The complete set of deterministic response error statistics for Models I-V are given in Table 10; in addition a relative measure is provided for the error variance as a percentage of the that Model I 1 offers, for this particular data set, a distinct improvement over the a priori model (Model I ) but that thereafter Models I 1 1 and I V provide only marginal increments in accuracy. the data; a reward for this effort in terms of a relatively large drop in the response error variance for BOD is thus, perhaps, only
A d i f f e r e n t f o r m o f model e r r o r s e q u e n c e t o t h a t g i v e n i n
Table 1 1 . Survey of one-step-ahead prediction error box model for BOD (Model Vc) is apparently remarkably better than that of the internally descriptive models. In the case of the black box model for DO the introduction of time-varying coeffi- cients for Model Vb also allows a substantial improvement upon the accuracy of Model Va. Yet the very use of words like "better"
and "substantial improvement" prompts the questions of how, in model assessment, one determines which model is "best" i n some
s e n s e , upon what criteria should this judgment be based, and can
Table 12. Covariance matrix specifications for generation of one-step-ahead prediction errors with a linear Kalman filter (Models I -+ IV only); all matrices are assumed diagonal and, where applicable, time- invariant. State
,
Diagonal Elements A Priori Estimation Error Covariance System Noise,
Observation Noise Covariance CovarianceT a b l e 1 3 . Numbers o f f o r m a l l y e s t i m a t e d p a r a m e t e r
a n d i t a l s o g e n e r a t e s a l o o k - u p t a b l e o f t h e a s s o c i a t e d model
s h o u l d it be c o n c l u d e d t h a t n i t r i f i c a t i o n i s r e a l l y r e s p o n s i b l e
the BOD decay rate constants respectively, should properly be accounted for as functions of flow rate (for a2 only) and tempera- ture. The estimated evaluation of the parameter a{(t) in Table underlying the development of Model IV from Model I is one that embodies a large measure of confidence in the assumptions of the the sources and sink terms of Model I are cast within a different set of assumptions about transport and mixing properties of the
Cam-1972 e x p e r i m e n t . Many q u e s t i o n s r e m a i n u n r e s o l v e d a n d i t i s
t h e i r h e l p . I s h o u l d t h e r e f o r e l i k e t o t h a n k members o f t h e A n g l i a n Water A u t h o r i t y ( t h e n t h e G r e a t Ouse R i v e r A u t h o r i t y ) a n d t h e Water R e s e a r c h C e n t r e , S t e v e n a g e ( t h e n t h e Water P o l l u t i o n R e s e a r c h L a b o r a t o r y ) f o r t h e i r a s s i s t a n c e i n l o a n i n g e q u i p m e n t a n d l a b o r a t o r y f a c i l i t i e s f o r t h e e x p e r i m e n t . From t h e p o i n t o f v i e w o f g u i d a n c e a n d d i s c u s s i o n i n t h e i n t e r p r e t a t i o n o f r e s u l t s a n d i n t h e a p p l i c a t i o n o f i d e n t i f i c a t i o n a n d e s t i m a t i o n a l g o r i t h m s I am e s p e c i a l l y g r a t e f u l t o P r o f e s s o r P e t e r Young a n d D r . P a u l W h i t e h e a d , b o t h now w i t h t h e 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 t t h e A u s t r a l i a n N a t i o n a l U n i v e r s i t y , and t o G u s t a f O l s s o n o f t h e D i v i s i o n o f A u t o m a t i c C o n t r o l , Lund I n s t i t u t e o f T e c h n o l o g y , Sweden. The c a r e f u l r e v i e w o f a n e a r l i e r v e r s i o n o f t h i s p a p e r by P r o f e s s o r Donald Harleman o f t h e M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y w i l l b e n e f i t t h e r e a d e r a n d i s much a p p r e - c i a t e d by t h e a u t h o r . The f i n a n c i a l s u p p o r t o f t h e R o y a l S o c i e t y , London, d u r i n g o n e y e a r i n Sweden a n d f o r a f u r t h e r t h r e e y e a r s i n C a m b r i d g e i s g r a t e f u l l y a c k n o w l e d g e d .
From t i m e t o t i m e I h a v e b e e n a s k e d t o p r o v i d e c o l l e a g u e s w i t h a c o p y o f t h e Cam-1972 f i e l d d a t a . T h i s h a s e n c o u r a g e d m e i n t h e hope t h a t s t i l l o t h e r s t o o w i l l f i n d t h e s e d a t a u s e f u l f o r t h e i r own p u r p o s e s . I f t h i s h o p e i s r e a l i z e d t h e n I s h o u l d b e p l e a s e d t o r e c e i v e d e t a i l s o f a n y a s s o c i a t e d r e s u l t s .
Appendix 1
Table A1 (contd.)
asampled d a t a t i m e , f o r m o d e l i n g p u r p o s e s ( d a y ) . b D a t e , w i t h r e s p e c t t o 1972.
C u p s t r e a m DO c o n c e n t r a t i o n ( q m -3 )
.
d ~ p s t r e a r n BOD c o n c e n t r a t i o n ( q m -3 )
.
e ~ o w n s t r e a m DO c o n c e n t r a t i o n ( q m -3 ) . - 3 f ~ o w n s t r e a r n BOD c o n c e n t r a t i o n (gm )
.
5 3 -1 g ~ t r e a r n d i s c h a r g e ( 1 0 m d a y )
.
T a b l e A1 (contd.) h ~ t r e a m t e m p e r a t u r e ( O C )
.
i S u n l i g h t i n c i d e n t upon l o c a l a r e a ( h r s p e r day) ' ~ a i n f a l l In l o c a l a r e a ( m m ) .
kThe u n d e r l i n e d v a l u e a t t14 d e n o t e s a v a l u e i n t e r p o l a t e d f o r a m i s s i n g downstream W c o n c e n t r a t i o n o b s e r v a t i o n .
Notes: The measured v a l u e s f o r BOD c o n c e n t r a t i o n , columns 4 and 6 , d u r i n g t o + t13 ( i n c l u s i v e ) a r e s u s p e c t e d t o be u n d e r e s t i m a t e s of t h e t r u e s t r e a m BOD c o n d i t i o n s . These measurements a r e d e r i v e d on t h e b a s i s of c a r r y i n g o u t t h e f i v e - d a y BOD b o t t l e t e s t on d i l u t e d samples of r i v e r w a t e r ; f o r t h e i n i t i a l p e r i o d of t h e e x p e r i m e n t it had been a n t i c i p a t e d t h a t s t r e a m BOD l e v e l s might be q u i t e high. I n t h e e v e n t t h i s p r e c a u t i o n a r y measure was u n n e c e s s a r y and s u b s e q u e n t comparisons of BOD'S o b t a i n e d from d i l u t e d and u n d i l u t e d samples i n d i c a t e d t h a t a n a l y s e s of t h e d i l u t e d samples gave c o n s i s t e n t l y low BOD r e a d i n g s . T h i s o b s e r v a t i o n i s p a r t i a l l y confirmed when BOD measurements of t h e sewage works e f f l u e n t a r e s u b s t i t u t e d f o r o t h e r modeling p u r - p o s e s ( s e e Appendix 3 ) .
These d a t a a r e a v a i l a b l e on r e q u e s t and c o n s i s t o f t h e f o l l o w i n g : a ) t h r e e - h o u r l y sampled measurements of u p s t r e a m and downstream DO c o n c e n t r a t i o n , and s t r e a m t e m p e r a t u r e , s t a r t i n g a t 1 2 . 0 0 h r s on
June 6 t h and f i n i s h i n g a t 18.00 h r s on August 2 5 t h ; b ) a c t u a l sampling t i m e s f o r upstream and downstream BOD c o n c e n t r a t i o n s , c o r r e c t e d t o t h e n e a r e s t t h r e e - h o u r l y sampling i n s t a n t ; c ) once d a i l y averaged v a l u e s f o r t h e v o l u m e t r i c f l o w r a t e and BOD c o n c e n t r a t i o n of t h e sewage work- d i s c h a r g e o v e r t h e e x p e r i m e n t a l p e r i o d .
Appendix 2 figure is the one given in Table A2. Note also that for modeling Table A2. Cross-sectional dimensions for channel of the River Cam
5 3
purposes, i.e. in Table 3, the value of 1 . 5 1 x 1 0 m has been substituted for the volume of water held in the reach; on the basis of the figures of Table A2, the volumetric holdup of water
5 3
is calculated as 1 . 4 8 x 1 0 m
.
This discrepancy arises because the more accurate details of Table A2 were not computed until after a more comprehensive experiment on the Cam in 1 9 7 5 . All dimensions in Table A2 are, of course, subject to the assumption of a nominal head of water in the reach.A p p e n d i x 3
e i t h e r t h e l i n e a r g r o w t h r a t e c o n s t a n t o r t h e s a t u r a t i o n c o n s t a n t
modeling, the paper reviews the complete conceptual process of deriving Model IV from the starting point of Model I. The paper does not enter into any detailed discussion of parameter esti- mation methods, nor does it attempt to formalize the notions of hypothesis testing and decisionmaking.
7 . Mode2 S t r u c t u r e I d e n t i f i c a t i o n f r o m E x p e r i m e n t a Z Data (Beck 1978d)
A theme clearly emerging from the above papers is that model structure identification is a problem central to success or failure in the modeling of complex, or poorly defined, systems.
This paper defines the context of model structure identification within the subject of system identification and parameter esti- mation--a subject which also includes the topics of experimental design, model verification, and model validation. The paper places considerable importance on the role of the EKF algorithms
(see also paper 3) in model structure identification; hence the paper presents a fairly detailed statement of how to apply the algorithms and how to interpret the results thereby obtained.
The paper complements the work reported in papers 3 and 6.
NOTES ON DIURNAL VARIATIONS, SEDIMENTATION, AND MODEL APPLICATIONS A consideration of these items has been kept separate because throughout the paper attention has been directed towards field data and models which do not deal with either diurnal variations or the sedimentation of particulate material from the sewage works effluent.
Diurnal Variations
Some brief remarks on the inclusion of these effects in the models are given in paper 2. The observed features in the data can be summarized as follows:
-
Distinct patterns of diurnal variations in the downstream DO concentration become established after day t30 and continue uninterrupted until the end of the experiment, except for the two days succeeding the thunderstorm;-
Prior to day t30 diurnal variations in the (downstream) DO are indistinct with an amplitude of probably little more than t0.25 gm-3;-
After day t30 the amplitude of the diurnal variations - .-
rises on occasion to a maximum value of k2.00 gm-';
-
At all times the diurnal variation in the upstream DO con- centration, where discernible, is significantly less than the variatons observed downstream.The p h a s e o f t h e downstream d i u r n a l v a r i a t i o n s , o r a l t e r n a -
TlME (days) TlME OF DAY
(24 hr clock) 18- 15 - 12 - 09 -
06 - 03 00
Figurc A3.1. Tinling ~ c a k diurnal DO concentration, o n a 24-hour clock basis.
for variorrs periods of the experi~ncnt.
MAXIMUM