Working Paper
A p p l i c a t i o n s o f System Identification and P a r a m e t e r E s t i n a t i o n i n Water Q u a l i t y I l o d e l i n g
rI. B . Beck
Septerrber 1979 WP-79-99
International Institute for Applied Systems Analysis
A-2361 Laxenburg, Austria
A p p l i c a t i o n s o f System Identification a n d P a r a m e t e r E s t i n a t i o n i n Water Q u a l i t y Modeling
P I . B . Beck
S e p t e r r b e r 1979 WP-79-99
M.B. BECK i s a r e s e a r c h s c i e n t i s t a t t h e I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s , S c h l o s s L a x e n b u r g , 2361 L a x e n b u r g , A u s t r i a .
I n r e c e n t y e a r s t h e r e h a s h e e n a c o n s i d e r a b l e i n t e r e s t i n t h e d e v e l o ~ m e n t o f m o d e l s f o r r i v e r a n d l a k e e c o l o a i c a l s y s t e m s . Much o f t h i s i n t e r e s t h a s b e e n d i r e c t e d t o w a r d s t h e d e v e l o p m e n t o f p r o g r e s s i v e l y l a r g e r a n d more c o m p l e x s i m u l a t i o n r . o d e l s . I n c o n t r a s t , r e l a t i v e l y l i t t l e a t t e n t i o n h a s b e e n d e v o t e d t o t h e p r o b l e n s o f u n c e r t a i n t l y a n d e r r o r s i n t h e f i e l d d a t a , o f i n a d e q u a t e n u n b e r s o f f i e l d d a t a , o f u n c e r t a i n t v i n t h e
r e l a t i o n s h i p s b e t w e e n t h e i m p o r t a n t s y s t e r v a r i a b l e s , a n d o f u n c e r t a i n t y i n t h e model n a r a n e t e r e s t i m a t e s . IIASF's R e s o u r c e s a n d Environlr.ent P r e a ' s Task on "proi!els f o r E n v i r o n m e n t a l
Q u a l i t y C o n t r o l a n d Management" a d d r e s s e s p r o b l e m s s u c h a s t h e s e .
A b r i e f s m - a r y o f t h e l i t e r a t u r e o n a p p l i c a t i o n s o f
s y s t e ~ . i d e n t i f i c a t i o n a n d p a r m e t e r e s t i m a t i o n i n w a t e r u u a l i t y m o d e l i n g i s p r o v i d e d i n t h i s p a p e r . The p a p e r i s t h e r e f o r e c o n c e r n e d w i t h s u m m a r i z i n g t h e s t a t u s o f c u r r e n t a n d r e c e n t s t u d i e s i n water q u a l i t y model c a l i b r a t i o n .
iii
A p p l i c a t i o n s of t e c h n i q u e s o f s y s t e m i d e n t i f i c a t i o n and p a r a m e t e r e s t i m a t i o n i n w a t e r q u a l i t y r . o d e l i n g a r e s u r v e y e d . T h i s s u r v e y of t h e l i t e r a t u r e c o v e r s t h r e e a r e a s : r i v e r w a t e r q u a l i t y , l a k e w a t e r q u a l i t y , and w a s t e w a t e r t r e a t m e n t p l a n t modelinq. The a p p l i c a t i o n s c i t e d a r e c l a s s i f i e d a c c o r d i n g t o t h e t y p e o f a l q o r i t h m u s e d f o r c a l i b r a t i o n , t h e t y p e o f model, and t h e f i e l d d a t a u s e d . Two b r o a d d i s t i n c t i o n s a r e made between: ( a ) o f f - l i n e and r e c u r s i v e methods o f p a r a m e t e r e s t i m a t i o n ; and ( b ) i n t e r n a l l y d e s c r i p t i v e ( s t a t e - s p a c e ) and b l a c k box ( i n p u t / o u t p u t ) model t y p e s . I n o r d e r t o a s s i s t
t h e c l a s s i f i c a t i o n , a number o f e s t i m a t i o n a l g o r i t h m s a r e v e r y b r i e f l y i n t r o d u c e d . Although t h e r e a r e c l e a r l y d i f f e r e n t
l i n e s of development i n e a c h a r e a of w a t e r q u a l i t y m o d e l i n g , i t i s p o s s i b l e t o i d e n t i f y problems common t o a l l t h r e e a r e a s . The m a j o r problems d i s c u s s e d c o n c e r n t h e a v a i l a b i l i t y of f i e l d d a t a , l e v e l s o f n o i s e i n t h e d a t a , and model s t r u c t u r e i d e n t i - f i c a t i o n .
1. Ib'TRODUCTIOF
C a l i b r a t i o n o f m o d e l s f o r w a t e r a u a l i t y i n r i v e r s , l a k e s , and w a s t e w a t e r t r e a t m e n t p r o c e s s e s i s , i n s e v e r a l i m p o r t a n t
r e s p e c t s , d i f f e r e n t f r o m t h e problem o f c a l i b r a t i n g , f o r e x a m p l e , r a i n f a l l - r u n o f f and f l o o d - r o u t i n g m o d e l s . R e c o r d s o f water q u a l i t v d a t a a r e o f t e n r e s t r i c t i v e l y s h o r t a n d i n a d e - q u a t e f o r t h e p u r p o s e s o f t h e - s e r i e s a n a l y s i s ; t h e d a t a a r e s u b j e c t t o p a r t i c u l a r l y h i g h l e v e l s o f e r r o r ; t h e s y s t e m t o b e d e s c r i b e d i s r a r e l y o f t h e m u l t i p l e - i n p u t / s i n g l e o u t p u t form ( a form which p e r m i t s s u b s t a n t i a l s i m p l i f i c a t i o n o f t h e a n l y s i s ) ; and s i g n i f i c a n t i n p u t p e r t u r b a t i o n o f t h e s y s t e m b e h a v i o u r , s u c h a s t h e s t o r m e v e n t , i s o f t e n a b s e n t f r o r . t h e r e c o r d e 8 . flat;.
.
I n d e e d , r e l s - t i o n s h i p s b e t w e e n " c a u s e s " and" e f f e c t s " a r e n o t a l w a y s s e l f - e v i d e n t p r i o r t o t h e a n a l y s i s o f t h e f i e l d d a t a . One may a r g u e , t h e r e f o r e , t h a t a p p l y i n g t e c h n i q u e s o f s y s t e n i d e n t i f i c a t i o n and p a r a m e t e r e s t i m a t i o n t o p r o b l e m s o f w a t e r q u a l i t y m o d e l i n g i s n o t t o b e t r e a t e d as a s t r a i q h t - f o r w a r d e x t e n s i o n o f t h e a p p r o a c h e s t y p i c a l l y u s e d i n t h e a n a l y s i s o f o t h e r f o r m s o f h y d r o l o g i c a l m o d e l i n g .
T h i s p a p e r s u r v e y s t h e l i t e r a t u r e o f w a t e r q u a l i t y model c a l i b r a t i o n . S i n c e t h e a p p l i c a t j . o n s c i t e d a r e c l a s s i f i e d . a c c o r d i n g t o t h e t y p e o f p a r a m e t e r e s t i m a t i o n a l q o r i t h m u s e d , t h e f o l l o w i n g s e c t i o n i n t r o d u c e s a minimum o f e x p l a n a t i o n f o r a number o f p o t e n t i a l l y a p p l i c a b l e a l g o r i t h m s . S e c t i o n 3 i s t h e p r i n c i p a l c o r n o n e n t o f t h e s u r v e y . I t i s n o t a n e x h a u s t i v e
r e v i e w ; s p a c e r e s t r i c t i o n s do n o t a l l o w more t.han j u s t a b r i e f s u r v e y o f t h e l i t e r a t u r e . S e c t i o n 4 d e a l s w i t h t h e s a l i e n t
p r o b l e m o f c u r r e n t a p p l i c a t i o n s o f p a r a m e t e r e s t i m a t i o n a l g o - r i t h s i n w a t e r q u a l i t y m o d e l i n g .
Fany a l g o r i t h m a r e a v a i l a b l e f o r ~ a r a m e t e r e s t i m a t i o n , a l t h o u g h t h e m a j o r i t y o f t h e s e a 1 q o r i t b . s a r e n o t s u b s t a n t i a l l y d i f f e r e n t from t h e b a s i c n o t i o n o f a l e a s t s q u a r e s e s t i r a t o r . C e r t a i n l y , t h e f u n d a m e n t a l r o l e o f l e a s t s q u a r e s a s t h e p o i n t o f d e p a r t u r e i n d e v e l o p i n g more complex a l g o r i t h m s i s u n d i s p u t e d ( D r a p e r a n d S m i t h , 1966; E y k h o f f , 1974; G e l h , 1 9 7 8 ; Young, 1.974;
Kashyap and Rao, 1 9 7 6 ; G r a u p e , 1 9 7 6 ) .
L e t u s d e f i n e , t h e r e f o r e , t h e f o l l o w i n g c r i t e r i o n f u n c t i o n f o r model p a r a m e t e r e s t i m a t i o n ( o r c a l i b r a t i o n ) a s ,
i n which & i s a v e c t o r o f model p a r a m e t e r e s t i m a t e s a n d E i s a
d
-
v e c t o r o f e r r o r s between model-based e s t i m a t e s o f t h e s y s t e m r e s p o n s e s and f i e l d o b s e r v a t i o n s o f t h o s e r e s p o n s e s . ?l i s a
I
m a t r i x o f w e i q h t . i n g c o e f f i c i e n t s , v z r i o u s c h o i c e s f o r which
d e f i n e d i f f e r e n t e s t i m a t i o n a l g o r i t h m s . When :C = I t h e i d e n t i t y
u .w'
m a t r i x , n i n i ~ i z a t i o n o f (1) w i t h r e s p e c t t o 6 y i e l d s t h e l e a s t
.I
s q u a r e s e s t i m a t e s . I n m o s t c a s e s o f p r a c t i c a l i n t e r e s t , t h e l e a s t s q u a r e s e s t i m a t e s w i l l b e b i a s e d b e c a u s e , i n g e n e r a l , t h e n o i s e ( o r random e r r o r ) s e q u e n c e s assumed t o b e p r e s e n t i n t h e o b s e r v e d f i e l d d a t a d o n o t conform t o w h i t e n o i s e s e q u e n c e s . Thus, i t c a n n o t be assumed t h a t t h e l e a s t s q u a r e s e s t i m a t e s w i i l e q u a l t h e s u p p o s e d l y " t r u e m v a l u e s o f t h e s y s t e m p a r a m e t e r s . One o f t h e most w i d e l y u s e d a l g o r i t h m s t h a t a v o i d s t h i s p r o b l e m
i s t h e method o f maximum l i k e l i h o o d (see, f o r example, R s t r o m
a n d B o h l i n , 1 9 6 6 ; Box a n d J e n k i n s , 1 9 7 0 ) . Faximum l i k e l i h o o d e s t i m a t i o n i s e q u i v a l e n t t o t h e s u b s t i t u t i o n W = R -1 i n t h e
Z H
c r i t e r i o n f u n c t i o n (11, w h e r e R i s e i t h e r t h e c o v a r i a n c e
N
m a x t r i x of t h e o u t p u t r e s p o n s e m e a s u r e m e n t errors ( G e l b , 1 9 7 4 ) o r t h e c o m p u t e d c o v a r i a n c e m a t r i x o f t h e e r r o r s E ( ~ g l l s t r 6 m
rY
e t a l . , 1 9 7 6 ) . P s s u m p t i o n s a b o u t t h e s t a t i s t i c a l p r o p e r t i e s o f t h e n o i s e s e q u e n c e s ( t h e i r mean a n d c o v a r i a n c e ) a r e n e c e s - s a r y i n o r d e r t o make t h i s s u b s t i t u t i o n , I f , i n a d d i t i o n , i t i s a s s u m e d t h a t e a c h e l e m e n t o f t h e n o i s e s e q u e n c e v e c t o r i s i n d e p e n d e n t o f a l l o t h e r e l e m e n t s , t h e n a somewhat s i m p l e r e s t i - m a t o r r e s u l t s . Under t h i s a s s u m p t i o n , W i s a d i a g o n a l m a t r i x
CI
a n d t h e e s t i m a t o r i s f r e q u e n t l y r e f e r r e d t o a s w e i g h t e d l e a s t
-
s a u a r e s . . .
An i n s t r u m e n t a l v a r i a b l e e s t i m a t o r ( K e n d a l l a n d S t u a r t , 1 9 6 1 ; J o h n s t o n , 1 9 6 3 ; Young, i 9 7 6 ) a l s o a v o i d s t h e p r o b l e m o f b i a s e d e s t i m a t e s . T h e m e t h o d s e e k s t o g e n e r a t e a s e q u e n c e o f v a r i a b l e s w i t h s p e c i f i c s t a t i s t i c a l p r o p e r t i e s
--
t h e i n s t r u - m e n t a l v a r i a b l e s--
t h a t may be s u b s t i t u t e d i n t o a n e s s e n t i a l l yl e a s t - s q u a r e s - l i k e a l g o r i t h m . F o r c e r t a i n f o r m s o f t h e i n s t r u - m e n t a l v a r i a b l e e s t i m a t o r ( e . g . , Young, 1 9 7 4 ) , t h e i n s t r u m e n t a l v a r i a b l e s a r e v i r t u a l l y e q u i v a l e n t t o s t a t e e s t i m a t e s . T h e r e a r e , t h e r e f o r e , s t r o n g s i m i l a r i t i e s b e t w e e n t h i s e s t i m a t o r a n d t h e e x t e n d e d Kalman f i l t e r ( J a z w i n s k i , 1 9 7 0 ) , a n a l g o r i t h m t h a t t r e a t s t h e p r o b l e m o f p a r a m e t e r e s t i m a t i o n a s a p r o b l e m o f c o m b i n e d s t a t e - p a r a m e t e r e s t i m a t i o n . I n t h a t s e n s e t h e m e t h o d o f q u a ~ i l i n e a r ~ z a t i o n i s s i m i ' l a r t o t h e e x t e n d e d Kalman
f i l t e r s i n c e i t t o o s e t s u p t h e p a r a m e t e r e s t i m a t i o n p r o b l e m by i n t e r p r e t i n g t h e m o d e l p a r a m e t e r s a s a d d i t i o n a l s y s t e m s t a t e v a r i a b l e s ( B e l l m a n a n d K a l a b a , 1 9 6 5 ; L e e , 1 9 6 8 ) .
Many of the above and closely related algorithms can be implemented as either off-line or recursive schemes of para- . .
meter estimation, The basic difference between the two schemes is that an off-line scheme assumes that a single, fixed set of estimates B may be substituted for computation of the re-
N
sponse errors ( E ) for all N field observations sampled from
C1
tlme tl
-
tN. With a recursive scheme it is possible to com- pute estimates 8(tk) for each kth instant of time, and thereforeU
it is possible to estimate time-varying parameter values.
3. SURVEY OF APPLICATIONS
Table 1 gives a broad survey of the literature on applications of parameter estimation to water quality modeling in streams, lakes, and wastewater treatment plants, Classification accord- ing to the type of model used is chosen partly because it is instructive to judge the size of the model being calibrated, and partly because the choice of model (internally descriptive, or black box; defines, to some extent, the nature of an appro- priate estimation algorithm. Unless otherwise indicated, as
either a "re?ressionn or "black box" model, all the rodels referenced j.n Table 1 are internally descriptive r?odels. Ry
"internally descriptive" it is ~ e a n t that the model is derived fror existin? theory and that it attempts to describe those internal chemical, bioloc~ical, and physical mechanisr.~ which are thouqht to govern svstern behaviour.
A few remarks are necessary in order to qualify the con- tents of Table 1. For example, the paper by Ivakhnenko et ai.
(1977) is primarily concerned with the problems of model
discrimination and model structure identification (see below)
a s o p p o s e d t o t h e p r o b l e m o f p a r a m e t e r e s t i m a t i o n ( w h i c h t h e GYDH a l g o r i t h m t r e a t s b y l e a s t s q u a r e s e s t i m a t i o n ) . O t h e r r e f e r e n c e s , S h a s t r y e t a l , ( 1 9 7 3 ) , Beck a n d Young ( 1 9 7 6 ) , Beck i 1 9 7 6 ) , J o l a n k a i a n d ~ z o l l o s i - ~ a g y ( 1 9 7 8 ) , a n d H a l f o n e t a l . ( 1 9 7 9 ) a r e s i m i l a r l y o r i e n t e d t o w a r d s t h e a n a l y s i s o f i d e n t i f y i n g m o d e l s t r u c t u r e ,
The l i t e r a t u r e q u o t e d f o r stream a n d l a k e w a t e r q u a l i t y m o d e l i n g s h o w s a p r e d o m i n a n t b i a s t o w a r d s t h e u s e o f i n t e r n a l l y d e s c r i p t i v e m o d e l s , w h e r e a s t h e p a p e r s a d d r e s s i n g w a s t e w a t e r t r e a t m e n t p l a n t m o d e l s t e n d t o e x h i b i t t h e o p p o s i t e b i a s t o w a r d s t h e u s e of h l a c k b o x time-series m o d e l s , T h i s
r e f l e c t s , i n t h e l a t t e r c a s e , a somewhat " r e t a r ? . e d n d e v e l o p - m e n t o f m o d e l c a l i b r a t i o n e x e r c i s e s i n w a s t e w a t e r t r e a t m e n t p l a n t m o d e l i n g . F o r s t r e a m water q u a l i t y m o d e l i n g T a b l e 1 i n f a c t r e f l e c t s a r a t h e r s e l e c t i v e s u r v e y o f t h e l i t e r a t u r e .
~ h e r e ~ h a v e b e e n s e v e r a l a p p l i c a t i o n s o f f r e q u e n c y r e s p o n s e , c o r r e l a t i o n a n a l y s i s , a n d time-series a n a l y s i s t e c h n i q u e s
i n s t r e a m q u a l i t y m o d e l i n g , f o r e x a m p l e , Thomann ( 1 9 6 7 , 1 9 7 3 ) , F u l l e r a n d T s o k o s ( 1 9 7 1 ) , E d w a r d s a n d T h o r n e s ( 1 9 7 3 ) , S c h u r r a n d R u c h t i ( 1 9 7 5 ) , a n d M e h t a e t a l . ( 1 9 7 5 ) . F u r t h e r a p p l i c a -
-
t i o n s o f time-series a n a l y s i s i n w a s t e w z t e r t r e a t m e n t p l a n t
..
m o d e l i n g c a n b e f o u n d i n B e r t h o u e x e t a l . ( 1 9 7 5 , 1 9 7 6 ) .4 . SALIENT PROBLEMS
I t i s a p p a r e n t f r o m t h e p r e v i o u s s e c t i o n ( a n d T a b l e 1) t h a t m o d e l c a l i b r a t i o n h a s d e v e l o p e d d i f f e r e n t l y i n t h e t h r e e c h o s e n a r e a s o f w a t e r q u a l i t y m o d e l i n g . T h i s i s p a r t l y a c o n s e q u e n c e o f d i f f e r e n t o b j e c t i v e s f o r t h e u s e o f m o d e l s , However, s i m i l a r i t i e s o f t h e p r o b l e m s e x p e r i e n c e d i n e a c h a r e a a r e more p r o n o u n c e d t h a n t h e i r d i f f e r e n c e s . T h u s t h r e e
g e n e r a l p r o b l e m s a r e d i s c u s s e d : ( a ) a v a i l a b i l i t y o f f i e l d
d a t a ; ( h ) n o i s e l e v e l s i n t h e d a t a ; a n d ( c ) d e g r e e o f a ~ r i o r i knowledge.
~ v a i l a b i l i t y o f f i e l d d a t a . An e s s e n t i a l d i f f e r e n c e b e t w e e n , f o r e x a m p l e , t h e calibration of r a i n f a l l - r u n o f f a n d
f l o o d - r o u t i n g mod.els a n d t h e c a l i k r a t i o n o f w a t e r q u a l i t y m o d e l s i s t h a t d a t a f o r t h e l a t t e r h a v e u s u a l l y b e e n s a ~ p l e d n o t o n l y a t i n a d e q u a t e l y l o w f r e q u e n c i e s b u t a l s o f o r i n s u f - f i c i e n t c o n t i n u o u s p e r i o d s o f t i m e . T t i s a c h a r a c t e r i s t i c f e a t u r e o f l a k e a n d b i o l o g i c a l w a s t e w a t e r t r e a t m e n t s y s t e m s t h a t t h e y e x h i b i t r e l a t i v e l y f a s t a n d r e l a t i v e l y s l o w compo- n e n t s o f d y n a m i c k ~ e h a v i o u r , b o t h o f w h i c h a r e i m p o r t a n t o r o b t a i n i n g a node1 o f t h e s y s t e m . P- l a k e e c o l o s i c a l model c a l i b r a t e d a g a i n s t s h o r t - t e r m r e c o r d s , u n d e r t h e i n e v i t a b l e a s s u m p t i o n t h a t l o n g e r - t e r m d y n a m i c p r o p e r t i e s a r e e s s e n t i a l l y a t s t e a d y - s t a t e , would c l e a r l y b e i n a p p r o p r i a t e f o r m a k i n g f o r e c a s t s o f l o n g - t e r n b e h a v i o u r p a t t e r n s . Two r e c e n t d e v e l o p - n e n t s , o n e o f a n a n a l y t i c a l n a t u r e a n d o n e r e l a t e d t o i n s t r u - m e n t a t i o n h a r d w a r e , may s i g n i f i c a n t l y a l t e r t h e s i t u a t i o n
r e g a r d i n g a v a i l a b i l i t y o f d a t a . F i r s t , S p e a r a n d H o r n b e r g e r ( 1 9 7 8 ) , i n t h e j r a n a l y s i s o f a l a k e e u t r o p h i c a t i o n p r o b l e m ,
p r o p o s e t h a t e v e n p a t c h y , i n a d e o u a t e f i e l d d a t a a n d q u a l i t a t i v e ob- s e r v a t i o n s p e r n i t a m e a n i n g f u l c a l i b r a t i o n e x e r c i s e ; l o q i c a l
c o n s t r a i n t s o n a c c e p t a b l e model ~ e r f o r m a n c e , r q . t h e r t h a n a
s q u a r e d e r r o r f u n c t i o n s u c h a s e v u a t i o n ( 1 1 , ~ r o v i d e t h e c r i t e r i o n f o r c a l i b r a t i o n . F e c o n d , i m p r o v e m e n t s i n s y e c i f i c - j . o n elec-
t r o d e s a n d t h e i n s t a l l a t i o n o f t e l e m e t r y n e t w o r k s f o r w a t e r q u a l i t y n o n i t o r i n g w i l l r a d i c a l l y a l t e r t h e q u a n t i t y a n d k i n d o f f i e l d d a t a a v a i l a b l e f o r a n a l y s i s .
? J o i s e l e v e l s i n t h e d a t a . 'This p r o b l e m i s p r o b a b l y m o s t e m p h a s i z e d i n d a t a c o l l e c t e d from r o u t i n e o p e r a t i o n s a t w a s t e - w a t e r t r e a t m e n t ~ l a n t s . The l a c k o f w e l l i d e n t i f i e d " d e t e r - m i n i s t i c " i n p u t d i s t u r b a n c e s , s u c h a s t h e s t o r m e v e n t , l e a d s t o f i e l d d a t a w i t h a p p a r e n t l y low s i g n a l / n o i s e r a t i o s . Con- s e q u e n t l y , i t i s d i f f i c u l t t o e s t i m a t e a c c u r a t e i n ~ u t / o u t p u t r e l a t i o n s h i ~ s a n d t h u s time-series m o d e l s w i l l t e n d p r e f e r e n -
t i a l l v t o i c ' e n t i f y a u t o r e y r e s s i v e q r w e r t i e s of t h e o u t r u t o b s e r v a t i o n s s e q u e n c e . T h e r e i s , t h e r e f o r e , v e r y l i t t l e
n a t u r a l e x ~ e r i m e n t s l b a s i s For s y s t e m i d e n t i f i c a t i o n . !.'oreover, e x t r e m e e v e n t s i n e c o l o g i c a l s y s t e m s , F o r i n s t a n c e , t h e s u d d e n p h y t o p l a n k t o n bloom, o c c u r b e c a u s e a s p e c i f i c b u t r e l a t i v e l y c o m ~ o n p l a c e c o m b i n a t i o n o f e n v i r o n m e n t a l c o n d i t i o n s f o r c e t h e
s t a t e o f t h e s y s t e m i n t o a r e g i o n i n w h i c h a n o n l i n e a r mode o f h e h a v i o u r i s e x c i t e d . Such s i g n i f i c a n t v a r i a t i o n o f t h e re- s ? o n s e s i s r a r e l y r e l z t e d t o e x t r e r e i n n u t d i s t u r b a n c e s .
Degree o f a p r i o r i knowledge. F t y p i c a l f e a t u r e of w a t e r q u a l i t y m o d e l i n g i s t h a t t h e a n a l y s t i s o f t e n u n c e r t a i n o f t h e b a s i c c a u s e - e f f e c t r e l a t i o n s h i p s i n t h e s y s t e ~ u n d e r i n v e s t i - g a t i o n . And e v e n when h e knows t5ese r e l a t i o n s h i p s i t i s n o t a l w a y s c l e a r w h a t form t h e y s h o u l d t a k e . Model s t r u c t u r e i d e n t i f i c a t i o n i s t h e p r o b l e m o f r e s o l v i n g s u c h i s s u e s by
r e f e r e n c e t o e x p e r i m e n t a l f i e l d d a t a ( B e c k , 1 9 7 8 , 1 9 7 9 a ) . More p r e c i s e l y , n o d e l s t r u c t u r e i d e n t i f i c a t i o n may b e d e f i n e e a s t h e problem o f i d e n t i f y i n g t h e way i n w h i c h t h e i n n u t d i s - t u r b a n c e s a r e r e l a t e d t o t h e s t a t e v a r i a b l e s , how t h e s t a t e s a r e r e l s t e d among t h e ~ s e l v e s , and how i n t u r n t h e n e a s u r e d o u t p u t r e s F o n s e s a r e r e l a t e d t o t h e s t a t e v a r i a b l e s . S o l u t i o n o f t h i s p r o b l e m n a t u r a l l y p r e c e d e s a c c u r a t e e s t i m a t i o n o f t h e
model p a r a m e t e r v a l u e s , a l t h o u g h t h e s o l u t i o n n a y i t s e l f de- p e n d upon t h e a p p l i c a t i o n o f a n e s t i m a t i o n z . l g o r i t h m . I f o n e
a c c e p t s t h a t t h e i s s u e of n o d e l s t r u c t u r e i d e n t i f i c a t i o n i s o f m a j o r i m p o r t a n c e
--
a n d t h e l i t e r a t u r e d o e s n o t s u g g e s t a w i d e s p r e a d r e c o g n i t i o n t h e r e o f--
t h e n it i s r e a s o n a b l e t o a r g u e t h a t c a l i b r a t i o n o f w a t e r q u a l i t y model-s s h o u l d c o n c e n - t r a t e on e s t a b l i s h i n g t h a t w h i c h i s e s s e n t i a l l y " d e t e r m i n i s t i c "a b o u t t h e o b s e r v e d s y s t e m h e h a v i o u r . I t i s , i n f a c t , p r e m a t u r e t o f o c u s a t t e n t i o n o n d e t a i l e d a s s u ~ p t i o n s a b o u t t h e d i s t r i h u - t i o n s a n d c o r r e l a t i o n p r o p e r t j e s o f t h e random c o m p o n e n t s o f t h e s y s t e m ' s b e h a v i o u r .
5 . COFCLUSIONS
The c a l i b r a t i o n o f w a t e r q u a l i t y m o d e l s i s s t i l l a t a p r i m i t i v e s t a g e o f d e v e l o p m e n t . T h e s e c o n c l u s i o n s s u m a r i z e t h e s t a t u s o f a n n l y i n g p a r a ~ . e t e r e s t i m a t i o n t e c h n i a u e s t o t h e t h r e e a r e a s o f l a k e w a t e r q u a l i t y , w a s t e w a t e r t r e a t m e n t p l a n t , a n d r i v e r q u a l i t y m o d e l i n g .
( a ) A d e s i r e t o c h a r a c t e r i z e a l l t h e d e t a i l e d f e a t u r e s o f a l a k e e c o l o ~ i c a l s v s t e m h a s I.ed t o t h e c?evelopment o f p a r t i - c u l a r l y c o m p l e x i n t e r n a l l y d e s c r i p t i v e m o d e l s o f s u c h s y s t e m s . T h e s e n o d e l s h a v e l i t t l e l i k e l i h o o d o f b e i n g r i q o r o u s l y c a l i b r a t e d a g a i n s t f i e l d d a t a ; i n d e e d , t h e i r l e v e l o f t h e o r e t i c a l complex- i t y seems d i s p r o p o r t i o n a t e l y h i g h when c o r . n a r e l ! w i t h t h e
s e v e r e l y r e s t r i c t e d r a n g e o f a v a i l a b l e f i e l d d a t a .
( b ) I n c o n t r a s t , t h e o b j e c t i v e s o f q u a n t i f y i n g a n d con- t r o l l i n q t h e v a r i a b i l i t y o f w a s t e w a t e r t r e a t m e n t p l a n t b e h a v i o u r h a v e l e 6 t y p i c a l l y t o t h e c a l i b r a t i o n o f l o w - o r d e r b l a c k box m o d e l s f o r t h e s e s y s t e p - s . S u c h r o d e l s , h o w e v e r , y i e l d l i t t l e
i n s i g h t i n t o t h e d o m i n a n t ( ~ . i c r o b i o l o g i c a l ) m e c h a n i s m s t h a t
1 I
govern the dynamics of waste removal processes.
(c) Tor stream quality modelina there has been a more balanced progress in both black box and internally descriptive approaches to model construction and its associated calibration prcblems. With present techniques and data it would be pos- sible to calibrate a dynaxvic lumped-parameter n-ode1 that
accounts for the basic properties of day-to-day variations in DO-ROD interaction, phytoplankton growth, and nitrification in rivers.
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Moore, R.Z. and Jones, D . A . (1978) Coupled Rayesian-Kalman filter estimation of paraneters and states of dynamic water quality models. In Applications of Kalman Filter
to Bydrology
,
Hydraulics and Yater Resources (edited by 1 C u : Cniversity of Pittsburgh, StochasticHydraulics Program, Pittsburgh, USA., pp. 559-635.
O l s s o n , G . a n d H a n s s o n , 0 . ( 1 9 7 6 ) ! ! o d e l i n g a n d i d e n t i f i c a t i o n o f a n a c t i v a t e d s l u d g e p r o c e s s . I n I d e n t i f i c a t i o n a n d S y s t e m Parameter E s t i m a t i o n ( P r e p r i n t s , I V t h IFAC Symposium.,
T b i l i s i , USPP., S e p t e m b e r , 1 9 7 6 ) : I n s t i t u t e o f C o n t r o l S c i e n c e s , !%scow, USSR, p p . 1 3 4 - 1 4 6 .
R i n a l d i , S . , S o n c i n i - S e s s a , R . , S t e h f e s t , E . a n d T a m u r a , P . ( 1 9 7 9 ) Y o d e l i n g a n d c o n t r o l o f r i v e r q u a l i t y : ?lcGraw- H i l l , Yew Yorl:, USA.
S c h u r r , J.M. a n d R u c h t i , J . ( 1 9 7 5 ) K i n e t i c s of o x y g e n e x c h a n g e , p h o t o s y n t h e s i s a n d r e s p i r a t i o n i n r i v e r s d e t e r m i n e d f r o m t i m e - d e l a y e d c o r r e l - a t i o n s b e t w e e n s u n l i g h t a n d d i s s o l v e d o x y g e n . S c h w e i z . 2. H y d r o l o g i e , v o l t 3 7 , n o . 1, 144-174.
S h a s t r y , J . S . , F a n L.T. a n d E r i c k s o n , I;.!?. ( 1 9 7 3 ) ! J o n - l i n e a r p a r a m e t e r e s t i m a t i o n i n w a t e r a u a l i t y m o d e l i n g . P r o c . m.. S o c . C i v . E n g r s . , J. Env. Eng. D i v . , v o l . 9 9 , n o . EE3, 315-331.
S p e a r , P.C. a n d H o r n b e r g e r , G.M. ( 1 9 7 8 ) Eutrophication i n P e e l I n l e t . R e p o r t N o . AS-P.19, C e n t r e f o r R e s o u r c e a n d E n v i r o n m e n t a l S t u d i e s , A u s t r a l i a n N a t i o n a l V n i v e r s i t y , C a n b e r r a , A u s t r a l i a .
S t e h f e s t , H . ( 1 9 7 8 ) On t h e m o n e t a r y v a l u e o f a n e c o l o g i c a l r i v e r q u a l i t y m o d e l . R e s e a r c h r e p o r t n o . RP.-78-1, I n t e r n a t i o n a l
I n s t i t u t e f o r A p p l i e d S y s t e r s A n a l y s i s , L a x e n b u r g , A u s t r i a . S v r c e k , Ts?.Y., E l ! - i o t t , R . F . , a n d Z a j i c , J . E . ( 1 9 7 4 ) The e x t e n d e d
K a l r a n f i l t e r a p p l i e d t o a c o n t i n u o u s c u l t u r e m o d e l . Aio- t e c h n o l o g y a n d g i o e n g i n e e r i n g , v o l . X V I , 827-846.
Tamura, H. ( 1 9 7 8 ) On s o m e i d e n t i f i c a t i o n t e c h n i q u e s f o r m o d e l - i n g r i v e r q u a l i t y d y n a m i c s w i t h d i s t r i b u t e d l a g s . I n Handbook o f L a r g e - S c a l e S y s t e r n s F n g i n e e r i n g A p n l i c a t i o n s
( e d i t e d b y F . G . S i n g h a n d A. T i t l i ) : N o r t h - H o l l a n d , A m s t e r d a m , H o l l a n d ( i n p r e s s ) .
~ h g , G . ( 1 9 7 8 ) P a r a m e t e r i d e n t i f i c a t i o n i n a n o d e l f o r t h e
c o n d u c t i v i t y o f a r i v e r b a s e d o n n o i s v ~ e a s u r e ~ e n t s a t two l o c a t i o n s . I n Y o d e l i n g , I d e ~ t i f i c a t i o n a n d C o n t r o l i n E n v i r o n m e n t a l S y s t e m s ( e d i t e d by C. C. r r a n s t e e n k i s t e ) : N o r t h - B o l l a n d , Amsterdam, H o l l a n d , pp. 823-830.
T h o n a n n , R . V . ( 1 9 6 7 ) Time-series a n a l y s i s o f w a t e r q u a l i t y d a t a . P r o c . Am. Soc. C i v . E n g r s . , J . ani it. Enq. D i v . , v o l . 9 3 , n o . SA1, 1-23.
Thomann, R . V. ( 1 9 7 3 ) E f f e c t o f l o n g i t u d i n a l d i s p e r s i o n o n d y n a m i c w a t e r q u a l i t y r e s p o n s e of streams a n d r e s e r v o i r s . V?ater R e s o u r c e s R e s e a r c h , v o l . 9 , n o . 2 , 355-366.
Y h i t e h e a d , P . G . a n d Young, P.C. ( 1 9 7 5 ) A d y n a m i c - s t o c h a s t i c m o d e l f o r w a t e r q u a l i t y i n p a r t o f t h e R e d f o r d O u s e r i v e r s y s t e m . I n C o m p u t e r S i m u l a t i o n o f Water R e s o u r c e s S y s t e m s ( e d i t e d b y G . C . V a n s t e e n k i s t e ) : P J o r t h - X o l l a n d , P m s t e r d a m , E o l l a n d ,
pp. 417-438.
Young, P.C. (1974) A recursive spproach to time-series analysis.
Bulletin Institute of )?athematics and its Applications, vol. 10, 209-224.
Younq, P.c. (1976) Some observations on instrumental variable methods of time-series analysis. Int. J. Control, vol.
23, 593-612.
Young, P.C. and Tqhitehead, P.G. (1977) A recursive approach to time-series analysis for multivariable systems. Int. J.
Control, vol. 25, 457-482.
TABLE 1. Summary of F.ecent Applications of Parameter Estimation A l g o r i t h s in Feter Quality Yodeling
Author (s 1 Field Data Algorithm
1
Type of Model Koivo & P h i l l i p s(1971)
Koivo & P h i l l i p s (1972)
Koivo & P h i l l i p s (1976) Koivo 8 Koivo
(1978)
Lee & Hwang (1971)
S h a s t r y e t a l . (1973) Huck & F a r q u h a r
(1974) Beck (1975)
Seck S Young
! 1976)
L%icehead & Young (1975)
Young & TJhitehead (1977)
L e t t e n m a i e r and Burges (1976) E r n i & R u c h t i
(1977)
Ivakhnenko e t a l . (1977)
--
--
-- --
S a c r a n e n t o R i v e r (1962) S t . C l a i r R i v e r
(1971) R i v e r Cam (1972)
R i v e r Cam (1972)
Bedf o r d d u s e R i v e r (1973)
R i v e r Cam (1972) B e d f o r d d u s e R i v e r (1973)
--
Aare R i v e r
R i v e r Cam (1972)
S t o c h a s t i c Approximation ( L e a s t S q u a r e s ) ; R*
L e a s t S q u a r e s ; 0 f
L i n e a r Kalman f i l t e r ; R L e a s t S q u a r e s ( s t a t e e s t i m a t i o n o n l y ) ; R O u a s i l i n e r a l i z a t i o n
( L e a s t S q u a r e s ) ; 0 Weighted L e a s t S q u a r e s ; Maximum L i k e l i h o o d ; 0
!laximum L i k e l i h o o d ; 0
Xaximun L i k e l i h o o d ; 0
Exrended Kalman F i l t e r ; R
X u l t i v a r i a b l e Instrumen- t a l Variable-Approxi- m a t e Xaximum L i k e l i h o o d
(MIVAML); R MTVAML; R
Extended Kalman F i l t e r ; R
D i f f e r e n t i a l Approxi- m a t i o n Method; 0
Group Method of Data Hand1 i n g (GMDH) ; 0
I
Time & s p a c e ; BOD, DO; a n a l y t i c a l s o l u t i o n t o 1 s t - o r d e r p a r t i a l d i f - f e r e n t i a l e q u a t i o n .
Space; BOD, DO; s t e a d y - s t a t e ana- l y t i c a l s o l u t i o n t o 1 s t - o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n . Time & s p a c e ; BOD, DO; d i f f e r e n c e e q u a t i o n s
Time & s p a c e ; BOD, DO; 1 s t - o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n . Space; BOD, DO; o r d i n a r y d i f f e r e n - t i a l e q u a t i o n .
Space; BOD, DO; o r d i n a r y d i f f e r e n - t i a l e q u a t i o n .
S i n g l e p o i n t s p a t i a l l o c a t i o n , time- v a r i a t i o n s ; DO, c h l o r i d e ; b l a c k box, t i m e - s e r i e s model.
Time; BOD, DO; o r d i n a r y d i f f e r e n - t i a l e q u a t i o n ; a l s o b l a c k box t i m e - s e r i e s model
Time; BOD, DO; o r d i n a r y d i f f e r e n - t i a l e q u a t i o n .
Time; BOD, DO; d i f f e r e n c e e q u a t i o n s
Time: BOD, DO; d i f f e r e n c e e q u a t i o n s
Space; BOD, DO; o r d i n a r y d i f f e r e n - t i a l e q u a t i o n s
S i n g l e p o i n t s p a t i a l l o c a t i o n ; t i m e - v a r i a t i o n s ; DO; d i f f e r e n c e e q u a t i o n s .
S i n g l e p o i n t s p a t i a l l o c a t i o n ; time v a r i a t i o n s ; BOD, DO; d i f f e r e n c e e q u a t i o n s .
TABLE 1. (Cont!.nued)
R ~ t h o r ( s )
[
Field D a t aI
P . l g o r i t h mI
Type o f ModelS t e h f e s t (1978)
S t e h f e s t (1978)
1
Rhine R i v e r (1971)Bowles & Grenney (1978a)
Moore & J o n e s (1978)
Q u a s i l i n e a r i z a t i o n ( L e a s t S q u a r e s ) ; 0 Rhine R i v e r
(1971)
S p a c e ; BOD, DO; o r d i n a r y d i f f e r e n - t i a l e q u a t i o n s
J o r d a n R i v e r , Utah
~ u a s i l j n e a r i z a t i o n ( L e a s t s q u a r e s ) ; 0
Extended Kalman F i l t e r ; R
Tamura (1978)
R i v e r Cam ( 1 9 7 2 j
L i n e a r Kalman F i l t e r ( a n d o t h e r s ) ; R
Coupled Bayesian-Kalman F i l t e r ; R
S p a c e ; e a s i l y d e g r a d a b l e o r g a n i c m a t t e r , s l o w l y d e g r a d a b l e o r g a n i c m a t t e r , b a c t e r i a l m a s s , p r o t o z o a n mass, DO; o r d i n a r y d i f f e r e n t i a l e q u a t i o n s .
S p a c e ; BOD, DO, NH3-N, NO3+, a l g a l - N , o r g a n i c + ; o r d i n a r y - d i f f e r e n t i a l e q u a t i o n s .
Time; BOD, DO; o r d i n a r y d i f f e r e n - t i a l e q u a t i o n s .
S p a c e ; SOD, DO; a n a l y t i c a l s o l u t i o n t o l s t - o r d e r o r d i n a r y d i f f e r e n t i a l e q u a t i o n s .
Time & s p a c e ; BOD, DO; d i f f e r e n c e e q u a t i o n s .
Gnauck e t a l . (1976)
Th< (1978)
J o l a n k a i and
~ z G l lSsi-b!agy ( 1 9 7 8 )
Lewis a n d Nir (1978)
R i v e r Rhine
H a l f o n , e t a l . (1979)
b
LP-KE b7P?TE l? PCALITY E!ODELING
S a i d e n b a c h Rese v o i r , GDR (1966 70) ;
K l i c a v a Reser- v o i r , CSSR
'
(1963-72)L i n e a r Kalman F i l t e r ; R
Lake B a l a t o n , Hungary
(1971-77)
Time and s p a c e ; c o n d u c t i v i t y ; 2nd- o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n
( f i n i t e d i f f e r e n c e a p p r o x i m a t i o n s o l u t i o n ) .
Time: a u t o t r o p h s , h e r b i v o r e s , c a r n i v o r e s ; o r d i n a r y d i f f e r e n t i a l e q u a t i o n s .
D i Cola e t a l . (1976)
G r e i f e n s e e , S w i t z e r l a n d (1973) S m a l l l a k e e c o s y s tern L e o p o l d ' s P a r k Pond, B r u s s e l s
(1973-75)
L e a s t S q u a r e s ; R L e a s t S q u a r e s ; 0
( s o l v e d a s a n o p t i m a l c o n t r o l problem)
Maximum L a k e l i h o o d ; R
Y e i g h t e d L e a s t S q u a r e s ; 0
L e a s t S q u a r e s ( a l s o f r e q u e n c y domain a n a l y - s i s ) ; 0
Time; DO, c h l o r o p h y l l - a , p a r t i c u - l a t e o r g a n i c m a t t e r ; r e g r e s s i o n r e l a t i o n s h i p .
Time; s o l u b l e r e a c t i v e phos- p h o r u s , c h l o r o p h y l l - a , exchange- a b l e p h o s p h o r u s i n s e d i m e n t ; o r d i n a r y d i f f e r e n t i a l e q u a t i o n s . Time; s o l u b l e r e a c t i v e p h s p h o r u s ,
p a r t i c u l a t e p h o s p h o r u s ; o r d i n a r y d i f f e r e n t i a l e q u a t i o n s .
Time; s o l u b l e p h o s p h o r u s , p a r t i - c u l a t e p h o s p h o r u s , a low m o l e c u l a r w e i g h t form o f p h o s p h o r u s , c o l -
l o i d a l p h o s p h o r u s ; o r d i n a r y d i f - f e r e n t i a l e q u a t i o n s .
-17- TABLE 1. ( C o n t i n u e d )
YASTE7JE TEF. TFEPTPf NT- PLAPT M~)DFT,IPC-
1 1
A u t h o r ( s 1
C
Benson (1979)
D i Toro and van S t r a t e n
(1979)
FOOTNOTES :
*
R denotes a r e c u r s i v e e s t i m a t i o n a l g o r i t h m t O d e n o t e s an o f f - l i n e e s t i m a t i o n a l g o r i t h mA l g o r i t h m L e a s t Squares; 0
Weighted L e a s t S q u a r e s ; 0
I
F i e l d D a t a Lake P l a c i d , B r i t i s h Colum- b i a , Canada Lake O n t a r i o
(1972)
Type o f M o d e l
A
Time; p h y t o p l a n k t o n biomass;
o r d i n a r y d i f f e r e n t i a l e q u a t i o n . Time; 16 s t a t e v a r i a b l e s d i v i d e d between e p i l i m n i o n and hypo- l i m n i o n l a y e r s ; o r d i n a r y d i f - f e r e n t i a l e q u a t i o n s .
S v r c e k , e t a l . (1974)
Olssorl and Hansson (1976) Crowther, e t a l .
(1976) Seck (1976)
Berthouex, e t a l . (1978)
Adayemi, e t a l . (1979)
Beck (1979b)
U a r s i l i - L i b e l l i (1.979)
Extended Kalnan F i l t e r ; R
Maximum L i k e l i h o o d ; 0
.Piaxinum L i k e l i h o o d ; 0
I n s t r u m e n t a l V a r i a b l e ; R
?laximum L i k e l i h o o d ; 0 llaximum L i k e l i h o o d ; 0
Extended Kalman F i l t e r ; R
L e a s t Squares ( w i t h
c u b i c s p l i n e s smoothing) ; 0
L
--
G e p p a l a Works Stockholm P h i l i p s h i l l
\,larks, Scot l a n d
N o w i c h ?-lorks ,
England
Madison !.larks, Wiscons i n Jones I s l a n d
!Jerks , Hilwaukee Wisconsin
Sorwich Works, England
P i l o t p l a n t , F l o r e n c e ,
I t a l y
Time; c e l l and s u b s t r a t e concen- t r a t i o n s ( g e n e r a l c o n t i n u o u s c u l - t u r e p r o c e s s ) ; o r d i n a r y d i f f e r e n - t i a l e q u a t i o n s
Time; DO ( a c t i v a t e d s l u d g e u n i t ) ; b l a c k box, t i m e - s e r i e s model.
Time; BOD, suspended s o l i d s (primary s e d i m e n t a t i o n t a n k s ) ; b l a c k box, t i m e - s e r i e s model.
T i n e ; gas p r o d u c t i o n r a t e ( a n a e r o b i c d i g e s t i o n u n i t ) ; b l a c k box, time- s e r i e s model.
Time ; BOD ( a c t i v a t e d s l u d g e u n i t ) ; b l a c k box, t i m e - s e r i e s model.
Time; t o t a l s o l u b l e phosphorus (phosphorus p r e c i p i t a t i o n u n i t ) ; b l a c k box, t i m e - s e r i e s mode!..
Time: NH?-N, NO -N, Xitrosom,onas, N i t r o b a c t e r ( a c 2 i v a t e d s l u d g e u n i t ) ; o r d i n a r y d i f f e r e n t i a l e q u a t i o n s .
T i n e ; BOD, b a c t e r i a l c o n c e n t r a - t i o n ( a c t i v a t e d s l u d g e u n i t ) ; o r d i n a r y d i f f e r e n t i ' a l e q u a t i o n s .
1
