UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

An application of decision theory to water quality management Hershman, Stanley 1974

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1974_A7 H47.pdf [ 3.92MB ]
Metadata
JSON: 831-1.0050515.json
JSON-LD: 831-1.0050515-ld.json
RDF/XML (Pretty): 831-1.0050515-rdf.xml
RDF/JSON: 831-1.0050515-rdf.json
Turtle: 831-1.0050515-turtle.txt
N-Triples: 831-1.0050515-rdf-ntriples.txt
Original Record: 831-1.0050515-source.json
Full Text
831-1.0050515-fulltext.txt
Citation
831-1.0050515.ris

Full Text

AN APPLICATION OF DECISION THEORY TO WATER QUALITY MANAGEMENT by." STANLEY HERSHMAN B.Eng., M c G i l l U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of C i v i l Engineering We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1974 In presenting t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada A B S T R A C T The t h e s i s presents an a l g o r i t h m f o r a p p l y i n g d e c i s i o n theory to the management of water q u a l i t y of a lak e that i s becoming c u l t u r a l l y e u t r o p h i c . The a l g o r i t h m was formulated using U t i l i t y theory and Bayesian d e c i s i o n s t r a t e g y as a t h e o r e t i c a l b a s i s . I t pays p a r t i c u l a r a t t e n t i o n to the f o l l o w i n g three elements of the management problem: (1) the i n t e r d i s c i p l i n a r y nature of the problem; the need to co-ordinate the e f f o r t s of b i o l o g i s t s , l i m n o l o g i s t s , engineers, e t c . , as w e l l as p r o v i d i n g a r o l e f o r the p u b l i c ; (2) the extent of u n c e r t a i n t y w i t h respect to the behavior of a n u t r i e n t enriched l a k e and the e f f e c t of v a r i o u s abatement measures upon the f u t u r e water q u a l i t y of the l a k e ; (3) the need f o r long-term planning s t r a t e g i e s so that the problem can be c o n t r o l l e d and not j u s t delayed. The technique provides the planners w i t h a systematic pro-cedure f o r e v a l u a t i n g the s c i e n t i f i c data. I t measures d e t e r i o r a t i o n of the lake i n non-economic u n i t s of u t i l i t y . A l t e r n a t i v e s are compared according to t h e i r "expected u t i l i t y " over a pla n n i n g p e r i o d . Skaha Lake, l o c a t e d i n the Okanagan Ba s i n i n B r i t i s h Columbia i s used as a case study f o r the development and a p p l i c a t i o n of the technique. The lake r e c e i v e s t r e a t e d sewage e f f l u e n t from a growing m u n i c i p a l i t y and has r e c e n t l y e x h i b i t e d a sharp i n c r e a s e i n b i o l o g i c a l p r o d u c t i o n . S e v e r a l p o s s i b l e management programs f o r the lake are com-pared using data obtained by l i m n o l o g i s t s and engineers. i i i A computer program was w r i t t e n i n WATFIV that can s t o r e and analyse data corresponding to a s p e c i f i c l a k e . The output of the program i s a g r a p h i c a l p l o t of an expected u t i l i t y f u n c t i o n over some planning p e r i o d f o r a p a r t i c u l a r p o l l u t i o n c o n t r o l measure. The ex-pected u t i l i t y s c a l e r e l a t e s d i r e c t l y to the water q u a l i t y and degree of e u t r o p h i c a t i o n of the lak e and can thus be used to compare the e f f e c t i v e n e s s of proposed a l t e r n a t i v e measures. TABLE OF CONTENTS Page LIST OF TABLES v l LIST OF FIGURES v i i CHAPTER I. INTRODUCTION , 1 I I . EUTROPHICATION AS A WATER QUALITY MANAGEMENT PROBLEM. . 5 A. EUTROPHICATION AS A NATURAL PHENOMENA 5 B. CULTURAL EUTROPHICATION 6 C. THE EFFECTS OF EUTROPHICATION 6 D. CONTROL OF EUTROPHICATION 8 E. EUTROPHICATION AS A LAKE MANAGEMENT PROBLEM . . . . 10 I I I . DECISION MAKING UNDER UNCERTAINTY 17 A. UTILITY THEORY 18 IV. DEFINING A UTILITY FUNCTION FOR A EUTROPHIC LAKE. . . . 22 A. DERIVING A UTILITY FUNCTION 23 B. DEFINING A TROPHIC STATE SCALE . 24 C. CALCULATION OF UTILITY FUNCTION USING CHOICE TABLE. 29 V. EUTROPHICATION IN SKAHA LAKE 35 A. GENERAL DESCRIPTION OF SKAHA LAKE REGION 35 B. WATER QUALITY PROBLEMS IN SKAHA LAKE 37 C. PHOSPHORUS AS A LIMITING NUTRIENT IN SKAHA LAKE . . 41 D. THE LAKE MANAGEMENT PROBLEM 42 VI. THE EXPECTED UTILITY FUNCTION 46 A. CALCULATING THE EXPECTED UTILITY FUNCTION 48 Phosphorus Loading P r o j e c t i o n s 49 Phosphorus Loading and Lake Trophic State 50 Skew Normal D i s t r i b u t i o n - . 51 i v V CHAPTER Page V I I . EXPECTED UTILITY AS A MANAGEMENT AID. . . . . . . . . . . . 59 A. MANAGEMENT OF SKAHA LAKE ' . 59 B. LONG RANGE PLANNING . 64 V I I I . DISCUSSION AND CONCLUSIONS 68 BIBLIOGRAPHY 71 APPENDIX A. THE AXIOMS OF COHERENCE 75 B. DERIVATION OF SKEW NORMAL DISTRIBUTION 77 C. PHOSPHORUS LOADING PROJECTIONS AND CONFIDENCE LIMITS. . . . 81 LIST OF TABLES TABLE Page 4.1 TROPHIC STATE SCALE CLASSIFICATION 27 4.2 ESTIMATED SECCHI DISC READINGS AND DISSOLVED OXYGEN LEVELS CORRESPONDING TO TROPHIC STATE SCALE 28 4.3 CHOICE TABLE FOR FINDING INDIFFERENCE POINTS BETWEEN CERTAIN TROPHIC STATE AND REFERANCE CONTRACT. . . 30 5.1 LOADINGS OF TOTAL PHOSPHORUS TO SKAHA LAKE FROM EXTERNAL SOURCES BASED ON 1969-71 DATA . . . . . . . 39 6.1 ESTIMATES OF TOTAL PHOSPHORUS LOADING VERSUS TROPHIC STATES FOR SKAHA LAKE .51 v l LIST OF FIGURES EXAMPLE OF DECISION MAKER'S UTILITY FUNCTION DERIVED FROM INDIFFERENCE TABLE 4.3 . . . . OKANAGAN RIVER SYSTEM ESTIMATES OF TOTAL PHOSPHORUS LOADINGS VERSUS TROPHIC STATES FOR SKAHA LAKE APPROXIMATE SHAPE OF POSTULATED PROBABILITY DENSITY FUNCTION OF TROPHIC STATES IN THE FORM OF A SKEW NORMAL CURVE. . . . EXt:'£iCijbi( UJ.TJ_IJ.AY CUixviiS rCi\ Si\±i£iA i__t\iUi. uuivixXiOrUi^ iiJxiNij TO 70%, 80% AND 90% PHOSPHORUS REMOVAL IN THE PENTICTON SEWAGE TREATMENT PLANT EXPECTED UTILITY CURVE FOR SKAHA LAKE CORRESPONDING TO LAND APPLICATION OF PENTICTON SEWAGE EFFLUENT. . PROJECTIONS OF PHOSPHORUS LOADING INTO SKAHA LAKE CORRESPONDING TO 70%, 80% AND 90% PHOSPHORUS REMOVAL AND LAND APPLICATION OF SEWAGE EXPECTED UTILITY CURVE WITH 10% AND 90% CONFIDENCE LIMITS CORRESPONDING TO 70% PHOSPHORUS REMOVAL EXPECTED UTILITY CURVE WITH 10% AND 90% CONFIDENCE LIMITS CORRESPONDING TO 80% PHOSPHORUS REMOVAL EXPECTED UTILITY CURVE WITH 10% AND 90% CONFIDENCE LIMITS CORRESPONDING TO 90% PHOSPHORUS REMOVAL . . . EXPECTED UTILITY CURVE WITH 10% AND 90% CONFIDENCE LIMITS CORRESPONDING TO LAND APPLICATION OF SEWAGE EFFLUENT. . . v i i A C K N O - W L E D G E M E N T S The author i s very g r a t e f u l to h i s s u p e r v i s o r P r o f e s s o r Samuel 0. R u s s e l l , f o r h i s constant encouragement and i n v a l u a b l e guidance throughout the development of t h i s t h e s i s . A s p e c i a l thanks i s extended to Dr. John Stockner who provided the l i m n o l o g i c a l exper-t i s e and who gave most generously of h i s time. The author would a l s o l i k e to thank B i l l Fleming f o r h i s a s s i s t a n c e and encouragement. v i i i CHAPTER I INTRODUCTION Research i n t o the n a t u r a l environment and i t s growing p o l l u t i o n problems has increased considerably w i t h i n the l a s t decade. Both aca-demic and i n d u s t r i a l i n s t i t u t i o n s are r a p i d l y developing a technology to deal w i t h the problem of abating p o l l u t i o n . This a c t i v i t y i s c l e a r -l y evident i n the f i e l d of water resources where annual expenditures f o r p o l l u t i o n c o n t r o l f a c i l i t i e s are i n the b i l l i o n s of d o l l a r s . However, i t i s becoming apparent that the management of water resources i s an extreme-l y complex problem and the s o l u t i o n i s not simply a matter of s t r a i g h t -p l i n e s i s necessary to deal w i t h water q u a l i t y problems and q u i t e o f t e n the u l t i m a t e d e c i s i o n s are made by laymen p o l i t i c i a n s or a d m i n i s t r a t o r s . Thus, the mechanics of approaching and d e a l i n g w i t h environmental problems be-comes a challenge i n i t s e l f . G i l b e r t White, an eminent geographer, a l l u d e d to t h i s c hallenge i n the f o l l o w i n g statement (White, 1969): " I t i s more l i k e l y that human w e l f a r e i n the United States w i l l be impaired through degra-d a t i o n of water q u a l i t y or through i n e p t manage-ment than from a p h y s i c a l s c a r c i t y of water." Although t h i s i s true, one should r e a l i z e that the i n e p t manage-ment i s l a r g e l y due to the very l i m i t e d understanding of the n a t u r a l en-vironment. Past experience has demonstrated that c o n v e n t i o n a l water q u a l -i t y management procedures are inadequate i n d e a l i n g w i t h the complex water 2 q u a l i t y problems. Optimum and e f f i c i e n t use of our l i m i t e d water resour-ces r e q u i r e s more s o p h i s t i c a t e d management t o o l s than have been employed to date. B e l l a et at. (1972) summarized q u i t e a p t l y the predicament of managing the environment: "Man's a b i l i t y to modify the environment w i l l i n c r e a s e f a s t e r than h i s a b i l i t y to forsee the e f f e c t s of h i s a c t i v i t i e s . . ," The authors of t h i s paper f e l t that because of t h i s i n a b i l i t y to comprehend the impacts on the environment, the d e c i s i o n making process should recognize the p o s s i b i l i t y of " u n c e r t a i n outcomes." They a l s o men-tioned the need f o r a s c a l e of " U t i l i t y . " This s c a l e would serve as a c r i t e r i a to measure the q u a l i t y of l i f e when as s e s s i n g outcomes of a de-c i s i o n . The concepts of u t i l i t y and u n c e r t a i n t y are both standard terms i n d e c i s i o n theory. The l a t e n t u n c e r t a i n t y i n water q u a l i t y management has always been . recognized, but seldom d e a l t w i t h e x p l i c i t l y . Engineers i n c l u d e s a f e t y f a c t o r s i n t h e i r design, but are u s u a l l y r e l u c t a n t to d i s c u s s the l i k e l i -hood of f a i l u r e . S i m i l a r l y , s c i e n t i s t s are q u i t e capable of producing ex-te n s i v e data on a given environmental s i t u a t i o n under study, but they are unable to make accurate p r e d i c t i o n s or refuse to speculate about the long term e f f e c t s of disturbances to the ecosystem. In the past , d e c i s i o n s con-cerning the environment were u s u a l l y made without any d e l i b e r a t e attempt to assess and in c o r p o r a t e the u n c e r t a i n t y element i n t o the environmental d e c i s i o n making process. In l i g h t of the stakes i n v o l v e d , both i n the im-pact on the environment and the money spent on treatment f a c i l i t i e s , t h i s 3 un c e r t a i n t y should not be ignored, D e c i s i o n making under u n c e r t a i n t y i s a su b j e c t that has r e c e i v e d c o n s i d e r a b l e a t t e n t i o n i n the f i e l d of management science. Various s t r a t e g i e s that deal w i t h u n c e r t a i n t y i n management problems have been developed. These d e c i s i o n making aids have been a p p l i e d i n f o r e s t management, g e o l o g i c a l i n v e s t i g a t i o n s ( H a l t e r and Dean, 1971), water q u a l i t y management and corporate investment, D e c i s i o n makers have found that when ap p l y i n g a c e r t a i n technique to a problem, they must analyze i t more c l o s e l y and i n a more o b j e c t i v e manner than would otherwise be foll o w e d . Thus, these s t r a t e g i e s serve to e l u c i d a t e the problem as w e l l as a i d i n g i n the f i n a l d e c i s i o n . The o b j e c t i v e of t h i s research was to apply d e c i s i o n theory, copCt.j .uix^ uiiau L. U UCLJ.OXUU Ilicl. L^ _L11£ UL1LC1 Laxi lLJ , L.U f i e l d of water q u a l i t y management, C u l t u r a l e u t r o p h i c a t i o n of a lake poses a water q u a l i t y problem that i s p a r t i c u l a r l y s u i t a b l e f o r t h i s type of a n a l y s i s . The management problem of a c u l t u r a l l y e u t r o p h i c lake contains a considerable element of u n c e r t a i n t y and i n v o l v e s s p e c i a l i s t s from many d i s c i p l i n e s . In a d d i t i o n , the p u b l i c i s d i r e c t l y a f f e c t e d by the water q u a l i t y d e t e r i o r a t i o n and, should t h e r e f o r e , be given a r o l e i n the d e c i s i o n making process. Many of the p o s s i b l e remedial a c t i o n s f o r e u t r o p h i c lakes are s t i l l i n the development stage and have yet to be t e c h n i c a l l y proven. In c e r t a i n s i t u a t i o n s , remedial a c t i o n can be p o l i t i c a l l y u n d e s i r a b l e even though i t might be t e c h n i c a l l y sound; As an i n c r e a s i n g number of lakes be-come n o t i c a b l y a f f e c t e d by c u l t u r a l e u t r o p h i c a t i o n , i t i s important t h a t s t r a t e g i e s be developed to deal w i t h the problem. 4 This t h e s i s describes a technique, based on d e c i s i o n theory, to a i d i n the d e c i s i o n making i n v o l v e d i n m a i n t a i n i n g a d e s i r a b l e water q u a l i t y i n a lake that i s becoming c u l t u r a l l y e u t r o p h i c . The manage-ment s o l u t i o n i n the past c o n s i s t e d of one d e c i s i o n that s e l e c t e d the "best" a l t e r n a t i v e among a s e r i e s of p o s s i b l e s o l u t i o n s . This t h e s i s approaches the problem as a s e r i e s of d e c i s i o n s over a long term pl a n n i n g p e r i o d . A growing community d i s c h a r g i n g e f f l u e n t i n t o a l a k e causes a d e t r i m e n t a l e f f e c t on the water q u a l i t y of the dake, and t h i s e f f e c t i n -creases w i t h time. A framework i s provided i n t h i s t h e s i s , that allows planners to manage the water q u a l i t y w i t h a sound understanding of the problem and to u t i l i z e most e f f i c i e n t l y the a v a i l a b l e e x p e r t i s e . The problems a s s o c i a t e d w i t h managing a l a k e that i s becoming euL.i-upuj_c a i c uxscuat>eci i i i owapter XX, Cliu.pLc.ir XXX z.z z. b r i e f 2xpl"r._i • t i o n of Bayesian d e c i s i o n theory which i s used as a b a s i s f o r the a l g o -r i t h m developed i n t h i s t h e s i s , Skaha Lake i s used as a case study f o r purposes of developing the model, Chapters IV and VI e x p l a i n the •: f o r m u l a t i o n of the model and Chapter V I I contains a d i s c u s s i o n of the technique and i t s i m p l i c a t i o n s f o r the management of Skaha Lake, CHAPTER I I • EUTROPHICATION AS A WATER QUALITY MANAGEMENT PROBLEM A. EUTROPHICATION AS A NATURAL PHENOMENA The concept of n a t u r a l e u t r o p h i c a t i o n r e f e r s to the aging pro-cess that occurs i n a l l l a k e s . In b r i e f , a lak e begins i n a s t a t e of low f e r t i l i t y or o l i g o t r o p h i c s t a t e . I t g r a d u a l l y increases i n f e r t i l i t y which i n turn s t i m u l a t e s b i o l o g i c a l p roduction. The lake passes through a mesotrophic s t a t e or intermediate l e v e l and f i n a l l y to a eut r o p h i c or n u t r i e n t r i c h s t a t e . This f i n a l phase of the aging process i s one of very high b i o l o g i c a l production but low species d i v e r s i t y . The major causes of the aging process are accumulation of n u t r i e n t s i n the lake brought i n by runoff and volume shrinkage due to sedimentation, the former being the most important. These changes a f f e c t the complex n u t r i -ent c y c l e s and hence, the b i o l o g i c a l organisms which r e q u i r e the n u t r i e n t s . I f t h i s a c t i o n continues unchecked f o r thousands of years, the l a k e f i l l s i n and becomes a marsh. Because t h i s phenomena i s a n a t u r a l , i r r e v e r s i b l e process which occurs g r a d u a l l y over many c e n t u r i e s , i t cannot r e a l l y be considered a water q u a l i t y problem. I t would r e q u i r e a p e r i o d of a t l e a s t f i f t y years to observe any detectable changes caused by n a t u r a l e u t r o p h i c a t i o n i n most l a k e s . (Lee, 1970). 5 6 B. CULTURAL EUTROPHICATION The c u l t u r a l e u t r o p h i c a t i o n of a l a k e , although s u p e r f i c i a l l y resembling an ac c e l e r a t e d aging process, i s a d i s t i n c t l y d i f f e r e n t pheno-menon. A c u l t u r a l l y eutrophic lake i s one that has been a r t i f i c i a l l y f e r t i l i z e d by the presence of man. An o l l g o t r o p h i c l a k e can become e u t r o -phic i n the p e r i o d of three to four years i f s u p p l i e d w i t h a s u f f i c i e n t amount of n u t r i e n t s . However, the process u s u a l l y occurs over a ten to twenty year p e r i o d and i s not e a s i l y d e t e c t a b l e i n the e a r l y stages. C u l t u r a l e u t r o p h i c a t i o n d i f f e r s p r i m a r i l y from i t s n a t u r a l counter-p a r t i n that i t i s a process that can be c o n t r o l l e d . A c u l t u r a l l y e u t r o -phic l a k e can be re s t o r e d to i t s n a t u r a l s t a t e . Milway (1968) f e l t that t h i s d i f f e r e n c e should be emphasized because people tend to be a p a t h e t i c about a s i t u a t i o n they see as i n e v i t a b l e . H e r e a f t e r i n t h i s t h e s i s , when r e f e r r i n g to a eutrophic l a k e or e u t r o p h i c a t i o n , c u l t u r a l e u t r o p h i c a t i o n i s i m p l i e d . E u t r o p h i c a t i o n as an environmental problem i s q u i t e common where populated centers have developed around freshwater l a k e s . (Edmondson, 1968). Human a c t i v i t y i n the watershed of a lak e gives r i s e to a number of n u t r i e n t sources which may add s u b s t a n t i a l l y to the e x i s t i n g n a t u r a l n u t r i e n t l o a d to the l a k e . Domestic sewage e f f l u e n t i s u s u a l l y one of the major c u l t u r a l sour-ces (Lee, 1970) and, because of i t s h i g h c o n c e n t r a t i o n of n u t r i e n t s , i t i s capable of generating a gr e a t e r biomass than an equal amount of n u t r i e n t s i n a more d i l u t e form (Edmondson, 1970). A g r i c u l t u r a l r u n o f f and i n d u s t r i a l wastewaters are a l s o a s i g n i f i c a n t source of n u t r i e n t s (Vollenweider, 1968). C. THE EFFECTS OF EUTROPHICATION As a lak e becomes f e r t i l i z e d , i t manifests a p r o g r e s s i v e decrease i n 7 water q u a l i t y . The degree of e u t r o p h i c a t i o n i s described l o o s e l y by t r o -phic, s t a t e s which r e f e r to the l e v e l and the nature of the b i o t i c a c t i v i -ty i n the lake . A v a r i e t y of nuisance c o n d i t i o n s a r i s e as the t r o p h i c s t a t e i n a lak e increases and these c o n d i t i o n s become much more pronounced i n l a t e r stages. E u t r o p h i c a t i o n r e s u l t s i n a reduced water q u a l i t y from the stand-p o i n t of r e c r e a t i o n a l , i n d u s t r i a l and domestic use. The f i r s t stages of e u t r o p h i c a t i o n cause a s u b s t a n t i a l i n c r e a s e i n the a l g a l p o p u l a t i o n r e s u l t -i n g i n a decrease i n water c l a r i t y . The l i t t o r a l zone, which i s the shallow region of a lake c l o s e to the s h o r e l i n e , begins to develop weed growths. The i n i t i a l e f f e c t s of e u t r o p h i c a t i o n are b e n e f i c i a l to the f i s h l i f e i n a lake, w i t h increased tintriftnt. supnl.v l e a d i n g to Larger - f i s h . As the t r o p h i c s t a t e i n c r e a s e s , the more favourable species begin to de-crease and more t o l e r a n t , l e s s d e s i r a b l e coarse species t h r i v e . Thus, salmon and t r o u t disappear as suckers and carp become more common. At the most advanced stages of e u t r o p h i c a t i o n , the lower l a y e r of a l a k e (hypo-limnion) i s depleted of oxygen and a l l species of f i s h die o f f . Diatoms and other p l a n k t o n i c algae f l o u r i s h as the n u t r i e n t l e v e l i n c r e a s e s . There i s a p o p u l a t i o n s h i f t to the more t o l e r a n t s p e c i e s , notably blue-green algae, which form massive a l g a l blooms during the e a r l y and l a t e weeks of the summer. Blooms begin to decompose a f t e r one or two weeks, causing unpleasant odors and consuming d i s s o l v e d oxygen i n the water. Advanced e u t r o p h i c a t i o n can l e a d to a r i s e i n i n s e c t p o p u l a t i o n and bathers i n h i g h l y e u t r o p h i c lakes may experience s k i n i r r i t a t i o n s (Mack-enthum et al. , 1964), These stages render a la k e v i r t u a l l y useless f o r 8 r e c r e a t i o n a l a c t i v i t i e s . E u trophic waters present s e r i o u s problems f o r domestic and indus-t r i a l users. The high c o n c e n t r a t i o n of algae i n the water creates s e v e r a l complications i n water treatment p l a n t s . T u r b i d i t y , caused by the algae, must be removed. This i s u s u a l l y done by c o a g u l a t i o n and f i l t r a t i o n . An excessive amount of algae causes f i l t e r c l o g g i n g to occur more f r e q u e n t l y , thus i n c r e a s i n g the operating cost of the water treatment p l a n t . The c h l o r i n e demand i n eutrophic waters i s increased c o n s i d e r a b l y by the high concentra-t i o n of organics. The presence of algae i n a water supply could p r o t e c t a pathogen from o x i d a t i o n which could l e a d to s e r i o u s d i s i n f e c t i o n problems (Lee, 1970). The t a s t e , odor and c o l o r of eu t r o p h i c water make i t aesthe-t i c a l l y very undesirable and they are most d i f f i c u l t to remove. AjjiJ.cuj.Luj.aI use i s ate aiSo s e i i o Usxy Hampered ay . eu-Lj.'opiiJ.o<i'CxOii, The blue-green algae may excrete h i g h l y t o x i c substances. In c e r t a i n cases t h i s has proven l e t h a l to l i v e s t o c k , waterfowl and other domestic animals. (Lee, 1970; Mackenthum et at. 1964), The t u r b i d water a l s o causes weed growth i n i r r i g a t i o n canals which reduces the t r a n s m i s s i o n of water. D. CONTROL OF EUTROPHICATION Although a great deal of research has been devoted to the study of l a k e e u t r o p h i c a t i o n , s c i e n t i s t s are unable to f u l l y e x p l a i n the process or a c c u r a t e l y p r e d i c t the behavior of a l a k e that i s being f e r t i l i z e d . A b a s i c concept of e u t r o p h i c a t i o n i s that the b i o l o g i c a l a c t i v i t y of a la k e i s not a f u n c t i o n of j u s t the water, but of the e n t i r e l a k e drainage and sediment b a s i n , Therefore, each l a k e has i t s own set of c h a r a c t e r i s t i c s and presents 9 a unique e u t r o p h i c a t i o n problem, Consequently, i t i s very d i f f i c u l t f o r s c i e n t i s t s to draw d e t a i l e d i n f o r m a t i o n obtained from one l a k e and apply i t d i r e c t l y to other l a k e s . V a r i a t i o n s i n the t r o p h i c s t a t e of a l a k e are determined by numerous b i o l o g i c a l , chemical and morphological f a c t o r s . A recent i n t e r -n a t i o n a l symposium (Milway, 1968) considered the f o l l o w i n g parameters most s i g n i f i c a n t . 1. S i z e , shape and l o c a l e of the l a k e b a s i n , w i t h an emphasis on depth of the l a k e . 2. Annual input of n u t r i e n t s , e s p e c i a l l y n i t r o g e n and phosphorus; s p r i n g c o n c e n t r a t i o n of n u t r i e n t s . 3. Thermal s t r a t i f i c a t i o n . 4. Rate of v o l u m e t r i c water change. "". . "loHjniprit- w?tf T t^y, cy* p-f* tT! "7 f> V* ? 1.5" 6. P e r i o d of i c e cover. The l i m i t i n g n u t r i e n t theory i s the commonly accepted e x p l a n a t i o n of the c o n t r o l l i n g mechanism of the process. The biomass i n a l a k e r e q u i r e numerous elements to s u s t a i n p r o d u c t i v i t y , According to the l i m i t i n g n u t r i -ent theory, one or p o s s i b l y two of the e s s e n t i a l elements i s present i n the l a k e i n a l i m i t i n g amount and thus c o n t r o l s the r a t e of growth of the b i o -mass. When an unnatural source s u p p l i e s an a d d i t i o n a l amount of the l i m i t -i n g n u t r i e n t , the b i o l o g i c a l organisms respond by i n c r e a s i n g the p r o d u c t i v i t y l e v e l u n t i l an e s s e n t i a l n u t r i e n t again becomes l i m i t i n g . This jump i n pro-d u c t i v i t y corresponds to an i n c r e a s e i n t r o p h i c s t a t e . Studies of v a r i o u s lakes have shown that a number of d i f f e r e n t elements may be l i m i t i n g b i o t i c growth. However, n i t r o g e n and phosphorus 10 are b e l i e v e d to be the most important elements i n n e a r l y a l l the lakes that have become eutrophi c (Hutchinson, 1969). The importance of phosphorus as the l i m i t i n g n u t r i e n t i n Lake Washington was demonstrated very c l e a r l y by Edmondsen (1970). A f t e r the sewage was d i v e r t e d from the l a k e , the b i o l o g i c a l a c t i v i t y dropped from a hi g h s t a t e of eutrophy to l e v e l s s i m i l a r to those that had e x i s t e d t h i r t y years e a r l i e r . In t h i s i n s t a n c e , the l a k e c o n c e n t r a t i o n of c h l o r o p h y l l decreased c l o s e l y i n accordance w i t h the phosphate c o n c e n t r a t i o n . Chloro-p h y l l c o n c e n t r a t i o n i s a parameter that estimates the a l g a l biomass. These r e s u l t s are s t r o n g evidence f o r phosphorus as the l i m i t i n g n u t r i e n t . S u b s t a n t i a l progress has been made i n a n a l y z i n g lakes f o r t h e i r p o t e n t i a l to become eutroph i c . Vollenweider (1968) has suggested that the nutrient: l o a d i n g race i s a s i g n i f i c a n t parameter, and compared the t t o p l i i c s t a t u s of v a r i o u s lakes by p l o t t i n g the annual l o a d i n g on an a r e a l b a s i s against the mean depth f o r s e v e r a l documented l a k e s . In another study, Vollenweider (1973) has shown how water r e t e n t i o n time and depth of a lake determine how s u s c e p t i b l e a lake i s to n u t r i e n t accumulation. Vollenweider's model .suggests a c r i t i c a l l o a d i n g l i m i t determined by the hydro-morphology of a l a k e , I f t h i s c r i t i c a l l i m i t i s exceeded, the b i o l o g i c a l growth rat e becomes unstable and the l a k e runs a h i g h r i s k of becoming e u t r o p h i c . In terms of managing the water q u a l i t y i n a l a k e , the models a v a i l a b l e now are r e l a t i v e l y crude but they do g i v e an i n d i c a t i o n of the order of magnitude f o r t o l e r a b l e and dangerous n u t r i e n t l o a d i n g l e v e l s . E, EUTROPHICATION AS A LAKE MANAGEMENT PROBLEM The c o n t r o l of e u t r o p h i c a t i o n f o r environmental reasons i s a complex 11 management problem. The main c o n s i d e r a t i o n s that must be d e a l t w i t h can be l o o s e l y d i v i d e d i n t o two c a t e g o r i e s : t e c h n o l o g i c a l and s o c i o l o g i c a l . The t e c h n o l o g i c a l problems stem from the f a c t that the r e a c t i o n s i n v o l v e d i n the e u t r o p h i c a t i o n of a l a k e are not w e l l understood. I t i s known that f e r t i l i z i n g a l a k e causes an i n c r e a s e i n the t r o p h i c s t a t e , but the nature and magnitude of the response i s u s u a l l y unknown u n t i l i t occurs. The problem i s a l s o complicated by the l a c k of one u n i v e r s a l c r i t e r i o n f o r a s s e s s i n g the t r o p h i c l e v e l i n a l a k e . This makes i t r a t h e r d i f f i c u l t to draw comparisons between d i f f e r e n t l a k e s . The f a c t that each lake's response i s determined to a great extent by the s p e c i f i c c h a r a c t e r i s t i c s of that par-t i c u l a r l a k e , increases t h i s d i f f i c u l t y . Another c o m p l i c a t i o n i n the lake management problem i s that of e s t i -mS Tr.„_!.£ "".I;"?. — .-.^ .-w-, J . . . L i i c o_4. i _V*t - - \--*-^>IL ^ J L w s e w a g e e f f l u e n t d i s c h a r g i n g i n t o a l a k e , determination of the n u t r i e n t l o a d i n g i s f a i r l y s t r a i g h t f o r w a r d . In some circumstances t h i s would account f o r the major p o r t i o n of the lake's n u t r i e n t budget. However, a g r i c u l t u r a l , urban and i n d u s t r i a l runoff a l s o c o n t r i b u t e s i g n i f i c a n t l y . The l o a d i n g s from these sources must be determined by i n d i r e c t methods which are q u i t e ques-t i o n a b l e i n accuracy (Te c h n i c a l Supplement IV, 1974), N a t u r a l n u t r i e n t sources such as r a i n f a l l , f a l l o u t and groundwater flow may a l s o have a s i g n i f i c a n t e f f e c t on the n u t r i e n t budget and are q u i t e d i f f i c u l t to d e t e r -mine a c c u r a t e l y . When combining these three aspects of the problem: the i n a c c u r a c i e s i n q u a n t i f y i n g the n u t r i e n t l o a d i n g , the i n a b i l i t y to a c c u r a t e l y assess the l a k e response and the l a c k of a standard to evaluate the t r o p h i c s t a t e of a 12 l a k e , i t becomes apparent that there i s a c o n s i d e r a b l e amount of uncer-t a i n t y in. the technology of the problem. This u n c e r t a i n t y i s a c r u c i a l i s s u e of the management r o l e and nrnst be taken i n t o account i n the d e c i s i o n making process. The s o c i o l o g i c a l c o n s i d e r a t i o n that poses d e f i n i t e problems i s the p a r t i c i p a t i o n of the p u b l i c i n the d e c i s i o n making process. The e x i s t -ance of p u b l i c involvement i n water resource management i s not new. There are c e r t a i n procedures a v a i l a b l e which a l l o w the community to play some part i n s e t t i n g p o l i c y f o r p u b l i c works. However, environmental problems d i f f e r s i g n i f i c a n t l y from other p u b l i c work i s s u e s because the p e r c e i v e d s e v e r i t y of the problem i s l a r g e l y dependent on personal p o i n t s of view. I t i s q u i t e u n l i k e l y that the p u b l i c would a l l agree on an i s s u e as vague ;mH complex <?'.'t.ronhi.potior1 • Managing an i n c i p i e n t e u t r o p h i c a t i o n problem i n a l a k e c o n s i s t s p r i m a r i l y of d e c i d i n g when and what type of c o n t r o l measure to implement. A c c e l e r a t e d e u t r o p h i c a t i o n i s caused only when an i n c r e a s i n g amount of n u t r i e n t s i s discharged i n t o the l a k e over a r e l a t i v e l y s h o r t p e r i o d of time. Although some b i o l o g i c a l organisms respond r e a d i l y to an i n c r e a s e i n a l i m i t i n g n u t r i e n t , the e f f e c t s on the l a k e are not immediately apparent. A s u b s t a n t i a l increase i n the b i o l o g i c a l a c t i v i t y must occur before the p u b l i c w i l l n o t i c e a change. In the i n i t i a l stages of e u t r o p h i c a t i o n , a l g a l blooms might only appear once during the summer. A l g a l blooms are very dependent upon sunshine which s t i m u l a t e s the p h o t o s y n t h e t i c r e a c t i o n s . There may be a p e r i o d of one or two years during which the t r o p h i c s t a t e i s i n c r e a s i n g , but the lake might not experience any a l g a l blooms because of weather or h y d r o l o g i c a l c o n d i t i o n s . This tends to reduce the s e v e r i t y of 13 the problem i n terms of public, o p i n i o n , even though c o n d i t i o n s might be d e t e r i o r a t i n g . An example of t h i s i s c i t e d by O'Riordan (1972) where the un-usual c l a r i t y of Skaha Lake at the time of a survey r e s u l t e d i n favour-able p u b l i c o p i n i o n on the s i t u a t i o n . The f a c t that a l g a l blooms had occurred on the l a k e during the previous two summers d i d not seem to i n -fluence the p o l l . In such a s i t u a t i o n , people are r e l u c t a n t to a l l o c a t e funds to s o l v e a problem that does not appear s e r i o u s . As the t r o p h i c s t a t e of a l a k e i n c r e a s e s w i t h n u t r i e n t l o a d i n g , the c o n c e n t r a t i o n of organic matter i n the water a l s o i n c r e a s e s . M i c r o -organisms, feeding on t h i s m a t e r i a l , consume oxygen i n the process arid g r a d u a l l y deplete the d i s s o l v e d oxygen from the water. This i s not too pfa^ -fon.c; f o r t":Vip imn^T" ipypt- o-f a I P V D T.I]I'O"I ,.ir"'1erg'?e~" some r.citur.T.1 c i c m t i o i i from the atmosphere. However, i n the summer months when a l a k e may be t h e r -mally s t r a t i f i e d , the lower l a y e r or hypolimnion does not r e c e i v e a c o n t i n -uous supply of oxygen. Consequently, high l e v e l s of m i c r o b i o l o g i c a l a c t i v -i t y o c c u r r i n g i n the advanced stages of e u t r o p h i c a t i o n e v e n t u a l l y cause c e r -t a i n regions of the hypolimnion to become anaerobic. This p o i n t i s very c r i t i c a l i n the e u t r o p h i c a t i o n of a l a k e because i t i s a precursor of a d r a s t i c d e c l i n e i n water q u a l i t y . Normally a l a r g e segment of the phosphorus load that enters a l a k e f a l l s to the bottom and gets trapped i n the sediment. This p o r t i o n , v a r y -i n g between 50 and 70 per cent of the t o t a l annual phosphorus load i s not a v a i l a b l e to the biomass i n the lake'. However, when-the hypolimnion i s anaer-o b i c , i t a f f e c t s the o x i d a t i o n - r e d u c t i o n - p o t e n t i a l (ORP) of the water such that phosphorus i n the bottom sediment r e d i s s o l v e s i n t o the water. At 14 t h i s stage the t o t a l annual phosphorus load to the l a k e as w e l l as some phosphorus from the sediment becomes a v a i l a b l e to the biomass. The pre-sence of t h i s a d d i t i o n a l amount of n u t r i e n t i n a phosphorus l i m i t i n g l a k e causes a very r a p i d i n c r e a s e i n the b i o l o g i c a l a c t i v i t y b r i n g i n g about a sharp decrease i n water q u a l i t y . Consequently, when managing a la k e b a s i n , the onset of anaerobic c o n d i t i o n s i n the hypolimnion i s considered c a t a s t r o p h i c . The s c i e n t i s t s can only crudely estimate what l o a d i n g w i l l p r e c i p i t a t e t h i s c o n d i t i o n . This i l l u s t r a t e s the importance of the time f a c t o r when dec i d i n g to implement abatement measures. The d e c i s i o n maker must be r e c e p t i v e to the o p i n i o n of the p u b l i c and a l s o be aware of the r i s k s of delay i n ta k i n g s u f f i c i e n t a c t i o n . rni^ _ „ _ 4_T 1 . . ,„ J! *- 1_ ,J - A- J - 1 _ 1_ ~ V. - -1 - - - - "» ',- —, • ~r~\ by Vollenweider (1973) are u s e f u l to the s c i e n t i s t i n understanding the theory. However, the d e c i s i o n maker who i s r e s p o n s i b l e f o r m a i n t a i n i n g an acceptable lake environment, r e q u i r e s a p r a c t i c a l l a k e management te c h -nique that i s more s e n s i t i v e to those c o n s i d e r a t i o n s discussed i n t h i s chapter.. The type of management problem t h a t i s considered i n t h i s t h e s i s i s that of a m u n i c i p a l i t y d i s c h a r g i n g i t s sewage i n t o a l a k e . The planner or d e c i s i o n maker i s faced w i t h a d e t e r i o r a t i n g water q u a l i t y problem. Once the d e c i s i o n i s made to take a c t i o n , there are s e v e r a l a l t e r n a t i v e s to be considered. One obvious s o l u t i o n i s to remove the o f f e n d i n g n u t r i e n t , most l i k e l y phosphorus, from the e f f l u e n t . At the present time there are a great many d i f f e r e n t phosphorus removal techniques a v a i l a b l e on the market. In a d d i t i o n there are a l s o designs i n the development stage which w i l l be-15 come a v a i l a b l e i n a short p e r i o d of time. A number of the a v a i l a b l e ad-vanced waste treatment techniques can be immediately d i s q u a l i f i e d from c o n s i d e r a t i o n because of c o s t s , design c o n s t r a i n t s , a v a i l a b i l i t y , e t c . However there would s t i l l be s e v e r a l a l t e r n a t i v e s from which too choose. The performance s p e c i f i c a t i o n s of a p a r t i c u l a r treatment method are sup-p l i e d r e a d i l y by engineers and manufacturers. However, consi d e r a b l e doubt e x i s t s on how the equipment w i l l a c t u a l l y perform, and of even gre a t e r u n c e r t a i n t y i s how competently w i l l the p l a n t be operated. Most engineers admit that sewage treatment p l a n t design i s not an exact s c i e n c e . Thus, the d e c i s i o n maker must deal w i t h t h i s a d d i t i o n a l area of u n c e r t a i n t y when con s i d e r i n g various a l t e r n a t i v e s . In order to place the management problem i n a s u i t a b l e perspec-t i v e - t h i s - t h e s i s wi 1.1. -f nriis nn the RHfnaHnn defined • 3i? f o l l o w ^ - A Tnimn o i — p a l i t y i s l o c a t e d i n the watershed of a r e l a t i v e l y l a r g e l a k e . The pre-sence of t h i s p o p u l a t i o n and i n p a r t i c u l a r , the m u n i c i p a l sewage e f f l u e n t , i s causing the t r o p h i c s t a t e of the l a k e to i n c r e a s e . For the purpose of developing the d e c i s i o n theory a l g o r i t h m , i t i s assumed that one or more persons are given the r e s p o n s i b i l i t y of managing the l a k e and hence the a u t h o r i t y to make or suggest the f i n a l d e c i s i o n s . The s i t u a t i o n faced by these people (the d e c i s i o n makers) i s that of a l a k e decreasing i n water q u a l i t y i n an u n c e r t a i n manner. Planning horizons i n the range of 40 to 50 years are being considered. Although the p o s s i b l e consequences are known, i t i s not c e r t a i n at what p o i n t i n time they w i l l occur. To minimize the r i s k of the l a k e d e t e r i o r a t i n g dras-t i c a l l y , the planners must consider implementation of abatement measures, The d e c i s i o n of when and what course of a c t i o n to take i s d i r e c t l y r e l a t e d 16 to the r i s k s and u n c e r t a i n t i e s i n v o l v e d . The d e c i s i o n maker's a t t i t u d e to r i s k i s a s i g n i f i c a n t aspect of the problem. In a d d i t i o n to the imme-d i a t e s i t u a t i o n , these people must consider the long-term events that w i l l occur i n the b a s i n and t h e i r e f f e c t on the Jake. This t h e s i s describes the development of an a l g o r i t h m that attempts to address the problems of d e c i s i o n making under u n c e r t a i n t y , c o - o r d i n a t i n g the v a r i o u s d i s c i p l i n e s and the long term planning f o r the management of water q u a l i t y i n a c u l -t u r a l l y eutrophic l a k e . . CHAPTER I I I DECISION MAKING UNDER UNCERTAINTY D e c i s i o n theory r e f e r s to the area o f management sc i e n c e t h a t i s concerned w i t h the d e c i s i o n making process. Various mathematical and s t a t i s t i c a l techniques have been formulated to d e a l w i t h s p e c i f i c types o f d e c i s i o n making problems. There i s u s u a l l y some c h a r a c t e r i s t i c e l e -ment i n the s t r u c t u r e of a problem that suggests a p a r t i c u l a r method or alg o r i t h m . The management problem of a lak e that i s becoming e u t r o p h i c , as described i n the previous chapter i s d i s t i n g u i s h e d by the element of u n c e r t a i n t y . Where tnere i s u n c e r t a i n t y i n d e c i s i o n making, i i . i s u e c - s . s e - t o consider both the p r o b a b i l i t i e s of the va r i o u s p o s s i b l e outcomes and t h e i r i n d i v i d u a l d e s i r a b i l i t i e s . Bayesian d e c i s i o n theory i s a s t r a t e g y t h a t i s used i n management s i t u a t i o n s where u n c e r t a i n t y i s a s i g n i f i c a n t f a c t o r . I t allows the p r o b a b i l i t i e s and the outcomes to be d e a l t w i t h s e p a r a t e l y . In d e c i s i o n theory, u n c e r t a i n t y i s defined as the circumstance where the " s t a t e of n a t u r e " i s unknown and the p r o b a b i l i t i e s f o r the various s t a t e s are unknown. T r a n s l a t e d i n t o the context of a e u t r o p h i c l a k e , t h i s means th a t i f a c e r t a i n a c t i o n i s taken towards ab a t i n g the problem, the f i n a l outcome might be one o f s e v e r a l p o s s i b l e s t a t e s . There are no w e l l defined p r o b a b i l i t i e s a s s o c i a t e d w i t h the occurrence of any one o f these s t a t e s . However, Bayesian d e c i s i o n theory permits the i n c o r p o r a t i o n of s u b j e c t i v e estimates of "undefined" p r o b a b i l i t i e s i n t o the d e c i s i o n making 17 18 process. An event such as lake e u t r o p h i c a t i o n must be considered non-r e p e t i t i v e f o r purposes of d e r i v i n g s t a t i s t i c a l d i s t r i b u t i o n s because of the time p e r i o d i n v o l v e d and the d e s t r u c t i v e nature of the event. Consequently, p r o b a b i l i t i e s can only be obtained by s u b j e c t i v e e v a l u a t i o n by experts. Under these circumstances, Bayesian d e c i s i o n theory provides a p o s s i b l e b a s i s f o r developing a management technique to dea l w i t h l a k e e u t r o p h i c a t i o n problems. A. UTILITY THEORY The d e c i s i o n making process c o n s i s t s of va r i o u s components that describe a p a r t i c u l a r s i t u a t i o n . For any given a l t e r n a t i v e , there i s as s o c i a t e d w i t h i t a " d e c i s i o n system" ( C a s t l e s et at., 1971) comprised o f : .... .... A l t e r n a t i v e > Outcome -> P r o b a b i l i t y Value .,, Bayesian d e c i s i o n theory uses a r a t i o n a l way of combining these components to a r r i v e at the optimal a l t e r n a t i v e . An o b j e c t i v e e v a l u a t i o n of each a l t e r n a t i v e r e q u i r e s that a l l p o s s i b l e outcomes be measured on the same "value s c a l e . " U t i l i t y theory provides t h i s common value s c a l e i n the form of a U t i l i t y Index. This i n -dex can measure a b s t r a c t q u a n t i t i e s such as " a t t i t u d e to r i s k " or " t r u e personal v a l u e " on a s i n g l e commensurable s c a l e . A u t i l i t y f u n c t i o n i s a perso n a l assessment of a p a r t i c u l a r prob-lem by the d e c i s i o n maker. I t i s i n the form of a monotonically i n c r e a s i n g f u n c t i o n measured on a c a r d i n a l s c a l e . The u t i l i t y of a c e r t a i n outcome i n d i c a t e s the r e l a t i v e d e s i r a b i l i t y of that event to the d e c i s i o n maker. U t i l i t y theory has been a p p l i e d i n s e v e r a l d i f f e r e n t types of de-c i s i o n making s i t u a t i o n s . I t i s most o f t e n used as a conversion s c a l e of 19 monetary value to p e r s o n a l preference and i s p a r t i c u l a r l y s u i t a b l e f o r the d e c i s i o n maker who i s faced w i t h f i n a n c i a l d e c i s i o n s i n v o l v i n g r i s k . ( H a l t e r and Dean, 19/1). S o c i a l ' u t i l i t i e s have Been used to evaluate non-monetary b e n e f i t s such as s o c i a l w e l f a r e schemes o r importance of s c i e n t i f i c data (Edwards, 1972). In a l l cases, a u t i l i t y f u n c t i o n i s designed to i n -d i c a t e the r e l a t i v e value of the outcomes to one s p e c i f i c d e c i s i o n maker. The b a s i s f o r the theory of u t i l i t y , as w e l l as f o r the j o i n t theory of u t i l i t y and s u b j e c t i v e p r o b a b i l i t y , i s provided by s i x assumptions, known as the "axioms of coherence." These axioms are l i s t e d i n Appendix A. They define the r a t i o n a l a c t i o n s that a person would take i n the face of u n c e r t a i n -t y . . I f a person s a t i s f i e s these axioms, then a u t i l i t y f u n c t i o n , i n the form of a s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n , w i l l e x i s t f o r him. (Winkler, 1972). This u t i l i t y f u n c t i o n , a l s o known as a r i s k or u t i l i t y curve, i s a, r e f l e c -t i o n of that person's judgement about u n c e r t a i n events and h i s preference f o r v a r i o u s payoffs (monetary and nonmonetary) i n a d e c i s i o n making problem. A technique t h a t a p p l i e s u t i l i t y theory, o r i g i n a t e d by Von Neumann and Morgenstern (1953), i s used to d e r i v e the u t i l i t y f u n c t i o n of a d e c i s i o n maker. The method u t i l i z e s the f o l l o w i n g axioms. (Winkler, 1972). R p R2 and R^ are three d i f f e r e n t p a y o f f s . U ( R 1 ) , i s the u t i l i t y of payoff R^. (1) I f payoff R 1 i s p r e f e r r e d to payoff R^, then U(R 1) > UCRj); i f ^ i s p r e f e r r e d to R^ then U(R 1) < I K I ^ ) ; and i f n e i t h e r i s p r e f e r r e d to the other, then U(R 1) = U ( R 2 ) . (2) I f one i s i n d i f f e r e n t between: (a) r e c e i v i n g a c e r t a i n payoff 20 or (b) t a k i n g a bet (or l o t t e r y ) i n which one r e -ceives payoff R, w i t h p r o b a b i l i t y "P" or pay-o f f R 3 w i t h p r o b a b i l i t y "1-P", then U.(R2) = P x U(R 1) + ( l - P ) x U(R 3) (3.1) (3) I f R^ i s p r e f e r r e d to R 2 > and R^ i s p r e f e r r e d to R^, then R-^  must be p r e f e r r e d to R^. The l a s t statement i s the axiom of t r a n s i s t i v i t y . Adherence to t h i s axiom which i m p l i e s consistency i n p r e f e r e n c e s , becomes important when d e a l i n g w i t h nonmonetary p a y o f f s . Axiom 2 i s used t o d e r i v e the u t i l i t i e s of v a r i o u s p o s s i b l e outcomes provided t h a t the best and worst outcomes are d e f i n e d . These two outcomes are a r b i t r a r i l y assigned the maximum and minimum u t i l i t y . Then, u s i n g Equation (3.1), the u t i l i t y of any i n t e r m e d i a t e outcome can be d e r i v e d . T h i s i s e x p l a i n e d f u r t h e r i n the next chapter. The s t r a t e g y of Bayesian d e c i s i o n theory i s to choose the a l t e r n a t i v e that has the "maximum expected u t i l i t y . " T his value i s computed f o r each a l t e r n a t i v e by summing the products of the p r o b a b i l i t i e s times the u t i l i t y f o r a l l p o s s i b l e outcomes. Mathematically, t h i s i s : EU = E 0 x P. (3.2) a j, i I where EU a i s the expected u t i l i t y o f a l t e r n a t i v e "a"; i s the p r o b a b i l i t y of outcome " i " ; and u\ i s the u t i l i t y of outcome " i " . The p r o b a b i l i t i e s are derived from the most recent i n f o r m a t i o n a v a i l -able and i n mostcases are s u b j e c t i v e assessments by the e x p e r t s . The v a l i d i t y of d e f i n i n g p r o b a b i l i t i e s i n t h i s manner i s questioned by some s t a t i s t i c i a n s . 21 However, management d e c i s i o n s based on personal judgement are o f t e n made without attempting to define the r i s k s , e x p l i c i t l y . I f the sub-j e c t i v e opinions can be kept e x p l i c i t , t h e i r r o l e i n d e c i s i o n making can be o b j e c t i v e l y examined. The a s s e r t i o n that i t i s not always p o s s i b l e to assess p r o b a b i l i t i e s i n c e r t a i n s i t u a t i o n s i s another c r i t i c i s m of Bayesian method. Coyle (1972) r e j e c t s t h i s idea by s t a t i n g : " I f - t h e p r o b a b i l i t i e s cannot be assigned to some p a r t i c u l a r set of outcomes then t h i s means that you are i n a s i t u a t i o n where you are about to launch i n t o the f u t u r e w i t h no idea of what i s l i k e l y to happen. This i s bad managerial p r a c t i c e , not i m p r a c t i c a l managerial theory." By r e q u i r i n g p r o b a b i l i t y assessments regarding a l l p o s s i b l e outcomes, d e c i s i o n theory forces the experts to analyze the negtiLive aspects c f a s i t u a t i o n which are too o f t e n ignored. The a p p l i c a t i o n of Bayesian s t r a t e g y as a d e c i s i o n making t o o l i n the f i e l d of water q u a l i t y management r e q u i r e s some m o d i f i c a t i o n of the conventional methods i n order to keep the c a l c u l a t i o n s manageable. However, wherever p o s s i b l e , the standard procedures of Bayesian d e c i s i o n theory are used. CHAPTER IV DEFINING A UTILITY FUNCTION FOR A EUTROPHIC LAKE The c r i t e r i o n that u s u a l l y c a r r i e s the most weight i n p r o j e c t e v a l -uations i s the c o s t . Understandably, when ideas or designs appear to be economically u n a t t r a c t i v e , they are u s u a l l y d i s c a r d e d f o r more p r o f i t a b l e ones. However, i n many a p p l i c a t i o n s and e s p e c i a l l y i n environmental matters the d o l l a r value alone cannot adequately d e s c r i b e b e n e f i t s and I t provides an incomplete assessment of the t o t a l worth of a p r o j e c t . I t i s n a i v e to expect concepts such as " p r e s e r v a t i o n of the n a t u r a l environment" or " q u a l -i t y of l i f e " to be d e s e r i b a b l e i n economic terms. What i s needed i s a d d i t i o n a l c r i t e r i a that evaluates environmental concerns i n a noneconomic l i g h t . U t i l i t y theory provides a technique i n which a b s t r a c t q u a n t i t i e s can be measured by a d e c i s i o n maker according to h i s p e r s o n a l a t t i t u d e s . Although t h i s i s somewhat a r b i t r a r y , i t does s h i f t the emphasis away from the economics, so that n o n p r o f i t a b l e but environmen-t a l l y sound ideas r e c e i v e more a t t e n t i o n . Furthermore, a u t i l i t y measure would not be intended to r e p l a c e economic e v a l u a t i o n s , only to provide a wider scope to the a n a l y s i s . The most common a p p l i c a t i o n of a u t i l i t y f u n c t i o n i s the conver-s i o n of monetary sums i n t o t h e i r t r u e r e l a t i v e v a l u e s w i t h respect t o one p a r t i c u l a r d e c i s i o n maker. ( H a l t e r and Dean, 1971). However, the p o t e n t i a l a p p l i c a t i o n of u t i l i t y theory extends as w e l l t o nonmonetary a p p l i c a t i o n s . Edwards (1972) d e f i n e s a procedural method f o r d e r i v i n g u t i l i t i e s f o r non-monetary q u a l i t a t i v e type c r i t e r i a , which he r e f e r s to as s o c i a l u t i l i t i e s . 22 23 P a t t a n a i k (1968) concludes thai: the use of " s o c i a l u t i l i t i e s " i s v a l i d f o r e v a l u a t i n g s o c i a l preferences and upholds the v a l i d i t y of u s i n g von Neumann-Morgenstem u t i l i t y indexes. The concept of an environmental u t i l i t y index i s a l o g i c a l p r o g r e s s i o n from s o c i a l u t i l i t i e s and has a n a t u r a l a p p l i c a t i o n i n environmental problems. A. DERIVING THE UTILITY FUNCTION When d e r i v i n g a u t i l i t y f u n c t i o n , i t i s e s s e n t i a l to i d e n t i f y : 1. The o r g a n i z a t i o n whose u t i l i t i e s are to be maximized. 2. The i s s u e or i s s u e s to which the u t i l i t i e s needed are r e l e v a n t . The i d e n t i f i c a t i o n of these two e n t i t i e s determines the o v e r a l l approach to i-1 — J - . - I J - l — A.1. j _ .. - JT J . _ J . J.,., •., »-7, •- ,— . • ? • > -1 -->.,....'.-.•••--In the context of t h i s t h e s i s , the o r g a n i z a t i o n under c o n s i d e r a t i o n i s the general p o p u l a t i o n l i v i n g w i t h i n the l a k e b a s i n or i n p r o x i m i t y to the b a s i n . This r e f e r s to the people who enjoy the b e n e f i t s of the lak e and/or whose a c t i v i t i e s are having a d e t r i m e n t a l e f f e c t on the l a k e . I t i s t h e i r q u a l i t y o f l i f e t h a t i s r e a l l y being considered i n the management of the l a k e . This p o p u l a t i o n i s made up of many s m a l l groups, a l l of whom may have d i f f e r e n t ideas and sentiments concerning the environmental q u a l i t y of the l a k e . I t would be f u t i l e to d e f i n e a u t i l i t y curve f o r each group, as i t would only complicate the problem. A simp l e r approach i s to con s i d e r the pe r -son or persons, presumably i n the l o c a l government, t h a t are u l t i m a t e l y r e s p o n s i b l e f o r managing the l a k e . T h e i r a t t i t u d e s are the most r e l e v a n t when c o n s i d e r i n g the r i s k s and u n c e r t a i n t y i n v o l v e d i n the management prob-24 lem. Therefore, these a d m i n i s t r a t o r s or p o l i t i c i a n s w i l l be regarded as the d e c i s i o n makers and the u t i l i t y f u n c t i o n w i l l r e f l e c t t h e i r judgement. The i s s u e to which the u t i l i t y curve i s r e l e v a n t i s the change i n water q u a l i t y o f the l a k e , s p e c i f i c a l l y , the e u t r o p h i c a t i o n of the l a k e . This i s the i s s u e t h a t i s causing a decrease i n the b e n e f i t s d e r i v e d from the l a k e . For the purpose o f s i m p l i f y i n g the for m u l a t i o n of the a l g o r i t h m , not a l l the l a k e b e n e f i t s are to be considered. The assumption i s made that the damage to the aquatic environment and l o s s of r e c r e a t i o n a l use are of foremost concern. Hence, they p r e v a i l over o t h e r i s s u e s such as l o s s o f i n d u s t r i a l or a g r i c u l t u r a l use. This assumption can l a t e r be changed to broaden the scope of the problem. The u t i l i t y f u n c t i o n , i n t h i s problem, i s to convert the v a r i o u s l e v e l s or b i o i o g i c a i a c t i v i t y i n tne laKe i n t o a u t i l i t y Index. Before t h i s can be done, a s c a l e i s needed th a t d e f i n e s the range of t r o p h i c l e v e l s t h a t might e x i s t i n the l a k e . B. DEFINING A TROPHIC STATE SCALE C l a s s i f y i n g the v a r i o u s stages that a n u t r i e n t enriched l a k e goes through i s a major area o f concern i n limnology. Present research i n t h i s f i e l d i s devoted mainly to assessing the trophic> s t a t e by mo n i t o r i n g a v a r i e t y of b i o l o g i c a l and chemical parameters. These parameters are c o r r e -l a t e d w i t h the broad lake c l a s s i f i c a t i o n s of o l i g o t r o p h y , mesotrophy and eutrophy. However, the magnitude of any one parameter can vary over a con-s i d e r a b l e range when observed i n s e v e r a l lakes that appear to be at the same t r o p h i c s t a t e . (Vollenweider, 1968). Thus, the c o r r e l a t i o n s can be r a t h e r vague or undefined. Brezonick and Shannon (1972) developed a c l a s s i f i c a t i o n 25 method by combining seven d i f f e r e n t l a k e parameters to form a t r o p h i c s t a t e index and attempted to i d e n t i f y the most s i g n i f i c a n t parameters through s t a t i s t i c a l c o r r e l a t i o n s . Although u s e f u l to the s c i e n t i s t s i n under-standing e u t r o p h i c a t i o n , most t e c h n i c a l c l a s s i f i c a t i o n s methods have l i t t l e meaning to the d e c i s i o n makers. Therefore, when d e f i n i n g a range of l a k e t r o p h i c l e v e l s that are to be used p r i m a r i l y by the d e c i s i o n makers, the d e f i n i t i o n s should be i n terms that are meaningful to them. E u t r o p h i c a t i o n i s a complex dynamic process that does not occur i n d i s c r e t e steps. Therefore, to c l a s s i f y i t i n t h i s manner i s somewhat a r t i f i c i a l . However, f o r purposes of c l a r i t y and i n order to develop a d e c i s i o n making technique, a d i s c r e t e s c a l e was constructed. Table 4,1 l i s t s a s c a l e w i t h seven d i f f e r e n t d e s c r i p t i o n s c o r r e s -oonding to seven la k e troDhic s t a t e s . The s c a l e i s designed to cover the e n t i r e range of b i o t i c a c t i v i t y that a l a k e w i l l e x h i b i t ; e i t h e r i n i t s nat-u r a l aging or i n i t s a r t i f i c i a l f e r t i l i z a t i o n , The d e s i g n a t i o n of "seven" s t a t e s i s not based on any s c i e n t i f i c r a t i o n a l e . I t i s an a r b i t r a r y c h o i c e that i s a compromise between the need f o r a l a r g e number of c l a s s i f i c a t i o n s to d i s t i n g u i s h a l l outcomes, and the need f o r a meaningful d i f f e r e n t i a t i o n between each t r o p h i c s t a t e . Trophic s t a t e (1) corresponds to a l a k e that has not been subjected to any unnatural n u t r i e n t l o a d i n g and i s i n an o l i g o t r o p h i c s t a t e . This s t a t e i s assumed to be. the most d e s i r a b l e c o n d i t i o n from the p o i n t of view of pre-s e r v i n g the n a t u r a l environment. Trophic s t a t e (7) describes a lake that i s experiencing e u t r o p h i c a t i o n i n i t s most advanced stage, r e s u l t i n g i n a t o t a l l o s s of a l l r e c r e a t i o n a l and environmental b e n e f i t s . Between these two ex-treme s t a t e s , f i v e intermediate t r o p h i c s t a t e s are described. Each t r o p h i c 26 s t a t e i s defined such that, i t denotes the major q u a l i t a t i v e changes that occur between that s t a t e and the preceding loweT t r o p h i c s t a t e . The t r o p h i c s t a t e s are defined p r i m a r i l y in. terms of e n v i r o n -mental and r e c r e a t i o n a l c o n s i d e r a t i o n s according to the problem d e f i n i t i o n mentioned e a r l i e r i n t h i s chapter. The d e s c r i p t i o n s could be expanded to in c l u d e other f a c e t s of the problem as w e l l . However, the c a t e g o r i e s can be more e a s i l y understood by the d e c i s i o n maker i f they are kept simple. The t r o p h i c s t a t e d e s c r i p t i o n s can be mod i f i e d f o r d i f f e r e n t s i t u a t i o n s o r i n l i g h t of new i n f o r m a t i o n . The t r o p h i c s t a t e s c a l e , given i n Table 4.1, i s not intended f o r s c i e n t i f i c use but can be s u b s t a n t i a t e d by s c i e n t i f i c data. I t s pur-pose i s to acquaint the d e c i s i o n maker w i t h i n f o r m a t i o n on e u t r o p h i c a t i o n tb*^t--is- s1--! ted t~o- n p p r j q . Tf>nQ. the p ^ ^ I P ~* Q ^ ^ f ""nee! ^irec*~1~*~ _'T*t terrns of data r e l e v a n t p r i m a r i l y to the d e c i s i o n maker. In order to r e l a t e the t r o p h i c s t a t e s c a l e to a more s c i e n t i f i c viewpoint,.estimates of Secchi d i s c readings and d i s s o l v e d oxygen concen-t r a t i o n i n the hypolimnion are made f o r the seven t r o p h i c s t a t e s . ( S t o c k n e r , 1974), These estimates are given i n Table 4.2. TABLE 4.1 TROPHIC STATE SCALE CLASSIFICATIONS TROPHIC STATE DESCRIPTION The lake is in its most naturally pure form, corresponding to an infertile oligotrophia lake. The clarity of the water is excellent end the shoreline is sparsely dotted with rooted aquatic plants. A stable fish population in the lake provides a fair angling catch of desirable spe-cies. In general, the lake is highly desirable for all -water based recreational and domestic uses. The water clarity in the lake has decreased slightly due to an increase in phytoplanktor. density. ' Otherwise, the lake is in the same condition as (1). The water clarity has decreased noticeably and a few email weed beds have enlarged close to the shoreline. The insect population is somewhat higher. A small in-crease in angling catch per unit effort and larger fish ocsi be expected. Depending upon climatic arid hydrolo-gical vutidliic.z ll-.crc iz a r : : : i ? " ' l i t y of one small algal bloom occurring during the summer. This would persist for one or two weeks causing a slight odor^ problem. The water clarity has decreased significantly and aquatic weed growth along the shoreline is becoming a nuisance. The desirable fish populations are beginning to decrease, although fish may now be larger. Insect population has shown a definite increase. Bathers may experience skin irriations. There is a high probability of one algal bloom erupting in early or late summer. The water has developed a slight taste and odor. The coarse fish population has increased, while salnonoids (white fish, trout, salmon) continue to decrease, result-ing in a sharply reduced catch per unit effort. Two major, algal blooms occm*ring during the summer are seriously affecting recreational benefits. Weed growth along the shoreline is hindering access to the lake. Taste and odor are definite problems. Major weed growths along most of the shoreline require harvesting of plants. Coarse fish are abundant in the lake and the salmotioids are extremely sparse. A prolonged blue-green algal bloom is sustained from June to September. Most species of fish have disappeared and massive weed patches are enlarging on the lake surface. The lake is . now completely wdesirable for any recreational or domes-tic use. TABLE 4.1 TROPHIC STATE SCALE CLASSIFICATIONS TROPHIC STATE DESCRIPTION The lake is in its most naturally pure form, corresponding to an infertile oligotrophia lake. The clarity of the water is excellent and the shoreline is sparsely dotted with rooted aquatic plants. A stable fish population in the lake provides a fair angling catch of desirable spe-cies. In general, the lake is highly desirable for all -water based recreational and domestic uses. The water clarity in the lake has decreased slightly due to an increase in phytoplankton density. ' Otherwise, the lake is in the same condition as (1). The water clarity has decreased noticeably and a few small weed beds have enlarged close to the shoreline. The insect population is somewhat higher. A small in-crease in angling catch per unit effort and larger fish cai be expected. Depending uvon climatic and hydrolo-gical conditions there is a poscibility of one small algal bloom occurring during the sv/tter. This would persist for one or two weeks causing a slight odor^ problem. The water clarity has decreased significantly and aquatic weed growth along the shoreline is becoming a nuisance. The desirable fish populations are beginning to decrease, although fish may now be larger. Insect population has-shown' a definite increase. Bathers may experience skin irriations. There is a high probability of one algal bloom erupting in early or late summer. The water has developed a slight taste and odor. The coarse fish population has increased, while salmonoids (white fish, trout, salmon) continue to decrease, result-ing in a sharply reduced catch per unit effort. Two major algal blooms occurring. during the summer are seriously affecting recreational benefits. Weed growth along the shoreline is hindering access to the lake. Taste and odor are definite problems. Major weed growths along most of the shoreline require harvesting of plants. Coarse fish are abundant in the lake and the salmonoids are extremely sparse. A prolonged blue-green algal bloom is sustained from June to September. Most species of fish have disappeared and massive weed patches are enlarging on the lake surface. The lake is now completely undesirable for any recreational or domes-tic use. TAJ LE 4.2 ESTIMATED SECCHI DISC READINGS AND DISSOLVED OXYGEN LEVELS CORRESPONDING TO TROPHIC STATE SCALE TROPHIC STATE NUMBER 1 2 3 4 5 6 7 SECCHI DISC 1 METERS 8 - 9 7 5 3 1.5 <1 <1 DISSOLVED 2 OXYGEN mg/1 12.0 11.0 !'i.O 7.0 5.0 1 - 3 . 0 0 'Mean growing season reading. In hypolimnion i n September. (S3 00 29 C. CALCULATION OF UTILITY FUNCTION USING CHOICE TABLE The d e r i v a t i o n of a x i t i l i t y f u n c t i o n f o r t h e . t r o p h i c s t a t e s c a l e requires that the two extreme outcomes of the s c a l e be assigned u t i l i t y v a l u e s . These two p o i n t s represent the maximum and minimum u t i l i t i e s on the s c a l e . The si m p l e s t arrangement i s t o ass i g n t r o p h i c s t a t e (1) a u t i l -i t y of 100 and t r o p h i c s t a t e (7) a u t i l i t y of 0. Now, a l l p o s s i b l e out-comes can be expressed on a 0 to 100 s c a l e . The choice of these two bound-ary values i s completely a r b i t r a r y and made only f o r convenience. D i f f e r -ent o r d i n a t e s w i l l not a f f e c t the shape of the u t i l i t y f u n c t i o n . Table 4.3 shows a choice t a b l e ( H a l t e r and Dean, 1971) that i s used to d e r i v e a d e c i s i o n maker's u t i l i t y f u n c t i o n , u s i n g the von Newmann Morgenstern technique. The s c a l e of t r o p h i c s t a t e s i s placed along the v e r -t"lC9_.' «? fl"">d VST"!""? rirntSah-n i t i p s rnrrpqnonrlinp tn c?-1"tflitl l o t t e r i e s o r Preference c o n t r a c t s " are placed on the h o r i z o n t a l s c a l e . Each c e l l i n the t a b l e represents a choice between two d i f f e r e n t a l t e r n a t i v e s . The d e c i s i o n makers are r e q u i r e d to express one o p i n i o n only f o r each c e l l . They can choose e i t h e r one of the a l t e r n a t i v e s o r , i f both a l t e r n a t i v e s are e q u a l l y d e s i r a b l e , then they i n d i c a t e t h e i r i n d i f f e r e n c e f o r that p a r t i c u l a r c e l l . The a l t e r n a t i v e s d e s c r i b e various r i s k s i t u a t i o n s p e r t a i n i n g to the manage-ment of the l a k e water q u a l i t y . The a l t e r n a t i v e s are defined as f o l l o w s . A. A reference c o n t r a c t whereby there i s a r i s k w i t h p r o b a b i l i t y " T T " t h a t the lake i s i n s t a t e (1) and w i t h a p r o b a b i l i t y of " l - u " that the l a k e i s i n s t a t e ( 7 ) . or B. The c e r t a i n t y o f the t r o p h i c s t a t e i n the lake corresponding to some p a r t i c u l a r s t a t e on the t r o p h i c s t a t e s c a l e . A L T E R N A T I V E B .. C E R T A I N TROPHIC STATE A L T E R N A T I V E A .REFERENCE CONTRACT WITH P R O B A B I L I T Y TT TT=I.O .9 .8 .7 .6 .5 .4 .3 .2 .1 0 Lake in i t s natural state, c l a r i t y high, ... no unpleasant taste or odor. Small density of rooted plants on shore. I B B B B B B . B B B B (2) Slight decrease in water c l a r i t y . A I B B B " B B B B B B Further decrease in water c l a r i t y ; large ,,. weed beds appearing on shoreline., i n -sect population up, p o s s i b i l i t y of one small algal bloom annually. A A A I B B B B B B. B Water c l a r i t y sharply reduced; weeds be-... coming a nuisance on shoreline, possible skin i r r i t a t i o n s for bathers; further i n -. crease in insect population. A ' A A A A I B B B B B Taste and odor noticeable; shoreline access more d i f f i c u l t , two major blue* green algal blooms per year. • Increase in coarse f i s h population-. A A A A A A A r B B B Taste and odor very noticeable; major .Weed development 'on shore; coarse f i s h •predominant; salmonoid very sparse; prolonged algal blooms May to Sept. A A A A A . A A A I B B Most species of f i s h gone; massive (7) weed development; generally highly undesirable for a l l uses. A A A A A A A A A A. I T A B L E 5.1 CHOICE T A B L E FOR FINDING- INDIFFERENCF- POINTS BETWEEN C E R T A I N TROPHIC STATE AND. REFERENCE CONTRACT 31 Each c e l l i n the choice t a b l e represents a unique combination of a p r o b a b i l i t y " T T " and t r o p h i c s t a t e . Beginning from the top l e f t hand corner of the t a b l e ; the d e c i s i o n maker i s asked to i n d i c a t e h i s preference between: A. The p r o b a b i l i t y of 1. of the l a k e being i n s t a t e (1) and the p r o b a b i l i t y of 0. of the l a k e being i n s t a t e (7) or B. The c e r t a i n t y of the l a k e being i n s t a t e (1). These are obviously i d e n t i c a l a l t e r n a t i v e s so t h e r e f o r e i t represents a p o i n t of i n d i f f e r e n c e and i s so i n d i c a t e d on the choice t a b l e . Proceed-i n g down the f i r s t column to the, next c e l l a l t e r n a t i v e A remains the same but a l t e r n a t i v e B corresponds to a c e r t a i n t y of t r o p h i c s t a t e (2). Between the two choices i n t h i s c e l l , a l t e r n a t i v e A i s more d e s i r a b l e and hence, the c e l l i s marked w i t h an "A". S i m i l a r l y , a l t e r n a t i v e A would be the obvious choice a l l the way down the f i r s t column. 0 The next column changes the reference c o n t r a c t to the p r o b a b i l -i t y of .9 of the lake being i n s t a t e (1) and a p r o b a b i l i t y of .1 of the l a k e being i n s t a t e ( 7 ) . The d e c i s i o n maker then proceeds to i n d i c a t e the preferences i n t h i s column. The choices are no longer obvious and must be determined by s u b j e c t i v e judgement. A c o n s i d e r a b l e amount of thought and i n t r o s p e c t i o n i s necessary when f i l l i n g out the t a b l e . I t u s u a l l y r e q u i r e s that the d e c i s i o n maker go over h i s answers s e v e r a l times u n t i l he i s s a t i s f i e d w i t h the r e s u l t s . The i n d i f f e r e n c e p o i n t s i n the choice t a b l e determine the d e c i s i o n maker's u t i l i t y f u n c t i o n . The u t i l i t i e s of titate (1) and (7) 32 have already been defi n e d . According to U t i l i t y theory, i f a person i s i n d i f f e r e n t between two separate payoffs (a payoff i s de f i n e d as the consequence to the d e c i s i o n maker of t a k i n g a p a r t i c u l a r a c t i o n ) then the u t i l i t i e s of the two payoffs are equal. This i s the f i r s t axiom presented i n the previous chapter. Therefore, the i n d i f f e r e n c e p o i n t s i n the choice t a b l e i n d i c a t e a l t e r n a t i v e s that are e q u i v a l e n t i n u t i l i t y to the d e c i s i o n maker. The u t i l i t y of each t r o p h i c s t a t e i s defined i n terms of i t s e q u i v a l e n t reference c o n t r a c t . Mathematically, t h i s i s expressed i n the f o l l o w i n g manner. I f the d e c i s i o n maker i s i n d i f f e r e n t between A. Certainty, of 1^; and B. P ^ ) = .9 and P ( T ? ) = .1. Therefore U(A) = U(B) by Axiom 1 which can be w r i t t e n as: U(T 2) = U{T |P(Tj) = .9; T ?|P(T 7) = .1} This reads as: the u t i l i t y of t r o p h i c s t a t e (2) i s equal to the u t i l i t y of a .9 p r o b a b i l i t y of t r o p h i c s t a t e (1) and a .1 p r o b a b i l i t y of t r o p h i c s t a t e (7). According to Axiom 2: U(T 2) =(.9) 11(1^) +(.1) U(T ?) = 0 From previous d e f i n i t i o n , UCT^ = 100 and U(T ?) = 0. Therefore' " " U(T 2) = 9 0 33 where T\ - lake i n t r o p h i c s t a t e ( i ) ; P(T ) ~ p r o b a b i l i t y of lake i n t r o p h i c s t a t e ( i ) ; U(T.) = u t i l i t y of T.. 1 1 The u t i l i t y of the other t r o p h i c s t a t e s are determined i n the same manner. The t r o p h i c s t a t e s are p l o t t e d a g a i n s t t h e i r r e s p e c t i v e u t i l i t 16S Slid, tl free hand curve can be drawn connecting a l l the p o i n t s . This curve d e f i n e s the u t i l i t y f u n c t i o n of the d e c i s i o n maker. A t y p i c a l response to a choice t a b l e i s shown i n Table 4.3, and the r e s u l t i n g u t i l -i t y f u n c t i o n i s drawn i n Figure 4.1. The d e c i s i o n maker should be aware of the consequences and repercussions of having the l a k e d e t e r i o r a t e . The u t i l i t y f u n c t i o n serves to e x p l i c i t l y summarize h i s . knowledge arid expelienee i n a man age ah i t form. The u t i l i t y curve ensures t h a t the d e c i s i o n maker w i l l be c o n s i s -tent p r o v i d i n g that a l l outcomes are assessed according t o h i s u t i l i t y f u n c t i o n . 34 Fig.4.1 EXAMPLE OF DECISION MAKER'S UTILITY FUNCTION DERIVED FROM INDIFFERENCE TABLE 4.3. CHAPTER V EUTROPHICATION IN SKAHA LAKE Skaha Lake, l o c a t e d i n the Okanagan Basin of B r i t i s h Columbia, i s a good example of a lake c u r r e n t l y e x h i b i t i n g c u l t u r a l e u t r o p h i c a t i o n . Skaha Lake i s one of a chain o f f i v e lakes that form the Okanagan R i v e r system (Figure 4.1). The major water q u a l i t y problem i n the Okanagan basin i s lake e u t r o p h i c a t i o n . I t has already s e r i o u s a f f e c t e d three of the l a k e s . Skaha Lake has e x h i b i t e d a s u b s t a n t i a l i n c r e a s e i n t r o p h i c s t a t e w i t h i n the l a s t decade and has experienced n o t i c e a b l e en-vironmental d e t e r i o r a t i o n . In terms of an imminent water q u a l i t y prob-l a k e s . S e v e r a l s c i e n t i f i c and economic s t u d i e s have been conducted on the Okangan b a s i n region. A j o i n t f e d e r a l - p r o v i n c i a l study, j u s t com-p l e t e d , obtained an extensive amount of i n f o r m a t i o n on the area. Because of the a v a i l a b i l i t y of data, Skaha Lake was used as a case study f o r de-v e l o p i n g the d e c i s i o n theory technique to d e a l w i t h e u t r o p h i c a t i o n prob-lems. A. GENERAL DESCRIPTION OF SKAHA LAKE REGION Recent economic growth i n the Skaha Lake b a s i n has been c l o s e l y l i n k e d w i t h the r e s t of the Okanagan region . A favourable combination of t e r r a i n , c l i m a t e and economic opportunity has made the Okanagan r e g i o n 3 5 Fig.5.1 OKANAGAN RIVER SYSTEM 37 h i g h l y d e s i r a b l e f o r r e s i d e n t i a l , a g r i c u l t u r a l and i n d u s t r i a l purposes. The pop u l a t i o n of the e n t i r e b a s i n r e g i o n has doubled w i t h i n the l a s t 10 years and t h i s h i g h r a t e o f growth i s expected to p e r s i s t at l e a s t u n t i l 1980. The l a k e s are i n t r i n s i c to the p o p u l a r i t y o f Okanagan r e g i o n and are the resource l a r g e l y r e s p o n s i b l e f o r i t s growth. The t o u r i s t i n -* d u s t r y , w i t h an estimated revenue of $50 m i l l i o n a nnually (O'Riordon, 1972) has demonstrated i t s s e n s i t i v i t y to the water q u a l i t y i n the l a k e s . Losses i n excess o f s e v e r a l m i l l i o n d o l l a r s were a t t r i b u t e d to p u b l i c i t y concerning a l g a l blooms on Skaha Lake d u r i n g the summer of 1968 (O'Riordon, 1972). The c i t y of P e n t i c t o n , which i s the major p o p u l a t i o n centre bor-d e r i n g on Skaha Lake, has always discharged i t s sewage e f f l u e n t i n t o the lak e . The c i t y has a po p u l a t i o n of about 18,000 (Brown et a l . t 1973) which increases to 25,000 during the summer months. Although Skaha Lake has d e t e r i o r a t e d i n water q u a l i t y , i t i s s t i l l used f o r r e c r e a t i o n a l purposes by both r e s i d e n t s and v i s i t o r s to the area. B. WATER QUALITY PROBLEMS IN SKAHA LAKE E u t r o p h i c a t i o n i s the major water q u a l i t y problem i n Skaha Lake. This c o n d i t i o n has developed w i t h the growth of domestic and i n d u s t r i a l This i n c l u d e s the Okanagan re g i o n and.the neighbouring Shuswap region which i s a smaller area. 38 a c t i v i t i e s i n the Skaha b a s i n which has caused a l a r g e i n c r e a s e i n the n u t r i e n t budget of the l a k e . Phosphorus and n i t r o g e n are b e l i e v e d to be the n u t r i e n t s c h i e f l y r e s p o n s i b l e f o r the i n c r e a s e i n b i o t i c a c t i v i t y i n Skaha l a k e . P a t a l a s and S a l k i (1973) estimated that 88 per cent to 90 per cent of the t o t a l phosphorus load to Skaha Lake o r i g i n a t e s from c u l t u r a l sour-ces. The va r i o u s sources of phosphorus are l i s t e d i n Table 5.1. The present c o n d i t i o n of Skaha Lake i s considered moderately e u t r o p h i c . According to the t r o p h i c s t a t e s c a l e defined i n the previous chapter, Skaha Lake would be halfway between Sta t e (4) and State ( 5 ) . The recent d e t e r i o r a t i o n i n the lake environment i s c l e a r l y evident when examining the f o l l o w i n g l i m n o l o g i c a l parameters. designed 22 cm d i s c , when lowered i n t o the water, disappears from s i g h t f o r an observer watching the descent. I t i s a d i r e c t measure of the t r a n s -parency o f the water and i s a good i n d i c a t o r f o r observing changes i n the b i o l o g i c a l a c t i v i t y o f a l a k e . In 1939, the Secc h i d i s c r e a d i n g was 11.9 metres (O'Riordon, 1972). In 1966, the average reading f e l l to 4.4 metres and i n 1972 was 2.5 metres (Patales and S a l k i , 1973). Since 1967, Skaha Lake has had two a l g a l blooms o c c u r r i n g a n n u a l l y . A minor bloom u s u a l l y occurs i n l a t e s p r i n g and a l a r g e r one i n l a t e summer or e a r l y f a l l . The blooms p e r s i s t f o r a two to three week p e r i o d . The d i s s o l v e d oxygen c o n c e n t r a t i o n i n the hypolimnion of Skaha lake i s i n the range of 6.3 mg/1 to 8.1 mg/1 ( P a t a l a s , 1972). During p e r -TABLE 5.1 LOADINGS OF TOTAL PHOSPHORUS TO SKAHA LAKE FROM EXTERNAL SOURCES BASED ON 1969-71 DATA (In Kilograms per Year) ( F i n a l Report, 1974) S O U R C E ESTIMATED LOADING (kg/yr) SL T r i b u t a r y Streams 1,910 Okanagan R i v e r from Ground Water*3 1,000 D u s t f a l l and P r e c i p i t a t i o n 760 M u n i c i p a l Sewage Treatment P l a n t 13,180 TOTAL 21,650 kg Includes a g r i c u l t u r a l source loadings to streams. Includes c u l t u r a l and n a t u r a l l o a d i n g s . 40 iods f o l l o w i n g the d i e o f f of a l g a l blooms, the d i s s o l v e d oxygen concen-t r a t i o n has f a l l e n as low as 2 mg/1 ( W i l l i a m , 1973). D i s s o l v e d oxygen l e v e l s below 5 mg/1 are considered d e t r i m e n t a l to c e r t a i n salmonoids. Phosphorus concentration i n the lak e during s p r i n g over-t u r n i s a measure of the average amount of phosphorus a v a i l a b l e f o r the b i o t i c a c t i v i t y i n a l a k e . In a l a k e where phosphorus i s the l i m i t i n g n u t r i e n t (which appears to be the case f o r Skaha Lake) the s p r i n g con-c e n t r a t i o n of phosphorus i s a very s t r o n g i n d i c a t o r of the t r o p h i c s t a t e . Vollenweider (1968) suggests a l e v e l of .020 mg/1 t o t a l phos-phorus as dangerous. The recorded l e v e l i n Skaha Lake was .070 mg/1 (Stockner, 1972). Another b i o l o g i c a l i n d i c a t o r that has d e f i n i t e i m p l i c a t i o n s f o r recj-eaclonal users of the lake i s p o p u l a t i o n changes among the l i s h s p e c i e s . Northcote (1972) has shown that i n the time p e r i o d from 1948 to 1972, there has been a s i g n i f i c a n t d e c l i n e i n salmonoids accompanied by an in c r e a s e i n coarse species of f i s h l i f e . V ollenweider (1968) compared the t r o p h i c s t a t e s of d i f f e r e n t lakes by e s t i m a t i n g the annual phospohrus l o a d i n g c a l c u l a t e d on an a r e a l b a s i s and p l o t t i n g i t a g a i n s t mean depth. P a t o l a s (1972) estimated the 2 lo a d i n g to Skaha Lake to be i n the range of 1.0 to 2.5 gm/m / y r . Lake E r i e , which i s considered a eutrophic l a k e (Zwick and Benstock, 1971) 2 has an estimated l o a d i n g of 1.06 gm/m / y r . However, e u t r o p h i c a t i o n i n Skaha Lake has been i n h i b i t e d by two important f a c t o r s . The lak e has a comparatively s h o r t f l u s h i n g time, on the average of 1.2 years, and i s r e p l e n i s h e d w i t h a water supply that i s r e l a t i v e l y low i n n u t r i e n t content. Thus, Skaha Lake has not yet shown 41 a d r a s t i c decline, i n water q u a l i t y which would occur i f the hypolimnion were to become anaerobic. The major question t h a t the d e c i s i o n makers must con s i d e r i s how c l o s e i s the present n u t r i e n t l o a d i n g l e v e l to the c r i t i c a l l i m i t . C. PHOSPHORUS AS A LIMITING NUTRIENT IN SKAHA LAKE Recent evidence from l a b o r a t o r y experiments has i n d i c a t e d that phosphorus may not now be the l i m i t i n g n u t r i e n t i n Skaha Lake (Stockner, 1973). Tests have shown th a t at c e r t a i n times of the year, n i t r o g e n and/or t r a c e elements may be r e g u l a t i n g the b i o t i c growth r a t e . However, t h i s f i n d i n g does not a f f e c t the importance of phosphorus l o a d i n g as being the key f a c t o r i n l i m i t i n g the b i o t i c growth i n Skaha Lake. This apparent The f a c t t h a t phosphorus does n o t , at the present time e x e r t a r a t e c o n t r o l l i n g i n f l u e n c e i m p l i e s t h a t there i s an over abundance of the element i n Skaha Lake. T h i s i s q u i t e possible.because the l a k e r e c e i v e s a phosphorus l o a d at approximately 10 times i t s n a t u r a l l o a d i n g r a t e . I t i s b e l i e v e d that before c u l t u r a l n u t r i e n t loadings f e r t i l i z e d Skaha Lake, phosphorus was the l i m i t i n g n u t r i e n t (Stockner, 1974). In order to con-t r o l the b i o t i c a c t i v i t y i n the l a k e i t i s necessary to make one n u t r i e n t l i m i t i n g and c o n t r o l the l o a d i n g r a t e o f t h a t n u t r i e n t . Phosphorus i n the form of phosphates i s the one n u t r i e n t whose i n -put to water can be most e a s i l y c o n t r o l l e d (Zwick and Benstock, 1972). Ex-perience w i t h l a b o r a t o r y s t u d i e s , p i l o t p l a n t and f i e l d s c a l e experiments has shown phosphorus removal from waste waters i s economically f e a s i b l e . 42 Although n i t r o g e n removal has a l s o been used as a form of n u t r i e n t con-t r o l , i t i s more c o s t l y and not as f a r advanced t e c h n o l o g i c a l l y as phos-phorus removal (Lee, 1970). Furthermore, n i t r o g e n f i x i n g b a c t e r i a i n l a k e s present an u n c o n t r o l l a b l e source of n i t r o g e n . Consequently, phosphorus l o a d i n g i n t o Skaha Lake remains the primary mechanism f o r c o n t r o l l i n g e u t r o p h i c a t i o n . I f Skaha Lake returns to a s t a t e where phosphorus again e x e r t s a r a t e c o n t r o l l i n g i n f l u e n c e and the phosphorus input to the l a k e i s r e g u l a t e d , the t r o p h i c s t a t e can then be maintained at a d e s i r a b l e l e v e l . D. THE LAKE MANAGEMENT PROBLEM The P e n t i c t o n sewage treatment p l a n t i s the primary source of phosphorus e n t e r i n g skana Latce, c o n t r i b u t i n g about fc>U. per cent ot the t o t a l load {Technical Supplement IVt 1974). The p l a n t u t i l i z e s an a c t i -vated sludge process of approximately 1.8 m i l l i o n g a l l o n s per day. In 1971, the p l a n t was expanded to i t s present c a p a c i t y and a phosphorus r e -moval system was added. Phosphorus removal i n the p l a n t i s achieved by chemical p r e c i p i t a t i o n . The primary and secondary f a c i l i t i e s i n the p l a n t are operating e f f i c i e n t l y and there i s thus, no BOD removal problem. However, the o p e r a t i n g cost of the t e r t i a r y f a c i l i t i e s , which i s due c h i e f l y to the chemical c o s t s , i s p r o v i n g to be c o n s i d e r a b l y h i g h e r than o r i g i n a l l y a n t i c i p a t e d i n the design stage. This h i g h o p e r a t i n g cost i s a t t r i b u t e d to the f o l l o w i n g f a c t o r s : the recent p r i c e i n c r e a s e of l i m e ; the i n e f f i c i e n t operation of the p r e c i p i t a t i o n techniques ( t h i s problem has apparently been solved to the extent where i t no longer represents a major 43 f a c t o r ) ; and the h i g h phosphorus removals c u r r e n t l y being obtained i n the p l a n t . The o r i g i n a l plans d i d not c a l l f o r the h i g h degree of phosphorus removal now b e l i e v e d to be necessary. Consequently, the phosphorus r e -moval f a c i l i t y which cost $195,000 t o i n s t a l l (O'Riordan, 1972), has an estimated annual operating budget between $50,000 to $90,000 to achieve phosphorus removals i n the range of 80 to 90 per cent. D i f f e r e n t combinations o f chemicals and v a r i o u s a p p l i c a t i o n techniques are s t i l l being t e s t e d i n the p l a n t i n an e f f o r t to improve the oper-a t i o n . The d e c i s i o n to i n s t a l l chemical p r e c i p i t a t i o n f a c i l i t i e s f o r phosphorus removal was made by the P e n t i c t o n C i t y C o u n c i l i n 1969. O'Rior-don (1972) d e s c r i b e s the v a r i o u s events t h a t l e d to the f i n a l d e c i s i o n , <3pvo-i-o] t-u^rin P . Y S " t.~' i U ' e c t r e t e " q u i t e clear• l y the problems ef d e c i s i o n " making under u n c e r t a i n t y t h a t a r i s e when managing lakes that are becoming n u t r i e n t enriched. In 1966 a report on the water q u a l i t y of the Okanagan la k e s was i s s u e d (0'Riordon,1972). I t had been prepared by three l o c a l s c i e n t i s t s who had conducted s t u d i e s on the lakes d u r i n g the previous year. Skaha Lake had not yet experienced any a l g a l blooms at the time t h i s r e p o r t was r e l e a s e d . However, the r e p o r t s t a t e d Skaha Lake had shown a s i g n i f i c a n t i n c r e a s e i n b i o t i c a c t i v i t y beli.eved to be caused by the domestic sewage discharged i n t o the l a k e . The r e p o r t c r i t i c i z e d the v a r i o u s a u t h o r i t i e s f o r t h e i r inadequate e f f l u e n t standards and i t a l s o contained s e v e r a l strong recommendations concerning waste discharges i n t o a l l Okanagan l a k e s . 44 The r e p o r t was q u i t e alarming t o the l o c a l community because the se r i o u s e u t r o p h i c a t i o n problem i t professed was not very apparent. The p u b l i c became f u r t h e r confused when government o f f i c i a l s , f e a r i n g the r e -po r t would cause adverse p u b l i c i t y f o r the area, attempted to d i s c r e d i t the f i n d i n g s of the r e p o r t . P e n t i c t o n was the. only c i t y i n the re g i o n that was prepared to consider some of the recommendations. However, the P e n t i c t o n C i t y C o u n c i l d i d not decide to look at t e r t i a r y f a c i l i t i e s u n t i l one year l a t e r . By then, Skaha Lake had erupted i n t o a massive a l g a l bloom, thus co n f i r m i n g the f i n d i n g s of the r e p o r t . The t e c h n i c a l accuracy of the rep o r t was never contested, but i t s extreme conclusions reduced i t s c r e d i b i l i t y . No mention was made of the un-c e r t a i n t y f a c t o r s i n v o l v e d and the p u b l i c , along w i t h the C i t y C o u n c i l , were faced w i t h two nBnnsinc views- T h p oonf"Pior- generated by t h i c s i t ' -u a t i o n was evident i n l a t e r years. In 1969, O'Riordan p o l l e d both the c i t y aldermen and the r e s i d e n t s of P e n t i c t o n on the question of present and f u t u r e water q u a l i t y i n Skaha Lake. The two summers previous to the summer i n which the p o l l was taken, the l a k e had e x h i b i t e d major a l g a l blooms. The r e s u l t s of the p o l l show a considerable discrepancy i n the views among those questioned. There was no c l e a r consensus among the aldermen on the present q u a l i t y o f the l a k e and a l l the aldermen except one, expected the f u t u r e water q u a l i t y to im-prove. F i v e years l a t e r t h i s improvement has not r e a l l y occurred. The r e s u l t s of the r e s i d e n t s ' p o l l shown an even wider d i v e r s i t y i n t h e i r o p inion on the water q u a l i t y i n Skaha Lake. This d i v e r s i t y could be due to v a r y i n g s e n s i t i v i t y to the p o l l u t i o n problem, but i s juke.'as l i k e l y 45 a t t r i b u t a b l e to the l i m i t e d understanding of the complex i s s u e o f e u t r o -p h i c a t i o n . Under these circumstances, i t i s d i f f i c u l t f o r the d e c i s i o n maker ( i . e . , C i t y Council) to come up w i t h sound d e c i s i o n s . The f i n a l d e c i s i o n by the C o u n c i l to implement a chemical p r e c i p i t a t i o n technique was based to a c o n s i d e r a b l e extent on the b e l i e f t h a t the sewage treatment p l a n t o p e r a t i o n would then meet the requirements s e t f o r phosphorus removal by the P o l l u t i o n C o n t r o l Board. However, t h i s was a blanket standard of 70 per cent set f o r a l l the Okanagan lakes and was not s t r i n g e n t enough f o r Skaha Lake. CHAPTER VI i THE EXPECTED UTILITY FUNCTION The d e c i s i o n maker r e s p o n s i b l e f o r the water q u a l i t y management i n a/eutrophic lake must make a s e r i e s of d e c i s i o n s over a p e r i o d of time aimed at keeping the lake i n a d e s i r a b l e s t a t e . Each d e c i s i o n made should take i n t o account not only the present circumstances, but the f u t u r e as w e l l . In a d d i t i o n , the d e c i s i o n maker should r e a l i z e he i s o p e r a t i n g i n an environment of considerable u n c e r t a i n t y . A framework i s needed that allows the d e c i s i o n maker to p e r c e i v e the problem i n t h i s manner and provide him w i t h a mechanism t o a i d i n the chapter attempts to create t h i s framework. The technique c o n s i s t s p r i m a r i l y of generating a mathematical f u n c t i o n that p r e d i c t s the most probable value o f the l a k e over a given planning h o r i z o n . The s t r a t e g y f o r the d e c i s i o n maker i s to choose the d e c i s i o n s t h a t tend to maximize t h i s mathematical f u n c t i o n throughout the planning p e r i o d . S e v e r a l c o n s i d e r a t i o n s of the management problem of e u t r o p h i c lakes were i d e n t i f i e d i n Chapter I I . The three c o n s i d e r a t i o n s which t o -gether determine the t r o p h i c s t a t e i n a lake and hence i t s r e l a t i v e d e s i r -a b i l i t y , can be expressed by three separate m a t h e m a t i c a l / r e l a t i o n s . These are: 46 47 (1) the p r o j e c t e d annual n u t r i e n t loadings f o r the f u t u r e planning p e r i o d . This can be considered as a set of numbers corresponding to the n u t r i e n t load up to year n. L = £(T) (5.1) where T = year, L = annual n u t r i e n t l o a d i n g . (2) the r e l a t i o n between the annual n u t r i e n t l o a d and the expected t r o p h i c s t a t e that would r e s u l t i n the l a k e . S = 5(L) (5.2) where S = t r o p h i c s t a t e a c c o r d i n g to Table 4.1. (.3) the r e l a t i o n between the l a k e i n a c e r t a i n t r o -p h i c s t a t e and the corresponding v a l u e of t h a t s t a t e to the d e c i s i o n maker measured i n u t i l i t y . This i s simply the u t i l i t y f u n c t i o n as d e r i v e d i n Chapter IV. U = U(S) (5.3) These three expressions can be combined i n a l o g i c a l mathematical manner to y i e l d a f u n c t i o n t h a t i s e q u i v a l e n t to the expected u t i l i t y of the lake over the designated time p e r i o d . The c a l c u l a t i o n of the expected u t i l -i t y f u n c t i o n r e q u i r e s that i n f o r m a t i o n and e x p e r t i s e be a v a i l a b l e to d e f i n e the three mathematical r e l a t i o n s as they p e r t a i n to the p a r t i c u l a r lake under qu e s t i o n . This technique then provides the d e c i s i o n maker w i t h a systematic procedure to process the i n f o r m a t i o n . The numerical r e l a t i o n s h i p s i n v o l v e d i n the f i r s t two mathematical ex-pressions can be s u p p l i e d by the s c i e n t i s t s and/or engineers who are f a m i l i a r 48 w i t h these p a r t i c u l a r areas of concern. The d e r i v a t i o n of these two ex-pressions are t e c h n o l o g i c a l problems t h a t are d i f f i c u l t to d e f i n e . i n exact mathematical terms, owing to the u n c e r t a i n t y . However, the extent of the u n c e r t a i n t y f o r each r e l a t i o n s h i p can be estimated by d e f i n i n g a most prob-able range of values f o r each v a r i a b l e . For example, i n the f i r s t e x p r e s s i o n , the n u t r i e n t loadings i n any year are d i f f i c u l t to p i n p o i n t . Y e t , i t i s p o s s i b l e to estimate h i g h and low annual l o a d i n g f i g u r e s t h a t would bound the range of p o s s i b l e l o a d -ings f o r t h a t given year. This range of loadings can be thought of as the area of u n c e r t a i n t y f o r that p a r t i c u l a r expression. In t h i s manner, the d e c i s i o n maker can begin to a p p r e c i a t e the extent of the u n c e r t a i n t y i n the d e c i s i o n s t h a t he must f a c e . Furthermore, the cXpteasious fepi'esa'ic utaiuly the e s s e u c i a l i n forma Lion r e q u i r e d by the d e c i s i o n maker w i t h regard to t h a t p a r t i c u l a r aspect of the problem. Thus, he i s able to pose d i r e c t questions to the e x p e r t s , r a t h e r than have to search through i r r e l e v a n t data i n order to make h i s d e c i s i o n s . A. CALCULATING THE EXPECTED UTILITY FUNCTION Phosphorus i s recognized as the key n u t r i e n t t h a t must be r e g u l a t e d i n order to manage the t r o p h i c s t a t e i n Skaha Lake. Therefore, only the phosphorus lo a d i n g w i l l be used i n e v a l u a t i n g the expected u t i l i t y of the l a k e . I t i s not necessary t h a t phosphorus be the b i o l o g i c a l l i m i t i n g nut-r i e n t i n the lake at a l l times i n order to apply t h i s technique. However, i t i s imperative that a r e a l i s t i c c o r r e l a t i o n e x i s t between the phosphorus loa d i n g and the lake t r o p h i c s t a t e s . In the case of Skaha Lake t h i s r e l a t i o n -49 ship i s b e l i e v e d to be d e f i n a b l e . PHOSPHORUS LOADING PROJECTIONS The Canada-British Columbia Okanagan Basin study has produced an extensive amount of in f o r m a t i o n on the Skaha Lake b a s i n . The study has o u t l i n e d p o s s i b l e economic p r o j e c t i o n s to the year 2020. Phosphorus l o a d -ings i n t o Skaha Lake have been estimated f o r the years 1980, 2000 and 2020. These p r o j e c t i o n s were made by m u l t i p l y i n g the estimated p o p u l a t i o n growth r a t e of the basin times the present phosphorus l o a d i n g v a l u e s . The u n c e r t a i n t y i n v o l v e d i n the p r o j e c t e d l o a d i n g r a t e s a r i s e from two sources: (1) the q u a n t i f y i n g of the phosphorus l o a d i n g sources i n t o the la k e ; and (2) the e s t i m a t i o n of the economic or p o p u l a t i o n growth r a t e s . The e v a l u a t i o n o f both these areas o f u n c e r t a i n t y i s q u i t e d i f f i -c u l t . Q u a n t i f y i n g a lake's phosphorus l o a d i n g sources i n v o l v e s examin-i n g a l l the sources i n d i v i d u a l l y . For each source, there are v a r i o u s mathe-m a t i c a l modeling techniques that are used to estimate the phosphorus con-t r i b u t i o n (Vollenweider, 1968). Every method has a d i f f e r e n t accuracy asso-c i a t e d w i t h i t and i t i s not always p o s s i b l e to estimate the magnitude of e r r o r . Consequently, d e f i n i n g the u n c e r t a i n t y or confidence l i m i t s i n the t o t a l phosphorus l o a d i n g to a lak e can be very complicated and the r e s u l t s may not j u s t i f y the e f f o r t . However, the pop u l a t i o n growth p r o j e c t i o n s are e a s i e r to handle. The planning study f o r the Skaha Lake b a s i n has devised a high and a low economic growth p r o j e c t i o n f o r the p l a n n i n g p e r i o d . These two growth 50 p r e d i c t i o n s ' can serve as the upper and lower l i m i t s f o r p o s s i b l e growth p a t t e r n s . Thus, i t i s p o s s i b l e to d e f i n e two phosphorus l o a d i n g pro-j e c t i o n s and the a c t u a l phosphorus l o a d i n g f o r the next 50 years can be expected to l i e somewhere i n between. The d i f f e r e n c e i n these two l o a d -i n g p r o j e c t i o n s i s considerably l a r g e r than the u n c e r t a i n t y t h a t might a r i s e from q u a n t i f y i n g the l o a d i n g sources. PHOSPHORUS LOADING AND LAKE TROPHIC STATE The r e l a t i o n s h i p between an annual phosphorus l o a d i n g i n t o Skaha Lake and the expected t r o p h i c s t a t e response to the load can only be evaluated by an expert who i s f a m i l i a r w i t h the l a k e . This r e l a t i o n -s h i p was d e r i v e d by e s t i m a t i n g the annual t o t a l phosphorus load that s c a l e . For each t r o p h i c s t a t e on the s c a l e , three l o a d i n g estimates were obtained corresponding t o : (1) the annual phosphorus l o a d t h a t would most l i k e l y give r i s e to the t r o p h i c s t a t e ; (2) the minimum annual phosphorus l o a d t h a t could cause the lake to reach the t r o p h i c s t a t e ; ( 3 ) the maximum annual phosphorus loa d t h a t the l a k e c o u l d withstand and s t i l l be w i t h i n the t r o p h i c s t a t e . Therefore, the r e l a t i o n s h i p i s defined f o r the seven t r o p h i c s t a t e s by 21 l o a d i n g e s t i m a t e s . The h i g h and low estimates f o r each t r o p h i c s t a t e d e f i n e the area of u n c e r t a i n t y . 51 The phosphorus l o a d i n g estimates f o r Skaha Lake were provided by Dr. John Stockner of the P a c i f i c Environment I n s t i t u t e of the F i s h e r -i e s Research Board of Canada. Dr. Stockner was r e s p o n s i b l e f o r co-ordina-t i n g the l i m n o l o g i c a l s t u d i e s f o r the j o i n t F e d e r a l - P r o v i n c i a l Okanagan Basin Study. The estimates are shown i n Table 6.1. TABLE 6.1 ESTIMATES OF TOTAL PHOSPHORUS LOADING VERSUS TROPHIC STATES FOR SKAHA LAKE TROPHIC STATE MINIMUM LOAD (kg) MOST LIKELY LOAD (kg) MAXIMUM LOAD (kg) 1 i ooo o nnr\ •a cnr\ 2 4,000 6,000 8,000 3 8,000 10,000 13,000 4 13,000 15,000 18,500 5 18,000 22,000 25,000 6 23,000 28,000 33,000 7 25,000 31,000 39,000 Estimates are i n phosphorus as t o t a l phosphorus SKEW NORMAL DISTRIBUTION Bayesian d e c i s i o n theory s t a t e s that the expected u t i l i t y o f an outcome i s equal to the p r o b a b i l i t y of t h a t event times i t s u t i l i t y . 52 However, the p r o b a b i l i t i e s a s s o c i a t e d w i t h the outcomes of a n u t r i e n t enriched lake are not e x p l i c i t . Given any one p a r t i c u l a r phosphorus l o a d i n g , there are no defined p r o b a b i l i t i e s to p r e d i c t the p o s s i b l e out-come of a t r o p h i c s t a t e i n the l a k e . There i s , however, a range of t r o -p h i c s t a t e s that could occur i n response to any one p a r t i c u l a r l o a d i n g . By p o s t u l a t i n g a p r o b a b i l i t y d e n s i t y f u n c t i o n f o r the range of t r o p h i c s t a t e s , a set of p r o b a b i l i t i e s can be e s t a b l i s h e d f o r v a r i o u s t r o p h i c s t a t e outcomes. Figure 6.1 shows three curves t h a t correspond to the maximum,, most l i k e l y and minimum l o a d i n g estimates that were made f o r e s t a b l i s h i n g r e l a t i o n 5.2, S = £>(L). The curves are drawn by p l o t t i n g the data i n Table 6.1 and i n t e r p o l a t i n g the p o i n t s . O r i g i n a t i n g at any p o i n t on phorus l o a d , a v e r t i c a l l i n e i s drawn that i n t e r s e c t s each of the three curves at one p o i n t . This i s i n d i c a t e d on F i g u r e 6.1. The upper and lower p o i n t s on the v e r t i c a l l i n e , S and S , d e f i n e the range of t r o p h i c U Li s t a t e s that could r e s u l t from that p a r t i c u l a r annual phosphorus l o a d i n g , L^. The middle p o i n t , S^, corresponds to the most probable t r o p h i c s t a t e t h a t might occur. A frequency d i s t r i b u t i o n f u n c t i o n , that i s bounded by the upper and lower p o i n t s on the v e r t i c a l l i n e , i s assumed to e x i s t f o r the range of p o s s i b l e . t r o p h i c s t a t e s . The frequency d i s t r i b u t i o n i s d e f i n e d such that the p r o b a b i l i t y d e n s i t y f u n c t i o n from to e i t h e r or f o l l o w s a normal curve, but the e n t i r e curve i s not n e c e s s a r i l y symetric about S^. The v a r i a n c e on the upper and lower bounds of the curve are not equal. The skew normal frequency d i s t r i b u t i o n curve was d e r i v e d by Dr. Rabab Ward. The mathematical d e r i v a -Fig.6.1 ESTIMATES OF TOTAL PHOSPHORUS LOADINGS VERSUS TROPHIC STATES FOR SKAHA L A K E . 54 t i o n i s given i n Appendix 2. The approximate shape of the p r o b a b i l i t y d e n s i t y f u n c t i o n f o r Load L^ i s i n d i c a t e d i n Figure 6.2. The skew normal curve i s used to assign a p r o b a b i l i t y value to the p o s s i b l e occurrence of an outcome. An annual phosphorus loa d p ro-j e c t i o n w i l l produce an outcome described by the p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n of t r o p h i c s t a t e s . Although the t r o p h i c s t a t e s c a l e i s d e f i n e d i n i n t e g e r s , the increments between the s t a t e s are s t i l l meaningful. A t r o p h i c s t a t e of 4.5 can be thought of as an intermediate s t a t e between (4) and (5), One must keep i n mind that the a c t u a l t r o p h i c changes t h a t a lake experiences are m a n i f e s t a t i o n s of many continuous r e a c t i o n s and that the t r o p h i c s t a t e s are only a r b i t r a r y c l a s s i f i c a t i o n s . The matter of assuming the e x i s t e n c e of the skew-normal d i s t r i b u -«-xoi.i i v / i . ^ ; i . c ^ X w i — t u g L . i i c i- upuj-^ o I*g L c ox. ci x CIIYC: c o i l C U i a i u c t c u q u e t i t-xuiZ— a b l e . Hdwever, there i s no way of d e r i v i n g an a c t u a l p r o b a b i l i t y f u n c t i o n because the data does, not e x i s t . The p o s t u l a t i o n of the f u n c t i o n i s an i n -t u i t i v e method th a t allows one to work i n an environment o f l i m i t e d knowledge. The shape of the p r o b a b i l i t y d i s t r i b u t i o n i s not as c r i t i c a l as the p o i n t s which d e f i n e i t , i . e . , S T , S w a n d STT. These p o i n t s determine the extent of the u n c e r t a i n t y recognized by the d e c i s i o n maker. The expected u t i l i t y f u n c t i o n i s c a l c u l a t e d s e p a r a t e l y f o r each year and i s determined by the p r o j e c t e d l o a d i n g d a ta f o r each year. The phosphorus load f o r one p a r t i c u l a r year d e f i n e s a range of p o s s i b l e t r o p h i c s t a t e s . The expected u t i l i t y f o r that year i s equal to the sum of the p r o b a b i l i t i e s of the p o s s i b l e t r o p h i c s t a t e s times t h e i r corresponding u t i l i t i e s . This i s an a p p l i c a t i o n of Equation (3.2). The u t i l i t y values are d e f i n e d according 55 L, Annual Phosphorus Load Fig.6.2 APPROXIMATE SHAPE OF POSTULATED PROBABILITY DENSITY FUNCTION OF TROPHIC STATES IN THE FORM OF A SKEW NORMAL CURVE . 56 to the d e c i s i o n maker's u t i l i t y f u n c t i o n . In mathematical terms, the c a l c u l a t i o n s are as f o l l o w s . The annual phosphorus loa d to the lake f o r year " i " i s L., which gives r i s e to the t r o p h i c s t a t e estimates ,SL> SM ^ V These d e f i n e a range of p o s s i b l e t r o p h i c s t a t e outcomes S^ to S^ . w i t h a most l i k e -l y outcome at S w. M A p r o b a b i l i t y d e n s i t y f u n c t i o n , G(S) e x i s t s over the range of p o s s i b l e outcomes S^ to S^. G(S) f o l l o w s the skew normal f u n c t i o n . G(S) M % ^ n + i For each t r o p h i c s t a t e v a l u e , there i s a corresponding u t i l i t y v a l u e , determined by the d e c i s i o n maker's u t i l i t y f u n c t i o n , U = F ( S ) . The expected u t i l i t y corresponding to the outcome of an incremental t r o p h i c s t a t e AS between S and S i s equal t o AEU n+^~ * n n+1 n n where s + S , " s + s AEU = {G --2 S+i} A S . {p n n+1, 2 2 57 which i s Expected U t i l i t y = p r o b a b i l i t y x U t i l i t y The expected u t i l i t y f o r a given year " i " , which i s c a l c u l a t e d over the e n t i r e range of p o s s i b l e outcomes, i . e . , S to S , i s found by Lt u summing up the incremental expected u t i l i t i e s , S EU(L.) = E U F(S.) • G(S.) • AS Therefore, f o r each year there i s a phosphorus l o a d estimate i n which i n turn can be t r a n s l a t e d i n t o an expected u t i l i t y . An expected u t i l i t y value can be c a l c u l a t e d f o r each year over a 40 to 50 year time span and the r e s u l t s p l o t t e d a g a i n s t a time a x i s . This y i e l d s an expected u t i l i t y curve. Each expected u t i l i t y curve corresponds to some phosphorus l o a d i n g r a t e s u s t a i n e d over a c e r t a i n p e r i o d o f time. • A computer program was w r i t t e n i n WATFIV to do the computations r e q u i r e d f o r c a l c u l a t i n g the expected u t i l i t y f u n c t i o n . The program a l s o s t o r e s the data required to d e f i n e the v a r i o u s f u n c t i o n s and a l s o c a l c u -l a t e s the phosphorus l o a d i n g s . The program can be adapted to d i f f e r e n t lake problems by s u b s t i t u t i n g the p a r t i c u l a r i n f o r m a t i o n r e q u i r e d i n t o the data f i l e . Expected u t i l i t y curves f o r Skaha Lake corresponding to v a r i o u s management programs were c a l c u l a t e d and are shown i n Chapter V I I I and Appendix C. The phosphorus l o a d i n g p r o j e c t i o n s were c a l c u l a t e d a n n u a l l y , s e p a r a t i n g the t o t a l l o a d i n t o c o n t r i b u t i o n s ' f r o m two sources; the sewage 58 treatment p l a n t at P e n t i c t o n and a l l other n a t u r a l and c u l t u r a l sources combined. The phosphorus loadings from the treatment p l a n t were c a l c u -l a t e d by i n c r e a s i n g the present i n f l u e n t l o a d of phosphorus i n p r o p o r t i o n to the p r o j e c t e d p o p u l a t i o n estimates and then de c r e a s i n g t h i s amount by the removal e f f i c i e n c y o f the p l a n t . The remaining sources are assumed to i n c r e a s e at a lower ra t e and were c a l c u l a t e d u s i n g t o t a l phosphorus l o a d i n g estimates. Data r e g a r d i n g growth p r o j e c t i o n s and phosphorus l o a d i n g estimates were obtained from the Final Report of the Canada-B r i t i s h Columbia Okanagan Basin Agreement. CHAPTER V I I EXPECTED UTILITY AS A MANAGEMENT AID Bayesian d e c i s i o n theory s t a t e s t h a t one should choose the a l t e r -n a t i v e which maximizes the expected u t i l i t i e s . T h i s i s b a s i c a l l y the s t r a -tegy followed when using t h i s a l g o r i t h m . However, the a n a l y s i s i s s l i g h t -l y more i n v o l v e d when c o n s i d e r i n g long-term expected u t i l i t i e s and when a t t a c h i n g economic c o n s t r a i n t s to the outcomes. The al g o r i t h m , i n i t s p r e -sent stage of development, cannot t e l l the d e c i s i o n maker which a l t e r n a t i v e i s best. This task i s s t i l l the r e s p o n s i b i l i t y of the d e c i s i o n maker and i t i s u n l i k e l y that any algorithm could t o t a l l y s u b s t i t u t e f o r human judg-ment . What the expected u t i l i t y curve can do i s give the planner a method of comparing a l t e r n a t i v e s according to the long term e f f e c t s . A t the same time i t accounts f o r the u n c e r t a i n t y i n the problem. A. MANAGEMENT OF SKAHA LAKE The expected u t i l i t y curves corresponding to three p o s s i b l e management a l t e r n a t i v e s f o r Skaha Lake are shown i n Figure 7.1. The com-pute r program was w r i t t e n to p l o t both the annual phosphorus l o a d i n g e s t i -mate and the r e s u l t i n g expected u t i l i t y f u n c t i o n f o r the p r o j e c t e d p l a n n i n g p e r i o d . The a l t e r n a t i v e s p e r t a i n to the degree of phosphorus removal i n the P e n t i c t o n sewage treatment p l a n t and correspond t o a 70%, 80% and 90% r e -duc t i o n of t o t a l phosphorus i n the e f f l u e n t . 59 100 T 80 19 70 1980 1990 2000 Ye a r Trophic State I 2010 2020 Fig.7.1 EXPECTED UTILITY CURVES FOR SKAHA LAKE CORRESPONDING TO 7 0 % , 8 0 % AND 9 0 % PHOSPHORUS REMOVAL IN THE PENTICTON SEWAGE TREATMENT PLANT. O 61 At the present time, removal e f f i c i e n c y i n the p l a n t , t a k i n g i n t o account both the b i o l o g i c a l treatment and the chemical treatment f a c i l i t i e s , i s averaging about 70% (Brown et al.t 1973). A p l a n n e r , d e a l i n g w i t h the water q u a l i t y problems i n Skaha Lake, must decide i f t h i s degree o f , removal i s s u f f i c i e n t . This d e c i s i o n i s t y p i c a l of the type of judgements that water q u a l i t y managers must make. I t i n v o l v e s a l l the c o n s i d e r a t i o n s of l a k e e u t r o p h i c a t i o n as w e l l as i s s u e s of f u t u r e growth trends, economics and p u b l i c involvement. A comprehensive study of these i s s u e s would r e q u i r e a n a l y z i n g a v a s t amount of data which might not make the question any l e s s d i f f i c u l t f o r the d e c i s i o n maker. However, when the problem i s put i n t o the format of the a l g o r -ithm developed i n the previous chapters, i t i s s i m p l i f i e d c o n s i d e r a b l y . t a i n a proper p e r s p e c t i v e on the problem. According to the curve i n F i g u r e 7.1 corresponding to a 70% phosphorus removal, the l a k e i s shown to d e c l i n e i n u t i l i t y at a f a i r l y r a p i d pace. One must bear i n mind th a t t h i s r e p r e -sents the most l i k e l y outcome and i s not meant to be an accurate p r e d i c t i o n of f u t u r e events. The exact slope of the curve i s not p a r t i c u l a r l y impor-t a n t . The most s i g n i f i c a n t c o n c l u s i o n to be drawn from the expected u t i l i t y curve i s that under the present c o n d i t i o n s of n u t r i e n t l o a d i n g , the l a k e can only decrease i n i t s v a l u e ; i . e . , the t r o p h i c s t a t e w i l l i n c r e a s e w i t h time. Thus, the planners can see that f u r t h e r c o n t r o l s w i l l have t o be implemented. Figure 7.1 a l s o i n d i c a t e s , on the expected u t i l i t y a x i s , the u t i l -i t i e s corresponding to the v a r i o u s t r o p h i c s t a t e s . The d e c i s i o n makers 62 can r e l a t e these values back to the a c t u a l water q u a l i t y by r e f e r r i n g back t o Table 4.1. Thus, they can get some i d e a of what the t r o p h i c s t a t e c o n d i t i o n s of the lake w i l l be f o r each a l t e r n a t i v e A f t e r conducting v a r i o u s t e s t s w i t h the chemical p r e c i p i t a t i o n f a c i l i t i e s i n the P e n t i c t o n sewage treatment p l a n t , the c o n s u l t i n g e n g i -neers now b e l i e v e t h a t the f a c i l i t i e s are capable of removing up to 90% of the t o t a l phosphorus from the i n f l u e n t . In t h i s s i t u a t i o n , u n c e r t a i n t y i s not a major f a c t o r . I t i s simply a matter of i n c r e a s i n g the chemical dosage u n t i l the d e s i r e d removal i s obtained. The expected u t i l i t y curves corresponding to phosphorus removals of 80% and 90% beginning i n the year 1976, are a l s o shown i n F i g u r e 7.1. The management a l t e r n a t i v e s f o r Skaha Lake, i n terms of the sewage t r e a t -able cost f i g u r e s f o r the v a r i o u s phosphorus removal l e v e l s would be ob-t a i n a b l e . With t h i s i n f o r m a t i o n , the planners can then c o n s u l t w i t h p u b l i c r e p r e s e n t a t i v e s to decide upon an a l t e r n a t i v e . .The d i f f e r e n c e i n cost between 70% and 90% phosphorus removal may be i n the order of $25,000 per year. Before approving such an expenditure, the d e c i s i o n makers would have to be convinced that i t can be j u s t i f i e d . With the a i d of the expected u t i l i t y curves, they can get a greater understanding of the d i f -ference between the two options i n terms of the outcome on Skaha Lake. Thus, they would be i n a b e t t e r p o s i t i o n to decide. An 80% phosphorus removal e f f i c i e n c y i n the p l a n t can be expected to improve the q u a l i t y of the lake so t h a t i t w i l l remain c l o s e to s t a t e (4) u n t i l about 1985. A 90% l e v e l of phosphorus removal shoul d s u b s t a n t i a l l y 63 improve the water q u a l i t y of the l a k e . The nature of the d e c i s i o n at t h i s p o i n t i s not one of attempting to j u s t i f y , on a c o s t - b e n e f i t b a s i s , the spending of p u b l i c funds to operate a treatment p l a n t . Nor i s i t one of implementing c e r -t a i n measures to the p l a n t discharge so i t w i l l meet e f f l u e n t standards. Rather, the d e c i s i o n i s d i r e c t l y r e l a t e d to the o b j e c t i v e of improving the water q u a l i t y of the l a k e . The planners might a l s o want to decide upon some s p e c i f i c u t i l i t y to serve as a minimum acceptable design c o n s t r a i n t . They could then deduce from the expected u t i l i t y curves what degree of phosphorus r e -moval would be necessary and hence, how much they should be prepared to spend. Thus, the planners can see immediately the i m p l i c a t i o n of t h e i r r l p i ; - 1 «5-i n n o . - • ••— . - -Other f a c t o r s would a l s o have to be considered by the planners such as when and what type of expansions are r e q u i r e d to i n c r e a s e present p l a n t c a p a c i t y . These c o n s t r a i n t s would have some e f f e c t on the manage-ment d e c i s i o n s and might e l i m i n a t e some a l t e r n a t i v e s . However, even when ta k i n g these i n t o account, the problem i s s t i l l r e l a t i v e l y uncomplicated. The d e c i s i o n makers are given v a r i o u s a l t e r n a t i v e s , the most probable out-comes expected from each a l t e r n a t i v e and the cost estimates. They are spared the task of a n a l y z i n g vast q u a n t i t i e s of data on s u b j e c t s beyond t h e i r e x p e r t i s e and can concentrate d i r e c t l y on the problem. The computer program i s a l s o capable of showing areas of un-c e r t a i n t y i n the form of confidence l i m i t s . Thus, a l i n e can be p l o t t e d that t e l l s the d e c i s i o n makers t h a t there i s a c e r t a i n percent chance 64 t h a t the expected u t i l i t y w i l l be above that l i n e . I n t h i s manner, they can see the extent of u n c e r t a i n t y . Expected u t i l i t y curves showing the 10% and 90% confidence l i m i t s are shown i n Appendix C. B. LONG RANGE PLANNING. The a l g o r i t h m has a p o t e n t i a l of becoming a u s e f u l t o o l f o r l o n g range planning of e u t r o p h i c a t i o n c o n t r o l measures. The expected u t i l i t y curves f o r Skaha Lake, i l l u s t r a t e q u i t e c l e a r l y the n e c e s s i t y f o r c o n s i d e r i n g l o n g term events f o r c u l t u r a l l y e u t r o p h i c l a k e s . I t i s becoming very apparent that i n order to p r o t e c t n a t u r a l o r d e s i r a b l e en-vironments, c l e a r o b j e c t i v e s are necessary. Long term comprehensive planning i s needed to co-ordinate the development of populated areas so c h a t u e S x g i i o b j e c t i v e s c a n b e ctu L c t X u e u • L>xiux x a i x y , C u e xuip l e i u c u L t V C i u i i Ox v a r i o u s p o l l u t i o n c o n t r o l measures should be c a r r i e d out such that the pre-sent a c t i o n s are compatible w i t h l o n g term goals. A l t e r n a t i v e s should be judged according to t h e i r u l t i m a t e u t i l i t y i n the eventual scheme. At t h i s l e v e l of planning f o r a populated l a k e b a s i n , the procedure i s j u s t an extension of the short range p l a n n i n g . One could i d e n t i f y a de-s i r a b l e t r o p h i c s t a t e to serve as an o b j e c t i v e . Working on the assumption t h a t the t r o p h i c l e v e l i n a l a k e i s determined by the annual phosphorus l o a d , the o b j e c t i v e c o n s i s t s p r i m a r i l y of m a i n t a i n i n g the l o a d i n g below a c e r t a i n maximum amount. Th i s amount i s found by the r e l a t i o n (6.1), S = S ( L ^ ) . The planners can then consider the v a r i o u s methods of keeping the phosphorus l o a d i n g a t t h i s l e v e l . Factors to be taken i n t o account ar e : r a t e of growth of the annual l o a d i n g ; design, c a p a c i t y and costs of the 65 various a l t e r n a t i v e s . E q u a l l y important i s the a b i l i t y to expand a given phosphorus removal f a c i l i t y when i t s design c a p a c i t y i s reached, so t h a t i t s t i l l remains u s e f u l and i s capable of combining w i t h a new updated f a c i l i t y . The planners can get some i d e a of p o s s i b l e long term events by observing the expected u t i l i t y curves correspon.ding to d i f f e r e n t management schemes. A management scheme may c o n s i s t of s e v e r a l p o s s i b l e a l t e r n a t i v e s to be introduced p e r i o d i c a l l y over the long range p l a n n i n g p e r i o d . A num-ber of schemes would be p o s s i b l e and v a r y i n g the sequence of the a l t e r -n a t i v e s could s i g n i f i c a n t l y a f f e c t the c o s t s . Thus, planners could com-pare schemes and look f o r the optimum system to meet t h e i r d e s i r e d g o a l . The planners could p o s s i b l y take advantage of sequencing techniques used *„'.- ^ j ... •_• . ~ .-. ... - 0 •-- ~ • •-. — j.- c — - . the one developed by Tsou et at.. (1973). A very simple demonstration of how l o n g term p l a n n i n g c o n s i d e r -a t i o n s can a f f e c t the e v a l u a t i o n of a l t e r n a t i v e s can be seen i n the Skaha Lake example. Land d i s p o s a l o f the sewage"effluent from the P e n t i c -ton sewage treatment p l a n t was suggested i n 1967 as an a l t e r n a t i v e to t e r -t i a r y treatment (O'Riordan, 1972). Although the technology behind t h i s type of method was not very f a r advanced, the i d e a s t i l l m e rited i n v e s t i -g a t i o n . However, the proposal.vwas defeated and s i n c e then has not been s e r i o u s l y considered. F i g u r e 7.2 shows the expected u t i l i t y curve of the l a k e t h a t could r e s u l t i f l a n d a p p l i c a t i o n of sewage e f f l u e n t would be im-plemented i n 1976. In t h i s p e r s p e c t i v e , the a l t e r n a t i v e appears very a t t r a c -t i v e . The l a k e would not be r e c e i v i n g any phosphorus load from the C i t y of Trophic State 1 IOO Fig.7.2 EXPECTED UTILITY CURVE FDR SKAHA LAKE CORRESPONDING TO LAND APPLICATION OF PE.NTICTON SEWAGE EFFLUENT. 67 P e n t i c t o n and consequently would not be a f f e c t e d by i t s growth. The phos-phorus loa d c o n t r i b u t e d by other sources does not i n c r e a s e by a s i g n i f i -cant r a t e and would not have a severe e f f e c t on the water q u a l i t y o f the lake i n the f u t u r e . There are, of course, many other f a c t o r s to con s i d e r such as land a c q u i s i t i o n , sewage t r a n s m i s s i o n , a p p l i c a t i o n methods and above a l l economics. But, other a l t e r n a t i v e s to l a n d a p p l i c a t i o n s h o u l d be weighed according to the cost of p r e s e r v i n g the l a k e a t the e q u i v a l e n t t r o p h i c s t a t e f o r an equal p e r i o d of time. An a l t e r n a t i v e such as chemical p r e c i p i -t a t i o n i s sim p l e r i n design and implementation and l e s s c o s t l y i n terms of c a p i t a l investment than l a n d a p p l i c a t i o n . However, when the two are compared on a lon g term b a s i s according to the c o n d i t i o n s c i t e d above, laud a p p l i c a t i o n c o u l d very l i k e l y appear to be mure xavuiable. CHAPTER V I I I DISCUSSION AND CONCLUSIONS The a l g o r i t h m developed i n t h i s t h e s i s i s an a i d f o r d e c i s i o n makers i n the management of c u l t u r a l l y e u t r o p h i c l a k e s . I t i s not intended to replace the e x p e r t i s e and i s not capable of g i v i n g the d e c i s i o n makers the p e r f e c t s o l u t i o n . The a l g o r i t h m c o n s i s t s of a r e l a t i v e l y simple mathe-m a t i c a l r e l a t i o n t h a t c a l c u l a t e s the expected u t i l i t y from a given annual phosphorus load to a l a k e . U t i l i t y i s a non-monetary measure of worth o f the l a k e d e r i v e d by e v a l u a t i n g the v a r i o u s t r o p h i c s t a t e s of the l a k e against the r i s k s l - ! - t _ • ~ ~ - C J U 1 . _ - 1 - . 1 - . - » 1 - 1 . - ( „ - . - t -1 - . - - -»-< J- i i U * x u g v j _ i . v / i _ i i u ' v i i i g i»o c : v> t u c J - C U ^ C I r i o - C i i w u g i i k.iijL£> m c L i i u u c i p p t _ ci . J . o somewhat a r b i t r a r y , i t i s able i n t h i s case to reduce a complex phenomena onto a one dimensional s c a l e . Furthermore, the v a r i a t i o n i n o p i n i o n t h a t would produce d i f f e r e n t shaped u t i l i t y curves, would not s i g n i f i c a n t l y e f f e c t the usefulness of the expected u t i l i t y f u n c t i o n . At t h i s stage, the a l g o r i t h m does not account d i r e c t l y f o r major f a c t o r s such as annual v a r i a t i o n s i n h y d r a u l i c flow, phosphorus r e t e n t i o n i n the l a k e , c l i m a t i c c o n d i t i o n s or p o s s i b l e e f f e c t s of other n u t r i e n t s . Instead, the impact of these f a c t o r s are presumably i n c l u d e d w i t h i n the estimated areas of u n c e r t a i n t y . These estimates can be e a s i l y changed i n l i g h t of any new i n f o r m a t i o n . Because of the r e l a t i v e l y s t r a i g h t f o r -ward method of c a l c u l a t i n g the expected u t i l i t y v a l u e s , i t would not be d i f f i c u l t to i n c l u d e some of these elements more d i r e c t l y i n t o the a l g o r -ithm. Phosphorus r e t e n t i o n could be included by having p a r t of the annual 68 69 phosphorus l o a d c a r r y over to the f o l l o w i n g year. The r e t e n t i o n f a c t o r might vary according to the t r o p h i c s t a t e i n the lak e and thus be c a l c u -l a t e d from that v a l u e . C l i m a t i c c o n d i t i o n s which i n tu r n determines h y d r a u l i c flow as w e l l as the photosynthetic r e a c t i o n s i n a lak e would be more d i f f i c u l t to i n c l u d e d i r e c t l y i n t o the model. Accurate p r e d i c t i o n s of c l i m a t i c changes fo r p e r i o d s i n t o the future are s c i e n t i f i c a l l y i m p o s s i b l e . However, p r o b a b i l -i t y estimates w i t h regard to the h y d r a u l i c f l o w and hours o f sunshine are obtainable and might p o s s i b l y be inc o r p o r a t e d i n t o the model. Water q u a l i t y management of a c u l t u r a l l y e u t r o p h i c l a k e i s a com-plex problem f o r d e c i s i o n makers. The advantage of the technique developed i n t h i s t h e s i s i s that i t p r o v i d e s the d e c i s i o n makers w i t h a systematic piOCcuUie Lo f o l l o w . j-iie uiecauCi gives Liieui a C l e a r perspective o i che prob-lem t h a t does not i n v o l v e an excessive amount of data o r s c i e n t i f i c theory. Environmental c o n s i d e r a t i o n s are not placed on a d o l l a r s c a l e i n an attempt to j u s t i f y c o s t s . Instead, d e c i s i o n s are d i r e c t e d a t the i s s u e o f managing the water q u a l i t y i n the l a k e . The v a r i o u s d i s c i p l i n e s i n v o l v e d i n the problem a r e co-ordinated such that the d e c i s i o n maker can u t i l i z e the s u b j e c t i v e o p i n i o n of the v a r -ious experts. Estimates regarding the u n c e r t a i n t i e s are made i n an e x p l i c i t manner and can be changed i n l i g h t of new i n f o r m a t i o n . Each expert c o n t r i -butes to the d e c i s i o n making process i n t h e i r area of e x p e r t i s e only and the f i n a l d e c i s i o n s can be made by management. The emphasis i s s h i f t e d from r e g a r d i n g the problem as one time d e c i s i o n event to the need f o r long term management p o l i c i e s . The te c h -70 nique permits the d e c i s i o n makers to r e a l i z e the l o n g term i m p l i c a t i o n s of a c u l t u r a l l y e u t r o p h i c l a k e and to manage the l a k e on that b a s i s . B I B L I O G R A P H Y B e l l a , D.A. and W, S. Overton (1972). "Environmental P l a n n i n g and Eco-l o g i c a l P o s s i b i l i t i e s , " Proc. A.S.C.E., J. Sanit. Eng. Div., 98 (SA3): 579-592. Brown, Murray, Alex F o r s y t h and Don Smith. Experiences with Tertiary Treatment at Penticton, B.C., Canada. Presented a t P a c i f i c North-west P o l l u t i o n C o n t r o l A s s o c i a t i o n Conference, Vancouver, B.C. (October, 1973), mimeo. Canada - B r i t i s h Columbia Okanagan Basin Agreement. D r a f t copy of The Final Report of the Consultative Board, November, 1973. P u b l i s h e d by the O f f i c e of the Study D i r e c t o r , P e n t i c t o n . B.C. C a s t l e s , F. G., D. J . Murray and D. C. P o t t e r (1971). Decisions, Organi-zations and Society. Middlesex, England: Penguin Books. Edmondson, W. T. (1968) i n Eutrophication: Causes, Consequences, Correc-tives, P u b l . 1700. Washington, D.C.: N a t i o n a l Academy of Sc i e n c e s , 124-129. , (1970). "Phosphorus, N i t r o g e n and Algae i n Lake Washing-ton A f t e r D i v e r s i o n of Sewage," Science, V o l . 169: 690-691. Edwards, Ward. (1972). " S o c i a l U t i l i t i e s , " Decisions and Risk Analysis, Powerful New Tools for Management, Ed. Arthur L e s s e r , J r . Hoboken, New Jersey: The Engineering Economist, Stevens I n s t i t u t e of Technology. H a l t e r , A l b e r t N. and G. W. Dean. (1971). Decisions Under Uncertainty with Research Application. C i n c i n n a t i : South-Westem P u b l i s h i n g Co. l Lee, G. F. (1970). Eutrophication. Madison, W i s e : U n i v e r s i t y of Wis-consin Water Resources Center. Mackenthum, K. M., W. M. Ingram and Ralph Porges. (1964). Lirmological Aspects of Recreational Lakes. Washington, D.C.: U.S. Department of H e a l t h , Education and Welfare. 71 "72 Milway, C. P. (1968). Eutrophication in Large Lakes and Impoundments. Symposia D'Uppsala, Organ. Econ. Coop. Develop. Northcote, T. G., T. G. Halsey and S. J . McDonald. (1972). Fish as Indicators of Water Quality in the Okanagan Basin Lakes, British Columbia, P r e l i m i n a r y Report No. 22, Can a d a - B r i t i s h Columbia Okanagan Basin Agreement. O'Riordan, Jonathan and Timothy O'Riordon. (1972). Okanagan Water De-cisions. U n i v e r s i t y o f V i c t o r i a , Western Geographical S e r i e s , V o l . 4. P a t a l a s , Kazimierez and Alex S a l k i . (1972). Crustacean Plankton and the Eutrophication of Lakes in Okanagan Valley, B.C. Winnipeg, Man.: F i s h e r i e s Research ;Board o f Canada, Freshwater I n s t i t u t e . P a t t a n a i k , P. K. (1968). " R i s k , Impersonality and S o c i a l Welfare F u n c t i o n , " Journal of Political Economy, 76: 1152-1169. Trophic State and N i t r o g e n and Phosphorus Loading Rates," Environ-mental Science and Technology, 6(8): 719-725. Stockner, J . G., W. Carney and G. McKenzie. (1972). The Phytobenthos in Skaha Lake, British Columbia, P r e l i m i n a r y Report, C a n a d a - B r i t i s h Columbia Okanagan Basin Agreement. . (1973). Phosphonrus as a Continuous Element Regulating Growth in Okanagan Basin Lakes. Vancouver, mimeo. . (1974). P e r s o n a l c o n v e r s a t i o n s . Swalm, R. 0. (1972). " I n t r o d u c t i o n t o U t i l i t y Theory," decisions and Risk Analysis, Powerful New Tools for Management, Ed. Arthur L e s s e r , J r . Hoboken, New Jersey: The Engineering Economist, Stevens I n s t i -t u t e of Technology. ' Te c h n i c a l Supplement IV. (1974). C a n a d a - B r i t i s h Columbia Okanagan Basin Agreement,. Water Quality and Waste Loadings in the Okanagan Basin. 73 Tsou, C. A., L. G. M i t t e n and S. 0. R u s s e l l . (1973). "Search Technique f o r P r o j e c t Sequencing," P r o c , A.S.C.E., J. Hyd. Div., 99 (HY5): 833-839. Vol l e n w e i d e r , R. A. " S c i e n t i f i c Fundamentals of the E u t r o p h i c a t i o n of Lakes and Flowing Waters, With P a r t i c u l a r Reference to N i t r o g e n and Phosphorus as Factors i n E u t r o p h i c a t i o n , " Organ. Econ. Coop. Develop. Rep. DAS/CSI, P a r i s , Frances, 68-27, May, 1968. . (1973). Input-Output Models. (Mimeo). Von Neumann, J . and 0. Morgenstern. (1953). Theory of Games and Economic Behavior. P r i n c e t o n , N.J.: P r i n c e t o n U n i v e r s i t y P r e s s . White, G. F. (1969). Strategies of American Water Management. Ann Arbor, Mich.: U n i v e r s i t y of Michigan P r e s s . Mains tern Lakes, P r e l i m i n a r y Report No. 18 (Suppl. 1 ) , C a n a d a - B r i t i s h Columbia Okanagan B a s i n Agreement. Wi n k l e r , Robert L. (1972). An Introduction to Bayesian Inference and Decision. New York: H o l t , Rinehart and Winston, I n c . Zwick, David and Marcy Benstock. (1971). Water Wasteland. New York: Bantam Books and Grossman P u b l i s h e r s Inc. A P P E N D I C E S 74 APPENDIX A THE AXIOMS OF COHERENCE These axioms provide a s t r a t e g y on how a " r a t i o n a l person" makes a d e c i s i o n when f a c i n g u n c e r t a i n t y . This s e t of axioms i s not unique. Other set s of axioms a l s o imply the e x i s t a n c e of u t i l i t y and s u b j e c t i v e p r o b a b i l i t y , but are b a s i c a l l y a m o d i f i c a t i o n of the s i x axioms that are presented below (Winkler, 1972). AXIOM 1: Given any two payoffs and R^ (including monetary and/or nonmonetary factors, the payoffs can be thought of as potential consequences in a decision making pro-cess), one can decide whether they prefer R to R , R^ and AXIOM 2: If one prefers payoff R^. to payoff R2 and one also pre-fers payoff R0 to payoff R , then they must prefer R to R^. AXIOM 3: If one prefers R to R^ and i?„ to R„, then some value of P exists such that a P-mixture of R and R~ is pre-ferred to R^; one can also find some other value of P such that i?„ is preferred to a P-mixture of R^ and R^l and finally, one can find s t i l l another value of P such that they are indifferent between R and P-mixture of R and R^. AXIOM 4: If Rj is preferred to i?^  cmd R^ is some other payoff, then any P-mixture of i?7 and R„ is preferred to the T-i • _|_ 7") -h n & same P-mixture of R2 and R^. AXIOM 5: If one is indifferent between R^ and then they may be substituted for each other as payoffs in a decision making problem. 75 76 AXIOM 6: If R %s preferred to R , then a P-mixture of R and R 0 i s preferred to a Q-mixture of i?w and Rn if P > Q. 1 2 APPENDIX B DERIVATION OF SKEW NORMAL DISTRIBUTION This d i s t r i b u t i o n was devised by Dr. Rabab Ward f o r use i n a water flow f o r e c a s t i n g technique. The shape of the den s i t y f u n c t i o n i s shown i n Figure B . l . 0 A AE = 3oj EB = 3cr F i g u r e B . l For x <_ u, the d e n s i t y f u n c t i o n i s F< x> - < y j b </2,rd^:> ( e " 2 o i v 2 J -4.5. e ) f o r x > u .x-y.2 -4.5, e ) 77 78 where a = .0027 + 6e •4.5 /2TT ( i ) I n t e g r a t i n g the above from A to B, the area under the curve can be shown to equal one. D e r i v a t i o n ; Two h a l f normal curves, both w i t h mean values u i b u t having standard e r r o r s and a^, are shown i n F i g u r e B.2. F i g u r e B.2 The area under the curve from -» to u i s h and the area from u to +«> i s a l s o k. The f u n c t i o n i s given as and •2TF O X 2a 1 - ( ^ ) e v„ 2 f o r x < u /2TT a 2 2 a 2 f o r x > u ( i i ) ( i i i ) 79 °1 °2 M u l t i p l y i n g ( i i ) b y ( — r — ) and ( i i . i ) by( ; ) y i e l d s a curve R J \ J J 0 ^ + A ^ v •> o +0"^  — — 0 l as i n F i g u r e B.3, where the area under the curve up to u i s ; a +o„ " °2 1 2 and the area under the curve beyond u i s ; . The t o t a l area i s ° l + a 2 equal to 1. Fi g u r e B,3 The values of the density f u n c t i o n s a t A and B where (u-A) = 3a.. and (u-B) = 3a,,, are both equal to = o /2ir aj*~a2 6 -4.5 In F i g u r e B.3, the area under the curve from -» to A, where (u-A) = 3a^, i s given i n the normal d e n s i t y t a b l e as .0027 and s i m i l a r l y , the .0027 area from B to +°° i s a l s o equal to '""^. Thus i n F i g u r e B.3, the t o t a l area from -<» to A and that from B to » i s shown to be .0027. S h i f t i n g the X-axis upwards so that the curve cuts the a x i s a t A and B, as i n F i g u r e B.4, the t o t a l area under the curve would be diminished by: 6e -4.5 .0027 + p • • 3(a.+a.,) which equals .0027 + o l Z /2TT = a. 80 F i g u r e B.4 The area new i s equal to (1-a). D i v i d i n g by (1-a), the area under the curve from A to B would now equal 1, and the equation c o r r e s -ponds to equation ( i ) of the skew normal f u n c t i o n . A P P E N D I X C PHOSPHORUS LOADING PROJECTIONS AND CONFIDENCE LIMITS 81 o - j , , 1 i — — — — n 1970 1980 1990 2000 2010 2 0 2 0 Y e a r Fig. CI PROJECTIONS OF PHOSPHORUS LOADING INTO SKAHA LAKE CORRESPONDING TO 7 0 % , 8 0 % AND 9 0 % PHOSPHORUS REMOVAL AND LAND APPLICATION OF SEWAGE . oo ~~ Trophic State I U U T F ig .C2 EXPECTED UTILITY CURVE WITH IO%AND 9 0 % CONFIDENCE LIMITS CORRESPONDING TO 7 0 % PHOSPHORUS REMOVAL. 00 Co Trophic State I 1970 1975 1980 1985 1990 Y e a r 1995 2000 2005 1 2010 Fig. C 3 EXPECTED UTILITY CURVE WITH I0%AND 9 0 % CONFIDENCE LIMITS CORRESPONDING TO 8 0 % PHOSPHORUS REMOVAL. 1970 1975 1980 1985 1990 1995 2000 2005 2010 Ye a r F i g . C 4 EXPECTED UTILITY CURVE WITH 10% AND 9 0 % CON Fl DENCE LIMITS CORRESPONDING TO 9 0 % PHOSPHORUS REMOVAL. )—•*! — " | ' • .yTanmiiMii» I || ii i — i n i p i i m i i m M M j .-.HUTM.MM nnvimnuM.^ 1970 1975 1980 1985 1990 1995 2000 2005 2010 Y e a r F ig .CS EXPECTED UTILITY CURVE WITH 10% AND 9 0 % CONFIDENCE LIMITS CORRESPONDING TO LAND APPLICATION OF-SEWAGE EFFLUENT. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0050515/manifest

Comment

Related Items