Bayesian belief networks as a meta-modelling tool in

EC O LO G IC A L E C O N O M ICS 6 6 ( 2 00 8 ) 9 1 –1 0 4

a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m

w w w. e l s e v i e r. c o m / l o c a t e / e c o l e c o n

ANALYSIS

Bayesian belief networks as a meta-modelling tool in integrated river basin management — Pros and cons in evaluating nutrient abatement decisions under uncertainty in a Norwegian river basin D.N. Bartona,⁎, T. Salorantaa , S.J. Moea , H.O. Eggestadb , S. Kuikkac a

Norwegian Institute for Water Research (NIVA), Gaustadalléen 21, NO-0349 Oslo, Norway Norwegian Institute for Agricultural and Environmental Research (Bioforsk), Fr. A. Dahlsvei 20, NO-1432 Ås, Norway c FEM Group, University of Helsinki, Department of Biological and Environmental Sciences, P.O. Box 56, FI-00014 University of Helsinki, Finland b

AR TIC LE I N FO

ABS TR ACT

Article history:

A Bayesian network approach is used to conduct decision analysis of nutrient abatement

Received 23 November 2006

measures in the Morsa catchment, South Eastern Norway. The paper demonstrates the use

Received in revised form

of Bayesian networks as a meta-modelling tool in integrated river basin management (IRBM)

5 February 2008

for structuring and combining the probabilistic information available in existing cost-

Accepted 5 February 2008

effectiveness studies, eutrophication models and data, non-market valuation studies and

Available online 18 April 2008

expert opinion. The Bayesian belief network is used to evaluate eutrophication mitigation costs relative to benefits, as part of the economic analysis under the EU Water Framework

Keywords:

Directive (WFD). Pros and cons of Bayesian networks as reported in the literature are

Decision analysis

reviewed in light of the results from our Morsa catchment model. The reported advantages

Influence diagrams

of Bayesian networks in promoting integrated, inter-disciplinary evaluation of uncertainty

Bayesian networks

in IRBM, as well as the apparent advantages for risk communication with stakeholders, are

Benefit–cost analysis

offset in our case by the cost of obtaining reliable probabilistic data and meta-model

Eutrophication

validation procedures.

Uncertainty

© 2008 Elsevier B.V. All rights reserved.

Water Framework Directive

1.

Introduction

In the last decade Bayesian networks have increasingly been applied to environmental management problems under uncertainty, and recently also to integrated water management issues (see for example Varis et al., 1990; Varis and Kuikka, 1999; Borsuk et al., 2001; Varis and Lahtela, 2002; Borsuk et al., 2004; Henriksen et al., 2004; Ames et al., 2005; Bromley et al., 2005; Labiosa et al., 2005). Common to these studies is the use

of Bayesian networks to integrate probabilistic information derived from data sets, model simulations and expert opinion in the study of water allocation or pollution problems. As an alternative to extensive scenario analysis using deterministic models (e.g. Hein, 2006) Bayesian networks hold the promise of a more complete accounting of integrated model uncertainty. In some cases, Bayesian networks are used to study the properties of integrating a number of sub-models for purposes of targeting data collection or joint risk analysis. In other

⁎ Corresponding author. Tel.: +47 924 42 111; fax: +47 22 18 52 00. E-mail address: [email protected] (D.N. Barton). 0921-8009/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2008.02.012

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cases, Bayesian networks are extended to include uncertainty regarding the costs and benefits of management decisions in what is known as influence diagrams. The main aim of this paper is to evaluate the advantages and disadvantages of using Bayesian networks for integrated assessment of the uncertain costs and benefits of eutrophication abatement measures through its application in a case study. The Vansjø lakes in the Morsa catchment in SouthEastern Norway (Fig. 1) are ‘at risk’ of not meeting the EU Water Framework Directive's (WFD) requirement for ‘good ecological status’ (GES) by 2015 due to eutrophication. Given the continued algal blooms, even several years after implementing nutrient mitigation measures in upstream agriculture and wastewater treatment, managers wonder whether current and supplementary measures will reduce nutrient concentrations as predicted in previous impact assessments

(e.g. Lyche Solheim et al., 2001). Decision-makers are also interested in the relative uncertainty of mitigation costs versus benefits. We evaluate how well suited Bayesian networks are to deal with this task. The WFD requires that a cost-effectiveness analysis of a programme of measures for attaining GES be conducted. The environmental objectives can be lowered (objective derogation) or delayed in time (time derogation) if the costs of these measures can be shown to be ‘disproportionate’ (Sections 3–7, art. 4 WFD). As a first step in the assessment of cost disproportionality we conduct a simple test of whether expected costs exceed expected benefits under uncertainty in our study. Consistent evaluation of the joint uncertainties underlying the cost and benefit estimates in the economic analysis has been identified as one of the main research challenges of WFD implementation (Brouwer, 2005). Bayesian

Fig. 1 – Morsa catchment draining to the Vansjø lakes including Lake Storefjorden.


networks offer a framework for documenting and assessing the probability of non-compliance with GES against uncertain abatement costs, as well as the probability that expected costs exceed expected benefits, by jointly assessing conditional probability distributions for abatement costs, water quality effects and benefits. The paper is structured as follows. Section 2 reviews limitations and advantages found in other applications of Bayesian networks to water management problems. Section 3 provides a brief introduction to the use of Bayesian networks in a driver-pressure-state-impact (DPSI) approach to integrated river basin modelling, as well as introducing the objectoriented model for the Morsa catchment. Section 4 presents the Morsa catchment in more detail, including a description of the data sets. Simulation models, surveys and expert evaluations are used to populate the network nodes with probability distributions. Section 5 presents results of the application to cost-effectiveness and cost–benefit analysis. Section 6 discusses some of the limitations of the Bayesian network approach in the Morsa catchment in assessing costs and benefits in relation to some of the previous findings in the literature. Section 7 provides some conclusions and directions for future work.

2. Bayesian networks in river basin modelling and management Bayesian networks (BNs) have a number of generic features which make them well suited for integrating the results of diverse models for the purpose of river basin management. The handful of studies that have focused on applications of BNs to watershed management are summarised in the next section. Varis and Kuikka (1999) report their lessons learned using belief networks in nine case studies, several of which include water resource management (restoration of a temperate lake; real time monitoring system for a river; costeffective wastewater treatment for a river). The authors note that while methodological development in Bayesian decision analysis is progressing rapidly, the way to empirical application tends to be long. They see the lack of acceptance of Bayesian approaches among established scientific specialists and in institutions with established management approaches as the main barrier to application. While research novelty may be a limitation to policy application, previous studies have argued that BNs applied to model integration in water management also have limitations inherent to the method itself. Borsuk et al. (2001, 2004) used belief networks to integrate a combination of process-based models, multivariate regressions and expert opinion of river eutrophication to predict probability distributions of policy-relevant ecosystem attributes. In addition to the strengths of BNs in eliciting expert information, the authors point out the tendency of experts to ‘overload’ a network with “pet processes” related to their own research, and the problems this can lead to in terms of model complexity and information costs in defining a stochastic model. For parsimonious network design they recommend evaluating whether variables are (1) controllable, (2) observable, and (3) predictable at the scale of the management

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problem, before inclusion in the network. Other limitations identified include BNs inability to explicitly represent system feedback relationships — BNs describe relationships as oneway causal influences at a particular instant in time or as net influences on eventual steady-state conditions. Also BNs face model validation problems, given that uncertainty may be in the causal structure itself, as well as in parameter uncertainty and natural variation that are captured by probability distributions. For checking uncertainty in model structure, the authors recommend Bayesian model averaging, learning from additional data, as well as rigorous model testing. Varis and Lahtela (2002) use BNs to conduct scenario analysis for basin-wide policy impacts on different user groups in the Senegal River. They show how BNs can deal with the fact that benefit–cost analysis of basin-wide measures do not generally account for non-quantifiable externalities, and often do not conduct risk or uncertainty analysis. However, the authors reach the conclusion that little change can be expected relative to the baseline scenario in the river basin even with rather strong management actions. Their model is quite big with 45 variables and 840 linkages. Low policy effectiveness may have been due to the large number of linkages combined with a low degree of discretization of only 6 qualitative probability classes. A large number of model nodes does not automatically lead to higher uncertainty, if each step is correctly evaluated, but does increase the probability that some processes in the overall network will be evaluated incorrectly. Ames et al. (2005) use BNs to model watershed management decisions with a case study application to phosphorus management in a small catchment in Utah. Their BNs integrated headwater and reservoir state variables with cost of wastewater treatment and revenues from recreational lake use. Only a 1% increase in the probability of improved recreational conditions in the target lake is observed under the most effective scenario to reduce non-point source nutrient loading. Here too the authors point out the effect that discretization has on the potential loss of information in a BN. On the other hand, discretization is also seen as a particularly useful property in modelling variables with breakpoints or thresholds relevant to management. The authors emphasise the need to validate completed BNs using independent information, but admit that this can be a challenge when the BNs probability distributions are derived from sources other than observed data, or when no new data becomes available for assessing the BN model. They therefore recommend testing the model against adaptive management, third party expert opinion or at a minimum sensitivity analysis. The issues pointed out in the literature which are also raised in the present case study, include (i) limitations of an acyclical causal structure, (ii) a tendency of over-complexity of network structure relative to the scale of the management problem, (iii) sensitivity to discretisation of probability distributions, (iv) cumulative uncertainty and resulting insensitivity of environmental objective variables to measures (v) selecting model validation approaches. In addition, the paper addresses (vi) the implicit assumptions of geographical and temporal scale in BN modelling, and (vii) correct specification of correlation between probability distributions.

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3.


Methodology

Discrete BNs represent factorisations of joint probability distributions over finite sets of discrete random variables. Variables are represented by nodes of the BN (describing variables of the problem at hand), and the links of the network represent the properties of conditional dependencies and independencies among the variables as dictated by the distribution. Each variable is specified as a probability distribution conditional on the configuration of its conditioning parent variables (Kjærulff and Madsen, 2005). The conditional probabilities are used to model how precisely we can describe the relationships of the variables. If uncertainties are high (e.g. low correlations in statistical analysis) and the problem (model) is complex, the end result may be a model that does not promise marked changes in the system impact variables due to management of driver or pressure variables. The nodes and links of a BN form a directed acyclic graph. The acyclical property is required to carry out the probability calculus that make Bayesian decision analysis software so efficient, but implies that feedback effects are not modelled in one and the same network (Jenssen, 2001). Fig. 2 illustrates the difference between a Bayesian network and influence diagram in the context of a generic driverpressure-state-impact model (OECD, 1993; Eurostat, 1999), for example for water quality management. In this stylised example the management context is made up of the states of an exogenous variable X conditioning water quality state S, and the decision D on whether the pressure P mitigating measure is implemented or not. In this framework prior knowledge of water quality can be expressed as a probability of a state S given pressure P and exogenous variable X: Pr (S | P, X). Similarly the probability of nutrient loading pressure P dependent on the decision D is Pr (P | D). In an impact analysis a manager may be interested in determining the posterior probability for a state given a pressure and the states of

context variables c = c(D, X): Pr (S | P, c), or conversely a likelihood, expressed as a probability of pressure given the state of given context variables: Pr (P | S, c). Bayes' rule (Eq. (1)) expresses the relationship between the prior, likelihood and posterior probabilities. PrðSjP; cÞ ¼

PrðPjS; cÞ PrðSjcÞ PrðPjcÞ

ð1Þ

In this case study we use commercially available software (Hugin Expert®: www.hugin.com) to implement Bayesian calculus using influence diagrams to evaluate the expected value of utility functions of decision alternatives. Whereas a BN is a model for reasoning under uncertainty, an influence diagram (ID) is a probabilistic network for reasoning about decisionmaking under uncertainty (Kjærulff and Madsen, 2005). Influence diagrams are close relatives of decision trees (e.g. Clemen, 1996), but their more explicit graphical presentation of cause–effect linkages offer non-specialists a better chance to understand integrated model structures. Referring back to the generic influence diagram in Fig. 2, the software depicts decision nodes as rectangles (exogenous policy drivers). Utility nodes representing impacts of decisions (costs and benefits) are depicted as diamonds. Chance nodes (ovals) are used to depict exogenous variables described by unconditional probability distributions, as well as endogenous variables described by joint probability distributions conditional on the states of one or more parent nodes. Influence diagrams with decision and utility nodes estimate expected (net) utility of decisions accounting for all probability distributions of the network. The BN structure is driven by the conceptual understanding of the problem, but also the detail of existing studies or models that are used to describe each link of the DPSI chain. When multiple sources of information at different scales and levels of abstraction (model simulations, data-correlations and expert judgement) need to be combined, an object-

Fig. 2 – Influence diagrams and Bayesian networks in the context of DPSIR (Driver–Pressure–State–Impact–Respons). Note: Decision nodes are represented by rectangles, utility nodes by diamonds and chance nodes by ovals.


oriented Bayesian network (OOBN) is convenient (Kjærulff and Madsen, 2005). This is a hierarchically specified probabilistic network where the objects can be either BNs or IDs, and can be considered as sub-models of the overall network. In this study we use OOBNs to organise information from a number of models, data sets and expert evaluations. The various OOBNs can be developed by different experts and then linked together to describe the expected behaviour of the whole system. As discussed in the literature review, model validation is a challenge in Bayesian decision models. This is especially the case if we are interested in predicting the outcomes of management options which have never been applied to the given area. We use the information analysis1 function in Hugin Expert® to determine which variables contribute most to uncertainty in the OOBN. Uncertainty is reduced with additional information defined in terms of changes in the probability distribution of a hypothesis variable of management interest (e.g. willingness to pay for increased bathing suitability). If P(T) is the probability density function of the hypothesis variable T then an indicator of information (V) — or conversely uncertainty (H) — can be defined as: VðTÞ ¼ HðTÞ ¼

X

PðTÞ log ðPðTÞÞ

ð2Þ

T

A higher value of H(T) is interpreted as more uncertainty about the true state of, in our example above, willingness to pay. An indicator of the reduction in uncertainty about T from observations of a conditional variable X (e.g. background nutrient loading) can be defined as I(X,T) where:2 VðTjXÞ ¼ ðHðTÞ IðX; TÞÞ

ð3Þ

In the example above I(X1,T) N I(X2,T) would tell us that uncertainty about the expected value of willingness to pay is reduced more by new observations on background nutrient loading (X1) than some other conditional variable such as nutrient-reduction effect of tillage (X2). This is a convenient approach to sensitivity analysis in large OOBNs, indicating which variables should be observed first to decrease the uncertainty in any hypothesis variable of interest. Fig. 3 shows the highest level of the OOBN for the Morsa catchment, with an indication of some of the network objects at more disaggregate levels. The OOBN can be understood as a metamodel (Varis, 1997) — probability distributions of the network nodes may be thought of as ‘response surfaces’ that summarise underlying model simulation results, data distributions or expert opinion, without having to process the model code, data or experts themselves each time the network is evaluated. The overall network shows the decision nodes for the four management measures under consideration — changed tillage practices, sedimentation dams, vegetation buffer strips, and individual residential waste water treatment. As an example of an OOBN, the ‘tillage cost/effect’ object in turn contains an object representing a phosphorus-loss regression model. Each abatement measure is represented by an object 1 Not to be mistaken with value-of-information analysis, which indicates the utility (monetary or otherwise) of decreasing the uncertainty in a variable that specifically conditions a management decision entailing costs and benefits (Clemen, 1996). 2 The Hugin Expert® manual refers to H as an “entropy indicator” and I as a “mutual information statistic”.

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containing the BNs used to calculate its cost and phosphorus loading effect, which is output to two effect nodes and a cost/ effect node. These effect nodes are aggregated to non-point and point source reduction nodes for dissolved (DIP) and particulate (PIP) inorganic phosphorus. The cost/effect node is used as such for cost-effectiveness analysis (Fig. 3). DIP and PIP effects are input to Lake Storefjorden object, which contains the BN that represents eutrophication and algal bloom models and criteria for bathing suitability. Suitability for bathing is determined by the water quality of the lake, which in turn depends on baseline loading and reductions in loading due to measures (Fig. 4). The temperature of the lake has an impact on bathing suitability as well, i.e. if water temperature happens to be less than is suitable in a given bathing season, a higher probability of suitable water quality does not lead to a higher probability of recreational benefits. Bathing suitability is in turn input into the ‘economic impact recreation’ object which contains a BN describing household willingness-to-pay (WTP) for bathing suitability. Notice that this impact does not cycle back as a response or feedback effect to the decision node due to the acyclic requirement of BNs (the R is missing from the well-known DPSIR framework for river basin modelling).

4.

Case study area and available data

4.1.

Catchment description

The Morsa catchment area is approximately 700km2, located in south-eastern Norway, including outskirts of Oslo in the north and a number of smaller lakes draining into the River Hobøl which runs into Lakes Storefjorden and Vanemfjorden (collectively known as Vansjø) in the south, which in turn drain into the Oslofjord through the town of Moss (Fig. 1). Land-use in the catchment is mainly forest and agriculture with the urban areas found in the southern end of the catchment. Population in the headwaters is mainly found in dispersed residential areas and farm housing. Lake Vanemfjorden is a highly eutrophic lake with frequent cyanobacteria blooms coinciding with the bathing season in June–August. Water enters Lake Vanemfjorden from Lake Storefjorden which has been the focus of upstream management measures to date. Total phosphorus (Tot-P) loading into Storefjorden in the period 1984–2000 varied between 5000 and 25,000kg Tot-P/ year (median 12,000kg/yr) (Lyche Solheim et al., 2001). In 2000 the main sources of nutrient loading were agriculture (57%), septic tanks from individual households (11%), municipal wastewater (6%) and natural background run-off (26%). The limiting nutrient in freshwaters, and focus of abatement efforts in the upper watershed, is phosphorus. The catchment has little industry of significance and other water quality issues are marginal compared to nutrient loading. Lake Storefjorden provides drinking water supply for 60,000 people through the MOVAR water supply utility. Thirty one thousand people live within the catchment municipalities and constitute a conservative estimate of the potential recreational user population of Lakes Storefjorden and Vanemfjorden . Due to algal blooms and heavy turbidity in spring floods, MOVAR has recently invested approximately 10million Euro in

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Fig. 3 – Object-oriented Bayesian network model for nutrient abatement in Morsa catchment. Note: grey ovals represent underlying sub-networks; orange boxes represent decisions on implementation of measures(true/false); white ovals represent nodes with conditional probability distributions.

additional active coal, flushing and ozone treatment of drinking water. Basic nutrient abatement measures that have been implemented in the catchment in the last decade have increased annual costs in agriculture (i.e. yield loss, maintenance of buffer strips, sedimentation dams, grassy water courses), and for households (septic tank upgrades, connection to municipal treatment). Monitoring data for 2005 show a reduction in Tot-P concentrations in Lake Storefjorden relative to 2000 (Bjørndalen et al., 2006), but this reduction falls within the range of variation in Tot-P concentrations in the lake since 1984. The most recent measurements of Chlorophyll a (ChlA) concentrations are as low as the lowest value since 1984, but variability relative to Tot-P concentrations also suggests that it is too early to tell whether management

measures taken so far actually have had an impact on water quality. Our analysis focuses on Lake Storefjorden using available monitoring data for upstream measures until the year 2000. Conclusions in this study therefore do not reflect trends or variability after this point in time. In the following sections the data is summarised along with some features of the OOBN. A fuller and more detailed presentation of the network and the data can be found in Barton et al. (2006).

4.2.

Abatement measures' costs and effects

Upstream abatement cost data are based on an original costeffectiveness analysis conducted by Lyche Solheim et al. (2001) in the catchment. In that study the authors estimated the total


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Fig. 4 – Lake Storefjorden state (instance) describing eutrophication and bathing suitability. Note: DIP:dissolved inorganic P; PIP: particulate inorganic P; SIS: suspended inorganic sediment; ChlA: Cholophyll A; %cyano: cyanobacteria as % of algal biomass. Nodes with grey rings indicate input and output nodes to other sub-networks.

costs of implementing agricultural and individual wastewater treatment measures comparable to the ones considered here in the order of 11–13million NOK/yr.3 The sub-networks for agricultural measures have been revised in the study reported here compared to Barton et al. (2005) based on more recent discussions with experts on run-off. Probability distributions reflect a variety of data sources including expert opinion, empirical data and regression model results. Where available, the minimum–maximum range for costs of measures were evaluated as uniform probability distributions. Where only point estimates are available, additional studies were examined and data points assessed as discrete distributions. Where no other studies are available the authors were advised by experts to use triangular distributions (min, median, max). Abatement costs are financial costs to farmers, corrected for any transfers, and are a first order assessment of economic opportunity costs common to feasibility analyses carried out at farm level. Conditional probabilities predicting Tot-P in runoff as a function of soil P, run-off and erosion-risk were adapted from the USLE-based regression model applied in 3 In 2000 1 NOK equalled approximately 0.11 US$. For the purpose of future mitigation decisions, downstream investments in drinking water treatment are considered sunk costs in this case study. Because water treatment operating costs depend only marginally on the severity of an algal bloom thanks to the water treatment upgrades, additional benefits of nutrient mitigation are furthermore expected to be small and therefore left out of the analysis.

Eggestad et al. (2001), including parameter distributions (mean, std. error) obtained from these authors. Uncertainty regarding erosion risk in Morsa was modelled as the variability in erosion risk classes for a statistical region, which includes municipalities inside and outside the Morsa catchment. A flat distribution of variation in soil P was used in the model as the distribution across the catchment is unknown. Changes in the land use erosion factors (C-factors) define the effectiveness of changed tillage practices. Uncertainty regarding C-factors was based on expert opinion.

4.3.

Water quality simulation

The ‘Storefjord lake state’ object, describing eutrophication and bathing suitability, is driven by changes in PIP and DIP loading as shown in Fig. 4. The main model forcing in our case study are the daily time series of meteorological, hydrological and nutrient loading data for the 16-year simulation period 1985–2000. The dynamic process-based model MyLake (Saloranta and Andersen, 2007) was used to simulate the water surface temperature, as well as the relationship between different PIP and DIP loads and the daily concentration of TotP and ChlA in Storefjord Lake. The corresponding conditional probability tables (CPTs) for these relationships were produced by running the model repeatedly with different parameter and input factor values in a Monte Carlo simulation. MyLake is a one-dimensional model code for the simulation of the daily vertical distribution of lake water temperature and thus

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density stratification, the evolution of seasonal lake ice and snow cover, sediment–water interactions, and phosphorus– phytoplankton dynamics. The basic idea behind MyLake has been to include only the significant physical, chemical and biological processes in a well-balanced and robust way. The model code, its calibration and application to Lake Storefjorden is described in more detail in Saloranta (2006) and Saloranta and Andersen (2007). Water quality criteria (Tot-P, ChlA, the proportion of cyanobacteria and temperature) were used as a proxy for recreational suitability in general. This ‘lake state’ object is central to the whole network and allows a comparison of costs of measures and benefits of improvements in recreational suitability. The impact of reductions in loading on bathing suitability is given as the difference in probability of suitable conditions between the state of Lake Storefjorden before and after implementation of a programme of measures. The effectiveness of measures to improve water quality in Lake Storefjorden is measured against the water authorities' operational limit values for ‘good ecological status’ (GES) for (i) cyanobacteria as a percentage of algal biomass (10%) and (ii) Tot-P (11–14μg Tot-P/l) (pers. com. H. Gunnarsdottir , Morsa project). The conditional probabilities for Secchi depth reflecting bathing water transparency and the percentage of cyanobacteria and ChlA were computed using data from Norwegian lakes, all collected by NIVA. The sampling period spans 1972 to 2002, but a large proportion of the samples were taken during the national eutrophication survey in 1988. Earlier analyses of these data are reported by Lyche Solheim et al. (2004). The proportion of cyanobacteria is calculated as the biomass of all cyanobacteria (except the genus Merismopedia) divided by the total phytoplankton biomass. The proportion of cyanobacteria generally increases with ChlA concentration, but the phytoplankton community may also be affected by factors such as alkalinity and humic content (Lyche Solheim et al., 2003). Lake Storefjorden belongs to the low-alkalinity, high-humus lake group. However, the lake chemistry is close to the limit for both typology parameters (4mg/L Ca and 5mg/L TOC). We have therefore included two states (high and low) for both of these two parameters, so that the network can cover all four lake groups assuming an equal probability for each. In order to obtain a data set that is representative for Lake Storefjorden, and at the same time contains enough samples to parameterise the probability tables, we first compared the relationship between ChlA and the proportion of cyanobacteria for different combinations of geographic ranges and lake types.4 This relationship was estimated by a non-parametric generalised additive regression model. The smallest data set (Eastern Norway, Lake Storefjorden type) was not sufficient to cover the whole eutrophication gradient of interest, while the full data set (Nordic countries, all lake types) resulted in more noise (data obtained from www.rbm-toolbox.net/rebecca). The best fit (n = 481, R2 = 0.50) was obtained with data from lowland lakes (b 200m above sea level) of all types in Eastern Norway, including alkalinity and humus levels as categorical covariables. Next, the entries for the CPT for the percentage of 4 Based on the Lakes database taken from the EU project REBECCA (www.rbm-toolbox.net/rebecca).

cyanobacteria were calculated as the number of observations per cyanobacteria state for each combination of states of ChlA, alkalinity and humic content. Likewise, the CPT for Secchi depth states was calculated as the number of observations per Secchi state for each state of ChlA. Note that the CPTs for cyanobacteria and Secchi depth are based on raw observations, contrary to the CPT for ChlA, which is based on model simulations. The empirically observed CPTs for the former contain more natural variability than the model-based CPT for ChlA.

4.4.

Willingness to pay for bathing suitability

Mean household WTP as an indicator of the economic benefits of improvements in the suitability of water in the Vansjø lakes was obtained from Magnussen et al. (1995). WTP was determined through a representative contingent valuation (CV) survey conducted in 1994 targeting 300 randomly selected households in the Morsa catchment. Households were asked for their incremental WTP over and above their current water bill for a programme of measures that would improve lake quality from a currently “poorly suited” state to a “well suited” state for bathing, boating, fishing and drinking water. A water quality ladder was used to depict the different states, where the baseline situation showed water quality suitable for boating and fishing, but not for bathing and drinking. While incremental change in suitability is related to multiple water uses in the CV scenario, the upgrade of drinking water treatment since the original study was conducted means that the main use and user benefits in the future will be related to water-based recreation, with bathing requiring the highest water quality (higher even than water to be treated for drinking). Non-user values for ecosystem improvements of households living outside the catchment were not sampled. However, the large number of lakes in the region and results from other valuation studies of fjord water quality improvement suggest that for local water bodies in Norway both user and non-user values are predominant (Magnussen and Bergland, 1996). While the benefit transfer literature suggests that WTP estimates may not be ‘time stable’ for periods of over some 5years (Brouwer and Bateman, 2005), the available WTP values were not corrected for any possible changes in contextual circumstances. Given that water quality has deteriorated since the mid-nineties in the Vestre Vansjø and Storefjorden lakes in the catchment it is possible that the transfer of the WTP values from 1994 results in an underestimation of current WTP for recreational water quality. Overall, the transferred WTP may on the one hand be an underestimate of the worsened conditions during the past decade since the original study was carried out and fail to capture user and non-user values of households outside the catchment, while on the other hand it may overestate the recreation use value in Lake Storefjorden as part of the wider Vansjø lakes. While the standard error of the original WTP estimate is incorporated in our BN, the uncertainties related to the benefits transfer described above have not. While it is rare to have an on-site water quality valuation study available for benefits assessment — even one that is more than tenyears old — the explicit temporal and geographical context of the abatement and water quality models in our BN highlight some


un-quantified uncertainties inherent in the explicit transfer and integration of economic value estimates. In our network uncertainty about the total benefits estimation is determined jointly by the error of the average WTP estimate in the original study (E(WTP) = 2269NOK per household per year, standard error (WTP) = 200NOK per household per year 5), as well as the error in the aggregation procedure, i.e. the total number of households over which the average benefit value is to be aggregated (between 10,000 and 11,000 households by 2015 (Norwegian Bureau of Statistics, 2007). The expected aggregate WTP for water quality if suitability passed from 100% unsuitable to 100% suitable for bathing would be in the order of magnitude of 24million NOK/yr. In the model object “Economic impact on recreation” expected WTP for improvements in bathing suitability is determined by the probability that bathing in the lake goes from “unsuitable” in state 0 without measures to “suitable” in state 1 with measures. We calculate expected aggregate WTP as the joint probability of bathing suitability with measures being true and bathing suitability without measures being false, multiplied by WTP aggregated across the number of households in municipalities neighbouring the lake. In other words, the WTP results only play a role if the mitigation measures lead to an increase in the probability of a positive binary change in suitability. While bathing suitability could also have been defined on a stepwise scale, suitability was defined to fit the binary nature of the original WTP results. If P is interpreted as the proportion of days in a season likely to have bathing conditions, then we are assuming that WTP is proportional to the length of the season. Ideally, a CV study conducted specifically for application in a BN would provide a continuum of WTP estimates for different probabilities of bathing conditions in any given season, but this is not the case for the results taken from Magnussen et al. (1995). Finally, due to natural variability of processes in the catchment and water body there are random increases in water suitability between two periods, i.e. not related to the implementation of measures. In other words, the BN also calculates expected benefits of a water quality improvement which cannot be attributed to the implementation of the measures. We subtract such “windfall” benefits from expected benefits “with measures” in order to get a correct estimation of the incremental benefits.

5.

Results

5.1. Cost-effectiveness of measures accounting for uncertainty A convenient feature of Hugin software for model evaluation is the viewing of probability distributions within the BN. In Fig. 5, the nodes called “kr/kg” (NOK per kilogram) shown previously in Fig. 3, have been expanded to show each measure's cost-effectiveness. Cost-effectiveness is measured here as kr/kg phosphorus on-site at the ‘end-of-pipe’ for each

5 WTP values were adjusted to the year 2000 using the consumer price index (CPI) to make them comparable with the costs of measures in the same year.

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nutrient abatement measure, i.e. without regard to the eutrophication effect downstream in Lake Storefjorden. Visual inspection of the probability distributions at the top of Fig. 5 indicate that the measure “buffer strip” is the most costeffective, followed by similar cost-effectiveness for tillage and sedimentation dams, while individual wastewater treatment is least cost-effective. While the ranking of measures is similar to that of Lyche Solheim et al. (2001), the difference from the original deterministic cost-effectiveness analysis is that an unequivocal ranking of measures is more difficult due to the uncertainty incorporated both in the cost and effect assessment. If budget constraints forced managers to choose between the similarly cost-effective tillage and sedimentation dam measures, inspection of probability distributions in their underlying network nodes help identify and visualise what cost or effect assumptions dominate cost-effect uncertainty and where to gather more information.

5.2. Costs and benefits of measures accounting for uncertainty In the scenario presented in Fig. 5, the expected net benefits of all the measures taken together is − 4.9million NOK per year. The net benefits due to the incremental effect of implementing all measures while accounting for the random water quality improvements caused by natural variation in the absence of measures (2.7million NOK/yr) is − 7.6million NOK per year. In a deterministic benefit–cost analysis, i.e. where the costs of measures are assumed certain and abatement measures are expected to be 100% effective in reducing eutrophication, and there is no uncertainty regarding household WTP, net benefits would be around 16million NOK per year. Hence, the implementation of the proposed measures results in a welfare improvement assuming there is no uncertainty. When uncertainty in the DPSI chain is modelled explicitly, the probability of bathing suitability increases from 18% without to 28% with abatement measures (i.e. in a 100day bathing season abatement measures would provide an expected 10 additional days suitable for swimming). This low probability (multiplied with the aggregate WTP value) is the main reason for the net benefits to turn out negative under uncertainty. The uncertainty in the nodes for P-loading, Pconcentrations, algal biomass and cyanobacteria blooms is propagated through the network and substantially reduces the expected value of the benefits involved. In a deterministic analysis, this low probability of reaching suitable bathing conditions would simply be ignored and the aggregate WTP value applied unconditionally in the CBA. An inspection of the expected benefits of each measure on its own reveals a more varied picture of net benefits (buffer strips: 0.9million kr/yr; reduced tillage: − 0.1million kr/yr; sedimentation dams: 0.5million kr/yr; and individual wastewater treatment: − 6.0million kr/yr). Due to non-linearities in the environmental part of the network, expected net benefits of the whole programme of measures is not equal to the sum of the expected benefits for each individual measure. Of interest here is furthermore how costs and benefits relate to reaching GES. When only agricultural measures are implemented there is a 49% probability that Tot-P concentrations, and a 19% probability that cyanobacteria as a percentage of

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Fig. 5 – Evaluating the uncertain benefits of a programme of measures. Note: nodes in networks are presented with selected probability density functions (first row probabilities; second row discretisation intervals). Square decision nodes display expected utilities of decisions. Expected net benefits of implementing all the measures is − 4. 9 million kroner (decision nodes at top are set “true”). This is due to an increase in bathing suitability from 17.8% to 28.4% of summer season with full implmentation of measures.

algal biomass exceed the water authorities' limit values for GES (see Section 4.3). With all measures, including individual wastewater treatment plants, this drops only a little to 44 and 18% respectively. Hence, agricultural measures dominate the model outcomes under uncertainty. However, while some of the agricultural measures have positive expected net benefits when implemented individually they are insufficient to reach GES. After the first agricultural measure has been implemented, successive measures do little in the model to increase bathing suitability. Net benefits of implementing all agricultural measures together are − 0.6million NOK per year. Apart from non-linearities in the overall network, the coarse

discretization of the baseline nutrient loading nodes (intervals of 3000kg P/yr) relative to the effect of the individual measures (median effectiveness of 100–500kg P/yr) is the principle explanation for the lack of sensitivity of bathing suitability in the model.

6.

Discussion

Which variables contribute most to the reduction of uncertainty about a hypothesis variable such as WTP? Of all nodes in the main network in Fig. 3, the information-analysis in


Hugin reveals that the “baseline PIP loading” has the highest information statistic for the hypothesis variable ‘WTP total’. We evaluated the effect of using different probability distributions and discretizations in the ‘baseline PIP loading’ node (Table 1). This addresses some of the concerns raised by authors in previous case studies regarding assumptions about probability distributions in chance nodes. Referring to the results in Table 1, a uniform distribution has the highest information score — new observations of “baseline PIP loading” have the highest impact on WTP, because a uniform distribution contains the least information about the true value of the variable. Finer discretization intervals have a higher information score. An empirically derived distribution has a higher information score than a parametric normal

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distribution (holding discretization constant). Finally, evidence in the form of a 100% probability of a given interval provides most information of all because we assume knowledge of the discrete interval in which the true state of the ”baseline PIP loading” node lies. While increasing discretization and empirical distributions are preferred to estimated distributions in terms of the mutual information score, the impact on the uncertainty of the hypothesis variable (WTP) does not follow directly. Table 1 shows that expected WTP is bimodal as it is conditional on threshold definitions of bathing suitability. When a reduction in “PIP baseline loading” leads to an increase in expected WTP (from a large probability mass at zero) this also increases bimodality as probability at zero is shifted to a probability of

Table11–Value information analysis example Table – Value ofof information analysis example

Note: Entropy indicator H(WTP) is score for the information in WTP — a higher score is interpreted as more uncertainty about the true state of the node. The mutual information statistic (I(WTP|PIP baseline load)) scores how sensitive WTP is to new observations of “PIP baseline load”. The higher the score, the more information is gained about WTP by getting new observations on “PIP baseline load”. The analysis in Table 1 has been run without any other instantiations than those shown — the results therefore differ from the scenario analysed in Fig. 5.

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positive aggregate WTP. For example, uncertainty about expected WTP is greater when the interval for “PIP baseline loading” is known but relatively low (H = 0.4214; last row in Table 1) compared to when we use the empirical distribution with relatively high but variable historical PIP loading (H = 0.2687; third row in Table 1). Propagated uncertainty in our linked models is large enough to cancel out net benefits of the programme of measures in a deterministic model. While this is in part due to discretization assumptions in the network, the low effectiveness of measures in our model is in accordance with the lack of effectiveness of eutrophication measures observed in Lake Storefjorden (Bjørndalen et al., 2006). Hysteresis has been suggested as an explanation for this kind of lack of response to measures in shallow lakes (Scheffer and Carpenter, 2003). This implies that the ecological degradation is not simply reversible — degradation of a lake will start when the phosphorus pressure has reached an ecological threshold, but the ecological restoration process will not start until the phosphorus pressure is reduced to a level much below this threshold. A hysteresis situation can be indicated by multiple thresholds and alternative stable states, given sufficient data. The BN approach is in principle suitable for detecting alternative stable states (as peaks in different categories). For our case study, however, there is currently no sufficient data to demonstrate such multiple states and to support the hypothesis of hysteresis. In addition to uncertainty due to lacking data, causal structures and natural variability, our study revealed a number of modelling challenges that will likely be common to any application of BNs (Uusitalo, 2007) some of which have been addressed in studies reviewed earlier.

6.1.

Model complexity

There may be too many nodes in the network relative to an optimal problem formulation (Borsuk et al., 2004; Varis and Lahtela, 2002). A further evaluation of network techniques is therefore needed (e.g. parent-divorcing versus simplification of discretization intervals). In our case, some nodes may be redundant due to a lack of abatement effect (e.g. dissolved organic phosphorus is not affected by agricultural measures).

6.2.

Discretization

Relative to a continuous probability function, there is some information loss at each node due to discretization assumptions (Varis and Lahtela, 2002; Ames et al., 2005). How much information is lost overall in the network, and how discretization of a particular node is propagated to the variables of interest to decision-making will depend inter alia on whether the resolution is increasing or decreasing in the causality direction of the network, and particularly the resolution of the most sensitive variables. In our case, especially the coarse discretization of nutrient loading nodes in the lake water quality model significantly affected the estimated effectiveness of certain abatement measures compared to the baseline situation.

6.3.

Parametrization of conditional probability tables

Simple empirical data correlations (e.g. ChlA and percentage of cyanobacteria) embody more uncertainty than model predictions (e.g. erosion risk run-off regression model) and simulations (e.g. MyLake water quality model). The choice between empirical data correlations and model simulations in the meta-model is a topic that belongs to scientists, but the uncertainties caused by these choices should be evaluated and discussed with decision-makers. What best represents “current knowledge” about the system and whether it can be independently verified using other data or third party expert opinion will determine the model's credibility as a basis for decision-making (Ames et al., 2005).

6.4. Implicit temporal and geographical scale and variability The BN practitioner has to make sure that the variability described in chance nodes reflects the same temporal and spatial scales and resolution throughout the network. This consistency requirement is often hard to meet for a model integrating different disciplines with different sampling regimes. An example in our case study concerns the number of assumptions required to couple existing WTP estimates to predictions of bathing suitability from the water quality models.

6.5. Probabilistic analysis of the models used to estimate some of the input–output relationships As we used for instance published regression models in some parts of the model and a validated and calibrated lake model in another part of the model, it is crucial that the joint uncertainty analysis of these models is carried out correctly. If, for example, the correlation of the regression parameters is not included in the analysis using the variance–covariance matrix results, it may be that the probabilistic model predictions show too high uncertainties. Markov chain Monte Carlo (MCMC) techniques (e.g. Gamerman (1999)) enable a more correct analysis of parameter uncertainties, but this technique was not used in our model applications (e.g. MyLake).

6.6.

Modelling response variables and feedback

For basin-wide modelling of eutrophication problems the acyclical properties of BNs may not be a major limitation, because of the predominantly one-way causality of hydrological processes, the fact that eutrophication is dominated by intra-annual processes (except where for example sediment Psource is involved), and the fact that the economic analysis of abatement measures is undertaken on a progressive annual basis. In other words, a single network object can capture the most important eutrophication processes and economic considerations. For the analysis of the abatement of persistent pollutants or inter-annual water allocation problems, a timeslicing approach (Jenssen, 2001) will need to be used to capture some of the relevant dynamics underlying these water management problems. Another alternative is to use a probabilistic simulation model that includes loops and time steps, and to use only the input–output combinations of these


models in the meta-model of BN. Because each OOBN is replicated for each time period of interest, parsimony in model specification is crucial.

7.

Conclusions

In this paper we demonstrated the use of Bayesian networks for the evaluation of the costs and benefits of the implementation of the EU Water Framework Directive. A number of limitations of Bayesian networks identified in the literature are elaborated through our application to benefit–cost analysis of nutrient abatement measures in the Morsa catchment. While a deterministic calculus shows the annual net benefits of a programme of measures to be positive, explicitly accounting for the uncertainty across integrated models in a BN, i.e. probability based network, shows that expected costs exceed the expected benefits. The relative lack of effectiveness of the programme of measures shown by our model may be counter-intuitive to managers used to working with deterministic models. This underlines the point that the integration and multi-disciplinary process of defining a network, determining its probability distributions and conducting sensitivity analysis may be more important than the results of the analysis itself. In the review process for this paper the Bayesian network was quality controlled by external reviewers, who spotted potential drawbacks and errors in a matter of a few hours, showing that a Bayesian network can portray a complex management problem in an easily accessible fashion. While we see Bayesian decision analysis as an important addition to river basin managers' toolbox, we feel that further work is needed on the limitations identified above before Bayesian networks gain wider appeal in integrated management of water resources.

Acknowledgements This work was supported by the EU research projects “North Sea Regional and Local Implementation of the Water Framework Directive” (NOLIMP-WFD) and “Benchmark Models for the Water Framework Directive (BMW)” and “Bayesian network integration of nutrient loading and lake eutrophication models in cost-effectiveness analysis of abatement measures” (EutroBayes), supported by the Norwegian Research Council. We are grateful to two anonymous referees for their useful comments on an early draft of the paper and in particular Jan Vermaat and Roy Brouwer from the Institute for Environmental Studies, Vrije Universiteit Amsterdam for their extensive critical review of the various paper versions.

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