A Bayesian hierarchical framework for calibrating aquatic biogeochemical models

Model practitioners increasingly place emphasis on rigorous quantitative error analysis in aquatic biogeochemical models and the existing initiatives range from the development of alternative metrics for goodness of fit, to data assimilation into operational models, to parameter estimation techniques. However, the treatment of error in many of these efforts is arguably selective and/or ad hoc. A Bayesian hierarchical framework enables the development of robust probabilistic analysis of error and uncertainty in model predictions by explicitly accommodating measurement error, parameter uncertainty, and model structure imperfection. This paper presents a Bayesian hierarchical formulation for simultaneously calibrating aquatic biogeochemical models at multiple systems (or sites of the same system) with differences in their trophic conditions, prior precisions of model parameters, available information, measurement error or inter-annual variability. Our statistical formulation also explicitly considers the uncertainty in model inputs (model parameters, initial conditions), the analytical/sampling error associated with the field data, and the discrepancy between model structure and the natural system dynamics (e.g., missing key ecological processes, erroneous formulations, misspecified forcing functions). The comparison between observations and posterior predictive monthly distributions indicates that the plankton models calibrated under the Bayesian hierarchical scheme provided accurate system representations for all the scenarios examined. Our results also suggest that the Bayesian hierarchical approach allows overcoming problems of insufficient local data by “borrowing strength” from well-studied sites and this feature will be highly relevant to conservation practices of regions with a high number of freshwater resources for which complete data could never be practically collected. Finally, we discuss the prospect of extending this framework to spatially explicit biogeochemical models (e.g., more effectively connect inshore with offshore areas) along with the benefits for environmental management, such as the optimization of the sampling design of monitoring programs and the alignment with the policy practice of adaptive management.     

Impact of time series data on calibration and prediction uncertainty for a deterministic hydrogeochemical model

Model calibration is fundamental in applications of deterministic process-based models. Uncertainty in model predictions depends much on the input data and observations available for model calibration. Here we explored how model predictions (forecasts) and their uncertainties vary with the length of time series data used in calibration. As an example we used the hydrogeochemical model MAGIC and data from Birkenes, a small catchment in southern Norway, to simulate future water chemistry under a scenario of reduced acid deposition. A Bayesian approach with a Markov Chain Monte Carlo (MCMC) technique was used to calibrate the model to different lengths of observed data (4–29 years) and to estimate the prediction uncertainty each calibration. The results show that the difference between modelled and observed water chemistry (calibration goodness of fit) in general decreases with increasing length of the time series used in calibration. However, there are considerable differences for different time series of the same length. The results also show that the uncertainties in predicted future acid neutralizing capacity were lowest (i.e. the distribution peak narrowest) when using the longest time series for calibration. As for calibration success, there were considerable differences between the future distributions (prediction uncertainty) for the different calibrations.     

An integrated modeling framework for performing environmental assessments: Application to ecosystem services in the Albemarle-Pamlico basins (NC and VA, USA)

The U.S. Environmental Protection Agency uses environmental models to inform rulemaking and policy decisions at multiple spatial and temporal scales. As decision-making has moved towards integrated thinking and assessment (e.g. media, site, region, services), the increasing complexity and interdisciplinary nature of modern environmental problems has necessitated a new generation of integrated modeling technologies. Environmental modelers are now faced with the challenge of determining how data from manifold sources, types of process-based and empirical models, and hardware/software computing infrastructure can be reliably integrated and applied to protect human health and the environment.In this study, we demonstrate an Integrated Modeling Framework that allows us to predict the state of freshwater ecosystem services within and across the Albemarle-Pamlico Watershed, North Carolina and Virginia (USA). The Framework consists of three facilitating technologies: Data for Environmental Modeling automates the collection and standardization of input data; the Framework for Risk Assessment of Multimedia Environmental Systems manages the flow of information between linked models; and the Supercomputer for Model Uncertainty and Sensitivity Evaluation is a hardware and software parallel-computing interface with pre/post-processing analysis tools, including parameter estimation, uncertainty and sensitivity analysis. In this application, five environmental models are linked within the Framework to provide multimedia simulation capabilities: the Soil Water Assessment Tool predicts watershed runoff; the Watershed Mercury Model simulates mercury runoff and loading to streams; the Water quality Analysis and Simulation Program predicts water quality within the stream channel; the Habitat Suitability Index model predicts physicochemical habitat quality for individual fish species; and the Bioaccumulation and Aquatic System Simulator predicts fish growth and production, as well as exposure and bioaccumulation of toxic substances (e.g., mercury).Using this Framework, we present a baseline assessment of two freshwater ecosystem services-water quality and fisheries resources-in headwater streams throughout the Albemarle-Pamlico. A stratified random sample of 50 headwater streams is used to draw inferences about the target population of headwater streams across the region. Input data is developed for a twenty-year baseline simulation in each sampled stream using current land use and climate conditions. Monte Carlo sampling (n = 100 iterations per stream) is also used to demonstrate some of the Framework's experimental design and data analysis features. To evaluate model performance and accuracy, we compare initial (i.e., uncalibrated) model predictions (water temperature, dissolved oxygen, fish density, and methylmercury concentration within fish tissue) against empirical field data. Finally, we ‘roll-up’ the results from individual streams, to assess freshwater ecosystem services at the regional scale.     

Predicting Bird Species Distributions in Reconstructed Landscapes

Abstract:  Landscape optimization for biodiversity requires prediction of species distributions under alternative revegetation scenarios. We used Bayesian model averaging with logistic regression to predict probabilities of occurrence for 61 species of birds within highly fragmented box–ironbark forests of central Victoria, Australia. We used topographic, edaphic, and climatic variables as predictors so that the models could be applied to areas where vegetation has been cleared but may be replanted. Models were evaluated with newly acquired, independent data collected in large blocks of remnant native vegetation. Successful predictions were obtained for 18 of 45 woodland species (40%). Model averaging produced more accurate predictions than "single best" models. Models were most successful for smaller-bodied species that probably depend on particular vegetation types. Predictions for larger, generalist species, and seasonal migrants were less successful, partly because of changes in species distributions between model building (1995–1997) and validation (2004–2005) surveys. We used validated models to project occurrence probabilities for individual species across a 12,000-km² region, assuming native vegetation was present. These predictions are intended to be used as inputs, along with landscape context and temporal dynamics, into optimization algorithms to prioritize revegetation. Longer-term data sets to accommodate temporal dynamics are needed to improve the predictive accuracy of models.     

Sensitivity of ecological models to their climate drivers: statistical ensembles for forcing.

Global and regional numerical models for terrestrial ecosystem dynamics require fine spatial resolution and temporally complete historical climate fields as input variables. However, because climate observations are unevenly spaced and have incomplete records, such fields need to be estimated. In addition, uncertainty in these fields associated with their estimation are rarely assessed. Ecological models are usually driven with a geostatistical model's mean estimate (kriging) of these fields without accounting for this uncertainty, much less evaluating such errors in terms of their propagation in ecological simulations. We introduce a Bayesian statistical framework to model climate observations to create spatially uniform and temporally complete fields, taking into account correlation in time and space, spatial heterogeneity, lack of normality, and uncertainty about all these factors. A key benefit of the Bayesian model is that it generates uncertainty measures for the generated fields. To demonstrate this method, we reconstruct historical monthly precipitation fields (a driver for ecological models) on a fine resolution grid for a climatically heterogeneous region in the western United States. The main goal of this work is to evaluate the sensitivity of ecological models to the uncertainty associated with prediction of their climate drivers. To assess their numerical sensitivity to predicted input variables, we generate a set of ecological model simulations run using an ensemble of different versions of the reconstructed fields. We construct such an ensemble by sampling from the posterior predictive distribution of the climate field. We demonstrate that the estimated prediction error of the climate field can be very high. We evaluate the importance of such errors in ecological model experiments using an ensemble of historical precipitation time series in simulations of grassland biogeochemical dynamics with an ecological numerical model, Century. We show how uncertainty in predicted precipitation fields is propagated into ecological model results and that this propagation had different modes. Depending on output variable, the response of model dynamics to uncertainty in inputs ranged from uncertainty in outputs that matched that of inputs to those that were muted or that were biased, as well as uncertainty that was persistent in time after input errors dropped.     

Parameter uncertainties in the modelling of vegetation dynamics—Effects on tree community structure and ecosystem functioning in European forest biomes

Dynamic vegetation models are useful tools for analysing terrestrial ecosystem processes and their interactions with climate through variations in carbon and water exchange. Long-term changes in structure and composition (vegetation dynamics) caused by altered competitive strength between plant functional types (PFTs) are attracting increasing attention as controls on ecosystem functioning and potential feedbacks to climate. Imperfect process knowledge and limited observational data restrict the possibility to parameterise these processes adequately and potentially contribute to uncertainty in model results. This study addresses uncertainty among parameters scaling vegetation dynamic processes in a process-based ecosystem model, LPJ-GUESS, designed for regional-scale studies, with the objective to assess the extent to which this uncertainty propagates to additional uncertainty in the tree community structure (in terms of the tree functional types present and their relative abundance) and thus to ecosystem functioning (carbon storage and fluxes). The results clearly indicate that the uncertainties in parameterisation can lead to a shift in competitive balance, most strikingly among deciduous tree PFTs, with dominance of either shade-tolerant or shade-intolerant PFTs being possible, depending on the choice of plausible parameter values. Despite this uncertainty, our results indicate that the resulting effect on ecosystem functioning is low. Since the vegetation dynamics in LPJ-GUESS are representative for the more complex Earth system models now being applied within ecosystem and climate research, we assume that our findings will be of general relevance. We suggest that, in terms of carbon storage and fluxes, the heavier parameterisation requirement of the processes involved does not widen the overall uncertainty in model predictions.     

Quantifying parameter uncertainty in a coral reef model using Metropolis-Coupled Markov Chain Monte Carlo

Coral reefs are threatened ecosystems, so it is important to have predictive models of their dynamics. Most current models of coral reefs fall into two categories. The first is simple heuristic models which provide an abstract understanding of the possible behaviour of reefs in general, but do not describe real reefs. The second is complex simulations whose parameters are obtained from a range of sources such as literature estimates. We cannot estimate the parameters of these models from a single data set, and we have little idea of the uncertainty in their predictions.We have developed a compromise between these two extremes, which is complex enough to describe real reef data, but simple enough that we can estimate parameters for a specific reef from a time series. In previous work, we fitted this model to a long-term data set from Heron Island, Australia, using maximum likelihood methods. To evaluate predictions from this model, we need estimates of the uncertainty in our parameters. Here, we obtain such estimates using Bayesian Metropolis-Coupled Markov Chain Monte Carlo. We do this for versions of the model in which corals are aggregated into a single state variable (the three-state model), and in which corals are separated into four state variables (the six-state model), in order to determine the appropriate level of aggregation. We also estimate the posterior distribution of predicted trajectories in each case.In both cases, the fitted trajectories were close to the observed data, but we had doubts about the biological plausibility of some parameter estimates. We suggest that informative prior distributions incorporating expert knowledge may resolve this problem. In the six-state model, the posterior distribution of state frequencies after 40 years contained two divergent community types, one dominated by free space and soft corals, and one dominated by acroporid, pocilloporid, and massive corals. The three-state model predicts only a single community type. We conclude that the three-state model hides too much biological heterogeneity, but we need more data if we are to obtain reliable predictions from the six-state model. It is likely that there will be similarly large, but currently unevaluated, uncertainty in the predictions of other coral reef models, many of which are much more complex and harder to fit to real data.     

A comparison of four process-based models and a statistical regression model to predict growth of Eucalyptus globulus plantations

In forest management and ecological research, consideration of the impacts and risks of climate change or management optimisation is complex. Computer models have long been applied as tools for these tasks. Process-based forest growth models claim to overcome the limitations of empirical statistical models, but the capacity of different process-based models and modelling approaches have rarely been compared directly. This study evaluates stepwise multiple regression models in comparison to four process-based modelling approaches (3-PG, 3-PG+, CABALA and Forest-DNDC) for greenfield predictions of Eucalyptus globulus plantation growth from 2 to 8 years after planting throughout southern Australia.     

Development of the gap model ZELIG-CFS to predict the dynamics of North American mixed forest types with complex structures

When the development of gap models began about three decades ago, they became a new category of forest productivity models. Compared with traditional growth and yield models, which aim at deriving empirical relationships that best fit data, gap models use semi-theoretical relationships to simulate biotic and abiotic processes in forest stands, including the effects of photosynthetic active radiation interception, site fertility, temperature and soil moisture on tree growth and seedling establishment. While growth and yield models are appropriate to predict short-term stemwood production, gap models may be used to predict the natural course of species replacement for several generations. Because of the poor availability of historical data and knowledge on species-specific allometric relationships, species replacement and death rate, it has seldom been possible to develop and evaluate the most representative algorithms to predict growth and mortality with a high degree of accuracy. For this reason, the developers of gap models focused more on developing simulation tools to improve the understanding of forest succession than predicting growth and yield accurately.In a previous study, the predictions of simulations in two southeastern Canadian mixed ecosystem types using the ZELIG gap model were compared with long-term historical data. This exercise highlighted model components that needed modifications to improve the predictive capacity of ZELIG. The updated version of the model, ZELIG-CFS, includes modifications in the modelling of crown interaction effects, survival rate and regeneration. Different algorithms representing crown interactive effects between crowns were evaluated and species-specific model components that compute individual-tree mortality probability rate were derived. The results of the simulations were compared using long-term remeasurement data obtained from sample plots located in La Mauricie National Park of Canada in Quebec. In the present study, three forest types were studied: (1) red spruce-balsam fir-yellow birch, (2) yellow birch-sugar maple-balsam fir, and (3) red spruce-balsam fir-white birch mixed ecosystems. Among the seven algorithms that represented individual crown interactions, two better predicted the changes in basal area and individual-tree growth: (1) the mean available light growing factor (ALGF), which is computed from the proportion of light intercepted at different levels of individual crowns adjusted by the species-specific shade tolerance index, and (2) the ratio of mean ALGF to crown width. The long-term predicted patterns of change in basal area were consistent with the life history of the different species.     

Optimization of ecosystem model parameters using spatio-temporal soil moisture information

Parameters in process-based terrestrial ecosystem models are often nonlinearly related to the water flux to the atmosphere, and they also change temporally and spatially. Therefore, for estimating soil moisture, process-based terrestrial ecosystem models inevitably need to specify spatially and temporally variant model parameters. This study presents a two-stage data assimilation scheme (TSDA) to spatially and temporally optimize some key parameters of an ecosystem model which are closely related to soil moisture. At the first stage, a simplified ecosystem model, namely the Boreal Ecosystem Productivity Simulator (BEPS), is used to obtain the prior estimation of daily soil moisture. After the spatial distribution of 0–10 cm surface soil moisture is derived from remote sensing, an Ensemble Kalman Filter is used to minimize the difference between the remote sensing model results, through optimizing some model parameters spatially. At the second stage, BEPS is reinitialized using the optimized parameters to provide the updated model predictions of daily soil moisture. TSDA has been applied to an arid and semi-arid area of northwest China, and the performance of the model for estimating daily 0–10 cm soil moisture after parameter optimization was validated using field measurements. Results indicate that the TSDA developed in this study is robust and efficient in both temporal and spatial model parameter optimization. After performing the optimization, the correlation (r²) between model-predicted 0–10 cm soil moisture and field measurement increased from 0.66 to 0.75. It is demonstrated that spatial and temporal optimization of ecosystem model parameters can not only improve the model prediction of daily soil moisture but also help to understand the spatial and temporal variation of some key parameters in an ecosystem model and the corresponding ecological mechanisms controlling the variation.     

Response of tree growth to a changing climate in boreal central Canada: A comparison of empirical,process-based,and hybrid modelling approaches

The impact of 2 × CO₂ driven climate change on radial growth of boreal tree species Pinus banksiana Lamb., Populus tremuloides Michx. and Picea mariana (Mill.) BSP growing in the Duck Mountain Provincial Forest of Manitoba (DMPF), Canada, is simulated using empirical and process-based model approaches. First, empirical relationships between growth and climate are developed. Stepwise multiple-regression models are conducted between tree-ring growth increments (TRGI) and monthly drought, precipitation and temperature series. Predictive skills are tested using a calibration–verification scheme. The established relationships are then transferred to climates driven by 1× and 2 × CO₂ scenarios using outputs from the Canadian second-generation coupled global climate model. Second, empirical results are contrasted with process-based projections of net primary productivity allocated to stem development (NPP_s). At the finest scale, a leaf-level model of photosynthesis is used to simulate canopy properties per species and their interaction with the variability in radiation, temperature and vapour pressure deficit. Then, a top-down plot-level model of forest productivity is used to simulate landscape-level productivity by capturing the between-stand variability in forest cover. Results show that the predicted TRGI from the empirical models account for up to 56.3% of the variance in the observed TRGI over the period 1912–1999. Under a 2 × CO₂ scenario, the predicted impact of climate change is a radial growth decline for all three species under study. However, projections obtained from the process-based model suggest that an increasing growing season length in a changing climate could counteract and potentially overwhelm the negative influence of increased drought stress. The divergence between TRGI and NPP_s simulations likely resulted, among others, from assumptions about soil water holding capacity and from calibration of variables affecting gross primary productivity. An attempt was therefore made to bridge the gap between the two modelling approaches by using physiological variables as TRGI predictors. Results obtained in this manner are similar to those obtained using climate variables, and suggest that the positive effect of increasing growing season length would be counteracted by increasing summer temperatures. Notwithstanding uncertainties in these simulations (CO₂ fertilization effect, feedback from disturbance regimes, phenology of species, and uncertainties in future CO₂ emissions), a decrease in forest productivity with climate change should be considered as a plausible scenario in sustainable forest management planning of the DMPF.     

Scaling up of biomass simulation for Eucalyptus plantations based on landsenses ecology

ABSTRACT

Sustainable forest management on a regional scale requires accurate biomass estimation. At present, technologically comprehensive forecasting estimates are generated using process-based ecological models. However, isolation of the ecological factors that cause uncertainty in model behavior is difficult. To solve this problem, this study aimed to construct a meliorization model evaluation framework to explain uncertainty in model behavior with respect to both the mechanisms and algorithms involved in ecological forecasting based on the principle of landsenses ecology. We introduce a complicated ecological driving mechanism to the process-based ecological model using analytical software and algorithms. Subsequently, as a case study, we apply the meliorization model evaluation framework to detect Eucalyptus biomass forest patches at a regional scale (196,158 ha) using the 3PG2 (Physiological Principles in Predicting Growth) model. Our results show that this technique improves the accuracy of ecological simulation for ecological forecasting and prevents new uncertainties from being produced by adding a new driving mechanism to the original model structure. This result was supported by our Eucalyptus biomass simulation using the 3PG2 model, in which ecological factors caused 21.83% and 9.05% uncertainty in model behavior temporal and spatial forecasting, respectively. In conclusion, the systematic meliorization model evaluation framework reported here provides a new method that could be applied to research requiring comprehensive ecological forecasting. Sustainable forest management on regional scales contributes to accurate forest biomass simulation through the principle of landsenses ecology, in which mix-marching data and a meliorization model are combined.     

Predicting a ‘tree change’ in Australia's tropical savannas: Combining different types of models to understand complex ecosystem behaviour

In this study, key ecological modelling limitations of a process-based simulation model and a Bayesian network were reduced by combining the two approaches. We demonstrate the combined modelling approach with a case study investigating increases in woody vegetation density in northern Australia's tropical savannas. We found that by utilising the strengths of a simulation model and a Bayesian network we could both forecast future change in woody vegetation density and diagnose the reasons for current vegetation states. The local conditions of climate, soil characteristics and the starting population of trees were found to be more important in explaining the likelihood of change in woody vegetation density compared to management practices such as grazing pressure and fire regimes. We conclude that combining the strengths of a process and BN model allowed us to produce a simple model that utilised the ability of the process model to simulate ecosystem processes in detail and over long time periods, and the ability of the BN to capture uncertainty in ecosystem response and to conduct scenario, sensitivity and diagnostic analysis. The overall result was a model that has the potential to provide land managers with a better understanding of the behaviour of a complex ecosystem than simply utilising either modelling approach in isolation.     

Assessment of herbaceous plant habitats in water-constrained environments: predicting indirect effects with fuzzy logic

Herbaceous plant production plays a key role in determining the function of rangeland ecosystems in the semi-arid and Mediterranean regions. Therefore, assessment of herbaceous plant habitats is important for understanding the ecosystem functioning in these regions and for applied purposes, such as range management and land evaluation. This paper presents a model to assess herbaceous plant habitats in a basaltic stony environment in a Mediterranean region. The model is based on geographic information systems (GIS), remote sensing and fuzzy logic, while four indirect variables, which represent major characteristics of herbaceous habitats, are modeled: rock cover fraction; wetness index (WI); soil depth; and slope orientation (aspect). A linear unmixing model was used to measure rock cover on a per pixel basis using a Landsat TM summer image. The wetness index and local aspect were determined from digital elevation data with 25 m × 25 m pixel resolution, while soil data were gathered in a field survey. The modeling approach adopted here is process-based and assumes that water availability plays a crucial role in determining herbaceous plant production in Mediterranean and semi-arid environments. The model rules are based on fuzzy logic and are written based on the hypothesized water requirements of the herbaceous vegetation. The results show that on a polygon basis there is positive agreement between the model proposed here and previous mapping of the herbaceous habitats carried out in the field using traditional methods. Intrapolygon tests show that the use of a continuous raster data model and fuzzy logic principles provide an added value to traditional mapping. Moreover, herbaceous biomass measurements at two time intervals—mid- and peak winter season—corresponded with the habitat assessment predictions achieved using a new scenario that is proposed in this research. This scenario suggests that rockiness increases herbaceous production on south-facing slopes, while in other slope aspects the rock cover has lower impact on herbaceous growth. Due to its simplicity, the model suggested here can be used by planners and managers, to adjust range activities over large areas. The process-based approach should allow adaptation of the model to other regions more effectively than models that were formulated on a purely empirical basis. The model could also be used to study the relationship between water availability and ecosystem productivity on a regional scale.     

Multiple sources of predictive uncertainty in modeled estimates of net ecosystem CO2 exchange

Net ecosystem CO₂ exchange (NEE) is typically measured directly by eddy covariance towers or is estimated by ecosystem process models, yet comparisons between the data obtained by these two methods can show poor correspondence. There are three potential explanations for this discrepancy. First, estimates of NEE as measured by the eddy-covariance technique are laden with uncertainty and can potentially provide a poor baseline for models to be tested against. Second, there could be fundamental problems in model structure that prevent an accurate simulation of NEE. Third, ecosystem process models are dependent on ecophysiological parameter sets derived from field measurements in which a single parameter for a given species can vary considerably. The latter problem suggests that with such broad variation among multiple inputs, any ecosystem modeling scheme must account for the possibility that many combinations of apparently feasible parameter values might not allow the model to emulate the observed NEE dynamics of a terrestrial ecosystem, as well as the possibility that there may be many parameter sets within a particular model structure that can successfully reproduce the observed data. We examined the extent to which these three issues influence estimates of NEE in a widely used ecosystem process model, Biome-BGC, by adapting the generalized likelihood uncertainty estimation (GLUE) methodology. This procedure involved 400,000 model runs, each with randomly generated parameter values from a uniform distribution based on published parameter ranges, resulting in estimates of NEE that were compared to daily NEE data from young and mature Ponderosa pine stands at Metolius, Oregon. Of the 400,000 simulations run with different parameter sets for each age class (800,000 total), over 99% of the simulations underestimated the magnitude of net ecosystem CO₂ exchange, with only 4.07% and 0.045% of all simulations providing satisfactory simulations of the field data for the young and mature stands, even when uncertainties in eddy-covariance measurements are accounted for. Results indicate fundamental shortcomings in the ability of this model to produce realistic carbon flux data over the course of forest development, and we suspect that much of the mismatch derives from an inability to realistically model ecosystem respiration. However, difficulties in estimating historic climate data are also a cause for model-data mismatch, particularly in a highly ecotonal region such as central Oregon. This latter difficulty may be less prevalent in other ecosystems, but it nonetheless highlights a challenge in trying to develop a dynamic representation of the terrestrial biosphere.     

Modeling seasonal course of carbon fluxes and evapotranspiration in response to low temperature and moisture in a boreal Scots pine ecosystem

Environmental conditions act above and below ground, and regulate carbon fluxes and evapotranspiration. The productivity of boreal forest ecosystems is strongly governed by low temperature and moisture conditions, but the understanding of various feedbacks between vegetation and environmental conditions is still unclear. In order to quantify the seasonal responses of vegetation to environmental factors, the seasonality of carbon and heat fluxes and the corresponding responses for temperature and moisture in air and soil were simulated by merging a process-based model (CoupModel) with detailed measurements representing various components of a forest ecosystem in Hyytiälä, southern Finland. The uncertainties in parameters, model assumptions, and measurements were identified by generalized likelihood uncertainty estimation (GLUE). Seasonal and diurnal courses of sensible and latent heat fluxes and net ecosystem exchange (NEE) of CO₂ were successfully simulated for two contrasting years. Moreover, systematic increases in efficiency of photosynthesis, water uptake, and decomposition occurred from spring to summer, demonstrating the strong coupling between processes. Evapotranspiration and NEE flux both showed a strong response to soil temperature conditions via different direct and indirect ecosystem mechanisms. The rate of photosynthesis was strongly correlated with the corresponding water uptake response and the light use efficiency. With the present data and model assumptions, it was not possible to precisely distinguish the various regulating ecosystem mechanisms. Our approach proved robust for modeling the seasonal course of carbon fluxes and evapotranspiration by combining different independent measurements. It will be highly interesting to continue using long-term series data and to make additional tests of optional stomatal conductance models in order to improve our understanding of the boreal forest ecosystem in response to climate variability and environmental conditions.     

Nutrient criteria for lakes,ponds, and reservoirs: A Bayesian TREED model approach

We develop regional-scale eutrophication models for lakes, ponds, and reservoirs to investigate the link between nutrients and chlorophyll-a. The Bayesian TREED (BTREED) model approach allows association of multiple environmental stressors with biological responses, and quantification of uncertainty sources in the empirical water quality model. Nutrient data for lakes, ponds, and reservoirs across the United States were obtained from the Environmental Protection Agency (EPA) National Nutrient Criteria Database. The nutrient data consist of measurements for both stressor variables (such as total nitrogen and total phosphorus), and response variables (such as chlorophyll-a), used in the BTREED model. Markov chain Monte Carlo (McMC) posterior exploration guides a stochastic search through a rich suite of candidate trees toward models that better fit the data. The Bayes factor provides a goodness of fit criterion for comparison of resultant models. We randomly split the data into training and test sets; the training data were used in model estimation, and the test data were used to evaluate out-of-sample predictive performance of the model. An average relative efficiency of 1.02 between the training and test data for the four highest log-likelihood models suggests good out-of-sample predictive performance. Reduced model uncertainty relative to over-parameterized alternative models makes the BTREED models useful for nutrient criteria development, providing the link between nutrient stressors and meaningful eutrophication response.     

Improving multi-site benefit functions via Bayesian model averaging: A new approach to benefit transfer

A benefit function transfer obtains estimates of willingness-to-pay (WTP) for the evaluation of a given policy at a site by combining existing information from different study sites. This has the advantage that more efficient estimates are obtained, but it relies on the assumption that the heterogeneity between sites is appropriately captured in the benefit transfer model. A more expensive alternative to estimate WTP is to analyze only data from the policy site in question while ignoring information from other sites. We make use of the fact that these two choices can be viewed as a model selection problem and extend the set of models to allow for the hypothesis that the benefit function is only applicable to a subset of sites. We show how Bayesian model averaging (BMA) techniques can be used to optimally combine information from all models.The Bayesian algorithm searches for the set of sites that can form the basis for estimating a benefit function and reveals whether such information can be transferred to new sites for which only a small data set is available. We illustrate the method with a sample of 42 forests from U.K. and Ireland. We find that BMA benefit function transfer produces reliable estimates and can increase about 8 times the information content of a small sample when the forest is ‘poolable’.     

Combining functional data with hierarchical Gaussian process models

Gaussian process models have been used in applications ranging from machine learning to fisheries management. In the Bayesian framework, the Gaussian process is used as a prior for unknown functions, allowing the data to drive the relationship between inputs and outputs. In our research, we consider a scenario in which response and input data are available from several similar, but not necessarily identical, sources. When little information is known about one or more of the populations it may be advantageous to model all populations together. We present a hierarchical Gaussian process model with a structure that allows distinct features for each source as well as shared underlying characteristics. Key features and properties of the model are discussed and demonstrated in a number of simulation examples. The model is then applied to a data set consisting of three populations of Rotifer Brachionus calyciflorus Pallas. Specifically, we model the log growth rate of the populations using a combination of lagged population sizes. The various lag combinations are formally compared to obtain the best model inputs. We then formally compare the leading hierarchical Gaussian process model with the inferential results obtained under the independent Gaussian process model.