Bayesian inference for source determination with applications to a complex urban environment

The problem of determining the source of an emission from the limited information provided by a finite and noisy set of concentration measurements obtained from real-time sensors is an ill-posed inverse problem. In general, this problem cannot be solved uniquely without additional information. A Bayesian probabilistic inferential framework, which provides a natural means for incorporating both errors (model and observational) and prior (additional) information about the source, is presented. Here, Bayesian inference is applied to find the posterior probability density function of the source parameters (location and strength) given a set of concentration measurements. It is shown how the source–receptor relationship required in the determination of the likelihood function can be efficiently calculated using the adjoint of the transport equation for the scalar concentration. The posterior distribution of the source parameters is sampled using a Markov chain Monte Carlo method. The inverse source determination method is validated against real data sets acquired in a highly disturbed flow field in an urban environment. The data sets used to validate the proposed methodology include a water-channel simulation of the near-field dispersion of contaminant plumes in a large array of building-like obstacles (Mock Urban Setting Trial) and a full-scale field experiment (Joint Urban 2003) in Oklahoma City. These two examples demonstrate the utility of the proposed approach for inverse source determination.     

Extensions to minimum relative entropy inversion for noisy data

Minimum relative entropy (MRE) and Tikhonov regularization (TR) were compared by Neupauer et al. [Water Resour. Res. 36 (2000) 2469] on the basis of an example plume source reconstruction problem originally proposed by Skaggs and Kabala [Water Resour. Res. 30 (1994) 71] and a boxcar-like function. Although Neupauer et al. [Water Resour. Res. 36 (2000) 2469] were careful in their conclusions to note the basis of these comparisons, we show that TR does not perform well on problems in which delta-like sources are convolved with diffuse-groundwater contamination response functions, particularly in the presence of noise. We also show that it is relatively easy to estimate an appropriate value for epsilon, the hyperparameter needed in the minimum relative entropy solution for the inverse problem in the presence of noise. This can be estimated in a variety of ways, including estimation from the data themselves, analysis of data residuals, and a rigorous approach using the real cepstrum and the Akaike Information Criterion (AIC). Regardless of the approach chosen, for the sample problem reported herein, excellent resolution of multiple delta-like spikes is produced from MRE from noisy, diffuse data. The usefulness of MRE for noisy inverse problems has been demonstrated.     

Laboratory longitudinal diffusion tests: 2. Parameter estimation by inverse analysis

This study focuses on the verification of test interpretations for different state analyses of diffusion experiments. Part 1 of this study identified that steady, quasi-steady and equilibrium state analyses for the through- and in-diffusion tests with solution reservoirs are generally feasible where the tracer is not highly sorptive. In Part 2 we investigate parameter identifiability in transient-state analysis of reservoir concentration variation using a numerical approach. For increased generality, the analytical models, objective functions and Jacobian matrix necessary for inverse analysis of transient-state data are reformulated using unified dimensionless parameters. In these dimensionless forms, the number of unknown parameters is reduced and a single dimensionless parameter represents the sorption property. The dimensionless objective functions are evaluated for individual test methods and parameter identifiability is discussed in relation to the sorption property. The effects of multiple minima and measurement error on parameter identifiability are also investigated. The main findings are that inverse problems for inlet and outlet reservoir concentration analyses are generally unstable and well-posed, respectively. Where the tracer is sorptive, the inverse problem for the inlet reservoir concentration analysis may have multiple minima. When insufficient measurement data is collected, multiple solutions may result and this should be taken into consideration when inversely analyzing data including that of inlet reservoir concentration. Verification of test interpretation by cross-checking different state analyses is feasible where the tracer is not highly sorptive. In an actual experiment, test interpretation validity is demonstrated through consistency between theory and practice for different state analyses.     

An inverse Gaussian plume approach for estimating atmospheric pollutant emissions from multiple point sources

A method is developed for estimating the emission rates of contaminants into the atmosphere from multiple point sources using measurements of particulate material deposited at ground level. The approach is based on a Gaussian plume type solution for the advection–diffusion equation with ground-level deposition and given emission sources. This solution to the forward problem is incorporated into an inverse algorithm for estimating the emission rates by means of a linear least squares approach. The results are validated using measured deposition and meteorological data from a large lead–zinc smelting operation in Trail, British Columbia. The algorithm is demonstrated to be robust and capable of generating reasonably accurate estimates of total contaminant emissions over the relatively short distances of interest in this study.     

Inverse modeling to estimate LNAPL plume release timing

Methodologies are presented for dating releases of light nonaqueous phase liquids (LNAPLs) using an inverse modeling approach with simple analytical models. Models for LNAPL plume migration are presented to predict LNAPL plume velocity in the unsaturated and saturated zones as a function of basic soil and fluid properties. A relative mobility factor is introduced for LNAPL movement at the water table that depends primarily on the van Genuchten n parameter (related to the breadth of the soil pore size distribution) and the magnitude of water table fluctuations. Estimated LNAPL plume velocities compare reasonably with more rigorous numerical models, which may be used in cases where data availability warrant the greater effort entailed.Two methods of estimating release timing and its uncertainty are investigated. A direct estimation method is described that determines travel time for a single observed travel distance based on estimated soil and fluid properties. Release date uncertainty may be determined using the first order (FO) or Monte Carlo (MC) methods. The second method for estimating release date involves nonlinear parameter estimation utilizing distance vs. time measurements and other data.A case study is presented for a field site where independent estimates of release timing were obtained from a numerical modeling analysis. Release timing estimates based on direct inversion of the analytical timing model agree well with the numerical analysis. Results for a second field site indicate that release date confidence limits estimated by the FO method, assuming log-normally distributed travel times, are close to values determined by the MC method, which makes no assumption regarding the form of the travel time probability distribution.Results for a hypothetical problem indicate that LNAPL velocity and travel time may be accurately estimated if sufficient data on travel distance vs. time are available. Incorporating prior information on relevant soil and fluid properties into the objective function reduces the uncertainty in release date if prior estimates are accurate. However, biased prior estimates may lead to over- or underestimation of release date uncertainty. Simultaneous estimation of soil and fluid properties and release date is possible if prior information is available to condition the parameter estimates.     

The Fifth International Clean Air Congress

ABSTRACT

The main objective of this study was to investigate the capabilities of the receptor-oriented inverse mode Lagrangian Stochastic Particle Dispersion Model (LSPDM) with the 12-km resolution Mesoscale Model 5 (MM5) wind field input for the assessment of source identification from seven regions impacting two receptors located in the eastern United States. The LSPDM analysis was compared with a standard version of the Hybrid Single-Particle Lagrangian Integrated Trajectory (HYSPLIT) single-particle backward-trajectory analysis using inputs from MM5 and the Eta Data Assimilation System (EDAS) with horizontal grid resolutions of 12 and 80 km, respectively. The analysis included four 7-day summertime events in 2002; residence times in the modeling domain were computed from the inverse LSPDM runs and HYPSLIT-simulated backward trajectories started from receptor-source heights of 100, 500, 1000, 1500, and 3000 m. Statistics were derived using normalized values of LSPDM- and HYSPLIT-predicted residence times versus Community Multiscale Air Quality model-predicted sulfate concentrations used as baseline information. From 40 cases considered, the LSPDM identified first- and second-ranked emission region influences in 37 cases, whereas HYSPLIT-MM5 (HYSPLIT-EDAS) identified the sources in 21 (16) cases. The LSPDM produced a higher overall correlation coefficient (0.89) compared with HYSPLIT (0.55–0.62). The improvement of using the LSPDM is also seen in the overall normalized root mean square error values of 0.17 for LSPDM compared with 0.30–0.32 for HYSPLIT. The HYSPLIT backward trajectories generally tend to underestimate near-receptor sources because of a lack of stochastic dispersion of the backward trajectories and to overestimate distant sources because of a lack of treatment of dispersion. Additionally, the HYSPLIT backward trajectories showed a lack of consistency in the results obtained from different single vertical levels for starting the backward trajectories. To alleviate problems due to selection of a backward-trajectory starting level within a large complex set of 3-dimensional winds, turbulence, and dispersion, results were averaged from all heights, which yielded uniform improvement against all individual cases.

IMPLICATIONS Backward-trajectory analysis is one of the standard procedures for determining the spatial locations of possible emission sources affecting given receptors, and it is frequently used to enhance receptor modeling results. This analysis simplifies some of the relevant processes such as pollutant dispersion, and additional methods have been used to improve receptor-source relationships. A methodology of inverse Lagrangian stochastic particle dispersion modeling was used in this study to complement and improve standard backward-trajectory analysis. The results show that inverse dispersion modeling can identify regional sources of haze in national parks and other regions of interest.     

Numerical methods for improving sensitivity analysis and parameter estimation of virus transport simulated using sorptive-reactive processes

Using one- and two-dimensional homogeneous simulations, this paper addresses challenges associated with sensitivity analysis and parameter estimation for virus transport simulated using sorptive-reactive processes. Head, flow, and conservative- and virus-transport observations are considered. The paper examines the use of (1) observed-value weighting, (2) breakthrough-curve temporal moment observations, and (3) the significance of changes in the transport time-step size. The results suggest that (1) sensitivities using observed-value weighting are more susceptible to numerical solution variability, (2) temporal moments of the breakthrough curve are a more robust measure of sensitivity than individual conservative-transport observations, and (3) the transport-simulation time step size is more important than the inactivation rate in solution and about as important as at least two other parameters, reflecting the ease with which results can be influenced by numerical issues. The approach presented allows more accurate evaluation of the information provided by observations for estimation of parameters and generally improves the potential for reasonable parameter-estimation results.     

Rigorous uncertainty assessment in contaminant transport inverse modelling: a case study of fluoride diffusion through clay liners

Inverse methods used in assessing landfill liner design have not yet taken advantage of current developments in inverse procedures. Here, a method for inverting contaminant transport models is presented including a general error model and procedures for differentially weighted multiple response regression. General error models are employed in cases where the residuals are heteroscedastic and correlated, and lead to valid inference on model parameter and predictive uncertainty. The Shuffled Complex Evolution algorithm is used to optimise model parameters. Model parameter uncertainty is assessed by exploring the posterior probability distribution with the Metropolis algorithm, a Markov chain Monte Carlo sampling method. The inverse method is applied to simultaneously determine the sorption and diffusion parameters from laboratory diffusion cell experiments. In these experiments, fluoride migration through kaolin clays was measured by sampling the source and collector cells over time. To uniquely determine the transport model parameters, it was necessary to simultaneously fit the observed data from two independent diffusion cell experiments with different initial concentrations. The jointly fitted transport model parameters compared well with those fitted to independent batch experiments.     

Insights into the use of time-lapse GPR data as observations for inverse multiphase flow simulations of DNAPL migration

Perchloroethylene (PCE) saturations determined from GPR surveys were used as observations for inversion of multiphase flow simulations of a PCE injection experiment (Borden 9 m cell), allowing for the estimation of optimal bulk intrinsic permeability values. The resulting fit statistics and analysis of residuals (observed minus simulated PCE saturations) were used to improve the conceptual model. These improvements included adjustment of the elevation of a permeability contrast, use of the van Genuchten versus Brooks-Corey capillary pressure-saturation curve, and a weighting scheme to account for greater measurement error with larger saturation values. A limitation in determining PCE saturations through one-dimensional GPR modeling is non-uniqueness when multiple GPR parameters are unknown (i.e., permittivity, depth, and gain function). Site knowledge, fixing the gain function, and multiphase flow simulations assisted in evaluating non-unique conceptual models of PCE saturation, where depth and layering were reinterpreted to provide alternate conceptual models. Remaining bias in the residuals is attributed to the violation of assumptions in the one-dimensional GPR interpretation (which assumes flat, infinite, horizontal layering) resulting from multidimensional influences that were not included in the conceptual model. While the limitations and errors in using GPR data as observations for inverse multiphase flow simulations are frustrating and difficult to quantify, simulation results indicate that the error and bias in the PCE saturation values are small enough to still provide reasonable optimal permeability values. The effort to improve model fit and reduce residual bias decreases simulation error even for an inversion based on biased observations and provides insight into alternate GPR data interpretations. Thus, this effort is warranted and provides information on bias in the observation data when this bias is otherwise difficult to assess.     

A review of statistical methods for the meteorological adjustment of tropospheric ozone

A variety of statistical methods for meteorological adjustment of ozone have been proposed in the literature over the last decade for purposes of forecasting, estimating ozone time trends, or investigating underlying mechanisms from an empirical perspective. The methods can be broadly classified into regression, extreme value, and space–time methods. We present a critical review of these methods, beginning with a summary of what meteorological and ozone monitoring data have been considered and how they have been used for statistical analysis. We give particular attention to the question of trend estimation, and compare selected methods in an application to ozone time series from the Chicago area. We conclude that a number of approaches make useful contributions to the field, but that no one method is most appropriate for all purposes and all meteorological scenarios. Methodological issues such as the need for regional-scale analysis, the nonlinear dependence of ozone on meteorology, and extreme value analysis for trends are addressed. A comprehensive and reliable methodology for space–time extreme value analysis is attractive but lacking.     

Statistical Issues in Assessing Anthropogenic Background for Arsenic

Conceptual and statistical issues surrounding the estimation of a background concentration distribution for arsenic are reviewed. How background area is defined and samples collected are shown to impact the shape and location of the probability density function that in turn affects the estimation and precision of associated distributional parameters. The overall background concentration distribution is conceptualized as a mixture of a natural background distribution, an anthropogenic background distribution and a distribution designed to accommodate the potential for contamination site samples being included into the background sample set. This concept is extended to a discussion of issues surrounding estimation of natural and anthropogenic background distributions for larger geographic areas. Finally, the mixture model is formally defined and statistical approaches to estimating its parameters discussed.     

Statistical Issues in Assessing Anthropogenic Background for Arsenic

Conceptual and statistical issues surrounding the estimation of a background concentration distribution for arsenic are reviewed. How background area is defined and samples collected are shown to impact the shape and location of the probability density function that in turn affects the estimation and precision of associated distributional parameters. The overall background concentration distribution is conceptualized as a mixture of a natural background distribution, an anthropogenic background distribution and a distribution designed to accommodate the potential for contamination site samples being included into the background sample set. This concept is extended to a discussion of issues surrounding estimation of natural and anthropogenic background distributions for larger geographic areas. Finally, the mixture model is formally defined and statistical approaches to estimating its parameters discussed.     

A Monitoring Network Design Procedure for Three-Dimensional (3D) Groundwater Contaminant Source Identification

Finding the location and concentration of contaminant sources is an important step in groundwater remediation and management. This discovery typically requires the solution of an inverse problem. This inverse problem can be formulated as an optimization problem where the objective function is the sum of the square of the errors between the observed and predicted values of contaminant concentration at the observation wells. Studies show that the source identification accuracy is dependent on the observation locations (i.e., network geometry) and frequency of sampling; thus, finding a set of optimal monitoring well locations is very important for characterizing the source. The objective of this study is to propose a sensitivity-based method for optimal placement of monitoring wells by incorporating two uncertainties: the source location and hydraulic conductivity. An optimality metric called D-optimality in combination with a distance metric, which tends to make monitoring locations as far apart from each other as possible, is developed for finding optimal monitoring well locations for source identification. To address uncertainty in hydraulic conductivity, an integration method of multiple well designs is proposed based on multiple hydraulic conductivity realizations. Genetic algorithm is used as a search technique for this discrete combinatorial optimization problem. This procedure was applied to a hypothetical problem based on the well-known Borden Site data in Canada. The results show that the criterion-based selection proposed in this paper provides improved source identification performance when compared to uniformly distributed placement of wells.     

Determining ecoregional numeric nutrient criteria by stressor-response models in Yungui ecoregion lakes,China

The importance of developing numeric nutrient criteria has been recognized to protect the designated uses of water bodies from nutrient enrichment that is associated with broadly occurring levels of nitrogen/phosphorus pollution. The identification and estimation of stressor-response models in aquatic ecosystems has been shown to be useful in the determination of nutrient criteria. In this study, three methods based on stressor-response relationships were applied to determine nutrient criteria for Yungui ecoregion lakes with respect to total phosphorus (TP), total nitrogen (TN), and planktonic chlorophyll a (Chl a). Simple linear regression (SLR) models were established to provide an estimate of the relationship between a response variable and a stressor. Multiple linear regressions were used to simultaneously estimate the effect of TP and TN on Chl a. A morphoedaphic index (MEI) was applied to derive nutrient criteria using data from Yungui ecoregion lakes, which were considered as areas with less anthropogenic influences. Nutrient criteria, as determined by these three methods, showed broad agreement for all parameters. The ranges of numeric nutrient criteria for Yungui ecoregion lakes were determined as follows: TP 0.008–0.010 mg/L and TN 0.140–0.178 mg/L. The stressor-response analysis described will be of benefit to support countries in their numeric criteria development programs and to further the goal of reducing nitrogen/phosphorus pollution in China.     

Investigating weekend air quality observations with the aid of Fourier analysis in Athens, Greece

Weekend air quality is analysed with the aid of Fourier analysis, in order to reveal time-series structures for Athens, Greece. For this reason, an appropriate analysis of a 10 year record from 1990–1999 of air quality data from the monitoring network in Athens is carried out for two specific monitoring sites. The aim of the paper is to investigate the temporal pattern of observations in order to reveal relations and trends. Thus, it is expected that the estimation of environmental consequences related to various incentives aimed at the reduction of air emissions will be enriched. The aim of the current work is also to demonstrate that the analysis of weekend air quality monitoring data records is an appropriate method for estimating the temporal variation of air pollution in urban agglomerations, based on appropriate approximations.     

Contaminant mass discharge estimation in groundwater based on multi-level point measurements: a numerical evaluation of expected errors

Different methods for the field-scale estimation of contaminant mass discharge in groundwater at control planes based on multi-level well data are numerically analysed for the expected estimation error. We consider "direct" methods based on time-integrated measuring of mass flux, as well as "indirect" methods, where estimates are derived from concentration measurements. The appropriateness of the methods is evaluated by means of modelled data provided by simulation of mass transport in a three-dimensional model domain. Uncertain heterogeneous aquifer conditions are addressed by means of Monte-Carlo simulations with aquifer conductivity as a random space function. We investigate extensively the role of the interplay between the spatial resolution of the sampling grid and aquifer heterogeneity with respect to the accuracy of the mass discharge estimation. It is shown that estimation errors can be reduced only if spatial sampling intervals are in due proportion to spatial correlation length scales. The ranking of the methods with regard to estimation error is shown to be heavily dependent on both the given sampling resolution and prevailing aquifer heterogeneity. Regarding the "indirect" estimation methods, we demonstrate the great importance of a consistent averaging of the parameters used for the discharge estimation.     

Evaluating the sensitivity of a subsurface multicomponent reactive transport model with respect to transport and reaction parameters.

The input variables for a numerical model of reactive solute transport in groundwater include both transport parameters, such as hydraulic conductivity and infiltration, and reaction parameters that describe the important chemical and biological processes in the system. These parameters are subject to uncertainty due to measurement error and due to the spatial variability of properties in the subsurface environment. This paper compares the relative effects of uncertainty in the transport and reaction parameters on the results of a solute transport model. This question is addressed by comparing the magnitudes of the local sensitivity coefficients for transport and reaction parameters. General sensitivity equations are presented for transport parameters, reaction parameters, and the initial (background) concentrations in the problem domain. Parameter sensitivity coefficients are then calculated for an example problem in which uranium(VI) hydrolysis species are transported through a two-dimensional domain with a spatially variable pattern of surface complexation sites. In this example, the reaction model includes equilibrium speciation reactions and mass transfer-limited non-electrostatic surface complexation reactions. The set of parameters to which the model is most sensitive includes the initial concentration of one of the surface sites, the formation constant (Kf) of one of the surface complexes and the hydraulic conductivity within the reactive zone. For this example problem, the sensitivity analysis demonstrates that transport and reaction parameters are equally important in terms of how their variability affects the model results.     

The effect of temperature on the gas–particle partitioning of reactive mercury in atmospheric aerosols

Measurements of gas–particle-partitioning coefficients for reactive mercury in dry urban and laboratory aerosol were found to strongly depend on ambient temperature. Samples of atmospheric and laboratory aerosols (defined as both the gas and particle phases) were collected using filter and absorbent methods and analyzed for reactive mercury using thermal desorption combined with cold vapor atomic fluorescence spectroscopy. Synthetic ambient aerosols were generated in the laboratory from ammonium sulfate and adipic acid mixed with mercuric chloride in a purpose-built aerosol reactor. The aerosol reactor was operated in a temperature-controlled laboratory. Linear relationships between the logarithm of inverse gas–particle partitioning and inverse temperature were observed and parameterized for use in the atmospheric modeling of reactive mercury. Reactive mercury was observed to partition from the particle to the gas phase as ambient temperature increased. Good agreement between measurements made using urban and laboratory aerosols was seen after gas–particle-partitioning coefficients were normalized for surface area instead of mass. Thermodynamic analyses of the urban and laboratory gas–particle-partitioning measurements revealed that the strength of interaction between reactive mercury and particle surfaces was suggestive of chemisorption. Gas–particle-partitioning coefficients made with the Tekran ambient mercury analyzer (AMA) also showed a dependence on temperature. However, the Tekran AMA partitioning coefficients did not agree well with partitioning coefficients measured using the filter-based methods. The disagreement is consistent with the 50 °C operational temperature of the Tekran AMA.     

Application of the method of temporal moments to interpret solute transport with sorption and degradation

In this note, we applied the temporal moment solutions of [Das and Kluitenberg, 1996. Soil Sci. Am. J. 60, 1724] for one-dimensional advective-dispersive solute transport with linear equilibrium sorption and first-order degradation for time pulse sources to analyse soil column experimental data. Unlike most other moment solutions, these solutions consider the interplay of degradation and sorption. This permits estimation of a first-order degradation rate constant using the zeroth moment of column breakthrough data, as well as estimation of the retardation factor or sorption distribution coefficient of a degrading solute using the first moment. The method of temporal moment (MOM) formulae was applied to analyse breakthrough data from a laboratory column study of atrazine, hexazinone and rhodamine WT transport in volcanic pumice sand, as well as experimental data from the literature. Transport and degradation parameters obtained using the MOM were compared to parameters obtained by fitting breakthrough data from an advective-dispersive transport model with equilibrium sorption and first-order degradation, using the nonlinear least-square curve-fitting program CXTFIT. The results derived from using the literature data were also compared with estimates reported in the literature using different equilibrium models. The good agreement suggests that the MOM could provide an additional useful means of parameter estimation for transport involving equilibrium sorption and first-order degradation. We found that the MOM fitted breakthrough curves with tailing better than curve fitting. However, the MOM analysis requires complete breakthrough curves and relatively frequent data collection to ensure the accuracy of the moments obtained from the breakthrough data.