NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR
SENSITIVITY TO UNCERTAINTY IN A PHYTOPLANKTON-OXYGEN MODEL FOR LOWLAND STREAMS
Gerrit van Straten Bart de Boer
April 1979 WP-79-28
Working P a p e r s are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
A-2361 Laxenburg, Austria
GERRIT VAN STRATEN i s w i t h Twente U n i v e r s i t y of Technology, Enschede,
The N e t h e r l a n d s . A t p r e s e n t , h e 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 Applied Systems A n a l y s i s , S c h l o s s Laxenburg, 2361 Laxenburg, A u s t r i a .
BART DE BOER i s w i t h t h e P r o v i n c i a l Waterboard, Arnhem, The N e t h e r l a n d s .
PREFACE
During t h e p a s t d e c a d e , t h e r e h a s been 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 p - ment o f models f o r r i v e r and l a k e e c o l o g i c a l s y s t e m s . Much of t h i s i n t e r e s t h a s been d i r e c t e d t o w a r d s t h e c o n s t r u c t i o n of p r o g r e s s i v e l y l a r g e r and
more complex d e t e r m i n i s t i c s i m u l a t i o n models. However, w h i l e o u r e x p e r i e n c e i s growing, we b e g i n t o r e a l i z e t h a t o u r c o m p u t a t i o n a l c a p a b i l i t i e s
exceed by f a r t h e l e v e l of o u r knowledge o f , t h e complex p r o c e s s e s i n t h e r e a l world. F u r t h e r m o r e , i n e c o l o g i c a l modeling we seldoxn, i f e v e r , en- c o u n t e r d a t a of s u f f i c i e n t amount a n d a c c u r a c y t o a l l o w f o r a r i g i d c a l i - b r a t i o n o f o u r models. IIASA's R e s o u r c e s a n d Environment Area t a s k on Models f o r Environmental Q u a l i t y C o n t r o l and Management r e c o g n i z e s t h e problems o f c o p i n g w i t h model, d a t a , and p a r a m e t e r u n c e r t a i n t i e s a s a
c h a l l e n g i n g i s s u e f o r f u t u r e r e s e a r c h . T h i s p a p e r c a n be s e e n a s a p r e l i m i n a r y and modest c o n t r i b u t i o n i n t h i s d i r e c t i o n .
The s t u d y was done a t IIASA i n a l i m i t e d amount of t i m e and was f i n a l i z e d d u r i n g a s h o r t s t a y o f t h e second a u t h o r a t t h e I n s t i t u t e . The r e s u l t s were a l s o p r e s e n t e d a t t h e j o i n t colloquium o f t h e Commission f o r t h e S t u d y o f Water Management i n t h e P r o v i n c e o f G e i d e r l a n d , t h e N e t h e r l a n d s , and t h e S p e c i a l R e s e a r c h Area 79 o f t h e F e d e r a l Republic of Germany (Hannover Univer- s i t y of Technology) h e l d a t Wageningen, The N e t h e r l a n d s , from 14
-
1 6F e b r u a r y 1979.
ABSTRACT
The a p p l i c a b i l i t y of water q u a l i t y models depends upon t h e q u a l i t y of t h e parameter e s t i m a t e s . A phytoplankton-oxygen model developed f o r c a n a l i z e d
lowland streams i s t e s t e d a g a i n s t data from a l i m i t e d plug following measure- ment program. The accuracy of t h e parameter e s t i m a t e s i s l i m i t e d by t h e
inaccuracy of t h e BOD measurement i n t h e presence of a l g a e . Other s o u r c e s of parameter u n c e r t a i n t i e s a r e : ( i ) s i t e dependency of parameters lumping complex subsystems, such a s t h e BOD decay r a t e c o e f f i c i e n t , having a higher value d i r e c t l y a f t e r a waste discharge, and ( i i ) time dependency of
lumped parameters, such a s t h e a l g a l death r a t e c o e f f i c i e n t . A s e n s i t i v i t y a n a l y s i s , based on t h e s o l u t i o n of t h e s e n s i t i v i t y equations of t h e model,
i s then performed t o provide some i n s i g h t i n t o t h e e f f e c t s of parameter u n c e r t a i n t i e s on model r e s u l t s . I t appears t h a t t h e growth and d e a t h r a t e s of algae dominate t h e phytoplankton, BOD and oxygen behaviour, and t h a t a s e p a r a t e e s t i m a t e i n t h e absence of a c c u r a t e BOD measurements i s d i f f i c u l t t o o b t a i n without a d d i t i o n a l information.
INTRODUCTION
The development o f d e c i s i o n making t o o l s i n t h e f i e l d of w a t e r q u a n t i t y a s w e l l a s q u a l i t y i s t h e p r i n c i p a l aim o f t h e i n t e g r a t e d r e g i o n a l w a t e r management s t u d y , c o n d u c t e d i n r e c e n t y e a r s by t h e Committee f o r t h e S t u d y o f Water Management i n t h e P r o v i n c e o f G e l d e r l a n d , t h e N e t h e r l a n d s (CWG).
The i n c l u s i o n o f w a t e r q u a l i t y problems i n t h e s t u d y was i n p a r t due t o t h e r e a s o n i n g t h a t a p o s s i b l e l i n k e x i s t s between s u r f a c e w a t e r q u a l i t y and i t s s u i t a b i l i t y f o r d r i n k i n g w a t e r p u r p o s e s . But even i f s u r f a c e w a t e r i s n o t t o b e u s e d , ground w a t e r b e i n g i n s u f f i c i e n t s u p p l y , t h e r e i s s t i l l a s t r o n g i n c e n t i v e t o s t u d y w a t e r q u a l i t y i n r e l a t i o n t o q u a n t i t y problems, s i n c e q u a l i t y i s a f f e c t e d by w a t e r management i n s e v e r a l ways
(van S t r a t e n , 1979)
.
The development of t h e w a t e r q u a l i t y m o d e l l i n g e l e m e n t i n t h e i n t e g r a t e d s t u d y i s b a s e d on s e v e r a l y e a r s of f i e l d r e s e a r c h i n t h e G e l d e r l a n d a r e a (mainly conducted a s s t u d e n t p r o j e c t s ) . T h i s work h a s f i n a l l y l e d t o t h e computer package GELQAM ( G e l d e r l a n d Water Q u a l i t y A n a l y s i s Model), a major p r o d u c t of t h e w a t e r q u a l i t y work. The package w i l l b e a v a i l a b l e f o r
i n t e r e s t e d u s e r s . GELQMI f o c u s e s on d i s s o l v e d oxygen a s r e l e v a n t
c h a r a c t e r i s t i c o f w a t e r q u a l i t y . A d e t a i l e d d e s c r i p t i o n o f GELQAM and i t s u n i q u e n u m e r i c a l f e a t u r e s i s g i v e n e l s e w h e r e ( d e Boer, 1 9 7 8 ) .
T h i s p a p e r d i s c u s s e s some o f t h e problems and d i f f i c u l t i e s m e t , and p a r t l y overcome, d u r i n g t h e s t u d y . I t i s o u r aim t o h i g h l i g h t u n c e r t a i n t i e s which are s t i l l a s s o c i a t e d w i t h r i v e r w a t e r q u a l i t y m o d e l l i n g nowadays. We d e v e l o p t h e i s s u e i n two ways: f i r s t w e d e a l w i t h u n c e r t a i n t i e s i n t h e e v a l u a t i o n o f model p a r a m e t e r s , and s e c o n d l y w e d i s c u s s t h e consequences o f s i m p l i f i c a t i o n s i n t h e p r o c e s s d e s c r i p t i o n . The e f f e c t o f inadequacy o f t h e d a t a i s a l s o i l l u s t r a t e d . We t h e n f o c u s on s e n s i t i v i t y a n a l y s i s a s a h e l p f u l t o o l i n t h e a p p r e c i a t i o n o f model r e s u l t s under t h e
u n c e r t a i n t i e s g i v e n .
SYSTEM CHARACTERISTICS
The r i v e r s and b r o o k s i n t h e G e l d e r l a n d a r e a a r e lowland streams, which are t y p i c a l o f r e l a t i v e l y f l a t a r e a s . Most o f them have been c a n a l i z e d f o r r e a s o n s o f w a t e r l e v e l c o n t r o l , mainly f o r a g r i c u l t u r a l p u r p o s e s . Only
some o f t h e u p p e r b r a n c h e s have r e t a i n e d t h e i r o r i g i n a l , more n a t u r a l shape.
From t h e h y d r o b i o l o g i c a l p o i n t o f view c a n a l i z a t i o n b r i n g s c o n s i d e r a b l e
changes. Water p l a n t s and s e s s i l e a l g a e c h a r a c t e r i z e t h e s h a l l o w , r e l a t i v e l y f a s t f l o w i n g n a t u r a l b r o o k s . I n t h e c a n a l i z e d s e c t i o n s less f a v o u r a b l e
c o n d i t i o n s p r e v a i l b e c a u s e of t h e h i g h e r l i g h t a t t e n u a t i o n i n t h e d e e p e r w a t e r column. I n s t e a d , e s s e n t i a l l y h i g h e r r e s i d e n c e t i m e s a l l o w f o r a n o t o r i o u s development o f p l a n k t o n i c a l g a e , e s p e c i a l l y d u r t n g d r y summer p e r i o d s . A l s o , a r e m a r k a b l e change o f macrophauna h a s been o b s e r v e d
(Tolkamp, 1975)
.
For s e v e r a l r e a s o n s t h e q u a l i t y work h a s been r e s t r i c t e d mainly t o t h e c a n a l i z e d s t r e a m s e c t i o n s o f b r o o k s and r i v e r s o f m o d e r a t e s i z e . One i m p o r t a n t j u s t i f i c a t i o n i s t h a t t h o s e s e c t i o n s a r e p r o b a b l y o f more
s i g n i f i c a n c e f o r p o t e n t i a l w a t e r u s a g e f u n c t i o n s t h a n t h e s m a l l e r , n a t u r a l brooks.
I n many r e s p e c t s c a n a l i z e d r i v e r s e c t i o n s behave s i m i l a r t o l a k e s . I t s h o u l d b e p o i n t e d o u t , however, t h a t t h e r i v e r system i s more s e n s i t i v e t o hydrodynamical v a r i a t i o n s . During s t o r m w a t e r p e r i o d s t h e c o n t e n t s of t h e r i v e r may b e r e f r e s h e d w i t h i n a few d a y s o r even h o u r s , s o t h a t t h e p l a n k t o n i c a l g a e a r e washed o u t . During s u b s e q u e n t d r i e r p e r i o d s a new d e v e l o p m e n t w i l l s t a r t , b u t t h e p o p u l a t i o n may b e d i f f e r e n t due t o a d i f f e r e n t s e e d f l u s h e d i n from t h e u p s t r e a m w a t e r c o u r s e s .
SHORT MODEL DESCRIPTION
I n g e n e r a l terms t h e r e s u l t s of o u r f i e l d s t u d i e s d i d n o t deny t h e p a t t e r n u s u a l l y r e p o r t e d f o r t h e oxygen b e h a v i o u r i n s t r e a m s . Thus, p r o c e s s e s i n f l u e n c i n g t h e d i s s o l v e d oxygen c o n t e n t a r e : decay o f b o t h c a r b o n a c e o u s and n i t r o g e n e o u s o x i d i z a b l e m a t t e r , consumption by t h e sediment l a y e r ,
consumption by a l g a l r e s p i r a t i o n , r e a e r a t i o n a n d , d u r i n g d a y l i g h t , p r o d u c t i o n by p h o t o s y n t h e s i s . C o n s e q u e n t l y , t h e s t a t e v a r i a b l e s i n t h e model a r e
d i s s o l v e d oxygen, C-BOD, N-BOD and a l g a e ( e x p r e s s e d a s c h l o r o p h y l l - a ) . I n a d d i t i o n , s o l u b l e r e a c t i v e phosphorus h a s been i n c l u d e d i n t h e s t a t e v e c t o r b e c a u s e of i t s dominant r o l e i n t h e c o n t r o l of a l g a l blooms. The pathways
f o l l o w e d i n t h e development o f t h i s s t r u c t u r e a r e o u t l i n e d i n van S t r a t e n (1977) and w i l l n o t b e r e p e a t e d h e r e .
The b a s i c e q u a t i o n s a r e g i v e n i n T a b l e 1. The model combines e l e m e n t s o f u s u a l d i s s o l v e d oxygen r i v e r models ( e . g . , O'Connor and D i T o r o , 19701,
w i t h e l e m e n t s frcm well-known a l g a l dynamics models f o r l a k e s ( e . g . , D i T o r o , O'Connor a n d Thomann, 1 9 7 4 ) .
The model a c c o r d i n g t o T a b l e 1 c o n s t i t u t e s a s e t o f s e c o n d 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 s . The l o n g i t u d i n a l d i s p e r s i o n term i s n o t shown b u t h a s been i n c l u d e d , b e c a u s e d i s p e r s i o n e f f e c t s c a n b e s i g n i f i c a n t e s p e c i a l l y d u r i n g low f l o w c o n d i t i o n s . A more d e t a i l e d a n a l y s i s o f t h e r o l e o f d i s p e r - s i o n c a n b e f o u n d i n van S t r a t e n ( 1 9 7 9 ) .
The n u m e r i c a l s o l u t i o n o f t h e s e t o f e q u a t i o n s r e q u i r e s s p e c i a l a t t e n t i o n . D i s c r e t i z a t i o n i n b o t h t i m e a n d s p a c e i s n e c e s s a r y . U s u a l d i f f e r e n c e schemes w i t h f i x e d t i m e a n d s p a c e g r i d a p p e a r less p r o f i t a b l e f o r r i v e r s y s t e m s . The r e a s o n i s t h a t t h e r e l a t i v e l y low d i s p e r s i o n ( a s opposed t o e s t u a r i e s ) r e q u i r e s a n u n e c o n o m i c a l number o f g r i d p o i n t s f o r a c c u r a t e r e s u l t s . T h e r e f o r e , f o r GELQAM, a moving c e l l method h a s b e e n d e v e l o p e d
( d e Boer, 1977, 1 9 7 9 ) . The b a s i c f e a t u r e o f t h i s i s a c o o r d i n a t i o n t r a n s - f o r m a t i o n i n s u c h a way t h a t t h e s y s t e m i s s o l v e d a l o n g t h e s t r e a m - f l o w t r a j e c t o r i e s . I n o t h e r words, t h e model f o l l o w s a number o f c o n s e c u t i v e
' p l u g s ' o f w a t e r o n t h e i r t r a v e l downstream.
PARAMETER ESTIMATION
B e f o r e t h e model c a n b e u s e d it i s n e c e s s a r y t o h a v e estimates f o r t h e p a r a m e t e r s . F o r most o f them a f i r s t o r d e r e s t i m a t e c a n b e o b t a i n e d from t h e l i t e r a t u r e . However, p a r a m e t e r s f o r which t h e model i s v e r y s e n s i t i v e h a v e t o b e e v a l u a t e d i n t h e f i e l d . Sometimes it i s p o s s i b l e t o i s o l a t e p a r t o f t h e p r o c e s s e s b y d e s i g n i n g s p e c i a l e x p e r i m e n t s . An example o f t h i s
i s t h e well-known d a r k a n d l i g h t b o t t l e e x p e r i m e n t . H e r e r e a e r a t i o n , v e r t i c a l m i x i n g a n d , i n t h e d a r k b o t t l e s , p h o t o s y n t h e s i s are e x c l u d e d . The v e r t i c a l p r o f i l e o f t h e n e t oxygen p r o d u c t i o n i n t h e b o t t l e s e n a b l e s t h e i d e n t i f i c a t i o n o f t h e a l g a l g r o w t h s y s t e m a n d t h e e s t i m a t i o n o f i t s p a r a m e t e r s ( i . e . , I o r I a n d t h e p r o d u c t Ycakpa. Note t h e u n c e r t a i n t y
s k
i n t h e g r o w t h r a t e i t s e l f as a c o n s e q u e n c e o f t h e wide r a n g e o f oxygen y i e l d Yca: from 50 t o 250 mg oxygen p e r mg c h l o r o p h y l l - a , c f . van S t r a t e n , 1978)
.
-
algae C-BOD
N-93D dissolved
OSYgen
a l g a l a l g a l algal C-DOD N-EOD
respiration death dea?f d~%ay reaeration Sedirrrent
interaction
phosphorus
Oxygen l i m i t f u n c t i o n
Ck + C
L i g h t f u n c t i o n
e x t i n c t i o n a n d s e l f s h a d i n g : k = ko + at
I ( t ) = I o ( t ) e x p ( - keH) A
d a i l y l i g h t p a t t e r n :
s i n ( 2 n ( t d l - 1 2 ) / 4 8 ) + s i n ( 2 n ( t - 6 1 / 2 4 ) I ( t ) = 0 . 5 I
tot 48 c o s ( 2 n ( l 2 - t ) / 4 8 ) + tdl s i n ( 2 n ( t . -12)/48)
2n d l a1
where tdl = d a y l e n g t h ( h )
and Itot = daysum o f t o t a l r a d i a t i o n ( ~ / c m ~ ) Phosphorus l i m i t f u n c t i o n Pm =
-
Pk P + PR e a e r a t i o n k r = c r Dm4 (Q/A)' H-3/2 ( 1 . 0 2 4 ) ~ - ~ '
where Q = f l o w r a t e ( m 3 / h ) 2 A = c r o s s s e c t i o n a l a r e a ( m ) H = d e p t h (m)
T e m p e r a t u r e f u n c t i o n s
S = S ( 1 . 0 6 5 ) ~ - ~ ' 20
C = 13.97 e x p ( - 0.021 T) where T = t e m p e r a t u r e (C)
TABLE 1 . hlodc 1 d e s c r i p t i o n
;?'st of si/mbols (:I:' parameter values given i n Table 2
In addition, information about the processes and the parameters can be obtained by following the course of the state variables in a plug of water moving downstream, a method most profitable if dispersion is relatively low.
If the system is simple, such as in the case of the classical BOD-DO model without algae, the evaluation of some of the parameters is straightforward.
For instance, the BOD-balance reduces to dL -
=-kbL
d'r
where
'ris the travel time, and k follows simply from b
LT
=L o e x p (-kb'r)
However, if algae are present these simple relationships no longer hold firstly because the algae contribute to the measured BOD and secondly because detritus produced by the death of algae constitutes an additional term in the mass balance. Therefore, essentially, the ultimate parameter estimates are obtained by tuning the model to the measurements.
River Berkel Example
A
data set is available for the Berkel River in the eastern Gelderland area where a plug following program was conducted from 21st-25th June, 1976
(Hiemstra, 1978). Starting fromknownvalue ranges for the parameters, obtained from earlier measurements in other Gelderland regions, and with additional information from a dark and light bottle experiment, a parameter set was found that could explain the experimental results. Table
2lists the full parameter set, with the appropriate changes to accommodate slight differences in model description between Table 1 and Hiemstra's original mode 1,
The River Berkel is a canalized river. The study reach covered two sections between weirs and was about 5.8
kmlong.
Adischarge of mechanically treated sewage water (approx. 15 kg c-bod/h) is located
1
kmupstream from the study reach. Table
3summarizes the main character-
istics. For reasons pointed out below, travel time and distance are
Nominal
Value U n i t
No P a r a m e t e r
growth r a t e a l g a e
Michaelis-Menton c o e f f i c i e n t f o r P
Pk e x t i n c t i o n c o e f f i c i e n t k
0
s e l f - s h a d i n g c o e f f i c i e n t
a
o p t i m a l l i g h t i n t e n s i t y I
S
a l g a l r e s p i r a t i o n r a t e c o e f f i c i e n t
a l g a l d e a t h r a t e c o e f f i c i e n t
k d a C-BOD decay r a t e c o e f f i c i e n t
kb 20 Michaelis-Menton c o e f f i c i e n t
gen Ck
d e t r i t u s p r o d u c t i o n e f f i c i e n c y
.,
ABOD/algae y i e l d Y
l a n i t r i f i c a t i o n r a t e
kn 20
N/algae y i e l d Y
na
r e a e r a t i o n f a c t o r c
r 0 / a l g a e y i e l d
2
0 /ammonium y i e l d 2
s e d i m e n t oxygen consumption
S20 P / a l g a e y i e l d
P r e c y c l i n g e f f i c i e n c y X P P s e d i m e n t a t i o n r a t e c o e f f i c i e n t k
s f r a c t i o n P t h a t might s e d i m e n t X
P*
- - - - - -
TABLE 2 .
P m m e t z r description and nominal values for BerkeZ River simulation
c o u n t e d from t h e b e g i n n i n g o f t h e d i s c h a r g e s e c t i o n , i . e . t h e s e c t i o n
b e f o r e
t h e s t u d y r e a c h .c r o s s e c t i o n d e p t h
3
t r a v e l t r a v e l r e a l t i m el o c a t i o n (m 1 (m) d i s t a n c e (m) t i m e ( h ) d a t e t i m e
w e i r 2/3 36 1.5 0 0 19-6 18.20
S TP 2050 26.5 20-6 20.50
53 2.0
w e i r 3/4 3090 42.7 21.6 13.00
55 . 1.9
7 0 2.3
w e i r 4/5
14 5380 84.7 23-6 7.00
1 . 2
w e i r 5/6 29 2.0 8920 105.9 24-6 4.20
TABLE 3.
C h a r a c t e r i s t i c s River BerkeZ/pZug foZZowing program
A q u a l i t a t i v e i d e a of t h e goodness o f f i t o b t a i n e d i n t h e p a r a m e t e r c a l i b r a t i o n p h a s e g i v e s F i g . 1 f o r a l g a e and F i g . 2 f o r d i s s o l v e d oxygen. C l e a r l y , q u i t e
l a r g e d i s c r e p a n c i e s between model and measurements e x i s t , s e e f o r i n s t a n c e t i m e 86h f o r a l g a e and t i m e 68h f o r d i s s o l v e d oxygen. I t i s i n t e r e s t i n g t o n o t e t h a t v e r y low f l o w c o n d i t i o n s p r e v a i l e d d u r i n g t h e measurement week, s u g g e s t i n g t h e p o s s i b i l i t y of o n l y l i m i t e d v e r t i c a l mixing e s p e c i a l l y
i n s t r e t c h e s f a r away from t h e w e i r . T h i s might e x p l a i n t h e h i g h oxygen c o n t e n t o b s e r v e d a t 68h, i . e . 14.00h c l o c k t i m e , b e c a u s e t h e sample i s t a k e n from t h e s u r f a c e where d u r i n g t h e day t h e h i g h e s t oxygen p r o d u c t i o n t a k e s p l a c e .
P a r a l l e l t o t h e p l u g f o l l o w i n g program an i n d e p e n d e n t i n p u t - o u t p u t program was a l s o done f o r 24 h o u r s . T h i s e n a b l e d t h e t e s t i n g o f t h e p a r a m e t e r s e t by comparing t h e o b s e r v e d o u t p u t t i m e series w i t h t h e o n e p r e d i c t e d by t h e model u s i n g t h e i n p u t series a s i n i t i a l c o n d i t i o n s . The r e s u l t was r e a s o n a b l e , a l t h o u g h i n t h e end t h e p r e d i c t e d c h l o r o p h y l l - a , and c o n s e q u e n t l y t h e d i s s o l v e d oxygen, were t o o high.
UNCERTAINTY
The B e r k e l R i v e r example i s a l s o s u i t a b l e f o r i l l u s t r a t i n g some u n c e r t a i n t i e s i n r i v e r q u a l i t y m o d e l l i n g . For t h i s p u r p o s e w e t a k e a l o o k a t b o t h model
BERKEL R I V E R MODEL
T R A V E L T I M E ( H I
F i g u r e 1.
Measurements (dots') and simulation for phytoplankton Arrow: waste discharge point
Bars
:w e i r s
BERKEL R I V E R MODEL
T R A V E L T I M E ( H I
F i g u r e 2 .
Measurements ( d o t s ) and simulation for d i s s o l v e d oxygen
and measurement r e s u l t s f o r carbonaceous BOD ( F i g . 3 ) . An immediate d i f f i c u l t y a r i s e s : what i s (C-)BOD? The model r e q u i r e s t h e u l t i m a t e BOD o f a l l o x i d i z a b l e o r g a n i c m a t t e r e x c e p t l i v i n g a l g a e . I n p r a c t i c e a 2?-day BOD-test was done. A p e r i o d of 20 days i s s u f f i c i e n t t o approach t h e BOD u l t i m a t e , b u t t h e r e s u l t i n c l u d e s b o t h t h e ammonium o x i d a t i o n a n d , p e r h a p s o n l y p a r t l y , t h e oxygen demand r e s u l t i n g from a l g a e d y i n g and r e s p i r i n g d u r i n g t h e t e s t . I n o r d e r t o e x c l u d e t h e a l g a e a BOD
2 0 f o r f i l t e r e d samples was a l s o done. However, t h i s p r o c e d u r e a l s o e x c l u d e s n o n - a l g a l p a r t i c u l a t e o x i d i z a b l e o r g a n i c m a t t e r . Thus, a l l t h a t can b e s a i d i s t h a t t h e a c t u a l BOD must be somewhere between t h e v a l u e s f o r t h e f i l t e r e d and u n f i l t e r e d samples, a s i n d i c a t e d by t h e open c i r c l e s i n F i g . 3 . A c o r r e c t i o n f o r t h e n i t r i f i c a t i o n was made by
m u l t i p l y i n g t h e measured ammonium n i t r o g e n c o n c e n t r a t i o n w i t h 4.3 (mg 0 p e r mg N) and s u b t r a c t i n g t h i s v a l u e from t h e BOD t e s t r e s u l t
2
( a g a i n i n t r o d u c i n g some u n c e r t a i n t y ) .
B E R K E L R I V E R MODEL
TRAVEL TINE ( H I
~ i g u r e 3.
Measurements for f i l t e r e d (lower c i r c l e ) and
t o t a l carbonaceous
BOD(upper c i r c l e ) and
two simulation r e s u l t s (see t e x t )
The model s i m u l a t i o n r e s u l t s ( s o l i d l i n e s F i g . 3 ; t h e o r i g i n o f t h e two c u r v e s i s e x p l a i n e d below) s u g g e s t a c l e a r i n c r e a s e i n c a r b o n a c e o u s BOD, c a u s e d by p r o d u c t i o n o f d e t r i t u s due t o a l g a l d e a t h . Although t h e
i n c r e a s e i s n o t v e r y c l e a r i n t h e measurements t h e t r e n d i s a l s o o b s e r v e d i n t h e i n p u t - o u t p u t r e s u l t s , where t h e C-BOD a t t h e o u t p u t i s s i g n i f i c a n t l y h i g h e r t h a n a t t h e i n p u t . S i m u l t a n e o u s l y t h e r e i s a r e m a r k a b l e d e c l i n e i n c h l o r o p h y l l - a ( F i g . 1 ) i n s p i t e of v e r y f a v o u r a b l e w e a t h e r c o n d i t i o n s f o r a l g a l growth, l i k e w i s e p o i n t i n g towards an u n u s u a l l y a c t i v e d e a t h p r o c e s s .
The a c c u r a c y o f t h e measurements d o e s n o t a l l o w f o r a p r e c i s e e v a l u a t i o n of t h e p a r a m e t e r s . L e t u s , t h e r e f o r e , f o r t h e t i m e b e i n g , a d o p t t h e p a r a m e t e r s e t o f T a b l e 2. Now, i f t h i s i s t h e v a l i d s e t , it must b e p o s s i b l e t o e x t r a p o l a t e t h e s i m u l a t i o n backward i n t i m e t o see whether a c c e p t a b l e v a l u e s a r e o b t a i n e d i n t h e d i s c h a r g e s e c t i o n b e f o r e t h e measurement s e c t i o n s . Doing t h i s it a p p e a r s t h a t t h e C-BOD j u s t b e f o r e
t h e w a s t e d i s c h a r g e must have been n e g a t i v e i n o r d e r t o p r o d u c e t h e r e s u l t s i n t h e measurement s e c t i o n s . T h i s , o f c o u r s e , i s i m p o s s i b l e , d e m o n s t r a t i n g t h a t an a c c e p t a b l e p a r a m e t e r s e t c a n become n o n - a c c e p t a b l e i f e x t r a p o l a t i o n s a r e made t o o t h e r s e c t i o n s o r o t h e r t i m e s .
An e x p l a n a t i o n c a n b e found by n o t i n g t h a t t h e BOD decay r a t e c o e f f i c i e n t r e p r e s e n t s a lumped p r o c e s s , c o v e r i n g a complex of b a c t e r i a l s u b p r o c e s s e s . O b s e r v a t i o n s i n d i f f e r e n t r i v e r s i n t h e G e l d e r l a n d a r e a i n d i c a t e t h a t t h e r a t e c o n s t a n t i s BOD d e p e n d e n t , a s i l l u s t r a t e d i n T a b l e 4.
BOD BOD2 C-BOD
Brook o r R i v e r f i l z g r e d t o t a ? c a l c u l a t e d
Groenlose S l i n g e 1972 40-180 15-130 1.5-2
Oude IJssel 1974 20-40 20- 60 5- 25 < O . 05
Oude I J s s e l 1975 8-15 20- 35 10- 25 0.1-0.6
B e r k e l 1974 10-20 20- 40 0 . 3
B e r k e l 1978 15-20 20- 25 0- 8 0.1-0.2
Eem ( U t r e c h t ) 1977 20-25 25- 40 0.1-0.2
TABLE 4. BOD
decay r a t e and a s s o c i a t e d BOD-ranges ( r e f . van S t r a t e n
and d e Boer,
1 9 7 9 )Obviously, t h e BOD decay r a t e c o e f f i c i e n t i s h i g h e r i n more p o l l u t e d s t r e a m s b e c a u s e u n t r e a t e d w a s t e w a t e r i s more e a s i l y d e g r a d a b l e t h a n e f f l u e n t s from sewage t r e a t m e n t p l a n t s . T h e r e f o r e , it may be h y p o t h e s i s e d t h a t t h e decay r a t e c o e f f i c i e n t h a s been h i g h e r i n t h e s e c t i o n d i r e c t l y a f t e r t h e w a s t e d i s c h a r g e t h a n i n t h e r e s t o f t h e r i v e r s t r e t c h .
S i m i l a r arguments a p p l y t o t h e d e a t h r a t e o f a l g a e . Again t h i s c o e f f i c i e n t lumps a v a r i e t y of u n d e r l y i n g p r o c e s s e s s u c h a s r e s p i r a t i o n , n a t u r a l d e a t h and d e a t h by zooplankton g r a z i n g . The l a t t e r p r o c e s s e s p e c i a l l y ,
i n t r o d u c e s a h i g h v a r i a b i l i t y o f t h e d e a t h r a t e i n t i m e . I n f a c t , i t i s known t h a t zooplankton was p r e s e n t d u r i n g t h e measurement p e r i o d . F o r t - n i g h t l y r o u t i n e measurements r e v e a l e d t h a t t h e measurement p e r i o d c o i n c i d e d w i t h t h e l o w e s t c h l o r o p h y l l - a (and t h e h i g h e s t p h e o p h y t i n e ) c o n c e n t r a t i o n i n t h e e n t i r e summer p e r i o d , and t h a t a s h a r p d e c l i n e had o c c u r r e d from a b o u t 140 pg c h l o r o p h y l l - a / l t o 50 p g / l w i t h i n t h e two p r e c e e d i n g weeks. Combined w i t h t h e low flow r a t e it i s most l i k e l y t h a t t h e zooplankton r e a c h e d i t s peak c o n c e n t r a t i o n s i n o r a f t e r t h e measurement week, r e s u l t i n g i n a s t i l l i n c r e a s i n g d e a t h r a t e c o e f f i c i e n t d u r i n g t h e measurement p e r i o d .
For d e m o n s t r a t i o n p u r p o s e s a s i m u l a t i o n was r u n based on t h e s e two h y p o t h e s i s . I t was assumed t h a t t h e a l g a l d e a t h r a t e was o n l y h a l f o f i t s nominal v a l u e d u r i n g t h e f i r s t 27 h o u r s , and t h a t t h e BOD decay r a t e was f o u r t i m e s i t s nominal v a l u e i n t h e r i v e r s e c t i o n between t h e d i s c h a r g e p o i n t and t h e f i r s t weir. The r e s u l t f o r C-BOD i s g i v e n i n F i g . 3 , where t h e z e r o i n i t i a l c o n d i t i o n i s t h e u l t i m a t e p o s s i b l e , though s t i l l n o t a s a t i s f a c t o r y assumption. I n F i g . 4 t h e e f f e c t on t h e a l g a l p a t t e r n i s shown.
The h y p o t h e s i s a s s u m p t i o n s s e r i o u s l y a f f e c t t h e b e h a v i o u r b e f o r e t h e measurement s e c t i o n . Whether o r n o t t h i s i s r e a l i s t i c c o u l d o n l y b e s a i d
i f d a t a had been a v a i l a b l e , a l t h o u g h t h e new i n i t i a l condition f o r a l g a e i s more l i k e l y . Note t h a t i n t h e absence o f d a t a , . numerous o t h e r
assumptions can b e made ( f o r i n s t a n c e , a lower e f f i c i e n c y of a l g a e i n t o BOD t r a n s f e r , p a r a m e t e r X , would l e a d t o e s s e n t i a l l y l e s s BOD produced by a l g a l d e c a y ) .
The problem o f i n a p p r o p r i a t e model d e s c r i p t i o n a s compared t o t h e complex r e a l i t y , d e m c n s t r a t e d by t h e r a t h e r academic e x e r c i s e s above, i s a funda- mental problem o f any m o d e l l i n g a c t i v i t y . S i n c e t h e model i s always a
s i m p l i f i c a t i o n one can e x p e c t t h a t t h e model p a r a m e t e r s a r e f u n c t i o n s
BERKEL RIVER MODEL
TRAVEL TINE (HI
~ i g u r e 4 .
E f f e c t o f parameter v a r i a b i l i t y on p h y t o p l a i k t o n simulation - constant parameters variable decay and death r a t e
c o e f f i c i e n t s ( s e e t e r t l
of time and space due t o t h e v a r i a b i l i t y i n t h e unobserved underlying processes.
A n a t u r a l way t o solve t h i s d i f f i c u l t y would be t o d e t a i l t h e d e s c r i p t i o n f u r t h e r , a s soon a s knowledge permits one t o do s o , i n t h e hope t h a t t h e new parameters a r e l e s s time and space v a r i a n t . However, t h i s implies a t t h e same time more s t a t e v a r i a b l e s and a considerable i n c r e a s e i n parameters t o be estimated. For t h i s reason such an approach i s not always t h e d e s i r a b l e one, e s p e c i a l l y not i f management problems have t o be solved. For i n s t a n c e ,
modelling zooplankton would c e r t a i n l y improve t h e p r e d i c t i o n of a l g a l minima, a t t h e expense of a considerable measurement e f f o r t , b u t would do l i t t l e f o r t h e a l g a l peaks. I f a l g a l peaks a r e t h e main concern of t h e management a model with a conservative estimate f o r t h e death r a t e , o r an empirical time function, could be very well acceptable.
Generally, it i s very u s e f u l f o r f u r t h e r model development t o examine t h e consequences of parameter u n c e r t a i n t i e s f o r t h e c e r t a i n t y o r u n c e r t a i n t y of model p r e d i c t i o n s . One way of doing t h i s i s by a s e ~ i s i t i v i t y anal.ysis.
SENSITIVITY ANALYSIS
An i n t e r e s t i n g way of analyzing t h e model s e n s i t i v i t y i s based on t h e s o l u t i o n of t h e s e n s i t i v i t y equations. Let t h e system equations be w r i t t e n i n s t a t e space form a s
where c i s t h e n-dimensional s t a t e vector ( v e c t o r of system v a r i a b l e s ) ,
-
p t h e r-dimensional parameter v e c t o r , u t h e m-dimensional input v e c t o r
- -
( f o r c i n g f u n c t i o n s ) and T t h e time. The i n i t i a l conditions a r e
Let
be defined a s t h e t r a j e c t o r y s e n s i t i v i t y vector of t h e s t a t e v e c t o r
-
c with r e s p e c t t o t h e parameter p around t h e nominal parameter s e t p 0.
Note t h a tj
-
t h e elements of s a r e time funcions. Then taking t h e t o t a l d e r i v a t i v e t o -j
p and interchanging t h e o r d e r of d i f f e r e n t i a t i o n l e a d s t o t h e so-called s e n s i t i v i t y equations (note: v e c t o r d i f f e r e n t i a l e q u a t i o n s ) :
w i t h
c f . F r a n k ( 1 9 7 8 ) . I t i s i n t e r e s t i n g t o n o t e t h a t t h e s e n s i t i v i t y system i s always l i n e a r , even i f t h e model f i s n o n l i n e a r . T h e r e f o r e , i n t h e c a s e of r e l a t i v e l y s i m p l e models such a s t h e S t r e e t e r - P h e l p s model, a n a l y t i c a l s o l u t i o n s f o r t h e s e n s i t i v i t y system c a n o f t e n be o b t a i n e d ( e . g . R i n a l d i and S o n s i n i - S e s s a , 1 9 7 7 ) . Formore complex models a n u m e r i c a l s o l u t i o n i s n e c e s s a r y b u t n o t d i f f i c u l t t o a c h i e v e .
The method was a p p l i e d t o t h e R i v e r B e r k e l example from t h e p r e v i o u s
s e c t i o n . A manageable s t a t e s p a c e form i s o b t a i n e d , a s b e f o r e , by c o n s i d e r - i n g o n l y one p l u g of w a t e r and n e g l e c t i n g d i s p e r s i o n e f f e c t s . I n t h i s a p p l i c a t i o n t h e J a c o b i a n m a t r i c e s a f / a c
- -
and a f / a p- -
were d e r i v e d a n a l y t i c a l l y . The s o l u t i o n o f t h e s e n s i t i v i t y system i s done s i m u l t a n e o u s l y w i t h t h esystem e q u a t i o n s ( i n v o l v i n g t h e n u m e r i c a l s o l u t i o n of a system of n * ( r
+
1) s i m u l t a n e o u s d i f f e r e n c e e q u a t i o n s ) .The r e s u l t s a r e shown f o r a l g a e i n Fig.5. The s e n s i t i v i t y on t h e v e r t i c a l a x i s h a s been t r a n s f o r m e d such t h a t t h e a b s o l u t e change of t h e s t a t e v a r i a b l e t o a r e l a t i v e i n c r e a s e of 1 0 % i n t h e p a r a m e t e r v a l u e i s shown. I n t h i s way t h e r e a d i n g s have t h e dimension o f c o n c e n t r a t i o n and can b e compared
d i r e c t l y w i t h F i g s . 1-4. The c u r v e s s e l e c t e d r e p r e s e n t t h e most s i g n i f i c a n t s e n s i t i v i t y f u n c t i o n s .
From F i g . 5 it i s s e e n t h a t b o t h t h e growth r a t e (Curve 1) and t h e d e a t h r a t e (Curve 7 ) govern t h e a l g a l system. The day and n i g h t p a t t e r n i s
a l s o r e f l e c t e d i n t h e s e n s i t i v i t y . The jump i n t h e d e a t h - r a t e s e n s i t i v i t y a t 27 h i s caused by t h e d o u b l i n g o f t h i s p a r a m e t e r beyond t h i s t i m e a s
mentioned p r e v i o u s l y . Note t h a t an i n c r e a s e of o n l y 1 0 % i n t h e d e a t h r a t e would c a u s e t h e a l g a l c o n c e n t r a t i o n t o d r o p by 0.8 mg dw/l a t t h e end of t h e s t u d y r e a c h , i . e . f 40% of t h e a c t u a l c o n c e n t r a t i o n . However, t h e e f f e c t could b e c o u n t e r b a l a n c e d by a n i n c r e a s e i n growth r a t e o f a b o u t 12%. Because t h e s e n s i t i v i t y p a t t e r n of t h e growth r a t e and t h e d e a t h r a t e c o e f f i c i e n t s a r e s i r i l a r , a p p a r e n t l y more t h a n one combination o f b o t h p a r a m e t e r s w i l l d e s c r i b e t h e o v e r a l l a l g a l p a t t e r n . T h i s i s t o be e x p e c t e d s i n c e t h e a l g a l e q u a t i o n r e a d s
S E N S I T I V I T Y BERKEL RIVER MODEL
TRAVEL TIME (HI
Figure
5. S e n s i t i v i t y functions for phytoplankton
2 =
growth r a t e c o e f f i c i e n t k
'Pa
3 =
e x t i n c t i o n c o e f f i c i e n t k
0
5 =
s a t u r a t i o n l i g h t i n t e n s i t y I s
7 =
death r a t e c o e f f i c i e n t kiia
so that in fact K =
k
F- k
is the rate constant of interest, and a Pa daseparation of
k
andk
on the basis of observations of the algaePa da
concentration alone will be practically impossible. This demonstrates how sensitivity analysis can help in indicating the principal components in parameter space, which is extremely useful for parameter estimation procedures.
F i g . 6 shows t h e s e n s i t i v i t y p l o t f o r d i s s o l v e d oxygen. The m i t i g a t i n g e f f e c t of t h e w e i r s on t h e d e f i c i t i s r e f l e c t e d i n t h e c u r v e s a s a r e d u c t i o n i n s e n s i t i v i t y d i r e c t l y a f t e r t h e w e i r . Not s u r p r i s i n g l y d i s s o l v e d oxygen i s s e n s i t i v e t o e s s e n t i a l l y t h e same p a r a m e t e r s a s t h e a l g a e . The s e n s i t i v i t y t o t h e BOD decay r a t e and n i t r i f i c a t i o n r a t e c o e f f i c i e n t s i s i n t h e o r d e r o f -0.1 mg 0 /1 p e r 1 0 % i n c r e a s e , and t o t h e
2
r e a e r a t i o n r a t e a p p r o x i m a t e l y 0 . 1 mg 0 /1 p e r 1 0 % i n c r e a s e ( n o t shown). '
2
T h i s d e m o n s t r a t e s t h e dominant r o l e of t h e a l g a l system i n t h e o v e r a l l oxygen b e h a v i o u r .
By comparing F i g s . 5 and 6 it can b e s e e n t h a t a s i m u l t a n e o u s 10% i n c r e a s e o f b o t h t h e growth and d e a t h r a t e c o e f f i c i e n t s w i l l l e a d t o somewhat l o w e r a l g a e c o n c e n t r a t i o n s and somewhat h i g h e r d i s s o l v e d oxygen c o n t e n t . T h i s i s b e c a u s e t h e growth o f a l g a e and t h e p r o d u c t i o n o f oxygen a r e d i r e c t l y c o u p l e d , whereas t h e d e a t h of a l g a e r e s u l t s i n BOD p r o d u c t i o n f i r s t , w i t h o n l y r e t a r d e d oxygen consumption. I n t h i s c o n n e c t i o n a comparison w i t h F i g . 7 , showing t h e C-BOD s e n s i t i v i t y i s i n t e r e s t i n g . A h i g h e r a l g a l d e a t h r a t e c o e f f i c i e n t (Curve 7 ) would i n i t i a l l y l e a d t o a h i g h e r BOD, b u t i n t h e l o n g r u n t o l e s s a l g a e and t h e r e b y t o a lower BOD c o n t r i b u t i o n from d e t r i t u s p r o d u c t i o n . Thus, a change i n t h e d e a t h c o e f f i c i e n t e f f e c t s t h e
shape o f t h e BOD c u r v e i n t i m e . A s i m i l a r argument a p p l i e s t o t h e e f f e c t o f t h e growth r a t e (Curve 1 ) : i n i t i a l l y t h e r e i s no e f f e c t , a s e x p e c t e d . b u t l a t e r t h e h i g h e r a l g a e c o n c e n t r a t i o n l e a d s t o more d e a t h and t h e r e b y t o more BOD p r o d u c t i o n .
The BOD i s a l s o q u i t e s e n s i t i v e t o t h e amount of BOD r e l e a s e d p e r u n i t d y i n g a l g a e ( X and Y Curve A ; t h e y have t h e same s e n s i t i v i t y b e c a u s e t h e y
l a '
a p p e a r a s a p r o d u c t ) . The e f f e c t on t h e d i s s o l v e d oxygen b a l a n c e i s i n t h e same o r d e r o f magnitude a s t h e e f f e c t s of t h e BOD decay r a t e
c o e f f i c i e n t ( n o t shown). The jump i n t h e s e n s i t i v i t y t o t h e BOD d e c a y r a t e c o e f f i c i e n t ( F i g . 7 , Curve 8 ) i s c a u s e d by t h e a s s u m p t i o n of a h i g h e r r a t e i n t h e s e c t i o n d i r e c t l y a f t e r t h e w a s t e d i s c h a r g e p o i n t .
From a l l g r a p h s it i s a p p a r e n t t h a t t h e s e n s i t i v i t y h a s s t i l l an i n c r e a s i n g t r e n d a t t h e end of t h e measurement r e a c h . T h i s i m p l i e s t h a t s m a l l e r r o r s i n p a r a m e t e r e s t i m a t e s o b t a i n e d from t h i s l i m i t e d measurement p e r i o d may l e a d t o c o n s i d e r a b l e p r e d i c t i o n e r r o r s i f e x t r a p o l a t i o n s a r e b e i n g made.
T h i s s u g g e s t s t h a t t h e measurement p e r i o d s h o u l d have been l o n g e r f o r more a c c u r a t e p a r a m e t e r e s t i m a t e s .
SENSITIVITY BERKEL RIVER MODEL
TRAVEL T I M E ( H I
F i g u r e 6.
S e n s i t i v i t y functions for dissolved oxygen parameters a s i n Figure
5 .SENSITIVITY BERKEL RIVER MODEL
-
1-00 TF i g u r e
TRAVEL T I M E (H)
7 .
S e n s i t i v i t y functions for
C-BODAdditional parameters:
8 = BODdecay r a t e c o e f f i c i e n t kbZo
A = BOD
released per u n i t dying algae
Xor
Yl a
L i m i t a t i o n s
C l e a r l y , s e n s i t i v i t y a n a l y s i s l e a d s t o c o n s i d e r a b l e i n s i g h t i n t o t h e systems b e h a v i o u r . However, we s h o u l d l i k e t o p o i n t o u t some of t h e
l i m i t a t i o n s o f t h e a p p l i c a t i o n above. One o f t h e s e i s t h a t t h e e f f e c t s o f p a r a m e t e r changes on t h e s t a t e v a r i a b l e s a r e compared on t h e b a s i s of
e q u a l p a r a m e t e r v a r i a t i o n s . T h e r e f o r , f o r u n c e r t a i n t y a n a l y s i s t h e method must b e extended t o cope w i t h d i f f e r e n t d e g r e e s of u n c e r t a i n t y i n t h e p a r a m e t e r s i n v o l v e d . For example, t h e e f f e c t o f v a r i a t i o n s i n t h e
e x t i n c t i o n c o e f f i c i e n t on a l g a e and d i s s o l v e d oxygen ( F i g s . 5 and 6 , Curve 3 ) can be even more d r a m a t i c t h a n s u g g e s t e d i n t h e g r a p h s , because t h e
e x t i n c t i o n c o e f f i c i e n t may v a r y by 50-100% r a t h e r t h a n 10%. S i m i l a r arguments a p p l y , o f c o u r s e , t o every p a r a m e t e r . I t s h o u l d a l s o be n o t e d t h a t t h e l i n e a r e x t r a p o l a t i o n of t h e s t a t e a c c o r d i n g t o Ac = s Ap. i s
j I o n l y allowed f o r s m a l l e x c u r s i o n s from t h e nominal p a r a m e t e r v a l u e s , because t h e s y s t e m s model i s n o n - l i n e a r . F i n a l l y , t h e system i s a l s o
s e n s i t i v e t o i n i t i a l c o n d i t i o n s and f o r c i n g f u n c t i o n s ; however, t h e s e c o u l d b e handled i n a s i m i l a r f a s h i o n w i t h o u t d i f f i c u l t y .
CONCLUSIONS
-
Models such a s GELQAM c a n produce a f a i r d e s c r i p t i o n o f t h e oxygen and p h y t o p l a n k t o n b e h a v i o u r i n ( c a n a l i z e d ) lowland s t r e a m s .-
Due t o u n c e r t a i n t y i n t h e u n d e r l y i n g p r o c e s s e s it may be e x p e c t e d t h a t p a r a m e t e r s a r e s i t e dependent ( e . 9 . BOD decay r a t e c o n s t a n t ) o r time dependent ( e . g . r a t e c o e f f i c i e n t f o r a l g a l d e a t h ) . I t i s n o t p o s s i b l e t o d e t e c t t h e s e f u n c t i o n a l r e l a t i o n s h i p s by measurements i n a l i m i t e d r i v e r s t r e t c h and f o r a s h o r t time.-
An a c c u r a t e measurement o f one of t h e model s t a t e v a r i a b l e s , carbonaceous BOD, i s n o t p o s s i b l e . T h i s u n c e r t a i n t y r e d u c e s t h e p r e c i s i o n of p a r a m e t e r e s t i m a t e s .-
A s e n s i t i v i t y a n a l y s i s i s a u s e f u l element i n t h e e v a l u a t i o n o f t h e e f f e c t s of p r o c e s s and p a r a m e t e r u n c e r t a i n t y . I n s i t u a t i o n s w i t h e x t e n s i v e a l g a l growth and moderate o r g a n i c waste d i s c h a r g e , t h e growth and d e a t h p r o c e s s e s of a l g a e govern t h e oxygen b e h a v i o u r . The s e n s i t i v i t y a n a l y s i s a l s or e v e a l s t h e d i f f i c u l t y of e s t i m a t i n g t h e growth and d e a t h r a t e c o e f f i c i e n t s i n d e p e n d e n t l y , i f no a c c u r a t e d a t a a r e a v a i l a b l e .
- The inadequacy of the model caused by lumping complex processes poses
the problem of predictive power, a fundamental difficulty of any model
activity. It is worthwhile to investigate the use of sensitivity analysis
in connection with a parameter variability study in assigning a 'reliability
factor' to the effectiveness of management decisions expected from the model.
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