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An algorithm for retrieval of monthly rainfall over the oceans from the TRMM microwave imager (TMI)

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AN ALGORITHM FOR RETRIEVAL OF MONTHLY RAINFALL OVER THE
OCEANS FROM THE TRMM MICROWAVE IMAGER (TMI)
A Dissertation
by
JUN HUANG
Submitted to the Office o f Graduate Studies o f
Texas A&M University
in partial fulfillment o f the requirements for the degree o f
DOCTOR OF PHILOSOPHY
December 2001
Major Subject: Atmospheric Sciences
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AN ALGORITHM FOR RETRIEVAL OF MONTHLY RAINFALL OVER THE
OCEANS FROM THE TRMM MICROWAVE IMAGER (TMI)
A Dissertation
by
JUN HUANG
Submitted to the Office o f Graduate Studies o f
Texas A&M University
in partial fulfillment o f the requirements for the degree of
DOCTOR OF PHILOSOPHY
Approved as to style and content by:
Gerald R. North
(Member)
Thomas T. Wilheit
(Chair o f Committee)
Michael I. BiggeCstaff,
(Member)
Richard E. Orville
(Member)
Kai Chang
(Member)
Gerald R. North
(Head o f Department)
§
December 2001
Major Subject: Atmospheric Sciences
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ABSTRACT
An Algorithm for Retrieval o f Monthly Rainfall
over the Oceans from
the TRMM Microwave Imager (TMI). (December 2001)
Jun Huang, B.S., University o f Science and Technology o f China;
M.S., Institute o f Atmospheric Physics,
Chinese Academy o f Sciences;
M.S., University o f Wyoming
Chair o f Advisory Committee: Dr. Thomas T. Wilheit
One o f the principal goals o f the Tropical Rainfall Measuring Mission (TRMM)
is to measure rainfall in the tropical and subtropical regions. In this study and through
collective efforts, a multichannel algorithm has been developed for retrieval o f monthly
rain totals for 5° by 5° grid boxes over the oceans from the TMI data. This multichannel
algorithm is based on relationships between rain rates and brightness temperatures (R-T)
generated by a physically-based microwave radiative transfer model (RTM) for the
different TMI channels. This RTM is based on the Wilheit et al. (1977) model with an
updated water vapor absorption procedure, a water vapor absorption correction factor
and reduced non-precipitating cloud liquid water (NCLW) contents which are consistent
with many field experimental results.
The combination o f R-T relationships for 19 and 21 GHz are used to obtain
freezing level information for each pixel. The freezing level information is then used to
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solve for rain rates corresponding to the 10,19 and 37 GHz brightness temperatures,
respectively. Those rain rates are adjusted according to their corresponding beamfilling
corrections and then accumulated to form rain histograms for each channel over each 5°
by 5° grid box.
The rain rate histogram for each channel is processed so as not to have a cut-off
at zero rain rate. The most probable rain rate retrieved by this algorithm is very close to
zero. Any remaining offset o f rain rate is shifted to zero accordingly.
In order to utilize the rain rates resulting from those three different channels, the
rain rates derived from 37 and/or 19 GHz channels are “smoothed” to the resolution o f
the 10 GHz channel so that the rain rates retrieved from different channels represent the
same area. The algorithm chooses the highest frequency possible among the 37,19 and
10 GHz channels. This gives us the greatest usable sensitivity to rain.
Global rainfall mappings clearly show tropical large-scale features such as heavy
rainfall over the Intertropical Convergence Zone (ITCZ). Quantitative comparisons with
the Pacific atoll rain data indicate that the current algorithm performs very well at least
in the tropical regions.
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V
ACKNOWLEDGEMENTS
I would like to thank my advisor, Dr. Thomas T. Wilheit, for his guidance and
patience during this study, the friendly atmosphere he has created for the Microwave
Remote Sensing Group and the opportunities he has provided me. Thanks also go to my
other committee members: Drs. Michael I. Biggerstaff, Kai Chang, Gerald R. North,
Richard E. Orville for taking their time to be on my committee. Thanks go to Dr. Renyi
Zhang for taking his time to be a substitute on short notice for Dr. Michael I. Biggerstaff
at my defense.
Many thanks go to my fellow group-mates, both past and present: Dong Heon
Lee, for sharing his research results with me; Jody L. Thomas-Stahle. for all the help she
has provided; Stepheni Moore, for her interesting discussions with me about Chinese
culture; Clay B. Blankenship, William Manning and Shaohua Alex Wang, for helping
me get started when I was just joining the group; Daniel J. Redmond, for discussions
with me about the research; Ye Hong and Jeffery R. Tesmer, for their communications
with me through emails that helped me understand their research better; Richard Weitz,
for re-formatting the original TMI data to make them easy to be used; Christopher T.
Bellows, for his nice results; Kyung-Wook Jin, for sharing his knowledge with me about
the field observations.
Special thanks to my wife Hong, my son Kaiwen (Kevin) and my parents for
their love, understanding and encouragement. Without their support, I could not have
completed this study.
This work was supported by the NASA TRMM and AQUA programs.
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TABLE OF CONTENTS
Page
A B ST R A C T .......................................................................................................................... iii
ACKNOWLEDGEMENTS ..................................................................................................
TABLE OF C O N TEN TS....................................................................................................
v
vi
LIST OF TABLES ................................................................................................................ viii
LIST OF FIGURES...............................................................................................................
ix
CHAPTER
I
INTRODUCTION ..................................................................................................
I
n
PRINCIPLES OF MICROWAVE RADIATIVE TR A N SFE R .........................
5
2.1 Principles o f microwave remote sensing for rainfall retrieval over the
o cea n s............................................................................................................. 7
2.2 Microwave radiative transfer m odel........................................................... 13
2.3 Rainrate-brightness temperature (R-T) relationships................................ 16
m
REVIEW OF PREVIOUS WORK AND METHODOLOGY............................ 19
3.1 Emission approach and some theoretical studies......................................... 20
3.2 Scattering approach.........................................................................................23
3.3 Other empirical approaches............................................................................26
3.4 Statistical physical ap p ro ach ......................................................................... 27
3.5 Application to area-time-averaged rainfall ra te ........................................... 28
3.6 Previous work directly related to the current research and methodology
for this stu d y ................................................................................................... 3 1
IV THE TRMM MICROWAVE IMAGER AND PRECIPITATION RADAR ......38
V ANALYSIS ................................................................................................................. 41
5.1 Impact o f DSD on R-T relationships..............................................................41
5.2 Freezing level retrieval.................................................................................... 43
5.3 Offset o f rain ra te .............................................................................................44
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CHAPTER
Page
VI ALGORITHM............................................................................................................ 53
6.1
6.2
6.3
6.4
6.5
6.6
R-T relationships.............................................................................................53
Beamfilling error.............................................................................................. 64
Selection o f saturation criteria....................................................................... 65
Smoothing o f rain rates................................................................................... 67
Combined rain rate histogram ..................................................................... 70
Average rain rate and monthly rain to ta l...................................................... 70
V n RESULTS AND VALIDATION............................................................................ 72
VUI SUMMARY AND CONCLUSION.........................................................................83
REFERENCES....................................................................................................................... 87
V ITA ....................................................................................................................................... 93
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LIST OF TABLES
TABLE
Page
1 Spatial resolution for each TMI c h an n e l...............................................................40
2
Constants in R-T relationships............................................................................... 59
3
Variances o f Gaussian for 10,19 and 37 GHz vertical chnnels............................ 69
4
Statistical results o f validation.............................................................................. 81
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LIST OF FIGURES
FIGURE
Page
1
Emissivity o f a smooth water surface at 20°C (from Wilheit and Chang, 1980) 8
2
Schematic diagram o f Wilheit etal. (1977) radiative transfer m odel................ 14
3
Calculated brightness temperatures at 19 GHz for nadir viewing as a function
o f rain rate for five different freezing levels (from Wilheit et al. (1977)........... 18
4
R-T relationships calculated by the TAMU model with the M-P DSD (solid
lines) and with the DSD proposed by Tesmer and Wilheit (1998) (dashed
lines) for 19 GHz vertical polarization channel at an incidence angle o f 52.8°
(TMI incidence angle).............................................................................................. 42
5
Two-dimensional histogram o f brightness temperatures from 19 and 21 GHz
vertical polarization channels over a 5° by 5° grid box in the tropical central
Pacific Ocean. Isolines o f freezing level (solid lines) and rain rate (dashed
lines) based on R-T relationships are overlaid...................................................... 45
6
Histogram o f occurrence o f rain rates derived from the 37 GHz vertical
polarization channel o f TMI. The non-precipitating cloud liquid water
(NCLW) content in the underlying radiative transfer model is the same
as that in the Wilheit et al. (1977) model............................................................... 46
7
Same as in figure 6 except from the 19 GHz vertical polarization channel o f
TMI............................................................................................................................. 47
8
Same as in figure 6 except from the 10 GHz vertical polarization channel o f
TM I............................................................................................................................. 48
9
Histogram o f occurrence o f rain rates derived from the 37 GHz vertical
polarization channel o f TMI. The non-precipitating cloud liquid water
(NCLW) content in the underlying radiative transfer model is only one
fifth o f that in the original Wilheit et al. (1977) model........................................ 50
10 Same as in figure 9 except from the 19 GHz vertical polarization channel o f
TM I............................................................................................................................. 51
11 Same as in figure 9 except from the 10 GHz vertical polarization channel o f
TM I............................................................................................................................. 52
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X
FIGURE
Page
12 R-T relationship calculated by the TAMU model for the 37 GHz vertical
polarization channel at an incidence angle o f 52.8° (TMI incidence angle)
54
13
Same as in figure 12 except for the 21 GHz vertical polarization channel
56
14 Same as in figure 12 except for the 19 GHz vertical polarization channel
57
15
58
Same as in figure 12 except for the 10 GHz vertical polarization channel
16 Best-fit (dashed lines) for the R-T relationship for the 37 GHz vertical
polarization channel. The poor fit at high rain rates does not affect rain
retrieval in the current algorithm............................................................................. 60
17 Same as in figure 16 except for the 21 GHz vertical polarization channel
61
18 Same as in figure 16 except for the 19 GHz vertical polarization channel
62
19 Same as in figure 16 except for the 10 GHz vertical polarization channel
63
20 Monthly rainfall over 5° by 5° grid boxes for January, 1998.............................. 73
21
Same as
in figure 20 except for February, 1998............................................... 74
22
Same as
in figure 20 except for March, 1998.................................................... 75
23 Map o f western Pacific atoll geography................................................................. 77
24 Scatter plot o f the TMI inferred rainfall versus the atoll rainfall data for Jan.,
1998............................................................................................................................ 78
25
Same as
in figure 24 except for Feb., 1998....................................................... 79
26
Same as
in figure 24 except for Mar., 1998...................................................... 80
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1
CHAPTER I
INTRODUCTION
Rainfall measurements are very important in meteorology, hydrology and
climatology. The latent heat released through the formation o f rainfall in the tropical
regions is the driving force o f a thermally directed Hadley cell. Knowledge about the
global precipitation can help us to understand intra-seasonal phenomena such as the 4050 day Madden and Julian oscillations and inter-annual phenomena such as El NinoSouthern Oscillation (ENSO) and may eventually help us to predict large-scale climate
events like El Nino and La Nina. To understand the global water and energy budget it is
necessary to measure the rainfall over the oceans accurately. For General Circulation
Models (GCM) rainfall over the oceans is an important input parameter and can be also
used to test consistency between reality and model output.
While the measurements o f rainfall over the oceans are so important,
conventional means (rain gauge, precipitation radar) over islands and ships have proven
to be impractical. Islands over the oceans occupy small ratio and they cannot represent
typical oceanic environments. Rain gauges on board ships are subject to pitching and
rolling all the time. On the other hand, rainfall over the oceans is highly variable both
temporally and spatially. To solve this problem spacecraft technologies are used.
Satellites offer suitable spatial and temporal coverage over the oceans and are the only
reasonable solution for the measurements o f oceanic rainfall.
This dissertation follows the style and format o f the Journal o f Applied Meteorology.
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2
Many efforts have been made for estimation o f rainfall from visible and infrared
measurements from satellite platforms (Barrett and Martin, 1981) since those
measurements are available on both the geo-synchronous satellites and other longer time
series polar-orbiting satellites and cover a larger fraction o f the earth surface. However,
these satellite-borne visible and infrared sensors can only measure physical properties,
such as cloud temperature, at the top o f the cloud. Therefore the estimation o f rainfall
based on visible and infrared measurements is generally based on empirical relationships
relating rain rate to cloud properties, such as cloud top temperature and cloud fraction.
These empirical relations for rainfall estimation from visible/IR measurements must be
re-produced for each new satellite program and this process takes many years for the
collection and analyses o f the data. The passive microwave technique offers an
alternative that overcomes these disadvantages.
Unlike the weak physical connection between visible and infrared measurements
and rainfall, microwave radiation interacts with hydrometeors much more strongly.
Therefore retrieval o f rainfall based on microwave observations has a firmer physical
basis. The single-channel nadir-viewing 19 GHz Electrically Scanning Microwave
Radiometer (ESMR) on board the Nimbus 5 satellite in 1972 was the first microwave
radiometer used for global rainfall mapping. This was followed by the 37 GHz dual­
polarization ESMR on Nimbus 6 satellite in 1975 and the Scanning Multichannel
Microwave Radiometer (SMMR) on Nimbus 7 satellite in 1978. The Special Sensor
Microwave/Imager (SSM/I), a well designed and well calibrated instrument, on the
Defense Meteorological Satellite Program (DMSP) polar-orbit satellite series (F-8 to
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3
F-15) has been put on operation since 1987 and has enabled us to produce high-quality
global rainfall estimates.
Recently, the Tropical Rainfall Measuring Mission (TRMM) satellite was
launched in late November o f 1997. It flies at a height o f 350 km with an inclination o f
35°. The low inclination and resultant orbital precession o f the TRMM satellite reduce
the sampling errors and the impact o f the diurnal circle on the rainfall retrieval. The
primary rainfall measuring instruments on board o f the TRMM are the TRMM
Microwave Imager (TMI) and Precipitation Radar (PR). The TMI is based on the SSM/I
flown on the DMSP satellite series. The key differences are the addition o f a pair o f
10.65 GHz channels with vertical and horizontal polarizations and the shift o f the water
vapor channel from the absorption line center at 22.235 GHz to 21.30 GHz to avoid
saturation o f this channel in the tropics. Compared with the SSM/I, the TMI has higher
spatial resolution due to the lower orbit o f the TRMM satellite.
The principal goals o f TRMM (Simpson et al., 1988) are to measure rainfall and
latent heat released through condensation in the tropical and subtropical regions. The
combination o f space-bome passive and active microwave sensors in the TRMM
satellite provides critical information such as geometry o f precipitation for algorithm
developers.
The objective o f this study is to develop a multichannel algorithm, which will
utilize both the 10 GHz and the 37 GHz channels along with the 19 GHz channel, for
retrieval o f monthly rain totals for 5° by 5° grid boxes over the oceans from the TMI
data. This multichannel algorithm is based on relationships between rain rates and
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4
brightness temperatures (R-T) generated by a physically-based microwave radiative
transfer model for the different TMI channels. According to these R-T relationships, the
10 GHz channel has the ability to measure very high rain rates (as high as 50 mm/hr)
while the 37 GHz channel is sensitive to the low rain rates and can differentiate these
low rain rates from non-precipitating cloud liquid water (NCLW). Through proper
handling, rain rates retrieved from these three different channels can be used in a directly
comparable manner thus achieving high sensitivity to the low rain rates where needed
and also able to reliably measure the high rain rates up to 50 mm/hr.
The remaining o f this dissertation is organized as followings. The scientific
background for retrieval o f rainfall over oceans from space-bome passive microwave
sensors is introduced in chapter II. The previous work and methodology for this research
are reviewed in chapter HI. In chapter IV, the TRMM instrumentation and the data
description are given. Several issues that affect rainfall retrieval algorithm are analyzed
in chapter V. Chapter VI describes the algorithm for retrieval o f monthly rainfalls. Those
monthly rainfall results are validated in chapter VH. Finally, a summary and conclusion
are presented in chapter vm.
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5
CHAPTER II
PRINCIPLES OF MICROWAVE RADIATIVE TRANSFER
All substances radiate electromagnetic energy as a result of constant motion of
molecules and/or atoms. The radiance, h(T), emitted by a substance at a wavelength A
and an absolute temperature T, is the product o f the emissivity, £x (0 < £*< 1), o f the
substance and the blackbody radiation Bx {T) at the wavelength A and the absolute
temperature T. Bx (T) is described by Planck function as follows (Liou, 1980):
exp(hc/ XkT) - 1
(2.1a)
where h is Planck’s constant, k is Boltzmann’s constant, c is velocity o f light.
Sometimes the Planck function is defined in terms of frequency v o r
wavenumber v (= A-1) rather than frequency. Different forms of Planck function are
related by:
Bv (T )dv = -B. k(T)dX = Bv- (T)dv
i.e., the energy integrated over the same spectral domain be equivalent. From the
relationship that c=Xv, we have d v = -c d X / X2 and d v = c d v , thus:
(2.1b)
cz
e x p ( h v /k T ) - l
Bf (T) = 2hc2v 3 ■
exp(/tcv / kT) -1
(2.1c)
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In the microwave region, where I GHz < v <300 GHz (1 GHz = 109 Hz) or I
m m < A < 300 mm, and at temperatures typical o f the Earth’s surface and its atmosphere
(T > 200 K), h v /k T «
1, the Planck function can be approximated by the Rayleigh-Jeans
formula such that
->hv} kT ~>vl kT
Bv (T) = —
cfiV
c-
(2. 2b)
kT
BV
- (T ) = 2/ic:v 3 ■— = 2 c v 1kT
hcv
(2.2c)
Under those conditions, microwave radiance is proportional to the absolute
temperature of a substance, and the brightness temperature ( TB) is defined such that it is
the product of the emissivity and the absolute temperature of a substance, i.e..
=
2ck
7» = £ ' r = W
a . 3a)
2ck
M ’,r)= ^
' ' (n
,1 3 b )
Thus TBcan be used directly to represent microwave radiance.
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7
2.1 Principles of microwave remote sensing for rainfall retrieval over the oceans
The brightness temperature observed from space is the microwave radiance
emitted and reflected from the Earth’s surface and absorbed, emitted and/or scattered
from its atmosphere. According to the theory o f radiative transfer (Chandrashekar, I960)
as applied to microwave remote sensing, if a microwave radiance TB, at a wavelength X,
is incident on a surface with a reflectivity r and a temperature
, the reflected
brightness temperature Tgf is given by:
T?f = r T f + e T ,
where s is the emissivity o f the surface and r + e = I by KirchofFs Law.
The emissivity, or equivalently the reflectivity, o f the surface is described by the
Fresnel relations (Jackson, 1962) and is a function o f the incidence angle, the
polarization and the complex index o f refraction o f the surface. The polarization is called
“horizontal” if the electric field vector o f the electromagnetic (EM) wave is in the
horizontal plane. On the other hand, a vertically polarized EM wave has its electric field
perpendicular to both the directions o f the propagating EM wave and the horizontal
polarization. In case o f the nadir or the zenith view, the definition o f polarization breaks
down.
Over the Earth’s surface, the oceans have a relatively uniform and low emissivity
(0.4 to 0.5 at nadir) in the microwave region, thus they appear cold in a microwave
image. Figure 1 shows a calculated typical smooth oceanic surface emissivity at a
temperature o f 20°C (Wilheit and Chang, 1980). In this calculation, the reflection from
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s
.6
-
.5 -
3.5% NaCL
>i>
(0
ot
I
u
FRESH
3.5% NaCL
WAVELENGTH (CM)
0.8
0
5
10
(5
20
25
30
35
FREQUENCY (GHz)
0.6
40
45
SO
Figure 1. Emissivity o f a smooth water surface at 20°C (from Wilheit and Chang, 1980).
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9
the ocean surface was treated as specular (mirror-like) and the dielectric constant (square
o f the complex refraction) data were from the Lane and Saxton (1952) measurements. It
is clear from figure I that the emissivity increases with frequency and the emissivity for
vertical polarization has a higher value than that o f the horizontal polarization as
required by the Fresnel relations (Jackson, 1962). In contrast, land has a highly variable
and high emissivity (0.8 to 1.0, depending on soil moisture and vegetation) ( Liou, 1980)
thus appears warm.
Liquid precipitation has a high absorptivity (thus a high emissivity by KirchofFs
Law). If there is rain over the ocean surface, the upwelling Tg appears warm and it can
be identified easily against the cold and uniform oceanic background.
The change of brightness temperature Tb over an infinitesimal distance ds in a
direction specified by the conventional polar angles, ( 9.<p), is given by Wilheit (1994)
and can be written in another form as follows:
dTg(9,0)
=G+L
(2.4)
ds
where
O = y ^ T is) + y,cu$ P{94,9'4')Ta(0'.0 W
£ -~(Yab, + rm )TB(0,4)
T(s) is thermodynamic temperature of medium, Tg(9.<p) is the microwave radiance in the
direction specified by the conventional polar angles ( 9, <j>), s is distance in the ( 9, <t>)
direction,
is the absorption coefficient and y,cu is the scattering coefficient.
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10
P (0 ,0 ,0 ',0 ') is the phase function describing the probability o f scattering from a
direction (d\<p') to the direction (0, <p) and is normalized such that:
The term G represents the microwave radiance gained by emission of medium along the
beam and by scattering of microwave radiance propagating in other directions being
scattered into the direction of the beam. While the term L represents the microwave
radiance lost by absorption along the beam and by scattering out of the beam.
Equation (2.4) is the Equation o f Radiative Transfer (ERT) for microwave
radiance. With the ERT organized this way, it is stated that the change of TB along the
path. 5. is decided by the balance of the microwave radiance gained and that lost.
If the scattering in the atmosphere (caused by rain drops and ice particles) is
omitted and assuming horizontal homogeneity, the ERT can be directly integrated and
Tg exiting the atmosphere is:
TB( \ ,9 .p ) = TBl+TBZ+TB}
(2.5)
where
T„= £ y (* U )ru )e x p [-r4(c,«)/i]M<fc
TB2 = e x p [ - r i (O,oo)/x]e.(0,p)r,
TBi = A - r ( A , 0 , p ) - 0
and
A = e x p [-r l (0,°°)/i]
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11
B = Tm e x p [-rA(0,<*>)p] + [
p [ - r A(0 ,z )/i]/^ z
or
B = T„ A + £
(z)7*( z) A/fc/z
where r- (z,, z , ) = j ' y ^ , iz')pdz' is the optical thickness of the atmospheric
M
layer between the height z/ and z:, Tc„, is the 2.7 K cosmic background brightness
temperature, Tt is the surface temperature, and e k ( 6. p) is the surface emissivity for the
wavelength, A, at the incidence angle, 9, and polarization, p (p is either vertical or
horizontal polarization), p = I/cos 6. The term Tbi represents the radiation emitted
upwards by each incremental layer o f the atmosphere and attenuated by the intervening
layers. The term Tb2 represents the emission from the surface as attenuated by the
atmosphere. The last term, Tbj. which results from radiation reflected by the surface,
consists of the downwelling emission attenuated by the portion of the atmosphere
between the emitting layer and the surface and by the entire atmosphere on the upward
trip plus the 2.7 K comic background as attenuated by a two-way pass through the entire
atmosphere (Wilheit, 1994).
In the atmosphere there are three important absorbers: molecular oxygen, water
vapor and liquid water. At frequencies between 50 and 70 GHz and near 119 GHz. the
absorption due to molecular oxygen is important and these frequencies are used for
temperature sounding but are not often used for rainfall sensing. For the purpose of this
work, the absorption due to molecular oxygen is a minor correction needed for
quantitatively accuracy but not o f conceptual importance.
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12
Water vapor has two important resonant absorption lines (22.235 GHz and
183.310 GHz) along with other minor resonant lines in the microwave region. Although
22 GHz line is relatively weak compared to the very strong absorption line near 183
GHz, it affects all the TMI frequencies below 85.5 GHz, Experimental results from both
pure water vapor and moist air show more absorption than that contributed by the
microwave resonances alone. Therefore, it is necessary to add a so-called “continuum”
term to make up the difference between measured and calculated values for water vapor
absorption. Some hypotheses have been suggested to explain the origin o f the absorption
represented by the continuum term, but none of them has solid theoretical basis. It is
agreed upon all the hypotheses that the continuum term comes from two parts, i.e..
interaction between two water molecules (so-called “self-broadening” component) and
the interaction of a water molecule with a nitrogen or oxygen molecule (so-called
“foreign-broadening” component).
Liquid water can be divided into two categories: non-precipitating cloud liquid
water (NCLW) and rain. NCLW droplets have diameters less than 50 |im , much smaller
than microwave wavelength. Therefore, the Rayleigh approximation (Gunn and East.
1954) applies. In such case, the scattering effect is negligible and the absorption
coefficient is proportional to the cube of the diameter therefore is a function of the total
mass of the NCLW independent o f the cloud droplet size distribution.
On the other hand, rain drops have comparable size (0.2 mm to 3.8 mm in
diameter) with microwave wavelength, thus the Rayleigh approximation cannot be
applied. The scattering effect caused by the rain drops is large enough that it may no
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13
longer be ignored. Mie (1908) was the First to develop a method to calculate absorption
and scattering coefficients of dielectric spheres. Gunn and East (1954) applied Mie’s
theory to the context o f rain and cloud. With the scattering effect playing a role in the
microwave radiation transfer, mathematical solution for radiative transfer problems
becomes very complicated.
Wilheit et al. (1977) was the first to develop a radiative transfer model to
calculate the upwelling brightness temperatures under raining conditions. Using this
model, the R-T relationships for different microwave frequencies can be obtained.
Details about the model will be given in the next section.
2.2 Microwave radiative transfer model
The Wilheit et al. (1977) radiative transfer model, as illustrated schematically in
Figure 2, assumes a Marshall-Palmer drop size distribution (M-P DSD) (Marshall and
Palmer, 1948) from the surface to the freezing level. The freezing level is a selectable
parameter inside the model. The temperature lapse rate is a constant 6.5 K/km to
resemble the U. S. Standard Atmosphere. The relative humidity is assumed to be 80% at
the ocean surface and linearly increases to 100% at freezing level and remains at 100%
up to II km. Immediately below the freezing level, there is a non-precipitating cloud
layer o f 0.5 km which contains 0.5 g/m3 cloud liquid water.
Wilheit et al. (1977) divided the model atmosphere into 200 layers, each o f them
100 m thick. Integration of radiative transfer equation begins from the top o f the model
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14
ii
100% RH
Non-Precipitating
Cloud 0.5 g/m3
i i
1/2 km
Marshall Palmer
drop size distribution
Lapse Rate
6.5°C/km
80% RH
Ocean surface
Figure 2. Schematic diagram o f Wilheit et al. (1977) radiative transfer model.
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15
atmosphere with 2.7 K cosmic background and works its way down to and off the ocean
surface and then returns to the top o f the atmosphere.
In Wilheit et al. (1977), the effect o f oxygen absorption was evaluated using the
model o f Meeks and Lilley (1963) as modified by Lenoir (1968). The water vapor
absorption was calculated using the model o f Staelin (1966). An updated oxygen and
water vapor absorption procedure o f Rosenkranz (1993,1998) will be used in this study.
The emissivity o f the ocean surface is given by Fresnel relations using the Lane and
Saxton (1952) dielectric data.
According to Gunn and East (1954), the extinction (absorption and scattering)
cross section o f liquid water drop (or droplet) is given by
o>« = ~ 7 - R e i ( 2 /! + l)(a, +bn)
Ik
( 2 .6 )
<i” i
and the scattering cross section by
(2.7)
where a„ and bn are the magnetic and electric 2n pole coefficients. In the limit o f cloud
droplets and for the microwave wavelength, only the electronic dipole term b[ is
significant, so the scattering is negligible and the extinction cross section, equivalent to
the absorption cross section here, is reduced to
(2 .8)
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16
where r is the radius o f the droplet and k is the dielectric constant of liquid water. The
absorption cross section is proportional to the volume of the droplet therefore is a
function o f the total mass of the water.
The extinction coefficient of rain drops in the atmosphere can be related to the
extinction cross section of a single rain drop by
(2.9)
where N(r) is the number density of rain drops per unit size interval. The MarshallPalmer (M-P) drop size distribution (DSD) is the most widely used rain DSD and has the
form:
A (r) = (V0exp(-A r)
( 2 . 10)
where No = 0.16 cm4 and A = 81.56 R ^ 21. here R is the nominal rain rate in mm/hr.
The rain rate R . for a given rain DSD of N(r), is calculated ignoring the vertical
movement o f the air as
R ' ( r ) = ^ l t j V ( r ) r }N{r)dr
( 2 . 11 )
where V(r) is the speed of a rain drop with radius, r. aloft and can be obtained from the
speed at sea level by multiplying an adjustment factor (Beard, 1976. 1977).
2.3 Rainrate-brightness temperature (R-T) relationships
Using the model discussed in the last section, the upwelling brightness
temperatures for a set o f five atmospheres and a range of rain rates for nadir view at
19.35 GHz (wavelength 1.55 cm) were calculated by Wilheit etal. (1977). The results
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17
are shown in figure 3. The freezing levels were 1 ,2 ,3 ,4 and 5 km, respectively and the
surface temperatures were specified according to the lapse rate (6.5 K/km) in the model.
As rain rates increase the brightness temperatures for each freezing level increase rapidly
before they reach the saturation (maximum) values. The descending portions of the
curves are due to the strong backward scattering from large rain drops as predicted by
the M-P DSD.
The strong dependence of brightness temperatures on freezing levels indicated
the importance o f accurate estimation of rain depth. Observation from ground based
radar, aircraft and satellites have shown that the Wilheit et al. (1977) model generates
reasonable R-T relationships at 19 and 37 GHz at least for the freezing level around 4
km, which is very common in the tropical and mid-latitude regions.
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18
*
250
<
c
iu
a.
S
ui
5 km
r
o
c
o
4 km
3 km
150
2 k m ___ — '
1 km
--------- — . rjA.
100
1000
RAINFALL RATE (mm/hr)
Figure 3. Calculated brightness temperatures at 19 GHz for nadir viewing as a function
o f rain rate for five different freezing levels (from Wilheit et al., 1977).
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19
CHAPTER m
REVIEW OF PREVIOUS WORK AND METHODOLOGY
There have been many efforts to estimate rainfall from visible and infrared
measurements on satellite platforms (Barrett and Martin, 1981) because those
measurements are available on both the geo-synchronous satellites and other longer time
series polar-orbiting satellites and cover a larger fraction o f the earth surface. However,
these satellite-bome visible and infrared sensors can only measure physical properties at
the top o f the cloud.
Unlike the weak physical connection between visible and infrared measurements
and rainfall, microwave radiation interacts directly with the hydrometeors. Active
microwave measurements (radar) can provide information such as height o f freezing
level in precipitating clouds. Passive microwave radiometers measure the microwave
radiance integrated along the sensor’s view path.
Passive microwave rainfall estimations are generally based on either microwave
absorption (i.e., emission followed by KirchofFs law) or scattering (Wilheit, 1986). At
frequencies below 22 GHz, absorption is the dominant process. Above 60 GHz,
scattering dominates. Between 22 and 60 GHz either can be dominant depending on the
specific situation (Wilheit, 1986).
Emission-based methods relate rainfall to brightness temperature through the
absorption and emission by liquid phase o f cloud droplets and rain drops, and require a
radiometrically cold background such as oceans. Scattering-based methods, on the other
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20
hand, depend on the general cooling o f brightness temperatures in the high frequency
channels caused by the strong scattering o f ice in the upper part o f the precipitation over
land or oceans. Many scattering-based methods utilize empirical or semi-empirical
relationships through field measurements o f surface rain rates from coincident groundbased radar and upwelling brightness temperatures from airborne or space-bome
microwave radiometry.
3.1 Emission approach and some theoretical studies
As mentioned in the sections 2.1 to 2.3, Wilheit et al. (1977) was the first to
explore the emission-based method and develop a plane-parallel microwave radiative
transfer model quantitatively relating microwave radiance (i.e., brightness temperature)
to rainfall over oceans. In this model a Marshall-Palmer rain drop size distribution (M-P
DSD) (Marshall and Palmer, 1948) is assumed from the ocean surface to the freezing
level. The freezing level is a selectable parameter from I to 5 km and can specify the
water vapor content with the assumption o f the temperature lapse rate and relative
humidity in the model. The model output was verified against the measurements from
the Electrically Scanning Microwave Radiometer (ESMR) at 19 GHz on the Nimbus 5
satellite, the ground-based radar and the up-looking radiometers o f 19 and 37 GHz. It
was proved that the ESMR data were successfully mapped into rain rates.
Weinman and Guetter (1977) used a plane-parallel, ice-free microwave radiative
transfer model to calculate the upwelling brightness temperatures at 37 GHz above land,
rough and calm water surfaces. Their results showed that the microwave radiances
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21
emerging from rain clouds were weakly polarized. These weakly polarized microwave
radiances could be used to discriminate themselves from the markedly polarized
microwave radiances emerging from open water. They concluded that if polarization
effects were taken into account, the brightness temperatures at 37 GHz observed from
the ESMR on Nimbus 6 satellite could be interpreted as rain rates.
Wilheit et al. (1982) modified the model in Wilheit et al. (1977) by adding an ice
layer above the freezing level to account for the very low brightness temperatures they
observed at 92 and near 183 GHz from airborne radiometers flying over tropical storm
Cora. Their results show that the brightness temperatures at 19 GHz are not sensitive to
the ice layer thickness at low to medium rain rates. This insensitivity is consistent with
the assumptions o f Wilheit et al. (1977).
Instead o f using a Rayleigh scattering approximation in microwave radiative
transfer model as in Weinman and Guetter (1977), Huang and Liou (1983) considered
Mie scattering polarization effects in their model calculation for a precipitating
atmosphere. They computed the microwave radiances and the degree o f polarization for
microwave window frequencies o f 19.35,37.0 and 85.5 GHz over both land and ocean
surfaces. They showed that the models using simple Rayleigh scattering approximation
underestimate the upwelling microwave radiances at 85.5 GHz to a significant degree.
Also, they investigated the degree o f polarization as a function o f microwave
frequencies, rainfall intensity, rain layer thickness, emerging angle and surface
properties.
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22
Wu and Weinman (1984) computed multiple-frequency brightness temperatures
ranging from 6.6 to 183.0 GHz emerging from precipitating clouds. They also
investigated the effects o f non-spherical hydrometeors on microwave radiances
emerging from precipitating clouds. They concluded that the lower frequencies (< 37
GHz) microwave radiances interacted more with water drops at lower parts o f the clouds
while the radiances at higher frequencies were more sensitive to ice near the cloud top.
They also concluded that the differences in microwave radiances from vertical and
horizontal polarizations at 37 GHz at high rain rates as observed by the Scanning
Multichannel Microwave Radiometer (SMMR) on Nimbus 7 satellite (Spencer et al.,
1983c) could be attributed to non-spherical ice in upper parts o f precipitating clouds.
The above models are all assumed a plane-parallel geometry. The plane-parallel
assumption is unable to address extreme variability, both horizontally and vertically, o f
rainfall. Weinman and Davies (1978) developed a finite cloud model to account for the
horizontal variability but assumed vertical homogeneity. Kummerow and Weinman
(1988) developed a model for horizontally finite clouds containing both ice and liquid
hydrometeors. They found that footprint-averaged brightness temperatures from finite
clouds were considerably different from that o f the plane-parallel approximation.
Recently, it has been a trend to use more sophisticated cloud models to study
microwave radiative transfer. Mugnai and Smith (1988) and Smith and Mugnai (1988)
used results generated by a time-dependent, two-dimensional cumulus model as input for
microwave radiative transfer calculation. Their model provided water drop spectra in the
horizontal and vertical directions, and in the hydrometeor-size domain. The model
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generated large amount o f cloud water that reduced radiation from rain layer and surface
below. They showed that the importance o f cloud liquid water and cloud evolution on
the upwelling microwave radiances.
Adler et al. (1991) used output from a three-dimension, time-dependent cloud
model as input into a microwave radiative transfer model to study the relationships
between the upwelling microwave radiances and the surface rainfall rates and the
hydrometeor structures for a tropical oceanic squall line. They found that in the retrieval
o f rainfalls from passive microwave observations it needed to consider the effects o f the
evolution o f non-precipitating cloud water and precipitation-sized ice.
3.2 Scattering approach
Although scattering-based methods are a more indirect estimation o f rainfall than
that o f emission-based methods, different forms o f scattering-based methods have been
used over land and/or ocean since the seventies.
Based on a passive microwave technique developed by Savage and Weinman
(1975), Rodgers et al. (1979) used 37 GHz channel data from ESMR on Nimbus 6
satellite to delineate synoptic rainfall over land despite some ambiguity in distinguishing
between rainfall over land and wet land surfaces.
Wilheit et al. (1982) observed extremely low brightness temperatures, as low as
140 K, at 92 and 183 GHz when flying over decaying rain cells o f tropical storm Cora.
They showed that this feature could be explained by large content o f precipitation-sized
ice in the upper part o f the rain cloud.
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Rodgers and Siddalingaiah (1983) improved their algorithm using data from
SMMR on Nimbus 7 satellite by adding information from a lower frequency channel
(18.0 or 10.7 GHz) to discriminate rain from wet ground.
Spencer et al. (1983a) first examined the very cold brightness temperatures at 37
GHz from SMMR on Nimbus 7 satellite. They found that in every case o f very cold
brightness temperature at 37 GHz, a heavy thunderstorm was observed over the surface.
They suggested that the very cold brightness temperatures at 37 GHz observed by
satellite be caused by a layer o f ice particles in the upper part o f the storm and these very
cold brightness temperatures at 37 GHz were an indicator o f heavy rain over land.
Spencer et al. (1983b) then compared the microwave radiometer data from
SMMR on the Nimbus 7 satellite with WSR-57 weather radar in US National Weather
Service network. They found that brightness temperatures over land at 37 GHz were
approximately linearly related to rain rates, with the lower brightness temperatures
corresponding to the heavier rain rates, and this linear relationship held for rain rate up
to at least 40 mm/hr.
Spencer et al. (1983c) made comparisons between 37 GHz brightness
temperatures from SMMR on Nimbus 7 and rain rates derived from the WSR-57 radars
over the G ulf o f Mexico. They found that the differences between the observed
temperatures and those predicted by a plane-parallel model could be explained in most
cases by the inhomogeneity within the footprint o f the 37 GHz channels (so called
footprint-filling problem or beamfilling problem). They also observed 10 to 20 K
differences for brightness temperatures between vertical and horizontal polarization for
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37 GHz channel, even for large obscuring storms, over both land and sea. They
tentatively hypothesized that emission and scattering by non-spherical hydrometers were
responsible for the observed brightness temperature polarization differences.
Spencer (1986) developed a more quantitative scattering-based method for
measuring rainfall over the ocean with 37 GHz brightness temperatures in both vertical
and horizontal polarizations. Their technique requires an estimate o f brightness
temperatures for a hypothetical totally opaque cloudy sky, and brightness temperatures
for a cloud-free sky over ocean surface at 37 GHz for both vertical and horizontal
polarizations. They showed that rain rates derived by this approach for five cases o f
convective storm over G ulf o f Mexico were related to those inferred from coincident
radar.
Spencer et al. (1989) used higher 85.5 GHz frequency channels o f SSM/I to
retrieve precipitation over both land and ocean. The 85.5 GHz channels are highly
sensitive to scattering by precipitation, especially ice above the freezing level. This ice
scattering effect results in SSM/I 85.5 GHz brightness temperatures occasionally below
100 K. They showed that the polarization difference available at 85.5 GHz channels
from the SSM/I could be used to distinguish the low brightness temperatures from water
surface versus those from precipitation. They utilized 85.5 GHz polarization corrected
temperature (PCT), with a threshold o f 255 K, to delineate the precipitation.
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3.3 Other empirical approaches
Along with the emission-based and scattering-based methods mentioned before,
other empirical or semi-empirical methods have been developed for retrieval o f rainfall
from satellite microwave observations. Some examples are given below.
Prabhakara et al. (1986) derived an empirical relationship to estimate seasonal
precipitation over the global oceans using SMMR data on Nimbus 7 satellite. They first
developed a technique to infer liquid water content in the atmosphere using brightness
temperatures o f two low frequencies at 6.6 and 10.7 GHz. Since seasonal mean patterns
o f liquid water content in the atmosphere derived from SMMR data over global oceans
were closely related to climatological patterns o f precipitation, the seasonal precipitation
over global oceans was then inferred from the retrieved liquid water content.
Petty and Katrsaros (1990) attempted to use a non-dimensional, normalized form
o f 37 GHz brightness temperature polarization difference observed by SMMR on Seasat
and Nimbus 7 satellites to establish a semi-empirical relationship between passive
microwave observations and rainfall for the same type o f tropical cloud clusters they
encountered during winter MONEX in South China Sea. The SMMR data for three cases
o f tropical mesoscale cloud clusters were compared with visible and infrared imagery
from a geostationary satellite. For two o f the three cloud cluster cases, they made
quantitative comparisons between SMMR observations and radar reflectivity
observations. They concluded that the results o f comparisons were unsatisfactory due to
the strong spatial inhomogeneity in the tropical cloud clusters.
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Petty and Katsaros (1992) reported a much better agreement between rainfall
indicated by index o f normalized 37 GHz brightness temperatures differences, as
described in their previous paper, and by the radar reflectivity data. They suggested that
a larger amount and better quality o f radar data, lower rain rate, as well as characteristics
o f remnant stratiform precipitation occupying high portion o f rainfall events were keys
to the good agreement.
Prahakara et al (1992) developed an empirical method to use brightness
temperatures at 37 GHz channels from SMMR and SSM/I for retrieving rainfall over
oceans. They showed that it could be rectified for underestimation o f rain rate due to
saturation o f 37 GHz with aid o f two parameters that depend on the total water vapor
content in the atmosphere.
3.4 Statistical physical approach
Instead o f using a “traditional” emission or scattering approach or other empirical
approach, Kummerow et al. (1989) developed a multi-channel statistical physical
approach for retrieving rainfall. This approach utilizes the regression relationships
between multi-channel brightness temperatures and rain rates based upon theoretically
calculated data by a cloud radiative transfer model. This approach requires consistency
between the observed and the calculated brightness temperatures. The model simulates
the upwelling brightness temperatures in each pertinent channel for a variety o f
conditions such as characteristics o f precipitation and surfaces, cloud microphysics and
vertical hydrometeor profiles. Kummerow et al. (1989) and Kummerow et al. (1991)
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28
used this approach to retrieve rainfall rates and precipitation profiles, respectively, from
brightness temperatures observed by passive microwave radiometers flown on a high
altitude aircraft.
Kummerow and Giglio (1994a) improved this statistical physical approach by
explicitly selecting for diverse hydrometeor profiles. The residual error between
observed and calculated brightness temperatures is used to assess the uniqueness o f the
solution and to derive the expected accuracy from this retrieval technique. Kummerow
and Giglio (1994b) retrieved rainfall and the vertical structure o f rainfall over both land
and oceans from the SSM/I data based on the retrieval technique developed in
Kummerow and Giglio (1994a).
Using the similar statistical inversion techniques, Olson (1989) quantitatively
estimated rainfall rates o f two tropical cyclones using brightness temperature data from
the SMMR channels. Smith et al. (1992) and Mugnai et al. (1993) also retrieved rainfall
rates from space measured brightness temperatures by using a different threedimensional, time-dependent cloud model with detailed cloud microphysics.
3.5 Application to area-time-averaged rainfall rate
For the purpose o f climatological studies and improving our understanding o f the
global weather system, algorithms for estimating area-time averaged rainfall from
satellite measurements have been developed. These algorithms are generally based on
rain rate-brightness temperature (R-T) relationships derived from the aforementioned
methods and on other statistical analyses from area-time averaged rainfall data.
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29
Shin et al. (1990) retrieved seasonal 5° by 5° area-averaged rain rate over the
tropical oceans by using single channel microwave measurements horn the ESMR on
Nimbus 5 satellite. They first collected brightness temperatures into histograms for each
season and in each 5° by 5° cell. They showed that each histogram is composed o f a
normally distributed background noise plus a skewed rain distribution and they filtered
out non-raining background brightness temperatures from the raw brightness
temperature histograms. After removing the non-raining background brightness
temperatures, the rain rates for the remaining raining histogram were obtained based on
the R-T relationships o f 19 GHz horizontal channel generated from a radiative transfer
model very similar to that o f Wilheit et al. (1977).
Wilheit et al. (1991) developed an algorithm for retrieval o f monthly rain totals
over oceans from microwave radiometric measurements. In their algorithm, histograms
o f brightness temperature and linear combinations o f brightness temperature are
accumulated for the month over each box. The histograms o f 19.35 GHz and 22.235
GHz brightness temperatures are used to generate an average freezing level for the
raining portions o f the month. Using this freezing level and a first guess set o f
parameters describing a mixed lognormal distribution o f rain, a theoretical histogram o f
a particular linear combination o f brightness temperatures is generated. The lognormal
parameters are adjusted until satisfactory agreement between the computed and observed
histogram is obtained. The rain total can be derived directly from the lognormal
parameters. The comparison with climatological and GATE radar observations
suggested that the estimates o f rainfalls using this algorithm were reasonable.
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30
Berg and Chase (1992) estimated monthly, seasonal, and annual oceanic rainfall
using data from SSM/I. The instantaneous rainfall rates were obtained by using a linear
regression o f brightness temperatures from 19.35 (H), 22.235 (V) and 37 (V, H)
channels. This linear regression relationship was derived from the Hughes D-matrix
algorithm developed by Environmental Research and Technology, Inc. The
instantaneous rainfall rates for each 1° by 1° cell over oceans for each month was fitted
into a lognormal distribution by using a maximum likelihood method to get the mean
rainfall rate. This mean rainfall rate was easily converted into a monthly, seasonal and
annual oceanic rainfall amount.
Hong (1994) and Hong et al. (1997) modified the Wilheit et al. (1991) algorithm
by directly applying the R-T relationships based on the Wilheit et al. (1977) model to
obtain histogram o f monthly rain rates for a grid box over a month period. The
histogram is then fitted into a lognormal distribution by using the maximum likelihood
estimation method (MLEM) when there is enough raining samples. When there is not
enough samples, simple summation method is used. The monthly totals o f rain for a grid
box over a month period are obtained by multiplying the total hours o f a month and the
average rain rates. The average o f rain rates is either the arithmetical mean, when there is
not enough raining samples, or the statistically calculated from the fitted lognormal
distribution.
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31
3.6 Previous work directly related to the current research and methodology for this
study
Although the algorithm o f Wilheit et al. (1991) has been very successful, it has
many shortcomings. The use o f a single freezing level for the entire month ignores dayto-day variability. It does not use the 37 or 10 GHz channels that are available on TMI.
These channels, if used properly, will expand the measurement dynamic range to higher
and tower rain rates.
Hong (1994) refined the algorithm o f Wilheit et al. (1991) while using the same
underlying radiative transfer model and the beamfilling correction. The algorithm o f
Hong (1994) differs from that o f Wilheit et al. (1991) in three ways. First, it converted
brightness temperatures into rain rates and freezing levels on a pixel-by-pixel basis
before accumulation into histograms. Secondly, it took limited dynamic range o f the
measurements into account rather than assuming availability o f rain rates over whole
dynamic range. Last, the 37 GHz observations were used, even if in an approximate
manner, to extend the dynamic range to somewhat lower rain rates.
In the Third Precipitation Inter-comparison Project (PIP-3) workshop, the
monthly rain totals submitted by TAMU (contributed partially by this study) were about
40% lower compared with the results from operational algorithm based on the algorithm
o f Wilheit et al. (1991) and from the Pacific atoll data. The TAMU algorithm was
grossly based on Hong (1994), except that the beamfilling correction formulas were
from Wang (1996). However, Wang (1996)’s beamfilling corrections should not
introduce large discrepancies. It is noticed from the processing o f PIP-3 data set that
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32
there were some cases that the MLEM was numerically unstable when the rain rates
distributions were not close to the lognormal distribution.
In order to find the reason(s) why the outcome o f the TAMU algorithm differs so
much from that o f the operational algorithm based on Wilheit et al. (1991) and that o f
the Pacific atoll data, A series tests for the algorithm were taken.
Redmond (1998) used a minimum chi-square method (MCSM) to replace the
MLEM in the statistical component o f Hong (1994) algorithm. It is found that the
lognormal fit with the MCSM is more stable numerically than that with the MLEM, but
the two different fits give almost the same results.
Although it did not find a reason in the statistical part o f Hong (1994) algorithm
for the underestimation o f monthly rain totals, Redmond ( I998)’s work pointed to a
direction that potential problems in the underlying radiative calculation and/or SSM/I
instrumentation calibration.
In terms o f the microwave radiative transfer calculation, in the atmosphere only
molecular oxygen, water vapor and liquid water interact with microwave radiation at
frequencies below 22 GHz water vapor absorption line to a significant degree. At
frequencies between 50 to 70 GHz and near 119 GHz, the absorption due to molecular
oxygen is important and these frequencies are used for temperature sounding but are not
often used for rainfall sensing. For the purpose o f this work, the absorption due to
molecular oxygen is a minor correction, which is needed for quantitative estimation o f
rainfall, thus will not lead to big discrepancies.
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33
Several water vapor absorption procedures are available and are tested for
calculating water vapor absorption o f microwave radiation in this study. Waters (1976)
used a kinetic line shape for the resonance for water vapor absorption calculation. The
empirical correction for the difference between the measured and the calculated values
for water vapor absorption in the widow regions is from Gaut and Reifenstein (1971).
Rosenkranz (1993,1998) gave a water vapor absorption model that uses the Van Vleck
and Weisskopf line shape for the resonance. The line strengths are from the Pickett et al.
(1998) compilation. The line widths are from Leibe and Dillon (1969) and Bauer et al.
(1989). The continuum, which corrects the difference between the measured and the
calculated values, is from Liebe (1987, 1989).
Results from the Texas A&M precipitable water (PW) retrieval algorithm
(Thomas-Stahle, 2001) showed that a water vapor absorption correction factor o f 0.95 is
needed to account for the overestimation o f the retrieved PW content compared with that
from the radiosonde data.
Based on the R-T relationships generated from this study, Bellows (1999)
examined the effects o f different water vapor absorption procedures and the water vapor
absorption correction factor on the freezing level retrieval and thereby on the overall rain
retrievals. The freezing level is a very important parameter in the estimation o f the rain
rate from brightness temperatures and is one o f the parameters in Wilheit et al. (1977)
model. The launch o f the TRMM satellite provided an opportunity to check the freezing
levels retrieved from the TMI data based on the Wilheit et al. (1977) model against those
inferred from the bright-band in the PR data. Bellows (1999) showed that the two
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34
freezing levels were in close agreement and that the two water vapor procedures
(Rosenkranz and Waters) and the precipitable water correction factor had little impact on
the overall rain retrievals.
Liquid water can be divided into two categories: non-precipitating cloud liquid
water (NCLW) and rain. According to the results o f many field experiments (TCM-90,
TOGA- COARE) (McGaughey and Zipser, 1996), the NCLW contents are too high in
the Wilheit et al. (1977) model. The effect o f this high NCLW content on rainfall
retrieval will be discussed later. In this study, the updated model will reduce NCLW to
the extent that is consistent with the field observations.
The most widely used rain drop size distribution (DSD) is the M-P DSD. Wilheit
et al. (1977) showed that the upwelling brightness temperatures at 19 GHz was relatively
insensitive to the DSD used in the model. Tesmer and Wilheit (1998) suggested that an
inverse exponential DSD similar in form to the M-P DSD. The new DSD contains a
greater number o f small precipitation particles and fewer larger precipitation particles
than that o f the M-P DSD. For example, the new DSD has an intercept (No) value that is
five times as that o f the M-P DSD. The impact o f the new DSD proposed by Tesmer and
Wilheit (1998) on the R-T relationships for each TMI channel will be examined in this
study later.
Using the Wilheit et al. (1977) model with the updated water vapor absorption
procedure (Rosenkranz, 1993,1998) and the water vapor absorption correction factor o f
0.95 or its equivalent (Thomas-Stahle, 2001), a series o f R-T relationships for each TMI
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35
channel have been generated (this study). The upgraded model will be referred hereafter
as TAMU model.
In this study, the combination o f R-T relationships for 19 and 21 GHz will be
used to obtain freezing level information for each pixel. The freezing level information
will be used to solve for rain rates corresponding to the 10,19 and 37 GHz brightness
temperatures, respectively, on a pixel-by-pixel basis. Those rain rates will be adjusted
according to their corresponding beamfilling corrections (Wang, 1996), again, on a
pixel-by-pixel basis, and then accumulated to form rain histograms for each channel over
each 5° by 5° grid box.
The rain histogram for each channel will be processed with a new rainfall
retrieval algorithm (this study). This new algorithm is modified so as not to have a cut­
off at zero rain rate. Even if physically nonsense, negative rain rates are mathematically
possible and could result from biases in the instrument calibration and/or the radiative
transfer model (RTM) used. However, in reality the most probable rain rate is zero even
in very rainy areas. The most probable rain rate retrieved by this modified algorithm
should have a negative offset since too much non-precipitating cloud liquid water
(NCLW) was assumed in the model. The NCLW will be reduced in the model according
to the results o f many field experiments (TCM-90, TOGA-COARE) and then retrieved
most probable rain rate should be very close to zero. Any remaining offset o f rain rate
will be shifted to zero accordingly.
There are many advantages o f using the three different channels, i.e., 37,19 and
10 GHz channels, together to retrieve rainfall. According to the R-T relationships
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36
generated by the TAMU model, the 10 GHz channel has the ability to measure very high
rain rates (as high as 50 mm/hr) while the 37 GHz channel is sensitive to the low rain
rates. This sensitivity can be used to differentiate low rain rates from non-precipitating
cloud liquid water (NCLW). By using the rain rates retrieved from these different
channels together, we achieve high sensitivity to the low rain rates where needed and
also obtain reliable measurements at high rain rates up to 50 mm/hr. This will expand the
measurement dynamic range to the higher and lower rain rates and potentially reduce or
eliminate the need to use the assumption o f lognormality (i.e., lognormal fit) to fit for the
low and high rain rate portions.
However, before we can utilized the rain rates resulting from those three different
channels, the rain rates derived from 37 and/or 19 GHz channels need to be “smoothed”
to the resolution o f the 10 GHz channel so that the rain rates retrieved from different
channels represent the same area. The work o f smoothing the rain rates retrieved from
the higher resolution channels to that o f the 10 GHz channels has been done by Lee
(2001). A brief description o f the smoothing procedure will be given later.
The algorithm will choose the highest frequency possible among the 37,19 and
10 GHz channels. This will give us the greatest usable sensitivity to rain. If any o f the
brightness temperature at 37 or 19 GHz channel exceeds a threshold value, typically
around 255 K (Lee, 2001), the measurement is considered to be saturated and therefore
unreliable. Saturated values at 10 GHz are extremely rare and, in any case, there is no
lower frequency channel available on TMI.
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37
Finally, a single combined rain histogram, with offset being corrected (a method
for correcting offset will be discussed later in section 6.6), will be obtained for each 5°
by 5° grid box. The average o f the histogram, obtained either by simple arithmetical
calculation or by a statistically fitted lognormal distribution depending on the specific
situation, will be easily converted into monthly rain total by multiplying the number o f
hours in that month. In fact, Lee (2 0 0 l)’s work has shown that the lognormal assumption
has no advantage over the simple arithmetical summation (direct summation) for the
monthly rainfall estimation using the TMI data. Therefore, the direct sum method will be
used in this multichannel rainfall retrieval algorithm.
The new algorithm will be ‘Validated”, in the last part o f this study, against the
rain gauge data from atoll. The new algorithm should provide us more accurate rainfall
totals.
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38
CHAPTER IV
THE TRMM MICROWAVE IMAGER AND PRECIPITATION RADAR
The Tropical Rainfall Measuring Mission (TRMM) satellite was launched in
November o f 1997. The primary goals o f TRMM are to measure rainfall and latent heat
released through precipitation in the tropical and subtropical regions. The TRMM
satellite flies at a height o f 350 km with an inclination angle o f 35 degrees. The rainfall
measuring instruments on TRMM are the TRMM Microwave Imager (TMI) and the
Precipitation Radar (PR). The Visible and Infrared Radiometer System (VIRS) is also
included in TRMM. In addition, the TRMM satellite carries two related Earth Observing
System (EOS) instruments in the Clouds and Earth’s Radiant Energy System (CERES)
and the Lightning Imaging System (LIS).
The TMI is a five-frequency, nine-channel passive microwave radiometer based
on the Special Sensor Microwave/Imager (SSM/I) flown on the Defense Meteorological
Satellite Program (DMSP) satellite since 1987. The SSM/I has seven different channels:
19.35 GHz vertical and horizontal, 22.235 GHz vertical, 37.0 vertical and horizontal,
85.5 vertical and horizontal. Changes from the SSM/I to the TMI include the addition o f
two channels (10.65 GHz vertical and horizontal polarization) and a shift o f the water
vapor channel from the absorption line center at 22.235 GHz to 21.30 GHz to avoid
saturation o f this channel in the tropics. Compared with the SSM/I, the TMI has higher
spatial resolution due to the lower orbit o f the TRMM satellite (350 km altitude
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39
compared to 850 km o f the DMSP satellite).
The TMI antenna is an offset paraboloid and it continuously rotates about its
nadir axis at a constant speed o f 31.6 rpm and has a conical scanning geometry. The
rotation draws a “circle” on the Earth’s surface. It takes data during either the forward or
backward 130° arc and has a swath width o f 759 km. The remaining sector is used for
calibration and other instrument housekeeping purposes. The TMI receives microwave
radiation from 49° off its nadir. Due to the Earth’s spherical shape the local incidence
angle on the surface is 52.8°.
Table I presents spatial resolution for each TMI channel. The spatial resolution is
important for beamfilling problem, which will be explained later, therefore important for
the algorithm.
The PR is the first precipitation radar flown on a spacecraft. It is a 13.8 GHz
radar that electronically scans a swath width o f 215 km every 0.6 s. The swath width o f
the PR is about one third o f that o f the TMI.
Detailed information on TMI, PR and the other instruments on board the TRMM
satellite can be found in Kummerow et al. (1998) and can also be referred to web site at:
http://trmm.gsfc.nasa.gov/overview_dir/instrumentfacts.htmI.
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40
Table 1. Spatial resolution for each TMI channel
Channel
Number
Center
Frequency (GHz)
Polarization
Spatial Resolution
(km)
I
10.65
V
63x37
2
10.65
H
63x37
3
19.35
V
30 x 18
4
19.35
H
30 x 18
5
21.3
V
23 x 18
6
37.0
V
16x9
7
37.0
H
16x9
8
85.5
V
7x5
9
85.5
H
7x5
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41
CHAPTER V
ANALYSIS
In this chapter, several issues that affect rainfall retrieval algorithm are
addressed. For instance, the impact o f DSD on R-T relationships will be examined. It
will be illustrated how freezing level is retrieved. Reason(s) responsible for the offset of
the most probable retrieved rain rate away from zero will be investigated and a method
for correcting this offset will be given.
5.1 Impact of DSD on R-T relationships
As mentioned before, the most widely used rain drop size distribution (DSD) is
the M-P DSD. Wilheit et al. (1977) showed that the upwelling brightness temperatures at
19 GHz was relatively insensitive to the DSD used in the model. Tesmer and Wilheit
(1998) suggested that an inverse exponential DSD similar in form to the M-P DSD in a
hybrid cloud-microwave radiative model they developed. The new DSD contains a
greater number o f small precipitation particles and fewer larger precipitation particles
than that o f the M-P DSD. For example, the new DSD has an intercept (N0) value that is
five times as that o f the M-P DSD. In this section, the impact o f this new DSD on R-T
relationships is examined by replacing the M-P DSD in TAMU model with this new
DSD.
Figure 4 shows a set o f R-T relationships generated by the TAMU model with
this new DSD and with the M-P DSD at 19 GHz vertical polarization for an incident
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42
R - T R e l a t i o n s h i p f o r 1 9 v GHz
280
260
km
240
3 km
220
200
180
0.1
1.0
10.0
100.0
Ram Rate ( m m / h r )
Figure 4. R-T relationships calculated by the TAMU model with the M-P DSD (solid
lines) and with the DSD proposed by Tesmer and Wilheit (1998) (dashed lines) for 19
GHz vertical polarization channel at an incidence angle o f 52.8° (TMI incidence angle).
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43
angle o f 52.8 degrees (TMI incident angle). It clearly shows that the TAMU model with
this new DSD produces warmer maximum upwelling brightness temperatures and
saturated slower than that with the M-P DSD for all freezing levels. Similar patterns can
be seen in the other channels (not shown) also. Note that this new DSD is an extreme
change from the M-P DSD and at most valid in the upper part o f the rain column in a
few cases. However, even if this new DSD is assumed through the whole rain column,
the change o f rain rate for a given brightness temperature is still within a factor o f two.
The insensitivity o f the calculated upwelling brightness temperature to the assumed DSD
is consistent with the pertinent result o f Wilheit et al. (1977). The new DSD proposed by
Tesmer and Wilheit (1998) is necessary, in some cases, to account for the facts that the
observed brightness temperatures are greater than the maximum brightness temperatures
calculated with the M-P DSD for a given freezing level in the model. But the new DSD
will not be used instead the M-P DSD will be used in the underlying radiative transfer
calculation in the TAMU model.
5.2 Freezing level retrieval
As point out before, the freezing level (FL) is a very important parameter in the
estimation o f rain rate from brightness temperatures. For example, for a raining-cloud
with a height o f FL at 5 km, underestimation (overestimation) o f FL by 0.5 km will
result in a 10% overestimation (underestimation) o f the rain rate. Following the approach
used by Hong (1994) but using 21 GHz vertical channel o f TMI instead o f 22 GHz
vertical channel o f SSM/I, the combination o f the R-T relationships for 19 and 21 GHz
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vertical channel will be used to obtain the FL information on a pixel-by-pixel basis. This
approach can be illustrated graphically with figure 5, which is a Freezing Level - Rain
Rate - Brightness Temperature (FLRRT) chart portraying a two-dimensional brightness
temperature histogram for 19 and 21 vertical channels over a 5° by 5° grid box in the
tropical central Pacific Ocean. It is shown clearly that most pixels do not contain any
rain and the raining pixels are mainly located along the freezing levels between 4 and
5 km.
5.3 Offset of rain rate
According to the results o f many field experiments (TCM-90, TOGA-COARE),
the NCLW content in the Wilheit et al. (1977) model is too high. The NCLW content
affects the upwelling brightness temperatures at low rain rates for each channel,
especially for the 37 GHz channel.
In order to show the effect o f the NCLW on the rainfall retrieval, we modified
the rainfall retrieval program so as not to have a cut-off at zero rain rate, i.e., even if
physically nonsense, negative rain rates are mathematically possible. The negative rain
rates could result from biases in the instrument calibration and/or the radiative model
used as well as from instrumental or geophysical noise. Figure 6 shows that many
negative rain rates occur in the rain rates retrieved from the 37 GHz vertical channel. In
fact, the most probable rain rate appears to be about -0.3 mm/hr. Similar offsets can be
found in the histograms o f retrieved rain rates from the 19 and 10 GHz vertical channels,
as showed in figure 7 and figure 8. In reality the most probable rain rate is zero even in a
very rainy area. Wilheit et al. (1991)’s algorithm absorbed this bias by solving for the
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45
280 —r
FLRRT for the TAMU Model
260
3(km)
240
>
C4
ft
220
0 ( m m /h r )
200
180 L_
180
165 - 170 W
Feb. 1 9 9 8
200
240
220
260
280
B19V
Occurrence
Figure 5. Two-dimensional histogram o f brightness temperatures from 19 and 21 GHz
vertical polarization channels over a 5° by 5° grid box in the tropical central Pacific
Ocean. Isolines of freezing level (solid lines) and rain rate (dashed lines) based on R-T
relationships are overlaid.
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46
HISTOGRAM OF RAIN RATE: 37v
0
4
165 -
170 W
Feb., 1998
0
0°
-5
0
5
10
15
20
Rain Rate (m m /hr)
Figure 6. Histogram o f occurrence o f rain rates derived from the 37 GHz vertical
polarization channel o f TMI. The non-precipitating cloud liquid water (NCLW) content
in the underlying radiative transfer model is the same as that in the Wilheit et al. (1977)
model.
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47
HISTOGRAM OF RAIN RATE: 19v
0
4
165 -
170 W
O ccurence
Feb., 1998
0
10°
-5
0
5
10
15
20
Rain Rate (m m /h r)
Figure 7. Same as in figure 6 except from the 19 GHz vertical polarization channel o f
TMI.
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48
HISTOGRAM OF RAIN RATE: 10v
-5
0
5
10
15
20
Rain Rate (m m /hr)
Figure 8. Same as in figure 6 except from the 10 GHz vertical polarization channel o f
TMI.
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49
non-raining brightness temperatures as part o f the fitting process. In a rain-rate based
algorithm it must be solved for the offset by finding the apparent most probable rain rate
and then shifting the retrievals accordingly. As mentioned before in this section, the
negative rain rates could result from biases in the instrument calibration and/or the
radiative model used. However, the common offsets among all three channels are
unlikely to be an instrument calibration error thus the most likely reason for this offset
error is the non-precipitating cloud liquid water (NCLW) assumption in the model.
The updated TAMU model will reduce the NCLW from 0.5 g/m3 to 0.1 g/m3 to
be consistent with the field observations (McGaughey and Zipser, 1996). The reduced
NCLW content has little impact on the retrieved FL since the updated FLRRT chart will
follow the general trend o f the original FLRRT chart and the retrieved FL will remain
the same as the original one (not shown). After reducing the NCLW content, the
retrieved most probable rain rates for all three channels are very close to zero (figure 9
through figure 11). Any remaining offset o f the rain rate will be shifted to zero
accordingly in the algorithm.
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50
HISTOGRAM OF RAIN RATE: 3 7 v
5 - 10 S
165 -
170 W
Feb., 1998
0
10
0
-5
0
5
10
15
20
Rain Rate (m m /h r)
Figure 9. Histogram o f occurrence o f rain rates derived from the 37 GHz vertical
polarization channel o f TMI. The non-precipitating cloud liquid water (NCLW) content
in the underlying radiative transfer model is only one fifth o f that in the original Wilheit
et al. (1977) model.
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51
HISTOGRAM OF RAIN RATE: 19v
1 0 5
1
'
'
1
1
!
■
■
■
■
i
■
1
1
1
i
^
1
5 -
1
■'
i
1
1
10 S
165 - 170 W
O ccurence
Feb., 1998
-5
0
5
10
15
20
Rain Rate (m m /h r)
Figure 10. Same as in figure 9 except from the 19 GHz vertical polarization channel o f
TMI.
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52
HISTOGRAM OF RAIN RATE: 10v
0
4
165 -
170 W
O ccurence
Feb., 1998
0
10
0
-5
0
5
10
15
20
Rain Rate (m m /hr)
Figure I I. Same as in figure 9 except from the 10 GHz vertical polarization channel o f
TMI.
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53
CHAPTER VI
ALGORITHM
In this chapter, an algorithm developed here at the Microwave Remote Sensing
Group (MRSG) o f Texas A&M University (TAMU) will be presented. It is worth
noting that this algorithm is developed through collected efforts at the MRSG at TAMU.
Results from many people’s research, i.e., Lee (2001), Thomas-Stahle (2001), Bellows
(1999) and Redmond (1998), especially Lee (2001), along with this research, have been
incorporated into this algorithm.
6.1 R-T relationships
Using the TAMU model, i.e., Wilheit et al. (1977) model with the updated water
vapor absorption procedure (Rosenkranz, 1993,1998) and the water vapor absorption
correction factor o f 0.95 (Thomas-Stahle, personal communication) or its equivalent, a
series o f R-T relationships for each TMI channel are generated in this study. It is worth
noting, and has been pointed out in section 5.3, that this TAMU model has the NCLW
content only one fifth o f that in the original Wilheit et al. (1977) model. This reduction
o f the NCLW content in the TAMU model is forced by and consistent with the field
observations.
Figure 12 shows how brightness temperatures vary with rain rate and freezing
level for 37 GHz vertical polarization at an incidence angle o f 52.8 degrees (the TMI
incidence angle). It can be seen that the brightness temperatures saturate at around
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54
R —T R e l a t i o n s h i p f o r 3 7 v GHz
280
260
lb
(K)
240
220
200
180
0.1
1.0
10.0
100.0
Rain R a t e ( m m / h r )
Figure 12. R-T relationship calculated by the TAMU model for the 37 GHz vertical
polarization channel at an incidence angle o f 52.8° (TMI incidence angle).
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55
4 mm/hr for a freezing level o f 4 km for the 37 GHz vertical channel. Similar pattern
can be found in figures 13,14 and 15 with different channels.
Following Wilheit et al. (1991), but with slight modification to add a constant 7/,
the best-fit analytical function for the R-T relationships generated by the TAMU model
for the four different vertical channels (10.65,19.35,21.0 and 37.0 GHz vertical
channels) is given as:
r ( r ) = r 0 +(7; - r o) [ l - e x p ( - - ) ] - a V7
rc
(6.1)
where T(r) is brightness temperature, r is rain rate, rc=b/Fc, To is background brightness
temperature which can be expressed as:
f 0 =ta + tb -F Jr t c - F l
The parameters ta, tb, tc, Ti, a. b and c are constants which vary with frequency,
polarization and view angle. Those constants are listed in Table 2. Furthermore, figures
16,17,18 and 19 show the best-fit for the R-T relationships for the 37, 21, 19 and 10
GHz vertical polarization as shown in figures 12 through 15, respectively. Note that the
poor fit at very high rain rates for each channel does not affect our retrieval since we do
not use it for our retrieval algorithm and also radiative transfer models (including the
TAMU model used here) are very uncertain in that range o f rain rates.
The best fits for the R-T relationships are used in this study to generate two
look-up tables: freezing level table and rain rate table, respectively. These look-up tables
can save a lot o f computation time when used in the rainfall retrieval algorithm.
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56
R - T R e l a t i o n s h i p f o r 21 v GHz
280
260
Tb
(K )
240
220
200
180
0.1
1.0
10.0
100.0
Rain Rate ( m m / h r )
Figure 13. Same as in figure 12 except for the 21 GHz vertical polarization channel.
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57
R - T R e l a t i o n s h i p f o r 1 9 v GHz
280
5 km.
260
Tb
(K )
240
3 km
220
200
180
0.1
1.0
10.0
100.0
Rain Rate ( m m / h r )
Figure 14. Same as in figure 12 except for the 19 GHz vertical polarization channel.
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59
Table 2. Constants in R-T relationships
10.65 GHz
vertical channel
19.35 GHz
vertical channel
21.3 GHz
vertical channel
37.0 GHz
vertical channel
ta
160.0
185.0
183.0
217.0
tb
1.75
-0.40
10.70
-4.00
tc
0.45
1.79
0.90
1.75
T,
320.0
295.0
292.0
284.0
a
4.96
5.40
5.44
9.06
b
52.36
20.59
20.77
7.20
c
0.819
1.13
1.30
1.35
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60
R - T R e l a t i o n s h i p f o r 3 7 v GHz
280
260
;m
240
n
i—
220
200
180
0.1
1.0
10.0
100.0
Rain Rate ( m m / h r )
Figure 16. Best-fit (dashed lines) for the R-T relationship for the 37 GHz vertical
polarization channel. The poor fit at high rain rates does not affect rain retrieval in the
current algorithm.
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61
R - T R e l a t i o n s h i p f o r 21 v GHz
280
260
240
I—
220
200
180
0.1
1.0
10.0
100.0
Rain Rate ( m m / h r )
Figure 17. Same as in figure 16 except for the 21 GHz vertical polarization channel.
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62
R - T R e l a t i o n s h i p f o r 1 9 v GHz
280
5 km.
260
Tb
(K )
240
'3 km
220
200
180
0.1
1.0
10.0
100.0
Rain Rate ( m m / h r )
Figure 18. Same as in figure 16 except for the 19 GHz vertical polarization channel.
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&
64
6.2 Beamfilling error
BeamfiUing (or footprint-filling) error refers to the retrieved rainfall bias arising
from the combined effects o f non-uniform rainfall distribution within the field o f view
(FOV) o f a microwave radiometer and the non-linear relationship between rain rate and
brightness temperature over oceans. This beamfilling error causes an underestimation of
retrieved footprint-averaged rainfall rate compared to the true rain rate. Although the
beamfilling error for each FOV varies, the ensemble mean beamfilling error is rather
stable.
Previous studies (Short, 1990, Chiu et al., 1990) suggested a multiplicative
constant that depends on the spatial resolution o f FOV be correction o f beamfilling error.
However, the beamfilling error corrected this way is based on a two-dimensional rainfall
field and ignored the variability o f vertical structure and the structure along a sloping
radiometer view path. Recent study o f Wang (1996), based on a three-dimensional radar
rainfall data collected during TOGA-COARE, suggested a much lower beamfilling error
than that o f the previous two-dimensional studies. Furthermore, Wang (1996) gave
beamfilling error correction as function o f the given radiometer parameters (i.e.,
frequency, polarization, view angle and spatial resolution) and the freezing level. The
beamfilling error correction formula o f Wang (1996) based on path averaged rainfall rate
is given as:
BCF = 1.0+(0.478 In(5) - 0.787) / rf
where BCF is the beamfilling correction factor, S is the longer radius o f the footprint
size in km, and rc is the characteristic rainfall rate representing the dependence o f
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65
brightness temperature on the frequency, view angle and freezing level as given in
equation 6.1.
6.3 Selection of saturation criteria
This section is based on the work o f Lee (2001). It is worth to point out that there
are no pre-determined saturation criteria for a FOV for any specific channel. Near the
peaks o f the R-T curves for different freezing levels o f each channel, the sensitivity o f
brightness temperature to rain rates is greatly reduced. The algorithm will have to avoid
retrieving rain rate near those peaks o f R-T curves as much as possible. Furthermore,
when rain rate exceeds the peak o f the R-T curve, the brightness temperatures, due to
back scattering o f large rain drops, will drop below its maximum value. Therefore, it is
hard to tell the correct rain rate by simply looking at the R-T curve since one brightness
temperature would have two corresponding rain rates. One potential solution for this
double-value problem is to use polarization difference since theoretical model studies
(Weinman and Guetter, 1977, Huang and Liou, 1983, Wu and Weinman, 1984) indicates
effective de-polarization o f brightness temperatures from vertical and horizontal
polarizations when precipitating cloud is over a highly polarized ocean surface. Huang
and Liou (1983) also gave the degree o f polarization as a function o f microwave
frequencies, rainfall intensity, rain layer thickness, emerging angle and surface
properties (section 3.1).
Redmond (1998) used polarization difference, 5 K. for the 19 GHz and 10 K. for
the 37 GHz, with the SSM/I data to determine whether those channels are saturated. He
found virtually the same monthly rain totals as without using polarization difference.
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66
Recall the observation o f Spencer (1983c) that there were 10 to 20 K. differences for
brightness temperatures between vertical and horizontal polarization for 37 GHz
channel, even for large obscuring storms, over both land and sea. This brightness
temperature polarization difference made Spencer (1983c) tentatively hypothesized that
the emission and scattering by the non-spherical hydrometers were responsible for the
observed brightness temperature polarization differences (section 3.2).
For developing the current algorithm, brightness temperatures were selected as
saturation criteria (Lee, 2001). Although it is hard to distinguish whether or not
saturation occurs by simply looking at the brightness temperatures since the R-T
relationships for each microwave channel has limited dynamic range and these R-T
relationships are also a function o f freezing level. The final version o f the algorithm, the
operational one, will use both polarization difference and brightness temperatures as
saturation criteria since it will be slightly more robust than that o f brightness
temperatures along. However, the saturation criteria based on brightness temperatures
are adequate for the present exploration.
Lee (2001) used the 37 GHz brightness temperature values 2 5 5 ,2 6 0 ,2 6 5 ,2 7 0
and 275 K and the 19 GHz brightness temperature values 255,260 and 265 as threshold
brightness temperatures and tested eight different combinations o f brightness
temperatures to determine the most reasonable saturation criteria. He found that the 255255 K. combination (37 and 19 GHz vertical channels) gave the highest monthly rain
total among the eight different combinations. Also in a very heavy raining case, the
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67
255-255 K combination handles the saturation correctly while others do not. Therefore,
the 255-255 K combination will be chosen as saturation criterion for the current
algorithm.
6.4 Smoothing of rain rates
This section is based on the work o f Lee (2001). There are many advantages o f
using three different channels, 37,19 and 10 GHz channels, together to retrieve rain
rates. In particular, it will expand the measurement dynamic range to the higher and
lower rain rates and potentially reduce or eliminate the need to use the assumption o f
lognormality (i.e., lognormal fit) to fit for the portions o f the low and high rain rates
(section 3.6).
However, before we can utilize rain rates resulting from those three different
channels, there are some issues need to be resolved. In particular, the spatial resolution is
inversely proportional (proportional) to the frequency (wavelength) which varies from
37.0 to 10.65 GHz (0.8 to 2.8 cm) over TMTs low frequency channels. It would be
erroneous simply to combine the rain rates from these three different channels since the
rain rates derived from these three channels represent different areas o f the Earth.
It is a temptation to make low resolution higher. However, this will cause
excessive noise that makes the prospect upgrading unlikely. To handle this problem, the
higher resolution (37 or 19 GHz) channel can be reduced to the same resolution as that
o f the 10 GHz channel. This process is referred as smoothing. Since the rainfall is highly
variable within the FOV even for the highest resolution channel used (37 GHz channel),
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68
smoothing the brightness temperatures straightforward to the resolution o f the 10 GHz
channel would exacerbate the largest source o f error, the beamfilling error, in the
measurement. In the current algorithm, we have chosen to retrieve the rain rate at the
highest resolution possible for the three channels and then smooth the resulting rain rates
to the resolution o f the 10 GHz channel. The work for smoothing the resulting rain rates
from the high resolution channels to that o f the 10 GHz channel is done by Lee (2001)
and is explained in detail also in Lee (2001). A brief description o f the smoothing
procedure is given as follows. First, local latitude and longitude o f each pixel (37 or 19
GHz channel) are converted into the TMI antenna coordinates. These can be derived
from the TMI scan geometry, i.e., conical scan with 52.8° incident angle and elliptic
shape o f footprint on the surface o f the Earth for each channel (37,19 and 10 GHz).
Secondly, the data are examined to make sure all the coordinate-converted pixels (source
channel’s footprints) falling within the 20 dB contour o f the 10 GHz channel are present.
The 20 dB (1 % power relative to the center o f the pixel), rather than the nominal
footprint (half power), domain is chosen to achieve the maximum accuracy. If even a
single pixel inside that 20 dB contour is missing, then smoothing will not proceed for
that particular 10 GHz pixel. Thirdly, the source (either 37 or 19 GHz channel) data are
convolved with a smoothing function to get the same weighting as the target (10 GHz
channel) data, and then the area averaged rain rate at 37 and 19 GHz, R, is calculated for
every 10 GHz pixel as follows:
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69
iw&
_
R = ~..V
W ,
where R, is the rain rate o f the source channel and W, is the smoothing (weighting)
function. The maximum weight goes to the center pixel o f the source channels and those
pixels in the rim have the least weight. The weighting function. W,, is given as:
v, =«cp[-i(£-+£>]
2 A~ A'
•
v
* -* T
where
»2
^
2
_2
r ^ ^\QOHHx)
A-
—
^v
~
2
® " l0 < /7 fc ( > |
wiwrift j )
s
t
2
^w urcety}
^ ioghz and aSOurce are the standard deviations of the Gaussian o f the 10 GHz channel and
the source channel, respectively, since the antenna patterns are assumed to be Gaussian
without side lobes for those three channels. Those standard deviations are determined by
each channel’s resolution and are listed in table 3.
Table 3. Variances o f Gaussian for 10.19 and 37 GHz vertical channels
10 GHz
19 GHz
37 GHz
Oy
628 Km2
167
46
(Tx
234 Km2
62
17
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70
6.5 Combined rain rate histogram
Alter smoothing, the rain rates o f the three channels over the oceans are put
together into one combined histogram for each 5° by 5° grid box. This combined
histogram is constructed such that the rain rates from the highest possible non-saturated
channel are chosen. For instance, for saturation criteria o f 255-255 (37-19 GHz
combination), if none o f the 37 GHz pixels has a brightness temperature higher than 255
K. within the 20 dB contour, the 37 GHz channel will be considered as unsaturated. The
smoothed rain rates from the 37 GHz channel will be chosen and put into the combined
histogram. The 37 GHz channel is considered as saturated if any 37 GHz pixel within
the 20 dB domain has a brightness temperature higher than the threshold value o f 255 K.
The same procedure will be applied to the 19 GHz channel. If both the 37 and 19 GHz
channels are saturated, then the rain rate from the 10 GHz channel will be selected and
inserted into the combined histogram. Saturation at the 10 GHz channel is extremely rare
and there is no lower frequency channel available on TMI. This will give us the greatest
sensitivity to the low rain rates where needed and also provide reliable measurements at
high rain rates up to 50 mm/hr (section 3.6).
6.6 Average rain rate and monthly rain total
Finally, a single combined rain histogram, with the offset being corrected, will be
obtained for each 5° by 5° grid box. The offset is corrected by finding the apparent most
probable rain rate and then shifting the retrievals accordingly. Furthermore, the
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71
non-precipitating noises are assumed to be symmetrical about the zero rain rate and
should be cancelled out each other when we fold the negative side o f noise to the
positive side in the rainfall calculation.
The average o f the histogram can been obtained either by simple arithmetical
summation o r by a statistically fitted lognormal distribution. Lee (2001) has shown that
the lognormal assumption has no advantage over the simple arithmetical summation
(direct summation) for the monthly rainfall estimation using the TMI data. Therefore,
the simple arithmetical summation (direct sum) will be used here. The average o f each
rain rate histogram is obtained by dividing the summation o f the rain samples with the
total numbers o f the valid retrieval. The average o f each rain histogram is then converted
into monthly rainfall by multiplying the number o f hours in that month.
Note that in very dry areas, if the non-precipitating noise on the negative side is
larger than its counterpart on the positive side, the monthly rainfall totals could be
negative when applying the symmetrical noise assumption directly to the rainfall
calculation. In such extreme cases, the monthly rainfall totals would be assigned to zero
to avoid retrieving negative rainfalls in these very dry areas.
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72
CHAPTER VII
RESULTS AND VALIDATION
Recall that the current algorithm and essentially all passive microwave
techniques for quantitative rainfall retrieval only work well over water surface. A series
o f tests have been taken to get a reasonable land/water ratio for classifying a 5° by 5°
grid box as land or water. It is concluded that a land/water ratio o f 3 to 1 is a reasonable
balance between maximizing amount o f grid boxes with rainfall retrieval over oceans
while at same time controlling the land contamination. Therefore, if no more than 75%
o f the pixels in one grid box for a month are located over land, this grid box will be
considered to be over water surface for rainfall retrieval purposes and the rainfall amount
for that month will be calculated.
Global monthly rainfall totals for January, February and March, 1998, are
computed and displayed in figures 20,21 and 22, respectively. The global rainfall
mappings in these figures clearly show the tropical large-scale feature such as heavy
rainfall over the Intertropical Convergence Zone (ITCZ). Thus, the current algorithm is
at least able to identify large-scale precipitation feature and provides useful rainfall
information for hydrological and climatological studies.
To quantitatively validate the current algorithm, it needs to compare these
retrieved monthly rainfalls with ground truth data. The ground truth data used in this
study are from the Pacific Rainfall Database (PACRAIN). The PACRAIN has been
supported by the National Oceanic and Atmospheric Administration (NOAA) and its
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73
Monthly Rainfall ( J a n . , 1 9 9 8 )
40
Latitude
20
0
-20
-4 0
100
20Q
Longitude
300
m o n th ly rainfall ( m m )
Figure 20. Monthly rainfall over 5° by 5° grid boxes for January, 1998.
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74
Monthly Rainfall ( F e b . , 1 9 9 8 )
Latitude
20
0
-2 0
-4 0
100
.200
Longitude
300
m o n th ly rainfall ( m m )
Figure 21. Same as in Figure 20 except for February. 1998.
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75
Rainfall
-*;—1--(Mar.,
|--t --f- , 1r 9 9•8■*)"■,r
*■Monthjy
■r
?
40
*
20
’U
o
n
3
.
”
1 *w
1
M. . .
0
..... .j
-2 0 "I -4 0
100
0
200
.200
Longitude
4Q0
.
300
%600
800
m o n th l y rainfall ( m m )
Figure 22. Same as in Figure 20 except for March. 1998.
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_
76
data analysis and quality-assurance technique development is provided by the
Environmental Verification and Analysis Center (EVAC) at the University o f Oklahoma.
The PACRAIN consists o f daily and monthly rainfall records beginning in 1971 and are
updated approximately monthly from many sites located on atolls and islands. However,
only atoll rainfall data with complete monthly records are used here. The atoll rainfall
data are the rainfall measured by rain gauges, upon which there is little topographic
effect, and can represent rainfall over open ocean environments. These atolls are located
in the tropical central Pacific and are illustrated in figure 23.
There are sixteen 5° by 5° grid boxes which consist o f at least one atoll station
that have complete monthly rainfall records for the months o f January, February and
March o f 1998. The atoll rainfall data are first averaged if there are more than one atoll
in any 5° by 5° grid box. Then rainfall estimation retrieved from the TMI versus the atoll
rainfall data are plotted in figures 24,25 and 26 for January. February and March o f
1998, respectively. In addition, simple statistical calculations, i.e., means, differences o f
these means and correlation coefficients between the satellite retrieved rainfall data and
the atoll rainfall data, are listed in table 4.
The correlation coefficients, for all three cases, are very high, ranging from 0.93
to 0.98, and the differences between the two means are within a reasonable range. This
indicates that the current algorithm performs very well at least in the tropical regions.
The differences between the satellite retrieved rainfalls and the atoll rain data are
understandable since satellites measure rainfall at a specific time but area-averaged in
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77
Figure 23. Map o f western Pacific atoll geography.
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78
Jan., 1998
600
c
o
E
^
E
E
400
0
C7>
3
O
o
• §
200
O
<
400
600
200
Satellite Estimation of Rainfall (mm/mon.)
0
Figure 24. Scatter plot o f the TMI inferred rainfall versus the atoll rainfall data for Jan.,
1998.
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79
Feb., 1998
600
c£
o
E
^
E
E
400
o
<
200
400
600
Satellite Estimation of Rainfall (mm/mon.)
0
Figure 25. Same as figure 24 except for Feb., 1998.
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80
Mar., 1998
600
c
o
E
^ 4-00
E
0
cj>
D
G
0
1
cr
200
o
<
600
400
200
Satellite Estimation of Rainfall (mm/mon.)
Figure 26. Same as figure 24 except for Mar., 1998.
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81
Table 4. Statistical results o f validation
Difference
O f the means
(mm)
-21.1
Correlation
Coefficient
Jan., 1998
130.6
Mean o f rainfall
from satellite
(mm)
109.5
Feb., 1998
91.1
68.4
-22.7
0.98
Mar., 1998
90.8
69.3
-21.5
0.93
Month
Mean o f Atoll
data (nun)
0.98
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82
space while the atoll rain gauges measure rainfall at a single point in space but
continuous in time. The spatial and temporal sampling differences between satellite and
atoll rain gauge measurements demonstrate some systematic differences between
satellite retrieved rainfalls and those recorded by atoll rain gauges.
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83
CHAPTER VIII
SUMMARY AND CONCLUSION
Rainfall measurements over oceans are very important in meteorology and
hydrology, as well as in understanding o f global climate change. However, conventional
means (rain gauge, precipitation radar) for measuring rainfall over oceans have proven to
be impractical. Measurements o f rainfall from satellites offer suitable spatial and
temporal coverage over the oceans and are the only reasonable solution for global
oceanic rainfall mapping.
Many efforts have been made for estimation o f rainfall from visible and infrared
measurements from satellite platforms since those measurements are available on both
the geo-synchronous satellites and other longer time series polar-orbiting satellites and
cover a larger fraction o f the earth surface. However, these satellite-bome visible and
infrared sensors can only measure physical properties, such as cloud temperature, at the
top o f the cloud. Unlike the weak physical connection between visible and infrared
measurements and rainfall, microwave radiation interacts with hydrometeors much more
strongly. Therefore retrieval o f rainfall based on microwave observations has a firmer
physical basis.
In this study and through collective efforts, a multichannel algorithm has been
developed for retrieval o f monthly rain totals for 5° by 5° grid boxes over the oceans
from the TMI data. This multichannel algorithm is based on relationships between rain
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84
rates and brightness temperatures (R-T) generated by a physically-based microwave
radiative transfer model for the different TMI channels.
First, a series o f R-T relationships for each TMI channel are generated using
Wilheit et al. (1977) model with an updated water vapor absorption procedure and a
water vapor absorption correction factor. The combination o f R-T relationships for 19
and 21 GHz are used to obtain freezing level information for each pixel. The freezing
level information is then used to solve for rain rates corresponding to the 10,19 and 37
GHz brightness temperatures, respectively, on a pixel-by-pixel basis. Those rain rates
are adjusted according to their corresponding beamfilling correctors, again, on a pixelby-pixel basis, and then accumulated to form rain histograms for each channel over each
5° by 5° grid box.
Secondly, the rain histogram for each channel is processed so as not to have a
cut-off at zero rain rate. Even if physically nonsense, negative rain rates are
mathematically possible. However, in reality the most probable rain rate is zero even in
very rainy areas. The most probable rain rate retrieved by the above-mentioned method
has a negative offset since too much non-precipitating cloud liquid water (NCL W) was
assumed in the model. The NCLW contents in the model are reduced to only one fifth o f
the original value according to many field experimental results and then retrieved most
probable rain rate is very close to zero. Any remaining offset o f rain rate is shifted to
zero accordingly.
There are many advantages o f using the three different channels, i.e., 37,19 and
10 GHz channels, together to retrieve rainfall. According to the R-T relationships
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85
generated by the model, the 10 GHz channel has the ability to measure very high rain
rates (as high as 50 mm/hr) while the 37 GHz channel is sensitive to the low rain rates.
This sensitivity is used to differentiate low rain rates from non-precipitating cloud liquid
water (NCLW). By using the rain rates retrieved from these different channels together,
we achieve high sensitivity to the low rain rates where needed and also obtain reliable
measurements at high rain rates up to 50 mm/hr. This expands the measurement dynamic
range to the higher and lower rain rates and eliminates the need o f using the assumption
o f lognormality (i.e., lognormal fit) to fit for the low and high rain rate portions.
However, in order to utilize the rain rates resulting from those three different
channels, the rain rates derived from 37 and/or 19 GHz channels need to be “smoothed”
to the resolution o f the 10 GHz channel so that the rain rates retrieved from different
channels represent the same area.
The algorithm chooses the highest frequency possible among the 37,19 and 10
GHz channels. This gives us the greatest usable sensitivity to rain. If any o f the
brightness temperature at 37 or 19 GHz channel exceeds a threshold value, typically
around 255 K, the measurement is considered to be saturated and therefore unreliable.
Saturated values at 10 GHz are extremely rare and, in any case, there is no lower
frequency channel available on TMI.
Finally, a single combined rain histogram, with offset being corrected, is
obtained for each 5° by 5° grid box. The offset is corrected by finding the apparent most
probable rain rate and then shifting the retrievals accordingly. Furthermore, the
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86
non-precipitating noises are assumed to be symmetrical about the zero rain rate and they
should be cancelled out each other in the rainfall calculation.
Result o f Lee (2001) shows that the lognormal assumption has no advantage over
the simple arithmetical summation (direct summation) for the monthly rainfall
estimation using the TMI data. Therefore, the direct sum method is used in this
multichannel rainfall retrieval algorithm. The average o f the histogram, obtained by
simple arithmetical summation, is easily converted into monthly rain total by
multiplying the number o f hours in that month.
Global monthly rainfall totals are computed for January, February and March o f
1998. These global rainfall mappings clearly show the tropical large-scale feature such
as heavy rainfall over the ITCZ. Quantitative comparisons with Pacific atoll rain data
show that the correlation coefficients, for all three months, are very high, ranging from
0.93 to 0.98, and the differences o f the means are within a reasonable range. This
indicates that the current algorithm performs very well at least in the tropical regions.
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87
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VITA
Jun Huang was bom in Liuzhou, Guangxi Zhuang Autonomous Region, People's
Republic o f China. He obtained his Bachelor o f Science in geochemistry in July 1986
from the University o f Science and Technology o f China. He received a Master o f
Science in atmospheric physics in July 1989 from the Institute o f Atmospheric Physics,
Chinese Academy o f Science for "Measurements o f methane (CH 4) flux from rice
paddies in southwestern China". He received another Master o f Science in atmospheric
sciences in December 1994 from the University o f Wyoming for "Measurements o f
oxygen (Oi) and hydrogen peroxide (H 2O 2) retention in rime ice". He was admitted to
the Department o f Meteorology (now the Department o f Atmospheric Sciences) and
joined the Microwave Remote Sensing Group in the fall o f 1995 at Texas A&M
University.
Correspondence may be addressed to a continuing email address,
huang@tamu.edu or via U.S. mail:
Jun Huang
Department o f Atmospheric Sciences
Texas A&M University
TAMUS 3150
College Station, TX 77843-3150.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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