Chow-analisis_model.pdf

WRC RESEARCH REPORT NO. 26

STOCHASTIC ANALYSIS OF HYDROLOGIC SYSTEMS

Ven Te Chow Principal Investigator

F I N A L

R E P O R T

P r o j e c t No. A-029- ILL

The work upon which t h i s p u b l i c a t i o n i s based was s u p p o r t e d by funds p r o v i d e d by t h e U.S. Department o f t h e I n t e r i o r as a u t h o r i z e d under t h e Water Resources Research A c t o f 1964, P.L. 88-379 Agreement No. 14-0 1-000 1 - 1632

UNIVERSITY OF ILLINOIS WATER RESOURCES CENTER 3220 C i v i l E n g i n e e r i n g B u i l d i n g Urbana, l l l i n o i s 61801 December, 1969

ABSTRACT STOCHASTIC ANALYSIS OF HYDROLOGIC SYSTEMS

H y d r o l o g i c phenomena a r e i n r e a l i t y s t o c h a s t i c i n nature; t h a t i s , t h e i r behavior changes w i t h t h e t i m e i n accordance w i t h t h e law o f p r o b a b i l i t y as w e l l as w i t h t h e s e q u e n t i a l r e l a t i o n s h i p between t h e occurrences o f t h e phenomenon. I n o r d e r t o analyze t h e h y d r o l o g i c phenomenon, a mathem a t i c model o f t h e s t o c h a s t i c h y d r o l o g i c system t o s i m u l a t e t h e phenomenon must be formulated. I n t h i s study, a watershed i s t r e a t e d as the s t o c h a s t i c h y d r o l o g i c system whose components o f p r e c i p i t a t i o n , r u n o f f , storage and e v a p o t r a n s p i r a t i o n a r e s i m u l a t e d as s t o c h a s t i c processes by time s e r i e s models t o be determined b y correlograms and s p e c t r a l a n a l y s i s . The h y d r o l o g i c system model i s then formulated on t h e basis o f t h e p r i n c i p l e of conservation o f mass and composed o f the component s t o c h a s t i c processes. To demonstrate the p r a c t i c a l a p p l i c a t i o n o f t h e method o f a n a l y s i s so developed, the upper Sangamon River b a s i n above M o n t i c e l l o i n e a s t c e n t r a l I l l i n o i s i s used as the sample watershed. The watershed system model so formulated can be employed t o generate s t o c h a s t i c streamflows f o r p r a c t i c a l use i n the a n a l y s i s o f water resources systems. This i s o f p a r t i c u l a r value i n t h e economic planning o f water supply and i r r i g a t i o n p r o j e c t s which i s concerned w i t h t h e long-range water y i e l d o f the watershed.

Chow, Ven Te STOCHASTIC ANALYSIS OF HYDROLOGIC SYSTEMS Research Report No. 26 , Water Resources Center, U n i v e r s i t y o f I l l i n o i s a t Urbana-Champaign, December 1969, 3 4 pp. KEYWORDS--systems a n a l y s i s / s t o c h a s t i c processes/synthetic hydrology/ water resources development/watershed studies/precipitation/streamflow/ evapotranspiration/storage/water y i e l d / h y d r o l o g i c models/hydrology

CONTENTS

. I1.

I

.

I11

IV

.

V. V I

.

. V I I I. V I I

......................... 1 F o r m u l a t i o n o f t h e H y d r o l o g i c System Model . . . . . . . . . . 3 Mathematical Techniques ................... 5 A . Mathematical Models f o r Time S e r i e s . . . . . . . . . . . 5 1 . Moving-Average Model . . . . . . . . . . . . . . . . . 5 2 . Sum-of-Harmonics Model . . . . . . . . . . . . . . . . 5 3 . A u t o r e g r e s s i o n Model . . . . . . . . . . . . . . . . . 6 B. TheCorrelogram . . . . . . . . . . . . . . . . . . . . . 6 C . The Spectrum A n a l y s i s . . . . . . . . . . . . . . . . . . 8 Analysis o f t h e H y d r o l o g i c S y s t e m . . . . . . . . . . . . . . 1 1 A . The Watershed under Study . . . . . . . . . . . . . . . . 1 1 B. TheHydrologicData . . . . . . . . . . . . . . . . . . . 1 1 1. Precipitation . . . . . . . . . . . . . . . . . . . . 11 2 . Streamflow . . . . . . . . . . . . . . . . . . . . . . 1 1 3 . Temperature . . . . . . . . . . . . . . . . . . . . . 12 4 . 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 . . . . . . . . . . . . . 12 C . E s t a b l i s h i n g t h e Records f o r Conceptual Watershed 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 . . . . . . . . . . 13 D . A n a l y s i s o f t h e H y d r o l o g i c Processes . . . . . . . . . . . . 15 E. D e t e r m i n a t i o n o f t h e System Model . . . . . . . . . . . . 17 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . Acknow 1 edgments . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction

'

Figures

...........................

I.

l NTRODUCT I ON

I t i s g e n e r a l l y n o t e d t h a t t h e n a t u r a l h y d r o l o g i c a l system and h y d r o l o g i c process a r e t r u l y "s tochas t i c";

t h a t i s , the behavior o f the

system o r t h e process v a r i e s w i t h a s e q u e n t i a l t i m e f u n c t i o n o f t h e probab i l i t y o f o c c u r r e n c e [1,2].9:

I n o t h e r words, t h e h y d r o l o g i c phenomenon

changes w i t h t h e t i m e i n accordance w i t h t h e law o f p r o b a b i l i t y as w e l l as w i t h t h e s e q u e n t i a l r e l a t i o n s h i p between i t s occurrences.

For example,

t h e o c c u r r e n c e o f a f l o o d i s c o n s i d e r 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 h e antecedant f l o o d c o n d i t i o n . Most c o n v e n t i o n a l methods f o r h y d r o l o g i c designs a r e " d e t e r ministic,"

t h a t i s , t h e b e h a v i o r o f t h e h y d r o l o g i c system o r process i s

assumed independent o f t i m e v a r i a t i o n s . '

F o r example, a u n i t hydrograph

derived f o r a given r i v e r basin f o r flood-control on h i s t o r i c a l f l o o d records.

p r o j e c t d e s i g n i s based

Once d e r i v e d , t h e u n i t hydrograph i s used

f o r analysis o f f u t u r e design floods.

Thus,

i t i s a u t o m a t i c a l l y assumed

unchanged w i t h t i m e ( f r o m t h e p a s t t o t h e f u t u r e ) and t h e r e f o r e i s deterministic. Some c o n v e n t i o n a l methods employ t h e concept o f p r o b a b i l i t y t o t h e e x t e n t t h a t no s e q u e n t i 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 . For example, t h e f l o o d r e c o r d i s analyzed and f i t t e d w i t h a c e r t a i n probab i l i t y d i s t r i b u t i o n t o determine t h e r e c u r r e n c e i n t e r v a l s o f t h e f l o o d o r the f l o o d frequencies.

Such methods a r e "probabi 1 i s t i c " b u t n o t i n t h e

t r u e sense " s t o c h a s t i c."

:

Numbers i n parentheses r e f e r t o r e f e r e n c e s l i s t e d a t t h e end o f t h e report.

The s t o c h a s t i c method, t h a t i s t o employ t h e concept o f probab i l i t y as w e l l as i t s s e q u e n t i a 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 of h y d r o l o g i c p r o j e c t s , because such methods have n o t been f u l l y developed.

While t h e n a t u r a l h y d r o l o g i c

phenomenon i s s t o c h a s t i c , i t i s i m p o r t a n t t o develop t h e s t o c h a s t i c method 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 system design.

Conventional methods,

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 , which do no't conform more c l o s e l y t o t h e n a t u r a l phenomenon, w i l l produce r e s u l t s t h a t d e p a r t from t h e t r u e b e h a v i o r o f t h e h y d r o l o g i c phenomenon and hence have 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 underdes i g n t h e h y d r o l o 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 d y i s t o f o r m u l a t e t h e mathematical model o f a s t o c h a s t i c h y d r o l o g i c system and t h e mathematical models o f t h e h y d r o l o g i c processes i n t h e system, p l e o f t h e h y d r o l o g i c system.

u s i n g t h e watershed as an exam-

I n t h i s s t u d y , i n o t h e r words, t h e frame-

work o f a method was developed t o u t i l i z e mathematical models 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 watershed as t h e h y d r o l o g i c system.

The

mathematical models s o developed should have 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 systems i n t h e w a t e r resources p l a n n i n g and development. 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 comprehensive review o f 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 process i n h y d r o l o 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 t e l y as "Water Resources Systems A n a l y s i s

-

Annotated B i b l i o g r a p h y on S t o c h a s t i c

Processes" [4.] and "Water Resources Sys tems Ana l y s i s Processes''

[5].

-

Review o f S t o c h a s t i c

11.

FORMULATION OF THE HYDROLOGIC SYSTEM MODEL

I n t h e f o r m u l a t i o n o f t h e h y d r o l o g i c system model, a watershed i s used as t h e h y d r o l o g i c system a l t h o u g h t h e mathematical approach would b e e q u a l l y a p p l i c a b l e t o o t h e r k i n d s o f h y d r o l o g i c systems w i t h some modif i c a t i o n s depending on t h e n a t u r e o f t h e system.

The watershed i s t r e a t e d

as a h y d r o l o g i c system w h i c h has an i n p u t , m a i n l y r a i n f a l l , and an o u t p u t , m a i n l y r u n o f f and e v a p o t r a n s p i r a t i o n .

The i n p u t and o u t p u t a r e t o b e

t r e a t e d as t i m e s e r i e s o r s t o c h a s t i c processes which d e s c r i b e t h e stochast i c b e h a v i o r o f t h e i n p u t and o u t p u t processes.

The amount o f w a t e r

s t o r e d i n t h e watershed 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 process w h i c h d e s c r i b e s 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 b s u r f a c e r u n o f f and t h e s o i l m o i s t u r e and groundwater s t o r a g e s . To f o r m u l a t e a mathematical model f o r t h e watershed h y d r o l o g i c system, t h e r u n o f f i s c o n s i d e r e d as t h e i n t e g r a l p r o d u c t o f t h r e e compon e n t s t o c h a s t i c processes ; namely,

( 1 ) a "conceptual watershed s t o r a g e "

a t t h e end o f t h e t - t h t i m e i n t e r v a l r e p r e s e n t i n g t h e s t o r a g e o f w a t e r on t h e ground s u r f a c e , such as l a k e s , ponds, swamps and streams, as w e l l as below t h e ground s u r f a c e , such as s o i l m o i s t u r e and groundwater r e s e r voirs,

( 2 ) t h e t o t a l r a i n f a l l i n p u t d u r i n g t h e t - t h t i m e i n t e r v a l , and

(3) t h e t o t a l l o s s e s , m a i n l y e v a p o t r a n s p i r a t i o n , d u r i n g t h e t - t h t i m e interval .

These t h r e e component s t o c h a s t i c processes can be mathemat i-

c a l l y r e p r e s e n t e d r e s p e c t i v e l y by t i m e s e r i e s f u n c t i o n s [ ~ ( t )tET], ; [ ~ ( t ;) tET] and [ E ( t ) ; ~ G T where ] T i s t h e t i m e range under c o n s i d e r a t i o n o r the length o f t h e h y d r o l o g i c record. b e s i m p l y denoted by S t ,

X t and E t ,

These s t o c h a s t i c processes can

respectively.

as independent b u t as a s t o c h a s t i c v e c t o r [ S ( t )

They a r e n o t c o n s i d e r e d

, x(t) ,

.

E ( t ) ; ~ C T ] The

t h e o r y o f t i m e s e r i e s can t h e r e f o r e b e used t o f o r m u l a t e t h e s t o c h a s t i c

model o f t h i s v e c t o r .

A r i g o r o u s mathematical a n a l y s i s o f t h i s v e c t o r

would r e q u i r e t h e use o f t h e t h e o r y 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 [61. I n view o f t h e accuracy 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 purpose 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 s s i v e mathematical involvement, t h e s t o c h a s t i c v e c t o r i s t o be analyzed by t h e s i n g l e t i m e s e r i e s a n a l y s i s techniques o f c o r r e l o g r a m and spectrum i n combination w i t h t h e cross-spectrum t h e o r y which 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 analysis o f m u l t i p l e time series. By t h e b a s i c concept o f system c o n t i n u i t y , t h e r u n o f f , which i s a s t o c h a s t i c process 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 i m e i n t e r v a l . as denoted by [ ~ ( t ) ; t € ~o ]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

component s t o c h a s t i c processes o f t h e h y d r o l o g i c system as f o l l o w s :

where S t m l

i s t h e conceptual watershed s t o r a g e a t t h e b e g i n n i n g o f t - t h

time i n t e r v a l .

111. A.

MATHEMATICAL TECHNIQUES

Mathematical Models f o r Time S e r i e s I n t h i s s t u d y t h r e e models o f t i m e s e r i e s which have been used

i n h y d r o l o g i c s t u d y were reviewed.

These models o r t h e i r combinations

would be employed 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 processes. h y d r o l o g i c t i m e s e r i e s i s denoted by [ u t ;

tET] where u

t

The

i s the hydrologic

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 i m e i n t e r v a l and T i s t h e l e n g t h o f t h e h y d r o 1 og ic record.

1.

where

E

Moving-Average Model.

i s a random v a r i a b l e ; al,

T h i s model may be expressed as

..., am a r e

a2,

t h e e x t e n t o f t h e moving average.

t h e w e i g h t s ; and m i s

T h i s e q u a t i o n may be taken as t h e

model r e p r e s e n t i n g t h e r e l a t i o n between, say, annual r u n o f f u and, say, annual 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 e n t o f t h e c a r r y o v 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 watershed. a model, t h e w e i g h t s al, unity.

a2,

..., am

For such

must be a l l p o s i t i v e and sum t o

By v i r t u e o f t h e moving average on t h e

E'S,

the simulated time

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-of-Harmonics Model.

T h i s model may be expressed as

N

Ut=

where A

j

1

( A . J cos

2IT't -J-+ T

BJ. s i n

2IT-t +)

+

Et

and 0 . a r e t h e amplitudes; 2 r j t / T i s t h e p e r i o 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 number o f r e c o r d i n t e r v a l s i n months,

y e a r s o r o t h e r u n i t s used i n t h e a n a l y s i s ; and

E~

i s a random v a r i a b l e .

T h i s e q u a t i o n may be t a k e n as a model r e p r e s e n t i n g a r e g u l a r o r o s c i l l a t o r y form o f v a r i a t i o n s , such as d i u r n a l , seasonal and s e c u l a r changes t h a t e x i s t f r e q u e n t l y i n h y d r o l o g i c phenomena.

Such v a r i a t i o n s a r e o f

n e a r l y c o n s t a n t p e r i o d and t h e y may be assumed s i n u s o i d a l as s i m u l a t e d i n t h e model.

3.

A u t o r e g r e s s i o n Model.

The g e n e r a l f o r m o f t h i s model may

be expressed as

u

where f (

= f(ut-l,

t

U t-2'

9

U

t-k

)

+

Et

) i s a mathematical f u n c t i o n , k i s an i n t e g e r , and

dom v a r i a b l e .

E~

i s a ran-

A s p e c i a l case o f t h i s model i s t h e l i n e a r a u t o r e g r e s s i v e

model o f t h e n - t h o r d e r :

where a I, a2,

. - a ,

a

n

are the regression c o e f f i c i e n t s .

For n = 1 , t h e

above e q u a t i o n becomes t h e f i r s t - o r d e r Markov process:

where a i s t h e Markov-process c o e f f i c i e n t

.

The a u t o r e g r e s s i o n model may be used as a model r e p r e s e n t i n g h y d r o l o g i c sequences whose nonrandomness i s due t o s t o r a g e i n t h e hydrol o g i c system, such as a watershed.

B.

The C o r r e l o g r a m The c h o i c e o f an a p p r o p r i a t e t i m e s e r i e s model f o r a g i v e n

h y d r o l o g i c process i s n o t an easy t a s k because t h e above-mentioned t h r e e

models a l l e x h i b i t o s c i l l a t i o n s resembling t h e f l u c t u a t i o n s which one u s u a l l y observes on most h y d r o l o g i c d a t a by v i s u a l i n s p e c t i o n .

A well-

known a n a l y t i c a l approach which can h e l p one t o s e l e c t t h e b e s t model i s t h e a n a l y s i s o f t h e sample correlogram. The c o r r e l o g r a m 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 correlation coefficient r

k

as a f u n c t i o n o f t h e l a g k where t h e values rk

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 values o f k as abscissas I n order t o reveal the features o f the correlogram b e t t e r , the 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 computed by

.

where c o v ( u t ,

u

t+k

) i s t h e sample a u t o c o v a r i a n c e and v a r ( u t ) and ~ a r ( u ~ + ~ )

a r e t h e sample v a r i a n c e ; o r

and

The c o r r e l o g r a 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 t h r e e types o f o s c i l l a t o r y t i m e s e r i e s mentioned p r e v i o u s l y .

I t

has been proved a n a l y t i c a l l y t h a t i f t h e t i m e s e r i e s i s s i m u l a t e d by a moving-average model f o r random elements o f e x t e n t m, then t h e c o r r e l o gram w i l l show a d e c r e a s i n g l i n e a r r e l a ' t i o n s h i p and vanishes f o r a l l v a l u e s o f k > m.

For a sum-of-harmonics

model, t h e c o r r e l o g r a m i t s e l f i s

a harmonic w i t h p e r i o d s equal t o those o f t h e harmonic components 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 case o f

an a u t o r e g r e s s i o n model, t h e c o r r e l o g r a m w i l l show a damping o s c i l l a t i n g curve.

I n t h e case o f a f i r s t - o r d e r Markov process 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 rl,

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 above t h e a b s c i s s a

w i t h a decreasing b u t nonvanishing a m p l i t u d e i f r l i s n e g a t i v e

[7].

I t may be n o t e d t h a t , when 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 computed c o r r e 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 sampling v a r i a t i o n s and thus may conceal i t s a c t u a l form.

C.

The Spectrum A n a l y s i s T h i s method

i s another 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 frequency domain, which can h e l p develop an a p p r o p r i a t e t i m e s e r i e s model f o r t h e h y d r o l o g i c process. A l l s t a t i o n a r y s t o c h a s t i c processes can be r e p r e s e n t e d i n t h e form

where i =

J-i- and

z(w) i s a complex,

g e n e r a t i n g process,

,

U s i n g t h i s as a

i t can 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 -

[a]

t i o n a r y process i s

where i =

random f u n c t i o n .

k i s t h e t i m e l a g , w i s t h e a n g u l a r frequency, and F(w)/yo

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 and bounded between F(-IT) = 0 and F(IT) =

= o2 where o i s . t h e s t a n d a r d d e v i a t i o n .

The func-

t i o n ~ ( w )i s c a l l e d t h e "power s p e c t r a l d i s t r i b u t i o n f u n c t i o n . "

For k = 0,

Yo

Eq. (12) g i v e s

,

which shows t h a t dF(w) r e p r e s e n 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 frequency band (w, w+dw)

.

Thus, dF (w) = f (w) dw where f (w) i s ca 1 1 ed t h e "power

spectrum" o f t h e process. 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 theory t h e processes a r e r e a l and t h e imaginary component i s dropped o f f , Eq.

thus

(12) becomes

k

= 2

/IT

coskw f (w)dw

0

The mathematical i n v e r s i o n o f t h e above e q u a t i o n g i v e s t h e power spectrum

For a f i n i t e amount o f d a t a [ u t ;

~ E T ]an e s t i m a t e o f t h e power spectrum i s

where C

k

i s t h e a u t o c o v a r i a n c e f o r a t i m e l a g k. The e s t i m a t e o f t h e power spectrum by Eq.

(16) i s c a l l e d t h e

"raw s p e c t r a l e s t i m a t e " because i t does, n o t g i v e a smooth power s p e c t r a l diagram.

To a d j u s t f o r t h e smoothness,

i t i s common t o use t h e "smoothed

s p e c t r a l e s t i m a t e ' ' i n t h e form

where h k (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 number t o be chosen much l e s s than T. weights

A commonly used w e i g h t i n g f a c t o r i s t h e "Tukey-Hamming"

[91: hk(w) = 0.54

+ 0.46 cos 57k

where m i s taken as l e s s than T/10. The s i g n i f i c a n c e o f t h e spectrum i s t h a t i t e x h i b i t s l e s s sampling v a r i a t i o n s than t h e corresponding correlogram.

Consequently, t h e

e s t i m a t e d spectrum would 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 parame t e r s i n v o l v e d i n a model.

I f t h e g e n e r a t i n g process c o n t a i n s p e r i o d i c

terms, t h e f r e q u e n c i e s o f these terms w i l l appear as h i g h and sharp peaks i n t h e e s t i m a t e d spectrum and t h e h e i g h t o f t h e peaks w i l l g i v e a rough e s t i m a t e o f t h e amp1 i t u d e .

IV. A.

ANALYSIS OF THE HYDROLOGIC SYSTEM

The Watershed under Study The watershed chosen as t h e h y d r o l o g i c system t o be analyzed i n

t h i s s t u d y i s t h e upper Sangamon R i v e r b a s i n o f 550 sq. m i . Monticello,

I l l i n o i s , and l o c a t e d i n e a s t c e n t r a l I l l i n o i s .

i n s i z e , above 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 watershed 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 d a t a such as t h e p r e c i p i t a t i o n , s t r e a m f low and temperature records have a reasonably c o n c u r r e 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 f needed 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 c o n v e n i e n t access t o i t s l o c a t i o n and t o i t s d a t a c o l l e c t i n g agencies.

F i g u r e 1 shows t h e map o f t h e Sangamon R i v e 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 stream gaging s t a t i o n a t M o n t i c e l l o and t h e p r e c i p i t a t i o n gages where d a t a were observed f o r use i n t h e a n a l y s i s .

B.

The H y d r o l o g i c Data 1.

Precipitation.

The monthly p r e c i p i t a t i o n s i n inches were

used i n t h e a n a l y s i s as t h e h i s t o r i c a l h y d r o l o g i ' c i n p u t s t o t h e watershed system.

The d a t a were taken from 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 by t h e U.S. Weather Bureau f o r I l l i n o i s .

The p e r i o d

o f records used i n t h e a n a l y s i s extends from October 1914 through September 1965 f o r s t a t i o n s a t Urbana, C l i n t o n , Bloomington and Roberts, f r o m March 1940 through September 1965 f o r t h e s t a t i o n a t R a n t o u l , and f r o m June 1942 through September 1965 a t M o n t i c e l l o .

The average monthly

p r e c i p i t a t i o n s o v e r t h e watershed were computed by t h e Thiessen polygon method. 2.

Streamflow.

The monthly s t r e a m f l o w records 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 , were used 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 watershed system i n t h e a n a l y s i s .

The U.S.

G e o l o g i c a l Survey, i n i t s c o o p e r a t i v e program w i t h t h e I l l i n o i s S t a t e Water Survey and o t h e r s t a t e ,

l o c a l and f e d e r a l agencies, c o l l e c t s long-

t e r m s t r e a m f low records t o determine t h e performance of r i v e r s and streams. The gaging s t a t i o n on t h e Sangamon R i v e r about one-half m i l e west o f M o n t i c e l l o had p u b l i s 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 i o d s o f February 1908 t o December 1912 and June. 1914 t o September 1968.

The monthly stream-

f l o w s from September 1914 through September 1965 were used i n t h e a n a l y s i s .

3.

Temperature.

I n t h e a n a l y s i s , t h e average monthly tempera-

t u r e s from October 1914 through September 1965 were taken from 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 by t h e U.S. Bureau f o r I l l i n o i s .

Weather

The mean o f t h e monthly average temperatures a t t h e

s t a t i o n s i n Urbana and Bloomington was c o n s i d e r e d as t h e average monthly '

temperature o f t h e watershed.

The r e l a t i v e l o c a t i o n o f these two s t a t i o n s

w i t h r e s p e c t t o t h e watershed has suggested t h i s c h o i c e .

4.

P o t e n t i a l Evapotranspi r a t i o n .

Necessary t o t h e a n a l y s i s o f

t h e watershed h y d r o l o g i c system i s t h e e s t i m a t i o n o f t h e monthly p o t e n t i a l evapotranspiration.

There a r e s e v e r a l methods f o r t h e computation o f t h e

p o t e n t i a l evapotranspi r a t ion.

The method proposed by Hamon [ I 0 1 was used

because i t has been t e s t e d i n l I 1 i n o i s [ I I ] 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 computation and t h e d a t a requirement a r e r a t h e r s i m p l e . The f o r m u l a proposed by Hamon i s

where E

P

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 r a n s p i r a t i o n i n inches, D i s t h e

p o s s i b l e hours o f sunshine i n u n i t s o f 12 hours and P t i s t h e s a t u r a t i o n

vapor 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 grams p e r c u b i c meter a t t h e d a i l y mean temperature.

The 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 watershed

and t h e month o f t h e year.

The v a l u e o f Pt depends on t h e temperature.

Tables f o r e u a l u a t i n g t h e values o f D and P t a r e p r o v i d e d 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 monthly daytime c o e f f i c i e n t of t h e Hargreaves e v a p o t r a n s p i r a t i o n formula [ 1 3 ] .

The v a l u e of P t can be found

f r o m t h e Smithsonian M e t e o r o l o g i c a l Tables.

For t h e watershed under con-

s i d e r a t i o n , i t s average l a t i t u d e i s 40" N . t w e l v e months a r e 0.64 (Jan.), 1.44 (May),

The v a l u e s o f D~ f o r t h e

0.79 ( ~ e b . ) , 0.99 (Mar.),

1.56 ( ~ u n e ) , 1.51 ( ~ u l y ) , 1.31 (Aug.),

1.22 ( A p r . ) ,

1.08 ( s e p t . ) ,

0.86

( ~ c t . ) , 0.69 ( N O V . ) , and 0.61 ( ~ e c . ) . The monthly 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 can then be computed by

Epm = 0.0055 ~ K D ~ P ~

(20)

where n i s t h e number o f days f o r each month and K i s a c o r r e c t i o n f a c t o r equal t o 1.04 because P t i s e s t i m a t e d f o r t h e monthly mean temperature i n s t e a d o f t h e d a i l y mean temperature.

C.

E s t a b l i s h i n g t h e Records f o r Conceptual Watershed Storage 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 R e w r i t i n g Eq. (1) g i v e s

Since t h e values o f monthly p r e c i p i t a t i o n X t and monthly r u n o f f Y t a r e known from t h e h i s t o r i c a l records, i t i s obvious f r o m t h e above e q u a t i o n t h a t i f t h e r e c o r d f o r t h e conceptual watershed s t o r a g e S t were known then t h e r e c o r d f o r t h e a c t u a l monthly 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 be

e a s i l y established.

On t h e o t h e r hand, i f t h e r e c o r d o f E t were known and

an i n i t i a l v a l u e o f S t were assumed, then t h e r e c o r d o f S t c o u l d a l s o be established. manner.

Unfortunately neither S

t

nor E

t

can be computed i n a d i r e c t

r

I t i s known, however, t h a t i n l a t e September and e a r l y October o f each y e a r i n I l l i n o i s t h e amount o f s u r f a c e w a t e r on t h e watershed and t h e s o i l m o i s t u r e a r e a t a minimum.

E s p e c i a l l y i n t h e case o f v e r y low

amount o f p r e c i p i t a t i o n d u r i n g t h e months o f August, September and October, t h e watershed s t o r a g e must be t h e lowest.

T h i s lowest amount o f s t o r a g e can

be considered as t h e r e f e r e n c e p o i n t o f t h e conceptual watershed s t o r a g e . . I n o t h e r words, t h e conceptual watershed s t o r a g e i s taken as z e r o a t t h e b e g i n n i n g o f t h e October o f t h e y e a r h a v i n g v e r y low p r e c i p i t a t i o n d u r i n g t h e months o f August, September and October.

I n the present analysis,

t h i s happens t o be t h e case 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 conceptual watershed s t o r a 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 procedure may be f o l l o w e d t o e s t a b l i s h t h e records o f conceptual watershed s t o r a g e and a c t u a 1 evapot ransp i r a t i o n . I f St-l

+

Xt

-

Y t 2 Ept where E

Pt

i s t h e p o t e n t i a l evapotran-

s p i r a t i o n f o r the t - t h time i n t e r v a l , then the a c t u a l evapotranspirat i o n Et

--

Ept.

Thus, t h e i n i t i a l s t o r a g e S t f o r t h e n e x t t i m e i n t e r v a l

can be computed by Eq. ( 1 ) .

If

+

Xt

-

Y t < Ept,

then E t = S t - l

+

Xt

-

Y t and Eq.

(1)

g i v e s S t = 0. The mass curves o f X t ,

Yt,

The d i f f e r e n c e between C X t and C Y t C(st

-

St-l)

E t and S t

-

S t -1

a r e shown i n F i g . 2.

i s e s s e n t i a l l y equal t o C E t s i n c e

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 l a r g e d s c a l e .

The

mass c u r v e f o r S t

-

St-1

r e p r e s e n t s t h e v a r i a t i o n i n conceptual watershed

s t o r a g e w i t h a mean o f 3.5

D.

inches.

A n a l y s i s o f t h e H y d r o l o g i c Processes 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 processes of p r e c i p i t a t i o n ,

conceptual watershed 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 a r e n o t t o be t r e a t e d independently o f each o t h e r b u t they a r e considered as a three-dimensional vector o r a multiple-time series.

Without i n t r o d u c i n g the theory o f

m u l t i p l e - t i m e s e r i e s , which has y e t t o be f u r t h e r developed and r e f i n e d , t h e f o l l o w i n g assumptions 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 process c o n s i s t s o f two p a r t s ; namely, one

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 and t h e p a r t s o f o t h e r processes. (b)

The d e t e r m i n i s t i c p a r t o f each s t o c h a s t i c process c o n s i s t s

a l s o o f two p a r t s ; one p a r t depending o n l y on t i m e and t h e o t h e r p a r t depending on t h e v e c t o r o f t h e s tochas t i c processes o f p r e c i p i t a t i o n , conceptual watershed 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 time intervals. Based on t h e above assumptions, t h e f i r s t s t e p i s t o determine t h e d e t e r m i n i s t i c p a r t o f each process which depends on time.

From t h e

e x p e r i e n c e i n h y d r o l o 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 data, t h e d e t e r m i n i s t i c p a r t appears 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 than a p o l y nomial o f time.

Hence, t h e sample correlograms can be computed f o r each

process t o t e s t t h e e x i s t e n c e o f harmonic components i n t h e process. 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 processes o f p r e c i p i t a t i o n , conceptual watershed s t o r a g e and t h e evapot r a n s p i r a t i o n were computed by Eqs.

(7), ( 8 ) , (9)

and (10) f o r t = 1,2,.

. . ,T.

I n t h e p r e s e n t s t u d y , T i s t h e l e n g t h o f t h e records equal t o 612 months and k i s from z e r o t o n/lO,say plots of r

k

60. The correlograms, o r t h e

versus k , f o r p r e c i p i t a t i o n , conceptual watershed 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 a r e shown i n F i g s . 3, 4 and

5 respectively.

For a l l

t h r e e processes these correlograms 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 damping, thus r e v e a l i n g t h e presence o f harmonic components i n a l l t h e processes

. I n o r d e r t o determine t h e p e r i o d s o f t h e harmonic components

which w i l l be i n c l u 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 processes and t h e h y d r o l o g i c system, t h e power spectrum f o r each o f t h e processes s h o u l d be computed. From Eqs.

(16) and ( 1 7 ) , t h e raw and smoothed 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

Ub,)

=

1

("c0

S u b s t i t u t i n g Eq. and s i m p l i f y i n g ,

c o s r-m k+t X C m m cos IT^)

(18) f o r t h e Tukey-Hamming w e i g h t s i n E q .

(23)

S i nce

COS

Trkt COS "k m m

cos

Trt

cos -~( tr+kl ) m

+

~rk cos -(t-1) m

and

Eq.

=

-.

1 2

[cos Tr(t+l)

+ cos T r ( t - l ) ]

(24) becomes

1

Ck

-Trk (t+l) m

+

Cm cos " ( t + l ) ]

1

Trk Ck cos -(t-1) m

+

Cm cos ~ ( t - 1 ])

m- 1

+

0. 23 [ c 0 2Tr

+ 2

,,,

+= [C

2Tr

0

+ 2

.

COS

m

.

As t h e raw s p e c t r a l e s t i m a t e s can be r e p r e s e n t e d by Eq.

(27)

( 2 2 ) , Eq. (27) may

be w r i t t e n as u(wt) = 0.23 L ( U ~ - ~+ ) 0.54 L(wt) + 0.23 L(ot+,)

Computer programs were w r i t t e n t o compute t h e a u t o c o v a r i a n c e by Eq.

(8) and t h e raw and smoothed s p e c t r a l e s t i m a t e s by Eqs

.

(22) and (28)

.

The smoothed s p e c t r a f o r p r e c i p i t a t i o n , conceptual watershed 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 a r e shown i n F i g s . 6, 7 and 8, r e s p e c t i v e l y .

The sharp

peaks e x h i b i t e d i n these s p e c t r a 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 , t h e p e r i o d i c i t i e s o f 12-month and 6-month which a r e a p p r o p r i a t e f o r use i n t h e model.

E.

D e t e r m i n a t i o n o f t h e System Model The proposed model f o r t h e h y d r o l o g i c processes i s a combination

o f the sum-of-harmoni cs and. t h e autogress i o n time s e r i e s models.

$ ince

t h e r e s u l t s o f t h e correlogram and s p e c t r a l analyses i n d i c a t e t h e presence of the 12-month and 6-month p e r i o d i c i t i e s , t h e general model f o r t h e h y d r o l o g i c s t o c h a s t i c processes under study may be w r i t t e n i n the form

Ut = c1

+

c

2

2lT t sin -+

+ where cl,

c4 s i n

c2, c3, c4 and c

c3

4lT t

5

COS-

2lT t 12

+ c5 cos

4lT t 12

+

U;

a r e t h e c o e f f i c i e n t s t o be estimated and u ' t

i s t h e r e s i d u a l s t o c h a s t i c process w i t h zero mean.

This model was there-

f o r e used t o f i t t h e h y d r o l o g i c processes o f p r e c i p i t a t i o n , conceptual watershed storage, and e v a p o t r a n s p i r a t i o n by t h e least-square method such as the one described by Brown [14].

The c o e f f i c i e n t s o f t h e model d e t e r -

' mined f o r p r e c i p i t a t ion, conceptual watershed s t o r a g e and evapotranspi ra-

t i o n a r e as f o l l o w s :

The f i r s t f i v e terms i n t h e time s e r i e s model represented by Eq. (29) a r e a p o r t i o n o f t h e d e t e r m i n i s t i c p a r t o f t h e simulated hydrol o g i c s t o c h a s t i c processes.

The f i r s t term i s a constant w h i l e t h e second,

t h i r d , f o u r t h and f i f t h terms a r e d e t e r m i n i s t i c harmonics as f u n c t i o n s o f time.

The l a s t term u;

represents t h e r e s i d u a l s t o c h a s t i c process which

may c o n s i s t o f a d e t e r m i n i s t i c p o r t i o n and the random p a r t o f t h 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 processes o f precipitation, conceptual watershed 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 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 process may be s i m u l a t e d by a r e p r e s e n t a 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 .

The determina-

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 process w i l l r e q u i r e further investigation.

i t may be suggested

I n further investigation,

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 processes be analyzed by t h e cross-spect rum t h e o r y

[el.

A 1 though t h e res idual s tochas-

t i c process i s a s i g n i f i c a n t component o f t h e model, i t s magnitude i s o f r e l a t i v e l y low o r d e r .

As a f i r s t a p p r o x i m a t i o n t h e r e s i d u a l s t o c h a s t i c

processes i n t h e watershed system may be considered c o m p l e t e l y random w i t h t h e i r means equal t o zero.

Thus, f o r t h e p r e s e n t study, X;=E;=S;=O

and t h e i r v a r i a n c e s were found t o be 2.754,

0.465 and 4.136 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 as normal 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 models a r e t o be found i n f u t u r e investigation. W i t h t h e h y d r o l o g i c processes o f p r e c i p i t a t i o n , conceptual watershed 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 determined, t h e r u n o f f process may be f o r m u l a t e d f r o m Eqs. (1) and' (29) as

Y t = 0.8036

+

0.5024 s i n

+ 0.6064 cos

+

nt T +1.7778

nt+ 0.5786 3

0,5583 s i n Tl(t-l) 3

-

cos -g nt

s i n 'm(t-l)

0.1366 cos

3

-

0.0303 s i n

Tl t 3

2.3821 cos n ( t -61 )

+

X;

- E; -

(5;

-

S;-l)

(31)

T h i s i s t h e system model expressed f o r t h e r u n o f f process of t h e upper Sangamon R i v e 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 .

T h i s model can be

employed t o g e n e r a t e s t o c h a s t i c monthly s t r e a m f l o w v a l u e s f o r use i n t h e a n a l y s i s o f w a t e r resources systems.

It i s o f p a r t i c u l a r value i n the

economic p l a 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 which i s concerned w i t h t h e long-range w a t e r y i e l d o f t h e watershed.

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 r c h on t h e s t o c h a s t i c a n a l y 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 systems i s t o f o r m u l a t e t h e mathematical 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 system f o r which a watershed i s considered.

The upper Sangamon R i v e 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 , i s

taken as an example o f t h e watershed.

T h i s s t u d y has demonstrated t h a t

such a model 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 problem i s workable. For 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 processes and t h e i r a p p l i c a t i o n i n h y d r o l o g y were reviewed.

I t was found 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 processes i n h y d r o l o g y has b a r e l y begun 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 processes b u t n o t t o composite h y d r o l o g i c systems.

The mathematical t h e o r y o f s t o c h a s t i c processes i s

v e r y e x t e n s i v e , b u t u n f o r t u n a t e l y most o f i 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 engineers and h y d r o l o g i s t s .

Furthermore, 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 system model i s u n a v a i l a b l e because 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 knowledge on 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 process and t h e system which i s u s u a l l y l a c k i n g on t h e p a r t o f t h e mathematician. i n t r o d u c e t h e use

bf

This study t h e r e f o r e attempts t o

a t h e o r e t i c a l model t o the. s i m u l a t i o n o f a p r a c t i -

c a l h y d r o l o g i c system. 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 mass, t h e watershed system i s represented by t h e mass balance e q u a t i o n i n which t h e system components o f p r e c i p i t a t i o n , conceptual watershed 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 considered as s t o c h a s t i c processes.

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 runoff a r e g i v e n , a method was developed t o estab-

l i s h t h e unknown r e c o r d s o f conceptual watershed s t o r a g e and evapotranspiration.

A 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 system component process i s analyzed by t h e t h e o r y o f c o r r e l o g r a m and spectrum.

Computer s u b r o u t i n e s

were programmed f o r t h e computation o f correlograms and s p e c t r a o f a discrete time series o f f i n i t e length.

The expected values o f t h e system

components o f p r e c i p i t a t i o n , conceptual watershed s t o r a g e and evapotrans p i r a t i o n were thus found t o be b e s t s i m u l a t e d b y harmonics o f 12-month and 6-month 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 t t e m p t o f c o n s i d e r i n g t h e n o n s t a t i o n a r i t y o f t h e processes i n v o l v e d i n t h e h y d r o l o g i c system because t h e expected values a r e taken as f u n c t i o n s o f time b u t n o t constants. The h y d r o l o g i c system model so f o r m u l a t e d f o r t h e upper Sangamon

.

R i v e r b a s i n can be used t o generate s t o c h a s t i c s t r e a m f l o w s f o r t h e use i n t h e p l a 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 i n t h e b a s i n .

The

method developed i n t h i s s t u d y i s t h e r e f o r e formed t o be 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 resources systems;

VI.

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 s e a r c h p r o j e c t on " S t o c h a s t i c A n a l y s i s o f H y d r o l o g i c Systems" sponsored by t h e U.S. O f f i c e o f Water Resources Research, which began i n J u l y 1968 and was completed i n June 1969.

Under t h e d i r e c t i o n o f t h e P r o j e c t I n v e s t i g a t o r , t h e h y d r o l o g i c

d a t a used i n t h i s s t u d y were m a i n l y c o l l e c t e d by M r . Gonzalo CortesR i v e r a , Research A s s i s t a n t i n C i v i l E n g i n e e r i n g , and t h e mathematical a n a l y s i s and computations were l a r g e l y performed b y M r . S o t i r i o s J. K a r e l i o t i s , Research A s s i s t a n t i n C i v i 1 Engineering.

VII.

1.

REFERENCES

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 a n a l y s i s o f h y d r o l o g i c data: P a r t 1 . Frequency a n a l y s i s , S e c t i o n 8 i n Handbook o f A p p l i e d ed. by V . T. Chow, McGraw-Hi 1 1 Book Co., New York, 1964,

:yd;r-;;gy, -

2.

Chow, V . T., A g e n e r a l r e p o r t on new ideas and s c i e n t i f i c methods i n h y d r o l o g y ( S i m u l a t i o n of t h e h y d r o l o g i c b e h a v i o r of watersheds) , Proceedings,, F o r t C o l l i n s , Colorado, 6-8 September 1967, pp. 50-65.

3.

Chow, V . T., H y d r o l o g i c systems f o r w a t e r resources management, Conference Proceedi ngs o f H y d r o l o g y i n Water Resources Management, Water Resources Research I n s t i t u t e Report No. 4, Clemson U n i v e r s i t y , Clemson, South C a r o l i n a , March 1968, pp. 8-22.

4.

Chow, V . T., and M e r e d i t h , D. D . , Water resources systems a n a l y s i s ~ n n o t a t e db i b 1 iography on s t o c h a s t i c proce;ses, ~ i v1 i ' ~ n ~ i part. I . n e e r i n g S t u d i e s , H y d r a u l i c E n g i n e e r i n g S e r i e s No. 19, U n i v e r s i t y o f Illinois, Urbana, I l l i n o i s , J u l y 1969.

5.

Chow, V . T., and M e r e d i t h , D. D., Water r e s o u r c e s systems a n a l y s i s P a r t I l l . Review o f s t o c h a s t i c processes, C i v i l E n g i n e e r i n g S t u d i e s , H y d r a u l i c E n g i n e e r i n g S e r i e s No. 21, U n i v e r s i t y of I 1 1 i n o i s , Urbana, I l l i n o i s , J u l y 1969.

6.

Q u e n o u i l l e , M. H., The a n a l y s i s o f m u l t i p l e t i m e s e r i e s , Hafner P u b l i s h i n g Co., New York, 1957.

7.

Dawdy, D. R., and M a t a l a s , N. C . , A n a l y s i s o f v a r i a n c e , c o v a r i a n c e , and t i m e s e r i e s , S e c t i o n 8-111, P a r t I l l i n Handbook o f A p p l i e d H y d r o l o g y , ed. b y V. T. Chow, McGraw-Hill Book Co., New York, 1964,

8.

Granger, C. W. J . , and Hatanaka, M., S p e c t r a l a n a l y s i s o f economic t i m e s e r i e s , P r i n c e t o n U n i v e r s i t y P.ress, P r i n c e t o n , New J e r s e y , 1964.

9.

Blackman, R. B . , and Tukey, J . W., The measurement of power s p e c t r a , Dover P u b l i c a t i o n s , I n c . , New York, 1959.

10.

Hamon, W. R., E s t i m a t i n g 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 , Proceedings, American S o c i e t y o f C i v i l Engineers, J o u r n a l o f H y d r a u l i c s D i v i s i o n , V o l . 87, No. H Y 3 , pp. 107-120, May 1961.

11.

Jones, D, M. A., V a r i a b i l i t y o f e v a p o t r a n s p i r a t i o n i n I l l i n o i s , l 1 1 i n o i s S t a t e Water Survey C i r c u l a r 89, 1966.

12.

Hamon, W. R., E s t i m a t i n g 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 , Massachusetts I n s t i t u t e o f Technology Department o f C i v i l and S a n i t a r y E n g i n e e r i n g , u n p u b l i s h e d M.S. t h e s i s , 1960.

-

-

13.

Veihmeyer, F. J., E v a p o t r a n s p i r a t i o n , S e c t i o n 1 1 i n Handbook o f 11-30, A p p l i e d Hydrology, ed. by V. T. Chow, McGraw-Hill Book Co., p. 1964.

14.

Brown, R. C . , Smoothing, f o r e c a s t i n g and p r e d i c t i o n o f d i s c r e t e time s e r i e s , P r e n t i c e H a l l , Inc., Englewood C l i f f s , N.Y., 1962.

VIII. Fig.

1.

F l GURES

Sangamon R i v e r b a s i n above M o n t i c e l l o ,

Illinois

F i g . 2.

Mass curves o f p r e c i p i t a t i o n , e v a p o t r a n s p i r a t i o n , and conceptual watershed s t o r a g e

F i g . 3.

Correlogram f o r p r e c i p i t a t i o n

F i g . 4.

Correlogram f o r conceptual watershed s t o r a g e

F i g . 5.

Correlogram fqr e v a p o t r a n s p i r a t i o n

F i g . 6.

Spectrum o f p r e c i p i t a t i o n

Fig.

7.

F i g . 8.

Spectrum o f conceptual watershed s t o r a g e Spectrum o f e v a p o t r a n s p i r a t i o n

runoff

39VtlOlS 03HSt131VM 1 V f l l d 3 3 N 0 3 ONV JJONfltl JO S3AUfl3 SSVW

' NOIlVUldSNVtllOdVA3 ' NOllVlld133Ud

I '9IJ