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Passive microwave remote sensing of rainfall in mountainous regions

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PASSIVE MICROW AVE REMOTE SEN SIN G OF RAINFALL IN
M O U N T A IN O U S R EG IO N S
A D issertation
P resen ted to the F aculty o f the G ra d u a te School
of C o rn ell U n iv ersity
in P a rtia l F ulfillm ent o f th e R eq u irem en ts fo r the D egree of
D octor of P h ilo so p h y
by
K irk R ay m o n d H a se lto n
M ay 2001
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© 2001 Kirk R aym ond H a se lto n
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BIOGRAPHICAL SKETCH
K irk H aselton w as b o m o n 31 May 1962 a n d sp e n t h is first 22 years in
C alifo rn ia, before co m in g to Ith aca and C ornell U n iv ersity in 1984 to enter
a Ph.D . program in A p p lied Physics. After ta k in g a b reak to h ike the US
C o n tin en tal D ivide T rail in 1985 he retu rn ed to C o rn ell a n d p u rs u e d first
m icrom echanical fab ricatio n u sin g sem iconductor p ro cessin g m ethods
follow ed by a year w o rk in g w ith femtosecond lasers. A one y ear pause from
academ ics then follow ed b efo re returning to th e A p p lied P hysics Ph.D.
p ro g ra m in 1991, this tim e seek in g an ap p licatio n o f physics w h ich d id n 't
in v o lv e w o rk in g in a clean ro o m or in the d a r k w h ic h w as fo u n d w ith
B ryan Isacks an d the EOS p ro ject in the D ep a rtm en t of G eological Sciences.
T h ere is still ro o m for h im to im prove on h o w to se t m o d e st goals w hich
are realizable in reaso n ab le p erio d s of time.
iii
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ACKNOW LEDGEM ENTS
I w ould like to th a n k th e sponsors of different stages of m y g raduate
w o rk especially N A SA th r o u g h s u p p o rt of a NASA G ra d u ate S tu d e n t
R esearch P rogram (GSRP) Fellow ship a n d the Fannie an d Jo h n H ertz
F o u n d atio n for th eir s u p p o r t w ith a fellow ship betw een 1984 a n d 1988.1
th a n k B ryan Isacks fo r g iv in g m e the o p p o rtu n ity to w o rk in th e EOS
project, p ro v id in g a w e ll e q u ip p e d co m p u ter laboratory a n d rev iew in g this
thesis after a long tim e co m in g . I th a n k a n d acknow ledge the M arshall
Space F light C enter (M SFC) for kindly p ro v id in g all S S M /I d a ta through
the WETNET project a n d th e G lobal H y d ro lo g y Research C e n ter (GHRC ),
O m ar a t SENAHMI for. In La Paz, Bolivia, Sophie M o reau (ABTEMA),
Jean V acher (ORSTOM) a n d O m ar (SENAHM I) p ro v id ed the necessary
contacts and d iscu ssio n s to enable acquisition of the Lake T iticaca lake level
data. D avid Legates k in d ly p ro v id ed th e locations of gauge sta tio n s in Asia
an d S o u th A m erica w h ic h h e u sed in h is p recip itatio n clim ato lo g y . Back in
the Snee Graphics Lab C h ris D uncan h elp ed me to a d a p t the OLS
reprojection so ftw are to u se w ith the S S M /I data, gave tre m e n d o u s
co m p u ter assistance a n d th o u g h tfu l scientific discussions. W o rk stu d y
stu d e n ts Brian B osart a n d C harles 'W es' M alsbury h elp ed w ith th e SSM /I
p ilo t projects an d Lake T iticaca level analysis, respectively. M an fred
S trecker graciously a llo w e d m e som e ad d itio n al m o n th s to co n tin u e
w o rk in g on this th esis a t C ornell.
Sigrid Rofiner h a s co n sisten tly g iv en the stro n g est p u s h in b ringing
this thesis to co m p letio n a n d su p p o rtin g me in all w ays.
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TABLE OF CONTENTS
B iographical S k etch
iii
A ck n o w led g em en ts
iv
Table of C o n ten ts
v
List of Tables
v iii
L ist of Figures
rx
CHAPTER 1: R ain fall in the A ndes a n d H im alaya d e riv e d from satellite
passive m ic ro w a v e observations
1
Abstract
1
In tro d u c tio n
3
R ainfall a n d sto rm s in m o u n ta in o u s areas
6
S outh A m e ric a
7
C o n v e ctiv e system s of th e A m a z o n Basin
9
C o n v ectiv e system s - A ltip la n o an d E astern A ndes
11
C e n tra l a n d S outhern C h ile: W esterly S to rm Tracks
16
El N in o a n d the S o u th ern O scillatio n
19
H im a lay a n ran g es an d Tibet - A s ia n M onsoon
21
Existing clim atologies
24
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Rem ote S e n sin g of R ain fall
30
In d ire c t M easures - V isib le/In frared
35
D ire c t satellite m e a su re s - M icrow ave
36
M ic ro w a v e ra d ia tiv e transfer - B rightness te m p e ra tu re
39
P a ss iv e m icro w av e em issio n of the e a rth
42
A tm o sp h eric effects - Scattering
46
SSM /I - P a ssiv e m icrow ave satellite o b serv atio n s
51
S a m p lin g errors
57
R a in ra te retriev als a n d v alid atio n
58
P ro cessin g
62
Rain rate s calculated fro m S S M /I d ata
63
B ack g ro u n d stu d y o f th e central A ndes
63
C e n tra l A ndes
70
S o u th A m erica
73
A s ia n m o n so o n
84
D iscussion o f results
93
C lim atological featu re s - filling in the g ap s
93
S c a tte rin g surfaces o n the A ltiplano
94
D iu rn a l rainfall m a x im a
95
E le v a tio n of m a x im u m rain rate
95
R a in fall an d e ro sio n
96
S u m m a ry
97
B ibliography
99
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CH A PTER 2: C onstraints o n the rainfall in p u ts to Lake Titicaca d eriv e d
from satellite passive m icro w av e o b serv atio n s
108
A bstract
108
In tro d u c tio n
108
S S M /I Data
112
Identifying p recip itatio n
113
E stim atin g rainfall in p u t to the b asin
117
D iscussion an d C onclusions
122
Bibliography
124
CHAPTER 3: Fractal F lood Statistics
125
A bstract
125
In tro d u c tio n
126
A nalysis
128
R esu lts
132
C onclusions
145
Bibliography
148
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LIST OF TABLES
Table 3.1
M ean flood in te n sity factor, F, b y US H y d ro lo g ic R egion
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142
LIST OF FIGURES
F ig u re 1.1
S o u th Am erica, m o is tu re tra n sp o rt a n d the ITCZ....................... 8
F ig u re 1.2
M CC's in the A m ericas....................................................................... 14
F ig u re 1.3
M CS's in South A m eric a.................................................................... 17
F ig u re 1.4
M CS's in A sia.........................................................................................18
F ig u re 1.5
W arm ENSO a n o m a lie s in S outh A m eric a................................. 20
F ig u re 1.6
Features of A sian clim ate an d m o is tu re tran sp o rt.................... 22
F ig u re 1.7
W M O p re c ip ita tio n clim atology fo r S o u th A m erica.................25
F ig u re 1.8
L egates-W illm ott p recip itatio n for S o u th A m erica................... 27
F ig u re 1.9
P recipitation s ta tio n locations in S o u th A m erica...................... 28
F ig u re 1.10
G H C N p rec ip ita tio n clim atology fo r South A m erica............... 29
F ig u re 1.11
Precipitation s ta tio n locations in A sia ........................................... 31
F ig u re 1.12
W M O p recip itatio n clim atology fo r A sia..................................... 32
F ig u re 1.13
Legates-W illm ott p recip itatio n clim atology for A sia................33
F ig u re 1.14
Transm ittance o f a clear atm o sp h ere..............................................34
F ig u re 1.15
R adar-derived r a in rate and VIS-tR clo u d area...........................37
F ig u re 1.16
R ain estim ates fro m h o u rly VIS-IR clo u d area........................... 38
F ig u re 1.17
M icrow ave tra n sm itta n c e of v a rio u s atm o sp h eres................... 44
F ig u re 1.18
Satellite view ing g eo m etry an d su rfa ce em ission...................... 45
F ig u re 1.19
M icrow ave w av elen g th s and freq u en cies.................................... 47
F ig u re 1.20
Scattering and a b s o rp tio n efficiencies for rain d ro p s................. 48
F ig u re 1.21
Ice particle m icro w av e scattering efficiency................................. 49
F ig u re 1.22
R aindrops, ice p a rtic les and p recip itatin g clouds.......................52
F ig u re 1.23
Global SSM /I co v erag e.......................................................................54
F ig u re 1.24
S SM /I tem poral sa m p lin g ................................................................. 55
F ig u re 1.25
Satellite orbit a n d v iew in g sw ath .................................................... 56
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Figure 1.26
Tv(85), F a n d SI for th e cen tral A n d es o n January 29,1995....60
Figure 1.27
Radar ra in rate an d scattering in d ex ............................................. 61
Figure 1.28
M ean m icro w av e b ac k g ro u n d o f the central A n d e s................65
Figure 1.29
Spectrally d efin ed regions o f th e central A ndes........................66
Figure 1.30
Tv (85) vs. TV(22) for cen tral A n d ea n regions............................67
Figure 1.31
F vs. Tv (85)bg for lan d reg io n s in the central A n d e s................ 69
Figure 1.32
SSM /I cen tral A ndes rainfall, D ecem ber '94 - A p ril '95........ 71
Figure 1.33
Locations of sw ath profiles for 1994-95 stu d y .............................74
Figure 1.34
Elevation a n d rain-rate sw ath profiles: 1994-95.........................75
Figure 1.35
C entral A n d es accum ulated rain -rates for 1993-1994.............. 76
Figure 1.36
E levation a n d rain-rate sw ath profiles: 1993-94.........................77
Figure 1.37
A ccum ulated rain -rates for S o u th A m erica, 1993 a n d 1994....78
Figure 1.38
C entral A n d es accum ulated rain -rates b y season..................... 82
Figure 1.39
C entral A n d es accum ulated rain -rates b y time of d a y ............ 83
Figure 1.40
H istogram of in stan tan eo u s ra in rate values............................. 85
Figure 1.41
Elevation d istrib u tio n of ce n tral A n d es rain -rates.................. 86
Figure 1.42
June-A ugust, 1992-94, rain -rates for H im alaya..........................88
Figure 1.43
H im alayan accum ulated rain -rates b y tim e of d a y ...................89
Figure 1.44
H im alayan o rographic p eaks, sw a th profile lo cations............90
Figure 1.45
Elevation a n d rain -rate sw a th profiles for H im alaya..............91
Figure 1.46
Elevation d istrib u tio n of rain -rates for the H im alay a.............92
Figure 2.1
Central A ltiplano an d Lake T iticaca............................................109
Figure 2.2
Level of Lake Titicaca an d rates of level change......................I l l
Figure 2.3
SSM /I d eriv ed rain-rates for 1993 an d 1994.............................. 115
Figure 2.4
Rain-rates according to tim e o f d a y ............................................. 116
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F ig u re 2.5
Lake Titicaca basin as resolved by 12.5 k m p ix e ls..................... 118
F ig u re 2.6
S S M /I d e riv e d in p u t to Lake Titicaca: 1994-95 rain y season.119
F ig u re 2.7
C o rrelatio n o f rain rates a n d ground sta tio n to ta ls.................. 121
F ig u re 3.1
Rescaled range (R/S) analysis......................................................... 130
F ig u re 3.2
O b serv ed floods and statistical distributions: exam ples......... 133
F igure 3.3
R escaled ran g e (R/S) for exam ples in F igure 3.2.......................135
F ig u re 3.4
D istrib u tio n of 1009 US stream flow sta tio n s.............................. 137
F igure 3.5
H y d ro lo g ic regions of th e conterm inous U n ite d S tates..........138
F ig u re 3.6
Ratios o f observed to p red icted m axim um d isch arg es............139
F ig u re 3.7
Fractal fits o f norm alized flood frequency d a ta ..........................141
F ig u re 3.8
Flood in te n sity factor, F, across US H ydrologic R egions.........143
F ig u re 3.9
Flood in te n sity factor, F, for subregions in C alifo rn ia.............144
xi
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CH APTER 1:
RAINFALL IN THE A N D ES A N D HIM ALAYAS DERIVED FROM
SATELLITE PASSIVE M ICROW AVE OBSERVATIONS
A b stra c t
Rainfall is a p o o rly m e a s u re d q u a n tity in rem o te regions of the
e a rth . The lo n g est ru n n in g re c o rd s are av ailab le fro m co n v en tio n al
clim atologies as m easu red b y ra in g auges a t clim ate stations. Rain gauges
sa m p le the rain fall field in a lim ite d w ay - m e a su rin g th e total
ac cu m u latio n o f rain fall o v e r so m e p e rio d o f tim e a t a single point.
S atellite observations are esp ecially v alu ab le in rem o te areas lacking
su rface d ata a n d offer u n iq u e in fo rm atio n b y p ro v id in g sp atially
c o n tin u o u s estim ates of in sta n ta n e o u s, area -a v erag e d rain fall rate.
In this ch ap ter th e b e st available surface clim atologies an d the
characteristics of the m o st im p o rta n t rain fall-p ro d u cin g m o nsoonal a n d
cyclonic w eath er system s are rev ie w e d for S outh A m erica a n d Asia.
P assiv e m icrow ave estim ates of th e in stan ta n eo u s, area -a v erag e d rain fall
ra te are in tro d u ced an d d iscu ssed for b o th regions. Surface precipitation
clim atologies a n d the m icro w av e rain fall estim ates are com pared,
esp ecially w ith reg ard to o ro g rap h ic p recip itatio n in p o o rly in stru m e n te d
reg io n s of the A ndes a n d H im alay as.
Satellite p assiv e m ic ro w a v e o b serv atio n s fro m th e Special S ensor
M icrow ave Im ag er (SSM /I) are u se d for th e estim a tio n of in stan ta n eo u s
ra in rate based o n p rev io u sly p u b lish ed stu d ies a n d algorithm s. A nalysis of
la n d surface b rig h tn ess te m p e ra tu re s in th e cen tral A n d es indicates th a t
th e se retriev als rem ain v a lid in th is h ig h elev atio n re g io n w ith a s tro n g ly
1
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h etero g en eo u s m ic ro w a v e b ack g ro u n d . The in sta n ta n e o u s ra in rate
retriev als u se S S M /I d a ta for th ree years, 1992 th ro u g h 1994, from two p o la r
o rb itin g satellites, F10 a n d F l l , o f th e D efense M eteorological Satellite
P ro g ram (DMSP). R ain rates are calculated fro m S S M /I d a ta a t the orig in al
sp atial reso lu tio n of th e 85 GFIz channel (ab o u t 12.5 km ) in ord er to b e st
observe the g e o g rap h ic v a ria tio n of o ro g rap h ic p recip itatio n . The use o f
d a ta from tw o satellites g reatly im proves te m p o ra l coverage com pared to
th a t of ju s t o n e sa tellite b y ro u g h ly d o u b lin g th e to ta l n u m b er of
o b serv atio n s w ith in a g iv en tim e p erio d a n d f u rth e r allow s an assessm en t
o f th e tim e o f d a y w h e n rain fall m axim a occur.
M ax im u m ra in fa ll is o b se rv e d to o ccur in area s w ith elevation
ap p ro x im a tely tw o k ilo m eters in b o th the c e n tra l A ndes a n d H im alaya.
A lth o u g h th e te m p o ra l d ynam ics of ra in -p ro d u c in g w eath er system s d iffe r
g reatly b etw een th ese tw o ranges, in b o th cases v e ry h igh m ountains
o b stru ct the flo w o f m o ist tropical air. As a re s u lt rising air cools and its
d ew -p o in t in creases u n til tem p eratu re d ro p s b e lo w d ew -p o in t and
co n d e n satio n can occur. R ainfall occurs w h e n c o n d e n sa tio n is great
e n o u g h for larg e d ro p le t g ro w th w ith in a clo u d to exceed evaporation. In
areas of p ro n o u n ced to p o g rap h y dynam ic effects su ch as u p w in d
tu rb u len ce o r co n v erg en ce can also influence p recip itatio n . Since
m ax im u m ra in fa ll o ccu rs a t th e sam e e le v a tio n in b o th regions these
d y n am ic effects to n o t a p p e a r to shift the e le v a tio n of m axim um rainfall.
In b o th reg io n s th e elev atio n o f m axim um ra in fa ll is d eterm in ed by m o ist
tro p ical air w h ic h is p re d o m in a n tly near th e su rfa ce an d p roduces the sam e
m ean clo u d base.
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3
T em p o ral b eh av io r of in d iv id u a l w eather sy ste m s can also be
in v estig ated since u p to four observations per d a y a re potentially o b tain ed
from th e tw o satellites u sed in this stu d y . On the e a s te rn slopes of the
central A n d es the g reatest rain fall is observed at 5 A M local time.
A n alo g o u s n o ctu rn a l w ea th er sy stem s responsible fo r m o st of the rain fall
are k n o w n to occur in the m id -la titu d e central U n ite d S tates and in m id ­
la titu d e A rg en tin a, su g g estin g th a t som e form of sm a ll Mesoscale
C onvective C om plex (MCC) o r larg e M esoscale C o n v e ctiv e System (MCS)
d o m in a te s the rain fall-p ro d u cin g w ea th er on th e A m a z o n basin side o f th e
central A ndes. In the H im alaya th e greatest rainfall is observed at 5 PM
local tim e as is expected for n o rm al convective ra in fa ll w hich, alth o u g h
w id e sp re a d as the m onsoon is, does n o t organize itse lf in to larger system s
such as M CS's or M CC's.
In tro d u ctio n
P recip itatio n is a vital co m p o n en t of the e a rth ’s w a te r cycle a n d p lay s
an im p o rta n t role in sh ap in g lan d scap es. Over m u c h o f the world it is in
the fo rm o f rainfall, frozen form s o f p recip itatio n p red o m in a tin g a t h ig h
elevations a n d latitudes. Practical n eed s for k n o w led g e o f the spatial
d istrib u tio n an d in ten sity of rain fall include w a te r m anagem ent, flood
p red ictio n , an d stu d ies of soil erosion. A t longer tim e scales models o f
tectonics a n d erosion have su g g ested the im p o rtan ce o f orographic
p re c ip ita tio n at low elevations for th e m aintenance o f h ig h relief a t th e
edges o f plateaus such as the A ltiplano and Tibet [Isacks, 1992; Masek et al.,
1994]. Yet available p recip itatio n clim atologies d iffe r in th eir details w h e n
ex am in ed a t local scales w h ere m easurem ents m a y b e sparse. This is
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4
especially tru e for estim ating o ro g rap h ic precip itatio n acro ss the h ig h e st
a n d m ost ex ten siv e m o u n tain ra n g e s of the w o rld .
C o n v e n tio n a l m e asu rem e n t of p recip itatio n is w ith rain g au g e s
w h ich p ro v id e accum ulation a t a p o in t over tim e. In te rp o la tio n o f ra in
gauge m e asu rem e n ts to area estim a te s involves co n sid erab le erro r d u e to
th e high s p a tia l a n d tem poral v aria b ility of p recip itatio n . Satellite estim ates
are areally co m p lete (over the field of view of the se n so r) a n d p ro v id e
global coverage, b u t suffer fro m in ad eq u ate sam p lin g in tim e. E ven in
areas of lo w to p o g rap h ic relief, satellite estim ates o f a re a p recip itatio n can
perform b e tte r th a n g ag e-in terp o lated estim ates w h e n th e distance to the
nearest g ro u n d statio n is g reater th e n 40 km. [Bellon a n d A ustin, 1986]. For
the case of S o u th America, ex istin g netw orks are e sp ecially in a d eq u ate,
disp lay in g h ig h variability a n d g ro ss u nderestim ates e v e n of co n tin en tal
scale p recip itatio n d u e to the lack of ground stations in th e w ettest areas of
the continent [W illm ott et al., 1994], G round-based r a d a r estim ates of
rainfall are lim ite d to a sm all n u m b e r of countries w h ic h alread y c o n tain
the d en sest clim ate station n e tw o rk s. There are no r a d a r system s for the
p arts of th e w o rld th at are m o st sparsely in stru m e n te d b y clim ate stations.
In m o u n ta in o u s regions, precip itatio n is in flu en c ed by to p o g ra p h y
through th e fo rced ascent of m o ist air and the d is ru p tio n of vertical
eq u ilib riu m in the atm osphere [Barros and L etten m aier, 1994]. D u ra tio n or
intensity m a y b o th be increased b y topography [Barrett, 1981]. A t the largest
spatial scale, th e Tibetan p la te a u serves as an elev ated h e a t source w h ich
drives the In d ia n m onsoon [H a h n an d M anabe, 1975]. A t sm aller sp a tia l
scales to p o g ra p h y influences th e spatial an d te m p o ra l d istrib u tio n of
p recip itatio n a n d is p articularly responsible for o b se rv e d peaks in
I
I
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p recip itatio n alo n g w in d w a rd slopes a n d th e w ell-know n rain s h a d o w
effect o n the lee sid e o f to p o g rap h ic b arriers. U p to a height of 1-2 km . a
sm all a m o u n t o f liftin g o f m o ist air fo rm s sta tio n a ry cap clouds w h ic h
enhan ce d ro p le ts fallin g fro m higher p re c ip ita tin g clouds, re su ltin g in a
'seeder-feeder' effect [Barry, 1987]. Above 1-2 km ., topography destabilizes
the v ertical s tru c tu re o f th e atm osphere, liftin g m oist air w ell a b o v e the
co n d e n satio n level a n d fo rm in g deep c lo u d s capable of co n tin u o u s
p recip itatio n [Barros a n d L ettenm aier, 1994].
The m a g n itu d e a n d lo n g term p o sitio n s of orographic p re c ip ita tio n
p eak s can b e im p o rta n t to m aintenance o f th e to p o g rap h y w h ic h influences
those peaks. In th e tectonic-clim atic m o d el o f M asek et. al. [M asek e t al.,
1994], o ro g rap h ic p recip itatio n in the p rese n ce of high to p o g rap h y ( > 4 km.)
confines h ig h ero sio n ra te s a n d d e n u d a tio n to th e plateau m a rg in s. This
m o d el d em o n strate s th e p lau sib ility th a t h ig h orographic p re c ip ita tio n
m ain tain s steep to p o g ra p h y th ro u g h h ig h ra te s of erosion. In a re a s of low
o ro g rap h ic p re c ip ita tio n su ch as the P ilco m ay o Basin gentle to p o g ra p h y
reflects lan d fo rm s less m o d ified by ero sio n . T his hypothesis m a d e use of
d ata from a few ex istin g clim ate stations s itu a te d in their stu d y a re a s of
e astern Bolivia a n d N ep al. K now ledge o f th e sp atial variability o f rainfall
in th ese reg io n s w as u n av ailab le.
In this p a p e r w e ex ten d p rev io u sly d e v e lo p e d m icrow ave satellite
m e a su re m e n ts to m a p ra in fa ll in the v ic in ity o f h igh m o u n ta in ra n g e s at
g reater sp a tial re so lu tio n th a n currently a v a ila b le satellite or g ro u n d based
clim atologies for th e se m o u n tain o u s re g io n s. First, we use d e ta ile d
m e asu rem e n ts of th e m icro w av e b a c k g ro u n d signal in the c e n tra l A ndes to
show th a t th ese ra in ra te s are n o t co n tam in ate d b y the cold b a c k g ro u n d
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6
su rface an d ex trem e h etero g en eity o f th e A ltip lan o a n d A n d e a n ranges,
in c lu d in g the p rese n ce o f sn o w and ice.
Second, ch aracteristics of rainfall a n d storm s in th e th ree regions are
rev iew ed . The e le v a tio n o f m ax im u m p re c ip ita tio n is fo u n d to be
ap p ro x im ately 1 k m . in th e A ndes a n d e a ste rn H im alay as. W e find no
evidence for h y p o th e siz e d secondary p eak s in o rographic precipitation in
the n o rth ern A n d es [L auscher, 1976], n o r is th e region o f h ig h , ~4
m e te rs/y e a r, o ro g ra p h ic p recip itatio n in so u th eastern C o lu m b ia m apped by
this m ethod. In th e H im alay as, b ac k g ro u n d snow cover lim its the ability of
this stu d y to assess rain fall m axim a a t h ig h e r elevations (> 3 km .).
S ignificant v a ria tio n of o rographic m ax im a are m a p p e d a lo n g the
H im alay an front a n d S hillong P lateau, as w ell as along th e T sangpo valley
a t 29°N and from 85-90°E.
R a in fa ll and s to r m s in m o u n ta in o u s areas
In some area s a few large storm s a re responsible fo r m o st annual
rainfall. In co n trast to th is, to p o g rap h y can influence th e d istrib u tio n of
rainfall, focusing lo n g term peaks in ce rta in areas. In th is sectio n both
aspects of rainfall a re rev iew ed for m o u n ta in o u s regions.
The g eo g rap h ic areas in this stu d y w e re chosen fo r h av in g the
h ighest, large-scale to p o g ra p h y on e a rth a n d for their ra n g e in climates.
T he n o rth -so u th tr e n d o f th e A ndes in c lu d e s regions in b o th n o rth ern an d
so u th e rn h em isp h eres. T he style of p re c ip ita tio n ran g es fro m th a t
associated w ith larg e-scale, deep co nvection in the tro p ical regions to m id­
la titu d e storm sy stem s in c e n tra l/s o u th e rn Chile. The H im a la y a n ranges
receive m ost of th e a n n u a l rainfall d u r in g th e su m m er m o n so o n except
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7
fo r p a rts of the n o rth w e ste rn H im alayas, P am ir m o u n ta in s and K ash m ir
reg io n . In those areas n o rth w este rly co n tin en tal air fro m the polar fro n t
co n v erg es w ith In d ian tra d e w in d s cau sin g som e a d d itio n a l n o n -m o n so o n
re la te d precipitation. A n ad d itio n al oro g rap h ic c o m p o n en t to p recip itatio n
is p rese n t in all these ranges. Existing precip itatio n clim atologies for th e
A n d es a n d H im alayas a n d other p u b lish ed know ledge o f storm s a n d
rain fall in these areas are review ed next.
S o u th A m erica
The m o st strik in g physical featu res of South A m erica are the A n d e s
m o u n ta in s, ru n n in g th e en tire le n g th o f th e w estern sid e of the c o n tin e n t,
a n d the A m azon basin, covering m u ch o f Brazil a n d th e eastern p a rts o f
C olum bia, E cuador, P e ru an d Bolivia (Figure 1.1). S easo n al convection in
th e A m azo n peaks in th e so u th e rn h em isp h ere s u m m e r d u rin g th e
an n u a l so u th w a rd m ig ratio n of th e Inter-T ropical C o nvergence Z one
(ITCZ) [Horel et al., 1989]. This large area of A m azo n ian convection
p ro v id es a h ea t source responsible for th e d e v e lo p m e n t o f a region o f h ig h
p re ssu re over the center of the co n tin en t, the B olivian H ig h [Lenters a n d
C ook, 1995], C irculation d u e to th e B olivian H igh ad v e c ts m oisture fro m
th e A m azon b asin so u th w a rd to w a rd n o rth ea stern A rg e n tin a [Schm it e t
al., 1990] as w ell as the A ndes of s o u th e rn P eru a n d B olivia [Johnson, 1976;
Virji, 1981], w h ere a com bination of convection, co n v erg en ce and
o ro g rap h ic u p lift p ro d u ce an n u al rain fall greater th a n 6 m eters in so m e
areas [WMO, 1975]. Rainfall decreases sh arp ly from e a st to west across th e
A ltip lan o of the central A ndes, to the extrem ely d ry d e se rts of coastal
so u th e rn P eru an d n o rth e rn Chile.
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8
Figure 1.1: South A m erican topography, m oisture transport and
the an n u a l m igration of the Inter-Tropical Convergence Zone (ITCZ).
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9
In a d d itio n to th e g en eral atm o sp h eric circu latio n , im p o rta n t
d y n am ic in flu en ces o n rain fall in the A m azo n b asin a n d c e n tra l/s o u th e rn
S o u th A m erica in c lu d e eq u ato rw ard -m o v in g cold fronts, larg e scale
convective sy stem s a n d w esterly sto rm tracks a t the m o st so u th e rly
la titu d e s. The la rg e s t a n d m o st im p o rta n t convective sy ste m s in South
A m erica as w ell a s th e im p act of the S o u th ern O scillatio n (SO) on South
A m erican p re c ip ita tio n are rev iew ed next.
C o n v e c tiv e s y s te m s o f th e A m a zo n b a sin
R ainfall in A m a z o n ia form s b a n d s w ith a NW -SE o rie n ta tio n
[Figueroa an d N o b re , 1990; N egri et al., 1994]. N egri et al. fin d alternating
p a tte rn s w ith th is o rie n ta tio n are associated w ith m o rn in g (AM ) vs.
afte rn o o n (PM) ra in fa ll. O f in terest to the p rese n t stu d y is a n AM
m ax im u m for th e b a n d o n the n o rth e a ste rn slopes of th e A n d e s along
s o u th e rn P eru a n d n o rth e rn Bolivia. O th er th a n this fe a tu re , th ey do n o t
o bserv e an y o b v io u s alig n m en t w ith to p o g ra p h y of the o th e r b an d s of
rain fall. F igueroa o b serv es th a t this o rien ta tio n is sim ilar to th a t of the
S o u th A tlantic C o n v e rg e n ce Z one (SACZ), b rin g in g lo w lev el m oisture
from m id -la titu d e fro n ta l system s p e n e tra tin g from th e s o u th [Figueroa
a n d N obre, 1990; L en ters, 1996]. These frontal system s e n h a n c e convection
o v er tro p ical re g io n s o f S o u th A m erica d u rin g au stra l s u m m e r [Figueroa
a n d N obre, 1990; M o lio n , 1987; O liveira an d N obre, 1986; V irji an d K ousky,
1989].
N egri et al. also find tw o A M /P M couplets in p re c ip ita tio n associated
w ith stro n g e le v a tio n g rad ien ts along the w estern ed g e o f th e Sierra
Pacaraim a (3°N , 67.5°W ) an d the Serra G era d e Golas (11°S, 48°W ),
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10
co n c lu d in g th at th ese r e s u lt from m o u n ta in /v a lle y circu lation. D u rin g the
d ay, confinem ent of h e a te d , ex p an d in g a ir in th e valley causes a n up-valley
flow ev en tu ally p ro d u c in g a PM m a x im u m alo n g the slo p es. A t n ig h t, cold
d en se air at higher ele v a tio n s d rain s d o w n slo p e o b tain in g a n A M
m ax im u m speed a n d co n v erg en ce in th e valley. N eg ri e t al. co n clu d e that
the d iu rn a l character o f ra in fa ll in the A m a z o n b asin is a com plex
co m b in atio n of effects in c lu d in g a t le a st th e m o u n ta in /v a lle y circulation
a n d la n d /s e a d istrib u tio n .
C onvective sy ste m s affecting th e ce n tral A m azon B asin d u rin g
A p ril/M a y 1987 have b e e n sep arated b y G reco into 3 classes [Greco e t al.,
1990] o n the basis of th e ir in itial d e v e lo p m e n t a n d ev o lu tio n . L in early
o rien ted Coastal O cc u rrin g System s (COS) originate in n o rth e a st Brazil
d u rin g the afternoon a n d fo rm large m esoscale to synoptic-scale system s up
to 3500 km . in length. T h ese system s m o v e across the A m azo n b a sin
reach in g the central re g io n 20-24 h o u rs afte r form ing o n th e coast (about
1400 to 1800 local tim e). T h ey reach th e ir m axim um rain fall b e tw e e n 1430
a n d 1630 local time. B asin O ccurring System s (BOS) are also m esoscale to
synoptic-scale in size (1,000 to 100,000 km ^), fo rm in th e A m az o n b a sin
itself a n d m ove so m e w h a t slo w er th a n COS. BOS reach th e ir p ea k rainfall
earlier in the day b e tw e e n 1130 an d 1230. D u rin g the p erio d of stu d y in
G reco et al., BOS a n d CO S accounted fo r 82% o f the rain fall in the central
A m azo n basin (41% each), 38% of the to ta l b ein g attrib u te d to o n ly 4
system s. The rem ain in g 18% of rainfall o v er th a t p erio d w as p ro d u c e d by
Locally O ccurring S y stem s (LOS) w h ich are m u ch sm aller convective
system s (< 1,000 km ^) la stin g only an h o u r o r so, g en erally p e a k in g around
1800 local tim e [Swap e t al., 1989]. A lth o u g h the total rain fall contribution
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11
of LOS w as sm all th e y w ere th e m o st c o m m o n system o ccu rrin g 42% o f the
tim e d u rin g th e stu d y p erio d . The classificatio n of these sy stem s is
co n sisten t w ith th e o b serv atio n th a t th e co n trib u tio n of re -e v a p o ra te d
m o istu re to b a s in rain fall increases fro m th e n o rth eastern co a st in la n d
[Salati et al., 1979]. M ore th an h alf of th e b a s in precipitation is re tu rn e d to
th e atm o sp h e re , th e re st d isch arg in g th r o u g h the A m azon riv er. T his
recycling w o u ld occur p red o m in an tly for BOS and LOS w h ich to g eth er are
resp o n sib le fo r 59% of the rain fall in th e se areas.
C o n v e c tiv e sy stem s - A ltip la n o a n d e a ster n A ndes
O n th e A ltip lan o in ten se solar ra d ia tio n an d surface h e a tin g d riv es
d e e p co nvection d u rin g the a u s tra l su m m e r, pro d u cin g in te n se
th u n d e rs to rm s w h ich co n trib u te m o st o f th e an n u al p re c ip ita tio n a n d
w h o se sev erity has been recorded b y se v e ra l authors as su m m arized b y
S chw erdtfeger [Schw erdtfeger, 1976]. T otal precipitation d u rin g a u stral
w in te r is m u c h less th a n in su m m e r a n d is p rim arily associated w ith cold
fro n ts from th e so u th [M olion, 1987; P aeg le, 1987; Ronchail, 1989]. The
ep iso d ic n a tu re o f d eep convection o n th e p la tea u gives w id e ly v ary in g
p recip itatio n fields over tim e scales of se v eral days [Aceituno an d
M ontecinos, 1989]. C onvection is p ro b ab ly enhanced by local top o g rap h ic
featu res a n d p ersists w h en in d iv id u a l co nvective cells are o rg a n iz e d into
la rg e r convective system s.
The la rg e st su ch convective sy stem s over the A ltip lano an d
n o rth e rn A rg e n tin a have th e ir a n a lo g u e in sim ilar, w ell-stu d ied m id ­
la titu d e sy stem s occurring o v er th e ce n tral plains of the U n ite d S tates, as
first in tro d u ced b y M addox [C am petella a n d Velasco, 1989; M addox, 1980;
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12
V elasco, 1989; Velasco an d F ritsch, 1986; Velasco a n d F ritsch, 1987]. M addox
id e n tified and d efin ed the M eso scale Convective C o m p lex (MCC) and
c o n tra ste d it w ith o th er co n v ectiv e system s on the b a s is o f th e ir in frared
sig n a tu res as o bserved in satellite im agery. M CC's are th e la rg e st class of
M esoscale C onvective System s (M C S's), d istin g u ish ed b y th e ir size,
d u ra tio n an d focused, central re g io n of vertical m o tio n . A s su c h they are
d istin c t from large scale tropical sy stem s, as occur in th e A m az o n basin for
ex am p le, w hich are sm aller, s h o rte r-liv e d a n d e x h ib it a m o re ran d o m m ix
o f convective elem ents an d a d ja c e n t areas of su b sid en ce. V ery recently,
M o h r a n d Z ipser classified M CS's u sin g passive m ic ro w a v e o b servations
[M ohr a n d Z ipser, 1996]. M addox d efin e d an MCC as a c o n tin u o u s cloud
sh ie ld o f area g reater th an 100,000 k m ^ and IR te m p e ra tu re less than -32°C
co n tain in g an in terio r region o f te m p e ra tu re less th a n -5 2 °C a n d area
g re a te r th a n 50,000 kmA In c o m p a riso n , convective ra in fa ll estim ates from
in fra re d im agery u su ally sta rt a c cu m u latin g rain fall a t in fra re d
te m p eratu res less th a n -32°C a n d approxim ate area > 700 km ^. M CC's form
w h e n several in d iv id u a l th u n d e rs to rm s p ro d u ce a m eso scale region of
w a rm in g stro n g e n o u g h to affect th e larger-scale e n v iro n m e n t. M idtro p o sp h eric inflow to the reg io n develops, p ro d u c in g co n v ergence, a
cen tral region of risin g m o tio n a n d localized h eav y ra in fa ll w ith in a larger
a rea of steady rain. P robably d u e to their long-life a n d slo w m ovem ent,
M C C 's have b een sh o w n capable of generating flash flo o d s in the U nited
S tates [M cGinley, 1986] an d s e v e re flooding in so u th e rn S o u th A m erica
[C am petella an d Velasco, 1989; V elasco, 1989]. In th e ce n tral p lain s of the
U n ite d States, M CC's account fo r 30% to 70% of th e p re c ip ita tio n from
A p ril th ro u g h Septem ber, som e y e a rs as great as 100% fo r certain areas,
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often w ith o n ly a few sy stem s con trib u tin g 1 /3 to 1 /2 of the to tal rainfall
[Fritsch e t al., 1986; T o lleru d a n d C ollander, 1993]. M o st system s begin w ith
the first th u n d e rs to rm s in th e ev e n in g (1900 lo c al tim e), organize into
M CC's in a few h o u rs tim e a n d reach their m a x im u m extent after
m id n ig h t.
V elasco a n d F ritsch d o c u m e n te d M CC's in S o u th A m erica (Figure
1.2) an d sh o w ed th a t w h ile th e y exhibited som e o f th e characteristics of
N orth A m erican sy stem s sig n ifican t differences e x iste d [Velasco a n d
Fritsch, 1987]. M id -la titu d e M C C's in A rgentina a re o n average 60% larger
th an th o se in th e c e n tra l p la in s of th e U nited S tates. A t m axim um extent
the S outh A m erican sy stem s h av e a cold clo u d s h ie ld a t less th a n -40°C of
nearly 500,000 km ^ c o m p a re d to ab o u t 300,000 k m ^ in th e US. V elasco an d
Fritsch id e n tified a se p a ra te p o p u la tio n of tro p ical M C C ’s, occurring alm ost
as frequently as th e m id -la titu d e system s b u t a b o u t th e same size as system s
in the U n ite d S tates. M id -la titu d e S outh A m eric an M C C ’s g en erally have
sim ilar tim es of in itia tio n a n d life cycles as in th e U n ite d States, w hile the
tropical M C C's b eg in a b o u t 4 h o u rs later (first sto rm s 22:30 local tim e) w ith
lifetim es a b o u t 3 h o u rs sh o rte r. T ropical M C C’s m o v e very little, o nly 15%
traveling fa rth e r th a n 200 km . Som e ap p a re n t m o v e m e n t is created by
expansion fro m o r m e rg in g w ith o th e r storm s. S o u th A m erican M CC's are
m ost likely re sp o n sib le fo r th e n ig h ttim e m ax im a o f convection an d
th u n d e rsto rm s o v e r c e n tra l a n d eastern A rg e n tin a , alth o u g h no account
has y et b e e n m a d e o f th e ir c o n trib u tio n to th e s u m m e r rainfall in South
America. O lascoaga [O lascoaga, 1950] has sh o w n th a t in the foothills of the
A rgentine A n d es, 50% of th e an n u a l rainfall co m es d u rin g only 5-6 days (23 w eather system s) d u rin g th e year. W hile O lasco ag a w as considering all
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100°W
80°W
60°W
40°W
Figure 1.2: M esoscale C onvective C lusters (MCC's) in the A m ericas,
a d ap ted from Velasco an d Fritsch (1987). In the m id-latitudes they
are essentially a su m m er phenom ena. They also occur in the tropics
w ith a concentration aro u n d n o rth w estern Columbia. A distinct
p o p ulation of M CC's also seem s to occur in northeast Bolivia at the
southw estern edge o f the A m azon Basin.
<
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15
rain fall, not n ecessarily convective sy stem s, h is stu d y sh o w e d that across
all p a rts of A rg en tin a 75% of the rainfall c o u ld be accounted for by less th a n
30% o f the days w ith rain , em p h asizin g th e episodic n a tu re o f rainfall d u e
to th e predom inance o f a few large w e a th e r system s.
Low-level jets a n d m o u n tain s p la y a n im p o rta n t ro le in the genesis
o f M CC's [Stensrud, 1996]. Gentle, slo p in g to p o g rap h y ea st o f the Rockies
h a s b ee n show n to b e responsible for th e d ev e lo p m e n t o f a nocturnal, low lev el jet [M cNider a n d Pielke, 1981], w h ich is also o b serv ed in South
A m erica [Paegle, 1987; Virji, 1981]. O b serv atio n s [Wallace, 1975] and
m o d elin g studies [Paegle, 1987] have d e m o n stra te d the im portance of the
low -level jet to th e d iu r n a l cycle of c o n v e ctio n an d d e v e lo p m e n t of severe
w ea th er. Tripoli a n d C o tto n [Tripoli a n d C o tto n , 1989] sim u lated an
o b se rv e d MCS in th e U.S. an d sh o w ed th a t th e rm a lly d riv e n m ountainv alley circulation p ro v id e s the basis for to p o g rap h ically generated
convection over th e R ocky M ountains. T h o u g h they d id n o t sim ulate a
m a tu re MCC, th ey sh o w e d th a t the m o im tain -v alley circu lation, through
th e genesis of the M CS, p ro v id ed "a su itab le trigger" to th e region of the
n o ctu rn a l jet for g e n e ra tio n of the o b se rv e d M CC. A b o u t 30% of the S outh
A m erican M CC's o rig in a te over the e a ste rn slopes of th e A n d es sim ilar to
th e genesis of 20-30% of U.S. MCC's o v er th e eastern R ocky M ountains
[M addox, 1983]. T he q u estio n of w h eth er th e higher to p o g rap h y and greater
slo p es of the A ndes a re responsible for th e g reater size of m id-latitude
S o u th A m erican M C C 's has not b een an sw e re d .
The A ltip lan o in p articu lar m a y focus MCC d ev e lo p m e n t through
it's action as an e le v a te d h ea t source a n d o ro g rap h ic liftin g o n adjacent
slo p es [Velasco a n d Fritsch, 1986; V elasco a n d Fritsch, 1987]. The limited
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16
evidence from the s tu d y b y Velasco a n d F rits c h su g g e st Bolivian M C C 's
occur as a d istinct g eo g rap h ic p o p u la tio n (F ig u re 1.2), alth o u g h they tre a t
th e Bolivian c o n c e n tra tio n as a n ex ten sio n cof th e m id -la titu d e sy stem s a n d
p o in t o ut the n eed fo r a larger sam ple size Ln o rd e r to establish them as a n
in d ep en d en t p o p u la tio n . M ohr an d Z ip ser [ M o h r a n d Z ipser, 1996]
identified tropical M C S's for four m o n th s, J a n u a r y , A p ril, July an d O cto b er.
T heir m aps also sh o w a clear se p aratio n befltween ce n tral A ndean M C S’s
a n d those so u th o f th e sem i-arid reg io n of s u b s id e n c e over n o rth e rn
A rgentina (Figure 1.3), as w ell as c o n c e n tra tio n s of M C S’s related to th e
topography of the H im a lay a in A sia (Figure? 1.4).
C entral and S o u th ern C hile: W e s te r ly S to r m T rack s
The h ig h e st m o u n ta in of S o u th A m etrican, M t. A concagua (la titu d e
33°S), lies in the n o r th e r n region of tem p errate c e n tra l C hile, so u th o f th e
sem i-arid zone. F ro m th is area so u th to a b o u t 42°S, fro n tal system s w ith in
the southern h e m isp h e re w esterlies b rin g s te ad ily in creasin g p re c ip ita tio n
d u rin g the au stral w in te r, from ab o u t M ay th ro u g h A ugust. C oastal ra n g e s
experience a re la tiv e m axim um in rain fall, w h ic h d ecreases in lan d in th e
C entral Valley of C h ile. P recipitation th en s tr o n g ly increases further e a st,
reaching a m ax im u m n e a r the A n d ean cres;t [M iller, 1976].
In so u th e rn C h ile, so u th o f ab o u t 38*°S, th e re is little seasonal
variation in p re c ip ita tio n as the w esterlies Ibecom e ex trem ely stro n g a n d
persistent. The A n d e s cause b o th liftin g a n d a s o u th w a rd tu rn in g of th e
w in d s causing co n v erg en ce and extrem ely h ig h co astal rainfall, u p to 6
m eters an n u ally in places. Rain tu rn s to sruow b e tw e e n 1 an d 2 km
l
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
17
M ohr a n d Z ipser
1996
MCS's by Tb(85V)
MCS's b y area
O ctober
^ -rr —
^ y -X
T r . . .*+^
^;; :-t.‘ + “ :V«' 1
Jan u ary
,%*£> o
-r+ v
w ^!
w
A pril
Ao-fV ° ° &
i "O
®
J u ly
Q
OO
Figure 1.3: M esoscale Convective S ystem s (MCS's) plotted b y area a n d T^,
(brightness tem perature) in M ohr a n d Z ipser (1996) show ing, am o n g other
features, focused MCS activity in N E Bolivia except in austral w in te r (July).
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
18
M ohr and Zipser
1996
MCS's by Tb(85V)
MCS's by area
+
* .4
Q)
April
O
<
a
July
o WM
'*
O
O
"^
M
October
Figure 1.4: Similar plots as in Figure 1.3 for southern Asia
show ing, am ong other features, a peak in occurence of
MCS's, especially as m easured by brightness temperature.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
19
elevation, d e p e n d in g o n la titu d e , th e elevation c h a n g in g little th ro u g h o u t
the year a t a n y g iv e n latitude. T h e h ig h spatial v a ria b ility of terrain a n d
difficulty o f p a s siv e m icrow ave m easurem ents to d istin g u ish b etw een ra in
an d snow on th e g ro u n d m ake in terp retatio n s o f p re c ip ita tio n in these
areas difficult a n d w ill not be ad d ressed here.
EL N in o a n d th e S ou th ern O scilla tio n
The 'n o rm a l' rainfall clim atologies d escrib ed abo v e v ary interan n u ally in asso cia tio n w ith th e El N in o /S o u th e rn O scillation (ENSO)
phenom ena. El N in o events, re c u rrin g episodes o f ex trem e rainfall a n d
severe flo o d in g a lo n g the n o rm ally arid coasts o f n o rth e rn P eru, have b een
connected w ith u n u su a lly w a rm sea surface te m p e ra tu re s an d a so u th w a rd
flow ing c u rre n t o ff the P eru v ian coast. A t a m u ch la rg e r scale, the
S outhern O scillatio n refers to in te r-an n u a l p re s s u re flu ctu atio n s over the
Indian O cean a n d eastern tropical Pacific. El N in o occurs d u rin g that p h ase
of the S o u th ern O scillation w h e n atm ospheric p re s s u re is low over the
eastern Pacific a n d h ig h over th e w estern Pacific. EN SO has b een the object
of considerable s tu d y su m m arized in several rev ie w s [Deser an d W allace,
1987; Enfield, 1989; Philander, 1990]. El N ino co n d itio n s have been
postulated to b e eq u iv alen t to av e ra g e clim ates a t certain tim es d u rin g the
quaternary [M artin et al., 1993], as opposed to b ein g an exceptional
occurrence c o m p a re d to the p re s e n t climate.
ENSO re la te d p recip itatio n fluctuations h a v e b een d o cum en ted in
m any areas of S o u th A m erica (Figure 1.5), in c lu d in g the p reviously
m entioned co astal rain s and flo o d in g of n o rth e rn P e ru [A ceituno, 1988]. In
this p ap er, 'ENSO years' refers to those years w h e n atm o sp h eric p ressu re is
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
20
Dry June - A u g u st
Wet December - February
D ry D ecem ber - February
o
O
or
Wet December - February
CO
CO
to
-O
o
Wet June - A ugust \
o
O
■st*'
CO
L iO
W80c
W60c
W40c
Figure 1.5: Broad precip itatio n anom alies in S outh America d u rin g
W arm ENSO episodes (u n u su ally strong "El Nino" events).
(A d ap ted from C lim ate P rediction C enter Fact Sheet - NOAA, 2000.)
R eprodu ced with perm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
21
a n o m a lo u sly lo w over th e eastern Pacific, a llo w in g the trad e w in d s to
m o v e w a rm su rface w a te rs so u th alo n g th e P e ru v ia n coast, a n 'El N in o ’ as
d escrib ed above.
C aviedes [Caviedes, 1973] d o cu m en ted episodes of d ro u g h t in
n o rth e a st Brazil d u rin g El N ino. Rao a n d H a d a [Rao and H a d a , 1990]
su g g e s t th at th ese d ro u g h ts are related to b lo ck in g action w h ic h hold
fro n ta l sy stem s in the a re a o f so u th ern B razil, P arag u ay a n d n o rth e a ste rn
A rg e n tin a , ca u sin g h ig h e r rainfall in th e P a ra n a riv er b asin a n d p rev en tin g
th e ad v e c tio n o f m o istu re in to Brazil.
In th e A concagua re g io n of cen tral C h ile, an n u al p re c ip ita tio n
in creases d u rin g ENSO y ea rs, causing a 35% a n d 145% increase in peak
s u m m e r a n d w in te r riv er discharges re sp e c tiv e ly [W aylen a n d C aviedes,
1990]. These increases re s u lt from a co m b in atio n of increased rainfall an d
w a rm e r tem p eratu res. T h e 1982-83 ENSO e v e n t w as p ro b ab ly th e strongest
in th is cen tu ry [Q uinn e t al., 1987; R asm u sso n a n d W allace, 1983],
accen tu atin g these effects a n d bringing d ro u g h t to the A ltip lan o [Goldberg
e t al., 1987; H o re l and C om ejo-G arrido, 1986; N obre and O liv eira, 1986].
H im a la y a n ranges and T ib e t - A sian M o n so o n
The g e n e ra l atm o sp h eric circulation in the greater H im a lay a n an d
T ib etan reg io n is greatly influenced by th e h ig h to p o g rap h y a n d elevated
p la te a u , red irectin g the w esterly jet stre a m to the n o rth o r s o u th (Figure
1.6). In a d d itio n , h eatin g a n d cooling o f th e p la te a u surface co n tro ls
p re v a ilin g w in d s of th e area . The p r e d o m in a n t sum m er m o n so o n
c irc u la tio n is d riv e n by th e rising m o tio n o f h ea ted air over th e plateau,
d ra w in g m o istu re lad en a ir from the s o u th to w a rd the H im alay as.
with p erm ission of the copyright owner. Further reproduction prohibited without perm ission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E60°
N40°
N40
m
IIM III w
i
>
"*••••
.'*>• *\
...
••*'.'?a- *
■
B H rag m IraSlS|
BSHHnIgflja
M
N35'
|9
N35°
81M
W
N 30c
N30'
N25'
(Summer m onsoon
pressure trough
(northern ITCZ
Elevation (km)
- -'• \.
'i i ' v ! •*'*' “
-’ .V v ..'.'''-. 'V.-i'
-■>\
,'vK- ! * v -.
'\^-vk\i '■•
N 25c
summer winds
at 1 km.
^ftanuB aK M aaaM saasaaausauaaiau
E70°
E80°
E90°
E100°
Figure 1.6: Features of Asian climate, moisture transport and the ITCZ. The Westerly Jet m igrates from
n o r th to s o u t h b e t w e e n s u m m e r a n d w in t e r r e s p e c t iv e ly . L o w -le v e l s u m m e r w i n d s b r in g m o is t u r e fro m
the Bay of Bengal northward to the Shillong Plateau, turning w estw ard along the Him alaya range front.
Id
23
W esterly w inds are m o stly forced n o rth of the p lateau a lth o u g h th e y
som etim es ap p ear so u th of the H im alayas. The ITCZ is a t it's m ost
p o le w a rd p o sitio n d u rin g the sum m er,, the m onsoon p re ss u re tro u g h
reaching across n o rth e rn P akistan, In d ia an d the Bay of Bengal. E asterly
low -level flow n o rth o f the tro u g h d u rin g su m m er b rin g s m o istu re fro m
the Bay of Bengal alo n g b o th sides o f the H im alayan front, along the
G anges low land as w ell as N ep al a n d so u th ern Tibet. M on so on ra in fa ll is
responsible for m o st of the a n n u a l p recip itatio n in these areas. In c lu d e d in
the com plex dynam ics of the en tire H im a lay a n /T ib etan reg io n is the
su g g estio n th at the ex ten t of w in te r snow cover o n the p la te a u in flu en ces
the stre n g th of the follow ing su m m e r m onsoon seaso n [B arnett e t al.,
1988].
D uring w in ter the w esterly flow of air is split, b ran ch es flow ing
n o rth an d so u th of th e T ibetan P late au converging on the lee side of T ibet.
The h eig h t of the p la tea u now cau ses cold, dense air to sin k an d flow o ff
the p la tea u , fu rth er p e rtu rb in g low -level circulation. In th e n o rth w e s te rn
area o f the K ashm ir basin, P ak istan an d the P am ir m o u n ta in s, w in te r
p recipitation is p ro v id ed b y convergence betw een the n o rth w este rly flo w
from th e polar fro n t an d In d ian tra d e w inds. W inter sn o w fall is m o st
ev id en t in the n o rth w este rn areas a lth o u g h it extends across the
H im alayan crest a n d to som e ex ten t o n the p lateau itself. For this s tu d y , w e
d id n o t separate sn o w cover fro m rainfall in the scatterin g sig n atu res.
R ather, w e rely o n the seasonal se p a ra tio n of snow fall a n d m o n so o n ra in s
to iso late o ro g rap h ic m o n so o n rain fall along the m o u n ta in front. T h is w ill
be discussed in m ore detail later.
R eprodu ced with perm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
24
E x istin g clim atologies
Legates a n d W illm ott [Legates a n d W illm ott, 1990] produced a
clim ato lo g y (h ereafter referred to as LW ) w hich m a d e u se of previously
co m p ile d global arch iv es in d e p e n d e n t o f rem otely se n se d data, corrected
fo r m ajo r biases of ra in gauge m e a su re m e n ts, and in c lu d e d data from
d iffe rin g tim e-p erio d s in order to a c h ie v e as dense a sp a tia l resolution as
po ssib le. A p u re ly objective g rid d in g p ro ced u re w as u s e d w hich d id n o t
a c co u n t for to p o g rap h ic variation b e tw e e n stations. L egates [Legates, 1995]
c o m p a re d LW w ith clim atologies b a s e d o n "subjectively d raw n m aps" a n d
s h o w e d th a t w h ile th e y exhibited s im ila r spatial p a tte rn s in general th ey
d iffe re d su b stan tia lly in the m a g n itu d e a n d seaso n ality o f precipitation.
C o n v ersely , g o o d ag reem en t w as o b s e rv e d w ith o th e r observation-based
clim ato lo g ies ev e n th o u g h those w e re u n co rrected for sta tio n biases.
L egates also fo u n d th a t the tim e p e rio d o f o b serv atio n d id not seem to be
sig n ifican t for co m p ariso n s b etw een L W an d o bservation-based
clim atologies a t th e scale of m o n th ly estim ates a n d fo r b o th 0.5° and 5 °
d e g re e grids, co n clu d in g that a d d itio n a l stations c o v e rin g disparate tim e
p e rio d s are im p o rta n t in resolving s p a tia l variability. T h e W orld
M eteorological O rg an izatio n (W M O) h a s p u b lish ed clim atic atlases of
S o u th A m erica a n d A sia [WMO, 1975; W M O , 1981]. F or S outh A m erica,
th e o ro g rap h ic p e a k in p re c ip ita tio n in th e eastern P e ru v ia n and B olivian
A n d e s is reso lv ed in th e WMO c lim a to lo g y w ith th e h e lp of som e 23
a d d itio n a l statio n s n o t available in o th e r datasets (F igure 1.7). The ed ito r,
Jo se Floffm ann, sta tes th a t "due to h etero g en eity a n d v ariatio n s in d e n sity
o f th e available d a ta , in p articu lar in m o u n ta in areas, it h a s been necessary
to g en eralize th e d ra w in g of the iso lin e s in m any p a rts o f the region". In
with perm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
25
W70°
W50°
0
1000 2000 3000 >4000
WMO A nnual Precipitation (mm)
Figure 1.7: A n n u al precipitation, g rid d ed from digitized contours of the W M O
C lim ate A tlas for South A m erica (WMO, 1975). M axim um annual rainfall
of u p to 6 m eters occurs w here the Bolivian A m azon m eets the Andes
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
26
lig h t o f th e subjective n a tu r e o f these maps, th e v alid ity of th e co n to u rs in
these a re a s is unclear. In. g e n e ra l the WMO clim atology depicts b ro ad er,
so m e w h a t sm oother p e a k s th a n LW (Figure 1.8). H ow ever, v isu a l
o b serv atio n of station lo catio n s as plotted in L egates & W illm ott [Legates
and W illm ott, 1990] an d L egates [Legates, 1995] indicate that sta tio n density
is su b s ta n tia lly low er in S o u th A m erica for L W th a n for the W M O
clim ato lo g y (Figure 1.9). In th is respect the q u alita tiv e n atu re of th e WMO
clim atology produces re aso n ab le results w ith o u t artifacts o u tsid e the
d o m a in o f co n trib u tin g s ta tio n observations.
LW agrees q u alita tiv ely w ith simple g rid d in g of station d a ta from
the G lobal H istorical C lim atological Network (G H CN ) [Vose e t al., 1992]
(Figure 1.10), consistent w ith the conclusions o f Legates [Legates, 1995] for
co m p ariso n of LW w ith o b serv atio n based clim atologies. G lobally the
G H C N u se s 7,533 sta tio n s w h ereas LW makes u se of 24,635. S im ilar to LW,
the G H C N is u n d e rre p re se n te d in the eastern A n d es w ith 99 P e ru v ia n
statio n s com pared to 165 u se d in the WMO a tla s. The GHCN th erefo re is
largely u n ab le to resolve th e p eak in orographic precipitation, h a v in g only
one s ta tio n in the v ic in ity o f th e rainfall m ax im u m . E m p h asizing the
p a u c ity o f available s ta tio n observations in m o u n ta in o u s areas, R onicke
[Ronicke, 1965] used d irec tio n al location of ra d io disturbances d u e to
th u n d e rsto rm s to sh o w th a t thunderstorm fre q u e n c y in n o rth w e ste rn
A rg e n tin a w as u n d e re stim a te d by up to 800% (th is was a m easu re of
th u n d e rsto rm frequency, n o t rainfall total).
In co n trast to th e situ a tio n for South A m erica, visual in sp e c tio n of
sta tio n s u se d in the W M O C lim ate Atlas for A sia [WMO, 1981] ind icates an
ex trem ely low sam pling d e n s ity (Figure 1.11). Precipitation c o n to u rs seem
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without p erm ission.
27
W70°
W50°
0
1000 2000 3000 >4000
Legates-W illmott Annual Precipitation (m m)
Figure 1.8: Legates-W illmott annual precipitation clim atology for South
America. Fewer stations are used as for the W MO precipitation climatology
in Figure 1.7, as can be seen in Figure 1.9.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
W70°
o<N
W70°
O
ts)
ro
CO
o
•
W M O Stations
•
Legates-W illm ott Stations
•
G H C N Stations
Figure 1.9: Locations of stations used for each of the three clim atologies, in order
of decreasing available data. The WMO clim atology uses precipitation data from
more stations than are available to the other tw o clim atologies.
29
W70°
W50°
2000
>3000
GHCN A nnual Precipitation (mm)
Figure 1.10: A n n u al precipitation as g rid d ed from the G H C N station
precipitation d ata for South Am erica. Far few er station records are available
th an are u se d in either the Legates-W illm ott or W M O climatologies.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without p erm ission.
30
to follow to p o g rap h ic co n to u rs w ith h ig h freq u en cy v a ria tio n s im possible
to o b tain w ith the a p p a re n t statio n d en sity , in d icatin g th a t so m e
assu m p tio n s ab o u t p re c ip ita tio n v a ria tio n w ith elevation w e re likely m ade
in co n stru ctin g this atlas. LW m a d e use o f extensive d ata in N e p a l from
the datab ase o f W e m ste d t [W em stedt, 1972] an d show s a re g io n of
increased p recip itatio n along the T san g p o v alley w hich is u n re so lv e d in
any of the other clim atologies (Figures 1.12 & 1.13).
R em o te Sensing o f R a in fa ll
W ith a n ap p ro x im ate te m p e ra tu re ran g e of 250-300 K th e earth's
th erm al em ission p eak s a t ap p ro x im ately 15 to 20 pm (pm = 10“^ m). This
rad ia n t energy is m o d ified by surface d e p e n d e n t em issivities, surface and
atm ospheric scattering, a n d a b so rp tio n a n d em ission w ith in th e
atm osphere. Satellite sen so rs specialize in m easu rin g rad ian ce a t different
p arts o f the sp ectru m w h ere tran sm ittan ce o f a clear a tm o sp h e re is
relatively high. Figure 1.14 show s the tran sm ittan ce of a clear atm osphere
as a function of w av elen g th an d frequency. The earliest satellites observed
the e a rth in the visible a n d in fra re d reg io n s of the sp ectru m , w h ile
d ev elo p m en t of m icro w av e rad io m eters follow ed later. E stim ates of
rainfall b eg an th erefo re w ith the v is ib le /in fra re d channels, techniques
w hich are still in use d u e to th e h ig h sp a tia l an d tem poral sa m p lin g of
satellites observing a t these p arts o f the sp ectru m . M icrow ave estim ates
w ere d ev elo p ed later a n d are o nly recently becom ing o p eratio n al.
Passive m icrow ave d ata m ay also be com bined w ith v isib le /in fra re d
im ages to use the advantages of each type of d ata [Adler et al., 1994].
Rainfall is m ore p h y sically related to th e scatterin g and e m issio n of the
R eprodu ced with perm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
E100'
E90
□*
oOLN
E80
o
-
oS£N
^
in
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° a. * *
oOSlSE
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* / ° *A 4•
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Pa D*
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E70
□4
□A '
□a
□A
□ AQL
Bk
E70°
E80°
E l 00°
Figure 1.11: Locations of stations used for the WMO and Legates-W illmott precipitation
A Legates-W illm ott
clim atologies for Asia. Legates-W illmott had additional data primarily from W em stedt (1972). D WM0
Figure 1.12: WMO precipitation clim atology for Asia as gridded from digitized contours.
Precipitation contours on the original m aps closely follow elevation contours, suggesting
an assum ption of significant topographic control during contour interpolation.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E70°
E80°
j'
■■:.V*'
E100c
E90c
o
O
‘NX ;V
oo
•:
■
'v
o v ^ v *t .\VV, ,v-.'
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O0Jl
2000 A nnual Precipitation
1000 Legates-W illm ott
(mm)
E70°
...;:.
E80°
E90°
E100°
Figure 1.13: Legates-W illmott precipitation clim atology for Asia. Features are
broadly similar to the WMO precipitation clim atology (Figure 1.12).
S SM /I frequency bands
19
85 37 22
w avelength (m)
100
Frequency (Hz)
UV
visible near-IR
thermal IR
m icrow ave
Figure 1.14: Transmittance of a clear atm osphere to electrom agnetic radiation. The 85 GHz band is located
w here scattering and absorption by water vapor, liquid water and ice crystals increases, thereby reducing
transm ission through the atm osphere and providing a physical basis for u sing passive m icrow ave radiation
to discern the presence of H 2O in any of these forms. (Adapted from Grody [1993], Fig. 6.1.)
35
m icrow ave rad iatio n , w h ich a t b est is sam p led o n ly a few tim es p er day.
V IS /IR ob serv atio n s are o n ly in d irectly re la te d to rainfall, b u t h av e the
ad v an tag e of m u ch b etter te m p o ra l sam pling (3 hours) a n d sp a tial
reso lu tio n (4 km .). A dler calculates rainfall u sin g V IS /IR d a ta first at the
te m p o ra l sam p lin g of th e m icro w av e, calculates a correction factor w hich is
the ra tio of this V IS /IR e stim a te to the m icrow ave-only estim a te, then uses
th a t correction factor to m o d ify th e V IS/IR estim ate at the full tem poral
sam pling. A ttem p ts to m erg e all available p recip itatio n m easu res use rain
g auge d a ta , m icrow ave a n d in fra re d satellite estim ates, a n d o u tp u t from
nu m erical w e a th e r p red ic tio n m odels, av erag ed onto m o n th ly , 2.5°x2.5°
grids [H uffm an e t al., 1995].
In d irect M easu res - V isib le/In fra red
Several d ifferen t p h y sical m easures are u se d to rem o te ly estim ate
rain fall, each w ith a u n iq u e tran sfo rm atio n fro m the o rig in al physical
m e a su re m e n t to the rain fall p ro d u c t. The rem o te sen sin g o f rainfall
includes g ro u n d based ra d a r, w id ely used in th e U.S. an d E urope, w hich are
n o t available in th e areas of th is study. Satellite observations p ro v id e
sn a p sh o ts of u niform co v erag e over the ea rth , an d are esp ecially valuable
in u n in s tru m e n te d areas.
The first satellite-based rainfall retrievals u sed o b serv atio n s of
clo u d s in visible an d in fra -re d channels to in d irec tly estim ate rainfall.
Som e of the m ethods re m a in actively used to d ay . The basic id ea is to relate
rain fall to cloud height, te x tu re a n d /o r m o rp h o lo g y . Both single
w a v e len g th an d m u lti-sp ectral techniques u se a v ariety of algorithm s
in c lu d in g area -tem p e ra tu re relatio n sh ip s a n d th resh o ld s, life h isto ry
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
36
m e th o d s, and indexing of cloud ty p es [Barrett, 1981] (Figures 1.15 and 1.16).
T he G eostationary P re cip itatio n In d ex (GPI) uses a n a lg o rith m developed
b y A rk in and M eissner [A rkin a n d M eissner, 1987] for e stim a tin g rainfall
o n a 2.5 deg. by 2.5 d eg . g rid in the tro p ics an d m id -latitu d es a n d is available
for th e p eriod 1986 to p resen t. The a d v a n ta g e of v is ib le /IR estim ates is th a t
sa tellite s observing a t th ese p arts o f th e sp ectru m p ro v id e v e ry good
te m p o ra l sam pling o f c lo u d fields, a t le ast every 3 h o u rs b y G eostationary
sa tellite s and often a t 1 / 2 h o u rly in te rv a ls. V isib le/IR o b serv atio n s are the
lo n g e s t running re m o te sen sin g -b ased rainfall estim ates. T he d isad v an tag e
of th e se m ethods is th a t th ey are in d irect. C loud ch aracteristics are used
s im p ly as indicators o f th e occurrence o f p recipitation. A v erag es over space
a n d tim e are necessary to b etter co rrelate these estim ates w ith ground
sta tio n data.
Besides the g lo b al ap p licatio n of in frared tech n iq u es (using GPI),
so m e regional stu d ies h a v e b een m a d e in A rg en tin a to tim e these m ethods
to th e southw estern p a r t of th a t co u n try [G onzalez a n d V elasco, 1989;
G o n zalez and Velasco, 1995]. These stu d ie s have b een lim ite d to a few dates
d u r in g daylight h o u rs , becau se of th e u se of visible sa tellite channels in th e
alg o rith m . They h av e s o u g h t to d efin e ra in areas b u t n o t d e p th s. They are
n ev e rth ele ss p ro m isin g as a first ste p to w a rd d e v e lo p m e n t o f a regionally
tu n e d v isib le /in fra re d tech n iq u e for m id -la titu d e S o u th A m erica.
D irect sa te llite m ea su res - M ic ro w a v e
M ore direct m e a su re s of ra in ra te are m ade a t m icro w av e
fre q u en cies w here th e th e rm a l m ic ro w a v e em ission o f th e e a rth is directly
affected by p recip itatin g h y d ro m eteo rs. R ain rate estim ates are m ade at low
with perm ission of the copyright owner. Further reproduction prohibited without p erm ission.
37
fro m B a rrett [1981]
V olum etric rainrate
(103 m 3 s"1)
30
D
<Z
u
m*
Visible area
\ ( 1 0 3 km 2)
20
0>
Infrared area
\ ( 1 0 3 km 2)
<z
<v 10
1
1
2
3
4
5
6
Time (hours)
Figure 1.15: Relationship betw een radar-derived rain rate an d
visible a n d infrared cloud areas. Plots of area a n d volumetric
rain rate show sim ilar shapes b u t cloud area lags behind
rainfall as seen in this figure. Rainrate retrievals from visible
a n d infrared data u se b o th cloud area an d it's rate of change,
(from B arrett [1981], Fig. 5.1)
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
38
fro m Barrett [1981]
visible infrared rainrate
ra d a r rainrate
2.0
1.5
1.0
0.5
0.0
10 15 5
d ay 1
10
15 20 0
5 10 15
d ay 2
day 3
time of day (hours)
20 5
10 15
day 4
Figure 1.16: Satellite rain rate based on v isib le/in frared cloud area at
one-hourly intervals. R adar-derived rain rate s are show n for com parison.
In each im age every cum ulonim bus is identified and it's area and location
are m easured. Their life histories are then tracked from image to image,
(from Barrett [1981], Fig. 5.2)
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without perm ission.
39
rain rates u sin g lo w frequem cy m icro w av e e m issio n fro m ra in d ro p s an d a t
h ig h ra in rates u sin g th e sc a tte rin g of h ig h fre q u e n c y m icro w av e radiation
by ice particles in c lo u d to p s . Im ages from a sin g le p o la r o rb itin g satellite
sam ples the rain field a t m o s t o n ly tw ice d aily , co n sid erab ly aliasing total
rain rate estim ates m a d e fro m th ese d ata. N ev e rth e less, th e se estim ates are
quite valuable d u e to th e m u c h b etter sp a tia l sa m p lin g o v e r available
g ro u n d statio n d a ta . T he tra n s fe r o f m icro w av e ra d ia tio n th ro u g h the
atm o sp h ere a n d in terp reta-fio n of p assiv e m ic ro w a v e ra d io m e try for
estim ating ra in ra te s is d e v e lo p e d in the fo llo w in g sections.
M icrow ave r a d ia tiv e tra n sfer - B r ig h tn e ss te m p e r a tu r e
Electrom agnetic e n e rg y is rad iated b y a n y b o d y w ith a tem perature
above 0° K (absolute zero). The energy m e a su re d b y a satellite radiom eter
results from all ra d ia tin g s ources from th e e a rth 's su rface th ro u g h the
atm o sp h ere. M ic ro w a v e e m is sio n of th e e a rth is a th e rm a l em ission
d eterm in ed b y th e te m p e ra tu re a n d em issiv ity of the su rface. In addition,
absorption, reflection a n d sc a tte rin g m o d ify th is ra d ia tio n a s it passes
th ro u g h the atm o sp h e re . In te n sity , p o la riz atio n , a n d fre q u en cy of the
rad iatio n as m e a su re d a t tlhe satellite a re u s e d to id e n tify d o m in a n t
processes g iv in g rise to th e m easured sig n al. R eview s of th e general
processes of rad iativ e tra n s fe r as d ev elo p ed b y C h a n d ra se k h a r
[C handrasekhar, 1950] a n d it's ap p licatio n to atm o sp h eric m icrow ave
radiom etry are g iv e n in Ba_rrett [Barrett, 1981], Jan ssen [Janssen, 1993], and
U laby, M oore a n d F u n g [UJlaby e t al., 1981]. T h e d isc u ssio n in the follow ing
p arag rap h s covers th e partts of this th eo ry im p o rta n t for r a in rate
estim atio n a n d so m e s im p lify in g a ssu m p tio n s.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
40
T he specific in te n sity , Iv, is d e fin e d as the in stan ta n eo u s ra d ia n t
p o w er p e r u n it area, p e r u n it-freq u en cy in terv al at a specified frequency v,
a n d in a given d ire c tio n p e r u n it so lid an g le. A d ifferen tial fo rm o f the
ra d ia tio n tran sfer e q u a tio n
dX.
ds
= -i,.a + s
(1.1)
describes the change in specific in te n sity d ly over a d istan ce ds along a line in
the d irectio n of p ro p a g a tio n w h ere a is a n ab so rp tio n coefficient an d S a
source te rm [Janssen, 1993]. If scatterin g is neglected th e n the source term S
describes only local ra d ia tio n co n trib u tio n s and a is a scalar. F u rth er, in the
case of local th e rm o d y n a m ic e q u ilib riu m (satisfied in th e trop o sp h ere) each
p o in t can be d escrib ed b y a te m p e ra tu re T a n d therm al ab so rp tio n equals
em issio n so th a t th e so u rce term , S, can b e w ritten S = ccBv(T ). By(T) is the
sp e ctral in te n sity o f th e ra d ia tio n (W m “2 s r 'l H z 'l), a n d is g iv e n by Planck's
ra d ia tio n law:
_
2/zv3
B ( T ) = — ------ w
C
i
( e„kT
k T
)
(12)
w h ere h is P lanck's c o n sta n t (6.63xl0-34 joules), v is frequency (Hz), k is
B oltzm ann's c o n s ta n t (1 .3 8 x 1 0 "^ jo u le K-l), T is ab so lu te tem p eratu re (K)
an d c th e velocity of lig h t (3x10^ m s'^ ). W ith the ab o v e a s su m p tio n s and
neg lectin g scatterin g , all term s in E q u atio n 1.1. d e p e n d o n ly on the
in te n sity along th e p a th o f p ro p a g a tio n . The optical thickness,
T (s x , So) = J3 a ( s ) d s , w ill be u se d in d eriving a so lu tio n to this sim ple
rad iativ e transfer eq u a tio n . M u ltip ly in g b o th sides b y e*(0/S) an d
rearran g in g gives
_ V O , s) J T
------= = * - + I va e T(0's> = S e T(0,sl
ds
.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
(1.3)
41
U sing d ifferen tiatio n b y parts a n d n o tin g th at a d s = d t this becom es
d I X e * 0' 51]
= Se
ds
.
(1.4)
In teg ratin g from th e source, s = 0, to the satellite s = s0, m u ltip ly in g
b y e-t(0'so), su b stitu tin g fo r S an d n o tin g th a t e T(0's,_T{0' 3,1 = e x(s'So) gives
= I u (0) e T(0' S3> + JJ0 ( X B ^ T je '^ '^ ’d s
(1-5)
E quation 1.5 assu m es th at atm o sp h eric scattering is negligible. The
specific in ten sity m e asu red at the satellite, Iv(so), is eq u al to the su m of tw o
term s. The first term rep resen ts the su rface brig h tn ess, lv(0), reduced in
m ag n itu d e by ab so rp tio n over the p ath , so, fro m surface to satellite. The
second term re p re se n ts th e rm a l em issio n fro m the atm o sp h ere itself and is
th e sum o f c o n trib u tio n s fro m in fin itesim al thicknesses ds w ith em itted
in ten sity aBy(T) red u ce d in m ag n itu d e b y ab so rp tio n in the in tervening
m aterial from s to so- A tm ospheric a b so rp tio n an d em ission can be
neglected w h en u sin g th e scattering cau sed b y hydrom eteors as the basis for
rainfall retriev als, as w ill be discussed in th e n ex t section.
In the lo w freq u en cy lim it (h v /k T «
1), the P lanck function, By(T),
m ay be ap p ro x im ated b y the Rayleigh-Jeans law:
2 V 2k T
2k T
B V(T) = ----- — = -rrc
AT
w here A. is the w av elen g th . (A t v=100 G H z, X = 3 m m , h v /k T = 0.015.) In
this lim it the P lan ck fu n ctio n is lin ear w ith te m p e ra tu re , an d th e
m icrow ave b rig h tn ess tem p eratu re, Tfc>, is d efined as
T (V ) = ----- I
;
2k v
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42
in o rd er to p la y th e ro le of in ten sity in the rad ia tiv e tran sfer equation. Tb is
d im en sio n ally scaled so th at it has u n its of d eg rees Kelvin. U sing this
d efin itio n th e ra d ia tiv e transfer eq u a tio n becom es
(1 -6 )
This b rig h tn e ss tem p eratu re is also called the Rayleigh-Jeans
eq u iv alen t b rig h tn e ss tem perature [Janssen, 1993]. Errors d u e to the
Rayleigh-Jeans ap p ro x im atio n are o n the o rd er of 1° K for earth
o b serv atio n a t th e se frequencies. B rightness te m p eratu re th u s defined is
th e b rig h tn ess te m p e ra tu re observed a t t h e s a t e l l i t e . The e m i t t e d brightness
tem p eratu re im m ed ia tely above a surface (or ig n o rin g atm ospheric effects)
is defined as
T e - esT s
(1.7)
w h ere es is th e su rfa ce em issivity. These d efinitions of brightness
te m p eratu re w ill be u se d extensively in the d iscu ssio n below on the
m icrow ave sig n a tu re s of surface an d atm o sp h eric conditions. In practice,
m icrow ave ra d ia n c e m easu red at the satellite is co n v erted in to eq u iv alen t
b rig h tn ess te m p e ra tu re for in te rp retatio n an d rain -ra te calculation.
P a ssiv e m ic r o w a v e em issio n o f th e e a rth
Figure 1.17 d ep icts the total m icrow ave tran sm ittan ce for a cloudfree, sta n d ard atm o sp h ere [Grody, 1993]. The n a rro w regions of low
tran sm ittan ce are d u e to resonant ab so rp tio n o f 0 2 and H 2 O (water vapor).
T he m ain d ifferen ces in transm ittance for the th re e atm o sp h eres is d u e to
n o n -reso n an t a b s o rp tio n by w ater vapor. The effects of clouds m ust be
considered a t all frequencies b u t are n o t rep resen ted here. Fortunately, little
R eprodu ced with perm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
43
absorption b y cirru s clouds (co n sistin g of ice crystals) a n d clouds co n sistin g
of liquid w a te r occurs below 20 G H z , allow ing su rfa c e observations a t these
frequencies. A b o v e 20 GHz a b s o rp tio n by clouds a n d la rg e r liquid d ro p s in
rain are u s e d to o b ta in e stim ates o f liquid w ate r in th e atm o sphere. A t 85
GHz a n d h ig h e r frequencies la rg e ice particles fo u n d in p recip itatin g clouds
scatter v e ry efficiently and are re a d ily identified b y d e p re sse d b rig h tn ess
tem p eratu res. T his is the b asis fo r retrievals of h ig h e r ra in rates a n d w ill b e
discussed in m o re detail below .
T h e su rfa ce em issivity in tro d u c e d in E q u a tio n 1.7, es, ranges in
value fro m ap p ro x im ately 0.4 - 0.6 a t nadir for w a te r to ap p ro x im ately 0.9
for d ry la n d . W e t la n d falls so m e w h e re in b etw e en d e p e n d in g o n th e
degree o f w e tn e ss an d surface ty p e . The em issivity c a n be expressed as
es(9p) = 1 - | R(6,p) | w here R is th e polarization d e p e n d e n t reflectivity for
p o larizatio n p , 0 is the in cid en ce o r view ing an g le. T h e p o la riz atio n
dependence o f R (and es) re s u lts fro m the v iew in g g e o m e try and
electrom agnetic b o u n d a ry c o n d itio n s at the surface as d epicted in F ig u re
1.18. W h e n th e v iew in g an g le is o th e r than 0 (n a d ir), vertically p o la riz e d
b rig h tn ess te m p e ra tu re s are g re a te r th an h o riz o n ta lly p o larized Tb's as a
result of th e electrom agnetic b o u n d a r y conditions a t th e surface a n d S nell's
law - th e rm a l em issio n of th e g r o u n d w ith p o la riz a tio n in the p la n e of
incidence (v ertically p o larizatio n ) is preferentially tra n sm itte d a t th e airground b o u n d a ry , tran sm issio n reach in g 100% a t B rew ster's A ngle (the
value o f B rew ster's angle can b e calculated from th e dielectric constant).
Total tra n s m is s io n does n o t o c c u r fo r h orizontal p o la riz a tio n .
R eprodu ced with p erm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
44
re s o n a n t a b so rp tio n features
h 2o
O,
o2
h 2o
adapted fro m
G rody [1993]
1.0
—\
0.8
<v
KJ
c*
0.6
4—»
4-*
r-
rz
0.4
£
0.2
0.0
0 A 40
A A
80
A
19
22 37
85
120
160
200
240
280
Frequency (GHz)
Figure 1.17: Total m icrow ave transm ittance of tropical, sta n d ard and
po lar atm ospheres. Locations of the S SM /I frequencies are indicated
along w ith the m ain reso n an t absorption features. N on-resonant
absorption by w a te r v a p o u r accounts for the m ain differences betw een
the transm ittance characteristics of the three atm ospheres, (adapted
from G rody [1993], F igure 6.2)
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
45
an ten n a tem perature
Ta
adapted fro m U laby [1981]
A n ten n a Pattern
T in
atm ospheric d ow n w ard em ission
Tsc
scattered radiation
TUp
atm ospheric upw ard em ission
lb
terrain em ission
atm osphere
e a rth ’s surface
gro u n d footprint
of satellite
Figure 1.18: Satellite view ing geom etry and surface em ission.
The strongest scattering surfaces are snow and sa n d w hile ice
crystals at the tops of precipitating clouds are th e p rim ary
scatterers for rainrate retrievals. A ntenna Tem peratures, Ta, are
initially recorded an d th en corrected for the an ten n a pattern
according to W entz [1991]. (figure after Ulaby [1981], Fig. 4.8)
R eprodu ced with perm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
46
A tm osp h eric e ffe c ts - S ca tterin g
The brightness te m p e ra tu re sensed b y th e satellite rad io m eter
rep rese n ts the m o d ificatio n o f Te by atm ospheric effects. The m o st
im p o rta n t m odifications in th e m icrow ave re g io n besides re so n a n t
ab so rp tio n peaks are d u e to liq u id w ater a n d ice. Liquid w ater p rim arily
absorbs (and emits) m icro w av e radiation, sc atte rin g only slightly. Ice is
p red o m in an tly a scatterer. B oth effects increase w ith increasing frequency
an d ra in rate, alth o u g h ice scatterin g increases m u c h more th a n liq u id
scattering, being the d o m in a n t effect above 60 G H z [Kidder an d H aar, 1995].
To see this, liq u id particle sizes in n o n p recip itatin g clouds typically
h av e ra d ii on the o rd e r o f 50 p m , m uch sm a lle r th a n the m icro w av e
w avelengths (Figure 1.19) so th a t the R ayleigh approxim ation ap p lies a n d
scattering is very w eak (Figure 1.20). In p recip itatin g clouds liquid drop
sizes can be m uch g reater, u p to 1 mm in ra d iu s for typical rain d ro p s
(Figure 1.21). Separations b etw e en cloud d ro p s are on the o rder of 1 cm.,
increasing to 10 cm. for falling raindrops. B ecause of the large se p aratio n s
betw een cloud drops h y d ro m eteo rs are co n sid ered to scatter and absorb
in dependently, so th a t single-particle scatterin g cross sections are a useful
indicator of hydro m eteor effects. Because o f the larg er drop sizes for
precip itatin g system s, ra d ia tiv e transfer stu d ie s usu ally use the M ie th e o ry
of scattering and ab so rp tio n , w hich is a rig o ro u s solution of the scatterin g
p ro b lem for hom ogenous sp h e re s of any size. T he validity of th ese
assu m p tio n s depends of course on the actual size an d phase d istrib u tio n of
h y d ro m eteo rs w hich is larg ely unknow n for precip itatin g system s.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without p erm ission.
47
S S M /I frequencies (GHz)
22 23
37.00
19.35
Frequency: 3 G H z
W avelength: 10 cm
10 G H z
30 G H z
85.50
100 G H z
300 G H z
1 mm
l cm
i . .
.
1.550
0.8108 0.3509
1.350
S S M /I w avelength s (cm)
strongest scattering of 85 GHz
radiation for ice particles 2-f mm
Figure 1.19: M icrow ave w avelengths an d frequencies.
SSM /I w avelengths and frequencies are indicated together
w ith typical ice crystal sizes im portant for rainrate retrievals.
i
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
48
fro m Janssen [1993]
0
<w
a
t/s
a
bO
O ■3
cum ulus
stratu s
4
■5
100
300
F requency (GHz)
Qs —> scattering efficiency
Qa —> ab so rp tio n efficiency
r ~ > d ro p size radius
Figure 1.20: Ratio of scattering to absorption efficiency for raindrops
of various sizes as related to precip itatin g or non-precipitating clouds.
Scattering of w ater drops starts to becom e im portant relative to
absorption only for the largest d ro p sizes at or above 1 m m , but is
still insignificant relative to the scatterin g caused by larg er ice particles,
(from Janssen [1993], Figure 1.7)
|
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
49
fro m Grody [1993]
6 ---------------------------------------------------------------
85 G H z
37 G H z
19 G H z
a
o
W
bO
a>
2
a
CD
1
0
0
2
4
6
14
12
8
10
D rop Diam eter (mm)
16
18
20
Figure 1.21: Spherical ice particle scattering efficiency at 3 SSM/I
frequencies. Typical diam eters of ice particles at raining cloud tops are
shorter th an the w avelengths of all but the 85 G H z b an d (3.509 mm)
where they cause significantly greater scattering, (from Grody [1993],
Figure 6.29)
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
50
M ost m odeling stu d ie s a n d radiative tra n sfe r calculations h a v e u se d the
M arshall-Palm er size d istrib u tio n [M arshall a n d Palm er, 1948] for
ra in d ro p s o u t of c o n v en ien ce, w hich v a rie s ex p o n en tially w ith, ra in rate
a n d rain d ro p diam eter. Sim ilarly, little d a ta exists for the size a n d shape
d istrib u tio n s of ice p articles, so that m o st stu d ie s assum e e ith e r spheres or
sp h ero id s [Gasiewski, 1993]. Errors d u e to th ese assum ptions h a v e been
ad d ressed in m odeling stu d ie s and are u n im p o rta n t at the s c a le o f the
SSMI sensor.
W u an d W e in m a n n [W u and W e in m a n , 1984] w ere t h e firs t to
in clu d e frozen h y d ro m e te o rs in m o d elin g th e relationship b e tw e e n
b rig h tn ess tem p eratu re a n d surface p re c ip ita tio n rate, a s su m in g
precip itatin g clouds co n sistin g of liquid, m ix ed phase and froz:en
hydrom eteors. They sh o w e d th at b rig h tn ess tem p eratu re d e p re s sio n s were
d u e to the presence of ice hydrom eteors. H ig h e r frequencies w e r e m ore
sensitive to ice near th e c lo u d top w hile lo w e r frequencies re s p o n d e d more
to w ater drops near th e clo u d bottom s. A sp h erical ice particles p ro d u ced
sm aller brightness te m p e ra tu re s at h o riz o n ta l p o larizatio n th a n a t vertical
polarizatio n . They also sh o w ed that the e rro rs in assum ing sp h e ric a l
versus aspherical ice a n d w ate r droplet sh a p es w ere less th a n 12 degrees at
37 G H z a n d less th a n 5 d eg rees at low er frequencies. W u a n d W einm an
assu m ed a stratified atm o sp h eric stru c tu re wdth only v ertical v ariatio n s in
h y d ro m eteo r stru ctu re, p h a se and te m p e ra tu re .
A dler and oth ers [A dler et al., 1991a; A d ler et al., 1991b] com bined a
three-dim ensional c lo u d m o d el w ith a p la n e-p aralle l ra d ia tiv e tran sfer
m odel. The rad iativ e tra n s fe r m odel also d iv id e d the a tm o s p h e re into
sep arate regions, each w ith a uniform d is trib u tio n of several co ex istin g
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
51
classes o f h y d ro m eteo rs. The clo u d m o d e l u se d varying h o riz o n ta l
reso lu tio n s ran g in g fro m 230 m . a t th e lo w e st level to a b o u t 1000 m. at the
h ig h e st level. R esults o f the m o d el w e re av erag ed to la rg e r spatial
re so lu tio n s to sim u late th e resp o n se o f d iffe re n t passive m icrow ave
rad io m eters. Their resu lts sh o w ed th a t a t 85 GHz the h ig h d egree of scatter
b e tw e e n b rig h tn ess te m p e ra tu re a n d ra in rate was n e a rly elim in ated at the
12 km . scale of the SSMI sensor. In terestin g ly , they also sh o w e d that
su p e rc o o led w ater coexisting w ith th e ice increased the u p w e llin g rad iatio n
a t 85 GHz. b y about 80° K th ro u g h a b s o rp tio n and re-em ission, reducing
th e b rig h tn ess te m p e ra tu re d e p re ssio n d u e to scattering alone. The vertical
s tru c tu re of the clo u d is an im p o rta n t co n tro l on b rig h tn e ss tem p eratu res
(Figure 1.22), w here convective c lo u d s p ro d u c e higher co n cen tratio n s of
c lo u d -to p ice th an stra tifo rm p re c ip ita tio n . M icrow ave m e a su re s therefore
can u n d erestim ate p re c ip ita tio n in a re a s w h ere stratifo rm p recip itatio n is
d o m in a n t. O rographic e n h a n ce m en t o cc u rs w ith both p re c ip ita tio n types,
th o u g h convection is m u c h m o re im p o r ta n t than stra tifo rm p recip itatio n
in th e areas of this stu d y .
S S M /I - P assive m icrozvave sa te llite o bservations
M icrow ave o b se rv a tio n s o f th e e a rth from satellite rad io m eters h av e
b e e n m ad e for over 20 years. The m o st re c e n t m icrow ave rad io m eter is the
Special Sensor M ic ro w a v e /Im a g e r (S S M /I) w hich has flo w n on polar
o rb itin g satellites of th e D efense D e p a rtm e n t M eteorological Satellite
P ro g ra m (DMSP) since 1987. The W ETN ET program a t N A SA 's M arshall
S pace F light C enter (MSFC) w as fo rm e d to facilitate m eteorological
ap p licatio n s of S S M /I d ata. M ore recen tly , SSM /I data h a v e been
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85 GHz radiation scattered by ice crystals in
cloud tops, other frequencies not affected
F = f(19,22,37 GHz.)
simulated 85 GHz
background (no rain)
from Barrett [1994]
Scattering Index,
> SI = F ■Tb(85) — ►Rain Rate = f(SI)
Rain if SI > 10
linear
A
Ice
Tb(22) ~ 250 -280 °K
A
Tb(85) ~ 170-180 °K
(rain)
i
Tb(85) ~ 230 -260 °K
(no rain)
Freezing
Layer
U1
to
o 6:p m 'a a
Rain
*
»
“ *
8V > / o
8 »
0 6 8 0 0 8
84
(0 « 4
a
'
4
j
|
a
emission of microwave radiation
I
I
Ground Surface
Figure 1.22: M odel of precipitating clouds, raindrops and ice particles (left) (from Barrett [1994], p. 43, Fig. 4.).
A schem a of how this is used for SSM /I rainrate retrievals is show n on the right.
53
d istrib u te d th ro u g h the E arth O bserv atio n System (EOS) Data A ctive
A rchive C enter (DAAC) a t th e N atio n al A eronautics a n d Space
A d m in istra tio n 's (NASA) M a rsh a ll Space F light C e n te r (MSFC) in
H u n tsv ille , A lab am a.
The polar o rb itin g DM SP satellites sam ple h ig h latitudes d aily b u t
low latitu d es d isco n tin u o u sly (Figure 1.23). A t th e eq u a to r 3 to 4 days of
o bservation are se p arate d b y 3 to 4 day s w ith o u t observation (Figure 1.24).
O rbits are su n -sy n ch ro n o u s w ith eq u ato r crossing tim es for the F l l
satellite, lau n ch ed in 1991, a t ap p ro x im ately 5 A M a n d 5 PM local tim e a n d
for th e F10 satellite (launched in 1990) a t ap p ro x im ately 10 AM a n d 10 PM
local tim e. (Because the F10 satellite d id n o t achieve p ro p er orbit, the actual
o rb it is m ore ellip tical th a n d e sig n e d an d the e q u a to r crossing tim e
increases by ab o u t 30 m in u tes p er year.) O rbital p e rio d is approxim ately 100
m in u tes for b o th satellites, th erefo re local o b serv in g tim es can be u p to 25
m in u tes before o r after the e q u a to r crossing tim es.
T he S S M /I in stru m e n t senses m icrow ave ra d ia tio n at 4 frequencies,
19.35, 22.23, 37.00 an d 85.50 G H z (w avelengths o f 1.550,1.350, 0.8108, and
0.3509 cm respectively). G eom etries of the orbit, observational sw aths an d
p o la riz atio n co nventions are d ep icte d in (Figure 1.25). "H" p o larization
co n tain s b o th h o rizo n ta lly a n d vertically p o la riz ed com ponents because
the scan n er v iew s the e a rth a t a n angle. M easu rem en ts are m ade a t dual
p o larizatio n at all frequencies except 22.23 G H z w h ic h m easures only
v ertical p o larizatio n . S p atial reso lu tio n ranges fro m 15 km x l3 km for the
85.50 G H z b an d s to 69 km x 43 k m for the 19.35 G H z bands. The spatial (or
an g u lar) reso lu tio n is lim ite d b y diffraction at th e receiving a n ten n a (as
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without p erm ission.
54
fro m Kidder a n d H aar [1995]
1400 k m sw ath w id th
Figure 1.23: Global S SM /I coverage (from K idder a n d H aar [1995])A scending sw aths trend w est as th ey go from so u th to north.
D escending sw aths trend west as th ey go from n o rth to south.
Sw aths for one d a y are depicted here, so that the black regions
indicate the areas o n this exam ple d ay which are n o t observed
at all b y the S SM /I instrum ent.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
55
f r o m W e n g e t a l. [1 9 9 4 ]
C/3
jy 20
a,
£ 16
«
12
o
8
<u
JO
£
4
2
0
0.5°x0.5°
C/5
QJ
a ,
r-
CZ
cc
C/3
C/5
<U 250
'a.
£ 200
S
c/j
150
C
oj 100
JO
£ 50
3
0
2
O
s
r-"
r—
10
15 20
D ay
2.5°x2.5°
25
30
10 15 20
Day
25
30
2
20
16
12
8
4
0
0.5°x0.5c
15
20
D ay
tropical grids
high-latitude grids
Figure 1.24: S S M /I observation sam pling characteristics u sin g two
satellites (F10 an d F ll) d u rin g A u g u st 1993 for tropical (d ashed lines)
an d high-latitude (solid lines) grids w ith cell sizes of either 0.5° or 2.5°
(from W eng e t al. [1994]). Published global SSM /I rain rate retrieval
datasets are averaged a t 0.5° or greater. Sam pling d ensity can be
im proved b y using d ata from tw o or even three satellites w h e n available.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
56
a d a p te d fro m H ollinger [1990]
satellite
833 k m
a ltitu d e
53.1° view ing angle
at the surface
Ground
Track
1394 km.
sw a th w idth
19 and 22 G H z
footprint
12.5 k m
^
37 G H z footprint
85 G Hz footprint
(observed g ro u n d signal)
Figure 1.25: Satellite orbit, view ing geom etry an d sw aths. Because the
S S M /I instrum ent does not observe at nadir, there is no p ure vertically
polarized signal - "V" polarization is actually a mix of b oth horizontal
a n d vertical p o la riz atio n w hereas "H" polarization represents truly
horizontal po larizatio n a t the surface. All frequencies are sam pled on
"A" scans w hile o n ly the 85 G H z frequency is sam pled o n the additional
"B" scans due to it's sm aller surface footprint.
(ad ap ted from H o llin g er [1990])
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57
d eterm in ed b y the w av elen g th s o f observation) a n d n o t b y the signal to
n o ise ratio for p assiv e m icro w av e rad io m etry .
S am p lin g errors
Because a satellite sam p les o n ly an in stan ta n eo u s rain field a few
tim es every 24 h o u rs, there is a s tro n g alias in an y a tte m p t to estim ate to tal
rainfall from th o se sn ap sh o ts. T h e success of satellite m easures rely o n a n
accu m u latio n o f m easu rem en ts av e ra g ed over space, tim e or both. This
problem is n o t u n iq u e to satellite m easu res only, g ro u n d totals of
p recip itatio n h av e b ee n sh o w n to g ain reasonable accuracy in estim atin g
area-av erag ed rain fall o n ly w h e n accu m u lated o v er tim e scales in c lu d in g
several rain fall ev en ts [Bellon a n d A u stin , 1986; Seed an d A ustin, 1990].
The d istrib u tio n of ra in rates h a s b e e n m od eled theoretically and fit to
observed d a ta , p rim a rily using m e asu rem e n t areas larg e enough to sam p le
the full size a n d life cycle d istrib u tio n of storm s [Atlas et al., 1990; Bell e t al.,
1990; K edem et al., 1990; R osenfeld et al., 1990]. These studies used
observations fro m the GARP (G lobal A tm ospheric R esearch Program )
A tlantic T ropical E xperim ent (GATE), co n d u cted in th e tropical A tlantic
O cean d u rin g 1974 o ver an area ab o u t 350 km. x 350 km . Raingauge
m easu rem en ts fro m research v essels an d ra d a r o b serv atio n s m ade e v e ry 15
m inutes a n d a v e ra g e d to 4 km . X 4 km . pixels form th e observational
dataset, from w h ic h the sam p lin g of p o lar o rbiting satellite retrievals w ere
sim ulated. T he stu d ies sh o w ed th a t rainfall (m ean ra in rate) could be
accurately m e asu red over large areas from a single polar-orbiting satellite
over periods as sh o rt as 3-4 w eeks w ith low error [Bell e t al., 1990; K edem et
al., 1990]. F u rth e r, a u to co rre latio n for m ean rain rate o v er the GATE area
R eprodu ced with perm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
58
w as sh o w n to be about 6 h o u rs, so that a single satellite sn ap sh o t
ap p ro x im ates the rainfall fo r a few hours. In th e case o f tw o polar-oarbiting
satellites, if sam pling in creases to 4 x /d ay , m e a n rate rates m ight t h e n have
low erro r for periods sh o rte r th a n 3 weeks [A tlas e t al., 1990; R osenheld et
al., 1990].
R ain ra te retrievals an d validation
As p a r t of the N A SA W etN et Project, th e F irst P recip itation
In terco m p ariso n Project (PIP-1) evaluated se v eral ex istin g SSMI raizn rate
alg o rith m s in relation to each o th er and rain g au g e d a ta u sin g sevearal areas
a ro u n d the w o rld over th e p e rio d A u gust-N ovem ber 1987 [Barrett, 1994;
W ilheit a n d al, 1994]. Som e of th e algorithm s w e re p rev io u sly calib ra te d
u sin g ra d a r d a ta as w ell. N o o n e algorithm o u tp e rfo rm e d all the o th e r s a n d
som e w ere m ore a p p ro p ria te in different reg io n s th a n others. Somee
ag ree m en t w as m ade a b o u t th o se algorithm s w h ic h m o st closely
a p p ro x im a te d non-satellite rain fall estim ates.
In p articu lar, m o u n ta in o u s areas an d e le v a te d p la tea u s such_ as th e
A ltip lan o a n d Tibet h ave b e e n excluded from som e global algorithrm s
because of the difficulty of sep aratin g the effects of rainfall versus c«old
surfaces (the high plateaus) o n depressed b rig h tn ess tem p eratures [ A dler et
al., 1994]. W e use an ap p ro ach introduced by G ro d y [G rody, 1991] w hich
attem p ts to separate the sc atte rin g an d n o n -scatterin g c o n trib u tio n s to the
85 G H z. b rig h tn ess te m p e ra tu re , w hich we w ill sh o w is n o t se n sitiv e to the
cold b ack g ro u n d surface o f the Altiplano.
G ro d y defined a scatterin g index SI
SI(85v) = F - Tv (85),
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without perm ission.
(1.7)
59
w h e re Tv (85) is th e v ertically p o larized b rig h tn ess te m p e ra tu re a t 85 G H z
a n d F estim ates th e n o n sc a tte rin g o r e m issio n c o n trib u tio n to Tv (85) as
d e te rm in e d from lo w e r freq u en cy c h a n n e ls
F (G rody, 1991) = 450.2 - 0.506*TV(19) - 1.874*TV(22) + 0.00637*(TV(22))2. (1.8)
W eng [W eng et al., 1994] revised F u sin g a global d a ta se t o v e r lan d an d
o cean for n o n p recip itatin g conditions. T h ey em pirically co rrelated Tv (85)
w ith th e vertically p o la riz e d b rig h tn ess te m p eratu res a t 19 a n d 22 GH z,
Tv (19) an d Tv (22), fo r n o n p re c ip ita tin g co n d itio n s o v er la n d a n d ocean
o b ta in in g
F (W eng, 1994) = 256.2 - 0.375*TV(19) - (0.2 - 0.00237*TV(22))*TV(22). (1.9)
Examples o f Tv (85), F an d SI(85v) for January 29, 1995, are given in
F igure 1.26.
W eng et al. also rela te SI(85v) to av erag e rain rate th ro u g h the
e q u a tio n
R ain Rate ( m m /h r ) = -1.7 + 0.29 * SI(85v),
w h ich w as developed u sin g rad ar rain fall d ata, surface g au g e
m easu rem en ts a n d S S M /I d a ta over J a p a n a n d th e U n ite d K ingdom
(Figure 1.27). The re la tio n sh ip in E q u a tio n 1.10 is u se d in all ra in rate
calculations d escribed below . First th e 'B ackground' S cattering Index is
d e fin e d an d ex am in ed to ascertain it's u tility , or not, in ra in rate
calculations in m o re lim ite d regions, as d escrib ed b elow .
R eprodu ced with p erm ission o f the copyright owner. Further reproduction prohibited without perm ission.
(1.10)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Tb(85v)
F
SI
Figure 1.26: Vertically polarized brightness temperature Tb(85V), non-scattering estim ation F and the scattering
index SI for the central A ndes on January 29,1995, illustrating the identification of convective rainfall. The
banding seen in F is along the original (swath projection) scan lines at the low er 25 km. spatial resolution of the
non-85 Ghz. bands. All calculations are m ade with non-projected brightness temperature data follow ing w hich
the results are reprojected (and resampled) to the Lambert A zim uthal Equal Area map projection sh ow n here.
61
fro m
W e n g e t a l. [1 9 9 4 ]
40
O
g 3° T3
Q
S
cn
cn
1
0
i
2
l
i
4
6
R ad ar Rain Rate (m m /h r)
i
8
10
Linear fit: R adar rain rate = -1.7 + 0.29*SI
Figure 1.27: Relationship b etw een radar rain rate a n d SSMI scattering
index from Weng et al. [1994], w ho used a linear fit to the data in deriving
a relationship to be u sed in SSMI-based rainfall retrievals. The linear fit
is used in the present stu d y to m aintain consistency w ith other published
w ork. Differences to n o n lin ear fits are insignificant over typical scattering
index values (see Figure 1.40).
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
62
P ro c e ssin g
D ata from the S S M /I in stru m e n t a b o a rd th e F10 an d F l l satellites for
1992 —1994 is u sed , acq u ired th ro u g h the D a ta A ctive A rchive C en ter
(DAAC) a t MSFC. The lo w er reso lu tio n b a n d s a re o v ersam p led to th e 85
G H z re so lu tio n in o rd e r to m a in ta in the fu ll re so lu tio n of th e 85 G H z
b a n d , w h ich supplies the p h y sical sc atte rin g sig n a tu re used in identifying
p recip itatio n . Therefore, sp a tia l v ariability a t 85 G H z is g reater th an for
lo w er freq u en cy m easu rem en ts.
T he source d ata w as o b ta in ed fro m th e M arshall Space F lig h t C enter
(MSFC) a n d consisted of N a tio n a l E n v iro n m e n ta l Satellite D a ta a n d
In fo rm atio n Service (NESDIS) S S M /I L evel l b d a ta w hich co n sists of
in s tru m e n t counts for each S S M /I ch an n el. T h ese w ere first c o n v e rte d to
a n te n n a tem p eratu res a n d th e n to b rig h tn e ss te m p eratu re u s in g code
s u p p lie d b y MSFC an d w ritte n b y D on M o ss a t th e U niversity o f A labam a.
The a n te n n a correction su b ro u tin e s u s e d to m a k e the c o n v e rsio n to
b rig h tn e s s tem p eratu re are p u b lish e d in W e n tz [W entz, 1991]. R ain rate
estim a tes in the original satellite p ro jectio n w e re calculated ac co rd in g to
E q u atio n s 1.7, 1.9-10, th en re g rid d e d to a c o m m o n grid w ith a 10 km .
sp a c in g in a L am bert A zim u th a l Equal A re a p ro jectio n u sin g co d e from the
N a tio n a l Snow an d Ice D ata C en ter (NSIDC) orig in ally d ev e lo p ed for
a n o th e r DMSP in stru m e n t (O ptical Line S can n e r) w hich w as m o d ifie d to
w o rk w ith the S SM /I ra in rate s. Finally, th e s e g rid s w ere su m m e d to
p ro d u c e m o n th ly totals b y p a ss an d a raste r-to -v ec to r co n v ersion w as m ade
crea tin g a separate d atase t o f n onzero g rid v a lu e s w ith their resp ectiv e date,
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
63
tim e (pass) a n d grid locations. Since m o st g rid p o in ts are zero this g reatly
red u ce s th e volum e o f th e d a ta se t fo r su b se q u en t analysis.
R a in rates calculated fr o m S S M /I D A T A
B ack grou n d stu d y o f th e c e n tr a l A n des
For th e central A n d es a d e ta ile d s tu d y w as first u n d e rta k e n o f tw o
rain y seasons, D ecem ber-A pril o f 1993-94 an d 1994-95. The stu d y area is
sm all e n o u g h to allow careful sc re e n in g of false p recip itatio n sig n a tu re s
th ro u g h th o ro u g h e x a m in a tio n o f th e b a c k g ro u n d terrestrial em issio n
p a rtic u la r to the central A ndes.
The global classification schem e defined b y G rody [G rody, 1991] w as
later im p ro v ed upon [Ferraro a n d G ro d y , 1994] a n d form s the p o in t of
d e p a rtu re for this w ork. This classification is u sed to co n stru ct rain -ra te
estim ates u sin g sw ath d a ta fro m tw o d ifferen t satellites a n d m a in ta in in g
th e te m p o ra l an d sp atial re so lu tio n of the original S S M /I sw a th d ata.
Som e features of the classificatio n schem e o f G rody a n d F erraro our
ex clu d ed since the s tu d y area d o es n o t include oceanic regions, sea ice or
larg e areas of snow cover. This allo w s m ore d etailed screening of the
la n d /w a te r / salt b ack g ro u n d of th e A ltiplano to identify a n d elim inate false
sig n atu res d u e to scattering surfaces, p rim arily the coastal d eserts a n d
s o u th e rn A ltiplano. T h e a lg o rith m in c lu d es a th resh o ld for th e p rese n ce of
scatterin g b y requiring SI(85v) > 10. Screening for snow cover a n d ice is n o t
m a d e since n o regions o f the cen tral A n d es are classified as su ch a t the
re so lu tio n o f the S SM /I.
D ifferent rain rate alg o rith m s fo r one rain y season, D ecem ber 1994
th ro u g h A p ril 1995, are ex am in ed to d eterm in e th e ir se n sitiv ity to false
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without perm ission.
64
scattering signatures. The ce n tra l A ndes p rese n t co ld er a n d m ore scattering
b ac k g ro u n d surfaces th a n lo w elevation, m ore h o m o g en o u s regions. A
m ean m icrow ave b ack g ro u n d w as constructed by av erag in g all days during
January, 1994, w hich d id n o t sh o w scattering effects at 85 G H z due to
p recip itatio n (Figure 1.28). To co n stru ct the av erag e in d iv id u al images
sh o w in g n o rainfall o v e r t h e e n t i r e r e g i o n w ere id en tified and then
averaged, pixel for pixel, so th a t the final im age d ep icts average, non-rainy
conditions for each pixel. If a n im ag e show ed rain fall an y w h ere in the
en tire region it w as ex clu d ed in o rd er to en su re a ro b u st average result.
P recipitation is readily id e n tified b y depressed b rig h tn ess tem peratures at 85
G H z and b y clear-sky co n d itio n s as observed w ith in fra-red satellite images.
The low er frequency b a n d s re c o rd features d eterm in ed prim arily by
tem p eratu re and em issivity. D u e to the low er em issiv ity of w ater (~0.4)
com pared to that of lan d (-0.95), w ater surfaces ap p ear cooler (darker) as is
the case for the ocean a n d Lake Titicaca. The salars of th e so u thern
A ltiplano also show d e p re sse d b rig h tn ess te m p eratu res, d u e either to the
presence of surface w ater, a lo w er em issivity of the sa lt surface, or both.
The lan d surface generally sh o w s low er b rig h tn ess tem p eratu res w ith
elevation, co rresp o n d in g to d ecreasin g land surface tem p eratu re w ith
elev atio n .
A t 85 GHz scattering effects an d the low er em issivity of w ater are
m u ch m ore evident. W e u n d e rta k e a system atic su rv e y of scattering
surfaces in order to id en tify th e p o ten tial for m isclassification of these
surfaces as precipitation. A b so rb in g surfaces such as w ate r generally behave
in the opposite sense fro m sc atte rin g surfaces, w ith Tb(85) greater than
Tfc>(22). Several regions w e re id e n tified on the basis o f th e ir spectral
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
ion prohibited without permission.
o
37 GHz Horiz
37 GHz Vert,
85 GHz Horiz
85 GHz Vert.
Figure 1.28: M ean m icrowave brightness temperature of the central A ndes for the seven SSM /I bands, averaged
over the non-rainy sw aths of the 1994-1995 study period (ie. - sw aths with no scattering index, SI).
65°W
70°W
75° W
a t 85 G H z, vertical p o larizatio n (°K)
230
250
270
290
Figure 1.29: Spectrally defined regions of th e central A ndes, based on
their average 85 G H z. brightness tem p eratu re (background image)
for non-rainy conditions.
R eprodu ced with perm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
67
Altiplano
S a la r s a n d L ake T itic a c a
280 r
260 r
290 r
'0- Salar d e Uyurn
Solar de- Coipnsa
Lake Ttacaca
280 r
~
CM
CM
§j 2 7 0 1
>
240-
2 6 0 1-
240
260
250 L
250
280
Central Attiptano
Northern Altiplano
260
270
v85
v85
>
290
M ountains
C o a s ta l/D e s e rt
CM
CM
280
285f
280 f
280 r
270
275 r
CM
_ _ _
E
260 r
270 b
•2
265 b
260 L
260
S. Atoptano/Pima
W estern Cordillera
250 f
South Atacama
North A tacam a
Eastern ConbOera
C oastal Peru
265
270 275
v85
280
285
240 L
240
25 0
260
v85
270
280
270 r
260-
2S0h
SaltLakes
Lake Tmcaca
Altiplano
Coastal.Desert
Mountains
240 r
230!-
220 :
20
40
60
100
Frequency(GH z)
F igure 1.30: Tv (85) vs. Tv (22) for different surfaces in the central A ndes.
Scattering regions have lo w er brightness tem p eratu res at 85 G H z th a n at
20 G H z. a n d are potential contam inating signals fo r rainfall identification.
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without p erm ission.
68
characteristics a n d know n g eo g rap h y (Figure 1.29). The com parison o f 22
G H z and 85 G H z brightness tem p eratu res d eterm in es surfaces as scatterers
or absorbers. Figure 1.30 show s scatterplots of Tv (85)bg vs. Tv (22)bgfor
sev eral of th e d efin ed regions. The salars h av e Tv (85)bg> Tv (22)bg, w ith
b o th Tv (85)bg a n d Tv (22)bg ab o u t 20 °C w arm er th a n Lake Titicaca. The
decreased b rightness tem peratures of the salars m a y be caused by the
presence of som e surface w ater, since it show s a n increase in brightness
tem p eratu re w ith frequency, characteristic of w ate r, an absorber. For th e
eastern a n d w estern C ordillera, Tv (85)bg < Tv (22)bg identifying these
regions as slig h tly scattering surfaces. A lth o u g h snow and ice cover is
p rese n t in local ranges, the surface area is too sm a ll to generate the
scattering signal in these regions. The coastal area s of Chile an d Peru are
also scattering surfaces, a fact ■which co n trib u tes to their potential
m isclassification as rain. Finally, the A ltiplano h a s relatively com parable
Tv (85)bg a n d Tv (22)bg, con sisten t w ith ro u g h ly eq u al emissivities for d ry
la n d , and is n o t strongly scattering. Figure 1.31 sh o w s the correlation
b etw een Fbg, calculated u sin g Tv (19) bg a n d Tv (22) bg and Equation 1.8, and
the average Tv (85) bg for la n d regions in the b ack g ro u n d image.
A new scattering in d ex can be defined, SIbg(85), using only th e
vertically p o larized 85 G H z channel, as
SIbg(85) = T v (85)bg-Tv (85)
(1-11)
w h ere Tv (85)bg is the m ean b ack g ro u n d as d escrib ed above. The use of this
index m ay red u ce aliasing effects at b o u n d aries su ch as the perim eter of
Lake Titicaca b y u sin g only one b a n d at fixed resolution. O n the other h an d ,
it m ay lead to increased e rro r b y n o t ad ju stin g to daily variations in
b rig h tn ess tem p eratu res fro m the m ean values. F igure 1.31 shows a
R e p ro du ced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
300
1
. T - I ...r
.I
T
r
. r
r
j...| T
j . .j . y - r i
r -T . r - r T
7
n
r -|-
r . r . r - r
. r - r .-r _ r T - j . . T - r - r
^
Background Scattering Index, BG-SI = F - Tv(85)bg
,r _T
.
dashed lines for BG-SI = +/-10 °K, +/- 5 °K
,
<£
xQ?
aSt*
.
Rain Rate is calculated w hen the
scattering index is greater than +10°K.
280
Pi
i i t ii 1
l( t - * *
1
t 1
260
' largest n eg a tiv e o u tliers
from m o u n ta in o u s areas
so u th of 22°S la titu d e in
th e A ltiplano.
•f1'i
'
240
240
. -i- I
1
I . I
250
1
I...I
I.
I .1.1
i
260
.1 . . I . i - j
-I...I.
O
N
vO
th e ~-4° bins is d istrib u te d
acro ss all lan d areas ex cep t
m o u n ta in s so u th of 22°S
I-
I.
270
Tv(85)bg
i. .1
.1. . 1 . 1
280
I
i
I
.1 . i . . 1 . . l — 1 . . . 1 - 1 — i . i .
290
i — i. _ i — i .
i
300
°K
Figure 1.31: F vs. Tv(85)bg for average, non-rainy brightness temperatures over land areas of the
central A ndes during January, 1994. F, as used here, is defined from global data in G rody (1991).
70
system atic n eg a tiv e b ia s of th is d ata from the fu n ctio n F as d e fin e d b y
G rody (1991), w h ich m ig h t b e expected to cause a n low er or m o re
co n serv ativ e ra in -ra te .
C en tra l A n des
F igure 1.32 sh o w s resu lts from tw o scatterin g indexes, th e first as
described b y G ro d y (Eq. 1.9) an d the second as d efin ed above u sin g the
m ean b ac k g ro u n d (Eq. 1.11). O n the left in F igure 1.32 is the s u t t i o f rain
rate calculations from SSMI sw ath d ata b etw e en D ecem ber 1994 a n d A pril
1995 usin g E q u atio n s 1.7-8 a n d 1.10 in calculating ra in rates for each sw ath.
O n the rig h t is a sim ila r su m o f rain rate calcu latio n s u sin g th e average
b ack g ro u n d as sh o w n in F igure 1.28 u sin g E q u atio n s 1.8 a n d 1.10-11 w here
eq u atio n 1.11 uses a sc atte rin g index calculated fro m the m e a n Jan u ary 1994
n o n p recip itatio n b a c k g ro u n d instead of 19 a n d 22 G H z d a ta fro m the samesw ath as is a ssu m e d in E q u atio n 1.7. The sw ath d a ta is ac q u ired four tim es
daily, 10 A M /P M a n d 5 A M /P M local tim e, respectively, for th e F10 and
F l l satellites. C o v erag e is incom plete so th a t for a given p o in t o n the earth
there are re g u la r o b se rv a tio n al gaps. The in d iv id u a l rain rate s as presented
in F igure 1.32 are su m m e d o v er all sw ath d a ta b u t do n o t y e t rep resen t
p o in t totals fro m w e a th e r statio n s or clim ate m ap . As areally av erag ed rain
estim ates th e y can be co m p ared w ith p o in t rain fall totals b y m ak in g
a ssu m p tio n s a b o u t h o w to co n v ert from rates to totals. O ne sim ple
assu m p tio n is to m u ltip ly these rates b y 12 h o u rs, the av erag e tim e
b etw een ea ch o b se rv a tio n (sw ath), to o b tain rain fall totals. F u rth e r
assu m p tio n s m u st th e n be m ad e to account for sw a th gaps as depicted in
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Sum s of rain rates calculated for each sw ath for satellites F10 and F ll, Decem ber 1994 - April 1995
W70°
o W75°
o
accum ulated
rain rate (m m /h r)
100
W65°
accum ulated
rain rate (m m /h r)
600
is
u sin g G ro d y S cattering Index, E q u atio n s 1.7-8
u sin g B ack g ro u n d S catterin g In d ex , Eqns. 1,8 & 1.11
Figure 1.32: Sum s of rain rates calculated from sw ath data using tw o versions of a Scattering Index (SI), on the left
u sin g SI as defined by Grody (1991) (Equations 1.7-8), w hich uses all SSMI bands at the sam e tim e of observation
from each sw ath to calculate a reference 'no-rain' field, and on the right using the average non-rainy background
w hich falsely identifies daily variations in Tb as rainfall-induced scattering and therefore overestim ates rainfall.
U nits for these accum ulated, areally averaged rain rates (m m /h r) are discussed in the text.
72
F igure 1.23 an d discussed in H u ffm an et al. [H u ffm an e t al., 1995]. N one of
th ese a ssu m p tio n s affect h o w ev e r th e regional v a ria tio n in rainfall to ta ls
su ch as locations of m ax im u m orographic p re c ip ita tio n . In this section th e
b ack g ro u n d scattering index, SIbg{85), is exam ined o n ly for its plausibility
in re p ro d u c in g regional p a tte rn s of precipitation. T he regionally
in co n sisten t p a tte rn seen o n th e rig h t side of F ig u re 1.32 w as determ ined
th ro u g h visual in sp ectio n o f th e en tire series o f s w a th im ages to be
p h y sically unrealistic an d to re su lt from false p re c ip ita tio n signals in th e
m o u n ta in s so u th of 22°S a n d alo n g the coast. T he n eg a tiv e bias seen in
F ig u re 1.31 m ig h t have in d ic a te d a m ore co n serv ativ e rain fall estim ate
u sin g SIbg(85), h ow ever th e fu n ctio n F as calcu lated sep arately for each
sw ath is ap p aren tly m u ch m o re accurate by acco m m o d atin g sw ath to s w a th
v ariab ility in surface b rig h tn ess tem p eratu re a t all SSMI frequencies a n d
thus p ro d u cin g a scattering index better related to rain fall an d less rela ted
to d ev iatio n s of surface co n d itio n s from a m o n th ly m ean .
The m axim um accu m u lated rain rates o ccu r a t lo w er slopes of th e
e a ste rn C ordillera an d are d iscu ssed in a follow ing section. The results fo r
SIbg(85) clearly pick up false sig n atu res in the co astal regions and so u th e rn
A ltip lan o an d am plifies differences in the d aily v a ria tio n s of background
b rig h tn e ss tem p eratu res relativ e to the seasonal m ean . B oundary effects
d u e to the com bination of d a ta of different re so lu tio n d oes n o t result in
sig n ifican t erro r d u e to aliasin g an d therefore th e sc atte rin g index as
d efin ed b y G rody relates b etter to rain rate th a n the b ack g ro u n d scattering
as d efin ed here. The b ack g ro u n d scattering index d o es n o t account for d a ily
v a ria tio n s in surface em ission as is m ost e v id e n t a lo n g th e w estern coast,
A tacam a d e se rt a n d m o u n ta in s of the so u th e rn A ltip la n o , accounting fo r
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
73
m any m o re false positive id e n tifica tio n s of ra in a n d , hence, a la rg e r a n d
unrealistic ra in rate sum as p lo tte d on the rig h t sid e in F igure 1.32. F a n d SI
as d efin ed in E quations 1.7-8 w ill b e used for ra in rate retriev als o v er a
much lo n g er p erio d as d escrib ed below .
C ross sectional profiles o f these estim ates for th e 94-95 rain y season
and to p o g ra p h y w ere m ad e a t th e locations sh o w n in F igure 1.33. The
profiles re p re se n t values a v e ra g e d over a sw ath 100 km w id e w ith a
running av erag e every 10 k m a lo n g the swath. T he p ro files (Figure 1.34)
show b ro a d orographic p eak s cen tered at ro u g h ly 1 km. elevation, an d
extending fro m 500 m. to 2 km . in elevation. A c c u m u lated rain rates
decrease to the south.
S o u th A m erica
B ased o n the results ab o v e a m uch larger d a ta se t co n sistin g of SSMI
observations from the F10 a n d F l l satellites w as u n d e rta k e n for th e years
1992-1994. Rain rates w ere calcu lated according to the G ro d y scattering
index fo r all of S outh A m erica a n d th e H im a la y a n /T ib e ta n region. Because
large d a ta gaps existed early in 1992, results for S o u th A m erica are discussed
only for th e years 1993-94. A ccu m u lated rain rates for th e central A ndes are
show n in F igure 1.35 w ith th e ir corresponding cross-sectional profiles w ith
to p ography in Figure 1.36. A ccu m u lated rain rates for all of S outh A m erica
are sh o w n in F igure 1.37.
T hese d a ta are ac cu m u late d from in d iv id u al satellite im ages (also
called p asses or sw aths) w h e re each image rep resen ts a sn a p sh o t o f the
passive m icro w av e rad ia tio n a t th a t time w hich is co n v erted as d escribed
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without perm ission.
74
W 75°
W70°
*1 G H C N Stations
W65°
14
I L egates-W illm ott Stations K
I
*■I W M O Stations
Figure 1.33: Locations o f the profiles for the 1994-95 study.
C lim ate station locations are show n a n d colored according to source.
The background is shaded-relief to p o g rap h y (GTOPO30).
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
75
4000h
SSM /I P ro file 1
j
H150
-iioo
2000j-
—50
0L
50
100
150
200
250
—0
300
SSM/I (mm/hr,
Elevation (m ), solid line
4000!-
SSM /I P ro file 2
^
2000i-
-10 0
_ _
0L~
0
.
—50
—0
400
200
40001
SSM /I P ro file 3
-150
-50
4000h
—0
300
W M O tran sect near
SSM /I p r o file 2
20001
r
-
0
0I
250
200
200
100
-
SSM /I P rofile 4
-150
-1 0 0
-50
2000
0
—
100
200
4000 u
300
400
0
500
W M O transect near -4000
SSM /I p r o file 4
2000
0c
2000
400
300
4000
4000
-2000
100
200
300
400
SSM/1 (mm/hr)
150
100
r,« T n Z
50
SSM/I
-1 0 0
2000L
0L
0
SSM/I (mm/hr)
Position along transect (km)
500
Figure 1.34: Profiles of elevation (solid lines) and rainfall (dashed).
A ccum ulated ra in rates from D ecem ber, 1994, th ro u g h A pril, 95, are
show n w ith tw o W M O annual precip itatio n profiles for comparison.
(Elevation d a ta o n the WMO p recip itatio n m ap is coarser than the 1 km .
GTOPO30 d a ta u se d in the other profiles, although still appropriate at
th e spatial reso lu tio n at which the W M O data can be used.)
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without perm ission.
76
W75°
W70°
W65°
A ccum ulated rain rate (m m /h r)
0
10
20
i
30
I
40
50
60 >70
Figure 1.35: Rain rates in the central A ndes for 1993-1994, accum ulated
from all available F10 and F ll passes. Locations of the four profiles
(following figure) across the rainfall peaks are d isplayed as w hite. The
units of accum ulated rain rate (m m /h r) are discussed in the text.
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without perm ission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ccum ulated rain
rate (m m /h r)
(min to max, solid shading)
150
Elevation (km)
(max, m ean and min)
6
4
2
50
0
0
6
150
4
WEST
(A ltiplano and
high peaks)
100
Profile 1
100
Profile 2
50
2
o
0
50
100
150
East
(A m azon basin)
250
200
Profile 3
150
Profile 4
100
50
100
150
200
0
Distance along transect (km)
Figure 1.36: Swath profiles of rain rates and elevations in the central A ndes for the profiles show n in the
previous figure. The profiles indicate that orographic precipitation is m ostly forced by elevations b elow 2 km.
78
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
79
above to a n in s ta n ta n e o u s rain fall ra te a t th a t tim e o f observation w ith
u n its of m m /h r . T h ese sn a p sh o ts o f rain fall rate a re th e n reprojected fro m
the o riginal sa te llite p ro jectio n to a co m m o n g rid u s in g the Lam bert
A zim u th al E q u al A re a P rojection. T h ese in d iv id u a l g rid s, each
rep resen tin g a s n a p s h o t of the ra in fa ll field a t th a t p artic u lar time of
satellite o v erp ass, a re th e n sim p ly a d d e d to g eth er fo r m onthly, seasonal or
an n u a l to tals of r a in rate, still w ith u n its of m m / h r . O n e can then m ak e
assu m p tio n s a b o u t h o w to m u ltip ly th ese ra in ra te s b y tim e in o rd er to
o b tain rainfall to ta ls, th e sim p lest o f w h ich is to m u ltip ly by the tim e
betw een each o b se rv a tio n . For th e d a ta p resen ted h e re tw o satellites w e re
u sed so th a t the tim e b etw een o b se rv a tio n s of a g iv e n area is 6 hours
(excluding sa m p lin g alias as seen in F igure 1.23). M u ltip ly in g the
cum ulative ra in ra te s in the p re se n t w o rk b y 6 h o u rs a n d dividing by th e
nu m b er of y ears o f o b serv atio n (2 y e a rs for South A m erica, 3 years for th e
H im alaya) gives p h y sically reaso n ab le totals for s u c h areally averaged
rainfall w h ich are h ig h e r th a n th o se fro m 4-d a n a ly sis fields averaged o v e r
larger areas b u t lo w e r th a n the p o in t totals as m e a su re d at climate statio n s.
For exam ple, in s o u th e a ste rn P e ru m a x im u m a re a lly averaged rainfall
using this a s su m p tio n is as h ig h as 39 c m /y r as c o m p a re d to 2 to 4 m / y r for
p o in t m e asu rem e n ts. D u rin g the ra in y season fro m D ecem ber-February
accum ulated ra in ra te s u p to 48 m m / h r are o b se rv e d w hich using the
sim plistic a s su m p tio n above gives a n average d a ily rainfall of 3.2 m m . For
January this rises to o v er 5 m m /d a y . These n u m b e rs are com parable to or
higher th an are a lly av erag ed o b se rv a tio n s of up to 3 m m /d a y from clim ate
m odels a n d clim ato lo g ies [H u ffm an e t al., 1995]. S ince this assum ption
only applies a m u ltip lic ativ e factor, leav in g the re la tiv e spatial d istrib u tio n
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without perm ission.
80
of the d a ta u n ch a n g ed , the resu lts are left as accu m u lated rain rate s w ith
u n its m m / h r .
M arked v a ria tio n in accu m u lated ra in rates is ap p a re n t fro m Figure
1.35, w ith peak ra in rates occurring along those areas facing ro u g h ly
n o rth e a st. P rofiles are n u m b e re d 1 th ro u g h 4 fro m n o rth to so u th
respectively. In p ro file 1 the Rio Pachitea valley seem s to be a focal point of
o ro g rap h ic p recip itatio n . It is in terestin g to n o te th a t the Rio P achitea is a
relativ ely sm all trib u ta ry of the Rio U culayi. The U culayi d rain s a large,
w ell-dissected a rea to th e so u th of the Pachitea w hile the P achitea is
p rese n tly d ra in in g a m ore b a sin w a rd p a rt of th e m o u n tain front.
Sim ilarly, a t th e n ex t p ea k to the so u th w h ere profile 2 is located, the
Rio U ru b am b a d o es n o t ex ten d far into th e A n d es b u t captures m o st of the
o ro g rap h ic rainfall. The Rio A p u rim ac d rain s a m uch larger a re a further
w est, o n the o th e r sid e of a to p ographic rise of o nly 2,500 m. b u t does not
have a large o ro g rap h ic p eak . C onsistent w ith all cross-sectional profiles of
this stu d y , p recip itatio n is efficiently induced b y topographic rises of only a
few th o u san d m eters (Figure 1.36).
The o th er tw o large peaks rep resen ted b y profiles 3 an d 4 seem to be
characterized o n ly b y a slig h tly concave sh ap e o f the m o u n tain fro n t facing
to the n o rth east. A t th e cen ter o f the 'dry' zone b etw een profiles 3 an d 4 the
Rio Beni exits fro m th e h ig h A ndes.
In Figure 1.37 no significant p recip itatio n differences ex ist betw een
1993 a n d 1994. The v ery larg e peaks along the C olum bian coast a n d in
w estern P atag o n ia are co n sisten t w ith k n o w n clim atologies. In P atagonia
h o w ev er the u se of p assiv e m icrow ave is v ery difficult to in te rp re t due to
the v ery m ixed b ac k g ro u n d of w ater, land a n d ice. N otably a b sen t is a peak
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without perm ission.
81
in rain fall a t th e n o rth w e ste rn corner o f th e A m azo n b a sin a it the
m o u n ta in fro n t o f so u th e a ste rn C olum bia, as sh o w n in g ro u in d station
clim atologies. It is p ro b ab le th a t this rain fall m axim um is d u ie to shallow
liftin g u n ac co m p a n ied b y convection a n d h ig h ice concentra- tions and
therefore m o re d ifficu lt to d etect w ith p a ssiv e m icrow ave d a ta. Elsew here
alo n g the A n d es su c h as in P eru an d B olivia no such d is a g re e m e n t w ith
k n o w n clim atologies is to be found, su g g e stin g th a t in th e se areas shallow
lifting does n o t p lay an im p o rtan t role a n d th a t, as is discuss*ed in all
clim atologies of th e area, convection is far m o re im p o rta n t f o r rainfall.
From D ecem ber to M ay all areas a lo n g the m o u n ta in tfro n t are
accum ulating ra in (Figure 1.38), the stro n g e st b ein g from De*cember to
F ebruary. O u tsid e of th e u su a l rainy season, significant accu n n u latin g of
oro g rap h ic p re c ip ita tio n occurs also from S eptem ber to N o v e m b e r. The d ry
season, June to A u g u st, is in d eed dry , a lth o u g h m inor orogr. aphic peaks
n ev erth eless c o n tin u e to contribute to a n n u a l totals.
D iu rn al v ariatio n s in p recip itatio n are clearly a p p a re m t in Figure
1.39. A m azon b asin p recip itatio n characteristically is h ig h est! in the
aftern o o n (5 p m local tim e) w h en h e a tin g a n d convection is: strongest. T he
o ro g rap h ic peaks h o w ev er are clearly stro n g e st d u rin g th e m ig h t, the m o st
w id esp re ad occurrence b ein g at 5 am local tim e. C o n s id e rin g the discussion
o f M CC's earlier, th is im p lies that the p rim a ry co n trib u tio n to these
o ro g rap h ic p eak s is n o c tu rn a l convection, ju s t as it is in th e plains of th e
US east of th e Rockies. The observations o f V elasco an d F ritrsch [Velasco
a n d Fritsch, 1987] su g g ested th at Bolivian M CC activity w a s sim ilar to
A rg en tin a b u t w as d ifferen t enough in ch aracter to su g g e st tth ey form ed a
d istin ct p o p u latio n . The p rese n t stu d y su g g ests th a t B oliviam MCC's are
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
82
Dec —> Feb
Mar —> May
W 70°
SIS
oOSS
Jun —> A ug
5°
W 70°
Sep —> N ov
W65°
oOLS
oStS
oOSS
Figure 1.38:1993-94 accum ulated rain rates in the central A ndes by
season. H eav iest precipitation is during austral w inter.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
83
10 AM, F10 desc.
5 AM, F ll desc.
W75°
W75°
W65°
W70°
W70°
W 65°
ms
^
-^>v: **
oQi-s
J-f ar~u.
ooss
5 PM, F ll asc.
W75°
W70°
W 65°
10 PM, F10 desc.
W 75°
W 70°
W65°
oOts
oSts
oOss
Figure 1.39:1993-94 accum ulated rain rates in the central Andes by satellite
pass. H eaviest precipitation is at night due to m esoscale convection.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without p erm ission.
84
very long-lived, events to g e n e ra te th e greatest accu m u lated rainfall at 5 am
local tim e.
F ig u re 1.40 show s a h isto g ra m of in d iv id u al ra in rate values. The
d ata p ro d u cin g the averages b y season, pass and y ear filtered points
exceeding 15 m m /h r, w hich a re d u e to b ad scan lines w hich passed earlier
checks on v a lid b rig h tn ess te m p e ra tu re values. D etailed exam ination of
the points w h ich w ere filtered a n d the filtered d a ta sh o w ed that they w ere
random ly d istrib u te d an d th a t th e su m m ary d ata w a s unchanged in all
areas of in te re st. A ccum ulated rain -rates (each g rid value) are plotted
against elev atio n in Figure 1.41, sh o w in g a m arked preference for th e
h ig h est v a lu e s a t elevations b e lo w 2 km.
A sia n m o n so o n
A lo n g the H im alay an ra n g e s the p rim ary rain fall occurs d u rin g the
m onsoon p e rio d of June to A u g u st. Since no d ata g a p s occurred after M ay,
1992, these resu lts are ac cu m u late d over all three y ears, 1992-94. Since a
separate screen in g for snow w a s n o t included, care m u st be taken in
in te rp re tin g th e results since s n o w cover and p recip itatio n have sim ilar
scatterin g sig n atu res. N ev erth eless, orographic rain s d u rin g the m onsoon
a n d the g re a te st areal cover o f snow fall in the H im alayas occur at sep arate
tim es of th e y ear an d at d iffe re n t elevations so th a t it is n o t difficult to
separate th e signals sp atially a n d tem porally. In the central A ndes snow
cover rem ain s below the su b p ix e l scale w hile fu rth e r south, from 25°S to
40°S in A rg e n tin a an d C hile, s n o w cover is a p p a re n t a t the highest
elev atio n s.
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Figure 1.40:Histogram of instantaneous rain rate values in the central Andes, showing the distribution of
valid rain rates mostly below 5 mm/hr and their separation from bad scan lines greater than 20 mm /hr.
85
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Accumulated
rain rate (m m /hr)
1993-1994
120-i
100-
:
*
ft
**
80; a . ■*
60- -- 'T -M:
f *
: |
t
it:
40: ssJi1'
'*
I;
l
*
*
if,
200- - ....... .......0
■
—
i.
1000
CO
Os
t
_
2000
3000
i.
s-t
4000
Elevation (m)
Figure 1.41: Elevational distribution of the rain rate values plotted in the sw ath profiles
for the central Andes. These are only those points along the sw aths plotted against the
mean elevation for that sw ath, show ing a clear clustering below 2000 m elevation.
a
*
+ ,-.F-
5000
87
F igure 1.42 sh o w s ac cu m u late d ra in rates fo r th e th ree m o n th s, Ju n e
to A ugust, from 1992-94. O bvious fe a tu re s are the v e ry large area over the
Bay of B engal a n d th e large sn o w c o v e r signal in the n o rth w e ste rn
H im alaya. O f in te re st in this stu d y a re the orographic p eak s w h ich are also
clearly to b e seen a lo n g the so u th e rn H im alayan front.
E x am ination o f the in d iv id u a l satellite passes ag a in show s a
p referred tim e for m axim um ra in ra te s (Figure 1.43). H ig h est accu m u lated
ra in rates o v er the flatter p a rts of In d ia are over the n ig h t, for the ascen d in g
F10 pass (10 p m local time) an d the descending F l l p ass (5 am local tim e).
O rographic peaks o n the other h a n d a re greatest for the ascending F l l pass
(5 p m local tim e). T his situ atio n is re v e rse that o f th e cen tral A ndes a n d
A m azo n b a sin , n o c tu rn a l co n v ectio n is im p o rtan t in th e In d ia n lo w la n d s
w h ile d ay tim e oro g rap h ic p re c ip ita tio n prevails alo n g th e H im alaya.
The S hillong P lateau (91°W , 2 6 °N ) is sufficiently h ig h to force
p re c ip ita tio n o n its so u th ern , w in d w a rd side, form ing a n effective rain fall
sh ield to th e p a rt of th e H im alaya im m ediately n o rth . F igure 1.44 show s a
closer look a t the H im alay an fro n t a n d the locations o f 9 cross-sectional
profiles sim ilar to th o se of the A n d es. For these p ro files a w id th of 50 km .
a n d av erag in g le n g th of 110 km w as u sed . Again, o ro g rap h ic rainfall is
forced at elevations below 2,000 m (Figure 1.45). E specially for the Shillong
p la te a u su c h a b a rrie r clearly p ro d u c e s a rainfall m ax im u m w ell in fro n t of
th e larg er m o u n ta in front. W hile th e T ibetan p la tea u d riv e s the In d ia n
m onsoon, rain fall d o es n o t n ec essarily focus at th e m o u n ta in front. A ny
sm aller to p o g rap h ic b arriers of o n ly 2,000 m. m ig h t h av e b een sufficient in
th e p ast to cap tu re m onsoonal p recip itatio n . Profile 9 show s a p eak along
th e n o rth e a ste rn P ah k ain g Bum ra n g e at 300 km. w h ic h d o e sn 't m a tch th e
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Accum ulated rain rate (m m /h r)
N40'
I
So"!
-N 4 0 c
>30
BMW
N 35°-
N35°
N 30°-
-N 3 0 c
hfi
-.
S
-N 2 5 c
N 25
m
S m
E100°
Figure 1.42: Accum ulated rain rate for three rainy seasons, June to A ugust, 1992-1994 for all F10 and F ll sw aths.
Scattering due to snow is not filtered out and is therefore responsible for the strong signal at high elevations
(ie. - not rain). This does not affect the object of this study w hich are the orograhic peaks along the m ountain front.
00
00
10 AM, F10 desc.
5 AM, F ll desc.
E70°
E80°
E90°
E110°
E100'
I
m
10 PM, F10 desc.
5 PM, F ll asc.
m
El 10°
N40'
itrwwfc
N35°
(m m /hr)
N30'
N25°
Figure 1.43: Accum ulated rain rate for three rainy seasons, June to A ugust, 1992-1994 by satellite pass. Unlike
in the central A ndes, orographic precipitation is strongest in the late afternoon but doesn't continue through
the night as m esoscale-organized convection.
South
c
N orth
H im alayan front —100
Shillong Plateau
U—
»
CC
>
J0J
100 3 r
s
0
100 200 300 400 500
D istance along the Profile (km)
Figure 1.44: H im alayan orographic rainfall maxima and locations of
swath profiles in the following figures. Two profiles are show n near the
Shillong Plateau illustrating how this 2 km. barrier forces m ost rainfall.
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91
A ccum ulated
rain rate (m m /h r)
Elevation (km):
5 r m in-mean-max
-5 0
Profile 1
o
50
100
150
JO
200
250
D istance along profile (km)
43 :r Profile 2
Profile 3
Profile 4
43 ir Profile 5
Profile 6
3
f Profile 9
-5 0
H25
—
300
^0
600
Figure 1.45: Swath profiles o f rain rates an d elevations in the H im alaya
for the rem aining profiles show n in the previous figure. A ll profiles,
including Profiles 7 a n d 8 in the previous figure, indicate th at orographic
precipitation is m ostly forced by elevations below 2 km.
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Accumulated
rain rate (m m /hr)
60 n ---------
1000
2000
3000
Elevation (m)
Figure 1.46: Elevational distribution of the rain rate values plotted in the sw ath profiles
for the Himalaya. These are only those points along the sw aths plotted against the
m ean elevation for that swath. Som e clustering of points b elow 2000 m is still evident.
4000
93
sta n d a rd p ic tu re o f southerly flow im p in g in g alo n g a topographic b a rrie r
a n d d ro p p in g p recip itatio n . It m ig h t be th a t local circulation in this a re a
focuses ra in fa ll o n th e so u th ern p a r t of the B ra h m a p u tra valley. T his
w o u ld b e c o n siste n t w ith the o b serv atio n th a t a n o th e r rainfall p e a k o ccu rs
o n ju s t the o th e r s id e of P ahkaing Bum , at a b o u t 130 km . along p ro file 9. If
large-scale circ u la tio n w ere forcing rainfall in th is a re a it w o u ld b e u n lik e ly
to p ro d u ce it o n ly o n both sides o f the range.
As w ith th e central A ndes, th e larg est accu m u latio n s of o ro g ra p h ic
p re c ip ita tio n a lo n g all the profiles in F igure 1.44 o ccu r below elev atio n s o f
2,000 m eters, th o u g h the sp read to h ig h er elev atio n s is so m ew hat g re a te r
for the H im alay a (Figure 1.46). C o m p arin g th e to p o g rap h ic profiles of
F igures 1.36 a n d 1.45 it can be seen th a t along the profiles for this stu d y
(chosen at th e lo catio n s of larg est o ro g rap h ic p recip itatio n ) the A ndes rise
m u ch m ore s h a rp ly w hile the H im alay as p re s e n t a b ro a d e r foothill re g io n
w ith m ore la n d a re a aro u n d an d slig h tly abo v e 2,000 m.
D iscussion o f resu lts
C lim a to lo g ic a l fea tu res - fillin g in th e g a p s
O nly tw o y e a rs of satellite observations suffice to describe the sp a tia l
d istrib u tio n o f rain fall, a valuable d atab ase for rem o te areas lacking o th e r
rain fa ll m e a su re s. W hile the m e asu rem e n ts ag ree in general w ith e x istin g
clim atologies, sig n ifican t differences exist. The v a lid ity of either m e a su re
m ig h t be e sta b lish e d w ith eith er ad d itio n a l g ro u n d statio n d ata o r satellite
d ata. E specially fo r the d iu rn al v ariatio n s in p recip itatio n it w o u ld be v e ry
im p o rta n t to ex a m in e the life cycle of convective sto rm s. In the case o f
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94
g ro u n d sta tio n s hourly re c o rd in g devices w o u ld b e id eal b u t a d d itio n a l
stations o f a n y k in d seem u n lik e ly in the n ea r fu tu re . A d d itio n a l sa tellite
observations a re m uch m o re lik ely to ad d ress this, especially 3 -h o u rly
v isib le /in fra re d ob serv atio n s fro m the GOES-8 satellite, w h ich fo r S o u th
A m erica is co n v en ien tly lo c a te d over the e a ste rn US. A d d itio n a l
m icrow ave ob serv atio n s b e g a n in 1997 w ith th e la u n c h o f the T ro p ical
Rainfall M e a su rin g M ission (TRMM) w hich in c lu d es tro p ical a n d s u b ­
tro p ical S o u th A m erica in it's o bservations.
S c a tte r in g su rfaces o n th e A ltip lan o
M icro w av e b ack g ro u n d b rig h tn ess te m p e ra tu re s allow a ssessm en t of
areas p ro n e to false scatterin g sig n atu res a n d p ro v id e d in fo rm atio n ab o u t
these surfaces, including th e p resen ce of w ate r on sa lt lakes. The
identification o f p recip itatio n is facilitated b y the relativ e absence of false
scattering sig n a tu re s in th e c e n tra l A ndes, e v e n th o u g h the id e n tific a tio n
of p recip itatio n scattering sig n a tu re s is com plicated b y the extrem e sp atial
h etero g en eity o f the lan d su rfa ce . F urther so u th in th e A ndes a n d in th e
H im alayan re g io n snow c o v e r w as ap p a re n t in the scatterin g sig n a tu re b u t
is easily se p a ra te d from ra in fa ll th ro u g h it's d iffe re n t spatial a n d tem p o ral
d istrib u tio n .
The accu m u lated S S M /I ra in rate estim ates p ro v id e m u c h b e tte r
spatial d e ta il th a n available fro m g ro u n d sta tio n m e asu rem e n ts. P eak s in
the sp a tial d istrib u tio n o f ra in rates show a rain fall m ax im u m a t lo w e r
slopes of th e e a ste rn A n d es, w h e re m o istu re la d e n a ir of the A m a z o n basin
is forced o v er th e to p o g rap h y . These peaks are m a p p e d in g reater d etail
than is av ailab le from p u b lis h e d m aps o r g ro u n d statio n s, b u t n e e d to be
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95
a c cu m u late d over se v eral seasons in o rd e r to d istinguish th e m as true local
m a x im a .
F u tu re im p ro v e m e n ts to these e stim a te s will com e fro m extending
th e reco rd back to 1987, w h e n the first S S M /I instrum ent w a s launched, as
w e ll as including d a ta fro m other sources s u c h as infra-red im ages that can
fill m issin g gaps in th e S S M /I coverage.
D iurnal ra in fa ll m axim a
O rographic rain fall o n the lo w er slo p es of both the A n d e s and
H i m a l a y a s clearly occurs a t preferred tim es. For the A ndes th e tim ing of
th e se m axim a su g g e st th e im portance o f M C C 's in co n trib u tin g to the
su m m ertim e rain fall of th is region. For m id -la titu d e this w a s established
b y Velasco and F ritsch [Velasco a n d F ritsch, 1987] b ut until th is study the
im p o rta n c e of n o c tu rn a l rainfall m ax im a in th e central A n d e s w as
u n k n o w n . The o b se rv a tio n s of this s tu d y a re consistent w ith Velasco a n d
F rits c h ’s o b serv ation th a t MCC's o v er B olivia occur som e 4 h o u rs later
th a n th e ir co u n terp arts in the central p la in s o f the US o r in A rgentina,
th o u g h the SSMI d a ta alo n e is in su fficien t to establish th e ir tim in g any
b e tte r. As m en tio n ed ea rlier, infra-red ob serv atio n s from G eostationary
satellites could be u se d in this regard.
E levation o f m axim u m rain rate
A long the lo w er slopes of th e e a s te rn Cordillera, th e m axim um rain
ra te s occur at ro u g h ly 1 km . elevation (Figure 1.36), as w o u ld b e expected
fo r m oisture la d en air forced to rise o v er th e topography. T h ese peaks are
co n siste n t w ith m ap s p u b lish ed b y th e W o rld M eteorological O rganization
[W M O , 1975]. T hey h av e also been o b se rv e d usin g SSMI d a ta b y Negri
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96
[N egri et al., 1994] alth o u g h no m e n tio n w a s m ade o f th e elevation o f th e
p eak in rain rate a n d it w as only a lim ited aarea in th e c o m e r of their stu d y .
N egri [Negri e t al., 1993] did find a sim ilar p e a k in th e S ierra M adre
O ccidental of M exico centered a t a p p ro x im a te ly 1 km . elevation.
W hile sig n ific an t variations a lo n g th e eastern A n d e s m ay be re la te d
to the single-season d u ratio n of th is stu d y , th e sam e p e a k s appear in b o th
the 1993 and 1994 to tals, suggesting th a t it is a stable pheno m enon, re p e a te d
for a t least these tw o years. The sig n ifican t m o istu re s u p p lie d from the
A m azo n b asin is resp o n sib le fo r su c h h ig h rainfall, w h ic h dim inishes
so u th w a rd in th e a rid climate o f n o rth e rn A rg e n tin a .
R ain fall a n d erosion
Looking a t Figures 1.35 a n d 1.36 an d -considering profiles 3 a n d 4,
som e interesting suggestions can b e m a d e re g a rd in g in te n sity of ero sio n
alo n g the m o u n ta in front. S u sp e n d e d s e d im e n t lo a d s in th e Beni (b etw ee n
pro files 3 an d 4) are m u ch h ig h e r th a n in th e Rio P ilco m ay o further so u th
a t 20°. D ifferences in average rain fall based, on tw o clim ate stations hav e
b een suggested as th e prim ary re a so n for th is [M asek e t al., 1994], W hile th e
d etailed relatio n sh ip betw een rain fall a n d ero sio n in th is o r other areas is
unclear, the h ig h er accum ulated ra in rates o f this s tu d y suggest th at
ero sio n rates m ig h t possibly be e v e n h ig h e r in these a rea s than riv er lo ad s
indicate. For this to be true, the in creased ero sio n in th e se areas w o u ld
eith er be carried in th e rivers b u t u n re p re se n te d for so m e reason or m a y b e
g o in g into sto rag e in the foreland. T oo fe w m e asu rem e n ts of sedim ent
loads or storage ex ist a t present to d ecid e th is for certain.
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97
Sum m ary
S atellite o b serv atio n s se rv e as a v ery u s e fu l com plem ent to lo n g ru n rd n g records of p re c ip ita tio n from g ro u n d sta tio n s, especially in areas
w ith a lo w density of g ro u n d statio n s. B eg in n in g w ith v isib le /in fra re d
o b se rv a tio n s from e a rly G eo statio n a ry satellites, rem o te sensing m e th o d s
h a v e b ee n u sed for e s tim a tin g rain fall over v e ry la rg e areas. For
v is ib le /in fra re d p recip itatio n retrievals it is n e c e ssa ry to average o v e r large
a re a s in o rd e r to o b tain reaso n ab le co m p ariso n s w ith ground sta tio n s. The
in fre q u e n t tem poral s a m p lin g of m ore d ire c t m ic ro w a v e retrievals also has
n e c e ssita te d averages o v e r la rg e spatial areas. In rece n t w ork h o w e v e r the
e m p h a s is h a s been o n p r o d u c in g m e asu rem e n ts fo r com parison w ith
clim ate m odels, typically g rid d e d a t 2.5°x2.5° a n d accum ulated a t m o n th ly
in te rv a ls .
T he p u rp o se of th is s tu d y w as to sh o w th e u tility of m icrow ave
m e a su re s a t the o rig in al s p a tia l reso lu tio n o f th e satellite, accu m u latin g
p re c ip ita tio n retrievals o v er a lo n g er p erio d o f tim e in order to b u ild u p
accep tab le spatial statistics. T his is especially im p o rta n t in rem ote,
m o u n ta in o u s regions w h e re relativ ely s ta tio n a ry orographic p re c ip ita tio n
p e a k s h a v e been u n d e te c te d b y trad itio n al g ro u n d statio n m easu rem en ts or
th e la rg e -a re a satellite retriev als.
A dditionally, th is s tu d y show ed the v a lid ity of precipitation
re trie v a ls over the elev ated su rfaces of the c e n tra l A ndes. Because th e
m ic ro w a v e techniques w e re orig in ally d e v e lo p e d for use over o ceans, an d
w e re th e n later ad a p te d to la n d , less effort h a s b e e n m ad e in e x ten d in g
m ic ro w a v e p recip itatio n retriev a ls. The re su lts o f th is study c o n c ern in g the
m ic ro w a v e b ack g ro u n d of th e central A n d es can b e extended in fu tu re
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
98
w o rk to p a rts of the H im alay as an d T ib etan P lateau . In th o se areas snow
w ill b e a p ro b lem (w here it w as n o t in th e A n d es), b u t c u rre n t w o rk
su g g ests th a t sep aratio n o f surface snow scatterin g from p recip itatio n
sc atte rin g m ay be p ossible w ith the a d d itio n a l in stru m e n ts o n the sam e
satellite p latfo rm as the SSMI [Bauer an d G ro d y , 1995].
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without perm ission.
99
BIBLIOGRAPHY
A ceitu n o , P., O n th e F unctioning o f th e S o u th ern O scillatio n in the S o u th
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Ulaby, F.T., R.K. M oore, and A.K. F ung, M i c r o z o a v e R e m o t e S e n s i n g , 456
pp., A ddison-W esley, R eading, 1981.
Velasco, I., M C C s in S outh A m erica. A rev iew ., 3 ' r d I n t e r n a t i o n a l
C o n feren ce on S o u th ern H e m isp h e r e M e te o ro lo g y a n d
O c e a n o g r a p h y , 280-283,1989.
Velasco, I., a n d J.M. Fritsch, M esoscale C onvective C o m p lex es over South
A m erica, in S e c o n d I n t e r n a t i o n a l C o n f e r e n c e o n S o u t h e r n
H e m i s p h e r e M e t e o r o l o g y , e d ite d b y AMS, pp. 453-456, A m ericana
M eteorological Society, W ellin g to n , N ew Z ealand, 1986.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
106
V elasco, I., and J.M. F ritsch , M esoscale C onvective C om plexes in the
A m ericas, J G R , 9 2 (D8), 9591-9613,1987.
Virji, H ., A P relim in ary S tu d y o f S u m m ertim e T ro p o sp h eric C irculation
P attern s over S o u th A m erica E stim ated from C lo u d w in d s, M W R ,
1 0 9 (March, 1981), 599-610,1981.
Virji, H ., a n d V.E. K ousky, R egional a n d global aspects of a low latitude
fro n tal p e n e tra tio n in A m azo n as a n d associated tropical activity,
3 'r d In tern a tio n a l C o n feren ce on S o u th ern H e m isp h e re
M e t e o r o l o g y a n d O c e a n o g r a p h y , 215-220,1989.
V ose, R.S., R.L. S chm oyer, P.M . S teurer, T.F. P eterson, R. H eim , T.R. Karl,
a n d J.K. E ischeid, T he G lobal H istorical C lim atology N etw ork: Long­
te rm m o n th ly te m p e ra tu re , p recip itatio n , sea level p re ssu re , and
statio n p ressu re d a ta ., C arb o n D ioxide Inform ation A nalysis Center,
O ak Ridge N atio n al L aboratory, 1992.
W allace, J.M., D iu rn al V a ria tio n s in P recip itatio n a n d T h u n d e rsto rm
F requency o v er th e C o n te rm in o u s U n ited States, M o n t h l y W e a t h e r
R e v i e z o , 103,406-419,1975.
W ay len , P.R., a n d C .N . C a v ied e s, A n n u al an d Seasonal F lu ctu atio n s of
P recip itatio n a n d S tream flo w in the A concagua R iver Basin, Chile,
J o u r n a l o f H y d r o l o g y , 1 2 0 (1990), 79-102,1990.
W eng, F., R.R. F erraro, a n d N .F. G rody, Global p recip itatio n estim ations
u sin g Defense M eteorological Satellite P rogram F10 an d F l l special
sen so r m icrow ave im a g e r d ata, J G R , 9 9 (D7), 14,493-14,502,1994.
W en tz, F.J., U ser's M a n u a l, S S M /I A n ten n a T em p eratu re T apes, Revision
1, Rem ote S en sin g S ystem s, 1991.
W e m ste d t, F.L., W o r l d C l i m a t i c D a t a , Clim atic D ata Press, Lem ont, 1972.
W ilh eit, T., an d e. al, A lg o rith m s for the R etrieval of R ainfall From
Passive M icrow ave M easu rem en ts, R e m o t e S e n s i n g R e v i e z u s , 1 1 ,
163-194,1994.
W illm o tt, C.J., S.M. R obeson, a n d J.J. F eddem a, E stim ating C ontinental and
T errestrial P re c ip ita tio n A verages From R ain-G auge N etw orks,
I n t e r n a t i o n a l J o u r n a l o f C l i m a t o l o g y , 1 4 (4), 403-414,1994.
W M O , C l i m a t i c A t l a s o f S o u t h A m e r i c a , 28 pp., W M O, G eneva, 1975.
W M O , C l i m a t i c A t l a s o f A s i a , 28 p p ., WMO, G eneva, 1981.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
107
W u , R., an d J.A. W ein m an , M icrow ave R adiances from P recipitating
C louds C o n tain in g A spherical Ice, C om bined Phase, an d Liquid
H ydrom eteors, J G R , 8 9 (D5), 7170-7178,1984.
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CH A PTER 2:
CO N STRA IN TS O N TH E RAINFALL IN P U T S TO LAKE TITICACA
DERIVED F R O M SATELLITE PASSIVE M ICR O W A V E OBSERVATIONS
A bstract
S atellite p assiv e m icro w av e o b se rv a tio n s fro m th e Special S en so r
M icrow ave Im a g e r (SSM /I) are u se d to estim a te tw ice-daily rain rates o v e r
the Lake T iticaca basin. E stim ates com piled fo r th e rain y season, D ecem ber,
1994 th ro u g h A p ril, 1995, a re in te g ra ted over th e Lake Titicaca d rain ag e
b asin to y ield estim ates of th e rain fall in p u t to th e w ater b u d g et of th e lake.
These estim a te s track the a n n u a l rise in lake le v e l m ore closely th a n
records of 24 h o u r p recip itatio n to tals av ailab le fro m the only two g ro u n d
stations in th e area. A lth o u g h te m p o ra l sa m p lin g is still insufficient for
m easuring a ll rain fall in p u t, resu lts in te g ra te d o v e r the b asin are b e tte r
th an the e s tim a tio n of b a sin rain fall from g r o u n d statio n s alone.
In tro d u ctio n
The A ltip lan o com prises a closed d ra in a g e b asin feeding Lakes
Titicaca an d P oopo. Lake Titicaca ap p ro x im a tely integrates the hydrological
balance in th e n o rth e rn A ltip lan o b asin (F ig u re 2.1). In p u ts to the lake
com e from s n o w an d ice m e ltw a te r from th e s u rro u n d in g A ndean
cordilleras, g ro u n d w a te r d isch arg e, and p re c ip ita tio n o v er su rro u n d in g
drainages a n d especially o v er th e lake itself. L osses are dom inated b y
ev ap o ratio n fro m the lake [K essler an d M o n h eim , 1968; Roche et. al, 1991]
w ith only a sm a ll surface d isc h arg e via the D e sa g u a d e ro River s o u th to
Lake Poopo. D u rin g the five year (1957-1961) d u ra tio n of the study b y
108
perm ission of the copyright owner. Further reproduction prohibited without p erm ission.
109
70°W
Lake^fg
T itic a c a
100
0
100
200
300
I i i i i i i i i i I__________ _L___________I___________ I K ilometers
Lakes
P erm an en t snow an d ice
—-
International borders
Rivers
Figure 2.1: C entral A ltiplano and Lake Titicaca. Most of the drainage
basin for Lake Titicaca (pastel colored regions around the lake) lies in
Peru. Juliaca an d La P az are the only clim ate stations in the area w ith
readily available records for the 1994-95 ra in season.
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110
av erag e discharge v ia th e D esaguadero R iver was 8 m 3/ s , w h ic h is 31 m m
o f lak e level lo w erin g o v e r a year, w h ile th e evap o rativ e loss fro m the lake
w as n e a rly 50 tim es g re a te r th an this w ith a n equivalent 1.48 m o f lake
level low ering. Lake le v el from 1939 to 1995 show s the a n n u a l cycle as w ell
as in ter-an n u al v a ria tio n s (Figure 2.2). N o tab le features in c lu d e the
e x te n d e d drops in lev el d u rin g the e a rly 1940's and late 1980's, b o th periods
of e x te n d e d El N in o e v e n ts. The ex trem e El N ino ev en t o f 1983 resu lted in
the co m plete absence o f th e an n u al rise in lake level for th a t y e a r, hence
the continuos dro p fo r tw o consecutive years. In 1984 an d 1986 th e lake
ex p erien ced u n u su a lly h ig h increases in level due to p recip itatio n .
A nalysis of th e a v e ra g e rates of rise an d fall in lake level o v er the
le n g th of record (F igure 2.2) show s the relativ ely constant seaso n al
lo w erin g during this p e rio d of record. T his analysis w as m ad e b y separately
p lo ttin g each rising a n d falling p art of th e y ear and g rap h ically selecting
a n d recording the lak e le v el change a n d d u ra tio n of each risin g a n d falling
p erio d . The typical lo w e rin g rate of 3 m m / d a y is the sam e as th e average
lo w erin g rate d u rin g M a y th ro u g h S ep tem b er obtained b y K essler and
M o n h eim [Kessler a n d M onheim , 1968] fo r the years 1957-1961, w hich
balances average e v a p o ra tio n of 4.3 m m /d a y and discharge 0.10 m m /d a y
a g a in st com bined p re c ip ita tio n over th e lak e and river inflow o f 1.4
m m /d a y .
As in the A m a z o n b asin , co n v ectiv e precip itatio n p re d o m in a te s on
the A ltiplano. This co n v e ctio n is p ro b ab ly enhanced b y local topographic
featu re s an d persists w h e n in d iv id u al convective cells are o rg a n iz e d into
la rg e r convective sy ste m s. D eep co n v ectio n d u rin g the a u s tra l sum m er
c o n trib u tes m ost o f th e a n n u a l p re c ip ita tio n , w ith sm aller a m o u n ts of
R eprodu ced with perm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
I ll
800
Level of L ak e Titicaca
E
“2 600
CD
>
<
D
eo
400
X
CO
■g 200
co
£3
£
c
1940
cs
“O
E
o
o
*5
1950
1960
1970
1980
1990
1980
1990
2.0
R a te s of rise a n d fall, Lake T iticaca
1.5
§> 1-0
D)
§
0.5
CO
u.
"O
c03 0.0
o
Q_
O)
0.5
CO
CC - 1 .0
1940
1950
1960
1970
Figure 2.2: The top curve is the level of Lake Titicaca from 1938 to 1995.
Besides secular changes in lake level, the higher frequency annual cycle
is obvious. Lake levels usually start to rise around the beginning of
December continuing until about A pril. The rest of the year they steadily
fall. The low er curve show s average rates of rising an d falling of lake level
as determ ined graphically from the rising and falling p arts of the annual
cycle - d eterm in ed separately for each season and taking account of
different onset an d end dates of the rainy season for each year. C onstant
rates of falling each year indicate steady evaporation, the prim ary
m echanism for w ater loss from the lake.
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112
w in te r p re c ip ita tio n associated w ith cold fronts fro m th e south. The
ep iso d ic n a tu re o f d eep co n v ectio n o n the p la te a u gives w idely v a ry in g
p recip itatio n field s an d ra d ia tio n balances over tim e scales of several d ay s.
D etailed k n o w le d g e of the s p a tia l a n d tem poral d is trib u tio n of rain fall in
th is area is lim ite d because o f th e sm all n u m b er o f clim ate g ro u n d sta tio n s
a n d a p re d o m in a n c e of m o n th ly records. L im ited s tu d ie s have in d ic a te d
th a t Lake T iticaca itself ex p erien ces the highest ra in fa ll in it's co n trib u tin g
b asin [Schw erdtfegger, 1976].
S S M /I Data
The W E T N E T p ro g ra m a t N A SA 's M arsh all S pace Flight C e n te r
(MSFC) w as fo rm e d to facilitate th e application of p assiv e m icrow ave
o b serv atio n s m a d e b y the S p ecial Sensor M ic ro w a v e /Im a g e r (SSM /I). This
in stru m e n t h a s flo w n aboard satellites of the D efense M eteorological
Satellite P ro g ra m (DMSP) fro m Ju ly , 1987 to the p re se n t. The polar o rb itin g
DM SP satellites sam ple h ig h la titu d e s daily b u t lo w la titu d e s
d isc o n tin u o u sly 3 to 4 days o f o bservation sep arated b y 3 to 4 days w ith o u t
o b serv atio n . T h e F l l satellite h a s a n orbit w h ich is su n -sy n ch ro n o u s w ith
e q u a to r cro ssin g tim es at a p p ro x im a tely 5 AM a n d 5 PM local tim e. Because
convection p e a k s in the a fte rn o o n , after solar h e a tin g h as taken place, th e
aftern o o n p a s se s record g re a te r a n d m ore freq u en t ra in rates.
The S S M /I in stru m e n t se n ses m icrow ave ra d ia tio n a t 4 freq u en cies,
19.35, 22.3, 37.00 an d 85.50 G H z. M easurem ents are m a d e a t dual
p o la riz atio n a t all frequencies ex cep t 22.3 G Hz w h ic h m easures o n ly
vertical p o la riz a tio n . Spatial re so lu tio n ranges fro m 15 k m x 13 k m for th e
85.50 G H z b a n d s to 69 km x 43 k m for the 19.35 G H z b an d s. Data from the
with perm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
113
S S M /I in stru m e n t ab o ard the F l l satellite for D ecem ber 1,1994 to A pril 22,
1995, w as o b tain ed from th e D ata A ctive A rchive C enter (DAAC) a t MSFC
th ro u g h th e W ETNET p ro g ra m fo r this w ate r b alan ce stu d y w h ich
ex am in es one season in d etail to assess the b asic u tility of S S M /I d a ta for
w a te r b alan ce estim ates. T he sam e m eth o d c a n b e ex ten d ed to in clu d e
o th e r satellites an d longer p e rio d s w h ich w as n o t d o n e here since these
d a ta w e re n o t available fro m th e W ETNET p ro g ra m at the tim e o f this
stu d y . T he lo w er reso lu tio n b a n d s are o v ersa m p led to the 85 G H z
re so lu tio n in o rd e r to m a in ta in th e full re so lu tio n o f th a t b a n d , w h ic h
su p p lie s the physical scatterin g sig n atu re u s e d in identifying p recipitation.
It is p ossible therefore th a t sp a tial v ariation reflected in the 85 G H z b a n d
w ill n o t b e rep resen ted b y the lo w er frequency m easurem ents.
Id e n tify in g P recipitation
The basis for id en tificatio n of p recip itatio n is the scattering of
terrestrial rad iatio n at 85 G H z b y ice particles a t the top of precipitating
clo u d s. L ow er frequencies are le ast affected b y scattering. We use the
m eth o d d evised by G rody [G r o d y , 1991], w h ere a scattering index is defined
as
SI(85v) = F - T v(85) ,
(2.1)
w h e re T v (85) is the vertically p o la riz ed b rig h tn ess tem perature a t 85 G H z
a n d F estim ates the n o n sc atte rin g c o n trib u tio n o f th e 85 GHz
m e a su re m e n ts a n d is d e fin e d as
F = 450.2 - 0.506*TV(19) - 1.874*TV(22) + 0.00637*TV(22)2
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
(2.2)
114
T his eq u atio n represents the b e s t fit to the 85 G H z brightness
te m p e ra tu re for nonscattering c o n d itio n s over b o th la n d and w ate r u sin g
the lo w er frequency b an d s. SI(85v) is co n v erted to average rain rate
th ro u g h th e eq u atio n
Rain Rate (mm/hr) = -1.7 + 0.29 * SI(85v),
(2.3)
w hich w as developed b y W eng [W eng e t al., 1994] u sin g radar rainfall d a ta ,
surface g au g e m easurem ents an d S S M /I d a ta over Jap an and the U n ited
K ingdom . T he screening proced u re as described in G ro d y (1991) is used,
w hich id en tifies the presence of sc a tte rin g by SI(85v) > 10. This th resh o ld
for d e te rm in in g rain from the sc a tte rin g index p rev en ts m isclassification of
m ixed pixels a t the lake boun d ary .
F ig u re 2.3 show s rain rates su m m e d over 1993 an d 1994 d eriv e d
from sen so r co u n t data from the F10 a n d F l l satellites in a later stu d y
w hich is discussed in C hapter 1. B ecause the lake surface has cold
b rig h tn ess tem p eratu res, it is d ifficu lt to resolve an y b u t the strongest
co n v ectio n o v er the lake. D etection o f lesser rainfall over the lake w o u ld
in v o lv e lo w e rin g the th resh o ld a t w h ic h rainfall is d eterm ined from the
scatterin g index and w o u ld then create false signatures of rainfall a t the
b o u n d arie s of the lake.
T he satellite estim ates d o h o w e v e r capture rain fall in the
s u rro u n d in g basin, these long te rm accu m u latio n s sh o w an im p o rta n t
p eak to th e sou th east of the lake. F ig u re 2.4 breaks d o w n the 1993-94 totals
into each satellite pass. O nly the a sc e n d in g F10 pass accum ulates rainfall
over the lake itself, consist w ith th e o b serv atio n s describing a n o ctu rn al
convergence over the lake itself d u e to d rain ag e from the adjacent valleys
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115
L ake s h o re
71 °W
70°W
K ilom eters
B asin B oundary
69°W
68°W
A ccum ulated rainrate
m m /h r
Figure 2.3: S SM /I derived precipitation rates for tw o fu ll years,
1993 an d 1994. Because of the low brightness tem p eratu re of the
lake surface, SSM /I derived rainrates are likely u n d erestim ated
over the lake surface. While over-ocean algorithm s m ig lit be used,
they w o u ld result in significant errorsfor the m ixed shoreline pixels
w hich m ake u p m ost of the lake region. The tw o peaks aro u n d the
sou th ern en d of the lake suggest m o u n tain /v alley circulations in
those areas contributing to larger total accum ulated rairuates.
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ion of the copyright owner. Further reproduction prohibited without permission.
70°W
70°W
68°W
68°W
1
A ccu m u lated
rain rate
F10 D escen d in g (10 AM local tim e)
F l l D escen d in g (5 AM local tim e)
70°W
70°W
68°W
68°W
a>
m m /h r
F l l asc en d in g (5 PM local tim e)
F10 A scen d in g (10 PM local tim e)
Figure 2.4: SSM /I derived precipitation rates for tw o full years, 1993 and 1994, separated according to
satellite pass (time of day). The tw o southern peaks are seen to occur at different tim es - the southeastern
peak in the late m orning and the southw estern peak in the evening.
117
[Schw erdtfegger, 1976]. This also seem s to favor th e so u th e astern p a r t o f th e
basin. S chw erdtfeger [Schw erdtfegger, 1976] also describes intense h eatin g
d u e to clear early m o rn in g skies. T he d escen d in g F10 p a ss in Figure 2.4
sh o w s th a t co n v ectio n d u rin g th is tim e is fav o red s o u th e a st of the lake.
E stim a tin g ra in fa ll in p u t to the ba sin
To exam ine in d etail the rain fa ll in p u t to th e L ake Titicaca b a sin
o v er a single ra in y seaso n rain -ra te estim ates from th e F l l satellite are
su m m e d over th e b asin for D ecem ber 1,1994, th ro u g h A p ril 22,1995,
(Figure 2.5) to fo rm a ru n n in g to ta l of accu m u lated p recip itatio n o v er th e
seaso n (Figure 2.6). A lgorithm re su lts in m m /h r are m u ltip lie d b y 12 h o u rs
(tim e b etw een passes) an d scaled to sp re a d the rain fall in each pixel (156
km.2) u n ifo rm ly o v er the lake a rea (8,167 km^). In c lu d e d in the ru n n in g
to tal is a co n stan t loss of 2.9 m m /d a y as d eterm in ed fro m Figure 2.2. The
m e asu red lake level is also p lo tte d for co m p ariso n (solid black line).
O bservations from tw o g ro u n d sta tio n s n e a r the b a sin (Figure 2.1) are also
su m m e d an d s h o w n o n the sam e p lo t in o rd e r to e v a lu a te their
effectiveness in c a p tu rin g the sam e w e a th e r system s w h ic h contribute
w ate r to the Lake Titicaca d rain ag e b asin , alth o u g h d a ta from only these
tw o stations is lik ely insufficient to a d e q u ate ly reco rd all basin in p u t since
th e ir separation, 240 km , and th e scale of th e basin, ro u g h ly 400 b y 150 km
is m u ch larger th a n typical size scales of rainfall sy stem s w hich are w ell
u n d e r 100 km. T h ere is no reaso n ab le b asis for c o n v e rtin g these p o in t to tal
m easurem ents to area averages since th e y are so far a p a rt, so they are scaled
to ro u g h ly m atch en d p o in ts in th e lake level record. T he SSM /I d e riv e d
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
118
K ilom eters
0
100
200
L ake sh o re
B asin B o u n d a ry
° utlet to th e Rio D e s a g u a d e ro
Figure 2.5: O utline of the Lake Titicaca basin a n d the lake shoreline
(shaded relief background) w ith the 85 GHz vertically polarized
SSM /I no-rain background displayed in color a t its spatial resolution
of 12.5 km. The locations of the only two climate stations w ithin o r
near the basin o r show n. These tw o stations alone are inadequate
for capturing rainfall events around the entire basin.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
119
Lake Titicaca Basin, SSM/I derived rainfall
and Ground Observations
400
m easu red lake level
380
380
oE
©
> 360
©
—i
CD
b oth ground stations
(Juliaca and La Paz) / ^
1360
SSM /I derived level !
©
©
_l
340
La Paz only
Juliaca only
l
320
5 121926 2 9 162330 6 1 3 2 0 2 7 6 132027 3 1017
D ecem ber
1994
Jan u ary
February
March
320
April
1995
Figure 2.6: The solid b lack curve represents the m easured changes in lake
level over the 1994-95 rain y season. The s o lid cyan curve starts at the
m easured lake level a t the beginning of the season, is steadily decreased
by the rate of low ering from Figure 2.2 (2.9 m m /d a y ) an d increases b y the
accum ulation of rainfall over the con trib u tin g basin as estim ated by SSM /I
rain rates from the F l l satellite, m ultiplied "by 12 hours (time betw een
passes) to convert to rainfall volum e and in stantaneously d istributed to
the lake. Since this sim ple m odel doesn't acco u n t for ro u tin g o r storage
the m easured actual lake level is expected to be m uch sm oother than the
coarse an d sim ple w ate r balance. The d a s h e d and dotted curves represent
the accum ulation of rainfall u sin g the Juliaca and La Paz g ro u n d stations
both separately and su m m ed together, an d arbitrarily scaled for
com parison w ith the o th er curves since th e se two stations alone are
insufficient for determ ining areally av erag ed rainfall. They are show n
for com parison only to gauge their effectiveness in cap tu rin g rainfall
events w h ich contribute to lake level rise.
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120
ra in fa ll in p u t to th e b a s in follow s la k e level rise over th e ra in y season
m o re closely th a n th e tim e e v o lu tio n o f the g ro u n d s ta tio n m e asu rem e n ts.
O n ly d u rin g Ja n u a ry 26-29 is there a p erio d ic gap in S S M /I coverage w hich
fails to capture ra in fa ll over the b a s in w h ich the g ro u n d sta tio n s do record.
A lth o u g h the sa te llite estim ates h a v e occasional o b se rv a tio n a l gaps, they
still p ro v id e an a d e q u a te m easure o f b a s in input to th e lak e w h ich follow s
th e tim e ev o lu tio n o f lake level b e tte r th a n sparse g a u g e s as sh o w n in
F ig u re 2.6.
This in te rp re ta tio n is s u p p o rte d b y Seed and A u s tin [Seed and
A u stin , 1990] w h o d escrib e the v e ry la rg e variability a n d in term itten cy of
rain fall fields in sp a c e a n d tim e, d e te rm in in g a co rrela tio n le n g th of 14 k m
fo r d aily convective rain fall an d th a t m o n th ly rainfall d e c o rre la te s at
distances of 20 km o r greater. E stim ates of clim atological a re a lly averaged
rain fall from g ro u n d statio n s re q u ire averaging over s e v e ra l rainfall
ev e n ts an d n e tw o rk s o f stations m u c h d en ser than th e tw o statio n s used
h ere. Bellon an d A u s tin show th a t sa te llite rainfall e s tim a te s are better
th a n in terp o latio n s fro m gauge e s tim a te s in areas w h e re th e n earest g au g e
is m o re than 40 k m a w a y [Bellon a n d A ustin, 1986].
The in su ita b ility of co rrelatin g rainfall gauge d a ta w ith satellite
rain fall estim ates is illu strate d in F ig u re 2.7 which c o m p ares tw o ground
sta tio n records w ith th e SSM /I p ix els corresponding to th o se g round
sta tio n positions. A s th e size of th e re g io n is increased fro m 1 pixel to a 5x5
p ix el area cen tered o n each station, th e S SM /I rain ra te p ic k s u p more
v alues. The S S M /I m easu rem en ts a re aliased because th e y a re capturing
o n ly u p to 4 s n a p s h o t s of rainfall ra te each day, p re c lu d in g close correlation
w ith daily g ro u n d s ta tio n m e asu rem e n ts since d e c o rre la tio n tim es of
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
Ground station 24 hour totals (in.)
One pixel box centered on Juliaca, Peru
w
w
3
2.5
75 2.0
solid lines - ground station totals
^
(all plots)
100 §
c
w
3
o'
50 3
(D
V)
1
20 5"
5oai
15§(0
3
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■C
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'28 5 12 19 26 2 9 16 23 30 6 13 20 27 6 13 20 27 3 1017
April
December January
February March
1994
1995
150 —
w
w
o
dashed lines - SSM/I rain rates
\ (all plots)
0
100 §
c
(!)
3
01
5'
50 3
1
.\i\J
28 5 12 19 26 2 9 16 23 30 6 13 20 27 6 13 20 27 3 10 17
December January
February March
April
1994
1995
!\ l X
28 5 12 19 26 2 9 16 23 30 6 13 20 27 6 1320273 10 17
December January
February March
April
1994
5 x 5 pixel box centered on Juliaca, Peru
Ground station 24 hour totals (in.)
One pixel box centered on La Paz, Bolivia
150 5-
5w
>
?
0 I
1995
5 x 5 pixel box centered on La Paz, Bolivia
2.5
20 5"
5a
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.,u
28 5 12 19 26 2 9 16 23 30 6 13 20 27 6 13 20 27 3 10 17
December January
February March
April
1994
1995
Figure 2.7: Satellite rainrates correlate poorly w ith daily ground station data since ground stations record total
accum ulation over 24 hours only at a point and in this case are very sparse (tw o stations separated by 240 km),
Correlation w ith ground station point totals im proves only w ith larger grids (2.5°) over longer tim es (months).
Satellite estim ates o f areally averaged rainfall are more accurate than raingauge netw orks w hich are this sparse.
122
rainfall are on th e o r d e r o f 2 to 3 hours [Seed a n d A ustin, 1990]. If
significant b u t s h o rt-liv e d rainfall events o ccu r a t the g au g e stations
b etw een satellite p a sse s th e n they will n o t b e cap tu red b y th e satellite, b u t
do n o t alw ays re p re s e n t significant in p u ts to the Lake Titicaca drainage
b asin either. If th e sy ste m s are long-lived a n d therefore contribute
significant w ater to th e b a s in then they w ill b e cap tu red b y the satellite
an y w ay w h ich ca n e s tim a te rain rate of the en tire raining area. Figure 2.7 is
show s the in su itab ility o f correlating g au g e d a ta w ith satellite rain
estim ates o v er su c h s h o r t tim e scales w h ile F ig u re 2.6 sh o w s th a t ev en the
S S M /I overpasses o f o n ly u p to four o b serv atio n s per d ay cap tu re rainfall
events m ore c o n siste n tly th a n the two g a u g e stations at Juliaca an d La Paz.
In this s tu d y o n ly a few rain system s o v er the Lake Titicaca basin in
m id D ecem ber a n d la te Ja n u ary are not c a p tu re d by the S S M /I coverage,
ad d itio n al satellite p a s se s a n d hybrid tech n iq u es using in frared satellite
observations can g re a tly im p ro v e this [H u ffm an et al., 1995]. The g ro u n d
statio n d ata is too sp a rs e for capturing th e sp a tia l v ariation of precipitation
an d d aily totals d o n ’t a llo w estim ation o f th e ev o lu tio n o f rainfall system s
over the course of a d a y , b o th of which are b e tte r ad d ressed w ith m ultiple
satellite sn ap sh o ts o f rain fall each day.
D iscussion and C o n clu sio n s
W hen in te g ra te d o v e r the drainage b a s in co n trib u tin g to Lake
Titicaca, the a c c u m u la te d S S M /I rain ra te estim ates m ore closely follow the
rising, w et season la k e level th an is possible from sparse g ro u n d stations
alone, d u e to th e in c lu s io n of rainfall in p u ts in u n in stru m e n te d areas of
the basin. The estim a te s can b e im proved, especially for th e few m issed
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123
rain fall events, b y th e b e tte r te m p o ra l coverage of n e w e r in stru m e n ts su c h
as o n th e Tropical R ainfall M e asu rin g M ission (TRMM) sa tellite an d by
in c lu d in g data from o th e r sources s u c h as infra-red (IR) im ag es that can fill
m issin g gaps in the S S M /I coverage. E ven in this s tu d y w h ic h is lim ited to
o n ly tw o satellites th e S S M /I in stru m e n ts consistently c a p tu re m ore
rainfall events w h ich co n trib u te to th e Lake Titicaca w a te r balance th an th e
n e a re st tw o gauge statio n s. As the rain fall in p u t to th e L ake Titicaca b asin is
fu rth e r refined b y in c lu d in g IR ra in fa ll estim ates a n d d a ta from the TRMM
satellite, it will m ake possible th e co n stra in t of o th er in p u ts to the lake,
su ch as snow m elt, w h ich d e te rm in e th e w ater b alance o f th e Lake Titicaca
system .
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124
BIBLIOGRAPHY
Bellon, A ., a n d G.L. A ustin, O n th e Relative A ccu racy o f Satellite a n d
R a in g ag e Rainfall M e a su re m e n ts o v er M id d le L atitu des d u r in g
D ay lig h t H ours, J o u r n a l o f C l i m a t e a n d A p p l i e d M e t e o r o l o g y , 25,
1712-1724,1986.
G rody, N .C ., Classification of S n o w Cover a n d P recip itatio n U sing th e
S p ecial Sensor M icrow ave Im ager, J G R , 9 6 (D4), 7423-7435,1991.
H u ffm an , G.J., R.F. A dler, B. R u d o lf, U. S chneider, a n d P.R. K eehn, G lo b al
P re c ip ita tio n E stim ates B ased on a T ech n iq u e for C om bining
S atellite-B ased E stim ates, R aingauge A n aly sis, a n d NW P M o d e l
P re c ip ita tio n In fo rm atio n , J o u r n a l o f C l i m a t e , 8 , 1284-1295,1995.
Kessler, A ., a n d F. M onheim , D er W a sse rh a u sh alt d es Titicacasees n a c h
N e u e re n M essergebnissen, E r d k u n d e , X X I I , 275-283,1968.
Roche, M .A ., J. Bourges, J. C o rtes, a n d R. M attos, C lim atologia e h id ro lo g ia
d e la cuenca del lago T iticaca, in E l L a g o T i t i c a c a , edited by ORSTOM ,
p p . 83-104,1991.
Seed, A ., a n d G.L. A ustin, V a ria b ility of S u m m er F lo rid a Rainfall a n d its
Significance for the E stim atio n of Rainfall b y G ages, R adar, a n d
S atellite, J G R , 9 5 (D3), 2207-2215,1990.
S chw erdtfeger, W., H igh th u n d e rs to rm freq u en cy o v e r the su b tro p ic al
A n d e s d u rin g the su m m e r; cause a n d effects, in C l i m a t e s o f C e n t r a l
a n d S o u t h A m e r i c a , e d ite d b y W. Schw erdtfeger, pp. 192-195,
E lsevier, A m sterdam , 1976.
W eng, F., R.R. Ferraro, an d N .C . G rody, G lobal p recip itatio n estim a tio n s
u sin g D efense M eteorological Satellite P ro g ram F10 and F l l sp ecial
se n so r m icrow ave im a g e r d ata, J G R , 9 9 ( 0 7 ) , 14,49-14,502,1994.
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CHAPTER 3:
FRACTAL FLOOD STATISTICS
A b stra ct
This ch ap ter rep rese n ts w ork w h ic h w as done together w ith Dr. D. L.
T u rc o tte a t C o rn ell U n iv ersity . The o rig in a l fram in g of the p ro b lem ,
d iscu ssio n of H u rs t a n d the first look a t th e benchm ark statio n s is d u e to
D r. T urcotte. T here w a s an evident n eed h o w e v e r to apply this analysis to
m a n y m o re o b se rv a tio n s th an the o rig in a l few b enchm ark statio n s.
W o rk in g w ith Dr. T u rc o tte I ad o p ted th is fram ew o rk an d ap p lie d it to the
m u c h larg er d a ta se t fo r the United States a s described below , w ith new
p lo ts for ev a lu atin g th e effectiveness o f th is technique w ith so m u c h d ata.
This w o rk also a p p e a rs in a conference p ro ceed in g s [Turcotte a n d H aselton,
1995].
A v ariety of em p irical statistical d istrib u tio n s have b een u se d for
flood-frequency forecasting. The objective is to extrapolate historical
reco rd s in o rd er to q u a n tify the flood h a z a rd . In the U nited S tates the log
P earso n type HI h as b e e n adopted for u se. A variety of com plex n a tu ra l
p h e n o m e n a have b e e n sh o w n to obey fra c ta l extrem e valu e statistics,
exam ples include ea rth q u ak es an d v o lcan ic eruptions. It is the p u rp o se o f
this p a p e r to exam in e w h eth er fractal sta tistics can be ap plied to floods.
The flood in te n sity factor F is in tro d u c e d as the ratio of th e ten-year
flood to the o ne-year flood. If floods o b ey fractal statistics th en F is also the
ratio of the 100-year flood to the ten y ear flo o d an d the ratio of th e 1000-year
flood to the 100-year flood. In order to te s t th e validity of the fractal flood
frequency h y p o th esis w e consider fractal fits to 41 year records for 1009
125
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126
USGS stream flow statio n s th a t are u n affected by flood co n trol projects,
g o o d fits are o b ta in e d . C o n sisten t reg io n a l variations in th e flood in ten sity
factor are also fo u n d . The physical ju stificatio n for the applicability of
fractal statistics is th a t river flows can be ap p roxim ated b y fractional
B row nian w alks w h ic h are kno w n to ex h ib it fractal ex trem al statistics.
In tro d u ctio n
V arious g eo statistical d istrib u tio n s have been a p p lie d to historical
flood-frequency reco rd s. Examples in clu d e pow er-law (fractal), log norm al,
gam m a, G um bel, log G um bel, H azen a n d log P earson ty p e III. A
com parison of th e se d istrib u tio n s (except the pow er-law ) w as given by
Benson [Benson, 1968] for ten b en c h m ark stream flow statio n s. This
com parison w as e x te n d e d to include th e pow er-law d istrib u tio n by
T urcotte and G reene [Turcotte an d G reene, 1993]. O n th e basis of such
com parisons the log P earson type III w as ad o p ted as th e official U.S. basis
for flood-frequency forecasting [Council, 1981]. T urcotte a n d Greene
[Turcotte an d G reene, 1993] show ed th a t the pow er-law (fractal) d istribution
consistently estim a tes a m ore severe 100-year (or lo n g er period) flood th a n
the log Pearson ty p e III distribution. H ow ever, on th e b asis of ten stations it
is clearly im possible to establish th e v a lid ity of one d istrib u tio n over
another. The p u rp o se of this p ap er is to stu d y the v alid ity of the pow er-law
d istrib u tio n on th e basis of data from 1009 stations.
Before co n tin u in g , a brief d iscu ssio n of the p h y sical basis for a
pow er-law d istrib u tio n w ill be given. T he flow in a riv e r can generally b e
co n sid ered a tim e series; the extrem e v alu es in the tim e series constitute
floods. H ow ever, d efin in g a flood is n o t a trivial p ro b lem . The basic
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127
q u estio n is w h eth er tw o p ea k s in the time series c o n s titu te two
in d e p e n d e n t floods o r are p a r t o f th e sam e flood. T h e an n u al flood is
d efin ed to be the m ax im u m flo w d u rin g a w ate r y e a r. B ut d urin g m a n y
w ater years the 2nd (or 3rd) statistically in d e p e n d e n t flood may b e la rg e r
th a n other yearly floods. W e w ill consider floods to b e statistically
in d e p e n d e n t if they are se p a ra te d b y m ore th a n tw o m o n th s.
A basic question is w h e th e r there is a p h y sic al b asis for a n a ly z in g the
flood-frequency problem . In p artic u lar, w hether fra c ta l statistics w o u ld b e
expected to be applicable. In d icatio n s that such a n a p p ro a c h m ight be
applicable comes from the w o rk o f H u rst [H urst, 1951; H u rst, 1956]. H u r s t
sp e n t his life stu d y in g the flo w characteristics o f th e N ile and in tro d u c e d
the rescaled range (R /S) analysis [Feder, 1988]. H e fo u n d th at the v a ria tio n s
in the reserv o ir storage (the ran g e) scaled w ith th e tim e considered as a
p o w er law . M andelbrot a n d W allis [M andelbrot a n d W allis, 1968;
M an d elb ro t an d W allis, 1969a; M andelbrot an d W a llis, 1969b; M a n d e lb ro t
a n d W allis, 1969c] in tro d u c e d th e concepts of fra c tio n a l G aussian n o ise s
and fractional B row nian w alk s, b o th are reco g n ized a s fractal d istrib u tio n s.
The fractional B row nian w alk s are the integrals o f th e fractional G a u s s ia n
noises an d yield a p o w er-law rela tio n betw een R /S a n d T, the p e rio d fo r
w h ich R is obtained. In g en eral th e noises do n o t h a v e a pow er-law
d istrib u tio n s of extrem e v alu es alth o u g h the w a lk s d o . If river flow s a re
eq u iv alen t to noises (re serv o ir volum es e q u iv a le n t to w alks), w h y s h o u ld
the extrem es of river flow s (floods) have a p o w e r la w d istrib utions? B u t
the riv er flow s them selves re p re s e n t the a d d itio n o f in d iv id u al r a in e v e n ts
(storm s), th u s it is n o t u n reaso n ab le to consider r iv e r flow s as w a lk s ra th e r
th a n noises. As w alks, a p o w er-law d istrib u tio n o f e x tre m e values (floods)
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128
w o u ld b e expected. A tim e series w ith o n ly sh o rt range co rrelatio n s (for
ex am p le ARM A series) m a y b e statio n ary . H o w ev er, w alk s b y defin itio n
h a v e lo n g ran g e co rrelatio n s an d are n o t sta tio n a ry . M a n d e lb ro t a n d W allis
[M an d elb ro t and W allis, 1969c] co n sid ered noises an d w alk s w ith nonG a u ssia n d istrib u tio n s a n d o b ta in e d sim ila r results. T hey also in tro d u ce d
th e N o a h a n d Joseph effects. The N o a h effect is the sk ew n ess o f th e
d is trib u tio n o f flows in a riv e r (or th e sk e w n e ss of a n o n -G a u ssia n
d istrib u tio n ) and the Jo sep h effect is the p ersisten ce o f the flow s.
A n a ly sis
If floods are fractal th e n th e p eak d isch arg e d u rin g a tim e interv al T
sh o u ld scale w ith the re la tio n [Turcotte, 1992]
(3.1)
w h e re H is know n as the H a u sd o rff m e asu re. Since (3-1) is an alo g o u s to the
s ta n d a rd deviation of a tim e series it is a p p ro p ria te to relate H to the fractal
d im e n sio n D by
D =2-H .
(3-2)
In o rd e r to avoid p ro b le m s w ith a n n u a l v ariab ility , th e in te rv a l T is
g en e ra lly tak en as an in te g e r n u m b e r of y e a rs. The scale in v a ria n t
d istrib u tio n can also be ex p ressed in term s of the ratio F o f th e p eak
d isch arg e over a 10-year p e rio d to the p ea k discharge over a 1-year period.
W ith self-sim ilarity the p a ra m e te r F is also the ratio of th e 100-year peak
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129
discharge to th e 10-year peak discharge a n d the ratio o f th e 1000-year p e a k
discharge to th e 100-year peak discharge. In term s of H a n d D we have
F = 1 0 h = \ 0 2~ d
(3.3)
C om bining (3.1) an d (3.3) a n d in tro d u cin g the te n -y e ar flood as a
reference flood g iv es
(3.4)
This r e la tio n w ill be te sted w ith flo o d d a ta a n d v alu es of F w ill b e
obtained.
Before co n sid erin g actual ex am p les w e w ill also in tro duce rescaled
ran g e (R/S) analysis. H u rst [Hurst, 1951; H u rst, 1956] p ro p o sed this
em pirical a p p ro a c h to the statistics of floods an d d ra u g h ts . The m ethod is
illu strated in F ig u re 3.1. C onsider a re se rv o ir b eh in d a d a m that never
overflow s or e m p tie s, the flow in to th e rese rv o ir is d e fin e d by
V (T )= IjoTV(t)dt
(3.5)
The v o lu m e o f w ater in the re se rv o ir V(t) is g iv e n b y
T .
V /t= V (0 )+ j V it^ d t'-tV C T )
(3.6)
0
a n d the range is d e fin e d by
R(T)=V max ~ V min
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t3-7)
Figure 3.1: Rescaled range (R/S) analysis.
130
>
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131
w h e re Vmax is th e m a x im u m v o lu m e a n d V niin th e m in im u m v o lu m e
sto re d d u rin g the interv al T. T he rescaled ra n g e is defined as R /S w h e re S
is th e sta n d a rd d ev iatio n o f th e flow d u rin g th e p eriod T
T
_
1.
S (D = [ij ( V ( t ) - V ) 2 d t] 2
0
(3.8)
H u rs t [H u rst e t al., 1965] found th a t fo r m an y time series th e rescaled
ra n g e satisfies th e em p irical relation
<3.9)
w h e re H Lis k n o w n as the H u rs t exponent. E xam ples in clu d ed riv e r
d isch arg es, rainfall, v arv es, tem p eratu res, s u n s p o t num bers, a n d tre e rings.
In m a n y cases th e valu e o f th e H urst e x p o n e n t is near 0.7.
If a G au ssian w h ite n o ise sequence o f n u m b ers is su m m e d th e result
is a B ro w n ian w alk. A n R /S analysis of th e w h ite noise sequence gives a
H u rs t ex p o n en t Hi=0.5, th u s the H urst e x p o n e n t is equal to the H a u sd o rff
m e a su re o f the in te g ra ted signal, a B ro w n ian w alk w ith H=0.5. M an d elb ro t
a n d W allis [M andelbrot a n d Wallis, 1968; M a n d elb ro t and W allis, 1969a;
M a n d elb ro t a n d W allis, 1969b; M andelbrot a n d Wallis, 1969c] in tro d u c e d
th e co n c ep t o f fractional G au ssian noises a n d their integrals, fractio n al
B ro w n ian w alks. They sh o w e d that the H u r s t exponent H i of a fractional
G au ssian noise is equal to th e H au sd o rff m e a su re of the c o rre sp o n d in g
fra ctio n a l B row m an w alk.
If 0.5 < H i < 1 the o rig in a l time series is said to be persistence;
ad jace n t v alu es are m ore stro n g ly c o rrela ted th a n if they w ere ra n d o m . The
h ig h e r th e v alu e of H i, th e greater the p ersisten ce. If 0 < H i < 0.5 th e
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132
o rig in al tim e series is sa id to be an tip ersisten t; adjacent v a lu e s are less
c o rre la te d th a n i f th e y w ere random .
R esu lts
W e now tu r n to the analysis of flo o d -freq u en cy reco rd s. As o u r first
ex am p le tw o b en c h m ark statio n s co n sid ered b y B enson [Benson, 1968] w ill
b e stu d ie d . B enson a p p lied a variety of geostatistical d istrib u tio n s to the
d a ta fro m these statio n s, these w ill be co m p ared w ith th e fractal ap p ro ach
d iscu ssed above. For o u r analysis w e co n sid er floods th a t are separated by
m o re th a n tw o m o n th s as d iscu ssed ab o v e.
The larg est floods for each record a re ord ered , th e la rg e st flood is
a ssig n ed a p erio d eq u al to the length of th e record T0, th e second largest
flood a p erio d of T0/2 , the 3rd largest flood a period of T0/3 , a n d so forth.
The log of the p e a k discharge for each flo o d is p lo tted a g a in st the log of its
assig n e d p erio d . R esults for station 1-1805 o n the M id d le B ranch of the
W estfield River in G oss H eights, M assach u setts are g iv e n in Figure 3.2a fo r
the p e rio d 1911-1960 [G reen, 1964]. The so lid line is th e least squares fit of
E q u atio n 3.1 w ith the d a ta over the ran g e 50< V <200 m V s; large floods are
o m itte d from th e fit b ecau se of their sm all n u m b er. T he so lid line
co rresp o n d s to H=0.51 a n d from E quation 3.3 w e h av e F=3.3. Results for
s ta tio n 11-0980 in the A rroyo Seco near P asad en a, C alifo rn ia are given in
F ig u re 3.2b for th e p e rio d 1914-1965. The so lid line c o rresp o n d s to (2) w ith
th e d a ta over th e ran g e 10< V <100 m 3/ s . T he solid lin e co rresp o n d s to
H =0.87 an d from E q u atio n 3.3 w e have F= 7.4. In b o th cases th e fit to the
scale-in v arian t (fractal) relatio n is quite g o o d . The v alu es of H a n d F in
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without p erm ission.
133
V (m3/sec)
1000
500 -
(a)
Fractal
•2 p aram eter Gamma
G um bel
- log G um bel
loe N o rm al
log P earso n type HI
1 0 0 r-
H = 0.87, F= 7.4
100
V (m3/sec)
1000 r
800
--------------- 2 p aram eter Gamma
600 ------------- G um bel
- l o g G um bel
log N o rm al
400
H azen
log P earson type IE
300
H = 0.51, F= 3.3
200
100
Figure 3.2: O bserved floods an d statistical distributions for
(a) station 1-1805, M iddle Branch, Westfield River, Goss Heights,
M assachusetts an d (b) statio n 11-0980, Arroyo Seco near Pasadena,
California.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
134
California are considerably larger th a n in M assachusetts. L arge floods are
relatively m ore p ro b ab le in th e arid clim ate than in the tem p erate c lim a te .
W e h av e also d e te rm in e d th e H u rs t exponent for th e tw o
benchm ark stations. V alues of R /S for T = 5 ,10,25, an d 50 years (R /S = l f o r
T=2 by d efinition) are g iv en in Figure 3.3a for station 1-1805 (W estfield,
MA) an d in F igure 3.3b for station 11-0980 (Pasadena, CA). G ood
correlations are obtained w ith E q u atio n 3.9 taking H ^O .67 for station 1-18-05
and H 1=0.68 for statio n 11-0980. O u r studies of all ten o f th e b enchm ark
stations [Turcotte an d G reene, 1993] indicate that there is considerable
v ariatio n of F b u t v ery little v a ria tio n in Hi. A sim ple ex p lan atio n is t h a t
the form er is sensitive to th e N o ah effect w hile the la tte r is sensitive to tlie
Joseph effect. The relative scaling of floods is sensitive to th e skew ness o f
the statistical d istrib u tio n b u t is n o t sensitive to the p ersisten ce of flow s o r
floods. A n im p o rta n t co n clu sio n is th a t R /S analysis is n o t relev an t to
flood frequency h azard assessm ents.
M any statistical d istrib u tio n s h av e been ap p lied to historical re c o rd s
of floods. Benson has g iv en six statistical correlations for each of his te n
benchm ark stations. H is resu lts for th e 2-param eter g am m a (Ga), G u m b e 1
(Gu), log G um bel (Lgu), log norm al (LN), H azen (H), a n d log P earson typ-e
□I (LP) are given in Figure 3.2a for statio n 1-1805 an d in Figure 3.2b for
station 11-0980. For large floods the fractal prediction (F) correlates best w d th
the log G um bel (Lgu) w h ile the o th er statistical techniques p red ict lo n g e r
recurrence tim e for v ery serious floods. The fractal a n d log G um bel are
essentially pow er-law correlations w hereas the others are essentially
exponential.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without p erm ission.
135
R /S
10
5
2
1
50
X, yrs
R /S
10
(b)
5
2
1
X, y rs
Figure 3.3: Rescaled range (R /S) for several intervals T w ith H u rst
exponents H i for (a) statio n 1-1805 a n d (b) station 11-0980.
R eprodu ced with p erm ission of the copyright owner. Further reproduction prohibited without perm ission.
136
W hile a few b e n c h m a rk statio n s p ro v id e a b a s is fo r com paring statistical
approaches, th ey h a rd ly m ad e a convincing case th a t fractal statistics are
p referab le to altern ativ es. In o rd e r to o v erco m e th is difficulty w e h av e
an aly zed a large n u m b e r of reco rd s and su p e rim p o se d the resu lts. O u r
system atic stu d y h as considered 41 year reco rd s for 1008 USGS stream flow
stations th at are unaffected b y flood control p ro jects [Wallis et al., 1991]. A
d ig itized record o f d aily m ean discharge d ata fo r the forty years has been
g iv en for each sta tio n . T he locations of the sta tio n s are given in F igure 3.4.
The best fit stra ig h t lines have b een o b ta in e d for the 1008 sta tio n
records considered. The resu lts w ill be d iv id e d in to the 18 h y d ro lo g ic
reg io n s of the c o n te rm in o u s U n ited States illu s tra te d in F igure 3.5. W e first
co n sid er the q u a lity of the fit o f the d ata to th e fractal relation in E quation
3.4. The ratios o f th e m e asu red p eak flows to th e v alu e p red icted b y the best
fractal fit are d e te rm in e d for p erio d s T=1 yr. (40th largest flood in the
series), 5 yrs. (8th larg est flood), 10 yrs. (4th la rg e st flood), 20 yrs. (2nd largest
flood) an d 40 yrs. (largest flood). Results for th e 111 stations in h ydrologic
reg io n 3 are g iv e n in F igure 3.6a, for th e 123 sta tio n s in hy d rologic region 7
in F igure 3.6b, for th e 18 statio n s in hydrologic re g io n 14 in F igure 3.6c, and
for the 100 statio n s in hy d ro lo g ic region 17 in F ig u re 3.6d. T hese re su lts are
typical of the 18 h y drologic regions. If all fits w e re perfect then all d a ta
p o in ts w ould b e u n ity . The scatter is greatest fo r the largest floods (T = 40
yrs.) as expected since th ey are based o n a sin g le d a ta point. In g en eral the
scatter is quite sm all, a sta n d a rd deviation of less th a n 5%.
It is also o f in te re st to com pare the fractal fits for the sta tio n s w ith in
a hydrologic reg io n . For each station th e flow associated w ith th e 10-year
flood Um (10) w a s n o rm alized b y the d ra in a g e a re a upstream o f th e station.
with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
Figure 3.4: Distribution of the 1,009 streamflow
stations analyzed for the conterminous US.
137
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3.5: Hydrologic regions of the conterminous United States.
138
W
0 M
w m ,
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
10.0
________
........... ,J a ) _____
o* 2*
o g
S
3
CD
<d «
1.0
. t
.
£>
)
•«
W'T! 1.0
T43 t3
i I j
QJ
1
' *
13
> 0.1
"ra
0. 1
10
i
10.0
......................
jb)._
..................................... _
> 2
o* J
2rf
100
P erio d (years)
_
id
O b serv ed flo o d s/fra c ta l p re d ic tio n
for h y d ro lo g ic reg io n 7
td
10
100
P eriod (years)
Jd)__
10.0
_____ _
O b se rv e d flo o d s /fra c ta l p re d ic tio n
for h y d ro lo g ic reg io n 17
§ g
<d 43
<d 43
i5 £
Sx^ 1.0
a
JL'S 1.0
ti
3 <
1C0J uU
I
I
I
'v t!
d uS
l/l
s *Ol
M
2 J3
73
>
IX
0» ID
2 3
13
>
i
S'S
<
d0)4 uCu
QJ
l_
>
_
O b serv ed flo o d s /fra c ta l p re d ic tio n
for h y d ro lo g ic reg io n 14
<d
O b serv ed flo o d s /fra c ta l p re d ic tio n
for hy d ro lo g ic reg io n 3
*1
<n <c
ra £>
ix^l
TO3J u
tl *3
3 <u
l/l
p,
(c)
10.0
0.1
10
P eriod (years)
100
0.1
I
.
|
10
|
100
P erio d (years)
Figure 3.6: Ratios of the observed m axim um daily average discharges to the values predicted by the fractal
fit to the data as a function of the assigned period (a) for the 111 stations in Region 3, (b) for the 123 stations
in Region 7, (c) for the 18 stations in region 14, and (d) for the 100 stations in region 17.
140
T h e straig h t-lin e co rrela tio n s fo r th e 111 statio n s in h y d ro lo g ic re g io n 3 are
g iv e n in F igure 3.7a, fo r th e 123 stations in h y d ro lo g ic reg io n 7 in Figure
3.7b, for th e 18 stations in h y d ro lo g ic region 14 in Figure 3.7c, a n d for the
100 sta tio n s in hydrologic reg io n 17 in Figure 3.7d. If all sta tio n s in a
h y d ro lo g ic region h a d th e sam e flood intensity factor F a n d if p e a k flows
w e re sim p ly p ro p o rtio n al to u p strea m d rain ag e area th en all p lo ts for the
h y d ro lo g ic region w o u ld lie o n a com m on line. T h ere is so m e v ariab ility
in th e flood intensity facto r F (slope) and from o n e to tw o o rd e rs o f
m a g n itu d e v ariab ility in th e n o rm alized flow s.
The m ean v alu e o f th e flo o d in ten sity facto r for each o f th e eighteen
h y d ro lo g ic regions is g iv e n in T able 3.1. The s ta n d a rd d e v ia tio n for each
re g io n is also given. R eg io n al variations of the flood in te n sity facto r F are
clearly illu strated in th is table. T hey are also illu strate d in F ig u re 3.8 w ith
v e ry h ig h (F > 4.8), h ig h (3.5 < F < 4.8), low (2.6 < F < 3.5) a n d v e ry low (F <
2.6) reg io n s illustrated. V ery h ig h regions are the a rid s o u th w e st a n d the
w e ste rn G ulf Coast (Texas). The latter is relativ ely arid an d p ro n e to
h u rric a n e s. The low est v a lu e is the Pacific n o rth w e st w ith a s tro n g
m a ritim e clim ate. A lso v e ry low are N ew E n g lan d an d the G re a t lakes
reg io n s. The flood in te n sity factor appears to v ary sy stem atically w ith
clim ate.
In som e cases th e s ta n d a rd deviations of th e values o f F in a
h y d ro lo g ic region are larg e. The largest sta n d a rd d ev iatio n is for region 18.
In F ig u re 3.9, the v alu es o f F for the 39 stations in this reg io n (principally
C alifornia) are p lo tted as a fu n ctio n of u p strea m d rain ag e a re a A. For this
p u rp o s e the region h as b e e n d iv id e d into th ree su b reg io n s: s o u th e rn and
c e n tra l coastal C alifornia (SC), th e Sierra N e v a d a (SN), a n d N o rth e rn
R eprodu ced with perm ission of th e copyright ow ner. Further reproduction prohibited without perm ission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1000
a>
(a)......
O)
N o rm alized fractal fit
for h y d ro lo g ic reg io n 3
60
Ic
P .J
x
l
zr
u ^
<u ^
N o rm a liz ed fractal fit
for h y d ro lo g ic reg io n 14
60
100
P
S
'£ *
5 ^
bo
(c)
100
<6
10
10
<u H
bo
13
13
A<j
•6
.a
p
10
100
1
P eriod (years)
1000
a
a>
1
(b )
q
N o rm a liz ed fractal fit
for h y d ro lo g ic r e g io n J Z
bo
■S„
100
i t
c ’x
i l
5 ^
5 ^
0) -ii-
100
\
10
60
100
10
U-t
a?
&
60
13
13
( d ) _____
1000
0ro)
N o rm a liz ed fractal fit
for h y d ro lo g ic reg io n 7
60
10
P eriod (years)
Serttu*?
1.0
■6
•6
.a
p
0.1
10
P eriod (years)
100
10
100
P eriod (years)
Figure 3.7: Fractal fits of the norm alized flood frequency data (a) for the ll'l stations in Region 3, (b) for
the 123 stations in Region 7, (c) for the 18 stations in region 14, and (d) for the 100 stations in region 17.
142
Table 3.1: A verage values an d standard deviations of the
flood intensity factor F for the 18 hydrologic regions.
H ydrologic Region F
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
2.369
2.998
2.758
2.183
2.396
2.505
2.782
3.021
4.7
3.557
3.897
4.848
4.104
2.283
6.066
2.778
2.076
5.134
S t. dev.
0.377
1.313
0.617
0.289
0.509
0.324
0.738
0.979
1.586
1.677
1.801
1.559
2.121
00.51
1.08
0.752
0.357
2.4
N um ber of
54
147
111
57
129
38
123
22
13
64
46
13
14
18
11
10
100
39
R eprodu ced with p erm ission o f the copyright owner. Further reproduction prohibited without perm ission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
s u m
H i Very low, F < 2.6
d H Low, 2.6 < F < 3.5
C m H igh, 3.5 < F < 4.8
I H Very high, F < 2.6
Figure 3.8: H ydrologic regions of the conterm inous United States, colored
according to the values of the flood frequency factor, F, for each region.
Flood intensity factor, F vs. Drainage Area for California
10
T '
T
1~ "T ' 1 " T ' T
1|
-
'1 ............. 1........ v -
-r •
- j - - 1—
J
............ v
I '' ' I
r ” 1*- T ’ I ~T
H;
■M
*
ft
ft
ft
ft
¥
PL,
ft
□
/\
□□
□
ft Southern/C oastal California
< Sierra N evada
□ Northern California
0
. _ . i ......... _ 1 ____ l -
l-
I -i - L. l_ 1...
. -1 .. ____1.... . - i. .. I _~l.
I..1------------
10
100
Drainage Area (sq. mi.)
I
. . _ I.
—X
»—
1000
Figure 3.9: Flood intensity factor, F, as a function of upstream drainage areas for the 39 stream flow
stations in hydrologic region 18 (California). Data are given for three subregions:
southern and central coastal California (SC), the Sierra N evada (SN), and northern California (NC).
145
C alifornia (NC). It is seen th a t th e SC values are system atically h ig h
w h ereas the SN a n d N C v alu es are system atically low . In general it is
fo u n d th at th e flood in ten sity factor F is low if a su b stan tia l fraction o f th e
d rain ag e area h as a w in ter sn o w pack. This is co n sisten t w ith low v alues
for the Sierra N e v a d a reg io n . N o rth e rn C alifornia h as a m aritim e clim ate
sim ilar to w e s te rn O reg o n a n d W ash in g to n a n d the lo w values of F a re
expected.
C onclusions
H istorical flood-frequency records have been ex am in ed to d eterm in e
w h eth er fractal (pow er-law ) statistics are applicable. A lth o u g h it m u st be
recognized th a t th e relativ ely sh o rt d u ra tio n o f historical records restricts
th e validity of conclusions; n ev erth eless, quite good ag ree m en t is o b ta in ed
b etw een fractal statistics a n d observations for 10 b en ch m ark stations an d
for 1009 o th er sta tio n s in th e U n ited States. The basic q u estio n in te rm s of
flood h azard assessm en t is w h e th e r extrem e floods d ecay ex p o n en tially in
tim e or as a p o w e r law . If th e p o w er-law b ehavior is applicable then th e
likelihood o f sev ere floods is m u c h h ig h er an d m ore co n serv ative d esig n s
for dam s a n d la n d use restrictions are indicated.
For fractal b eh av io r the ratio of the 10 y ear to th e one year flood F is
also the ratio of th e 100 y ear to the 10 year flood an d th e ratio of the 1000
year flood to th e 100 year flood. W e find large regional v ariations in v alues
of F. In arid reg io n s such as th e so u th w e stern U n ited States the v alu es of F
are nearly th re e tim es the v alu es in m ore tem p erate reg io n s such as th e
n o rth w e ste rn a n d n o rth e a ste rn co m ers of the co u n try . S m aller v a lu e s of F
are also fo u n d if u p strea m d rain a g e areas have large sn o w packs.
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146
The relevance o f R /S an aly sis to flo o d frequency fo recastin g has also
b e e n ad d ressed . For th e ten b e n c h m ark statio n s w e find th e H u rs t
ex p o n en t to be H ^ O .7 + / - 0.03. This value indicates m oderate persistence
for th e floods b u t also sh o w s th a t d eterm in atio n s of H u rst ex p o n e n ts are
n o t u sefu l fo r flood h a z a rd assessm en ts. T he H u rs t ex p o n en t H Ldoes n o t
correlate w ith the fractal flo o d p aram eter F. In the terms in tro d u c e d by
M an d elb ro t an d W allis [M an d elb ro t a n d W allis, 1968] the H u rs t ex p o n en t
H i is sensitive to the Jo se p h effect o r p ersisten ce of events w h erea s the
fractal flood p aram eter F is se n sitiv e to the N o ah e f f e c t o r sk ew n ess of the
statistical d istrib u tion s of flo o d s.
A p rim ary objective o f th is p ap e r h a s b een to show th a t th e tw o
p aram ete r fractal d is trib u tio n is in as goo d ag reem en t w ith o b servations,
o n average, as em pirical la w s w ith th ree o r m ore free p aram ete rs.
H o w ev er, a m ore fu n d a m e n ta l objective is to arg u e that th e re is a physical
b asis for p referrin g th e scale-in v arian t fractal relation to em p irical
a lte rn a tiv e s th at in v o lv e o n e o r m ore scalin g param eters. A v a lid q u estio n
is w h eth e r all the co m p lex ities associated w ith floods could be con sisten t
w ith one statistical law . P h y sical processes associated w ith floods include:
(1) the am o u n t of rain fall p ro d u c e d by the sto rm or storm s in q uestion, (2)
the u p stre a m d rain ag e area,, (3) the sa tu ra tio n of the soil in th e d rain ag e
area, (4) th e topography, so il type, an d v eg etatio n in the d ra in a g e area, an d
(5) w h e th e r snow m e lt is in v o lv e d . In a d d itio n dam s, strea m
ch an n elizatio n , a n d o th e r m a n -m a d e m o d ificatio n s can affect th e severity
o f floods. H ow ever, a v a rie ty of com plex n a tu ra l p h en o m en a h a v e been
sh o w n to obey fractal sta tistics including earth q u ak es an d volcanic
e ru p tio n s.
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147
A n im p o rta n t a s p e c t of fractal statistics is th e relation o f fractio n al
n o ises to fractional w alk s. In general fractio n al noises d o n o t h a v e p o w erlaw d istrib u tio n s of ex tre m e values a lth o u g h th e w alks do. If riv e r flow s
are e q u iv a le n t to n oises (reserv o ir v o lu m es e q u iv a le n t to w alk s), w h y
sh o u ld th e extrem es o f riv e r flows (floods) h a v e a p o w er law d istrib u tio n s?
B ut th e riv e r flow s th e m selv es re p re s e n t the a d d itio n of in d iv id u a l ra in
ev en ts (storm s), thus it is n o t u n reaso n ab le to co n sid er river flow s as w alks
ra th e r th a n noises. As w a lk s, a p o w er-law d istrib u tio n of ex trem e v alu es
(floods) w o u ld be expected.
R eprodu ced with perm ission of the copyright owner. Further reproduction prohibited without p erm ission.
148
BIBLIOGRAPHY
B enson, M .A ., U n ifo rm flo o d -freq u en cy e stim a tin g m eth o d s for fed eral
agencies. W a t e r R e s o u r c e s R e s e a r c h , 4, 891-908,1968.
C o u n cil, U.S.W .R., F lood flow frequencies, H y d ro lo g y Com m ittee,
B u lletin , 1981.
Feder, J., F r a c t a l s , P len u m Press, N ew York, 1988.
G reen, A.R., M ag n itu d e a n d Frequency o f Floods. P a rt 1-A, U . S . G e o l o g i c a l
S u r v e y , W a t e r S u p p l y P a p e r , 1 6 7 1 , 212-213,1964.
H u rst, H .E., L ong-term storage capacity of reservoirs, A m e r i c a n S o c i e t y o f
C i v i l E n g i n e e r s - T r a n s a c t i o n s , 1 1 6 , 770-779,1951.
H u rs t, H .E., M ethods of u sin g lo n g -term sto rag e in reservoirs, I n s t i t u t e o f
C i v i l E n g i n e e r i n g - P r o c e e d i n g s , 5 (P art 1), 519-590,1956.
H u rs t, H .E., R.P. Black, a n d Y.M. Sim aika, L o n g - t e r m S t o r a g e , C onstable,
L o n d o n , 1965.
M a n d e lb ro t, B.B., a n d J.R. W allis, N o ah , Jo sep h , a n d operational
h ydrology, W a t e r R e s o u r c e s R e s e a r c h , 4,909-918,1968.
M a n d e lb ro t, B.B., an d J.R. W allis, C o m p u te r ex p erim en ts w ith fra ctio n a l
G au ssian noises. P arts I, H, HI., W a t e r R e s o u r c e s R e s e a r c h , 5 , 228-267,
1969a.
M a n d e lb ro t, B.B., an d J.R. W allis, Som e lo n g -ru n p ro p erties of g eo p h y sical
records, W a t e r R e s o u r c e s R e s e a r c h , 5,32 1 -3 4 0 ,1969b.
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