The incorporation of stochasticity in risk analysis and management: a case study

 The selection of optimal management strategies for environmental contaminants requires detailed information on the risks imposed on populations. These risks are characterized by both inter-subject variability (different individuals having different levels of risk) and by uncertainty (there is uncertainty about the risk associated with the Yth percentile of the variability distribution). In addition, there is uncertainty introduced by the inability to agree fully on the appropriate decision criteria. This paper presents a methodology for incorporating uncertainty and variability into a multi-medium, multi-pathway, multi-contaminant risk assessment, and for placing this assessment into an optimization framework to identify optimal management strategies. The framework is applied to a case study of a sludge management system proposed for North Carolina and the impact of stochasticity on selection of an optimal strategy considered. Different sets of decision criteria reflecting different ways of treating stochasticity are shown to lead to different selections of optimal management strategies.     

Probabilistic risk and uncertainty analysis for bioremediation of four chlorinated ethenes in groundwater

Groundwater contamination risk assessment for health-threatening compounds should benefit from a stochastic environmental risk assessment which considers the effects of biological, chemical, human behavioral, and physiological processes that involve elements of biotic and abiotic aquifer uncertainty, and human population variability. This paper couples a complex model of chemical degradation and transformation with movement in an aquifer undergoing bioremediation to generate a health risk analysis for different population cohorts in the community. A two-stage Monte Carlo simulation has separate stages for population variability and aquifer uncertainty yielding a computationally efficient and conceptually attractive algorithm. A hypothetical example illustrates how risk variance analysis can be conducted to determine the distribution of risk, and the relative impact of uncertainty and variability in different sets of parameters upon the variation of risk values for adults, adolescents, and children. The groundwater example considers a community water supply contaminated with chlorinated ethenes. Biodegradation pathways are enhanced by addition of butyrate. The results showed that the contribution of uncertainty to the risk variance is comparable to that of variability. Among the uncertain parameters considered, transmissivity accounted for the major part of the output variance. Children were the most susceptible and vulnerable population cohort.     

2D Monte Carlo versus 2D Fuzzy Monte Carlo health risk assessment

Risk estimates can be calculated using crisp estimates of the exposure variables (i.e., contaminant concentration, contact rate, exposure frequency and duration, body weight, and averaging time). However, aggregate and cumulative exposure studies require a better understanding of exposure variables and uncertainty and variability associated with them. Probabilistic risk assessment (PRA) studies use probability distributions for one or more variables of the risk equation in order to quantitatively characterize variability and uncertainty. Two-dimensional Monte Carlo Analysis (2D MCA) is one of the advanced modeling approaches that may be used to conduct PRA studies. In this analysis the variables of the risk equation along with the parameters of these variables (for example mean and standard deviation for a normal distribution) are described in terms of probability density functions (PDFs). A variable described in this way is called a second order random variable. Significant data or considerable insight to uncertainty associated with these variables is necessary to develop the appropriate PDFs for these random parameters. Typically, available data and accuracy and reliability of such data are not sufficient for conducting a reliable 2D MCA. Thus, other theories and computational methods that propagate uncertainty and variability in exposure and health risk assessment are needed. One such theory is possibility analysis based on fuzzy set theory, which allows the utilization of incomplete information (incomplete information includes vague and imprecise information that is not sufficient to generate probability distributions for the parameters of the random variables of the risk equation) together with expert judgment. In this paper, as an alternative to 2D MCA, we are proposing a 2D Fuzzy Monte Carlo Analysis (2D FMCA) to overcome this difficulty. In this approach, instead of describing the parameters of PDFs used in defining the variables of the risk equation as random variables, we describe them as fuzzy numbers. This approach introduces new concepts and risk characterization methods. In this paper we provide a comparison of these two approaches relative to their computational requirements, data requirements and availability. For a hypothetical case, we also provide a comperative interpretation of the results generated.     

Influential input classification in probabilistic multimedia models

 Monte Carlo analysis is a statistical simulation method that is often used to assess and quantify the outcome variance in complex environmental fate and effects models. Total outcome variance of these models is a function of (1) the variance (uncertainty and/or variability) associated with each model input and (2) the sensitivity of the model outcome to changes in the inputs. To propagate variance through a model using Monte Carlo techniques, each variable must be assigned a probability distribution. The validity of these distributions directly influences the accuracy and reliability of the model outcome. To efficiently allocate resources for constructing distributions one should first identify the most influential set of variables in the model. Although existing sensitivity and uncertainty analysis methods can provide a relative ranking of the importance of model inputs, they fail to identify the minimum set of stochastic inputs necessary to sufficiently characterize the outcome variance. In this paper, we describe and demonstrate a novel sensitivity/uncertainty analysis method for assessing the importance of each variable in a multimedia environmental fate model. Our analyses show that for a given scenario, a relatively small number of input variables influence the central tendency of the model and an even smaller set determines the spread of the outcome distribution. For each input, the level of influence depends on the scenario under consideration. This information is useful for developing site specific models and improving our understanding of the processes that have the greatest influence on the variance in outcomes from multimedia models.     

Sensitivity analysis of a no-crossflow model for the transient flowmeter test

 Logarithmic sensitivities and plausible relative errors are studied in a simple no-crossflow model of a transient flowmeter test (TFMT). This model is identical to the model of a constant-rate pumping test conducted on a fully penetrating well with wellbore storage, surrounded by a thick skin zone, and situated in a homogeneous confined aquifer. The sensitivities of wellbore drawdown and wellface flowrate to aquifer and skin parameters are independent of the pumping rate. However, the plausible relative errors in the aquifer and skin parameters estimated from drawdown and wellface flowrate data can be proportionally decreased by increasing the pumping rate. The plausible relative errors vary by many orders of magnitude from the beginning of the TFMT. The practically important flowrate and drawdown measurements in this test, for which the plausible relative errors vary by less than one order of magnitude from the minimum plausible relative errors, can begin approximately when the dimensionless wellface flowrate exceeds q _D =q/Q≈0.4. During most of this stage of the test, the plausible relative errors in aquifer hydraulic conductivity (K _a ) are generally an order of magnitude smaller than those in aquifer specific storativity. The plausible relative errors in the skin hydraulic conductivity (K _s ) are generally larger than the plausible relative errors in the aquifer specific storativity when the thick skin is normal (K _s >K _a ) and smaller when the thick skin is damaged (K _s <K _a ). The specific storativity of the skin zone would be so biased that one should not even attempt to estimate it from the TFMT. We acknowledge Wiebe H. van der Molen for recommending the De Hoog algorithm and sharing his code. This research was partially supported by the US Geological Survey, USGS Agreement #1434-HQ-96-GR-02689 and North Carolina Water Resources Research Institute, WRRI Project #70165.     

Sampling-based flood risk analysis for fluvial dike systems

A dike system of moderate size has a large number of potential system states, and the loading imposed on the system is inherently random. If the system should fail, in one of its many potential failure modes, the topography of UK floodplains is usually such that hydrodynamic modelling of flood inundation is required to generate realistic estimates of flood depth and hence damage. To do so for all possible failure states may require 1,000s of computationally expensive inundation simulations. A risk-based sampling technique is proposed in order to reduce the computational resources required to estimate flood risk. The approach is novel in that the loading and dike system states (obtained using a simplified reliability analysis) are sampled according to the contribution that a given region of the space of basic variables makes to risk. The methodology is demonstrated in a strategic flood risk assessment for the city of Burton-upon-Trent in the UK. 5,000 inundation model simulations were run although it was shown that the flood risk estimate converged adequately after approximately half this number. The case study demonstrates that, amongst other factors, risk is a complex function of loadings, dike resistance, floodplain topography and the spatial distribution of floodplain assets. The application of this approach allows flood risk managers to obtain an improved understanding of the flooding system, its vulnerabilities and the most efficient means of allocating resource to improve performance. It may also be used to test how the system may respond to future external perturbations.     

Can global sensitivity analysis steer the implementation of models for environmental assessments and decision-making?

 We illustrate a method of global sensitivity analysis and we test it on a preliminary case study in the field of environmental assessment to quantify uncertainty importance in poorly-known model parameters and spatially referenced input data. The focus of the paper is to show how the methodology provides guidance to improve the quality of environmental assessment practices and decision support systems employed in environmental policy. Global sensitivity analysis, coupled with uncertainty analysis, is a tool to assess the robustness of decisions, to understand whether the current state of knowledge on input data and parametric uncertainties is sufficient to enable a decision to be taken. The methodology is applied to a preliminary case study, which is based on a numerical model that employs GIS-based soil data and expert consultation to evaluate an index that joins environmental and economic aspects of land depletion. The index is used as a yardstick by decision-makers involved in the planning of highways to identify the route that minimises the overall impact.     

Modeling of transport in groundwater for environmental risk assessment

This paper presents the principles underlying a recently developed numerical technique for modeling transport in heterogeneous porous media. The method is then applied to derive the concentration mean and variance, the concentration CDF, exceedance probabilities and exposure time CDF, which are required by various regulatory agencies for risk and performance assessment calculations. The dependence of the various statistics on elapsed travel time, location in space, the dimension of the detection volume, natural variability and pore-scale dispersion is investigated and discussed.     

Robustness of spatial ranges of environmental chemicals to changes in model dimension

 A comparison is made between a circular and a more adequate spherical reaction-diffusion multi-media mass balance model. This comparison adds new aspects to ongoing debates about more effective assessments of potentially harmful substances including persistent organic pollutants (POPs). The circular model serves as a paradigm in investigations of persistence and spatial range of non-polar chemicals. An analytic solution of the spherical model is presented. It is utilized in order to establish circular spatial ranges as versatile approximations of their spherical counterparts for most cases. Deviations in the few exceptions are demonstrated as playing a minor role compared to sensitivities against parameter uncertainties which characterize these exceptional cases as well. The sensitivities are fundamentally linked to the multi-scale nature of the underlying system. Finally, the insensitivity of spatial ranges with respect to dimension – circle versus sphere – is further secured by extensive studies of the role of the mode of entry for which a set of rules is established. Present address: Potsdam Institute for Climate Impact Research P.O. Box 60 12 03, D-14412 Potsdam, Germany e-mail: held@pik-potsdam.de The author would like to thank B. H. Hawkins, H. A. Schweers, and M. Str?be for helpful comments.     

Toxicity testing,risk assessment,and options for dredged material management

Programs for evaluating proposed discharges of dredged material into waters of the United States specify a tiered testing and evaluation protocol that includes performance of acute and chronic bioassays to assess toxicity of the dredged sediments. Although these evaluations reflect the toxicological risks associated with disposal activities to some degree, analysis activities are limited to the sediments of each dredging project separately. Cumulative risks to water column and benthic organisms at and near the designated disposal site are therefore difficult to assess. An alternate approach is to focus attention on the disposal site, with the goal of understanding more directly the risks of multiple disposal events to receiving ecosystems. Here we review current US toxicity testing and evaluation protocols, and describe an application of ecological risk assessment that allows consideration of the temporal and spatial components of risk to receiving aquatic ecosystems. When expanded to include other disposal options, this approach can provide the basis for holistic management of dredged material disposal.     

A risk assessment model for produced water discharge from offshore petroleum platforms-development and validation

Produced water discharge accounts for the greater portion of wastes arising from offshore oil and gas production operations. Development and expansion of Canada’s offshore oil and gas reserves has led to concerns over the potential long-term impacts of produced water discharges to the ocean. To examine this emerging environmental issue at a regional scale, an integrated risk assessment approach was developed in this study based on the princeton ocean model (POM), a random walk (RW) and Monte Carlo simulation. The use of water quality standards arrayed in a Monte Carlo design in the developed approach has served to reflect uncertainties and quantify environmental risks associated with produced water discharge. The model was validated against field data from a platform operating off Canada’s east coast, demonstrating its usefulness in supporting effective management of future produced water discharge.     

The A.D. 79 eruption as a future explosive scenario in the Vesuvian area: evaluation of associated risk

 Due to the lack of an effective policy of planning and prevention, over the past decades the area around Mt. Vesuvio has undergone a steady increase in population and uncontrolled housing development. Consequently, it has become one of the most hazardous volcanic areas in the world. In order to mitigate the damage that the impact of an explosive event would cause in the area, the Department of Civil Defense has worked out an Emergency Management Plan using the A.D. 1631 subplinian eruption as the most probable short-term event. However, from 25 000 years B.P. to present, the activity of the Somma-Vesuvio volcano has shown a sequence of eight eruptive cycles, which always began with a strong plinian eruption. In this paper we utilize the A.D. 79 eruption as an example of a potential large explosive eruption that might occur again at Vesuvio. A detailed tephrostratigraphic analysis of the eruption products was processed by a multivariate statistical analysis. This analysis proved useful for identifying marker layers in the sequences, thus allowing the recognition of some major phases of synchronous deposition and hence the definition of the chronological and spatial evolution of the eruption. By combining this reconstruction with land-use maps, a scenario is proposed with time intervals in the eruptive sequence similar to those reported in Pliny's letter. Thus, it was calculated that, after 7 h from the start of the eruption, a total area of approximately 300 km² would be covered with the eruption products. In the following 11 h, a total area of approximately 500 km² would be involved. The third and last phase of deposition would not cause significant variation in the total area involved, but it would bring about an increase in the thickness of the pyroclastic deposits in the perivolcanic area. Received: 30 November 1996 / Accepted: 29 May 1997     

Dispersion modelling of air pollution caused by road traffic using a Markov Chain–Monte Carlo model

 Although the strict legislation regarding vehicle emissions in Europe (EURO 4, EURO 5) will lead to a remarkable reduction of emissions in the near future, traffic related air pollution still can be problematic due to a large increase of traffic in certain areas. Many dispersion models for line-sources have been developed to assess the impact of traffic on the air pollution levels near roads, which are in most cases based on the Gaussian equation. Previous studies gave evidence, that such kind of models tend to overestimate concentrations in low wind speed conditions or when the wind direction is almost parallel to the street orientation. This is of particular interest, since such conditions lead generally to the highest observed concentrations in the vicinity of streets. As many air quality directives impose limits on high percentiles of concentrations, it is important to have good estimates of these quantities in environmental assessment studies. The objective of this study is to evaluate a methodology for the computation of especially those high percentiles required by e.g. the EU daughter directive 99/30/EC (for instance the 99.8 percentile for NO₂). The model used in this investigation is a Markov Chain – Monte Carlo model to predict pollutant concentrations, which performs well in low wind conditions as is shown here. While usual Lagrangian models use deterministic time steps for the calculation of the turbulent velocities, the model presented here, uses random time steps from a Monte Carlo simulation and a Markov Chain simulation for the sequence of the turbulent velocities. This results in a physically better approach when modelling the dispersion in low wind speed conditions. When Lagrangian dispersion models are used for regulatory purposes, a meteorological pre-processor is necessary to obtain required input quantities like Monin-Obukhov length and friction velocity from routinely observed data. The model and the meteorological pre-processor applied here, were tested against field data taken near a major motorway south of Vienna. The methodology used is based on input parameters, which are also available in usual environmental assessment studies. Results reveal that the approach examined is useful and leads to reasonable concentration levels near motorways compared to observations. We wish to thank Andreas Schopper (Styrian Government) for providing air quality values, M. Kalina for providing the raw data of the air quality stations near the motorway and J. Kukkonen for providing the road site data set from the Finish Meteorological Institute (FMI). The study was partly funded by the Austrian science fund under the project P14075-TEC.     

Risk tolerance measure for decision-making in fuzzy analysis: a health risk assessment perspective

In risk assessment studies it is important to determine how uncertain and imprecise knowledge should be included into the simulation and assessment models. Thus, proper evaluation of uncertainties has become a major concern in environmental and health risk assessment studies. Previously, researchers have used probability theory, more commonly Monte Carlo analysis, to incorporate uncertainty analysis in health risk assessment studies. However, in conducting probabilistic health risk assessment, risk analyst often suffers from lack of data or the presence of imperfect or incomplete knowledge about the process modeled and also the process parameters. Fuzzy set theory is a tool that has been used in propagating imperfect and incomplete information in health risk assessment studies. Such analysis result in fuzzy risks which are associated with membership functions. Since possibilistic health risk assessment studies are relatively new, standard procedures for decision-making about the acceptability of the resulting fuzzy risk with respect to a crisp standard set by the regulatory agency are not fully established. In this paper, we are providing a review of several available approaches which may be used in decision-making. These approaches involve defuzzification techniques, the possibility and the necessity measures. In this study, we also propose a new measure, the risk tolerance measure, which can be used in decision making. The risk tolerance measure provides an effective metric for evaluating the acceptability of a fuzzy risk with respect to a crisp compliance criterion. Fuzzy risks with different membership functions are evaluated with respect to a crisp compliance criterion by using the possibility, the necessity, and the risk tolerance measures and the results are discussed comparatively.     

Convergence of realization-based statistics to model-based statistics for the LU unconditional simulation algorithm: some numerical tests

 In geostatistics, stochastic simulation is often used either as an improved interpolation algorithm or as a measure of the spatial uncertainty. Hence, it is crucial to assess how fast realization-based statistics converge towards model-based statistics (i.e. histogram, variogram) since in theory such a match is guaranteed only on average over a number of realizations. This can be strongly affected by the random number generator being used. Moreover, the usual assumption of independence among simulated realizations of a random process may be affected by the random number generator used. Simulation results, obtained by using three different random number generators implemented in Geostatistical Software Library (GSLib), are compared. Some practical aspects are pointed out and some suggestions are given to users of the unconditional LU simulation method.     

On the choice of the optimal temporal control in renewable resource management

 This paper addresses the following hitherto unstudied question in renewable resource management: How should a resource manager set the temporal control optimally for renewable resources such as rangelands and fisheries that are managed with spatial and temporal controls? We use a dynamic and stochastic framework to first derive the resource manager's long run average net cost function. We then demonstrate how the temporal control can be chosen to minimize this objective function. We thank George Christakos and two anonymous referees for their comments on a previous version of this paper. Batabyal acknowledges financial support from the Gosnell endowment at RIT. The usual disclaimer applies.     

Assessment of the prediction error in a large-scale application of a dynamic soil acidification model

 The prediction error of a relatively simple soil acidification model (SMART2) was assessed before and after calibration, focussing on the Al and NO₃ concentrations on a block scale. Although SMART2 is especially developed for application on a national to European scale, it still runs at a point support. A 5×5 km² grid was used for application on the European scale. Block characteristic values were obtained simply by taking the median value of the point support values within the corresponding grid cell. In order to increase confidence in model predictions on large spatial scales, the model was calibrated and validated for the Netherlands, using a resolution that is feasible for Europe as a whole. Because observations are available only at the point support, it was necessary to transfer them to the block support of the model results. For this purpose, about 250 point observations of soil solution concentrations in forest soils were upscaled to a 5×5 km² grid map, using multiple linear regression analysis combined with block kriging. The resulting map with upscaled observations was used for both validation and calibration. A comparison of the map with model predictions using nominal parameter values and the map with the upscaled observations showed that the model overestimated the predicted Al and NO₃ concentrations. The nominal model results were still in the 95% confidence interval of the upscaled observations, but calibration improved the model predictions and strongly reduced the model error. However, the model error after calibration remains rather large.     

An annual wet sulfate deposition index

 Averages of annual wet deposition data are often used as an indicator of acid amounts in the atmosphere. From the view point of statistics, the average is a meaningful estimator only for identically distributed data with specific types of probability distribution. Wet deposition data usually carry seasonal trends. To learn about year-to-year concentration changes, a more accurate summary information accommodating the trend is thus necessary. This paper suggests a quantity (index) describing the annual wet deposition of SO₄ and formulas for its calculation. The formulas are in some sense optimal in case the data can be considered independent, lognormally distributed and have a time dependent trend. Generalization of these assumptions is also discussed. The index is derived from specific features of the average applied to data with lognormal distribution. The theory is utilized to analyze a set of SO₄ precipitation concentrations observed in the Ontario region. A work sponsored by the Atmospheric Environment Service in Toronto and the Canadian NSERC. I want to thank the Ontario Ministry of the Environment for provision of the data used in this study and the anonymous referee for a number of comments that resulted in improvement of the final presentation.     

Quantitative microbial risk assessment: uncertainty and measures of central tendency for skewed distributions

In the past, arithmetic and geometric means have both been used to characterise pathogen densities in samples used for microbial risk assessment models. The calculation of total (annual) risk is based on cumulative independent (daily) exposures and the use of an exponential dose–response model, such as that used for exposure to Giardia or Cryptosporidium. Mathematical analysis suggests that the arithmetic mean is the appropriate measure of central tendency for microbial concentration with respect to repeated samples of daily exposure in risk assessment. This is despite frequent characterisation of microbial density by the geometric mean, since the microbial distributions may be Log normal or skewed in nature. Mathematical derivation supporting the use of the arithmetic mean has been based on deterministic analysis, prior assumptions and definitions, the use of point-estimates of probability, and has not included from the outset the influence of an actual distribution for microbial densities. We address these issues by experiments using two real-world pathogen datasets, together with Monte Carlo simulation, and it is revealed that the arithmetic mean also holds in the case of a daily dose with a finite distribution in microbial density, even when the distribution is very highly-skewed, as often occurs in environmental samples. Further, for simplicity, in many risk assessment models, the daily infection risk is assumed to be the same for each day of the year and is represented by a single value, which is then used in the calculation of p _Σ, which is a numerical estimate of annual risk, P _Σ, and we highlight the fact that is simply a function of the geometric mean of the daily complementary risk probabilities (although it is sometimes approximated by the arithmetic mean of daily risk in the low dose case). Finally, the risk estimate is an imprecise probability with no indication of error and we investigate and clarify the distinction between risk and uncertainty assessment with respect to the predictive model used for total risk assessment.     

Geodetic methods for detecting volcanic unrest: a theoretical approach

In this paper we study the application of different geodetic techniques to volcanic activity monitoring, using theoretical analysis. This methodology is very useful for obtaining an idea of the most appropriate (and efficient) monitoring method, mainly when there are no records of geodetic changes previous to volcanic activity. The analysis takes into account the crustal structure of the area, its geology, and its known volcanic activity to estimate the deformation and gravity changes that might precede eruptions. The deformation model used includes the existing gravity field and vertical changes in the crustal properties. Both factors can have a considerable effect on computed deformation and gravity changes. Topography should be considered when there is a steep slope (greater than 10°). The case study of Teide stratovolcano (Tenerife, Canary Islands), for which no deformation or gravity changes are available, is used as a test. To avoid considering topography, we worked at the lowest level of Las Cañadas and examined a smaller area than the whole caldera or island. Therefore, the results are only a first approach to the most adequate geodetic monitoring system. The methodology can also be applied to active areas where volcanic risk is not associated with a stratovolcano but instead with monogenetic scattered centers, especially when sites must be chosen in terms of detection efficiency or existing facilities. The Canary Islands provide a good example of this type of active volcanic areas, and we apply our model to the island of Lanzarote to evaluate the efficiency of the monitoring system installed at the existing geodynamic station. On this island topography is not important. The results of our study show clearly that the most appropriate geodetic volcano monitoring system is not the same for all different volcanic zones and types, and the particular properties of each volcano/zone must be taken into account when designing each system.