Identifying Influences on Model Uncertainty: An Application Using a Forest Carbon Budget Model

Uncertainty is an important consideration for both developers and users of environmental simulation models. Establishing quantitative estimates of uncertainty for deterministic models can be difficult when the underlying bases for such information are scarce. We demonstrate an application of probabilistic uncertainty analysis that provides for refinements in quantifying input uncertainty even with little information. Uncertainties in forest carbon budget projections were examined with Monte Carlo analyses of the model FORCARB. We identified model sensitivity to range, shape, and covariability among model probability density functions, even under conditions of limited initial information. Distributional forms of probabilities were not as important as covariability or ranges of values. Covariability among FORCARB model parameters emerged as a very influential component of uncertainty, especially for estimates of average annual carbon flux.     

USE OF STATE ESTIMATION TECHNIQUES IN WATER RESOURCE SYSTEM MODELING1

ABSTRACT. A relatively straightforward illustration of the potential uses of State Estimation techniques in water resources modeling is given. Background theory for Linear and Extended Kalman Filters is given; application of the filter techniques to modeling BOD and oxygen deficit in a stream illustrates the importance of model conceptualization, model completeness, uncertainty in model dynamics and incorporation of measurements and measurement errors. Potential applications of state estimation techniques to measurement system design; model building, assessment and calibration; and data extension are explored.     

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.     

UNCERTAINTY ANALYSIS OF BMP EFFECTWENESS FOR CONTROLLING NITROGEN FROM URBAN NONPOINT SOURCES1

ABSTRACT: The effectiveness of urban Best Management Practices (BMPs) in achieving the No-Net-Increase Policy (NNTP), a policy designed to limit nonpoint nitrogen loading to Long Island Sound (US), is analyzed. A unit loading model is used to simulate annual nitrogen exported from the Norwalk River watershed (Connecticut) under current and future conditions. A probabilistic uncertainty analysis is used to incorporate uncertainty in nitrogen export coefficients and BMP nitrogen removal effectiveness. The inclusion of uncertainty in BMP effectiveness and nitrogen export coefficients implies that additional BMPs, or BMPs with a greater effectiveness in nitrogen removal, will be required to achieve the NNIP. Even though including uncertainty leads to an increase in BMP implementation rates or BMP effectiveness, this type of analysis provides the decision maker with a more realistic assessment of the likelihood that implementing BMPs as a management strategy will be successful. Monte Carlo simulation results indicate that applying BMPs to new urban developments alone will not be sufficient to achieve the NNIP since BMPs are not 100 percent effective in removing the increase in nitrogen caused by urbanization. BMPs must also be applied to selected existing urban areas. BMPs with a nitrogen removal effectiveness of 40–60 percent, probably the highest level of removal that can be expected over an entire watershed, must be applied to at least 75 percent of the existing urban area to achieve the NNIP This high rate of application is not likely to be achieved in urbanized watersheds in the LIS watershed; therefore, additional point source control will be necessary to achieve the NNIP     

WATER QUALITY MODEL CALIBRATION: A COMPARISON OF INPUT AND OUTPUT ERROR CRITERIA1

ABSTRACT: A common framework for the analysis of water resources systems is the input-parameter-output representation. The system, described by its parameters, is driven by inputs and responds with outputs. To calibrate (estimate the parameters) models of these systems requires data on both inputs and outputs, both of which are subject to random errors. When one is uncertain as to whether the predominant source of error is associated with inputs or outputs, uncertainty also exists as to the correct specification of a calibration criterion. This paper develops and analyzes two alternative least squares criteria for calibrating a numerical water quality model. The first criterion assumes that errors are associated with inputs while the second assumes output errors. Statistical properties of the resulting estimators are examined under conditions of pure input or output error and mixed error conditions from a theoretical perspective and then using simulated results from a series of Monte Carlo experiments.     

PROBABILISTIC METHODS IN STREAM QUALITY MANAGEMENT1

ABSTRACT. Recent advances in water quality modelling have pointed out the need for stochastic models to simulate the probabilistic nature of water quality. However, often all that is needed is an estimate of the uncertainty in predicting water quality variables. First order analysis is a simple method of providing an estimate in the uncertainty in a deterministic model due to uncertain parameters. The method is applied to the simplified Streeter-Phelps equations for DO and BOD; a more complete Monte Carlo simulation is used to check the accuracy of the results. The first order analysis is found to give accurate estimates of means and variances of DO and BOD up to travel times exceeding the critical time. Uncertainty in travel time and the BOD decay constant are found to be most important for small travel times; uncertainty in the reaeration coefficient dominates near the critical time. Uncertainty in temperature was found to be a negligible source of uncertainty in DO for all travel times.     

Nonparametric Monte Carlo Simulation for Flood Frequency Curve Derivation: An Application to a Korean Watershed1

Abstract: A mix of causative mechanisms may be responsible for flood at a site. Floods may be caused because of extreme rainfall or rain on other rainfall events. The statistical attributes of these events differ according to the watershed characteristics and the causes. Traditional methods of flood frequency analysis are only adequate for specific situations. Also, to address the uncertainty of flood frequency estimates for hydraulic structures, a series of probabilistic analyses of rainfall‐runoff and flow routing models, and their associated inputs, are used. This is a complex problem in that the probability distributions of multiple independent and derived random variables need to be estimated to evaluate the probability of floods. Therefore, the objectives of this study were to develop a flood frequency curve derivation method driven by multiple random variables and to develop a tool that can consider the uncertainties of design floods. This study focuses on developing a flood frequency curve based on nonparametric statistical methods for the estimation of probabilities of rare floods that are more appropriate in Korea. To derive the frequency curve, rainfall generation using the nonparametric kernel density estimation approach is proposed. Many flood events are simulated by nonparametric Monte Carlo simulations coupled with the center Latin hypercube sampling method to estimate the associated uncertainty. This study applies the methods described to a Korean watershed. The results provide higher physical appropriateness and reasonable estimates of design flood.     

Probabilistic fugacity analysis of Lake Pontchartrain pollution after Hurricane Katrina

After Hurricane Katrina passed through the US Gulf Coast in August 2005, floodwaters covering New Orleans were pumped into Lake Pontchartrain as part of the rehabilitation process in order to make the city habitable again. The long-term consequences of this environmentally critical decision were difficult to assess at the time and were left to observation. In the aftermath of these natural disasters, and in cases of emergency, the proactive use of screening level models may prove to be an important factor in making appropriate decisions to identify cost effective and environmentally friendly mitigation solutions. In this paper, we propose such a model and demonstrate its use through the application of several hypothetical scenarios to examine the likely response of Lake Pontchartrain to the contaminant loading that were possibly in the New Orleans floodwaters. For this purpose, an unsteady-state fugacity model was developed in order to examine the environmental effects of contaminants with different physicochemical characteristics on Lake Pontchartrain. The three representative contaminants selected for this purpose are benzene, atrazine, and polychlorinated biphenyls (PCBs). The proposed approach yields continuous fugacity values for contaminants in the water, air, and sediment compartments of the lake system which are analogous to concentrations. Since contaminant data for the floodwaters are limited, an uncertainty analysis was also performed in this study. The effects of uncertainty in the model parameters were investigated through Monte Carlo analysis. Results indicate that the acceptable recovery of Lake Pontchartrain will require a long period of time. The computed time range for the levels of the three contaminants considered in this study to decrease to maximum contaminant levels (MCLs) is about 1 year to 68 years. The model can be implemented to assess the possible extent of damage inflicted by any storm event on the natural water resources of Southern Louisiana or similar environments elsewhere. Furthermore, the model developed can be used as a useful decision-making tool for planning and remediation in similar emergency situations by examining various potential contamination scenarios and their consequences.     

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.     

A dynamic hydrological Monte Carlo simulation model to inform decision-making at Lake Toolibin,Western Australia

Lake Toolibin, an ephemeral lake in the agricultural zone of Western Australia, is under threat from secondary salinity due to land clearance throughout the catchment. The lake is extensively covered with native vegetation and is a Ramsar listed wetland, being one of the few remaining significant migratory bird habitats in the region. Currently, inflow with salinity greater than 1000 mg/L TDS is diverted from the lake in an effort to protect sensitive lakebed vegetation. However, this conservative threshold compromises the frequency and extent of lake inundation, which is essential for bird breeding. It is speculated that relaxing the threshold to 5000 mg/L may pose negligible additional risk to the condition of lakebed vegetation. To characterise the magnitude of improvement in the provision of bird breeding habitat that might be generated by relaxing the threshold, a dynamic water and salt balance model of the lake was developed and implemented using Monte Carlo simulation. Results from best estimate model inputs indicate that relaxation of the threshold increases the likelihood of satisfying habitat requirements by a factor of 9.7. A second-order Monte Carlo analysis incorporating incertitude generated plausible bounds of [2.6, 37.5] around the best estimate for the relative likelihood of satisfying habitat requirements. Parameter-specific sensitivity analyses suggest the availability of habitat is most sensitive to pan evaporation, lower than expected inflow volume, and higher than expected inflow salt concentration. The characterisation of uncertainty associated with environmental variation and incertitude allows managers to make informed risk-weighted decisions.     

Spatial assessment of the economic feasibility of short rotation coppice on radioactively contaminated land in Belarus, Ukraine, and Russia. II. Monte Carlo analysis

A Monte Carlo analysis of two sequential GIS-embedded submodels, which evaluate the economic feasibility of short rotation coppice (SRC) production and energy conversion in areas contaminated by Chernobyl-derived (137)Cs, was performed to allow for variability of environmental conditions that was not contained in the spatial model inputs. The results from this analysis were compared to the results from the deterministic model presented in part I of this paper. It was concluded that, although the variability in the model results due to within-gridcell variability of the model inputs was considerable, the prediction of the areas where SRC and energy conversion is potentially profitable was robust. If the additional variability in the model input that is not contained in the input maps is also taken into account, the SRC production and energy conversion appears to be potentially profitable at more locations for both the small scale and large scale production scenarios than the model predicted using the deterministic model.     

Human actions, the survival of keystone species, and the resilience of ecological–economic systems

Human (managerial) actions affect the survival probabilities of the keystone species of an ecological–economic system. In turn, the well-being of these keystone species translates into the well-being or the resilience of the underlying ecological–economic system. What are the theoretical connections between human actions, keystone species survival, and the resilience of ecological–economic systems? In this note, we construct a simple stochastic model to draw out the links between this trinity.     

Bayesian Nonstationary Frequency Analysis of Hydrological Variables1

Ouarda, T.B.M.J. and S. El‐Adlouni, 2011. Bayesian Nonstationary Frequency Analysis of Hydrological Variables. Journal of the American Water Resources Association (JAWRA) 47(3):496‐505. DOI: 10.1111/j.1752‐1688.2011.00544.x Abstract: The present paper provides a discussion of nonstationary frequency analysis models in hydrology with a focus on the Bayesian approach. The Bayesian model provides an efficient estimation framework of hydrological quantiles in the presence of nonstationarity. In nonstationary frequency analysis models, the parameters are functions of covariates, allowing for dependent parameters and trends. The use of the nonstationary Generalized Maximum Likelihood Estimation method in hydrologic frequency analysis is discussed. This model allows using prior information concerning the variables under study and considering a number of models (linear, quadratic, etc.) of the dependence of the parameters on covariates. A discussion is also provided concerning the use of the reversible jump Monte Carlo Markov Chain procedure which allows carrying out the estimation of the posterior distributions of the parameters and the selection of the Bayesian model at the same time. An application to a case study is presented to illustrate the potential of the model.     

Ecological Impact Assessment in Data-Poor Systems: A Case Study on Metapopulation Persistence

Legislation on the protection of biodiversity (e.g., European Union Habitat and Bird Directives) increasingly requires ecological impact assessment of human activities. However, knowledge and understanding of relevant ecological processes and species responses to different types of impact are often incomplete. In this paper we demonstrate with a case study how impact assessment can be carried out for situations where data are scarce but some expert knowledge is available. The case study involves two amphibian species, the great crested newt (Triturus cristatus) and the natterjack toad (Bufo calamita) in the nature reserve the Meinweg in the Netherlands, for which plans are being developed to reopen an old railway track called the Iron Rhine. We assess the effects of this railway track and its proposed alternatives (scenarios) on the metapopulation extinction time and the occupancy times of the patches for both species using a discrete-time stochastic metapopulation model. We quantify the model parameters using expert knowledge and extrapolated data. Because of our uncertainty about these parameter values, we perform a Monte Carlo uncertainty analysis. This yields an estimate of the probability distribution of the model predictions and insight into the contribution of each distinguished source of uncertainty to this probability distribution. We show that with a simple metapopulation model and an extensive uncertainty analysis it is possible to detect the least harmful scenario. The ranking of the different scenarios is consistent. Thus, uncertainty analysis can enhance the role of ecological impact assessment in decision making by making explicit to what extent incomplete knowledge affects predictions.