... ! ... ........ ......: .... 1....... i : ......i...... i...... 1 ! .. i r : ! 1 I ! i ! : i s 1 120.0 ! j ... t .... r .... —■ __ j RNGLE i ! ..... J..._.! .. i i ; 90.0 ! ; 1 7! • _.J..... I.......... •: ..... 1...... ZENITH Figure 9.25. i ...!...... • i : • ! i 0.0 .... i i [ : 1 ..... : : 1 -------- --...... 1...... 1 .. ----- ------I T :----- ...... ... ! ..■ ’ " J ” ' ! ----- r - ! I • | . ..... r ... i...... _ i .. __ ; 1_ > ; ......... .1 ... | ! i ......j ..... i i - ...r - ...i•...... ■ j ! I ; ..... ; 1 ' " i ...... ! i ! i . cc o : j ; TO I ; ! i ..... ._ !....L J o i i ..... i...... u ... ..... i ....... T - 150.0 ! ! ■ 180.0 (DEGREES) Boresight G/T ratio for the rain example of antenna temperature. Vertical polarization and 36,000 km range. Terrestrial source for 9z>90.0°. 240 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. airborne sources within the rain, the surface projection (D) of the range (R) between antenna and target is used in the Global Model to obtain the attenuation. The attenuation of the range path is then found from the attenuation of the surface path through the use of Equations (5.12) and (5.13) where in this case H is the altitude of the target. The curves for the horizontally polarized antenna in this environmental condition are given in Figures 9.26, 9.27, 9.28 and 9.29. F. SUMMARY This chapter has given the results of antenna temperature calculations for three different environmental conditions. A clear sky, clear sky with a cloud layer and a rain condition were considered. These examples have demonstrated the utility of an antenna temperature model for the analysis and design of microwave antenna systems. Time and space limitations prevent utilizing the model to reveal general properties of antenna temperature. However, the calculated antenna temperature of the antenna considered in this chapter was able to demonstrate the importance of earth emissions and the importance of the main beam in determining the total noise temperature. The relationship between the antenna and the environment could be investigated more thoroughly in the future by concentrating on a single antenna design and using the Antenna Temperature Code to simulate a wide variety of operational conditions. 241 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CLIMATE ZONE - D 2 LATITUDE (DEG) - H O . 00 FREQUENCY (GHZ) - 11.00 STATION HEIGHT (KM) * 0.238 ANTENNA HEIGHT. IM) - 2.000 RAIN RATE IMM/HR) = 12.70 PROBABILITY ■*= 0.1357 E X0-ATM0SPHER1C TEMPERATURE (K) = 3.00 SURFACE TEMPERATURE (K) - 238.15 SURFACE PERMITTIVITY * 10.00 -J 0.00 HORIZONTAL POLARIZATION o* (KELVIN) oo- ff* (V BRIGHTNESS TEMPERRTURE CTV CO CO- in to- O CD- CO CO in to cu 0.0 30.0 60.0 ZENITH Figure 9.26. 90.0 ANGLE 120.0 (DEGREES) 150.0 180.0 Brightness temperature profile for the rain example. Horizontal polarization. 242 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CLIMATE ZONE * D2 LATITUDE IDEG) - M O . 00 FREQUENCT (GHZ) - 11.00 STATION HEIGHT (KM) - 0.238 ANTENNR HEIGHT. (M) - 2.000 RAIN RATE (MM/HR) - 12.70 PROBABILITY - 0.1357 EXO-RTMOSPHERIC TEMPERRTUR E (K) » 3.00 SURFACE TEMPERATURE (K) - 288.15 SURFACE PERMITTIVITY - 10.00 -J 0.00 CRIN (OBI) - 48.28 TRUP (DEG.) - 0.00 c* CD' CD- total id - RNTENNfl TEMPERRTURE (KELVIN) MAIN BEAM em eu cn co CO ID emCU- cn CO- ID- CO OJ 0.0 30.0 60.0 ZENITH Figure 9.27. 90.0 RNGLE 120.0 150.0 180.0 (DEGREES) Antenna temperature for the rain example. Horizontal polarization. Total antenna temperature with main beam contribution. 243 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CLIMRTE ZONE - D2 LATITUDE (DEG) - 40.00 FREQUENCT (GHZ) - 11.00 STATION HEIGHT (KM) * 0.238 ANTENNA HEIGHT. (M) «• 2.000 RAIN RATE (MM/HR) = 12.70 PROBABILITY *= 0.1357 EXO-ATMOSPHERIC TEMPERATURE (K) = 3.00 SURFACE TEMPERATURE (K) = 288.15 SURFACE PERMITTIVITY * 10.00 -J 0.00 GAIN (OBI) •= 48.28 o TAUP (DEG.) = 0.00 in . G/T (dBK~ cn . . ...r .. j. 0.0 30.0 60.0 90.0 ZENITH ANGLE Figure 9.28. 120.0 150.0 180.0 ( DEG REE S) G/T ratio for the rain example of antenna temperature. Horizontal polarization. 24A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CLIMATE ZONE - D2 LATITUDE IDEG) - 40.00 FREQUENCY (GHZ) = 11.00 STATION HEIGHT (KM) = 0.238 ANTENNA HEIGHT. (M) ■= 2.000 RAIN RATE (MM/HR) = 12.70 PROBABILITY = 0.1357 EXO-ATMOSPHERIC TEMPERATURE IK) - 3.00 SURFACE TEMPERATURE (K) - 288.15 SURFACE PERMITTIVITY = 10.00 -J 0.00 o GAIN (DBI) - 48.28 RANGE (KM) - 3SOOO.OO TAUP (DEG.) - 0.00 o in BORESIGHT G/T (dBK“ o o o (M . I... .. !- • o 30.0 60.0 90.0 ZENITH ANGLE Figure 9.29. 120.0 150.0 180. Cl (DEGREES) Boresight G/T ratio for the rain example of antenna temperature. Horizontal polarization and 36,000 km range. Terrestrial source for 9 >90.0°. z 245 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER X SUMMARY AND CONCLUSIONS This study has presented a model for the calculation of antenna temperature, for earth based microwave antennas in the 1-40 GHz frequency range. The antenna temperature is calculated by simulating the operational environment of the antenna. This simulation provides the brightness temperature profile of the environment which is integrated with the antenna pattern to produce the antenna temperature. The antenna pattern is obtained from another source which could be measured data or calculations from an antenna code. The environmental model created in this study has been implemented to work in conjunction with the OSU Reflector Antenna Code [1], The antenna temperature calculation is done using the far-field pattern of the antenna. Therefore it is implicitly assumed that there is no near field region of the antenna. Further study may wish to investigate the influence of the environment in the near field on the antenna temperature. The environmental model obtains the brightness temperature profile by taking into account the constituents of the environment which interfere with microwave propagation. The fundamental component is the atmosphere because it is the propagation medium used by the antenna. This study has emphasized the modeling of the atmosphere for that reason. 246 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tvo separate models have been used to simulate the atmosphere in the environmental model. The first model that was presented was for an atmosphere in which rain is not present. This model allows the antenna temperature calculation to include the effects of atmospheric oxygen and water vapor. included. The capability for simulating clouds and fog has also been The absorption coefficients of these three constituents are calculated using methods that have been established in the literature. The absorption coefficients depend on the distribution of the density of the constituents, temperature and total pressure, with respect to altitude. The dry air pressure and temperature variation can be provided by any one of the standard or supplemental atmospheres that are available. study. The U.S. Standard Atmosphere, 1976 was the choice for this Further investigation of the supplemental atmospheres and compensation for local variations in meteorological parameters is needed in the future. The brightness temperature for the atmosphere without rain was generated by a technique that avoids doing a rigorous radiative transfer calculation for most antenna pattern angles. This technique uses the radiative transfer integration at two observation angles, but then uses algebraic equations to find the brightness temperature for the remaining observation angles. This method was adopted as a computational time saving measure in order to keep the time and cost involved with an antenna temperature calculation practical. This technique is an approximation and has no theoretical justification at present. However, comparison with brightness temperatures obtained by integration show 247 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. favorable results, especially for the systems design application being contemplated for this study. Future work on this aspect of the environmental model may wish to focus on establishing the theoretical basis for this approach, if one exists. Further, it would be prudent to establish the limits of accuracy on this technique. Those who wish to exceed the limits can be provided with the option to numerically integrate along each observation path to avoid any error that may be present in the approximate technique. The other atmospheric model, that has been included, is for treatment of the atmosphere during rain. Rain requires separate treatment because as a meteorological event, it does not lend itself well to prediction or deterministic models, especially from the point of view of systems design. The approach to generating the brightness temperature due to rain was to include a statistical prediction of rain attenuation in the environmental model. The Global Model of Crane was used mainly because of the sensitivity to world climate variations that has been built into the model. The brightness temperature is calculated from the attenuation by assuming the atmospheric temperature is constant along the observation path through the rain. scattering into the observation path. This approach ignores The significance of rain scattering, with regard to the accuracy of the antenna temperature calculation, should be established in the future. This approach also ignores the gaseous absorption and cloud absorption of the atmosphere 248 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. during rain, although these components will be insignificant during moderate to heavy rains. Further development of the rain model should also concentrate on developing a technique for combining the rain model with the clear atmosphere model. Such a technique would allow the effects of all significant atmospheric constituents to be included in the brightness temperature calculation. This is especially important when trying to determine the brightness temperature during a light rainfall. Under this condition, the contribution of the rain is on the same order as the contribution from clouds and gases. Therefore a more accurate calculation could be achieved by using a single model which includes all of these effects. One approach that has been suggested is to use the Global Model to generate a statistically averaged path rain rate [47]. This rain rate could then be used to find the absorption coefficient of the rain along the path. This coefficient can be added to the absorption coefficients of the gases and clouds to produce the total absorption coefficient of the atmosphere. The brightness temperature results from a radiative transfer calculation using this total absorption coefficient. This approach would allow for a single atmospheric model and it would avoid the assumption of a constant rain medium temperature. The environmental model contains a provision to include exoatmospheric sources that are uniformly incident on the atmosphere. Generally, the only radiation that is like this is the cosmic background radiation. Future development of the model in this area should include 249 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the capability to define angular regions of space that have equivalent temperatures which are different from the background. In this manner, the contributions of the sun or other strong exo-atmospheric emissions could be included in the brightness temperature calculations. The environmental model simulates the surface of the earth by assuming that it is smooth, flat and homogeneous. This is a simple, first approximation but it allows calculations to be performed without requiring that a detailed description of the local terrain, surrounding the station, be given. For observation paths that include the earth, the environmental model includes the reflection from the surface, emission from the surface and it accounts for the atmospheric absorption and emission that occurs between the surface and the observation point. The surface model can be improved in the future by allowing for some variability in the surface characteristics. The permittivity of the surface could be made a function of the coordinates in the station coordinate system. Rough terrain could be accounted for by including a scattering mechanism for the surface. A ray tracing capability could be included in the model to account for the curvature of the earth and for large terrain features such as mountains and valleys. Future study should also investigate the brightness temperature for near horizon and horizon observation paths. The present model is capable of calculating the antenna temperature for linearly polarized antennas only. The extension of the model to circularly polarized antennas should be addressed in the future. 250 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The model developed in this study also includes the effects of the antenna network in the calculation of antenna temperature. Consideration is given to both the VSVR of the network as well as the ohmic losses which occur in the network. The model relies on some outside source to provide the VSWR information but contains a simulation of networks which allow for ohmic loss calculations. No improvements are anticipated in this area. In conclusion, this model should provide the designer with a versatile and practical tool for system design. With it, the designer can simulate a variety of environmental conditions to determine their effect on communications system performance. This information will allow the designer to analyze the environment and take steps to minimize the environmental noise that enters the system. Further, this model provides a means by which noise problems in existing communications links may be investigated. Antenna temperature measurements are anticipated in the future. These measurements will provide data for verification of the accuracy of the model. The comparison of the measured data with the calculations should indicate the areas of the model which require improvement. The improvements could follow the suggestions of this chapter or developments, not anticipated in this study, could be required. A description of the computer code which implements the environmental model and performs the antenna temperature calculation is given in the Antenna Temperature Code — User's Manual [61]. This report also contains instructions on the use of the code. 251 with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES [1] T.H. Lee, R.C. Rudduck, "Microwave Antenna Technology, Final Report Volume III: Reflector Antenna Code — User's Manual," Final Report 717822-3, The Ohio State University ElectroScience Laboratory, May 1987. [2] R.C. Rudduck, K.M. Lambert and T.H. Lee, "Microwave Antenna Technology, Final Report, Volume I: Antenna Designs, Calculations and Selected Measurements," The Ohio State University ElectroScience Laboratory, Report 717822-1, May 1987. [3] T.H. Lee, K.M. Lambert and R.C. Rudduck, "Microwave Antenna Technology, Final Report, Volume II: Complete Set of Antenna Pattern Measurements," The Ohio State University ElectroScience Laboratory, Report 717822-2, May 1987. [4] H.C. Ko, "Temperature Concepts in Modern Radio," The Microwave Journal, Vol. 4, pp. 60-65, June 1961. [5] H.C. Ko, "Radio Telescope Antennas," in Microwave Scanning Antennas, Vol. 1, edited by R.C. Hansen, Chapter 4, Academic Press, New York, 1964. [6] J.D. Kraus, Radio Astronomy, McGraw-Hill, New York, 1966. [7] R.E. Collin, Antennas and Radiowave Propagation, McGraw-Hill, New York, 1985. [8] H. Nyquist, "Thermal Agitation of Electric Charge in Conductors," Physical Review, Vol. 32, pp. 110-113, 1928. [9] A.C. Ludwig, "The Definition of Cross Polarization," IEEE Transactions on Antennas and Propagation, Vol. AP-21, pp. 116-119, January 1973. [10] S. Chandrasekhar, Radiative Transfer, Dover, New York, 1960. [11] A. Ishimaru, Wave Propagation and Scattering in Random Media, Academic Press, New York, 1978. [12] E. Novotny, Introduction to Stellar Atmospheres and Interiors, Oxford University Press, New York, 1973. — [13] J.W. Waters, "Absorption and Emission by Atmospheric Gases," in Methods of Experimental Physics, Vol. 12B, edited by M.L. Meeks, Chapter 2.3, Academic Press, New York, 1976. 252 with permission of the copyright owner. Further reproduction prohibited without permission. [14] K.T. Lin, "Site-Diversity Attenuation Measurements at 28 GHz by Radiometers for an Earth-Space Path,” Dissertation for Ph.D., The Ohio State University, Department of Electrical Engineering, Columbus, Ohio, 1986. [15] R.E. Leonard, "Calculation of Mean Path Temperature Involved in Radiometrically Inferred Attenuation," Thesis for M.Sc., The Ohio State University, Department of Electrical Engineering, Columbus, Ohio, 1984. [16] L.J. Battan, Fundamentals of Meteorology, Prentice-Hall, New York, 1979. [17] E.K. Smith, "Centimeter and Millimeter Wave Attenuation and Brightness Temperature Due to Atmospheric Oxygen and Water Vapor," Radio Science, Vol. 17, No. 6 , pp. 1455-1464, Nov.-Dec. 1982. [18] P.W. Rosenkranz, "Shape of the 5-mm Oxygen Band in the Atmosphere," IEEE Transactions on Antennas and Propagation, AP-23, pp. 498-506, 1975. [19] N.E. Gaut and E.C. Reifenstein, "Interaction Model of Microwave Energy and Atmospheric Variables: Atmospheric Model for Effects of Water Vapor, Liquid Water, and Ice Upon Radiative Transfer Process at Microwave Frequencies and in the Far Infrared," Rep. NASA-CR61348(13), Environmental Research and Technology, Lexington, Mass., 1971. [20] URSI Commission F Working Party," URSI Working party report: Attenuation by Oxygen and Water Vapor in the Atmosphere at Millimetric Wavelengths," Radio Science, Vol. 16, No. 5, pp. 825829, Sept.-Oct. 1981. [21] "U.S. Standard Atmosphere, 1976," NDAA-S/T 76-1562, U.S. Government Printing Office, Washington, D.C., October 1976. [22] S.L. Valley (Ed.), Handbook of Geophysics and Space Environments, McGraw-Hill, New York, 1965. [23] E.K. Smith and J.W. Waters, Microwave Attenuation and Brightness Temperature Due to the Gaseous Atmosphere, JPL Publication 81-81, Jet Propulsion Laboratory, Pasadena, California, 1981. [24] P. David and J. Voge, Propagation of Waves, Pergamon Press, New York, 1969. [25] L.J. Ippolito, Jr., Radiowave Propagation in Satellite Communications, Van Nostrand Reinnold, New York, 1986. 253 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [26] S.D. Slobin, "Microwave Noise Temperature and Attenuation of Clouds: Statistics of These Effects at Various Sites in the United States, Alaska and Hawaii," Radio Science, Vol. 17, No. 6 , pp. 1443-1454, Nov.-Dec. 1982. [27] D.C. Hogg and T.S. Chu, "The Role of Rain in Satellite Communications," Proceedings of the IEEE, Vol. 63, pp. 1308-1331, September 1975. [28] R.K. Crane, "Attenuation Due to Rain — A Mini-Review," IEEE Transactions on Antennas and Propagation, Vol. AP-23, pp. 750-752, September 1975. [29] R.K. Crane, "Propagation Phenomena Affecting Satellite Communication Systems Operating in the Centimeter and Millimeter Wavelength Bands," Proceedings of the IEEE, Vol. 59, No. 2, February 1971. [30] J.O. Laws and D.A. Parsons, "The Relation of Raindrop Size to Intensity," Transactions of the American Geophysical Union, Vol. 24, pp. 452-460, 1943. [31] J.S. Marshall and V.M. Palmer, "The Distribution of Raindrops with Size," Journal of Meteorology, Vol. 5, pp. 165-166, 1948. [32] J. Joss, J.C. Thams, and A. Waldvogel, "The Variation of Raindrop Size Distributions at Locarno," Proceedings of the International Conference on Cloud Physics, Toronto, Canada, pp. 369-373, 1968. [33] D.J. Fang and C.H. Chen, "Propagation of Centimeter/Millimeter Waves Along a Slant Path Through Precipation," Radio Science, Vol. 17, No. 5, Sept.-Oct. 1982. [34] J. Goldhirsh and I. Katz, "Useful Experimental Results for EarthSatellite Rain Attenuation Modeling," IEEE Transactions on Antennas and Propagation, Vol. AP-27. No. 3, pp. 413-415, 1979. [35] J.W. Ryde and D. Ryde, "Attenuation of Centimetre and Millimetre Waves by Rain, Hail, Fogs, and Clouds," Rep. No. 8670, Research Laboratories of the General Electric Company, Wembley, England, 1945. [36] J.W. Ryde, "The Attenuation and Radar Echoes Produced at Centimetre Wavelengths by Various Meterological Phenomena," in Meteorological Factors in Radio Wave Propagation, the Physical Society, London, pp. 169-188, 1946. [37] K.L.S. Gunn and T.W.R. East, "The Microwave Properties of Precipitation Particles," Quarterly J. Royal Meteor. Soc., Vol. 80, pp. 522-545, 1954. 254 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [38] G. Hie, Ann. Physik, Vol. 25, p. 377, 1908. [39] Van De Hulst, Light Scattering by Small Particles, John Wiley & Sons, New York, 1957. [40] R.L. Olsen, D.V. Rogers and D.B. Hodge, "The aR^ Relation in the Calculation of Rain Attenuation," IEEE Transactions on Antennas and Propagation, Vol. AP-26, No. 2, pp. 318-329, March 1978. [41] CCIR, Report 721-1, "Attenuation by Hydrometeors in Particular Precipitation, and other Atmospheric Particles," in Volume V, Propagation in Non-Ionized Media, Recommendations and Reports of the CCIR— 1982, International Telecomm. Union, Geneva, pp. 167-181, 1982. [42] A. Ishimaru and R.L.T. Cheung, "Multiple-Scattering Effect on Radiometric Determination of Rain Attenuation at Millimeter Wavelengths," Radio Science, Vol. 15, No. 3, pp. 507-516, May-June 1980. [43] L.J. Ippolito, R.D. Kaul and R.G. Wallace, Propagation Effects Handbook for Satellite Systems Design -- A Summary of Propagation iinpairments on 10 to lOO GHz Satellite links with Techniques f o ~ System Design, Report NASA PR-1082(03), NASA Headquarters, Washington, D.C., June 1983. [44] R.K. Crane, "Prediction of Attenuation by Rain," IEEE Transactions on Communications, Vol. C0M-28, No. 9, September 1980. [45] R.K. Crane, "A Two-Component Rain Model for the Prediction of Attenuation Statistics," Radio Science, Vol. 17, No. 6, pp. 13711387, 1982. [46] R.K. Crane, personal communication, 1986. [47] C.A. Levis, personal communication, 1987. [48] W.H. Peake, "Interaction of Electromagnetic Waves with Some Natural Surfaces," IRE Transactions on Antennas and Propagation, Vol. AP-7, pp. S324-S329, December 1959. [49] V.M. Papadopoulos, "Propagation of Electromagnetic Waves in Cylindrical Waveguides with Imperfectly Conducting Walls," Quart. J. Mech. and Appl. Math, Vol. 7, pp. 325-334,Sept. 1954. [50] R.E. Collin, Field Theory of Guided Waves, Sec.5.3, McGraw-Hill, New York, 1960. [51] R.A. Waldron, Theory of Guided Electromagnetic Waves, Van Nostrand Reinhold, London, 1969. 255 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [52] R.E. Collin, Foundations for Microvave Engineering, McGraw-Hill, New York, 1966. [53] S.A. Schelkunoff, Electromagnetic Vaves, D. Van Norstrand, Princeton, 1943. [54] R.F. Harrington, Time Harmonic Electromagnetic Fields, McGraw-Hill, New York, 1961. [55] M.S. Narasimhan and B.V. Rao, "Transmission Properties of Electromagnetic Vaves in Conical Waveguides," Int. J. Electron., Vol. 27, pp. 119-139, August 1969. [56] P.J.B. Clarricoats and A.D. Olver, Corrugated Horns for Microwave Antennas, Peter Peregrinus, London, 19847 [57] P.J.B. Clarricoats, A.D. Olver and S.L. Chong, "Attenuation in Corrugated Circular Waveguides, Part I, Theory," Proc. IEEE, 1975, Vol 122, p. 1173. [58] P.J.B. Clarricoats and P.K. Saha, "Propagation and Radiation Behavior of Corrugated Feeds, Pt. I, Corrugated Waveguide Feeds," Proc. IEE, 1971, 118, p. 167. [59] P.J.B. Clarricoats and P.K. Saha, "Attenuation in Corrugated Circular Waveguide," Electron. Lett. 1970, Vol. 6 , p. 370. [60] P.J.B. Clarricoats and A.D. Olver, "Low Attenuation in Corrugated Circular Waveguides," Electron. Lett, 1973, Vol. 9, pp. 376-377. [61] K.M. Lambert, "Microwave Antenna Technology, Final Report, Volume V: Antenna Temperature Code — User's Manual," The Ohio State University ElectroScience Laboratory, Report 717822-5, May 1987. 256 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A EVALUATION OF THE PHI INTEGRAL IN THE ANTENNA TEMPERATURE CALCULATION The purpose of this appendix is to provide the details behind the analytic evaluation of the phi integral in the antenna temperature calculation. In the text, the integral was presented as, N • e i=l ■ i:* J9Q. ’Jl T(0,4>) E(0,4>) sin|@| d0 d <#> . (A.l) • «-J l where ♦q 1+1 . r*p.i i*N , (A.2) (A. 3) V ° ’ (A.4) and the summation is over all the phi cuts calculated. ^ represents the lower $ boundary at which points are calculated and <|>p the upper i boundary. The temperature is calculated along those contours of constant phi as well. Since no functional information is available between calculated phi points, an interpolation can be used- to represent 257 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the function there. Linear interpolation is a scheme which will allow analytic integration in phi. Using linear interpolation between and i 4>q , the functions become, *~*Q. E<e’*> - *~*P. 9Qi E(0’*P*i> " VT ^ ~ V E(0’*O > ui (A’5) and <Mq T < e >+> = *-4»p -f ~ A ~ 9Pi 9Q. T < 9 ’ +P > *i " 9?i 9q4 T < e >+Q ui > • ( A - 6) Taking the magnitude squared of (A.5) and multiplying it with (A.6) gives, Qi |E(e,<J») |2 T(6,*) = |E(0, <#» )|2 T(0,+_ ) + *P -+Q i i ( < M P )2(<h*n ) i i ( V y N 9 ,+q .)I! T(e,*Pi) T(e, l - [e *(0, ^>£(0, +q ^)+E(0, +p^)E*(0, +Qi)] l <*-4>n )2(*~4>p ) wi *i (9p -*q )3 *i i |E(0,«J»p )|2 T(0,*Q ) 258 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p. i + p " *Q . i i |e(©, ♦ ) |2T(e,4> i i ) + i )2(<h*p )1 i T (0, 4^) [e *(o ,4 ^ )E(e, ^ ) + E ( e , ♦p )e *(e, ♦ )]. (♦P.-+Q.>3 l l (A.7) When Equation (A.7) is substituted into Equation (A.l), the order of the integrations may be interchanged and the phi integration can be carried out. Doing so will require four basic integrals due to the various phi variations in Equation (A.7). They are, v * Qi ■+pi *A = d<|> = (A.8) [V * '^P. 2( 4~ 4’q s ( VV IC = V-+Q. l l 1 + 9Qi ) 4>p -+Q (A.9) d+- ^ L. d+ (*P -+Q >’ i i ?i Qi 12 and 259 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.10) Using the integrals IA , Ig, IQ, and Ip, Equation (A.l) can be shown to reduce to, JL *p -*Qj [i -i=l E - QjO) sin|e| d0 -n (A.12) where 0.(0) = | E ( e , ) |2 T(0,<|> ) + |E(0,+ )|2 T(0,*n ) + i i ui Qi 1 + 3 |E(0,*P )|2 T(0,<f> ) + |E(0, i i )|* T(0,4>p ) + i Fi + (T<0,^) + T(0,+Q )[E*(e,^)E(0,^)+ E(0f*^)E*(0,* )] (A.13) which is the result given in Equations (2.33) and (2.34) in the text. One final comment should be made concerning the term, C(9,<fr ,♦ ) = E*(0,4>p )E(0,4> ) + E(0,* )E*(0,+n ) *i Ui Fi Qi Pi Qi 260 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.14) which appears in Equation (A.13). Since most pattern data are provided in phasor format, some manipulation is required to evaluate C(0,*p i Let i E(e,*p ) = A/ a i (A.15) E<e,*Q ) = B/ b , Then C(e,<fp ,*n ) = AB/b-a + AB/-(b-a) i i Expanding the phasors, C(0,*_ i ) = AB [cos(b-a) + jsin(b-a) + cos(b-a) - jsin(b-a)] i and c(0,*p ,*Q ) = 2AB cos(b-a) l l This allows evaluation of C(9,4>p l (A.16) ) from the phasor format. i 261 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX B DERIVATION OF SLANT PATH LENGTHS IN THE ATMOSPHERE Consider the spherical earth and atmosphere geometry shown in Figure B.l. |**L cos 6 — L sin 6 Figure B.l. Geometry for the calculation of slant path length. 262 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is desired tofind the slant path lengthL to a height H, along the direction defined by the elevation angle 9. R is the radius of the earth. To account for atmospheric refraction of the electromagnetic propagation, the effective earth radius, Rg should be used. the effective path length traveled through the atmosphere. Hence, L is In this appendix, R will be used to represent the radius for notational simplicity. of Since this is a mathematical development, the actual values the variables are not important. An equation for L can be found by using the the large triangle in the figure. Pythagorean theorem to This action results in the equation, (R + L sin(0))2 + (L cos(0))2 = (R+H)2 (B.l) Rearranging this equation produces, L 2 + 2RL sin(0) - (2HR+H2) = 0 which can be solved by the quadratic equation. (B.2) Doing so gives the solution, L = -R sin(0) ± lR2sin2(0) + 2HR + H 2 . The proper root is found by considering the case where 0=0°. (B.3) For this angle, Equation (B.3) reduces to 263 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0=0° L = ± 42HR + H2 from which it is obvious that the positive root is correct. (B.4) Hence, the slant path length can be written as L = R -sin(9) + jsin2(0) + + (|) (B.5) which is the form which appears in Equation (4.53) of the text. For heights within most of the atmosphere, H « R and hence iY « 1 So for the atmosphere, this term can be ignored in Equation (B.5). Doing this and multiplying Equation (B.5) by a factor equal to unity, L = R -sin(0) + sin2(0) + 2H sin(9) + isin2(8) + 2H/R .sin(0) + 4sin2(0) + 2H/R . produces a commonly stated form of the slant path length, 264 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B. 6) L = ________ ?H____________ _ (B7) sin(0) + 4sin2(0) + 2H/R For the lower atmosphere, where most effects to microwave propagation occur, the condition 2H . -R « 1 is satisfied. In this case, Equation (B.7) reduces to L ' HH(e) ' H csc<9) <B -8> which is the cosecant law. 265 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX C HORIZONTAL PROJECTIONS OF SLANT PATHS FOR THE GLOBAL HODEL Consider the spherical earth and atmosphere geometry shown in Figure C.l. Figure C.l. Geometry for the calculation of the horizontal projection of the slant path L. 266 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is desired to find the horizontal projection D of the slant path L. The slant path follows the direction defined by the elevation angle 0, from the local surface elevation Hq, to an altitude H in the atmosphere. The effective earth radius is represented by R for notational simplicity and ip is the subtended angle of the slant path, as seen from the center of the earth. The law of cosines allows the following expression for L to be written directly as, L = j(R+H)2 + (R+Hq)2 - 2(R+H)(R+Ho)cos(tp) (C.l) which is used in Equation (5.13) of the text. From the law of sines it is possible to write L________ R+H_______ R+H sin(»p) sin(9O°+0) cos(0) (c sin(«) _ cos(O) R+H " R+H • o (r ( a = 90° - 0 - <p (C.4) and ' Since then 267 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I (C.5) sin(a) = cos(0+<|>) and from Equation (C.3), R+H cos(0+<J») = cos(6) . (C.6) Expanding the cosine of a sum, R+H cos(0)cos(<J») - sin(0)sin(<J/) = cos(0) . (C.7) Now using cos(tp) = - sin2(<(/) (C.8) and defining R+H B = cos(0) ^ (C.9) Equation (C.7) can be written as, cos(0) 4l-sin2(iJ/) = sin(0)sin(i|O + B . Squaring both sides, 268 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.10) I cos2(0)[l-sin2(»J/)] = [sin(0)sin(i|/) ]2 + 2B sin(0)sin(<J>) + B 2. (C.ll) Collecting terms according to powers of sin(«j»), [sin2(0)+cos2(0)]sin2(\j>) + 2B sin(0)sin(\j<) + B 2 - cos2(0) = 0 (C.12) or sin2(<J/) + 2B sin(0)sin(^) + [B2 - cos2(0)] = 0 which can be solved by the quadratic equation. (C.13) Doing so gives sin(t|/) = -B sin(0) + iB2sin2(0) - b2+cos29 (C.14) sin(ij/) = -B sin(0) ± Jl - B 2 cos(0) (C.15) or Substituting Equation (C.9) into Equation (C.15) giver. sin(Y) = -Sin(0)cos(0) ( ^ ] ± l_Cos2(0) ( ^ ) cos(0) 269 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.16) Equivalently, (R+H 'j ( (R+H y sin(^) = -sin(9)cos(9) | ^ J ± Jl-(l-sin2(9)) [ ^ J cos(9) (C.17) and sin(«J/) = ^ < 3 ) ^T(R+H)2 + (sin2(9)-l) (R+Hq)2 - (R+Ho)sin(9)} . (C.18) Finally, sin(*) = ^(R+HQ)zsin2(9) + 2R(H-Ho)+H2-H2 - (R+HQ)sin(9)J (C.19) from which t|/ = sin~1^ g J ^ [ i ( R + H o)2sin2(9) + 2R(H-Hq)+H2-H2 - (R+Ho)sin(9)]]» (C.20) which is used in Equation (5.11) of the text. 270 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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