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Applications of System Identification and Parameter Estimation in Water Quality Modeling

Beck, M.B.

IIASA Working Paper

WP-79-099

September 1979

Beck, M.B. (1979) Applications of System Identification and Parameter Estimation in Water Quality Modeling. IIASA

Working Paper. WP-79-099 Copyright © 1979 by the author(s). http://pure.iiasa.ac.at/1084/

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Working Paper App l ica t ions of System Identification and Parameter E s t i n a t i o n i n Water Q u a l i t y I lodeling

rI. B . Beck

Septerrber 1979 WP-79-99

International Institute for Applied Systems Analysis A-2361 Laxenburg, Austria

App l i ca t i ons o f System Identification and Parameter 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

Septerrber 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 Appl ied Systems Ana l ys i s , Sch loss Laxenburg, 2361 Laxenburg, 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 heen 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 models f o r r i v e r and l a k e e c o l o a i c a l systems. Much o f t h i s i n t e r e s t h a s been d i r e c t e d towards t h e development of p r o g r e s s i v e l y l a r g e r and more complex s i m u l a t i o n r .ode ls . 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 has been devo ted t o t h e p rob lens o f uncer ta in t l y and e r r o r s i n t h e f i e l d d a t a , o f i nadequa te nunbers 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 between t h e impor tan t s y s t e r v a r i a b l e s , and 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 estimates. IIASF's Resources and Environlr.ent P r e a ' s Task on "proi!els f o r Envi ronmenta l Q u a l i t y Con t ro l and Management" a d d r e s s e s problems such 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 on 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 and p a r m e t e r e s t i m a t i o n i n wa te r u u a l i t y model ing i s p rov ided i n t h i s paper . The paper i s t h e r e f o r e concerned w i t h summariz ing t h e s t a t u s o f c u r r e n t and 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

App l ica t ions of techniques of system i d e n t i f i c a t i o n and parameter es t ima t ion i n water q u a l i t y r .odel ing a r e surveyed. Th is survey of t h e l i t e r a t u r e covers t h r e e a reas : r i v e r water q u a l i t y , l a k e water q u a l i t y , and wastewater t rea tment 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 according t o t h e type of a lqor i thm used f o r c a l i b r a t i o n , t h e type o f model, and t h e f i e l d d a t a used. Two broad 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 of parameter es t ima t ion ; and (b ) i n t e r n a l l y d e s c r i p t i v e ( s ta te -space ) and black box ( inpu t /ou tpu t ) model types. I n o rde 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 es t ima t ion a lgor i thms a r e very b r i e f l y in t roduced. 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 each a r e a of water q u a l i t y modeling, it 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 major problems d iscussed concern 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 of models f o r wa te r a u a l i t y i n r i v e r s , l a k e s , and

wastewater t r e a t m e n t p rocesses is , i n s e v e r a l impor tan t

r e s p e c t s , d i f f e r e n t from t h e problem o f c a l i b r a t i n g , f o r

example, r a i n f a l l - r u n o f f and f l ood - rou t i ng models. Records

of 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 and inade-

q u a t e f o r t h e purposes 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 gh l e v e l s o f e r r o r ; t h e system t o

be desc r i bed 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 pe rm 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 system

behav iour , such a s t h e s torm 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

recorde8. flat;. . Indeed, r e l s - t i o n s h i p s between "causes" and

" e f f e c t s " a r e n o t a lways 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

of 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 app l y i ng

techn iques of s y s t e n i d e n t i f i c a t i o n and parameter e s t i m a t i o n

t o problems of wa te r q u a l i t y model ing is n o t t o be t r e a t e d as

a s t ra i qh t - f o rwa rd e x t e n s i o n o f t h e approaches t y p i c a l l y used

i n t h e a n a l y s i s o f o t h e r forms of hyd ro log i ca l model ing.

Th i s paper su rveys t h e l i t e r a t u r e of wa te r q u a l i t y model

c a l i b r a t i o n . S ince t h e app l i ca t j . ons c i t e d a r e c l a s s i f i e d .

accord ing t o t h e t y p e o f parameter e s t i m a t i o n a l qo r i t hm used,

t h e f o l l ow ing s e c t i o n i n t r o d u c e s a minimum o f exp lana t i on f o r

a number of p o t e n t i a l l y a p p l i c a b l e a l go r i t hms . S e c t i o n 3 i s t h e

p r i n c i p a l co rnonen t o f t h e survey. I t i s no t an e x h a u s t i v e

rev iew; space r e s t r i c t i o n s do n o t a l l ow more t.han j u s t a b r i e f

su rvey of t h e l i t e r a t u r e . S e c t i o n 4 deals w i t h t h e s a l i e n t

p r o b l e m of c u r r e n t a p p l i c a t i o n s of parameter e s t i m a t i o n a lgo-

r i t h s i n wa te r q u a l i t y model ing.

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 though

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 squa res e s t i r a t o r . C e r t a i n l y ,

t h e fundamental r o l e of l e a s t squa res a s t h e p o i n t of d e p a r t u r e

i n deve lop ing more complex a lgo r i t hms i s undisputed (Draper

and Smith, 1966; Eykhof f , 1974; Gelh, 1978; Young, 1.974;

Kashyap and Rao, 1976; Graupe, 1976) .

Let u s d e f i n e , t h e r e f o r e , t h e fo l low ing c r i t e r i o n f unc t i on

f o r model parameter 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 of model parameter e s t i m a t e s and 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 of t h e system

responses and f i e l d o b s e r v a t i o n s o f t hose responses. ?l is a I

m a t r i x of weiqht. ing 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 lgor i thms. 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

squares e s t i m a t e s . I n most c a s e s of 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 squa res e s t i m a t e s w i l l be b iased because, 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 ) sequences assumed t o be p r e s e n t i n

t h e observed f i e l d d a t a do n o t conform to whi te n o i s e sequences.

Thus, it canno t be assumed t h a t t h e l e a s t squa res e s t i m a t e s w i i l

equal t h e supposedly " t r u e m v a l u e s of t h e system paramete rs .

One of t h e most w ide ly used a lgo r i t hms t h a t avo ids t h i s problem

i s t h e method of maximum l i k e l i h o o d (see, f o r example, R s t r o m

and B o h l i n , 1966; Box and J e n k i n s , 1 9 7 0 ) . Faximum l i k e l i h o o d

-1 e s t i m a t i o n is 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 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, where R is 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 measurement errors (Gelb, 1974)

o r t h e computed 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 and c o v a r i a n c e ) are neces -

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 , it

i s assumed 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 is

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 est 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

and t h e e s t i m a t o r is f r e q u e n t l y r e f e r r e d t o as w e i g h t e d least - 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 ,

1961; J o h n s t o n , 1963; Young, i 9 7 6 ) a l s o a v o i d s t h e p rob lem o f

b i a s e d estimates. The method s e e k s t o g e n e r a t e a sequence 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 y

l 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 fo rms 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 are v i r t u a l l y e q u i v a l e n t t o s ta te estimates. There

are, t h e r e f o r e , s t r o n g s im i la r i t i es be tween t h i s e s t i m a t o r and

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 rea ts t h e p rob lem o f p a r a m e t e r e s t i m a t i o n as a prob lem

o f combined 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

method o f q u a ~ i l i n e a r ~ z a t i o n is 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 it t o o sets 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 rob lem

by i n t e r p r e t i n g t h e model p a r a m e t e r s as 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 (Bel lman a n d Ka laba , 1965 ; Lee , 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 therefore U

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 opposed t o t h e p rob lem o f p a r a m e t e r e s t i m a t i o n (which t h e

GYDH a l g o r i t h m t r e a t s by 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 and Young ( 1 9 7 6 ) ,

Beck i 1 9 7 6 ) , J o l a n k a i and ~ z o l l o s i - ~ a g y ( 1 9 7 8 ) , and Ha l fon

e t a l . (1979) a r e s i m i l a r l y o r i e n t e d towards t h e a n a l y s i s o f

i d e n t i f y i n g model 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 and l a k e w a t e r q u a l i t y

mode l ing shows a p redominan 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 mode ls , whereas 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 mode ls 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

towards t h e u s e of h l a c k box time-series mode ls , 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 deve lop -

ment o f model 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 mode l ing . F o r s t r e a m water q u a l i t y mode l i ng 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 been 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 , and 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 mode l i ng , f o r example, Thomann (1967, 1 9 7 3 ) ,

F u l l e r and Tsokos ( 1 9 7 1 ) , Edwards and Thornes ( 1 9 7 3 ) , S c h u r r

and R u c h t i ( 1 9 7 5 ) , and Mehta 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

.. model ing can b e found i n Be r thouex e t a l . (1975, 1 9 7 6 ) .

4 . SALIENT PROBLEMS

I t is a p p a r e n t f rom t h e p r e v i o u s s e c t i o n (and T a b l e 1) t h a t

model c a l i b r a t i o n h a s deve loped d i f f e r e n t l y i n t h e t h r e e

chosen a r e a s o f w a t e r q u a l i t y mode l ing . T h i s is p a r t l y a

consequence 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 mode ls ,

However, s i m i l a r i t i e s o f t h e p rob lems 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 pronounced t h a n t h e i r d i f f e r e n c e s . Thus t h r e e

g e n e r a l problems 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 ; and ( 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

between, f o r example, t h e calibration of 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 mod.els and 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

models i s t h a t d a t a for t h e l a t t e r have u s u a l l y been 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 and b i o l o g i c a l was tewa te r t r e a t m e n t sys tems

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 and r e l a t i v e l y s l o w compo-

n e n t s o f dynamic k ~ e h a v i o u r , b o t h o f which 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 of t h e sys tem. 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 , under t h e i n e v i t a b l e

assumpt ion t h a t l onger - te rm dynamic p r o p e r t i e s are 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 for making

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 deve lop-

n e n t s , one o f a n a n a l y t i c a l n a t u r e and one r e l a t e d t o i n s t r u -

m e n t a t i o n hardware , 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 , Spear and Hornberger

( 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 prob lem,

p ropose t h a t even 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 and 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 mean ing fu 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 , rq . ther 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 . Fecond, improvements i n s y e c i f i c - j . o n elec-

t r o d e s and 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 ne tworks 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 te r t h e q u a n t i t y and k ind

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 .

?Joise l e v e l s i n t h e d a t a . 'This problem i s probab ly most

emphasized 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 waste-

wa te r t r ea tmen t ~ l a n t s . The l a c k of 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 , such a s t h e s torm 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-

sequen t l y , it 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 and t h u s time-series models w i l l t end 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 qrwert ies of t h e o u t r u t

o b s e r v a t i o n s sequence. There is , t h e r e f o r e , ve ry 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 system i d e n t i f i c a t i o n . !.'oreover,

ext reme e v e n t s i n e c o l o g i c a l sys tems, For i n s t a n c e , t h e sudden

phytop lankton bloom, occur because a s p e c i f i c b u t r e l a t i v e l y

com~onp lace combinat ion o f env i ronmenta 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 system i n t o a r e g i o n i n which a n o n l i n e a r mode

o f hehav iour 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?onses 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 of a p r i o r i knowledge. F t y p i c a l f e a t u r e of water

q u a l i t y model ing 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 of t h e

b a s i c cause -e f f ec t r e l a t i o n s h i p s i n t h e s y s t e ~ under i n v e s t i -

g a t i o n . And even when he knows t5ese r e l a t i o n s h i p s i t i s

no t a lways c l e a r what form they shou ld 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 is t h e problem of r e s o l v i n g such i s s u e s by

r e f e r e n c e t o expe r imen ta l f i e l d d a t a (Beck, 1978, 1979a) . More

p r e c i s e l y , node 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 be 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 which t h e i n n u t d i s -

t u rbances 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 neasured

o u t p u t resFonses 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

of t h i s problem n a t u r a l l y p recedes a c c u r a t e e s t i m a t i o n o f t h e

model pa ramete 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 se l f de-

pend 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 . lgor i thm. I f one

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 is

o f ma jor impor tance -- and t h e l i t e r a t u r e does n o t s u g g e s t

a widespread r e c o g n i t i o n t h e r e o f -- t h e n it is 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 concen-

t r a t e on e s t a b l i s h i n g t h a t which is 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 sys tem hehav iou r . I t i s , i n f a c t , p rematu re

t o f o c u s a t t e n t i o n on 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 and 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 components o f

t h e s y s t e m ' s behav iou 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 models i s st i l l a t a p r i m i t i v e

s t a g e o f development . These 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 , was tewa te r t r e a t m e n t p l a n t , and r i v e r

q u a l i t y model ing.

( 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 of

a l a k e e c o l o ~ i c a l svs tem 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 complex i n t e r n a l l y d e s c r i p t i v e models o f s u c h sys tems.

These n o d e l s have 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 cor.narel! 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 and 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 was tewa te r t r e a t m e n t p l a n t b e h a v i o u r

have le6 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 low-order b l a c k box

models f o r t h e s e sys tep -s . Such r o d e l s , however, 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 dominant ( ~ . i c r o b i o l o g i c a l ) mechanisms 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.

Adayemi, S .O., T.?u, S . V . , and Berthouex, P .In.. (1979) Nodeling and control of a phosphorus removal process by multi- variate time series method. water Research vol. 13, 105-112.

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Bdwards, A.Y.C. and Thornes, J.B. (1973) >nnual cycle in river water quality: a time-series approach. Xater Resources Res., vol. 9, 1286-1295.

Erni, P.E. and guchti, J. (1977) Der Sauerstoffhaushalt von Fliessgewaessern: kritische Pruefung eines ~athematischen Sauerstoffmodells Anhand der Identifikation seiner Para- meter in Flussen und kuenstlichen Rinnen. Schweiz Z. Hydrol., vol. 39, no. 2, 261-276.

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Fuller, F.C. and Tsokos, C.P. (1971) Time-series analysis of water pollution data. Biometries, vol. 27, 1017-1034.

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Graupe, D. (1976) Identification of Systems: Pobert E. Krieger Publishinq Co. , Euntington, PTew York, U S A .

Halfon, E., Unbehauen, E., and Schmid, C. (1979) Model order estimation and system identification theory and application to the modeling of 3 2 ~ kinetics within the tro~hogenic zone of a small lake. !?,cological 'Flodeling, vol. 6, 1-22.

Huck, P.M.. a.nd Farquhar, G.T. (1974) Vater quality nodels using the Box-Jenkins method. ?roc. Am. Soc. Civ. En~rs., J. Env. Eng. Div., Vo1.100, no. EF3, 733-752.

Ivakhnenko, A . G . , Krotov, G.I., Cheberkus, V.L. and Vysotskiy, V.M. (1977) Identification of dynamic equations of a complex plant on the basis of experinental data using self-organization cf rr.odels, Part 1 - one dimensional problems. Soviet Automatic Control, vol. 10, no. 2, 24-30.

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Johnston, J. (1363) Econometric methods: ?!&raw-Hill, New York, C'SA.

Jolankai, G . , and Fz6llosi-~?acjy, A. (1973) A simnle eutrophication model for the bay of I<eszthely, Lake Salaton. In Modeling the Water Quality of the Hydrological Cycle (Proceedings of the Baden Symposium, September, 1978), pp. 137-150: IAHS Puhl. no. 125.

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TABLE 1. Summary of F.ecent Applications of Parameter Estimation A lgor i ths 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 8 Koivo (1978)

Lee & Hwang (1971)

Shas t ry e t a l . (1973)

Huck & Farquhar (1974)

Beck (1975)

Seck S Young ! 1976)

L%icehead & Young (1975)

Young & TJhitehead (1977)

Lettenmaier and Burges (1976)

Ern i & Ruchti (1977)

Ivakhnenko e t a l . (1977)

--

--

--

--

Sacranento River (1962)

S t . C l a i r River (1971)

River Cam (1972)

River Cam (1972)

Bedf o r d d u s e River (1973)

River Cam (1972) Bed fo rdduse River (1973)

--

Aare River

River Cam (1972)

S tochas t i c Approximation (Least Squares) ; R*

f Leas t Squares; 0

L inear Kalman f i l t e r ; R

Leas t Squares ( s t a t e es t ima t i on on ly ) ; R

O u a s i l i n e r a l i z a t i o n (Least Squares) ; 0

Weighted Least Squares; Maximum L ike l ihood; 0

!laximum L ike l ihood; 0

Xaximun L ike l ihood; 0

Exrended Kalman F i l t e r ; R

Xu l t i va r i ab l e Instrumen- t a l Variable-Approxi- mate Xaximum L ike l ihood (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- mat ion Method; 0

Group Method of Data Hand1 ing (GMDH) ; 0

I Time & space ; BOD, DO; a n a l y t i c a l s o l u t i o n t o 1st -order p a r t i a l d i f - f e r e n t i a l equat ion.

Space; BOD, DO; s teady-s ta te ana- l y t i c a l s o l u t i o n t o 1st -order p a r t i a l d i f f e r e n t i a l equat ion.

Time & space; BOD, DO; d i f f e r e n c e equa t ions

Time & space; BOD, DO; 1st -order p a r t i a l d i f f e r e n t i a l equat ion.

Space; BOD, DO; ord inary d i f f e ren - t i a l equa t ion .

Space; BOD, DO; o rd ina ry d i f f e r e n - t i a l equat ion.

S ing le po in t s p a t i a l l o ca t i on , time- v a r i a t i o n s ; DO, c h l o r i d e ; b lack box, t ime-ser ies model.

Time; BOD, DO; ord inary d i f fe ren- t i a l equat ion; a l s o b lack box t ime-ser ies model

Time; BOD, DO; o rd inary d i f f e ren - t i a l equa t ion .

Time; BOD, DO; d i f f e rence equa t ions

Time: BOD, DO; d i f f e rence equat ions

Space; BOD, DO; ord inary d i f f e ren - t i a l equa t ions

S ing le po in t s p a t i a l l oca t ion ; t ime-var ia t ions; DO; d i f f e rence equa t ions .

S ing le po in t s p a t i a l l o ca t i on ; time v a r i a t i o n s ; BOD, DO; d i f f e rence equa t ions .

TABLE 1. (Cont!.nued)

R ~ t h o r ( s ) [ Field Data I P.lgor i thm I Type of Model

S t e h f e s t (1978)

Stehf e s t (1978) Rhine River 1 (1971)

Bowles & Grenney (1978a)

Moore & Jones (1978)

Q u a s i l i n e a r i z a t i o n (Leas t Squares) ; 0

Rhine River (1971)

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

Jo rdan R ive r , Utah

~ u a s i l j n e a r i z a t i o n (Least s q u a r e s ) ; 0


Tamura (1978)

River Cam (1972j

L inear Kalman F i l t e r (and o t h e r s ) ; R

Coupled Bayesian-Kalman F i l t e r ; R

Space; e a s i l y degradable o rgan ic m a t t e r , s low ly degradab le o r g a n i c m a t t e r , b a c t e r i a l mass, pro tozoan mass, DO; o r d i n a r y d i f f e r e n t i a l equa t ions .

Space; BOD, DO, NH3-N, NO3+, algal-N, organ ic+; o r d i n a r y - d i f f e r e n t i a l equa t ions .

Time; BOD, DO; o r d i n a r y d i f f e r e n - t i a l equa t ions .

Space; 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 equa t ions .

Time & space ; BOD, DO; d i f f e r e n c e equa t ions .

Gnauck e t a l . (1976)

Th< (1978)

J o l a n k a i and ~ z G l lSsi-b!agy ( 19 7 8)

Lewis and Nir (1978)

River Rhine

Hal fon, e t a l . (1979)

b

LP-KE b7P?TE l? PCALITY E!ODELING

Saidenbach Rese v o i r , GDR (1966 70) ; Kl icava Reser- v o i r , CSSR

' (1963-72)

L inear Kalman F i l t e r ; R

Lake Bala ton, Hungary (1971-77)

Time and space; c o n d u c t i v i t y ; 2nd- o rde 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 approximat ion 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 equa t ions .

D i Cola e t a l . (1976)

Gre i fensee , Swi tzer land (1973)

Small l ake ecosy s tern

Leopold 's Park Pond, B r u s s e l s (1973-75)

Leas t Squares; R

Leas t Squares; 0 ( so lved a s an opt imal c o n t r o l problem)

Maximum Lakel ihood; R

Yeighted Leas t Squares; 0

Leas t Squares ( a l s o f requency domain analy- s i s ) ; 0

Time; DO, ch lo rophy l l -a , p a r t i c u - l a t e o rgan ic 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 phosphorus, ch lo rophy l l -a , exchange- a b l e phosphorus i n sediment; o rd ina ry d i f f e r e n t i a l equa t ions .

Time; s o l u b l e r e a c t i v e phsphorus , p a r t i c u l a t e phosphorus; o r d i n a r y d i f f e r e n t i a l equa t ions .

Time; s o l u b l e phosphorus, p a r t i - c u l a t e phosphorus, a low molecu lar weight form of phosphorus, co l - l o i d a l phosphorus; o r d i n a r y d i f - f e r e n t i a l equa t ions .

-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 ra ten (1979)

FOOTNOTES :

* R denotes a recurs ive es t imat ion a lgor i thm

t O denotes an of f - l ine est imat ion a lgor i thm

A l g o r i t h m

Least Squares; 0

Weighted Least Squares; 0

I

F i e l d D a t a

Lake P lac id , B r i t i s h Colum- b i a , Canada

Lake Ontar io (1972)

Type o f Mode l A

Time; phytoplankton biomass; ord inary d i f f e r e n t i a l equat ion.

Time; 16 s t a t e va r i ab les div ided between epi l imnion and hypo- l imnion l aye rs ; ord inary d i f - f e r e n t i a l equat ions.

Svrcek, 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)

Ua rs i l i - L i be l l i (1.979)

Extended Kalnan F i l t e r ; R

Maximum Likel ihood; 0

.Piaxinum Likelihood; 0

Inst rumenta l Variable; R

?laximum Likelihood; 0

llaximum Likel ihood; 0


Least Squares (with cub ic sp l i nes smoothing) ; 0

L --

Geppa la Works Stockholm

P h i l i p s h i l l \,larks, Scot land

Nowich ?-lorks , England

Madison !.larks, Wiscons i n

Jones Is land !Jerks , Hilwaukee Wisconsin

Sorwich Works, England

P i l o t p l an t , Florence, I t a l y

Time; c e l l and subs t ra te concen- t r a t i o n s (genera l continuous cul- t u r e p rocess) ; ord inary d i f fe ren- t i a l equat ions

Time; DO (ac t i va ted sludge un i t ) ; black box, t ime-ser ies model.

Time; BOD, suspended s o l i d s (primary sedimentat ion tanks) ; black box, t ime-ser ies model.

Tine; gas product ion r a t e (anaerobic d iges t i on u n i t ) ; black box, time- s e r i e s model.

Time ; BOD (ac t i va ted sludge u n i t ) ; black box, t ime-ser ies model.

Time; t o t a l so lub le phosphorus (phosphorus p r e c i p i t a t i o n u n i t ) ; b lack box, t ime-ser ies mode!..

Time: NH?-N, NO -N, Xitrosom,onas, Ni t robacter (ac2ivated s ludge u n i t ) ; ord inary d i f f e r e n t i a l equat ions.

Tine; BOD, b a c t e r i a l concentra- t i o n (ac t i va ted sludge u n i t ) ; ord inary d i f f erent i 'a l equat ions.

1

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