Journal of Environmental Radioactivity 49 (2000) 319}344
The application of the lake ecosystem index in multi-attribute decision analysis in radioecology Lars Ha kanson *, Eduardo Gallego, Sixto Rios-Insua Uppsala University, Institute of Earth Sciences, Villav. 16, 752 36 Uppsala, Sweden Nuclear Engineering Department, Polytechnic University of Madrid, Jose& Gutie& rrez Abascal, 2, 28006 Madrid, Spain Artixcial Intelligence Deptartment, Polytechnic University of Madrid, Boadilla del Monte, 28660 Madrid, Spain Received 13 August 1999; received in revised form 19 November 1999; accepted 24 November 1999
Abstract This work gives a summary of multi-attribute analysis (MAA) and its use in decision support systems for radiological and environmental contamination problems and presents a modi"cation of the lake ecosystem index (LEI) as a tool to give an holistic account for the environmental (and not just radiological) consequences of chemical remedial measures (lake and wet land liming, potash treatment and lake fertilisation) carried out to reduce radionuclide levels in water, sediments and biota. The "rst step in determining a LEI-value is to set normal or initial values of two important limnological state variables, pH and total-P. The second step involves predicting state indices describing the abundance of key functional groups (the "sh yield and biomasses of phytoplankton and bottom fauna). The next step concerns the de"nition of a lake ecosystem index based on the state indices. The "nal step is the derivation of the utility function to be used in the multi-attribute analysis to compare environmental, economical and social attributes of di!erent dimensions (ECU, kg, Bq/kg, etc.). The ecosystem index characterises the entire lake over longer periods of time (months), and not speci"c sites in lakes or speci"c sampling events. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Radioecology; Lakes; Models; Management; Lake ecosystem index; Multi-attribute analysis; Decision support system
1. Introduction The basic aim of this work is to present a new version of the lake ecosystem index (LEI; see Ha kanson & Peters, 1995; Ha kanson, 1996 for further information about * Corresponding author. Tel.: #46-18-471-3897; fax: #46-18-471-2737. E-mail address:
[email protected] (L. Ha kanson). 0265-931X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 5 - 9 3 1 X ( 9 9 ) 0 0 1 2 6 - 5
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LEI) to be used within the framework of a multi-attribute decision support system in aquatic radioecology (see Rios-Insua, Mateos & Gallego, 1997; Monte, Hakanson & Brittain, 1997; and Gallego, Rios-Insua, Mateos & Rios-Insua, 1998 for further information about this multi-attribute analysis system). The work is carried out within the framework of the MOIRA-project, (an EU-project in Radioecology) the objective of which is to construct a model-based decision support system (DSS) to identify optimal remedial strategies for restoring radionuclide-contaminated aquatic ecosystems (see Appelgren et al., 1996; Monte et al., 1997). Within this overall objective, there are three speci"c goals of this paper: Firstly, to give a rationale for this work by presenting a summary of MAA and its applications to radiological and ecological problems; secondly, to motivate the modi"cations and simpli"cations of the existing LEI-model and to illustrate the practical use of the lake ecosystem index; and "nally, to de"ne an utility function, u(x), which is needed in the MAA to compare environmental, economical and social attributes of di!erent dimensions (ECU, Bq/kg etc.). Within the MOIRA-project, new models have been developed to predict realistic water chemistry changes (e.g., in lake pH) from lake liming (see Ottosson & Ha kanson, 1997), changes in K-concentration from potash treatment (Abrahamsson & Ha kanson, 1997) and changes in total-P concentrations from lake fertilisation (Ha kanson, Abrahamsson, Ottosson & Johansson, 1998). By means of the overall lake model for radiocesium (the modi"ed VAMP-model; see Ha kanson, Heling, Brittain, Monte, Bergstrom & Suolanen, 1996), one can then also predict how these changes in water chemistry in#uence the concentrations of radiocesium in water, sediments and biota ("sh). However, there may also be negative e!ects on a lake ecosystem from these remedial measures, e!ects that should be taken into account before arriving at an optimal decision. The lake ecosystem index (LEI) is utilised in this context as a tool to account for such potential problems in a simple, rational manner from a radioecological perspective. The lake ecosystem index is based on the fact that changes in phosphorus concentrations and lake pH in#uence biomasses of key functional organisms, and hence cause changes to the structure and function of lake ecosystems (Ha kanson & Peters, 1995). By de"nition, the ideal LEI-value should be one. This is when the actual pH and total phosphorus (TP) values are equal to the lake characteristic (normal) reference values. Then the actual biomasses of the selected key functional organisms (phytoplankton, "sh and benthos) are also equal to the lake characteristic reference values. Any departure from the normal reference values for pH and TP, e.g. related to remedial measures (liming, potash treatment and fertilisation) will create changes expressed in the LEI-value. It should be stressed that the approach to determining the lake ecosystem index is intentionally very simple. Ecosystems are very complex (see Ha kanson & Peters, 1995) since many chemical, physical and biological factors and processes interact in such a way that even very extensive models are simplistic. Large models are comparatively easy to build but are very di$cult to validate, especially for many individual ecosystems. The greatest challenge is to build models yielding good predictions of target variables based on the smallest possible number of driving variables (see Ha kanson, 1995 for a discussion about the optimal size of predictive models).
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This new and simpli"ed version of the original LEI-model is meant to relate directly to the listed chemical remedial measures used in the MOIRA project, lake and wet land liming, potash treatment and fertilisation. This means that the original submodels in LEI for lake colour and mercury in "sh (see Ha kanson, 1996) are omitted in this approach. The structure of this paper is as follows. First, there is a section describing MAA. The aim is to give the overall framework for how LEI is used as an environmental attribute together with economic and social attributes in the search for a holistic and practical evaluation tool for the consequences of various remedial actions (doing nothing is also an action). Then, there is a section explaining why pH and phosphorus can be regarded as limnological state variables and can be utilised as the basis for the LEI. Section 4 de"nes the LEI and develops the equations relating changes in pH and TP to changes in LEI. In MAA, it is fundamental to be able to compare attributes of di!erent dimensions, e.g. monetary units and kg of "sh. The tool to obtain dimensionless comparative units is called the `utility functiona. In Section 5, we describe how the utility function for LEI is de"ned. Finally, in Section 6, we give a scenario meant to illustrate the practical applicability of these concepts in radioecology and water management.
2. Multi-attribute analysis (MAA) MAA is a technique recommended by the International Commission on Radiological Protection (ICRP) for use in optimisating problems (ICRP, 1989). It has also been applied to evaluate di!erent problems in which environmental impact was one of the main issues to consider. For instance, Merkhofer and Keeney (1987) used MAA for the evaluation of alternative sites for the disposal of nuclear waste; Seip, Ibrekk and Wenstop (1987) for the analysis of phosphorus abatement measures to improve lake water quality. The accident at the Chernobyl reactor in 1986 showed the complexity of the decisions both during the emergency and after. Since then, several applications of MAA in decision support systems (DSS) have been reported, e.g. by the International Chernobyl Project, a collaborative project undertaken by seven international agencies, co-ordinated by the IAEA. The structuring of the decision problems into key socio-economic factors, together with the physical, radiological and medical information allowed a more objective and clearer evaluation of the protective measures taken in the Republics a!ected by the accident (see French, 1991). The RADE-AID Project (of the European Commission) discussed the development of a MAA-based DSS to be used after radiological accidents (Wagenaar, 1990). MAA has been used more recently in RODOS, a comprehensive system developed by an European Consortium, which is intended to provide support from the moment of an accidental release, to months or years after an accident, at all distances from the release point. Multi-attribute decision methods have proved to be very helpful in balancing the many con#icting objectives under such circumstances (French, Papamichail, Ranyard & Smith, 1998). So, MAA is an activity with a long history. It consists of a decomposition technique for structuring and solving multi-attributed decision-making problems, where a given
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set of available actions must be evaluated with regard to multiple, often con#icting objectives. This means that doing well with regard to one objective means normally doing poorly with regard to another. The decision-maker (DM) is required to select what he/she considers the `besta action based on trade-o!s between objectives and on the relative importance given to them. The theory of such an approach has been formalised (see, e.g. Keeney & Rai!a, 1976) and many applications have been reported in di!erent "elds. MAA breaks down a decision problem into three general areas: problem structure, uncertainties in the outcomes of the actions, and DM's preference. The structuring process includes the identi"cation of options or alternative actions and the development of a hierarchy of objectives. Objectives are often hierarchical in nature and the overall objective can be divided into sub-objectives. These sub-objectives can be further divided until attributes are identi"ed on which the accomplishment of the objectives can be measured. This structuring process of decision problems is often more `arta than science since there is no formal theory guiding this step. However, creating a hierarchy is always bene"cial, since it allows the disaggregation of highly complex problems into their components. The second stage is related to the probability model, where the DM identi"es possible uncertain variables and assesses the probability distributions for the outcomes of the alternative actions. Finally, a preference model is constructed, represented through a multi-attribute utility function, of additive type in the case of the MOIRA-system, where the DM is "rst elicited to give scores for each of the actions with respect to each attribute in order to assess the individual utility functions. Next, he/she assigns relative weights to the attributes and objectives that express the trade-o!s among attributes. By means of the individual utility functions each action receives a given score in the interval [0,1] with respect to a given attribute. The shapes of these individual utility functions depend very much on the nature of the attributes. Linear and non-linear, monotonically increasing or decreasing functions, step-wise functions etc. are acceptable choices if they adequately represent the DM's preferences with respect to that attribute; score 1 always meaning the most preferred option, score 0 the least. Figs. 1 and 2 give a schematic layout of the MOIRA-system for MAA. The display in Fig. 1 works interactively to de"ne a hierarchical tree of objectives. The di!erent branches may be easily activated or deactivated, depending on the nature of the problem analysed or on the availability of data for the attributes of the top level: radiation doses, economic costs, health impact etc. At the root there are three de"ned objectives, environmental, social and economic impacts. The environmental impact has never before been incorporated as an objective in similar DSS-systems. Two attributes are considered to represent the environmental impact: the LEI and the radiation dose to biota ("sh). The social impact is handled by two objectives: minimising impacts on health and on living conditions. The radiation dose is an obvious factor concerning health; in our case, we focus on dose to critical individuals, who should never receive levels above the thresholds for early health e!ects, and collective dose, which, according to the usually accepted Non-Threshold Linear hypothesis (ICRP, 1991), induces a linear increase on the risk of developing serious latent e!ects, mainly cancer. The overall health status of people can also be
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323
Fig. 1. Main display of the module for multi-attribute analysis (MAA) used in the MOIRA decision support system system, with the hierarchy tree of objectives considered. The overall objective can be expressed as three sub-objectives, minimising the environmental, social and economic impacts. These objectives are quanti"ed by di!erent attributes. The aim of this MAA is to obtain a holistic picture of bene"ts and drawbacks of di!erent remedial strategies in aquatic radioecology.
a!ected by an accident situation and the countermeasures and by several less speci"c stress-related ewects. For the **living restrictions++, other impacts are taken into consideration, e.g. those of countermeasures a!ecting drinking water and water used by the food industry, the direct use of "sh for food or its processing in the food industry (canned, smoked, salted, etc.), the use of water for irrigation of crops and the recreational uses of water bodies. For all these objectives, the attributes will be the number of persons, area of crops or amount of "sh a!ected by restrictions and the duration of such restrictions. Finally, the economic impact will be decomposed into direct e!ects, more amenable to quanti"cation, and the intangible e!ects, like loss-of-image and adverse market reactions for the concerned area. Within the direct e!ects, the costs caused by the di!erent bans or restrictions to normal living conditions can be sub-divided into costs to the economy and the more subjective costs of recreation lost, and the costs of chemical and physical remedial countermeasures. An example has been prepared to illustrate the basic principles of MAA, evaluating eight alternative remedial measures. This example uses data from a Norwegian lake ("vre Heimdalsvatn; see Ha kanson, 1996), but it could have been any lake, river or coastal area. The measures in this scenario are: no action (which is also an action with de"ned social and economic consequences from a radiological perspective) a "sh ban,
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Fig. 2. Example of results of MAA for a given lake scenario. The upper display shows the overall utility and the ranking of eight alternative strategies being evaluated. The lower display includes the values of attributes for a certain strategy and the intermediate weights assigned to each objective of the hierarchy tree.
lake liming, lake liming plus "sh ban, potash treatment, potash plus "sh ban, fertilisation and fertilisation plus "sh ban. These actions in#uence the environmental, social and economic attributes, such as the individual and collective doses to man. They can cause stress and discomfort to a certain number of persons for a given time in the given area and they can in#uence the income of "shermen, cause economic losses for the tourist industry etc. (see Fig. 1).
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The "rst display in Fig. 2 is the result of the full MAA for the Norwegian lake and shows the relative ranking of the eight strategies analysed, with their respective overall utilities. The "rst place was taken by the strategy consisting of fertilisation plus "sh ban, closely followed by liming plus "sh ban and fertilisation. The last one was the no action strategy. The second display in Fig. 2 contains the attribute values for a given strategy and the average weights assigned to the objectives in each branch junction. This kind of information will be of great help to the DM in understanding the real meaning of a given decision and the reasons behind it. The interactive software allows an easy sensitivity analysis and multiple consistency checks through the decisionmaking process. Some of the measures, such as a "sh ban in this example, have little or no environmental impact. All the chemical remedial measures may in#uence the concentrations of radionuclides (e.g. radiocesium) in lake water and "sh. This is considered by the overall lake model (the VAMP-model for radiocesium) and by the sub-models for the remedial measures (e.g. the potash treatment sub-model). The chemical remedial measures may also in#uence the structure, reproduction and biomasses of key functional groups of organisms. The lake ecosystem index is included in the system to handle this aspect. A detailed approach would be to consider the growth and reproduction of all individual species and their interactions with other species and with the physical and chemical environments (see Hynes, 1970; Vannote et al., 1980; Craig, 1980). This is, however, quite unrealistic in the context of practical radioecology, both in modelling terms and in terms of our present ecosystem knowledge (Appelgren et al., 1996). Within radioecology, the LEI-approach is meant to be simple, but rational. It focuses on the same three trophic levels as the original LEI-model, "sh (at the top of the trophic system) phytoplankton (at the bottom of the trophic system), and benthos (at the bottom of the lake).
3. The state variables * total-P and pH The most important nutrients in aquatic ecosystems are phosphorus and nitrogen. Total phosphorus (TP) has long been recognised as the nutrient most likely to limit lake primary productivity (Schindler, 1977,1978; Chapra, 1980; Wetzel, 1983). Several compilations of models, theories and approaches to the role of phosphorus in lake eutrophication exist (Chapra & Reckhow, 1979,1983; Vollenweider, 1968,1976). Both experimental and comparative studies of whole lake ecosystems have been used to derive loading models for lake management (Dillon & Riegler, 1974; Schindler, 1978; OECD, 1982). The concentration of total phosphorus can be linked to most other biological lake characteristics (Table 1). Thus, the concentration of total-P in the lake water is a powerful predictor in limnology and one of the traditional key variables used to classify lakes (see Table 2). There is a signi"cant overlap between the classes ultra-oligotrophic, oligotrophic, mesotrophic, eutrophic and hypertrophic in Table 2. This is further shown in Fig. 3 for total-P and a key operational variable for eutrophication e!ects, lake chlorophyll-a
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Table 1 Selected regressions illustrating the key role of lake total phosphorus in predictive limnology y-value
Equation
Range for TP
r value
n
Units
Chlorophyll Max. prim. prod. Mean prim. prod. Nanoplankton Phytoplankton Fish yield Macrozooplankton Zooplankton Bacteria Net plankton Fish Crustacean plankton Microzooplankton Blue}greens Benthos
y"0.073 TP y"20 TP-71 y"10 TP-79 y"17 TP y"30 TP y"7.1 TP y"20 TP y"38 TP y"0.90 TP y"8.6 TP y"590 TP y"5.7 TP y"17 TP y"43 TP y"810 TP
3}300 7}200 7}200 3}80 3}80 10}550 3}80 3}80 3}60 3}80 10}550 3}300 3}80 8}1300 3}100
0.96 0.95 0.94 0.93 0.88 0.87 0.86 0.86 0.83 0.82 0.75 0.72 0.72 0.71 0.48
77 38 38 23 27 21 12 12 12 23 18 49 12 29 38
mg/m mg C/m d\ mg C/m d\ mg ww/m mg ww/m mg ww/m yr\ mg ww/m mg ww/m mill./ml mg ww/m mg ww/m mg dw/m mg ww/m mg ww/m mg ww/m
Many biological variables whose determinations normally require extensive and expensive "eld and laboratory work may be estimated or predicted from one key, abiotic state variable, total-P (in mg/m). Some variables may be predicted with great precision, others with much less. n"number of lakes used in the regression. ww"wet weight. dw"dry weight. Modi"ed from Peters (1986) and Ha kanson and Peters (1995)
Table 2 Characteristic features in lakes of di!erent trophic categories Trophic level
Primary prod. Secchi (m) Chl-a (g Cm yr\) (mg/m)
Algal vol. (g/m)
Total-P (mg/m)
Total-N (mg/m)
Dominant "sh
Ultraoligotrophic Oligotrophic
(6
'10
(1.5
(0.4
(5
(200
Char, Trout
5}30
12}5
1}3
0.2}0.8
4}15
Mesotrophic
25}60
7}2
2}10
0.5}1.9
10}30
Eutrophic Hypertrophic
40}200 130}600
4}1 (2
6}35 30}400
1.2}2.5 2.1}20
20}100 '80
150}350 Trout, White"sh 300}500 White"sh, Perch 350}600 Perch, Roach '600 Roach, Bream
Mean value for the growing period (May}Oct.) Note that there is a great overlap between the di!erent categories, such that in oligotrophic lakes the concentrations of total-P may vary within a year from very low to high values (modi"ed from OECD, 1982; Ha kanson & Jansson, 1983).
concentration. The information about the variability within the di!erent classes in Fig. 3 for total-P will be used in this study as an important input for the derivation of the utility function of the LEI to handle changes caused by chemical remedial measures in individual lakes. The information given in OECD (1982) is also compiled
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Fig. 3. Probability distributions for di!erent trophic categories. Modi"ed from OECD (1982).
in Table 3, which gives the actual data on the spread (the standard deviation, SD) for each trophic class. One can note from Table 3 that for oligotrophic lakes (there are data from 21 lakes with a range in total-P from 3 to 17.7 lg/l), 1 SD corresponds to $0.22 for the logarithmic values or, for a mean value (MV) of 8 lg/l, one has 10 } "4.8 to 10 > "13.2 lg/l. From Table 3, it can be seen that the mean absolute di!erence for the three classes is about 0.25. This means that 95% of all
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Table 3 Compilation of results from OECD (1982) concerning mean values and standard deviations for lakes belonging to di!erent trophic classes Total-P (lg/l)
Absolute values
Oligotr. Mean 8 #1 SD 13.3 !1 SD 4.85 Range 3.0}17.7 n 21
Log. values
Mesotr.
Eutr.
Oligotr.
26.7 49 14.5 10.9}95.6 19
84.4 0.90 189 1.12 48 0.69 16.2}386 71
Di!. for log. values
Mesotr.
Eutr.
Oligotr.
Mesotr.
Eutr.
1.43 1.69 1.16
1.93 2.28 1.68
0.22 0.22
0.26 0.27
0.35 0.25
Mean abs. di!. of log. values"0.26
"0.25
The di!erences (Di!.) are calculated for the logarithmic values (e.g. 0.22"1.12}0.90 for oligotrophic lakes)
total-P values for all the given classes fall approximately within an order of magnitude, since the lower 95% con"dence limit is given by 10V\H and the upper 95% con"dence limit by 10V>H and the range by 10V>H /10V\H or 10V>H \V\\H "10"10. This means that one can be quite certain (95%) that if an action causes a change in TP-concentration from x to 10*x for a given lake, then it is very likely that there will also be a change in lake trophic level, which entails a change in the abundance and/or biomass of several key functional organisms, like the key "sh species illustrated in Table 2. This is crucial information in the de"nition of the requested utility function, which should, ideally, be set to 0 if there is a total change in the ecosystem structure in such a way that the normal (or reference/initial) key functional groups are replaced by other groups which would be typical of another trophic level. Lake pH is also a limnological state variable which in#uences the entire ecosystem. Fig. 4 shows that many animals accustomed to a circum-neutral pH (pH&7) cannot reproduce and survive in acidi"ed lakes. Some, like crustaceans and snails, are very sensitive to changes in pH, whereas other animals, like perch (Perca yuviatilis) and pike (Esox lucius), are less sensitive. The literature on anthropogenic acidi"cation of land and water, its ecological damage and its economical consequences is extensive (e.g. Likens, Wright, Galloway & Butter, 1979; Ambio, 1976; Overrein, Seip & Tollan 1980; Monitor, 1981,1991; Merilehto, Kenttamies & Kamari, 1988). There are many models and modelling approaches to address the acidi"cation of aquatic and terrestrial environments and to propose remedial measures for acidi"cation (Eliassen & Saltbones, 1983; Sverdrup, 1985; Warfvinge, 1988). There are even entire books of literature references on acid precipitation (e.g. Seip, 1989). In Sweden lake acidi"cation has caused more harm for "sh biology, in terms of number of lakes with reduced or extinct number "sh species, than all of the other listed threats (Fig. 5; see also Ha kanson, 1999). Liming as a countermeasure will not solve the acidi"cation problem but it can prevent severe damage to ecosystems until the sulphur emissions are reduced to levels that natural ecosystems can withstand in the
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329
Fig. 4. Illustration of target organisms using the example of biological and ecological e!ects of lake acidi"cation. The "gure shows examples of key functional groups and target organisms for acidi"cation. Crustaceans react rapidly to changes in pH, whereas certain "sh such as brook trout and eels do not die until acidi"cation is far advanced. White moss (e.g. Spagnum) and "lamentous algae should not normally be found in these lakes and the abundance of such species indicates ecological e!ects of acidi"cation. Modi"ed from Ha kanson and Peters (1995).
long term. About 8000 lakes in Sweden have been limed and the total annual cost for liming in Sweden was about SEK 200 million (K20 million ECU) in 1994 (Henrikson & Brodin, 1995). Since there are great costs connected with liming, it is very important that the liming is performed e!ectively and that the amount of lime in each lake is optimised so that it will give the most suitable change in water chemistry (pH, alkalinity, etc.) at a minimum cost. Di!erent lakes will not respond in the same manner when limed. The response depends very much on the environmental
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Fig. 5. Di!erent causes for the reduction and/or extinction of "sh communities in Swedish lakes. Note that acidi"cation is most important among these causes, but these also eutrophication due to agriculture has given rise to signi"cant changes in the structure of "sh communities in Swedish lakes. Modi"ed from EPA (1997).
properties of the lake. This means that it is very important to have a method to calculate the amount of lime needed to achieve a certain pH in a certain lake (see lake liming model by Ottosson & Ha kanson, 1997). Several remedial measures, either chemical or biological, have been tested to try to decrease the concentrations of radiocesium in lake "sh (see Ha kanson & Andersson, 1992 for further details) and all these measures can in#uence pH and TP: E Wetland liming and full-scale catchment area liming have been tested to reduce the transport of cesium from land to water through the modi"cation of soil chemical processes to reduce cesium mobility. E Lake liming was tested to increase the proportion of cesium deposited on the lake bottom, thereby preventing or delaying its bio-uptake. The hypothesis is that the binding to particulate matter and the #occulation tendency and sedimentation of the cesium-carrying particles are increased by increasing the pH, alkalinity and hardness of the lake water. E Potash treatment has been carried out to increase the potassium concentration of the lake waters thereby blocking and reducing the proportion of cesium taken up by "sh. Potassium and cesium are taken up by "sh in a similar manner (Black, 1957; Fleishman, 1963; Carlsson, 1978). Other ions, such as Ca, Mg and Na, may also
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participate in di!erent blocking processes. This implies that the di!erent liming measures, which give a general increase in the ionic strength of the water, may also have a positive e!ect in this manner. E Reducing the "sh stock (intensive "shing) or changing the predation pressure so that the nutrient web is changed in such a way that the phytoplankton biomass increases, which could result in lower concentrations of radiocesium in biota at all trophic levels. E Lakes can be treated with di!erent types of fertiliser (discharges from aquaculture, commercial fertiliser and P-enriched lime). The intention is to increase the biomass and thereby disperse the given amount of cesium. By means of `biological dilutiona this will decrease the concentration of cesium in each individual "sh. The method is based on theories involving biological bu!ering (Jansson, Hayman & Forsberg 1981; Ha kanson, 1999).
4. The lake ecosystem index (LEI) * de5nitions and set-up The complexities involved in establishing simple, practical and meaningful ecological indices sometimes seem insurmountable. Still, the bene"ts of even crude environmental indices are so great that they are worth pursuing. So long as one clearly states the criteria, theories and evidence, then these components can be discussed, tested and improved. The "rst step in LEI is to establish normal values (corresponding to natural, initial, pre-industrial or reference conditions) for the state variables. These values may be predicted from empirical models (see Ha kanson & Peters, 1995). In the MOIRAsystem they may be derived from algorithms based on GIS-information (e.g. land use, bedrock, soils, vegetation, precipitation and topography) or obtained from databases. It is assumed that normal or initial values for lake total-P concentration and lake pH are available (they will be used as reference constants); the actual values of lake total-P concentration and lake pH are calculated from the MOIRA sub-models for liming and fertilisation using a calculation time (dt) of one month (see Fig. 6, which gives an outline of the modi"ed LEI-model). The index re#ecting the status of "sh, the "sh yield ratio (FYR), is de"ned from the following arguments: 1. Actual and normal "sh yields (in mg ww/m*yr) are "rst calculated from the equation given in Table 1. FYR "7.1TP, where TP is the mean monthly concentration of total-P (mg/m). The ratio between the measured or predicted actual "eld yield and the de"ned normal "sh yield constitutes the basis of this equation. 2. Lake pH will in#uence this ratio. For example, the reproduction and abundance of roach (Rutilus rutilus) are very sensitive to alterations in lake pH (Fig. 4). This is described by a dimensionless moderator acting upon the "sh yield ratio. The moderator is 1 when the measured or predicted actual pH is equal to the normal pH. Since sensitive species like roach cannot reproduce at pH lower than 5, the moderator is calibrated to be zero for pH"5 when normal pH is set to 7. This
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Fig. 6. Upper part: Interactions among the basic components of the modi"ed lake ecosystem index and its relationship to the state variables; Lower part: Panel of driving variables. The data used for open land, percentage of rocks and mires in the catchment and relief may be obtained from GIS-maps in the MOIRA-system. The data on pH and total phosphorus, lake water retention time and lake area are generally obtained from data bases. The panel lists all the necessary driving variables to calculate LEI.
gives an amplitude value of #3.5. The moderator is > pH"(1#3.5 (Actual}pH/Normal}pH!1)).
(1)
So, the "sh yield ratio is "rst set to (see also Table 4) FYR"(Actual}TP/Normal}TP)> pH.
(2)
It is, however, unrealistic to assume that the changes in "sh biomass can be directly related to mean monthly changes in lake TP-concentrations * it takes much longer to change "sh biomasses than lake TP-concentrations. To describe this in detail is a most complicated matter related to the size (length/weight/age) and species of the "sh, lake temperature and many food-web characteristics. In this simpli"ed approach, this problem is handled by means of an exponential smoothing function (see Ha kanson & Peters, 1995). The smoothing function is written as: SMTH(input, averaging time, initial value) or using a di!erential equation dx/dt"x(t)#(z(t)!x(t))/¹ ,
(3)
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where z(t) is the input (a time series of data), x(t) is a selected start value plus a time series of data. ¹"the mean, characteristic age (a selected time constant in months) of the given organism. The mean age of all types of "sh is set to 18 months as a default value. As a comparison, the following "gures could be used as typical values for other organisms: zooplankton 3 months, macrozooplankton and benthos 6 months, planktivorous "sh (like small perch) 12 months, benthivores, omnivores and herbivores 18 months, piscivores 24 months, and large piscivores 36 months. Fig. 7 illustrates how this delay function approach works in a hypothetical case for the "sh yield ratio if the age of the "sh is set to 1, 3, 6, 12, 24 and 36 months and if there is a sudden decrease in lake total-P concentration from 60 to 20 lg/l. The idea of this example is to illustrate how Eq. (3) works and the realism of such a sudden drop in total-P concentration will not be elaborated. In this approach, the input as well as the initial value is given by FYR and the default value for the characteristic age of the entire "sh stock in the lake is set to 18 (months). This will smooth the response in the "sh yield to changes in lake TPconcentrations. This change (or reaction time) is assumed to be much shorter for the benthic community than for the "sh community (this is related to the smaller size/age/weight of the benthos as compared to "sh) and the default characteristic age of benthos is set to 6 months. No smoothing function is used for phytoplankton (or for bacteria or microzooplankton). Since phytoplankton generally has a life span much shorter than 1 month (which is the calculation time used in this model), there is no need for a smoothing function.
Fig. 7. Illustration of the smoothing function for the "sh yield ratio using di!erent "sh ages (from 1 to 36 months) in a hypothetical scenario when lake total-P concentration is set to decrease from 60 to 20 lg/l in month 30.
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The interpretation of the "sh yield ratio is that the value is 1 for a normal lake. A ratio of 2 means that the "sh yield is 2 times the normal. The same approach is used for the plankton biomass index and the bottom fauna biomass index. The phytoplankton biomass ratio (PBR) is based on the quotient between the actual (from measured or predicted data) and the given normal biomass of phytoplankton (from Table 1), i.e. 30 actual}TP /30 normal}TP . This ratio is in#uenced by a dimensionless moderator for pH using an amplitude value of 2 (see Ha kanson, 1996), such that a lower pH implies a lower phytoplankton biomass. The bottom fauna biomass ratio (BFBR) "rst gives the ratio between actual and normal biomass of benthos (in mg ww/m from Table 1), i.e. 810 actual}TP / 810 normal}TP . This ratio is in#uenced by a dimensionless moderator for pH using an amplitude value of 3, such that a lower pH implies a lower benthic biomass. The amplitude values describe the strength of the change in the index when the actual values depart from the normal values. The rationale for setting the highest amplitude values for the in#uence of pH for the "sh yield ratio and lowest for the plankton biomass ratio is that "sh (like roach) biomass is assumed to be the most sensitive to changing pH. These assumed values describe the methodological foundation of establishing ecosystem indices. Any change in state variables pH and TP will a!ect the given key functional groups in the ecosystem and also the given ratios. Such changes would also a!ect other important conditions in a lake, like the concentration of toxins in "sh. Alterations in lake pH and productivity would also in#uence the concentration of radiocesium in "sh (Ha kanson, 1996). However, this is not accounted for by this index but by the overall lake model for radiocesium, the VAMP-model, in the MAA-analysis. Finally, the lake ecosystem index (LEI) is de"ned as LEI"(FYR#PBR#BFBR)/3.
(4)
Note that if FYR (or PBR or BFBR) is (1, then the model sets the value to 1/FYR. Thus, the LEI-value is always *1. LEI is never allowed to attain negative values but any departure from 1 is always negative from an ecological point of view because it represents a departure from normal. A compilation of equations used to calculate the LEI-value is given in Table 4.
5. The utility function One of the main points of interest in economics, and more recently in other areas, has been the choice behaviour of people. The utility concept has played a central role in this theory of choice that may be traced back to Bernoulli (1738, 1954). The development of the axiomatic basis for the utility theory by von Neumann and Morgenstern (1944) provided the support for most subsequent analysis of economic behaviour under uncertainty. If the decision maker's (DM's) preference can be represented by a utility function, individuals should maximise the expectation utility, which is a cardinal function of outcomes, normally assumed to be levels of the considered attribute. Thus we shall be able to predict the selection of measures.
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Table 4 A compilation of equations used for the Lake Ecosystem Index, LEI 1. Fish yield ratio FYR"SMTJ(FYR , 18 FYR) FYR "(Actual}"sh}yield/Normal}"sh}yield) Y pH Fish yield"7.1 TP (in mg ww/m*yr) Moderator 1 for pH: Y pH"1#3.5 (Actual}pH/Normal}pH-1) Averaging time set to 18 months for "sh as a default value for the entire "sh community. SMTH"smoothing function. 2. Phytoplankton biomass ratio"PBR PBR"(Actual}biomass/Normal}biomass) Y pH Phytoplankton biomass"30 TP (in mg ww/m) Y pH"1#2 (Actual}pH/Normal}pH-1) No smoothing is used for phytoplankton when dt is set to 1 month. 3. Bottom fauna biomass ratio BFBR"SMTH(BFBR , 6, BFBR ) BFBR "(Actual}biomass/Normal}biomass) Y 'pH Bottom fauna biomass"810 TP (in mg ww/m) Y pH"1#3 (Actual}pH/Normal}pH-1) Averaging time set to 6 months as a default value for the entire benthic community. Lake ecosystem index, LEI"(FYR#BFBR#PBR)/3 If FYR (BFBR or PBR) (1, then 1/FYR instead of FYR. Thus, LEI is always *1. A departure from LEI"1 is negative.
We shall de"ne the concept of utility as follows: Given the set X of all possible levels of an attribute which the DM can choose, a utility function is a function u : XP[0,1], which is a faithful representation of a preference relation on X. Therefore, by means of a utility function, a real number in the closed interval [0,1] is assigned to each level x3X, such that, given two levels x, y3X, level x is preferred to y if and only if the utility function indicates a higher value for x than for y (i.e. u(x)"u(y)). Thus a utility function is simply a very concise way of representing a preference relation on a certain set X. There are many possible ways to assess a utility function for the MAA-analysis (see Farquhar, 1984; Kirkwood, 1997) and several of them have been incorporated in the MOIRA-system (see Rios-Insua, Gallego, Mateos & Rios-Insua, 2000). In this paper, the basic idea of the utility function is to give a scienti"c rationale to implement LEI in the MAA-analysis in such a way that environmental, social and economic objectives can be compared. Di!erent users would then arrive at the same results if they follow the given manual of the MAA-analysis. The utility function, u(x), should be directly related to the LEI-value and, following the standard use of the utility theory, it should be a value between 0 and 1. Utility means in this MAA-analysis a measure of how good a countermeasure would be to minimise the ecological impact expressed by LEI. Therefore, if the optimal value of LEI is 1, this should correspond with the maximum `utilitya u(x)"1.0; in the other extreme u(x) should be 0. But at which value for LEI should u(x) approach 0? The information in Table 3 and Fig. 3 can then be used to determine an equation describing the relationship between
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LEI and u(x). It is evident, as stated, that, if lake TP-concentrations change by a factor of 10, i.e. if the ratio between actual and normal TP changes by a factor of 10, from !2 SD to #2 SD, then, with a 95% certainty, there will be a shift from one trophic category to another, e.g. from oligotrophic to mesotrophic. There will also be associated changes in abundance and biomass of key functional groups. If one starts with an Actual/Normal ratio of 1 for lake TP, and, as all empirical data suggest, the log-normal frequency distribution for lake TP-concentrations (see OECD, 1982; Ha kanson & Peters, 1995) is used, then one can de"ne the utility function directly from the probability function for logarithmic TP-values (see Fig. 8A). This means that there is a 95% probability that TP-values lie within 4 and 40 lg/l if the normal TP-value is set to 20; and there is a 52% probability that the TP-values would fall between 13.3 and 30 (see Fig. 8B). One can then determine the probability function, set equal to (1!u(x)), for di!erent ratios of Actual/Normal. In this study, the following ratios 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.2, 2.4, 2.6, 2.8 and 3.0 were tested in two ways. Firstly, according to Figs. 8A and B, i.e. for symmetric divergences from the mean value for log-transformed TP-concentrations and secondly, for asymmetric values according to Fig. 8C, i.e. for situations where the ratio Actual/Normal for the absolute (untransformed) TP-concentrations were used. The "rst utility function is given by u(x), the second by u (x). As expected, there is a very close relationship between the two cases (Fig. 9), and in the following u(x) will be implemented in the LEI-model. The relationship between LEI and u(x) is given in Fig. 10. It is evident from Fig. 10 that u(x) can describe LEI almost perfectly (r"0.9998). The values in Fig. 10 emanate from the following algorithm and the calculated data are derived from the Monte Carlo simulations: u(x)"(1!1.21((LEI!1)/LEI) ).
(5)
The basic approach, (LEI-1)/LEI, is logical since u(x) should be 1 for LEI"1. From Fig. 10B one can note that the utility function, u(x), approaches 0 as LEI approaches 5.
6. Practical application Figs. 11 and 12 illustrate how the model works, showing change in lake total-P concentrations (fertilisation as a remedial measure) and changes in lake pH (liming as a remedial measure), respectively. The driving variable in Fig. 11A is the given change in actual total-P (TP) concentrations (pH is kept constant at 7). First, there is a phase where actual TP is equal to normal TP, then there is a phase of comparatively slow increases in TP (to illustrate `creepinga eutrophication), then there is an instant reduction in TP back to normal values, then a period of adjustment, followed by a period with very low actual TP-values, a period of normal TP and, "nally, a period where actual TP is set to 60 lg/l. The normal TP is kept at 20 lg/l throughout the simulation. Note that this simulation is NOT intended to illustrate realistic changes in lake TP-concentrations but rather how the model works. The results for the "sh yield ratio (curve 1), the
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Fig. 8. Monte Carlo simulations to de"ne the utility function, u(x), which is set to be complementary to the probability function: (A) Symmetric de"nition for the log-normal frequency distribution. Example when lake total-P is set equal to 20. Then there is a 95% certainty that TP-values would fall between 4 and 40 lg/l. u(x) is 1!0.95"0.05; (B) Symmetric de"nition for the log-normal frequency distribution. u(x)"1!0.52"0.48 if TP-values fall between 13.3 and 30 lg/l; (C) Asymmetric de"nition. Example of u (x) for lake total-P set equal to 20. Then there is a 80% certainty that values would fall between 6 and 34 lg/l. u (x)"0.2.
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Fig. 9. The relationship between the two utility functions, u(x) and u (x) de"ned according to the symmetric and the asymmetric presuppositions in Fig. 8.
bottom fauna biomass ratio (curve 2) and the phytoplankton biomass ratio (curve 3) are given in Fig. 11B. The instant changes in the driving TP-variable (curve 4 in Fig. 11A) cause much slower changes in the "sh yield ratio, similar slow changes for the bottom fauna biomass ratio, but rapid changes for the phytoplankton biomass. Also note that a change of 2 in any of these ratios means a double (or halved) change in the biomass as compared to the normal. The corresponding changes in LEI and the utility function, u(x), are shown in Fig. 11C. The natural or reference value for the lake ecosystem index (LEI) is 1 when lake TP is normal (20 lg/l in these simulations). If LEI is greater than 2, u(x) is smaller than 0.26; if LEI '4, then u(x)'0.03. If LEI '4, this implies a very signi"cant alteration of the entire lake ecosystem. Fig. 12 gives an example for lake pH when TP is kept constant at 20 lg/l. First, there is a phase where actual pH is equal to normal pH ("7), then there is a phase of comparatively slow decreases in pH (to illustrate the acidi"cation process), then there is an instant increase in pH back to normal values (e.g. from lake or wet land liming), then a period of adjustment, followed by a period with very high actual pH-values. The implications of such pH-changes are given in Fig. 12B for the "sh yield ratio (curve 1), the bottom fauna biomass ratio (curve 2) and the phytoplankton biomass ratio (curve 3). In this case, the instant changes in pH also cause slow changes in the "sh yield ratio and the bottom fauna biomass ratio. The corresponding changes in LEI and the utility function, u(x), are shown in Fig. 12C.
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Fig. 10. The relationship between the lake ecosystem index (LEI) and its utility function, u(x): (A) The regression; (B) The utility curve.
7. Concluding remarks These simulations demonstrate how variability in the two state variables, total-P and pH, in#uence the lake ecosystem index and the utility function. Very high values of LEI and low values of u(x) are obtained if the actual values of these two state variables depart signi"cantly from the normal (initial or reference) values. These examples show that the approach has potential in multi-attribute analysis, water
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Fig. 11. Simulations and sensitivity tests with the model for the lake ecosystem index for changes in actual lake total-P concentration. Normal values for total-P"20 lg/l and pH"7: (A) Illustration of the assumed actual lake TP-concentrations (curve 4; the driving function in these simulations): (B) Response for the "sh yield ratio (curve 1), the bottom fauna biomass ratio (curve 2) and the phytoplankton biomass ratio (curve 3). (C) Response for the lake ecosystem index (LEI, curve, 1) and the utility function (u(x), curve 2).
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Fig. 12. Simulations and sensitivity tests for the lake ecosystem index for changes in actual lake pH. Normal values for total-P"20 lg/l and pH"7; (A) Illustration of the actual pH-values (curve 2; the driving function in these simulations); (B) Response for the "sh yield ratio (curve 1), the bottom fauna biomass ratio (curve 2) and the phytoplankton biomass ratio (curve 3); (C) Response for the lake ecosystem index (LEI, curve, 1) and the utility function (u(x), curve 2).
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management and radioecology because it would facilitate overview and provide an aim and direction to evaluating the ecological impact of chemical remedial measures.
Acknowledgements This work is a part of a project funded by the European Union (contract FI4PCT96-0036), MOIRA (A Model computerised system for management support to identify optimal remedial strategies for restoring radionuclide contaminated aquatic ecosystems and drainage areas). We would like to express our gratitude to all colleagues in this project for help and support during this work and especially to John Brittain who has contributed with know-how and ideas.
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