Development of a fuzzy-stochastic nonlinear model to incorporate aleatoric and epistemic uncertainty

Site uncertainties significantly influence groundwater flow and contaminant transport predictions. Aleatoric and epistemic uncertainty are both identified in site characterization and represented using proper uncertainty theories. When one theory best represents one parameter whereas a different theory may be more suitable for another parameter, the hybrid propagation of aleatoric (random) and epistemic (nonrandom) uncertainties will occur. The computational challenges of joint propagation of aleatoric and epistemic uncertainty through groundwater flow and contaminant transport models are significant. A fuzzy-stochastic nonlinear model was developed in this paper to incorporate these two types of uncertain site information and reduce the computational cost. The results show that (1) the computational cost using the nonlinear model is reduced compared with that of using the sparse grid algorithm and Monte Carlo methods; (2) the uncertainty of hydraulic conductivity (K) significantly influences the water head and solute distribution at the observation wells compared to other uncertain parameters, such as the storage coefficient and the distribution coefficient (K_d); and (3) the combination of multiple uncertain parameters substantially affects the simulation results. Neglecting site uncertainties may lead to unrealistic predictions.     

Simulating uncertainty in climate-pest models with fuzzy numbers

Inputs in climate-pest models are commonly expressed as point estimates ('crisp' numbers), which implies perfect knowledge of the system in study. In reality, however, all model inputs harbor some level of uncertainty. This is particularly true for climate change impact assessments where the inputs (i.e., climate projections) are highly uncertain. In this study, uncertainties in climate projections were expressed as 'fuzzy' numbers; these are uncertain numbers for which one knows that there is a range of possible values and that some values are 'more possible' than others. A generic pest risk model incorporating the combined effects of temperature, soil moisture, and cold stress was implemented in a fuzzy spreadsheet environment and run with three climate scenarios: (1) present climate (control run); (2) crisp climate change; and (3) fuzzy climate change. Under the crisp climate change scenario, winter and summer temperatures and precipitation were altered using best estimates (averaged predictions from the 1995 assessment report of the Intergovernmental Panel on Climate Change [IPCC]). Under the fuzzy scenario, climate changes were expressed as triangular fuzzy numbers, utilizing the extremes (lowest and highest predictions from the IPCC report) in addition to the best estimates. Under each scenario, environmental favorability was calculated for six locations in two geographical regions (Central North America and Southern Europe) with two hypothetical pest species having temperate or mediterranean climate requirements. Simulations with the crisp climate change scenario suggested only minor changes in overall environmental favorability compared with the control run. When simulations were conducted with the fuzzy climate change scenario, however, important changes in environmental favorability emerged, particularly in Southern Europe. In that region, the possibility of considerably increased winter precipitation led to increased values of environmental favorability. However, the simulations also showed that this result harbored a very broad range of possible outcomes. The results support the notion that uncertainty in climate change projections must be reduced before reliable impact assessments can be achieved.     

Sensitivity to change for low-ANC eastern US lakes and streams and brook trout populations under alternative sulfate deposition scenarios

A weight-of-evidence approach was used by the US National Acid Precipitation Assessment Program (NAPAP) to assess the sensitivity of chemistry and biology of lakes and streams to hypothesized changes in sulfate deposition over the next 50 years. The analyses focused on projected effects in response to differences in the magnitude and the timing of changes in sulfate deposition in the north-eastern United States, the Mid-Appalachian Highlands, and the Southern Blue Ridge Province. A number of tools was used to provide the weight of evidence that is required to have confidence in an assessment that has many uncertainties because of the complexity of the systems for which the projections of future conditions were made and because of limited historical data. The MAGIC model provided the projections of chemical changes in response to alternative deposition scenarios. Projected chemical conditions were input into biological models that evaluate effects on fish populations. The sensitivity of water chemistry and brook trout resources to the hypothesized changes in deposition was found to be greatest in the Adirondacks and Mid-Atlantic Highlands. Under the hypothesized sulfur deposition reduction scenarios, chemical conditions suitable for fish were projected to improve 20-30 years sooner than with the scenario that assumed no new legislated controls. Other lines of evidence, e.g. other models, field observations, and paleolimnological findings, were used to evaluate uncertainty in the projections. Model parameter/calibration uncertainty for the chemical models and population sampling uncertainty were explicitly quantified. Model structural uncertainties were bracketed using model comparisons, recent measured changes, and paleolimnological reconstructions of historical changes in lake chemistry.     

A generalized fuzzy linear programming approach for environmental management problem under uncertainty

In this study, a generalized fuzzy linear programming (GFLP) method was developed to deal with uncertainties expressed as fuzzy sets that exist in the constraints and objective function. A stepwise interactive algorithm (SIA) was advanced to solve GFLP model and generate solutions expressed as fuzzy sets. To demonstrate its application, the developed GFLP method was applied to a regional sulfur dioxide (SO2) control planning model to identify effective SO2 mitigation polices with a minimized system performance cost under uncertainty. The results were obtained to represent the amount of SO2 allocated to different control measures from different sources. Compared with the conventional interval-parameter linear programming (ILP) approach, the solutions obtained through GFLP were expressed as fuzzy sets, which can provide intervals for the decision variables and objective function, as well as related possibilities. Therefore, the decision makers can make a tradeoff between model stability and the plausibility based on solutions obtained through GFLP and then identify desired policies for SO2-emission control under uncertainty.     

Uncertainty in Nonpoint Source Pollution Models and Associated Risks

The usefulness of water quality simulation models for environmental management is explored with a focus on prediction uncertainty. The specific objective is to demonstrate how the usability of a flow and transport model (here: MACRO) can be enhanced by developing and analyzing its output probability distributions based on input variability. This infiltration-based model was designed to investigate preferential flow effects on pollutant transport. A statistical sensitivity analysis is used to identify the most uncertain input parameters based on model outputs. Probability distribution functions of input variables were determined based on field-measured data obtained under alternative tillage treatments. Uncertainty of model outputs is investigated using a Latin hypercube sampling scheme (LHS) with restricted pairing for model input sampling. Probability density functions (pdfs) are constructed for water flow rate, atrazine leaching rate, total accumulated leaching, and atrazine concentration in percolation water. Results indicate that consideration of input parameter uncertainty produces a 20% higher mean flow rate along with two to three times larger atrazine leaching rate, accumulated leachate, and concentration than that obtained using mean input parameters. Uncertainty in predicted flow rate is small but that in solute transport is an order of magnitude larger than that of corresponding input parameters. Macropore flow is observed to contribute to the variability of atrazine transport results. Overall, the analysis provides a quantification of prediction uncertainty that is found to enhance a user's ability to assess risk levels associated with model predictions.     

Uncertainty in Nonpoint Source Pollution Models and Associated Risks

The usefulness of water quality simulation models for environmental management is explored with a focus on prediction uncertainty. The specific objective is to demonstrate how the usability of a flow and transport model (here: MACRO) can be enhanced by developing and analyzing its output probability distributions based on input variability. This infiltration-based model was designed to investigate preferential flow effects on pollutant transport. A statistical sensitivity analysis is used to identify the most uncertain input parameters based on model outputs. Probability distribution functions of input variables were determined based on field-measured data obtained under alternative tillage treatments. Uncertainty of model outputs is investigated using a Latin hypercube sampling scheme (LHS) with restricted pairing for model input sampling. Probability density functions (pdfs) are constructed for water flow rate, atrazine leaching rate, total accumulated leaching, and atrazine concentration in percolation water. Results indicate that consideration of input parameter uncertainty produces a 20% higher mean flow rate along with two to three times larger atrazine leaching rate, accumulated leachate, and concentration than that obtained using mean input parameters. Uncertainty in predicted flow rate is small but that in solute transport is an order of magnitude larger than that of corresponding input parameters. Macropore flow is observed to contribute to the variability of atrazine transport results. Overall, the analysis provides a quantification of prediction uncertainty that is found to enhance a user's ability to assess risk levels associated with model predictions.     

Ambient Air Quality and Automotive Emission Control

This paper is concerned with uncertainties involved in projecting ambient air quality. Ambient air quality was projected by assuming a linear dependence on estimated future emissions. Future automotive emissions were estimated by a method recommended by EPA. Projections were made for the locations reported to have the highest ambient air concentrations of each pollutant; Chicago for carbon monoxide and the California South Coast Air Basin for hydrocarbon and oxidant. The sensitivity of the projections to several input parameters was determined.

The uncertainty in projection of air quality due to the use of a maximum, once-per-year concentration is large. For example, the reduction in total CO emissions in Chicago in 1975, necessary to meet the air quality standard, was as high as 68% or as low as 26%, depending on whether the historic high, 8 hr average concentration of 44 ppm or the 1970 maximum of 21 ppm was used. The effects of uncertainties in growth rates and fraction of emissions attributed to the automobile were also sizeable. Differences in automotive growth rate had a large near-term effect on projected concentrations, while differences in nonautomotive growth rate or fraction of emissions attributed to the automobile had a large long-term effect. The effect of 1975 interim automotive emission standards on projected air quality was negligible when compared with projected air quality based on the previous Federal automotive emission standards for 1975.     

Projecting Climate Change Effects on Forest Net Primary Productivity in Subtropical Louisiana,USA

This study projected responses of forest net primary productivity (NPP) to three climate change scenarios at a resolution of 5 km × 5 km across the state of Louisiana, USA. In addition, we assessed uncertainties associated with the NPP projection at the grid and state levels. Climate data of the scenarios were derived from Community Climate System Model outputs. Changes in annual NPP between 2000 and 2050 were projected with the forest ecosystem model PnET-II. Results showed that forest productivity would increase under climate change scenarios A1B and A2, but with scenario B1, it would peak during 2011–2020 and then decline. The projected average NPP under B1 over the years from 2000 to 2050 was significantly different from those under A1B and A2. Forest NPP appeared to be primarily a function of temperature, not precipitation. Uncertainties of the NPP projection were due to large spatial resolution of the climate variables. Overall, this study suggested that in order to project effects of climate change on forest ecosystem at regional level, modeling uncertainties could be reduced by increasing the spatial resolution of the climate projections.     

A multimedia environmental model of chemical distribution: fate, transport, and uncertainty analysis

This paper presented a framework for analysis of chemical concentration in the environment and evaluation of variance propagation within the model. This framework was illustrated through a case study of selected organic compounds of benzo[alpha]pyrene (BAP) and hexachlorobenzene (HCB) in the Great Lakes region. A multimedia environmental fate model was applied to perform stochastic simulations of chemical concentrations in various media. Both uncertainty in chemical properties and variability in hydrometeorological parameters were included in the Monte Carlo simulation, resulting in a distribution of concentrations in each medium. Parameters of compartmental dimensions, densities, emissions, and background concentrations were assumed to be constant in this study. The predicted concentrations in air, surface water and sediment were compared to reported data for validation purpose. Based on rank correlations, a sensitivity analysis was conducted to determine the influence of individual input parameters on the output variance for concentration in each environmental medium and for the basin-wide total mass inventory. Results of model validation indicated that the model predictions were in reasonable agreement with spatial distribution patterns, among the five lake basins, of reported data in the literature. For the chemical and environmental parameters given in this study, parameters associated to air-ground partitioning (such as moisture in surface soil, vapor pressure, and deposition velocity) and chemical distribution in soil solid (such as organic carbon partition coefficient and organic carbon content in root-zone soil) were targeted to reduce the uncertainty in basin-wide mass inventory. This results of sensitivity analysis in this study also indicated that the model sensitivity to an input parameter might be affected by the magnitudes of input parameters defined by the parameter settings in the simulation scenario. Therefore, uncertainty and sensitivity analyses for environmental fate models was suggested to be conducted after the model output was validated based on an appropriate input parameter settings.     

A framework for error analysis of a long-range transport model with emphasis on parameter uncertainty

A comprehensive framework for model error analysis is applied to the EMEP-W model of longrange transport of sulfur in Europe. This framework includes a proposed taxonomy of model uncertainties. Parameter uncertainties were investigated by Monte Carlo simulation of two source-receptor combinations. A 20% input parameter uncertainty (expressed as a coefficient of variation = standard deviation/mean) yielded a 15–22% output error of total sulfur deposition. The relationship between output error and input uncertainty was approximately proportional. Covariance between parameters can have an important effect on computed model error, and can either exaggerate or reduce errors compared to the uncorrelated case. Of the model state variables, SO₂ air concentration and wet deposition had the highest error, and total sulfur deposition the lowest. It was also found that it is more important to specify the dispersion of the input parameter frequency distributions than their shape. The results of the model error analysis were applied to routine calculations of deposition in Europe. An error (coefficient of variation) of 20% for transfer coefficients throughout Europe yielded spatial variations in the order of a few tens to a few hundreds of km in computed deposition isolines of 2 and 5 g sulfur m⁻²a⁻¹.     

Uncertainties in a Baltic Sea Food-Web Model Reveal Challenges for Future Projections

Models that can project ecosystem dynamics under changing environmental conditions are in high demand. The application of such models, however, requires model validation together with analyses of model uncertainties, which are both often overlooked. We carried out a simplified model uncertainty and sensitivity analysis on an Ecopath with Ecosim food-web model of the Baltic Proper (BaltProWeb) and found the model sensitive to both variations in the input data of pre-identified key groups and environmental forcing. Model uncertainties grew particularly high in future climate change scenarios. For example, cod fishery recommendations that resulted in viable stocks in the original model failed after data uncertainties were introduced. In addition, addressing the trophic control dynamics produced by the food-web model proved as a useful tool for both model validation, and for studying the food-web function. These results indicate that presenting model uncertainties is necessary to alleviate ecological surprises in marine ecosystem management.     

Relating error bounds for maximum concentration estimates to diffusion meteorology uncertainty

The intent of this paper is to relate the magnitude of the error bounds of data, used as inputs to a Gaussian dispersion model, to the magnitude of the error bounds of the model output, which include the estimates of the maximum concentration and the distance to that maximum. The research specifically addresses the uncertainty in estimating the maximum concentrations from elevated buoyant sources during unstable atmospheric conditions, as these are most often of practical concern in regulatory decision making. A direct and quantitative link between the nature and magnitude of the input uncertainty and modeling results has not been previously investigated extensively. The ability to develop specific error bounds, tailored to the modeling situation, allows more informed application of the model estimates to the air quality issues.In this study, a numerical uncertainty analysis is performed using the Monte-Carlo technique to propagate the uncertainties associated with the model input. Uncertainties were assumed to exist in four model input parameters: (1) wind speed, (2) standard deviation of lateral wind direction fluctuations, (3) standard deviation of vertical wind direction fluctuations, and (4) plume rise. For each simulation, results were summarized characterizing the uncertainty in four features of the ground-level concentration pattern predicted by the model: (1) the magnitude of the maximum concentration, (2) the distance to the maximum concentration, and (3) and (4) the areas enclosed within the isopleths of 50% and 25% of the error-free estimate of maximum concentration.The authors conclude that the error bounds for the estimated maximum concentration and the distance to the maximum can be double that of the error bounds for individual model input parameters. The model output error bounds for the areas enclosed within isopleth values can be triple the error bounds of the input. It was not our intent to cover all possible combinations for the error in the input parameters. Ours was a much more limited goal of providing a lower bound estimate of model uncertainty in which we assume the input is reasonably well characterized and there is no bias in the input. These results allow estimation of minimum bounds on errors in model output when considering reasonable input error bounds.     

Meteorological Considerations in Air Quality Planning

Results from dispersion models which are routinely used for regulatory purposes do not reflect the uncertainties which are Inherent In the input data. To remedy this, a formula for propagating measurement uncertainties of emission rate, wind speed, wind direction, horizontal dispersion parameter, vertical dispersion parameter, effective emission height, and mixing depth is derived for EPA’s Industrial Source Complex Short Term (ISCST) Gaussian dispersion model for the simple case of a single stack-type source and nonbuoyant plume. Values for the uncertainties of the input variables are chosen and used to calculate ambient concentration uncertainties. These calculated uncertainties are compared with the standard deviation of ambient concentrations calculated from 2500 input data sets for each of four stability classes and three downwind distances, which were randomly altered to simulate the effects of measurement uncertainty. The calculated uncertainties do not differ significantly from the standard deviations of the randomized calculations for input data uncertainties as high as 30 percent and Stability Classes A-C. The calculated uncertainties overestimate the actual uncertainty of model calculations for input data uncertainties greater than 20 percent for Stability Class D.     

Evaluating Environmental Parameters for Minimum Ammonium Losses during Composting of Trimming Residues

Abstract

A neural fuzzy system was used to investigate the influence of environmental variables (time, aeration, moisture, and particle size) on composting parameters (pH, organic matter [OM], nitrogen [N], ammonium nitrogen [NH₄ ⁺-N] and nitrate nitrogen [NO₃ ^--N]). This was to determine the best composting conditions to ensure the maximum quality on the composts obtained with the minimum ammonium losses. A central composite experimental design was used to obtain the neural fuzzy model for each dependent variable. These models, consisting of the four independent process variables, were found to accurately describe the composting process (the differences between the experimental values and those estimated by using the equations never exceeded 5–10% of the former). Results of the modeling showed that creating a product with acceptable chemical properties (pH, NH₄ ⁺-N and NO₃ ^--N) entails operating at medium moisture content (55%) and medium to high particle size (3–5 cm). Moderate to low aeration (0.2 L air/min · kg) would be the best compromise to compost this residue because of the scant statistical influence of this independent variable.     

Priority assessment of toxic substances in life cycle assessment. Part II: assessing parameter uncertainty and human variability in the calculation of toxicity potentials

Toxicity potentials are standard values used in life cycle assessment (LCA) to enable a comparison of toxic impacts between substances. This paper presents the results of an uncertainty assessment of toxicity potentials that were calculated with the global nested multi-media fate, exposure and effects model USES-LCA. The variance in toxicity potentials resulting from input parameter uncertainties and human variability was quantified by means of Monte Carlo analysis with Latin Hypercube sampling (LHS). For Atrazine, 2,3,7,8-TCDD and Lead, variation, expressed by the ratio of the 97.5%-ile and the 2.5%-ile, ranges from about 1.5 to 6 orders of magnitude. The major part of this variation originates from a limited set of substance-specific input parameters, i.e. parameters that describe transport mechanisms, substance degradation, indirect exposure routes and no-effect concentrations. Considerable correlations were found between the toxicity potentials of one substance, in particular within one impact category. The uncertainties and correlations reported in the present study may have a significant impact on the outcome of LCA case studies.     

Uncertainty of exposure assessment of consumers to pesticide residues derived from food consumed

The sources of quantitative and qualitative uncertainties of the exposure of a consumer to a pesticide residue are identified. The contribution of quantifiable uncertainties of input parameters of deterministic model to the combined uncertainty of the estimated exposure is shown with detailed calculation using the pesticide residue content of food consumed during two days. The daily intakes of bifenthrin residues calculated for the 60 kg bodyweight of the reporting person are 0.00257 mg/kgbw and 0.00281 mg/kgbw for day 1 and day 2, respectively with 27-28% combined uncertainty. The major contributors were fruits and whole meal bread. The contribution of the individual steps to the combined uncertainty depends on the particular food item. In general, the variability of recipes, estimation of the mass of consumed food, sampling, processing of raw commodities and analysis of pesticide residues influenced most, in decreasing order, the combined uncertainty of the estimated daily exposure.     

Database uncertainty as a limiting factor in reactive transport prognosis

The effect of uncertainties in thermodynamic databases on prediction performances of reactive transport modeling of uranium (VI) is investigated with a Monte Carlo approach using the transport code TReaC. TReaC couples the transport model to the speciation code PHREEQC by a particle tracking method. A speciation example is given to illustrate the effect of uncertainty in thermodynamic data on the predicted solution composition. The transport calculations consequently show the prediction uncertainty resulting from uncertainty in thermodynamic data. A conceptually simple scenario of elution of uranium from a sand column is used as an illustrating example. Two different cases are investigated: a carbonate-enriched drinking water and an acid mine water associated with uranium mine remediation problems. Due to the uncertainty in the relative amount of positively charged and neutral solution species, the uncertainty in the thermodynamic data also infers uncertainty in the retardation behavior. The carbonated water system shows the largest uncertainties in speciation calculation. Therefore, the model predictions of total uranium solubility have a broad range. The effect of data uncertainty in transport prediction is further illustrated by a prediction of the time when eluted uranium from the column exceeds a threshold value. All of these Monte Carlo transport calculations consume large amounts of computing time.     

A comparison of numerical solutions of partial differential equations with probabilistic and possibilistic parameters for the quantification of uncertainty in subsurface solute transport

Traditionally, uncertainty in parameters are represented as probabilistic distributions and incorporated into groundwater flow and contaminant transport models. With the advent of newer uncertainty theories, it is now understood that stochastic methods cannot properly represent non random uncertainties. In the groundwater flow and contaminant transport equations, uncertainty in some parameters may be random, whereas those of others may be non random. The objective of this paper is to develop a fuzzy-stochastic partial differential equation (FSPDE) model to simulate conditions where both random and non random uncertainties are involved in groundwater flow and solute transport. Three potential solution techniques namely, (a) transforming a probability distribution to a possibility distribution (Method I) then a FSPDE becomes a fuzzy partial differential equation (FPDE), (b) transforming a possibility distribution to a probability distribution (Method II) and then a FSPDE becomes a stochastic partial differential equation (SPDE), and (c) the combination of Monte Carlo methods and FPDE solution techniques (Method III) are proposed and compared. The effects of these three methods on the predictive results are investigated by using two case studies. The results show that the predictions obtained from Method II is a specific case of that got from Method I. When an exact probabilistic result is needed, Method II is suggested. As the loss or gain of information during a probability–possibility (or vice versa) transformation cannot be quantified, their influences on the predictive results is not known. Thus, Method III should probably be preferred for risk assessments.     

A two-stage interval-stochastic programming model for waste management under uncertainty

This study introduces a two-stage interval-stochastic programming (TISP) model for the planning of solid-waste management systems under uncertainty. The model is derived by incorporating the concept of two-stage stochastic programming within an interval-parameter optimization framework. The approach has the advantage that policy determined by the authorities, and uncertain information expressed as intervals and probability distributions, can be effectively communicated into the optimization processes and resulting solutions. In the modeling formulation, penalties are imposed when policies expressed as allowable waste-loading levels are violated. In its solution algorithm, the TISP model is converted into two deterministic submodels, which correspond to the lower and upper bounds for the desired objective-function value. Interval solutions, which are stable in the given decision space with associated levels of system-failure risk, can then be obtained by solving the two submodels sequentially. Two special characteristics of the proposed approach make it unique compared with other optimization techniques that deal with uncertainties. First, the TISP model provides a linkage to predefined policies determined by authorities that have to be respected when a modeling effort is undertaken; second, it furnishes the reflection of uncertainties presented as both probabilities and intervals. The developed model is applied to a hypothetical case study of regional solid-waste management. The results indicate that reasonable solutions have been generated. They provide desired waste-flow patterns with minimized system costs and maximized system feasibility. The solutions present as stable interval solutions with different risk levels in violating the waste-loading criterion and can be used for generating decision alternatives.     

Identification of transport processes in column experiments using a frequency domain approach

When a solute transport process is viewed as a dynamic system with input and output, a system identification technique can be used to study it from input-output data. According to the design of excitation signals in the system identification approach, the commonly used rectangular pulse as input signal for column experiments is not optimum because it does not simultaneously meet the requirements for exciting the studied transport process, i.e., possessing frequency components with high enough amplitude and wide enough passband. In this article, stepped sine signals were proposed to replace the rectangular pulse because their amplitude and passband can be independently chosen. The stepped sine signals of concentration were generated by a High Performance Liquid Chromatography (HPLC) and used as the input for the column experiments to identify parameters of the convection-dispersion equation (CDE) and mobile-immobile model (MIM). In order to test the effect of noise on the identification of transport process, numerical experiments were carried out to identify the CDE under white noise when the input was designed as stepped sine functions and rectangular pulse. The results of the numerical experiments showed that the input signal of a sine function was more robust and accurate in process identification than that of a rectangular pulse.