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Estimation of rainfall rates from passive microwave remote sensing

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Order Number 8813315
E stim a tio n o f rainfall rates from p assive m icrow ave rem ote
sen sin g
Sharma, Awdhesh Kumar, Ph.D.
University of Wyoming, 1987
U MI
300 N. Zeeb Rd.
Ann Arbor, M 48106
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Estimation of Rainfall Rates from Passive
Microwave Remote S ensing
by
Awdhesh Kumar Sharma
A D i s s ertation
Submitted to the
Department of Physics a nd Ast r o n o m y and
The Graduate School of the Unive r s i t y of Wyoming
in Partial Fulfillment of Requirements
for the Degree of
Doctor of P h i l o s o p h y
University of W y o m i n g
Laramie, W y o m i n g
December,
1987
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This thesis, having been approved by the
special Faculty Committee, is accepted
by the Graduate School of the
University of Wyoming
in partial fulfillm ent of the requirements
for the degree of
._EQCfcQE-Qf-EMl-QS-QpbLy-.
Dean of the Graduate School
Date...I
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S h a r m a , Awdhesh K ., ESTIMATION OF RAINFALL RATES
FROM PASSIVE MICROWAVE REMOTE
SENSING.
Ph.D., Department of
Physics and Astronomy,
December,
1987.
Rainfall rates have been estimated
microwave
and
visible/infrared
D a t a of September 14,
Mi c rowave
1978 from
Radiometer
(SMMR)
Scanning
board
SEA
V isible and Infrared Spin Scan Radiometer
G O ES-W
the
passive
remote sensing techniques.
the
on
using
Multichannel
SAT-A and the
(VISSR)
on
board
(Geostationary Operational Environmental Satellite -
West) w a s obtained and analyzed for rainfall rate retrieval.
Mic r o w a v e brightness temperatures
the
mi c r o w a v e
radiative
of
rates
of
rainfall
due
Gamma
to
drop
and
a
from precipitating clouds
and
water.
Microwave
size
distributions.
Microwave
oxygen a nd water vapor are based on the
schemes given by Rosenkranz,
s cat t e r i n g
(MRTM)
due to ice and liquid water are calculated using
M i e - t h e o r y and
a bso r p t i o n
model
using
These MBT were computed as
w h i c h are in a combined phase of ice
exti n c t i o n
are simulated,
transfer
atmospheric scattering models.
function
(MBT)
a nd
Barrett
and
Chung.
The
phase matrix involved in the MRTM is found using
Eddington's two-stream approximation.
The
surface
effects
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due to winds and foam are included through the ocean surface
e miss i v i t y model.
Rainfall
o p t i mization
linear
rates
are
technique
regression
data
the
has
been
estimating
estimates.
to
inverted from MBT u s i n g the
and
a
Bounds"
and
multiple
relationship b e t w e e n the
This relationship has been
oceanic rainfall rates from SMMR data.
inverted
Griffith's scheme.
of
"Leaps
leading
rainfall rates and MBT.
infer
then
for
the
rainfall
used
to
The VISSR
rates
using
This scheme provides an independent means
rainfall
rates
for
cross
checking
SMMR
The inferred rainfall rates from both
techniques
have been plotted on a w o r l d m a p for comparison.
A resonably
good
correlation
has
been
obtained
between
the
estimates.
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two
AC K N O W LED G M EN TS
The author wishes to
express
sincere
thanks
to
the
faculty and staff of the Department of Physics and Astronomy
for
their
assistance.
and
constributions
to
the
graduate
program
Special a p p r eciation is due to Dr. D.J.
Dr. R.K. Kakar
contributed
their
(NASA Headquarter,
Washington,
time,
and
knowledge,
d i r ecting exposition of the thesis.
and
Hofmann
D.C.) who
support
while
I would like to thank my
colleagues Arun V. Kulkarni a n d Robert F. Gutmaker for their
contributions in proof reading and editing the manuscript.
wish
to thank B.H. L a m b r i g h t s e n
for providing the SMMR and
v aluable
(JPL, Pasadena,
VISSR
and
California)
for
his
many
suggestions in locating the earth co-ordinates for
a given IR image of GOES satellite.
C.G.
data
I
Griffith
(ERL,
NOAA,
Special
Boulder)
thanks
for
to
Dr.
the trouble she
u n d e r t o o k to explain to me the details of her algorithm
for
rainfall estimation from IR data.
This
research w as m a inly supported by a grant from the
National Aeronautics and Space Administration
contract
(Project
#
956954)
(NASA)
under
from JPL to the Department of
Physics and Astronomy U n i v e r s i t y of Wyoming.
Finally,
I
wou l d
like
to
thank
the
Meteorology
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iii
Division,
Space
Applications
R esearch Organization,
Centre
Ahmedabad,
(SAC),
INDIA
for
Indian Space
granting
me
leave to conduct this research.
Appreciation
Manish,
is given to my wife,
and Vikash for their
support,
Sunita,
love,
and our sons
and
patience
throughout this period of research.
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TABLE
OF
CONTENTS
CHAPTER
I
Page
INTRODUCTION
1.1
...................................
1
Statement of the P r o b l e m .............. 4
Review of Past S t u d i e s ................ 6
1.2
II
Rainfall Measurements from Space . . .
RAINFALL MEASUREMENTS
2.1
2.3
............................
Remote Sensing ofRainfall
..................
12
..................
13
Passive Remote Sensing of
Precipitation
III
10
Limitations of Satellite Remote Sensing
in Rainfall Studies
2.5
.............
9
Advantages of Satellite Remote Sensing
in Rainfall Studies
2.4
9
In Situ Techniques and Their
Limitations
2.2
........................
6
..........................
MICROWAVE RADIATIVE TRANSFER MODEL
...........
14
19
D E F I N I T I O N S ........................ 19
3.1
Intensity and F l u x ....................... 19
3.2
Brightness Temperature
3.3
Polarization and Stoke's Vector
3.4
Microwave Radiometer
3.5
Emission and E x t i n c t i o n ........... 28
3.6
Mie Scattering and the Phase Matrix
.................
. . .
...................
.
21
23
26
30
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V
THE RADIATIVE TRANSFER EQUATION
3.7
. ..
33
Microwave Interaction with the
A t m o s p h e r e ................................33
3.8
Microwave Interaction with the
Ocean-Surface
IV
.........................
41
MICROWAVE EMISSION AND ABSORPTION BY THE
SURFACE AND THE A T M O S P H E R E .................... 50
4.1
Microwave Emission of the Model Ocean
S u r f a c e ................................... 50
A
Dielectric Constant of Sea Water
. . .
50
B
Microwave Emission by a Calm Sea . . .
52
C
The Effect of Sea Surface Wind and
F o a m ....................................... 53
4.2
Microwave Absor p t i o n b y Atmospheric
G a s e s ......................................63
A
Oxygen Absorption Coefficient
B
Water Vapor A b s o r p t i o n Coefficient
4.3
. . 67
............................
74
Absorption Coefficient due to
non-raining Clouds
B
63
Microwave Extinction by Liquid
Hydrometeors
A
. . . .
.....................
75
Extinction Cross-Section of a
R a i n - d r o p ................................. 77
C
Drop Size Distributions of Rain . . .
81
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D
V
VI
Extinction Coefficient of Rain Volume
87
MODEL ATMOSPHERES AND SIMULATION OF
BRIGHTNESS TEMPERATURES
.....................
107
5.1
Atmospheric Models
......................
107
5.2
Brightness Temperature Simulation
.
.
112
STATISTICAL METHODS AND OPTIMIZATION
T E C H N I Q U E S ....................................... 131
6.1
P r ecipitation using Microwave
(SMMR)
Data from SEA SAT S a t e l l i t e ........136
6.2
Precipi t a t i o n using Vis/IR
(VISSR)
Data from GOES S a t e l l i t e ................ 141
VII
RESULTS A ND D I S C U S S I O N .........................155
7.1
Maps of Rainfall Rates using
M i c r o w a v e D a t a ........................... 155
7.2
Maps of Rainfall Rates using
Infrared D a t a ........................ 159
7.3
C o mparison of Rainfall Rates Inferred
from Microwave
(MW) and Infrared
(IR)
D a t a ................
7.4
VIII
160
Sensitivity Study of Rainfall Rates
.
175
R E T R O S P E C T .......................................181
C onclusions and Comments
BIBLIOGRAPHY
.....................
183
...................................
188
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LIST OF FIGURES
Figure
Page
3.1
Geometry of specific intensity of radiation
3 .2
Geometry of a Polarization Ellipse
3 .3
Geometry of a directional microwave antenna
3.4
Geometry of reflection and refraction at the
air-sea interface
4 .1
.
.........
21
. 24
.
..............................
27
43
Plots of Ocean Horizontal and Vertical
E missivity versus Frequency and Angles
for S a l i n i t y = 3 5 .,S u r f .T e m p . = 300.,Wind=0. . . . 57
4 .2
Plots of Ocean Horizontal and Vertical
Emissivity versus Frequency and Angles
for S a l i n i t y = 3 5 .,S u r f .T e m p . = 300.,Wind=5.0 . .
4.3
58
Plots of Ocean Horizontal and Vertical
E m issivity versus Frequency and Angles
for S a l i n i t y = 3 5 . ,S urf.Temp.= 3 0 0 . ,Wind=10.0
4.4
. . 59
Plots of Ocean Horizontal and Vertical
Emissivity versus Frequency and Angles
for S a l i n i t y = 3 5 . ,Surf.Temp.= 3 0 0 . ,Wind=15.0
4 .5
. . 60
Plots of Ocean Horizontal and Vertical
E missivity versus Frequency and Angles
for S a l i n i t y = 3 5 . ,S u rf.Temp.= 3 0 0 . ,Wind=20.0
4 .6
. . 61
Plots of Ocean Horizontal and Vertical
Emissivity versus Frequency and Angles
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v iii
for S a l i n i t y = 3 5 S u r f .T e m p .=300.,Wind=25.0 . . 62
4.7
Plots of Integrated O x ygen and Water Vapor
Absorption Coefficients versus Frequency
4.8
...
69
. . . .
83
Plots of Extinction Coefficients at SMMR
Frequencies versus Rainfall Rates using
Marshal1-Palmer Drop size Distribution
4.9
Total number of Drops for Gamma Distribution
versus Rainfall Rates
.........................
86
4.10 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate u s i n g the Shape Parameters
p=-3 .42 , 6=. 80 , €=.013,
and N g = 1 . 2 9 ............ 89
4.11 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate u s i n g the Shape Parameters
jj= - 1 .79
, 6 =. 35 , €=.095,
and N Q= 9 1 . 3 ............ 90
4.12 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate u s i n g the Shape Parameters
U=~l
.34,
6=.30,
€=.069,
and N Q=1310
.........
91
4.13 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate usi n g the Shape Parameters
jj=-0
.01,
6=. 22,
€=.081,
and N Q=.109E+06
. . .
92
4.14 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate u s i n g the Shape Parameters
jli= 1. 63
, 6= .16 , €=.106,
and N Q=.754E+07
. . . .
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93
ix
4.15 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate us i n g the Shape Parameters
jj= 5
.04 , 6= .10 , €=. 129 , and N Q=.920E+11
. . . .
94
4.16 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate vising the Shape Parameters
jli= —
.79 , 6=. 26 , €=.077,
and N Q=.724E+04
. . . .
95
4.17 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate usi n g the Shape Parameters
y = 0 .18, 6=.21,
€=.082,
and N Q=.196E+06
. . . .
96
4.18 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate usi n g the Shape Parameters
jj= 1 .01,
6=. 18 , €=.110,
and N 0=.753E+06
. . . .
97
4.19 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate usi n g the Shape Parameters
y =4.65,
6=.11,
€=.114,
and N Q= .6 4 0 E + 1 1 . . . .
98
4.20 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate usi n g the Shape Parameters
jj= 0
.40 , 6=. 20 , € = .118,
and N Q=.705E+05
. . . .
99
4.21 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate us i n g the Shape Parameters
y = l .01, 6=. 18 , €=.090,
and N Q=.246E+07
. . . .
100
4.22 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate us i n g the Shape Parameters
jj= 1. 01,
6=. 18 , e= .101,
and N Q=.124E+07
. . . .
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101
X
4.23 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate using the Shape Parameters
= 1 .63 , 6=. 16 , €=.130,
jj
and N Q=.205E+07
. . . .
102
4.24 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate using the Shape Parameters
y = - l .39,
6=.31,
€=.031,
and N 0=.159E+05
. . .
103
4.25 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate using the Shape Parameters
jj
= - 1 .03,
6 =. 28,
€=.055,
and Ng=.982E+04
. . .
104
4.26 Extinction Coefficient at SMMR Frequencies
versus Rainfall Rate using the Shape Parameters
jj= -
5.1
.27,
6 =. 23,
€ = .080, and N Q=.427E+05
. . . .
105
Frequency Distribution of Sea Surface Temperature
(K) of 600 Model A t m o s p h e r e s ................. 113
5.2
Frequency D i s t ribution of Sea Surface W i n d
(m/sec)
of 600 Model A t m o s p h e r e s ..................... 113
5.3
Frequency D i s t ribution of Water Vapor
(gm/cm
3
)
of 600 Model A t m o s p h e r e s ..................... 114
5.4
Frequency D i s t ribution of Liquid Water Content
2
(kg/m ) of 600 Model A t m o s p h e r e s ............ 114
5.5
Frequency Distribution of Rain Water
2
(gm/m )
of 600 Model A t m o s p h e r e s ..................... 115
5.6
Frequency Distribution of Ice Water
2
(kg/m )
of 600 Model A t m o s p h e r e s ..................... 115
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xi
5.7
Frequency Distribution of Rainfall Rate
(mm/hr)
of 600 Model A t m o s p h e r e s ......................... 116
5.8
Flow Chart of Microwave Radiative Transfer Model
for Brightness Temperature Simu l a t i o n . . . .
5.9
122
SMMR Brightness Temperatures versus Rainfall Rate
(mm/hr)
for a Cloud Model of Liquid Water Content
LW = .33 .....................
5 .10 SMMR Brightness Temperatures versus Rainfall Rate
(mm/hr)
for a Cloud Model of Liquid Water Content
LW = .53
.....................
. . . . .
124
5 .11 SMMR Brightness Temperatures versus Rainfall Rate
(mm/hr)
for a Cloud Model of Liquid Water Content
LW = 1.04
...................
5 .12 SMMR Brightness Temperatures versus Rainfall Rate
(mm/hr)
LW = .42
for a Cloud Model of Liquid Water Content
.....................
5 .13 SMMR Brightness Temperatures versus Rainfall Rate
(mm/hr)
for a Cloud Model of Liquid Water Content
LW = .40 .....................
5 . 14 SMMR Brightness Temperatures versus Rainfall Rate
(mm/hr)
LW = .28
for a Cloud Model of Liquid Water Content
.....................
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5.15 SMMR Brightness Temperatures versus Rainfall Rate
(mm/hr)
for a Cloud Model of Liquid Water Content
LW = . 3 2 .......................................
129
5.16 SMMR Brightness Temperatures versus Rainfall Rate
(mm/hr)
for a Cloud Model of Liquid Water Content
LW = . 5 9 .......................................
6.1
Frequency Distribution of Estimated Rainfall Rate
using four SMMR
6.2
best channels s u b s e t ........... 142
Frequency Distribution of Estimated Rainfall Rate
using five SMMR
6.3
130
best channels s u b s e t ........... 142
Frequency Distribution of Estimated Rainfall Rate
using six SMMR best channels s u b s e t ............ 143
6.4
Frequency Distribution of Estimated Rainfall Rate
using seven SMMR best channels subset
6.5
143
Frequency Distribution of Estimated Rainfall Rate
using eight SMMR best channels subset
6.6
. . . .
. . . .
144
Frequency Distribution of Estimated Rainfall Rate
using nine SMMR best channels s u b s e t ........... 144
6.7
Frequency Distribution of Estimated Rainfall Rate
using full ten SMMR channels s e t ................145
6.8
Frequency Distribution of Estimated Rainfall Rate
from IR image of 14 September
6.9
1978 at time T.^
152
Frequency Distribution of Estimated Rainfall Rate
from IR image of 14 September 1978 at time T 2
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152
xiii
6.8
Frequency Distribution of Estimated Rainfall Rate
from IR image of 14 September 1978 at time T
7.1
World Map of Rainfall Rates using SMMR Data
from Lat.=-50°
7.2
Long.=-180°
to -80°157
to
+50°and
Long.=120°
to 180°158
to
*-50°and
Long.=-180°
to -80°161
World Map of Rainfall Rates using IR Data
from L a t .=-50°
7.5
+50°and
World Map of Rainfall Rates using IR Data
from Lat.=-50°
7.4
to
World Map of Rainfall Rates using SMMR Data
from Lat.=-50°
7.3
153
Plot of
jj
to
+50°and
Long.=120°
versus Simulated Rainfall Rates
to 180°162
(mm/hr)
using best five channels of SMMR Frequencies
. 177
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LIST OF TABLES
Table
4.1
Page
Resonant Frequencies of Molecular Oxygen and
Amplitude Factors
................................
4.2
Atmosphere Model after Cole et a l .
(1965)
4.3
Coefficient used in Water Vapor Absor p t i o n
5.1
Cloud Statistics of 600 Model Atmosphere
5.2
Statistics of atmospheric parameters of 600
66
. .
.
.
.
68
. 73
. 110
model a t m o s p h e r e s ................................. 110
5.3
Cloud M o d e l s ........................................121
6.1
Result of Leaps and Bounds Technique for
Selecting 5 Best Subsets of Size 4 to 9
C h a n n e l s ............................................135
6.2
Multiple Linear Regression Coefficients for
Rainfall Rate Estimation using 600 Simulated
Brightness Temperatures at SMMR Frequencies.
6.3
Statistics of Estimated Rainfall Rates from
SMMR Data using Different sets of Channels
6.4
. . 140
Constants for the Empirical W e i ghting
Coefficients
6.5
. 137
.....................................
148
The Statistics of Inferred Rainfall Rates from
IR Data at times T
, T
1
, T
^
......................151
U
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;:v
7.1
Measures of difference between the microwave
rainfall estimates
estimates
7.2
(IRR)
(MWR) and Infrared rainfall
for a sample of size N . . . . 164
Measure of Difference B etween M W and IR
Rainfall Estimates Whose Saptial Difference
is less than Equal to 0 . 5 ° ....................... 166
7.3
Measure of Difference Between MW and
IR
Rainfall Estimates Whose Saptial Difference
is less than Equal to 0 . 1 ° ....................... 167
7.4
Measure of Difference Between M W and
IR
Rainfall Estimates Whose Saptial Difference
is less than Equal to 0 . 0 5 ° ....................... 168
7.5
Measure
of Difference
Between
MW
and IR
Rainfall Estimates Whose Saptial Difference
is less than Equal to 0 . 5 ° .........................170
7.6
Measure of Difference Between M W and
IR
Rainfall Estimates Whose Saptial Difference
is less than Equal to 0 . 1 ° ....................... 171
7.7
Measure of Difference Between M W and
IR
Rainfall Estimates Whose Saptial Difference
is less than Equal to 0 . 0 5 ° ....................... 172
7.8
Measure
of Difference
Between
MW
and IR
Rainfall Estimates Whose Saptial Difference
is less than Equal to 0 . 5 ° ....................... 173
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xvi
7.9
Measure of Difference Between MW and IR
Rainfall Estimates Whose Saptial Difference
is less than Equal to 0.1°
................
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174
CHAPTER
(I)
INTRODUCTION
Precipitation plays a
very
significant
dynamics of the earth's atmosphere.
role
in
the
The latent heat released
by precipitation is a major source of atmospheric heating in
the
tropics.
therefore,
Understanding
demands
precipitation.
the
high
However,
earth's
quality
atmosphere,
measurements
of
precipitation is highly variable in
both space and time.
Conventional methods for estimating the
rainfall rates
inadequate
regions
of
are
most
desired,
be
values,
considered
based
on
calibration problem.
in
most
political,
Over the vast ocean surfaces, where
there
p recipitation at all.
can
inaccessible
the earth's surface due to economic,
and terrain factors.
is
and
are
no
Island data
to
radar
be
close
direct
on
to
measurements
measurements
precipitation,
it
of
whic h
the oceanic rainfall
suffer
from
the
In practice the calibration of radar is
done w i t h the help of rain gauges which are within the sight
of
the radar.
In the absence of gauge rainfall measurements
ever the oceanic regions the calibration of
difficult
problem.
In
the
face
radar
poses
a
of increasing demand for
rainfall estimates over the land and oceans surfaces,
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remote
2
sensing of precipitation commands a great deal of attention.
Applications of remote sensing data of infrared sensors
to
the
problem
of
rain
estimations
on
synoptic
or
climatological scales have shown very encouraging results in
the
past.
But
the point measurements like rain gauges and
satellite estimates of rainfall averaged
over
an
hour
or
half an hour are very poorly correlated or not correlated at
all.
This
is
due
to
measurements whereas the
values
of
the fact that rain gauges are point
satellite
estimates
are
average
rainfall signature over the area of the sensor's
field of view.
M a n y schemes using visible
from
both
and
infrared
observations
geostationary and polar orbiting satellites have
been applied to the precipitation estimation problem
and
Thiele,
1981;
Barrett
and
Martin,
(Atlas
1981).
These
techniques make use of information obtained from the cloud's
t o p - s u r f a c e s . They must be tuned for specific locations
and
are
the
thus
difficult
to
apply
globally.
a c c u m ulated rainfall obtained from such
g ood
agreement
However,
techniques
are
in
with the ground measurements, when averaged
over a time scale greater than 3-6 hours and a
space
greater
x longitude)
than
Griffith
measurable
or
(1987).
equal
The
parameters
if the scale is reduced.
to
l°xl°
(latitude
relationship
between
scale
satellite
and rainfall is weakened drastically
Therefore the results obtained from
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3
these techniques are good e n ough to be
used
only
for
the
study of climatological or synoptic scale problems.
Problems
like
weather
flood control etc. are
increasingly
rainfall measurements.
rates
from
a
forecasting,
river management,
demanding
which
can
passive
microwave technique appears to be a
through
i n t ensity of rainfall
Pa s s i v e
microwave
a
thick
through
techniques
cloud
and
absorption
are
a nd
thus
measure
and
very
ones
the
scattering.
useful over ocean
surfaces because the ocean emissivity lies
0.6
Despite their
passive microwave techniques are the only
see
term
Estimations of instantaneous rainfall
v i able source which can fulfill these demands.
limitations,
short
between
0.4
to
provides a cold background for detecting any
physical changes of atmospheric conditions lying
above
it.
Over land areas, passive microwave results are substantially
less quantitative because the land emissivity varies between
0.8
to
1.0
and
surface emission
mi c r o w a v e
spheres)
provides
as
scattering
allows
background,
but
the
direct
due
the
same order of magnitude of
atmosphere
to rain particles
observation
of
and
it.
rainfall
over
than that in the case of absorption.
ice
The
(water and ice
the relationship to the rain rate
radiances are more physically related
(water
above
any
because the scattering is primarily due to
the ice spheres aloft,
less
does
the
spheres)
a nd
to
thereby
the
is
Microwave
hydrometeors
represent
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a great
4
p otential
for
improved
satellite
observations
of
precipitation.
(1.1) STATEMENT OF THE PROBLEM
The thermal m i c rowave emis s i o n of the atmosphere due to
a tmospheric gases
have
been
empirical a nd observational
thorough
i nvestigations
theoretical and computational
rainfall
and liquid
water
well u n d e r s t o o d in the case of no rain from past
theoretical,
study,
(oxygen a nd water vapor)
rates
using
have
schemes
microwave
M u l t ichannel Microwave Radiometer)
microwave
brightness
studies.
for
e s timating
at
SMMR
frequencies.
the
(Scanning
Theoretical
have b e e n calculated at
these frequencies u s i n g the radiative transfer equat i o n
a
this
been carried out for
data
temperatures
In
for
given model atmosphere. A number of model atmospheres are
g e n erated from the m e t e o r o l o g i c a l observations
rawinsonde data)
by
parameters.
m i c rowave
The
in t r o d u c i n g
rain
emission
a nd
(radiosonde /
other
surface
of the ocean has been
s tu d i e d in detail and its d ependence on surface temperature,
surface salinity,
and surface roughness has been calculated.
The ocean surface e m i s s i v i t y model developed b y
Kakar
(1982)
(Hollinger,
from
the
1971; Strogryn,
m o d i f i e d by Kakar has
been
results
of
previous
1972; Wilheit 1979)
used.
The
Pandey
and
studies
and later on
radiative
transfer
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5
problem is solved for the case of scattering atmosphere,
the
values of microwave brightness temperatures at all SMMR
frequencies are obtained for each of the model
The
effects
of
multiple
scattering
on
atmospheres.
the
transfer of
thermal microwave radiation in rain are evaluated using
Mie
theory
and
particles.
the
The
brightness
Gamma
relationships
temperatures
Bounds"
and
between
the
microwave
and rainfall rates have been found
statistical
"Multiple Linear
these
Regression".
techniques
are
inversion
The
then
methods,
results
employed
by
Leaps
such as
obtained
usi n g
the
measurements to estimate the rainfall rates. Finally,
rainfall
rates
from
SMMR
these
are compared w i t h those inferred from VISSR
(Visible and Infrared Spin Scan Radiometer)
algorithm
developed
collected
onboard
Environmental
the
drop size distribution of rain
usi n g optimization techniques such as "Regression
and
and
b y Griffith et al.(1978).
the
GOES
satellite)
algorithm,
are essentially
microwave
data
and
data
using
The IR data,
(Geostationary
satellite,
different
in
the
a nd
nature
Operational
Griffith's
from
the
the a l g orithm and therefore provide an
independent source for comparison.
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6
R E V I E W OF PAST STUDIES
:
(1.2) Rainfall Measurements from Space
The present state-of-art of rainfall measurements
space
is
Thiele
(1981). Basically,
limited
described
to
by
Atlas et a l . (1982);
identifying
cover.
pr i marily for convective rainfall
latitude).
Microwave
and Atlas and
visible and infrared
cloud
rainfall
at
methods
They
low
can
be
latitudes
estimates
from
over
are
used
(<30°
oceans are
limited to the sensor's field of view.
Several
d e v eloped
visible
during
and
the
GARP
infrared
techniques
(Global
have
Atmospheric
Research
Program)
under the GATE
(Arkin,
1979; Augustine et a l ., 1981; H u dlow a nd Patterson,
1979; Stout
Richards
and
et
a l .,
Arkin
(GARP Atlantic Tropical
been
1979;
(1981)
and
have
relationship between p r ecipitation
quantities.
Satellite-techniques
Woodley
analyzed
and
have
a l .,
in
1979). Although none of these
applied
all
(Griffith et al.,
methods
ocean,
borne
can
be
types of precipitation and to all regions,
they can be improved usi n g complementary estimates
space
observed
b e e n extended to an
1981; Wylie,
from
1980).
detail the
satellite
empirical estimation of rainfall over land
to
et
Experiment)
passive
microwave
obtained
radiometers over the
and suitable radar systems over the land or ocean.
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7
Rodgers and Ad l e r
(1981) have analyzed tropical cyclone
rainfall characteristics data
data.
Earlier,
the
usi n g
estimation
microwave
of rainfall rates over the
oceans was p e r formed b y Wilheit et
al.
(Electrically
Radiometer)
Scanning
Nimbus-5 satellite.
assumed
to
be
M i c rowave
a ssumption
is
usi n g
The
not
true
p a r ticles
transparency
at
ESMR
onboard the
a n d the Marshall-Palmer
drop size d i s t ribution was used.
particles
(1977)
In their scheme the ice
transparent
radiometric
higher
were
(1948)
of
ice
microwave
f requencies and Marshall-Palmer drop size dist r i b u t i o n
over
estimates the number of drops at low rainfall rates.
Savage a nd Wei n m a n
technique
for
rain
w h i c h w as later
technique
was
(1975)
developed a p a s s i v e microwave
measurements
tested
by
over land u s i n g 37.0 GHz
Rodgers
et
al.
be
marginal
microwave
it
proved
The s c attering m e c hanism is enhan c e d
frequencies
fact was u s e d by Wilheit et
tropical
and
for mapping rain and could not provide the
rainfall intensity.
higher
This
based on the scattering of m i c r o w a v e s by the
hydrometers near the top of the rain column,
to
(1979).
sto r m data.
92 GHz and 183 GHz, a nd this
al.
(1982)
in
analyzing
and
Lewis
(1986)
in
synoptic scale features of Nor t h
Their
results
the
A linear relationship b e t w e e n rainfall
rate and the brightness temperature at 37.0 GHz was used
Katsaros
at
analyzing
Pacific
m e soscale
weather
by
and
systems.
were in good agreement with the coastal rain
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8
gauge measurements, when the terrain effects were
for.
Spencer
et
al.
(1983-a)
compared
accounted
SMMR
brightness
temperatures to the radar derived rain rates over
of
Mexico.
in
freezing
the
18-37
level
in
distribution.
None
GHz
the
of
frequency
cloud
the
and
past
extinction
range
on
studies
varia t i o n s in these quantities in
detail.
st u d y
to
microwave
extinction
due
rain
by
depend
the
rain
on the
drop
size
have considered
In
the
drops
present
and
different drop size distributions are considered in
S c attering
Gulf
What all the studies of rainfall rate retrieval
from microwave data have shown is that
drops
the
the
detail.
due to hydrometeors is taken into account in the
mi c r o w a v e radiative transfer calculations u s i n g
Mie
a n d the Gamma drop size distribution.
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theory
CHAPTER
(II)
RAINFALL MEASUREMENTS
(2.1)
In situ Techniques and Their Limitations
Rain gauges are most commonly utilized
in
determining
rainfall over the land surfaces. Measurements of rain gauges
are
generally
topography,
located
affected
site,
on
by
wind,
an
island
the
and
interrelated
gauge
are
design.
not
likely
representative values of rainfall over the
vicinity.
Since
factors
Rain
to
ocean
of
gauges
give
in
the
their
ships do not provide a stable platform for
rain gauges either,
no direct source of rainfall measurement
exists over the oceans.
Another instrument employed in rainfall monitoring over
land
and
coastal
instrument
suffers
regions
from
is
the
problems
scattered microwave energy to the
partial
filling of the radar beam,
beam by intervening drops,
ground
weather
of
drop
radar.
relating
size
the
This
back
distribution,
attenuation of the radar
absorption and reflection by
(anomalous p r o p a g a t i o n ) , and signal calibration.
calibration of a radar requires another source
measurements.
In
the
absence
of
of
oceanic
the
The
rainfall
rainfall
measurements by any other mea n s the radar measurements of
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10
rainfall
data over the ocean surfaces can not be assumed to
be 100% reliable. However,
gauges
of
p r o viding
a
the radar h as an
spatially
advantage
continuous
view.
e mployed extensively in support of rain gauge data.
the
cost
of
a
radar
and
technical and engineering
the
requirement
support,
the
over
It is
Due
to
of continual
operational
world
w i d e u se of radar for rainfall m onitoring is limited.
In
situ measurements of rainfall b y conventional means
are deficient in m a n y areas,
are
required
in
near
real
parti c u l a r l y if
rainfall
data
time for weather forecasting,
river management a nd flood control. A l s o the
rainfall
data
o btained from conventional mea n s are too inadequate in space
and too infrequent in time to be useful to satisfy the large
amount of requirements of users.
de m a n d i n g
situation
for
Therefore in this extremely
rainfall
s ensing from satellite p l ateforms is
method
by
measurements,
potentially
remote
the
only
w h i c h the n e c c e s s a r y measurements can be made in
space and t i m e .
(2.2) Remote Sensing of Rainfall
Remote sensing of rainfall rate measurements beg a n w i t h
the development of meteorological radar after w o r l d
(Barrett
and
Martin
war
II
(1981)). Measurements from space were
not possible until the Environmental Satellites were
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placed
11
in
orbits.
These
satellites
have been well equipped with
s ens ors capable of yiel d i n g data for rainfall
measurements.
Th e s e sate l l i t e s have been c ategorized into two types namely
(i) L o w Earth-Orbiting,
The
first
type
and
(ii) Geosta t i o n a r y satellites.
of satellites o c cupy low-level orbits
(usually b e t w e e n 500-1500 km
above
the
earth's
surface),
u s u a l l y p a s s i n g over the earth's poles w i t h the time periods
of
the
order of 100 minutes.
Therefore,
are r e q u i r e d for each satellite to cover
per day.
over
the
14-15 orbits
entire
globe
U s u a l l y such satellites v i e w one-half of each orbit
the
night
time
side of the globe,
globe is thus v i ewed twice per day,
once
about
at
night
time.
The
current
once in
every area on the
day
light
operational
and
series of
A m e r i c a n p o l a r - orbiting environmental satellites are NIMBUS,
TIROS,
and N 0 A A series.
Program
which
(DMSP)
The Defence Meteorological Satellite
is the operational m i l i t a r y satellite system,
comprises
two
or
more
polar-or b i t i n g
satellites
c a r r y i n g meteorological sensors.
The
s e c o n d type of satellites are p l a c e d into orbit at
ap p r o x i m a t e l y 35400 k m high above the earth's center.
orbital
p l anes
are in the plane of the earth's equator and
they move in the same
earth.
Their
d i r ection
as
the
rotation
This type of orbit is called geosynchronous,
of
the
in whi c h
the satellite kee p i n g pace w i t h the rotation of the earth on
its
polar
axis,
a nd a satellite occupying it appears to be
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12
s t ationary w i t h respect to the earth's m o tion and is
ge o s t a t i o n a r y
satellite.
Such
a
satellite
constant geographic field of v i e w through
out
called
provides
its
a
motion,
and the data frequently o b t a i n e d from this has been utilized
in
rainfall study.
The current geosta t i o n a r y meteorological
satellite systems are METEOSAT,
(2.3) Advan t a g e s of Satellite
G M S , GOES,
R e mote
and INSAT.
Sensing
in
Rainfall
satellites
in
rainfall
Studies
The advantages of usi n g e a r t h
study are following:
(a)
S a t ellite
systems p rovide global coverage of data, a nd
thus provide access to remote regions.
(b) S a t ellite imaging
data,
co n t r a s t i n g
systems
with
those
yield
spatially
continuous
o b t a i n e d from the irregular
netwo r k s of surface wea t h e r stations.
(c)
Satellite
o b s e rvations
are
distributed
more
h o m o g e n e o u s l y than in situ observations.
(d)
Geostationary
satellites
can provide information more
frequently than is commonly o b t a i n e d from surface and
upper
air we a t h e r stations.
(e)
S a t ellite
sensors
me a s u r e the radiances emerging from
the u n d e r l y i n g atmosphere and the surface therefore contains
the
i n f o r mation
on
the
para m e t e r s
integrated
over
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the
13
atmospheric column and the surface.
These radiances are then
inverted to retrieve the information contained in them.
(f)
Satellite
or near real
data can be obtai n e d for large area,
time
for
many
uses
in
in real
meteorological
and
oceanographic studies.
(2.4) Limitations of Satellite Remote
Sensing
in
Rainfall
Studies
Despite the great potential for use of
in
atmospheric
and
oceanographic
satellite
studies,
data
these
investigations are often limited in practice by a number
problems as described below
(a)
Satellite
data
are
of
:
mostly
synoptic.
Thus there are
difficulties in correcting and/or calibrating the data for a
number of factors which m ay influence them.
(b) Transformation
of
images
from
satellite
co-ordinate
frame to earth locating frame is not straight forward.
(c)
Conversion
desired
of
information
satellite-observed
requires
the
radiances
knowledge
into the
of
the
relationship between the two.
(d)
Processing
of
remotely
sensed raw data may eliminate
some of its important features and thus
may
not
reproduce
all the information contained in the raw data.
(e)
Extracting
the
data of interest from a hugh satellite
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14
data base is technically difficult.
(f)
Developing
the
detailed
interpretation,
and
use
of
methods
for
satellite
the
analysis,
data for specified
operational applications have been difficult due to
numbers
of parameters required b y the methods.
(g)
been
The
types
and
sufficient
resolutions of available data have not
for
most
of
the
atmospheric
and
oceanographic studies.
(h) A multichannel-system operating at different frequencies
(SMMR observing at different frequencies)
instantaneous
field
of
view
(IFOV)
can have different
therefore it is very
cumbersome to extract the data of all channels corresponding
to the same geographic location for their simultaneous use.
(i) Retrieval of atmospheric and/or oceanographic parameters
from satellite observations require a realistic model of the
atmosphere and the ocean
and
a
very
efficient
inversion
technique.
(j)
Existing
operational procedures require changes if n e w
data types are to be employed.
(2.5) Passive Remote Sensing of Precipitation
Satellite estimated rainfall rates
using
data
provided
Spin-scan radiometers.
by
have
been
(a) Scanning radiometers,
derived
and
(b)
The w avelengths most commonly used in
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15
these instruments for rainfall studies are
y m - 0.7 ym) ; (ii)
12.5 y m ) ; a nd
Infrared,
(i) Visible,
(3.5 y m - 4.2 ym)
(iii) Microwave,
n a t urally
(0.5
or
(10.5 ym -
emitted
radiation
(0.3 cm - 4.5 c m ) .
(a)
A scanning radiometer on board p o lar-orbiting satellite
consi s t s of a revolving mirror,
filters.
As the
revolving
satellite
mirror
the
advances
along
its
the
radiances
which
are
r e volution
of
the
m i rror
adjacent to the one before,
Some
of
these
Resolution
visible,
e.g.
are
(A V H R R )
near infrared,
(6.6,
10.69,
o p e r a t i n g at two polarizations
capable
o n T iros-n
been
derived
track.
of
having
(including
18.0,
The SMMR consists
21.0,
and
37.0
of
of
GHz)
(horizontal and v e r t i c a l ) .
rates
over
the
u s i n g SMMR data of SEASAT-A.
S M M R is a dual p o l arized co n i c a l l y s canned radiometer
angle
Very
and water vapor channels),
In this study estimations of rainfall
have
wavelengths.
the four channels A d v a n c e d
Radiometer
infrared,
channels
ocean
passed
produces a n e w scan line,
a n d the SMMR on Nimbus-7 a nd SEASAT-A.
five
then
across the sub-satellite
radiometers
multis p e c t r a l systems,
High
the
beam splitter a n d then the spectral filters to
give the intensity of r a d iation at the desired
Each
orbit
scans the target across the sub-satellite
track a nd collects
thr o u g h
a b e a m splitter and spectral
incidence approximately 48.8°
The
which
mak e s
an
nadir.
The SMMR has been d e s c r i b e d in detail by Gloer s e n and
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from the
16
B a rath
(b)
(1977) a nd Njoku et a l . (1980).
Spln-scan radiometers, have b e e n flown on
satellites and consist of a scan n i n g device,
IR/VIS
(reflected
light)
sensors.
these instruments consists of a
m e c h anically
to
provide
spinn i n g motion of
scanning.
Thus
the
an
a telescope and
The scanning system in
mirror
whi c h
is
advanced
n o r t h to south viewing, while the
satellite
provides
west
to
east
image of the entire v isible disc of the
earth is built u p over a p e riod of about 20
which
geostationary
minutes,
after
the mirror is returned to its initial p o s i t i o n and is
ready for another image.
VIS S R
of
GOES
(Visible a nd Infrared Spin Scan Radiometer)
satellite
have
been
data
obtained for e stimating the
rainfall rates using the al g o r i t h m given b y Grif f i t h et
(1976).
This
rainfall rate
with
the
algorithm
provides an independent source of
measurement w h i c h can be
used
for
comparing
estimated rainfall rates obtained from SMMR data.
The V I S S R instrument has b e e n de s c r i b e d in detail b y
(1984).
clouds
are
r e latively
bright
in
cloud
tops
at
d i f ferent
altitudes
br ightness temperatures in the IR region,
interpreted
in
ice,
visible
observations are the r a d iation temperatures of
Since
Gibson
Visible data is most strongly related to the albedo
of the target i.e. higher reflective surfaces of
and
al.
snow
image.
the
IR
target.
have different
this data
may
be
terms of clo u d top temperatures or heights.
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17
From the cloud top temperatures
determined
using
an
rainfall
empirical
Griffith et a l . (1976), Martin,
rates
have
been
relationship
obtained
by
Stout and Sikdar
(1975)
and
many o t h e r s .
In
the
visible images,
the probability
rainfall
of
rainfall
relationship
is
as cloud brightness increases,
increases,
strongly
this
brightness-
time dependent.
correspondence between cloud brightness and rainfall at
earth's
surface
to
frequency,
time.
relate
cloud
intensity
and
Sikdar
brightness
to
any
precipitation
and extent must consider changes w i t h
This scheme has
Stout
the
is better w h e n a cloud system is young and
vigorous than when it is old and decaying. Consequently,
attempt
The
been
(1975)
successfully
used
by
Martin,
for deep convective clouds in the
tropics.
In
the
IR
relationship
satellite
between
images,
the
the
cloud's
simple
empirical
top temperature
rainfall has been obtained by Scofield
and
Oliver
and
(1977).
Their scheme can be used for estimating rain from convective
storms.
The
scheme
and
tuning for local terrain
Microwave
radiometers
data
the estimates from it need a fine
effects and for the climate.
observed
It
satellite-borne
have been shown to reveal not clouds,
cases of VIS/IR schemes, but
clouds.
from
is
in
this
rain
region
areas
that
as in the
embedded
rain
in
the
has been most
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18
directly evidenced from satellite data available until
Schemes
involving
V I S/IR
data
depend
now.
on less physically
direct relationships between clouds and rain.
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CHAPTER
(III)
MICROWAVE RADIATIVE TRANSFER MODEL
In the study
theory
of
microwave
radiative
(MRT)
and its applications to remote sensing one generally
encounters a few special physical terms.
applicability of these terms
(MRS)
transfer
in
The definitions and
microwave
remote
sensing
are described here.
DEFINITIONS
(3.1)
:
Intensity and Flux
The specific intensity
spectral
intensity),
(also known as monochromatic
I ,
or
is the flux of energy in a given
direction per second per unit frequency range per unit solid
angle per unit area normal to the given direction
M.
(Goody, R.
1964) .
Consider the flow through a point P
(3.1),
surrounded
by
a
small
element
in
space,
Figure
2
of area d A s (m ),
•—+
normal to the direction of s. From each point on d A s a
of
solid
angle
bundle of rays,
d O s (steradians)
originating on
df>s , transports in time dt
(sec)
cone
is drawn about the s. The
dAs ,
and
contained
within
and in the frequency range
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20
v
to v+dv
(hertz), the e n e r g y is given by
dE^fP)
where Iy(p,s)
= IV (P,S) d A g d O g dv dt
is the specific intensity
(3.1)
(watt m
—2
st
—1
hz
—1
)
at the point P in the s-direction.
The flux in the direction of d, F
is
defined
as
j(p)*
1r / Cl
at a point
the total e n e r g y flowing per second,
unit area normal to d, per unit frequency interval
dFv , d (P) =
dEv (P)
/
^ d
dv dt
P,
across
(dv»)
(3'2)
where
dA
s
= d A , cos e
d
(3.3)
e is the angle between s and d
dA^,
. The energy flux across
integrated over all s - d i r e c t i o n is given by
V P' S>
Js
F v,d =
where the integral extends
system
of
polar
cos 0 d0s
over
co-ordinates
all
solid
with
direction of the o utward normal to d A
(3.4)
s
the
angles.
z-axis
In
in
, the solid angle
a
the
do
s
is defined as
dO
s
= sin e de d$
(3.5)
'
'
and the expression for the net flux is
r2iT
Fv (0,e)
=
0
'e
I
cos e 1 sin e 1 d e ' d ^ 1
0 v
(3.6)
or
F i;(0,e) = nr I
s i n 2e
where 1^ is the m e a n value of
b etween
zenith
angles
0
and
(3.7)
, averaged over the
e.
surface
A black body or lambert
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21
surface radiates isotropically
(i.e.
is independent of
and <f>) and has the flux of emission F
v
(0,
tt/ 2)
or
ttI
e
in the
V
upper hemisphere.
d0„
cJA
5
Geometry of specific intensity of rodiotion
Figure (3.1)
(3.2)
Brightness Temperature
Instead of intensity,
used
in
thermal
brightness
microwave
temperature
studies.
equivalent temperature of a black body
amount
of
energy
same frequency band.
temperature".
The
as
(BT)
is
It is defined as the
radiating
the
same
that received by a radiometer in the
This
is
also
known
as
"radiometric
relationship between BT and intensity is
derived from the properties of a black
body
whose
thermal
e miss i o n is given by Planck's law, mathematically stated as
B (v ,T) =
w he r e
T
(2
h v 3/C2 )
is the temperature
(1/{e x p (hi// k T )-1})
(in degree kelvin)
(3.8)
of the black
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22
__QI
body,
h is Planck's constant
(6.623 x 10~
Joule sec), C is
Q
the speed of light in vacuum
Boltzman's constant
(2.99793 x 10
(1.38 x 10
-23
meter/sec), k is
Joule/Kelvin),
and
v
is
the frequency in hertz of electromagnetic radiation.
At
microwave frequency and terrestrial temperature the
quantity hv/kT << 1, therefore the Planck's formula
reduces
to the Rayleigh-Jeans law, w h i c h is given by
B(v,T) =
Thus,
at
a
(2 k v 2/C2 ) T
given
frequency interval,
proportional
frequency the energy flux per unit
B(v,T)
to
the
(3.9)
(Watt
m
_o
temperature.
radiation of a given frequency,
v,
sec),
For
is
linearly
electromagnetic
and known intensity,
1^,
one can associate a unique temperature known as BT such that
B (v,T)|_
= I . Thus equation
(3.9) c an be wr i t t e n as
B
Iv
A
=
(2 k v 2/C2 ) Tb (v)
microwave
radiometer
detects
po l a r i z e d components at a time,
incident
(3.10)
only
one of the two
therefore only half
of
the
unpolarized radiation is received by a radiometer.
Hence
p =
T
a
(v,p) =
where p denotes
s ubsituting which in
I/2
the
(C2/ kv2 ) I
(p=v)
gives
(3.11)
V ,p
vertical
(3.10)
or
horizontal
polarization of radiation.
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(p=h)
23
(3.3) Polarization and Stoke's Vector
The vertical or horizontal plane of polarization is the
plane parallel or perpendicular to that containing the local
normal
vector
(normal to the plane of the
(line
detector)
and
Poynting
of s i g h t ) . The electric vector of a polarized
electromagnetic wave lies in its plane of polarization.
term
polarized
component
refers
total stream of
radiation
for
to
The
one component of the
electric
field
vectors oscillate in one given plane of polarization,
on the
other
hand
the
which
the
term p o l arized wave is meant to imply that
the wave is polarized w i t h fixed
phase
any two orthogonal components of
its electric field vector.
Four
parameters
known
as
1960; Vandehulst,
between
are required to describe the state of
an electromagnetic wave.
1852,
differences
They were introduced
S t o k e 1s vector,
1957; Kerker,
and m ay be
1969)
by
Stoke
in
(Chandrasekhar,
expressed as
I = {I,Q , U ,V)
(3.12)
I =
Iv + Ih
(3.13)
Q =
Iv - 1^
U =
Q tan 2x
(3.15)
V =
I sin 2 1
(3.16)
where
= I cos 2 Y cos 2x
(3.14)
1^ and Iv are the horizontal a nd vertical intensities of the
two polarized components of the beam, x is the angle
between
the vertical axis and major axis
is
of the
ellipse.
i
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the
24
angle of w h i c h the tangent is the ratio of the minor axis to
the
major axis of the ellipse as shown in the Figure
At a ny point,
Ip is proportional to
EE*
where
E*
(3.2).
is
c o m p l e x conjugate of E. Thus o ne obtains
(3.17)
xh = C ' E hE h
i
I
v
<= C
*
E E
v v
(3.18)
LI-HHIS
I SINS
H -n illS
H
H
FIGURE ( 3 .2 ) GEOMETRY OF A POLARIZATION ELLIPSE
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the
25
E x pressing the Stoke's
vector
in
terms
of
electric
field vectors
I
= c'
[Ev E* + EhE*]
(3.19)
Q =
c'
[EvE ^ -
Eh E*]
(3.20)
U =
C*
[E E* +E.E*] = 2 C '
L v h
h vJ
V =
ic'[E E* - E.E* ] =
L v h
h vJ
Re
(E E*)
' v h'
-2C' Re
(E E*)
4 v h'
w h e r e C 1 is the constant of p r o p o r t i o n a l i t y and
the
bar
electric
fields.
The
Stokes v e ctor due to scattering,
c an
(3.22)
'
i=-F-l,
and
indicates that time a verage is taken over the time
intervals m u c h larger than the p e r i o d of
respective
(3.21)
'
'
v i b ration
change
reflection
of
the
occurring in the
and
refraction
be d e t e r m i n e d from the change occurring in the electric
field components E.. and E. .
v
h
The most important p r o p e r t y of the
the
additivity
beam. However,
the
of
S t o k e 1s
vector
is
its comp o n e n t s for incoherent polarized
the state of p o l a r i z a t i o n may be
processes of reflection,
altered
by
t r a n s m i s s i o n and scattering.
p a r t i a l l y p o l a r i z e d beam of r a d i a t i o n
is
obtained
by
A
the
s u p e r p o s i t i o n of incoherent be a m s a nd can be decomposed into
a
fully
polarized
a nd
unpolarized
(natural)
parts.
Stokes vector for fully p o l a r i z e d and u n p o l arized parts
(>T(Q2+U 2+ V 2 ) , Q, U, V) and
(I, 0, 0, 0} respectively.
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The
are
26
(3.4) Microwave Radiometer
Microwave radiometer is a device used
microwave
radiation
radiometer uses
power,
an
P(e,4>),and
a
gi v e n
antenna
the
factor known as gain,
specific
in
direction
whi c h
to
measure
direction.
receives
or
A
the
typical
transmits
pattern of this power is given by a
G(e,$),
whe r e
e
and
$
define
the
and a mathematical relationship between
g a i n and power is given by
(3.23)
0 ( e . ♦) = --- ;—
i f Jo" JS
8' <58
d<t>
Remote sensing antennas are designed to have
beam
width
(main lobe)
defined
as
the
is
of G(0,0),
linearly
Beam
width
angle subtended at the antenna b etween
lines that intersect G(e,4>), where
valu e
narrow
so that unwanted contributions from
side lobes in the measurements can be minimized.
is
a
the
as shown in the Figure
proportional
to
its
gain
is
half
the
(3.3). A ntenna gain
effective
apperture,
A e (e,$), at a given w avelength X.
A e (6 ,<t>) =
( X 2 / 4 tt)
G (e ,<t>)
(3.24)
The total power per unit frequency interval incident on
the antenna is related to the effective apperture by
2 kT
r—
A e (e,4>) sin e de d<l>
(3.25)
d P in - "I
A
where
1/2 is introduced because linearly polarized antennas
T2tr
rtt
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27
receive only half the incident randomly polarized
radiation,
k
is
Boltzman's
constant,
A
is
blackbody
the
center
wavelength of the incident radiation and dv is the frequency
wi d t h of the radiation.
In
T^,
the
is
measured
and
practice,
incident
antenna
power
temperature,
received
radiometer is then computed from the equations
The antenna temperature,
a
hypothetical
by a
(3.23-3.25).
is defined to be the temperature of
resistor,
and
is
related to the incident
power as given by
Tft = (dPin/kdy)
(3.26)
y
Antenna
Effective Aperture
Ae (9t4>)
Antenna
Power
Pattern
6 (8,<f>)
Side iobes
Main lobe
c35.
■30
20;
Intensity Scale
(d B )
Beamwidth —
'o'"'
Ocean's Surface
G e o m e t r y of a directional m i c r o w a v e antenna
F igure (3.3)
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28
(3.5)
Emission and Extinction
The interaction of radiation w i t h matter
processes
of
emission
and
extinction.
r a d iation travels through a m e d i u m
weakened
due
respectively.
to
emiss i o n
it
or
involves
When
is
the
a pencil of
strengthened
extinction
or
processes
These processes are governed b y Lambert's
law
w h i c h is m a t h e m a t i c a l l y s t ated as
dl
dl
w he r e
the
V
v
(extinction)
= - B
(emission)
=
constants
e x tinction
e x tinction
t ravels
the
process
through
(3.27)
ds
(3.28)
T^ e
in
an<*
•
in
in
due
in
physical
(temperature.pressure.composition)
.absorption
(energy
good
matter.
is
matter is held constant.
(3.27)
to
order as the radiation
ds(m),
a nd
sign
volume
emission
proportionality
state
a p p r o ximate
the
volume
change
r everse
distance,
are
negative
differential
is
a
. J
emi
v
®e x t^m
^emi^”1 ^ ’
that
ds
V
proportionality
coefficient,
coefficient,
indicates
of
a
6X1
only
This
when
the
of
the
Extin c t i o n is defined as the sum of
c o n verted
into
heat)
and
scattering
(energy redistributed in different d irections without change
in f r e q u e n c y ) . Since all e x t i n c t i o n processes are linear,
it
can be defined as
J3 ^ =
ext
where a a b s (m
and
B
SGcl
(m *)
.
abs
ct
+ B
) *s t*16
(3.29)
1
sea
volume
a bsorption
coefficient
is the v o lume s c attering coefficient. More
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29
conveniently a parameter is d efined as optical
path-length,
d T , w h i c h is ex p r e s s e d as
dT =
The
- B . ds
ext
negative
sign
(3.30)
'
in
(3.30)
is
due
to convention
because the optical path-length is defined to be a
definite
is zero
where
quant i t y
'
positive
whose va l u e at the top of the atmosphere
(t = 0 ) . In the case of a plane
parallel
atmosphere,
z is defined along the z e n i t h direction,
the distance
traveled by radiation in the d i r e c t i o n
e
degree
from
the
nadir is given by
ds = sec e dz
The
source
function,
the units of intensity,
(3.31)
, giv e n in equation
(3.28) h as
is d efined as the r a d iation emerging
in the d i r ection specified by the z e nith angle, e,
and
the
a zimuth angle, $, due to s c attering and emission processes.
For
a
plane
parallel
horizontally
homogeneous
scattering atmosphere the source function is given by
w
Jv=
4 ^
[2tt
JO
Ptt
V
*
6 '*'6 ''*')
s i n e 1 de'd<t>'
+(1- cov )
The first term in
(3.32)
the
right
hand
Bv
(3.32)
side
of
equation
defines the radiation emer g i n g due to scattering a n d
the second term is a
P (e ,<J>,e 1 ,$ 1)
is
contribution
due
to
self
emission.
the phase m a trix characterized b y a pencil
of radiation incident
in the d i r e c t i o n
(e1, ^ 1) and scattered
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30
in the
direction
e miss i o n
B^(T),
whose
(e,4>).
Stokes
By
is
vector
the
unpolarized
is given by
thermal
[1/2 By (T), 1/2
0, 0], where B y (T) is the Planck function at a
frequency,
v,
and
at
scattering albedo, u^,
scattering
is
the
(e',$')
given temperature,
is
coefficient
1^(0'#$')
di r ection
a
defined
to
incident
the
as
the
T. The single
ratio
extinction
radiation
given
of
the
coefficient.
emerging
in
the
at the point of scattering.
(3.6) Mie Scattering and the Phase Matrix
Mie scattering theory has been used to
ab sorption
determined
the
and scattering properties of homogeneous spheres
of water having the same optical properties.
Drops of liquid
wat e r in the form of clouds and rain, known as hydrometeors,
affect the propagation of microwaves in the atmosphere.
intensity
of
incoming
e x p ( - B ext ds)
ds,
where
The
radiation is reduced by a factor of
in traversing the medium
through
a
distance
the extinction coefficient due to rain particles
B e x t , will be computed later in chapter IV using Mie theory.
The solution of
conditions
by
Maxwell
this
for
a
boundary
theory
(1908) and is known as Mie
theory.
one can obtain the relationship between
incident and scattered electric field vectors
the
given
a sphere of radius r and index of refraction
m, was treated by Mie in
Usi n g
equations
intensity components.
in
terms
of
This relationship is defined by a
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31
4x4 transformation m a trix kn o w n as the phase matrix. The
elements
of
the
phase
16
matrix are functions of scattering
angle e, the index of refraction m,
the particle
radius
r,
and the incident wavelength A. For isotropic and homogeneous
p a r ticles like hydrometeors these 16 elements are reduced to
4
independent
elements.
This reduction is possible by the
virtue of the principles of reciprocity,
spherical symmetry.
mirror symmetry and
The reduced phase matrix
for
a
single
homogeneous sphere is then given by
P(e)
P..
P, a
0
o
=
P.*
0
0
P, ,
0
0
0
p,, - p s<
0
P 3<
P 33
(3.33)
where
P lf
= (A2 / 2TT2r 2 Q s c a ) (i,
+ ia )
(3.34)
P ia =
(A2 / 2TT2r 2 Q s c a ) (ia - i.)
(3.35)
P 33 =
(A2 / 2Tr2r 2 Q s c a ) (i3 .+ i4 )
(3.36)
P 34 = (A2 / 2Tt2r 2 Q s c a ) (i3 - i4 )
(3.37)
i, = S, S* =
I
|s, I2
(3.38)
ia = S 2S* =
|S2 |2
(3.39)
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32
x
i, = S 2S
(3.40)
X
i., — Sj S
(3.41)
a
(i,
Q sea
+ i2 ) sin e de
wh e r e stars denote complex conjugation and
are
amplitude
functions.
Since
sufficiently far from each other,
the
the
it is
S,
and
particles
possible
to
S2
are
study
scattering by one particle wi t h o u t reference to others.
Consequently,
intensities sc a t t e r e d b y various particles m a y
be added without regard
waves.
for
(3.42)
0
to
the
phases
of
the
scattered
The independent elements of the phase m a t r i x obtained
a sample of particles in the particle range
(r1#r2 ) are
then given by
(3.43)
sea
(3.44)
sea
(3.45)
2
2 tt Q
sea
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33
(3.46)
The size distribution of the particles,
the
amplitude
functions,
dn(r)/dr,
and
s, , and sa , will be described in
the next chapter IV.
THE RADIATIVE TRANSFER EQUATION
Mi c rowave
understanding
remote
of
the
:
s ensing
problems
interaction
atmo s p h e r e and the earth's surface.
of
In
require
m i c rowave
this
an
with the
section
these
interactions will be studied.
(3.7) Microwave Interaction w i t h the Atmosphere
The
total
re s u l t i n g
from
differential
the
equal to the sum of
change
Equation
radiative
the
(3.27) a n d
(3.28),
+ dl^(emission)
is known as S c h w a r z s c h i l d 1s equation
transfer
considered
(the
emission).
For
in
energy
a
(3.47)
(3.43)
(I
(3.48)
intensity
interaction of matter and radiation is
dl y = d l y (extinction)
dl
in
which
lost
by
plane-parallel
energy
conservation
exti n c t i o n
reappears
of
is
as
horizontally homogeneous
s c a t t e r i n g atmosphere the equation of radiative transfer
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34
using
dl
v ,p
(3.32) m a y be expressed as
(t ,e
'
,<!>)'
w
VpCr.e,*)
dT
polarized
and
components
v
P (e ,<t>,e 1 ,<t>')
0
-
I„(T,e' ,<t>') d (cos e ') d<J> * -
where p=h or v; h
2vr '1
(T)
(1 -
denote
wv
(t )) B v (T(t ))
horizontal
respectively,
the
(3.49)
and
bar
vertical
denotes
average over an ensemble of particles of the quantity, v
the
frequency
of
scattering albedo (B
depth
given
by
SCcl
observations,
w v (t)
/B
the
.), t
6Xt
is
is
the
vertical
the
is
single
optical
(3.30), B (T(t)) is the unpolarized thermal
microwave emission given by (3.9), e and 4> are the nadir and
azimuth angles of the radiometer
is
the
diffuse
radiance
respectively.
I (t ,e1,$1)
averaged over both polarizations
incident on the scattering volume from zenith angle
azimuth
angle
e1
and
P(e ,<J>,e ' ,$ 1) is the phase function of the
particles that describe scattering from zenith angle e 1 to e
and azimuth angle $ 1 to 4>. This can be approximated by
P (e ,<t>,e ' ,<J>1) = 1 + 3g(T)(cos e cos e 1
+ sin e sin e 1 cos
w h e r e g(T)
(<t>-<t>')} (3.50)
is the as y m m e t r y factor of the phase function for
s c a t t e r i n g particles.
For the purpose of this work,
tobe
plane-parallel,
the atmosphere is assumed
horizontally
homogeneous
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and
35
vertically
non-isotheraial.
Therefore transfer of microwave
radiation in the atmosphere will be azimuthally
and
the
independent
intensity will be only a function of optical depth
and zenith angle. Using Eddington's
streams
of radiances
(Liou,
1980)
approximation
of
two-
the intensity is expanded
into its components as
Iv (T,e ',<J>') = Iq (t) + I 1 (t) cos e'
(3.51)
where IQ and 3^ are the two components of the intensity
(3.50)
Substituting
source term of
0
TT
= w v (t)
(3.49), yields
[ Iq(t) + g(-r)
and equation
f i (T'e)
of
jj
I^t)
(3.52)
cos e 1]
(3.49) can be rewritten as
=I(t,9) - w (t ) [
- {1 - w
where
in the scattering part of
1_
P (e ,<J>,e 1 ,$ 1) I (T.e1, ^ 1) d(cos e 1) d<t>
n
v
2 tt
4
(3.51)
and
.
I0 (t) + 5(t) I ^ t )
»
]
(t )> B{T( t )>
(3.53)
= cos e 1 a nd radiances are assumed to
frequency and p o l a r i z a t i o n
emission
explicitly
now
and
radiances,
albedo,
depend
on
functions
(not shown explicitly now and
h e r e a f t e r ) . Also the single scattering
microwave
be
the
hereafter).
It
albedo
frequency
may
be
and
thermal
(not
shown
noted
that
a s y mmetry factor and temperature of the
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36
atmosphere all are functions of optical depth
be transformed
to
another
convenient
vertical height
(z) of the atmosphere.
(t ), which can
variable
known
as
The transformation is
given by
d-r(z) = - B e x t (z) sec e dz
t
J" fie x t (z
(z ) = - sec e
(3.54)
) dz
(3.55)
where B g x t (z) is the total volume extinction coefficient
of
the atmospheric layer of thickness dz and at a height z from
the
earth surface.
This total volume extinction coefficient
is the sum of the coefficients of absorption due to
water vapor,
liquid water of clouds and of extinction due to
rain.
These
wh i c h
dominate
frequencies.
oxygen,
are
the
main
absorption
constituents of the atmosphere
a nd
scattering
at
microwave
Thus the total volume extinction coefficient at
an altitude z is given by
B e x t (Z) = “o. (Z) + “rH . O (Z) + “ c d ' 2 ’ + B r a i n |Z)
where
aQ
is the oxygen absorption,
absorption,
clouds
and B
is
ram
the
liquid
Q
water
is the water vapor
absorption
frequency
in
the
is the extinction coefficient of rain. All
these absorption and extinction coefficients
of
(3'56)
and
the
atmospheric
are
functions
parameters
pressure,
temperature and humi d i t y at the height z.
Inserting
explicit
(3.51)
dependence
and
(3.54)
into
(3.53)
and
on
altitude in the parameters for the
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deleting
37
subsequent discussion,
yields
(1 - »> B e xtB(T)
In order to separate 1^
(3.57),
integrating
and
(3.57)
I
components
of
I
in
(3.57) w i t h respect to y dy and dy over
-l<y<l, yields
(3.58)
dz
dl
1
dz
3 B e x t (1 - w)
where B
. the extinction
ext
particles,
The
(3.59)
(I0" B(T))
coefficient
is c a lculated from
of
an
ensemble
of
(3.56).
bound a r y conditions are d e t e r m i n e d by the downward
flux at the top of the cloud and u p w a r d flux at
of the cloud as given by W u a nd W e i n m a n
the
bottom
(1984).
(3.60)
and
where
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38
and €v are the horizontal and
vertical
emissivities
of
the surface and Tg is the surface temperature.
Equation
(3.58)
and
(3.59)
are
differential equations in IQ a nd I
first order coupled
w h i c h can be transformed
into second order uncoupled differential
equations
for
IQ
and I 1 .
d2lo
ri r
i __
dl 0
° + B _ „ ( l - w g ) 5 - [=■
1
ext
w M ' dz , 5
- dz
L BB e x -t (1
a "
- w
« g)
g]
dz
L
. 2
-
= 3 B^x t (l - Z g)
-
(1 - Z)
(3.63)
,2 .
+ B e x t (1 " “ »> Hz
dz
text'1
3 B e x t (1 "
The
Z)
solutions
equations
(3.63)
because the parameters B e x t » w
height
water.
w) -1
d f {T) + 3 B e x t (1 ~
numerical
differential
~
during
the
Therefore,
phase
equations
w 5)
11
of
these
and
(3.64) become unstable
and
g
s e cond
vary
abrup t l y
transitions
from
(3.58),
(3.59)
and
using
a
finite
difference m e t h o d
order
with
ice to liquid
numerically w i t h the boundary conditions given by
(3.61)
(3.64)
are
solved
(3.60) and
(Wu and Weinman,
1984) .
Mu l t i p l y i n g
(3.49)
2
2
by a factor C /kv
and u s i n g
and the relative component of
given by
(3.9),
(3.11)
one obtains
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39
the
expression
terms
of
for
the
brightness
horizontally
equation of radiative transfer in
temperature
homogeneous
and
for
a
plane-parallel
axially symmetric atmosphere
whi c h is
^ TB v (t
'P)
w
= TB v (T'*J'p) "
d T ~
(t ) fi
P(T,y,y')
- 2 ~
- (1 -
where
p
denotes
temperature,
and
the
T
the
atmosphere at optical depth,
T
Dl/
and
is
wv
(t )) T( t )
of
ambient
t
(T,y,p)
dy1
Bv
polarization
is
T
0
(3.65)
the
brightness
temperature
. The brightness
of
the
temperature,
a function of frequency, optical depth, nadir angle
polarization.
The
equation
(3.65)
m ay
be
directly
integrated to solve for Tg^
r
T
Bv (t ,y ,p) = eT
[ " |
Jv (T',y,p)
The constant of integration
initial
condition
that
is
at
j
(3.66)
calculated
by
e T dT1 + C
C
is
t
= 0 ,
the
temperature is given by T _ ( 0 , y , p )
and therefore,
T B v (T,y,p) = T g y (0,y,p) eT - eT
J v (T',y,p)
aV
Rearranging
T g y (0 , y » P)
(3.67)
=
the
brightness
e “T 'd r ' (3.67)
in the right order
T b v (t 'jj,p)
e T + Jo
e_T ' dT 1
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(3.68)
40
where
J „(T'f*i,p> = (1 - wv (r')) T( t ')
P( t ',p ,p ')
Equation
(t 1,p 1,p) dp
(3.69)
(3.68).states that at a point P, outside the medium
the intensity of radiant energy is the sum of
emissions
all points upstream of P reduced by a factor e
for extinction by the intervening medium.
purposes,
where
radiometer,
the
equation
—T *
at
to account
For remote sensing
up w e l l i n g radiances are detected by a
(3.68)
is simplified and
rewritten
in
an appropriate form.
The
upward radiances of polarization p
(generally h or
v) received b y a radiometer at an altitude H, viewing at
angle e from nadir is given by
where
T = exp
dz' ]
(3.71)
exp
dz']
dz
exp
d z 1 ] dz
(3.72)
(3.73)
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an
41
J(z,e)
T
s
= {1 - w(z)}T(z)+w(z)[i(z)+g(z)I(z)
and € (e) represent surface
p
respectively.
equation
that
The
(3.70)
is
three
temperature
terms
represent
on
the
(3.74)
and
emissivity
the right hand side of
(i) the emission
the
surface
attenuated by the intervening atmosphere,
(ii) the
downward emission of the atmosphere,
by
cos e]
surface,
with
attenuated
in
its
atmosphere,
r ,
and
atmosphere,
T
.
a will
, that is reflected
reflectivity
upwelling
(iii)
The
the
surface
by
(1-
path
(€
by
u p ward
the
(e)),
intervening
emission
brightness
and
by
the
temperature is
given by the product of its temperature and emissivity.
The
surface emissivity has been calculated in the next section.
(3.8) Microwave Interaction w i t h the Ocean-Surface
When
microwave
interface,
and
some
Snell's equation.
s ea
of
it
by
the
air-sea
is
transmitted
according
to
The transmitted radiation gets absorbed b y
wa t e r because sea wat e r has a large imaginary part
of the dielectric constant
complex
encounters
it is partially r e f lected according to Fresnel's
equations,
the
radiation
at
microwave
Fresnel reflection coefficients
(Kerker,
frequencies.
and
are giv e n
1969)
Rv =
{ cos e,
- m cos e*
} / { cos e,
Rh =
( m cos e t - cos ea } / { m cos e,
The
+ m cos es } (3.75)
+ cos e 2 } (3.76)
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42
w h e r e e,
and e a
respectively
are
as
angles
shown
in
of
the
incidence
Figure
and
refraction
(3.4) and m is the
complex index of refraction of sea water which is
given
by
the square root of its dielectric constant, k.
whe r e
m = m 1 - i m"
(3.77)
k - k 1 - i k"
(3.78)
m =
(3.79)
i=>T-l;
-f
k
and m ! , k 1 are the real parts and - m " , k" are
the imaginary parts
dielectric
of
constant
complex
index
respectively.
The
of
refraction
and
angles are related
t hrough Snell's law as
Sin e,
= m Sin e a
(3.80)
Reflection of microwaves from the ocean surface can
understood
in
r e f lected
waves
components.
The
d enoted by the
terms of S t o k e 's vectors of the incident and
by
finding
the
relation
between
subscript
i
and
r
respectively
a nd
I.
(3.81)
I
(3.82)
r
their
incident and reflected Stoke's vectors are
d e s c r i b e d as
l
be
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are
\ !
/
'/:
'/ :
/■■■
/
/
/•
nil”
••<;:;
\
;: 5 e 3 : W 3 t S r
•
• •>.. •
■'
’
v . ■i •
■ 2 : ' i
v!
Nacrr
Figure 3.4 Geometry cf reflection one refraction ai the air-sea interface
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44
where
*
E. E. +
IV IV
E ihE ih 3
(3.83}
*
E. E. IV IV
E ihE ih ]
(3.84)
*
*
(E. E., + E -. E .
ah iv
' av lh
(3.85)
*
<E ivE ih
*
E .. E .
ah iv
(3.86)
= [ I
+ I , ] = C ' [ e
E* + E ,E*. ]
L rv
rh J
L rv rv
rh rh J
(3.87)
*
!. E.. )
i v
I
r
r
L
•4c
Q
=
U
= 2 C 1 Re
r
lh
~
I
V = -2C'
r
rv
In
«4c
- I u 1= c 1 r E
E
- E . E . l
rh J
L
rv rv
rh rh J
(3.88)
v
'
*
*
*
(E E . ) = C 1 (E E . + E . E
)
' rv rh'
' rv rh
rh rv'
(3.89)
7
'
(E E* ) = i c 1 (E E . - E . E )
' rv rh'
' rv rh
rh rv'
where C 1 ds the constant of proportionality
EE*
at
the interface.
(3.90)
'
I
between
Re (E rh ) >
Im
are
respectively.
Re (E r v )
are related to the components of the incident
radiation at the interface,
and
and
The components of the electric-field
vector of the reflected radiation at the interface,
and
'
real
and
R e ( E i v ) and
imaginary
parts
^E i h ^ '
of
the
where
Ra
quantity
These components are related by
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45
E rv
(3.91)
R v E iv
(3.92)
E rh “ R h E ih
The relations between the components
reflected
(3.83)
I
Stoke's
to
vectors
are
of
the
determined
incident
from equations
(3.92) which are expressed as
=1.
R R* + I..R.R*
iv v v
ah h h
(3.93)
Q = I . R R* - I .. R, R*
^r
av v v
ah h h
(3.94)
r
U r = U i Re
< R vR h > + V i Im
V r ° V i Re
( RvR h ) - u i ^
Equations
r Jr 1
and
to
(3. 93)
'
^r
Ur
V
r
R1
R2
R2
0
R1
0
0
0
(3.96)
< RvR h >
>
(3.95)
(3.96)
can be
0
0
0
0
R3
R4
ui
~R 4
R3
vi
r 11
(3.97)
or
t3.98)
where
R_ =
R„ =
(Rv Rv + R h R h )/2
(Rv Rv " R h R h )/2
R„ = Re
(Rv R*)
(3.99)
(3.100)
(3 .101)
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46
R 4 = Im
The bar
h as
(Rv R*)
been
independent
dropped
quantities.
(3.102)
because
The
R^,
and
R^
are
time
reflection matrix
[R]
for
de t e r m i n i n g the Stoke's vector of reflected radiation is the
same for the radiation originating inside the water as it is
for the radiation originating in the air so long
of
incidence
l aw
(3.80).
incident
Since the energy is
intensities of
1^,
r e f lected
ra d iation
must
be
radiation,
I , and
r
of
transmitted
(ocean-surface)
frequency
that
(3.103)
according
is also a good emitter
to
Kirchoff's
law.
The
arrives at the top of the air-sea interface
gets absor b e d b y a few centimeters of the top
depth)
of
equal to the sum of the
= Ir + 1 t
good absorber
at the same
conserved, the intensity
1^.
Ii
A
angles
and refraction are those described by Snell's
radiation,
radiation,
as
layer
(skin-
the sea. This is then re-emitted from sea to air
after bei n g abso r b e d by the skin-depth.
The intensity of the
ra d iation e mitted b y this top layer of the sea must be equal
to the intensity of emission of a blackbody having the
temperature
same
as that of the layer. At the air-sea interface,
a portion of the
r a d iation
which
arrives
from
below
is
reflected back into the sea and the rest is transmitted into
air,
and is given by
It = I. -
[ R ] I. =
( [ I ] -
[ R ] ) I.
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(3.104)
47
where
[I] is an identity 4x4 m a t r i x given by
[13
1
0
0
0
=
0
1
0
0
0
0
1
0
0
0
0
1
(3.105)
The transmission m a t r i x can be defined as
[ T ] =
[ I ] -
[ R ]
1 - R,
[ T ] =
- R,
(3.106)
0
0
0
- R2 1 " R1
0
0
1 - R,
0
0
(3.107)
0
1 - R
and
It =
[ T ] Ii
(3.108)
The general transmission m a t r i x
[T]
may
be
used
for
ra d i a t i o n h a ving ar b i t r a r y polarization emerging from either
side
of
the
air-sea interface provided that the Fresnel's
coefficients used in
(3.102)
(3.107)
are those giv e n
by
(3.99)
to
and radiation travelling along the path dictated by
the angles e, and e 2 given by
(3.80).
The
Stoke's
vector
d e f i n e d for the intensity of thermal radiation emerging from
b e l o w the air-sea interface is
Id =
and
the
{ B(v,Ts ) , 0 , 0 , 0 }
(3.109)
transmitted thermal emission into the air from the
sea is
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48
(1 - R x) B ( v ,T g )
- R
0
B(v,T
)
(3.110)
s
In a local thermodynamic equilibrium,
emissivity
water.
the
sea
surface
is derived from reflective properties of the sea
The emissivity of a body is defined as the ratio of a
polarized component of the radiation emerging from the
body
to that of a blackbody whose temperature is the same as that
of
the body.
The sea surface emissivities of horizontal and
vertical polarizations,
eh and €v , respectively are defined
as
€h = 1
Rh Rh
(3.111)
= { It h <v.Ts ) } / { B h (v,Ts ) }
e
V
= 1 - R R*
V
V
(3.112)
= { I. (V ,T ) } / { B (V ,T ) }
tv
' S
v s
Expressing the various intensities in
equivalent
brightness
temperatures
given
terms
by
corresponding microwave emission of a blackbody,
vector
for
the
temperature,
Tg
B
where
the
of
their
(3.11)
the
and
Stokes
emission of the sea in terms of brightness
, can be expressed as
< <t b v + t b h >'
vertical
and
<t b v - t b h >' °'
horizontal
° >
components
(3.113)
of
the
brightness temperature of the sea are given by
T
= € T
aBV
v
s
(3.114)
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These
represent
components
the
of
thermal
ocean
brightness
emission
of the ocean given by the
first term in the right hand side of
m ult i p l i e d
temperature
(3.70), which
is
by a factor r the so called transmittance of the
under lying atmosphere.
Thus the surface contribution to the
total radiances received by a radiometer is the
transmittance
calculated
from
product
and
radiative
by
(3.115). The polarization of radiances enter the
transfer
w h i c h involves the
specific
of
(3.71) and the corresponding
component of thermal emission of the ocean surface given
(3.114)
then
calculations
calculation
polarization.
This
through the surface term,
of
surface
calculation
emissivity
of
emissivity will be dealt w i thin the next chapter.
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of
surface
CHAPTER (IV)
MICROWAVE EMISSION AND A B S O R P T I O N BY
THE
SURFACE
AND
THE A T MOSPHERE
Theoretical
the
formulation of m i c rowave
interaction
with
ocean and the atmosphere was considered in Chapter III.
However the numerical c a l c ulations of the parameters used in
the theory was deferred.
will
be
performed
In this chapter these
using
specific
ocean
calculations
surface
and
atmospheric models.
(4.1) Microwave Emission of the Model Ocean Surface
M icrowave emission of the ocean surface depenus on
e m issivity
surface
and
temperature.
(specular surface)
The
its
emissivity of a calm sea
is a function of its
dielectric
constant which in turn is a function of sea surface salinity
and t e m p e r a t u r e ,
(A) Dielectric Constant of Sea Water
The theory of dielectric constant of pure liquid
has b e e n derived by Debye in 1929.
The real
water
(K1) and the
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51
imaginary
(K") parts of
the
dielectric
constant
(K)
are
given by
K = K +
---------------------------- (4.1)
(1 + w t
)
and
qt
(K - K )
K" = ------- §---------------------------- (4.2)
(1 + w
wh e r e K
is
s
dielectric
the
constant
to infinity and
static
static
t
T
)
dielectric
constant,
as the angular frequency
is the r e l a x a t i o n
time
de termined
form
is
«
the
(u=2IIv) tends
(sec).
Both
the
dielectric constant and the relaxation time for pure
w a t e r are functions of temperature,
water.
K
value
of
the
T.
term
The
is 4.9 for pure liquid
W h e n pure water is repla c e d b y sea
of
the
dependence
may
m o d i f i c a t i o n in the form of k"
e x p e r i mentally
be
water,
a ssumed
with
0-
(4.3)
w
€q
because the conductivity of impure water
the
impurity.
conductivity
of
the
medium
temperature,
T,
and salinity,
slight
+
The second term on the right hand side of
with
a
same
:
U T (K C - K tf )
K" = ------3
v - - (1 +
increases
the
which
S,and
is added
(containing
Here
<r
is
a
c.^
(4.3)
is
is
the
function
salt)
ionic
of
the perm i t t i v i t y
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
52
of
vacuum
represents
(8.854x10
the
-12
damping
Farad/meter).
of
The
We
ppt
have chosen the values of K ,
s
on the measurements made by Saxton and
these
measurements Hollinger,
<t / u € q
the electric field vector. Sea
w at e r salinity is usually between 30 to 40
thousand) .
term
Lane
(parts
per
and <r
based
(1952) .
Using
t
(1970) obtained the following
empirical equations;
Kg =
[88.0 - 4.339X10-01 S + 1.71xl0_° 3 S2 - 4.035xl0_01 T
+ 8.065X10-04 T 2 + 6.170X10-03 S T -
8.910xl0-05 S 2 T
- 6.934X10-05 S T 2 + 1.439X10-06 S2 T 2 ]
T =
(4.4)
[18.70 - 7.924X10-02 S + 6.35xl0“°4 S2 - 5.489xl0_01 T
+ 5.758X10-03 T 2 + 1.889X10-03 S T -
- 5.299x10
-07
ST
2
-07
2
- 2.101x10 u/ S^ T
7.209xl0-06 S2 T
2
-12
] 10 ^
(4.5)
or = [7.788xl0~03 S - 1.672xl0_06 S2 - 8.570xl0_15 T + 2.996
Xl0-16 T 2 + 4.059X10-04 S T -
3.215xl0-06 S 2 T - 1.423
X10-06 S T 2 + 3.229X10-08 S 2 T 2 ] 1011
where the frequency,
parts
per
thousands
v,
is in hertz,
(4.6)
the salinity,
and the temperature,
S, is
in
T, is in degrees
centigrade.
(B) Microwave Emission by a Calm Sea
Microwave emission of a calm sea can be
the
theory
understood
by
developed in chapter III for the reflection and
transmission of
electromagnetic
radiation
by
an
air-sea
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53
interface.
sea
The polarized components of microwave emission of
are given by
the sea
surface
emissivity
of
(3.114)
and
(3.115) whi c h are dependent on
temperature
the
ocean.
a nd
horizontal
or
vertical
The s ea surface emissivities are
functions of Fresnel's coefficients whi c h in turn depend
on
the index of refraction of sea water and the zenith angle of
the
transmitted
radiation.
The complex index of refraction
of sea water is a function of its temperature,
mi c rowave
frequency.
The
horizontal
emissivities of sDecular sea surface,
ca lculated
using
Fresnel's
e .,
sh
equations
salinity
and
e
vertical
have
sv
which
and
relate
been
the
specular emissivities to the dielectric constant as given by
€gk = 1 - [{Cos e - >r(K-Sin2 e)}/{Cos e
+ -J"(K-Sin2 e )} ] 2
(4.7)
and
€sv = 1 - [{K Cos e - >T(K-Sin2 e)}/{K Cos e
+ >T(K-Sin2 e )} ] 2
The complex dielectric constant,
the
relation
given
by
(4.1)
to
K,
(4.8)
is calculated
from
(4.6) and e is the zenith
angle.
(C) The Effect of Sea Surface W i n d and Foam
The
affected
ocean emissivity at all microwave frequencies
by the
sea foam
and
surface
waves
is
which are
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54
g e n e r a t e d by the surface winds. This effect can be explained
in terms of the
surface
distribution
of
the
slopes
of
the
whi c h is a function of sea surface temperature,
salinity, wind stress,
fetch and duration of wind.
of microwave wavelength,
surface
radiation.
order
the surface can not be treated as a
(specular
surface)
for that w a velength of
The exact effect of surface winds and foam on the
s ea microwave emission is not known so far because too
environmental
conditions
d u r a t i o n of w i n d speed,
have
not
sea
Since the
radius of curvature of any sea surface w a v e is of the
smooth
sea
been
such
as
atmospheric
sea temperature,
and
many
stability,
sea
salinity
adequately known c o n c urrently over an ocean
surface.
The d i s t ribution of the sea slopes suggested by Cox
a nd M u n k
(1954a ,''54b) was u s e d in the
study
of
microwave
e miss i o n by the ;,ea of Stogryn (1967).
The effect of ocean surface roughness and foam is taken
into
account
by
coupling
the expressions of specular sea
s urface emissivities w i t h the empirical
by
Pandey
and
Kakar
p o l a r i z e d emissivity,
(1982)
as
e (e), of the
relations
modified
rough
by
sea
obtained
Kakar.
The
surface
is
hr
given b y
«p(e)
- « sp + 4 € p
(4.9)
where p = h or v
This can be d e composed into two parts the first part is
the
emissivity
due
to
the
foamless rough water surface,
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
55
r e p r e sented by €
, and the s e cond part is due to
the
foam
* Jr
covered sea surface,
denoted by € fp / ant* the total polarized
emis s i v i t y can be wr i t t e n as
€ p ( e)
whe r e
= (1 - F)
€rp + F € fp
(4.10)
F is the fractional foam cover whi c h ranges from zero
for foam free surface
to
one
for
100%
foam
cover.
The
foamless rough sea water e missivity is considered to be
e n =
rp
Subs t i t u t i n g
e^ + A €
sp
rp
(4.11)
into
(4.11)
(4.10) and comparing with
(4.9),
yields
A£p = A£rp + F <£ f p - £ rp>
Equation
(4.12)
e m issivity
is
states
due
to
that
two
the
'4 ' 12>
total
reasons.
in
the
The first of which is
change in the foamless rough sea w a t e r
second
change
e m issivity
and
the
being the difference of the foam emissivity from the
rough wat e r emissivity.
This difference is superimposed only
over the foam covered area.
The change in
two
polarized
components
of
foamless
rough sea surface emissivities are given by
A £ rh =
( W ^ / T g ) (!• 15xl0_01 + 3.80X10-05 e 2 )
A€rv =
(WV/T
(4.13)
and
) (1.17x10
-01
- 2.09x10
E x p ( 7 . 32x10
whe r e
-02
-09
©))
(4.14)
W is the surface wind speed in m/s,
t emperature in degree kelvins,
a nd
v
is
Tg is the surface
the
frequency
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
in
56
GHz.
The
two
polarized
components of foam emissivity are
giv e n by
€fh =
{(208+1.29v)/Tg }
(1 - 1.748X10-03 0 - 7.336X10-05 0 2
—
+ 1.044x10
efv =
07
^
)
0
( (208+1.29v)/Ts )
(1 - 9.946 X10
—04
9 + 3.218x10
- 1.187X10-06 e 3
The
(4.15)
fraction
of
foam
—
0*1
+ 7.Oxio-20
cover
empirical relation obtained by Wu,
is
9
6
0 10)
calculated
and Fung
the
relations
(4.7)
to
from
an
(1972)
F = 7.751X10-06 W 3 *231
Using
(4.16)
(4.17)
(4.17),
both
polarized
components of rough sea water emissivity can be obtained.
The graphs of horizontal and vertical
and
€v
of ocean surface verses frequency
ten zenith angles 0, 30,
d e gree
are
plotted
35, 40,
in Figures
sp e e d v arying from 0 to 25
a nd
at
a
salinity
surface
45,
(4.1)
50,
to
(1 to 300 GHz)
at
55,
85
60,
65,
for the wind
(m/s) w i t h a interval of 5
temperature
300
and
(m/s)
a fixed value of
(35 0 /0 0 ). These Figures show that the emissivities
temperature.
At
the
values
than nadir,
linear
functions
nadir looking case, e=0, both the
horizontal and vertical components
Their
€,
n
(4.6)
are non-linear functions of frequency but
of
emissivities
and
are
equal.
increase w i t h the w i n d speed. At angles other
increases and ev decreases non linearly with
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
OCEAN EMISSIVITY VS FREQUENCY AND ANGLES
SAUNITY=35.,SURF.TEMP.=300.0,WIND =
.0
1.0
.9
,8
E>
CO
tn
s
w
.7
<
y
6
L - 5 0 .0 V
a
ca
>
.5
Q
Z
<
►J
.4
Z
o
K
o
W
N
M
.3
.2
1
0
0
20
40
60
80
100 120 140 160 1B0 200 220 240 260 280 300
FREQUENCY (GHZ)
USING S.T. WU MODEL FOR FOAM
Figure
(4.1)
R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w itho ut perm ission.
OCEAN EMISSIVITY VS FREQUENCY AND ANGLES
SALLNTTY=35.,SURF. TEMP.=300.0,WIND = 5.0
1.0
tn
tn
L - SO.0 V
2
w
►j
<
u
H
K
W
R - 6 5 .0 V
>
1 - 6 5 .0
Q
2
<
iJ
<
E-
2
O
N
M
K
O
X
0
20
40
60
80
100 120 140 160 180 200 220 240 260 280 300
FREQUENCY (GHZ)
USING S.T. WU MODEL FOR FOAM
Figure
(4.2)
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
V
OCEAN EMISSIVITY VS FREQUENCY AND ANGLES
SALINITY=35.,SURF.TEMP.=300.0, WIND =10.0
1.0
.9
■J*; C - 3 0 .0 M
.8
>•
E-
>
tn
co
.7
i—i
s
w
<
.6
>
.5
u
H
K
Cd
Q
Z
<!
.4
<!
.4
EZ
o
N
5
O
K
.3
.2
.1
0
0
20
40
60
80
100 120 140 160 180 200 220 240 260 280 300
FREQUENCY (GHZ)
USING S.T. WU MODEL FOR FOAM
Figure
(4.3)
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60
OCEAN EMISSIVITY VS FREQUENCY AND ANGLES
SALINITY=35.,3URF.TEMP.=300.0, WIND =15.0
EMISSIVITY
1.0
VERTICAL
L- 60.0 7
HORIZONTAL
AND
0 * 6 5 .0 H
0
20
40
60
80
100 120 140 160 180 200 220 240 260 280"300
FREQUENCY (GHZ)
USING S.T. WU MODEL FOR FOAM
Figure
(4.4)
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
61
OCEAN EMISSIVITY VS FREQUENCY AND ANGLES
SALINITY=35.,SURF.TEMP.=300.01WIND =20.0
1.0
EMISSIVITY
C- 30.0 H
0- 30.0 V
I- -4S.0 H
L- 60.0 V
VERTICAL
N- 66.0 V
0- 65.0 H
R- 65.0 V
HORIZONTAL
AND
65.0 V
FREQUENCY (GHZ)
USING S.T. WU MODEL FOR FOAM
Figure
(4.5)
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
OCEAN EMISSIVITY VS FREQUENCY AND ANGLES
SALINITY=35.,SURF.TEMP.=300.0,WIND =25.0
1.0
ANGLES
L - 50.0 V
3
w
5S .0 V
o - so.o h
U
H
w
;.o v
>
Q
$
Z
o
N
M
K
O
X
o
20
40
80
80
100 120 140 160 180 200 220 240 260 280 300
FREQUENCY (GHZ)
USING S.T. WU UODEL FOR FOAM
Figure
(4.6)
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63
w i n d speed
whereas
difference
of
both
decrease
with
temperature.
vertical and horizontal emissivities
The
(e -€. )
' v
h'
increases with angle at all frequencies but at higher angles
like 85°
speed,
this difference starts to decrease.
ev
and
values
of
wind
both get saturated at higher frequencies.
The behaviour of the difference
the
At higher
the
ocean
(sv _€h)
was
analysed
from
emissivity calculated from this
model. The Quantity (e -€. ) decreases with
1
v
h'
wind
SDeed
and
increases with temperature for all frequencies and angles up
to 65 degrees.
(4.2) Microwaves Absorption by Atmospheric Gases
Oxygen
microwaves
do
not
and
water
absorption
scatter
vapor
significantly
in the atmosphere.
microwave
r a d iation
dominate
Since these gases
significantly,
the
emissive properties are directly related to their absorption
properties,
the v o lume absorption coefficients of the gases
are to be calculated as functions of the variables of state.
(A) Oxygen A b sorption Coefficient
There is a complex band of
lines
in
stron g l y
absorbing
oxygen
the atmosphere centered around 60 GHz. The oxygen
molec u l e has a permanent magnetic dipole moment.
The changes
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64
in the orientation of the electronic spin
orientation
magnetic
of
the
dipole
transition
at
molecular
transitions
relative
near
60
GHz
and
with
and
angular
momentum
J
=
N
frequencies
at
(4.1).
zero
oxygen
vN+
In addition,
by
b r oadening
there is
using
Rosenkranz
the w o r k of Gorden
(1975)),
of
states
a b sorption
and are listed in
non-resonant
absorption
Microwave absorption coefficient due to
calculated
s u g gested
line
(Rosenkranz
frequency.
is
single
J = N ± 1. The
transitions between these states permit resonant
Table
a
118.75 GHz. Coupl i n g of electronic spin w i t h
triplet
at
the
rotation produces a band of
the rotational angular momentum forms a
total
to
(1967)
due
the
computational
scheme
(1975) whi c h relies principally on
and
Van
Vleck
(1947).
Pressure
to interactions among atmospheric gas
m o l ecules and temperature dependence of this line broadening
are taken into account
relat i n g
to
the
means
of
R o s e n k r a n z 1s
o x y g e n absorption coefficient
frequency
temperature
by
v
T
(GHz),
pressure
P
a
formula
(Neper/Km)
(millibars),
and
( K ) . The a bsorption coefficient is thus given
by
cr0 j (v) =
(CP2 v 2 /T2 )(Z n +N [fJ(v)
+ f N (_v)
+ f N (v) + fN (-v)]
+
where C
=
0.330
is
(.70 W b /tv2 +
a
(PWb )2 ])}
constant.
N
is
(4.18)
the
fractional
p opu l a t i o n of state N at temperature T
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
65
<t>N = { ( 2 N + D / 0 . 7 2 5 T) exp (-2 .0685 N(N+1)/T)
(4.19)
The shape factor for the transition line is given by
f*(W
= <WH < 4 )2+(‘'-V H t )2-Y N )
/ <<v - v n ± )2+(PWn )2 >
where
(4-20)
vN+ are the resonant frequencies given in Table
(4.1)
a nd the summation is over odd rotational states N from 1
to
39. The amplitudes of positive and negative transition lines
at v* and v N are given by
d* =
[N(2 N + 3 ) / (N+l)(2 N + 1 )]° ’5
(4.21)
d^ =
[(N+l)(2N-1)/N(2N+1)]0,5
(4.22)
The non-resonant line width is given by
W b = 0.48X10-03
(4.23)
(300/T)° *89 GHz/mb
and the resonant line broadening half width is computed from
W N = 1.16X10-03
The
interference
(4.24)
(300/T)0 '85 GHz/mb
coefficient
for
coupling
between
near
states due to molecular collisions is given by
Y N = dN
^ 2dN+2 “ W N (up)^/{v,N “ VN+2*
+ {2dl-2 V
where the line
dn)}/{vN - v * _ 2 )
" <W b /V N } " {Wb /(VN + 6 0 *>>J
(4.25)
widths
coupling
for
the
collisional
are
computed in a sequence from the relations
W N (dn) = W b ~ W N " W N (up)
(4*26)
W N - 2 (UP) = W N (dn)
(4.27)
and
Microwave
absorption
(4’n /4>N - 2 )
above
40
Km
was
considered
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
TABLE
(4.1)
R e s o n a n t F r e q u e n c i e s of M o l e c u l a r Ox y g e n a n d A m p l i t u d e
Frequencies
N
+
VN
(GHZ)
,'n
Factors
Amplitude Factors
dN
dN
1
56.2648
118.7503
0.9129
0.8165
3
58.4466
62.4863
0. 9 8 2 0
0.9759
5
59.5910
60 . 3 0 6 1
0.9924
0.9909
7
60.4348
5 0 .1642
0. 9 9 5 8
0.9952
9
61.1506
58.3239
0.9974
0.9971
11
61.8002
57.6125
0. 9 9 8 2
0.9980
13
62.4112
5 6 . 9682
0. 9 9 8 7
0.9986
15
62.9980
56.3634
0.9990
0.9989
17
63.5685
5 5 . 7838
0. 9 9 9 2
0.9992
19
64.1278
55.2214
0 .9994
0.9993
21
64.6789
5 4 . 6711
0 .9995
0.9994
23
65.2241
54 . 1300
0 .9996
0.9995
25
65.7647
53.5957
0.9996
0.9996
27
66.3020
53.0668
0.9997
0.9997
29
66.8367
5 2 . 5422
0 .9997
0.9997
31
67.3694
5 2 . 0212
0.9998
0.9997
33
67.9007
51 . 5 0 3 0
0. 9 9 9 8
0.9998
35
68.4308
50 . 9873
0.9998
0.9998
37
68.9601
50.4736
0.9998
0.9998
39
69.4887
49 .9618
0.9998
0.9998
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
67
negligible.
Zeeman splitting effects
were
ne g l e c t e d
below
this altitude where the temperatures are too low.
The
integrated
atmosphere
(Table
frequencies
oxygen absorption for a representative
4.2)
between
has
1
been
to
300
calculated
GHz.
The
for
all
variation
integrated oxygen absorption versus frequency is shown by
solid
curve
in
Figure
(4.7).
absorption between 50 and 70 GHz,
GHz.
The
frequencies
This
curve
and between
shows
115
of
a
strong
and
12 2
be t w e e n 50 and 70 GHz have different
we ightings of oxygen abso r p t i o n at different altitude in the
atmosphere.
The
concentration
w e ightings
and
the
are
determined
by
oxygen
temperature dependance of the cross
section.
(B) Water Vapor Absorption Coefficient
The
water
molecule
dipole
moment
which
region.
It has a single,
possesses
a
permanent
electric
causes an absorption in the microwave
weak,
pure
rotational
resonance
line w h i c h arises from the transition 5_ -6,_ at 22.235 GHz.
23
lo
The next resonance frequency is at 183.3 GHz.
The resonance frequency 22.235 GHz has been extensively
studied
by
Becker and Autler
(1946)
and Townes and Shallow
(1955). For the frequencies below 170 GHz
ab sorption
has
been
the
water
given b y Barrett and Chung
vapor
(1962)
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
and
68
TA3LE (4.2)
Atsosphere Model after Cole et al. (1355, p.2-3)
-I-
Height
(Ka>
Pressure
(3b)
3.0
1313.25
-I-
TeisperaturelDew Point Illat. Vapor
( K)
ITeap. ( K) ICen.(gn/nf’)
------------------------------ T ------------------------------- J -------------------------------
330.4
I
------------------
3.5543
353.3
237.1
------------------
1.3324
300.3
234.4
------------------
1•\iLLu
353.3
231.5
235.5
I 232.4
j
I 283.7
833.8
283.'
.5
753.3
235.3
I
237.3
I
-I-
I 5.5224
I 275.4
»------- 1----278.2 I 263.7 I 3.6835
----- j--------1----231.3
3.7611
3
I 11.7712
-I- - - - I 3.3135
-I------
I
1
3.1532
1
I 13.8641
J------------------- T-------------------
1
2.5331
I 15.2322
------------------- -------------------
-I
2.0373
I 13.4353
1------------------- j------------------
3
«
533. C
,1
*
A** I I
T
'“t
£/ *t• 4
1
iQ't.i
1
i..7u£u
inrift
-------- T----------- 1--------
18
273.3
11
.3
6.6542
12
450.3
1I 257.2
1-
2.3
I
!-
1--------------
243.7
253.3
1---
13
7.5545
14
8.5434
OCC ^
403.3
i.Jj.4
243.3
9.6432
330.0
1 u,533B
-I-
T
i 0
ot'Anrc
TuOO
1
ftj.SA T
1---------I--------
1 237.8
1---
15
I 1.4112
!- - - - I 3.3341
241.7
I 233.2
I 0.2337
T
_
"i-------
i
i/iiibJ
1------------------ 1-----------
16
18.3168
233.1
250.3
I 213.2
I
3.3353
1------------------ 1-----------
17
f n
10
12.4354
I
1 4 .2 2 5 1
233.3
I
1 5 3 .3
222.7
I 153.3 I4.3174E-33
- - - - - 1- - - - - - - j- - - I
I 1
•-VOQ •”>
L U i.L
co
n
iJ O .U
1 3 . 12 G 2 E -C 3
|C ,1 o
I \J«J « O
I3 .4 S 3 3 E -G 3
« en
.■»
l diii'iJ
Tc
♦ •*n n r
nn
i O « 't O O c C ” U O
T
19
t
1
«r
f n « ■>
iC iC iO li
23
I
2 3 .6 1 3 4
—
?
1
1 3 3 .0
I
1 9 6 .2
1
33.e
I
195.2
-T
I
I
-I-
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
too
WATER VAPOR ABSORPTION
90
OXYGEN ABSORPTION
80
70
i
—
z
U
HJ
O
M
t i­
50
ll.
lii
o
o
2 40
o
M
J
0.
0
1
o
to
CD
<
20
O
O
—
O
O
O
O
O
O
O
O
Q
C s J C 0 r i / 3 ( D h - C 0 0 ) O
o o o o o o o o o o o o o o o o o o o
-cvjcoTiiiior^coojo- c j c o ’T i n t o r ^ c o o )
C 'J C M O J C M C v J C V J C N J C 'J
300
to
FREOUENCY CGHZ>
Figure (4.7)
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70
u s e d by Staelin
1-centimeter
(1966)
to interpret the microwave data
wavelength.
The
water
vapor
near
absorption
at
microwave frequency below 170 GHz is
a„ _(v) = ,2P105 [(3.24x 10-04/T3 ‘125)
n2U
e x p (-644./T)(P+0.147PT)
( 1 / { ( v - v 0 )2 + a v 2 > + 1 / { ( v + v 0 )2 + a v 2 } )
+2.55 x 10_ 0 8 (a v /T1 *5 ) ]
where
water
Nepers/Km
vapor
at
a
absorption
frequency
v
(4.28)
coefficient
GHz,
P
is
aH Q
T is the temperature in degree kelvin, v 0
resonant
frequency
GHz
and
p
in
the pressure in
millibar,
22.235
is
is
the
is the water vapor
density in gm/m3 .
The line width a v
is given by
& v = 2 .58x l 0 ~ ° 3 [P+0.0 1 4 7 P T ] ( T / 3 1 8 )“ ° ’625
(4.29)
Water vapor absorption for all microwave frequencies as
described below w as found by Kakar
(Kakar,
R.K.;
personal
communication 1986).
0 (v ) =
[Resonance+{(1+n)/n)10 9 p1 , 1 5 (P/P0 )
(31 8 / T ) (a+b)V 2 ]103
(4.30)
where
a = 1.5
P Q = 1013.25
n =
L
(mb)
(number of terms between L,
and L a )
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71
b =
A d
Resonance =
TXP
"2
A it A
O
■— ■- —
-a
(L) / n
2
,,
'
■
----
SP(L)
T,
Z_a
Li
2
(v
S T ( L ) DEL FACT vR (L) i
2
- vR
(L) ) + 4
2
v"1 DEL
(4.31)
2
Del = C 1 (L)(P/P0 )(318/T)TXP(L)
(1 + C 2 (L)(PT/P))
If {(1.43879/T)
(4.32)
(VR (L)/29.9793) < 0.1}
then
Fact = V_.(L) (1.43879/(30 T) }
K
exp ( - l . 43879
TE1(L)/T)
(4.33)
otherwise
Fact = exp(-TE1(L)
1.43879/T)
- e x p [-(TE1(L)+v_(L)/29.9793)
K
(1.43879/T)]
(4.34)
The coefficients used in the equations (4.30)
are given in Table
(4.3) where L,
to
(4.34)
and L2 are calculated from
the relations given by
L, = L - 3 (1 + P / P 0 )
(4.35)
La = L + 3 (1 + P / P 0 )
(4.36)
where L defines the number of terms in the given arrangement
satisfying the condition that the frequency of
v,
is
less
frequency,
than
or
equal
a
particular
resonance
v D , w h e n the resonant frequencies are arranged in
£\
a monotonically increasing
(4.3).
to
observation,
When
L,
order
as
shown
in
the
Table
is less than 1 it is considered 1 and if L 2
is greater than 9 it is considered 9. The
integrated
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water
72
vapor abso r p t i o n for a representative atmosphere
(Table 4.1)
has b e e n calculated at all frequencies
1
between
and
300
G H z . The v a r iation of this integrated water vapor absorption
with
frequency
is shown in Figure
(4.7) by a dashed curve.
The total a b sorption of microwaves by the atmospheric
is
d o m inated b y water vapor over a large range of microwave
frequencies.
atmosphere
The
is
distribution
highly
variable
of
water
quantity
interest for meterological research.
study
this
quant i t y
then
surface
ph e n o m e n a
frequencies w h i c h
resonant bands.
22.235
one
are
and
and
If one
On the
would
far
away
other
require
from
Staelin et a l . (1976),
GHz
vapor
and Duff
31.4
would
the
of most
like
to
hand
to
to
study
choose
water
and Grody
the
vapor
(1976) have
GHz to infer total water vapor
over
oceans.
(1979) derived the total liquid water content
of the atmosphere from Nimbus-6
Spectrometer)
is
these
content of the atmosphere u s i n g Nimbus-5 data
Liou
in
one must select the frequencies
near 22.235 GHz or 183.3 GHz.
used
gases
SCAMS
(Scanning
Microwave
data over land.
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CO
CM
rH
rH
c-«
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CD
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to
CO
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to
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o
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r*
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|
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(4.3)
rH
m
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TABLE
rH
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tl
■
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DU
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10
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74
(4.3) Microwave Extinction by Liquid Hydrometeors
The liquid water drops in clouds and rain are known
hydrometeors
which
absorb
radiation.
In
1908,
Mie
scattering
properties
and
scatter
described
thermal microwave
the
absorption
and
of a homogeneous spherical particle.
His theory may be used to calculate
and
as
the
volume
absorption
scattering coefficients of a polydispersive medium like
rain and clouds if the particle size distribution is
Since
extinction
particles
are
individual
processes
randomly
extinction
are
linear
distributed
coefficient
in
can
in
the
nature
and
medium,
the
be summed over the
particle size distribution in order to calculate
volume
extinction
coefficient
calculation assumes that
distances
the
of
particles
known.
the
are
the
total
medium.
This
separated
by
large compared to their respective diameters.
The
extinction properties of cloud particles smaller than 0.1 mm
in diameter are determined us i n g the Rayleigh
in
approximation
which the scattering processes are neglected compared to
the absorption processes. For this case Goldstein
(1951) has
derived
absorption
a
coefficient
content.
calculate
simple
of
a
expression
cloud
Gunn and East
for
the
volume
as a function of its liquid water
(1954)
have
applied
Mie
theory
the attenuation of microwaves by hydrometeors.
to
In
all of these studies regression equations have been obtained
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75
relat i n g
the
atmospheric
volume
e x tinction
parameters.
properties
considered
these
of
to
coefficient
medium
the
the
calculate
as
(S.S.),
the
have
not
been
In this study exact Mie
the
volume
extinction
us i n g three different particle
size distributions for the medium.
known
medium
calculations.
theory has been used
of
to
The distinction between absorption
and scattering
in
coefficient
(i) Marshall Palmer
These
(M.P.),
distributions
are
(ii) Sekhon Srivastava
and (iii) Gamma distribution.
(A) Absorption Coefficient due to non-raining Clouds
Clouds are assumed to be a collection
dielectric
spheres
(me t e r ) .
These
described
by
per unit
of
pure
spheres
dn(a)/da
volume,
N(a),
are
liquid
of
water
distributed
homogeneous,
of
in
—4
(m
). The total number
between diameters 0 to a
N(a) - I ? f i ,a)
diameter a
the
of particles
is given b y
da
The extinction cross-section,
medium
‘ 4 - 37>
o
^ e x ^.ia )
(m ) , of
a
drop
of
diameter
a
is the sum of its absorption cross-section,
o
o’ t^(a) (m )>
and
its
scatt e r i n g
cross-section,
cr
(a)
aos
sea
(m2 ) .
" e x t 131 =
In
a
tra b s (a> +
non-rainin g
r sca'a >
cloud whose maximum
<4 '3 8 >
drop
diameter,
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76
amax'
*s of t*ie order of
-0001
(m) ,
the
scattering
cross-
s ection is negligible compared to the absorption coefficient
w h i c h is given by
cra b s (a) =
w he r e
(ir2v a 3 /C)
Im
[ (1-m2 )/ (2+m2 ) ]
(4.39)
Im represents the imaginary part of the bracket term,
m is the complex index of r e fraction of pure water,
s pe e d of light,
r efractive
3.79)
and
and v is the
index
for
frequency
(4.4-4.6).
radiation.
pure wat e r is calculated from
(4.1-4.6) where the salinity,
to zero in
of
C is the
The
(3.78-
S, was as s u m e d equal
Therefore the refractive index of pure
w at e r is a function of frequency a nd temperature.
Thus the volume abso r p t i o n coefficient of a non-raining
cloud is given by
amax
Q
“ cld
S u b s tituting
(4.39)
, . dn(a) ,
abs(
di
da
into
a cld =
(4.40)
(6Trv/Pc >
. ...
(4‘40)
and simplifying one obtains
[ (1-m2 )/ ( 2 + m 2 )] M c
Im
(4.41)
where
M
famax
c = J 0—
tt
g
p
3
a
N(a)
__
(4.42)
da
is the mass of liquid water content of the
v o lume
3
(gm/m ) and
Si m p l i f y i n g
the
P
is
the
e x pression
final
per
of imaginary part of
and
expression
evaluating
for
the
unit
3
(gm/m ).
the d ensity of liquid water
terms of dielectric constant
factor,
cloud
(4.41)
in
constant
absorption coefficient
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77
(Np/Kra) due to n on-raining clouds is given by
aC
where K 1
j.Vui
and
dielectric
= 0.0629
K"
the
constant
[3K'/ { ( 2 + K 1)2+ K " 2 }]
real
of
a nd
pure
(4.2) and M c is calculated from
M
imaginary
C
(4.43)
v
parts
wat e r are given by
of
the
(4.1) and
(4.42).
(B) Extinction Cross-section of a Rain-drop
In the study of theoretical absorption
p roperties
of
atmosphere,
two important facts should
B eca use
a re
of
polydispersive
media,
and
such
be
as rain in the
considered.
the finite size of the particles,
assumed
characteristics
spherical,
can
not
be
the
(i)
even if these
angular
approximated
scattering
scattering
w i t h sufficient
a c c u r a c y by asymptotic e xpressions based on geometric optics
or Green's function approximations for the
internal
field.
T he complete Mie-series must be u s e d for each particle,
T he
(ii)
size distribution function u s e d for computing the total
extin c t i o n cross-section must be v e r y close to the real size
spect r u m of the particles in the medium.
Mie-theory,
as
a
Before applying the
it is convenient to define few
drop size parameter,
Q e x t * a bsorption e fficiency
e f f i c i e n c y factor,
Q
parameters
such
X, e xtinction efficiency factor,
factor,
Qabs '
and
scattering
, which are related as follows:
SCcl
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78
X = c ircumference of the sphere
w a velength of the radiation
ext
<ext
tt
r
tt
0
,
xabs
w h e r e r is
the
^
X
r
=0
-
ext
radius
S ^ 1 (2n+1)
0
(1957)
scattering
sec 6 9.22)
s
a
n
=
m
X
J Yn
[ Am i z,+ _ sX_ ]J
r m A (z )
n
=
m
z = m X =
tt
w h e r e m is the complex
diameter
a
(m).
-i
X
-j
J
J
tz;
—
+ —
m a
v
and
(Abramowitz and Stegun
v
„
?
the
(Van-
(X) - Y n _ 1 (X)
(4.48)
V
x)
i
(X) n4 '
(4.49)
S
, (X)
n - 1' '
(1972)
index
of
sphere
of
a
are Riccati Bessel functions
sec 6 10.3)
(X) = X j (X)
1&
is
X
(4.50)
refractive
Yn
sphere,
n
[
[ m a
(4.46)
as follows:
L L
b
|bn |2 >
an and bn are given
[ _ V 2)+ — n_ ] T
L
(4.45)
(4.47)
w a v e l e n g t h of incident plane wave,
de Hulst
(|an |2+
(an +bn )
"
“
sea
of
44>
X
S ^ 1 (2n+1) Re
” ‘
X“
sea
sea
_ 2 Trr
(4.51)
11
?n (X) = T n (X) + ixn (X)
(4.52)
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79
where
x n (X) = - X y n (X)
Jn (x) and Y n (x) are
(4.53)
spherical
Bessel
initial two terms of these functions are
The
given by
(X) = Sin
(X)
(4.54)
x Q (X) = Cos
(X)
(4.55)
y q
All
functions.
Y 1 (X)
=
(1/X) y Q (X)
- x Q (X)
(4.56)
x ^(X)
=
(1/X) x Q (X)
+ y Q (X)
(4.57)
the
higher
terms of the functions are calculated
using the recurrence relations given as
fn + l (X) =
C ( 2 n + D / X ] ?n (X) - fn _ 1 (X)
(4.58)
T
[Y (X)
X
n n+1
(4.59)
.(X) =
n+1
(X)
The logarithmic derivative
function, An
, used in
(4.48)
and
-1 ]
of
Yn
/x(X)
n
is
(4.49)
prime
value
of
denotes
the
significant
(4.60)
the derivative w i t h respect to Z. The
function
A
n
isobtained
digits using Lentz method.
N, used in the Mie- series
theoretical
grounds
a
and is given by
An (z) = Y n (z) / Y n (z)
where
denoted by
(4.45)
depends
on
and
to
at
least
5
The number of terms,
(4.46),
calculated
on
the size parameter in the
following manner
1 +
X +
N = 2 +
X +
2 +
X +
Once
the
4
X 1/3
4.05 X 1/3
4
X 1/3
if .02 < X < 8
if 8 < X < 4200
(4.61)
if 4200 < X < 20000
extinction and scattering cross-sections, cr
ext
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80
and
o-
S C a
. are known from
e xtinction
13 __
SCa
and
scattering
(per unit length) ,
these
(4.45)
coefficients
(4.46)
a nd
coefficients
may
over
be
the
then
the
total
of rain,
obtained
by
complete range
and
integrating
of the
size
distribution
= l a " 3"1 <re x t (a-” >v)di | a)
min
da
(4’62)
Bs c a (m' v) = Ja” 3* <rs c a ( a -m' * ' )dM a>
da
(4 ' 63>
m m
where
am .
and
m m
a
are
max
the
mi n i m u m
and
diameters of rain drops reaching the surface.
and
scattering
diameter,
index
of
a,
cross-sections
frequency of
refraction
of
are
radiation,
The extinction
and
of
the
drop
complex
pure water, m, w h i c h in turn,
are functions of temperature,
S C a
maximum
functions
v,
function of temperature and frequency.
B
the
Therefore,
frequency,
B
is a
.
and
and drop size
distribution.
The
size
determined
by
of
precipitation
by
the
p rocesses
of
collision
humid i t y
prevail in the atmosphere
distance
before
is
p a rtly
the strength of the up draught producing the
cloud and
The
particles
in
the
subcloud
and coalescence
layer.
among drops
also influence the particle
The
that
size.
w h i c h a drop can fall through unsaturated air
complectly
e v a p o rating
increases
rapidly
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with
31
i ncresing
drop size;
in an atmosphere of 90% humidity drops
of 100 pm and 1 m m would fall
respectively.
at
least
(Mason
150 m and 40 km
Since the bases of dense clouds generally
lie
a few hundred meters above the ground a radius of
100 ym may be regarded as a lower
precip i t a t i o n elements.
a
The
(1975))
. = .0001
man
max i m u m
drop
( m m/h), and was
for
the
size
of
Hence
(m)
(4.64)
diameter
given
limit
by
depends
St e p h e n
on
the
(1962)
rain rate,
from
R
empirical
studies to be
o 913
amax = ° - 0023 R
The
rainfall
coefficient"
(m)
rate,
calculations
m a x i m u m drop diameter
R,
through
(4.65)
enters
the
the
extinction
definition
of
the
(4.65).
(C) Drop Size Distributions of Rain
T he distribution most commonly used in
of
rainfall
(1948)
study
is
that
giv e n
(here after called M.P.
exponential
of drop
different
the
literature
by Marshall and Palmer
distribution).
This
has
an
form obtained e m p i r ically from the measurements
diameters
at
the
surface
for
intensities and time durations.
steady
rains
Mathematically it
is e xpressed by
dn(a)/da = NQ exp(-^a)
of
(4.66)
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82
The quantity dn(a)/da is the
d iameter
(m
—4
a
rate,
R
of
drops
(m)
and
8.0
the
x
10
6
—4
(m
),
variation
of ^
a
is
the
b e e n c a lculated from
and
rainfall
(4.67)
SMMR
(4.62) u s i n g M.P.
frequencies
distribution given by
rates are shown in the Figure
(18.0,21.0,37.0
(4.8).
GHz)
The variation
is
significant
due
to scattering processes.
GHz)
the extinction coefficient can be assumed to
be
to
the
not
absorption
At lower frequencies
coefficient
which
s i g n i f i c a n t l y w i t h the rainfall rate.
been
have
(4.67). The variations of these coefficients with
at higher frequencies
s u c c essfully
used
orig i n a t e s from snow flakes
over space and time.
fails
drop
(mm h 1 ) , is d e termined from
The extinction coefficients at
has
a
(m- 1 ) with rainfall
^ = 4100 R - 0 '21
(4.66)
of
per unit volume per unit drop-diameter interval
). N q is given by
diameter
number
for
a nd
does
The M.P.
vary
rain, which
sufficiently
On the other hand the M.P.
equal
distribution
subtropical
is
(6.6,10.7
averaged
distribution
to reproduce the drop size distribution for w a r m rain
in the tropics.
The vari a t i o n s in N Q for different
rainfall
types were found to be independent of those that occurred in
/s.
Therefore,
to
describe the drop size distribution more
p r e c i s e l y Sekhon and S rivastava
(1970)
found it necessary to
s p e c i f y both N Q and ^ as independent variables of
size
distribution.
Using
the
data
of
the
drop
Gunn and Marshall
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MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
6.0
HHU-
.00
0.6GHZ
J0.7GHZ
l«.OSHZ
21 .CCHZ
OCL- .00
5.5
37.0CH2
5.0
4.5
4.0
H
2 3.5
a
u
E
fe.
w
o
u
3.0
Z
2.5
o
2.0
•B-4
0
10
20
30
40
50
60
70
80
90
100
RAIN FALL RATE (MM/HR)
USING M.P. DISTRIBUTION
Figure
(4.8)
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84
(1958)
and others,
which
were
found
they provided the
to
fit
a
parameters
and
NQ
large number of observations
adequately,
—
^ = 2290. R
o 46
(4.68)
N q = 2.5 x 106 R - 0 '94
(4.69)
For ice, above freezing level in
the
atmosphere,
parameters take the values given by Gunn and Marshall
the
(1958)
as
^ = 2550. R ~ ° ' 48
(4.70)
N Q = 3.8 x 106 R - 0 *87
(4.71)
Ulbrich and Atlas
rainfall
rate
distribution
(1984) have shown that improvement in
estimation
can be achieved if the drop size
(DSD) is assumed to
be
a
gamma
distribution
gi v e n by
dn(a)/da =
The
N
ay exp
U
coefficient
(-^a)
n ow has
; 0 < a < a
the
(4.72)
ulaX
unit m 4 y and the
exponent y can have any positive or negative value given
Ulbrich
and
Atlas
(1984). The r e have been large and sudden
changes in N Q from moment to m o ment with
type
as
noticed
by Waldvogel
These changes were found to be
occurred
facts,
in
^
Ulbrich
^ =
by
given
rainfall
(1974) and Donnadieu
independent
from moment to moment.
(1983)
a
of
(1982).
those
In response to these
gave v a lues of the parameters
(3.67 + y
+
10- 0 *3 (^+ 9 )) / Dq
N Q = 6.0 x 1 0 6 e x p (3.2 y )
that
(4.73)
(4.74)
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85
Inserting
(4.74)
into
(4.72)
one obtains
dn(a)/da = 6.0 x 106 ay exp (3 .2y-/*a)
The m e dium volume diameter,
(4.74-a)
(m), has been found from
an empirical result obtained b y Ulbrich
(1983). He has
also
shown that €, 6, and D Q are expressible in terms of rainfall
rate and the parameter y , according to
DQ = € R 6
€ =
(4.75)
[(3.67+y)
x
102 / (33.31 N Q T l / 6 ) 6 ]
(4.76)
6 = 1/(4.67+y)
Thus
from
for
(4.73)
a ~-,-„
min
to
a given val u e of y , /* and N Q can be obtained
and
(4.74).
The drop diameter,
(4.62)
m ax
fixed and is given by
droos
lying
and
(4.64)
rainfall rate as given by
of
(4.77)
(4.63).
v
'
where
(4.65).
varies
from
The value of a
. is
man
as
a,
a
varies
max
with
Therefore the total number
be t w e e n the diameters a
. to a
for the
min
max
g a m m a drop size d i s t ribution is given by
N(y,R)
ro.0023 R 0 *213
a^e x p (3.2y-^a)
J O . 0001
independent
param e t e r s
y a nd
R
= 6.0 x 10
Two
evalu a t e
the
v a r i a t i o n of
total
this
number
total
of
number
drops
of
da
(4.78)
are
from
drops,
needed
(4.78).
N(A),
to
The
verses
rainfall rate for different v a l u e s of y usi n g the gamma drop
size
F i gure
(1983)
dist r i b u t i o n
(as d e s c r i b e d by
(4.9). The different v a l u e s of
have
been
used
(4.78))
y
is shown in the
given
by
Ulbrich
for the curves shown in the Figure
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GAMMA DROP SIZE DISTRIBUTION VERSES
RAINFALL RATES
-.27
5.04
-1.70
2.00
o
10-
0
10
20
30
40
50
60
70
80
90
100
RAINFALL RATES (M U /H R )
USING CARLTON W.ULBRICH PAPER
F ig u r e
(4.9)
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(4.9). These curves represent various types of rain
ranging
from a very light rain to a heavy thunder storm. For a large
positive value of y, the total number of drops,
with
rainfall
rate,
R,
N(A), varies
very rapidly u p to 25 mm/h. For a
smaller positive value or negative value of y , the variation
of N(A) with R is only up to 5 mm/h after which it does
v a r y much.
N(A)
For a fixed value of rainfall rate,
with
y is very fast, as y increases
positive value)
for
N(A),
the
the N(A)
negative
the change in
(from negative to
decreases very rapidly.
value
of
Therefore,
y, the total number of drops,
is larger than that for the positive value of
sources
of
different
not
types
of
rain
and
parameters y, €, 6 and N Q are given b y Ulbrich
y.
The
corresponding
(1983) but he
did not report any extinction coefficients for the different
values of those parameters.
(D) Extinction Coefficient of Rain Volume
The volume extinction coefficient of rain for
DSD
is
obtained
by
subsituting
(4.45) and
a
gamma
(4.74-a)
into
(4.62), which results in
a
max ( R )
a 2+yexp (3. 2y-/^a)
(4.79)
a . Qe x t (m'a 'v) da
m m
where
a .
m m
and
a^_,
max
are
given
by
1
(4.64)
and
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(4.65)
88
respectively.
from
Q ext is given by
(4.73-4.77).
Qext
(4.45),
volume
function
of
radiation,
extinction
rainfall
is
calculated
depends on index of refraction,
whi c h is a function of temperature
Therefore
a nd ^
rate,
within
the
coefficient
atmosphere.
of
temperature,
rain
for a gamma DSD using
from
in Figures
different
the
Showers,
and
6
and N Q are c alculated from
(4.72)
to
(4.77)
and
(4.62)
rainfall
(4.10)
sets
bottom
to
(4.26)
against
rainfall
rate
of
these
(4.15)
Figures.
n o n - linearly
negative or
curves
is
or
positive.
that
the
coefficient
linearly
Another
(iii)
beyond
which
varies
according
fact
(i)
Thunderstorm,
with
Figures
Showers,
rainfall
to whether p is
observed
from
these
extinction coefficient increases with
frequency and p, and attains a
GHz,
shown
Four types of rain
correspond to the first category,
in whi c h the e x tinction
rate
for
of values p, €, 6 and N Q . These values were
(ii) Wide spread or Stratiform,
to
rate
1 to 100 mm/h. These coefficients are plotted
(iv) Orographic are categorized usi n g the data.
(4.10)
of
for the 17
observed by a number of investigators whose names are
at
a
and p.
of values of p, e,
var y i n g
is
frequency
Extinction coefficients at SMMR frequencies,
sets
m,
no
value 6.5
at
calculations were performed.
case of Orographic rain
(Blanchard,
e xtinction
is
coefficient
(Np/Km)
6.9
1953),
(Np/Km)
the
37.0
In the
value
at 37.0 GHz.
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of
In the
MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
-V'- A- O.OGKZ
Z B- I0.7GHZ
.65
c- ii.omz
D- 21.0GHZ
C- 37.0GHZ
.60
.50
.45
.40 .35
.30
.25
.20
.10
.05
100
RAIN FALL RATE (M M /HR)
USING GAMMA DISTRIBUTION
SHOWERS HIGGS (1952)
Figure
(4.10)
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MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
2.0
g.esHZ
KHU— 1.78
do.-
.a s
I0.7GHZ
It.OCHZ
2I.0GKZ
37.0CH2
1.4
E- 1.2
2
td
CJ
Ed
1.0
O
U
2
O
E->
U
2
EX
Ed
100
RAIN FALL RATE (M M /HR)
USING GAMMA DISTRIBUTION
SHOWERS FOOTE (1966)
Figure
(4.11)
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MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
7ITH AN INTERVAL OF
GHZ, ANGLE =48.8
4.5
4.0
3.5
/—N
2
z
\
a,
z
y 2.5
Cs,
W
o
u
z
o
100
RAIN FALL RATE (MM/HRt
USING GAMMA DISTRIBUTION
SHOWERS MUCHNIK (1961)
Figure
(4.12)
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MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
TOTH AN INTERVAL OF
GHZ, ANGLE =48.3
6.5
A« S.6GHZ
3- I0.7GH2
6.0
I>21.COC
C- 37.0GHZ
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
:go
RAIN FALL RATE (M U/HR )
USING GAMMA DISTRIBUTION
SHOWERS IMAI (1960)
Figure
(4.13)
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
6.5
6.0
E- 37.0GHZ
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
100
RAIN FALL RATE (M M /HR)
USING GAMMA DISTRIBUTION
SHOWERS FUJIWAHA (1965)
Figure
(4.14)
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
6.5
rlNU- 5.04
A- C.6GXZ
3- I0.7GHZ
6.0
D- 2I.0GHZ
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
100
RAIN FALL RATE (MM/HR)
USING GAMMA DISTRIBUTION
SHOWERS JONES (1956)
Figure
(4.15)
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
MIE CALCULATIONS FROM 6.6TO37.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE = 48.8
5.0
DU-
4.5
4.0
£
3.0
o 2.0
1.0
100
RAIN FALL RATE (M U/HR)
USING GALIMA DISTRIBUTION
WIDESPREAD(STRATIF0RM RAIN) U. P. (1948)
Figure
(4.16)
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =46.8
6.5
6.0
E- 37.0GHZ
5.5
5.0
r— .
2
2
\ 4.5
PU
2
E-* 4.0
2
Cd
5 3.5
fc.
2
W
3 30
2
I
—i
P
E- 2.5
U
2
P
X
H
2.0
1.5
1.0
100
RAIN FALL RATE (M U/HR )
USING GAMMA DISTRIBUTION
WIDESPREAD(STBATIFORM RA1N)FUJIWARA, 1965
Figure
(4.17)
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
6.0
HHU- I.01
»« O.OGHZ
5.5
_ o
la.GCHT
E- 37.0GHZ
5.0
4.5
4.0
3.5
3.0
2.5
2.0
100
RAIN FALL RATE (M U/HR )
USING GAMMA DISTRIBUTION
WIDESPREAD(STRATIFORM RAIN) ATLAS (1957)
F ig u r e
(4 .1 8 )
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
6.0
KHU- 4.65
A- 6.6GHZ
8- I0.7GHZ
5.5
D“ 21.OGKZ
E- 37.0GHZ
5.0
4.5
4.0
3.5
y
fc. 3.0
w
o
u
2
o
H
U
2.5
2 2.0
1.5 -
r*—
100
RAIN FALL RATE (M U /H R )
USING GAMMA DISTRIBUTION
WIDESPREAD (STRATIFORM RAIN) JONES (1956)
Figure
(4.19)
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MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =40.8
5.0
A- o.oxz
0.7CHZ
s .ochz
4.5
I.OGMZ
7.0GHZ
4.0
2
3.5
a.
COEFFICIENT
3.0
EXTINCTION
Z
2.0
2.5
1.0
lao
RAIN FALL RATE (MM/HR)
USING GAMMA DISTRIBUTION
THUNDERSTORM RAIN FUJIWaRA (1965)
Figure
(4.20)
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MIE CALCULATIONS FROM 8.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.0
B
A“ fl.GCHZ
B" I0.7SXZ
7
E- 37.0GHZ
6
w
5
4
3
2
1
•A—
0
100
RAIN FALL RATE (MM/HR)
USING GAMMA DISTRIBUTION
THUNDERSTORM RAIN SAVARAMAKRISHNAN(1956)
Figure
(4.21)
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MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
A- 8.0GHZ
8- I0.7GHZ
6.0 f-
5.5
5.0
4.5
H
4.0
Z
I—I
H
U
z
H 2.0 X
H
1.0
■A-
100
RAIN FALL RATE (MM/HR)
USING GAMMA DISTRIBUTION
THUNDERSTORM RAIN BLANCHARD (1953)
Figure
(4.22)
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
5.5
A- 6.6GHZ
B- 10.7GHZ
5.0
_ 0- 21.OGHZ
- E- 37.OGHZ
4.5
4.0
3.5
2.5
2.0
100
RAIN FALL RATE (MM/HR)
USING GAMMA DISTRIBUTION
THUNDERSTORM RAIN JONES (1956)
Figure
(4.23)
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
M E CALCULATIONS FROM 5.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
7.0
0.6GHZ
I0.7GKZ
18.0042
6.5
.OGHZ
’.CCHZ
6.0
5.5
5.0
4.0
y
fa 3.5
fa
o
u
2
3.0
o
t—«
1 2-5
H
X 2.0
Dd
1.5
1.0
100
RAIN FALL RATE (MM/HR)
USING GAMMA DISTRIBUTION
OROGRAPHIC RAIN BLANCHARD (1953)
F ig u r e
(4 .2 4 )
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
6.0
A- 6.6GHZ
8- 10.7GHZ
5.5
E- 37.0CHZ
5.0
_
4.5
2
a
fa 4.0
2
B2
fa
y
3.5
k,
fa 3.0
fa
o
u
2
2.5
§
2.0
o
S
u
H
X
fa
IQO
RAIN FALL RATE (MM/HR)
USING GAMMA DISTRIBUTION
OROGRAPHIC RAIN RAMANA,MURTY,GUPTA(1959)
Figure
(4.25)
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission.
MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ
WITH AN INTERVAL OF
GHZ, ANGLE =48.8
6.0
6.QGHZ
0.7GHZ
5.5
i.o o a
I .o tx z
7.0GKZ
5.0
4.5
2
2
\
ft,
2
4.0
3.5
y
Cz*
fe. 3.0
w
o
u
2 2.5
o
2.0
1.5
100
RAIN FALL RATE (MM/HR)
USING GAMiiA DISTRIBUTION
OROGRAPHIC RAIN WEXLER (1948)
F ig u r e
(4.26)
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106
radiative transfer the volume extinction coefficient of rain
is calculated for the value of y which was randomly selected
from the values given by Ulbrich
y
(1983).
This randomness
in
will generate the all possible types of rain found in the
atmospheres.
Volume extinction
using
coefficient
of
ice
is
the distribution given by Gunn and Marshall
shown in equations
3.8
4
B I c e (R'm,v)
(4.70-4.71)
tt
(1958)
as
and is given by
max
1q6 r~0.87
a
The altitude,
calculated
min
2
a exp(-Aa)
Qe x t (m'a 'v) da
(4.80)
z, enters the calculation of Qe x ^. through
the definition of the temperature of the atmosphere at
that
altitude.
Whenever the cloud is found above the 0° isotherm,
the
volume extinction coefficient has been used in the
ice
radiative transfer calculations.
The atmospheric models used
in the simulations of microwave brightness
discussed
in
the
next
chapter
where,
temperature
the
are
computational
procedure will also be described in detail.
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CHAPTER (V)
M O D E L ATMOSPHERES AND SIMULATION OF BRIGHTNESS TEMPERATURES
In order to simulate m i c rowave brightness
was d i s cussed in the preceding chapters,
surface
and
the
atmospheric models.
o ce a n surface model are surface
surface
wind.
one has to know the
Inputs needed for the
temperature,
salinity
and
The atmospheric model requires the knowledge
of vertical distribution of its
temperature,
temperatures
water
clouds and rain,
the ice clouds.
vapor
parameters
density,
like
pressure,
liquid water density in
and water content in form of ice present in
H o w the
values
of
these
parameters
were
o b t a i n e d is described in the next section.
(5.1) Atmospheric Models
The
humidity
vertical
are
profiles
given
of
pressure,
temperature
by r a d iosonde/rawinsonde measurements.
These m e a s urements are routinely obtained
located
all
over
by
stations
whi c h
stations,
whi c h are distributed in latitude
w ere selected.
are
Also in selecting the data,
the mon t h were varied,
and
meterological
the globe.
and
Several
longitude,
both the year and
so as to have a representative amount
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108
of climatology.
Six years of data
in
(from J anuary to December)
all
season
(from 1966 to 1972) v a ried
and in all region
(by including the stations sprea d e d all over the globe) were
c ollected and a careful s e l ection was
measurements.
profi l e s of
levels
These
meas u r e m e n t s
temperature
(1013.25,
and
1000.0,
620.0,
570.0,
300.0,
250.0,
200.0,
150.0,
135.0,
60.0,
50.0,
30.0,
25.0,
20.0,
3.0,
2.0,
1.5,
1.0,
0.5,
475.0,
0.2,
only
430.0,
15.0,
600
up to 20 lower levels.
780.0,
100.0,
0.1 millibar).
but
pressure
700.0,
400.0,
10.0,
350.0,
85.0,
70.0,
7.0, 5.0,
4.0,
The temperature
the
humidity
was
The humidity was then
e xtr a p o l a t e d for upper 21 v a l u e s from its
values.
select
fixed
850.0,
115.0,
w a s observed at all pressure levels
measured
at
920.0,
670.0,
to
describe by the vertical
humidity
950.0,
500.0,
made
last
5
measured
F r o m the knowledge of vertical profiles of pressure
+•v»
a n d temperature one can find the
altitude
of
the
(i+1)
layer in k i lometers using the relat i o n
H i - H i+1 = l o g ( P . + 1 / P i ) [(Ti+ T i + 1 )/68.2831]
The
liquid water in these profiles was introduced from
the knowledge of its
clouds.
(5.1)
The
distribution
distribution
of
in
liquid
different
types
water in a cloud w as
a s s u m e d to be uniform from its base to the top. Only in
a tmosp h e r e s were clouds and rain introduced;
t hem
were
assumed clear.
of
200
and the rest of
These 200 cloudy atmospheres w e r e
c a t e g o r i z e d into 8 cloud model
types of 25 atmospheres each,
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109
according to their base and top altitudes and
content
as
shown in Table
liquid water contents
of
the
Table
water content
(5.1).
(ILWC)
liquid
water
Their integrated columnar
are shown in the fourth
column
(5.1). This ILWC is the product of the liquid
(LWC) and the cloud
thickness
(CL(2)- C L (1)).
The rainfall rates for these clouds were calculated from the
relation obtained by Paris
R = 18.05
Aft e r
calculating
(1971)
(ILWC)1 ’19
(5.2)
the rainfall rate from this equation,
w as then super imposed by a random number between
generated
by
a random number generator.
of a random number is equivalent
error
in
the
rainfall rate.
to
0
it
and
5
This superposition
adding
a
statistical
The relative humidity was set
equal to 100% in the clouds or rain and the drop temperature
was assumed
to
vicinity.
The
altostratus,
shown
in
mentioned
in
the
most
temperature
probable
of
the air
thickness
last
4
cumuli form
the
clouds
literature.
may
go
extrapolated from the lowest
The
its
of altocumulus,
m o dels of the Table
km,
as
(5.1). However,
up
to
8
km
as
The sea surface temperature
corresponding to the surface pressure level
measurements.
in
and stratocumulus clouds was assumed 2
the
thickness of
be
2
values
of
1013.25
the
(mb) w as
radiosonde
sea surface wind was randomly introduced
in the model
between 0 and 25
atmospheres
contained
a
(m/s).
certain
Each
amount
of these
model
of water v a p o r ,
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110
TABLE (5.1)
Cloud Statistics of 600 Model Atmospheres
I Number of
IAtmospheres
I
I
1
1
I
I
1
1
I
400
25
25
25
25
25
25
25
25
Cloud Base
(km)
CLC1)
Cloud Top
(km)
CL(2)
0.0
1.0
1.0
0.0
7.0
1.0
1.0
2.0
6.0
0.0
2.0
2.0
8.0
8.0
3.0
3.0
4.0
8.0
Liq. Hater Content 1
of Cloud (gra/ar') 1
LHC
I
0.0
0.1
0.3
0.1
0.2
0.04
0.08
0.02
0.2
1
I
I
1
I
I
I
I
I
TABLE (5.2)
Statistics of atmospheric parameters of 600 model atmospheres
IStatistics Surface I Surface
Iof ataosp- temperaturl Uind
( K) I (m/s)
Ieric models
Rainfall Ilntegrated Ilntegrated {Integrated Ilntegrated I
IWat. Vapor ILiq. Hater IRain Hater lice Hater I
Rate
(mo/h) I (gm/cm3 ) I (kg/tf) I (kg/nr) I (kg/a1) I
2BB.43 I
12.67
2.03
I
1.76
1
0.61
I
0.22
I
0.01
I
I Kedian
230.80 I
12.76
0.25
I
1.56
I
0.06
I
0.04
I
0.005
I
I
233.0
I
8.0
0.0
1
1.0
1
0.0
I
0.0
I
0.0
1
1 Minimum
270.0
1
0.0
8.0
I
0.0
1
0.0
I
0.0
I
0.0
I
I Maximum
305.0
I
25.0
18.0
I
6.0
I
18.0
I
6.0
I
1.0
I
I 1055.0
I
368.0
I
133.0
I
6.0
I
I
I
Kean
Kode
Sum
172480.0 I 7600.0
1218.0
I Std.Error
0.42
I
0.23
0.16
I
0.06
I
0.10
I
0.04
I
0.304
I
I Std.DEv
10.24
I
7.10
3.85
1
1.43
I
2.32
I
0.86
I
0.100
I
I Skeuness
-0.311 I -0.004
2.441
I
0.573
I
4.623
I
4.337
I
3.874
I
I Kurtosis
-1.402 I -1.133
6.140
I -0.411
I 95.817
I
I 23.363
I 13.325
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Ill
liquid water content in the clouds and rain
water in the form of ice clouds.
was
considered
form,
and
ice
The ice water of the clouds
to be the equivalent amount of liquid water
that exists in the form of ice. All phases of water existing
in the atmosphere can be expressed in
depths
of
terms
of
equivalent
liquid wat e r denoted by WV, LW, R W and IW. WV is
the equivalent depth of the total amount of water
the
atmosphere,
vapor
LW is the total equivalent depth of liquid
w at e r contained in the cloud,
RW
depth of liquid water present
in the form of rain,
the
total
clouds.
equivalent
depth
is
of
total
areal
, and 400 w e r e clear.
content
a
.524 N q
m ax
of liquid water.
a
is
given
by
(4.73)
(4.65).
_2
(RW) was
(1983)
and
(4.74)
The value of
respectively,
p
was randomly
(1983).
The
e quivalent depth of liquid water in the form of rain,
—2
),
was
obtained
1.5
(5.3)
selected from the v a lues gi v e n by Ulbrich
m
The
in terms
The rain water
3+ jj
.
. ,
exp(-^a) da
where ^ and N Q are given by
am ax
18, 6 and
One m m corresponds to one km m
d eter m i n e d from the relation given by Ulbrich
and
and IW is
water present in ice
LW, R W a nd IW are from 0 to 60,
respectively.
RW =
equivalent
There were a total of 600 model atmospheres obtained
ranges of WV,
of
the
ice
in w h i c h 200 were cloudy and rainy
(mm)
in
from integrating
total
RW
(kg
(5.3) over surface to
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112
zero degree Isotherm.
The frequency distribution of sea surface
SST
(°K),
vapor, W V
cloud,
sea
(gm cm
LW
(km m
—
2
, RW
(kg m
and
rainfall
(5.7).
-2
-2
-2
),
wind,
SSW
integrated
liquid
integrated water
water
content
rate
). Out of 200
(mm/h)
are
shown
IW
(kg
in Figures
m
cloudy
which
atmospheres
contained
of
clouds
and
rainfall
rainfall
rain,
rate
rate,
were
),
to
141
only
(kg
45
more than 0.5
liquid
water
and ice water of clouds are
h i g h l y skewed towards small values.
all
there
—2
(5.1)
liquid water content was less than 0.5
(mm/h). The distributions of
content
of
), integrated liquid water content of rain
), integrated ice water content,
the
a tmospheres
for
(m/s),
In 541 atmospheres there was no ice cloud and in
atmospheres
m
surface
temperature,
The complete
statistics
600 model atmospheres is given in the Table
(5.2).
These model atmospheres were then employed by the
microwave
radiative
transfer
equation
to
calculate
the
microwave
brightness temperatures at SMMR frequencies.
The
algorithm
u s e d for the calculations is giv e n in the next section.
(5.2) Brightness Temperature Simulation
A
microwave
program),
p ola r i z a t i o n
which
radiative
takes
effect
into
of
transfer
account
model
the
non-isothermal
(computer
M i e-scattering
clouds
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in
113
Frequency Distribution of SST
o f 600 m odel atm ospheres
110
100
-
90 60 70 60 SO -
40
-
30 -
2010
-
270 2 7 2 2 7 4 2 7 6
2 7 6 280
2 8 2 2B4 2 8 6 288 290 292* 294- 296
2 9 8 3 00 302 3 0 4
SST (degrees Kelvin)
Figure
(5.1)
Frequency Distribution of SSW
o f 600 m odel atm ospheres
60
50 -
40 -
30 -
20
-
10
-
2
4
6
8
10
12
14
16
18
20
22
SSW (m /s e c )
Figure
(5.2)
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114
F re q u en cy D istribution of W ater Vapor
o f 600 model atmoaoherea
160
140
-
130 -
(N )
120
of Cases
100
Number
-
110
-
90 80 70 -
50 -
30 20
-
10
-
1
2
3
4
6
5
Water Vapor (gm/cn?)
Figure
(5.3)
Frequency D istribution o f LWC
oi 600 model atmospheres
12
11 10
9
8
-
7 -
6
5
4
2
1
0
11 I
!Z
3
-
21 &
6
7
B
9
10
11
12
13
14
15
16
17
13
Liquid Water Content (kg/m*-)
Fimire
Ia .4
1
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115
F re q u e n cy D istrib u tio n o f Rain Water
of 600 modol atmoaoheres
20
-
4 -
87-
4 -
3
2
1
5
4
6
R a in W o t e r ( k g / r r T )
Figure
(5.5)
F re q u en cy D istrib u tio n o f Ice Water
o f 600 model atmospheres
26 24 -
22
-
Number
of Coaes
(N )
20
4 -
0.5
1
1.5
2
Ice Water above freezing level ( k g / r r f )
Figure
(5.6)
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116
F re q u en cy D istrib u tio n o f R ainfall Rate
of 600 model ctmosoherea
40
25 -
20
-
N um ber
of
Coses
(N )
35 -
10 -*7-.
1
2
3
4
5
6
7
B
9
10
11
12
13
14
15
16
17
18
Rainfall Rate ( m m /h )
Figure
(5.7)
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117
inhomogeneous atmospheres has b e e n developed.
general enough to compute the b r ightness
given
frequency
at
any
h e ight
The program is
temperature
at
a
of a given atmosphere.
It
assumes the Rayleigh s c attering phase function
no n - r a i n i n g
cloudy
atmosphere.
into a number of layers;
pressure,
temperature
The
case
of
atmosphere is divided
each of which is
and humidity.
in
characterized
by
This atmospheric model
(radiosonde/rawinsonde data together w i t h the cloud and rain
model)
is required by the computer program as an input.
geophysical
parameters required b y the computer program are
the surface temperature and wind.
(GHz),
and
(degree)
and
the
look
angle
The microwave frequency
of
model
requires
the
variation
of
at
the
surface,
R
surface
emissivity
cloud
and
rainfall
( m m / h ) . An important part of the
p r o g r a m is to calculate the s urface
o ce a n
e
liquid water
content w i t h altitude w i thin the cloud extent
rate
v
observation from nadir,
are also needed as input by the program. The
rain
The
is
emissivity
derived
from
e (e).
P
The
the Fresnel's
refl e c t i o n coefficients as d e s c r i b e d earlier in chapters III
a n d IV,
and is a function of
surface temperature,
surface
wind,
salinity,
frequency and angle of observation.
sea
The
atmospheric transmittance of e a c h layer of the atmosphere is
calcu l a t e d
clo u d
(3.71).
from
abso r p t i o n
The
oxygen absorption,
and
rain
water vapor absorption,
e x tinction
us i n g
(3.56)
and
upward and d o w n w a r d emission of the atmosphere
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118
known as T
(3.73)
by
and
atm
and
T sky
(3.72)
respectively
are
calculated
in whi c h the source term J(z,e)
from
is given
(3.74).
The
calculation
knowledge
of
the
asymmetry
factor
particles
g(z) ,
of
the
source
single
of
the
the
scattering
phase
scattering
requires
albedo
function
the
w(z) ,
of
the
scattering
intensity components IQ (z) and I ^ z ) ,
the temperature profile T(z)
single
term
and
the
look
angle
9.
The
albedo is defined as the ratio of volume
scattering coefficient of rain at an altitude z to the total
volume extinction coefficient
altitude.
described
The
by
(3.56),
coefficients
extinction
total
due
coefficients
of
calculated from
volume
is
the
atmosphere
extinction
composed
to oxygen,
coefficient
of
of
of
rain.
The
oxygen,
water
vapor,
(4.18),
(4.28),
and
(4.72)
and
respectively.
The
are
calculated
volume
asymmetry
absorption
and
absorption
clouds
(4.41) respectively.
from
factor
(4.62)
of
the
involve
and
ensemble
a
g(z)
=
2 + jj
amax
exp(-^a)
0.00013
(5.4)
da
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are
The
the
(4.63)
particles is given by
max
0.0001
as
clouds and volume
extinction and scattering coefficients of rain
DSD
that
coefficient,
volume
water vapor,
at
of
119
where
Is obtained from
*
(4.73)
and the asymmetry factor
of
individual drops, g ( a ) , is calculated from the Mie-theory.
The
Kerker
g(a)
asymmetry
(1969)
factor of individual drops is given by
to be
= (4/X2 Q s c a }2:C{n(n+2)/(n+l)}Re (ana*+1+ b nb* + 1 )
+ { (2n+l)/n(n+l)}Re (an b * ) ]
where an and b n are given by
and
of
Q_„
is calculated from
S C d
(5.5)
which
the summation over
is
given
by
(5.5)
(4.48)
and
(4.46).
(4.49)
respectively
In the right hand side
n takes the value from
0
to
N
(4.61), a nd X is size parameter given in
(4.44).
The intensity components 1^ and 1^ are the solutions of
first
order
coupled
differential
equations
(3.58)
and
(3.59). These coupled differential equations are then solved
numerically
(3.60)
and
Weinman,
w i t h the boundary conditions given by equations
(3.61) using a finite difference method
1984).
in the specific direction making an angle
e from the nadir, and is then u s e d in
and
Ts k y '
The
u p w elling
the
radiances
calculations
received
radiometer at an
altitude H a nd at a looking angle
nadir
calculated from
is
then
transmittance
and
and
The source term is calculated at each layer
of the atmosphere,
T atm
(Wu
(3.69) which uses
( D , the downward and
upward
of
by
e
a
from
the total
radiances
Tg-tm'
T ,
sky
polarized component of emissivity e (e) and
P
surface temperature (T ). The various steps in
this
scheme
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120
are
clearly
illustrated
in
a
flow chart shown in Figure
(5.8).
The SMMR brightness temperatures are calculated
model
atmosphere given in Table
by
Ulbrich
(1983).
jj,
(5.16)
and
6
Figures
(5.9)
for different cloud models given in Table
(5.3).
The surface parameters wind,
held
constant.
high
frequencies
variation
NQ , €
The variation of these brightness
temperatures with rainfall rate are shown in
to
a
(4.2) us i n g Gamma drop size
d i s t ribution and a particular set of values
given
for
at
These
for
low
temperature and
variations
small
frequency
salinity
were
are h i g h l y non-linear at
liquid
water
(6.6 GHz)
content.
The
is linear for small
liquid water density but becomes non-linear for
both
large
den s i t y and high altitude clouds.
The
simulated b rightness temperatures are then inverted
to retrieve the rainfall rates
used
i nversion
optimization technique and a
technique
uses
an
in
the
models.
This
statistical m e thod wh i c h is e x p lained in the next chapter.
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121
TABLE (5.3)
Cloud Models
1 Bottoa
I (ka)
TOP
(ka)
Density
(ga/o3)
Hax.Liquid Hat.
(ng/go2)
1
I
I
1
2
0.1
10
I
I
1
2
0.3
30
I
I
0
8
0.1
80
I
I
7
8
3.2
20
1
I
1
3
0.04
8
I
I
1
3
0.08
16
I
I
2
4
0.02
4
1
I
6
8
0.2
40
I
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122
CLOUD
& RAIN
MODELS
ATMOSPHERIC
RADIOMETER
GEOPHYSICAL
SYSTEM MODEL
MODELS
^
^
Polarization
SST . SW , and S
X 7
CL 1M A T 0L 0 6 I C A L
DA TA
P , T.
and H
MICROWAVE RADIATIVE TRANSFER MODEL FOR
BRIGHTNESS TEMPERATURES SIMULATION
T * ( V O
z x ------------
z x -----
OCEAN SURFACE
UP and DOWN WELLING
EMISSIVITY MODEL
ATMOSPHERIC RADIANCES
CALCULATIONS
OXYGEN ABSORPTION
7^7
WATER VAPOR ABSORPTION
LIQUID WATER ABSORPTION
RAIN
DROP SIZE
DISTRIBUTION
EXTINCTION
','T.'
MIE THEORY
Figure (5.3)
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123
BRIGHTNESS TEMPERATURE VS RAINFALL RATES
MHU=4.65, N0=6.4E10, EPS=.114, D E L = .ll
W=0.0, SST=300.4, WV.= 5.01 . LW.= .33
300
6 .8 H
280
0- 10.7 V
£• 16.0 H
F- 16.0 V
260 J- 37.0 V
~
240 - y j j
g
220 - / >
5
-7 5
/
| 200 - /
05
I
• 7
i 180 ^
K
u
i— .
y
A|
160 I - / — 'a‘
05
m
140
120
100
100
RAIN FALL RATE (MM/HR)
USING GAMUA DROP SIZE DISTRIBUTION
F ig u r e
( 5 . 9)
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124
BRIGHTNESS TEMPERATURE VS RAINFALL RATES
MHU=4.65, N0=6.4E10, E?S=.114. D E L = .ll
W=0.0, SST=3Q0.4, WV.= 5.01 . LW.= .53
300
s.s H
6.6 V
10.7 H
280
10.7 V
E- 16.0 H
16.0 V
21.0
260
H
21.0 V
07.0 n
37.0 V
240
g 200
m 180
S
160
140
120
100
100
RAIN FALL RATE (MM/HR)
USING GAMMA DROP SIZE DISTRIBUTION
Figure (5.10)
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BRIGHTNESS TEMPERATURE VS RAINFALL RATES
MHU=4.65, N0=6.4E10, EPS=.114, D E L = .ll
W=0.0, SST=300.4. WV.= 6.96 . LW.= 1.04
300
0.6 H
280
e- la.o h
r- la.o v
O 21.0 H
260
*>• 37.0 v
240
g 220
S
« 200
2
wh
E
w 180
(0
w
a
u
160
K
G
140
1201-
100
0
10
20
30
40
50
60
70
80
90
100
RAIN FALL RATE (MM/HR)
USING GAMMA DROP SIZE DISTRIBUTION
Figure
(5.11)
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126
BRIGHTNESS TEMPERATURE VS RAINFALL RATES
MHU=4.65, N0=6.4E10, EPS=.114, DEL*. 11
W=0.0, SST=300.4, WV.= 4.67 , LW.= .42
300
6.0 H
6.6 V
10.7 H
280
10.7 V
E- 18.0 H
18.0 V
21.0 H
260
21.0 V
37.0 n
37.0 V
240
a
K 220
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K
£ 200
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w
a
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160
140
120
0
10
20
30
40
50
60
70
BO
90
100
RAIN FALL RATE (MM/HR)
USING GAMMA DROP SIZE DISTRIBUTION
F ig u r e
(5.12)
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127
BRIGHTNESS TEMPERATURE VS RAINFALL RATES
MHU=4.65, N0=6.4E10, EPS=.114, D E L = .ll
W=0.0, SST=300.4, WV.= 5.40 , LW.= .40
300
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:=£!=
280
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120
100
0
10
20
30
40
50
60
70
80
90
100
RAIN FALL RATE (MM/HR)
USING GAMMA DROP SIZE DISTRIBUTION
Figure
(5.13)
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123
BRIGHTNESS TEMPERATURE VS RAINFALL RATES
MHU=4.65, N0=6.4E10, EPS=.114, D E L = .ll
W=0.0, SST=300.4, WV.= 5.36 , LW.= .28
300
280
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260
r- i o .o v
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0
10
20
30
40
50
60
70
80
90
100
RAIN FALL RATE (MM/HR)
USING GAMMA DROP SIZE DISTRIBUTION
Figure
(5.14)
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129
BRIGHTNESS TEMPERATURE VS RAINFALL RATES
MKU=4.65, N0=6.4E10, EPS=.114, D E L = .ll
W=0.0, SST=300.4, WV.= 5.40 , LW.= .32
300
S.S H
280
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20
30
40
50
60
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an
100
RAIN FALL RATE (MM/HR)
USING GAMMA DROP SIZE DISTRIBUTION
Figure
(5.15)
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130
BRIGHTNESS TEMPERATURE VS RAINFALL RATES
MHU=4.65, N0=6.4E10, EPS=.114, D E L = .ll
W=0.0, SST=300.4, OT.= 4.77 , LW.= .59
300
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RAIN FALL RATE (MM/HR)
USING GAMMA DROP SIZE DISTRIBUTION
Figure
(5.16)
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
CHAPTER
(V I)
STATISTICAL METHODS AND OPTIMIZATION TECHNIQUES
In order to retrieve rainfall rates from the
brightness
temperatures
(TBs), the optimization technique
"Regressions by Leaps and Bounds",
Furnival
and Wilson
wh i c h
was
developed
by
(1974), has been used for selecting the
radiometric channels,
Linear
simulated
Regression"
and the statistical
has
been
used
method
"Multiple
for the inversion. The
Leaps and Bounds technique has been successfully applied
Pandey
and
Kakar
(1983)
temperature and by Kakar
in
and
by
the retrieval of sea surface
Lambrightsen
(1984)
in
the
retrieval of atmospheric water vapor profiles.
The
problem of selecting the best subset of predictor
variables in a linear
tedious.
the
regression
model
is
usually
quite
The number of subsets increases exponentially with
number
of
(multiplications
channels
and
N.
The
divisions)
number
of
required
to
operations
invert
the
moments matrix associated with each subset is of
the order
3
N . The
computer time required for p erforming such task is
enormous even for a moderate value of N.
past
studies,
However,
in
the
the best possible subset of a given size has
been determined using a forward selection method.
Stepwise
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132
regression
is
a
forward
selection
method which has been
commonly used in selecting the best subset of a
from a set of channels.
the
residual
have
been
sums
and
size
In the technique of leaps and bounds
of squares for all possible regressions
computed
operations
given
with
then
the
minimum
best
w ithout examining all possible
number
of
arithmetic
subset has been determined
subsets.
The
advantage
of
this optimization technique is in the reduction of number of
operations
by
several
other techniques.
that
it
provides
in
of magnitude as compared to
Another privilege
information
channels subsets of
useful
orders
a
desired
on
using
to
technique
several
which
is
is
next best
extremely
deciding the optimum set of channels for quality
For example,
one
would
retrieve nearly the same values of rainfall rate
different
temperatures
subsets
of
corresponding
SMMR
scatter
measured
brightness
to the same geographic location
and time. B ad radiometric measurements
large
this
the
size,
control of radiometric measurements.
expect
of
will
give
rise
to
in these retrieved rainfall rates and may be
rejected on the basis of this
large
radiometric
identified
channel
can
be
scatter.
on
Also
the
a
bad
basis of
consistent b ad retrievals and the process of elimination.
A two step statistical
inversion
algorithm
has
been
adopted for retrieving the rainfall rates from SMMR data.
the
first
step,
the
In
best radiometric channels subsets of
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133
different sizes have been obtained from the Leaps and Bounds
technique
using
temperatures
a
and
simulated
data
rainfall rates.
base
brightness
In the second step,
subsets have been used to invert the
Multiple
of
rainfall
rates
these
using
Linear Regre ssion method. This method provides the
relationship between rainfall
brightness
temperatures.
rate
Each
a nd
of
the
the
corresponding
subsets
yield
regression equation wh i c h has been employed to retrieve
rainfall
rates
temperatures.
from
These values
of
measu r e d
SMMR
rainfall
rates,
which
from
compared.
This comparison provides a self consistency
the
regression
the
brightness
obtained
on
different
the
a
equations,
are
are
then
check
SMMR measurements and is used for rejecting the bad
measurements.
If the values of
given
regressi on equation,
by
a
rainfall
rates,
which
are
are consistently bad then
that particular subset has been dropped
from
the
rainfall
rates retrieval scheme.
The
criterion
for
selecting the best possible subset
can be made from any of the three criteria viz
adjusted
R
2
,
and
penalty.
When the
search
continues
other
two
(ii)
(iii) Mallows c(p) with F r a n e 1s variable
first
until
criterion,
the
R
2
,
desired
is
criteria,
adjusted R
2
employed
number
regressions are found for each size of subset.
the
o
(i) R ,
of
Whereas
and Mallows c(p),
search terminates after locating the desired number of
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the
best
for
the
best
134
regressions
regardless
of
subset size.
the best subset were based on
the
R
2
All selections for
criterion
which
is
d efined as
R 2 = 100
where
R
the
)
(6.1)
is the residual sum of squares of the dependent
sss
variable
( 1 - R sss / T ss
(rainfall rate) for a subset of size s and
'
'
total
variable.
T
is
ss
sum of squares about m e a n value of the dependent
The maximum value that R
2
can a t tain in
principle
is 100.
The
results
obtained
p r e sented in the Table
with
four
to
(6.1)
from
Leaps
and
Bounds
in w h i c h the best five
are
subsets
nine predictor channels are shown along w i t h
the corresponding R
2
values.
The value of R
2
increases
with
the number of channels used in the subset from four to nine.
However,
the
the change in the R
subset
of
size
six.
2
value is not significant after
The value of R
2
for the full ten
channels set is 90.62 which is not much larger
the
R
o
value
for
subset
of
size six.
than
This implies that
adding a channel after six is not very advantageous
retrieval
of
channels
(6.6H,
be
optimum
the
rainfall
rates retrieval.
obtained
6.6V,
set
rate.
18.0V,
89.36,
in
the
Therefore subset of size six
21.0V,
37.OH,
37.0V)
seems
to
whi c h is most appropriate in rainfall
The next best subset of
size
six
from
can
be
the
last best subset
by replacing 21.0V by
O
21.OH where the R “ does not change
significantly but
this
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135
TABLE (6.1)
Result of Leaos ano Bounds Technique for Selecting
5 Best Subsets of Size 4 to 3 SMMR Channels
Size of
I
the best I
subsets
V
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136
set can be useful when 21.0V has an
erroneous
behavior.
A
set of retrieval equations of different sizes using the best
subset
is
generated
from
the
multiple linear regression
model.
The
coefficients
rainfall
rate
of
obtained
summarized in Table
the
retrieval
from
(6.2)
the
for best
equations
simulated data base are
channels
of
through nine and for the full ten channels set.
coefficients
are
linearly
for
related
to
size
four
These
the
brightness
temperatures in the following form
R(s)
= A Q + A. T B (vi )
(6.2)
where s is the size of the subset, A Q is the
A^ 1s
are
the
intercept
and
regression coefficients corresponding to the
brightness temperatures T_(v.)
D
X
at frequencies
v ..
3.
R(s)
is
the estimated rainfall rate corresponding to the size s.
(6.1)
Precipitation using Microwave
(SMMR)
Data
from
SEA-
SAT Satellite
The rainfall
values
of
rates
are
from
the
measured
SMMR brightness temperatures onboard SEA-SAT and
using the relationship given by
rates,
inferred
corresponding
to
the
(6.2).
Different
rainfall
same geographic location and
time,
are obtained for different set of channels
size)
as given in Table
(varying in
(6.2). The discrepancy found in
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE (6.2)
Multiple inear Regression Coefficients for Rainfall Rate Essi'.iation
Using 00 Simulated Brightness Temperatures at SiiflE Frequencies.
- t -
-I-
-I-
Keqressici Coefficients and Associated Standard Errors
Degree oil Intersect!
multiple ; ■fIS Error
Poly- I end Std. I- - - - - - 1- - - - - 1-. . . . . — ... I- - - - - 1- - - - - 1- - - - - - 1- - - - - !. -— I- - - - - - - ICorr.' IS) 1 in Pain F- Test
nonial I Error of I 6.5 H I 6.6 V 1 10.69 10.63 V I18.0 H I
IS.0V I 21.0 H I 21.0 V 1
27.0 H I37.0 V land adj.
1 fall
(p) I
AO ! At
A4
I A5
I A6
I A7
I A3
I A3
I A10 I R-2
!
- - - - - r- - - - - - r-1
-I-I-I-I-II -4.07506 I
I 0.03626 1-0.10373 I 30.320 I
! 0.05533 I
I
I
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x
x
I
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I 1.5330
1(0.00431)1
1(0.00651)1(0.00753)1(31.453) I
1(0.00563)1
1(1.80772) I
j----------- j
-I- - - - - 1-I-I-I--------- j------------I--I1
1-0.07714 I
I 0.13520 1-0.13554 I 33.624
1-12.33633 1-0.16541 1 0.32433 I
344
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--- 1------ j.
I. . . . . . r-—
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1-0.03356 I 0.16317 1-0.16652 1 33.603 I
I -4.2337 1-0.17326 I 0.31333 I
639
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I
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(0.00531)
1(0.00637)I(0.00735)1(37.431)
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-i1-I-I-!- - - - - - I-I-I•I-I-II 0.20721 1-0.20043 I 33.703 I
[-0.01360 I 0.00424 1-0.07320 I
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609
7 1
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x
I
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I(0.01054)1<0.01133)I(37.669) I
1(0.01443)1(0.01570)1(0.00446)1
1(1.35257) 1(0.01561)1(0.02116)1
-I-I-I-I-I-I-I-I-I-II 0.20571 1-0.20710 ! 33.773 I
1-0.00715 1-0.00223 1-0.07373 I
1-11.52130 1-0.15443 1 0.32654 1-0.00173
53B
1
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x
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1
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1(0.01051)1(0.01136)1(37.770) 1
1(1.34315) 1(0.01557)1(0.02111)1(0.00070)
1(0.01461)[(0.01536)I(0.00444)I
1-I-I-I-I-i-I-I-II
0.13337
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33.302
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480
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i------ r
r_
.. 1 .
-I-I-I-I-I-II -3.73243 1-0.14523 1 0.23661 1-0.04714 0.0-1133 1-0.01025 I 0.00146 1 0.00332 1-0.03242 I 0. 13057 1-0.13227 1 33.313 I
440
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------1.... — i......i-..... i— -... i... -— i... -— I.....
-
-
-I-
133
estimated rainfall rates are
due to the different
functions
in the channels contributing to
the
of the atmosphere
various
rainfall
regression
rates
equations
should have
used.
However,
the same set of
and
time).
Therefore
a
careful
channels m ay reduce this discrepancy in the
measurements.
selection
case
of
threshold,
then
that
measurement is rejected.
value of F-test was obtained for the
channels.
error
subset
a
preset
The maximum
of
size
five
This subset gives the multiple correlation 93.62%
w h i c h is as good as the full set
five-channels
subset
(6.6H,
93.91%
6.6V,
.
rainfall
rates
was employed in
m easu r e d
retrieval
of
the
SMMR data.
(Julian day 257)
of
this
channels
at SMMR frequencies.
estimating
values
Therefore
21.OH, 3 7.OH, 37.0V)
the best optimum subset with least number
1973
of
If the difference in the rainfall rates
d erived from the best two subsets is greater than
14,
SEA-SAT
(SMMR brightness temperatures corresponding to the same
location
free
these
nearly the same numerical value
because they have been d erived from
data
response
rainfall
is
for
This subset
rates
from
the
This SMMR data of September
obtained over
the
Pacific
ocean
corresponds to the satellite pass at starting time 17 hours,
0
minutes
and
9
seconds,
m inutes and 55 seconds.
occur r e d
Figures
along
(7.1) and
the
ending
A total of 2285
five
(7.2).
and
sub
time 18 hours,
measurement
satellite
Each event
tracks
consisted
of
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52
events
shown in
radiance
139
detection
at
five frequencies and two polarizations,
total of ten channels.
r eferred
to as "cases".
e s t imated us i n g
Table
These
(6.2).
A
events
rainfall
be
different
regression
equations
given
complete
statistics
of
these
estimated
Negative
estimates
rates
were
(6.3).
omitted
in
the
c alculations and are listed as wild cases
Mi c r o w a v e
henceforth
Rainfall rates for these cases were
rainfall rates is given in Table
of
shall
for a
rest
in
of
Table
in
the
(6.3).
estimated rainfall rates for different footprints
along the subsatellite track are plotted on the world map in
F igures
that
(7.1) and
all
These
high
represent
(7.2). F r o m these figures
is
evident
rainfall rates higher than 3.5 mm/hr lie on land.
rainfall
the
actual
rates
however
rainfall
estimation
over
land.
do
rates
technique is not desig n e d to work
rate
it
very
not
accurately
since the microwave
well
for
rainfall
The maximum estimated rainfall
rate over the ocean for this data set is 3.5 mm/hr at
longitude and 10°
-155°
latitude location.
To get a qualitative picture of the microwave estimates
the
full
data
set
corresponding
to points over land and
ocean have been grouped into frequencies for different class
intervals in rainfall rate.
to
14
mm/hr.
The classes range from
The class interval is 2 mm/hr.
0
Rainfall rates
wh i c h correspond to the lower limit of a class interval
not
included in that class interval.
Figures
mm/hr
(6.1)
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to
are
(6.7)
140
TABLE (5.3)
Statistics o f Estimated Rainfall Ratesfroa SilMR Data
llsir.q Different Sets of Channels
12(5)
Stati sties I R(4)
of Rainfall I
I
Rates
1 (aa/h) I (aa/h)
1Hean
I aZ. r nn »i
-I-IMedian I 3.357 I 3.
-Iilude
I
9
1--Minimum 1 3
. -II 3(7) I 2(3)
I
vmiaJ n) I (aiii/h) I (raij/’h)
-i-1
R(o)
•iu*T
3.513
3
.
^
?
i
4
»ni
i ■*r£1
-I
.18
11
3.257
I 3.273
1.321 : 1.234
. . . . . I. . . . . -II -3.174 I
I -3.323
-I-i......
Mild
I
13
I
-I-I-IIValiu casesi 2275 I 2357 I 2273 I 2127
I. . . . . . . I-IjrvcWitebs
(riei/ii) I
I
-I-
n
(na/fi)
R CIS)
3.333
I 3.333
Maxi mum I
13
-ISun
I 5313
td.Error I 3.33
-IStd.Dev. I 4.232
R(3)
■1773
4327
4637
I 3.378
.373
3.35
I 1.331
*i•'\-l
u*i
-I-
3.331
I
143
71
I 2137
-I. . . .
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141
show the frequency distributions of these rainfall rates for
estimations using four to ten channel
represents
sets.
The
number
N
the number of cases where the estimated rainfall
rates were less than or equal to zero.
Two observations from these histograms can be made viz
(i) The distributions are highly skewed
values,
(ii)
towards
the
small
There are two peaks in all the distributions,
one at the lower end between 0 mm/hr and 2 mm/hr,
and
at
It must be
the
higher end between 8 mm/hr and 10 mm/hr.
noted that the data corresponding to the first peak
other
is
the
data over ocean while higher rainfall rates belonging to the
second
peak
m ay
be
regarded
represent rainfall over land.
as
unimportant
The fact
that
the
since they
different
histograms resemble each other implies that there is some
consistency
between
estimations that use different channel
sets.
(6.2) Precipitation using
Vis/IR
(VISSR)
Data
from
GOES
Satellite
The estimation of precipitation from
data
is
(1978).
based
on
the
In their scheme,
temperatures.
satellite
scheme provided by Griffith et a l .
thermal infrared
the VISSR from GOES was used to infer
cloud-top
Vis/IR
(10-12 ym)
rainfall
data of
rates
from
Initially a technique was developed
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FREQUENCY DISTRIBUTION OF RAINFALL RATE
Using Four SMMR Channels R(4)
400
350
0 f o r N « 1327
250 -
200
-
150 -
100
-
50
2
4
8
6
10
14
12
Figure (6.1)
FREQUENCY DISTRIBUTION OF RAINFALL RATE
Using Five SMMR Channels R(5)
260
240 -
for N =
220
-
200
-
1285
180 160 -?
140 120
-
100
-
eo 60 40 -
20
2
6
8
10
12
H
Figure (6.2)
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143
FREQUENCY DISTRIBUTION OF RAINFALL RATE
Using Six SMMR Channels R(6)
6CQ
1320
400 -
300 -
Number
of
C a sas
500
200
-
100
VZ£
2
4
6
8
10
12
14
Figure (6.3)
FREQUENCY DISTRIBUTION OF RAINFALL RATE
Using Seven SMMR Channels R(7)
260
240
220
0 for N = 1245
200
of Coaoa
160 -
N um ber
180 - .
120
-
100
-
140 -
80 60 40
20
2
4
6
8
10
12
14
Figure (6.4)
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FREQUENCY DISTRIBUTION OF RAINFALL RATE
Using Eight SMHR Channels R(8)
260
240 -
220
0 for N = 1131
-
200
180
160 140 120
-
100
-
80 60 40 - f
20
2
4
6
8
10
12
Figure (5.5)
FREQUENCY DISTRIBUTION OF RAINFALL RATE
Using Nine SMMR Channels R(9)
450
400 -
350 -
300 ~
250
200
-
150
100
-
50 -
2
4
6
8
10
m
.
12
Figure (6.6)
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FREQUENCY DISTRIBUTION OF RAINFALL RATE
Using Ten SMMR Channels R(10)
800
700
€00
O
m
o
O
9
i.
o
a
£
3
Z
ECO-'
400 “ ,
300 - ,
200
100
2
4
6
8
10
12
14
Rgure (6.7)
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146
w h i c h required a sequence of images to determine cloud
h i s tories
life
and the change in the cloud area with time. Later
the life history aspect of the technique was by
p a ssed
estimates
were
streamlined
technique.
This technique uses a digital array
temperatures
made
based
firstly
on
to
single
produce
volumetric output of the convection,
rainfall
rates.
image
an
of
infrared
estimate
and secondly
and
of
to
the
infer
Raining convective clouds are identified by
/
the threshold temperature of -20®C
threshold was
chosen
to
(degree centigrade).
maximize
the
This
determination
of
precipitating clouds, while m i n i m i z i n g the inclusion of n o n ­
r aining
The inferred rainfall, expressed as either
Q
total volumetric output (m ) or
area
averaged
rain
depth
(mm),
clouds.
is
calculated
as a function of both areal extent of
the storm at -20°C as well as the fractional coverage of the
storm
by
satellite
v o lume
colder
temperatures.
Rainfall
rates
for
each
pixel are derived b y apportioning this calculated
over
the
temperatures.
storm
For
as
a
tropical
function
storms
the
of
cloud-top
computation
rainfall rate for each satellite picture element
(pixel)
of
in
the grid is expressed by the relationship
j = b i5 R v 10"3 /
R ij = 0.557
2.g± .
(Am / g...)
Zb
(6.3)
(b... Z a ^
where
R^ . is the inferred rainfall rate
pixel
of
the
grid,
b^
is
the
/ Zb)
(mm/h)
empirical
(6.4)
in the
(i,j)
we i g h t i n g
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147
coefficient of the
volumetric
(i,j) pixel,
output
is the area
Rv is the satellite-inferred
2
(km ) of the
(i,j) pixel.
the denominator of the right h a n d
side
The factor of 2 in
of
equation
aries from the rainfall apportionment scheme.
factor
10
_3
un i t s from m
measured
in the numerator of equation
3
to mm km
from
3 -1
(m h
),
rate of the storm for one image
the
2
. A
image
a nd
is
The conversion
(6.3) converts the
2
(km ) of
is the area
m
(6.3)
the
'
storm
the sum of areas of the
p i xels whose temperature is less than or equal to -20°C,
a^
is the fraction of the storm c overed by the kth temperature,
is
the empirical w e i g h t i n g coefficient corresponding to
the kth temperature,
and Z a b
runs over all pixels
K ££
in
the
storm that are -20°C or colder.
The
factor
of
storm rainfall rate
fractional
(0.0667)
0.557
is
(16.7x10
coverage
of
3
the product of the tropical
3 —1
2
m hr
km )
echo
area
and
for
the
inferred
tropical
storms
multiplied by the product of the apportionment
c onv e r s i o n
constant
(5x10
-4
).
a nd
The weighting coefficient b
enters in the calculations by which the inferred rainfall is
r e l a t e d to the colder temperatures of the storms.
exponential
function
function
is
0
and
255
of temperature.
and
is
an
inverse
The coefficient is of
the form
b = exp
(^
an
of the GOES digital count wh i c h takes
on integer values between
exponential
It
+ c 2 V)
/ 11.1249
(6.5)
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148
wh e r e V is either GOES digital count
or
temperature
(°C),
the value of b ranges from 1 at -20°C to 4.55 at -110°C.
For
temperatures
The
warmer
than
value of the constants
-20°C,
and C 2
the values are zero.
are
given
in
the
Table
(6.4) .
TABLE
(6.4)
Constants for the Empirical Weighting Coefficients
Digital Counts
154 < D C < 176
Temperature
(DC)
177<DC<255
-20° C<T<-310C
(T)
-32° C<T< — 110° C
Cl
0.026667
0. 11537
1.784059
2.278682
C2
0.01547
0.01494
-0.03094
-1.1494
Altho u g h the total area of the storm canopy at -20° C is used
in the rain computation of equation
necessarily
rain
does
not
fall from the entire canopy. Rain is assumed to
fall only from
distributed
(6.4),
so
the
coldest
that
half
of
the
canopy
and
is
half of the rainfall from the coldest
10% of the canopy area, w i t h the remaining half occurring in
the next coldest 40% of the canopy.
50/40-50
rain,
apportionment
etc.)
(10%
of
This is called
the
10-
storm area has 50% of the
and is based on radar studies intra
cloud
rain
rate distributions made by W oodley et a l . (1980). The factor
of
2
in
the
denominator
of
equation
required to halve the rain volume,
(6.3)
is therefore
Rv . The digital counts of
the pixels are converted to b values through equation
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(6.5).
149
The fractional coverage of the
(a^)
is
determined
and
storm
of the two values.
each
(bk ) . These products are
( S a ^ b ^ . The sum of b's
(Zb) assumes one
The decis i o n criteria for the appropriate
value are based on a ranking of the b values.
values
for
the
storm
(which define the top 10
determined.
For example,
250th
to
correspond
and
50%
b values.
are
temperature,
to
the
top
10%
of
the
b
values
the coldest 10% of the storm. Similarly,
coldest
half
of
the
the
The b values in the top 50%, but not in the top 10%,
top
10%
interest,
u s e d in
10%,
two
Because the b values are inversely
are summed and are referred to as Z< 0 0. b. The
the
values)
b
if there were 500 pixels comprising
top 50% of the b values comprise the
storm.
the
the break points w o u l d be defined by the 50th and
largest
related
After
are ranked in descending order,
breakpoints
the storm,
temperature
is then multiplied b y the b value
that corresponds to the temperature
subsequently summed
at
are
summed
for
if b.. is in the top
iJ
(6.4).
I!« o ox
''0
b
S l00,
'0
b
For
(i.e .,coldest)
values
the
in
pixel of
10%, 2 t0o b is
SQ
If b^. is in the top 50%,
is used.
but
not
the
If b. . is not in the top 50%,
iJ
top
R. . is
ij
set to zero.
A rain computation begins with the navigational digital
data from which the clouds are isolated according to
20°C
isotherm.
Then
the
rain
volume
the
for every cloud is
computed from the component relationship of
the
technique,
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150
and
an
empirical
algorithm
(the
rainfall apportionment)
subsequently maps volumetric rainfall on
g rid
squares
constituting the cloud,
of rain depths.
as
the
to
the
satellite
resulting in an array
The resolutions of the array data
spatial
scale
of
are
same
the navigated satellite digital
imagery.
The VISSR data of 14th September 1978
(Julian day 257),
is obtained for three images starting at different times
(17 hours,
hours,
47 minutes,
47 minutes,
hours,
47
24 seconds,
20 seconds,
minutes,
540 m i l l i s e c o n d s ) ,
620 milliseconds),
and T_
17seconds, 980 milliseconds).
minutes and 35 seconds.
(21
v
Each of
these images are constructed from the data of time
13
(20
interval
All three images are then merged
t o g e t h e r , in order to have the
maximum
number
of
overlap
w i t h the SMMR derived rainfall rates.
The statistics of rainfall rates corresponding to these
images
Table
at
starting
times
T^
(6.5). The combined IR
these
times
F igures
T 2 and T 3 are shown in the
rainfall
(7.3) and
for
The
this
to
for
(7.4). Ag a i n some data points are found to
maximum IR
data
longitude and 10°
close
estimates
T , T 2> and T g are plotted on the w o r l d map in
be on land where the rainfall rates
mm/hr.
rate
set was
were
as
high
as
6.7
estimated rainfall rate over the ocean
found
to
latitude location.
be
6.1 mm/hr at -152°
This location
is
very
that at which the microwave technique estimates a
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151
TABLE (5.5)
The Statistics of Inferred Rainfall Rates froa IR Data
at tines Tl, T2, and 13
IStatistics of IR
I Rainfall Rates
I
I
I
At tine T1
R1
(oim/h)
I
I
I
At tine T2
R2
(mn/h)
I
I
I
At tine T3
R3
(na/h)
I
I
I
I
Mean
I
2.898
I
2.884
I
2.885
I
I
Median
I
1.938
I
1.915
I
1.945
I
I
Mode
I
2.00
I
2.00
I
2.00
I
I
Minicun
I
0.00
I
0.00
I
0.00
I
I
Maxinua
I
50
I
49.00
I
31
I
I
Sum
I
200113.0
I
255921.0
I
199923.0
I
I
Std.Error
I
0.010
I
0.010
I
0.010
I
I
Std.Dev.
I
2.747
I
2.377
I
2.531
I
I
Skeuness
I
1.877
I
2.806
I
1.667
I
I
Kurtosis
1
4.973
I
18.333
I
1.344
I
I
Mild
I
4
I
89
I
3
I
I
Valid cases
I
59287
I
88748
I
69290
I
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FREQUENCY DISTRIBUTION o f RAINFALL RATE
Using (IR) data at Time T t , R(1)
60
50 -
40 - /
n^
o«
O
tt
c
50
*£
EC
20
10
-
2
4
6
10
8
16
12
Figure (6.8)
FREQUENCY DISTRIBUTION of RAINFALL RATE
Using (IR) data a t Tim© T2, R(2)
60
2629
0 for N
50 - /r r.
•*o©
s-s
a «
O
TT
_ c
o
u°
w
U3
oo
tt
50
20
-
10
-
2
m
8
10
12
14
16
Figure (6.9)
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FREQUENCY DISTRIBUTION o f RAINFALL RATE
Using (IR ) data at Time T3, R(3)
50
45 40 35 30 25
20 -,
10-,
E72L
2
4
6
8
10
12
14
16
Figure (6.10)
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154
maxima for the rainfall rate over the ocean.
is
better
than
close,
spatial resolution
longitude)
in
because
these
This
agreement
owing to the limit on the
figures
(2°
latitude
X
2°
a 2° difference in latitude or longitude might as
well correspond to the same position on the earth's surface.
Frequency
versus
class
interval histograms have been
constructed in a fashion analogous to that of the
data
for
(6.10))
the
three
sets
R 2 , and R 3
microwave
(Figures
(6.8)
-
which are the rainfall rates for entire images taken
at times T^,
T^,
and T g
respectively.
from 0 mm/hr to 16 mm/hr.
rainfall
rates
that
classes
range
The class interval is 2 mm/hr. The
are
limit of an interval are
The
less
not
than or equal to the lower
included
in
that
interval.
there are two peaks in all histograms just as in the case of
microwave
data.
The
first
peak
is between 0 mm/hr and 2
mm/hr and the second peak is between 8 mm/hr and
The
resemblance
10
mm/hr.
between the microwave and IR distributions
is good.
Conclusions drawn from
this
resemblance
must
be
limited
to only the first peak since the second peak in the
microwave data corresponds to points on land where microwave
estimates based on ocean emissivity are
next
chapter
not
good.
In
the
VII comparisons are made between the rainfall
rates derived from infrared and microwave data.
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CHAPTER
(VII)
RESULTS AND DISCUSSION
The estimated rainfall rates obtained
sensors
and
September,
VISSR
techniques
are
described
1978,
from the
SMMR
onboard
GOES-W
w as
Corresponding
ship
data
Atmosphere Data Set,
Boulder,
Colorado).
of
using
here.
onboard
Data
SEASAT-A
obtained
COADS
different
of 14
and
and
the
analyzed.
(Comprehensive
Ocean-
Release 1) w as obtained from NOAA
(ERL,
Although this data set did not provide
the rainfall rate measurements which could be used as ground
truth for comparison it did provide the weather condition at
the time of observation.
was
coded
in
qualitative
success
a
variable
comparison.
because
two data sets
The weather condition at that
time
na m e d PW w h i c h was employed for
This
attempt
did
not
meet
any
the space and time differences between the
(ship and microwave)
were found to
be
larger
than that optimal for rainfall rate comparison.
(7.1)
Maps of Rainfall Rates Using Microwave Data
The processed data of SMMR w as
This
provided
P ropulsion
Laboratory).
contained
observation,
the latitude and the longitude of
by
JPL
the
the
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time
(Jet
of
sensors
1 56
field
of
view,
and the ten sets of brightness temperatures
at SMMR channels.
these
The rainfall
radiometric
rates
were
estimated
temperatures us i n g equation
(6.2).
estimated rainfall rates were then averaged over a
2°
latitude
x
2°
longitude.
The
rainfall
generated from these averaged values are
(7.1)
and
shown
and the second part from 120°
world
These
grid
rates
in
of
maps
Figures
(7.2). The world map was divided into two parts,
the first part exhibits the longitude from
the
from
map
from
-80°
-180°
to
-80°,
to 180°. The remaining part of
to 120°
is not shown because the
SMMR data over this region did not exist in the data set.
It is evident from these maps that rainfall over
v a ried up to 3.5 mm/hr.
Rainfall rates higher than 3.5 mm/hr
were
scattered
land.
The data over these areas will be referred to as
contaminated
over
data.
land
areas
or regions very close to
be
expected
contaminated data.
from
land
are
to
true
the
model
estimates
for
land
Only rainfall rates for points
far
away
considered
give
emissivity
as true estimates and these are
used for comparison with rainfall rate
techniques.
land
Since radiative transfer calculations in
this model involv o n l y ocean surface
cannot
ocean
estimates
of
other
For this data set therefore microwave e s t imated
rainfall rates higher than 3.5 mm/hr are irrelevant and
are
not considered for comparison purposes.
In
Figure
(7.1)
estimated
rainfall
rates
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for five
157
MAP OF RAIN FALL RATES
USING SMMR DATA
- A Tm
t —
-I
—7—Z
r--"
1=
.3
2s
.6
3= 1.0
4= 1.3
5s 1.6
6= 1.9
7= 2 .2
8= 2 .5
9= 2 .9
n= 3 .2
2 3 2
3= 3 .5
L_
C= 3 .8
0= 4 .1
E= 4 .5
n
F= 4 .0
G= S . l
4
P
4 fl S
H= S. 4
1= 5 .7
J= 6 .1
K= 6 .4
L - 6 .7
H= 7 .0
Z 212
I I
N= 7 .3
0= 7 .6
P= 8 .0
0= 8 .3
R= 8.6
S= 0 .9
4 4 4 3
4 3 414
3 4 3 3
m i
,
T= 9 .2
U=9.6
V= 9 .9
W=I0.2
3343
ttet
X -1 0 .5
Y=1Q.B
Z=ll.l
-=ll.S
■ = 11.8
* = 12.1
== 12.4
0= 1 2 .7
LRT=-5Q.Q TQ 5Q.Q . L0NC=-18Q.OT0 -6 0 . Q
NUM 0F P01NT5= 319. URT LPN PV=RRGE=
2
MIN =
.0 0 MRX= 1 2 .7 4 INCREMENT*
.3 2
Figure
(7.1)
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
153
MAP OF FAIN FALL RATES
■
USING SMMR DATA
/■
\-
.3
2=
.5
3= 1 .0
4= 1.3
5= 1 .5
6 - 1 .9
lL 1 1 J
f' r ' / i
i vI
M L i
• i " i
6= 2 .5
9 - 2 .3
■f
fi= 3 .2
B= 3 .5
' —r
l U - l J - h J —
fi i im
7= 2 .2
r | «?> \ \
,/■■v>i
l
II
C= 3.8
D= 4 .1
E= 4 .5
F= 4 .8
r r ; i , i i■i
A
H Ay~ArfA 4:-A-7-J i 1“
11
-
A l l
' r n
A
l
„
.
—
,
..
"12II 1
~
F "T
G= 5 .1
H= 5 .4
1= 5 .7
J= 6 .1
h1
K= 6 .4
L= 6 .7
Ms 7 .0
K- 7 .3
0= 7 .5
P= 8 .0
D= 8 .3
l
J - l l M
,W \ A \
\ \ A_- '
I
\
Q
.
i
4_ _ _ I '
y a _ i
83 i,
I
II41
i
i ^
L 7
; /
/
7
/
/ / /
1 L -U J ./
R= 8 .6
S= 8 .9
T= 9 .2
U= 9 .6
V= 9 .9
H=!Q-2
X =10.5
7 = 1 0 .8
X \X & T M U
'j L u
?
L flT = -5 0 .O T0 SO.O . L J ‘iu= 120.0TA
IBO.O
MUM 0F P31NTS= 13S. LflT L0N flVERRGE=
2
M1N=
.0 0 MflX= 1 2 .7 4 INCREMENTS
.3 2
Figure
(7.2)
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
2= 11.1
-=11.S
i= 1 1 .8
♦ = 12.1
= = 12 .4
0 = 1 2 .7
159
orbits of SMMR data are
seperated
by
shown.
Each
of
these
orbits
is
a fixed distance on the globe and is observed
at different times of the day.
It can be observed from
this
figure that maximum rainfall over the ocean is found at
-155°
at
longitude,
-138°
10°
latitude in the northern hemisphere and
longitude,
-36®
latitude
in
the
southern
hemisphere.
In Figure
orbits
of
(7.2) m i c rowave estimated
SMMR
rainfall
latitude and at 164°
in
the
longitude,
southern
32°
latitude.
longitude,
Most
of
30°
the
h e misphere is land contaminated and
therefore irrelevant for comparison purposes.
figures
two
passes is shown. Maximum rainfall over the
ocea n in northern hemisphere is found at 146°
d ata
for
In
both
the
no rainfall is found at the equator wh i l e rain rate
is found to increase w i t h latitude in both the hemispheres.
(7.2) Maps of Rainfall Rates Usi n g Infrared Data
The inferred rainfall rates from
d e s cribed
in
latitude x 2°
plo t t e d
on
chapter
longitude.
VI,
are
These
VI S S R
averaged over a
averaged
values
grid of 2°
are
then
The w o r l d m ap has again been divided into
two parts in whi c h the longitudes vary from
from
as
a wor l d map in the same w ay as des c r i b e d in the
case of SMMR data.
and
data,
120°
to 180°
-180°
to
-80°
and IR estimated rainfall rates for
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
160
these regions have been pl o t t e d in Figures
In
Figure
whi c h
m i c rowave data.
is
an
at
agreement
-152°
inferences.
In
in
Figure
-36°
agreement
with
IR estimates in the
land
contaminated
areas
they are not useful
for
invalid microwave inferred rainfall rates.
In the nothern hemisphere IR estimates
far
10°
(7.4)
and although they are valid results,
with
longitude,
reasonable
s outhern hemisphere correspond to
points
longitude,
In the southern hemisphere IR estimates over
latitude which fact is also
comparison
(7.4).
w i t h that inferred from
the ocean indicate a maximum at -132°
m i c rowave
and
(7.3) maximum rainfall rate over the ocean in
the northern hemisphere is found
latitude,
(7.3)
from
are
found
only
at
the SMMR subsatellite track and therefore
cannot be compared.
(7.3) C o mparison of Rainfall Rates Inferred
(MW) and Infrared
of
and M W data
defining
the
Microwave
(IR) Data
The coincident rainfall
(VISSR)
from
rates
derived
from
(SMMR) h a v e been compared.
coincident
data
The criterion
rainfall rates depends on the
distance between the two data points of IR and MW,
if
IR
that
is
the differences between the latitudes a nd the longitudes
of a IR and a M W data point are less
p r e d e f i n e d threshold,
than
or
equal
to
a
then these two data points are grouped
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161
MAP OF RAIN FALL RATES USING
IR DATA
1=
.S
2=
.5
3=
.3
4=
.3
5= t . l
r
/
I
u
r
^
\
> v
6= 1.3
7= 1.4
0 = 1 .6
~ i ,—
JG [ 3 ' 1
I—
/SC
U
r.
~HC Bi
MpP^
I
r N B!M)G T 3
1
, U
I
p
1
i
i C.K
i M iHu 0a 9 /
L o
9 f E L 5 S - l E 3 R 0 pig
9= 1 .7
A
i
.1 F JM.K3
9 I ViP
•
10 0 9
C
L LIQ
i uFr Jj nK
^gst'
,riT
A
3 3 IK 2
[3 .1
I
B= 2 .1
C* 2 .2
2 0 3 ,0 b
9 J C
_ € ,3 S K 9 J J H fi
F E Y*B
>3 R I C H G L B
s u M H Gl7 7 3 5 I K S O E 3
-6 0
2 C S P 8 9 B_ I 9 2 V j is H [ 0 ?\r F C 3 2, 1T2‘-
E lfJ F B R M J W Q M H - »K J R C C(6
3 5 ' 7 7 S 1.0 fl F 0 C /6 0
0 2.
J fl B 5 2
1
R= 1.3
I M £ J ia E K 8 13
G 19 P ,0 H 0 0 914 6
I
3 2iC 6
D= 2 .4
E= 2 .5
F= 2 .7
C= 2 .3
H= 3 .0
1= 3 .2
J= 3 .3
K= 3 .5
-1
!
_L
L= 3 .7
K= 3 .8
N= 4 .0
0= 4 .1
P= 4 .3
3
. OE
1 3 1.
2
0= 4 .5
2 2
R= 4 .6
fl}HR I Z 2\D __2
D I f i’c D ? rlfi8 2 2
B NF 9 I P C C i
B 0 5'.4 G I 1 \
9C 9 0‘f H 6 9
_ R
JE 19 2
1 \
f l 9 K EV* 2
9 T273P5T
B S C fl
\
S= 4 .8
2 1
2RI23H
1
‘
T= 4 .9
U= 5 .1
. ,
F * I J G 7 |3
2 2 1 I C iI H F Q C j3 _ f l 2
V= 5 .3
N J ?F 2~t\l
ITDT R/TTB
9 2 10 M K K O il G H 8 L
M= 5 .4
Xs 5 .6
G 0 3 7 S S I GO H R 4 0/9
B 0 R 213 0 P H Mfc 3 E f l 7 E
Y= S. 7
LhCib
Z= 5 .9
-=
6 .1
i = 5 .2
+= 6 .4
== 6 .5
0= 5 .7
“
7
^
LRT=-5G.O T0 5 0 .0 . L0NG=-18O.OT0 - 3 0 .0
NMH Bp POINTS: 569. LRT LBN RVERR3E=
2
h iN = .3 0 0 HflX= 6 .6 9 7 IHCREKENTs .160
Figure
(7.3)
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
163
together and considered for comparison.
A common
longitudes
data
and
set,
which
predefined threshold
latitude and in longitude.
a
sub
the
latitudes,
inferred rainfall from both IR and MW data,
was obtained for a
value
contained
data
set
equal
to
0.5°
in
Then by decreasing this threshold
was constructed.
This sub data set
contained the observations w h i c h are more closely spaced and
are therefore expected to be better correlated.
The frequency distribution
general
Figures
not
a
normal
(6.1-6.10).
of rainfall
distribution
rate
is
in
as already noticed in
The measures of difference that are used
to describe the discrepancy between M W and IR estimates
listed
in
Table
(7.1).
The sample ratio,
w hi c h indicates that the accumulated M W
sample
is
larger
(Rg>l)
or
accumulated IR estimates for the sample.
sample
ratio,
estimates
sample
(MWR)
RM ,
are greater than 1
which
the
than
the
mean
(IRR)
averaged
over
the
measure,
a
(MWR/IRR), which
can offset R ^ from those values which
ED ,
is
are
defined
effect of the ratio greater than one
and is called the "factor
always
the
Similarly the
The values of this ratio
Therefore another
eliminates
(Rg <l)
for
average of the ratio of the M W
to the IR estimates
of size N.
less than one.
is
R g , is a measure
estimates
smaller
are
of
difference"
whose
value
is
less than one. The relationship between ER and R^ is
analogous to that between
|x|
and x. The
mean
error
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of
a
164
TABLE
(7.1)
Measures of difference between the microwave rainfall
estimates
(MWR)
and Infrared rainfall estimates
(IRR)
for a sample of size N
D ifference
Definition
Measure
Sample Ratio
(?-g )
Mean Sample Ratio
ZM W R/ZIRR
(R^)
Factor of Difference
(E )
ri
Z(MWR/IRR)/N
(ZR/N)
[R = MWR/IRR if M W R < IRR
R = IRR/MWR if IRR<MWR]
Average Error
(p )
[ (MWR-IRR)2/ N ] 1/2
Root-Mean-Square
Error
(Ep_M S )
Normalized Root-Mean-Square
Error
Z(MWR-IRR)/N
(Norm.
[Z{(MWR-IRR)/lRR}2/ N ] 1/2
E_,M S )
Normalized Bias
(Norm.3)
Normalized Standard Deviation
(Norra.STDEV)
Correlation C o e f f i c i e n t ^ 0 }
( 1 / I R R ) [(Z(MWR-IRR)}2/ { N ( N - l ) } ] 1/2
(1 / I R R ) [(Z(MWR-IRR)2R ) 2
N ( M W R - I R R ) 2/ ( N - l ) } ] 1 / 2
(MWR.IRR - MWR
. IRR)
/
[(MWR" - M W R 2 )(IRR2 - IRR2 )]1/2
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165
sample of size N is calculated from the
M W and IR estimates.
m easures
the
The root-mean-square
absolute
difference
estimates of M W and IR pairs,
error
(norm.
er m s
^
measures
bias
be
the
the
E_.„0 ,
RMS
rainfr.ll
rate
normalized
absolute
The
of
the
sample
and
Comparisons
the
square
rate.
of
error
the squared
the
sampling
are made for the linear quantities
(bias a nd standard deviation)
rainfall
RMS
difference as
mean
variance
be t w e e n
error,
decomposed into two squared quantities,
difference.
IR
(RMS)
between
whereas
n o r m a l i z e d by the IR rainfall rate.
can
difference
All
n ormalized by a sample average
the
measures
of
difference
are
calcu l a t e d for the data of M W and IR images corresponding to
the
star t i n g
times
T^,
T 2 , and T g . C o m bining these three
images of IR data a combined data set
measures
of
difference
the present study
d i fference
as
these
well
was
obtained.
are defined in Griffith
measures
depend
on
These
(1987).
the
In
spatial
as the temporal difference of the data
points.
I nitially the time dependence was
the
spatial
differences
0.5°,
0.1°
and 0.05°
considered
and
(the differences of latitudes and
longitudes of M W and IR data)
to
not
whi c h were less than or
equal
were considered for calculating the
m eas u r e s of difference as sho w n in the Tables
best co r r e l a t i o n between the M W
and
IR
(7.2-7.4).
inferred
The
rainfall
rates is obtained for the spatial difference equal to 0.05°.
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1 66
TABLE (7.2)
Maasurs of Difference Between ffiJ and IR Rainfall Estimates
Whose spatial Difference is less than equal to 0.5
I. . . . . . . - ■ ! - - - - - - - - - - !- - - - - - - - - - 1- - - - - - - - - - 1- - - - - - - - I Parameter
I
AT Tine T1 I
AT Tine T2 I
AT Tine T3 I
Combined
j---------- j__..........r-----------1---------- 1--------I Humber of data I
9S4
I
940
I
372
I
2736
I Sample (N)
I
I
I
I
J----------------- T------------------- J------------------ J------------------ J---------------
JAverage IR Rain T
I fall'Estisate I
I
(IR)
I
i
l
I
3.15
I
3.13
I
I
T-------------------T-------------------- T------------------- J------------------- J---------------lAvsraoe HH Rain I
I
I
!
I fall Estimate I
1.32
I
1.33
I
2.03
I
1.33
I
(Mi)
I
I
I
!
3.2
I-----------------
I
I
l
3.05
r------------------ :------------------ t ---------------
I Sample Ratio I
0.5
!
0.53
I
0.55
I
0.53
I
(Rs)
I
I
I
I
i— -.........
i...........i---------- 1--------I Hean Sample I
0.34
•
0.30
I
0.S5
I
0.90
I Ratio (Res) I
I
I
I
I- - - - - - - - - - 1- - - - - - - - - - !- - - - - - - - - - 1- - - - - - - - - - 1- - - - - - - Factor of Biff. I
0.47
I
0.34
I
0.51
I
0.51
I
(Er)
I
I
I
I
I- - - - - - - - - - 1- - - - - - - - - - !- - I- - - - - - - - - - 1. . . . . . . .
I Averaae Error I
-1.23
! -1.07
I -1.12
I -1,15
I
(E)
I
I
I
I
T---------- j----------- ;-----------j-----------t--------I RHS-Error
I
3.7051
I
2,3734
I
2.322
I
3.1733
I
(Eras)
I
I
I
I
J -----------------------------------1-------------------------------------- T------------------------------------ T------------------------------------ r.............................. ..
INorn. RHS-Error I
1.0131
I 1.1325
I 0.B135
I 1.0241
I (Horn.Eras) I
I
I
I
I- - - - - - - - - 1- - - - - - - - - - 1- - - - - - - - - - 1- - - - - - - - - - 1- - - - - - - I Norn. Bias I
0.4
I
0.35
I
0.35
I
0.37
I
(Norn. 3) I
I
I
I
J ---------------------------------- T-------------------------------------- T------------------------------------ T------------------------------------ r __________________
I Horn. Sid.Dev. I
I (NORM. STDEV) I
1.0331
I
I
0.3752
I
I
0.S233
I
I
0.3449
T----------------- T------------------- i------------------ 1------------------ J---------------
I Correlation ! -0.1725
I coeff. (f) I
I
I- - - - - - - - - - 1- - - - -
!
0.1439
I
0.2312
I
!
!- - - - - - - - - - 1. . . . . .
I
0.043S
1 - ......
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167
TABLE (7.2)
Measure if Difference Between Til! and IE Rainfall Sstisates
T'hcss spatial Difference is less than equal to 0.1
Parameter
AT Tiae T1 I
1-
AT Tins T2 I
1-
-
Nuaber of data I
Saapie (N) !
Average IR Rain I
fall Estiaaie I
f'l'i
r
4. OS
AT Tine T3 I
I37
I
Ccibined
119
.S3
Average ill* Rain I
faH'Sstizate !
(Mil)
I
2.31
2.03
Saapie Ratio I
CEs)
I
0.43
0.59
(lean Saapie
Ratio (Ez)
I
I
0.93
0.73
Factor of Diff. 1
(Er)
I
0.29
Average Error I
’(E)
I
on
-o.se
-1.57
4.359S
1.9542
2.532S
3.5339
1.1046
0.5461
0.5208
0.3037
0.57
0.S7
0.3?
0,53
0.5!
--------------- j_
RMS-Errcr
(Eras)
Mora. RHS-Error I
((forz.Erzs) I
1Nora. 31 as I
(((ora. 3) !
(fora. Std.Dev. I
I (NORM. STDEV) I
-----------------------------T I------I Correlation I
I coeff, (f) I
0.41
1.0925
0.5197
0.5343
0.3721
-0.4707
O.aiJV
0.6/Ql
0.131
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168
TABLE (7.4)
'ieaaure of Diff=rsnce Between ;!W 2nd IR Rainfall Estimates
Whose spatial Difference is lass than equal to 0.05
-------r... ....... J-
Faraneter
)T Time T1 I
AT Time 12 I
AT Time T3 I
Co,shined
--------------------------- T-
75
I Himbar of data I
I Sample (?!) r
IIAverage IR Rain
I fall i-jtiaata
I
(IS)
n
co
J 1 uU
lAverags !!H Rain
I fall Estimate
2.02
I---------I Sample Ratio
I
CEs)
I
--- I Hean Sample
I Ratio (Ra)
IFactor 0 ? Biff.
I
fEr)
I. . . . . . . . . .
I Average Error
I
(E)
I---------I SHS-Error
I
(Eras)
T-------IKora. RUS-Error
I (Nora.Eras)
r----------
2.37
0.57
0.31
0.33
-
0.7
0.45
0.55
1.00
-0.44
3.3545
0.57
-1.32
a
Li
1.1373
-I- - - - - - I
0.4301
I
0.9513
r
ro o
i
\ 7nnn
V* / VUi.
0.34
I
(Nora. 2)
I
----I liara. Std.Dev,
r
i
/Mi-'r.M
ilUlttl.
1.2003
0.48S5
ornc»M
-JIUL.V
7__________
I Ccrrslaticn
I ccsff. (f)
-0.241!
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169
The M W rainfall rates are directly correlated
with
rainfall
T3
rates
at
starting
times
T2
and
the
but
IR
anti
c orrelated with the IR image at starting time T 1 .
Time
is investigated by considering time differences of
1 hour,
hours,
3
hours,
2
and 5.5 hours between M W and IR estimates.
The exact time was calculated for each pixel
from the starting time of the image.
(7.7),
lag
in the IR image
In the Tables
(7.5)
to
the measures of difference are given as functions of
time difference and for spatial differences 0.5°,
0.05°
respectively.
All
data
points
c o mparison of oceanic rainfall rates in
0.1°,
and
considered
for
these
Tables
were
not constrained in any way.
Since
the
IR
technique
is
tuned for estimating the
tropical rain a constrained data set is
obtained
from
the
full data set based on the c o n dition that the latitude of IR
and
MW data should lie w i t h i n the 30°
c omparisons are made for
d ifference
are calculated.
g i v e n in the Tables
0.5°
and 0.1°
(7.8)
data
set
3 hours,
set
coincident
not
spatial
measures
of
These measures of difference are
and
(7.9)
for spatial
and 5.5 hours.
data
points
In
this
were
d i fference less than or equal to 0.05°.
are
and
The
differences
respectively for the time differences 1 hour,
2 hours,
no
this
from the equator.
constrained
found
for
data
spatial
U n fortunately
there
enough coincident data points even for the case of
differences
0.5°
and
0.1°
wh i c h
can
give
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any
170
TABLE (7.5)
Measure of Difference Between Mr) end IR Rainfall Estimates
Nhose spatial Difference is less t'.ian equal to 3.SB
I
I
Parameter
ITime difference Time difference ITime difference ITime difference I
2 hours
I 3 hours
i
1 hour
I 5.5 hours I
I Number of data I
I Sa£ipl& (N) I
‘37
374
I
1331
I
2376
I
2.'35
3.12
?
1
n «a
0. i L
I
3.14
I
3.33
1.32
I
1.37
I
1.35
I
0.3
3.52
I
0.65
I
3.62
I
3.63
3.35
I
0.37
I
0.3
I
IFactor of Diff. I
I
(Er)
I
3.52
0.52
t
i
n cn
U .J j
I
3.51
I
I Average Error I
I
(E)
I
-2.37
-1.2
I
-1.14
I
-1.13
I
5.443
3.1334
j
4
0» ii u4 .u
I
3.2317
I
3.7224
Iliorn. RMS-Error I
I (Nora.Eros) I
t _ __ _______________ r _____
i
—
------------------- 1------I Nora. Bias I
3.7
I
(Nora. B) I
n nn « c
tJ.O Oi J
I
1.0211
I
1.3181
I
3.33
T
T
1
fl 70
t l i JU
T
1
I Nora. Std.Dev. I
I (NORM. STDEV) I
n
IAverage IR Rain I
I fall Estimate I
I
(IR)
I
r. —
— T— —
IAverage MU Rain I
I fall Estimate I
I
(P!W)
I
I Sample Ratio I
I
(Rs)
I
IT-_ —
—
—
— —— T
j . -. —
I Mean Saspie
I Ratio (Rn)
I
I
T --— - - - - - - - - —-T --—
r
t
I
I
RMS-Error
(Eras)
I
I
t _____________________r
I Correlation
I coeff. ( f )
______
I
I
1.7233
-3.3337
fl
0 .3 7 /0
I
0.3113
T
1
r» ^ 7 A n
ii« Z l 30
?
i
n «n n r
0 .1000
I
0.1614
t
1
a
O .U O //
Ti.
n n o
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
171
TABLE (7.S)
Measure of Difference Between KW and IR Rainfall Estimates
Ufiose spatial Difference is less than equal to S.IQ
I - - - - - - - - - - 1- - - - - - - - - -
I
Parameter
ITima Difference ITima Difference Tiae Difference ITine Difference I
I
1 Hour
I 2 Hours
3 Hours
I 5.5 Hours
I
INumber of data I
I Sample (If)
I
I
I
43
73
I
123
I
IAverage IF: Sain
Ifall Estimate
I
(IR)
1r
I
I
1r
I
I
3.7
nu« nc
vlU
T1
hj.DO
rn
T1
IAverage MW Rain
Ifall Estimate
I
(HU)
1f
I
I
Ti
I
I
2.13
2.14
I
2.37
I
I
1
T
I Mean Sample
I
Tl
I Ratio (Rs)
T1
IFactor of Diff. I
I
(Er)
I
I
I
T
2
Tt,
T
i
I
I
0.53
3.64
I
0.57
I
3.£7
0.73
I
3.83
I
3.53
3.53
I
9.51
I
I Sample Ratio
I
(Rs)
___
IAverage Error
I
'(E)
I
TI
I
T1
-1.51
-1.22
I
-1.56
I
I RMS-Error
I
(Eras)
I
I
I
I
2.5
2.23
I
3.52
I
lilora. RMS-Error 1
I (Nora.Eras) I
1
I
3.52
0.53
I
9.81
I
I Norm. Bias
I (Nora. 3)
I
T1
I
iT
3.41
3.37
I
3.43
1
INorm. Std.Dev. I
I (KORN. STDEV) I
I
I
3.54
n
a .ju
Ti
U$UI
Ti
I Correlation
I coeff. (f)
I
T
3.53
CC
u. uO
I
0.13
I
I
T
i
en
I
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission.
Measure of Difference 3etueen MH and IR Rainfall Estimates
Hhose spatial Difference is less than equal to 0.35
T
I
I
T
Parameter
T
ITioe difference
I
t hour
r
T
_ T
T
'iBe difference ITime difference ITime difference I
I
5.5 hours 1
2 hours
I 3 hours
T
"li
I Number of data I
I Sample (N) I
I
23
I
29
I
*3
cn
0 1 uo
I
2.31
I
2.37
I
1.31
1
1.88
I
1.93
I
I Sample Ratio I
I
(Rs)
I
3.52
I
3.67
I
3.57
I
I Mean Sample
I Ratio (Rni)
1
I
3.73
I
3.78
I
3.85
I
IFactor of Diff. I
I
(Er)
I
3. £3
I
3.53
I
3.54
I
I Average Error I
I
(E)
I
-1.32
I
-0.92
I
-0.95
I
I
I
I
I
3.2124
I
2.1232
I
2.5533
I
INorm. RMS-Error I
I (Norm.Erms) I
3.4331
I
0.5433
I
3.7C32
c*
cJ.u*T
T
i
U ig - T
T
1
iJ .w l'T
I Norm. Std.Cev. I
I (NORM. STBEV) I
3.7377
I
'3.7015
T
fl n n r * «
I Correlation
I coeff. (rt
3.1125
I
T
I
3.4733
I
IAverage IR Rain I
I fall Estimate I
I
(IR)
I
T ___________________
T ____
IAverage MU Rain I
I fall Estimate I
I
(MU)
I
I
I
RMS-Error
(Eras)
Norm. Bias
(Norm. 3)
I
I
n
o,i
L
i
I
I
A
T
T
l
r
U .O O tll
**
- - - - - -
0.159
1
- -
^
1
T
r
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
173
TABLE (7.3)
Measure of Difference Between MH and IR Rainfall Estimates
Hhose spatial Difference is less than equal to 2.58
I
Parameter
I Nuaber of data
I Sample (H)
I- - - - - - - - - Time difference ITiae difference ITime difference ITime difference
1 hour
I 2 hours
I 3 hours
I
5.5 hours
I
66
I
157
I
167
T
IAverage If: Rain
I fall Estimate
I
(IR)
I
2.17
I
2.65
i
I
j
*> £5
IAverage MU Rain
I fall Estimate
I
(MU)
I
1.43
I
1.42
I
1.42
I Sample Ratio
I
(Rs)
I
8.63
I
3.53
I Mean Sample
I Ratio (Ra)
I
8.33
I
8.75
T
I
f* ir
U.
/b
IFactor of Diff.
I
(Er)
I
3.65
I
3.64
I
8.64
I Average Error
I
(E)
I
-0.58
I
-1.24
I
-1.24
1
I
RMS-Error
(Er sis)
I
1.7443
T
a
nn«n
I
2.3313
INorn. RMS-Error
I (Horn.Eros)
I
0.5631
I
8.5111
I
3.5111
I
I
I
3.32
I
3.47
I
8.47
I Norm. Std.Dev.
I (NORM. STDEV)
I
3.746
I
8.7637
I
8.7537
I Correlation
I coeff. (p)
i
I
3.3733
I
3.8453
I
3.3458
Norm. Bias
(Norm. B)
J
«•* j j
8.53
T
I
r
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
174
TABLE (7.9)
Measure of Difference Between MW and IE Rainfall Estimates
Hhose spatial Difference is less than equal to 3.13
I
I
Paraaeter
ITiae difference Tiae difference ITiae difference ITiae difference I
I
3 hours
I
5.5 hours I
I
1 hour
2 hours
I Nuaber of data I
I Saaple (N) I
7
I
I
11
I
11
I
IAverage IR Rain I
I fall Estiaate I
I
(IR)
I
1.84
I
i
1.S4
1
1.64
I
1.44
I
1.44
I
IAverage MU Rain I
I fall Estiaate I
I
(MU)
I
I................!................
I Saaple Ratio I
I
(Rs)
I
1.48
i
!
3.89
1
I
3.83
I
3.83
I
I
I
3.S3
I
1
0.33
I
3.83
I
IFactor of Diff. I
I
(Er)
I
3.71
1
I
8.78
I
3.76
I
I Average Error I
I
(E)
I
-3.13
I
1
-3.28
I
-3.23
I
I
I
I
I
3.5239
1
I
0.4679
I
3.4679
I
IMora. RMS-Error I
I (Nora.Eras) I
0.3213
I
1
0.2359
I
3.2359
I
0.12
1
1
8.13
I
3.13
I
0.3233
I
I
0.272
I
3.272
I
1.83
I
1
3.6181
I
3.8131
I
I Mean Saaple
I Ratio (Ra)
I
I
RMS-Error
(Eras)
Nona. Bias
(Nora. 8)
I
I
I Nora. Sid.Dev. I
I (NORM. STDEV) I
I Correlation
I coeff. (/)
I
I
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
175
definite information regarding the time lag between
IR
estimates.
However,
MW
and
in the absence of data for the time
difference 1 hour the best correlation is obtained
spatial difference equal to 0.1°
for
the
and for the time difference
equal to 2 hours.
(7.4)
Sensitivity Study of Rainfall Rate
Rainfall rates have been simulated for various types of
atmospheres to study the sensitivity of rainfall rate to the
variation of parameters such as am i n , am a x , p,
atmospheric
a
temperature.
The
drop
and
parameter
refractive
p.
index,
Since
is
the
very
mb
spectrum is defined by
. , a
and the shape of the distribution is
nun
m ax
r
the
500
microphysics,
temperature
decided
by
especially the
sensitive,
the
rainfall rate variation w i t h the 500 mb temperature has been
investigated.
Microwave
brightness
temperatures at 10 SMMR channels
have been simulated for 48 different types
In
atmospheres.
these simulations one parameter at a time was varied and
the rest of them were assumed constant.
for
of
each
of
the
10
SMMR
brightness
Therefore 48
temperatures
values
were
obtained to study the effect of variation in each parameter.
Rainfall
rates
temperatures
were
using
calculated
the
from
relation
these
(6.2)
brightness
where
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
the
176
corr esponding
regression
channels
given
are
coefficients
in
the Table
for
the
best
5
(6.2). From the detailed
a nalysis of this study following inferences are drawn.
(a) It was observed that the variation of a . from 50 pm to
v '
man
100 pm produces a change in rainfall rate of
3.0
X
the
fact
_2
10
(mm/hr) which is insignificant.
that
at
the
lower
end
of
the
the
order
of
This is due to
spectrum
the
e xtinction and scattering cross-sections which are functions
of the drop diameter are extremely small.
(b)
The
Darameter
relation
(4.65)
incremented
a
given
in
was calculated from the empirical
max
r
by
5 steps.
Stephens
It
was
then
In each step the increment was 10%
of the calculated value. For a
rainfall
(1962).
the
_3
changes by a factor of the order of 9 X 10
rate
50
%
change
in
a
max
(mm/hr). The explanation for this lies in the fact that
total
number
of
drops
decreases
with
the
the
diameter and
therefore the extinction and scattering coefficients do
not
change significantly with am a x *
(c)
The variation of the shape parameter p in the drop size
distribution was found to
change
in
produce
as
much
as
3
(mm/hr)
rainfall rate. The bulk of this change was found
to occur between the values of p from -2.0 to -1.0 as
in
the
Figure
parameter
orographic
p
for
rain
(7.5).
Ulbrich
different
shown
(1983) has categorized the
types
of
rain;
p
<
0
for
(indicating a broad DSD with large numbers
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
177
MI IU VERSES RAINFALL RATES
FOR ATMOSPHERE NO.
1
P. 5
HO
-
1.0
0
.2
.4
.0
.0
1.0
1.2
1.4
1.0
1.0
2 .0
2 .2
2.4
2 .0
2 .0
3 .0
3 .2
RAINFALL HA'IBS (M M/M R)
USING FIVE SMMR CHANNELS
P lg u re ( 7 . 5 )
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
3 .4
3.0
178
of small drops)
(corresponding
and
to
0
a
<
<
jj
2
narrower
for
thunderstorm
rain
DSD w i t h reduced numbers of
small d r o p s ) . For wides p r e a d or stratiform rain
the
of
For showers
are more variable but tend to be positive.
jj
it is not possible to define
exact
value
of
p.
values
However,
U l b r i c h has shown from analysis of past studies that
m o s t l y within the range -2 <
The
jj
simulation of b rightness temperatures requires the
values of
jj
particular values of
known
and
perhaps
jj
may not be fixed.
reduce
the
rain
rainfall
rate
and
the
Nevertheless this is the best that
present
ranges
circumstances.
are
not
Random v a r i a t i o n of
type
was
used.
jj
This
accu r a c y of the simulated brightness
temperatures and consequently of
the
the
for given rain types
over the known ranges for each
to
Although
for different rain types are reas o n a b l y well
known,
tends
lies
< 4.
k n o wledge of jj for various rain types.
of
jj
the
relationship
brightness
one
can
be t w e e n
temperatures.
do
under
the
If the values of jj were to be kno w n
more precisely then the inversion of rainfall rates w o u l d be
more accurate.
however,
data
In the absence of such precise kn o w l e d g e of
any c o r roborating measurement that yields
jj
as
jj
a
input must of course m a t e r i a l l y improve the reliabilty
of estimations of the model because it will reduce the range
of u n c e rtainity in
jj
.
(d) At 500 mb and temperatures below 0°C only ice exists.
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
A
179
variation
of the temperature at this altitude from 0°C to -
40°C results only in the
There
change
is no change of phase.
of
the
ice
temperature.
The effect of this temperature
variation on the rainfall rate was found to be of the
of
3.0
X
10
—2
mm/hr
order
which may be considered negligible.
Therefore one concludes that ice temperatures do not
rainfall rate estimations significantly.
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
affect
CHAPTER
(VIII)
RETROSPECT
In this study a microwave
which
takes
effect
into
for
the
microwave
precipitation
basic
physical
raaiometery
been
calculated
program
clouds
atmosphere has been developed.
and
have
been
as
a
function
the
of
an
concepts
developed.
and
in
In order to do
mathematical
emission and absorption from the ocean
have
transfer
the Mie-scattering polarization
nonisothermal
inhomogeneous
this
account
radiation
of
Microwave
atmosphere
the appropriate
variables of state using classical theories and experimental
data obtained by
particular,
microwave
different
several
ocean
investigators
emissivity
frequencies
surface
has
ranging
conditions
been
from
(see
Extinction properties of microwaves by
have
1
the
past.
In
calculated at all
to
300
Figures
liquid
GHz
for
(4.1-4.6)).
hydrometeors
been considered in detail using the Mie-theory and the
Gamma drop size distribution.
the
in
Gamma
drop
size
rate and a
size
parameter
graphically in Figure
The total number of
drops
in
distribution depends on the rainfall
(4.9).
jj
.
This
dependence
is
shown
Extinction coefficients at SMMR
frequencies for liquid hydrometeors have been calculated for
different values of the size parameters
jj
, N Q>
€, and S.
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
181
These size parameters decide the
distribution
extinction
curve.
The
coefficient
graphically
transfer program has
microwave
600 scattering,
model
rainfall
(4.10-4.26).
been
brightness
used
the
rate
temperatures
for
both
5.16).
(6.6
theoretical
homogeneous,
coefficient of liquid water
as
and
functions
vertical
of
to
0.1
mm
was
GHz)
increase
where
temperatures
as
linearly
at
high
polarizations
were
rainfall rate in Figures
The brightness temperatures
clouds,
shown
The brightness temperatures at SMMR frequencies
horizontal
presented
size
at SMMR frequencies for
h o r i zontally
Scattering
is
calculate
drops whose diameters were less than equal
negelected.
drop
The microwave radiation
to
plane-parallel,
atmospheres.
of
variation of liquid h y d r o m e t e o r s 1
with
in Figures
shape
at
with
low
SMMR
rainfall
frequencies
(5.9-
frequency
rate for low
the
brightness
are non-linear functions of rainfall rate. The
rate of change of brightness temperatures w i t h rainfall rate
were positive initially and
their
saturation
points.
then
SMMR
frequency
37.0
brightness temperatures at the two
with
rainfall
rate.
effects
reaching
rainfall
GHz.
The
rate
for
difference
polarizations
the
in
decreases
This can be explained to be due to the
underlying surface being obscured
polarization
after
The saturation point varies with
frequency and has minimum value of
largest
negative
are
by
eliminated.
hea v y
rain
so
that
The ice hydrometeors
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
182
present in the form of ice clouds scatter more
h igh
frequencies
temperatures.
hydrometeors
frequency.
On
and
result
the
other
increases
brightness
which
induces
darkening
b rightening at low frequencies.
frequency radiances respond
bottom
the
presence
of
of
the
ice
temperature at low
These results can be explained to be due
scattering
at
in decreasing the brightness
hand
the
strongly
to
the
at high frequency and
It was observed that the low
more
to
water
drops
at
the
clouds while the high frequencies radiances
were more sensitive to ice near the cloud top.
It must be
particles
m e n tioned
were
that in
assumed spherical
this study,
and
emission and extinction coefficients
any changes
due
to
particle density and shape were not included.
between the two phases
their
extinction
refractive
index
ice
the
in the
variable
The difference
(ice and rain) was considered through
coefficients
and
particle
which
size
are
functions
distribution.
ice
and
Gamma
drop
size
of
The
particle size distribution given by Marshall and Gunn
for
and
(1958)
distribution for rain were
assumed.
The best possible subset of SMMR channels for
retrivals
was
obtained (6.6H,
6.6V,
21.OH,
rainfall
37.OH,
37.0V)
using the optimization technique "Leaps and Bounds" given by
Furnival
and
temperatures
Wilson
of
these
(1974).
The
frequencies
computed
brightness
were then inverted for
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
rain
183
rainfall rates using multiple linear regression method.
rainfall
rates
September,
were
inferred
from
the SMMR data of 14
1978 over the Pacific ocean.
These estimates were
m a p p e d on the World map for comparing the rainfall
elsewhere.
Another
means
of
of
rainfall
(7.4).
rainfall
World
GOES
satellite.
al.
(1976).
The
The
computer
maps are shown in Figures
The comparisons made
o btained
the
from VIS/IR data are generated from
the scheme given by Griffith et
generated
obtained
estimating rainfall rate was
from infrared data obtained from
estimates
The
for
estimated
rainfall
(7.1)rates
u s i n g these two different types of data have shown
promising results Tables
(7.2-7.9).
Conclusions and Comments
The
main
ingredients
in
rainfall
rate
estimation
through microwave remote sensing are the microwave radiative
transfer
surface.
model
Most
scattering
and models for the atmosphere and the ocean
radiative
and
hence
e stimation problem.
large
rainfall
transfer
When
rates
or
models
are
not
the
radiative
do
suitable
not
include
for
rainfall
tranfer
.involves
higher microwave frequencies the
scattering plays a dominant role and
cannot
The
to a three dimensional
inhomogeneity
of
space
leads
be
neglected.
radiative transfer equation w h i c h is very difficult to solve
and consequently requires an exhorbitant amount of
computer
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
184
time.
It
is impractical to deal with such an equation even
though this would be ideal for treating the
problem
in
an
exact
way.
If
remote
one assumes that the space
inhomogeneity lies only in the vertical direction
solution
of
the
radiative
sensing
transfer
equation
greatly in both conception and application.
This
then
the
simplifies
assumption
does not severly affect the solution of the problem at hand.
The polarized ocean surface emits
whose
intensity
is
zenith-angle
dependent
horizontally
homogeneous;
is
considered
intensity
of
the
emergent
considered azimuthally
the
second
phase
validity
two-streams
order
scattering
of
radiances
independent.
matrix
radiative transfer equation.
Eddington's
radiances
only
atmosphere
approximate
microwave
can
This
in
atmospheric
one
needed
This assumption is
It
the
therefore
enables
elements
approximation.
and the
be
to
in
the
called
the
involves
only
particles.
The
this approximation has already been tested for
rainy atmosphere and is severely limited only when there are
big hailstones
rough.
The
or
the
polarized
surface
reflectors
components
of
are
extremely
microwave
ocean
emissivity are calculated from the Fresnel equations and the
Debye equations for dielectric constant. For
surface,
various
temperature,
factors
such
as
a
surface
the frequency of observation and the
calm
ocean
salinity,
angle
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
of
185
incidence
influence
the
surface wind generates
surface
whose
ocean surface emissivity.
foam
emission
and
produces
a
known,
rough
The physics of
(emission and reflection of microwaves)
and only empirical estimations have
estimation
ocean
and reflection properties are very
different from those of the specular surface.
the foam
The sea
is not well
been
used.
The
of rough ocean surface emissivity can perhaps be
improved by a composite model in
which
rough
given
by
a two-scale scattering
emission
is
given
water
theory
surface
and
the
is
foam
emission
by
from
a
the
layered
dielectric theory.
The theoretical basis for estimating the rainfall rates
depends
upon
particles,
attenuation
(i)
(ii)
the
the
size,
index
properties
shape
of
of
the
and phase of the rain
refraction
and
radiation.
(iii)
The estimation
becomes better depending upon how accurately the
signature
is
e s timation
of
different
drop
seperated
the
size
distributions used,
latter
rain
DSD.
drop
the
size
distributions
rain
is
background
signature.
provided
(DSD).
Of
Marshall Palmer and the Gamma
The
by
the
the
two
DSD,
the
has been found to closely simulate the actual DSD in
different types of rain.
types
from
the
can
be
The behaviour for various
parametrized
by the exponent
jj
rainfall
in the Gamma
Ice particle distributions for various rainfall are not
kno w n and are perhaps not unique.
However,
better
knowledge
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
186
about
ice
the ice particle distribution and their shapes in the
clouds
can
improve
the
estimation
of
microwave
ex tinction coefficient of rain.
The
passive
method for estimating rainfall rate over
the ocean at SMMR frequencies relies
relationship
between
m i c rowave
rainfall rate. At high
through
upon
the
fundamental
brightness temperature and
frequencies
the
penetration
depth
the precipitation becomes shorter due to scattering
and the rain becomes opaque even at very low rain rate.
can be observed in 37.0
gets
saturated
d e creasing w i t h
the
spatial
GHz
at very
brightness
low
rate
rain rate v e r y rapidly.
On
which
and then starts
the other
hand
resolution of a radiometer for a given antenna
size increases with frequency.
two
rain
temperature
This
competing
influences
A
compromise
has
between
these
to be found which will have
better resolution and estimation.
Microwave rainfall estimates are
infrared
estimates
when
time interval and within
differences.
with
comparable
0.1°
of
latitude
and
longitude
Correlation bet w e e n these two estimates reduces
increase
microwave
in
the time a nd spatial differences.
underestimates
the
c omparison w i t h the IR. The d iscrepancy
estimates
the
they were found w i t h i n 2 hours of
estimates needs to be tuned for local effects.
time
with
can
be attributed
to
Most
rainfall
between
various
The IR
of
rates
the
in
these
two
factors.
The
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
187
principle employed
different
in
in
nature.
these
two
methods
directly
related
Microwave estimates
filling,
(ii)
of the sensors,
completely
In IR technique the rainfall is based
on the cloud's top temperature whereas
are
are
to
suffer
the
from
microwave
hydrometeors
problems
like
radiances
themselves.
(i)
different instantaneous field of view
(iii) height of the
rain
column
and
beam
(IFOV)
(iv)
inhomogeneities within a radiometer's IFOV.
If
one were to ask w h i c h technique might best estimate
the rainfall rate,
it
is required.
the answer to this w o u l d depend on
where
If one needs to know it over the ocean then
passive microwave would be the best choice.
Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.
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