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

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O rd er N u m b er 8717667
A n environm ental model for calculating the antenna
tem perature o f Earth based microwave antennas
Lambert, Kevin Michael, Ph.D.
The Ohio State University, 1987
U MI
300N. Zeeb Rd.
Ann Arbor, M I 48106
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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
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To my family
ii
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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(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
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(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
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(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
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(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
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(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
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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 ^
,$p
are the boundaries of each region and
18
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The summation is over all regions.
The advantage of writing the
integral in this manner is that the phi integration can be eliminated.
As mentioned previously, antenna patterns are typically slowly varying
functions of phi.
The brightness temperature is also a relatively
smooth angular function.
This will be demonstrated when the brightness
temperature profiles are presented in Chapter VI.
Exploiting the
characteristics of these two factors, linear interpolation in phi may be
used on the terms in the integrand and the phi integration solved
analytically.
In this manner, the evaluation of Ij requires a single
numerical integration instead of a double integration.
Using linear
interpolation, the number of terms N needed to evaluate 1^ in phi is far
fewer than the number required by a numerical integration.
The details
of the analytic evaluation of the phi integral are tedious and are
therefore removed to Appendix A.
The results are all that are required
here.
Evaluation of 1^ results in,
(2.33)
where
19
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I
0.(9) =
E(0,*p ) 2T(0,*
1
i
+ 3
) +
E(9f+n ) 2T(0,*n ) +
E(0,* ) 2T(0,*n ) + E(0,* )|2T(0,+p )
I
i I
i
'
i •
*i
(t (0,*p ) + T(0,*
)][e V , ^
)E(0,*Q )+E(0,^) e \0,* q )]
(2.34)
Although formidable, this equation is readily implemented on the
computer.
Note, the term involving conjugate values of the field is
real as shown in Appendix A.
By setting all the temperature functions to unity in Equations
(2.33) and (2.34) a similar result is obtained for I2,
rfi- V - V
Q2(0) sin|0| d0
-It
i=l
(2.35)
where
Q„(0) =
E(0,<> )
'
i
E(0,$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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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(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
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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
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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
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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
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(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
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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
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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
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(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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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(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
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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
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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
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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
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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
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Q.
O
3
"O
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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
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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
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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.
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-75.0
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-«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
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!
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/ '
i/ :
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i
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:
7
i
i!
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V
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/
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-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---------------
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----- ■
i
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j
•
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:
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I
I
1
!
i
i
i
1
!
;
i
|
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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
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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
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;
.
1 /
*
*
I
S ' ,
i
i
I
1
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|
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i
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------------------- ■------- ^
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i
1
I
I:
<i
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i
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j
j
i1i
j
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i
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j
i
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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= 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
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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
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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
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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
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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
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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
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<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
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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
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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
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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
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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
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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
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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
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{
i
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i
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1
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-- MMIIN DC.A M
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1 i
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t |
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1
i
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30.0
60. 0
i
i
i
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!
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
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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
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j
;
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!
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i
... J........
o
i
1
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i
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r
i
[
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..... L _ J _
i
m i
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!
:
jz
0 <
f=
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CO
IxJ
i
CO o
:
V
;
i
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.....i
I
:
:
•
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i
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!
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..... :....... J . ...... ___ 4 ... ....
' I .........
:
5
i
:
30.0
60.0
... 4
^ ...
;
i
J ..
i
i
i
i
;
?
......!__ ;
r!
’i—
;
!
!
i
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......: .... 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
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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
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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
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REFERENCES
[1]
T.H. Lee, R.C. Rudduck, "Microwave Antenna Technology, Final Report
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1987.
[2]
R.C. Rudduck, K.M. Lambert and T.H. Lee, "Microwave Antenna
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[3]
T.H. Lee, K.M. Lambert and R.C. Rudduck, "Microwave Antenna
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[13] J.W. Waters, "Absorption and Emission by Atmospheric Gases," in
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[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
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[19] N.E. Gaut and E.C. Reifenstein, "Interaction Model of Microwave
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[20] URSI Commission F Working Party," URSI Working party report:
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[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.
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[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.
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[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
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[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.
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[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.
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[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
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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
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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
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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
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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
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(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.
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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.
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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
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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,
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(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)
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(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
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