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Bayesian retrieval of complete posterior PDFs of rain rate from satellite passive microwave observations

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Graduate School Form 9
(Revised 6/03)
PURDUE UNIVERSITY
GRADUATE SCHOOL
Thesis Acceptance
This is to certify that the thesis prepared
Jui-Y uan Chiu
By
E n titled
BAYESIAN RETRIEVAL OF COMPLETE POSTERIOR PDFS OF RAIN RATE
FROM SATELLITE P A SSIV E MICROWAVE OBSERVATIONS
Complies with University regulations and meets the standards of the Graduate School for originality
and quality
For the degree o f
D o c to r o f P h ilo so p h y
Signed by the final examining committee:
Approved by:
\ t 4-&^'a JA ai'-AaA I^j
_____________________ A i/J U i
Head o f the Graduate Program
This thesis
(_ /
is not to be regarded as confidential.
<m 0 3
Date
—
Major Professor
Format Approved by:
or
Chair, Final Exam ining Committee
/}
D epartm ent T hesis Form at A dvisor
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
BAYESIAN RETRIEVAL O F C O M PLETE PO ST E R IO R PD FS OF RAIN RATE
FROM SATELLITE PASSIVE MICROWAVE OBSERVATIONS
A Thesis
Subm itted to th e Faculty
of
Purdue University
by
Jui-Y uan C. Chiu
In P artial Fulfillment of the
Requirem ents for th e Degree
of
D octor of Philosophy
August 2003
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ACKNOW LEDGM ENTS
I would like to express my gratitude to Drs. Grant W. Petty, Harshvardhan,
Lawrence W. Braile, and Sonia Lasher-Trapp for serving on my Com m ittee. Their
helpful advice and insightful comm ents have been beneficial to my research. I p ar­
ticularly wish to th an k my adviser, Dr. G rant Petty. His expertise, encouragement,
support, and counsel added considerably to my graduate experience. I doubt th a t
I will ever be able to express my appreciation fully, bu t I owe him th e m ost over­
whelming debt of gratitude.
A very special thanks goes out to Dr. W illiam S. Olson, who has been always
a patient and kind m entor to me. I m ust also acknowledge K athy Kincade for her
assistance in adm inistration.
I am also grateful to my office m ate Benjamin T.
Johnson for his advice in com puter and editing, and especially for his friendship.
Finally, b u t not least, I would also like to thank my family for providing me
unconditional support and love through my entire life. W ithout their encouragem ent,
I would not have finished this dissertation.
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iii
TABLE OF CO N TEN TS
Page
LIST OF T A B L E S ............................................................................................................
vi
LIST OF F I G U R E S ........................................................................................................
vii
A BSTRACT
1
2
IN TRO D U CTIO N
...................................................................................................
1
1 .1
Im portance of precipitation m e a s u re m e n ts .............................................
1
1.2
Microwave rem ote s e n s i n g ............................................................................
2
1.3
Overview of Bayesian algorithm s
..............................................................
5
1.4
Objectives and p r o c e d u r e s ............................................................................
8
DATA D E S C R IP T IO N .............................................................................................
11
2.1
11
TM I d a t a ..........................................................................................................
2.1.1
Ideal response to surface rain r a t e ..................................................
11
2 1 .2
A ttenuation in d e x .................................................................................
13
2.1.3
D istributions of attenuation index of T M I ..................................
16
2.2
P R rain r a t e .......................................................................................................
19
2.3
PR -T M I m atch-up procedure
.....................................................................
21
.............................................................................................
25
3.1
Prior d is tr ib u tio n .............................................................................................
25
3.2
Conditional likelihood
...................................................................................
26
3.3
E stim ators of th e posterior d is trib u tio n ....................................................
29
P R O O F - O F - C O N C E P T ..................................................................................................
31
.
3
4
......................................................................................................................... xiii
ALGORITHM BASIS
4.1
Rain rate estim ates from NWS W SR - 8 8 D n e tw o rk ................................
31
4.2
Polarization c a lc u la tio n s ...............................................................................
32
4.3
Comparison w ith PR -T M I m atch-up d a ta
36
..............................................
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iv
5
4.4
Bayesian algorithm from sim ulations
........................................................
39
4.5
Retrieval based on s im u la tio n s ......................................................................
40
SENSITIVITY T E S T ................................................................................................
48
5.1
T he prior and conditional d is tr ib u tio n s .....................................................
49
5.2
Designed e x p erim en ts.......................................................................................
51
5.3
Results for theoretical t e s t ............................................................................
57
5.3.1
Intrinsic uncertainty of th e a lg o r ith m .............................................
57
5.3.2
Sensitivity to th e prior k n o w le d g e ...................................................
64
5.3.3
Sensitivity to th e conditional d is trib u tio n ......................................
69
5.4
6
7
Sum m ary
...........................................................................................................
85
..........................................
86
6.1
G P R O F ..............................................................................................................
86
6.2
P E T T Y TM I algorithm
................................................................................
89
6.3
P E T T Y HIST4 a l g o r i t h m .............................................................................
91
6.4
Linear model
....................................................................................................
91
REAL-W ORLD A PPLIC A TIO N AND V A LID A T IO N ....................................
93
7.1
Bayesian a lg o r ith m ..........................................................................................
93
7.2
D atasets for v a lid a tio n ....................................................................................
96
7.2.1
Selected global rain c a s e s .....................................................................
96
7.2.2 H eavy/w idespread rain e v e n t s ...........................................................
96
7.2.3 PR -T M I global m atch-up d ataset of 1998/04 .................................
99
Validation m e tric s ..............................................................................................
99
BENCHMARK A LGORITHM D ESCR IPTIO N S
7.3
7.3.1
B i a s .............................................................................................................100
7.3.2 Root-m ean-squared d iff e r e n c e ..............................................................100
7.3.3 Correlation coefficient
........................................................................... 100
7.3.4 Heidke Skill S c o re ..................................................................................... 101
7.3.5 Theoretical HSS d is trib u tio n ................................................................. 103
7.4
R e s u lts ..................................................................................................................... 108
7.4.1
Typhoon c a s e ............................................................................................ 108
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8
B auer’s 12 oceanic c a s e s ........................................................................ 114
7.4.3
Heavy/widespread rain e v e n t s ...............................................................119
7.4.4
April 1998 ............................................................................................... 123
7.4.5
S u m m a r y .................................................................................................. 127
UNCERTAINTY ANALYSIS........................................................................................ 134
8.1
Assessment of the posited conditional lik e lih o o d ...................................134
8.2
U ncertainty of the posterior m ean due to the prior distribution . . . .
8.3
Uncertainty of the posterior variance due to the prior distribution . . 143
8.4
R e s u lts ................................................................................................................ 145
8.5
9
7.4.2
141
8.4.1
IV m o d e l .................................................................................................. 145
8.4.2
2V m o d e l .................................................................................................. 148
Sum m ary
..............................................................................................................151
CALIBRATION OF MODEL SIM U L A T IO N ...........................................................152
9.1
C alibration of radar-sim ulated rain r a t e ................................................... 152
9.2
Verification of radiative transfer c a lc u la tio n s ......................................... 153
9.3
Effects of freezing level h e i g h t .................................................................... 154
9.4
Im plication
...........................................................................................................156
10 CONCLUSIONS AND FU T U R E W O R K ................................................................ 164
VITA
...................................................................................................................................... 174
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vi
LIST O F TABLES
Table
Page
1.1
C haracteristics of T R M M /P recipitation R ad ar..........................................
4
4.1
Param eters a and b in the approxim ation of liquid w ater extinction
coefficient...............................................................................................................
33
5.1
Coefficients of oM, b^ and cM.............................................................................
50
5.2
Inform ation of designed experim ents in sensitivity tests, including the
experim ent ID, the training dataset, and th e specifications of th e prior
and conditional likelihoods applied to th e Bayesian algorithm ...............
52
Inform ation of date, orbit number, and th e nadir locations for rain
events in the validation d a ta se t.......................................................................
97
7.1
7.2
S tandard 2 x 2 contingency table for evaluation of the skill of a binary
classification procedure......................................................................................... 1 0 2
7.3
Sum m ary of bias, root-m ean-square difference (RMS), and correlation
coefficients against validation datasets of orbit 336 and B auer’s cases
for each algorithm. The unit of bias and RMS is m m /h r........................... 109
7.4
Sum m ary of bias, root-m ean-square difference (RMS), and correla­
tion coefficients against validation datasets of heavy/w idespread rain
events and 1998/04 dataset for each algorithm . The unit of bias and
RMS is m m /h r........................................................................................................ 1 1 0
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LIST OF FIG U RES
Figure
Page
2.1 Idealized brightness tem peratures dependence on surface rain rate
(From Petty, 2001)..............................................................................................
12
2.2 Overall m ultichannel distributions of P index represented by th e num ­
ber of pixels. D ata were selected from 110 TM I orbits during 1999
and 2000. Contours are logarithm ically spaced; actual value is 10x
where x is th e contour label, x are p lotted for values of [0.5, 1, 2, 3,
4, 5] at a fixed P i 0 interval................................................................................
17
2.3 Same as figure 2.2, bu t at a fixed P 3 7 interval..............................................
18
2.4 Probability distributions of th e near-nadir P R 15-km interpolated rain
rate are depicted by lighter-colored dots. D ata were calculated from
the 2A25 P R near-surface rain ra te estim ates in January, April, July
and O ctober of 1998, and a cut-off value of 0.04 m m /h r was applied.
The resulting sample size is around 2.4 million. T he fitted lognorm al
distribution is also presented here by a dashed curve...............................
23
3.1 Com parisons of the prior rain ra te probability distributions used in
this study...............................................................................................................
27
4.1 C ontours of the num ber of pixels based on TM I d a ta (the first and
th ird columns) and radar-radiative sim ulations (the second and fourth
columns). Contours are logarithm ically spaced; actual value is 10x
where x is th e contour label, x are p lo tted for values of [0.5, 1, 2, 3,
4, 5]..........................................................................................................................
38
4.2 1km NWS W SR - 8 8 D network com posite reflectivies for 10:40am, Au­
gust 13, 2002, w ith suspicious ra d a r retu rn s removed...............................
41
4.3
Plots of sim ulated 15 x 15 km rain ra te based on the W SR- 8 8 D com­
posite reflectivity for 10:40am, A ugust 13, 2002, and a Z-R relation­
ship proposed by M arshall and Palm er (1948)
42
4.4
Sim ulated P w for the d a ta of 10:40am, A ugust 13, 2002.........................
43
4.5
Sim ulated Pig for the d ata of 10:40am, A ugust 13,
.........................
44
4.6
Sim ulated P 3 7 for the d ata of 10:40am, A ugust 13, 2002.........................
45
2 0 0 2
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viii
4.7
Plots of retrieved rain rate for th e d a ta of 10:40am, A ugust 13, 2 0 0 2 ,
based on the new Bayesian algorithm ............................................................
46
T he probability distribution of rain rates and the joint and m arginal
pdfs of the P vector based on analytical solutions of th e sophisticated
model. The joint pdfs are plotted for [0.05, 1, 5, 7, 10, 12].....................
53
5.2
Same as figure 5.1, b u t for the random ly generated D 2 d a ta se t
54
5.3
M ultichannel relationships by plotting th e num ber of pixels at a given
P 10 (left column) or given P 3 7 (right column) for D2 training dataset.
Contours are logarithm ically spaced; actual value is 1 0 x' where x is
th e contour label, x are plotted for values of [0.5, 1, 2, 3, 4, 5].............
56
2-D contours of th e stan d ard deviations of th e posterior probability
distribution in the P 3 7 vs. P i 9 space at a fixed P 10 value for experiment
R0. Contours are plotted in an interval of 5 m m /h r. Regions outside
th e zero value of th e contour indicate an impossible P vector due to
a zero m arginal density......................................................................................
58
Histograms of retrieved rain ra te at different rain ra te ranges for R0
experim ent. Titles contain inform ation about range of tru e values,
sample size, and th e estim ator. Numbers in parentheses represent the
m ean and standard deviation of the histogram . Percentages present
the fraction of retrieved rain rates located in the correct range.............
60
5.6
Same as figure 5.5, b u t using MLE estim ations.........................................
62
5.7
Examples of derived posterior rain rate distributions at some given P
vectors in experim ent R0. The observation vector (P 1 0 , P 1 9 , P 3 7 ) is
presented by the three num bers in parentheses...........................................
63
Plots of the prior probability distributions used in th e sensitivity tests.
To b etter illustrate th e differences, ranges of [0, 10] and [10, 100]
m m /h r are respectively shown in (a) and (b )..............................................
65
Same as
figure 5.5,
but
for experim ent
R 1................
66
5.10 Same as
figure 5.6,
but
for experim ent
R 1................
68
5.11 Same as
figure 5.5,
but
for experim ent
R 4................
70
5.12 Joint and m arginal probability distributions of th e physical model for
e x p e rim e n t SO (u p p e r tw o row s) a n d S5 (b o tto m tw o ro w s). C o n to u rs
are plotted for [0.05, 0.5, 1.0, 2.5, 5.0, 7.5, 10, 12.5, 15] in conditional
pdfs..........................................................................................................................
71
5.13 Same as figure 5.5, b u t for experim ent SO....................................................
72
5.1
5.4
5.5
5.8
5.9
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ix
5.14 Same as figure 5.13, b u t using m axim um likelihood estim ates as single­
pixel retrieved rain ra te s....................................................................................
73
5.15 Similar to figure 5.5, b u t for MEAN (left panel) and MLE (right panel)
estim ates of Experim ent S5 at ranges of heavy tru e surface rainfall
(greater th a n 30 m m /h r)...................................................................................
74
5.16 C onditional and m arginal pdfs of experim ents S 6 (first row), S7 (sec­
ond row), and S 8 (the bottom ). Note th a t different channels are shown
here for each experim ent to highlight th e change due to the difference
in the covariance m atrix ....................................................................................
76
5.17 Similar to figure 5.5, bu t for the MEAN (left panels) and M LE (right
panels) estim ates of experiment S 6 a t heavy precipitation rate ranges
(greater th a n 50 m m /h r)...................................................................................
77
5.18 Same as figure 5.5, b u t for experim ent S 8 ...................................................
79
5.19 Same as figure 5.18, b u t for MLE estim ates...............................................
81
5.20 Joint and m arginal probability distributions of th e physical model for
experim ent R5. T he joint pdfs are p lo tted for [0.05, 1, 5, 7, 10, 12]. .
82
5.21 Same as figure 5.5, b u t for experim ent R 6 ..................................................
84
6.1
6.2
7.1
Normalized histogram s of surface rainfall of cloud profiles included in
the old (solid curve) and new (dotted curve) G PR O F database. The
to ta l num ber of cloud profiles are 11069 and 3097 for th e old and new
database, respectively.........................................................................................
88
S catter plots of num ber of cloud profiles on th e rain rate vs. P i 9
dom ain a t a given interval of P 10 and P 3 7 in the old (left panel) and
new (right panel) G PR O F d atab ase...............................................................
90
Locations of heavy/w idespread rain events selected from TM I orbits
during January-D ecem ber 1998.......................................................................
98
7.2
Plots of th e sample mean of th e retrieval vs. the posterior mean
estim ated from th e synthetic dataset a t a given P if the corresponding
sample size is (a) greater th an 100; (b) greater th a n 30; (c) greater
th a n 10; and (d) less th an or equal to 10. Note change of range in
each case....................................................................................................................104
7.3
Plots of 2-D Heidke skill scores (HSS); th e m axim um of HSS w ith re­
spect to tru e rain ra te threshold; and th e best algorithm threshold vs.
tru e rain ra te threshold based on th e synthetic d a ta of the Bayesian
3V model, (a)-(c) are illustrated for M EAN estim ates, while (d)-(f)
are for M L E ..............................................................................................................106
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X
7.4
M aps of P R interpolated rain ra te w ith 15-km resolution and retrieved
rain ra te of G PR O F (old and new databases), P E T T Y TM I, P E T T Y
HIST4, and linear model algorithm s for T R M M /T M I orbit 336. . . . I l l
7.5
Same as figure 7.4, b u t for Bayesian 3V-MEAN, 3V-MLE, 2V-MEAN,
2V-MLE algorithm s............................................................................................... 112
7.6
S catter plot of retrieved rain rate vs. P R rain rate for all algorithm s,
based on th e validation d a ta of the TM I orbit 336.................................... 113
7.7
2-D distribution of Heidke skill scores (HSS) for th e 12 selected cases
from th e B auer et al. (2001). The value noted in th e bottom -right
corner of each plot indicates th e highest HSS of the algorithm ................. 115
7.8
Plots of th e best algorithm rain ra te threshold w ith respect to th e
threshold of P R rain rate for th e B auer’s cases. T he algorithm used
in th e retrieval is shown in th e title of each p lo t........................................... 116
7.9
P lots of th e m axim um Heidke skill score vs. the threshold of P R rain
ra te based on th e dataset of 12 B auer’s cases.................................................117
7.10 2-D distribution of Heidke skill scores (HSS) for the heavy/w idespread
rain events. The value noted in the bottom -right corner of each plot
indicates th e highest HSS of the algorithm ..................................................... 120
7.11 Plots of th e best algorithm rain ra te threshold w ith respect to th e
threshold of P R rain rate for th e cases w ith heavy and w idespread
precipitation............................................................................................................. 1 2 1
7.12 Plots of th e m axim um Heidke skill score vs. the threshold of P R rain
ra te based on the dataset of heavy/w idespread rain events........................ 1 2 2
7.13 2-D distribution of Heidke skill scores (HSS) in random ly selected 118
files in A pril 1998. The value noted in th e bottom -right corner of each
plot indicates th e highest HSS of th e algorithm .............................................124
7.14 P lots of th e best algorithm rain ra te threshold w ith respect to th e
threshold of P R rain rate for th e cases of April, 1998............................... 125
7.15 Plots of th e m axim um Heidke skill score vs. the threshold of P R rain
ra te based on th e dataset of April, 1998....................................................... 126
7.16 P lots of th e m axim um Heidke skill score vs. th e P R rain rate threshold
in th e ra n g e o f [0, 10] m m / h r b a s e d o n B a u e r ’s c a se s......................................128
7.17 Same as figure 7.16, bu t for th e heavy and widespread precipitation
cases............................................................................................................................128
7.18 Same as figure 7.16, bu t for th e d ataset of April, 1998............................. 130
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xi
7.19 Plots of th e best algorithm threshold vs. P R rain ra te threshold
in
th e rain ra te range of [0, 10] m m /h r for th e B auer’s cases..........................131
7.20 Same
as figure 7.19, b u t for heavy/w idespread precipitation cases.
7.21
as figure 7.19, b u t for th e d ataset of April, 1998........................ 133
8.1
Same
. .132
Plots of joint probability density functions / ( P 37, i?) based on th e
I V model (upper) and near-nadir PR -T M I d ata (bottom ). C ontours
are logarithm ically spaced; actual value is 10X where x is th e contour
label, x are plotted for values of [-5, -4, -3, -2, -1, 0, 1] and [-4, -3, -2,
-1, 0, 1] for th e I V model and observations, resp ectiv ely ...........................135
8.2 Same as figure 8.1, b u t for th e joint p d f of (P i 9 ,R) based on th e 2 V
model (u p p er)..........................................................................................................137
8.3
Same as figure 8 . 1 , b u t for th e joint p d f of (P i 0 ,R) calculated from
the 3 V m odel........................................................................................................... 138
8.4 Plots of conditional pdfs / ( P i o | P i 9 , P 3 7 , P ) based on th e PR -T M I d a ta
(dashed curve) and th e posited Bayesian model (dotted curve), a t a
given set ( P , P 19,P 37) of (a) (1.5, 0.70, 0.40); (b) (4.5, 0.35, 0.25);
(c) (6.5, 0.45, 0.20); (d) (11.5, 0.3, 0.1); (e) (15.5, 0.15, 0.05); and
(f) (19.5, 0.1, 0.05). N noted in figures represents th e sample size of
observed d a ta ........................................................................................................... 140
8.5
Plots of th e posterior m ean (m m /hr) vs. P 3 7 in the I V model under
the specification th a t the prior distribution is logiV( —2.8, 2.0). T he
uncertainty of the prior is sim ulated by th e symmetric contam ination
function w ith a factor of 0.25. Solid and dotted curves indicate the
suprem um and infimum, re s p e c tiv e ly ..............................................................146
8 .6
Range of posterior stan d ard deviation for the given posterior m ean in
th e I V model when P 3 7 is (a) 0.6, (b) 0.4, and (c) 0.2. T he square
and asterisk symbols present th e m inim um and m axim um values, re­
spectively. Thus, the area inside th e curves indicates th e region where
th e stan d ard deviation might be located..........................................................147
8.7
Range of posterior m ean in th e 2 V model for (a) the lower bound, (b)
upper bound, and (c) th e m aginitude of variation in units of (m m /h r).
T he contam inated function and factor used here are th e same as those
applied in th e I V m odel....................................................................................... 149
8 .8
Same as figure 8 .6 , bu t for 2 V model and given (P i9, P 37) = (a) (0.7,
0.4) (b) (0.6, 0.4), (c) (0.5, 0.2), (d) (0.4, 0.2), (e) (0.2, 0.1), (f) (0.2,
0.05), (g) (0.05, 0.05) and (h) (0.06, 0.07).................................................... 150
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xii
9.1
Cum ulative distribution functions of P R observed (solid curve) and
NWS W SR- 8 8 D netw ork-sim ulated (dotted curve) rain rate. P R curve
is based on the near-nadir P R interpolated rainfall intensity during
Jan., Apr., Jul., and Oct. of 1998. W SR - 8 8 D-simulated curve is calcu­
lated a) using a fixed Z-R relationship w ithout adjustm ents tow ard P R
m easurem ents, and b) w ith adjustm ents from a lookup table. Sample
sizes are shown in th e parentheses..................................................................... 157
9.2
2-D contours of num ber of pixels associated w ith rain rate less th an
1 m m /h r at a given Pio interval. Left and right panels are based on
W SR- 8 8 D simulations w ith a freezing level at 3 and 5 km, respectively.
The middle panel is derived from PR -T M I d ata in 1998. C ontours are
plotted for values of [1, 10 ,50, 100 , 500, 103, 104, 105]..................................158
9.3
Same as figure 9.2, b u t for rain ra te of [1, 5] m m /h r and different
intervals of P \ q.........................................................................................................159
9.4
Same as figure 9.2, b u t for rain ra te of [5, 15] m m /h r and different
intervals of Pio.........................................................................................................160
9.5
Same as figure 9.2, but for rain rate greater than 15 m m /hr and
different intervals of lower P w ...........................................................................161
9.6
Plots of fraction of d a ta vs. bright band height based on P R 2A25
algorithm retrieval of 55 orbits during August 1998 and July 1999.
Season inform ation is shown in titles. D JF represents th e w inter sea­
son, while MAM expresses th e spring. The vertical resolution of 2A25
product is 0.25 km ..................................................................................................162
9.7
Plots of fraction of d a ta vs. altitu d e of freezing level based on TM I
m easurem ents in January and July, 1998. (a) and (b) are derived from
the equation (9.1), while (c) and (d) are com puted from W ilh eit’s
approach. Note th a t the W ilheit’s m ethod constrains th e freezing
level to th e climatological altitu d e (around 4.9 km) at a given 302 K
of sea surface tem p eratu re ....................................................................................163
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A BSTRA CT
Chiu, Jui-Y uan C.. Ph.D ., P urdue University, A ugust, 2003. Bayesian Retrieval of
Complete Posterior PD F s of Rain R ate From Satellite Passive Microwave Observa­
tions. M ajor Professor: G rant W. Petty.
This dissertation presents a new Bayesian rain rate retrieval algorithm for th e
TRM M Microwave Im ager (TM I), along w ith associated error analysis of synthetic
sensitivity tests and real-world applications. The Bayesian approach offers a rig­
orous way of optim ally combining the actual m ultichannel observations w ith prior
knowledge. It has been applied in m any studies to retrieve instantaneous rain rates
from microwave radiances. However, this is believed to be the first self-contained
algorithm whose ou tp u t is not ju st a single rain rate, bu t rath er continuous posterior
probability distributions of th e rain rate.
The success of th e Bayesian algorithm depends on th e accuracy of b o th the
conditional probability density function (pdf) of microwave observations and the
prior pdf of rain rate, as well as on th e interpretation of th e posterior probability
distribution of rain intensities.
T he current study presents explicit functions to
reasonably approxim ate th e physical relationships between rain rates and microwave
radiance based on b o th model simulations and observations from TM I and TRM M
Precipitation R adar (PR). T he prior distribution is lognormal based on P R rainfall
measurements.
Two common statistical estim ators are tested for converting th e
posterior pdf of th e retrieved rain ra te to a single rain rate estim ate for a pixel.
To advance th e understanding of theoretical benefits of th e Bayesian approach,
sensitivity tests are conducted using two synthetic datasets for which th e ’’tru e ”
physical model and th e prior distribution are known. Sensitivity results have dem on­
strated th a t even when th e prior and conditional likelihoods have been applied per­
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xiv
fectly, an apparent bias in retrieval at high surface rain rate occurs. In addition, the
tests suggest th a t the choice of th e estim ators and the prior inform ation are b o th
crucial to th e retrieval.
In this study, the new Bayesian algorithm has also been applied to real TM I d a ta
over the ocean. Its estim ates are validated against independent d atasets and the
perform ance of the new Bayesian algorithm is compared w ith th a t of other bench­
m ark algorithm s. The results are satisfactory in th a t our algorithm has com parable
perform ance to other algorithm s while having th e additional advantage of providing
posterior rain ra te probability distribution.
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1
1. IN T R O D U C T IO N
1.1
Im p ortan ce o f p recip itation m easurem ents
M easurem ents of precipitation are essential for our understanding of ocean and
land-atm osphere interaction, and for th e q uantitative understanding of the global hy­
drological cycle. Ocean and land exchange freshwater and heat w ith the atm osphere
across the interface via precipitation and evaporation. The terrestrial exchange pro­
cess affects the soil m oisture and runoff, and is thus an im portant issue in w ater
balance and w ater m anagem ent.
In th e m arine environm ent, precipitation is also a crucial factor th a t determ ines
the salt balance in the upper layers of th e ocean, and thus influences th e oceanic circu­
lation on seasonal and inter-annual tim e scales. Ropelewski and H alpert (1996), Dettinger et al. (1998), and Paegle and Mo (2002) dem onstrated strong evidence relating
interannual to decadal variability of regional precipitation to El N ino-Southern Os­
cillation (ENSO). O ther studies from Fowler and Kilsby ( 2 0 0 2 ), and Lucero and
Rodriguez (2002) also suggested th a t precipitation at some areas is highly correlated
with th e N orth A tlantic Oscillation (NAO).
A num ber of studies have dem onstrated promising results th a t assim ilations of
precipitation inform ation lead to significant improvements in forecasts and simula­
tions (Treadon, 1996; K rishnam urti et al., 2001).
Hou et al. (2001) also suggested
th a t the assim ilations of surface rainfall improve not only th e hydrological cycle in
global analyses and short-range forecasts, b u t also key climate param eters such as
clouds and radiation. In addition, inform ation about rainfall helps to validate th e
current forecast/sim ulation ability of w eather prediction models and regional/global
climate models (Kleinn et a l, 2002; Wei et al., 2002).
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2
In short, the need for inform ation about spatial and tem poral distributions of
precipitation occurs in m any areas of earth science, and it points out th e im portance
of a regular basis of system atic and accurate rainfall measurem ents. In th e next
section, a num ber of techniques for m easuring precipitation will be discussed in
detail.
1.2
M icrowave rem ote sensing
The standard technique for m easuring rainfall has been to use surface rain gauges
at discrete points. It measures precipitation in th e most direct way, b u t suffers from
th e representativeness problem because rain gauges only observe a point quantity.
In contrast to the low spatial resolution of rain gauge observations, surface w eather
radars provide high tem poral and spatial resolution precipitation m easurem ents, bu t
their coverage is still limited. In particular, th e oceans rem ain largely unobserved
by surface radar. Unlike surface-based m ethods, satellite rem ote sensing offers rela­
tively frequent global coverage, and is an excellent tool for m easuring precipitation,
especially in the m arine environment.
Precipitation m easurem ents using spaceborne visible or infrared sensors are made
by inferring rainfall from the characteristics of cloud. They do not directly sense the
rainfall.
However, due to th e ability of microwave radiation to p en etrate clouds,
satellite passive microwave observations have been of interest for estim ating precip­
itation over the globe in the past decades. A num ber of passive microwave sensors
have flown on spaceborne platform s, including the earliest Electronically Scanning
Microwave Radiom eter (ESMR) aboard Nimbus 5 and Nimbus 6 , th e Scanning M ul­
tichannel Microwave Radiom eter (SMMR) on Seasat and Nimbus, th e Special Sensor
M icrowave/Imager (SSM /I) onboard th e Defense Meteorological Satellite Program
(DMSP) F - 8 to F-15 satellites, and th e recent Tropical Rainfall M easuring Mission
(TRM M ) launched w ith b o th passive and active microwave sensors in 1997.
TRM M operates in a circular orbit w ith a 35° inclination angle at an original
altitude of 350 km. The altitu d e was boosted to 402 km in A ugust 2001 in order
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3
to extend the mission. The prim ary instrum ents on TRM M include th e TRM M
Microwave Imager (TMI), th e Precipitation R adar (PR ) and th e Visible and Infrared
R adiom eter System (VIRS). T he em phasis of this study is on th e TM I and P R
measurements.
The TM I is a m ultichannel passive microwave radiom eter based on th e design
of the highly successful SSM /I. The TM I antenna scans conically, and the resulting
incident angle and swath w idth are about 52.8° and 758.5 km. TM I provides mi­
crowave measurem ents at 10.65, 19.35, 37.00, and 85.50 GHz vertical and horizontal
polarization, and at 21.3 GHz w ith vertical polarization. T he effective fields of view
(EFOV) in down-track and cross-track directions for th e above channels (ascending
frequencies) are 44 x 27 km, 30.4 x 19.9 km, 27.2 x 18.3 km, 16.0 x 12.6 km,
and 7.2 x 6.0 km (Bauer and B ennartz, 1998). Detailed descriptions of other TM I
characteristics can be found in Kummerow et al. (1998).
1
Precipitation R adar is an active microwave sensor th a t scans in a cross-track
strategy from nadir to 17°. T he sw ath w idth is about 215 km and th e num ber of
independent samples for each scan is 49. T he minimum detectable threshold of P R
reflectivity is about 17 dBZ (around 0.5-0.7 m m /h r) in th e absence of attenuation.
The horizontal resolution is ab o u t 4.3 km a t nadir, while the vertical resolution is
0.25 km. O ther detailed inform ation is sum m arized in table 1.1.
SSM /I and TM I m easurem ents have been extensively used to develop a num ber
of retrieval algorithms for rain intensities.
In th e developm ent of retrieval algo­
rithm s, th e key element is to find th e relationship between th e rain intensity and th e
observed brightness tem peratures (or some transform ed variables from microwave
m easurem ents). The observed variables have to show a great ability to reflect the
precipitation signal and have a lower degree of sensitivity to noise from instrum ents,
surface, and the atm ospheric variations. In addition, the construction of the physical
relationship can be either empirically derived via ground m easurem ents from rain
1N ote, however, that paper gave an incorrect characterization of the TM I’s effective field-of-view
(EFOV).
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4
Table 1.1
Characteristics of T R M M /P recipitation R adar.
PR
Frequency
13.8 GHz (K u band)
Beam w idth
0.71°
Peak transm it power
500W
Pulse duration
1 .6
Pulse repetition frequency
2776 Hz
Minimum detectable signal
~ 1 7 dBZ
A ntenna height above m ean sea level
350 km
Speed of one scan
34° in 0.6 s
Scan range (cross track)
±17°
Swath w idth
215 km
Horizontal resolution at nadir
4.3 km
Vertical resolution a t nadir
0.25 km
/is
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5
gauges or radar, or based on simulations from conceptual or explicit cloud models
and radiative transfer models, w ith or w ithout empirical calibration.
1.3
O verview o f Bayesian algorithm s
An algorithm th a t applies Bayes theorem is called a Bayesian algorithm . The
Bayes theorem states th a t given th e d a ta P th e distribution of the param eters is
proportional to th e conditional likelihood tim es th e prior distribution
ir(9|P) <x / ( P |9 ) jt(«).
(1.1)
The first term on th e right side specifies th e probability distribution of th e response
variables (or observation vector) P given th e values of covariates and param eters
0, which is a conditional probability density function (pdf). Since th e distribution
expresses statistical and physical inform ation ab o u t the relationship between P and
6, we will refer to this p art as th e physical m odel in the study. T he second term
on the right hand side summarizes our prior knowledge of the param eter before the
d ata are seen. The prior distribution of th e param eters is denoted as 7t(9). The
prior distribution may only represent our beliefs, or it can be estim ated from exper­
iments. T he interaction of the physical m odel and th e prior probability distribution
determ ines th e so-called posterior distribution 7r( 0 |P ), which tells us th e new pdf of
9 in light of th e observations P . Normally, th e effect of P is to reduce th e spread
of tt(0 |P ) relative to 7r(0); the degree of reduction is a m easure of th e inform ation
content of P .
The Bayesian approach has been applied in many studies to retrieve in stan ta­
neous rain rates and vertical hydrom eteor profiles from microwave radiance m easure­
ments. Evans et al. (1995) developed a Bayesian algorithm based on num erical cloud
models and forward radiative transfer modeling. The conditional pdfs in their algo­
rithm were constructed simply by assum ing th a t th e observation vector (brightness
tem peratures in their study) is norm ally d istrib u ted around th e sim ulated bright­
ness tem peratures w ith a fixed variance. In th eir assum ptions, these channels in the
observation vector were independent of each other. As to the prior distribution, a
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6
m ultivariate lognormal form based on cloud model sim ulations was used to approx­
im ate th e distributions of hydrom eteor profiles. Since th e focus of their study was
placed on finding the most likely atm ospheric state, they com puted th e maximum
likelihood estim ate of th e posterior p d f by optim izing a cost function. However, as
they mentioned, some relationships in the parameter vector did not quite follow a
lognormal behavior. In addition, due to th e high dim ensionality b o th in the obser­
vations vector and the param eters in their m ethod, it is h ard to trace the retrieval
errors and to understand th e sensitivity of th e algorithm to those specifications.
The G oddard Profiling algorithm (G PR O F) is a simplified Bayesian scheme, pro­
posed by Olson et al. (1996) and Kummerow et al. (1996). Similar to the previous
work, G PR O F is also based on cloud models and forward radiative transfer calcula­
tions. The assum ptions regarding th e observation vector are th e same as in Evans
et al. (1995): the conditional probability distributions are norm ally distributed, and
the observational and modeling errors at various channels are uncorrelated. In ad­
dition, the retrieved variables in G PR O F also include th e surface rain rate and ver­
tical hydrom eteor profiles. However, unlike th e Evans study, th e prior distribution
was not specified as a lognorm al distribution th a t contained covariance information,
and th e posterior probability density function was not directly com puted. Instead,
they introduced a database from cloud-radiative model sim ulations to serve as th e
prior distribution, representing th e occurrence frequency of cloud profiles. Then, th e
single-pixel retrieval was estim ated from an approxim ated m ean E of the posterior
distribution using the expression:
E’(x)
^ Xjexp{~°-5[y° ~ y*(xj)]T( ° + s )~1[y0 - y s(xi)]}
3
^
A
where x and y represent th e cloud profiles (covariates) and th e observation vector;
j is the index of candidate profiles in th e database; th e subscript of o and s indicate
th e observed and sim ulated quantity, respectively; O and S describe the diagonal
error m atrix caused from m easurem ent noises and model simulations, while in fact
th e simulation errors were assum ed zero; finally, A is th e norm alization factor stated
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7
as
A = ^ e x p { - 0 . 5 [ y o - y s (xi )]T( 0 + S )-1 ^
j
- y^x,-)]}.
(1.3)
Based on th e estim ator of equation(1.2), the retrieved rain rates and hydrom eteor
profiles in G PR O F are determ ined by how close th e observations are to th e candi­
dates of the database (the term of yQ—ys), and how often the selected cloud profiles
occur in th e database (j).
A nother TRM M standard retrieval algorithm (2B31) combines P R observations,
TM I m easurem ents at 10.65 GHz, and an adapted radar retrieval algorithm to es­
tim ate the profiles of rain clouds via th e Bayesian approach. Details are contained
in H addad et al (1997). The m ain response variables they used are th e rad ar re­
flectivities, one-way integrated atten u atio n s for each radar beam , microwave radi­
ance m easurem ents from TM I, and th e derived microwave brightness tem peratures
from the rad ar model, while th e covariates encompass the rain profiles and th e drop
size distribution (DSD) param eters. T he prior knowledge about SDS param eters
is obtained from the analysis of th e D arw in and TOGA-COARE d ata. The prior
distribution of the rain profiles is based on model calculations and represented in
a form of m ultiplications of Dirac 8 and G aussian functions via th e dependence of
radar beams.
A similar Bayesian algorithm for G PR O F was developed by Bauer et al. (2001).
Their retrieval technique was also based on combined cloud-radiative transfer sim­
ulations. In addition, the same functions as equations (1.2) and (1.3) were used
to com pute th e retrieved near-surface precipitation. However, there are three ma­
jor modifications in their work. First, they criticized the representativeness of the
database in G PR O F, and provided a m erged database th a t was constructed from
cloud model simulations and 130 orbits of TM I data. Second, th e dim ensionality of
th e observation variables were reduced by replacing brightness tem p eratu re informa­
tion w ith two or three empirical orthogonal functions (EOFs). T hird, the modeling
errors in forward transfer calculations were no longer assumed to be zero, b u t were
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8
assigned 2, 4,
6
, and 10 K for 10.65, 19.35, 37.00, and 85.50 GHz, respectively.
However, these channels are still assum ed uncorrelated.
In summary, the above Bayesian retrieval m ethods all rely on th e ability of inde­
pendent cloud-radiative databases to represent th e most possible solutions th a t have
to be close to the real observations. The retrieved variables for those algorithm s
are surface rain rate and vertical hydrom eteor profiles. In th e construction of the
conditional and the prior distributions, simplified assum ptions are made for both
elements.
In addition, to reduce th e com putational requirem ent, th e single-pixel
retrieval is desired instead of com puting th e complete posterior probability density
function.
The effects of imperfection of th e conditional likelihood and the prior inform ation
on the retrieval have not been fully investigated. Due to th e complexity in both
cloud and radiative transfer models, and the n atu ral uncertainties in th e atm ospheric
variables, it is difficult to evaluate how well th e Bayesian approach itself works
in the retrieval applications, and how sensitive th e Bayesian algorithm is to those
assum ptions and uncertainties. It is essential to know the behavior of the Bayesian
estim ator in the rain rate retrieval before adapting the Bayes rules to real-world
applications.
1.4
O b jectives and procedures
The success of the Bayesian algorithm depends on th e accuracy of both the
conditional probability density function (pdf) of microwave observations and the
prior p d f of rain rate, and the in terpretation of th e posterior probability distribution
of rain intensities.
In this study, we present a Bayesian algorithm for rain rate
retrieval over the ocean which has two unique characteristics:
1)
Unlike other Bayesian m ethods, ours is based on explicit functional models of
th e conditional likelihood f ( P \ 9 ) and th e prior distribution n(9) fit to actual d ata
derived from both observations and simulations.
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9
2) Unlike other methods, th e result of this m ethod is not a single ’’b e st” rain rate
b u t rath er the complete posterior probability distribution.
The objectives of this study are:
1) To present explicit, closed-form functions to reasonably approxim ate th e phys­
ical and statistical relationships between rain rates and microwave m easurem ents,
based on b o th observations and model simulations.
2) To provide a system atic sensitivity te st to show th e theoretical benefits of the
Bayesian estim ators in th e retrieval, and to examine w hether th e Bayesian algorithm
can be im plem ented into real-world applications w ithout losing its advantages when
im perfect inform ation is used.
3) To implement our Bayesian algorithm s to real-world applications and inter­
compare w ith other benchm ark algorithms.
4) To assure the adequacy of the specification of the conditional likelihoods th a t
describe the statistical and physical relationships between microwave radiance and
surface rainfall, and to examine th e robustness of th e retrieval algorithm w ith respect
to th e uncertainty of th e prior precipitation distribution. The results will provide
guidance for the quantitative uncertainty of the retrieved rain rate.
This thesis is organized as follows. C hapter 2 introduces th e microwave observed
variables used in th e study, and gives the detailed descriptions of their characteristics
and m ultichannel relationships from th e TM I m easurem ents, of surface rainfall infor­
m ation based on P R m easurem ents, and of th e PR -T M I m atch-up procedures. The
m athem atical and physical basis to specify probability distributions in our Bayesian
algorithm are sum m arized in C hapter 3. A fter th e presentation of th e theoretical
framework, C hapter 4 offers model simulations based on reflectivity d a ta of th e Na­
tional W eather Service’s (NWS) W eather Surveillance D oppler R adar (W SR- 8 8 D)
network and a simplified one-dimensional plane-parallel radiative transfer model to
prove th e applicability of th e Bayesian approach. Following th e proof-of-concept,
a thorough sensitivity test using two random ly generated d atasets are described in
C hapter 5.
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10
To p u t th e perform ance of th e new Bayesian algorithm into context, a num ber
of benchm ark algorithm s are introduced in C hapter
6
. C hapter 7 develops our new
Bayesian algorithm from the PR -T M I m atch-up dataset and employs the new algo­
rithm to real-world applications. This chapter also uses several validation m etrics to
evaluate th e perform ance of all algorithm s applied to identical validation datasets.
Then, when the com parable perform ance of th e new algorithm is found, a q u an tita­
tive analysis is conducted in C hapter
8
to com pute th e uncertainty of th e retrieval.
The purpose of C hapter 9 is to provide a basis to calibrate the NWS W SR- 8 8 D
netw ork-estim ated surface rainfall and to validate th e radiative transfer calculations
based on PR -T M I m atch-up data. It dem onstrates th a t once a more reliable dataset
of surface rainfall m easurem ents becomes available, th e Bayesian algorithm could be
modified from the model simulations. Finally, conclusions and future work will be
presented in C hapter 10.
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11
2. DATA D E SC R IP T IO N
The emphasis of this chapter is to introduce two prim ary microwave observation
datasets used in the study and their general characteristics. T he first d ataset is
microwave radiance m easurem ents from TM I. Due to the intrinsic ambiguities in the
physical relationship between microwave brightness tem peratures and rain rates, we
use a transform ed variable, attenuation index P, to construct our Bayesian models.
The other dataset is reflectivity m easurem ents from P recipitation R adar (PR). PR
offers detailed inform ation about vertical and horizontal structures of rain cloud,
b u t we will only focus on the near-surface rainfall rate in the study. Finally, the
coincident P R and TM I observations are combined together to provide a m atch-up
dataset for b o th th e development and validation of our Bayesian models.
2.1
T M I data
The purpose of th e section is to describe th e ideal response of microwave bright­
ness tem peratures to surface rain rate; to briefly introduce th e atten u atio n index; to
summarize the advantages of its use; and to explore the m ultichannel distributions
of P values from TM I observations.
2.1.1
Ideal response to surface rain rate
P etty (2001) dem onstrated th e dependence of th e idealized brightness tem per­
atures (T b ) on surface rain ra te (R) over th e ocean, when horizontal homogeneity
of rain cloud is assumed (Figure 2.1). The brightness tem peratures a t all channels
in cre ase as rain ra te becomes larger, due to th e emission of cloud and rain water.
However, beyond some satu ratio n point, the brightness tem p eratu res decrease w ith
increasing surface rain intensity as scattering effects become more significant. Higher
microwave frequency tends to satu rate at lower surface rain rate. Furtherm ore, the
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12
300
19 GHz
250
•vT
&
37 GHz
d. 200
I
H
U5
V
•u.
a
■
■da 150
85.5 GHz'
03
100
100
R ain R ate (m m /hr)
Figure 2.1. Idealized brightness temperatures dependence on surface rain
rate (Petty, 2001).
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13
ideal behavior of th e microwave signature in response to rain has revealed a strong
polarization difference in brightness tem peratures over the open ocean, and this po­
larization difference varies w ith frequencies. W hen surface rain rate increases, the
polarization difference decreases and theoretically, the vertically and horizontally
polarized brightness tem peratures will reach to th e same value for th e cases of heavy
precipitation.
The curve in figure 2.1 suggests an ambiguous relationship between microwave
signature and surface rainfall rate, in th a t two very different rain rates can be asso­
ciated w ith a single TB ■ To reduce the ambiguity, polarization differences a t various
channels have been used to eliminate the signal from ocean surface and to isolate th e
effects of the rain cloud. In the next section, a variable representing a normalized
polarization difference will be introduced.
2.1.2
A tten u a tio n index
D efin ition
The attenuation index at a given frequency is defined as (Petty, 1994a)
p = TT v ~ l H ,
J- V.O — 1 H ,0
(2.1)
where T y and T# are the vertically and horizontally polarized brightness tem per­
atures; Tyfl and TH q are th e clear-sky background brightness tem peratures. The
background brightness tem peratures are m ainly affected by w ater vapor, surface
wind speed, and sea surface tem perature, and th u s th e attenuation index reflects th e
inform ation about th e column liquid water. According to th e definition, theoreti­
cally, P index will be in a range of [0, 1], while 1 represents a pixel where th e cloud
is absent, and
0
depicts a very opaque atm ospheric condition.
The advantages to using the attenuation index in th e retrieval algorithm are: 1)
P value decreases m onotonically w ith increasing rainfall intensity. 2) Since P m ainly
reflects the signal coming from the rain cloud, th e atten u atio n index is not sensitive
to the background variabilities. 3) For th e special case of a horizontally homogeneous
rain layer, P and tran sm ittan ce t obey an approxim ate power-law relationship, P ~
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14
t a . Therefore, in this limiting case, P index yields a direct indication of th e rain
cloud transm ittance.
For convenience, we denote th e vertically polarized brightness tem peratures at
10.65, 19.35, 21.3, 37.0, and 85.5 GHz as Tlov, T\gy, T 2 1 V 1 Tyrvi &ftd T ^ y , and
replace V w ith H for horizontal polarization. Similar rules are applied for cloud-free
brightness tem perature, but O is added to present the background quantities. For
example, T i 0 y,o expresses the vertically polarized clear-sky brightness tem perature
at 10.65 GHz.
E stim a tio n o f T y o and T u ,o
In an a tte m p t to obtain the clear-sky background brightness tem peratures, col­
um n w ater vapor, surface wind speed, and sea surface tem p eratu re have to be esti­
m ated first. The estim ation of column w ater vapor is developed from a Radiosonde
observations (RAOBS)-TM I m atch-up process, similar to Alishouse et al. (1990)
and P e tty (1994b). The RAOBS were extracted from the N ational Centers for Envi­
ronm ental Prediction (NCEP) A utom ated D ata Processing (ADP) U pper Air Obser­
vation subsets in January and July, 1999. Since our purpose is to find th e m atch-up
d a ta w ith TM I measurements over th e ocean, only stations located w ithin [40°S,
40°N] and associated land area percentage less th a n 19% of th e 19 GHz footprint
were considered in the subsequent m atch-up procedure. T he m atch-up criteria is th a t
th e distance between RAOBS stations and TM I pixels have to be w ithin 100 km ra­
dius, and th e difference of their observation tim e is not greater th a n 4 hours. The
final sam ple size after m atch-up procedure is 39,360. Then, based on th e regression
analysis, our w ater vapor algorithm is given by th e following:
V = 128.57+ 33.941n(290-T i9y ) - 7 2 .1 3 1 n (2 9 0 -T 2iy) + 10.481n(290-T 37//), ( 2 .2 )
where th e w ater vapor content (V in k g /m 2) is estim ated by th e vertically polarized
brightness tem peratures (K) at 19.35 and 21.3 channels, and horizontally polarized
brightness tem perature at 37 GHz.
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15
Estim ations of surface wind speeds are based on th e m atch-up betw een the TM I
m easurem ents and the buoy wind d a ta from NOAA M arine Environm ental Buoy
database, collected by National D ata Buoy Center (NDBC). Sim ilar to th e estim ation
of w ater vapor, criteria in location and land area percentage were applied to screen
th e buoy stations. TM I d a ta from January and July 1999 were processed to find the
m atch-up pixels. The pixel was selected if it satisfied th a t th e distance was w ithin
100
km radius of the buoy location and th e observation tim e was w ithin
2
hours
of buoy measurements. In addition, TM I pixels w ith possible rain were excluded.
Due to various observed heights for th e surface wind speed a t buoy stations, all
wind speeds were converted to th e stan d ard height of
10
m based on a logarithmical
vertical wind profile. Through th e m atch-up procedure, a d ataset w ith a sample size
of 68,674 was obtained and utilized for regression analysis to derive th e following
relationship:
U = 130.908 + 0.170Tiov T 0.1287jo.fr —0.0347\gy—
0 .1 157W - 0.079T21V -
1 -1 2 1
T37V + 0.543T37H
,
(2.3)
where U is the surface wind speed (m /s); T indicates th e brightness tem peratures
(K) from TM I observations; and th e subscripts represent th e channels and polarized
directions used in the regression.
T he regression equations for estim ations of background brightness tem peratures
were derived from 489 TM I orbits in July 1999 w ith a final sample size of 419,272
pixels. The cloud-free 7g is approxim ated by w ater vapor, surface wind speed and
sea surface tem perature (7s), as th e following equations shown:
Tiw ,o
T Wh ,o
=
154.1 + 0.076V + 0.241/ + 0.477s
= 73.8 + 0.14 V + 0.901/ + 0.247s
ln(300 - T IW>0)
= 4.89 - 0.0072V - 0.001717 - 0.00257s
In(300 - T iqh o )
= 5.39 - 0.0078V - 0.006317 - 0.000527s
ln(300 - Tzrv>0)
= 4.65 - 0.0058V + 0.0005517 - 0.000697s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
16
ln(300 - T37Hi0) =
5.22 - 0.0065V - 0.0080P + 0.00031TS,
(2.4)
where the column w ater vapor content V (kg/m 2) and surface wind speed U (m /s)
are estim ated by equations ( 2 .2 ) and (2.3), and the sea surface tem p eratu re Ts (°C)
is obtained from a 0.5 x 0.5° climatological database (obtained from N A SA /G SFC).
Then, p u ttin g the estim ated background brightness tem peratures into equation 2.1,
estim ated P values for the T M I pixel can be obtained. For convenience, we hence­
forth denote the P value a t 10.65, 19.35, and 37 GHz channels as Pio, Pig, and P 3 7 ,
respectively, and P as th e vector (Pio, P 1 9 , P 3 7 ).
2.1.3
D istrib u tion s o f a tten u ation index o f TM I
To b e tte r understand th e overall distribution of P , 110 TM I orbits during 1999
and 2000 were selected to investigate th e m ultichannel relationship of the P values
a t 10.65, 19.35, and 37.00 GHz. Since th e overall 3-D d istribution is of interest, no
rain screen task is applied in th e following process. TM I m easurem ents were checked
every 10 pixels and only those classified as ’ocean’ pixel were picked. As a result,
th e final sample size is 4.6 x 106 pixels.
Figure 2.2 and 2.3 depicts th e 3-D structure of P values by plotting 2-D contours
of the num ber of pixels at a given P 10 and P37, respectively.
Theoretically, in a
homogeneous case of rain cloud, a unique non-linear relationship in P is expected.
However, in reality, due to various beam-fillings effects, P scatters in 3-D space and
its distribution forms a cloud on the 2-D contours (as shown in th e above figures).
There are some features noted in th e 3-D m ultichannel relationships from TM I mea­
surements. First of all, th e P value w ith th e m axim um num ber of pixels is near (1.0,
. , 1 .0 ), indicating th a t th e estim ations of background brightness tem peratures are
1 0
reasonable. Secondly, the shape of the 2-D contours is asym m etric and elliptical
w ith v a ry in g o rie n ta tio n s . It d e m o n s tra te s t h a t th e P v a lu e s a t th e s e th re e c h a n ­
nels are definitely correlated. An assum ption th a t the response vector elements are
uncorrelated is not appropriate. In addition, for a given smaller P i 0 value which is
often associated w ith a more significant rain rate, th e ellipse shape tends to be like a
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17
P 1 0 = 0 .3
P 1 0 = 0 .7
TMI
TMI
1.2
1.0
0.8
0.8
E 0.6
CL
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.0 0.2 0.4 0.6 0.8
P19
P10 = 0.4
0.0 0.2 0.4 0.6 0 .8 1.0 1.2
P19
1.0 1.2
TMI
P 1 0 = 0 .8
TMI
1.2
1.0
0.8
E
0.8
0 .6
0.4
0.4
0.2
0.2
0.0
0.0
0.0 0.2 0.4 0.6 0.8
P19
P 1 0 = 0 .5
1.0 1.2
0.0 0.2 0.4 0.6 0 .8
P19
P 1 0 = 0 .9
TMI
1.0 1.2
TMI
.2
.0
0.8
0.8
a
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.0 0.2 0 .4 0.6 0.8
P19
P 1 0 = 0.6
1.0 1.2
0.0 0.2 0.4 0.6 0 .8
P19
P 1 0 = 1.0
TMI
1.0 1.2
TMI
.2
.0
0.8
0 .6
0.8
a
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.0 0.2 0.4 0.6 0 .8 1.0 1.2
P19
0.0 0.2 0.4 0.6 0 .8
P19
1.0 1.2
Figure 2.2. Overall m ultichannel distributions of P index represented by
th e num ber of pixels. D ata were selected from 110 TM I orbits during
1999 and 2000. Contours are logarithm ically spaced; actual value is 10x
where x is the contour label, x are plotted for values of [0.5, 1, 2, 3, 4,
5] a t a fixed Pio interval.
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18
P 3 7 = 0 .0
O)
cl
TMI
P 3 7 = 0 .4
0.8
0.8
0 .6
cl
0.4
0.2
0.2
0.0
0.0
P 3 7 = 0.1
cn
1.0 1.2
TMI
0.8
cl
0 .6
0.4
0.4
0.2
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
P10
0.0
0.0 0.2 0.4 0.6 0.8
P10
P 3 7 = 0.8
TMI
0.8
1.0 1.2
TMI
0.8
cl
0 .6
0.4
0.4
0.2
0.2
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
P10
P 3 7 = 0 .3
cl
P 3 7 = 0.6
0.8
a 0.6
O)
0.0 0.2 0.4 0.6 0 .8
P10
TMI
0.6
P 3 7 = 0.2
cn
0 .6
0.4
0.0 0.2 0.4 0.6 0.8 1.0 1.2
P10
a
TMI
0.0 0.2 0.4 0.6 0.8 1.0 1.2
P10
TMI
P 3 7 = 1.0
0.8
0.8
0 .6
a. 0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.0 0.2 0.4 0.6 0 .8 1.0 1.2
P10
0.0 0.2 0.4 0.6 0.8
P10
TMI
1.0 1.2
Figure 2.3. Same as figure 2.2, bu t at a fixed P 37 interval.
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19
banana, showing the existence of non-linear relationships in th e P values at m ultiple
channels.
The TM I observations show th a t m ost P w values are greater th a n 0.4, pointing
out the scarcity of the extrem ely low Pio values. T he scarcity corresponds to the
fact th a t the polarization difference at 10.65 GHz is rarely lower th a n 30 K based on
TM I m easurem ents. A low polarization difference at 10.65 GHz requires widespread
intense precipitation throughout the field of view a t 10.65 GHz, which rarely occurs in
th e real world, even though typhoon and strong convective cases have been included
in the 110 TM I files.
At higher frequencies of 19.35 and 37.0 GHz, th e lowest polarization difference
is found to be a few Kelvin instead of zero, even under th e condition of very thick
rain cloud. Therefore, the num erator of equation 2.1 is seldom equal to zero as well
as P values a t 19.35 and 37.0 GHz. Liu and Simmer (1996) suggested th a t the low
polarization differences may be explained by spherical particles alone. In addition,
coupling the instrum ental noises, th e errors from th e estim ated w ater vapor and
surface wind speed, w ith the uncertainties in th e regression equations, P might be
slightly greater th a n 1, b u t smaller th a n 1.1 in m ost cases.
2.2
P R rain rate
Due to th e operating frequency of 13.8 GHz, P R suffers from strong attenuation
in heavier rain. Therefore, P R reflectivity observations have to be corrected before
they are used to estim ate rain rate. TRM M stan d ard product 2A25 from th e rainprofiling algorithm of the Precipitation R adar contains estim ations of th e vertical
rainfall rate profile, the atten u atio n corrected rad ar reflectivity, and m any other
interm ediate param eters, such as attenuation param eters and coefficients used in the
Z-R relationship. Among them , th e variable of near surface rain intensity is utilized
in b o th the development and validation of th e TM I Bayesian rain rate retrieval
algorithm . For convenience, th e near-surface rain ra te from 2A25 is referred as P R
retrieved (surface) rain intensity hereafter.
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20
The details of the P R retrieval algorithm for 2A25 were given in Iguchi et al.
(2000). T he key points are summarized here. First, due to strong atten u atio n ef­
fects, th e observed P R reflectivity is corrected using a hybrid of th e surface reference
m ethod and th e Hitschfeld-Bordan m ethod w ithout considering th e beam filling ef­
fect of th e rain cloud. The param eters (a and j3) representing th e relationship of
attenuation (k) and corrected reflectivity (Z e) 1 k = a Z @, are adjusted to find a m atch
in the path-integrated attenuation (PIA) between the two m ethods. Then, after ob­
taining th e prelim inary corrected reflectivity, a non-uniform ity index of th e rain is
estim ated from the variability of th e PIA. The atten u atio n derived from th e surface
reference technique will be modified again based on th e non-uniformity, and a corre­
sponding tru e reflectivity could be com puted. Finally, the estim ation of rain ra te is
based on a power-law relationship w ith attenuation-corrected reflectivity, specified
by R = aZg. T he param eters (a and b) are determ ined by rain type, th e height of
freezing level, and the height of the storm top. The ranges for a and b are constrained
from 0.005 to 0.2, and from 0.5 to 1.0, respectively.
Since there is no long-term and w ide-area dense rain gauge network or other
reliable d a ta to provide the true rainfall intensity in th e m arine environm ent, it is
im portant to keep in mind how the P R retrieved rain ra te behaves in comparison
w ith other surface measurem ents before a m eaningful comparison between the PR
(assumed tru e value) and the retrieved rain rates from TM I can be made. Bolen and
Chandrasekar (2000) made simultaneous comparisons w ith P R and S-band polarim etric rad ar (S-POL) and concluded th a t there was a good agreem ent between the
attenuation-corrected reflectivities at altitudes less th a n 2 km to w ithin 1 dBZ with
ground rad ar observations. In addition, due to th e low sensitivity of P R to light rain
rates, a 3.8% underestim ation is indicated in th e mean areal rainfall. Furtherm ore,
some cross-validations were made com paring P R rain rate estim ations w ith surface
m easurem ents over India, China, and A ustralia. Those results dem onstrated th a t
P R accum ulated rainfall was around 20% lower th a n th a t obtained from th e surface
m easurem ents (according to personal com m unications w ith Dr. Iguchi).
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21
The rad ar a t th e Kwajalein TRM M validation site was supposed to provide an
invaluable oceanic dataset for our validation purpose. However, Schumacher and
Houze (2000) dem onstrated th a t when the ’ground tr u th ’ ra d a r was adjusted up­
ward by 2 dBZ, then the rainfall estim ate of P R at an altitu d e of 3 km would be
consistent w ith th a t derived from th e Kwajalein radar.
Houze et al. (2001) also
pointed out th a t necessary calibration corrections have to be m ade in th e Kwajalein
radar m easurem ents. Oki et a l (1998) attem p ted to validate th e spaceborne P R d a ta
using ground-based radars, and their results ended up suggesting th a t P R could be
used as a space calibrator due to high reliability of P R data. In summary, P R rainfall
rate estim ates might not be able to represent th e absolute tru e rain intensity due
to the uncertainty in the attenuation correction and Z-R relationship. However, the
P R is b e tte r calibrated th a n a typical ground based rad ar (personal communications
w ith Dr. Y uter), and has an extraordinary advantage of th e sim ultaneous observa­
tions w ith TM I. Therefore, th e P R provides a unique way to validate TM I retrieval
algorithm s, which contains long term , various environments and precipitation sys­
tems. In chapter 9, we will show th a t th e Bayesian algorithm can be adjusted when
even more reliable m easurem ents of rainfall intensity become available.
Note th a t the 2A25 outputs include an estim ate of th e error in th e near-surface
rain rate, indicating th e uncertainties of param eters used and of th e m easurem ent
errors inherent to th e instrum ent or th e observations. W hen attem p tin g to use the
error estim ate as a quality control of P R rainfall intensity, it flags out a large fraction
of pixels associated w ith significant precipitation, in which more interest is placed.
Therefore, we simply use estim ates in th e study for now, b u t th e m axim um rain
intensity is not allowed to exceed 150 m m /h r.
2.3
P R -T M I m atch-up procedure
The 2A25 P R retrieved rain rates has a 4-km resolution on th e precipitation map.
For the purposes of developing or validating a TM I retrieval algorithm representing
rain intensity at a 15-km resolution, th e P R estim ated rain rates were averaged by
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22
Gaussian weighting functions and interpolated to the location of TM I pixels. Since
the P R swath is much narrower th a n th e TMI, the interpolation was only conducted
for the TM I pixels th a t were located ± 20 TM I high-resolution pixels from th e nadir
(around 160 km wide). In addition, the lightest rain intensity th a t P R is able to
detect is around 0.5—0.7 m m /h r, and hence th e smallest area-averaged P R rain rate
at a 15 km resolution should be ab o u t 0.035—0.049 m m /h r. Therefore, we assigned
a minimum threshold of 0.04 mm / h r for the interpolated P R rainfall intensity, i.e.,
any pixel w ith a lower interpolated rain rate th a n the threshold was excluded in the
study. For convenience, the rain ra te estim ated through this stage is referred to as
the P R (surface) rain rate hereafter.
Note th a t the P R reflectivity m easurem ents might suffer from th e sidelobe con­
tam ination when th e m ain beam of the P R is off-nadir, and the resulting errors
would propagate to the rain ra te estim ations and interpolations. To reduce th e un­
certainty, only near-nadir pixels were included in the developm ent of th e Bayesian
retrieval algorithm . In the study, th e near-nadir pixel is defined as th e one w ith a
TM I pixel index between 102 and 106 (i.e., 104 (nadir) ± 2 ) . As to the validation
process, while the qualitatively horizontal structures of precipitation system are of
interest, th e off-nadir inform ation will be shown as well. Otherwise, only near-nadir
d a ta will be used to calculate verification measures and to quantify th e retrieval
ability for all algorithms.
Figure 2.4 depicts the probability distribution of the near-nadir P R 15-km inter­
polated rain rates. D ata periods contain January, April, July, and O ctober of 1998.
It is clear th a t the distribution of rain rate is very close to th e lognorm al function
denoted as follows:
-
eXP
logN(r\fj., cr) = <
0
,
r =
0
5
^ (In r - g f
,
where logN expresses the lognorm al function and rain rate r is in m m /h r. /i and a
are th e param eters of the lognorm al distribution in unit of m m /h r, b o th determ ining
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23
R a i n R a t e D i s t r i b u t i o n — P R ( 1 9 9 8 , n e a r —n a d i r )
0.20
15
0.10
0 .0 5
0.00
0.001
0 .0 1 0
0 .100
1.000
10.000
100.000
Rain R a t e ( m m / h r )
Figure 2.4. Probability distributions of th e near-nadir P R 15-km interpo­
lated rain ra te are depicted by lighter-colored dots. D ata were calculated
from th e 2A25 P R near-surface rain rate estim ates in January, April, July
and O ctober of 1998, and a cut-off value of 0.04 m m /h r was applied. T he
resulting sam ple size is around 2.4 million. The fitted lognormal distri­
bution is also presented here by a dashed curve.
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24
th e mode and the variance of th e rain rate probability distribution. W hile trying
to fit th e distribution w ith a lognorm al function, th e pair of param eters, (/j, a), was
found to be around (-2.7, 2.2). p and a change w ith varying d atasets from -2.9 to
-2.6, and between 2 and 2.3, respectively.
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25
3. A LG O R ITH M B A SIS
As we m entioned in chapter 1, th e Bayesian posterior density function is de­
term ined by a prior rain rate distribution th a t represents our knowledge or belief,
and a conditional likelihood th a t statistically describes th e physical relationships
between rain ra te and microwave signal. Therefore, there are three key elements
in our algorithm : th e prior distribution, th e conditional likelihood, and th e esti­
m ator interpreting th e posterior distribution. In this chapter, th e generic forms to
model the prior and conditional distributions are introduced, and two commonly
used estim ators for th e posterior pdf to represent th e single-pixel retrieval are also
discussed. Note th a t th e generic forms for conditional pdfs in our algorithm are used
to fit b oth model sim ulations (chapter 4) and actual observations (chapter 7) in the
development of our Bayesian algorithm.
3.1
P rior d istrib u tion
The probability distribution of the surface rain rates has been estim ated by many
researchers using various forms for which the m ean and stan d ard deviation m ight
not vary a lot, b u t tail behavior differs appreciably. A lognormal function is widely
used to approxim ate th e rain rate distribution (Houze and Cheng, 1977; Kedem
and Chiu, 1987), and th e PR -TM I m atch-up d a ta (shown in th e previous chapter)
also dem onstrated th e adequacy of th e approxim ation. Therefore, in our algorithm ,
the prior rain ra te likelihood is modeled by a lognormal density function, which is
d e n o te d as log N( [i, cr) a n d d e sc rib e d b y e q u a tio n (2.5).
Since (p, a) varies w ith different rain ra te observation datasets, a num ber of
sets of (/a, cr) will be tested in the study. First, Kedem et al. (1990) docum ented
the estim ations for th e param eters using a m inim um chi-square estim ation based on
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26
the G A RP (Global Atmospheric Research Program ) A tlantic Tropical Experim ent
(GATE) observations. In their fitting processes, the GATE d a ta were tru n c a ted at
1 m m /h r and represented a 4 x 4 km averaged instantaneous rain rates. A lthough
retrieved rain rates from microwave m easurem ents are more often based on 15 x 15
km area, their study is still valuable for the investigation of uncertainty of th e prior
rainfall distribution.
Two sets of th e param eters from their study were adapted:
(1.140, 1.047) and (1.043, 1.205), which were estim ated from different tim e periods
and only had slight differences in //, and a.
In th e previous chapter, PR -T M I m atch-up dataset represented th e rain rate
measurem ents on th e basis of 15 x 15 km area. In addition to a given threshold of
0.04 m m /h r, th e m atch-up d ataset suggested a set of (-2.7, 2.2) for describing the
prior rain rate distribution. It dem onstrates th e applied cutoff value has dram atic
influence on th e fitted values of param eters, and thus the resulting characteristics of
tails will be different significantly. In addition to th e set of (-2.7, 2.2), th e param eters
of (-2.8, 2.0) will also be used due to th e variations of the /j and a. Furtherm ore, for
a purpose of theoretical tests, two sets of (0, 1) and (0, 2) were also employed in the
study. To b e tte r understand the difference in th e modes, means, variances, and the
tails between those prior specifications, figure 3.1 depicts the lognorm al distributions
with th e above various param eter com binations. The differences between th em will
be discussed in more detail in chapter 5.
3.2
C on d ition al likelihood
The conditional likelihood in our Bayesian algorithm is a m ultivariate probability
distribution since rain ra te and observation vector are all involved.
In this kind
of highly dim ensional condition, we would like to specify th e underlying density
functions in term s of some param etric families. A param etric form provides a simple
function to characterize the density function, and th e advantage of th e use is the
ability to obtain an analytical solution.
T he param etric forms have to be used
w ith caution to ensure th a t th e equations reflect the behavior of d a ta as close as
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27
1.2
1.0 ^
!l
logN( 0 . 0 0 , 1 . 0 0 0 )
0.8 .
logN( 1 . 1 4 ,1 .0 4 7 )
logN( 0 . 0 0 , 2 . 0 0 0 )
logN( 1 .0 4 ,1 .2 0 5 )
c 0 .6
<D
Q
J\
_______ l o g N ( - 2 . 8 0 , 2 .0 0 0 )
0.4
0.2
0.0
0
2
4
6
Rain r a t e ( m m / h r )
8
10
0.010
(b)
logN( 0 . 0 0 , 2 . 0 0 0 )
0 .0 0 8
logN( 0 . 0 0 ,1 .0 0 0 )
logN( 1 .1 4 ,1 .0 4 7 )
0 .0 0 6
*ccn
<D
logN( 1 .0 4 ,1 .2 0 5 )
0 .0 0 4
l o g N ( - 2 . 8 0 , 2 .0 0 0 )
■\
\ \ ..............
logN( —2 . 7 0 ,2 .2 0 0 )
0.002
20
40
60
80
Rain r a t e ( m m / h r )
100
Figure 3.1. Comparisons of th e prior rain rate probability distributions
used in this study.
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28
possible. A non-param etric approach w ith less rigid assum ptions is an alternative,
which allows d a ta to speak for themselves. However, it is beyond the scope of the
paper.
A univariate case is used first to illustrate th e approach of specifying th e condi­
tional pdfs. Let us take th e P 3 7 variable and model its relation to rain rate w ith a
closed-form function. Since m ost values of P index are w ithin around a range of [0,
. ] (see chapter 2 ), the conditional p df can be simply constructed by
1 1
/ ( P 3 7 |P ) oc P 37(a - P 37) exp
1
:(-P
2ct32
*37
~~
h
3
Y ,P
3 7
G [0,a],
(3.1)
where / ( P 3 7 |P ) is the likelihood of P 3 7 a t a given rain rate R; a is a constant. /i 3
and <r3 are functions of rain ra te and determ ined by the fitting to th e d ata. The
distribution basically follows a norm al distribution, bu t it is bounded in [0 , a] and
modified by the term of P 37(n — P 37). In addition, th e likelihood is exactly equal to
zero a t the two extreme boundaries. T he distribution is skewed to th e right if most
pixels achieve saturation associated w ith larger rain rates, while the distribution
skews to th e left under a condition th a t th e atm osphere is almost tran sp aren t to
this frequency. The distribution tends to be more sym m etric if th e mode is around
0.5 to 0.6. Unlike the norm al distribution, the param eters of /i 3 and <r3 here are
not exactly equal to th e m ean and th e stan d ard deviation of th e distribution, bu t
they are nevertheless related to th e location of th e mean and the variance of th e pdf.
Furtherm ore, based on th e actual observations from TM I, the a is assigned as 1.1 in
the study.
W hen the observation vector contains (Pio, P
i 9 ,
P 3 7 ) ,
the distribution of P at a
given rain rate can be approached hierarchically:
/ ( P |P ) = / ( P 3 7 |P ) / ( P 1 9 |P 37, P
) / ( P
i o | P i
9,
P 3 7 , R)
(3.2)
where / ( P i 9 |P 37, R ) represents th e distribution of P 1 9 when P 3 7 and R are fixed; and
th e / ( P 1 0 IP 1 9 , P 3 7 , R) describes th e p d f of P index a t 10.65 GHz while th e other
three variables are known. Similarly to th e univariate case, those conditional pdfs
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29
can be approxim ated as
/(P 3 7 I-R )
oc P37(a - P 37) exp
/ ( P 1 9 I-P3 7 , R )
oc P 19(a - P i9) exp
f (P 1 0 1P1 9 j P 3 7 >R)
oc P 10( a - P i o ) e x p
where /i 2 and
<72
1
( _
,;,p2
2 <7‘
'2' 2
:(Pio —A41 )2
(3-3)
2 cT !2
are functions of rain ra te and P 3 7 ; and
//1
and <7 i are functions of
rain rate, P 37 and P 1 9 . Again, these param eters are all determ ined by fitting the
d ata b oth from model sim ulations and observations.
Finally, the conditional likelihood of m ultivariate observation vector at a given
rate could be obtained combining th e equation set (3.3) w ith th e equation (3.2).
Note th a t the last term
set of
(P 1 9 , P 3 7 ,
/ ( P 1 0 I P 1 9 , P 3 7 ,
R) requires inform ation about P 10 a t a given
P ). Since three variables are fixed while fitting this conditional pdf,
the sample size has to be large to provide enough d a ta points for the fitting.
3.3
E stim ators o f th e p osterior distrib ution
Based on the Bayes theorem (equation 1.1), th e posterior distribution can be
derived as
rrfpin
( 1 >
fS41
Jf(P\R)w{R)dR’
( ■)
where th e denom inator is used to normalize th e posterior distribution, and the inte­
gration w ith respect to rain rate (R ) in th e four-dimensional space will be perform ed
numerically. Theoretically, th e upper lim it of rain rate in the lognormal distribution
goes to infinity, then the integration should be conducted to the same upper limit.
Practically, a reasonable upper bound R max has to be given to define the integration
interval during the num erical calculations. We set R max to be 80—100 m m /h r in
our study, depending on th e train in g dataset. In addition, th e interval of th e rain
ra te for integration is
increases to
0 .2
0 .0 1
a t a very light rain situation (less th a n
0 .2
m m /h r), and
m m /h r a t th e range of [0 .2 , P max] m m /hr.
P index ranges from 0 to 1.1 for all channels, and the resolution of 0.01 is used
in the calculation and norm alization of th e conditional pdfs.
Once th e posterior
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30
distribution is ready, two most common estim ators were taken: th e m ean and the
maximum likelihood estim ate of th e posterior probability distribution, denoted as
MEAN and MLE, respectively.
Then, th e corresponding Bayesian estim ates are
stored in a 3-dimensional lookup table w ith a resolution of 0.2 for each P value for
the single-pixel retrieval.
In th e next chapter, th e algorithm basis will be applied to model sim ulations to
dem onstrate the retrieval ability.
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31
4. P R O O F-O F-C O N C E PT
The purpose of the chapter is to prove th e concept of the Bayesian approach from
model simulations. This sim ulated training dataset was generated from NWS W SR88D network observations and a simplified one-dimensional plane-parallel radiative
transfer model. To statistically characterize the physical relationships between rain
rate and microwave signature shown in th e simulations, the conditional likelihood is
derived using explicit, closed-form functions to fit the data. The generic form has
been described in the previous chapter, and the associated coefficients fitted to the
model simulations will be given in th e chapter. Finally, the algorithm is applied to
one of the WSR-88D network d atasets to dem onstrate the retrieval ability of th e
Bayesian approach.
4.1
R ain rate estim ates from N W S W SR -88D network
Beam-filling errors arising from inhomogeneities of rain clouds have been a sig­
nificant problem in rain rate retrieval from passive microwave m easurem ents. The
coupling of non-uniform ity of precipitation w ith th e nonlinear relationship between
microwave radiance and rain ra te produces a footprint-averaged microwave signature
th a t is not uniquely related to th e footprint-averaged rain rate. T he errors can be
reduced if beam-filling effects are considered in the algorithm in either an implicit
or explicit way.
To b e tte r account for th e heterogeneous beam filling effects in our algorithm ,
we in c lu d e d re a listic a lly v a ry in g p r e c ip ita tio n s p a tia l s tru c tu re s in th e tra in in g d a ta
using th e NWS W SR-88D network product of 1 km US N ational Base Reflectivity.
T he reflectivity d a ta were com posited from the 154 sites in th e US w ith an hourly
tem poral resolution and a spatial resolution of 1 km. Observations in precipitation
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32
and clear air modes were b o th collected in the product.
Since characteristics of
precipitating systems are of interest, m easurem ents from th e clear air mode were
excluded in the study. However, some d a ta still showed th a t there were apparent clear
air returns (most of them less th a n 20 dBZ) even in the precipitation mode, especially
during the nighttim e, which might mislead us to the wrong precipitation p attern s
and then translated to incorrect probability distributions. To avoid th e errors from
those noises, the reflectivity values were cut off at 20 dBZ before performing the
subsequent rain intensity estim ations and forward microwave radiance calculations.
The surface rain rate for each rad ar pixel was modeled by th e most commonly
used Z-R relationship (M arshall and Palm er, 1948)
Z = 200.R1'6,
(4.1)
where Z is th e observed rad ar reflectivity factor (dBZ) and R is the rain rate in
m m /hr. Since reflectivities higher th a n about 55 dBZ (corresponding to a rain rate
of around 100 m m /hr) usually indicate the presence of hail, th e estim ates of rain
rates might be questionable in th a t case. Therefore, an arb itrary upper limit of the
radar pixel rain rate was set as 150 m m /h r, and estim ated intensities were averaged
over 15 x 15 km domains to serve as th e ’’tru e ” instantaneous rain rates.
22 radar reflectivity files were random ly picked during July and August 2002.
Each file includes 4736 x 3000 observations all over th e U nited States. After fil­
tering out th e suspicious returns, rainy pixels comprised 2.5% of d a ta for each file.
Furtherm ore, the strength of rad ar retu rn for precipitation mode was recorded in
particular reflectivity levels w ith an interval of 5 dBZ, resulting in an obviously dis­
crete property in the histogram of th e rain rates. To com pensate for this deficiency,
reflectivity values were system atically adjusted by a shift of ± 2 and ± 1 dBZ, so a
to ta l of 110 files were processed in th e study.
4.2
Polarization calculations
The mass extinction coefficient of suspended cloud w ater Ke i was com puted
from Liebe et al. (1991) assum ing a tem perature of 0°C, and th e corresponding
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33
Table 4.1
Param eters a and b in th e approxim ation of liquid w ater extinction
coefficient.
^e,l
a
b
10.65
0.0244
0.002956
1.18759
19.35
0.0785
0.01585
1.09403
37.00
0.261
0.06896
1.01876
85.50
0.932
0.2799
0.84693
Channel (GHz)
values are listed in Table 4.1 w ith unit of m 2/kg.
The relationship between the
volume extinction coefficients of rain water kej. and rain rates (R) was estim ated
from Mie theory, assuming spherical rain drops w ith a liquid w ater tem perature of
10°C, and using the M arshall-Palm er drop size distribution and microwave complex
index of refraction formula of Liebe et a l (1991). A power-law form was found to
approxim ate the relationship well (P etty, 1994b),
A;e>r = a R b.
(4.2)
Values of the coefficients a and b for each channel are also shown in Table 4.1.
The optical depth r r was m odeled as
Tr
~ Z f k etr(R),
(4.3)
representing the contribution of rain w ater only. Since there was no information
about th e values of the freezing height Z f in th e rad ar reflectivity product, a value
of 3 km was assumed for Z f here. In real-world applications, th e freezing height is
a free param eter in th e Bayesian retrieval algorithm , so a series of values of freezing
heights will be used to calculate th e microwave radiance, and th en the corresponding
relationships of P values and rain rates can be derived. More discussions are given
in C hapter 9.
The optical
following.
depth caused from th e suspended cloud
w ater was modeled as th e
The suspended cloud w ater content (L in k g /m 2)
was sim ulated as a
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34
random noise generated from a lognormal distribution lo g N (//£ , 0 l) .
The h l , ol
were given by
^
=
aL =
0 .1 5 + 0.03#
0.25 + 0.045#,
(4.4)
where R is th e surface rain ra te in m m /hr. Once th e cloud w ater content is specified,
th e optical depth a ttrib u te d to th e suspended cloud w ater (-p) can be given as
Tt = KejL.
(4.5)
A simplified 1-D plane-parallel model was used to com pute the brightness tem ­
peratures. T he radiative transfer equation is w ritten as
T b = (1 —t)Tj\ + etTs + (1 — t ) ( l —s)tTA,
(4-6)
where the sim ulated brightness tem perature TB is determ ined by th e transm ittance
t, the specular emissivity of th e surface (sea w ater in th e study) e, th e surface
tem perature Tg, and th e air tem perature # 4 . T he tran sm ittan ce t is defined as
t = e x p |- s b ]'
(4-7)
where 0 is th e incident angle and r is the optical depth. T he surface tem perature
was approxim ated from th e extrapolation of the tem p eratu re profile, assum ing a
tem perature at the freezing level of 0°C and a lapse ra te of 6.5 K /km .
T 4 was
estim ated by the air tem peratu re at the mid point between th e surface and the
freezing level. Since the emissivity of the ocean is polarized, b o th the vertically and
horizontally polarized radiances were obtained.
As mentioned in chapter
2
, th e attenuation index is used in th e study to serve
as the observation vector instead of brightness tem peratures . In order to com pute
P index, values of background brightness tem peratures are also needed. The cloudfree brightness tem peratures (TVp and TH,o) are approxim ated as follows. Based on
equation (4.6), we can derive th e polarization difference A TB as
A T b = Tv — T h = (ev —£fj)[t(Ts — T a ) + t 2TA}.
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(4.8)
35
Define th e t a, ti, and tr as th e tran sm ittan ce attrib u te d to th e atm osphere, the
suspended cloud water, and the rain w ater, respectively, and t\ as th e to ta l tran s­
m ittance (the product t a, ti, and tr). Then, th e atten u atio n index can be given
as
A Tb
T v - TH
A T b ,o
T v ,o ~ T h ,o
h { T s - T a ) + t \T A
t a{Ts ~ T a ,o ) + t l T A,o ’
where TAjo represents the air tem perature when th e suspended cloud w ater and rain
w ater are absent. We assume TA ~ TA}o in th e simulations. In addition to th a t the
order of th e first term for both the num erator and denom inator is much smaller than
the second term , the attenuation index can be approxim ated as
^
^
=
2
) 2 =«?£
(4.10)
It is found th a t t a will be cancel out in th e calculations of P, and th u s th e absolute
value of t a does not have a great effect on P. Note th a t P etty and K atsaros (1992)
also suggested a power-law relationship betw een P value and th e transm ittance,
(4.11)
where a « 2 ,and the transm ittance t in equation (4.11) presents th e contribution
from th e suspended cloud w ater (ti) and rain w ater (tr). Since th e atten u atio n index
considers th e ratio of the polarization differences, and th e contribution of th e atm o­
spheric transm ittance will be canceled out, th e cloud-free brightness tem peratures
can be easily approxim ated employing a tran sm ittan ce of unity (zero optical depth)
in equation (4.6), in which case TB = sTsIn th e radiative transfer equation (4.6), th e tran sm ittan ce a ttrib u te d to th e sus­
pended cloud w ater should be included to calculate the brightness tem peratures. In
our calculations, we com puted th e contribution of rain w ater first by replacing t w ith
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36
t r to yield microwave brightness tem peratures and prelim inary values of P index at
different channels. Then, using th e relationship described by equations (4.10) and
(4.11), the contribution from the suspended cloud w ater was added and th e modified
P value (P*) for each pixel can be estim ated from
P* = t}tl = t\P.
(4.12)
Finally, when th e P values were ready, a zero-mean G aussian random noise was
added to P value w ith standard deviations of 0.01,
.
0 0 2
, and 0.02 for 10.65, 19.35,
and 37.00 GHz channel, respectively. At this point, a sim ulated d ataset is obtained,
containing inform ation about rain rate, P w , Pw , P 3 7 for each d a ta point.
It has to be noted th a t since the rad ar com posite reflectivity d a ta only provide
horizontal rain rate estim ates, some assum ptions have to be m ade in our model about
other param eters, such as surface and air tem peratures th a t describe the atm ospheric
environment. One m ight argue th a t th e sim ulated microwave signature would be very
sensitive to those param eterizations. This concern points out th e rationale of the
usage of P index in th e study, which is able to distinguish th e m ajor signal of th e
rain cloud itself from the background variability, and thus P is not sensitive to the
assum ptions we made. In addition, the 3-D stru ctu re of rain cloud is ignored in the
radiative transfer calculations, and it is a potential source of errors for the com puted
attenuation index. Since there is no easy way to deal w ith th e issue, we will use this
1-D model to provide simulations for this study.
4.3
C om parison w ith P R -T M I m atch-up d ata
Before using the model simulations to construct th e Bayesian algorithm , two
more processes have to be conducted for th e sim ulated d ataset. One is th a t a rain
ra te cutoff has to be applied to th e dataset, and th e other is to ensure the repre­
sentativeness of the sim ulated microwave signatures. The m ajor reason for applying
a cutoff rain ra te is provided as the following. T he prior rain rate distribution will
always have th e largest probability a t zero rain ra te in th e real world. Theoretically,
th e posterior rain ra te distribution for non-rainy pixels should obey a d elta function
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37
with an impulse at zero rain rate. Practically, if th e zero rain rates are not excluded
in the constructions of the prior and conditional pdfs, and th e algorithm is applied
for every single pixel, th e com puted posterior rain ra te p d f for non-rainy pixels will
have finite probabilities a t non-zero rain rates, and it leads to widespread non-zero
estim ations everywhere. Therefore, we will separate the rain screening as another
issue and only include rainy d a ta points in th e development of our Bayesian algo­
rithm . Then, apply our retrieval algorithm only for the rainy pixels. In addition,
th e cutoff value is set as small as possible to make the lower lim it of th e retrieval
range smaller. Evans et al. (1995) used a cutoff value of 1 m m /h r for their model
simulations, b u t found out th a t th e cutoff led to an overestim ation in the areas of
light rain rate. Therefore, in this study a cutoff of 0.04 m m /h r was applied to keep
more inform ation and to obtain b e tte r estim ation for the regimes of light rain rate.
As a result, the final sample size was 4.7 million after th e cutoff.
To evaluate the representativeness of th e radar-radiative simulations, th e m ulti­
channel relationships of the sim ulated microwave signatures are com pared w ith those
of the TM I measurements. Figure 4.1 depicts th e 2-D contours of num ber of pix­
els at a given Pio or P 3 7 based on actual TM I observations (shown in th e first and
th ird column) and from radar-radiative sim ulations (the second and fourth colum n).
The contours are shown in a logarithm ic scale. Note th a t the dataset of the TM I
m easurem ents in figure 4.1 is same as th e one shown in figure 2.2 and 2.3. Plots
are drawn here again for the convenience of comparisons. T he 3-D stru ctu re of P
values from rad ar simulations dem onstrates a great sim ilarity to th e actual TM I ob­
servations in b oth locations and orientation in th e 2-D slices. T he distribution of the
radar sim ulations is generally broader th a n th e TM I data. This discrepancy might
be a ttrib u te d to the random noise added to th e P index. In addition, note th a t th e
sample sizes from TM I and rad ar sim ulations are on th e same order. However, TM I
d a ta include non-raining pixels, while rad ar d ataset was employed a rain rate cutoff.
T he difference might also contribute to th e broadness th a t th e sim ulated contours
have.
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38
P 1 0 = 0 .3
P 1 0 = 0 .3
0 .0 0 .2 0 .4 0 .6 0 .8 1.0 1.2
P 1 0 = 0 .4
P 3 7 = 0 .0
RADAR
0 .0 0.2 0.4 0.6 0.8 1.0 1.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2
P 1 0 = 0 .4
TMI
RADAR
P 3 7 = 0.2
1.2
1.2
TMI
TMI
P 3 7 = 0 .0
0.0 0.2 0.4 0.6 0.8
P 3 7 = 0.2
1.0
1.0
1.0
1.0
0.8
0.8
0.8
£
0.2
0.6
a
0.4
0.4
S 7 ..............
0.2
0.0
4)
0 .0 0.2 0.4 0.6 0 .6 1.0 1.2
P 1 0 = 0 .5
TMI
P 1 0 = 0 .5
RADAR
RADAR
1.2
1.2
0.8
2 0.6
RADAR
0.6
r #
Jffnix
CL 0.6
/ j )
0.4
0.4
0.2
0.2
0.0
0 .0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
P 3 7 = 0 .4
&
P 3 7 = 0 .4
RADAR
P 3 7 = 0 .6
RADAR
0.0 0.2 0.4 0.6 0.8 1.0 1.2
P 1 0 = 0 .6
TMI
P 1 0 = 0 .6
RADAR
P 3 7 = 0 .6
TMI
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0 .0 0.2 0 .4 0 .6 0 .8 1.0 1.2
P1 0 = 0.8
TMI
0 .0 0.2 0.4 0.6 0.8 1.0
P 1 0 = 0 .8
0.0 0.2 0.4 0.6 0.8 1.0 1
RADAR
P 3 7 = 0.8
1.2
1.0
SI 'tlvl
<n 0 8
TMI
0 .0 0.2 0.4 0.6 0.8 1.0 1
P 3 7 = 0 .8
RADAR
1.2
#
1.0
0.8
0.4
0.4
0.2
0.2
#
0 .0 0.2 0.4 0.6 0 .8 1.0 1.2
P 1 0 = 1 .0
TMI
P10 = 1 .0
RADAR
P 3 7 = 1.0
TMI
P 3 7 = 1.0
RADAR
Figure 4.1. Contours of the num ber of pixels based on TM I d a ta (the
first and th ird columns) and radar-radiative simulations (the second and
fourth columns). Contours are logarithm ically spaced; actual value is 10x
where x is the contour label, x are p lo tted for values of [0.5, 1, 2, 3, 4,
5].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
In this chapter, radar-radiative sim ulations are only used to prove th e concept
of the Bayesian approach, so the reflectivity-estim ated surface rainfall has not been
calibrated by any other precipitation m easurem ents yet. Therefore th e exam inations
of physical relationship between rain ra te and atten u atio n index are not conducted
here.
However, despite the uncertainty in th e rainfall estim ations from radar re­
flectivity, the simple forward radiative model simulations have shown a surprising
consistency w ith the TM I observations in 3-D atten u atio n index distributions. An­
other aspect of application of model sim ulations and further analysis of relationship
between rain ra te and microwave signatures will be provided in chapter 9.
4.4
B ayesian algorithm from sim ulations
The generic forms representing th e statistical and physical relationship between
surface rainfall and microwave observations have been presented by equations (3.2)
and (3.3). Using those analytical forms to fit to th e radar-radiative simulations, the
corresponding param eters describing each individual conditional probability distri­
bution are obtained as follows:
a
= 1.1
Hz
= 1.6exp(—0.24#) —0.62
0-3
= 10.3 • 0.07L88/T (1 .8 8 ).^(i- 8 8 -i) exp(_o.07.R) + 0.06
/U
= bo + b\P%j + 62
b0
= 0.25 exp(—0.024#)
bx
= 0.76 - 0.01# + 0.0005#2
b2
= 0.4 + 0.0035#
S2
= ( 0 . 5 - 1 . 0 8 ^ 7 + 3.11P3 - 0 . 3 \ / # + 0 . 0 5 # - 0 .0 0 3 # 15) -e x p (—10#37)
02
= 0.13 —0.084#37 — 0.004#37
Hi
= c0 exp [ - w ^ ( # i9 - cxf ] + c3 + <Si
ZC2
Co
= -0 .3 5 - 0.3#37+ 1.5 exp [ - 0 . 5 ( # 3 7 - 0 . 3 ) 2] + 0 . 0 1 # - 0 . 0 8 ^
Ci
=
- 0.000004#3
1. 1
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40
c2
= 1.08 + 0.0037? —0.27 exp(P37) —0.00487? exp(P37)
c3
= 0.99 + 0.03P37 - 0.002P + 0 . 0 3 3 ^ - 0.037P37\ / P - c0
S1
= 0.06 - 0.023P19 + 0.03P37 - 0.02 exp(P37) - 0.0002P
oi
= 0 .1 3 - 0 . 0 9 / p ^ + 0.015,
(4.13)
where the unit of rain rates (7?) is m m /h r, and P index is dimensionless. In those
equations, th e complexity gradually increases when more variables are considered.
Combining this set of fitted param eters w ith an assum ed prior surface rainfall
probability distribution, the posterior probability density function can be obtained.
In the next section, we will dem onstrate an example to assess th e applicability of
concept of th e Bayesian approaches.
4.5
R etrieval based on sim ulations
To evaluate the applicability of the Bayesian algorithm, th e physical model de­
rived from th e sim ulated dataset is applied to one of the NW S W SR - 8 8 D network
datasets to reveal the retrieval ability. It has to be made clear th a t even though
th e surface rainfall inform ation from over-land radars is used in th e simulations, our
Bayesian algorithm is an ocean algorithm th a t is being evaluated. Therefore, in the
following figures, only radar composite reflectivity is plotted over th e land w ith coast
lines. O ther sim ulation-related variables are assumed over th e ocean w ithout plot­
ting th e US continent. In addition, the aim of the section is to provide an overall idea
about the qualitative perform ance of sim ulated retrievals, and th u s th e quantitative
comparisons are not given here. D etailed discussions about th e perform ance of the
Bayesian algorithm derived from th e actual P R and TM I observations will be given
in C hapter 7.
Figure 4.2 shows the 1 km NWS W SR- 8 8 D network com posite reflectivities on
A ugust 13,
2 0 0 2
, after suspicious rad ar returns have been removed (see section 4.1).
T he estim ated rain rate w ithout adjusting reflectivity is m apped in figure 4.3, and
th e corresponding m odel-calculated attenuation index a t 10.65, 19.35, and 37.00
GHz are depicted in figures 4.4, 4.5 and 4.6, respectively. These plots show some
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(dBZ)
41
n0r_ 020 813 _1040
Figure 4.2. 1km NWS W SR- 8 8 D network composite reflectivies for
10:40am, A ugust 13, 2002, w ith suspicious rad ar retu rn s removed.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Simulated
Rain R a te ( m m / h r )
42
Figure 4.3. Plots of sim ulated 15 x 15 km rain rate based on th e W SR8 8 D composite reflectivity for 10:40am, A ugust 13, 2002, and a Z-R re­
lationship proposed by M arshall and Palm er (1948).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.4. Sim ulated P i 0 for th e d a ta of 10:40am, A ugust 13, 2002.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
:n ,
*• '•• '
A*
^ • **
. \X^L* .
-?(
w ^ fr j *^
..;i
V> ■*. * * *• "j
^5r?o
* P
* • • ‘ ■»- "t •
A /
i •*,
* V
■'
:
j?£W
^ X ?
a*/
V i '.
v .*' *•»'
Figure 4.5. Sim ulated P i 9 for th e d a ta of 10:40am, A ugust 13, 2002.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.6. Simulated P 3 7 for th e d a ta of 10:40am, A ugust 13, 2002.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Retrieved
Rain R a te ( m m / h r )
46
Figure 4.7. Plots of retrieved rain rate for the d ata of 10:40am, August
13, 2002, based on the new Bayesian algorithm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
strong precipitation systems associated w ith very low P\g and P 3 7 values in the
Midwest and th e east coast, and other scattered convection structures in th e west and
south. Figure 4.7 reveals th e retrieved rain rate using the prior rainfall distribution
of log 77(1.14, 1.047) and th e m ean estim ates.
The retrieval dem onstrates very
consistent horizontal precipitation distributions to those of th e reflectivity-estim ated
rainfall intensity. Since th e retrieval was only processed for th e pixels in which either
one of P 10, P i 9 , P 3 7 values is less th a n
. , some very light precipitation areas (~
0 8
0.01 m m /h r) w ith high P values were not retrieved and are not shown in th e 4.7.
Up to this point, the Bayesian rain ra te retrieval algorithm using explicit, closedform functions fitting to th e d a ta has shown high retrieval ability. M ost im portantly,
our algorithm is not only able to give the single-pixel retrieval, b u t also provides
a continuous posterior probability distribution. Note, however, th a t th e retrieval
yielded from this given prior rain ra te distribution and th e specified conditional
likelihood associates a bias in some degree. The errors m ight be introduced from 1)
imperfect prior o r/a n d 2) im proper conditional likelihood functions o r/a n d 3) the
m ean estim ator.
Instead of reducing th e errors tuning th e prior and conditional
probability distribution or using another estim ator to interpret th e posterior pdf, a
more fundam ental and im portant issue for the development of Bayesian algorithm s
is to understand how well th e Bayesian approach works when perfect inform ation is
applied; and w hether the Bayesian algorithm still preserves its retrieval ability when
im perfect functions are used. To clarify this issue, a detailed theoretical sensitivity
analysis will be introduced in th e following chapter.
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48
5. SE N SIT IV IT Y TEST
The success of a Bayesian retrieval m ethod depends on w hat assum ptions are
made in th e conditional likelihood, how th e prior precipitation distribution is speci­
fied, and how the posterior density function is interpreted. In order to system atically
analyze th e effects of th e aforementioned uncertainties on th e retrieval, theoretical
sensitivity tests were conducted using two synthetic datasets where th e ” tru e ” condi­
tional and th e prior distribution were idealized. One synthetic d ataset was generated
by a simple conditional pdf, which was characterized by a covariance m atrix and thus
only linear relations were included. The other dataset was generated by th e modelderived physical model th a t is m entioned in th e previous chapter to statistically
describe th e physical relationships between rain ra te and atten u atio n index. The
la tte r d ataset considers b o th linear and non-linear interactions of th e observation
variables.
One thing worthy of m ention is th a t a theoretical simulation te st was perform ed
as well in th e study of Evans et al. (1995). The m ain purpose of th eir test was to de­
term ine which covariates had th e greatest influence on th e accuracy of th e retrieval.
Nevertheless, their results still suffered from the imperfection of th e cloud model and
radiative transfer model, and the assum ptions in b o th conditional and prior distri­
butions. It is hard to trace the sources of errors due to the high complexity in b o th
models and dimensions. Since th e synthetic d a ta in the study were generated from
the condition where all of the probability density functions were exactly known, our
tests are able to evaluate th e contribution of each com ponent (the conditional, prior
and posterior probability distributions) w ithout th e influences of th e uncertainty in
th e assum ptions.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49
The aim of the theoretical exam inations here is to provide answers to th e fol­
lowing questions: 1) W hat is th e performance of th e Bayesian-approached retrieval
algorithm in different rain rate ranges, especially under extrem ely heavy precipitation
condition? 2) How do th e various interpretations (e.g., using different estim ators) to
th e posterior distribution affect the retrieved rain rates? 3) Is th e posterior density
function sensitive to b o th th e conditional and prior probability distributions? 4)
W hich p art, conditional distribution, prior pdf, or th e interpretation of th e posterior
function, will have more significant im pacts on the retrieval? 5) W hen th e uncer­
tainty of prior distributions and the imperfectness of the physical model exist, which
is the case in m ost retrieval applications, is th e retrieval still meaningful?
5.1
T h e prior and conditional distributions
Since one of the m ain purposes of the sensitivity tests is to b e tte r understand
the behavior of retrieval a t ranges of higher rain intensity, logN(0, 2) was served as
th e prior distribution to generate the random variates in an a tte m p t to prevent a
scarcity of high rain rates in the training dataset.
Two physical models will be used for the element of conditional likelihood. In
addition to th e complete conditional pdfs derived from th e m odel sim ulations (in
chapter 4), another way to specify the conditional likelihood is shown here as well. As
m entioned in chapter 1, th e m ultichannel relationships in other Bayesian algorithm s
were expressed by a linear covariance m atrix. To investigate th e influence of the
covariance m atrix on the retrieval, a similar specification for conditional p df was
built as well in th e study. Like th e univariate situation, th e conditional probability
density function representing the physical relationships between P and R can be
w ritten as
f ( P \ R ) oc P i o ( a - P 1 0 ) P 1 9 ( o - P
19
) P 3 7 ( o - P 37)-e x p
i ( p - p ) c r 1( p - p ) T , (5.1)
where th e param eter /j, and th e covariance m atrix C are similar to th e (p, a 2) of the
univariate case.
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50
Table 5.1
Coefficients of a^, bfl and cfl.
Channel(GHz)
K
10.65
0.75
0.03
0.30
19.35
1.35
0.05
-0.30
37.00
1.55
0.10
-0.50
fi is defined as
At = (1 1 1 , 1*2 , 1*3 )',
W = an,iexp [-b n,iR ] + cliii,
(5.2)
where th e subscripts 1, 2, and 3 describe th e quantities at 10.65, 19.35, and 37.00
GHz, respectively.
Values for th e param eters of a^, bfil and cfl are ta b u la ted in
Table 5.1, while i expresses th e corresponding channel. T he /q a t a given channel
exponentially decreases w ith respect to rain rates, bu t the decreasing ra te and the
achieved minim um vary w ith microwave frequencies. P 37 tends to decrease more
rapidly w ith R th a n P 19 and P i0 do, and th e lower minimum lim it can force th e peak
of
P 37
to be very close to zero if rain rates become large. On th e contrary,
P 10
are
seldom lower th a n 0.4 based on th e actual TM I observations.
In addition,
u
C 21
C31
C
\2
C 22
C 32
C
l
3
CO
C
C 33
The covariance m atrix defines the variance and correlations of P index a t th e three
channels. T he diagonal elements indicate th e variations associated w ith P i0,
and
P 3 7
P 1 9 ,
themselves, while th e non-diagonal elements express th e covariances of the
three variables. It is noted th a t th e arb itrary covariance m atrix in th e stu d y has to
be positive definite in order to keep th e exponential term in a stable mode, which is
expected physically and produces reasonable probabilities for th e conditional distri­
bution. Several sets of values for th e covariance m atrix are given and tested later.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
Finally, since the synthetic d a ta include rain rates up to 100 m m /h r, th e retrieved
range of R in th e sensitivity test is set to be [0, 100] m m /h r as well for th e purpose
of comparisons.
5.2
D esign ed exp erim ents
Various experim ents w ith different com binations of th e training d ataset (i.e., re­
trieval target), and the conditional and prior distributions used in th e retrieval calcu­
lations, are designed for the sensitivity te st (summarized in table 5.2). Experim ent
ID is nam ed by starting w ith a letter S or R, expressing the source of the training
dataset based on th e simple model (D1 dataset) and radar-radiative sim ulations of
the complicated model (D2 d ataset), respectively. Note th a t D1 and D2 b o th were
generated along w ith the a same prior distribution logN{0,2). The values of th e covariance m atrix elements given in D 1 d ataset are th e same as the experim ent SO. D1
was random ly generated by a rejection m ethod (von Neumann, 1963), and th e to tal
sample size was 1 million. For D2 dataset, the random variates were generated from
those conditional distributions of th e com plicated model (as equations (3.2), (3.3)
and (4.13) shown). First, generate rain rates based on the prior distribution, then
generate P 3 7 from / ( P 3 7 |P ) by using a rejection m ethod. Next, similarly, generate
P i 9 and P 10 from th e corresponding conditional density function, th e n a full vector
(P , P37, P 1 9 , P 1 0 ) can be returned. The sample size was 5 million in D2 dataset.
Before conducting th e sensitivity tests, it is essential to ensure th a t th e training
datasets indeed follow the designed probability density functions; th e random vari­
able generating process does not introduce any other significant errors into th e data;
and the physical model is able to represent th e behavior of actual observations to
some degree. Figures 5.1 and 5.2 illustrate th e distributions of the surface rain rate,
multichannel relationships, and m arginal distributions of P index a t three channels,
which are calculated from theoretical functions of the sophisticated m odel and from
the corresponding D2 training d ataset, respectively. Plots have revealed th a t the
analytical solutions and random ly-generated d a ta m atch w ith each other very well.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
Table 5.2
Inform ation of designed experim ents in sensitivity tests, including the
experim ent ID, the training d ataset, and th e specifications of the prior
and conditional likelihoods applied to th e Bayesian algorithm.
Exp ID
D ata
7r(R)
Cu
C 22
C 33
C\2
C l3
C 23
SO
D1
logN( 0 , 2 )
0 .0 1
0.04
0.06
0.015
0 .0 2 0
0.045
SI
D1
logN( 0,1)
0 .0 1
0.04
0.06
0.015
0 .0 2 0
0.045
S2
D1
l o g N ( l . U , 1.047)
0 .0 1
0.04
0.06
0.015
0 .0 2 0
0.045
S3
D1
l o g N ( l M 3 , 1.205)
0 .0 1
0.04
0.06
0.015
0 .0 2 0
0.045
S4
D1
U n if o r m [ 0,100]
0 .0 1
0.04
0.06
0.015
0 .0 2 0
0.045
S5
D1
logN (0, 2)
0 .0 1
0.04
0.06
0.000
0.000
0.000
S6
D1
logN{0, 2)
0 .0 2
0.04
0.06
0.015
0 .0 2 0
0.045
S7
D1
logN{ 0,2)
0 .0 1
0.0 8
0.06
0.015
0 .0 2 0
0.045
S8
D1
lo g N ( 0 , 2)
0 .0 1
0.04
0.12
0.015
0 .0 2 0
0.045
R0
D2
logN( 0,2)
Com plicated model from equation (4.13)
R1
D2
logN( 0,1)
Com plicated model from equation (4.13)
R2
D2
logN(1.14:, 1.047)
Com plicated model from equation (4.13)
R3
D2
logN ( I M S , 1.205)
Com plicated model from equation (4.13)
R4
D2
U n if o r m [ 0 ,100]
Com plicated model from equation (4.13)
R5
D2
l o g N (0 , 2 )
Param eterization from equation (5.4)
R6
D2
logN( 0,2)
0.01
0.04
0.06
0.015
0.020
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.045
53
(a ) Rain ro te
(c ) P19 vs. P37
(b ) P10 vs. P19
2.0
1.0
r- 0.6
c
V
o
0.4
0.4
0.2
0.2
0.5
0.0
0
10
5
20
15
0.0
0.0
0.0
0.2
0.4
0.6
0 .8
0 .0
1.0
0.2
0.4
0.6
0.8
1.0
R oin R o t e ( m m / h r )
6
(d ) M arginal p d f (P 1 0 )
( f) M arginal p d f (P 3 7 )
(e ) M arginal p d f (P 1 9 )
2.0
4
5
3
D e n sity
4
>,
3
a>
o
2
1
a>
o
1
n E. . . i .
0.0
2
0.2
.
0.4
0.6
0.8
0 .5
n F ^ - r - T '^ . . .
1.0
0.0
0.2
0.4
0.0
0.6
0.8
1.0
0.0
0.2
0.4
0 .6
0.8
Figure 5.1. T he probability distribution of rain rates and the joint and
m arginal pdfs of th e P vector based on analytical solutions of the sophis­
ticated model. The joint pdfs are plotted for [0.05, 1, 5, 7, 10, 12].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
54
( c ) P19 vs. P37
(b ) P10 vs. P19
(o ) Rain rate
2.0
0.8
0.8
E
o
a> 0.6
X
0.4
0 .4
0.2
0.2
0.5
0.0
0
10
5
20
15
0.0
0.0
0.0
0.2
0.4
0 .6
0.8
1.0
0 .0
0.2
0.4
0.6
0.8
1.0
R oin R o te ( r r > m /h r )
6
(e ) M arginal p df (P 1 9)
(d ) M arginal pdf (P 10)
2.0
4
( f) M arginal pdf (P 3 7 )
1 i ' ' ■- r ' " i ■ ■ ■ i « ' ■
5
3
4
2
2
1
0
0.0
1
0.5
oE^rrrT.
0.2
0.4
0.6
0.8
1.0
0 .0
0.2
......
0 .4
0.6
0.8
0.0
1.0
0.0
0.2
0.4
0.6
0.8
P10
Figure 5.2.
dataset.
Same as figure 5.1, b u t for the random ly generated D2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
55
This, in turn, implies a sufficient sample size and a proper generating m ethod for
the training dataset.
The m ultichannel relationships of P values in th e D2 d ataset are conveyed by
figure 5.3. Com pared to th e figure 4.1, the sim ilarity to each other reveals a proper
specification of the physical model. Recall th a t th e prior distribution of D2 d ataset
tends to generate more points w ith higher rain rates, so the contours of th e D2 dataset
are broader in 2-D space, and a little more d a ta are seen in th e lower P\ q values
th an the original radar-radiative simulations.
However, th e general m ultichannel
relationships are very consistent.
Control runs (SO and RO) represent the experim ents th a t th e Bayesian retrieval
m ethod uses the exactly same conditional and prior pdfs w ith th e training d ata. An
analysis of these two control experim ents provides insight into th e inherent uncer­
tainty of the retrieval, since each pdf is perfect and no assum ption is m ade in the
algorithm. The sensitivity of th e algorithm to th e specifications of th e prior d istri­
butions is evaluated by experim ents SI to S4, and R1 to R4. S5 to S8 are designed
to investigate the dependency of the retrieval on the assum ptions of th e correlations
of the response vector in th e case of the simple physical model. S5 uses a diago­
nal m atrix, expressing th a t th e three observed variables are uncorrelated w ith each
other. Note th a t S5 is sim ilar to w hat the previous work assumed. S6, S7, and S8
express the situations where one of the diagonal elements is twice th a t of th e control
run while the others rem ain th e same as th e control run.
To investigate th e sensitivity of th e Bayesian algorithm to varying param eterizations in the physical model, a different approach of conditioning / ( P 2|P 3, R) of the
complicated model is tested in experim ent R5. T he change is only m ade via the
param eterization of p 2 as follows:
Hi
=
exp(60 + hi In R) + 52
b0 =
-0 .4 5 1 + 0.365P37 - 0.14 exp [ - 0 .5 (- 37 ~ o°7 ° 8Q) ] + 0.07
b\
—0.44 exp(—14P37) —0.04 — 0.04 exp [ — 0.5( 37^ ^ — ) ]
p
=
n QO ^
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
P 1 0 = 0 .3
0.0
0 .2
0.4
0.6
P 1 0 = 0 .4
P37 = 0 .0
RADAR
0.8
1.0
0.0
1.2
1.2
1.0
1.0
N
a 0.6
cl
0.2
0.4
.......................................
0.6 0.8
1.0
1.2
P 1 0 = 0 .5
0.2
0.4
0.6
P 1 0 = 0 .6
0 .2
0.4
0.6
P 1 0 = 0 .8
0.0
0.8
1.0
1.2
RADAR
0.6
0.2
0.2
0 .0
0.6
0.4
0.4
0.0
0.4
P37 = 0 .2
RADAR
1.2
0.0
0.2
RADAR
0.2
0.4
0.6
P10 = 1 .0
0.0
0.0
1.0
1.2
0.0
1.0
1.2
0.0
1.0
0.2
0.4
0.6
0.2
0.4
0.6
P 3 7 = 0 .8
RADAR
0.6
0.6
P 3 7 = 0 .6
RADAR
0.8
0.4
P 3 7 = 0 .4
RADAR
0.8
0.2
1.2
0.0
0.2
0.4
0.6
P 3 7 = 1.0
RADAR
0.8
1.0
1.2
RADAR
0.8
1.0
1.2
RADAR
0.8
1.0
1.2
RADAR
0.8
1.0
1.2
RADAR
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 .0
0.2
0.4
0.6
0.8
1.0
1.2
Figure 5.3. M ultichannel relationships by plotting the num ber of pixels
at a given P w (left column) or given P 3 7 (right column) for D2 training
dataset. Contours are logarithm ically spaced; actual value is 1 0 r where
x is the contour label, x are plotted for values of [0.5, 1, 2, 3, 4, 5].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
57
62
=
R — 4
1.08 • exp(—3 OP3 7 ) exp [ - 0.5(— -— ) ]
O
(5.4)
Results from the test here will give an indication about th e degree of how accurate
th e physical model has to be and how much uncertainty th e algorithm is able to
tolerate.
Experim ent R6 is designed to apply th e P -R relationships of th e simple model
into the algorithm , while a complicated d ataset D2 is the targ et to be retrieved. It
helps to evaluate the adequacy of simple assum ptions of the conditional p d f in the
retrieval algorithm , when in fact th e d ataset has much larger complexities.
5.3
R esu lts for theoretical test
5.3.1
Intrinsic uncertainty o f th e algorithm
An im portant property of the Bayesian analysis is th a t a probability statem ent
can be m ade about the realization of th e retrieval based on th e derived posterior
distribution. The objective of the section is to assess the intrinsic uncertainty of th e
retrieval and the difference in inference m ade by th e MEAN and MLE, when th e ideal
physical model and exact prior distribution are known in th e Bayesian algorithm .
The inherent uncertainty of th e Bayesian retrieval algorithm is illustrated in
figure 5.4 by plotting the standard deviation (m m /h r) of the posterior p d f a t a given
P 1 0 value for th e experim ent RO. Since observed P 1 0 values are rarely lower th an
0.4, only results where P 10 is greater th a n 0.4 are shown in the figure. In th e case of
larger P 10 values, say 0.8 to 1.08, th e contour shows an overall small uncertainty in
th e retrieved rain intensity, except in some regions where P 3 7 is reaching saturation.
For instance, at a given Pw value of 0.9, th e com binations from [0.0, 0.6] of P 19
and [0.0, 0.1] of P 3 7 produce around 10 m m /h r deviations.
Note th a t this kind
of P vector is often seen in th e real-world satellite observations, and th e Bayesian
retrieval will be associated w ith a considerable uncertainty even when th e perfect
prior/conditional pdfs has been applied.
Figure 5.4 also reveals th a t the m axim um deviation for each slice increases w ith
decreasing P i0 values. Generally speaking, th e large variances occur in two regions
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58
P10 = 0 .7 0
P10 = 1.08
1.0
0.8
0.8
iv 0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.0
0.2
0.4
0.6
P19
0.8
1.0
-10
0 .0
0.2
0.4
0.6
P19
0.8
1.0
0.8
1.0
0.8
1.0
0.8
1.0
P10 = 0 .6 0
P10 = 1.00
0.8
0.8
^ 0.6
1-0
Q.
0.4
0.4
0.2
0.2
0.0
-10
0.0
0.2
0.0
0.4
0 .6
P19
0.8
1.0
0.0
0.2
0.4
0.6
P19
P10 = 0 .5 0
P10 = 0 .9 0
1.0
0.8
0.8
0.4
0.4
0.2
0.2
0.0
-10
0 .0
0.2
0.0
0.4
0.6
P19
0.8
1.0
0 .0
0.2
0.6
P19
P10 = 0 .4 0
P 10 = 0 .8 0
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.4
-10
0 .0
0 .2
0.0
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
Figure 5.4. 2-D contours of the stan d ard deviations of th e posterior
probability distribution in the F 3 7 vs. Pig space at a fixed Pig value for
experim ent RO. Contours are plotted in an interval of 5 m m /h r. Regions
outside the zero value of th e contour indicate an impossible P vector due
to a zero m arginal density.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
of th e 2-D contour space. One is associated w ith a low Pio, a satu rated P 3 7 , bu t
a m oderate Pig value. T he other region happens in th e com bination of a low P i0,
high Pig, b u t a m ediate P37. Those response vectors express precipitating systems
associated w ith particular horizontal structures, which are not commonly seen in
observations. Under the condition, a relatively large uncertainty m ight have existed
in the physical model, and then it m igrates into th e posterior distribution. Therefore,
when this kind of vector is observed, caution has to be exercised in th e corresponding
retrieval, and further additional inform ation or a more precise physical model might
be required.
Figure 5.5 depicts th e histogram of the retrieved rain rates for RO experiment.
The title for each subplot is made of three components. R R indicates th e specific
range of rain rates, where th e retrieved rain intensity is draw n if th e corresponding
rain rate of the training d a ta point is in this range. It is followed by a num ber
showing how m any d a ta points are picked to compose th e histogram . MEAN or
MLE describes which estim ate is used to interpret the posterior distribution, the
mean or th e m axim um likelihood value. T he pair of th e num bers on th e upperright corner in each subplot indicates th e m ean and th e stan d ard deviation of the
histogram , while th e percentage shown right below it tells us th e proportion th a t the
retrieved rain rates are in th e same range w ith th e actual rainfall intensities w ith
respect to th e whole histogram . Note th a t for th e tru e range R R lower th a n 30
m m /hr, th e histogram is plotted in a logarithm ic scale.
The results from RO experim ent using MEAN estim ates suggest th a t th e Bayesian
algorithm in the case is able to retrieve th e light rain ra te very well, although the
general m ean value of the histogram s m ight be slightly larger th a n th e range of
the training data. For the m oderate intensity (7—15 m m /h r), th e retrieval m ethod
captures around 46% d a ta points, and th e m ean value is about right. T he algorithm
tends to underestim ate the rain rates when the actual intensity in training d a ta
increases to a heavier range (15 to 75 m m /h r). Meanwhile, th e associated stan d ard
deviation of th e histogram starts to increase as well. Even so, th e retrievals still
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
R R = [ 0 .1 ,
0 .2 ]
430957 -
MEAN
RR = [ 1.0,
0.8
5
RO
( 0 .2 4 , 0 .1 5 )
670027 -
R R = [1 5 .0 , 3 0 .0 ]
0.30 f
MEAN
222467 -
MEAN
(1 4 .0 1 , 8 .6 9 )
( 2 .6 0 , 1 .9 3 )
32.4%
4 8 .1 % .
4
H is to g ra m
2 .0 ]
R0
28.6%
0.6
3
E
2
X
o
S’ 0.1 5
0.2
1
0
0.0
1.0
10.0
0.1
100.0
R R = [ 0 .2 ,
0 .4 ]
1.0
10.0
0.1
562461 -
= [ 2.0 ,
MEAN
4 .0 ]
597394 -
R R = [3 0 .0 , 5 0 .0 ]
0.20
R0
MEAN
3.0
ro:
( 5 .5 7 , 3 .
( 0 .3 7 , 0 .2 5 )
2.5
1.0
10.0
98415 -
MEAN
( 2 4 .2 7 ,1 3 .2 3 )
21.2% :
2 7 .9 7
39.6% '
100.0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
0.1 5
2.0
E
o
E
□
cn
o
g10.10
X
x
0 .05
0.5
0.0
0.1
1.0
10.0
100.0
0.1
0 .6 ]
100.0
1 0.0
0.00
0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R R = [ 0 .4 ,
1 .0
374641 -
2.0[---------------------
R = [ 4.0 ,
MEAN
7 .0 ]
40
MEAN
395442
R R = [5 0 .0 , 7 5 .0 ]
0.1 4 •
R0
21.6%
60
100
80
R e tr ie v e d RR ( m m / h r )
0.4
( 0 .6 2 , 0 .4 3 )
20
49896 -
MEAN
(45.8,6,17.58)
35.0%
34.1
0.3
E
o
g*0.2
^ 0.08
x
0.1
0.0
0.1
1.0
10.0
100.0
0.1
R R = [ 0 .6 ,
1 .0 ]
1.0
1 0 .0
498245 -
R R = [ 7.0, 1 5 .0 ]
MEAN
RO
R0
391911 -
MEAN
( 1 0 .0 9 , 6 .8 7 )
•
30.0%
20
40
60
80
100
R e tr ie v e d RR ( m m / h r )
0.30
1.4
0
10 0 .0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
- 46 .4 % -
R R = [7 5 .0 ,1 0 0 .0 ]
0.10
R0
24139 -
MEAN
( 6 9 .6 3 ,1 4 .2 0 )
42 .0%
0.08
J
0.20
1.0
E
E
o
o
o> 0.8
E 0.06
g* 0.15
■± 0.6
x
0.04
0.10
0.4
0.02
0.05
0.2
0.00
0.0
0.1
1.0
10.0
R e tr ie v e d RR ( m m / h r )
1.0
1 0.0
R e tr ie v e d RR ( m m / h r )
0.00
0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
Figure 5.5. Histograms of retrieved rain ra te a t different rain ra te ranges
for R0 experiment. T itles contain inform ation about range of tru e values,
sample size, and the estim ator. Numbers in parentheses represent the
m ean and standard deviation of th e histogram . Percentages present the
fraction of retrieved rain rates located in th e correct range.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
61
encompass around 30% d a ta of the tru e range. For th e extremely large rain rate,
there is around 42% of d a ta in the correct rain range, and th e mode of th e histogram
falls w ithin the tru e range as well. In th e control experim ent, the Bayesian algorithm
shows the ability of retrieving the rain intensity over all ranges, even for th e case
w ith extrem ely heavy precipitation (greater th a n 75 m m /hr).
The histogram of MLE for experim ent RO (as figure 5.6 shown) discloses a po­
tential problem when th e inference about th e posterior distribution is m ade by the
m aximum likelihood estim ate. Results from th e light rain regime (from 0.1 to 4
m m /hr) dem onstrate th a t the perform ance of th e algorithm is quite good. However,
the underestim ation for interm ediate intensities (4 to 30 m m /h r) is significant. More­
over, there are two distinct regimes in th e histogram when the tru e range R R goes
up greater th a n 30 m m /hr. T he one associated w ith a lower rainfall ra te dom inates
in the range of [30, 75] m m /h r, and it m ight result in retrieved rain rates at least 20
m m /h r smaller th a n the tru e rain intensity. T he other peak becomes dom inant w ith
th e increasing tru e rain rates, and reflects th e consistency of th e retrieved intensity
with the tru e rain rates, giving around 40% d a ta points in th e correct range. Under
th e extremely heavy precipitation, th e mode w ith th e lower rain intensity becomes
much smaller, and the algorithm produces m ost of th e retrieval a t an appropriate
intensity range.
It is im portant to understand th e physical m eaning of the two various regimes
in the M LE histogram and their associated im pact on th e overall perform ance of
th e Bayesian m ethod.
Figure 5.7 shows posterior distributions for some specific
P vectors in experim ent R0. Some posterior distributions have a single m aximum
over the entire rain ra te range, b u t some have two local maximums. T he single
maximum depicts a condition th a t th e P vector expresses very clear inform ation
about w hat rain rate will be m ost likely happened when b o th physical relationships
and prior knowledge are taken into account. T he double local maximums imply th a t
two com petitive rain intensities exist. T he peak on the left may indicate a scene
of widespread stratiform precipitation associated w ith a smaller rain rate, while the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
R R = [ 0.1 ,
4
RO
0 .2 ]
430957 -
R R = [1 5.0 , 3 0 .0 ]
MLE
( 0.09, 0.11)
RO
( 1.20, 0.61)
RO
56.1%
16.8%
o
cr>
o
0.4
I
1
0.5
0
0.2
0.0
0.0
1.0
1.0
1 0.0
R e tr ie v e d RR ( m m / h r )
0 .4 ]
: 2.5%
E 0.6
E
R R = [ 0 .2 ,
4
RO
MLE
10.0
100.0
1.0
MLE
RR = [ 2.0 ,
( 0.19, 0.19)
597394 -
RR = [3 0 .0 , 5 0 .0 ]
MLE
14.5%
98415
MLE
(14.92,18.95)
RO
10.7%
49.2%
0.8
3
?2
4 .0 ]
10 .0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
562461
( 6.17, 9.24)
0.8
3
0.1
222467 -
2.0
o
x 0-4
0.2
0
0.05
o.oo a
0.0
0.1
1 .0
1 0 .0
1.0
1 0 0 .0
R e tr ie v e d RR ( m m / h r )
RR = [ 0.4 ,
4
RO
0 .6 ]
20
10.0
374641 -
MLE
R R = [ 4 .0 ,
( 0.36, 0.26)
7 .0 ]
RO
27.6%
395442
R R = [5 0 .0 , 7 5 .0 ]
MLE
60
80
49896 -
12.6% .
0.6
MLE
(41.5[4,28.94)
2.99, 3.52)
0.8
3
40
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
43.4%
§ 0.15
o' 2
£ 0.4
1
0.2
0
0.0
0.1
1.0
1 0 .0
1 0 0 .0
0.1
R e tr ie v e d RR ( m m / h r )
R R = [ 0.6 ,
2.0
RO
1 .0 ]
1 .0
1 0 .0
20
1 0 0 .0
498245 -
MLE
R R = [ 7.0 , 1 5 .0 ]
( 0.62, 0.35):
RO
. %:
42 8
391911 -
MLE
R R = [7 5 .0 ,1 0 0 .0 ]
0.14
3.8!4, 5.50)
E
o
coh
60
80
241 3 9
(7 4 .7 5 .2 1 .4 4 )
RO
0.12
0.10
: 3.5% .
0.8
40
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
40.5%
:
E
%>0.08
o
i
I
0.4
x 0.06
0.04
0.5
0.2
0.0
0.0
0.1
1.0
10.0
R e tr ie v e d RR ( m m / h r )
0.1
1 .0
1 0 .0
0.02
0.00
0
R e tr ie v e d RR ( m m / h r )
20
40
60
80 100
R e tr ie v e d RR ( m m / h r )
Figure 5.6. Same as figure 5.5, b u t using MLE estim ations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
0.12
(0.8 0 , 0.50, 0.10)
(0 .8 0 , 0.20, 0.05)
(0 .6 0 , 0.02, 0.01)
• f 0.08
c
o0)
S '
a.
0.06
0.04
0.02
0.00
0
20
40
60
Roin Rate ( m m / h r )
80
100
0.035
(0 .6 8 , 0.28, 0.01)
0.030
(0 .6 0 , 0.28, 0.0 1 )
0.025
a
0.020
-9 0.015
0.010
0.005
0.000
0
20
40
60
80
100
R ain R a te ( m m / h r )
Figure 5.7. Examples of derived posterior rain rate distributions at some
given P vectors in experim ent R 0 . The observation vector (P 10, P i 9 . P 3 7 )
is presented by the three num bers in parentheses.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
64
second peak suggests th a t a strong convection precipitating system may occur in
th a t 15 x 15 km area. This plot suggests th a t th e M LE estim ate will increase from 5
to 60 m m /h r if th e Pio decreases from 0.64 to 0.60 and Pw and P 3 7 rem ain the same.
However, this kind of decrease in P value is possibly attrib u te d to th e noise from the
instrum ent, atm ospheric condition, and the calculations of brightness tem peratures,
not necessarily meaning a dram atic change in th e rain intensity of th e scene.
The fact th a t there are two very different regimes in th e M LE histogram under
high rain rate cases brings up a drawback for th e algorithm. It implies th a t a slight
change in the one of P values may make th e retrieved rain ra te ju m p from one regime
to the other, and introduce a discontinuity in th e retrieval product. Moreover, the
characteristics of two distinct regimes are also the reason th a t th e peaks in the
histogram of the MEAN estim ates are significantly underestim ated.
T he values
are around between 15 and 43 m m /h r for the d a ta range of [30, 50] and [50, 75]
m m /h r, respectively, since th e MEAN is trying to find th e averaged point of these
two regimes.
5.3.2
S en sitiv ity to th e prior know ledge
The purpose of the section is to evaluate th e sensitivity of th e Bayesian algorithm
to the prior surface rainfall distribution. Analysis of experim ents S I to S4 and R1 to
R4 helps to dem onstrate how th e retrieved rain rate changes to th e various specifi­
cations of the prior distribution when th e form ulation of physical model rem ains the
same. Since results of experim ents SI to S4 (simple physical model) indicate similar
inform ation to experim ents R1 to R4 (physical model derived from sim ulations), this
section only gives detailed outcome of R1 to R4. ’S’ series experim ents are needed
here since it m anifests th a t the property of being sensitive to th e prior inform ation
is not localized in a specific physical model, and thus th e following statem ents are
representative.
The prior probability density functions utilized in th e sensitivity test are all
plotted again in th e figure 5.8 to highlight their individual characteristics and for
th e convenience of illustration. Com pared to th e control prior density function, the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
P rio r PDF
logN(0.00,2.000)
logN(0.00,1.000)
0.8
logN( 1.14,1.047)
>.
c 0.6
<D
Q
logN( 1.04,1.205)
0 .4
0.2
0.0
0
2
4
6
Rain r a t e ( m m / h r )
8
10
P rio r PDF
0.012
logN(0.00,2.000)
0.010
logN(0.00,1.000)
togN(1.14,1.047)
0 .0 0 8
logN(1.04,1.205)
>,
c 0 .0 0 6
(D
O
0 .0 0 4
0.002
0.000
20
40
60
80
Rain r a t e ( m m / h r )
100
Figure 5.8. Plots of th e prior probability distributions used in th e sen­
sitivity tests. To b e tte r illustrate the differences, ranges of [0, 10] and
[10, 100] m m /h r are respectively shown in (a) and (b).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
RR = [ 1.0,
2 .0 ]
670027 -
R R = [1 5.0 , 3 0 .0 ]
MEAN
222467 -
MEAN
0.30
0.8
( 5.97, 2.70)
( 1.78, 0 .9 0 )
( 0.38, 0.12)
0.25
48.4%
0 .0% .
'
0.9% •
0.6
0 .20
0
CP
O
1 0.4
E
o
X
x
x
E
o
E
? 0.15
0 .1 0
0.2
0.05
0.0
0.1
1.0
10.0
100.0
0.00
0.1
R e tr ie v e d RR ( m m / h r )
R R = [ 0 .2 ,
0 .4 ]
1.0
1 0 .0
1 0 0 .0
0.1
562 4 6 1 -
RR = [ 2 .0 ,
MEAN
4 .0 ]
597394 -
MEAN
RR = [3 0 .0 , 5 0 .0 ]
0.4 r
3.0
1 0 .0
1 0 0 .0
98415 -
MEAN
0 .2 0
( 3.13, 1.34)
0.48, 0.18)
2.5
1 .0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
8 .4 8 , 4.0 8 )
56.8%
38.9% -
0.3%
0.15
6o
2.0
cr>
o
cn
I
X
0.05
0.5
0.0
0.00
0.1
1.0
10.0
100.0
0.1
R e tr ie v e d RR ( m m / h r )
R R = [ 0 .4 ,
0 .6 ]
374641
1.0
10.0
100.0
0
R e tr ie v e d RR ( m m / h r )
-
MEAN
R R - [ 4.0 ,
7 .0 ]
395442 -
( 0.66, 0.27)!
MEAN
60
80
100
R R = [5 0 .0 , 7 5 .0 ]
49896 -
MEAN
80
100
0.14
4.02, 1.66)
41.1%
3 1 . 1% :
40
R e tr ie v e d RR ( m m / h r )
0.4 f
2 .0
20
0 .1 2
0 .1 0
E
E
o
CP
o
o
x
X
0.08
0.06
0.04
0.5
0.02
0 .0
0.00
0.1
1 .0
1 0 .0
1 0 0 .0
0.1
R e tr ie v e d RR ( m m / h r )
R R =[ 0 .6 ,
1 .0 ]
1.0
1 0 .0
1 0 0 .0
498245 -
MEAN
R R =[ 7.0, 1 5.0 ]
391911
40
60
-
MEAN
R R = [7 5 .0 ,1 0 0 .0 ]
24139 -
MEAN
0 .1 0
(;4.68, 2.10)
0.25
41.3% .
E
o
20
R e tr ie v e d RR ( m m / h r )
0.30
( 0.96, 0.44)'
o
0
R e tr ie v e d RR ( m m / h r )
:
: 1 1 .2 % *
0.08
0.20
£ 0.06
E
o
0.8
S’ 0.15
x 0.6
0.04
x
0.10
0.4
0.02
0.05
0.2
0.0
0.00
0.1
1 .0
1 0.0
R e tr ie v e d RR ( m m / h r )
0.00
0.1
1 .0
1 0 .0
R e tr ie v e d RR ( m m / h r )
0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
Figure 5.9. Same as figure 5.5, bu t for experim ent R l.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
67
prior distribution in R1 experim ent, lo g N (0,1), has a larger proportion of probability
over th e lower rain rate (0.4—4 m m /h r) and relatively smaller probabilities at very
light rain rates and beyond 5 m m /h r. The prior probability is very close to zero
at a value of 30 m m /hr. The property of the prior pdf leads th e algorithm to miss
all points for the true d a ta range of [0.1, 0.2] m m /h r and to produce a peak at
around 0.35 m m /h r in the histogram , when the MEAN is th e Bayesian estim ator
(as Figure 5.9 shown). However, th e algorithm is still able to retrieve reasonably
for th e tru e intensities between 0.2 and 7 m m /h r. The characteristic th a t there is a
smaller probability a t higher rain rates obviously limits th e ability of th e algorithm
in retrieving heavy precipitation. Note th e cases th a t th e R R is above 15 m m /hr,
th e averaged rain intensities of those histogram s are around 6, 8.5, 14, 33 m m /h r for
th e range of [15, 30], [30, 50], [50, 75], and [75, 100] m m /h r, respectively, indicating a
significantly low bias in the retrieval. In addition, th e histogram s are highly skewed
to the right, encompassing a much lower proportion of true d a ta points th a n th a t of
experim ent R0.
Similar results are obtained in th e MLE estim ates for th e R1 experim ent, and the
change of th e histogram over th e higher rain rates is emphasized here (as figure 5.10
shown). Like experim ent R0, the MLE estim ates also reveal two distinct regimes
in the histogram , b u t it is only shown in th e case of extrem ely heavy precipitation.
The signal of th e one associated w ith a lower rain rate is much stronger in R1 th an
R0. It m eans th a t the algorithm has few chances to generate from interm ediate to
extrem ely heavy rain intensities, as indicated by th e zero proportion of d a ta points
in th e range of [30, 50] and [50, 75] m m /h r.
T he prior density function param eterized from observed rain rates is tested in
th e R2 experim ent. Com pared to R0, th e prior probability density function of R2
has a less ’spiky’ distribution w ith a less heavy tail (see figure 5.8 (a)). In other
words, th e probability density function is relatively small a t very light rain rates
(between 0.1 to 1 m m /h r) and a t large rain intensity (above 25 m m /h r), while being
larger for m oderate precipitation (from 1 to 25 m m /hr). Therefore, similar to R l,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
68
R R =[ 0 .1 ,
0 .2 ]
430957 -
RR = [ 1.0,
2.0
MLE
4
2 .0 ]
670027 -
MLE
R R = [1 5 .0 , 3 0 .0 ]
1.0
59.9%
34.8%
MLE
( 3.02, .1.07)
.17, 0.50)
0.24, 0.11)
222467 -
• 0 .0 %
0.8
3
e 0.6
E
o
O'
o
0.4
X
0.5
1
0
0.2
0.0
0.0
0.1
1.0
0.1
10.0
R R = [ 0 .2 ,
0 .4 ]
1.0
0.1
10.0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
MLE
562461
RR = [ 2.0 ,
4 .0 ]
597394
MLE
4
( 1.91, 0.69)
( 0 .3 2 , 0 .1 5 )
33.2%
10.0
100.0
RR = [3 0 .0 , 5 0 .0 ]
98415 -
MLE
0 .3 0 1---------------------------------------------( 3 .9 1 , 1 .3 4 )
46.7% .
0.8
3
1.0
R e tr ie v e d RR ( m m / h r )
X 0.4
1
0.2
0
0.1
1.0
10.0
0.0
100.0
0.1
R R = [ 0 .4 ,
0 .6 ]
1.0
10.0
20
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
374 6 4 1 -
MLE
RR = [ 4 .0 ,
7 .0 ]
40
60
80
100
R e tr ie v e d RR ( m m / h r )
395442
R R = [5 0 .0 , 7 5 .0 ]
MLE
4
49896 -
MLE
0.25
2.34, 0.78)
0.46, 0.21)
39.7%
( 4.97, 6.60)
: 0 .0%
0.20
3
6 0.1 5
o' 2
0 .1 0
1
0.05
0
0.1
1.0
10.0
1.0
R e tr ie v e d RR ( m m / h r )
0]
10.0
0
100.0
R e tr ie v e d RR ( m m / h r )
498245 -
MLE
R R =[ 7.0 , 1 5.0 ]
391 9 1 1
20
40
60
80
100
R e tr ie v e d RR ( m m / h r )
-
MLE
R R = [7 5 .0 ,1 0 0 .0 ]
24139 -
MLE
2.0
( 0.67, 0.30)
(14.79,28.93;
2.5>B, 0.91)
49.2%
• 0.1% .
0.8
1.5
0.12
0.10
E
o
0.6
0.08
O'
o
f
x
0.4
•■f 0.06
0.04
0.5
0 .2
0.02
0.0
0.00
0.0
0.1
1.0
10.0
R e tr ie v e d RR ( m m / h r )
1.0
10.0
R e tr ie v e d RR ( m m / h r )
20
40
60
80
R e tr ie v e d RR ( m m / h r )
Figure 5.10. Same as figure 5.6, bu t for experim ent R l.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
th e MEAN estim ates show th a t the algorithm overestim ates intensity at the true
range of [0.1, 0.4], and considerably underestim ates heavy rain intensities (figures
not shown). However, the skewness of th e histogram under heavy precipitation is
not as strong as R l, thus some good retrievals are yielded for th e high R R of rain
ranges. T he perform ance in the interm ediate rain rates is not affected much. The
MLE estim ates of experim ent R2 reveal a similarity to R l, having th e feature of two
distinct regimes, b u t the probability of the peak w ith a lower rain ra te is smaller
th a n R l and larger th a n RO (figure not shown).
E xperim ent R3 yields very similar retrieval to the experim ent R2 in either MEAN
or MLE histogram s, showing th a t the effects of th e small changes in th e param eter­
ization of th e prior distribution due to n atu ral variations on th e Bayesian algorithm
are negligible.
A non-inform ative prior distribution is used for the experim ent R4, which assigns
equal weight (0.01 for the experim ent) to all values over the param eter space. Based
on the Bayes theorem , one can know th a t in this experim ent th e posterior probability
density is proportional to the likelihood represented in the d a ta only, so th e retrieval
is dom inated by the behavior of th e physical model. Note th a t th e use of th e uniform
prior p d f leads to equivalent results in th e classical approach. Figure 5.11 reveals
th a t uniform prior knowledge yields much higher retrieved rain rates in light and
m oderate rain situations. Com pared to th e retrieval of RO experim ent, th e entire
histogram of R4 retrieval shifts to th e larger rain range by 7 to 15 m m /h r when the
tru e rain rates are greater th an 2 m m /h r. For th e true range of heavier rain rates,
experim ent R4 captures more d a ta points in th e correct range. Since th e experim ent
only contains th e d a ta information, results here suggest th a t th e P vector is sufficient
to reflect th e signal of intensity under th e heavy precipitating systems.
5.3.3
S en sitiv ity to th e conditional distribu tion
The m otivation of the section is to advance the understanding of th e sensitivity of
th e Bayesian retrieval to the specification of the conditional likelihood th a t provides
statistical inform ation about th e physical relationship between surface rainfall and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
RR = [ 0 .1 ,
0 .2 ]
430957 -
R R = [ 1.0,
MEAN
2 .0 ]
670027 -
R R = [1 5 .0 , 3 0 .0 ]
MEAN
222467 -
MEAN
0.8
5
F4
R4
( 0.54, 0.69)
( 6.46, 4.48)
57.7%
10.3%
13.6% .
4
(2 5 .i:5 ,i:i.1 5 )
0 .6
E
o
O'
g* 0.4
2
x
0.2
1
0
1.0
10.0
0 .0
100.0
0.1
R R = [ 0 .2 ,
0 .4 ]
1.0
10.0
100.0
0.1
562461 -
R R = [ 2 .0 ,
MEAN
4 .0 ]
597394 -
RR = [3 0 .0 , 5 0 .0 ]
MEAN
0.4
3.0
R4
R4
(12.81, 7.69)
MEAN
(38.49,13.80)
. %:
48 9
0.1 5
0.3
2.0
E
o
E
o
?
100.0
98415 -
5.3% :
26.2% ‘
H is to g r a m
10.0
0.20
R4
( 0.88, 1.06)
2.5
1.0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
g 4 0.1 0
0.2
x
x
0.05
0.1
0.5
0.00
0.0
0.0
0.1
1.0
0.1
10.0
RR = [ 0 .4 ,
0 .6 ]
1.0
10.0
100.0
0
20
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
374641 -
MEAN
R R = [ 4 .0 ,
7 .0 ]
395442 -
MEAN
R R = [5 0 .0 , 7 5 .0 ]
0.4
2.0 [--------------------------------------------
R4
( 1.56, 1.65)
(17.52, 9.61):
0 .14
5.8% :
0 .12
13.0%
40
60
100
80
R e tr ie v e d RR ( m m / h r )
49896 -
MEAN
R4
0.3
0 .10
E
H istogram
o
0 .08
S’ 0.2
i
1
0.06
0.04
0.1
0.02
0 .0 0
0.0
0.1
1.0
10.0
100.0
0.1
R R = [ 0 .6 ,
1 .0 ]
498245 -
1.0
10.0
100.0
0
20
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
MEAN
R R = [ 7.0 , 1 5 .0 ]
0.30 r
391911
40
60
100
80
R e tr ie v e d RR ( m m / h r )
-
R R = [7 5 .0 ,1 0 0 .0 ]
MEAN
24139 -
MEAN
0 .1 0
R4
R4
30.4%
(77.3:0, 9.79)
63.9%
0.08
.
Histogram
1.0
E 0.06
0.8
x
0.6
0.04
0.4
0.02
0.2
0.0
0.0 0
0.1
1.0
10.0
R e tr ie v e d RR ( m m / h r )
0.1
1.0
10.0
R e tr ie v e d RR ( m m / h r )
0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
Figure 5.11. Same as figure 5.5, b u t for experiment R4.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
71
P10 vs. P19
1.0
P10 vs. P37
P19 vs. P37
SO
SO
SO
0.8
0.8
0 .6
r. 0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.8
os 0 . 6
^
0.0
0.0
0.0
0.0
0.2
0.4
0 .6
P10
0.8
1.0
0.0
0.2
0.6
P19
0 .8
0.0
1.0
0.2
4
0.6
P10
0.8
1.0
2.0
so
SO
0.4
M arginal p df (P 3 7 )
M arginal pdf (P 1 9)
M arginal pdf (P 1 0)
6
5
0.4
SO
3
4
3
2
2
0.5
1
1
Of . . . i . . . ■ __ . . . . . . .
0 .0
0.2
0 .4
0.6
P10
0.8
n F — r-r-rT , .
1.0
0 .0
0.2
0 .6
P19
0.0
0.8
0 .0
1.0
•o*
0.4
0.2
0.0
0 .2
0.4
0.6
P10
0.8
0.2
1.0
0.8
0.4
0.4
0.2
0.2
0 .4
0.6
0.8
0 .0
1.0
0.2
0 .4
0.6
P10
0.8
M arginol pdf (P 3 7 )
M arginal p df (P 1 9 )
4
2.0
S5
S5
S5
0.8
3
4
3
10
c
a>
o
2
2
0.5
1
1
nf
0.0
-—
0.2
0 .4
....
0.6
0.8
i . . . . . . . . .
1.0
0.0
1.0
0.0
0.2
Marginal pdf (P 1 0 )
6
0.6
P37
P10 vs. P37
0.8
0.0
0.0
0.4
S5
S5
0.8
5
. ■_ .
P19 vs. P37
P10 VS. P19
S5
0 .0
0.4
0.2
0 .4
0.6
0.8
0.0
.
1.0
0.0
0.2
0.4
0.6
P37
0.8
Figure 5.12. Joint and m arginal probability distributions of th e physical
model for experim ent SO (upper two rows) and S5 (bottom two rows).
Contours are plotted for [0.05, 0.5, 1.0, 2.5, 5.0, 7.5, 10, 12.5, 15] in
conditional pdfs.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
72
R R = [ 0 .1 ,
2.0r
0 .2 ]
86024 -
R R = [ 1.0,
MEAN
2 .0 ]
so
0.94, 2.23)
MEAN
133861
i.or--------------------( 1.95,
:
R R = [ 15.0, 3 0 .0 ]
0 .1 4
2.92):
H is to g ra m
E 0.6
SO
MEAN
(13.31, 5.93}
0.12
0.10
29.0% .
44198 -
il: 24.5% :
E
o
a* 0 . 0 8
0
1
0 .0 6
0 .0 4
0.02
0.00
0.1
1.0
10.0
0.1
100.0
R R = [ 0 .2 ,
0 .4 ]
1.0
10.0
0.1
100.0
1.0
R R = [ 2 .0 ,
0.5 E
1 1 2 8 0 7 -M E A N
2.0 [-------------------------------------------------------
4 .0 ]
119 7 2 8 -
= [3 0 .0 , 5 0 .0 ]
MEAN
MEAN
34.8%
28.0%
2 1 .8 %
19853 -
(29.19,16.09}
( 3.70, 3.65)
( 1.02, 2.33)
10.0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
H istogram
0 .0 4
0.1
1.0
10.0
100.0
0.1
R R = [ 0 .4 ,
0 .6 ]
1.0
10.0
100.0
0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d R R ( m m / h r )
74899 -
MEAN
R R = [ 4 .0 ,
7 .0 ]
78879 -
RR = [5 0 .0 , 7 5 .0 ]
MEAN
0 .4
2.0
SO
0 .0 6
SO
.15, 2.41)
20
40
60
80
R e tr ie v e d RR ( m m / h r )
1
MEAN
(56.211,17.40}
6.82, 3.98)
59.4%
28.9%
27.2%
9881 -
100
0 .3
E
H istogram
o
?
0.2
x
0 .5
0.0
0.0
0.1
1.0
10.0
100.0
0.1
R R = [ 0 .6 ,
1 .0 ]
99679 -
MEAN
R R = [ 7.0 , 1 5 .0 ]
0.20
SO
2 .0
SO
1.0
10.0
100.0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
( 1.33, 2.49)
29.1%
78788 -
RR = [7 5 .0 ,1 0 0 .0 ]
MEAN
4780 -
100
MEAN
0 .0 6
SO
9.87, 3.42):
•
(71.30,12.60}
4.6.4% i
0 .0 5
7 5 .3 % :
0 .1 5
0 .0 4
o
E
?
0 .0 3
x
x
0.02
0.01
0.00
0
Histogrom
E
o
g 0.10
0.5
0 .0 5
0.00
0.0
0.1
1.0
10.0
R e tr ie v e d RR ( m m / h r )
100.0
0.1
1.0
10.0
R e tr ie v e d RR ( m m / h r )
20
40
60
80
R e tr ie v e d RR ( m m / h r )
Figure 5.13. Same as figure 5.5, b u t for experiment SO.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
73
R R =[ 1.0,
2.0
2 .0 ]
R R * [1 5 .0 , 3 0 .0 ]
MLE
133861
44198 -
MLE
2 .0
SO
( 8.71, 7.76)
( 0.35, 1.38):
SO
( 0.05, 0.68
1 2 .6 %
0.0 %
0 .0%
H is to g r a m
G
o
X
0.5
0 .5
0.0
0.0
0.1
1.0
1.0
10.0
R e tr ie v e d RR ( m m / h r )
RR = [ 0 .2 ,
0 .4 ]
0.1
10.0
1.0
R R =[ 2.0 ,
1 12 8 0 7 - M L E
4 .0 ]
RR = [30.Q , 5 0 .0 ]
119728 - M L E
19853 -
MLE
(15.80,19.54)
1.42, 2.49)
( 0.06, 0.6
10.0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
26.6%
Histogram
2 2 .8 %
0.1
1.0
0.1
10.0
1.0
10.0
100.0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R R = [ 4 .0 ,
7 .0 ]
7 88 7 9 -
R R = [5 0 .0 , 7 5 .0 ]
MLE
9881 -
MLE
2.0
2 .0
SO
SO
0.09, 0.76)!
(38.85,30.71)
3.96, 3.39)
0.0% :
41.5%
37.7%
Histogram
E
o
cn
o
«
X
0 .5
0 .5
0.0
0.0
0.1
1.0
).1
10.0
R R =[ 0.6 ,
1 .0 ]
99679
1.0
10.0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
MLE
= [ 7.0 , 1 5 .0 ]
78788
R R = [7 5 .0 ,1 0 0 .0 ]
MLE
2 .0
4780 -
100
MLE
0 .1 2
SO
5, 3.66)
( 0.13, 0.9
SO
0.10
40.9%
0.07
(63.85,30.05)
■31.9% :
Histogrom
1 .5
E
o
1.0
0 .0 8
g ' 0 .0 6
i
0 .0 4
0.5
0.02
0.0
0.1
1.0
10.0
R e tr ie v e d RR ( m m / h r )
100.0
0.00
0.1
1.0
10.0
R e tr ie v e d RR ( m m / h r )
0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
Figure 5.14. Same as figure 5.13, b u t using maximum likelihood estim ates
as single-pixel retrieved rain rates.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
74
RR = [3 0 .0 , 5 0 .0 ]
19853 -
R R = [3 0 .0 , 5 0 .0 ]
MEAN
19853 -
MLE
0 .0 6
( 1 4 .7 4 , 4 .
( 1 6 .9 7 , 5 .7 2 )
0 .0 5
0 .0 4
E
S ' 0 .0 3
i
0.02
0.01
0.00
0
100
20
40
60
80
R e tr ie v e d RR ( m m / h r )
R R = [5 0 .0 , 7 5 .0 ]
9881 -
20
40
60
80
R e tr ie v e d RR ( m m / h r )
R R = [5 0 .0 , 7 5 .0 ]
MEAN
0.20
0 .0 6
S5
(25.05, 8.11)
0 .0 5
• 0 .4 %
S5
100
9881 -
MLE
(21.1D, 6.44)
■ 0 .0 %
■
0 .1 5
0 .0 4
E
o
E
0
^
w
1
g1 o.io
0 .0 3
w
X
0.02
0 .0 5
0.01
0.00
0
20
R R = [7 5 .0 ,1 0 0 .0 ]
4780 -
0.00
0
100
80
40
60
R e tr ie v e d RR ( m m / h r )
S5
0.12
0.10
(31.55, 9.81
0 .0 5
•
60
80
40
R e tr ie v e d RR ( m m / h r )
RR = [7 5 .0 ,1 0 0 .0 ]
MEAN
0 .0 6
0 . 0%
0 .0 4
100
20
S5
4780 ( 2 5 .9 4 ,
•
MLE
7 .7 1 )
0 .0%
0 .0 8
E
E
o
o
S ’ 0 .0 3
S ' 0 .0 6
x
I
0.02
0 .0 4
0.01
0.00
0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
100
0.02
0.00
0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
100
Figure 5.15. Similar to figure 5.5, b u t for MEAN (left panel) and MLE
(right panel) estim ates of Experim ent S5 at ranges of heavy tru e surface
rainfall (greater th a n 30 m m /h r).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
microwave signature. R esults from simple model (S5 to S8) th a t specifies physical
model via a covariance m atrix will be given first, followed by those from R5 and
R6 whose training d ataset is generated from a more complete set of conditional
distributions (equation 4.13).
To b etter illustrate th e analysis, th e m ultichannel relationships of th e microwave
observation vector described by th e covariance m atrix of SO experim ent is shown in
the first two rows of figure 5.12, and th e corresponding retrieved MEAN and M LE
rain rate are depicted in figures 5.13 and 5.14. Note when using M LE as th e retrieved
rain rate, SO experim ent reveals th a t th e algorithm is not able to yield satisfactory
retrieval for the very light rain ra te (less th a n 2 m m /h r). It is due to th e com bination
of the simplified covariance m atrix and prior distribution, which tends to favor the
very small surface rainfall rate.
In experim ent S5, th e covariance m atrix th a t approxim ates the physical relation­
ship is a diagonal m atrix, implying no correlation between response vector elements.
The joint pdfs (as shown in th e b ottom two rows of the figure 5.12) of S5 exper­
iment reflect the assum ption of uncorrelation in th e elliptic contours.
There are
slight changes in th e m arginal pdfs of all three channels in experim ent S5.
The
tongue shape directed tow ard th e low P values is due to saturation, resulting from
the param eterization of /q in equation (5.2). For th e MEAN estim ates, substantial
differences occur between experim ent S5 and SO as th e true rain intensity is greater
th an 30 m m /h r (as figures 5.13 and 5.15 shown). The algorithm significantly under­
estim ates the tru e heavy rain rates, and only few percentages of retrievals are in th e
correct range. Under extrem ely heavy precipitating systems, th e m axim um retrieved
rain intensity is only up to 65 m m /h r or so. Similar behaviors are seen in th e M LE
retrieved surface rainfall (figure 5.14 and th e right panel of figure 5.15) as well.
The effect of assum ptions in diagonal elements of covariances of th e response
vector on the Bayesian retrieval is shown by the results of experim ents from S6 to
S8. As expected in th e experim ent S6, the double in the CYi element of covariance
m atrix produces a broader spectrum for th e m arginal probability distribution of P w
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
Morginol p df (P 1 0 )
P 10 vs. P37
P10 vs. P19
| i i i | -i—i—r ■f . * . f r
6
S6
5
S6
0.8
4
3
0 .4
0 .4
0.2
0.2
1
0.0
0.0
nE
0 .0
0 .2
0 .4
P10
0 .6
P10
0 .8
0 .0
1 .0
2
0 .2
P19
VS.
0 .4
0 .6
P10
0 .8
1.0
0 .0
. . . . .
0 .2
1.0
3
0.8
O'
0 .8
4
S7
0.8
0 .6
M arginal p df (P 1 9 )
P19 vs. P37
S7
0 .4
O'
2
Q.
0 .4
0 .4
0.2
0.2
1
0.0
0.0
0 .0
0 .2
0 .4
0 .6
P10
0 .8
1.0
0 .0
0 .2
P19 vs. P37
P37
1.0
0 .4
0 .6
P19
0 .8
1.0
0 .0
0 .2
0.8
0.8
0.6
r-. 0 . 6
0 .4
0 .4
0.2
0.2
0 .6
0 .8
1.0
Marginal pdf (P 3 7 )
P 10 vs. P37
2.0
S8
S8
0 .4
S8
vc
Q
0 .5
0.0
0 .0
0.0
0.0
0 .2
0 .4
0 .6
0 .8
1 .0
0 .0
0 .2
0 .4
0 .6
0 .8
1.0
0 .0
0 .2
0 .4
0.6
0.8
Figure 5.16. Conditional and m arginal pdfs of experim ents S6 (first row),
S7 (second row), and S8 (the bottom ). Note th a t different channels
are shown here for each experim ent to highlight th e change due to th e
difference in th e covariance m atrix.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
77
RR = [5 0 .0 , 7 5 .0 ]
0.06 ............................... ..... ■
S6
0.05
i
RR = [5 0 .0 , 7 5 .0 ]
0.20
9881 - MEAN
i i • 1 1
(50.1D, 18.13)
1 i ■
57.0%
9881 -
MLE
(25.83,27.B9)
26.9%
*
0.15
0.04
£
o
S’ 0.03
w
E
g“ 0.10
X
X
0.02
0.05
0.01
0.00
0.00
20
40
60
80
R e tr ie v e d RR ( m m / h r )
R R = [7 5 .0 ,1 0 0 .0 ]
4780 -
20
40
60
80
R e tr ie v e d RR ( m m / h r )
100
MEAN
RR = [7 5 .0 ,1 0 0 .0 ]
0.12
0 .0 6
S6
0 .0 5
0.10
0 .0 4
0 .0 8
E
S6
4780 -
100
MLE
( 3 7 . 1fc,32.85)
•11.5%
'
E
o
o
S’ 0 . 0 3
g ’ 0 .0 6
i
x
0.02
0.01
0
0 .0 4
0.02
20
40
60
80
R e tr ie v e d RR ( m m / h r )
100
0.00
0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
100
Figure 5.17. Similar to figure 5.5, b u t for th e MEAN (left panels) and
MLE (right panels) estim ates of experim ent S6 a t heavy precipitation
rate ranges (greater th a n 50 m m /h r).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
78
and a less dogmatic joint pdf of P 10 and P 19, as figure 5.16 shows in th e first row.
The m arginal properties of P i 9 and P 3 7 rem ained th e same (figures not shown). The
effect of th e increase in the uncertainty of P 10 value is particularly dem onstrated
in th e retrieval for the heavy rain rates, shown in figure 5.17. T he mean of the
histogram of the retrieved rain intensity in S 6 is around 61 m m /h r a t th e RR range
of [75, 100] m m /h r and only 11.4% of pixels are in the correct range, while control
ru n shows a mean of 71 m m /h r and contains 46% d a ta consistent w ith th e training
data. Similarly, the histogram for M LE estim ates shifts tow ards smaller rain rates
as well for th e case of heavy precipitation in experiment S 6 , implying a considerable
retrieval bias for the large rain rates. T he reason for the low bias is th a t P 10 is a very
im portant factor in the rain ra te retrieval under th e condition associated w ith heavy
rain, while P 19 and P 3 7 might reach satu ratio n at th a t point. In other words, P 10 has
a broader and higher dynamic range for th e rain rate retrieval. Once th e uncertainty
in P 10 increases, the inform ation from th e channel becomes more vague, and in turn,
a smaller rain rate is suggested by th e algorithm since th e prior distribution favors
th e relatively smaller rainfall intensities.
Com paring the joint and m arginal pdfs of S7 w ith the control ru n (SO) at three
channels, one can find similar broadness in th e marginal p df of P 19 in experim ent S7
(see figure 5.12 and the second row of figure 5.16). Surprisingly, in experim ent S 7, the
double in the uncertainty of the P 19 does not affect th e retrieval significantly in either
MEAN or MLE estim ations (figures not shown). The result m ight be a ttrib u ted to
fact th a t P 19 values represent inform ation of interm ediate rain intensities compared
to the other two channels. W hile th e correlations of the variables (i.e., non-diagonal
elements in the covariance m atrix) are well-defined and unchanged, th e combination
of P values still provide enough precise m ultichannel relationships in this retrieval
algorithm , so there are not significant changes shown in th e corresponding retrieval.
For experim ent S8 , the corresponding probability distributions a t three channels
are also illustrated in figure 5.16. N ote th a t the m arginal d istribution for P 3 7 is
more like a bell shape th an th e control run, showing th a t th e ’b u m p ’ at around 0.15
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
79
RR = [ 0.1 ,
0 .2 ]
86024 -
MEAN
R R =[ 1.0,
2 .0
2 .0 ]
S8
( 1.66, 1.82)
S8
0 .0%
133861 -
RR = [1 5.0 , 3 0 .0 ]
MEAN
( 2.18, 2.19);
0 .1 4
45.3% .
0 .1 2
0.8
S8
44198 -
MEAN
( 8.65, 4.50):
0 .1 0
F
E
H is to g ra
E 0.6
0 .0 8
o
V)
x
0 .4
0 .0 6
0 .0 4
0 .5
0.2
0 .0 2
0.0
0.0
0.1
1.0
10.0
100.0
0.1
0 .4 ]
10.0
0.00
100.0
0.1
1.0
112 8 0 7
R R = [ 2 .0 ,
MEAN
2.0
0 .5
58
0 .0 %
4 .0 ]
S8
119728 - M E A N
3 = [3 0 .0 , 5 0 .0 ]
100.0
19853 -
MEAN
(19.67,10.36)
( 2.82, 2.50)
13.6%
29.5% .
0 .4
10.0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R R = [ 0 .2 ,
1.0
Histogrom
E 0 .3
o
O'
o
x
0.2
0 .5
0.1
0.0
0.0
0.1
1.0
10.0
0.1
100.0
RR = [ 0.4 ,
0 .6 ]
1.0
10.0
100.0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
74899 -
R R = [ 4.0 ,
MEAN
2.0
7 .0 ]
78879 -
=[5 0 .0 , 7 5 .0 ]
MEAN
0 .0 6
0 .4 F
S8
3.84, 2.79)
S8
0.9%
•21.0% :
0 .0 5
22 .2%
MEAN
(38.07,14.65)
0 .0 4
Histogram
E
o
g 4 0 .0 3
X
0.02
0 .5
0.01
0.0
0.00
0.1
1.0
10.0
0.1
100.0
R e tr ie v e d RR ( m m / h r )
R R = [ 0.6 ,
1 .0 ]
10.0
100.0
0
99679 -
MEAN
R R = [ 7.0 , 1 5.0 ]
0.20
( 1.90, 1.99)
78788 -
40
60
60
R e tr ie v e d RR ( m m / h r )
RR = [7 5 .0 ,1 0 0 .0 ]
MEAN
O.Ub
S8
29.6%
4780 -
MEAN
(52.(31,14.51)
0 .0 5
• 24.8% :
1 .5
S8
5.27, 2.98)!
100
20
R e tr ie v e d RR ( m m / h r )
2.0
58
1.0
: 3.3%
’
0 .1 5
0 .0 4
E
E
Histogram
o
?
o
g
0.10
0 .0 3
I
x
0 .0 2
0 .5
0 .0 5
0.0
0.00
0.01
0.1
1.0
10.0
R e tr ie v e d RR ( m m / h r )
100.0
0.00
0.1
1.0
10.0
R e tr ie v e d RR ( m m / h r )
0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
Figure 5.18. Same as figure 5.5, b u t for experim ent S8.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
80
of P 3 7 in SO experiment due to high rain intensities has been sm oothed out w ith
increasing uncertainty in P 3 7 values. Unlike the results of th e experim ents S 6 and
S7, the increase in the C 3 3 element of th e covariance m atrix has notable im pacts on
th e retrieval over all ranges of the rain rates (see figure 5.18). For th e light rain rates
(less th a n 1.0 m m /hr), using MEAN of th e posterior distribution in S 8 overestim ate
th e rain intensity by around 0.7 m m /h r in general. Com pared w ith SO, the additional
positive bias decreases when the tru e rain rates are in the range of [1 , 2 ] m m /h r.
T he algorithm tends to underestim ate th e intensity when th e tru e rain rate range
increases above 2 m m /hr. T he d a ta proportion in the true range is reduced to almost
half, being only 3% for the case of extrem ely heavy precipitation, while SO has around
45% d a ta in the correct range. The m ode of th e histogram generally moves to the
smaller intensity by about 20 m m /h r due to th e double uncertainty in P 3 7 .
The MLE estim ates from S8 show a greater sensitivity to th e uncertainty of P 3 7
th a n th e MEAN. Figure 5.19 indicates th a t th e doubling in th e variations of P 37
results in a failure of the algorithm under th e situations of m oderate rain rates
(2
to 15 m m /h r), while SO experim ent contains at least 20% d a ta points in the correct
range. The algorithm of S 8 produces m uch lower rain rates com pared to the true
training data. For the heavy precipitation case, th e d ata proportion in the correct
range is only up to 9%, which is one-fifth of th a t SO can retrieve successfully.
W ith increasing uncertainty of P 37, th e variable provides less precise information,
implying th a t the probability is more com patible a t larger P 3 7 value w ith respect to
the value where the peak locates, if considering a condition at a given rain rate.
In addition to th a t P 3 7 m ainly provides inform ation about light rain, and th e prior
distribution always favors th e smaller rain rates, th e algorithm obtains much smaller
rain rates over all ranges.
Specifications of the physical model in b o th RO and R5 are derived from the same
radar-radiative simulation d a ta using different approaches, and they represent very
similar relationships between P values and surface rain rates. T he minor difference
is th a t the mode of the m arginal p d f of P 19 in R5 is located a t a slightly larger
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
81
RR = [1 5 .0 , 3 0 .0 ]
2.0
2.0
S8
H is to g ra m
I
X
0 .5
0 .5
0.0
1 0.0
0 .0
1 0 0 .0
1.0
R R = [ 0.2 ,
0 .4 ]
1 0 .0
1.0
112 8 0 7 -
MLE
R R = [ 2.0 ,
4 .0 ]
100.0
119728 - M L E
R R = [3 0 .0 , 5 0 .0 ]
19853 -
MLE
0 .4 1
S8
( 0 .0 3 , 0 .3 9 )
10.0
R e tr ie v e d RR ( m m / h r )
2.0
S8
0.1
1 0 0 .0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
2.0
3 .0 %
o
O'
o
1.0
•
E
o
o>
0.0
MLE
( 0 . 9 :1 , 3 . 7 2 )
0 .0 %
E
0.1
S8
( 0 .0 4 , 0 .5 4 )
0 .0 %
0 .5
44198 -
2 .0
S8
0 .0 3 , 0 .4 2 )
( 0 .0 4 , 0 .5 6 )
0 .0%
8 .3 4 ,1 2 .4 8 )
0 .0 %
E
Histogram
o
X
0 .5
0 .5
0.0
0.0
0.1
1.0
1 0.0
0.1
1 0 0 .0
R e tr ie v e d RR ( m m / h r )
R R = [ 0 .4 ,
0 .6 ]
1.0
10.0
100.0
74899 -
MLE
R R = [ 4 .0 ,
2 .0
7 .0 ]
78879 -
MLE
S8
0 .0 3 , 0 .3 8 )
RR = [5 0 .0 , 7 5 .0 ]
9881 -
100
MLE
0.20
2 .0
S8
20
40
60
80
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
S8
0 .0 7 , 0 .8 0 )
0 .0 %
( 2 5 .9 5 ,2 0 .7 8 ) :
: 9 .3 %
0 .0%
:
0 .1 5
Histogram
1 .5
S' 0.10
x
0 .5
0 .0 5
0 .5
0.0
0.00
0.0
0.1
1.0
10.0
).1
R e tr ie v e d RR ( m m / h r )
1.0
10.0
100.0
0
R e tr ie v e d RR ( m m / h r )
R R = [ 7.0 , 1 5 .0 ]
2.0
78788 -
MLE
20
40
60
80
R e tr ie v e d RR ( m m / h r )
R R = [7 5 .0 ,1 0 0 .0 ]
4 78 0
100
MLE
2.0
S8
S8
( 0 .0 3 , 0 .4 8 )
0 .1 2 ,
0 .0 %
:
( 4 2 .8 0 , 2 4 .2 7 V
1 .1 6 )
0 .7 %
E
Histogram
o
S' 1.0
X
0 .5
0 .5
0 .0
0.0
0.1
1.0
1 0 .0
R e tr ie v e d RR ( m m / h r )
10 0 .0
1 .0
1 0.0
R e tr ie v e d RR ( m m / h r )
20
40
60
80
R e tr ie v e d RR ( m m / h r )
Figure 5.19. Same as figure 5.18, b u t for M LE estim ates.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
P19 vs. P37
P10 vs. P19
Rain ra te : th e o re tica l
2.0
R5
R5
R5
0.8
0.8
a> 0 .6
c
QJ
^
0.6
a.
0
0 .4
0 .4
0.2
0.2
0 .5
0.0
0.0
0.0
0
5
10
20
15
R oin R o t e ( m m / h r )
0 .0
0 .2
0 .6
P10
0 .8
0 .0
1 .0
0 .2
2 .0
4
R5
R5
0 .4
0 .6
0 .8
1.0
M arginal p df (P 3 7 )
M arginal p df (P 1 9 )
M arginal pdf (P 1 0 )
6
5
0 .4
R5
3
D e n s ity
4
3
14)
Q
2
‘I
av
1.0
2
1
0 .5
1
n E.
0 .0
■<
0 .2
0 .4
0 .6
...... t .
0 .8
1 .0
OF
0 .0
. - ‘r T 'T ■
0 .2
0.0
0 .4
0 .6
0 .8
1.0
0 .0
0 .2
0 .4
0 .6
0 .8
Figure 5.20. Joint and m arginal probability distributions of th e physical
model for experim ent R5. The joint pdfs are plotted for [0.05, 1, 5, 7,
10 , 12 ].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
83
value th a n th a t of the control run (as figure 5.20 shown). Since results from MEAN
and M LE estim ates are similar, only outcomes for MEAN will be m entioned here
(figures not shown). The experiment reveals th a t the different param eterizations
of the conditional likelihoods in R5 still produce very consistent retrievals, except
a slight increase in the intensity over th e very light rain rate. It is partially due
to the aforem entioned shift of th e peak in the m arginal p df of P 19. However, the
difference is negligible. R esults of experim ent R5 suggest th a t th e retrieval is not so
sensitive to th e different ways of specifying the conditional distributions as long as
th e relationships represented by the physical model are not far from reality.
The adequacy of using a simplified conditional probability distribution to rep­
resent th e more com plicated behavior of d a ta is evaluated by experim ent R6. A
num ber of notable changes in the retrieval histogram are sum m arized in figure 5.21.
In general, th e linear physical relationships yield much larger rain rates, and the
variations of the retrieval histogram are almost twice th a t of th e control experim ent
RO. For very light rain rates (less th a n 0.2 m m /h r), R6 only produces 0.1% d ata
points in th e correct range, while th e control ru n produce alm ost half of th e d a ta
in the correct range. T he positive bias becomes even more obvious when the true
range is in m oderate intensities. Under the condition th a t th e tru e rain rates are
only 2 to 15 m m /h r, some retrieval from R6 are even greater th a n 50 m m /h r. The
large variations in the retrieved intensity at a certain true rain ra te range is seen
in the case of heavy precipitation as well. Retrieval from MLE estim ations behaves
similarly to th e MEAN (figures not shown).
It is not appropriate to draw a conclusion th a t a simplified physical model is
definitely not suitable for the use of a Bayesian algorithm, since th ere are considerable
differences in th e physical relationships between th e training d ataset and th e simple
model for th e experim ent R6. The retrieval bias might decrease if th e simple physical
model is improved to be more close to th e training dataset. However, in fact, the
param eters used to build th e simple model are estim ated from th e sam e radarradiative sim ulations. It implies th a t some behaviors of the d a ta can not be explained
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
84
R R =[ 1.0,
2 .0 ]
MEAN
670027
R R = [1 5 .0 , 5 0 .0 ]
o.8 r
0 .3 0
• •" r
( 3.99, 3.94)
0.51, 0.29)
0.1%
222467 -
"• • .......... • ....................
R6
0 .2 5
1 8 .9 %
MEAN
1
( 2 3 .2 4 ,1 :9 .5 3 )
2 2 .3 % :
0 .2 0
E
E
o
o
I t ° - 15
i
X
0 .1 0
0 .0 5
0 .0 0
0.1
1 .0
0.1
1 0 .0
0 .4 ]
562461
1 0 .0
1.0
1 0 0 .0
-
RR = [ 2 .0 ,
MEAN
4 .0 ]
597394 -
MEAN
R6
R R = [3 0 .0 , 5 0 .0 ]
( 8.52, 7.23)
H is to g ra m
E
o
a
S’ 0.2
g 0.10
x
x
0.1
0 .0 5
0.0
0.00
0.1
100.0
0 .6 ]
1.0
10.0
100.0
0
20
40
80
60
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R R = [ 0 .4 ,
374 6 4 1 -
:
0 .1 5
E
10.0
MEAN
27.3%
0 .3
1.0
98415 -
(39.97,22.71)
R6
11.9%
2 8 .5 %
0.1
1 0 0 .0
0.20
0 .4
( 0.67, 0.50)
1 0 .0
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
R R = [ 0 .2 ,
1.0
RR = [ 4.0 ,
MEAN
2.0
7 .0 ]
395442 -
MEAN
RR = [5 0 .0 , 7 5 .0 ]
49896 -
100
MEAN
0 .4
R6
0.99, 0.90)
R6
0 .1 4
(11.99,10.39)
•
24.8%
11.8%:
R6
0.12
0 .3
H istogram
0.10
E
E
o
o
?
1.0
ct>
0 .2
x
x
0.5
0.1
0.0
0.0
0 .0 8
0 .0 6
0 .0 4
0.02
1.0
0.1
0.00
0.1
10 .0
R e tr ie v e d RR ( m m / h r )
R R = [ 0 .6 ,
1 .0 ]
498245
1.0
10.0
100.0
MEAN
R R = [ 7.0, 1 5 .0 ]
391911 -
MEAN
0 .3 0
R6
0
20
40
60
80
R e tr ie v e d RR ( m m / h r )
R e tr ie v e d RR ( m m / h r )
( 1.68, 1.72)'
22.8% .
R R = [7 5 .0 ,1 0 0 .0 ]
o .io r
R6
:
(81.95,13.67)
: 50.2% '
Histogram
0 .2 0
E
E 0 .0 6
o
0.8
Jr
°-15
w
0.6
•-f 0 . 0 4
X
0 .1 0
0 .4
0 .0 5
0.2
0.0
0.00
0.1
1.0
10 .0
R e tr ie v e d RR ( m m / h r )
100.0
1.0
MEAN
'
(15.00,14.04)
0 .2 5
24139 -
100
1 0 .0
R e tr ie v e d RR ( m m / h r )
20
40
60
80
R e tr ie v e d RR ( m m / h r )
Figure 5.21. Same as figure 5.5, bu t for experim ent R6.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
by a simplified physical model if it is approached by assum ptions of fixed variances or
a simple covariance m atrix, even though th e m atrix is calculated from th e same data.
As shown in experiment R6, if th e physical model cannot show th e multichannel
relationships to a certain degree, th ere might be significant errors and bias in the
retrieval product.
5.4
Sum m ary
Results from the above sensitivity tests have revealed th e retrieval ability of the
Bayesian algorithm over various rain rate ranges when th e prior and conditional
likelihoods are bo th idealized, although th e single-pixel retrieved rain rate might be
associated w ith a bias due to th e inherent uncertainty in th e physical relationship
and th e interpretation using MEAN and MLE. The resulting posterior probability
might have m ultiple peaks, which could result in a significant difference in the re­
trieved surface rainfall when th e observation vector only has a slight change. This
deficiency might be improved via introducing more observation variables to offer
more additional inform ation to reduce the am biguity of relation of th e microwave
observations to rain rate.
The MEAN tends to provide small values of to ta l bias and b e tte r retrieval in the
m oderate rain rate range, while th e MLE has b etter retrieval ability in the range of
very light rain rate (less th a n 2 m m /h r) and extreme heavy intensity (greater th an
50 m m /h r). In addition, the specifications of both th e prior and th e conditional pdfs
are crucial to the posterior distribution. However, once th e prior distribution and
th e physical model represent adequate inform ation about th e tru e d ata, the Bayesian
algorithm is not so sensitive to either th e very small variation in th e param eters of
th e prior distribution or the way to approxim ate th e physical relationship between
microwave signature and rain rate. Therefore, when th e n a tu ra l variations of the
prior rain ra te distribution and th e imperfectness of th e physical model exist, the
Bayesian retrieval is still meaningful, b u t some degree of bias m ight exist.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86
6. B E N C H M A R K A LG O R ITH M D E SC R IP T IO N S
A num ber of current and new retrieval algorithm s are utilized and developed
in the study to provide inter-comparisons for th e purpose of validations. D etailed
descriptions for each algorithm will be given in th e chapter.
6.1
G PR O F
The G oddard Profiling algorithm (G PR O F), as m entioned in chapter 1, is a sim­
plified Bayesian retrieval algorithm. A database is introduced in the algorithm to
represent the presum ed probability distribution of cloud profiles; the possible cloud
profile whose microwave signature is close to th e satellite observations are selected
from the database; and then an averaged cloud and precipitation structure for each
pixel is obtained, based on the relative occurrences of each candidate profile in the
database.
Details about the formulations and calculations are referred to equa­
tions (1.2) and (1.3). Since th e database plays a m ajor role in G PR O F (Kummerow
et a l, 1996, 2000; L ’Ecuyer and Stephens, 2002), it is worthwhile to give some de­
scriptions here about th e old and new database of G PR O F. We will com pare our
retrieved rain ra te w ith G PR O F using b o th databases in chapter 7.
The old database of G PR O F for the m arine environm ent included three differ­
ent precipitation systems and was mainly based upon numerical model sim ulations
of the G oddard Cumulus Ensemble Model (G CEM ), th e University of W isconsin
Nonhydrostatic M odeling System (UW-NMS), and 1-D Eddington radiative transfer
model.
Cases contained a tropical squall line in T O G A /C O A R E sim ulated on a
128 x 128 km dom ain w ith 1-km resolution; a subsequent squall line sim ulation on
384 x 384 km w ith 3-km resolution; a tropical cyclone on 3.3-km resolution w ith
205 x 205 grid points; and a thunderstorm complex observed during th e C ooperative
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87
Huntsville Meteorological Experim ent (COHMEX) on 50 x 50-km dom ain w ith 1-km
resolution (Olson et a l, 1999). Note th a t the GCEM and UW-NMS have different
assum ptions of microphysics scheme, and various m ethods to deal w ith dynamics.
The new database of G PR O F for th e oceanic pixels was obtained from th e simula­
tions of the Pennsylvania S tate University-National C enter for A tmospheric Research
mesoscale model version 5 (MM5). It included th e following simulations: Hurricane
Bonnie in 1991 (158 x 158 km inner dom ain w ith 1.33-km resolution); a tropical
squall line system th a t occurred on 12 Septem ber, 1974 during GATE (140 x 140
km domain w ith 2-km resolution); and another tropical squall line on 22 February,
1993 in th e T O G A /C O A R E (140 x 140 grid points w ith 2-km resolution). Since the
simulations were ru n from the same mesoscale model, all were based on th e G oddard
explicit m oisture scheme.
To highlight th e difference between the old and new G PR O F databases, th e rain
rate distribution and its relation to the microwave signature are dem onstrated here.
In G PR O F, th e response vector y representing microwave signature includes the
attenuation index a t 10.65, 19.35, and 37.00 GHz
( P i o ,P i 9 , P 3 7 ),
and th e scattering
index at 85.5 GHz (Ass). The scattering index is proposed by P etty (1994a) and
defined as
S = P T Vfi + (1 - P )T C - Tv ,
(6.1)
where the scattering index S is determ ined by th e atten u atio n index P , vertically po­
larized clear-sky/observed brightness tem p eratu re T ^ o /T y , and the lim iting bright­
ness tem perature of a non-scattering and optically thick cloud layer T q - Here, T c is
assumed to be 273 K. High values of S represent significant scattering signals from
ice particles.
Figure 6.1 illustrates the normalized histogram of the rain ra te in th e old and
new database. These two databases have similar probabilities while th e surface rain
intensity is less th a n 3 m m /h r. However, it is obvious th a t the new d atab ase has
more cloud profiles w ith m oderate rain rate (3—10 m m /h r) and fewer profiles w ith
heavy precipitation (greater th an 20 m m /h r) th a n th e old database. T he largest
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
88
H istogram
o f r a i n r a t e in t h e G P R O F d a t a b a s e
0 .0 5
Old d a t a b a s e
0 .0 4
New d a t a b a s e
0 .0 3
x 0.02
0.01
0.00
0
20
40
60
80
100
120
140
Rain r a t e ( m m / h r )
Figure 6.1. Normalized histogram s of surface rainfall of cloud profiles in­
cluded in th e old (solid curve) and new (dotted curve) G PR O F database.
The to ta l num ber of cloud profiles are 11069 and 3097 for th e old and
new database, respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
89
rainfall rate in th e new database only reaches about 87 m m /h r, while 2.5% of the
rain rate in the old database exceeds
88
m m /h r and some go up to around 140
m m /hr.
There is no easy way to visualize a high-dimensional relationship by plotting.
Since the plot of rain rate vs.
S index is more scattered, it is h ard to identify
the change between old and new database, and how sensitive th e retrieval is to
the scattering index.
Therefore, we only attem p t to dem onstrate th e significant
difference in the 4-D relationship between rain rate and P values at 10.65, 19.35,
and 37 GHZ channels. Figure 6.2 depicts th e 2-D contours of th e num ber of cloud
profiles on the rain rate vs. P \ 9 domain, a t a given interval of P w and P 37. The
distribution shows some key features of these two databases. First, the two databases
have a similar pattern: the range of th e P i 9 value shifts to the left (decreasing P 1 9 )
while P 10 and P 3 7 b o th decrease. However, th e distribution of th e old database shows
a larger coverage and more scatter th a n th e new database. In addition, it is noted
th a t the cloud profiles clutter a t different locations, indicating th a t th e occurrence
frequency of cloud files in 4-D space differs between the old and new database. For
example, a t a given interval of 0.07—0.15 and 0.55—0.65 for P 3 7 and P 1 0 , respectively,
the new database has more cloud profiles associated with
0 — 10
m m /h r of rain rate
and 0.2—0.45 of P 1 9 , b u t the old database yields more profiles w ith heavier rain rate
(10—20 m m /h r) if the P 1 9 is between 0.2 and 0.45 as well.
6.2
P E T T Y T M I algorithm
The m ain concept of the P etty T M I retrieval algorithm is to find optim al rain
rate estim ates, reducing the discrepancy between observed and forward-calculated
microwave signatures.
In th e algorithm , th e attenuation index is used to adjust
th e rain rate field, and the scattering index is utilized to provide initial rain rate
information. The rain mask in th e retrieval algorithm is determ ined by three criteria
when either one of them is satisfied. First, rain is possible at a given pixel if significant
scattering signature is present (i.e., S 3 5 > 10/7). In addition, the pixel is assigned as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
New D a t a b a s e
Old D a t a b a s e
50
50
-
"
1
40
P 3 7 = [0 .0 7 ,0 .1 5 ]
P 3 7 = [0 .0 7 ,0 .15]
_
P 1 0 = [0 .6 5 ,0 .7 5 ]
P 1 0 = [0 .6 5 ,0 .7 5 ]
40
■
30 r
20
20
q:
10 r
0
0.0
_
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r--- r
|
■■
■
i
i
.
|
i
.
<
,
i
i
i
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50
P 3 7 = [0 .0 7 ,0 .1 5 ]
P 1 0 = [0 .5 5 ,0 .6 5 ]
^
a)
£ 20 L
£. 20
c
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^ 10
30
QL
:
s i r - . .
10 r
;
0E
0.0
.
0.2
0.4
0.6
P19
0.8
1.0
P 1 0 = [0 .5 5 ,0 .6 5 ]
h .
T l_
New D a t a b a s e
Old D a t a b a s e
100
P37 = [0 .0 0 ,0 .0 3 ]
P 3 7 = [0 .0 0 ,0 .0 3 ]
P 1 0 = [0 .4 5 ,0 .5 5 ]
P 1 0 = [0 .4 5 ,0 .5 5 ]
80
SI
|
I
60
a>
a)
o
oc
20
0.0
60
40
20
0.2
0.4
0.6
P19
0.8
1.0
0.0
0.2
J
\
0 ............... ................................................................................................ .
0.4
0.6
0.8
1.0
0.0
0.2
P19
1. 0
100
80
o.
' • ■ i ...’ i 1 1 1 i 1 ■ ■ i 1 1 ' i > .
P 3 7 = [0 .0 7 ,0 .1 5 ]
40
5 (m m /h i
OJ
o
4 0
_C
I
0.6
P19
New D a t a b a s e
Old D a t a b a s e
» - r
50
0.4
0.2
0.4
0.6
P19
0.8
1.0
Figure 6.2. Scatter plots of num ber of cloud profiles on the rain rate vs.
Pig dom ain a t a given interval of P w and P 37 in the old (left panel) and
new (right panel) G PR O F database.
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91
containing possible rain while th e absolute polarization difference a t 85 GHz is less
than 10.5 K. Furtherm ore, if th e atten u atio n index (Pss) is smaller th a n th e value
expected for the case associated w ith 0.5 k g /m 2 cloud w ater plus th e tolerance, then
the pixel is classified as a rainy pixel. Once the rain mask procedure is done, an
initial first-guess rain rate field is assigned for each rainy pixel by
R 0 ( m m /h r ) = S ^ / 7 . 0.
(6-2)
Based on the initial rain ra te field, corresponding attenuation indices at 10.65,
19.35 and 37 GHz can be com puted from forward calculations. Then, th e calculated
attenuation indices are com pared w ith the observed values, and then th e current
rain rate field is adjusted to achieve a b e tte r agreement between the observed and
com puted microwave signature. This procedure will be processed iteratively until
the differences at different channels are all w ithin their own assigned tolerance. In
the current im plem entation, th e tolerances are given as 0.2 for 85 GHz and 0.05 for
other TM I channels. More related details are found in P etty (1994b).
6.3
P E T T Y H IST 4 algorithm
A new algorithm based on th e histogram s of PR -TM I m atch-up d a ta was de­
veloped in th e study. The histogram analysis of the m atch-up d a ta was conducted
using values of P i0, Pig, P 3 7 , and S 3 7 w ith intervals of 0.05 and 1 for P and S
indices, respectively. Based on the histogram distribution in discrete intervals, the
corresponding m ean and stan d ard deviation were able to be com puted, and the num ­
ber of pixels could be counted as well. These statistics were stored in a 4-D lookup
table. W hile applying th e algorithm , th e retrieved rain ra te and associated devia­
tion could be found according to the location of th e m ultichannel observations in th e
lookup table.
6.4
Linear m odel
A fitting linear model was developed from PR -T M I m atch-up d a ta in Jan u ary and
July, 1998. The idea of th e algorithm is to use linear regression analysis to describe
the physical relationship between rain ra te and microwave signature.
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Regression
92
variables include th e P indices at 10.65, 19.35, and 37 GHz, and S values a t 37 and
85 GHz. Due to th e non-linearity between rain rate and P values, and the satu ratio n
at low P index, th e P values were transform ed first by Box-Cox analysis to obtain
a more linear relationship. As to the scattering index, its dependency on rain rate
was not as strong as th a t shown in th e atten u atio n index. Thus, no transform ation
was carried out for S index in th e linear model.
The coefficients in th e linear model were calculated from the R language statistical
package (mgcv), which was developed by th e departm ent of statistics, University of
Wisconsin at Madison. A to ta l of 1.2 million d a ta samples were used in th e regression
analysis, and the final value of multiple R-squared is 0.634. Variable transform ations
and corresponding regression equations in th e linear model are expressed as the
following:
P Xl — 1
n o
=
PI9
pA2 _ 1
=
----- 1A2 = —0.3
P'm
=
R=
----- , Ax = - 0 .8
Ra7! - 1
----- ,A3 = -0 .7
0.201 + 4. l O P ^ - 4.831P*9 + 0.182P3*7 + 0.110S37 + 0.017S85
+ 0 . 8 1 0 ^ * 9 + 0 .9 6 0 /^^ 3 7 - 0.234P*9P3*7 + O .IW P ^ P ^ P * ,,
(6.3)
where R is th e retrieved rain rate in m m /h r, and th e asterisk symbol represents the
transform ed variables.
In the linear regression model, all pixels were included to be com puted w ithout
rain screening. Therefore, some constraints have to be added on th e calculated rain
rate field in order to exclude unrealistic precipitation intensity and structure. We
assume the pixel is not rainy and a zero of final retrieved rain rate would be assigned
if the calculated rain intensity is less th a n 0.5 m m /h r.
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93
7. REAL-W O RLD A PPL IC A T IO N A N D VALIDATIO N
In the previous chapter, we have introduced a num ber of algorithm s. In this
chapter, those algorithm s will be used to estim ate th e surface rain rate from TM I
observations, and the estim ates will be validated against independent datasets. T he
purpose is to put the perform ance of the new Bayesian algorithm into context. A
satisfactory outcome would be th a t its performance is found to be com parable to
th a t of other algorithm s while also having the added advantage of posterior rain rate
probability density functions.
7.1
B ayesian algorithm
As we m entioned in chapter 1, th e Bayesian posterior density function is the
product of the conditional likelihood (or called physical model in th e study) th a t
statistically describes th e physical relationships between rain ra te and microwave
signal, and a prior rain ra te distribution th a t represents our knowledge or belief. In
chapters 3 and 4, in order to provide a theoretical basis, th e conditional likelihood
was derived from model sim ulations, and the prior rain rate distribution was based
on the intense observations from field experiments. In this section, in an a tte m p t to
construct an algorithm w ith calibrated precipitation m easurem ents, th e conditional
and prior pdfs were obtained from th e fit of the near-nadir PR -T M I m atch-up d a ta
of January and July, 1998.
The prior pdf was estim ated from th e fitting of the P R interpolated rain rate.
Since there were natu ral variations in th e distribution of rain ra te when d a ta from
different m onths were employed, we used (-2.8, 2.0) for the param eters (//, a) of
th e prior distribution in our Bayesian algorithm. In chapter 8, th e im pact of th e
uncertainty in the prior distribution on the retrieval will be discussed and quantified.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
94
The conditional probability distribution is denoted as f ( P \ R ) . Here, we would
like to construct the IV , 2V, and 3V models, where IV indicates th a t only informa­
tion about P 3 7 is used in th e retrieval algorithm; 2V represents th a t P 19 and P 3 7 are
included to establish th e physical model; and finally, the physical relationship in th e
3V model is represented by
P 1 0 ,
and
P 1 9 ,
P 3 7 .
W hen more th a n one observed variable
is used, th e conditional likelihood is built in a hierarchical way. The conditional pdf
in the IV , 2V, and 3V model are given as:
(7.1)
IV m o d el : / ( P |P ) = / ( P 3 7 |P )
2
V m odel : / ( P |P ) -
3V m odel : f ( P \ R ) =
/ ( P
1 0 , P
/ ( P
1 9 , P 3 7 | P )
1 9 , P 3 7 | P )
=
/ ( P
3 7 | P
) / ( P
i 9 |P 3 7 ,
= / ( P 3 7 |P ) / ( P i 9 |P 3 7 , P
) / ( P
R)
i o | P 1 9 ,
(
P 37,
7 .2
)
P )
(7.3)
The generic forms for
/ ( P 3 7 I P ) ,
/ ( P
i 9 | P 37 ,
R), and
/ ( P i o | P i 9 , P 3 7 ,
equation (3.3), and th e corresponding param eterization for
P ) are defined by
n 3, ciy, cr2, and
cr3 are summarized as follows:
P + 0.06 2-1
+ 0.05
AP =
1.02
P — 3 .3 4 \2'
0.5
+ 0.06
<73 = 0.33 exp
T39 ) .
1.07 e x p (-0 .6 1 P ) - 0 . 0 7
A^3a
R - 1.4 \ 2
0.1
— 0 . 1 exp - 0 . 5
0.8
1.5
fa =
0.93 exp
V2
=
0.5
bo + bi P 3 7 + S2
bo =
0.25 exp(—0.024R)
bl
=
0.77 - 0.028P + 0.0009P2 - 0.000009P3
b2
=
0.4 + 0 .0 0 3 5 P
^2 =
$21 =
^21 + $22 + <^23
(-0.0005 - 1.6202^
7 + 4.4076P37 - 0.0323/ R + 0.001P + 0.0006P15) •
(exp[—(6 .2 5 P 3 7 )2] + 0.02exp[—100(P 3 7 - 0.9)2])
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95
S2 2
=
_ o .0 4 ( e x p [ - 0 .5 ( ^ y ^ ) 2] + 1) • exp[-0.5(-— -■p 2 )2]
<523 =
+0.05[1 - exp(—0.5P)] e x p [-0 .5 (P37Q 3° '8 )2]
cr2 =
0.233 e x p (-0 .2 6 9 5 P 37) - 0.1487
Pl
Co =
C0 exp [ - —!-r (P 1 9 - c i)2] + c3 + 5i
ZC2
- 0 . 0 9 - 0.23P37 + 0.89 exp [ - 0.5(P37 - 0.3)2] - 0.023P + 0.14V P
Ci
1.1
=
=
c2 =
1.06 — 0.049P —0.33 exp(P37) + 0 .0 4 6 P ex p (P 37)
c3 =
1.04 - 0.022P37 - 0.0066P + 0.047V P - 0.056P37> /P - c0
=
^ii +
Sn
=
-0 .1 3 - 0 .0 0 0 5 P i 9 - 0.35P37 + 0.2 exp(P37) - 0.001P
S12
=
—0.1 exp[—0.5(——------:— )2] • (1 — 0 .5 ex p (—0.2P))
Si
=
0.1 • [1 — 0 .7exp(—0.1P)]
812
+ 6i3
P
___ A AIX
813 =
0.03exp[-0.5( - 1-^ 15° '4 )2] • e x p [-0 .5 (P37Q 2^ '° 5 )2] • (1 - 0 .5 e x p (-0 .2 R))
<
7 1
( 0 . 1 4 - 0 . 1 2 / ^ + 0.015) • (0.9e x p (-O .O lP )+ 0.1),
=
(7.4)
where the unit of rain rates (P ) is m m /h r, and P index is dimensionless. Note th a t
th e dependence on P is included in these param eters.
Recall th a t P37 saturates quickly w ith increasing rainfall intensity, and thus th e
dynamic range of retrieval in th e I V model is quite small. Therefore, the I V model
is not very useful in real-world applications, especially for heavy rain events. How­
ever, the I V model could be used to examine if th e specification of th e / ( P 37, P ) is
appropriate. In addition, it helps to dem onstrate how th e assum ptions of physical
model and prior distribution affect the retrieval since only one element is involved
in the observation vector.
Having the posterior probability distribution, th ere are two commonly used esti­
m ates to represent th e retrieved rain rate: one is th e m ean value (MEAN), and the
other is the m axim um likelihood estim ate (MLE). T he Bayesian 2 V model using the
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96
posterior m ean as the estim ate is denoted as the Bayesian 2E-M EAN algorithm , and
the 2V-M LE presents th a t th e retrieved rain rate is estim ated from th e maximum
likelihood value of the posterior pdf. Similar rules for the names are applied to the
Bayesian ZV algorithm s as well.
7.2
D a ta sets for validation
A num ber of cases were processed to evaluate overall perform ances for all algo­
rithm s described in the previous chapter. The cases include various precipitating
systems, such as typhoon/hurricane, frontal rain bands of extra-tropical cyclones,
heavy and widespread rain events, and some scattered strong convection cells. De­
tailed descriptions for each dataset are as follows.
7.2.1
S elected global rain cases
A typhoon case with TM I orbit num ber 336 and twelve other cases over the ocean
from Bauer et al. (2001) are ad ap ted to help th e validation of th e retrieved rain
rates. T he ’tru e ’ rain rate in th e validation d ataset is obtained from the coincident
P R 15 x 15 km averaged rain rate. T he key inform ation about cases is summarized
in Table 7.1. T he rationale of selections of these cases is to conduct prelim inary
exam inations to see if algorithm s work reasonably in each particular w eather system.
7.2.2
H ea v y /w id esp rea d rain even ts
P etty (2001) pointed out th a t th e effects of beam-filling and 3-D geometric cloud
structures are im portant sources of am biguity in the physical interpretation of mi­
crowave signature for the surface rainfall estim ations. To reduce th e im pact from
those problems, precipitating system s associated w ith widespread heavy rainfall were
of interest in his paper, which were likely free of 3-D structures and having uniform
precipitation throughout the field of view. We will use the same criteria for th e TM I
observations to isolate the heavy and w idespread rain events, in order to understand
how each algorithm performs in th e p articu lar precipitation system.
The special rain events were sam pled from over-ocean pixels of TM I m easure­
m ents during the period of January-D ecem ber 1998.
Pixels were selected if the
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97
Table 7.1
Inform ation of date, orbit num ber, and the nadir locations for rain events
in the validation dataset.
Description
Date
O rbit
Lat.
Long.
19971217
00336
15
136
Pacific Ocean
19971216
00295
12
144
Cyclone Paka, Pacific Ocean
19980210
01171
-15
60
19980216
01273
-5
-145
19980826
04283
33
-75
19980829
04328
-23
-142
19990121
06620
3
81
Deep convection, Indian Ocean
20000802
15438
6
-16
Squall line, W est African coast
20000825
15795
18
70
20000828
15835
13
-134
20000830
15876
34
-72
Frontal system , N orth American east coast
20000911
16059
35
-33
E xtra-tropical depression, east A tlantic
20000917
16151
3
70
Cyclone Anacelle, Indian Ocean
Deep convection, Pacific ITCZ
Hurricane Bonnie, west A tlantic
Frontal system, east Pacific
Monsoon, west Indian coast
Convection, central Pacific
Monsoon scattered convection, Indian Ocean
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98
L o c a tio n s fo r h e a v y /w id e s p re a d ra in e ve n ts in 1 9 9 8
45
••
%5.
-9 0
Figure 7.1. Locations of heavy/w idespread rain events selected from TM I
orbits during January-D ecem ber 1998.
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99
criterion T2i v - ? W < 5 K had been m et for b o th th e pixel itself and th e surround­
ing pixels in a 5 x 5 pixel array, as proposed by P e tty (2001). This ensures th a t the
rainfall in th e event is b oth heavy and widespread. Furtherm ore, in order to avoid
contam ination from the sidelobe in th e P R reflectivity a t off-nadir locations, only
rain events th a t occur close to the nadir of P R scans were included. As a result, a
to tal of 88 rain events were found, and th eir locations are shown in figure 7.1. Each
rain event comprises around 360 pixels for th e validation.
7.2.3
P R -T M I global m atch-up d ataset o f 1 9 9 8 /0 4
118 orbit files in April 1998 were random ly selected to validate th e overall per­
formance of all algorithms. Note th a t our Bayesian retrieval algorithm was derived
from the d a ta of January and July in 1998, and therefore, th e T M I-P R m atch-up
d a ta of A pril is an independent d ataset for th e validation. In addition, since the
dataset comprises enough large samples, in order to filter out the uncertainty in the
off-nadir P R interpolated rain rate, only near-nadir pixels were considered in the
comparison. T he validation d ataset contains a to ta l sample size of 142,100 pixels.
7.3
V alidation m etrics
There are a num ber of verification m easures for researchers to quantify statisti­
cal differences between calculated or retrieved estim ates (or a forecast) and the true
value. Among m any others, th e bias, th e root-m ean-square error, and th e correla­
tion coefficient are the three mostly commonly used measures, which are employed
to continuous underlying variables. T he Heidke Skill Score (HSS) is often used in
’categorical forecasts’ when only yes/no type forecasts are considered, and this ap­
plies prim arily to discrete variables. This section gives the definition and the usage
for each verification measure. Note th a t all validation m etrics are highly context
sensitive, depending on the characteristics of the validation d ataset as well as the
perform ance of the algorithm. Therefore, validation statistics cannot easily be in­
terpreted in isolation b u t rath er should be used as a basis for com paring different
algorithm s applied to the identical validation d a ta set.
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100
7.3.1
B ias
The bias is defined as
B IA S,,. =
- x.,),
(7.5)
-‘V1=1
where th e x j and x Q denote th e forecasts or retrieval ( /) and observations (o), re­
spectively, and i indicates each d a ta point. The to tal sample size is presented by
N . Bias is therefore a measure of th e m ean difference between th e retrievals and the
validations. Note th a t the validation dataset itself may contain biases of unknown
m agnitude.
7.3.2
R oot-m ean -sq u ared difference
The root-m ean-squared difference (RMS) is defined as
R M S ,,. =
(7.6)
RMS is th e positive square root of th e average of th e squared differences. It is a
measure of dispersion of the retrieved values from the validation data. As a second
order error measure, RMS is more strongly influenced by extrem e differences. In
addition, it is sensitive bo th to bias and linear correlation. Since th e absolute cali­
bration of the validation d a ta cannot be guaranteed, RMS difference m ust be treated
w ith caution.
7.3.3
C orrelation coefficient
T he correlation coefficient Rf,0 between / and o is com puted by
R
=
£ £ i ( g /<
-*o)
(7 7)
y / Z t i (xfi ~ x f )2y / Z h fa* - x o f
where x 7 and x Q are the sample m ean of estim ations and observations, respectively.
C orrelation coefficients represent th e strength of th e linear relationship between ob­
servations and retrievals/forecasts. This m easure is sensitive to th e finest details
of the errors between the tru e value and th e estim ates (B arnston, 1992). However,
since it only describes the linearity between observations and retrievals, retrieved
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101
values associated w ith a uniform or linear bias would still achieve a high correlation
w ithout showing the discrepancy. This characteristic brings interesting points for
the validation processes of the retrieval algorithms. It is th a t th e correlation coeffi­
cient is still a good measure to verify retrieval when no absolutely calibrated d a ta
are available. High correlations indicate th a t th e tru e and estim ated value behave
in the same direction.
However, caution m ust be exercised when evaluating retrieval algorithm s based
only on th e correlation coefficient. It is typical th a t each retrieval algorithm has
its own dynam ic range. W hen a large portion of the actual values are beyond the
dynamic range, th e correlation can underestim ate the retrieval skill substantially
because of th e satu ratio n at higher rain rates. In addition, due to th e quadratic
relationship between correlations and errors, th e correlation coefficient is affected
significantly by th e outliers. In the presence of outliers, the correlation coefficients
might m isrepresent th e tru e retrieval ability as well.
In summary, the correlation coefficient measures how strong th e linear relation­
ship is between th e tru e and estim ated value. We recommend making scatter plots
whenever feasible, because it will tell us if there is an obvious bias and if there is
any possible m isinterpretation in th e correlation coefficient.
7.3.4
H eidke Skill Score
W hile forecasts are expected to have rough accuracy or th e verification is focused
on measuring th e skill forecast of discrete num ber of events (so-called categorical
forecasts), the Heidke Skill Score (HSS) is a commonly used verification measure.
The skill score is com puted from a contingency table (as shown in table 7.2 for 2
categories) th a t sum m arizes th e occurrences of events for observations and forecasts
(or retrievals). The contingency table is constructed at a given threshold th a t dis­
tinguishes ’yes’ and ’no’ events and defines the categories. HSS not only m easures
the proportion of correct forecasts, including b o th correct hits and rejections, b u t
also takes into account th e expected skill obtained by chance in th e absence of any
forecasting skill (B arnston, 1992; G audet and C otton, 1998; Stephenson, 2000).
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102
Table 7.2
S tandard 2 x 2 contingency table for evaluation of th e skill of a binary
classification procedure.
Observed Yes
Observed No
Predicted Yes
A (hits)
C (false alarms)
Predicted No
B (misses)
D (correct rejections)
According to th e definition of th e proportion of correct forecasts, th e proportion
correct (P C ) can be w ritten as
PC =
A + D
= A + D
A +B +C +D
N
.
’
^
'
where N is the to ta l num ber of events. T he definition of HSS is sim ilar to P C , bu t
HSS su b tracts the correct forecasts due purely to random chance, E. Therefore, it
is defined as
H S S = (A+n D} e E ,
(7-9)
where E can be derived as follows. Based on random guess, th e expected num ber
for hits is given by
A!
=
N • P(o) ■P ( f )
where P ( o )
and P ( f )
c™ >
express th e probability th a t the event willoccur based on
observations and forecast, respectively. Similarly, the expected num ber for correct
rejections D' can be derived from
jy
where P(o)
and P ( f )
=
n
■p (o ) ■p (J)
=
n
.A ± £ 1 .(E ± A
N
N
’
(7 1 I)
[
’
describes th e probability th a t events willno t occur based
on observations and forecast, respectively. Then, th e expected num ber of correct
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103
forecasts by random chance E is equal to A' + D'. Finally, combining w ith equations
(7.9), (7.10), and (7.11), the H S S can be derived as th e following equation:
2(AD —BC)
B 2 + C 2 + 2AD + (B + C)(A + D)
'
'
Unlike the bias, RMS difference, and correlation coefficient, th e Heidke Skill Score
is based on binary classification and applied m ainly to discrete underlying variables.
In the present context, HSS can be applied to determ inations of rain vs. no rain by
both th e retrieved algorithm and th e validation. However, th e definition of rain and
no rain is not unique. Employed intensity thresholds, calibration errors and minim um
detectable rain ra te of b o th validation and retrieved d a ta will all result in different
definitions of rain vs. no rain. T he validation m etric can be applied unambiguously
only under the condition th a t b o th th e validation d a ta and retrieval have th e same
definition of rainy events. If th e condition is not satisfied, as is often th e case in
real-world applications, a single scalar HSS m ight m isrepresent th e retrieval ability of
algorithms. Therefore, instead of using a fixed rain ra te threshold, Conner and P etty
(1998) tre a ted th e intensity threshold as a continuous variable. W hen th e variation
of the threshold is applied, th e HSS can be considered as a function of separate
rain rate thresholds for b o th th e observed and retrieved d ata. T hen by reference to
2-D plots of this function, it is possible to evaluate retrieval skills of algorithm s at
delineating observed rain ra te as a continuous function of th e threshold of observed
rain intensity, to reflect the bias of th e retrieval from the position of th e m axim um
Heidke Skill Score axis, and to further calibrate th e algorithm s.
7.3.5
T h eoretical H SS d istribution
Before employing HSS into the validations of real-world cases, it would be use­
ful to understand th e two-dimensional HSS distributions for th e Bayesian retrieval
when the perfect conditional and prior likelihoods are used. To address this issue, a
synthetic dataset was produced from the Bayesian ZV model using random num ber
generation. In order to assure th a t th e generation process of th e synthetic d ataset is
appropriate and th e num erical solutions indeed follow th e Bayes rule, th e posterior
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104
7
1.0
6
0.8
5
0.6
4
3
0 .4
2
0.2
0.0
0
0 .0
0 .2 0 .4 0 .6 0 .8
D a ta s e t ( m m / h r )
1.0
0
1
2
3
4
5
6
D a ta s e t ( m m / h r )
7
100
25
80
E
E
60
o>
<D
<D
cr
40
20
0
5
10
15
20
D a ta s e t ( m m / h r )
25
0
20
40
60
80
D a ta s e t ( m m / h r )
100
Figure 7.2. Plots of th e sample m ean of th e retrieval vs. th e posterior
m ean estim ated from th e synthetic d ataset at a given P if th e corre­
sponding sample size is (a) greater th a n 100; (b) greater th a n 30; (c)
greater th a n 10; and (d) less th a n or equal to 10. Note change of range
in each case.
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105
mean of the synthetic dataset at a given P is com pared w ith th e sample mean of the
corresponding retrieval.
Figure 7.2 depicts the scatter plots of th e sample m ean of th e retrieval vs. the
posterior m ean of the dataset points a t various observation vectors P. For any given
P, a bin size of 0.01 was chosen for P 10, P 1 9 , and P 37, and th en the sample size
would be th e counts of d a ta points located w ithin the 3-D grid box. Different P
associates w ith various sample sizes, so th e d a ta points are grouped and plotted
by different sample size. It is found th a t th e posterior m ean of th e dataset has a
good agreem ent w ith the m ean of the retrieval when th e sample size is greater th an
100, as shown in figure 7.2(a). The shift from th e diagonal line at d ataset rain rate
of
0 .2
m m /h r and retrieved rain ra te of
0 .1
m m /h r is a ttrib u te d to the fact th a t
th e algorithm only applies to th e pixel where either one of th e atten u atio n indices
a t 10.65, 19.35 and 37.00 GHz is less th a n 0.8. Then, some of pixels associated
w ith non-zero tru e rain ra te (dataset) have zero retrieved surface rainfall, and thus
results in an underestim ation in this range of rain rate.
W hen th e sample size
decreases, th e means from the synthetic d ataset and retrieval reveal some scattering,
which is expected and reasonable from a dataset of random variates. If th e sample
size is less th a n or equal to 10 (as shown in figure 7.2(d)) and m ost of th e points
might represent only one pixel at given P, the relation dem onstrates very significant
scattering phenomenon. In short, th e good m atch-up for th e observation vectors
w ith larger sample sizes and th e substantial scattering for those w ith smaller sample
sizes axe expected in the synthetic d ataset and its corresponding retrieved rain rate.
Therefore, we do have the confidence th a t th e generation process of random variates
is proper, and th e algorithm follows th e Bayesian principles.
Figure 7.3(a) and (d) depict th e theoretical 2-dimensional HSS distributions for
th e Bayesian 3M-MEAN and 3M-MLE algorithm s, respectively. Results show th a t
there are two areas associated w ith high skill scores. One is located a t rain rates less
th a n 20 m m /h r w ith larger HSS, and th e other occurs w ith a skill score of around 0.4
a t larger rain rate. Moreover, th e coverage of th e 2-D contours is an indication of the
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106
BAYESIAN 3 V -M L E
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Figure 7.3. Plots of 2-D Heidke skill scores (HSS); th e m axim um of HSS
w ith respect to tru e rain rate threshold; and th e best algorithm threshold
vs. tru e rain ra te threshold based on th e synthetic d a ta of th e Bayesian
ZV model, (a)-(c) are illustrated for MEAN estim ates, while (d)-(f) are
for MLE.
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107
relative location between the tru e and retrieved rain rate. For example, at a given
PR -T M I threshold of 40 m m /h r, th e contours fall w ithin around 10—60 m m /h r using
the posterior m ean as the estim ator, indicating th a t the retrieved rain ra te varies in
a wide range even in th e synthetic test. Since th e 0.1 contour could extend to very
large tru e rainfall rate, it shows th e retrieval ability of the Bayesian 3M-MEAN and
3M-MLE algorithm s over the heavy precipitation systems if th e specifications of the
prior pdf and th e physical model are correct.
At a given tru e rain ra te threshold, a m axim um HSS over th e algorithm rain rate
dom ain could be found. The curve of m axim um HSS vs. tru e threshold represents the
discrim ination ability of the algorithm for a certain rain rate threshold. Figure 7.3(b)
and (e) dem onstrate th a t the score is very high in th e situation of very light rain rate
in b oth MEAN and M LE algorithms. The m axim um HSS decreases w ith increasing
tru e rain intensity, reaches a lowest point of 0.35 a t 20—40 m m /h r, th en increases
w ith a small degree while th e tru e rain rate is between 60—80 m m /h r. T he plots
indicate th a t th e 31/-MEAN and 3R-M LE algorithm s are able to differentiate th e
rain rate in areas w ith very light or heavy precipitation, b u t might have difficulty to
discrim inate the rain intensity of 20—40 m m /h r even though perfect specifications
were used.
The algorithm rain ra te threshold associated w ith the m axim um HSS a t a fixed
observed threshold is defined as th e best algorithm threshold. Thus, th e best algo­
rithm is able to provide th e inform ation about bias. The plot of th e best algorithm
threshold vs. th e tru e rain rate threshold (as shown in figure 7.3(c)) dem onstrates
th a t th e Bayesian 3P-M EA N algorithm yields very good estim ates for th e tru e rain
rate threshold of 0—20 m m /h r, b u t underestim ates higher rain rate. A notable bias
occurs a t higher tru e rain rate in th e retrieval even though perfect inform ation has
been applied. It indicates th a t the inform ation from the dataset is lim ited due to th e
ambiguity of th e physical relation to th e rain ra te and the satu ratio n of th e variables.
In addition to th a t th e prior gives more weights in th e light rain rate, th e Bayesian
3M-MEAN algorithm produces a posterior p d f skewed to th e left and th u s leads to
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108
an apparent underestim ation for the pixels w ith higher rain rate. Furtherm ore, fig­
ure 7.3(f) reveals th a t the perform ance of the Bayesian 3V-M LE algorithm in th e
regimes w ith 5—20 m m /h r rain ra te is not as good as th a t of th e 3Y-M EAN model,
and th e retrieval using MLE has a substantial negative bias when th e tru e rain rate
is between 20—50 m m /hr. However, it is evident th a t th e 3P-M L E algorithm has
more skills to capture the very heavy rain rate when the conditional likelihood and
th e prior distribution are perfectly modeled.
7.4
R esu lts
7.4.1
T yp h oon case
In order to obtain a direct sense of how different the retrieval from each algo­
rithm behaves, P R interpolated and algorithm -retrieved rain ra te for all pixels are
m apped for th e typhoon case (as figure 7.4 shows). Those rain m aps exhibit th a t all
algorithm s are able to retrieve the eye, the two separate rain bands, and the overall
cyclonic structure of the typhoon. Exceptionally, the Bayesian 2Y-M LE algorithm
failed in th e retrieval of th e eye. In addition, m ost algorithm s have similar m agni­
tude of rain ra te to the P R interpolated data. However, th e G PR O F w ith th e old
database displays higher rain intensity generally, while th e Bayesian algorithm s w ith
MLE b o th in th e 2 V and 3 P models dem onstrate noticeable underestim ations about
th e rainfall rate.
The scatter plot of th e retrieval vs. P R rain ra te for orbit 336 is presented in
figure 7.6. Apparently, th e old G PR O F overestimates th e intensity when the P R
rain rate is greater th a n 10 m m /h r. Furtherm ore, if th e M LE is used in th e Bayesian
algorithm , th e dynam ic retrieval range is only around 10 m m /h r, which is much
smaller th a n th a t derived from th e posterior mean. These results could be confirmed
by th e set of columns in table 7.3, showing a positive bias in th e old G PR O F and a
negative bias in the Bayesian-MLE type algorithm. Table 7.3 also indicates th a t the
new G PR O F, P E T T Y TM I, P E T T Y HIST4, linear model, and th e Bayesian-MEAN
type algorithm s have similar bias, RMS, and correlation coefficients.
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109
Table 7.3
Sum m ary of bias, root-m ean-square difference (RMS), and correlation
coefficients against validation datasets of orbit 336 and B auer’s cases for
each algorithm . The unit of bias and RMS is m m /hr.
O rbit 336
Bias
RMS
Corr
G PR O F-O LD DATABASE
1.84
5.55
0.87
G PR O F-N E W DATABASE
-0.22
2.68
0.88
P E T T Y TM I ALGORITHM
-0.15
2.98
0.84
P E T T Y HIST4 ALGORITHM
-0.51
2.72
0.90
LIN EA R M O D E L -PR /T M I DATA
-0.29
2.71
0.88
0.00
2.89
0.85
-1.04
3.75
0.85
0.22
2.85
0.86
-1.06
3.90
0.82
B auer’s
Bias
RMS
Corr
G PR O F-O LD DATABASE
0.23
1.94
0.78
G PR O F-N E W DATABASE
-0.22
1.64
0.76
P E T T Y TM I ALGORITHM
-0.12
1.70
0.74
P E T T Y HIST4 ALGORITHM
-0.11
1.45
0.80
LINEAR M O D E L -PR /T M I DATA
0.00
1.58
0.78
BAYESIAN 3Y-MEAN
0.08
1.84
0.71
-0.29
1.80
0.74
0.11
1.75
0.75
-0.29
1.78
0.75
BAYESIAN 3Y-MEAN
BAYESIAN 3E-M LE
BAYESIAN 2E-M EAN
BAYESIAN 2E-M LE
BAYESIAN 3E-M LE
BAYESIAN 2P-M EA N
BAYESIAN 2Y-MLE
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110
Table 7.4
Sum m ary of bias, root-m ean-square difference (RMS), and correlation
coefficients against validation datasets of heavy/w idespread rain events
and 1998/04 dataset for each algorithm. The unit of bias and RMS is
m m /hr.
H eavy/w idespread rain events
Bias
RMS
Corr
G PRO F-O LD DATABASE
1.85
5.59
0.74
G PR O F-N EW DATABASE
-0.15
4.08
0.75
P E T T Y TM I A LGORITHM
0.04
4.08
0.75
P E T T Y HIST4 A LGORITHM
0.54
3.68
0.80
LINEAR M O D E L -P R /T M I DATA
0.84
4.34
0.74
BAYESIAN 3E-M EA N
1.30
5.03
0.68
-0.19
4.49
0.69
1.74
5.30
0.71
BAYESIAN 2R-M LE
-0.22
4.54
0.69
1998/04 random events
Bias
RMS
Corr
G PRO F-O LD DATABASE
0.24
2.02
0.73
G PR O F-N EW DATABASE
-0.21
1.18
0.78
P E T T Y TM I A LG O RITH M
-0.18
1.18
0.78
PE T T Y HIST4 A LGORITHM
-0.14
1.06
0.82
LINEAR M O D E L -P R /T M I DATA
-0.08
1.09
0.81
0.03
1.26
0.78
-0.01
1.21
0.75
0.04
1.26
0.78
-0.01
1.22
0.75
BAYESIAN 3Y-M LE
BAYESIAN 2Y-M EAN
BAYESIAN 3E-M EA N
BAYESIAN 3R-M LE
BAYESIAN 2Y-M EAN
BAYESIAN 2Y-M LE
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Ill
PS2A25 Rain R ate (1 5 k m )
GPROF -
GPROF -
PETTY TMI ALGORITHM
NEW DATABASE
PETTY HIST4 ALGORITHM
OLD DATABASE
LINEAR MODEL - TMI/PR DATA
16.c
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Figure 7.4. M aps of P R interpolated rain rate w ith 15-km resolution and
retrieved rain ra te of G PR O F (old and new databases), P E T T Y TM I,
P E T T Y H IS T 4 , a n d lin e a r m o d el a lg o rith m s for T R M M /T M I o rb it 336.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 7.5. Same as figure 7.4, b u t for Bayesian 3V-MEAN, 3V-MLE,
2V-MEAN, 2V-MLE algorithm s.
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113
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Figure 7.6. S catter plot of retrieved rain ra te vs. P R rain ra te for all
algorithm s, based on th e validation d a ta of th e TM I orbit 336.
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60
114
7.4.2
B a u er’s 12 oceanic cases
For the twelve oceanic cases, th e P E T T Y HIST4 has th e best perform ance in
term s of th e least root-m ean-squared error and th e highest correlation coefficient (as
shown in the table 7.3). The linear model has zero bias and th e second largest corre­
lation coefficient in this case. The retrieval from the Bayesian 317-MEAN algorithm
associates w ith a bias of only 0.08 m m /h r, bu t have a larger root-m ean-squared er­
ror and a smaller correlation coefficient, implying th e lack of retrieval ability in the
heavy rain rate.
Figure 7.7 shows th e contours of 2-dimensional HSS of each algorithm for the
oceanic cases, and indicates th a t high skill scores occur in th e range of 0—10 m m /h r
for all algorithm s.
T he coverage of 0.1 contour suggests th a t th e new G PR O F,
PE T T Y HIST4, the linear mode, and th e Bayesian 2V model have b e tte r skills to
retrieve higher rain ra te th a n th e P E T T Y TM I and the Bayesian 3V algorithm do
in the twelve cases.
Moreover, th e relationship between th e best algorithm rain
rate and the P R threshold is depicted in figure 7.8.
The linear relation in the
plots dem onstrates th a t m ost algorithm s, especially P E T T Y TM I and the BayesianMEAN models, offer good agreem ents w ith PR d a ta when th e tru e rain ra te is less
th an 20 m m /hr.
Comparisons of th e HSS distributions between th e Bayesian 317-MEAN and th e
217-MEAN algorithm s dem onstrate th a t th e 3R model does offer additional infor­
m ation in th e retrieval procedure. For example, as figure 7.7 shows, the retrieval
from th e 317-MEAN algorithm varies between [2, 20] m m /h r at a given P R rain rate
threshold of 10 m m /h r, while th e 217-MEAN algorithm yields rain rate up to 26
m m /hr. However, th e additional variable of P 10 also constrains th e retrieved rain
rate significantly for th e cases associated w ith heavy precipitation. T he degradation
at higher rain rate in th e 317 algorithm m ight originate from th e poor fit to th e d ata
in the specification of conditional likelihood.
Recall in th e 2-D HSS distributions of the synthetic d a ta (as shown in fig­
ure 7.3(a)), the 0.4 and 0.2 contours of th e 317-MEAN algorithm should be able
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115
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Figure 7.7. 2-D distribution of Heidke skill scores (HSS) for the 12 se­
lected cases from the Bauer et al. (2001). The value noted in th e bottom right corner of each plot indicates th e highest HSS of th e algorithm .
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Figure 7.8. Plots of th e best algorithm rain ra te threshold w ith respect
to the threshold of P R rain rate for th e B auer’s cases. The algorithm
used in the retrieval is shown in the title of each plot.
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80
117
PLOT OF HSS_MAX VS. PR THRESHOLD
in
in
GPROF - NEW DATABASE
PETTY TMI ALGORITHM
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BAYESIAN
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PR THRESHOLD ( m m / h r )
Figure 7.9. Plots of the m axim um Heidke skill score vs. th e threshold of
P R rain ra te based on th e dataset of 12 B auer’s cases.
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118
to go up to th e P R rain ra te threshold of 20 and 95 m m /h r, respectively. Similar
HSS distributions to the synthetic d a ta could be obtained if th e physical relationships
and th e prior rain rate pdfs in the real-world applications are perfectly represented
by the Bayesian model. However, it is found th a t th e 0.4 contour for th e valida­
tion cases only approaches to around 10 m m /hr, when th e 0.2 contour is reaching
P R threshold of 30 m m /hr. The discrepancy between the theoretical te st and realworld applications might be be a ttrib u te d to the poor fit at high rainfall rate, and it
might also im ply th a t some physical relationships in real world applications are not
modeled in th e algorithm.
To identify the discrim ination ability of each algorithm w ith respect to th e true
rain rate, plots of the m axim um HSS vs.
in figure 7.9.
P R rain rate threshold are presented
The distribution of HSS shows th a t each algorithm generally has
great discrim ination ability for th e rain rate of 0 to 10 m m /h r, b u t has different
perform ance w ith respect to surface rainfall. For example, th e retrieval ability of
th e new G PR O F and P E T T Y TM I algorithm reaches th e weakest point a t rain
rate of ~ 10 m m /h r, then the HSS goes up w ith increasing rain intensity.
The
P E T T Y HIST4, linear model, and th e Bayesian 2Y-MEAN have sm aller HSS in
th e range of 12—18 m m /h r, while th e maximum skill of th e Bayesian 3 V model
continues to decrease w ith increasing rain rate. It is evident th a t th e 2R-M EAN
algorithm perform s superior to the 3 V model in term s of discrim ination ability.
Interestingly, th e discrim ination ability of the 3F-M LE algorithm is the highest
among th e Bayesian algorithm s (as figure 7.9(b) shows).
Note th a t th e curve of th e m axim um HSS based on the synthetic d ataset changes
from 0.8 to around 0.35 when th e P R rain rate threshold is betw een 0 and 40 m m /h r
(referred to figure 7.3(b)). For this validation dataset, th e m axim um HSS from the
3Y-M EAN algorithm has a similar tren d to the synthetic dataset.
However, the
curve in th e real-world application reaches 0.35 when the P R threshold is around 12
m m /h r, and has th e lowest point of 0.15 score at 40 m m /h r. O n th e contrary, for
th e 3Y-M LE algorithm , th e tren d of th e maximum HSS in th e validation dataset is
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119
very different from th a t seen in th e synthetic test. It is not quite clear yet w hat is
responsible for the discrepancy.
7.4.3
H ea v y /w id esp rea d rain events
The values of bias listed in tab le 7.4 for the heavy/w idespread rain events tell
th a t the P E T T Y HIST4 algorithm has the least bias, when th e G PR O F w ith the old
database and th e Bayesian-MEAN type have significant positive bias in the retrieved
rainfall rate. P E T T Y HIST4 has th e greatest performance regarding th e errors and
correlation coefficients. Validations for th e heavy/w idespread cases by Heidke skill
scores are shown in figures 7.10, 7.11 and 7.12.
Figure 7.10 shows th e 0.1 contours of 2-dimensional HSS for all algorithm s are
substantially flatter in the P R rain ra te threshold of 0—5 m m /h r, except the P E T T Y
TM I and new G PR O F algorithm . W hen P R threshold is around 5 m m /h r, th e re­
trieved rain rate from P E T T Y HIST4, linear model, and th e Bayesian-type algo­
rithm s ranges from 0 to around 20 m m /h r, bu t the variations in th e P E T T Y TM I
and new G PR O F are about 0—15 m m /h r. This result points out a very unique
feature of the P E T T Y TM I algorithm . In G PR O F, the beam-filling effects are con­
sidered by th e inclusion of th e areal fraction of convective rainfall in th e observation
vector. T he P E T T Y HIST4, linear model, and the Bayesian algorithm do not add
the inhomogeneity of precipitation structures into th e retrieval process. Those algo­
rithm s tre a t each pixel individually, and the connections between th e pixel and its
surrounding environm ent are not considered. In this way of retrieval, th e P E T T Y
HIST4, linear-model, and th e Bayesian algorithm s generate a more significant varia­
tion in th e retrieval while different spatial structures of precipitation are of interest.
In contrast to other algorithm s, th e P E T T Y TM I algorithm considers th e beam
filling factor implicitly. In th e iterations of adjustm ent between th e observed and
calculated attenuation index for all three channels at the same tim e, th e precipita­
tion variation in different footprints has been included in P E T T Y TM I algorithm in
order to find an optim al estim ate. As a result, the P E T T Y TM I algorithm has a
more consistent perform ance in any kind of validation datasets.
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120
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Figure 7.10. 2-D distribution of Heidke skill scores (HSS) for the
heavy/w idespread rain events. T he value noted in the bottom -right cor­
ner of each plot indicates th e highest HSS of th e algorithm .
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40
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Figure 7.11. Plots of th e best algorithm rain rate threshold w ith respect
to the threshold of P R rain ra te for th e cases w ith heavy and widespread
precipitation.
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80
122
PLOT OF HSS_MAX VS. PR THRESHOLD
GPROF - NEW DATABASE
PETTY TMI ALGORITHM
PETTY HIST4 ALGORITHM
LINEAR MODEL - TMI/PR DATA'
BAYESIAN 3V-MEAN
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Figure 7.12. Plots of th e m axim um Heidke skill score vs. th e threshold
of P R rain ra te based on th e dataset of heavy/w idespread rain events.
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123
Similarly to B auer’s oceanic cases, figure 7.11 dem onstrates th a t m ost algorithm s
provide reasonable retrieved rain ra te for th e P R rain rate less th a n 20 m m /h r.
Among the algorithm s, the P E T T Y HIST4 algorithm shows extraordinary skills in
the heavy precipitation.
In addition, th e results from the G PR O F w ith th e old
database suggest th a t there is th e m ost significant bias in th e comparison.
In the figure 7.11, the Bayesian 2 V and 3 V model are associated w ith a noticeable
positive bias in the range of m oderate rain rate. Explanations ab o u t th e source of
th e bias are given here. Recall th a t there are only 88 events satisfying the criteria
of heavy and widespread precipitation in th e entire year of 1998. T he fact indicates
how sporadic this kind of precipitating system is in real-world applications, and as
well in the training dataset. Therefore, the current Bayesian model has difficulty to
represent the physical relationship of th e special cases, especially in th e 3 V model.
Due to the sampled events th a t present areas w ith widespread rain, th e precipitation
systems discussed here are less beam-filling th a n the average. Thus, even if the tru e
rain rate is only 5—10 m m /h r, th e values of P index a t 19.35 and 37.00 GHz are
almost saturated, and the P \ q could be around 0.35—0.45. According to such a low
Pio value plus satu rated P i9 and P 37, the Bayesian 3V model yields an estim ate w ith
higher rain rate th a n the tru e intensity, and then results in a positive bias in th e
range of m oderate rain rate. More discussions w ith detailed pdfs are described in
section 7.1.
The plot of the m axim um HSS vs. P R rain rate threshold are represented in
figure 7.12.
The distribution of HSS shows th a t the P E T T Y TM I and P E T T Y
HIST4 algorithm s have overall high m axim um skill scores in all ranges, while th e
linear model and Bayesian m odel degrades substantially.
In addition, unlike th e
previous twelve cases, the discrim ination abilities in all of our Bayesian algorithm s
do not differ so significantly.
7.4.4
A pril 1998
In th e random ly selected cases in April 1998, the P E T T Y HIST4 and the linear
model still have th e highest correlation coefficients and least error com pared to other
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124
GPROF -
OLD DATABASE
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Figure 7.13. 2-D distribution of Heidke skill scores (HSS) in random ly
selected 118 files in A pril 1998. The value noted in the bottom -right
corner of each plot indicates the highest HSS of th e algorithm.
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40
125
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OLD DATABASE
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Figure 7.14. Plots of the best algorithm rain ra te threshold w ith respect
to the threshold of P R rain rate for the cases of April, 1998.
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80
126
PLOT OF HSS_MAX VS. PR THRESHOLD
GPROF - NEW DATABASE
PETTY TMI ALGORITHM
PETTY HIST4 ALGORITHM
LINEAR MODEL - TMI/PR DATA'
BAYESIAN 3V-MEAN
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PLOT OF HSS_MAX VS. PR THRESHOLD
BAYESIAN
BAYESIAN
BAYESIAN
BAYESIAN
3V-MEAN
3V-MLE
2V-MEAN
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PR THRESHOLD ( m m / h r )
Figure 7.15. Plots of the m axim um Heidke skill score vs. th e threshold
of P R rain ra te based on the d ataset of A pril, 1998.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
127
algorithm s. A lthough the correlation coefficients from th e Bayesian models are not
as high as the two aforementioned algorithm s, th e Bayesian 3R-M EAN retrieval
shows a smaller bias in the validation.
The 2-D HSS distribution (figure 7.13) and th e plot of th e best algorithm vs.
P R rain rate threshold (figure 7.14) reveal th e good skill of th e Bayesian-MEAN
algorithm s for the case of rain ra te less th a n 20 m m /h r. T he Bayesian 3P-M EA N
algorithm seems to have trouble when th e rain rate is around 20—30 m m /hr. Inter­
estingly, all algorithm s failed to retrieved th e pixels associated w ith rain intensity of
~ 60 m m /h r in the dataset.
Figure 7.15 depicts the change of th e m axim um HSS w ith respect to the P R rain
rate threshold. In the 1998/04 dataset, th e discrim ination ability for all algorithm s
reduces when the tru e rain threshold increases. However, th e P E T T Y HIST4 al­
gorithm still has b etter ability to differentiate rain ra te higher th a n 20 m m /hr. In
addition, all of the Bayesian algorithm s have similar m axim um HSS curves for the
random ly picked dataset.
7.4.5
Sum m ary
The above validation results have dem onstrated th a t our Bayesian algorithm is
able to yield retrieved surface rainfall com parable to other benchm ark algorithm s
in term s of the bias, root-m ean-squared difference, correlation coefficient, and the
Heidke Skill Score for the various precipitation systems, while our Bayesian algo­
rithm has the additional advantage of posterior rain rate probability distributions.
T he m axim um Heidke Skill Score (HSS) a t a given P R rain ra te threshold for an
algorithm is an im portant indicator of th e potential ability to differentiate true rain
rate. However, the proportion of th e num ber of hits to th e to ta l d a ta points in the
contingency table for all algorithm s drops dram atically (to less th a n 1%) in th e P R
rain ra te threshold higher th a n 10 m m /h r, and thus th e m axim um HSS might not be
so meaningful when the tru e rain rate is above 10 m m /h r. Therefore, the sum m ary
of this chapter will place more em phasis on th e retrieval ability between 0 to 10
m m /h r for the Bayesian 3P-M EA N , 3P-M L E and other benchm ark algorithms.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
128
PLOT OF HSS_MAX VS. PR THRESHOLD
f01
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X
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GPROF - NEW DATABASE
PETTY TMI ALGORITHM
PETTY HIST4 ALGORITHM
LINEAR MODEL - TMI/PR DATA
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BAYESIAN 3V-MLE
10
4
6
PR THRESHOLD ( m m / h r )
Figure 7.16. Plots of th e m axim um Heidke skill score vs. th e P R rain
ra te threshold in the range of [0, 10] m m /h r based on B auer’s cases.
PLOT OF HSS_MAX VS. PR THRESHOLD
co 0-6
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8
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Figure 7.17. Same as figure 7.16, b u t for th e heavy and widespread
precipitation cases.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
129
Figures 7.16, 7.17, and 7.18 depict th e m axim um HSS vs. P R rain rate threshold
plotted only for [0, 10 m m /hr] based on th e aforementioned three validation datasets.
Overall, regardless of the bias, th e Bayesian ZV type algorithm s dem onstrate b e t­
te r maximum retrieval skills th a n th e G PR O F-N E W DATABASE and th e P E T T Y
TM I algorithm, b u t their skills are not as high as those of P E T T Y HIST4 and the
linear fitting model. The Bayesian 3Y-M EAN and 3Y-M LE algorithm s have similar
maximum HSS distributions, b u t th e Bayesian 3Y-M EAN algorithm tends to have
slightly higher scores th a n th e 3Y-M LE algorithm a t th e rain rate range of 2—6
m m /h r, especially for th e dataset of 1998/04.
W hen the detailed inform ation about bias is concerned, plots of the best algo­
rithm threshold vs. P R rain rate threshold are used to point out the associated
bias for each validation dataset (as shown in figures 7.19, 7.20, and 7.21). Results
reveal th a t the retrieved rain rate from P E T T Y HIST4 has a very consistent linear
delineation w ith the P R rain rate, even though there is an apparent system atic bias
for the validation dataset of heavy/w idespread precipitation systems. The best algo­
rithm threshold of the Bayesian ZV algorithm dem onstrates a good linear relation to
th e PR rain rate threshold as well for th e B auer’s cases and th e d ataset of 1998/04,
b u t has significant non-linear delineation and bias in th e cases of widespread and
heavy rain rate. On the contrary, th e Bayesian 3Y-M LE shows th e linearity at th e
rain range of 1—6 m m /h r in th e p articu lar heavy/w idespread precipitation cases.
Surprisingly, the best algorithm threshold of the G PR O F-N EW DATABASE algo­
rithm shows perfect linear dependence on th e P R threshold for the heavy/w idespread
precipitation systems, while all other algorithm s suffer from a notable positive bias.
Results in this chapter have m anifested th e sensitivity of th e validation m etrics
to the context, the validation dataset, and th e perform ance of the algorithm itself.
Comparisons between our Bayesian algorithm s and other retrieval algorithm s only
provide a general basis for us to b e tte r understand th e strength and weakness th a t
each algorithm might have, and to have th e direction how the algorithm could be
used and further improved.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
130
PLOT OF HSS_MAX VS. PR THRESHOLD
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Figure 7.18. Same as figure 7.16, b u t for th e d ataset of April, 1998.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
131
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Figure 7.19. Plots of th e best algorithm threshold vs. P R rain rate
threshold in the rain ra te range of [0, 10] m m /h r for th e B auer’s cases.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
132
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Figure 7.20. Same as figure 7.19, bu t for heavy/w idespread precipitation
cases.
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133
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.
.
.
1 .
4
6
8
PR THRESHOLD ( m m / h r )
.
.
*
m'
■
2
PETTY HIST4 ALGORITHM
2
4
6
8
P R THRESHOLD ( m m / h r )
,
4
6
8
P R THRESHOLD ( m m / h r )
10
BAYESIAN 3V -M LE
2
4
P R THRESHOLD ( m m / h r )
Figure 7.21. Same as figure 7.19, b u t for th e d ataset of April, 1998.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
134
8. U N C E R T A IN T Y ANA LY SIS
Based on Bayes theorem , th e posterior probability is proportional to th e product
of the conditional likelihood and th e prior probability distribution, which are both
derived from the PR -T M I m atch-up d a ta in our Bayesian 2 V and ZV models. Given
this, th e most natural questions to ask might be: How do we know if the physical
model th a t we posited to statistically describe th e relationship between rain rate
and observation vector provides a good fit to th e d ata? How does the assum ption of
th e prior distribution affect th e posterior probability? These two questions concern
th e uncertainty of the posterior pdf, and they are directly related to the model
assessment and the robustness test of the Bayesian algorithm , respectively. In the
section, th e analysis for th e two questions will be presented.
8.1
A ssessm en t o f th e p osited conditional likelihood
There are qualitative and quantitative ways to assess th e fitness of th e condi­
tional likelihood in the model. Q uantitative means provide a hypothesis test and
corresponding discrepancy variables to suggest if th e posited model should be ac­
cepted or rejected under th e observed dataset. However, it is still controversial since
th e test results strongly depend on w hat kind of statistical variable is used, and
different variables might lead to completely opposite conclusions. In addition, if the
statistical value does not support rejection of th e model, it does not guarantee th a t
th e model provides a high goodness-of-fit. In contrast to q u an titativ e m ethods, qual­
itative approaches (e.g., visual graphs) help to bring a more intuitive image of how
th e observed/sim ulated d a ta behave. Although the graphical way has a drawback in
th a t th e difficulty increases in m ultivariate cases, it offers a m ore direct comparison
and avoids m isinterpretation assessments based on a single num ber.
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135
JOINT PDF ( 1 V MODEL)
14
12
10
8
6
4
2
OL
0.0
0.2
0.4
0.6
0.8
1.0
P37
JOINT PDF (OBSERVATIONS)
14
12
10
8
6
4
2
0
0 .0
0 .2
0 .4
0 .6
P37
0.8
1.0
Figure 8.1. Plots of joint probability density functions / ( F 3 7 , R) based on
th e I V model (upper) and near-nadir PR -T M I d a ta (bottom ). C ontours
are logarithm ically spaced; actual value is 1 0 x' where x is th e contour
label, x are plotted for values of [-5, -4, -3, -2, -1, 0, 1] and [-4, -3, -2, -1,
0, 1] for th e I V model and observations, respectively .
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136
In th e Bayesian I V model, since only two variables are involved, th e model
is assessed by the comparison between observed and sim ulated joint distribution.
Figure 8.1 depicts th e joint distribution of f ( P 37, R ) and th e p d f derived from th e
near-nadir PR -T M I m atch-up data. Contours of th e density function are logarith­
mically spaced, so the value of -2 presents the probability of 0.01. Note th a t since
th e sample size of the observations is around 1.2 million, the contour value less th an
-3 indicates th a t the grid box has a num ber of pixels less th a n 4. In th a t case, the
contours of -4 and -5 are not very meaningful for th e comparison, bu t rath er give an
indication of an overall coverage of th e data. The results in figure 8.1 dem onstrate
th a t th e conditional probability density function in th e I V model visually fits the
observations very well. However, there are some minor differences between those
two distributions. First, the contour w ith probability of 0.01 in the PR -TM I d a ta
goes up to 10 m m /hr, b u t only to 8 m m /h r in th e model. In addition, the nearly
saturated P37 is less frequent in the PR -T M I d a ta th a n th a t in th e simulations. Fur­
therm ore, some artificial effects are shown in th e -3 contour a t larger P37 values in
th e IV model.
The goodness-of-fit of the 2V model is evaluated from th e joint pdf, f ( P i 9,R ) ,
which is the m arginal pdf of f(Pig,P37, R) and could be com puted from
/(P i9 , R ) = f f ( P 37\R)f(Pi9\P37, R M R ) • d ( p 37).
JP37
(8.1)
Similarly, th e 3V model is checked by the f ( P w ,R ) , yielded from the following
equation:
f(P w,R )= [
[
f ( P 37\ R ) f ( P 19\P37, R ) f ( P w \P19P37)n( R) ■d(P37)d(Pl9). (8.2)
JPig J P 3 7
The joint probability distribution derived from th e 2 V model (as shown in figure 8.2)
reveals th a t the functions describing th e physical relationship between P 19, P37l and
R provide a good fit to th e observed data. The coverage and shape of the contour
w ith probability of 0.001 in the 2V model are b o th very consistent w ith those of the
PR -T M I data. However, th e contour of probability w ith 0.01 shows a discrepancy
when Pig is between 0.25 and 0.35.
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JOINT PDF ( 2 V MODEL)
40
30
E
E
<u 20
□
c
'o
cr
0.0
0.2
0.4
1.0
0.8
0.6
P19
JOINT PDF (OBSERVATIONS)
40
30
E
E
a) 20
o
~
0 .0
0.2
0.4
0 .6
P19
-T
1 I
I - ___
0 .8
0
1.0
Figure 8.2. Same as figure 8.1, but for th e joint p df of (P 19,R) based
th e 2 V model (upper).
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138
JOINT PDF ( 3 V MODEL)
40
30
20
10
oL
0.0
0.2
0 .4
0.6
0.8
1.0
P10
JOINT PDF (OBSERVATIONS)
40
30
20
;Vi-
10
oL
0.0
0.2
0 .4
0.6
0.8
1.0
P10
Figure 8.3. Same as figure 8.1, b u t for the joint pdf of (Pio,R) calculated
from th e 3 V model.
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139
The distribution of / ( P 10|P ) is illustrated in figure 8.3, and th e results suggest
th a t th e construction of th e 3 V model is appropriate as well to yield a good agreem ent
w ith th e probability distribution of th e PR -T M I data. It is notew orthy th a t the
coverages of th e contours w ith value of -4 and -5 are not quite th e same as th e
observations. At a given rain ra te of 40 m m /h r, the observations are distributed in
a P w range of [0.3, 0.8], while th e -5 contour in model intercepts th e rain ra te of 40
m m /h r at Pw = [0.35, 0.9].
Note th a t th e physical model overestim ates occurrences at P values greater th an
1.0 in figures 8.1, 8.2 and 8.3. T he discrepancy is introduced from th e shape and
skewness of the param etric function fitted to the d a ta do not m atch those of the
observed probability distribution at very high P values.
However, since large P
values often indicate no rain or very light surface rainfall, th e discrepancy here does
not have a great im pact on th e perform ance of our Bayesian algorithm .
One might argue th a t th e m arginal probability distribution used to assess the
3 V model is an integrated quantity, and it might not be able to provide the details
of the 4-D m ultivariate relationships. To clarify this issue, we com pare th e observed
histogram and the m odel-calculated probability distribution of Pw at a given interval
of P 19, P37, and R.
Figure 8.4 depicts th e histogram s from observations (dashed
curve) and model-derived conditional distributions (dotted curve) for six grid boxes
only in this 4-D variable space. It dem onstrates th a t the fitted conditional likelihood,
/(P io |P i 9 , -P3 7 , R), is very close to th e observed distribution if the sample size is
large enough. Figure 8.4(e) shows a shift of approxim ately 0.1 on th e P 10 value,
and th e associated variance of th e fitted distribution tends to be larger th a n th a t of
the observed d a ta under heavy precipitation (as shown in b o th (e) and (f)). It is
plausible to param eterize th e fitted functions w ith a reduced variance and thus to
make the /( P io |P l 9 , P 3 7 , R) more dogmatic. However, due to th e larger field of view,
observations a t th e 10.65 GHz channel have a more serious beam-filling problem th an
other channels. Thus, it might not be beneficial for the retrieval to p u t more weight
on th e conditional likelihood of P i0, and test results (not shown) proves th e point.
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140
12
(b) R =
P19=
P37=
N =
R = 1.50
P 1 9 = 0 .7 0
P 3 7 = 0 .4 0
N = 2401
10
>.
4 .5 0
0 .3 5
0 .2 5
8
Jl
•” ( I
'it
_o
5o 4
I it
CL
0 L
0 .0
0 .2
10
0 .4
0 .6
P10
0 .8
0.0
1.0
8
6 .5 0
0 .4 5
0 .2 0
158
6
_>N
6
15
0r> 4
O
CL
4
0 .6
P10
0 .8
1
i
i
-
2
2
0 .4
' ' 1 1 ...................
'
( d ) R = 1 1 .5 0
p i 9 = 0 .3 0
/'
P 3 7 = 0 .1 0
11
N =
90
1
'''/ 4.
■
8
R =
P19=
P37=
N =
0.2
.1
1.0
■
-
-
'l!
1
i
\i
'i
0
0.8
0
0 .0
1.0
P10
I 1.
:(e)
.
11 1. 11
/'R =
\ P19 =
1 P37=
/ 'n =
'
:i \ '
■'1 '■ l
_
:
: 1
• 1
• /
-
15 .5 0
0 .1 5 '
0 .0 5 43
0 .2
. . J . ............ V v . 1 .>.
0 .4
0 .6
0 .8
1.0
P10
6
R = 19 .5 0
P 1 9 = 0 .1 0
P 3 7 = 0 .0 5
N =
22
5
4
3
2
11
-t
u
-
0
0 .0
0 .2
0 .4
0 .6
P10
0 .8
1.0
0.6
0 .8
1.0
P10
Figure 8.4. Plots of conditional pdfs /( P io |P i 9 , -P3 7 , R) based on th e PR TM I d a ta (dashed curve) and th e posited Bayesian model (dotted curve),
a t a given set (P , P19, P37) of (a) (1.5, 0.70, 0.40); (b) (4.5, 0.35, 0.25); (c)
(6.5, 0.45, 0.20); (d) (11.5, 0.3, 0.1); (e) (15.5, 0.15, 0.05); and (f) (19.5,
0.1, 0.05). N noted in figures represents the sample size of observed data.
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141
If the sample size is too small from a statistical point of view, there m ight be some
problems in the fit using the param etric functions. For example, in th e figure 8.4(b),
th e sample size is only eight, and there are two very different regimes shown in th e
distribution. It is hard to determ ine if these two peaks are statistically different
due to the small sample size. However, it might give some im plications here. T he
peak a t a lower P w value m ight represent a scene associated w ith less beam-filling
(e.g., the widespread precipitation system). Under this kind of rain cloud, the P
values at three channels are around 0 .2 -0 .4 at a given rain rate of 4.5 m m /h r.
T he other peak on th e right side describes a case more like a convective system,
which is associated w ith a strong beam-filling effect and produces a much higher P w
value. Since the sample size is so small, it is difficult to fit th e distribution and to
further evaluate the adequacy of th e posited pdf. W hile th e model suggests fitting
th e distribution w ith th e do tted line, the lack of goodness-of-fit on th e left peak
will result in an overestim ate of rain rate in widespread rain events, as shown in
chapter 7. On th e other hand, th e shift of the right peak reduces th e retrieved rain
intensity. Considering figure 8.4(b) and (e) together, th e m ism atch in the fitting to
th e observed d a ta results in th e lack of th e ability to retrieve higher P R rain ra te in
th e Bayesian 3U algorithm .
8.2
U n certain ty o f th e p osterior m ean due to th e prior distrib u tion
The sensitivity of our Bayesian rain rate retrieval algorithm to th e prior is evalu­
ated to ensure th e robustness of the algorithm and to quantify th e uncertainty of th e
posterior mean. There are several ways to estim ate th e uncertainty caused from th e
specification of the prior, b u t the analysis here is adapted from th e work of Berger
(1990). First, a class of priors was selected to model prior uncertainty. This class
has to include as m any reasonable priors as possible, b u t meanwhile it should not
contain unreasonable priors. For com putational convenience and adequacy, we em­
ployed an e-contamination class of priors in th e study. Detailed descriptions and
related m athem atical proofs about the class are given in Berger (1990). T he class of
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142
priors is expressed as
r« = {IT = (1 - e K , + eq: q € L},
(8,3)
where ttq is th e base prior distribution used in our algorithm; e represents the am ount
of uncertainty in 7To and ranges between 0 and 1; q is th e contam inated function;
and L describes th e class of th e q.
The choice of th e L has a great effect on th e estim ation of th e uncertainty of
the posterior mean. Sivaganesan and Berger (1989) suggested th a t there were four
possible sets to define L. In our case, since th e distribution of rain ra te R is close
to lognormal, th e probability of In R would generally follow a norm al distribution,
which is symmetric and unim odal. Due to this characteristic of the base prior distri­
bution, contam inations associated w ith a set of sym m etric unim odal distributions are
reasonable and appropriate to model th e uncertainty of th e prior. Furtherm ore, the
contam ination was added w ith respect to In i? first, then th e probability d istribution
was transform ed back to th e rain rate domain.
Once we have chosen the symmetric unim odal subset for contam ination, q could
be represented as a m ixture of uniforms:
roo \
(8.4)
where I is the indicator function; G is an a rb itrary distribution on (0, oo); p is
the param eter describing th e mode of th e norm al distribution over Ini?; and po
represents the param eter used in th e base prior, 7r0. Note th a t the param eters (p,
a) of the lognormal distribution for rain rate are exactly th e same as the param eters
in the norm al distribution for In R.
From th e equations (8.3) and (8.4), the posterior mean can be w ritten as
q{tt)
Sr t
■f ( P \ r ) n ( r ) d r
(1 - e) J f ( P \ r ) n 0{r)dr + e f£° H 2(z)Gdz
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(8.5)
143
where g(7r) and m(ir) are the posterior mean and m arginal probability of P associated
w ith the prior ir. In addition,
1
f e+
,1
(8.6)
where
9- = expf^o - z)
;
6+ = exp(p<> +
2
).
(8.7)
By defining
f ( z)
=
(1 - e) /
+
' Hi ( z )
g(z)
=
( l - e ) J f ( P \ ( ) M m + t H 2(z),
(8.8)
equation (8.5) can be rew riten as
^ ff(z)Gdz
e M = lW ) G T z
7
(8.9)
'
Finally, for a quantity formed as equation (8.9), th e variations of the q u an tity could
be obtained by finding the m inimum and m axim um values of f ( z ) / g ( z ) (proof can
be found in Sivaganesan and Berger (1989)).
8.3
U n certain ty o f th e posterior variance due to th e prior d istrib u tion
It is im portant to not only evaluate th e uncertainty of the posterior mean, bu t
also provide an accuracy measure by considering th e change of the posterior vari­
ance. Sivaganesan and Berger (1989) proposed an elegant m athem atical framework
to estim ate th e sensitivity of th e posterior variance to the prior distribution. Let
( f and V w represent the posterior m ean and posterior variance w ith respect to the
prior 7r. For g0 £ [h^Tr L f ( r ), sup^ gn(r)}, if we let
r 0 = {tt = (1 - e)7r0 + eq: q E Q
and
g*{r) = p0},
(8.10)
where r 0 expresses th e class of the prior, and g0 represents the fixed posterior mean,
then we would like to find infT V n (r), sup^ V n(r) to examine the range of th e posterior
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144
variance at a given posterior mean. T he problem could be solved by a theorem (proof
was dem onstrated in Sivaganesan and Berger (1989)) th a t
sup V n(r) =
7r€ro
inf V ^(r) =
7rer0
sup V n (r)
7rero2
inf V*(r),
7rer02
(8.11)
where
r 0 2 = {tt — (1 — e)7To + eq G Tq : q € (?2 }>
(8.12)
and
Q 2 = {q = aU(/Jlo - z ,H o + z ) + ( l - a ) U ( f i 0—z*,/j,q+z*) : 0 < a < 1
and
z , z * > 0},
(8.13)
where U represents the uniform function. One of th e three quantities a , z , z* has to
be determ ined by the constraint th a t th e posterior m ean for n is g0, and thus the
m axim izations over the r 02 are effectively a two-dimensional problem. Here, we have
chosen to constrain the coefficient a , and a could be obtained by
Q0 ~~
(! - e) I r t ' f ( P \ r ) n 0(r)dr + e a ± //+ f ( P \ r ) d r + e(l - a ) ^ f r f ( P \ r ) d r
f.
o*
>
(! —e) I r f ( P |r)7T0(r)<fr +
/ #_+ f ( P \ r ) ± d r + e(1 - a ) ^ /„.* /(P |r ) ±<fr
(8.14)
where
9*_ = exp(//0 - z*)
;
9*+ = exp(/j0 + z*).
Once a is determ ined, the posterior variance V* over r
02
(8.15)
can be calculated from
V?m ( z , z ' ) =
(1-e) JR(r-So)2f(P\r)iro(r)dr+ea-^(r-g0)2f(P\r)-^dr+e(l-a)-fe} {r~eo)2f(P\r)^dr
--------------------------------------------- ------------------------------ 55— --------------------(8.16)
(l-«) f R f(P\r)iro(r)dr+ea-±
f ( P \ r ) - ^ d r + e ( l ~ a ) ^ r f g,+ f ( P \ r ) ± d r
Then, the ranges of the posterior variance could be found by searching th e m axim um
and minimum of the variance VpQ2 over th e 2-dimensional (z vs. z*) domain.
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145
8.4
R esu lts
Based on the framework in th e two previous sections, th e sensitivity of th e poste­
rior m ean and variance to the prior can be evaluated for th e Bayesian models. Since
th e Bayesian 3 V model has deficiency in th e fit at higher rain rate, the uncertainty
analysis for the 3 V model might not be so meaningful. Therefore, only results from
the I V and 2V models are shown in the section. In addition, to dem onstrate the
variations of posterior pdf in th e same units as rain rate, th e sensitivity of posterior
variance is depicted using th e posterior standard deviation.
8.4.1
IV m odel
Figure 8.5 depicts th e sensitivity of th e posterior m ean to the prior distribution
in the I V model when a sym m etric and unim odal contam ination function is applied
to model the uncertainty of th e prior. The am ount of uncertainty in the prior is
assumed to be 25% in th e study. Note th a t since th e tails of the prior drastically
affect the posterior distribution, th e uncertainty analysis here mainly concentrates on
adding a larger tail to the assum ed prior and on estim ating how much the posterior
m ean will increase due to th e higher rain rate of th e prior. The curves of lower
(dotted) and upper (solid) bounds in figure 8.5 reveal th a t the variation of the
posterior mean becomes more significant when th e value of P37 is less th an 0.2. If
th e P 3 7 is close to saturation, th e uncertainty of th e prior may result in a 3 m m /h r
difference (increasing from 8.5 to 11.5 m m /h r) in th e retrieval, which is about 35%
of change. The increase of th e posterior m ean comes from th e heavier tail of th e
contam inated prior, b u t beyond some point, P37 is satu rated and is no longer able to
distinguish rain rate greater th a n 12 m m /h r from others. T he result indicates th a t
other variables are necessary to increase the dynam ic range of th e retrieval.
The ranges of the posterior stan d ard deviations for given values of th e posterior
m ean are shown in figure
8 .6
a t P37 value varying from 0.6, 0.4, to 0.2. First, when
th e prior is not contam inated, th e posterior stan d ard deviation increases rapidly
from 0.5 to 3.5 m m /h r w ith decreasing P37 value. The result is expected since a
wide range of rain ra te could occur in th e case w ith low P37. In addition, when P37 is
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146
1V m o d e l
14
Choice = L - s u
12
epsilon = 0 .2 5
muR = - 2 . 8
10
8
6
4
2
0
0.0
0.2
0.4
0.6
P37
0.8
1.0
Figure 8.5. Plots of the posterior m ean (m m /hr) vs. P 37 in the I V model
under the specification th a t th e prior distribution is log N ( —2.8, 2.0).
The uncertainty of the prior is sim ulated by th e symmetric contam ination
function w ith a factor of 0.25. Solid and dotted curves indicate th e
suprem um and infimum, respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
147
P 3 7 = 0 .6 0
9
0 .9
0.6
0.3
0.60
0.80
0.7 0
0.75
0.65
POSTERIOR.MEAN ( m m / h r )
0.85
P37 = 0.40
1.4
;(b)
1.2
h-
1.0
LlJ
Q
0.8
0.6
a.
1.30
1.60
1.70
1.50
1.40
POSTERIOR.MEAN ( m m / h r )
1.80
P37 = 0.20
_c
8
o
i—
<
/>
o 2L
CL
3.0
4.0
4.5
3.5
POSTERIOR.MEAN ( m m / h r )
5.0
Figure 8.6. Range of posterior stan d ard deviation for th e given posterior
m ean in the I V model when P 37 is (a) 0.6, (b) 0.4, and (c) 0.2. The
square and asterisk symbols present th e minimum and m axim um values,
respectively. Thus, the area inside th e curves indicates th e region where
th e standard deviation m ight be located.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
148
0.6, the posterior mean varies from 0.64 to 0.82 m m /h r, and th e posterior standard
deviation changes between 0.5 and 0.8. If th e P 3 7 decreases to 0.2, the standard
deviation changes significantly from 3.5 to
6
m m /h r due to th e uncertainty of the
prior. T he result implies th a t cautions have to be given while th e pixel for retrieval
is associated w ith low P37.
8.4.2
2V m odel
The minimum, m aximum and range of th e posterior m ean in th e 2 V model over
th e P 19 and P 3 7 domains is shown in figure 8.7. Note th a t only th e pairs of (P19, P 37)
observed in the PR -T M I m atch-up d a ta are plotted in this figure. Points, for ex­
ample, having (P19, P 37) = ( 0 . 2 , 0.8), are unlikely to occur in real-world applications,
and are not discussed here. The distribution of th e lower bound indicates th a t the
dynam ic range of the retrieval extends to rain ra te greater th a n 15 m m /h r, and the
posterior m ean increases w ith decreasing P19 and P 37. In th e case th a t P 3 7 = 0 .2 ,
th e I V model only yields 3—5 m m /h r (as figure 8.5 shows), b u t a range between
3—15 m m /h r can be retrieved in th e 2 V model, depending on th e value of P19. In
addition, the range between th e m inim um and th e m axim um of th e posterior mean
(as shown in figure 8.7(c)) reveals th a t except some points, m ost variations of the
posterior m ean are less th a n
Figure
8 .8
6
m m /h r, indicating th a t the 2 V model is quite robust.
depicts the range of the posterior stan d ard deviation in th e 2 V model
while th e contam ination factor of 0.25 is added to the prior pdf. Results indicate
th a t th e percentage of change in stan d ard deviations is around 25% — 33%, and
goes up to around 45% when th e P values a t 19.35 and 37.00 GHz b o th are near
saturated. Similar to the I V model, th e 2 V model also shows the reduced accuracy
for the pixies associated w ith low P 19 and P 3 7 values.
Meanwhile, th e posterior
standard deviations for these pixels are th e m ost sensitive to th e uncertainty of the
prior distribution as well. Unfortunately, low P values usually indicate th e presence
of substantial precipitation, which is of particular interest. Therefore, to reduce the
posterior standard deviation and the sensitivity to the prior, more variables might
be needed to add more inform ation to make th e posterior p d f more informative.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
149
POSTERIOR.MEAN.MIN IN L-SU
1.0
RAIN RATE (M M /H R )
0.8
ps. 0.6
Q_
0.4
0 .2
0.0
0.0
0.2
0.4
0.6
P19
0.8
1.0
POSTERIOR.MEAN.MAX IN L-SU
1.0
RAIN RATE (M M /H R )
0.8
r-s. 0.6
to
Q_
0.4
0 .2
0.0
0
VARIATIONS OF POSTERIOR.MEAN IN L-SU
1.0
RAIN RATE (M M /H R )
0.8
^
to
a.
0.6
0.4
0.2
0.0
0
Figure 8.7. Range of posterior m ean in th e 2 V model for (a) th e lower
bound, (b) upper bound, and (c) th e m aginitude of variation in units of
(m m /h r). The contam inated function and factor used here are th e same
as those applied in the I V model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
150
P 1 9 = 0 .6 0 ,P 3 7 = 0 .4 0
P19 = 0 .7 0 ,P 3 7 = 0 .4 0
2.0
2.0
\£E
JZ
e
E
E,
o
g
z
z
<
>
<
o
O
>
ag
o
O'
S
£
0 .5
0 .5
o
Q.
0.0
2.0
0.0
1 .5 0
1 .6 0
1 .7 0
1 .8 0
1 .9 0
POSTERIOR.MEAN ( m m / h r )
2 .00
2.2
2 .4
2.8
2.6
POSTERIOR.MEAN ( m m / h r )
3 .0
3 .2
P19 = 0 .4 0 ,P 3 7 = 0 .2 0
P19 = 0 .5 0 ,P 3 7 = 0 .2 0
10
6
E
E
4
z
o
•i
>
iij
3
o
O'
o
2
CE
1
O F .........................
3 .2
3 .4
. . ....I
1
4
3 .6
3 .8
4 .0
4 .2
POSTERIOR.MEAN ( m m / h r )
4 .4
9
7
8
POSTERIOR.MEAN ( m m / h r )
6
4 .6
P 1 9 = 0 .2 0 ,P 3 7 = 0 .0 5
P 1 9 = 0 .2 0 ,P 3 7 = 0 .1 0
14
14
12
12
10
10
8
8
6
6
12
13
10
7 .0
14
16
15
POSTERIOR.MEAN ( m m / h r )
7 .5
8 .5
9 .0
8.0
POSTERIOR.MEAN ( m m / h r )
9 .5
10.0
23
24
P 1 9 = 0 .0 6 ,P 3 7 = 0 .0 7
P 1 9 = 0 .0 5 ,P 3 7 = 0 .0 5
18
16
O
O
14
14
10
15
16
17
18
19
POSTERIOR.MEAN ( m m / h r )
20
21
18
19
22
20
21
POSTERIOR.MEAN ( m m / h r )
Figure 8.8. Same as figure 8.6, b u t for 2 V model and given (Fig, F 37)=
(a) (0.7, 0.4) (b) (0.6, 0.4), (c) (0.5, 0.2), (d) (0.4, 0.2), (e) (0.2, 0.1), (f)
(0.2, 0.05), (g) (0.05, 0.05) and (h) (0.06, 0.07).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
151
8.5
Sum m ary
Analysis of the model assessment in this chapter indicates th a t th e specified
conditional likelihoods are able to provide a good fit to th e actual observations
and present the statistical and physical relationships between surface rainfall and
microwave variables. Furtherm ore, results of the robustness test suggest th a t our
Bayesian algorithm is robust even under th e existence of uncertainty in th e prior rain
ra te distribution. Note th a t a 25% contam ination factor in th e prior distribution is
added in the entire uncertainty analysis, and thus results here m ight present a larger
extent of variations in th e posterior mean and variances th a n th e reality. W hen
the contam ination factor can be estim ated in a more quantitative way, th e analysis
conducted in th e chapter can easily be applied again to com pute th e uncertainty.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
152
9. C A L IB R A T IO N OF M ODEL SIM ULATIO N
In chapter 3, we constructed a Bayesian retrieval algorithm from NWS W SR - 8 8 D
network 1-km composite reflectivity m easurem ents and a simplified 1-D radiative
transfer model. A fixed Z-R relationship ( Z = 200i?L6) was used in th e estim ations
of rain rate, which was not calibrated w ith any other rain rate m easurem ents. W hen
a reliable d ataset of observed rainfall intensity is available, how can th e model-derived
Bayesian retrieval algorithm be modified to yield calibrated rain ra te estim ates? On
th e other hand, since the PR -T M I m atch-up d ataset provides sim ultaneous observa­
tions of rain ra te and microwave signature, it offers a basis to validate th e radiative
transfer calculations in our simplified
1
-D plane-parallel model. Again, we assume
th a t P R interpolated rain rate presents tru th , and both th e calibration and valida­
tion of th e retrieval algorithm are based on the P R measurements. Once a b etter
d ataset becomes available, the same principles and procedures can be easily applied
to the new training dataset.
9.1
C alibration o f radar-sim ulated rain rate
In th e study, we suggest calibrating th e estim ates derived from NWS WSR-
8 8
D network reflectivity adjusting th e cum ulative density function (cdf) of model-
sim ulated rain rate to th a t of PR -T M I m atch-up data. Figure 9.1(a) presents the
cdfs of near-nadir interpolated rain ra te of P R in 1998, and the W SR - 8 8 D networksim ulated rain rate derived from a fixed Z-R function and an assum ed freezing level
of 3 km. It is evident th a t the sim ulated rainfall intensity from NW S W SR - 8 8 D
m easurem ents has a larger portion of d a ta associated w ith higher rain rate.
For
example, a t a given rain rate of 2 m m /h r, th e cumulative density has reached to 0.9
for the P R d ataset, meaning th a t only 10% of P R rain rate is greater th a n 2 m m /h r.
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153
However, 20% of the W SR - 8 8 D sim ulations are associated w ith rain rate higher th a n
2 m m /hr. It implies th a t the fixed Z-R relationship m ight not be always suitable for
every pixel.
In an atte m p t to force th e sim ulated rain rate d istribution to be close to th e PR TM I dataset, a lookup table based upon the cdf of th e observations was generated
to describe the conversion between the NWS W SR- 8 8 D network reflectivity-derived
rain intensity and calibrated rainfall rate. Then, th e sim ulated rain intensity was
adjusted at each grid point to produce a cumulative probability distribution closer to
th e cdf of P R rain rate. As a result, th e cdf of modified rainfall ra te from W SR- 8 8 D
simulations is able to converge to th e PR curve after two adjustm ent iterations (as
shown in figure 9.1(b)).
9.2
V erification o f rad iative transfer calculations
Once the adjusted W SR - 8 8 D network-simulated rain ra te is ready, th e assum p­
tions of radiative properties and radiative transfer calculations in our simplified
plane-parallel model can be verified by the m ultivariate relationship between rain
rate and microwave signature shown in Pw , Pig, and P37. T he 4-D distribution of
th e occurrence frequency is dem onstrated by plots of 2-D contours while the other
two variables are selected w ithin certain intervals. Figure 9.2 depicts the contours of
th e num ber of d a ta points where th e corresponding rain ra te is less th an
1
m m /h r.
T he left panel is the distribution from the W SR- 8 8 D network sim ulations a t a given
freezing height of 3 km, and th e middle panel is th e result draw n from the PR -T M I
m atch-up data. The orientations of the contour distributions from th e observations
and simulations are consistent. However, it is obvious th a t th e contours of PR -T M I
m atch-up d a ta has a long and narrow shape and the coverage of th e contours extends
further to lower .P3 7 values. T he model simulations indicate a broader distribution
in the direction of P 1 9 , b u t narrower along the P 3 7 axis.
W hile d a ta points having rain rate between 1 and 5 m m /h r are selected, similar
discrepancies shown in th e previous contour plots are also found here in th e com par­
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154
ison between th e observed and sim ulated d a ta (as shown in figure 9.3). Note th a t
the contour of PR -T M I m atch-up dataset has a tongue-like shape tow ard very low
Pw (less th a n 0.2) values, b u t the sim ulated P\g values are seldom smaller th a n 0.2.
In addition, if the Pio is in the range between [0.55, 0.65], some sim ulated points
w ith higher P37 values are not shown in the PR -T M I data. The existence of these
points might be due to th e Gaussian random noise we added in th e model.
For the case of larger rain rate (5 -1 5 m m /h r), figure 9.4 suggests th a t these
two datasets have a similar m ultichannel relationship, although th e contours of th e
simulations still cover a larger area at th e P37 vs. P 19 domain. Furtherm ore, it is
notable th a t the location of the m aximum occurrence from th e observations is a t P37
values of
0 .1
w ith a varying Pw, bu t th e sim ulations have more points associated
w ith almost zero in P37 values. Under very heavy precipitation (rain ra te greater
th an 15 m m /h r as shown in figure 9.5), the sim ulated d a ta produce more points w ith
much higher P37 values, and those points are not generated from th e observations.
The possible source of errors is given in the next section.
9.3
E ffects o f freezing level height
The discrepancy in th e 4-D distribution between th e PR -T M I m atch-up d a ta
and the W SR- 8 8 D sim ulations might be caused from th e assum ption of th e freezing
height. To examine th e adequacy of th e assigned value of th e freezing height, the
bright band inform ation is extracted from th e 55 orbits of th e P R 2A25 product to
clarify the issue. Again, in order to avoid th e uncertainty from off-nadir contam ina­
tion, only near-nadir pixels are included here. Figure 9.6 delineates th e histogram of
the height of P R bright bands in various seasons. T he distributions reveal th a t th e
bright bands are generally located at altitudes of 4—6 km in tropical environm ents.
If we assume th a t th e freezing level is about 0.25 km higher th a n th e peak of th e
bright band, then th e freezing heights estim ated from P R reflectivity are around 4.25
— 6.25 km. Note th a t th e histogram has a strong peak in altitude of
6
km for all
seasons, and it is not clear yet if an upper lim it has been applied to th e altitu d e of
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155
the bright bands in th e P R 2A25 algorithm . Since there is no independent d ataset to
help validate th e locations of th e bright band, it is not possible here to quantify th e
errors of th e estim ations. However, radiosonde observations rarely reveal freezing
level heights greater th a n 5 km.
To provide further inter-com parison w ith estim ated P R bright bands, freezing
heights based on TM I m easurem ents are also calculated using the m ethod W ilheit
et al. (1991), and th e following regression equation we proposed:
Z f = exp 3.469 -
7.6041
InV
(9.1)
where the freezing height Z f is in units of km, and V is the column w ater vapor
(kg/m 2). This regression model is derived from th e TM I and RAOBS m atch-up d ata,
same as those used to develop th e w ater vapor algorithm . The corresponding correla­
tion and root-m ean-square error for the estim ations are 0.86 and 0.4 km, respectively.
In W ilheit’s m ethod, th e freezing level heights are determ ined by brightness tem per­
atures at 19.35 and 22 GHz channels via a lookup table, which is based on model
simulations using a simplified atm ospheric condition.
The histogram s of freezing height for oceanic and rainy pixels in Jan u ary and
July of 1998 based on TM I observations are shown in Figure 9.7. O ur regression
model and W ilheit’s m ethod b o th yield m ost freezing levels at around 5 km, even
though the second m ode of th e distribution is located at different altitudes. Based
on the radiosonde and TM I observations, an upper limit of 5 km for th e freezing level
height seems more reasonable. Therefore, we modify the param eter of th e freezing
level height from 3 km to 5 km in our model, re-generate simulations, and re-com pare
with the PR -T M I m atch-up datasets.
The re-generated sim ulations w ith th e freezing level at 5 km are shown in the
right panels of figures 9.2, 9.3, 9.4, and 9.5. These results dem onstrate th a t th e
assum ption of higher freezing level yields very consistent 2-D contours w ith those of
the observations, in term s of contour extension, orientation, and the location of th e
maximum occurrence frequency. It suggests th a t th e assum ptions in our simplified
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156
radiative transfer model is appropriate. However, generally, the sim ulations still have
broader coverage in the direction of Pi 9 , and P 3 7 is still satu rated a t lower value th a n
the PR -T M I d a ta w ith increasing rain rate.
9.4
Im plication
Simulations based on the NWS W SR - 8 8 D reflectivity and the simplified
1
-D
plane-parallel radiative transfer model have shown a surprising good agreem ent w ith
the PR -TM I m atch-up d a ta in the m ultivariate physical relationship. In addition,
model simulations suggest th a t th e free param eter of heights of freezing levels has
a significant im pact on the 4-D (rain ra te and P vector) probability distribution.
To obtain b e tte r retrieval, th e Bayesian rain ra te algorithm has to be developed
according to various freezing levels, and th en th e free param eter can be applied to
select more appropriate Bayesian models to com pute the retrieval.
More detailed analysis has to be conducted for th e validation of heights of freezing
levels. It is still not quite understood w hat th e uncertainty of the estim ates of freezing
heights is, and how much error in freezing heights is acceptable in th e retrieval
application.
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157
CDF of Rain Rate
PR Da ta ( 2 3 7 6 3 0 8 )
S im u l a te d Da ta ( 4 7 8 4 5 1 9 )
0.6
U-
O
o
0 .4
0.2
0.0
0
2
4
6
8
10
12
Rain R a t e ( m m / h r )
CDF of Rain Rate
1.0
0.8
PR D at a ( 2 3 7 6 3 0 8 )
S im u l a te d D ata ( 3 6 5 4 9 9 7 )
0.6
0 .4
0.2
0.0
0
2
4
6
8
10
12
Ra i n R a t e ( m m / h r )
Figure 9.1. Cum ulative distribution functions of P R observed (solid
curve) and NWS W SR-88D network-sim ulated (dotted curve) rain rate.
P R curve is based on th e near-nadir P R interpolated rainfall intensity
during Jan., Apr., Jul., and Oct. of 1998. W SR-88D-simulated curve is
calculated a) using a fixed Z-R relationship w ithout adjustm ents tow ard
P R m easurem ents, and b) w ith adjustm ents from a lookup table. Sample
sizes are shown in th e parentheses.
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158
RR ' = [
1.0
O,
1]
1
P 1 0 = [ 0 .9 5 ,1 .0 5 ]
J
1.0
RR = [
O,
1]
'
p i o = [ o . 9 5 , i .0 5 ]
fffy k
1.0
ff/m
m
m
rv 0.6
l \U
0.4
0.2
0.2
on
0.0
0.2
0.4
0.6
P19
0 .8
0.0
0.0
1.0
M
i
K>
CL
.............................
0.6
P19
0.8
1.0
P1 0 = [0 .8 5 ,0 .9 5 ]
P37
0.2
0 .4
0.6
P19
0.8
.
1.0
NEXRAD s i m u a l t i o n s (5.0KM)
RR = [ 0, 1]
P 1 0 = [0 .8 5 ,0 .9 5 ]
P 1 0 = [0 .8 5 ,0 .9 5 ]
0.8
0.8
0.6
rv
0 .6
0 .6
0.4
0.4
0.2
0.2
0.0
0.0
0.0
0.2
0.4
0 .6
0.8
0.0
1.0
■
C
O
Q.
&
0.4
0.2
nn
0.0
0 .0
0.2
0.4
0.6
P19
i
0.8
1.0
0 .0
j
/
0.4
i■k9 r
0.2
0.4
0 .6
P19
1.0
1.0
:
■
:
$
0.2
.........
0.0
0.8
1.0
0 .0
PR ( 1 9 9 8 , n e a r - n a d i r )
NEXRAD s i m u a l t i o n s (3 .0KM)
RR = [ 0, 1]
P 1 0 —[0 .6 5 ,0 .7 5 ].
/
0 .8
J n l
f( u .
r*
ro. 0.6
a.
0.4
0.6
NEXRAD s i m u a l t i o n s (5.0KM)
0.6
0.2
0 .4
1.0 RR = [ 0, 1]
P 1 0 = [ 0 .7 5 , 0 .8 5 ] y ^ |‘
0.8
1.0 RR =[ 0, 1]
P 1 0 = [0 .7 5 ,0 .8 5 ]
0.8
Fit
0.6
0.2
P R (1 9 9 8 ,n ear-n ad ir)
NEXRAD s i m u a l t i o n s (3.0KM)
1.0 . RR = [ 0, 1]
P 1 0 = [0 .7 5 ,0 .8 5 l> fA
0.8
P37
......
0.0
0 .0
PR( 19 9 8 ,n e a r- n a d ir )
NEXRAD s i m u a i t i o n s (3 .0KM)
■
0.2
w
0 .4
m m
M W
0.4
i f f
0.2
i
rv 0.6
M m
CO
Q.
0.4
'(fflfrS§
0.8
m il M
0.6
RR
° ’ 11
P 1 0 = [ 0 .9 5 ,1 .0 5 ]
.
0.8
0.8
P37
NEXRAD s i m u a l t i o n s (5.0KM)
PR ( 1 9 9 8 , n e a r - n a d i r )
NEXRAD s i m u o l t l o n s (3.0KM)
0.2
0 .4
0.6
P19
0 .8
1.0
NEXRAD s i m u a l t i o n s (5 .0KM)
RR = [ 0, 1]
P 1 0 = [0 .6 5 ,0 .7 5 ]
1.0
P 1 0 = [0 .6 5 ,0 .7 5 ]
0.8
0.8
P37
r*. 0.6
0.4
0 .2
0.2
0.0
0.2
0.4
0.6
0 .8
1.0
0.0
0.0
0.0
0.2
0 .4
0.6
P19
0.8
1.0
0 .0
0.2
0.4
0.6
0 .8
Figure 9.2. 2-D contours of num ber of pixels associated w ith rain ra te less
th a n 1 m m /h r a t a given P \ q interval. Left and right panels are based on
WSR-88D sim ulations w ith a freezing level at 3 and 5 km, respectively.
The middle panel is derived from PR -T M I d a ta in 1998. C ontours are
plotted for values of [1, 10 , 50, 100 , 500, 103, 104, 105].
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
1.0
159
RR '=[ 1, 5]
P 1 0 = [0 .8 5 ,0 .9 5 ]
1 .0
'
NEXRAD s i m u a l t i o n s (5.0KM)
PR( 1 9 9 8 , n e a r - na d i r )
NEXRAD s i m u a l t i o n s (3.0KM)
RR = [ 1, 5]
P 1 0 = [0 .8 5 ,0 .9 5 ]
RR = [ i . 5]
P 1 0 = [0 .8 5 ,0 .9 5 ]
’ .
^
P37
0.8
0.6
r»
0 .6
0.4
0.2
0.2
0.0
0 .0
0 .2
0.4
0 .6
P19
0.8
o.o
0.0
1.0
P 1 0 = [0 .7 5 ,0 .8 5 ]
P37
P 1 0 = [0 .7 5 ,0 .8 5 ]
0.8
0 .8
0.6
(v. 0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 .0
0 .0
0.2
0.4
0.6
P19
0.8
1.0
0 .0
0 .0
PR ( 1 9 9 8 , n e a r - n a d i r )
NEXRAD s i m u a l t i o n s (3.0KM)
0.8
1.0
0.2
0.4
0.6
0.8
1.0
NEXRAD s i m u a l t i o n s (5.0KM)
RR = [ 1, 5]
P10 —[0.6 5 ,0 .7 5 ]
P 1 0 = [0 .65,0 .7 5 ]
0 .6
P19
P 1 0 = [0 .7 5 ,0 .8 5 ] . ■:
t.
0.8
0.0
0.4
NEXRAD s i m u o l t i o n s (5.0KM)
P R (1 9 9 8 ,n e a r-n a d ir)
NEXRAD s i m u a l t i o n s (3.0KM)
0.2
1.0
P 1 0 = [0 .6 5 ,0 .7 5 ]
P37
0 .8
0.4
0.2
0.0
0.0
. RR '=[ 1, 5]
P 1 0 = [0 .5 5 ,0 .6 5 ]
'
0.4
0.6
0.8
^
1.0
P 1 0 = [0 .5 5 ,0 .6 5 ]
RR = [ 1, 5]
P 1 0 = [0 .5 5 ,0 .6 5 ]
0.8
0.8
P37
0.6
0.4
0.2
0.2
o.o'
0.0
0 .0
0.2
0.4
0.6
0.8
1.0
NEXRAD s i m u o l t i o n s (5.0KM)
PR( 1 9 9 8 , n e a r - n a d i r )
NEXRAD s i m u a l t i o n s (3.0KM)
0.2
1.0
Figure 9.3. Same as figure 9.2, b u t for rain rate of [1, 5] m m /h r and
different intervals of P \ q.
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160
NEXRAD s i m u a l t i o n s (5.0KM)
PR (1998,neor-nadir)
NEXRAD s i m u a l t i o n s (3 .0K M)
RR =[' 5, 15]
P1 0 = [0 .7 5 ,0 .8 5 ]
1.0
P 1 0 = [0 .7 5 ,0 .8 5 ]
P 1 0 = [0 .7 5 ,0 .8 5 ]
P37
0.8
0.8
0.8
0.6
^
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.2
0.4
0.6
0.8
0.2
0.0
0.0
1.0
0.2
0.6
P19
0.8
1.0
o.o
0.0
0.2
0.4
0 .6
P19
0.8
1.0
NEXRAD s i m u a l t i o n s (5.0KM)
RR '=[ 5, 15]
P 1 0 = [0 .6 5 ,0 .7 5 ]
P 1 0 = [0 .6 5 ,0 .7 5 ]
P 1 0 = [0 .6 5 ,0 .7 5 ]
0.8
0.8
P37
0.4
P R (1998,n ear-n ad ir)
NEXRAD s i m u a l t i o n s (3 .0K M)
0.6
r- 0 .6
►
o
CL
0.4
0.4
0.2
0.2
o.o
0.0
0.2
0.4
0 .6
0.8
1.0
0.0
0.0
0.2
0.4
0.6
P19
0.8
1.0
NEXRAD s i m u a l t i o n s (5.0KM)
P R (1 9 9 8,near-nadir)
NEXRAD s i m u a l t i o n s (3 .0K M)
.RR '=[ 5, 15]
P 1 0 = [0 .5 5 ,0 .6 5 ]
1.0
P 1 0 = [0 .5 5 ,0 .6 5 ]
P 1 0 = [0 .5 5 ,0 .6 5 ]
0.8
0.8
P37
r*. 0.6
0 .2
0.0
.
0.0
0.2
0.4
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P19
0.8
1.0
0.4
0.4
0 .2
0 .2
o.o
0.0
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P19
0.8
1.0
1.0
1.0
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n2
0.2
0.2
0 .0
0.2
0.4
0.6
P19
0.8
1.0
0.0
0.0
0.6
0 .8
1.0
. RR = [ 5, 15]
P 1 0 = [0 .4 5 ,0 .5 5 ]
:
rv. 0.6
Q.
0.4
on
0 .4
0.8
P37
o
*01
0.6
0.2
NEXRAD s i m u a l t i o n s (5.0KM)
RR =[ 5, 15]
P 1 0 = [0 .4 5 ,0 .5 5 ]
0.8
0.8
0.0
0 .0
PR(19 9 8 ,n e a r- n a d ir )
NEXRAD s i m u a l t i o n s (3 .0KM)
. RR =[ 5, 15]
P 1 0 = [0 .4 5 ,0 .5 5 ]
P37
^
0 .2
0.4
0.6
P19
0.8
1.0
0.0
0.0
• »v
0.2
0.4
0.6
P19
0.8
Figure 9.4. Same as figure 9.2, b u t for rain rate of [5, 15] m m /h r and
different intervals of P iq.
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1.0
161
’
' :
0.8
0.8
P37
.RR =[ 15,100]
P 1 0 = [0 .6 5 ,0 .7 5 ]
.RR = [ 15,100]
P 1 0 = [0 .6 5 ,0 .7 5 ]
RR = [ 15,100]
P 1 0 = [0 .6 5 ,0 .7 5 ]
0 .6
0.4
0.4
0.2
0 .2
o.o
0.0
0.2
0.4
0 .6
0.8
1.0
o.o
0 .0
0.2
0.4
0.6
P19
0.8
0.0
1.0
0.2
0.4
0.6
P19
0 .8
1.0
NEXRAD s i m u a l t i o n s (5.0KM)
P R (1 9 9 8 ,n ear-n ad ir)
NEXRAD s i m u a l t i o n s (3 .0KM)
P37
NEXRAD s i m u o l t i o n s (5.0KM)
P R (1 9 9 8 ,near-nodir)
NEXRAD s i m u o l t i o n s (3.0KM)
. RR = [ 15,100]
P 1 0 = [0 .5 5 ,0 .6 5 ]
1 n .RR = [ 15,100]
P 1 0 = [0 .5 5 ,0 .6 5 ]
1.0
0.8
0.8
0.8
0.6
r- 0.6
Q.
r- 0.6
Q.
0.4
0.4
1.0
. r r =[ 1 5 , 1 0 0 ]
P 1 0 = [0 .5 5 ,0 .6 5 ]
:
0.4
,
no
.
0 .0
0.2
0.4
.................... ....................
0.6
0 .8
1.0
P19
NEXRAD s i m u a l t i o n s
........................................................
0.0
0 .0
(3.0KM)
P37
0.2
0 .4
0.6
P19
0.8
0.0
0.0
1.0
0.6
0.4
0.4
0.4
0 .2
0 .2
0.2
0.6
P19
0.8
1.0
0.0
0.0
1.0
0.2
0.4
0.6
P19
0 .8
1.0
RR = [ 15,100]
P 1 0 = [0 .3 5 ,0 .4 5 ]
0.2
0.2
0.4
0.6
P19
0.8
1.0
0 .6
0.0
0 .0
0.4
0.6
P19
0 .8
1.0
NEXRAD s i m u a l t i o n s (5.0KM)
1.0
0 .4
0.4
0.2
0.2
0.0
0 .0
0.2
. RR = [ 15,100]
P 1 0 = [0 .3 5 ,0 .4 5 ]
r-'. 0.6
fO
P37
o
0.6
1.0
0.8
0.8
0.8
0.0
0.0
rv
PR( 1 9 9 8 , n e a r —n a d i r )
NEXRAD s i m u a l t i o n s (3.0KM)
.RR = [ 15,100]
P 1 0 = [0 .3 5 ,0 .4 5 ]
0.8
0.8
0.6
0.4
0.6
P19
RR - [ 15,100]
P 1 0 = [0 .4 5 ,0 .5 5 ]
0.8
0.2
0.4
NEXRAD s i m u a l t i o n s (5. 0K M)
PR ( 1 9 9 8 , n e a r - n a d i r )
0.8
0.0
0.0
0.2
RR = [ 15,100]
P 1 0 = [0 .4 5 ,0 .5 5 ]
RR = [ 15,100]
P 1 0 = [0 .4 5 ,0 .5 5 ]
P37
0.2
0.2
0.2
0.2
.......................................................
0.4
0 .6
0.8
1.0
P19
^
0.0
0.0
0.2
.......................................................
0.4
0 .6
0.8
1.0
P19
Figure 9.5. Same as figure 9.2, b u t for rain ra te greater th an 15 m m /h r
and different intervals of lower P i0.
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:
162
PR ( n e a r - n a d i r , M A M )
PR ( n e a r - n a d i r , D J F )
0.20
0 .2 0
0 .1 0
0.10
0 .0 5
it 0 .0 5
o
o
c
o
0 .00
0.00
0
2
4
6
8
0
10
8
10
0 .2 0
O
oa
6
0 .2 0
0 .1 0
c
o
4
PR ( n e a r - n a d i r , S O N )
PR ( n e a r - n a d i r , J J A )
0 .2 5
D
2
Bri ght b a n d h e i g h t ( k m )
Bright b a n d h e i g h t ( k m )
0.10
t
Li -
0 .0 5
0 .0 5
0 .0 0
0.00
0
2
4
6
8
Bright b a n d he i g h t ( k m )
10
0
2
4
6
8
10
Br ight b a n d h e i g h t ( k m )
Figure 9.6. Plots of fraction of d a ta vs. bright band height based on
P R 2A25 algorithm retrieval of 55 orbits during August 1998 and July
1999. Season inform ation is shown in titles. D JF represents th e w inter
season, while MAM expresses th e spring. The vertical resolution of 2A25
product is 0.25 km.
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163
T M I- 19980'
TMI —1 9 9 8 0 '
0.20
0.4
GPROF-Wilheit
D
0 .3
c 0.10
0.2
TO
)
O
o
o
u: 0 .0 5
Li -
O.O
0
0.00
0
2
4
6
8
10
2
4
6
8
10
Freezing height (km )
Freezing height (km )
TMI—1 9 9 8 0 7
T M I - 19 9 8 0 7
0.4
0 .3 0
GPROF-Wilheit
o
-§
0 .2 5
o
0.20
o
o
c 0.2
co
oo
o
<
oj
L i-
L i-
0 .0 5
0.00
0.0
0
2
4
6
8
Freezing height (km )
10
0
2
4
6
8
10
Freezing height (km )
Figure 9.7. Plots of fraction of d a ta vs. altitude of freezing level based on
TM I measurem ents in Jan u ary and July, 1998. (a) and (b) are derived
from th e equation (9.1), while (c) and (d) are com puted from W ilheit’s
approach. Note th a t th e W ilheit’s m ethod constrains th e freezing level
to th e climatological altitu d e (around 4.9 km) at a given 302 K of sea
surface tem perature.
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164
10. C O N C LU SIO N S A N D F U T U R E W ORK
The emphasis of th e dissertation has been presenting a new Bayesian rain rate
retrieval algorithm which has two unique characteristics. One is th a t the conditional
likelihoods representing statistical and physical relationships between surface rainfall
and microwave signatures, and th e prior rain ra te probability distribution are derived
from both observations and model simulations.
T he other is th a t th e o u tp u t of
the algorithm is not ju st a single-pixel retrieved rain rate, bu t rath er a continuous
posterior probability distribution of surface rainfall.
Unlike other Bayesian algorithm s which are all based on simulations of cloudradiative models and simplified specifications in conditional likelihoods, this study
offers generic forms which are comprised of explicit, closed-form functions and are fit­
ted to both model sim ulations (based on NWS W SR-88D network reflectivity and a
simplified 1-D plane-parallel radiative transfer model) and real m easurem ents (from
PR-TM I m atch-up dataset). These generic forms make it possible to analytically
obtain a continuous posterior probability d istribution of surface rainfall w ith lim ited
com putational requirem ents. Various estim ators to the posterior pdf can be used
later to serve as the single-pixel retrieved value, depending on th e desired character­
istics of the retrieval.
Simulations based on W SR-88D reflectivity d a ta and a simplified one-dimensional
plane-parallel radiative transfer model are surprisingly close to actual observations
in term s of m ultichannel relationships in atten u atio n index. The sim ulated d ataset
helps to prove the prelim inary retrieval ability of th e Bayesian approach using th e
generic forms. W hile th e prior and conditional probability distributions show a great
effect on the Bayesian algorithm developed from the model simulations, a sensitivity
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165
test is needed to clarify the theoretical advantage of the Bayesian approach and the
retrieval ability if imperfect inform ation is applied to th e algorithm.
Two synthetic random ly-generated d atasets are used in th e theoretical sensitivity
tests. One of th e datasets has a physical model specified via a covariance m atrix
(same as other Bayesian algorithm s), and th e other dataset is generated from the
conditional pdfs based upon the radar-radiative model simulations. Results of the
sensitivity test suggest th a t we should be aware of th e inherent retrieval uncertainty
associated w ith some observation vectors and th a t significant bias occurs at higher
rain rate, even when the exact prior and conditional likelihoods were employed in the
Bayesian algorithm. The deficiency is a ttrib u te d partially to th e retrieval ambiguity
in some scenes because of beam-filling effects, and partially to th e lack of further
inform ation from d a ta for the cases of heavy rain rate, due to the satu ratio n of
microwave signatures.
The theoretical test also dem onstrates th a t the retrieved surface rainfall is sen­
sitive to the assum ptions in the prior rain ra te distribution. The effect of the prior
distribution on the retrieval is different a t various rain ra te ranges, m ainly deter­
mined by the characteristics of th e prior pdf. W hen two prior pdfs have very similar
distributions, the Bayesian algorithm is no t sensitive to th e slight change of th e pa­
ram eters used to express the distribution. Furtherm ore, th e test results show th a t use
of a simple covariance m atrix to describe th e conditional pdfs might not be able to
provide enough accurate inform ation ab o u t th e physical m ultichannel relationships,
and thus can lead to substantial errors and bias in th e retrieval.
A dataset from the m atch-up of P R and TM I m easurem ents is also used to
find another set of coefficients for those generic forms, and to develop Bayesian
3R-M EAN, 3E-M LE, 2E-M EAN and 2E-M LE algorithm s, where MEAN and MLE
indicate th a t the expectation value and m axim um likelihood estim ate of th e poste­
rior pdf are of interest, respectively; and ZV and 2 V present th e num ber of variables
used in th e retrieval algorithm. T he retrieved surface rainfall is com pared w ith th a t
of other benchm ark algorithm s and validated against independent datasets, includ­
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166
ing random cases and particular types of precipitation systems. Comparisons with
other algorithm s suggest th a t our Bayesian algorithm s perform com parably to oth­
ers b u t have th e additional advantage of continuous posterior rain rate probability
distribution. Validations have dem onstrated th a t th e retrieval from th e Bayesian
3V-M EAN algorithm has a low bias except for cases of heavy/w idespread precip­
itation systems, which are not random ly sampled. We found th a t th e heavy and
widespread precipitation systems are relatively rare in th e entire year of 1998, and
th a t their characteristics m ight not be included well in b o th th e training dataset and
th e Bayesian model.
The Bayesian-type algorithm shows good skill a t retrieving th e rain rate at 0
~ 20 m m /h r, b u t has difficulty obtaining heavy rain rate in all cases. The lack of
ability to yield the high rain rate increases th e corresponding root-m ean-square error
and reduces th e correlation coefficients. T he degradation of th e Bayesian algorithms
at the higher end of the tru e rain rate results from th e inherent lim ited inform ation
offered by th e microwave variables. Since th e microwave signature is satu rated in the
scene of heavy precipitation, and th e prior rain rate distribution always gives larger
probability for the light rain rate, th e Bayesian algorithm s ten d to underestim ate
higher rain intensity.
The M LE-type algorithm s have a smaller dynam ic retrieval range th a n th a t of
th e M EAN-type algorithms, b u t have b e tte r skills for th e retrieval of very light rain
intensity.
If the purpose of the analysis is th e overall (or global) rain rate w ith
the least bias, one might take th e expectation value of th e posterior probability
distribution as the retrieved surface rainfall for a single pixel. If th e precision in
th e areas of light rain rate is emphasized, one m ight use th e m axim um likelihood
estim ates for the retrieval.
A rigorous uncertainty analysis in th e study dem onstrates how the m ean and
variance of th e posterior density function change owing to th e uncertainty of the
prior rain ra te distribution, and indicates th a t our Bayesian algorithm is relatively
robust. In th e study, the additional variable of P 10 in the 3V -typc algorithm s pro­
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167
vides further inform ation in some scenes associated w ith heavy precipitation, bu t
the algorithm suffers from th e uncertainty introduced by th e larger field of view at
10.65 GHz. Results from the model assessment also point out an im perfect fit to the
PR -T M I m atch-up d a ta in 4-D space due to th e small sample size. These two factors
combine to make th e benefits of th e inclusion of P 10 questionable. In addition, the
distinguishable regimes shown in the 4-D conditional probability distribution suggest
th a t other variables m ight help to isolate out different signals.
Finally, th e four-dimensional rain rate and m ultichannel physical relationships
are examined by a comparison between PR -T M I m atch-up d ataset and th e radarradiative model simulations. The analysis helps to calibrate th e W SR-88D-derived
surface rainfall and to verify the adequacy of the model assum ptions about mi­
crowave properties and radiative transfer calculations. R esults reveal a very good
agreement between observations and simulations, b u t the 4-D relationships show a
great sensitivity to th e heights of freezing levels. T he la tte r statem ent points out
th e need for inform ation about freezing levels in b o th actual m easurem ents and the
development of our Bayesian algorithm. The verification conducted in th e study has
two crucial im plications in th e development of our Bayesian rain ra te retrieval algo­
rithm . One is th a t the verification provides a basis th a t th e estim ates of rain rate
can be calibrated and th e Bayesian algorithm can be modified when more reliable
surface rainfall m easurem ents are available. T he other is th a t th e assum ptions about
microwave property and radiative transfer calculations in th e radar-radiative model
could be used for further analysis if actual observations are no longer available.
The altitude of the freezing level is not included as either a variable or a free
param eter in the current version of our Bayesian algorithm . Since the model ver­
ification dem onstrates th e im portance of th e inform ation ab o u t th e freezing level,
reliable estim ates of th e freezing level heights are highly needed. The ideal approach
to address this issue for th e future is to quantify th e accuracy of th e retrieved freezing
level heights estim ated from the P R reflectivity by com paring w ith a large d ataset
of RAOBS, and to investigate th e effects of th e inclusion of freezing level heights on
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168
the retrieved surface rainfall. Furtherm ore, based on model sim ulations, th e freezing
level heights can be added as a free param eter. Then, depending on th e freezing
level of the pixel, various posterior p df and single-pixel retrievals will be given. The
corresponding perform ance can be further evaluated against th e current Bayesian
algorithm and other benchm ark algorithms.
While the attenuation index at 10.65 GHz is expected to provide more detailed
inform ation about th e heavy surface rainfall for the retrieval algorithm , b u t is as­
sociated w ith more uncertainty due to the larger field of view, th e scatter index at
85.50 GHz m ight be a favorable alternative. The PR -T M I m atch-up d ataset is ready
to offer the scattering inform ation, b u t th e radar-radiative transfer model is not able
to present the scattering signal due to th e simplicity of the model. Therefore, future
work should evaluate th e contribution of th e scattering index a t 85.50 GHz on th e
PR -T M I m atch-up d ataset first. Then th e radiative transfer model should include
more details on scattering effects.
More d a ta associated w ith heavy rain ra te should be included in th e future as well
to yield a b e tte r goodness-of-fit of the conditional likelihood
/(-P 1 0 I -P 1 9 , P 3 7 ,
R)- The
need will be more obvious if th e scattering index is added to th e Bayesian algorithm .
Fortunately, one of the advantages of our Bayesian algorithm is th a t conditional pdfs
are of interest, and therefore we could focus on th e pixels where th e heavy rainfall
rates occur to improve th e fit w ithout processing redundant d a ta points.
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LIST OF REFERENCES
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VITA
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174
VITA
Jui-Yuan C hristine Chiu was born in Ping-Dong, Kaohsiung, Taiwan. She re­
ceived her Bachelor and M aster of Science degrees in Atmospheric Physics from th e
N ational C entral University of Taiwan in 1992 and 1994, respectively. She passed
the national government exam ination in 1993 and had worked at the Environm ental
Protection A dm inistration of Taiwan for three years since 1994.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
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