Consistent with the decision analytic approach, our model building process started with the construction of a comprehensive objectives hierarchy concerning the goals of river rehabilitation (Reichert et al. 2007). This process was undertaken in collaboration with a collection of organized stakeholder groups for a river rehabilitation project in Switzerland, as described by Hostmann et al. (2005a, 2005b). Measurable attributes describing the achievement of the lower-level objectives of the hierarchy then provide the set of variables that should ideally be predicted by the probabilistic outcome model. The key variables resulting from this process, in our case, in addition to the obvious ones of flood protection level and rehabilitation cost, included: river morphology, hydraulic habitat characteristics, erosion potential, siltation of riverbed, benthic population diversity and biomass, riparian community abundance, fish population abundance and age structure, local employment potential, and adjacent land use impacts (see Hostmann et al. 2005a for details). These variables are related to each other and to input variables describing rehabilitation alternatives as shown in the schematic network model of Figure 1. As most of the submodels delineated by the dotted boxes in Figure 1 have been described elsewhere, we will describe them only briefly in the following subsections.
Benthic population submodel
Periphyton and invertebrates are important components of river ecosystems due to their ability to produce organic material, decompose detritus, and serve as a food source for organisms at higher trophic levels. We developed a set of simple data-based models to describe the biomass of periphyton, total invertebrates, and 3 invertebrate functional feeding groups (collector-gatherers, predators, and scrapers) as a function of time since the last bed-moving flood, mean water depth, grain size, mean flow velocity, and season (time within the year) (Schweizer 2006). Parameters were estimated from a statistical fit to survey data from as many as 8 sites in 3 rivers. The statistical approach made it easier to derive reliable relationships across different rivers and sites than would have been possible with a strictly mechanistic modeling approach.
Considering the diversity of data sets and the simplicity of the formulations, the models lead to a remarkably good agreement with time series of measurements (Schweizer 2006). Because of the larger data set available, this was particularly true for the periphyton model.
Riparian community submodel
Our riparian community model focuses on riparian arthropods, a significant contributor to overall riverine biodiversity and a functionally important component of river ecosystems. The model predicts the abundance of 3 major groups (spiders, ground beetles, and rove beetles) as a function of river morphology and riparian siltation. Both of these variables are outputs of the “Morphology and Hydraulic” submodel described above.
Using the data of Paetzold et al. (2008), we carried out multiple regression analyses to relate the variation in species abundance to the abiotic predictor variables (Figure 3). We found that for all species there were significant differences between natural and channelized morphologies. In addition, siltation reduced the abundance of all species similarly in both types of morphologies, except for spiders at channelized sites that were already so low that siltation had no further effect. Uncertainty was accounted for through the inclusion of a probabilistic error term, derived from the distribution of residuals from the regression model.
Figure 3. Predicted and observed abundance of (a) spiders, (b) rove beetles, and (c) ground beetles as a function of riparian siltation and river morphology. Squares represent data from channelized rivers and circles represent data from rivers with natural morphology.
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The core of our fish submodel is a dynamic, age-structured population simulation (Borsuk et al. 2006). Because our model was initially developed for application to midland rivers in Switzerland, our focal species is brown trout, a desirable sport fish and indicator of river ecosystem health (Lasne et al. 2007).
We represent 5 major stages of the brown trout life cycle in the model: eggs, newly emergent fry, late summer fry, immature juveniles, and adult spawners. The distinction between emergent and late summer fry was made to differentiate the period of greatest density dependence. The number of individuals in each life stage is determined by the number in the previous life stage, as well as relevant population parameters, such as survival and reproductive rates. These parameters are influenced, in turn, by intermediate variables, such as body size and growth rate, or by external controls relevant to rehabilitation measures, such as including substrate and habitat quality, stocking practices, angling, prey resources, and competing species (Figure 4).
Figure 4. Graphical probability network of brown trout population model. Dark shaded nodes indicate life cycle stages. Light shaded nodes indicate variables likely to be influenced by rehabilitation measures.
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Consistent with the Bayesian viewpoint that probabilities represent subjective degrees of belief, conditional probabilities in a Bayesian network may be based on any available information, including experimental or field results, process-based models, or the elicited judgment of experts. Many of the relationships in the fish submodel were based on expert elicitations (see Borsuk et al. 2006 for details).
The socioeconomic submodel estimates the direct and indirect effects of river rehabilitation on the local economy (Spörri et al. 2007). The basic model is expressed in terms of an input–output matrix (Miller and Blair 1985), tracking goods and service flows between various sectors of the economy. This matrix is then used to calculate the direct and indirect changes in economic output and jobs per sector resulting from the direct changes in demand associated with the rehabilitation project (e.g., in the construction, real estate, and other involved industries). We account for uncertainty in the data and in some of the model assumptions using a probabilistic formulation and propagating these probabilities through the model equations (see Spörri et al. 2007 for details).
Model users can input into the model a local input–output matrix, if available, or modify a national matrix using, for example, local employment statistics (the location quotient method; Miller 1998). The input–output method can also be used to estimate the longer-term effects on the local economy of changes in tourism and recreation expected to result from rehabilitation.
The submodels described in the above sections were implemented in Analytica, a commercially available software program for propagating uncertainty through models formulated as graphical networks (Lumina 1997). In principle, the model could be implemented using other software; we chose Analytica because it allows for a wide variety of probability distributions and functional model forms.
The inputs required to run the integrative river rehabilitation model consist of both site-specific characteristics (e.g., discharge statistics, reach slope, gravel size, water temperature patterns) and rehabilitation design criteria (e.g., project expenditures, dike height and spacing, land use changes, revegetation plans, fish stocking). The former can generally be determined from historical or site survey data for the river reach of interest, and the latter should be viewed as decision variables to be set to values corresponding to current conditions, specific design alternatives, or scenarios used for sensitivity analysis.
In Analytica, each input variable is specified by either a fixed value, a probability density function, or, in the case of a discrete variable, a probability table. Input nodes can also be specified as multivariate joint distributions if the uncertainty of 2 or more variables cannot be assumed to be independent. Each child node is then defined by a conditional probability table or, more generally, by a functional expression of the form of Equation 1, derived from submodels such as those described in the sections above. In the latter case, model parameters and error terms are represented explicitly as marginal nodes that serve as parents of a child node along with that variable's predictors. Covariance in model parameters is included by using appropriate multivariate distributions.
Once all model variables, parameters, and relationships are specified, a large sample of realizations is drawn for each marginal node using random Latin hypercube sampling (McKay et al. 1979). These realizations are then propagated through the functional expressions defining the conditional distribution of each child node. These children then serve as parents of the next set of nodes in the network, and their uncertainty is propagated accordingly. The distributions of final model endpoints thus convey the combined uncertainty and variability coming from their entire set of ancestors. This process of Monte Carlo–type simulation is implemented in many other software packages, but to our knowledge, Analytica is the only commercial software that provides a graphical diagramming interface consistent with the Bayesian network concept. Analytica also allows for the inclusion of decision and objective nodes, making it easy to implement a description of stakeholder preferences to yield decision analytic results (e.g., Hostman et al. 2005a; Reichert et al. 2007).
The process of integrating our various submodels into a cohesive network required some additional assumptions. For example, whereas sufficient knowledge and data exist to build the benthic population submodel at a flow–dynamic scale (days to weeks), the fish population model focuses on life cycle transitions at a longer time scale (months to years). Thus, for estimating brown trout prey availability, the seasonal average benthic biomass was used, neglecting shorter term fluctuations. In other network models, we have faced the opposite situation, needing to express changes in highly dynamic variables as functions of more slowly evolving variables (Borsuk et al. 2004), a technique referred to as “variable speed splitting” (Walters and Korman 1999). Either of these methods may be necessary for cross-scale modeling with Bayesian networks.
Another challenge comes with reconciling the spatial scales of the various submodels. For example, although the physical processes of siltation can be described mathematically at a fine spatial and temporal scale, the brown trout spawning model required the reach-wide seasonal average as input. Therefore, we decided that a reasonable simplifying procedure would be to: 1) assume spatially homogeneous conditions within each hydraulic unit (i.e., runs, pools, riffles), 2) simulate the siltation dynamics under each of these sets of conditions, then 3) calculate the seasonal average siltation level for each hydraulic unit, and finally 4) calculate the weighted average siltation level for the reach using the modeled proportions of each type of hydraulic unit. Similar assumptions concerning homogeneity within hydraulic units were also necessary for other variables, including water depth and velocity, gravel size, and habitat suitability.