Expériences in the development and Application of Mathematical Models in Hydrology and Water Resources in Latin America (Proceedings of the Tegucigalpa Hydromath Symposium, September 1983). fAHSPubl.No. 152.

A REVIEW OF RAINFALL-RUNOFF MODELING David R. Dawdy

Consulting Hydrologist Visiting 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 develop­ments. 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, money, 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 pro­blems previously unsolvable.The constraint inhydrology changed from the inability to solve a problem to the inability to collect suffi­cient and sufficiently accurate data to prove that a solution is co­rrect 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 whe­re 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 desi­red. A model is a mathematical statement of the response of a sys­tem which takes system inputs and transforms them into system out­puts. Even though the jargon is modern, the rational method for es­timating 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

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 is. 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 deter­mine 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 time invariant, 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 quan­tity. 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 mathema­tical 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. 99

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 = S0 - Sf (5) g t g X X

Where V = v e l o c i t y With t u r b u l e n t f l u c t u a t i o n s H = depth of water With t u r b u l e n t f l u c t u a t i o n s S0= s lope of channel bottom an ' a v e r a g e ' s lope of a reach Sf= f r i c t i o n s lope a conceptua l a b s t r a c t i o n

The va lue for Sf i s de r ived from a s o - c a l l e d ' f r i c t i o n fo rmula ' , such as Chezy, which i s ' t h e o r e t i c a l ' , or Manning, which 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 the e m p i r i c i s t s fo r using the 'wrfflng' f r i c t i o n fo rmula / However, the two can be shown to be almost 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 roughness i s cons ide red . For example, i f we were to assume t h a t we have a g rave l -bed stream wi th a ' c h a r a c t e r i s t i c g r a in s i z e ' of 2 cen t ime te r s and were t o assume a depth of 1/2, 1, 2 , 5 , and 10 m e t e r s , the P r a n d t l equat ion would give d i f f e r e n t va lues fo r Chezy C as depth i n c r e a s e d , because r e l a t i v e roughness would change. On the o the r hand, Manning's n would remain almost cons t an t , because the va lues of Manning's n inc lude changes of r e l a t i v e roughness . However one uses Equation 5 , i t e n t a i l s b lack magic i n the r e a l wor ld , even though i t i s a d i f f e r e n t i a l e q u a t i o n . Considerable ' e n g i n e e r i n g judgment' e n t e r s i n t o the choice of Sf, even wi th the aid of the e x c e l l e n t work of Barnes (1967) and o t h e r s i n the USGS, who have t r i e d t o r a t i o n a l i z e the de te rmina t ion of r e s i s t a n c e to flow for use in open channel flow problems.

The instantaneous unit hydrograph.- With the foregoing as a p r e l u d e , the IUH can be seen as a t o o l of l i n e a r systems a n a l y s i s . The 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 of 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 s t 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 of i n p u t a p p l i e d

•\00 D.R. Dawdy

instantaneously in time an abstract concept. Its mathematical statement is the convolution integral

t

y(tl = J* hft-t)x(X)dt (.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 based on the. concept of a linear storage resulting from a hypothetical line­ar reservoir. As stated earlier:

I-Q = dS/dt Continuity (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

uî( 0 t\ _ 1 -(t-t )/K single linear reservoir (8) ~~ (Clark and others) 1_ K

K

e" ( t"

t - t p

K

-t c

e~

,)/K

• ( t - t Q

(n-1) u (0 t\ _ 1 t-t Q e-(t-t 0)/K n equal linear n " 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. 101

Thus, the problem of l i n e a r s y n t h e s i s in IUH a n a l y s i s invo lves the assumption of a reasonable model and the development of a method t o e s t ima te the parameter values for t h a t model. This genera l approach leads in two major d i r e c t i o n s : the development of conceptual and of b lack-box models.

The Conceptual IUH - Conceptual models came f i r s t , with the g r e a t e s t amount of a c t i v i t y in the 1950 's . However, the work on conceptual models s t a r t e d e a r l i e r . The Muskingum method for f lood rou t ing (McCarthy, 1938) i s in the form of a l i n e a r s to rage model. Nash (1959) showed t h a t the Muskingum model rou tes flows through two l i n e a r r e s e r v o i r s , the f i r s t with nega t ive storage—which exp la in s the anomalous r e s u l t s of a decrease in flow ob ta ined a t the beginning of a r ou t i ng in many c a s e s .

An i n t e r e s t i n g approach t o channel r ou t i ng by l i n e a r a n a l y s i s was developed by Kal in in and Milukov (1958). They developed the. concept of a c h a r a c t e r i s t i c length over which the rou t i ng was a s i n g l e l i n e a r r e s e r v o i r . The parameters of the length and the s to rage were r e l a t e d to channel measurements. Once the c h a r a c t e r i s t i c length i s determined, longer r eaches , rou ted s e q u e n t i a l l y , develop a gamma d i s t r i b u t i o n s i m i l a r t o a Nash cascade for bas in r o u t i n g . Thus, the Sovie t s were working on s i m i l a r problems and developing s o l u t i o n s s i m i l a r to those desc r ibed e a r l i e r dur ing t h i s p e r i o d .

Most conceptual models of the IUH are based on the twin concepts of l i n e a r s to rage and l i n e a r channels . Linear s to rage was descr ibed e a r l i e r (Equation 3 ) . A l i n e a r channel i s one which passes an i n p u t hydrograph wi thout a t t e n u a t i o n . The l i n e a r channel i s used t o develop a t ime-a rea his togram (TAH) , which i s the outflow hydrograph from an ins t an taneous r a i n f a l l - e x c e s s appl ied uniformly over a bas in i f there were no s to rage a c t i n g to a t t e n u a t e the hydrograph. The s imp le s t form of 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 routed through a l i n e a r r e s e r v o i r with a s to rage c o e f f i c i e n t on the o rde r of the time base of the t r i a n g l e y i e l d s a response funct ion q u i t e s i m i l a r t o the usua l runoff hydrograph and to the gamma d i s t r i b u t i o n of the Nash cascade . 0 'Ke l ly (1955) in t roduced the i s o s c e l e s tr iangle TAH. This e a r l y formulat ion has some p h y s i c a l j u s t i f i c a t i o n . Consider t h a t overland flow genera tes a response funct ion of uniform flow for t ime, T1, i n t o a main channel system with a time of t r a v e l of T2. Thus, the response funct ion of each of t h e s e , t r e a t e d as l i n e a r channels , i s a r e c t a n g u l a r p u l s e . The outflow TAH i s the convolut ion of two r e c t a n g l e s . I f T1 = T2, the 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 . Mi tche l l (1962) showed t h a t most small s treams in I l l i n o i s could be modeled with such a TAH. However, he found t h a t some streams had a f l a t - t o p p e d IUH, and r equ i r ed the use of a t r a p e z o i d for a TAH. I f T1 / T2, the convolut ion produces a t r a p e z o i d of base lengths T1 + T2 and T1 - T2. Thus, once aga in , p h y s i c a l j u s t i f i c a t i o n may follow e 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 model 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 by 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 by i s o c h r o n e s . The a r e a s b e t w e e n i s o c h r o n e s determi_ n e s a t ime a r e a h i s t o g r a m (TAH). 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 on t h e b a s i s o f t h e TAH and 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 model 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 component o f t h e S t a n f o r d W a t e r s h e d Model (Crawford and 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 model by Dawdy, L i c h t y , and Bergmann (1972) and i s an a l t e r n a t i v e i n t h e Corps o f E n g i n e e r s HEC-1 ( 1 9 7 0 ) .

C o n c e p t u a l mode 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 had a common b a s e i n some form o f l i n e a r r e s e r v o i r r o u t i n g and i n t h e c o n ­c e p t of a l i n e a r , c h a n n e l . The l i n e a r c h a n n e l moves 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 . The l i n e a r s t o ­r a g e p r o v i d e s t h e means 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 ­mes 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 . The C l a r k TAH p r o ­v i d e s t h e means t o model b a s i n s f o r which t h e IUH h a s a complex s h a ­p e . The p a r a m e t e r s of 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 . The t h e o r y was summar ized i n D o o g e ' s e x c e l l e n t monograph ( 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-BOX IUH

The 1960's saw an o u t b u r s t of i n t e r e s t in b lack-box modeling of the IUH. The s i m p l i f i c a t i o n s of l i n e a r r e s e r v o i r models l ed t o a search for a l t e r n a t i v e a n a l y s i s . Simple harmonic a n a l y s i s were t r i e d by O'Donnell (1960). Truncat ion in the harmonic a n a l y s i s cau­sed problems o r r ing ing and smoothing. Chiang and Wiggert (1968) p laced harmonic a n a l y s i s for the IUH in the framework of genera l black-box a n a l y s i s as developed in e l e c t r i c a l eng inee r ing .

Matrix i n v e r s i o n techniques for the d e r i v a t i o n of the IUH were in t roduced s imul taneous ly by Nash (1961) and by o t h e r s , such as the TVA and Snyder. Each undoubtedly r e a l i z e d t h a t d i g i t a l computers o-p e r a t e most e f f i c i e n t l y in mat r ix m u l t i p l i c a t i o n , and t h a t an IUHis a l i n e a r mat r ix t r ans fo rma t ion . The r e a l i z a t i o n t h a t the IUH i s a l i ­near matr ix t r ans format ion i s d i s c r e t e time has d i r e c t i m p l i c a t i o n s in conceptual IUH modeling, so t h a t conceptual models gained by a sp in -o f f from black-box modeling, p a r t i c u l a r l y from the works of Nash and O'Donne 11. Not a l l models u t i l i z e t h i s p r i n c i p l e complete­l y , and t h e i r r e s u l t i n g computer program i s made more complex and t i me consuming than i s neces sa ry .

Some black-box modelers gained knowledge from conceptual models in the development of methods for i n v e r s i o n . An example of t h i s approach i s shown by Dooge (1965), who used Laguerre funct ions for the i nve r s ion of i n p u t - o u t p u t p a i r s t o develop the IUH. The r e s u l t ­ing IUH i s s i m i l a r t o the Nash cascade conceptual IUH.

Review of Rainfall - Runoff Modeling. 103

F i n a l l y , b lack-box modeling moved i n t o the non l inea r domain, with the work of Amorocho and Orlob (1961). They developed a method t o i s o l a t e and model the non l inea r elements in the response func­t i o n . L a t e r , Amorocho and B r a n d s t e t t e r (1971) developed a g e n e r a l , non l i nea r , - b lack-box i n v e r s i o n t echn ique . Ac tua l ly , b lack-box model ing impl ies a l i n e a r system. The non l inea r models might b e t t e r be c 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 models, r a t h e r than black-box models.

Been shown t h a t i f a s epa ra t e of s e t of events not used in the f i t i s used t o t e s t the accuracy of the r e s u l t i n g models, conceptual models perform b e t t e r than black-box models. The very c o n s t r a i n t s which make the f i t fo r conceptual models worse are what a l so cause them to p r e d i c t b e t t e r . There have been some a t tempts to b u i l d cons t r a i n t s i n t o b lack-box models in o rder t o p r e d s i c t b e t t e r a t the e x " pense of f i t t i n g worse . An example i s Eag leson ' s (1966) optimum re_a l i z a b l e IUH. He used a l i n e a r programming format with a non-zero c o n s t r a i n t on the o r d i n a t e s of the IUH.

A major drawback t o the use of black-box models i s t h a t they cannot be used t o model a changing system. Because black-box models are not concerned with the i n t e r n a l workings of the system they can­no t be modified e a s i l y to r e f l e c t the r e s u l t s of such changes. Many i f no t most uses of watershed models today are t o a s sess the e f f e c t of p a s t or p o t e n t i a l fu ture man-made changes on a watershed. Concep_ t u a l models are we l l s u i t e d for such u s e s , because the parameters in a conceptual model may be r e l a t e d t o p h y s i c a l parameters of a b a s i n .

That need for the modeling of the e f f e c t s of man-made changes has l ed t o developments in two major d i r e c t i o n s . Both developments are in conceptua l modeling. The f i r s t development i s i n the use of a non l inea r r ou t i ng model based on the k inemat ic wave e q u a t i o n s . The second development i s the b u i l d i n g of d i s t r i b u t e d parameter models to r ep lace the lumped parameter models of c l a s s i c a l IUH theo ry .

COMPARISON OF BLACK-BOX AND CONCEPTUAL IUH

Black-box model development has tended to move in the d i r e c t i o n of the use of the knowledge gained from the use of conceptual models However, t o the e x t e n t t h a t b lack-boxes remain b l ack , they are not concerned with the i n n e r wdrkings of the system which they model. Conceptual models are cons t r a ined so t h a t t h e i r shape w i l l "look r i g h t " in terms of r e a l world hydrographs .

As a r e s u l t of the lack of c o n s t r a i n t s in t h e i r s t r u c t u r e , b lack-box models tend to f i t a s e t of da ta b e t t e r than do conceptual models. I f a s i n g l e event i s used t o der ive a b lack-box IUH, the da_ t a can be f i t p e r f e c t l y . Conceptual models w i l l , in g e n e r a l , no t

104 D.R. Dawdy

f i t even a single event perfectly. If a set of events is used with least squares f i t t ing to derive an IUH, black-box models, in general will f i t the data bet ter . However, i t has

Kinematic Wave Models

The kinematic wave (KW) is one step away from the linear stora­ge 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 par t ia l differential terms, where q is the la tera l inflow, 9q/9x is the outflow per unit with, and 9 y / 3 t i s the change in

depth with time, which i s equal to change in storage per unit width. Equation 10 is combined with the kinematic assumption.

Q = «A1" (11a)

q = (Xym (11b)

where <X and m a r e t h e KW p a r a m e t e r s . E q u a t i o n s 10 and 11 a r e combi_ ned t o 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, le t us assume that Manning's equation applies over a reach of in teres t . Then

Review of Rainfall - Runoff Modeling. 105

1/2 1.5 2 / 3 S

Q = — - AR ( 1 3 )

where 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 , and 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 become e m p i r i c a l . I f t h e w i d t h i s much 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 (14a)

' 2 / 3 s 1 / 2

2 = ¥ A w 2 / 3 (14b)

1/2

n W ^ / J (14c)

m = 5/3

and s i m i l a r equa t ions may be der ived for o t h e r shapes of channe ls . Thus, (X i s a funct ion of p h y s i c a l measures of the reach,and both a and m are funct ions of the shape of the channel cross s e c t i o n and of the f r i c t i o n law assumed (Manning's equa t ion in t h i s example) .

The equa t ion i s q u i t e s i m i l a r t o the r e s u l t s of e a r l i e r a t tempts a t developing a n o n l i n e a r s to rage equa t ion . I f s to rage i s assumed d i r e c t l y r e l a t e d t o a power funct ion of flow depth or t o c r o s s - s e c ­t i o n a l a r e a , the two are i d e n t i c a l . However, the use of the KW equji t i o n has taken a s t ep away from the hydro log ie assumptions of l i n e a r and non l inea r s to rage and toward h y d r a u l i c r o u t i n g .

A major advantage of KW rou t ing i s t h a t i t s parameters r e l a t e t o the p h y s i c a l world. I f t h a t p h y s i c a l world i s modified, the e f f e c t on the rou t i ng parameters can be e s t i m a t e d , and r e s u l t i n g changes i n the bas in response can be p r e d i c t e d . A major s h o r t - c o ­ming of KW rou t ing i s t h a t Equat ions 14 assume t h a t a unique, s i n ­g l e -va lued , simple s tage d ischarge r a t i n g app l i e s wherever the equa­t ion i s used. The k inemat ic wave number can be used to screen out those cases where the equat ion does not apply because dynamic e f f e c t s cause s tage and d ischarge t o be r e l a t e d d i f f e r e n t l y on the r i s i n g and the f a l l i n g limb of the hydrograph. A more s e r i o u s conse_ quence of the k inemat ic assumption a r i s e s because Equations 13 and 14 apply b e s t a t c o n s t r i c t i o n s or con t ro l r eaches . The added storage

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 advan­tage 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 end in bas in response modeling i s t o use a d i s t r i ­buted parameter d e s c r i p t i o n of the b a s i n . A t y p i c a l d i v i s i o n of a bas in for d i s t r i b u t e d - p a r a m e t e r modeling i s shown i n Figure 1. F i r s t , the main channel system i s d e t a i l e d . Reaches are then determined which have 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 throughout t h e i r l e n g t h . The over land flow and channel segments o u t l i n e d in Figure 1A are then desc r ibed in such a manner as to develop the schematic diagram shown in Figure 1B.

The assignment of i npu t p h y s i c a l da ta to the bas in def ines the bas in response func t ion . Thus, t he r e are s e v e r a l major advantages which the d i s t r i b u t e d parameter model has over a lumped parameter model such as an IUH. The f i r s t major advantage i s t h a t the response funct ion can be developed d i r e c t l y from the inpu t p a r a ­meters i f an appropr i a t e model, such as KW, i s used. A t y p i c a l s e t of inpu t da ta for a d i s t r i b u t e d parameter model i s shown in Figure 2. A second major advantage i s t h a t nonuniform storms may be app l ied t o the b a s i n - t y p i c a l i s o h y e t a l s of mean annual r a i n f a l l are shown in Figure 1A, which may be used t o d i s t r i b u t e r a i n f a l l over the b a s i n .

The t h i r d , and compel l ing, major advantage of d i s t r i b u ­t ed parameter models i s t h a t the change in bas in response r e s u l t i n g from man-made changes over p a r t of the bas in may be a s se s sed . Any p a r t of the schematic in Figure 1B may be modeled wi th "before and a f t e r " p r e d i c t i o n s by changing the s e t of parameters for t h a t p a r t of the b a s i n .

One major disadvantage of d i s t r i b u t e d parameter models i s t h a t they gene ra l l y r equ i r e more da ta and much more computer time t o run than do lumped-parameter models.

Review of Rainfall - Runoff Modeling. 107

27 -

A. STREAM CHANNEL NETWORK OF BASIN

\ ;> / c-

\

V

\ /

7

/

7

\

i

/ „ \ 9 \

\ '

/ /

/ \

\

V ,± \

^

\

Y /

/

7 \ / /3\

\

B. DIVISION OF BASIN INTO STREAM CHANNEL

AND OVERFLOW SEGMENTS

! FIGURE 1. TYPICAL SCHEMATIC REPRESENTATION OF A BASIN FOR USE IN

DEVELOPING A DISTRIBUTED-PARAMETER RAINFALL-BUNOFF MODEL

As computers get larger and fas ter and cheaper that disadvari tage decreases in importance. With the advent of minicomputers in -every off ice, i t may reassume importance. An important point to consider i s tha t proper programming can greatly reduce computing time. Note in Figure 1B tha t there are 34 overland flow sections flowing into 20 channel reaches, but overland flow reaches are numbered to 7 (in the corners of the overland flow segments) and channel reaches to 13. Thus 54 segments have been modeled as 20 segments. If segment charac te r i s t i cs are suff ic ient ly s imilar , large savings in computer time can r e s u l t . Even so, the canned bulk-para-

108 D. R. Da'wdy

ROUTING COMPONENT

INPUT DATA:

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. Typical Set of Input Data Used To Define a Segment for A Distributee-Parameter Rainfall-Runoff Model.

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 mathe­matical 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 approximation may be r ep re sen t ed by a tank model. The s e t of t a n k s , each 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 be arranged in s e r i e s or in p a r a l l e l . The parameters for each tank may be e s t ima ted from p h y s i c a l parameters o r by o t h e r means app rop r i a t e for the given component. The i n p u t s and ou tpu t s fo r each tank are def ined and Voi la ! We have a tank model.

The c losed form s o l u t i o n of the response funct ion for some conf igu ra t ions of tank models can be de r ived . Nash (1958) obvious ly solved the case for a s e r i e s of n equal t a n k s . Sugawara (1961) solved many more complex c a s e s . In a d d i t i o n he d i scussed piecewise l i n e a r s o l u t i o n of k e r n e l s by use of complex geometry and m u l t i p l e o u t l e t t a n k s . Sugawara d i scussed the i n t e r p r e t a t i o n and e s t ima t ion of the tank parameters for d i f f e r e n t components. F i n a l l y , Sugawara p r e s e n ­ted 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 model development through the use of lumped parameter tank modeling of s u b - b a s i n s .

A most i n t e r e s t i n g fac t in the mathematical development of tank models i s t h a t most of the subsequent i n t e r e s t in t h i s d e t e r m i n i s t i c r a i n f a l l - r u n o f f model ou t s ide Japan comes from s t o c h a s t i c hydrology. A simple s e r i e s tank model wi th a s i n g l e i n p u t of white no ise and with a s i n g l e ou tpu t genera tes an au tore -gress ive-moving average (ARMA) model. Moss and Dawdy (1973) showed t h a t a conceptual r a i n ­f a l l - r u n o f f model e q u i v a l e n t t o a s i n g l e tank developed an ARMA (1,1) model for s t o c h a s t i c s imula t ion of monthly streamflow. Pegram (1977) showed the mathematical e q u i v a l e n t of a Clark IUH formulation and an ARMA model under c e r t a i n assumptions. Selvalingam (1977), a s tuden t of Sugawara ' s , showed the exac t e q u i v a l e n t of tank models and ARMA models . The f a s t f r a c t i o n a l Gaussian no ise model (Mandelbrot, 1971) i s , of cour se , a p a r a l l e l tank model, which should r e s u l t in summation of ARMA (1,1) models r a t h e r than a summation of a u t o r e -g re s s ive models. I n c i d e n t a l l y , s imu la t ion of average flows ( d a i l y , weekly, or monthly) adds one dimension t o the moving average p o r t i o n in r e l a t i o n to sampling a t d i s c r e t e i n t e r v a l s . Average flows a re d i s c r e t i z e d bu t not 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 should be kep t in mind when b u i l d i n g models fo r s t o c h a s t i c s i m u l a t i o n .

Thus, tank models seem to be a t o o l fo r drawing t o g e t h e r s t o c h a s t i c and d e t e r m i n i s t i c models, 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 and conceptual models, and emp i r i ca l and t h e o r e t i c a l mode_ l e r s . A gene ra l monograph i s i n orden which draws t o g e t h e r the work of Chiang and Wiggert (1968), Dooge (1959), Sugawara (1961), Moss and Dawdy (1973), Pegram (1977), and Selvalingam (1977). That monograph should become the c l a s s i c paper which Dooge's paper i s .

Today and Tomorrow — he t r e n d today in r a i n f a l l - r u n o f f modeling i s toward 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 models. However, t he re i s a t r e n d a t the same time toward i n t r o d u c i n g too many b e l l s and w h i s t l e s i n t o the models because the modeler or h i s employer "knows" t h a t a p a r t i c u l a r f ac to r i s impor tan t , and, t h e r e f o r e , t h a t f a c t o r should be modeled.

110 D.R.Dawdy

The conceptual modelers have shown t h a t very simple models perform as wel l as much more complicated models in de r iv ing the model of the runoff component (IUH). They have shown e m p i r i c a l l y how some of the e f f e c t s of man-made changes on the runoff hydrograph can be e s t ima ted (Car t e r , 1961). However, the model of the sur face runoff i s where the b e s t case can be made for p h y s i c a l modeling. The KW model i s a good example. There are problems with KW modeling which w i l l be mentioned l a t e r , b u t the parameters are easy to der ive and the e f f e c t s of man-made changes can be e s t ima ted .

The i n f i l t r a t i o n fuc t ion i s much more d i f f i c u l t t o model, and e r r o r s in r a i n f a l l i npu t da ta tend t o be passed d i r e c t l y i n t o the e s t i m a t i o n of parameter values for i n f i l t r a t i o n (Dawdy and Bergmann, 1969). Yet t he re i s where modelers tend t o p r o l i f e r a t e in d e t a i l of modeling. The e f f e c t s of man-made changes are assumed more than proven, and seldom are modeling r e s u l t s s-ubjected t o s p l i t - s a m p l e t e s t i n g or o t h e r r igorous a n a l y s i s . How does one e s t ima te parame— t e r s for an i n f i l t r a t i o n model which con ta ins s i x or seven or n s o i l l a y e r s ? Perhaps the conceptual modelers should concen t ra te on the modeling of i n f i l t r a t i o n so 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 t as i n sur face runoff modeling.

KW modeling s t i l l has problems, as mentioned. I n t r o d u c t i o n of the n o n - l i n e a r i t y i n t o the model of the sur face water component has o v e r - c o r r e c t e d the model. Flood v e l o c i t i e s are much too f a s t . The unique r a t i n g curve assumption holds f a i r l y w e l l because t he r e e x i s t i n most channels a s e r i e s of c o n t r o l l i n g r eaches . However the KW model assumes a p r i s m a t i c channel , and i t t he r e fo re does no t allow for s t o r age adequa te ly . That problem cannot be solved by changing t o dynamic r o u t i n g . I t i s the assumption concerning the p r i s m a t i c channel which i s a t f a u l t . Modeling overbank flow i s necessary for h igher f lows, bu t the assumptions of a p r i s m a t i c channel s t i l l holds and the b a s i c problem remains . How can the a t t e n u a t i o n of f lood peak as a r e s u l t of i r r e g u l a r i t i e s i n channel c ross s e c t i o n be i n t r o ­duced i n t o KW models?

More b a s i c a l l y , i s the Sugawara tank model a v a l i d s u b s t i t u t e for KW models for modeling the surface runoff component? Sugawara p r e s e n t s p iece -wise l i n e a r models. The parameters for h i s models may have as much p h y s i c a l meaning as those for KW models for l a r g e r d i scharges where overbank flow e x i s t s . I s t he re a syn the s i s of KW and l i n e a r s to rage models which i s more p h y s i c a l l y meaningful than e i t h e r alone?

De t e rmin i s t i c and s t o c h a s t i c models are drawing c l o s e r toge ther . Resu l t s concerning response funct ions for tank models are d i r e c t l y t r a n s f e r a b l e from one to the o t h e r , as shown by Pegram (1977). Results along these l i n e s have not been followed up a g g r e s s i v e l y . I f a p h y s i c a l l y based s t o c h a s t i c model can be developed for which many c losed form s o l u t i o n s are known, s t o c h a s t i c modeling of streamflow may take a s t e p forward toward wider acceptance and use .

Review of Rainfall - Runoff Modeling. 111

In conc lus ion , I w i l l end on a p e s s i m i s t i c note and hope to be proven wrong. The tendency i s for models t o cont inue to p r o l i f e r a t e and to become more complex. I p r e d i c t t h a t sur face water r o u t i n g w i l l cont inue to be f ine tuned and i n f i l t r a t i o n modeling w i l l c o n t i ­nue to r ece ive r e l a t i v e l y l e s s a t t e n t i o n . What a t t e n t i o n modeling of i n f i l t r a t i o n does rece ive w i l l be agency o r i e n t e d and w i l l tend to make i n f i l t r a t i o n models complex, d i s t r i b u t e d - p a r a m e t e r models wi thout i n t roduc ing r igorous e r r o r a n a l y s i s t o t e s t whether complexi_ ty improves p r e d i c t i o n . Furthermore, the commonality which tank models give to s t o c h a s t i c and d e t e r m i n i s t i c modeling of streamflow w i l l not be e f f i c i e n t l y e x p l o i t e d to solve some t o the as ye t unans­wered resea rch problems in s t o c h a s t i c modeling.

I s h a l l work hard the next few years t o prove my p r e d i c t i o n s wrong. I hope you do, a l s o .

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Amorocho, J . and Orlob, G.T., 1961. Nonlinear Analysis of Hydrologie Systems, Un ive r s i ty of C a l i f o r n i a , Water Res. Center Cont. No. 40.

Barnes, H.H., J r . , 1967. Roughness C h a r a c t e r i s t i c s of Natura l Elements , US Geological Survey Water Supply Paper 1849.

C a r t e r , R.W., 1961. Magnitude and Frequency of Floods in Suburban Areas , U. S. Geol Survey Prof. Paper 424-B, A r t i c l e 5, pp . B-9 t o B- 1 1 .

Chiang, T.T. and-Wiggert, J .M. , 1968. Analysis of Hydrologie Sys­tems, Bulletin 12, Water Resources Research Center, V i r g i n i a Po ly techn ic I n s t i t u t e .

Clark , C O . , 1945. Storage and the Unit Hydrograph, Trans. ASCE, Vol. 110, pp . 1419-1446.

Clarke , R.T. , 1973, Mathematical Models in Hydrology, FAO, Rome, Irrigation and Drainage Paper 19.

Crawford, N.H. and L ins l ey , R.K., 1962. The Synthes i s of cont inuous Streamflow Hydrographs on a D i g i t a l Computer, Stanford Un ive r s i ty Dept. Civ. Eng. Tech. Report No. 12.

Craw-Ford, N.H. and L ins l ey , R.K. , 1966. D i g i t a l Simulat ion in Hydrology, Stanford Watershed Model IV, Stanford Unive r s i ty Dept. Civ. Eng. Tech. Report No. 39.

112 D.R.Dawdy

Dawdy, D.R. and Bergmann, J .M. , 1969. Effec t of Rainfall V a r i a b i l i t y on Streamflow Simula t ion , Water Res. Res, Vol. 5, No. 5 , pp . 958-966.

Dawdy, D.R. and K a l i n i n , G.P. , 1970. Mathematical Modeling in Hydrology, Report to Mid-Decade Conferencer of IHD, Bulletin of IASH.

Dawdy, D.R., L ich ty , R.W. and Bergmann, J .M. , 1972. A R a i n f a l l -Runoff Simulat ion Model for Es t imat ion of Flood Peaks for Small Drainage Bas ins , USGS Prof. Paper 506-B,

Dawdy, D.R. and O'Donnell , Terence, 1965. Mathematical Models of Catchment Behavior, Proc. usee, Vol. 9 1 , No. HY4, Paper 4410, pp. 123-127.

Dooge, J . C . I . , 1959. A General Theory of the Unit Hydrograph, Jour. Geophys. Res. , Vol 64, No. 2, pp . 241-256.

Dooge, J . C . I . , 1965. Analys is of Linear Systems by Means of Laguerre Funct ions , Jour. SIAM Control, Ser . A, Vol. 2, No. 3, pp. 390-408

Eagleson, P . S . , Mejia, R. R. and March, F . , 1966. Computation of Optimum Real izab le Unit Hydrographs, Water Res. Res, Vol. 2, No. 4

Hydrologie Engineer ing Center , US Army Corps of Engineers , 1970. HEC-1 Flood Hydrograph Package, Users Manual, Davis, C a l i f o r n i a .

K a l i n i n , G.P. and Milukov, P . I . , 1958. Aproximate Ca lcu l a t i on of Unstable Movement of Streams (in Russ i a ) , Trudy SIP, I ssue 66, Gidrometeoizdat , Leningrad.

Mandelbrot, B .B. , 1971. A Fas t F r a c t i o n a l Gaussian Noise Generator, Water Res. Res., Vol. 7, No. 3, pp . 5 43-55 3.

McCarthy, G . J . , 1938. The Unit Hydrograph and Flood Routing, paper presented at a conference of North Atlantic Division, us Army Corps of Engineers .

Moss, M.E. and Dawdy, D.R., 1973. S t o c h a s t i c Simulat ion for Basins with Short or no Records of Streamflow, Proc. of IAHS Symp on Design of Water Res. Projects with Inadequate Data, Madrid.

M i t c h e l l , W.M., 1962. Ef fec t of Reservoir Storage on Peak Flow, US Geological Survey Water Supply Paper 1580 - C.

Nash J . E . , 1958. The Form of the Ins t an taneous Unit Hydrograph, Bull, IAHS, Vol. 3 , No. 45, pp . 114-121.

Nash, J . E . , 1959 A note on the Muskingum Flood-Routing Method, Jour. Geophys. Res. Vol. 64.

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Nash, J . E . , 1961. A Linear Transformation of a Discharge Record, Proceedings, Ninth Convention, IAHR, Vol. 3 , pp . 13-1 to 2.

O'Donnell , Terence, 1960. Ins t an taneous Unit Hydrograph Der iva t ion by Harmonic Ana lys i s , IASH, Publ. No. 51, p p . 546-557.

O'Kelly, J . J . , 1955. The Employment of Unit Hydrographs to "Determine The Flows of I r i s h A r t e r i a l Drainage Channels, Jour. Inst. Civ. Engrs:, Vol, 4, pp . 365-445.

Pegram, G.G.S, 1977. Phys i ca l J u s t i f i c a t i o n of a Continuous Stream-flow Model, Proc. Third Intl. Hydr. Sump., F t . C o l l i n s , Colorado.

Selval ingam, s . , 1977. ARMA and Linear Tank Models, Proc. Third Intl. Hydr. Sump. , F t . C o l l i n s , Colorado.

Sherman, L. K., 1932. Streamflow from R a i n f a l l by the Unit Hydrograph Method, Engr. News-Record, Vol. 108, pp . 501-505.

Sugawara, M., 1961. On the Analys is of Runoff S t r u c t u r e about Seve­r a l Japanese R ive r s , Jap. Jour. Geophys. , Vol 2, No. 4, pp. 1-76.