Uncertainty Analysis In Dissolved Oxygen Modeling in Streams

Uncertainty analysis in surface water quality modeling is an important issue. This paper presents a method based on the first-order reliability method (FORM) to assess the exceedance probability of a target dissolved oxygen concentration in a stream, using a Streeter–Phelps prototype model. Basic uncertainty in the input parameters is considered by representing them as random variables with prescribed probability distributions. Results obtained from FORM analysis compared well with those of the Monte Carlo simulation method. The analysis also presents the stochastic sensitivity of the probabilistic outcome in the form of uncertainty importance factors, and shows how they change with changing simulation time. Furthermore, a parametric sensitivity analysis was conducted to show the effect of selection of different probability distribution functions for the three most important parameters on the design point, exceedance probability, and importance factors. Note: This version was published online in June 2005 with the cover date of August 2004.     

DEVELOPMENT OF A BACTERIAL TRANSPORT MODEL FOR COARSE SOILS1

ABSTRACT: A bacterial transport model, developed to analyze bacterial translocation in coarse-grained soils, is presented. The complex governing equation is presented first, followed by analyses of each of the major processes influencing bacterial transport. These analyses suggest simplification of the governing equation is feasible when input data on specific processes are limited or unavailable. Model parameters, including bacterial die-off, bacterial distribution, input bacterial concentration, and saturated hydraulic conductivity, were randomly generated using a procedure known to produce either a normal or log-normal distribution of random numbers. Monte Carlo simulations were completed, and the resulting output was used to generate cumulative frequency distributions showing the probability of bacterial transport beyond various soil depths. Results from these simulations indicate that bacteria have a high probability of traveling through coarse-grained soils when low clay content and soil water temperatures limit bacterial retention and die-off.     

An approach to optimise nutrient management in environmental sanitation systems despite limited data

The material flow analysis method can be used to assess the impact of environmental sanitation systems on resource consumption and environmental pollution. However, given the limited access to reliable data, application of this data-intensive method in developing countries may be difficult. This paper presents an approach allowing to develop material flow models despite limited data availability. Application of an iterative procedure is of key importance: model parameter values should first be assessed on the basis of a literature review and by eliciting expert judgement. If model outputs are not plausible, sensitive input parameters should be reassessed more accurately. Moreover, model parameters can be expressed as probability distributions and variable uncertainty estimated by using Monte Carlo simulation. The impact of environmental sanitation systems on the phosphorus load discharged into surface water in Hanoi, Vietnam, is simulated by applying the proposed approach.     

ESTIMATING THE PROBABILITY OF EXCEEDING GROUNDWATER QUALITY STANDARDS1

ABSTRACT: A model for estimating the probability of exceeding groundwater quality standards at environmental receptors based on a simple contaminant transport model is described. The model is intended for locations where knowledge about site-specific hydrogeologic conditions is limited. An efficient implementation methodology using numerical Monte Carlo simulation is presented. The uncertainty in the contaminant transport system due to uncertainty in the hydraulic conductivity is directly calculated in the Monte Carlo simulations. Numerous variations of the deterministic parameters of the model provide an indication of the change in exceedance probability with change in parameter value. The results of these variations for a generic example are presented in a concise graphical form which provides insight into the topology of the exceedance probability surface. This surface can be used to assess the impact of the various parameters on exceedance probability.     

OPTIMIZING NONPOINT SOURCE CONTROLS IN WATER QUALITY REGULATION1

ABSTRACT: A stochastic programming framework is developed to evaluate the economic implications of reliability criteria and multiple effluent controls on nonpoint source pollution. An integrated watershed simulation model is used to generate probability distributions for agricultural effluents in surface and ground water resulting from agricultural practices. Results from the planning model indicate that reliability and multiple effluent constraints significantly increase the cost of nonpoint controls but the effects vary by control alternative. The analysis indicates that an evaluation of multiple water quality objectives can be an important planning tool for designing nonpoint source controls for innovative programs to promote cost-effective water quality regulation.     

Mapping the probability of exceeding critical thresholds for cadmium concentrations in soils in The Netherlands

The probability of exceeding critical thresholds of Cd concentrations in the soil was mapped at a national scale. The critical thresholds in soil were based on food quality criteria for Cd in crops or in organs of cattle (Bos taurus), and were calculated by inverting a regression model for the Cd concentration in the crop, with the Cd concentration in soil, soil organic matter (SOM) content, clay content, and pH as predictors. The probability of exceeding the critical threshold for Cd in soil per node of a 500- x 500-m grid was approximated by Monte Carlo simulation, using the estimated cumulative distribution functions (cdf) of SOM, clay, pH, and Cd as input. The cdfs were estimated by simple indicator kriging with local prior means. For SOM, clay, and pH, detailed maps of soil type and land use were used to define subregions with assumed constant local means of the indicators (a priori distributions). The cdfs were sampled by Latin hypercube sampling. We accounted for correlation between the actual and critical Cd concentrations in soil by drawing Cd values from cdfs conditional on SOM and clay. The estimated probability for grassland is negligible, even in areas with high Cd concentrations in soil, and for maize (Zea mays L.) land the probability is almost everywhere smaller than 5%. For arable soils, however, these probabilities commonly are larger than 5% when sugar beet (Beta vulgaris L.) or wheat (Triticum aestivum L.) is taken as a reference crop, and locally exceed 50%.     

CONTROLLING NONPOINT EXTERNALITIES WITH INPUT RESTRICTIONS IN AN IRRIGATED RIVER BASIN1

ABSTRACT: Irrigated agriculture is a major nonpoint source of surface water quality degradation. Nonpoint source discharges can be controlled by either output taxes or restrictions, or input taxes or restrictions. The economic theory of externality control with taxes or restrictions on input use is developed. The effectiveness of alternative input control policies in improving surface water quality is demonstrated with a mathematical model of the agriculture and water quality in Washington State's Yakima River Basin. Water quality parameters considered were river nitrogen concentration, water temperature, and cropland soil losses. Producer and social abatement costs were the most important measures of policy effectiveness.     

Spatial Variability of Groundwater Depth and Quality Parameters in the National Capital Territory of Delhi

The groundwater quantity and quality scenario is of much concern in the National Capital Territory of Delhi, India, which necessitates an investigation to envisage the extent of spatial variability of groundwater depth and pollutant concentration levels in this region. Therefore, in this study, an effort was made to generate the spatial variability map of groundwater depth and quality parameters (viz. chloride, electrical conductivity, fluoride, magnesium, and nitrate). Ordinary kriging was used to analyze the spatial variability of groundwater depth and quality parameters, whereas indicator kriging was used to analyze groundwater quality parameters equal to or greater than the pollution threshold values. It was observed that the semivariogram parameters fitted well in the exponential model for water depth and in the spherical model for water quality parameters. The generated spatial variability maps indicated that in 43% of the study area, groundwater depth was within 20 m. The salinity level was higher than 2.5 dS m⁻¹ in 69% of the study area and the nitrate concentration exceeded 45 mg l⁻¹ in 36% of the area. The probability maps showed that about 24% of the area had the highest probability (0.8–1.0) of exceedence of the threshold electrical conductivity value and an area of 2% exhibited the highest probability of exceedence of the threshold value of nitrate concentration in the groundwater. The generated spatial variability and probability maps will assist water resource managers and policymakers in development of guidelines in judicious management of groundwater resources for agricultural and drinking purposes in the study area.     

PRACTICAL IMPLICATIONS IN THE USE OF STOCHASTIC DYNAMIC PROGRAMMING FOR RESERVOIR OPERATION1

ABSTRACT: A stochastic dynamic programming model is applied to a small hydroelectric system. The variation in number of stage iterations and the computer time required to reach steady state conditions with changes in the number of storage states is investigated. The increase in computer time required to develop the storage probability distributions with increase in the number of storage states is reviewed. It is found that for an average of seven inflow states, the largest number of storage states for which it is computationally feasible to develop the storage probability distributions is nine. It is shown that use of the dynamic program results based on a small number of storage states results in unrealistically skewed storage probability distributions. These skewed distributions are attributed to “trapping” states at the low end of the storage range.     

Evaluating the impact of uncertainty and variability on the value chain optimization of a forest biomass power plant using Monte Carlo Simulation

Mathematical programming models have been used to optimize the design and management of forest bioenergy supply chains. A deterministic mathematical model is beneficial for making optimum decisions; however, its applicability to real-world problems may be limited because it does not capture all the complexities, including uncertainties in the parameters, in the supply chain. In this paper, a combination of Monte Carlo Simulation and optimization model is used to evaluate the impact of uncertainty in biomass quality, availability and cost, and electricity prices on the supply chain of a forest biomass power plant. The optimization model is a deterministic mixed integer non-linear model with monthly time steps over a 1-year planning horizon. Variability in biomass quality, i.e. moisture content (MC) and higher heating value (HHV), based on the historical data of a real case study is studied in detail and fitted probability distributions are used in the model, while for electricity prices different scenarios are considered. The results show that the impact of variability in the MC on profit is higher than that of uncertainty in HHV. It is observed that the annual profit ranges between $13.3 million and $17.9 million in the presence of all possible uncertainties while its average is $15.5 million. Uncertainty in biomass availability and cost and electricity price results in the risks of having annual profit of less than $14 million and low monthly storage levels.     

MODELING THE BIG BLACK RIVER: A COMPARISON OF WATER QUALITY MODELS1

ABSTRACT: The Mississippi Department of Environmental Quality uses the Steady Riverine Environmental Assessment Model (STREAM) to establish effluent limitations. While the U.S. Environmental Protection Agency has approved of its use, questions arise regarding the model's simplicity. The objective of this research was to compare STREAM with the more commonly utilized Enhanced Stream Water Quality Model (QUAL2E). The comparison involved a statistical evaluation procedure based on sensitivity analyses, input probability distribution functions, and Monte Carlo simulation with site‐specific data from a 46‐mile (74‐km) reach of the Big Black River in central Mississippi. Site specific probability distribution functions were derived from measured rates of reaeration, sediment oxygen demand, photosynthesis, and respiration. Both STREAM and QUAL2E reasonably predicted daily average dissolved oxygen (DO) based on a comparison of output probability distributions with observed DO. Observed DO was consistently within 90 percent confidence intervals of model predictions. The STREAM approach generally overpredicted while QUAL2E generally matched observed DO. Using the more commonly assumed lognormal distribution as opposed to a Weibull distribution for two of the sensitive input parameters resulted in minimal differences in the statistical evaluations. The QUAL2E approach had distinct advantages over STREAM in simulating the growth cycle of algae.     

A Sensitivity and Error Analysis Framework for Lake Eutrophication Modeling1

ABSTRACT: A framework for sensitivity and error analysis in mathematical modeling is described and demonstrated. The Lake Eutrophication Analysis Procedure (LEAP) consists of a series of linked models which predict lake water quality conditions as a function of watershed land use, hydrolgic variables, and morphometric variables. Specification of input variables as distributions (means and standard errors) and use of first-order error analysis techniques permits estimation of output variable means, standard errors, and confidence ranges. Predicted distributions compare favorably with those estimated using Monte-Carlo simulation. The framework is demonstrated by applying it to data from Lake Morey, Vermont. While possible biases exist in the models calibrated for this application, prediction variances, attributed chiefly to model error, are comparable to the observed year-to-year variance in water quality, as measured by spring phosphorus concentration, hypolimnetic oxygen depletion rate, summer chlorophyll-a, and summer transparency in this lake. Use of the framework provides insight into important controlling factors and relationships and identifies the major sources of uncertainty in a given model application.     

Impact of data quality and model complexity on prediction of pesticide leaching

Accurate input data for leaching models are expensive and difficult to obtain which may lead to the use of "general" non-site-specific input data. This study investigated the effect of using different quality data on model outputs. Three models of varying complexity, GLEAMS, LEACHM, and HYDRUS-2D, were used to simulate pesticide leaching at a field trial near Hamilton, New Zealand, on an allophanic silt loam using input data of varying quality. Each model was run for four different pesticides (hexazinone, procymidone, picloram and triclopyr); three different sets of pesticide sorption and degradation parameters (i.e., site optimized, laboratory derived, and sourced from the USDA Pesticide Properties Database); and three different sets of soil physical data of varying quality (i.e., site specific, regional database, and particle size distribution data). We found that the selection of site-optimized pesticide sorption (Koc) and degradation parameters (half-life), compared to the use of more general database derived values, had significantly more impact than the quality of the soil input data used, but interestingly also more impact than the choice of the models. Models run with pesticide sorption and degradation parameters derived from observed solute concentrations data provided simulation outputs with goodness-of-fit values closest to optimum, followed by laboratory-derived parameters, with the USDA parameters providing the least accurate simulations. In general, when using pesticide sorption and degradation parameters optimized from site solute concentrations, the more complex models (LEACHM and HYDRUS-2D) were more accurate. However, when using USDA database derived parameters, all models performed about equally.     

AN INTERACTWE MODELING ENVIRONMENT FOR NON-POINT SOURCE POLLUTION CONTROL1

ABSTRACT: Non-point source pollution cuntinues to be an important environmental and water quality management problem. For the moat part, analysis of non-point source pollution in watersheds has depended on the use of distributed models to identify potential problem areas and to assess the effectiveness of alternative management practices. To effectively use these models for watershed water quality management, users depend on integrated geographic information systems (GIS)-based interfaces for input/output data management. However, existing interfaces are ad-hoc and the utility of GIS is limited to organization of input data and display of output data. A highly interactive water quality modeling interface that utilizes the functional components and analytical capability of GIS is highly desirable. This paper describes the tight coupling of the Agricultural Non-point Source (AGNPS) water quality model and ARC/INFO GIS software to provide an interactive hybrid modeling environment for evaluation of non-point source pollution in a watershed. The modeling environment is designed to generate AGNPS input parameters from user-specified GIS coverages, create AGNPS input data files, control AGNPS model simulations, and extract and organize AGNPS model output data for display. An example application involving the estimation of pesticide loading in a southern Iowa agricultural watershed demonstrates the capability of the modeling environment. Compared with traditional methods of watershed water quality modeling using the AGNPS model or other ad-hoc interfaces between a distributed model and GIS, the interactive modeling environment system is efficient and significantly reduces the task of watershed analysis using tightly coupled GIS databases and distributed models.     

Propagation of uncertainties in soil and pesticide properties to pesticide leaching

In the new Dutch decision tree for the evaluation of pesticide leaching to groundwater, spatially distributed soil data are used by the GeoPEARL model to calculate the 90th percentile of the spatial cumulative distribution function of the leaching concentration in the area of potential usage (SP90). Until now it was not known to what extent uncertainties in soil and pesticide properties propagate to spatially aggregated parameters like the SP90. A study was performed to quantify the uncertainties in soil and pesticide properties and to analyze their contribution to the uncertainty in SP90. First, uncertainties in the soil and pesticide properties were quantified. Next, a regular grid sample of points covering the whole of the agricultural area in the Netherlands was randomly selected. At the grid nodes, realizations from the probability distributions of the uncertain inputs were generated and used as input to a Monte Carlo uncertainty propagation analysis. The analysis showed that the uncertainty concerning the SP90 is 10 times smaller than the uncertainty about the leaching concentration at individual point locations. The parameters that contribute most to the uncertainty about the SP90 are, however, the same as the parameters that contribute most to uncertainty about the leaching concentration at individual point locations (e.g., the transformation half-life in soil and the coefficient of sorption on organic matter). Taking uncertainties in soil and pesticide properties into account further leads to a systematic increase of the predicted SP90. The important implication for pesticide regulation is that the leaching concentration is systematically underestimated when these uncertainties are ignored.     

SPACE-TIME VALIDATION OF A THUNDERSTORM RAINFALL MODEL1

A probability model for predicting the occurrence and magnitude of thunderstorm rainfall developed in the southwestern United States was tested in the metropolitan Chicago area with reasonable success, especially for the moderate to the extreme runoff-producing events. The model requires the estimation of two parameters, the mean number of events per year and the conditional probability of rain given that an event has occurred. To tie in the data from more than one gage in an area, an event can be defined in several ways, such as the areal mean rainfall exceeding 0.50 inch and at least one gage receiving more than 1.0 inch. This type of definition allows both of the model parameters to be obtained from daily warm-season rainfall records. Regardless of the definition used a Poisson distribution adequately described the number of events per season. A negative binomial distribution was derived as representing the frequency density function for rainfall where several gages are employed in defining a storm. Chicago data fit both distributions very well at events with relatively high return periods. The results indicate the possibility of using the model on a regional basis where limited amount of data may be used to estimate parameters for extensive areas.     

An imprecise fuzzy risk approach for water quality management of a river system

Uncertainty plays an important role in water quality management problems. The major sources of uncertainty in a water quality management problem are the random nature of hydrologic variables and imprecision (fuzziness) associated with goals of the dischargers and pollution control agencies (PCA). Many Waste Load Allocation (WLA) problems are solved by considering these two sources of uncertainty. Apart from randomness and fuzziness, missing data in the time series of a hydrologic variable may result in additional uncertainty due to partial ignorance. These uncertainties render the input parameters as imprecise parameters in water quality decision making. In this paper an Imprecise Fuzzy Waste Load Allocation Model (IFWLAM) is developed for water quality management of a river system subject to uncertainty arising from partial ignorance. In a WLA problem, both randomness and imprecision can be addressed simultaneously by fuzzy risk of low water quality. A methodology is developed for the computation of imprecise fuzzy risk of low water quality, when the parameters are characterized by uncertainty due to partial ignorance. A Monte-Carlo simulation is performed to evaluate the imprecise fuzzy risk of low water quality by considering the input variables as imprecise. Fuzzy multiobjective optimization is used to formulate the multiobjective model. The model developed is based on a fuzzy multiobjective optimization problem with max–min as the operator. This usually does not result in a unique solution but gives multiple solutions. Two optimization models are developed to capture all the decision alternatives or multiple solutions. The objective of the two optimization models is to obtain a range of fractional removal levels for the dischargers, such that the resultant fuzzy risk will be within acceptable limits. Specification of a range for fractional removal levels enhances flexibility in decision making. The methodology is demonstrated with a case study of the Tunga–Bhadra river system in India.     

MULTIVARIATE EXTREME VALUE DISTRIBUTION WITH MIXED GUMBEL MARGINALS1

ABSTRACT: Bivariate and trivariate distributions have been derived from the logistic model for the multivariate extreme value distribution. Marginals in the models are extreme value type I distributions for two-component mixture variables (mixed Gumbel distribution). This paper is a continuation of the previous works on multivariate distribution in hydrology. Interest is focused on the analysis of floods which are generated by different types of storms. The construction of their corresponding probability distributions and density functions are described. In order to obtain the parameters of such a bivariate or trivariate distribution, a generalized maximum likelihood estimation procedure is proposed to allow for the cases of samples with different lengths of record. A region in Northern Mexico with 42 gauging stations, grouped into two homogeneous regions, has been selected to apply the models. Results produced by the multivariate distributions have been compared with those obtained by the Normal, log-Normal-2, log-Normal-3, Gamma-2, Gamma-3, log-Pearson-3, Gumbel, TCEV and General Extreme Value distributions. Goodness of fit is measured by the criterion of standard error of fit. Results suggest that the proposed models are a suitable option to be considered when performing flood frequency analysis.     

A two-stage inexact joint-probabilistic programming method for air quality management under uncertainty

A two-stage inexact joint-probabilistic programming (TIJP) method is developed for planning a regional air quality management system with multiple pollutants and multiple sources. The TIJP method incorporates the techniques of two-stage stochastic programming, joint-probabilistic constraint programming and interval mathematical programming, where uncertainties expressed as probability distributions and interval values can be addressed. Moreover, it can not only examine the risk of violating joint-probability constraints, but also account for economic penalties as corrective measures against any infeasibility. The developed TIJP method is applied to a case study of a regional air pollution control problem, where the air quality index (AQI) is introduced for evaluation of the integrated air quality management system associated with multiple pollutants. The joint-probability exists in the environmental constraints for AQI, such that individual probabilistic constraints for each pollutant can be efficiently incorporated within the TIJP model. The results indicate that useful solutions for air quality management practices have been generated; they can help decision makers to identify desired pollution abatement strategies with minimized system cost and maximized environmental efficiency.     

COMPUTING THE RISKS ASSOCIATED WITH WASTELOAD) ALLOCATION MODELING1

ABSTRACT: The risks associated with a traditional wasteload allocation (WLA) analysis were quantified with data from a recent study of the Upper Trinity River (Texas). Risk is define here as the probability of failing to meet an established in-stream water quality standard. The QUAL-TX dissolved oxygen (DO) water quality model was modified to a Monte Carlo framework. Flow augmentation coding was also modified to allow an exact match to be computed between the predicted and an established DO concentration standard, thereby providing an avenue for linking input parameter uncertainty to the assignment of a wasteload permit (allowable mass loading rate). Monte Carlo simulation techniques were employed to propagate input parameter uncertainties, typically encountered during WLA analysis, to the computed effluent five-day carbonaceous biochemical oxygen demand requirements for a single major wastewater treatment plant (WWTP). The risk of failing to meet an established in-stream DO criterion may be as high as 96 percent. The uncertainty associated with estimation of the future total Kjeldahl nitrogen concentration for a single tributary was found to have the greatest impact on the determination of allowable WWTP loadings.