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8/8/2019 A Review of Rainfall-runoff Modeling
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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
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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,
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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
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•\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.
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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 .
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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 .
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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
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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
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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
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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 .
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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
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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-
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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
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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 .
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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 .
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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 .
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