Bayesian approach to contaminant source characterization in water distribution systems: adaptive sampling framework

Bayesian analysis can yield a probabilistic contaminant source characterization conditioned on available sensor data and accounting for system stochastic processes. This paper is based on a previously proposed Markov chain Monte Carlo (MCMC) approach tailored for water distribution systems and incorporating stochastic water demands. The observations can include those from fixed sensors and, the focus of this paper, mobile sensors. Decision makers, such as utility managers, need not wait until new observations are available from an existing sparse network of fixed sensors. This paper addresses a key research question: where is the best location in the network to gather additional measurements so as to maximize the reduction in the source uncertainty? Although this has been done in groundwater management, it has not been well addressed in water distribution networks. In this study, an adaptive framework is proposed to guide the strategic placement of mobile sensors to complement the fixed sensor network. MCMC is the core component of the proposed adaptive framework, while several other pieces are indispensable: Bayesian preposterior analysis, value of information criterion and the search strategy for identifying an optimal location. Such a framework is demonstrated with an illustrative example, where four candidate sampling locations in the small water distribution network are investigated. Use of different value-of-information criteria reveals that while each may lead to different outcomes, they share some common characteristics. The results demonstrate the potential of Bayesian analysis and the MCMC method for contaminant event management.     

Comparison of ensemble-based state and parameter estimation methods for soil moisture data assimilation

Model parameters are a source of uncertainty that can easily cause systematic deviation and significantly affect the accuracy of soil moisture generation in assimilation systems. This study addresses the issue of retrieving model parameters related to soil moisture via the simultaneous estimation of states and parameters based on the Common Land Model (CoLM). The state-parameter estimation algorithms AEnKF (Augmented Ensemble Kalman Filter), DEnKF (Dual Ensemble Kalman Filter) and SODA (Simultaneous optimization and data assimilation) are entirely implemented within an EnKF framework to investigate how the three algorithms can correct model parameters and improve the accuracy of soil moisture estimation. The analysis is illustrated by assimilating the surface soil moisture levels from varying observation intervals using data from Mongolian plateau sites. Furthermore, a radiation transfer model is introduced as an observation operator to analyze the influence of brightness temperature assimilation on states and parameters that are estimated at different microwave signal frequencies. Three cases were analyzed for both soil moisture and brightness temperature assimilation, focusing on the progressive incorporation of parameter uncertainty, forcing data uncertainty and model uncertainty. It has been demonstrated that EnKF is outperformed by all other methods, as it consistently maintains a bias. State-parameter estimation algorithms can provide a more accurate estimation of soil moisture than EnKF. AEnKF is the most robust method, with the lowest RMSE values for retrieving states and parameters dealing only with parameter uncertainty, but it possesses disadvantages related to increasing sources of uncertainty and decreasing numbers of observations. SODA performs well under the complex situations in which DEnKF shows slight disadvantages in terms of statistical indicators; however, the former consumes far more memory and time than the latter.     

Characterization of groundwater contaminant source using Bayesian method

Contaminant source identification in groundwater system is critical for remediation strategy implementation, including gathering further samples and analysis, as well as implementing and evaluating different remediation plans. Such problem is usually solved with the aid of groundwater modeling with lots of uncertainty, e.g. existing uncertainty in hydraulic conductivity, measurement variance and the model structure error. Monte Carlo simulation of flow model allows the input uncertainty onto the model predictions of concentration measurements at monitoring sites. Bayesian approach provides the advantage to update estimation. This paper presents an application of a dynamic framework coupling with a three dimensional groundwater modeling scheme in contamination source identification of groundwater. Markov Chain Monte Carlo (MCMC) is being applied to infer the possible location and magnitude of contamination source. Uncertainty existing in heterogonous hydraulic conductivity field is explicitly considered in evaluating the likelihood function. Unlike other inverse-problem approaches to provide single but maybe untrue solution, the MCMC algorithm provides probability distributions over estimated parameters. Results from this algorithm offer a probabilistic inference of the location and concentration of released contamination. The convergence analysis of MCMC reveals the effectiveness of the proposed algorithm. Further investigation to extend this study is also discussed.     

Assessment of uncertainty sources in water quality modeling in the Niagara River

This paper presents a framework to quantify the overall variability of the model estimations of Total Polychlorinated Biphenyls (Total PCBs) concentrations in the Niagara River on the basis of the uncertainty of few model parameters and the natural variability embedded in some of the model input variables. The results of the uncertainty analysis are used to understand the importance of stochastic model components and their effect on the overall reliability of the model output and to evaluate multiple sources of uncertainty that might need to be further studied. The uncertainty analysis is performed using a newly developed point estimate method, the Modified Rosenblueth method. The water quality along the Niagara River is simulated by coupling two numerical models the Environmental Fluid Dynamic Code (EFDC) – for the hydrodynamic portion of the study and the Water Quality Analysis and Simulation Program (WASP) – for the fate and transport of contaminants. For the monitoring period from May 1995 to March 1997, the inflow Total PCBs concentration from Lake Erie is the stochastic component that most influences the variability of the modeling results for the simulated concentrations at the exit of the Niagara River. Other significant stochastic components in order are as follows: the suspended sediments concentration, the point source loadings and to a minor degree the atmospheric deposition, the flow and the non-point source loadings. Model results that include estimates of uncertainty provide more comprehensive information about the variability of contaminant concentrations, such as confidence intervals, and, in general offer a better approach to compare model results with measured data.     

Optimization of groundwater-monitoring networks for identification of the distribution of a contaminant plume

A new methodology is proposed to optimize monitoring networks for identification of the extent of contaminant plumes. The optimal locations for monitoring wells are determined as the points where maximal decreases are expected in the quantified uncertainty about contaminant existence after well installation. In this study, hydraulic conductivity is considered to be the factor that causes uncertainty. The successive random addition (SRA) method is used to generate random fields of hydraulic conductivity. The expected value of information criterion for the existence of a contaminant plume is evaluated based on how much the uncertainty of plume distribution reduces with increases in the size of the monitoring network. The minimum array of monitoring wells that yields the maximum information is selected as the optimal monitoring network. In order to quantify the uncertainty of the plume distribution, the probability map of contaminant existence is made for all generated contaminant plume realizations on the domain field. The uncertainty is defined as the sum of the areas where the probability of contaminant existence or nonexistence is uncertain. Results of numerical experiments for determination of optimal monitoring networks in heterogeneous conductivity fields are presented.     

Robust H∞ control for aseismic structures With uncertainties in model parameters

This paper presents a robust H∞ output feedback control approach for structural systems with uncertainties in model parameters by using available acceleration measurements and proposes conditions for the existence of such a robust ontput feedback controller.The uncertainties of structural stiffness,damping and mass parameters are assumed to be norm-bounded.The proposed control approach is formulated within the framework of linear matrix inequalities,for which existing convex optimization techniques,such as the LMI toolbox in MATLAB,can be used effectively and conveniently.To illustrate the effectiveness of the proposed robust H∞ strategy,a six-story building was subjected both to the 1940 El Centro earthquake record and to a suddenly applied Kanai-Tajimi filtered white noise random excitation.The results show that the proposed robust H∞ controller provides satisfactory results with or without variation of the structural stiffness,damping and mass parameters.     

Model identification for hydrological forecasting under uncertainty

Methods for the identification of models for hydrological forecasting have to consider the specific nature of these models and the uncertainties present in the modeling process. Current approaches fail to fully incorporate these two aspects. In this paper we review the nature of hydrological models and the consequences of this nature for the task of model identification. We then continue to discuss the history (“The need for more POWER‘’), the current state (“Learning from other fields”) and the future (“Towards a general framework”) of model identification. The discussion closes with a list of desirable features for an identification framework under uncertainty and open research questions in need of answers before such a framework can be implemented.     

Probability Theory as Logic: Data Assimilation for Multiple Source Reconstruction

Probability theory as logic (or Bayesian probability theory) is a rational inferential methodology that provides a natural and logically consistent framework for source reconstruction. This methodology fully utilizes the information provided by a limited number of noisy concentration data obtained from a network of sensors and combines it in a consistent manner with the available prior knowledge (mathematical representation of relevant physical laws), hence providing a rigorous basis for the assimilation of this data into models of atmospheric dispersion for the purpose of contaminant source reconstruction. This paper addresses the application of this framework to the reconstruction of contaminant source distributions consisting of an unknown number of localized sources, using concentration measurements obtained from a sensor array. To this purpose, Bayesian probability theory is used to formulate the full joint posterior probability density function for the parameters of the unknown source distribution. A simulated annealing algorithm, applied in conjunction with a reversible-jump Markov chain Monte Carlo technique, is used to draw random samples of source distribution models from the posterior probability density function. The methodology is validated against a real (full-scale) atmospheric dispersion experiment involving a multiple point source release.     

A Coupled Groundwater–Surface Water Modeling Framework for Simulating Transition Zone Processes

There is an identified need for fully representing groundwater–surface water transition zone (i.e., the sediment zone that connects groundwater and surface water) processes in modeling fate and transport of contaminants to assist with management of contaminated sediments. Most existing groundwater and surface water fate and transport models are not dynamically linked and do not consider transition zone processes such as bioturbation and deposition and erosion of sediments. An interface module is developed herein to holistically simulate the fate and transport by coupling two commonly used models, Environmental Fluid Dynamics Code (EFDC) and SEAWAT, to simulate surface water and groundwater hydrodynamics, while providing an enhanced representation of the processes in the transition zone. Transition zone and surface water contaminant processes were represented through an enhanced version of the EFDC model, AQFATE. AQFATE also includes SEDZLJ, a state‐of‐the‐science surface water sediment transport model. The modeling framework was tested on a published test problem and applied to evaluate field‐scale two‐ and three‐dimensional contaminant transport. The model accurately simulated concentrations of salinity from a published test case. For the field‐scale applications, the model showed excellent mass balance closure for the transition zone and provided accurate simulations of all transition zone processes represented in the modeling framework. The model predictions for the two‐dimensional field case were consistent with site‐specific observations of contaminant migration. This modeling framework represents advancement in the simulation of transition zone processes and can help inform risk assessment at sites where contaminant sources from upland areas have the potential to impact sediments and surface water.     

A Systematic Approach to Designing a Multiphase Unsaturated Zone Monitoring Network

A systematic approach is presented for the design of a multiphase vadose zone monitoring system recognizing that, as in ground water monitoring system design, complete subsurface coverage is not practical. The approach includes identification and prioritization of vulnerable areas: select ion of cost-effective indirect monitoring methods that will provide early warning of contaminant migration: selection of direct monitoring methods for diagnostic confirmation; identification of background monitoring locations; and identification of an appropriate temporal monitoring plan. An example of a monitoring system designed for a solid waste landfill is presented and utilized to illustrate the approach and provide details of system implementation. The example design described incorporates the use of neutron moisture probes deployed in both vertical and horizontal access tubes beneath the lcachate recovery collection system of the landfill. Early warning of gaseous phase contaminant migration is monitored utilizing whole-air active soil gas sampling points deployed in gravel- filled trenches beneath the subgrade. Diagnostic confirmation of contaminant migration is provided utilizing pore- liquid samplers. Conservative tracers can be used to distinguish between chemical species released by a landfill from those attributable to other (e.g. off-site) sources or present naturally in the subsurface. A discussion of background monitoring point location is also presented.     

A probabilistic framework for floodplain mapping using hydrological modeling and unsteady hydraulic modeling

ABSTRACT

Prediction of design hydrographs is key in floodplain mapping using hydraulic models, which are either steady state or unsteady. The former, which require only an input peak, substantially overestimate the volume of water entering the floodplain compared to the more realistic dynamic case simulated by the unsteady models that require the full hydrograph. Past efforts to account for the uncertainty of boundary conditions using unsteady hydraulic modeling have been based largely on a joint flood frequency–shape analysis, with only a very limited number of studies using hydrological modeling to produce the design hydrographs. This study therefore presents a generic probabilistic framework that couples a hydrological model with an unsteady hydraulic model to estimate the uncertainty of flood characteristics. The framework is demonstrated on the Swannanoa River watershed in North Carolina, USA. Given its flexibility, the framework can be applied to study other sources of uncertainty in other hydrological models and watersheds.     

An Idealized Ground-Water Flow and Chemical Transport Model (S-PATHS)

The number of studies on the actual and potential environmental consequences of contaminated ground water is growing. One means of studying these consequences is through an idealized flow and transport model, S-PATHS, which allows the hydrologist to determine the salient features of contaminant migration with a minimum of data. The transport of contaminants by ground water from many waste disposal sites can be geometrically idealized as flow between a line and a circle. The flow system adjacent to the disposal site can be represented as a contaminant line source, and a downgradient pumping well as a circular sink. To study waste disposal sites on a larger scale the model geometry is reversed and the disposal site is represented as a circular source, and a river or other convenient line of evaluation is represented as a line sink. This idealization allows S-PATHS to describe the flow and transport process directly by a single partial differential expression. S-PATHS considers transmissivity, effective porosity, sorption, source strength, source concentration, decay, potentiometric gradient, circle size, and distance to the line. Coding for the model is not lengthy and can be run on a large-capacity, hand-held calculator.     

Passive Samplers for Monitoring VOCs in Groundwater and the Prospects Related to Mass Flux Measurements

Measurement and interpretation of mass fluxes in favor of concentrations is gaining more and more interest, especially within the framework of the characterization and management of large-scale volatile organic carbon (VOC) groundwater contamination (source zones and plumes). Traditional methods of estimating contaminant fluxes and discharges involve individual measurements/calculations of the Darcy water flux and the contaminant concentrations. However, taken into account the spatially and temporally varying hydrologic conditions in complex, heterogeneous aquifers, higher uncertainty arises from such indirect estimation of contaminant fluxes. Therefore, the potential use of passive sampling devices for the direct measurement of groundwater-related VOC mass fluxes is examined. A review of current passive samplers for the measurement of organic contaminants in water yielded the selection of 18 samplers that were screened for a number of criteria. These criteria are related to the possible application of the sampler for the measurement of VOC mass fluxes in groundwater. This screening study indicates that direct measurement of VOC mass fluxes in groundwater is possible with very few passive samplers. Currently, the passive flux meter (PFM) is the only passive sampler which has proven to effectively measure mass fluxes in near source groundwater. A passive sampler for mass flux measurement in plume zones with regard to long-term monitoring (several months to a year) still needs to be developed or optimized. A passive sampler for long-term monitoring of contaminant mass fluxes in groundwater would be of considerable value in the development of risk-based assessment and management of soil and groundwater pollutions.     

Transition probability/Markov chain analyses of DNAPL source zones and plumes

At sites where a dense nonaqueous phase liquid (DNAPL) was spilled or released into the subsurface, estimates of the mass of DNAPL contained in the subsurface from core or monitoring well data, either in the nonaqueous or aqueous phase, can be highly uncertain because of the erratic distribution of the DNAPL due to geologic heterogeneity. In this paper, a multiphase compositional model is applied to simulate, in detail, the DNAPL saturations and aqueous-phase plume migration in a highly characterized, heterogeneous glaciofluvial aquifer, the permeability and porosity data of which were collected by researchers at the University of Tübingen, Germany. The DNAPL saturation distribution and the aqueous-phase contaminant mole fractions are then reconstructed by sampling the data from the forward simulation results using two alternate approaches, each with different degrees of sampling conditioning. To reconstruct the DNAPL source zone architecture, the aqueous-phase plume configuration, and the contaminant mass in each phase, one method employs the novel transition probability/Markov chain approach (TP/MC), while the other involves a traditional variogram analysis of the sampled data followed by ordinary kriging. The TP/MC method is typically used for facies and/or hydraulic conductivity reconstruction, but here we explore the applicability of the TP/MC method for the reconstruction of DNAPL source zones and aqueous-phase plumes. The reconstructed geometry of the DNAPL source zone, the dissolved contaminant plume, and the estimated mass in each phase are compared using the two different geostatistical modeling approaches and for various degrees of data sampling from the results of the forward simulation. It is demonstrated that the TP/MC modeling technique is robust and accurate and is a preferable alternative compared to ordinary kriging for the reconstruction of DNAPL saturation patterns and dissolved-phase contaminant plumes.     

Probabilistic-fuzzy health risk modeling

Health risk analysis of multi-pathway exposure to contaminated water involves the use of mechanistic models that include many uncertain and highly variable parameters. Currently, the uncertainties in these models are treated using statistical approaches. However, not all uncertainties in data or model parameters are due to randomness. Other sources of imprecision that may lead to uncertainty include scarce or incomplete data, measurement error, data obtained from expert judgment, or subjective interpretation of available information. These kinds of uncertainties and also the non-random uncertainty cannot be treated solely by statistical methods. In this paper we propose the use of fuzzy set theory together with probability theory to incorporate uncertainties into the health risk analysis. We identify this approach as probabilistic-fuzzy risk assessment (PFRA). Based on the form of available information, fuzzy set theory, probability theory, or a combination of both can be used to incorporate parameter uncertainty and variability into mechanistic risk assessment models. In this study, tap water concentration is used as the source of contamination in the human exposure model. Ingestion, inhalation and dermal contact are considered as multiple exposure pathways. The tap water concentration of the contaminant and cancer potency factors for ingestion, inhalation and dermal contact are treated as fuzzy variables while the remaining model parameters are treated using probability density functions. Combined utilization of fuzzy and random variables produces membership functions of risk to individuals at different fractiles of risk as well as probability distributions of risk for various alpha-cut levels of the membership function. The proposed method provides a robust approach in evaluating human health risk to exposure when there is both uncertainty and variability in model parameters. PFRA allows utilization of certain types of information which have not been used directly in existing risk assessment methods.     

The integrated effects of climate and hydrologic uncertainty on future flood risk assessments

This work examines future flood risk within the context of integrated climate and hydrologic modelling uncertainty. The research questions investigated are (1) whether hydrologic uncertainties are a significant source of uncertainty relative to other sources such as climate variability and change and (2) whether a statistical characterization of uncertainty from a lumped, conceptual hydrologic model is sufficient to account for hydrologic uncertainties in the modelling process. To investigate these questions, an ensemble of climate simulations are propagated through hydrologic models and then through a reservoir simulation model to delimit the range of flood protection under a wide array of climate conditions. Uncertainty in mean climate changes and internal climate variability are framed using a risk‐based methodology and are explored using a stochastic weather generator. To account for hydrologic uncertainty, two hydrologic models are considered, a conceptual, lumped parameter model and a distributed, physically based model. In the conceptual model, parameter and residual error uncertainties are quantified and propagated through the analysis using a Bayesian modelling framework. The approach is demonstrated in a case study for the Coralville Dam on the Iowa River, where recent, intense flooding has raised questions about potential impacts of climate change on flood protection adequacy. Results indicate that the uncertainty surrounding future flood risk from hydrologic modelling and internal climate variability can be of the same order of magnitude as climate change. Furthermore, statistical uncertainty in the conceptual hydrological model can capture the primary structural differences that emerge in flood damage estimates between the two hydrologic models. Copyright © 2014 John Wiley & Sons, Ltd.     

Reducing fluctuations in the sample variogram

In the analysis of regionalized data, irregular sampling patterns are often responsible for large deviations (fluctuations) between the theoretical and sample semi-variograms. This article proposes a new semi-variogram estimator that is unbiased irrespective of the actual multivariate distribution of the data (provided an assumption of stationarity) and has the minimal variance under a given multivariate distribution model. Such an estimator considerably reduces fluctuations in the sample semi-variogram when the data are strongly correlated and clustered in space, and proves to be robust to a misspecification of the multivariate distribution model. The traditional and proposed semi-variogram estimators are compared through an application to a pollution dataset.     

Simultaneous identification of the pollutant release history and the source location in groundwater by means of a geostatistical approach

This paper describes an innovative procedure that is able to simultaneously identify the release history and the source location of a pollutant injection in a groundwater aquifer (simultaneous release function and source location identification, SRSI). The methodology follows a geostatistical approach: it develops starting from a data set and a reliable numerical flow and transport model of the aquifer. Observations can be concentration data detected at a given time in multiple locations or a time series of concentration measurements collected at multiple locations. The methodology requires a preliminary delineation of a probably source area and results in the identification of both the sub-area where the pollutant injection has most likely originated, and in the contaminant release history. Some weak hypotheses have to be defined about the statistical structure of the unknown release function such as the probability density function and correlation structure. Three case studies are discussed concerning two-dimensional, confined aquifers with strongly non-uniform flow fields. A transfer function approach has been adopted for the numerical definition of the sensitivity matrix and the recent step input function procedure has been successfully applied.     

Contaminant remediation decision analysis using information gap theory

Decision making under severe lack of information is a ubiquitous situation in nearly every applied field of engineering, policy, and science. A severe lack of information precludes our ability to determine a frequency of occurrence of events or conditions that impact the decision; therefore, decision uncertainties due to a severe lack of information cannot be characterized probabilistically. To circumvent this problem, information gap (info-gap) theory has been developed to explicitly recognize and quantify the implications of a severe lack of information in decision making. This paper presents a decision analysis based on info-gap theory developed for a contaminant remediation scenario. The analysis provides decision support in determining the fraction of contaminant mass to remove from the environment. An info-gap uncertainty model is developed to characterize uncertainty due to a lack of information concerning the contaminant flux. The info-gap uncertainty model groups nested, convex sets of functions defining contaminant flux over time based on their level of deviation from a nominal contaminant flux. The nominal contaminant flux defines a best estimate of contaminant flux over time based on existing, though incomplete, information. A robustness function is derived to quantify the maximum level of deviation from nominal that still ensures compliance for alternative decisions. An opportuneness function is derived to characterize the possibility of meeting a desired contaminant concentration level. The decision analysis evaluates how the robustness and opportuneness change as a function of time since remediation and as a function of the fraction of contaminant mass removed.     

The Utility of Multiple-Completion Monitoring Wells for Describing a Solvent Plume

At a study site in the midwestern United States, multiple-completion wells demonstrated that a vertical hydraulic gradient was responsible for the contamination pattern exhibited by chlorinated solvent plumes. The typical pattern consisted of little or no contamination in the upper portion of the aquifer with concentrations increasing with depth. When ground water contamination was discovered in an unexpected portion of the site, water level elevations and contaminant distribution data obtained from multiple-completion wells resulted in identification of the source location. The well eventually determined to be located in the source area displayed contaminant levels much higher in the upper zone of the aquifer — the opposite contamination pattern of other on-site wells. Such results indicated that the spill had occurred near this location and that solvent residing along the capillary fringe was continuing to contaminate the aquifer.