A Review of Rainfall-runoff Modeling

8/8/2019 A Review of Rainfall-runoff Modeling

http://slidepdf.com/reader/full/a-review-of-rainfall-runoff-modeling 1/18

Expériences in the development andApplication of Mathematical Models inHydrology and Water Resources in LatinAmerica (Proceedings of the TegucigalpaHydromath Symposium, September 1983).fAHSPubl.No. 152.

A REVIEW OF RAINFALL-RUNOFF MODELING

David R. Dawdy

Consulting HydrologistVisiting Professor of Civil Engineering,

The University of Mississippi

Introduction Matematical modeling of the rainfall-runoff pro_

cess has a long history. However, progress was slow prior to about

the last half century. The decade of the 1930's saw an outburst of

activity which laid the groundwork for most of the present develop

ments. Hydrology advanced on all fronts during the 1930's. The con_

cept of physical hydrology was introduced and led to an understanding

of the physics of the hydrologie cycle. The tools developed during

the 1930's to solve practical problems were tailored to costs in

terms of time, mone y, and manpower, and they did not reflect the

level of understanding at that time Hydrology reached a point as a

result of the advances of the 1930's where the ability to state the

problem far exceeded the ability to solve it.

The Second World War brought a halt to the attention paid to the.

advencement of hydrology. However , the war led to the development

of digital computers. That was a tool with which to solve the pro

blems previously unsolvable.The constraint inhydrology changed from

the inability to solve a problem to the inability to collect suffi

cient and sufficiently accurate data to prove that a solution is co

rrect or more nearly correct or less incorrect than other solutions.

This paper will try to trace the developments outlined above, pla_

ce them in perspective, and trace the history of how we arrived whe

re we are today in hydrology. In addition, some suggestions will be

made about where we are, why we are there, and where we might be -

going.

The essence of hydrology is modeling. As a physical science,

hydrology is concerned with numbers quantitative numbers are desi

red. A model is a mathematical statement of the response of a sys

tem which takes system inputs and transforms them into system out

puts. Even though the jargon is mode rn, the rational method for es timating peak runoff used data available in the middle of the 19th

century with a model based on physical principles time response of

the basin, rainfall intensity, and proportion of excess precipitation

were used to determine the peak rate of funoff.

Linear Systems and Mathematical Hydrology. The modern burst of

development in deterministic modeling of rainfall-runoff processes

dates from the 1930's, and the unit hydrograph concepts of Sherman

(1932). Although not stated in those terms at that time, Sherman

assumed that the runoff process was linear and time invariant, the

basic assumptions of linear systems analysis..

The essence of a system is that it interrelates two things the

inputs to and the outputs from the system. The system is a model

which determines a system function, a set of parameter values which

97

http://fahspubl.no/

http://fahspubl.no/

http://fahspubl.no/



98 , D. R. Dawdy

determine the response function, and a set of values for the state

variables, which in hydrology describe how wet or how dry the system

i s . This model is an abstraction, a mathematical construct which,

we hope, acts somewhat similar to the way the real world does. It is

the modeler's conception of how the real world acts. The values of

the parameters of the model define a particular system. They deter

mine how the model reacts to inputs when they are applied to a parti

cular basin. The state variables are measures on the system which

change in response to inputs.

A linear system is one which can be described by a linear

differential equation. The coefficients of the equation may be

constant, as in Darcy's law for saturated flow in porous media,

or they may be variable, as in Darcy's law for flow in unsaturated

media, or they may describe a probability density function in a

stochastic differential equation. If the coefficients are timeinvariant, then superposition holds, which is the basic tool of

linear systems analysis. Superposition says that if an input is

doubled, the output also is doubled. Thus, superposition is the

property which places unit hydrograph theory in the realm of linear

systems analysis, and it is the property on which most-of linear

hydrologie modeling has been based.

Confusion introduced by models.- — A model is the choice of the

modeler. It is a conceptual abstraction. Parameters are a part of

the model, and they have no meaning outside the model. If the

modeler builds a physically based model, then the parameters are

abstractions which may approximate some physically meaningful quan

tity. In hydrology, approximations often are quite gross. That

fact cannot be ignored by the model user. Much of the confusion in

hyldrology results from the attempt by the user to give a physical

explanation to a rule of thumb without supplying a rigorous mathema

tical foundation.

An example in hydrology is the attempt to give physical meaning

to the time response of a basin. The concept of linear storage is

widely used and quite useful in hydrology. The assumption that

outflow from a reservoir varies linearly with storage:

S = KQ (1)

combined with an equation of continuity of mass :

I - Q = ds/dt (2)

leads to the relation:

I - Q =K dQ/dt (3)

to which the solution for no inflow is:

Qt = (̂ e-ft-toJ/K (4>

where Q is the outflow discharge, S is storage, I is inflow discharge,



Review of Rainfall • Runoff Modeling. 9 9

t is time, t is the starting time, and K es a coefficient. K has the

dimensions of time, and it has a meaningful interpretation in terms of

its use in. the model. Time of concentration, lag time, and other such

terms lead only to confusion unless presented and interpreted in such

a mathematical framework.

Storage in not a discrete quantity in modeling a basin by use

of an instantaneous unit hydrograph (ITJH), so that the logic of

equations 1 to 4 cannot be directly interpreted in a physically

based manner. The linear storage concept in IUH modeling must account

for all the storage attenuation of the hydrograph in a basin. Thus,

the parameter K must account for dynamic storage as well as discrete

storage distributed over a basin. K has been related empirically to

size of basin, length of basin, and slope of the basin and/or the

main channel, but it has no true physical definition.

On the other hand, much of the confusion in hydraulics results

from the use of a rigorous mathematical formulation which is treated

as if it were the real world. For example, the dynamic equation

for one-dimensional, steady flow in open channels is

i I + V V + H = S 0 - S f (5)

g t g X X

W here V = v e l o c i t y W i th t u r b u l e n t f l u c t u a t i o n sH = d e p t h o f w a t e r W i th t u r b u l e n t f l u c t u a t i o n s

S 0= s l o p e o f c h a n n e l b o t t o m a n ' a v e r a g e ' s l o p e of a r e a c hSf= f r i c t i o n s l o p e a c o n c e p t u a l a b s t r a c t i o n

The v a l u e f o r Sf i s d e r i v e d from a s o - c a l l e d ' f r i c t i o n f o r m u l a ' ,s u c h a s C h ez y, w h i ch i s ' t h e o r e t i c a l ' , o r M a n n in g , w h i ch i s ' e m p i r i c a l ' . The t h e o r e t i c i a n s c o n t i n u a l l y d e r i d e t h e e m p i r i c i s t s f o r u s in gth e 'wrf flng' f r i c t i o n f o rm u la / How e ve r , t h e two c a n be shown to bea l m o s t e q u i v a l e n t i f v a r i a t i o n i n r e l a t i v e r o u g h n es s i s c o n s i d e r e d .F o r e x a m p l e , i f w e w e r e t o as s u m e t h a t we h a v e a g r a v e l - b e d s t r e a mw i t h a ' c h a r a c t e r i s t i c g r a i n s i z e ' o f 2 c e n t i m e t e r s an d w e re t oa ss u m e a d e p t h o f 1 / 2 , 1 , 2 , 5 , a nd 10 m e t e r s , t h e P r a n d t l e q u a t i o n

w o u ld g i v e d i f f e r e n t v a l u e s f o r C hezy C a s d e p t h i n c r e a s e d , b e c a u s er e l a t i v e r o u g h n e s s w o u l d c h a n g e . On t h e o t h e r h a n d , M a n n i n g ' s nw o u ld r e m a i n a l m o s t c o n s t a n t , b e c a u s e t h e v a l u e s o f M a n n i n g ' s ni n c l u d e c h a n g e s o f r e l a t i v e r o u g h n e s s . H ow ev er o ne u s e s E q u a t i o n 5 ,i t e n t a i l s b l a c k m a g i c i n t h e r e a l w o r l d , e v e n t h o ug h i t i s ad i f f e r e n t i a l e q u a t i o n . C o n s i d e r a b l e ' e n g i n e e r i n g j u dg m e n t ' e n t e r si n t o t h e c h o i c e o f S f , e v e n w i t h t h e a i d of t h e e x c e l l e n t w o rk o fB a r n e s ( 1 9 6 7) an d o t h e r s i n t h e USGS, w ho h a v e t r i e d t o r a t i o n a l i z et h e d e t e r m i n a t i o n o f r e s i s t a n c e t o fl o w f o r u s e i n op en c h a n n e l f lo wp r o b l e m s .

The instantaneous unit hydrograph.- W ith t h e f o r e g o i n g a s ap r e l u d e , th e IUH c an b e s e e n a s a t o o l o f l i n e a r s y s t em s a n a l y s i s .T h e IUH i s t h e i m p u l s e ; r e s p o n s e f u n c t i o n o f a l i n e a r ,t i m e - i n v a r i a n t s y s t e m . An i m p u l s e r e s p o n s e f u n c t i o n i st h e r e s p o n s e of a s y s t e m t o a u n i t o f i n p u t a p p l i e d



•\00 D.R. Dawdy

i n s t a n t a n e o u s l y i n t i m e a n a b s t r a c t c o n c e p t . Its m a t h e m a t i c a l

s t a t e m e n t is the c o n v o l u t i o n i n t e g r a l

t

y ( t l = J* h f t - t ) x ( X ) d t (.6)

where h(T) is the impulse response function and x(t) is the input.

Equation 11 can be used to derive Sherman's T-hour unit graph. In hy_

drology, h(t) is conventionally denoted u(o,t) for the unit hydrog-

raph of duration o, and u (T,t), then is 'the T-hour unit hydrograph,

so that.

u(.T,t) = f u(o,t-x) S(T -T) dX (71

where S ( T - X ) = J_ for c < T-T < T

T

= o otherwise.

Most of the theory of the instantaneous unit hydrograph (IUH) is basedon the. concept of a linear storage resulting from a hypothetical line

ar reservoir. As stated earlier:

I-Q = dS/dt C ontinuity (2)

S = KQ Linear Reservoir (1)

with the same notation as Equations 3 to 6,

which leads to;

I-Q = K dQ/dt (3)

for which the IUH is

u î ( 0 t\ _ 1 -(t-t )/K single linear reservoir (8)

~~ ( C l a r k and o t h e r s )1_

K

K

e" ( t"

t - t p

K

- tc

e~

, ) /K

• ( t - t Q

(n -1 )

u ( 0 t\ _ 1 t - t Q e - ( t - t 0 ) / K n e q u a l l i n e a rn " j£ K (n~-T) ! cascaded reservoirs.

(Nash cascade) (9)

For the Nash cascade (Nash, 1958) the response function is a gamma

function. Although there are n "equal" reservoirs, n need not be

discrete, and the IUH may be a generalized gamma function. Nash has

shown that the parameters may be determined based on the gamma

function, and that nK is the first moment about the origin and nK

is the second moment about the origin.



Review of Rainfall - Runoff Modeling. 10 1

T h u s , t h e p r ob l em o f l i n e a r s y n t h e s i s i n IUH a n a l y s i s i n v o l v e st h e a s s u m p t i o n o f a r e a s o n a b l e m o d e l a n d t h e d e v e l o p m e n t o f a m e t h o dt o e s t i m a t e t h e p a r a m e t e r v a l u e s f o r t h a t m o d e l . T h i s g e n e r a l a p pr oa chl e a d s i n tw o m a j o r d i r e c t i o n s : t h e d ev e l o p m e n t o f c o n c e p t u a l an d o fb l a c k - b o x m o d e l s .

T he C o n c e p t u a l IUH - C o n c e p t u a l m o d e l s cam e f i r s t , w i t h t h eg r e a t e s t a m ou nt o f a c t i v i t y i n t h e 1 9 5 0 ' s . H o w e ve r, t h e w ork onc o n c e p t u a l m o d el s s t a r t e d e a r l i e r . The M uskingum m e t ho d f o r f l o o dr o u t i n g ( M c C a r t h y , 19 38 ) i s i n t h e fo rm o f a l i n e a r s t o r a g e m o d e l .Nash (1 9 5 9 ) sh owed t h a t t h e Mu sk in gu m mo d e l r o u te s f lo ws th ro u g h twol i n e a r r e s e r v o i r s , th e f i r s t w i t h n e g a t i v e s t o r ag e — w h i c h e x p l a i n st h e an o m al ou s r e s u l t s o f a d e c r e a s e i n fl ow o b t a i n e d a t t h e b e g i n n i n g

o f a r o u t i n g i n many c a s e s .

An i n t e r e s t i n g a p p ro a c h t o c h a n n e l r o u t i n g b y l i n e a r a n a l y s i s

w a s d e v e l o p e d b y K a l i n i n a n d M i l u k o v ( 1 9 5 8 ) . T he y d e v e l o p e d t h e .c o n c e p t o f a c h a r a c t e r i s t i c l e n g t h o v e r w h ic h th e r o u t i n g w as a s i n g l e

l i n e a r r e s e r v o i r . T he p a r a m e t e r s o f t h e l e n g t h a nd t h e s t o r a g e w e re

r e l a t e d t o c h a n n e l m e a s u r e m e n t s. O nce t h e c h a r a c t e r i s t i c l e n g t h i sd e t e r m i n e d , l o n g e r r e a c h e s , r o u t e d s e q u e n t i a l l y , d e v e l o p a gamma

d i s t r i b u t i o n s i m i l a r t o a N ash c a s c a d e f o r b a s i n r o u t i n g . T h u s , th e

S o v i e t s w e re w o r k i n g on s i m i l a r p r o b l e m s a nd d e v e l o p i n g s o l u t i o n s

s i m i l a r t o t h o s e d e s c r i b e d e a r l i e r d u r i n g t h i s p e r i o d .

M o s t c o n c e p t u a l m o d e l s o f t h e IUH a r e b a s e d on t h e t w i n c o n c e p t so f l i n e a r s t o r a g e a nd l i n e a r c h a n n e l s . L i n e a r s t o r a g e w as d e s c r i b e de a r l i e r ( E q u a t i o n 3 ) . A l i n e a r c h a n n e l i s o ne w h ic h p a s s e s an i n p u th y d r o g r a p h w i t h o u t a t t e n u a t i o n . The l i n e a r c h a n n e l i s u s e d t o d ev el opa t i m e - a r e a h i s t o g r a m (TAH) , w h i c h i s t h e o u t f l o w h y d r o g r a p h f ro m a ni n s t a n t a n e o u s r a i n f a l l - e x c e s s a p p l i e d u n i f o r m l y o v e r a b a s i n i f t h er ew e re n o s t o r a g e a c t i n g t o a t t e n u a t e t h e h y d r o g r a p h . T he s i m p l e s t

form o f a TAH i s an i s o s c e l e s t r i a n g l e . An i s o s c e l e s t r i a n g l e r o u t e dt h r o u g h a l i n e a r r e s e r v o i r w i t h a s t o r a g e c o e f f i c i e n t on t h e o r d e ro f t h e ti m e b a s e o f t h e t r i a n g l e y i e l d s a r e s p o n s e f u n c t i o n q u i t es i m i l a r t o t h e u s u a l r u n o f f h y d r o g r a p h a nd t o t h e gamma d i s t r i b u t i o no f t h e N as h c a s c a d e . 0 ' K e l l y (1 95 5) i n t r o d u c e d t h e i s o s c e l e s t r i a n g l eTAH. T h i s e a r l y f o r m u l a t i o n h a s some p h y s i c a l j u s t i f i c a t i o n . C o n s i d e rt h a t o v e r l a n d fl ow g e n e r a t e s a r e s p o n s e f u n c t i o n of u n i f o r m f lo w f o rt i m e , T 1 , i n t o a m a in c h a n n e l s y s t e m w i t h a t i m e o f t r a v e l o f T 2 .T h u s , t h e r e s p o n s e f u n c t i o n of e a c h o f t h e s e , t r e a t e d a s l i n e a rc h a n n e l s , i s a r e c t a n g u l a r p u l s e . The o u t f l o w TAH i s t h e c o n v o l u t i o no f tw o r e c t a n g l e s . I f T1 = T 2 , t h e r e s u l t i s an i s o s c e l e s t r i a n g l e .

M i t c h e l l (1 96 2) sh ow ed t h a t m o s t s m a l l s t r e a m s i n I l l i n o i s c o u l d b emo d e led w i th su ch a TAH. Ho wev er , h e fo u n d t h a t some s t r e am s h ad af l a t - t o p p e d IU H, a nd r e q u i r e d t h e u s e o f a t r a p e z o i d f o r a TAH. I fT1 / T 2 , t h e c o n v o l u t i o n p r o d u c e s a t r a p e z o i d o f b a s e l e n g t h s T1 +T2 a nd T1 - T 2 . T h u s , on ce a g a i n , p h y s i c a l j u s t i f i c a t i o n m ay f o l lo we m p i r i c a l o b s e r v a t i o n .



102 0. K. Dawdy

P e r h a p s t h e " b e s t " c o n c e p t u a l l i n e a r s t o r a g e m o d e l f o r r i v e r

b a s i n s i n t h a t d e v e l o p e d b y C l a r k ( 1 9 4 5 ) . C l a r k d i v i d e d t h e b a s i n

i n t o s u b - b a s i n s b y i s o c h r o n e s . T he a r e a s b e t w e e n i s o c h r o n e s d e t e r m i_

n e s a t i m e a r e a h i s t o g r a m (T AH ). E x c e s s p r e c i p i t a t i o n on t h e b a s i n

i s r o u t e d t o t h e o u t f l o w p o i n t o n t h e b a s i s o f t h e TAH a n d t h e n i s

r o u t e d t h r o u g h a l i n e a r r e s e r v o i r . T h a t m o d e l i s t h e b a s i s f o r t h e

s u r f a c e w a t e r r o u t i n g c o m p o n en t o f t h e S t a n f o r d W a t e r s h e d M o d e l

( C r a w f o r d a n d L i n s l e y , 1 9 6 2 ) , t h e U . S . G e o l o g i c a l S u r v e y m o d e l b y

D aw dy , L i c h t y , a n d B e r g m an n ( 1 97 2 ) a n d i s a n a l t e r n a t i v e i n t h e

C o r p s o f E n g i n e e r s H EC -1 ( 1 9 7 0 ) .

C o n c e p t u a l m o d e l s b l o s s o m e d f o r t h i n t h e 1 9 5 0 ' s . A l l h a d a

common b a s e i n s om e f or m o f l i n e a r r e s e r v o i r r o u t i n g a n d i n t h e c o n

c e p t o f a l i n e a r , c h a n n e l . T he l i n e a r c h a n n e l m o v es t h e p r e c i p i t a

t i o n e x c e s s t h r o u g h t h e b a s i n w i t h o u t a t t e n u a t i o n . T h e l i n e a r s t o r a g e p r o v i d e s t h e m e a n s t o a t t e n u a t e t h e h y d r o g r a p h s o t h a t i t a s s u

m es t h e t y p i c a l s h a p e ^ o f a d i s c h a r g e h y d r o g r a p h . T he C l a r k TAH p r o

v i d e s t h e m e a n s t o m o d e l b a s i n s f o r w h i c h t h e IUH h a s a c o m p l e x s h a

p e . T he p a r a m e t e r s o f t h e c o n c e p t u a l IUH u s u a l l y a r e r e l a t e d t o phy_

s i c a l m e a s u r e s o f t h e b a s i n . T he t h e o r y w a s s u m m a r i z e d i n D o o g e ' s

e x c e l l e n t m o no g r ap h ( 1 9 5 9 ) , b u t t h e t h e o r e t i c a l j u s t i f i c a t i o n h a d fo_

l l o w e d e m p i r i c a l d e v e l o p m e n t .

THE BLACK-BO X IUH

The 1 9 6 0 ' s saw a n o u t b u r s t o f i n t e r e s t i n b l a c k - b o x m o d e l i n g o ft h e IUH . The s i m p l i f i c a t i o n s o f l i n e a r r e s e r v o i r m o d els l e d t o as e a r c h f o r a l t e r n a t i v e a n a l y s i s . S im p le h a r m o n ic a n a l y s i s w e ret r i e d b y O ' D o n n e l l ( 1 9 6 0 ) . T r u n c a t i o n i n t h e h a r m o n i c a n a l y s i s c a u s e d p r o b l e m s o r r i n g i n g a n d s m o o t h i n g . C h i a n g a n d W i g g e r t ( 19 6 8)p l a c e d h a r m o n i c a n a l y s i s f o r t h e IUH i n t h e fr am e w or k o f g e n e r a lb l a c k - b o x a n a l y s i s a s d e v el o pe d i n e l e c t r i c a l e n g i n e e r i n g .

M a t r i x i n v e r s i o n t e c h n i q u e s f o r t h e d e r i v a t i o n o f t h e IUH w e rei n t r o d u c e d s i m u l t a n e o u s l y by N as h ( 19 61 ) a n d b y o t h e r s , su c h a s t h eTVA a nd S n y d e r . E ac h u n d o u b t e d l y r e a l i z e d t h a t d i g i t a l c o m p u t e rs o -p e r a t e m os t e f f i c i e n t l y i n m a t r i x m u l t i p l i c a t i o n , a nd t h a t an I U H is al i n e a r m a t r i x t r a n s f o r m a t i o n . The r e a l i z a t i o n t h a t t h e IUH i s a l i n e a r m a t r i x t r a n s f o r m a t i o n i s d i s c r e t e t i m e h a s d i r e c t i m p l i c a t i o n si n c o n c e p t u a l IUH m o d e l i n g , s o t h a t c o n c e p t u a l m o d e l s g a i n e d b y as p i n - o f f from b l a c k - b o x m o d e l i n g , p a r t i c u l a r l y f rom t h e w o r ks o fNash and O'Donne 1 1 . N ot a l l m o de ls u t i l i z e t h i s p r i n c i p l e c o m p l e t e l y , a nd t h e i r r e s u l t i n g c o m p u t e r p r o g r a m i s made m o re c om p le x a n d t ime c o n su m i n g t h a n i s n e c e s s a r y .

Some b l a c k - b o x m o d e l e r s g a i n e d k n o w l e d g e fro m c o n c e p t u a l m o d e l si n t h e d e v e lo p m e n t o f m e t h o d s f o r i n v e r s i o n . An e x a m p le o f t h i sap p ro ach i s sho wn b y Doo ge (1 9 6 5 ) , who u se d L ag u e r r e f u n c t i o n s fo rt h e i n v e r s i o n of i n p u t - o u t p u t p a i r s t o d e v e l o p t h e IUH . The r e s u l t i n g IUH i s s i m i l a r t o t h e N a sh c a s c a d e c o n c e p t u a l IU H .



Review of Rainfall - Runoff Modeling. 103

F i n a l l y , b l a c k - b o x m o d e l i n g m oved i n t o t h e n o n l i n e a r d o m ai n,w i th th e wo rk o f Amo ro ch o an d O r lo b (1 9 6 1 ) . Th ey d e v e lo p ed a me th o dt o i s o l a t e a nd m o d el t h e n o n l i n e a r e l e m e n t s i n t h e r e s p o n s e f u n c t i o n . L a t e r , A m oro cho a nd B r a n d s t e t t e r ( 19 71 ) d e v e l o p e d a g e n e r a l ,

n o n l i n e a r , - b l a c k - b o x i n v e r s i o n t e c h n i q u e . A c t u a l l y , b l a c k - b o x m od eli n g i m p l i e s a l i n e a r s y s t e m . The n o n l i n e a r m o d e ls m i g h t b e t t e r b ec a l l e d n o n - s t r u c t u r e i m i t a t i n g m o d e l s , r a t h e r th a n b l a c k - b o x m o d e l s .

B e en sho wn t h a t i f a s e p a r a t e o f s e t o f e v e n t s n o t u s e d i n t h ef i t i s u s e d t o t e s t t h e a c c u r a c y o f t h e r e s u l t i n g m o d e l s , c o n c e p t u a lm o d el s p e r fo r m b e t t e r t h a n b l a c k - b o x m o d e l s . The v e r y c o n s t r a i n t sw h i c h m ake t h e f i t f o r c o n c e p t u a l m o d e l s w o r s e a r e w h a t a l s o c a u s eth em t o p r e d i c t b e t t e r . T h e re h a ve b e e n som e a t t e m p t s t o b u i l d c o n st r a i n t s i n t o b l a c k - b o x m o de ls i n o r d e r t o p r e d s i c t b e t t e r a t t h e e x "

p e n s e o f f i t t i n g w o r s e . An e x a m p l e i s E a g l e s o n ' s ( 19 6 6) o p ti m u m re_al i z a b l e IU H. He u s e d a l i n e a r p r o gr a m m i n g f o r m a t w i t h a n o n - z e r oc o n s t r a i n t o n t h e o r d i n a t e s o f t h e IU H.

A m a j o r d r aw b a c k t o t h e u s e o f b l a c k - b o x m o d e l s i s t h a t t h e yc a n n o t b e u s e d t o m o d e l a c h a n g i n g s y s t e m . B e c a u s e b l a c k - b o x m o d e l sa r e n o t c o n c e r n e d w i t h t h e i n t e r n a l w o r k i n g s o f t h e s y s t e m t h e y c a n n o t b e m o d i f i e d e a s i l y t o r e f l e c t t h e r e s u l t s o f s u c h c h a n g e s . Manyi f n o t m o s t u s e s o f w a t e r s h e d m o d e l s t o d a y a r e t o a s s e s s t h e e f f e c t

o f p a s t o r p o t e n t i a l f u t u r e m a n-m a de c h a n g e s o n a w a t e r s h e d . C oncep_t u a l m o d e ls a r e w e l l s u i t e d f o r su c h u s e s , b e c a u s e t h e p a r a m e t e r s i na c o n c e p t u a l m o de l may b e r e l a t e d t o p h y s i c a l p a r a m e t e r s o f a b a s i n .

T h a t n e e d f o r t h e m o d e l i n g o f t h e e f f e c t s o f m an -m ad e c h a n g e s h a sl e d t o d e v e l o p m e n t s i n tw o m a j o r d i r e c t i o n s . B o th d e v e l o p m e n t s a r ei n c o n c e p t u a l m o d e l i n g . T he f i r s t d e v e l o p m e n t i s i n t h e u se o f an o n l i n e a r r o u t i n g m o de l b a s e d on t h e k i n e m a t i c w ave e q u a t i o n s . Thes e c on d d e v el o pm e n t i s t h e b u i l d i n g o f d i s t r i b u t e d p a r a m e t e r m o de lst o r e p l a c e t h e l um p ed p a r a m e t e r m o d e l s o f c l a s s i c a l IUH t h e o r y .

COMPARISON OF BLACK-BOX AND CONCEPTUAL IUH

B l a c k - b o x m o d el d e v e l o p m e n t h a s t e n d e d t o move i n t h e d i r e c t i o no f t h e u s e o f t h e k n o w l e d g e g a i n e d f r o m t h e u s e o f c o n c e p t u a l m o d e l sH o w ev er , t o t h e e x t e n t t h a t b l a c k - b o x e s r e m a in b l a c k , t h e y a r e n o tc o n c e r n e d w i t h t h e i n n e r w d r k i n g s o f t h e s y s t e m w h i c h t h e y m o d e l .C o n c e p t u a l m o d el s a r e c o n s t r a i n e d so t h a t t h e i r sh a p e w i l l " l o okr i g h t " i n t e rm s o f r e a l w o r l d h y d r o g r a p h s .

As a r e s u l t o f t h e l a ck o f c o n s t r a i n t s i n t h e i r s t r u c t u r e ,b l a c k - b o x m o d e ls t e n d t o f i t a s e t o f d a t a b e t t e r t h a n do c o n c e p t u a lm o d e l s . I f a s i n g l e e v e n t i s u s e d t o d e r i v e a b l a c k - b o x IU H, t h e da_t a c a n b e f i t p e r f e c t l y . C o n c e p t u a l m o de ls w i l l , i n g e n e r a l , n o t



104 D.R. Dawdy

fit even a single event perfectly. If a set of events is used with

least squares fitting to derive an IUH, black-box models, in general

will fit the data better. However, i t has

Kinematic Wave Models

The kinematic wave (KW) is one step away from the linear stora

ge assumption toward the use of a dynamic routing equation. I t has

long been known that as storms increased in intensity over a basin,

the response time of the basin tended to decrease. Thus, the IUH

was not identical for small and large storms. The kinematic wave

equation tends to overcome the shortcoming of the IUH.

The KW equation s t i l l is based on the continuity assumption

Q = dS/dt

( 2 )

qL ~ ax at ( 10 )

in partial differential terms, where q is the lateral inflow,

9q/9x is the outflow per unit with, and 9 y/3 t is the change in

depth with time, which is equal to change in storage per unit width.

Equation 10 is combined with the kinematic assumption.

Q = «A1" (11a)

q = (Xym

(11b)

w h e r e <X and m are t he KW p a r a m e t e r s . E q u a t i o n s 10 and 11 are combi_

n e d to y i e l d

mQ/y dj_ + dj_ = qL (12)

3x 9t

which is used in place of the linear reservoir routing equation.

The appealing feature of Equation 12 is that the KW parameters have

physical significance. For example, let us assume that Manning's

equation applies over a reach of interest. Then



Review of Rainfall - Runoff Modeling. 10 5

1 /21 . 5 2 / 3 S

Q = — - AR ( 1 3 )

w h e re n i s M a n n i n g ' s c o e f f i c i e n t , R i s h y d r a u l i c r a d i u s , S i s s l o p e ,

a n d o u r " t h e o r e t i c a l a p p r o a c h " h a s a l r e a d y b ec om e e m p i r i c a l . I f t h e

w i d t h i s m uch g r e a t e r t h a n t h e d e p t h ,

R = A/(W + 2D) = A/W = D (1 4 a)

' 2 / 3 s 1 / 2

2 = ¥ A w 2 / 3 (14b)

1/2

n W ^ / J ( 1 4 c )

m = 5 / 3

a nd s i m i l a r e q u a t i o n s may b e d e r i v e d f o r o t h e r sh a p e s o f c h a n n e l s .T h u s , (X i s a f u n c t i o n o f p h y s i c a l m e a s u r e s o f t h e r e a c h , a n d b o t h a

a n d m a r e f u n c t i o n s o f t h e s h a p e of t h e c h a n n e l c r o s s s e c t i o n a n d o ft h e f r i c t i o n law a ss um e d (M a n n i n g 's e q u a t i o n i n t h i s e x a m p l e ) .

The e q u a t i o n i s q u i t e s i m i l a r t o t h e r e s u l t s of e a r l i e r a t t e m p t sa t d e v e l o p i n g a n o n l i n e a r s t o r a g e e q u a t i o n . I f s t o r a g e i s as su m edd i r e c t l y r e l a t e d t o a p ow e r f u n c t i o n o f flo w d e p t h o r t o c r o s s - s e c t i o n a l a r e a , t h e tw o a r e i d e n t i c a l . H o w e v er , t h e u s e o f t h e KW e qu jit i o n h a s t a k e n a s t e p aw ay fro m t h e h y d r o l o g i e a s s u m p t i o n s o f l i n e a ra nd n o n l i n e a r s t o r a g e a nd t o w a rd h y d r a u l i c r o u t i n g .

A m a j o r a d v a n t a g e o f KW r o u t i n g i s t h a t i t s p a r a m e t e r s r e l a t et o t h e p h y s i c a l w o r l d . I f t h a t p h y s i c a l w o r l d i s m o d i f i e d , t h ee f f e c t on t h e r o u t i n g p a r a m e t e r s c an b e e s t i m a t e d , a nd r e s u l t i n gc h a ng e s i n t h e b a s i n r e s p o n s e c an b e p r e d i c t e d . A m a jo r s h o r t - c o m in g o f KW r o u t i n g i s t h a t E q u a t i o n s 14 a ss um e t h a t a u n i q u e , s i n g l e - v a l u e d , s i m p l e s t a g e d i s c h a r g e r a t i n g a p p l i e s w h e r e v e r t h e e q u a t i o n i s u s e d . T he k i n e m a t i c w av e n u m b er c a n b e u s e d t o s c r e e n o u tt h o s e c a s e s w h e r e t h e e q u a t i o n d o e s n o t a p p l y b e c a u s e d y n a m i ce f f e c t s c a u s e s t a g e a nd d i s c h a r g e t o b e r e l a t e d d i f f e r e n t l y on t h er i s i n g a nd t h e f a l l i n g l i m b o f t h e h y d r o g r a p h . A m ore s e r i o u s con se_q u e n c e o f t h e k i n e m a t i c a s s u m p t i o n a r i s e s b e c a u s e E q u a t i o n s 13 a n d14 a p p l y b e s t a t c o n s t r i c t i o n s o r c o n t r o l r e a c h e s . The a dd e d s t o r a g e



106 D.R.Dawdy

resulting from minor expansions and contractions of the channel system

is not accounted for. This is particularly true of overbank flows

at higher stages. Although overbank flow can be modeled by an

iterative procedure involving multiple ratings, a single rating is

assumed throughout a reach of stream channel. Such a case seldom

occurs. Therefore, KW models tend to over correct for the nonlinea—

rity in the routing function, and higher peaks tend to be overestima

ted, with the time of response of the basin decreasing with discharge

more rapidly than occurs in the real world. One final major advan

tage of KW models is that they are perfectly suited for use in

distributed parameter models. That fact may explain the widespread

acceptance and use of kinematic wave models.

Distributed Parameter Models

The l a t e s t t r e n d i n b a s i n r e s p o n s e m o d e li n g i s t o us e a d i s t r i b u t e d p a r a m e t e r d e s c r i p t i o n o f t h e b a s i n . A t y p i c a l d i v i s i o n o f ab a s i n f o r d i s t r i b u t e d - p a r a m e t e r m o d e li n g i s show n i n F i g u r e 1 . F i r s t ,t h e m a in c h a n n e l s y s t e m i s d e t a i l e d . R e a c h es a r e t h e n d e t e r m i n e dw h ic h h av e s i m i l a r r o u t i n g c h a r a c t e r i s t i c s t h r o u g h o u t t h e i r l e n g t h .T he o v e r l a n d fl ow a n d c h a n n e l s e g m e n t s o u t l i n e d i n F i g u r e 1A a r et h e n d e s c r i b e d i n s u c h a m a n n er a s t o d e v e l o p t h e s c h e m a t i c d i a g r a mshown in Figure 1B.

The a s s i g n m e n t o f i n p u t p h y s i c a l d a t a t o t h e b a s i n d e f i n e s t h eb a s i n r e s p o n s e f u n c t i o n . T h u s , t h e r e a r e s e v e r a l m a j o r a d v a n t a g e sw h i ch t h e d i s t r i b u t e d p a r a m e t e r m od el h a s o v e r a lu m pe d p a r a m e t e rm o d e l s u c h a s a n IU H . T he f i r s t m a j o r a d v a n t a g e i s t h a t t h er e s p o n s e f u n c t i o n c an b e d e v e l o p e d d i r e c t l y from t h e i n p u t p a r a m e t e r s i f a n a p p r o p r i a t e m o d e l , s u c h a s KW, i s u s e d . A t y p i c a l s e to f i n p u t d a t a f o r a d i s t r i b u t e d p a r a m e t e r m o de l i s show n i n F i g u r e

2 . A s e c o n d m a j o r a d v a n t a g e i s t h a t n o n u n i f o r m s t o r m s m ay b ea p p l i e d t o t h e b a s i n - t y p i c a l i s o h y e t a l s of m ean a n n u a l r a i n f a l l a r eshow n i n F i g u r e 1 A, w h i c h may b e u s e d t o d i s t r i b u t e r a i n f a l l o v e rt h e b a s i n .

The t h i r d , a nd c o m p e l l i n g , m a j o r a d v a n t a g e o f d i s t r i b u t e d p a r a m e t e r m o d e ls i s t h a t t h e c ha n ge i n b a s i n r e s p o n s e r e s u l t i n gfro m m a n-m ad e c h a n g e s o v e r p a r t o f t h e b a s i n may b e a s s e s s e d . Anyp a r t o f t h e s c h e m a t i c i n F i g u r e 1B may b e m o d e l e d w i t h " b e f o r e a n da f t e r " p r e d i c t i o n s b y c h a n g in g t h e s e t o f p a r a m e t e r s f o r t h a t p a r t

o f t h e b a s i n .

One m a j or d i s a d v a n t a g e of d i s t r i b u t e d p a r a m e t e r m o d e ls i s t h a tt h e y g e n e r a l l y r e q u i r e m or e d a t a an d m uch m o re c o m p u t e r t im e t o r u nt h a n d o l u m p e d - p a r a m e t e r m o d e l s .



Review of Rainfall - RunoffModeling. 107

2 7 -

A . S T R E A M C H A N N E L N E T W O R K OF B A S I N

\ ; >

/ c -

\

V

\/

7

/

7

\

i

/ „ \9 \

\ '

//

/ \

\

V ,± \

^

\

Y/

/

7

\/ /3\

\

B . D I V I S I O N OF B A S I N I N T O S T R E A M C H A N N E L

A N D OV E R F L O W S E G M E N T S

!FIGURE 1. TYP ICA L SCHEMATIC REPRESENTATION OF A BASIN FOR USE I N

DEVELOPING A DISTRIBUTED-PARAMETER RAINFALL-BUNOFF MODEL

As computers ge t l a rg er and fa s te r and cheaper th a t d i sadvar it age dec rea ses in im por tanc e . Wi th the advent of min icomputers in -every o f f i c e , i t may reassume im por ta nce . An im por ta n t p o in t toco ns id er i s th a t p ro per programming can g re a t ly reduce comput ing

t i m e . Note in F igure 1B t h a t th ere a re 34 ov er lan d flow se c t io n sf lowing in to 20 channel re ac he s , bu t ove r land flow reach es a renumbered t o 7 ( in the co rn er s of the ov er lan d flow segments) andcha nne l re ac he s to 13 . Thus 54 segm ents have been modeled as 20s e g m e n ts . I f se gm e nt c h a r a c t e r i s t i c s a re s u f f i c i e n t l y s i m i l a r , l a r g esavin gs in computer t ime can r e s u l t . Even so , the canned bu lk -p ar a-



108 D. R. Da'wdy

ROUTING COMPONENT

I N P U T D A T A :

NUMBER OF DIFFERENT SEGMENTS IN BASIN

UPSTREAM SEGMENTS

LATERAL SEGMENTS

TYPE OF SEGMENTS

SLOPE OF SEGMENT

FLOW LENGTH OF SEGMENT

ROUGHNESS (CORRESPONDS TO MANNING'S N)

CHANNEL DIMENSIONS, PROPORTION OF IMPERVIOUS AREA

THIESSEN COEFFICIENT

RAINFALL EXCESS

OUTPUT :

STREAMFLOW HYDROGRAPH

Figure 2. T ypical Set of Input Data Used To Define a Segmen t for A D istributee-Param eter Rainfall-RunoffModel.

meter model you replace must be grossly inefficient to overcome its

natural advantage. However, some do manage.

Tank Models - Off on another track a separate development has

taken place in basin rainfall-funoff modeling. Sugawara (1961)

introduced the concept of a tank model. A single tank yields a linear

storage model such as equations 3 to 6. A series of tanks yields a

Nash cascade. Therefore, tank models are very much in the spirit

of linear systems analysis for IUH analysis. However tank models

have a major advantage and a major disadvantage in terms of mathe

matical development. Interestingly, the advantage and the disadvan-taga are the same - the model can be physically visualized. For the

empiricist and the engineer that is an advantage. For the theoretician

and the mathematician that is a disadvantage.

Each component of the hydrologie cycle for which there is a



Review ofRainfall - Runoff Modeling. 109

l i n e a r a p p r o x i m a t i o n may b e r e p r e s e n t e d b y a t a n k m o d e l . T he s e t o ft a n k s , e a c h r e p r e s e n t i n g a l i n e a r s t o r a g e , may b e a r r a n g e d i n s e r i e so r i n p a r a l l e l . The p a r a m e t e r s f o r e a c h t a n k may b e e s t i m a t e d fro mp h y s i c a l p a r a m e t e r s o r b y o t h e r m ea ns a p p r o p r i a t e f o r t h e g i v e nc o m p o n e n t . T he i n p u t s a n d o u t p u t s f o r e a c h t a n k a r e d e f i n e d a n dV o i l a ! We h a v e a t a n k m o d e l .

T he c l o s e d fo rm s o l u t i o n o f t h e r e s p o n s e f u n c t i o n f o r som ec o n f i g u r a t i o n s o f t a n k m o d e l s c a n b e d e r i v e d . N a sh ( 19 58 ) o b v i o u s l ys o l v e d t h e c a s e f o r a s e r i e s o f n e q u a l t a n k s . S u g aw a r a (1 96 1) s o l v e dm any m ore c om p le x c a s e s . I n a d d i t i o n h e d i s c u s s e d p i e c e w i s e l i n e a rs o l u t i o n o f k e r n e l s b y u s e o f c o m pl ex g e o m e tr y a n d m u l t i p l e o u t l e tt a n k s . S u ga w ar a d i s c u s s e d t h e i n t e r p r e t a t i o n an d e s t i m a t i o n o f t h et a n k p a r a m e t e r s f o r d i f f e r e n t co m p o n e n ts . F i n a l l y , S ug aw a ra p r e s e n t e d a s e m i - d i s t r i b u t e d r a i n f a l l - r u n o f f m od el d e ve lo p m en t t h ro u g h t h eu s e o f lu m p ed p a r a m e t e r t a n k m o d e l i n g o f s u b - b a s i n s .

A m o s t i n t e r e s t i n g f a c t i n t h e m a t h e m a t i c a l d ev e l o p m e n t o f t a n km o d el s i s t h a t m o s t o f t h e s u b s e q u e n t i n t e r e s t i n t h i s d e t e r m i n i s t i cr a i n f a l l - r u n o f f m o de l o u t s i d e J a p an com es fro m s t o c h a s t i c h y d r o l o g y .A s i m p l e s e r i e s t a n k m o d el w i t h a s i n g l e i n p u t o f w h i t e n o i s e a ndw i t h a s i n g l e o u t p u t g e n e r a t e s a n a u t o r e - g r e s s i v e - m o v i n g a v e r a g e(ARMA) m od e l . Moss and Dawdy (1973) showed t h a t a c o n c e p tu a l r a i n f a l l - r u n o f f m o d e l e q u i v a l e n t t o a s i n g l e t a n k d e v e l o p e d a n ARMA( 1 , 1 ) m o d el f o r s t o c h a s t i c s i m u l a t i o n o f m o n t h ly s t r e a m f l o w . P eg ra m

( 19 7 7) s ho w ed t h e m a t h e m a t i c a l e q u i v a l e n t o f a C l a r k IUH f o r m u l a t i o na n d a n ARMA m o d e l u n d e r c e r t a i n a s s u m p t i o n s . S e l v a l i n g a m ( 1 9 7 7 ) , as t u d e n t o f S u g a w a r a ' s , sh ow ed t h e e x a c t e q u i v a l e n t o f t a n k m o d e l sa nd ARMA m o d e l s . T he f a s t f r a c t i o n a l G a u s s i a n n o i s e m o d e l ( M a n d e l b r o t ,1971) i s , o f c o u r s e , a p a r a l l e l ta n k m o d e l , w h i c h s h o u l d r e s u l t i ns u m m a t io n o f ARMA ( 1 , 1 ) m o d e l s r a t h e r t h a n a s u m m a t i o n o f a u t o r e -g r e s s i v e m o d e l s . I n c i d e n t a l l y , s i m u l a t i o n o f a v e ra g e f lo w s ( d a i l y ,w e e k l y , o r m o n t h l y ) a d d s o ne d i m e n s i o n t o t h e m o v in g a v e r a g e p o r t i o ni n r e l a t i o n t o s a m p l in g a t d i s c r e t e i n t e r v a l s . A v er ag e f lo w s a r ed i s c r e t i z e d b u t n o t d i s c r e t e v a r i a b l e s , and. t h a t f a c t s h o u l d b e k e p ti n m in d w hen b u i l d i n g m o d els f o r s t o c h a s t i c s i m u l a t i o n .

T h u s , t a n k m o d e l s s ee m t o b e a t o o l f o r d r a w i n g t o g e t h e rs t o c h a s t i c and d e t e r m i n i s t i c m o d e ls , p h y s i c a l l y - b a s e d , s t r u c t u r e -i m i t a t i n g a nd c o n c e p t u a l m o d e l s , a nd e m p i r i c a l a nd t h e o r e t i c a l mode_l e r s . A g e n e r a l m o no gr ap h i s i n o r d e n w h i ch d ra w s t o g e t h e r t h ework o f Ch ia ng a nd W igg e r t (1 96 8) , Dooge (1 95 9) , S uga wa ra (196 1) , Mossa nd Dawdy (1 97 3) , P e g ra m ( 19 77 ) , a nd S e lv a l i ng a m (1 97 7) . Tha t monographs h o u l d b eco me th e c l a s s i c p a p e r w h i ch D o o g e ' s p a p e r i s .

Today and Tomorrow — h e t r e n d t o d a y i n r a i n f a l l - r u n o f f m o d el in gi s t ow a rd p h y s i c a l l y - b a s e d d i s t r i b u t e d - p a r a m e t e r m o d e l s . H ow ev er ,t h e r e i s a t r e n d a t t h e sam e t i m e t o w a r d i n t r o d u c i n g t o o m any b e l l sa n d w h i s t l e s i n t o t h e m o d e l s b e c a u s e t h e m o d e l e r o r h i s e m p l o y e r"kn ow s" t h a t a p a r t i c u l a r f a c t o r i s i m p o r t a n t , an d , t h e r e f o r e , t h a tf a c t o r s h o u l d b e m o d e l e d .



110 D.R.Dawdy

T h e c o n c e p t u a l m o d e l e r s h a v e s h o w n t h a t v e r y s i m p l e m o d e l sp e r f o r m a s w e l l a s much m o re c o m p l i c a t e d m o d e l s i n d e r i v i n g t h e m o d e lo f th e ru n o f f co mp o n en t ( IU H) . Th ey h av e sho wn e m p i r i c a l ly h ow so meo f t h e e f f e c t s o f m an -m a de c h a n g e s o n t h e r u n o f f h y d r o g r a p h c a n b e

e s t i m a t e d ( C a r t e r , 1 9 6 1 ) . H o w e ve r, t h e m o d el o f t h e s u r f a c e r u n o f fi s w h e re t h e b e s t c a s e c a n b e m ade f o r p h y s i c a l m o d e l i n g . T he KWmo d e l i s a g o o d ex am p le . Th e re a r e p ro b le m s w i th KW m o d e l in g wh ichw i l l b e m e n t i o n e d l a t e r , b u t t h e p a r a m e t e r s a r e e a s y t o d e r i v e a n dt h e e f f e c t s o f m an -m ade c h a n g e s ca n b e e s t i m a t e d .

The i n f i l t r a t i o n f u c t i o n i s much m ore d i f f i c u l t t o m o d e l, a nde r r o r s i n r a i n f a l l i n p u t d a t a t e n d t o b e p a s s e d d i r e c t l y i n t o t h ee s t i m a t i o n o f p a r a m e t e r v a l u e s f o r i n f i l t r a t i o n (Dawdy a n d B e rg m an n,1 9 6 9 ) . Y e t t h e r e i s w h ere m o d e l e r s t e n d t o p r o l i f e r a t e i n d e t a i lo f m o d e l i n g . The e f f e c t s o f man -mad e ch a n g e s a r e a s su med mo re th anp r o v e n , a n d s e ld o m a r e m o d e l i n g r e s u l t s s - u b je c te d t o s p l i t - s a m p l et e s t i n g o r o t h e r r i g o r o u s a n a l y s i s . How d o e s o ne e s t i m a t e p a ra m e —t e r s f o r a n i n f i l t r a t i o n m o de l w h ic h c o n t a i n s s i x o r s e v e n o r n s o i ll a y e r s ? P e r h a p s t h e c o n c e p t u a l m o d e l e r s s h o u l d c o n c e n t r a t e on t h em o d e li ng o f i n f i l t r a t i o n s o t h a t , e v e n t u a l l y , a s y n t h e s i s may r e s u l ta s i n s u r f a c e r u n o f f m o d e l i n g .

KW m o d e l in g s t i l l h a s p r o b l e m s , a s m e n t i o n e d . I n t r o d u c t i o n o ft h e n o n - l i n e a r i t y i n t o t h e m o de l o f t h e s u r f a c e w a t e r co m po ne nt h a so v e r - c o r r e c t e d t h e m o d e l . F l o o d v e l o c i t i e s a r e m uch t o o f a s t . The

u n iq u e r a t i n g c u rv e a s s u m p t io n h o l d s f a i r l y w e l l b e c a u s e t h e r e e x i s ti n m o s t c h a n n e l s a s e r i e s o f c o n t r o l l i n g r e a c h e s . Ho we ve r t h e KWm o de l a s su m e s a p r i s m a t i c c h a n n e l , a nd i t t h e r e f o r e d o e s n o t a l l o wf o r s t o r a g e a d e q u a t e l y . T h a t p r o b l e m c a n n o t b e s o l v e d b y c h a n g i n gt o d yn am ic r o u t i n g . I t i s t h e a s s u m p t i o n c o n c e r n i n g t h e p r i s m a t i cc h a n n e l w h i c h i s a t f a u l t . M o d e l i n g o v e r b a n k f lo w i s n e c e s s a r y f o rh i g h e r f l o w s , b u t t h e a s s u m p t i o n s o f a p r i s m a t i c c h a n n e l s t i l l h o l d sa nd t h e b a s i c p r o b l e m r e m a i n s . How c a n t h e a t t e n u a t i o n o f f l o o dp ea k as a r e s u l t o f i r r e g u l a r i t i e s i n c h a n n el c r o s s s e c t i o n be i n t r o d u c e d i n t o KW m o d e l s ?

M ore b a s i c a l l y , i s t h e S u ga w a ra t a n k m o de l a v a l i d s u b s t i t u t efo r KW mo d e l s f o r m o d e l in g th e s u r f a ce ru n o f f co mp o n en t? Su g awarap r e s e n t s p i e c e - w i s e l i n e a r m o d e l s . The p a r a m e t e r s f o r h i s m o de lsm ay h a v e a s much p h y s i c a l m e a n i n g a s t h o s e f o r KW m o d e l s f o r l a r g e rd i s c h a r g e s w h e re o v e r b a n k f lo w e x i s t s . I s t h e r e a s y n t h e s i s o f KWa nd l i n e a r s t o r a g e m o d e ls w h ic h i s m ore p h y s i c a l l y m e a n i n g f u l t h a ne i t h e r a l o n e ?

D e t e r m i n i s t i c an d s t o c h a s t i c m o de ls a r e d r aw i n g c l o s e r t o g e t h e r .

R e s u l t s c o n c e r n i n g r e s p o n s e f u n c t i o n s f o r t a n k m o d e ls a r e d i r e c t l yt r a n s f e r a b l e fr om o n e t o t h e o t h e r , a s s ho wn b y P e g ra m ( 1 9 7 7 ). R e s u l t sa l o n g t h e s e l i n e s h a ve n o t b e e n f o l l o w e d up a g g r e s s i v e l y . I f ap h y s i c a l l y b a s e d s t o c h a s t i c m o de l ca n b e d e v e l o p e d f o r w h i ch manyc l o s e d fo rm s o l u t i o n s a r e k no wn , s t o c h a s t i c m o d e l i n g o f s t r e a m f l o wmay t a k e a s t e p f o r w a r d t o w a r d w i d e r a c c e p t a n c e a nd u s e .



Review of Rainfall - Runoff Modeling. 111

I n c o n c l u s i o n , I w i l l en d on a p e s s i m i s t i c n o t e a nd h op e t o b ep r o v e n w r o n g. The t e n d e n c y i s f o r m o d e ls t o c o n t i n u e t o p r o l i f e r a t ea nd t o b ec om e m ore c o m p l e x . I p r e d i c t t h a t s u r f a c e w a t e r r o u t i n gw i l l c o n t i n u e t o b e f i n e tu n e d a nd i n f i l t r a t i o n m o d e l in g w i l l c o n t i

nue t o r e c e i v e r e l a t i v e l y l e s s a t t e n t i o n . W hat a t t e n t i o n m o d e li ngo f i n f i l t r a t i o n d o es r e c e i v e w i l l b e a ge nc y o r i e n t e d an d w i l l t e n dt o make i n f i l t r a t i o n m o d els c om p le x, d i s t r i b u t e d - p a r a m e t e r m o de lsw i t h o u t i n t r o d u c i n g r i g o r o u s e r r o r a n a l y s i s t o t e s t w h e t h e r com plexi_t y i m p r o v e s p r e d i c t i o n . F u r t h e r m o r e , t h e c o m m o n a li ty w h i ch t a n km o d el s g i v e t o s t o c h a s t i c an d d e t e r m i n i s t i c m o d e l in g o f s t r ea m f l o ww i l l n o t b e e f f i c i e n t l y e x p l o i t e d t o s o l v e some t o t h e a s y e t u n a n s w e re d r e s e a r c h p r o b le m s i n s t o c h a s t i c m o d e l i n g .

I s h a l l w ork h a r d t h e n e x t few y e a r s t o p r o v e my p r e d i c t i o n s

w r o n g . I h o p e y o u d o , a l s o .

REFERENCES

A m oro ch o, J . a nd B r a n d s t e t t e r , A . , 1 9 7 1 . D e t e r m i n a t i o n o f N o n l i n e a rR e sp on se F u n c t i o n s i n R a i n f a l l - R u n o f f P r o c e s s e s , Water Res. Res. Vol 7 ,No. 5 , p p . 1 0 8 7 -1 1 0 1

A m o ro ch o, J . a n d O r l o b , G . T . , 1 9 6 1 . N o n l i n e a r A n a l y s i s o f H y d r o l o g i e

S y s t e m s , U n i v e r s i t y o f C a l i f o r n i a , Water Res. Center Cont. No. 40.

B a r n e s , H . H . , J r . , 1 96 7. R o ug hn es s C h a r a c t e r i s t i c s of N a t u r a l

E l e m e n t s , US Geological Survey Water Supply Paper 1849.

C a r t e r , R . W ., 1 9 6 1 . M a g n i t u d e a n d F r e q u e n c y o f F l o o d s i n S u b u r b a n

A r e a s , U. S. Geol Survey Prof. Paper 424-B, A r t i c l e 5 , p p . B -9 t o

B- 1 1 .

C h i a n g , T . T . a n d - W i g g e r t , J . M . , 1 9 6 8 . A n a l y s i s o f H y d r o l o g i e S y s t e m s , Bulletin 12, W ater Resources Research Center, V i r g i n i aP o l y t e c h n i c I n s t i t u t e .

C l a r k , C O . , 1 9 4 5 . S t o r a g e a nd t h e U n i t H y d r o g r a p h , Trans. ASCE,V o l . 1 1 0 , p p . 1 4 1 9 - 1 4 4 6 .

C l a r k e , R . T . , 1 9 7 3 , M a t h e m a t i c a l M o d e ls i n H y d r o l o g y , FAO, R om e,Irrigation and Drainage Paper 19.

C r a w f o r d , N . H . a nd L i n s l e y , R . K . , 1 9 6 2 . T he S y n t h e s i s o f c o n t i n u o u s

S t re a m f l ow H y d r o g ra p h s o n a D i g i t a l C o m p u te r, S t a n f o r d U n i v e r s i t yD e p t . C i v . E n g . Tech. Report No. 12.

C ra w-F ord , N .H . a n d L i n s l e y , R .K . , 1 9 6 6 . D i g i t a l S i m u l a t i o n i n

H y d r o l o g y , S t a n f o r d W a t e rs h e d M o del I V , S t a n f o r d U n i v e r s i t y D e p t .

C i v . E n g . Tech. Report No. 39.



112 D.R.Dawdy

D aw dy, D .R . a n d B e r g m a nn , J . M . , 1 9 6 9 . E f f e c t o f R a i n f a l l V a r i a b i l i t y

o n S t r ea m f l o w S i m u l a t i o n , Water Res. Res, V o l . 5 , N o . 5 , p p . 9 5 8 -

9 6 6 .

D aw dy, D .R . a n d K a l i n i n , G . P . , 1 9 7 0 . M a t h e m a t i c a l M o d e l i n g i nH y d r o l o g y , R e p o r t t o M i d - D e c a d e C o n f e r e n c e r o f I H D , Bulletin of IASH.

D aw dy, D . R . , L i c h t y , R .W . a n d B e r g m a nn , J . M . , 1 9 7 2 . A R a i n f a l l -R u no ff S i m u l a t i o n M o de l f o r E s t i m a t i o n o f F l o o d P e a k s f o r S m a l lD r a i n a g e B a s i n s , USGS Prof. Paper 506-B,

D aw dy , D .R . a nd O ' D o n n e l l , T e r e n c e , 1 9 6 5 . M a t h e m a t i c a l M o d e l s o f

C a t c h m e n t B e h a v i o r , Proc. usee, Vo l . 9 1 , No . HY4 , Pap e r 4 4 1 0 , p p .

1 2 3 - 1 2 7 .

D o o g e, J . C . I . , 1 9 5 9. A G e n e r a l T h e o r y o f t h e U n i t H y d r o g r a p h , Jour.Geophys. Res. , V o l 6 4 , N o. 2 , p p . 2 4 1 - 2 5 6 .

D o o g e, J . C . I . , 1 9 6 5 . A n a l y s i s o f L i n e a r S y s t e m s b y M e an s o f L a g u e r r e

F u n c t i o n s , Jour. SIAM Control, S e r . A , V o l . 2 , No . 3 , p p . 3 9 0 -4 0 8

E a g l e s o n , P . S . , M e j i a , R . R. a n d M a r c h , F . , 1 9 6 6 . C o m p u t a t i o n o fO ptim um R e a l i z a b l e U n i t H y d r o g r a p h s , Water Res. Res, Vol . 2 , No . 4

H y d r o l o g i e E n g i n e e r i n g C e n t e r , US Army C o r p s o f E n g i n e e r s , 1 9 7 0 .HEC-1 F lo o d Hy d ro g rap h Pack ag e , Users Manual, D a v i s , C a l i f o r n i a .

K a l i n i n , G . P . a nd M i l u k ov , P . I . , 1 9 58 . A p r o x im a t e C a l c u l a t i o n o fU n s t a b l e M ovem ent o f S t r e a m s ( i n R u s s i a ) , Trudy SIP, I s s u e 6 6 ,G i d r o m e t e o i z d a t , L e n i n g r a d .

M a n d e l b r o t , B . B . , 1 9 7 1 . A F a s t F r a c t i o n a l G a u s s i a n N o i s e G e n e r a t o r ,Wa ter Res. Res., V o l . 7 , No. 3 , p p . 5 4 3 -5 5 3 .

M c C a r t h y, G . J . , 1 9 3 8 . T he U n i t H y d r o g r a p h a n d F l o o d R o u t i n g , paper

presented at a conference of North A tlantic Division, us Army Corpso f E n g i n e e r s .

M o s s , M .E . a n d D aw dy, D . R . , 1 9 7 3 . S t o c h a s t i c S i m u l a t i o n f o r B a s i n sw i t h S h o r t o r n o R e c o r d s o f S t r e a m f l o w , Proc. of IAHS Symp on Design

of Water Res. Projects with Inadequate Data, Madrid.

M i t c h e l l , W .M ., 1 9 6 2 . E f f e c t o f R e s e r v o i r S t o r a g e o n P e a k F l o w , US

G e o l o g i c a l S u r v e y Water Supply Paper 1580 - C.

N as h J . E . , 1 9 5 8 . T he F orm o f t h e I n s t a n t a n e o u s U n i t H y d r o g r a p h , Bull,IAHS, V o l . 3 , N o . 4 5 , p p . 1 1 4 - 1 2 1 .

N a s h , J . E . , 1 959 A n o t e o n t h e M u sk in gu m F l o o d - R o u t i n g M e t h o d ,Jour. Geoph ys. Res. V o l . 6 4 .



Review of Rainfall - Runoff Modeling. 1 1 3

N a s h, J . E . , 1 9 6 1 . A L i n e a r T r a n s f o r m a t i o n o f a D i s c h a r g e R e c o r d ,

Proceedings, Ninth Conve ntion, IAHR , V o l . 3 , p p . 1 3 - 1 t o 2 .

O ' D o n n e l l , T e r e n c e , 1 96 0 . I n s t a n t a n e o u s U n i t H y d r o g ra p h D e r i v a t i o nb y H a r m o n i c A n a l y s i s , IASH, Publ. No. 51, p p . 5 4 6 - 5 5 7 .

O ' K e l l y , J . J . , 1 9 5 5 . T he E m p lo y m en t o f U n i t H y d r o g r a p h s t o " D e te rm i n eThe F lo w s o f I r i s h A r t e r i a l D r a in a g e C h a n n e l s , Jour. Inst. Civ.Engrs:, V o l , 4 , p p . 3 6 5 - 4 4 5 .

P e gr am , G . G .S , 1 9 77 . P h y s i c a l J u s t i f i c a t i o n o f a C o n t i n u o u s S t r e a m -f low Model , Proc. Third Intl. Hydr. Sump., F t . C o l l i n s , C o lo r a d o .

Se lv a l in ga m , s . , 1977 . ARMA and L in e ar Tank M ode ls , Proc. Third

Intl. Hydr. Sump . , F t . C o l l i n s , C o l o r a d o .

S h e r m a n , L . K . , 1 9 3 2 . S t r e a m f l o w f ro m R a i n f a l l b y t h e U n i tH y d r o g r a p h M e t h o d , Engr. News-Record, V o l . 1 0 8 , p p . 5 0 1 - 5 0 5 .

S u g a w a r a , M . , 1 9 6 1 . On t h e A n a l y s i s o f R u n of f S t r u c t u r e a b o u t S e v e r a l J a p a n e s e R i v e r s , Jap. Jour. Geophys. , Vol 2 , No. 4 , p p . 1-76.



A Review of Rainfall-runoff Modeling

Documents