chow, ven te. stochastic models
TRANSCRIPT
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WRC RESEARCH REPORT NO 26
STOCHASTIC ANA LYSIS OF HYDROLOGIC SYSTEMS
Ven Te Chow
P r i n c i p a l I n v e s t i g a t o r
F I N A L R E P O R T
P r o j e c t N o. A -0 29 - I L L
Th e w o r k u po n w h i c h t h i s p u b l i c a t i o n i s b a se d was s u p p o r t e d by f u nd s
p r o v i d e d b y t h e U.S. D e pa rtm e nt o f t h e I n t e r i o r as a u t h o r i z e d u n d e r
t h e W a t e r R e s o u r ce s R e s e a r ch A c t o f 1 9 64 P .L . 8 8 -3 7 9
Agreement No. 14-0 1-000
1
1632
UNIVERSITY OF IL LI NO IS
WATER RESOURCES CENTER
3220 C i v i l E n g i ne e r in g B u i l d i n g
U r b a n a
l l l i n o i s 61801
December 1969
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ABSTRACT
STOCHASTIC ANALYSIS OF HYDROLOGIC SYSTEMS
Hydro logic phenomena a re i n re a l i t y s tochas t i c i n na tu re; tha t i s th e i r
behav ior changes w i th the t ime i n accordance w i th the law o f p r ob ab i l i t y
as wel l
as w i t h the sequent ia l re la t i on sh ip between the occurrences o f
the phenomenon.
I n ord er t o analyze t he hyd ro lo gi c phenomenon a mathe-
mati c model o f the s to cha st i c hy dro log ic system t o s im ulat e t he phenom-
enon must be formulated. I n t h i s study a watershed i s tr ea te d as the
sto cha st i c hydrologi c system whose components of pr ec ip i t at io n runof f
storage and eva pot ran spi rat ion are simulate d as sto ch as ti c processes by
time se rie s models t o be determined by correlograms and spe ct ra l analy sis.
The hydr olo gic system model i s then formulated on the basis o f the p r i n c i -
p l e of con ser vat ion o f mass and composed o f the component st oc ha st ic proc-
esses.
To demonstrate the pr ac t i ca l a pp l i cat io n o f the method of analys is
so developed the upper Sangamon Riv er bas in above Mon t ic el lo i n ea st
ce nt ra l I l l i n o i s i s used as the sample watershed. The watershed system
model so form ulat ed can be employed t o generate s to ch as ti c streamflows
fo r pr ac t i ca l use i n the analys is of water resources systems. This i s
o f pa r t ic u l a r va lue i n the economic planning o f water supply and i r r i ga -
t io n pr o jec t s wh ich i s concerned w i th the long- range water y i e l d o f the
watershed.
Chow Ven Te
STOCHASTIC ANALYSIS OF HYDROLOGIC SYSTEMS
Research Report No.
6 Water Resources Center Un i ver s i t y o f I l l i n o i s
a t Urbana-Champaign December 1969
4
pp.
KEYWORDS--systems analysis/stochastic processes/synthetic hydrology/
water resources development/watershed studies precipitation streamflow
evapotranspiration storage water
y ie ld/hydrologic models /hydrology
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CONTENTS
I n t r o d u c t i o n
I Form ula t io n of the Hy dro lo gi c System Model 3
I Mathematical Techniques 5
A Mathemat ical Models f o r Time Ser ie s 5
. Moving Average Model
2. Sum of Harmonics Model 5
Autoregression Model
6
TheCor re logram 6
.
The Spectrum Analysis 8
I V
A n a l y s i s o f t h e H y d r o l o g i c S y s t e m
The Watershed under Study
T h e H y d r o l o g i c D a t a
. P r e c i p i t a t i o n
2 Streamflow
Temperature 12
. Po t en t i a l Evapo t r ansp i r a t i on 12
C
Es ta bl is h i ng t he Records f o r Conceptual Watershed
Storage and Actual Evap ot ra nspi rat i on 3
Ana lys is o f the Hyd ro log ic Processes 5
. Det ermi nat i on o f th e System Model 17
V Conclusions
V I Acknow edgments
V I I . References
V I I I
Figures
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I
l NTRODUCT
ON
t i s gene r a l l y no ted t h a t t he na t u r a l h yd r o lo g i ca l s ys tem and
hydro log i c p rocess a re t r u l y s tochas t c ; t h a t i s , t h e b e h av i o r o f t h e
sys tem o r t h e p rocess va r ie s w i t h a sequen t ia l t im e f unc t i on o f t he p roba -
b i l i t y o f occu r rence [1,2].9:
I n ot he r words, th e hy dr ol og ic phenomenon
changes w i t h t he t im e i n accor dance w i t h t he l aw o f p r o ba b i l i t y as we l l
as w i th the sequent ia l re la t i on sh ip between i t s occur rences . For example ,
t h e o cc u rr e nc e o f a f l o o d i s c on si d er e d t o f o l l o w t h e law o f p r o b a b i l i t y
and a l s o t h e r e l a t i o n s h i p w i t h t he a n te ce da nt f l o o d c o n d i t i o n .
Mos t convent iona l methods f o r hyd ro l og i c des igns a r e de te r -
m i n i s t i c , t h a t i s , t h e b e h av i o r o f t h e h y d r o l o g i c s ys te m o r p ro ce ss i s
assumed independent o f t ime va r i a t io ns . For example, a u n i t hydrograph
d e r i v e d f o r a g i v e n r i v e r b as i n f o r f l o o d - c o n t r o l p r o j e c t d e si g n i s based
on h i s t o r i ca l f l o od r eco rds . Once de r i ved, t he un i t hydr og r aph i s used
f o r a na l y s i s o f f u t u r e de si g n f l o o d s. Thus,
t
i s au tom at i ca l l y assumed
unchanged w i th t ime ( f rom the pas t t o the fu tu re ) and there fo re i s
d e t e r m i n i s t i c .
Some convent i onal methods employ t he concept of p r ob a b i l i ty t o
t he ex t e n t t h a t n o s e q ue n ti a l r e l a t i o n s h i p i s i n v o l v e d i n t h e p r o b a b i l i t y .
Fo r examp le , t he f l o od r ecor d i s ana lyzed and f i t t e d w i t h a ce r t a i n pr oba-
b i l i t y d i s t r i b u t i o n t o determi ne t h e r e cu rr en ce i n t e r v a l s o f t h e f l o od o r
the f l oo d f requenc ies . Such methods are p robab i i s t i c bu t no t i n the
t r u e sense s tochast c .
Numbers i n pa r en theses r e f e r t o r e f e r ences l i s t e d a t t he end o f t he
r e p o r t .
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T he s t o c h a s t i c m e th od , t h a t i s t o e m plo y t h e c o nc e p t o f p r o ba -
b i l i t y as w e l l as i t s se q u e nt ia l r e l a t i o n s h i p , has n o t been w e l l i n t r o -
duced i n t h e p r a c t i c a l
d e s i g n and p l a n n i n g o f h y d r o l o g i c p r o j e c t s , b e ca us e
s u c h m e th od s h a v e n o t be e n f u l l y d e v e l o p e d .
W h i le t h e n a t u r a l h y d r o l o g i c
phenom enon i s s t o c h a s t i c , t i s i m p o r t a n t t o d e ve lo p t h e s t o c h a s t i c m etho d
o f h y d r o l o g i c a n a l y s i s f o r h y d r o l o g i c s y st em d e s ig n . C o n v e nt io n a l m e th od s,
d e t e r m i n i s t i c and p r o b a b i l i s t i c , w h i c h d o n o ' t c o nf or m m ore c l o s e l y t o t h e
n a t u r a l phenom enon, w i l l p ro d uc e r e s u l t s t h a t d e p a r t f ro m t h e t r u e b e h a v io r
o f t h e h y d r o l o g i c p henomenon and h en ce ha ve t h e p o s s i b i l i t y t o e i t h e r o v e r -
d e s i g n o r u nd er de s i g n t h e h y d r o lo g i c p r o j e c t
[ 3 ]
The o b j e c t i v e o f t h i s s t u dy i s t o f o r m u l a t e t h e m at he m at ic a l
m od el o f a s t o c h a s t i c h y d r o l o g i c s y st em and t h e m a t h e m a ti c al m o de ls o f
t h e h y d r o l o g i c p r oc e s s es i n t h e s ys te m , u s i n g t h e w a t e r s h e d as an exam -
p l e o f t h e h y d r o l o g i c sys tem . I n t h i s s t u d y , i n o t h e r w ord s, t h e fra me -
w o rk o f a m eth od was d e ve lo p ed t o u t i l i z e m a th e m a t ic a l m od els t o s i m u l a t e
t h e s t o c h a s t i c b e h a v i o r o f a wa te r sh e d a s
t h e h y d r o l o g i c s y ste m .
The
m a t h e m a ti c al m o de ls s o de v el op e d s h o u ld h av e a p ' r a c t i c a l a p p l i c a t i o n t o
t h e a n a l y s i s o f h y d r o l o g i c sy ste ms i n t h e w a t e r r e so u r ce s p l a n n i n g an d
d e v e l o p m e n t .
The i n i t i a l s t e p o f t h e s t u d y i n v o l v e d a c om p re he ns iv e re v ie w
o f t h e a p p l i c a t i o n o f t h e t he o r y o f s t o c h a s t i c p ro ce ss i n h y d ro lo g y . The
r e s u l t s o f t h i s i n i t i a l s t e p o f i n v e s t i g a t i o n a r e r e p o r t e d s e p a r a te l y as
W a te r R es o ur ce s S ys te ms A n a l y s i s A n n o t a t e d B i b l i o g r a p h y o n S t o c h a s t i c
P r o c e s s e s
[4.] and Water Resources Sys tems Ana l y s s
-
R ev ie w o f S t o c h a s t i c
P rocesses ' ' [5]
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11
FORMULATION OF THE HYDROLOGIC SYSTEM MODEL
I n t he f o r m u la t i on o f t he hyd r o l og i c syst em model ,
a watershed
i s used as th e hy dr o l og ic system al tho ugh the mathematica l approach would
be equa l l y app l i c ab l e t o o t he r k inds o f hyd r o l og i c sys tems w i t h some mod i-
f i ca t i on s depending on the na ture o f th e sys tem. The watershed i s t rea ted
as a hyd ro l og i c system wh ich has an in pu t , ma in ly r a i n f a l l , and an ou tpu t ,
ma in l y runo f f and evapo t ransp i ra t ion . The inp u t and ou tpu t a re to be
t r ea t ed as t im e se r ie s o r s t och as t i c p rocesses which desc r ibe t he s tochas-
t i c behav io r o f t he inpu t and ou tpu t p rocesses . The amount o f water
s t o r e d i n t h e w at er sh ed i s a l s o t r e a t e d as a t i m e s e r i e s o r s t o c h a s t i c
p ro ce ss w h ic h de s cr i b es t h e s t o c h a s t i c n a t u r e o f i n f i l t r a t i o n , s u bs ur fa ce
ru no f f and the s o i l mo is tu re and groundwater s to rages .
To fo rmu la te a mathemat ical model f o r the watershed hydro log ic
sys tem, the ru no f f i s cons idered as th e i n t eg ra l p roduc t o f t h re e compo-
nent s tochas t ic p rocesses ; namely,
1 )
a conceptual watershed storage
a t t h e end o f t h e t - t h t i me i n t e r v a l r e pr e se n t in g t h e s t o r ag e o f w a te r
on the ground su rfa ce , such as la kes , ponds, swamps and streams, as w e l l
as below the ground sur face, such as s o i l mo is t ure and groundwater res er -
v o i r s , 2 ) t h e t o t a l r a i n f a l l i n p u t d u ri n g t h e t - t h t i me i n t e r v a l , and
3 ) t h e t o t a l l o ss es , m a i n l y ev a po tr an s pi r a t i o n , d u r i n g t h e t - t h t i me
i n t e r v a l . These three component stochastic processes can be mathemat i-
c a l l y r e p re se n te d r e s p e c t i v e l y b y t i m e s e r i e s f u n c t i o n s [ ~ t ) ;ET],
[ ~ ( t ) tET] and [ E t ) ; ~ G T ] here T is the t ime range under cons idera t ion
o r the leng t h o f th e hy dro lo g i c record . These s to ch as t i c p rocesses can
be s imply denoted by S t ,
X t
and E t r e s p e c t i v e l y .
They are not cons idered
as independent bu t as a s toc ha s t i c vec t o r
[
( t ) x ( t ) E ( t ) ; ~ C T ] The
t heo r y o f t im e se r i es can t he r e f o r e be used t o f o r m u la t e t he s t och as t i c
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m od el o f t h i s v e c t o r . A r i g o r o u s m a th e m a ti ca l a n a l y s i s o f t h i s v e c t o r
w o u l d
r e q u i r e
t h e us e o f t h e t h e o ry o f m u l t i p l e t i m e s e r i e s
a n a l y s i s [ 6 1 .
I n v ie w o f t h e ac c ur ac y o f t h e n a t u r a l h y d r o l o g i c d a t a and f o r t h e p u r -
p os e o f p r a c t i c a l a p p l i c a t i o n w i t h o u t r e s o r t i n g t o e x c e ss iv e m at he m at ic a l
i nv o lv e m en t , t h e s t o c h a s t i c v e c t o r i s t o be an a ly z ed by t h e s i n g l e t im e
s e r i e s a n a l y s i s t ec h ni qu e s o f c o r r e lo g r a m and s p ec tr u m i n c o m b i na t io n
w i t h t h e c r os s -s p ec tr um t h e o r y w h i ch p r o v i d e s a p o w e r f u l t o o l i n t h e
a n a l y s is o f m u l t i p l e t im e s e r i e s .
By t h e b a s i c c o nc e pt o f sy s te m c o n t i n u i t y , t h e r u n o f f , w h i ch i s
a s t o c h a s t i c p ro ce ss o f t o t a l r u n o f f o u t p u t d u r i n g t h e t - t h t im e i n t e r v a l
as d e n o te d b y [ ~ t ) ;
€ ~
r s i m p l y
Y t
can be r e l a t e d t o t h e o t h e r t h r e e
co mpone nt s t o c h a s t i c p ro ce ss es o f t h e h y d r o l o g i c s ys te m as f o l l o w s :
w h e re S t m l i s t h e c o n c ep t u al w a te r sh e d s t o r a g e a t t h e b e g i n n in g o f t - t h
t im e i n t e r v a l .
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111
MATHEMATICAL TECHNIQUES
A. M a t h e m a t i c a l M o d e ls f o r T im e S e r i e s
I n t h i s s t u d y t h r e e m o de ls o f t i m e s e r i e s w h ic h h a ve b een u se d
i n h y d r o l o g i c s t u d y w e re r e v ie w e d . T he se m o de ls o r t h e i r c o m b i n a ti o ns
w o u ld b e e m plo ye d t o s i m u l a t e t h e h y d r o l o g i c s t o c h a s t i c p r o c es s e s . The
h y d r o l o g i c t i m e s e r i e s i s d e n ot ed b y [ u t ; t E T] w he re
u
i s t h e h y d r o l o g i c
t
v a r i a b l e a t t r i b u t e d t o t h e t - t h t im e i n t e r v a l a nd T i s t h e l e n g t h o f t h e
h y d r o
1
og i r e c o r d .
1
Mov i ng -Ave rage Mode l . T h i s model may be exp ress ed as
whe re
E
i s a ra nd om v a r i a b l e ; a l , a 2,
...
am a r e t h e w e i g h t s ; a nd
m
i s
t h e e x t e n t o f t h e m o vin g a v er a ge . T h i s e q u a t i o n may be ta k e n as t h e
m od el r e p r e s e n t i n g t h e r e l a t i o n b e tw e en , s a y , a n n u a l r u n o f f u a nd , s a y ,
annua l e f f e c t i v e p r e c i p i t a t i o n
E
where m i s t h e e x t en t o f t h e c a r r y ov e r
due t o t h e w a t e r - r e t a r d a t i o n c h a r a c t e r i s t i c s o f t h e w a te rs he d. F o r s uc h
a m o d e l, t h e w e i g h t s a l , a 2 ,
...
a
m ust be a l l p o s i t i v e and sum t o
m
u n i t y . By v i r t u e o f t h e m o vin g a ve ra ge o n t h e
E S ,
t h e s i m u l a t e d t i m e
s e r i e s u i s n o t random b u t s t o c h a s t i c .
2 . Sum-o f -Harmon ics Mode l . T h is mode l may be ex pre ss ed as
N
2IT t 2 IT - t
t
= A.
J cos
-J-+
B . s i n
+) + E t
where
A
and
0
a r e t h e a m p li t u de s; 2 r j t / T i s t h e p e r io d o f c y c l i c i t y
J
w i t h j
=
1,2,
...
and
N
b e i n g t h e num ber o f r e c o r d i n t e r v a l s i n m o nth s,
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ye ars o r o th e r u n i t s u sed i n t h e a n a l ys i s ; and
E~
i s a random variable.
Th is equat ion may be taken as a model repres ent ing a r egu lar o r os c i l la -
to ry form of va r i a t io ns , such as d i ur na l , seasonal and sec ular changes
th a t ex i s t f requen t l y i n hydr o log ic phenomena. Such va r i a t io ns a re o f
ne ar ly c onstan t pe ri od and they may be assumed sinu so ida l
as s imulated
i n the model .
3. Aut ore gre ssi on Model. The genera l for m o f t h i s model may
be expressed as
t
U
=
f (u t - l , t -2
E t
t - k
where
f
i s a mathemat i ca l f unct ion , k i s an in t eger , and E~ i s a r an -
dom v ar ia bl e.
A
sp e c ia l case o f t h i s model i s t h e l i n e a r a u to re g re ssi ve
model o f t he n- t h ord er :
where a
I,
a2,
. - a
a a r e t h e r e g r e s s i o n c o e f f i c i e n t s . For n
=
1
t h e
n
above equat ion becomes the f i rst-order Markov process:
where a i s the Markov-process co ef f ic i e n t .
The au to re gr es si on model may be used as a model represent ing
hy dr ol og ic sequences whose nonrandomness i s due t o st ora ge i n t he hydro-
l o g i c system, such as a watershed.
B . The Correlogram
The choice o f an appr opr i a te t im e ser i es model f o r a g iv en
hy dr ol og ic process i s no t an easy task because the above-mentioned thr ee
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m od e ls a l l e x h i b i t o s c i l l a t i o n s r es em b li n g t h e f l u c t u a t i o n s w h ic h one
u s u a l l y ob se rv es on m os t h y d r o l o g i c d a t a b y v i s u a l i n s p e c t i o n .
A
w e l l -
known a n a l y t i c a l a pp ro ac h w h i ch can h e l p o ne t o s e l e c t t h e b e s t mo del i s
t h e a n a l y s i s o f t h e sa m ple c o r r e lo g r a m .
The c o r re l og r am i s a g r a p h i c a l r e p r e s e n t a t i o n o f t h e s e r i a l
c o r r e l a t i o n c o e f f i c i e n t
r as a f u n c t i o n o f t h e l a g k w he re t h e v a lu e s
r
k
a r e p l o t t e d as o r d i n a t e s a g a i n s t t h e i r r e s p e c t i v e va l ue s o f k as a bs c is s a s
I n o r d e r t o r e ve a l t h e f e a tu r e s o f t h e c o r r e l og ra m b e t t e r , t h e p l o t t e d
p o i n t s a r e j o i n e d each t o t h e n e x t by a s t r a i g h t l i n e . The s e r i a l c o r r e -
l a t i o n c o e f f i c i e n t o f l a g k i s c om puted by
w h e re c o v u t , u
)
i s t h e sam ple a u t oc o v a r ia n c e and v a r u t ) and ~ a r u ~ + ~
t + k
a r e t h e s am p le v a r i a n c e ; o r
and
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The c o rr e lo g ra m p r o v i d e s a t h e o r e t i c a l b a s i s f o r d i s t i n g u i s h i n g
among t h e th r e e ty p es o f o s c i l l a t o r y t i m e s e r i e s m e nt io ne d p r e v i o u s l y . I t
has bee n p ro v ed a n a l y t i c a l l y t h a t i t h e t i m e s e r i e s i s s i m u la t e d b y a
m o v i n g- a v er a g e m o de l f o r ra ndo m e l e m e n t s o f e x t e n t m, t h e n t h e c o r r e l o -
gram w i l l show a d e c re a s in g l i n e a r r e l a t i o n s h i p and v a n is h es f o r a l l
v a lu e s o f
k
> m F o r a s um - of -h ar m on ic s m o de l, t h e c o r r e l o g r a m i t s e l f i s
a h a rm o n i c w i t h p e r i o d s e q u a l t o t h os e o f t h e h a rm o n ic c om ponents o f t h e
model and i t w i l l t h e r e f o r e show t h e same o s c i l l a t i o n s . I n t h e c ase o f
an a u t o r e g r e s s i o n m o de l, t h e c o r r e lo g r a m w i l l s how a da mp ing o s c i l l a t i n g
c u rv e . I n t h e ca se o f a f i r s t - o r d e r M arkov p ro ce ss w i t h a s e r i a l c o r r e l a t i o n
c o e f f i c i e n t r l
i t
w i l l o s c i l l a t e w i t h p e r i o d u n i t y a bove t h e a b s c i s sa
w i t h a d e c re a s in g b u t n o n v a n is h in g a m p l i t u d e
i f r l
i s n e g a t i ve [ 7 ]
t may b e n o t e d t h a t , w hen t h e t i m e s e r i e s i s t o o s h o r t , t h e
c om pu te d c o r re l o g r a m may e x h i b i t s u b s t a n t i a l s a m p l i n g v a r i a t i o n s a nd t h u s
may c o n c e a l i t s a c t u a l f o rm .
C
T he S p e c t ru m A n a l y s i s
T h i s m ethod i s an o th e r d i a g n o s t i c t o o l f o r t h e a n a l y s i s o f
t i m e s e r i e s i n t h e f r e q u e n c y do ma in , w h i c h c an h e l p d e v e l o p an a p p r o p r i a t e
t i m e s e r i e s m odel f o r t h e h y d r o l o g i c p r oc e s s .
A l l s t a t i o n a r y s t o c h a s t i c p ro ce ss es can b e re p re s e n te d i n t h e
f o r m
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w h e re
i = J i
and z ( w ) i s a c om p le x, ra nd om f u n c t i o n . U s i n g t h i s a s a
g e n e r a t i n g p r o c e s s ,
i t
c an be shown t h a t t h e a u t o c o v a r i a n c e f o r a s t a -
t o n a r y p ro c es s i s [ a ]
w h e re
=
k i s t h e t im e l ag , w i s t h e a n g u l a r f r e q u e n c y , a nd F ( w )/ y o
i s a d i s t r i b u t i o n f u n c t i o n m o n o t o n i c a l l y i n c r e a s i n g a nd b ou nd ed b etw ee n
F( - IT )
=
0 and F( IT)
=
Y o
=
o2
w he re i s . t h e s t a n d a rd d e v i a t i o n . The f u nc -
t i o n ~ ( w ) s c a l l e d t h e power s p e c t ra l d i s t r i b u t i o n f u n c t i o n . F o r k = 0 ,
E q. ( 1 2 ) g i v e s
w h i ch shows t h a t d F(w ) r e p r e s en t s t h e v a r i a n c e a t t r i b u t e d t o t h e f re q ue n cy
band (w, w+dw)
Thu s, dF (w) = f (w) dw w he re f (w)
i
ca ed
th e p o w e r
s p ec tr u m o f t h e p r o ce s s .
I n t h e p r a c t i c a l h y d r o l o g i c a p p l i c a t i o n o f t h e s p e c t r a l t h e or y
t h e pr o c es s e s a r e r e a l an d t h e i m a g i n a r y c om po ne nt i s d ro p p ed o f f , t h u s
Eq. (1 2) becomes
k
= IT
coskw
f
(w)dw
0
T he m a t h e m a ti c al i n v e r s i o n o f t h e a b ov e e q u a t i o n g i v e s t h e p o we r s p e c tr u m
F o r a f i n i t e a mount o f d a t a [ u t ; ~ E T ] n e s t i m a t e o f t h e p ow er s p e c tr u m i s
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w he re i s t h e a u t o c o v a r ia n c e f o r a t i m e l a g k .k
T he e s t i m a t e o f t h e p ow er s p e c t r u m b y Eq.
( 16 ) i s c a l l e d t h e
ra w s p e c t r a l e s t i m a t e b ec au se
t
d oe s, n o t g i v e a s m oo th p o w er s p e c t r a l
d ia g ra m . To a d j u s t f o r t h e s m oo th n es s,
i t
i s common t o use th e smoo thed
s p e c t r a l e s t i m a te ' ' i n t h e fo rm
w h e r e h
(w) a r e s e l e c t e d w e i g h t i n g f a c t o r s and
m
i s a n um be r t o b e c h o se n
k
much l es s than
T.
A co mm only u s e d w e i g h t i n g f a c t o r i s t h e T uk ey -H am m in g
w e i g h t s
[91:
57k
hk (w ) 0 .54 + 0.46 cos
whe re m i s t a k e n as l e s s t h a n T /1 0 .
The s i g n i f i c a n c e o f t h e s p ec t r um i s t h a t i t e x h i b i t s l e s s
s a m p li n g v a r i a t i o n s t h a n t h e c o r r e s p o n d in g c o rr e l o g ra m . C o n s e q ue n t ly , t h e
e s t i m a t e d s p e ct ru m w o u ld p r o v i d e a b e t t e r e v a l u a t i o n o f t h e v a r i o u s param -
e t e r s i n v o l v e d i n a m od el. f t h e g e n e r a t in g p r oc es s c o n t a i n s p e r i o d i c
t er m s , t h e f r e q u e n c i e s o f t h e s e te rm s w i l l a p p e a r as h i g h and s h a rp pe ak s
i n t h e e s t im a t e d s p ec tr um and t h e h e i g h t o f t h e p eaks w i l l g i v e a ro ug h
e s t i m a t e o f t h e amp1 i t u d e .
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I V
ANALYSIS OF THE HYDROLOGIC SYSTEM
A.
T he W a te rshed unde r S tudy
T he wa te rshed chosen as t h e h y d r o l o g i c sy s t em t o b e a na ly z ed i n
t h i s s t u d y i s t h e u p p er Sangamon R i v e r b a s i n o f 55 0 s q. m i i n s i z e , a bo ve
M o n t i c e l l o , I l l i n o i s , and l oc a te d i n e a s t c e n t r a l I l l i n o i s . The c r i t e r i a
f o r s e l e c t i n g t h i s w a t er sh ed a r e t h a t t h e a v a i l a b l e h y d r o l o g i c da t a su ch
as t h e p r e c i p i a t o n , s t r e a m f l ow and t e m p e r a t u r e r e c o r d s h av e a r e a s o n a b l y
c o n c u r re n t p e r i o d and t h a t a d d i t i o n a l d a t a
i
nee ded can be r e l a t i v e l y
e a s i l y c o l l e c t e d due t o co n ve n ie n t access t o i t s l o c a t i o n and t o i t s
d a ta c o l l e c t i n g a ge nc ie s . F i g u r e
1
shows th e map o f th e Sangamon R i ve r
b a s i n above M o n t i c e l l o , I l l i n o i s w i t h t h e l o c a t i o n s o f t h e st re am ga g i n g
s t a t i o n a t M o n t i c e l l o a nd t h e p r e c i p i t a t i o n g ages w h ere d a t a w ere o b se rv ed
f o r u s e i n t h e a n a ly s i s .
B. Th e H y d r o l o g i c D a ta
1. P r e c i p i t a t i o n . The m o n th ly p r e c i p i t a t i o n s i n in ch es w ere
used i n t h e a n a l y s i s a s t h e h i s t o r i c a l h y d r o lo g i 'c i n p u t s t o t h e w a te rs he d
sys tem.
The d a t a w e r e ta k e n f r o m t h e C l i m a t i c Summ ary o f t h e U n i t e d
S t a t e s p u b l i s h e d b y t h e U.S. W e ath er B u re au f o r I l l i n o i s . Th e p e r i o d
o f r e c o r d s u s e d i n t h e a n a l y s i s e x t e n d s f r o m O c t o b er 1 91 4 t h r o u g h S ep-
te m be r 1965 f o r s t a t i o n s a t U rb an a, C l i n t o n , B l o om i n g to n and R o b e r t s ,
f r o m M ar ch 194 0 t h r o u g h S e pte m be r 1 965 f o r t h e s t a t i o n a t R a n t o u l , a nd
f r o m J u ne 1 9 42 t h r o u g h S e pt em b er 1 9 65 a t M o n t i c e l l o . T he a v e r a g e m o n t h l y
p r e c i p i t a t i o n s o v e r t h e w a te r s he d w e re c om pu te d by t h e T h i e s s e n p o ly g o n
method.
2. S t r e a m f l o w . The m o n t h l y s t r e a m f l o w r e c o r d s f o r t h e
Sangamon R i v e r a t M o n t i c e l l o , I l l i n o i s , w ere u se d as t h e h i s t o r i c a l
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h y d r o l o g i c o u t p u t s o f t h e w a te rs h ed s y s te m i n t h e a n a l y s i s . The U.S.
G e o lo g ic a l Su rv ey , i n i t s c o o p er a t i ve p ro gra m w i t h t h e I l l i n o i s S t a t e
W a te r S u rv e y a nd o t h e r s t a t e , l o c a l a nd f e d e r a l a g e n c ie s , c o l l e c t s l o n g -
t e r m s t r e a m f lo w r e c o r d s t o d e t e r m i n e t h e p e r f o rm a n c e of r i v e r s a nd s tr e a m s .
The g a g i n g s t a t i o n o n t h e Sangam on R i v e r a b o u t o n e - h a l f m i l e w e s t o f
M o n t i c e l l o had p u b l is h e d d a t a a v a i l a b l e f o r t h e p e r io d s o f F e br ua ry 1908
t o December 1912 and June. 1914 t o S eptember 1968.
T h e m o n t h l y s t r e a m -
f l o w s f r o m S e pt em b e r 1 91 4 t h r o u g h S e p te m be r 1 96 5 w e r e u s e d i n t h e a n a l y s i s .
3 T e m p er at u re . I n t h e a n a l y s i s , t h e a v e r a g e m o n t h l y t em p er a-
t u r e s f r o m O c t o b e r 1 91 4 t h r o u g h S e p te m be r 1 96 5 w e r e t a k e n f r o m t h e
C l i m a t i c Summary o f t h e U n i t e d S t a t e s p u b l i s h e d b y t h e U S Weather
B u r e a u f o r l n o i s . The mean o f t h e m o n t h l y av e r ag e te m p e r a tu r e s a t t h e
s t a t i o n s i n U rb an a a nd B lo o m i n g t o n was c o n s i d e r e d a s t h e a v e ra g e m o n t h l y
t e m p e ra t u re o f t h e w a te rs h ed . The r e l a t i v e l o c a t i o n o f th e s e tw o s t a t i o n s
w i t h r e s p e c t t o t h e w at er sh e d has s u g ge s te d t h i s c h o i c e .
4. P o t e n t i a l E v a p o tr a ns p i r a t o n . N e ce ss ar y t o t h e a n a l y s i s o f
t h e w a te rs h ed h y d r o l o g i c sy ste m i s t h e e s t i m a t i o n o f t h e m o n th ly p o t e n t i a l
e v a p o t r a n s p i r a t i o n . T he re a r e s e v e r a l m etho ds f o r t h e c o m p u ta t io n o f t h e
p o t e n t i a l e v a p ot r an s p i r a t o n .
The method propo sed by Hamon [ I 0 1 was used
because i t h as be en t e s t e d i n
l
I 1 n o i s
[ ]
w i t h s a t i s f a c t o r y r e s u l t s and
t h e c o m p u t a t io n an d t h e d a t a r e q u i r e m e n t a r e r a t h e r s im p l e .
T he fo rm u l a p roposed by Hamon i s
whe re E i s t h e d a i l y p o t e n t i a l e v a p o t ra n s p i ra t i o n i n in ch es , D i s t h e
P
p o s s i b l e h o ur s o f s u n s h in e i n u n i t s o f 12 h o ur s and
P t
i s t h e s a t u r a t i o n
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v ap or d e n s i t y ( a b s o l u t e h u m i d i t y ) i n gram s p e r c u b i c m e t er a t t h e d a i l y
m ean t e m p e r a t u r e . T he v a l u e o f
D
depends o n t h e l a t i t u d e o f t h e w a te rs h ed
an d t h e m on th o f t h e y e a r .
Th e v a l u e o f P t d ep en ds o n t h e t e m p e r a t u r e .
T ab le s f o r e u a l u a t i n g t h e va lu e s o f
D
and P t a re p r ov i d ed by Harnon [121 .
The v a l u e o f
D
i s e s s e n t i a l l y t h e m o n t hl y d ay ti m e c o e f f i c i e n t o f t h e
H a rg r ea v es e v a p o t r a n s p i r a t i o n f o rm u l a [ 1 3 ] .
T he v a l u e o f P t c a n b e f o u n d
f r o m t h e S m i t h s o n i a n M e t e o r o l o g i c a l T a b l e s . F o r t h e w a t e r s h e d u n d e r c on -
s i d e r a t i o n , i t s a ve ra ge l a t i t u d e i s 40 N
The v a lu e s o f
D~
f o r t h e
t w e l v e m o nt hs a r e 0 .6 4 ( J a n . ) , 0 . 79 ( ~ e b . ) , 0 . 9 9 ( M a r . ) , 1 .2 2 ( A p r . ) ,
1 .4 4 ( M ay ), 1.56 ( ~ u n e ) , 1.51 ( ~ u l y ) , .3 1 ( A ug . ), 1 .0 8 ( s e p t . ) , 0 .8 6
( ~ c t . ) , 0 .69 ( N O V. ) , and 0 .6 1 ( ~ e c . ) .
The m o n t h l y p o t e n t i a l e v a p o t r a n s p i r a t i o n c an t h e n be co mp ute d b y
Epm 0.0055 ~ K D ~ P ~ ( 20 )
w h er e n i s t h e n um ber o f d ay s f o r e ac h m o nth an d K i s a c o r r e c t i o n f a c t o r
e q u a l t o 1 . 04 b e c a us e
P t
i s e s t i m a t e d f o r t h e m o n t h ly mean t e m p e r a t u r e
i n s t e a d o f t h e d a i l y m ean t e m p e ra t u re .
C E s t a b l i s h i n g t h e R ec or ds f o r C o n c e pt u al W a te rs he d S t o r a g e
a nd A c t u a l E v a p o t r a n s p i r a t i o n
R e w r i t i n g E q. ( 1 ) g i v e s
S i nc e t h e v a lu e s o f m o nt h l y p r e c i p i t a t i o n
X t
and m o n t h l y r u n o f f Y t a r e
known f ro m t h e h i s t o r i c a l r e co r d s, i t i s o b v i o u s f r o m t h e a b ov e e q u a t i o n
t h a t
i
t h e r e c o r d f o r t h e c o n c e pt ua l w a t er sh e d s t o r a g e
S t
were known
t he n t h e r e co r d f o r t h e a c t u a l m o n th ly e v a p o t r a n s p i r a t i o n E t c o u l d b e
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e a s i l y e s t a b l i s h e d .
On th e o t h e r hand , i f t h e r e c o rd o f E t were known and
an i n i t i a l v a lu e o f
S t
w e r e assu med, t h e n t h e r e c o r d o f
S t
c o u l d a l s o b e
e s t a b l i s h e d . U n f o r t u n a t e l y n e i t h e r S n o r E c an b e c om pu te d i n a d i r e c t
t
t
manner.
I t
i s know n, h o w ev e r, t h a t i n l a t e S ep te m be r an d e a r l y O c t o b e r
o f e ac h y e a r i n I l l i n o i s t h e am ount o f s u r f a c e w a t e r o n t h e w a te rs h ed a nd
t h e s o i l m o i s t u re a re a t a m inim um . E s p e c i a l l y i n t h e c as e o f v e r y lo w
a mount o f p r e c i p i t a t i o n d u r i n g t h e mo nth s o f A u gu s t, S ep te mb er a n d O c t o b e r ,
t h e w a t e r s h e d s t o r a g e m u st b e t h e l o w e s t . T h i s l o w e s t am ount o f s t o r a g e c an
b e c o n s i d e re d as t h e r e f e r e n c e p o i n t o f t h e c o n c e p t u a l w a t e rs h e d s t o r a g e .
.
I n o t h e r w o rd s, t h e c o n c e pt u a l w a t e rs h e d s t o r a g e i s t a k en a s z e r o a t t h e
b e g i n n in g o f t h e Oc to be r o f t h e y e a r h a v in g v er y lo w p r e c i p i t a t i o n d u r i n g
t h e mo nth s o f A u g u s t , S ep te m be r a nd O c t o b e r . I n t h e p r e s e n t a n a l y s i s ,
t h i s ha ppens t o b e t h e c as e f o r t h e y e a r o f 1914.
Once t h e i n i t i a l s t a g e o f t h e c o n c ep tu a l w a te rs he d s t o ra g e i s
e s t a b l i s h e d , t h e f o l l o w i n g p r o c ed u r e may b e f o l l o w e d t o e s t a b l i s h t h e
r e c o r d s o f c o n c e p t u a l w a t e rs h e d s t o r a g e and a c t u a
1
e v a p o t r a n s p
i
a t i o n .
I f
S t - l
+
X t
Y t E p t where E
i s t h e p o t e n t i a l e v ap o tr an -
P
t
s p i r a t i o n f o r t h e t - t h t im e i n t e r v a l , t he n th e a c tu a l ev a p o t ra n s p ir a -
- E p t .
T hus, t h e i n i t i a l s t o ra g e
S t
f o r t h e ne x t t i m e i n t e r v a l
i o n E t -
can be computed by Eq. 1 ) .
I f
X t
- Y t
<
E p t t h e n E t = S t - l +
X t -
Y t and Eq. 1 )
g i v e s
S t
= 0.
T he mass c u r v e s o f
X t
Y t E t and
S t
St
a r e s hown i n F i g . 2 .
Th e d i f f e r e n c e b et we e n C X t and C Y t i s e s s e n t i a l l y e qu a l t o C E t s i n c e
C s t -
S t-l) i s r e l a t i v e l y s m a l l as p l o t t e d i n an e n la r ge d s c a l e .
The
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m ass c u r v e f o r
S t
- S t-1 r e p r e s en t s t h e v a r i a t i o n i n co n c e pt u a l w a te r sh e d
s t o r a g e w i t h a mean o f
3.5
i n c h e s .
D
A n a l y s i s o f t h e H y d r o l o g i c Pr oc es se s
I n t h i s a n a l y s i s , t h e s t o c h a s t i c p ro ce ss es of p r e c i p i t a t i o n ,
c o n c ep t u al w a te rs h ed s t o r a g e a nd e v a p o t r a n s p i r a t i o n a r e n o t t o b e t r e a t e d
i n d e p e n d e n tl y o f e a ch o t h e r b u t t h e y a r e c o n s i d e r e d as a t h r e e -d i m e n s i o n al
v e c t o r o r a m u l t i p l e - t i m e s e r i e s . W i th o ut i n t r o d u c i n g t h e t h e or y o f
m u l t i p l e - t i m e s e r i e s , w h i ch ha s y e t t o be f u r t h e r d e ve lo p ed and
r e f i n e d ,
t h e f o l l o w i n g a s su m pt io ns a r e t o be made i n t h e p r e s e n t a n a l y s i s :
a )
Each s t o c h a s t i c p r oc e s s c o n s i s t s o f t w o p a r t s ; n am e ly , o ne
d e t e r m i n i s t i c and t h e o t h e r random and u n c o r r e l a t e d t o t h e d e t e r m i n i s t i c
p a r t a nd t h e p a r t s o f o t h e r p ro ce ss es .
b )
The d e t e r m i n i s t i c p a r t o f e ac h s t o c h a s t i c p ro ce ss c o n s i s t s
a l s o o f tw o p a r t s ; o ne p a r t d ep en din g o n l y o n t i m e and t h e o t h e r p a r t
d ep en din g on t h e v e c t o r o f
t h e s o c h as
t
p ro ce ss es o f p r e c i p i a t o n ,
c o n c ep t u al w a te rs h ed s t o r a g e and a c t u a l e v a p o t r a n s p i r a t i o n a t p r e v i o u s
t im e i n t e r v a l s .
Based on t h e a bo ve as s um p ti on s , t h e f i r s t s t e p i s t o d e t e rm i n e
t h e d e t e r m i n i s t i c p a r t o f ea ch p ro c e s s w h i c h dep ends o n t i m e . From t h e
e x p e ri en c e i n h y d ro lo g y and t h e e x h i b i t i o n o f h y d r o l o g i c d at a , t h e
d e t e r m i n i s t i c p a r t ap pe ars t o be a p e r i o d i c f u n c t i o n r a t h e r t ha n a p o l y -
n o m i a l o f t i m e . H en ce , t h e s a m p le c o r r e l o g r a m s c a n b e c om p ut ed f o r e a ch
p ro c es s t o t e s t t h e e x i s t e n c e o f h a rm o n ic co mp onen ts i n t h e p ro c es s .
The s e r i a l c o r r e l a t i o n c o e f f i c i e n t s r k f o r t i m e l a g k f o r t h e
p r oc e ss e s o f p r e c i p i t a t i o n , c o n c e p t u a l w a t e rs h e d s t o r a g e an d t h e ev ap o-
t r a n s p i r a t i o n w e r e c om p ute d b y Eqs.
7),
8 ) , 9) and 10 ) f o r 1 ,2 ,. . T.
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I n t h e p re s e n t s t ud y , T i s t h e l e n g t h o f t h e r e co rd s e qu a l t o
6 12 m o n th s a nd k i s f r o m z e r o t o n / l O , sa y 6 0. T he c o r r e l o g r a m s , o r t h e
p l o t s o f v er su s k , f o r p r e c i p i t a t i o n , c o n c ep tu a l w a te rs he d s t o r a g e and
k
e v a p o t r a n s p i r a t i o n a r e shown i n F i g s .
3 ,
and 5 r e s p e c t i v e ly . F or a l l
t h r e e p ro ce ss es t h es e c o rr e l og ra m s a r e o s c i l l a t i n g w i t h o u t any i n d i c a t i o n
o f d am pin g, t h u s r e v e a l i n g t h e pr e se n ce o f h a r m o n i c c om po ne nts i n a l l t h e
p r o c e s s e s
I n o r d e r t o d e t e r m i ne t h e p e r i o d s o f t h e h a r m o n ic c om po ne nts
w h ic h w i l l b e i n c lu d e d i n t h e model t o s i m u l a t e t h e h y d r o l o g i c p r oc es se s
and t h e h y d r o l o g i c s y st em ,
t h e p ow er s p e c t ru m f o r e a ch o f t h e p r oc e ss e s
shou l d be compu ted .
F ro m E qs . ( 1 6 ) a nd ( 1 7 ) , t h e ra w a n d s m oo th e d s p e c t r a l e s t i m a t e s
may be w r i t t e n r e s p e c t i v e l y as
and
1
Ub,) ( c0 c o s - + X Ck t cos
IT^
m m m
S u b s t i t u t i n g Eq . ( 1 8 ) f o r t h e T uk ey -H am m in g w e i g h t s i n Eq . ( 23 )
a n d s i m p l i f y i n g ,
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i
ce
Trkt
COS
~ r k ~ r k
k c o s - ( t + l ) + cos - ( t - 1 )
OS
m m m m
and
1
cos T r t = . c o s T r ( t + l ) + c o s T r ( t - l ) ]
2
Eq. (2 4) becomes
m-
1
Trk
+
0 23
[ c +
2 k COS - ( t + l )
+
Cm c o s ( t + l ) ]
2Tr 0 m
Trk
=C
+
2
C k cos
- ( t - 1 ) + Cm cos ~ ( t - 1 ) (2 7 )
2Tr
0
. . m
As t h e r aw s p e c t r a l e s t i m a t e s c a n b e r e p r e s e n t e d b y E q. ( 2 2 ) , E q. ( 2 7 ) may
b e w r i t t e n as
u ( w t )
=
0.23 L ( U ~ - ~ ) 0 .54 L (w t )
+
0 .23 L (o t+ , )
C om pu te r p ro g ra m s w e r e w r i t t e n t o c om pu te t h e a u t o c o v a r i a n c e b y
Eq.
(8 ) and th e raw and smoothed s p e c t r a l es t i m a t es by Eqs (22 ) and (28 )
The s mo oth ed s p e c t ra f o r p r e c i p i t a t i o n ,
c o n c e p t u a l w a t e r s h e d s t o r a g e a n d
e v a p o t r a n s p i r a t i o n a r e shown i n F i g s . 6 , 7 a nd 8, r e s p e c t i v e l y . T he s h a r p
peaks e x h i b i t e d i n t he s e s p e c t ra i n d i c a t e a s i g n i f i c a n t amount o f t h e
v a r i a n c e w i t h , h e p e r i o d i c i t i e s o f 12-m onth a nd 6-m onth w h ic h a r e
a p p r o p r i a t e f o r u se i n t h e m od el.
E . D e t e r m i n a t i o n o f t h e S ys te m M od el
The p r o po s ed model f o r t h e h y d r o l o g i c p r o ce s s es i s a c o m b i n a t i on
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o f the sum-of-harmoni cs and. the autogress o n time se ri es models.
i ce
the re su l t s of the corre logram and spectr a l analyses in di ca te the presence
of the 12-month and 6-month pe ri od ic i ie s, t h e general model f o r t he
hyd rol ogi c s to cha st i c processes under s tudy may be wr i t t en i n the form
2lT t 2lTt
U t = c1 +
c
s i n
- +
c3 C O S -
2 12
4lTt 4Tt
+
c 4 s i n
+
c5
os
2 U
where cl, c2, c3, c4 and c are the co ef f i c i en t s t o be estimated and u
t
i s the res idu al st och ast ic process w i t h zero mean. This model was there-
fo re used to i t the hydro log ic processes o f p re c i p i ta t i on , conceptua l
watershed stora ge, and ev ap ot ra ns pi ra ti on by th e lea st-sq uare method such
as the one descr ib ed by Brown
[14 ] .
The coe f f ic ie n t s o f the model de ter -
mined f o r pr ec ip i t a t on, conceptual watershed stora ge and evapotranspi ra-
t i on a re as fo l l ows :
The f i r s t f i v e terms i n the t ime ser ies model represented by
Eq. (29) are a po r t io n o f the deter min is t ic par t o f the s imula ted hydro-
lo g i c s tochas t ic processes. The f i r s t term is a cons tant wh i l e the second,
th i r d , fo ur th and f i f t h terms are det erm in i s t i c harmonics as func t ions o f
t ime. The la s t te rm u represents the res idua l s tochas t ic process which
may consis t of a de te rm in is t i c po r t io n and
the random p a r t o f th e model.
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T h i s d e t e r m i n i s t i c p o r t i o n may be c o r r e l a t e d w i t h t h e v e c t o r o f t h e p ro c -
e s s e s o f precipitation c o n c e p t u a l w a t e rs h e d s t o r a g e a nd e v a p o t r a n s p i r a -
t i o n a t p r e v i o u s t i m e i n t e r v a l s , w h i l e t h e random p a r t o f t h e p ro c es s may
b e s i m u l a t e d by a r e p r e s e n ta t i v e p r o b a b i l i t y d i s t r i b u t i o n .
T h e d e t e r m i n a -
t i o n o f a s u i t a b l e model f o r t h e r e s i d u a l s t o c h a s t i c p ro ce ss w i l l r e q u i r e
f u r t h e r i n v e s t i g a t io n . I n f u r t h e r i n v e s t i g a t io n ,
t
may be suggested
t h a t t h e d e t e r m i n i s t i c p o r t i o n o f t h e r e s i d u a l s t o c h a s t i c p ro ce ss es be
a n a l y z e d b y t h e c r o s s - s p e c t rum t h e o r y
[ e l
A t h o ug h t h e r e s u a l s o c h a s -
t i c p ro ce ss i s a s i g n i f i c a n t com ponent o f t h e m od el , i t s m ag n i t ud e i s o f
r e l a t i v e l y lo w o r d e r . As a f i r s t a p p ro x im a t io n t h e re s i d u a l s t o c h a s t i c
p r o c e s s e s i n t h e w a t e r s h e d s y s t e m may b e c o n s i d e r e d c o m p l e t e l y ra nd om
w i t h t h e i r means equa l t o ze ro . T hus, f o r t he p r es en t s tu dy , X;=E;=S;=O
a nd t h e i r v a r i a n c e s w er e f o u n d t o b e 2 .7 54 , 0 .4 65 a nd 4 .1 36 r e s p e c t i v e l y .
T h e i r p r o b a b i l i t y d i s t r i b u t i o n s may be r o u g h l y assumed a s no rm a l a t p r e s e n t
u n t i l b e t t e r p r o b a b i l i t y d i s t r i b u t i o n m ode ls a r e t o b e fo un d i n f u t u r e
i n v e s t i g a t i o n .
W i t h t h e h y d r o l o g i c p ro ce ss es o f p r e c i p i t a t i o n , c o nc e pt ua l
w a t e rs h e d s t o r a g e and e v a p o t r a n s p i r a t i o n b e i n g d e t er m i ne d , t h e r u n o f f
p rocess may be fo rm u l a te d f r o m Eqs. (1 ) a nd (29 ) as
n t Tl t
n t 0 . 0 3 0 3 s i n
t = 0.8036 + 0 . 5 0 2 4 s i n
T
.7778 cos g
3
+
0.6064 cos
nt
0 . 5 7 8 6 s i n
n ( t - 1 )
3
m(t- l) 2.3821 cos 6
+ 0 ,5583 s i n Tl (t - l)
-
0.1366
cos
+ X; - E
-
5 ;
-
S;-l)
( 31 )
3 3
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T h i s i s t h e sy s te m m od el e x p re s s e d f o r t h e r u n o f f p r o c e s s o f t h e u p p e r
Sangamon R i v e r b a s i n a bo ve M o n t i c e l l o I l l i n o i s . T h i s m od el c an b e
e m plo ye d t o g e n e ra t e s t o c h a s t i c m o n t h ly s t r e a m f l o w v a l u e s f o r u s e i n t h e
a n a l y s i s o f w a t e r r e s ou r c es s y st em s .
t
i s o f p a r t i c u l a r v a l u e i n t h e
e co no mic p la n n i n g o f w a t e r s u p p l y and i r r i g a t i o n p r o j e c t s w h i ch i s
con-
c e rn e d w i t h t h e lo ng -r an ge w a t e r y i e l d o f t h e w a te rs h ed .
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V
CONCLUSIONS
The u l t i m a t e o b j e c t i v e o f t h e r e s e a rc h o n t h e s t o c h a s t i c a n a ly -
s i s o f s t o c h a s t i c h y d r o l o g i c sy ste ms i s t o f o r m u l a t e t h e m a th em a ti c a l
model
f o r a s t o c h a s t i c h y d r o l o g i c sy st em f o r w h i ch a w a te r sh e d i s c on -
s i d e r e d . The u p pe r Sangamon R i v e r b a s i n a bo ve M o n t i c e l l o I l l i n o i s i s
t a k e n a s a n e x am p l e o f t h e w a te r s h e d .
T h i s s t u d y h as d e m o n s tr a te d t h a t
su ch a mode l i s f e a s i b l e and i t s a p p l i c a t i o n t o a p r a c t i c a l p ro b l e m i s
w o r k a b l e .
F or t h i s s t u d y t h e 1 i t e r a t u r e on s t o c h a s t i c p ro ce ss es a nd t h e i r
a p p l i c a t i o n i n h y d r o lo g y w e re re vi ew e d. I t was fo u nd t h a t t h e a p p l i c a -
t i o n o f t h e t h e o r y o f s t o c h a s t i c p r oc es se s i n h y d r o l o gy has b a r e l y begu n
and t h e t h e o r y has a p p l i e d m o s t l y t o s i n g l e p r oc es se s b u t n o t t o c o m po si t e
h y d r o l o g i c s ys te m s. The m a t h e m a ti c al t h e o r y o f s t o c h a s t i c pr oc e ss e s i s
v e r y e x t e n s iv e b u t u n f o r t u n a t e l y m os t o f
t
i s w r i t t e n n o t f o r p r a c t i c -
i n g e n g in e e rs and h y d r o l o g i s t s . F u rt he rm o r e a s y s t e m a t i c t h e o r y f o r
t h e f o r m u l a t i o n o f a s t o c h a s t i c s y st em m od el i s u n a v a i l a b l e be ca us e t h e
f o r m u l a t i o n o f t h e model r e q u i r e s t h e p r a c t i c a l k no wle dg e o n t h e p h y s i -
c a l c h a r a c t e r i s t i c s o f t h e p ro ce ss and t h e sy st em w h i ch i s u s u a l l y l a c k -
i n g o n th e p a r t o f t h e m a th e m a ti ci an . T h i s s t u dy t h e r e f o r e a t te m p t s t o
i n t r o d u c e t h e u se bf a t h e o r e t i c a l m ode l t o th e. s i m u l a t i o n o f p r a c t i -
c a l h y d r o l o g i c sy ste m .
Based on t h e p r i n c i p l e o f c o n s e r v a t i o n o f m ass t h e wa te r sh e d
s y s te m i s r e p r e s e n te d b y t h e mass b a l a n c e e q u a t i o n i n w h i c h t h e s y s te m
co mpone nts o f p r e c i p i t a t i o n c o n c e pt u a l w a te rs h ed s t o r a g e e v a p o t r a n s p i r -
a t i o n and r u n o f f a r e c o ns i de r e d as s t o c h a s t i c p ro c es s es . Whi l e t h e d a t a
o f p r e c i p i t a t i o n and r u n o f f a r e g i v e n a m eth od was d e ve lo p ed t o e s t ab -
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l i s h t h e u nknow n r e c o r d s o f c o n c e p t u a l w a te r s he d s t o r a g e a nd e v a p o t r a n-
s p i r a t i o n .
d e t e r m i n i s t i c p o r t i o n o f t h e s ys te m com ponent p r oc e ss i s
a n a l y z e d b y t h e t h e o r y o f c o r r e l o g r a m a nd s p e c tr u m . C om p ute r s u b r o u t i n e s
w e re p rog ra mm ed f o r t h e c o m p u t a t io n o f c o r r e l o g r a m s a nd s p e c t r a o f a
d i s c r e t e t i m e s e r i e s o f f i n i t e l e n g t h . The e xp ec te d v a lu e s o f t h e s ys te m
c om po ne nts o f p r e c i p i t a t i o n c o n c e p t u a l w a t e rs h e d s t o r a g e and e v a p o t r a n -
s p i r a t i o n w e re t h u s f ou n d t o be b e s t s i m u l a t e d b y h a rm o n ic s o f 12-m on th
and 6-m onth p e r i o d i c i t i e s . T h i s a n a l y s i s c o n s t i t u t e s an i m p o r t a n t s t e p
i n t h e a tt e m p t o f c o n s i d e r in g t h e n o n s t a t i o n a r i t y o f t h e pr oc es se s i n v o l v e d
i n t h e h y d r o l o g i c s y st e m b ec au se t h e e x p e c t e d v a l u e s a r e t a k e n as f u n c -
t i o n s o f t im e b u t n o t c o n st a n ts .
The h y d r o l o g i c s y s te m m od el s o f o r m u l a t e d f o r t h e u p p e r Sangam on
R i v e r b a s i n ca n b e used t o g e ne r at e s t o c h a s t i c s t r e a m f lo w s f o r t h e u se i n
t h e p l a n n in g o f w a t e r s u p p ly and i r r i g a t i o n p r o j e c t s i n t h e b a s i n . The
m ethod de ve lo pe d i n t h i s s t u d y i s t h e r e f o r e f o rm e d t o b e o f p r a c t i c a l
v a l u e i n t h e a n a l y s i s o f w a t e r r es o ur ce s s ys te m s;
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V I
ACKNOWLEDGMENT
T h i s r e p o r t i s t h e r e s u l t o f a r e se ar ch p r o j e c t on S t oc h as t ic
Ana ly si s of Hyd rol ogi c Systems sponsored by the U.S. Of f i c e o f Water
Resources Research, which began i n Ju l y 1968 and was completed i n June
1969. Under t he d i r e c t i o n o f t he P ro j ec t I nves t i ga to r , t he hyd ro l og i c
da ta used i n t h i s s t udy were ma in l y co l l e c ted by
M r
Gonzalo Cortes-
Rivera, Research As sis tan t i n C i v i l Engineer ing, and the mathematical
ana ly si s and computat ions were la rg el y performed by M r S o t i r i o s J .
Ka re l i o t i s , Research Ass i s t an t i n C i v i
1
Engineering.
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V I I . REFERENCES
1 .
Chow,
V .
T., S t a t i s t i c a l and p r o b a b i l i t y an a l y s i s o f h y d r o l o g i c
da ta : Par t
1 .
Frequency anal ysi s, Sec t ion 8 i n Handbook o f Ap pl ie d
:yd;r-;;gy
ed. by
V .
T. Chow, McGraw-Hi
1 1
Book Co., New York , 1964,
2. Chow, V . T.
A
genera l r epo r t on new ideas and s c i e n t i f i c methods i n
hydro logy (S imulat io n o f the hydro lo g
i
beha vio r of watersheds) ,
Proceedings
o r t Co l l i n s , Colo-
rado, 6-8 September 1967, pp. 50-65.
3. Chow,
V .
T., Hy dr ol og ic systems f o r wa ter resour ces management,
Conference Procee di ngs o f Hyd rol ogy i n Water Resources Management,
Water Resources Research I n s t i t u t e Report No. 4 Clemson Universi ty,
Clemson, S outh Ca ro li na , March 1968, pp. 8-22.
4. Chow,
V .
T., and Mer edit h,
D . D .
Water resources systems analysis
-
p a r t . I
~ n n o t a t e d i b1 ography on s t oc h as t ic proce;ses, ~ i v i ~ n ~ i -
neer ing Stud ies, Hydr au l i c Engineer ing Ser ies No.
19
U n i v e r s i t y o f
Il l inois Urbana, I l l i n o i s , Ju ly 1969.
5. Chow,
V .
T., and Mer edit h,
D . D .
Water resource s systems ana ly si s
-
P a r t I l l . Review o f s toc has t i c p rocesses, C i v i l Eng ineer ing S tud ies,
Hyd rau l ic Engineer ing Ser ies No. 21, Un iv er si ty o f I1 in o i s, Urbana,
I l l i n o i s , J u l y 1969.
6 . Quenou i l l e ,
M . H .
The a n al ys i s o f m u l t i p l e t ime se r i e s , Hafne r
Pu bl i sh in g Co., New York , 1957.
7. Dawdy, D . R . and Matalas, N . C . Ana lys i s o f va r iance , covar iance ,
and t ime ser ies , Sect ion 8-111 , Par t I l l i n Handbook o f App l ied
Hydrology, ed. by
V .
T. Chow, McGr aw-H il l Book Co., New Yo rk , 1964,
8. Granger,
C .
W. J. , and Hatanaka,
M.
Spectra l an a ly sis o f economic
t im e se ri es , Pr inc et on Un iv er si ty P.ress, Prin cet on, New Jersey, 1964.
9. Blackman,
R.
B .
and Tukey, J. W . The measurement of power spectra,
Dover Pu bl ic at io ns , In c. , New York, 1959.
10. Hamon, W.
R.
Est imat ing po te n t ia l evapo t ra nsp i ra t i on , Proceedings,
American Soc ie ty o f C i v i l Eng ineers , Journa l o f Hydrau l i cs D iv i s i on ,
Vol. 87 No.
H Y 3
pp. 107-120, May 1961.
1 1 .
Jones, D M. A., V a r i a b i l i t y o f e va po tr an sp ir at io n i n I l l i n o i s ,
l 1 1
no is S ta t e Water Survey C i r cu la r 89
1966.
12. Hamon,
W.
R.
Est ima t ing po te n t ia l evapo t rans p i ra t ion , Massachuset ts
I n s t i t u t e o f Technology Depar tment o f C i v i l and San i ta ry Eng ineer ing ,
unpubl ished
M . S .
thesis, 1960.
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13
Veihmeyer
F J.
Evapot ransp i ra t ion Sect ion
1 1
i n Handbook of
11-30
pp lie d Hydrology ed. by V . T Chow McGraw-Hi ll Book Co. p.
1964.
14. Brown
R .
C .
Smoothing for eca s t in g and pr ed ic t io n o f d isc re te t ime
ser ies Prent ic e Ha l l Inc . Englewood C l i f f s
N . Y .
1962.
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VIII F GURES
Fig . 1
F ig .
2
F ig . 3
F ig . 4
Fig . 5
F ig . 6
Fig . 7
F ig . 8
Sangamon River bas in above Mon t ice l lo I l l i n o i s
Mass cur ves o f p r ec ip i t a t i o n evapo t r ansp i r a t i on r uno f f
and conceptual watershed storage
C o r r e l o g r a m f o r p r e c i p i t a t i o n
Corre logram f o r conceptual watershed storage
Correlogram
qr
e v a p o t r a n s p i r a t i o n
S p e c t r u m o f p r e c i p i t a t i o n
Spect rum o f conceptual watershed storage
Spect rum o f evapo t r ansp i r a t i on
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3O
VON
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