CHOW, Ven Te. Stochastic Models

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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



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



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



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 .



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]



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



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 .



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,



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



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



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



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



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 .



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



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



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



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



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.



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 ,



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



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.



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



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 .



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 -



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;



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.



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.



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.



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





3O

VON













Documents

CHOW, Ven Te. Stochastic Models