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UNCERTAINTY ANALYSIS OF TRANSPORT-TRANSFORMATION MODELSBY SASTRY S. ISUKAPALLI A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Chemical and Biochemical Engineering Written under the direction of Dr. Panos G. Georgopoulos and approved by New Brunswick, New Jersey January, 1999 c 1999 Sastry S. Isukapalli ALL RIGHTS RESERVED ABSTRACT OF THE DISSERTATION Uncertainty Analysis of Transport-Transformation Models by Sastry S. Isukapalli Dissertation Director: Dr. Panos G. Georgopoulos Characterization of uncertainty associated with transport-transformation models is often of critical importance, as for example in cases where environmental and biological models are employed in risk assessment. However, uncertainty analysis using conventional methods such as standard Monte Carlo or Latin Hypercube Sampling may not be efficient, or even feasible, for complex, computationally demanding models. This work introduces a computationally efficient alternative method for uncertainty propagation, the Stochastic Response Surface Method (SRSM). The SRSM approximates uncertainties in model outputs through a series expansion in normal random variables (polynomial chaos expansion). The unknown coefficients in series expansions are calculated using a limited number of model simulations. This method is analogous to approximation of a deterministic system by an algebraic response surface. Further improvements in the computational efficiency of the SRSM are accomplished by coupling the SRSM with ADIFOR, which facilitates automatic calculation of partial derivatives in numerical models coded in Fortran. The coupled method, SRSMADIFOR, uses the model outputs and their derivatives to calculate the unknown coefficients. The SRSM and the SRSM-ADIFOR are general methods, and are applicable to any model with random inputs. The SRSM has also been implemented as a black-box, ii web-based tool for facilitating its easy use. The SRSM and the SRSM-ADIFOR have been applied to a set of environmental and biological models. In all the case studies, the SRSM required an order of magnitude fewer simulations compared to conventional methods, and the SRSM-ADIFOR required even fewer simulations. In addition to their computational efficiency, these methods directly provide sensitivity information and individual contributions of input uncertainties to output uncertainties; conventional methods require substantially larger numbers of simulations to provide such information. Thus, the SRSM and the SRSM-ADIFOR provide computationally efficient means for uncertainty and sensitivity analysis. Finally, this research addresses uncertainties associated with model structure and resolution with application to photochemical air quality modeling. A three dimensional version of the regulatory Reactive Plume Model (RPM), RPM-3D, has been developed and applied to understand model uncertainty. iii This work has benefited from various researchers. I am also grateful to the members of my committee for taking the time to guide me through my dissertation. I would also like to acknowledge the following researchers and organizations who have provided models and input files for the evaluation of the methods developed here: Dr. Arlene Bicknell. Shengwei Wang for sharing their documentation on the UAM-IV and CB-IV (adapted here as Appendix C). Amit Roy for his continued support. I thank the countless authors of all the free software that I have used during my research work – their selfless efforts have significantly aided iv . Menner Tatang. and frank. Daewon Byun for his continued encouragement and support. and to Professor Henrik Pedersen. Shengwei Wang and Dr. and Professor David Kosson for their valuable comments and ideas. Julie Greenman for their continued help. and for providing me with excellent facilities to pursue my work. I am also grateful to Dr. valuable guidance. I am also thankful to Ms. Saravanan Arunachalam) for the input data for the UAM model.Acknowledgements I would like to express my sincere thanks and appreciation to my advisor. Special mention goes to Dr. Panos G. for guidance. Yee Chiew for introducing me to the concepts of Monte Carlo simulations. and Hydrogeologic Inc. for the input data for the EPACMTP model. Roberta Salinger. and for ensuring financial support throughout my studies. I am also grateful to Dr. I am also thankful to Dr. Ashwin Walia and Dr. Amit Roy for the PERC PBPK model and input data for the RPM model. the ozone modeling group (Dr. and Ms. I am grateful to my colleagues at the CCL for their valuable discussions and suggestions. and for his insightful remarks on this thesis. Georgopoulos. I also thank Dr. whose work served as the starting point for this thesis. critical comments throughout my research. Ms. I would like express my appreciation to Professor Alkis Constantinides for his insightful remarks. and for being there for me at all times. I would also like to acknowledge financial support provided by the U. and has contributed tremendously to this thesis. v .my work.S. and Ramana Isukapalli. as a colleague. and U. he has helped me continuously. NERL (National Exposure Research Laboratories). As a friend. I consider it as my fortune to be his friend. and as a mentor.S. for providing a constant source of encouragement and support. EPA. Special thanks go to my friends. and to have worked with him. DOE through CRESP (Consortium for Risk Evaluation with Stakeholder Participation). Ravi Bhupathiraju. This thesis would not have materialized but for the help of Amit Roy. Padma Isukapalli. Dedication to my parents and to Amit Roy vi . . . . . . . . . . . . . . . . . . vii . . . . . . . 1. . . . . . . . Dedication . . . . . Outline of the Thesis . . . . . . . . . . . . . . .1. . . . . List of Tables . . .3. Fuzzy Theory . . . 2. . . . .1. . . . Reducible and Irreducible Uncertainty . . . . . . . . . . . . . .1. . . . . . . . . . .1. . . . . . . . . . . . . . . . . . . . . . . Limitations in Performing Uncertainty Analysis . . . .3. . . . . . . . . . 2. . . . Model Uncertainty . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . xv 1 1 2 4 4 6 10 10 10 11 13 14 15 16 18 20 23 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probabilistic Analysis . . . . . . . . . . . . . . . . . . . . . . . .2. . . .Table of Contents Abstract . . . . . . . . 1. . . . . . . . . . Natural Uncertainty and Variability . . ii iv vi Acknowledgements . . . . . . . . Interval Mathematics . . . . . . . . Parametric/Data Uncertainty . . . . .3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv List of Figures . . . . . Sensitivity and Sensitivity/Uncertainty Analysis . . . . .1. . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . Uncertainty Analysis . . . . . . . . . . 2. . . . . . . . . . . Transport-Transformation Models . . . . . . . . . . . . .1. . . . . . .4. . . .2. .3. . Types and Origins of Uncertainty in Transport-Transformation Models . . . . . . . . . . . . . . . . . . . BACKGROUND . . . .5. . . . . . . . . . . . . . . . . 2. . . . . . 1. . . . . . . . . . . . . . .4. . . 2. .3. . . Approaches for Representation of Uncertainty . . . INTRODUCTION . Objective of the Thesis . . .1. . . . . . . . 1. . . . . . . . . . . . . .2. . . . . . . . . . . . . . . .2. . . 1. . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3. 1. . . .3. . . . . 3. .5. . . 2. . . . . . . . . . . . . . . . . . . . . . . . 3.2. . . . . . 3. . . . .2.1. . . . . . . . . . . . . Direct Transformations . . . . . . . . . .4. . . . . Step II: Functional Approximation of Outputs . . . . . . . . . . 3. . .2. . . . . . . . . . THE STOCHASTIC RESPONSE SURFACE METHOD: DEVELOPMENT AND IMPLEMENTATION . . . . . Need for Alternative Sensitivity/Uncertainty Analysis Methods . . . . . .3. . . 3. . . . . . Analytical Methods . . . . . . . . 2. . . . . .1. . . . . . . .3. . . .5. . . . 3. . . Step III: Estimation of Parameters in the Functional Approximation . . . . . . Sampling Based Methods . . . . . . . . . . . . . Monte Carlo and Latin Hypercube Sampling Methods . . . . . . . . . . . . . . . .2. .1. . . . . . .6. . 2. . . . . . . . . . Reliability Based Methods (FORM and SORM) . . . Transformation via Series Approximation . . . . . . . 3. . . . . . . . . .2. .4. . Limitations of DEMM/PCM . . . . . . Transformation of Empirical Distributions . Computer Algebra Based Methods . . . . . . . . . . Green’s Function Method . . . . . . . Response Surface Methods . . . . . . . . . . . . . 2. . . . . . Coupled/Decoupled Direct Method . . 3. . . . . 3. .5. . . . . . . . . . . . . . . 3. . . . . . . Deterministic Equivalent Modeling Method/Probabilistic Collocation Method (DEMM/PCM) . . . . . Fourier Amplitude Sensitivity Test (FAST) . . . . . Differential Analysis Methods . . . . . . . . . .5. . . . . . . Spectral Based Stochastic Finite Element Method . . . . . . . . . . Overview of the Method . . . . . . . . . . . . . . Sensitivity Testing Methods . . . . . . . . . . . . . .4. . . . . Specific Example: The Dirichlet Distribution . . Step I: Representation of Stochastic Inputs . . . . . . . . . . .4. . . . . . . . . . . . . . . . . . . . .3. . . . . . . . .2. . .5. . .2. . . . . . Conventional Sensitivity/Uncertainty Analysis Methods .1. . . . . . . . . . . . . .2. 2. . . . . . . . . . . . . . 24 25 26 26 26 26 27 27 28 29 30 32 33 35 36 36 38 40 41 41 42 44 44 46 47 48 viii . . . . . . . . . . . . . . . . . . . Transformation of Correlated Inputs . . . . . . . . . . . . . . . . . . Automatic Differentiation using ADIFOR . . . . . . . . . . . . 5.3. . . . . . . . . . 4. . . . . . . . . . . . . . . Applicability and Limitations of ECM . Specification of Parameter Uncertainty . . . 4. . Step IV: Estimation of the Statistics of the Output Metrics . . . . . . . . . . . . . . . . . .2. . .1.4. 53 56 3. . . . . . . . . . An Illustration of the Application of the SRSM-ADIFOR . . . . . . . . . . . . . . . . Regression Based SRSM . . . . 5. . 4. . . . . 4. Implementation of the PERC PBPK Model . COUPLING OF THE SRSM WITH SENSITIVITY ANALYSIS METHODS: DEVELOPMENT AND IMPLEMENTATION OF SRSM-ADIFOR . . . . . . . . CASE STUDIES FOR THE EVALUATION OF THE SRSM AND THE SRSM-ADIFOR . . Accuracy and Efficiency of Approximation 53 3. . . . . . . . . Implementation of a Web Interface to the SRSM . . Uncertainty Analysis of a Physiologically Based PharmacoKinetic Model for Perchloroethylene . . . . . .3.1. . .2. . . . . . . . .6. Coupling of SRSM and ADIFOR . . . . . . . . . . Implementation of the SRSM-ADIFOR . . . . . . . . Efficient Collocation Method (ECM) . . . . . . . . . . . . . 5. . . .2. . . . . . . . . . . . . .8. . . .8. . .1. . . . . . . .9. . . . . . . . . Description of the Case Study . . . . . . . . . . Case Study I: A Zero Dimensional Physiological System . . . . . . . . . . . . . . . . . . . . Computational Implementation . . . . Step V: Evaluation of Convergence. . . . .1. . . . 3. . . .4. . . . . Selection of Sample Points . . . . . . .3. . . . . Methods for Estimation of Partial Derivatives . . . . . .8.3. . . 3. . . . . . . Implementation of the Conventional Uncertainty Propagation Methods 56 3. . . . .4. .4.5. .1. . . . . . . . . 3. Implementation of the Stochastic Response Surface Method . . . . 5. .1. . . . . . . . 5. . . 3. . 49 50 50 51 3.3. . . .7. . . . .1. . . . . . . . . . . . . . . . . 4. . 77 77 79 82 83 76 77 64 66 69 70 72 73 74 59 60 ix . . . . . . . . . 4. . . . . . . . 5. . . . . . 4. . . . . . . . Results for the PBPK case study . . .1. . . .1. . . . . . . . . .2. . . .5. . . . . . . An Illustration of the Application of the SRSM . . . . . . . 104 5. . . . Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5. . . .4. .2. . . . . . . . . .4. . . . . . . . . . . . . . . . . EPACMTP Model . 5. . . . . .3. . . . . Implementation of the Monte Carlo Method in EPACMTP . . . .3. . . . . . . 5. . .1. . . . .3. . . .3. . . 5. . . . . . . . . . . . . . . . Uncertainties Associated with Biogenic Emission Estimates . . . . . . . . . . . . . . . . . . . . . . Case Study III: A Three-Dimensional Urban/Regional Scale Photochemical Air Quality Model . . . .4. . 112 5. . . . . . 102 Uncertainty Analysis of the Urban Airshed Model (UAM-IV) for the New Jersey/Philadelphia Domain . . . . . . . . . . . . . . . . . . .Stage I (Six Uncertain Parameters): Evaluation of DEMM/PCM and ECM . . . . . . . . . 102 5. . . . . . . . . . . . . . . . . Data Sources for the Application of EPACMTP . . Uncertainty Analysis of RPM-IV . . . . . . . . . .4. . . . . . . .2. . . . . 112 Uncertainty Analysis of Tritium Contamination in a Landfill using the EPACMTP Model . . . . . . . .3.2. . . . . .4. 5. . . . . . . . . Estimation of Uncertainties . . . 115 5.4.1. . . .3. . .5. . . . Uncertainty Analysis of the Reactive Plume Model (RPM-IV) . . . . . .4.1. . . . . . . . . . . . Case Study II: A Two-Dimensional Photochemical Air Quality Model . . . 5. . . . . . . Stage II (Eleven Uncertain Parameters): Evaluation of ECM.3. . . . . . . . . . . . . . . . . . Regression Based SRSM and LHS .4. . . . . . Uncertainty Analysis . .2. . 114 5. . . . . . 116 88 92 94 94 95 96 97 83 x . . . . . . . . . . . Results of the Case Study . . . . Uncertainty Analysis . . . . . . Description of RPM-IV . . . . . . . 113 5.2. . . . .2. . . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . Case Study IV: A Ground Water Model with Discontinuous Probability Distributions . . . . . . . . . . . . . . . . . . . . . 108 5. . . . . . . . . . . . . 105 5. . . . . . . . . . . . . . . . . . . . . Results and Discussion . . 115 5. . . . . . . . . . . Evaluation of SRSM-ADIFOR . . . . .3. 3. . . . . . . . . . . .2. . . . . . . 122 6. . . Random Variables . . . . . . . . . . . . . . . . . . . . 142 7. . . Mathematical and Numerical Formulation of PAQSMs . . Development and Application of the SRSM-ADIFOR . . . . Directions for Future Work . . . 120 6. . . . . . . . . . . . . Case Studies . .2. . . . . . . . .4. . . Continuous Random Variables . . . . . . 142 7. . . . . . . . 139 7. . . . . . . . . . . . . . 147 Appendix A. .4. .1. . . 127 6. . . . . . . . . .4. . . . . . . . . . . . . . . . . . Formulation of the RPM-3D . 145 7. . . .6. . . . 130 6. . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . Addressing uncertainty propagation under constraints . . . . . . . . . . . . .1. . . . . . 122 6. . . . . . . . . . . .3. . . . . . . . . . . . . . .1. . . . .2. . 143 7. . . . . . . . . . 143 7. PROBABILISTIC APPROACH FOR UNCERTAINTY ANALYSIS 165 A. . . . . . .4. . . . . . 125 6. 169 A. Discussion . . 142 7. Improvements to the Stochastic Response Surface Method . . . .4. . . . . . .1. Initial Physical Dimensions of the Plume . . . . . . . . . . . . . .3. . . . . . .3. . . . . . . .3. . . . . . . . . Moments of a Random Variable . 120 6. .1. . . . . . . . . . . . . . Photochemical Air Pollution Modeling .1. . . . . . . . . . . . . . . . .5. . 141 7. . .3.2. . . .2. . . . 145 References . . 171 xi . . . . . . . . . . . . CONCLUSIONS AND DISCUSSION . . Random processes and random fields . . . . . . Development and Application of the SRSM . . . CHARACTERIZATION AND REDUCTION OF MODEL UNCERTAINTY: AN ATMOSPHERIC PLUME MODEL STUDY . . . .4. . . . . . . . . . 129 6. . . . . . . . . 131 6. . . . Formulation of The Reactive Plume Model (RPM) . . Mode and Percentiles of a distribution . . . . . . . . . . . . . 166 A.3. . . Uncertainties Associated with Evaluation Data . . . . . RPM-IV Model Equations .1. . . . . . . . . . . . Introduction and Background . . 139 7.1.1. . . .3. . . Consideration of Uncertainties Beyond Parametric/Data Uncertainty . . . . . . . . . . . . . . Further evaluation of the SRSM and the SRSM-ADIFOR . 167 A. . . . . . . . . . . .4. . . . . .6. . . 138 7. . . . . . . . . . . . .4. . . . . . . . . . . . . Median. . . Limitations of the RPM-IV . .1. . . . . . . . . .5. 127 6. . . . . . . . . . . THE EPACMTP MODEL . . . . . . . . . . 192 D. .2. . . . . . . . . . . . . . . . . . . . Moments of Jointly Distributed Random Variables . . . . . 176 C. . . . . . . . . .1. . . . . . . . . . . . . . . . . .3. . . . . . Formulation of the Saturated Zone Contaminant Migration Module 195 Solution Method for the Transport Module . . . . . . . . . . . . .1. . . . . . . . . . . . . . . Applications of the UAM . . . . . . 172 A. . . . . URBAN AIRSHED MODEL (UAM-IV) AND CARBON BOND MECHANISM (CB-IV) . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . . Treatment of Physical/Chemical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1. . . . . . . . BASIC PBPK MODEL EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . 197 xii . . . . . . . 180 C. . . . . . . . . . Jointly Distributed Random Variables . 194 D. . . . . . . . . . .1. .2. . . . . . . 178 Advective pollutant transport . . . . . Description of the FECTUZ Module . . . PROGRAMS INCLUDED IN THE CDROM . . 176 C. . . . 189 D. . . .2. . . . . . . . . . . . . 188 D. .2. . . The Urban Airshed Model – UAM-IV . . . . . . . . . . . . . . . . . . . 173 A. . . . Dependence of Random Variables . . . . . . . . . . . . . . . 179 C. . . . . . . . . . 175 Appendix C. . . . 191 Solution Method for Solute Transport Module . . . . . . . . 179 Surface Removal Processes . . . . . . . . . . . . Introduction .3. . . . . . . . . .2. . . . . . .2. . . . .1. . .A. . .2. . . . . .3.1. . . . . . . . . . .2. Steady State Flow Module Formulation .2. . . . . . . 188 D. Description of the CANSAZ-3D Module . . . . 192 D. . . . . . . . Formulation of the Groundwater Flow Module . . . . . . . . 179 Atmospheric Chemistry . 181 Appendix D. . . . . . . . . . 174 Appendix B. . . . . .1. .2. . . . . . . . . . . . . 196 Appendix E. . . . .3. . . Solute Transport Module Formulation . . . . 178 Turbulent diffusion . . . Carbon Bond Mechanism (CB-IV) . . . 190 Solution Method for Flow Module . . . . . . . . . . . . . .1. . . . . . . . 190 D. . . Conceptual Overview . . . . . . . . . . . . 176 C. .2. . . . . . . . . . . . . . . . . . . .Vita . . . . . . . . . . . . . 200 xiii . . . . . . . . . . . . . . . . Programs required for the implementation of the SRSM . .2. . . . . . . . . 3 13 2. . . 80 40 22 24 33 5. . 199 E. . Selected probability density functions useful in representing uncertainties in environmental and biological model inputs . . . . . . . .3. Programs required for the implementation of the SRSM-ADIFOR . 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contribution of individual parameters to the overall uncertainty in the UAM-IV112 5. . . . A representative list of packages available for automatic differentiation . . . .4. . . . . . . . . . . . . . . . .1. . . . . 199 xiv .4. .3. . . . . . . . .List of Tables 1. . . . Chemical Species in the CBM-IV mechanism . . . . . . Representation of common univariate distributions as functionals of normal random variables . . . . . . . . . . . . Deterministic and uncertain parameters used in the uncertainty analysis of the PERC PBPK model . . . . . . . . . Deterministic parameters in the EPACMTP model . . . . . . Uncertain parameters in the EPACMTP model . . . . 185 E. . . . . . . Programs required for the web based SRSM tool . Sources of uncertainty in transport-transformation models . A summary of sensitivity measures employed in sensitivity analysis . .1. . . . . . . . . . . . . . . .2. . . 5. . . . . . Chemical Reactions in the Carbon Bond IV Mechanism . . . Transport-transformation components in biological and environmental models 2. 2. . 117 C.1. . . . .1. 3. . .2. . . . . . .4.1. . . 198 E. . 198 E. .2. 116 5. . . . . . . . . 184 C. . . . . . . . . . . . . . .1. . Other Programs and Input Files . . . .3. . . 3. . . . . . Application of the web based SRSM tool: recommended sample points and the corresponding model outputs . . . . . .1. . 4. . . . . . . . . . . . . . . . .1. . . . Outline of the Thesis . . . .1.1. . . . . Evaluation of DEMM/PCM: uncertainty in CML (6 uncertain inputs) . 62 3.5. . .4. . . .3. A schematic depiction of the propagation of uncertainties in transporttransformation models . .1. . . . . . . . 5.5. 3. . . .3. . . . . . . . . . . .2. .3. . . . . . . Schematic illustration of the Automatic Differentiation method . . . . . . . Evaluation of DEMM/PCM: uncertainty in AUCA and AUCL (6 uncertain inputs) . . . . . . . Rationale for coupling of the SRSM with a sensitivity analysis method . . . . . . . . . . Application of the web based SRSM tool: specification of input uncertainties 61 3. . . . . . .2. . . . Schematic depiction of the Stochastic Response Surface Method . . . . .2. . . . . . .An Example . . . . .List of Figures 1. . . . . Schematic depiction of the steps involved in the application of the SRSMADIFOR . . . . . . . . . . . . . . 5. . 2. FORTRAN derivative code generation using ADIFOR . . . . . . . . . . . . . 4. . . . . . . . . . . . 4. . .6. . . . . . 3. . . . Schematic depiction of the steps involved in the application of the Stochastic Response Surface Method . . Application of the web based SRSM tool: estimates of the pdf s of the outputs 63 4. . . . 5. . . . 84 86 87 75 79 83 65 69 71 xv . Schematic representation of a PBPK model for PERC . . . . . . . . . Types of uncertainty in transport-transformation modeling . . .4. . . . . . . Evaluation of ECM: uncertainty in CML (6 uncertain inputs) . . . . . . . Schematic depiction of the steps involved in the application of conventional Monte Carlo method . . . . . . . . . . . . . . . . .4. . 2. . . . . . . . Evaluation of ECM: uncertainty in AUCA and AUCL (6 uncertain inputs) . . . . 5. .2. . . . . . . . . . . . . . . 5. . . . . 9 11 21 39 56 59 3. . . . . 5.22. . . . . .9. . Probability distribution of the pervasiveness of the ozone episode . . . . . . . . Evaluation of ECM: uncertainty in AUCA and AUCL (11 uncertain inputs) . . . . . . .19. . . . 5. . The Philadelphia/New Jersey modeling domain used for uncertainty analysis 107 5. . 126 6. . . . . . . . . . . .20. . . . Evaluation of ECM: uncertainty in CML (11 uncertain inputs) . . Evaluation of SRSM and LHS: uncertainty in CML (11 uncertain inputs) . . . . . . . Evaluation of SRSM-ADIFOR: uncertainty in AUCA and AUCL .7. . 103 5. . . . . . . . . . . . . . . . . . . . . . . . Uncertainty in estimated maximum tritium concentration in a receptor well 5. . . .16. . . . . . . . . .3. . 118 6. 5. . Evaluation of SRSM: uncertainty in downwind ozone concentrations . . . . . . . . 88 89 90 91 92 93 98 5.18. . .13.11. . .10. .1. . . . .14. 108 5. .2. . . . . . 110 5. . . . Dependence of plume average O3 concentration on vertical resolution of the RPM-3D in a VOC dominant regime (4 horizontal cells) . Discretization of the plume cross section by RPM-IV and RPM-3D . 5. . Probability distribution of the daily average ozone concentration . Origins and types of uncertainty present in Photochemical Air Quality Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .15. . 133 xvi . Uncertainty in estimated time of occurrence of maximum tritium concentration in a receptor well . . . . Evaluation of SRSM and LHS: uncertainty in AUCA and AUCL (11 uncertain inputs) .21. .6. . . . .8. . . . . . Probability distribution of the daily maximum ozone concentration . . . Schematic depiction of the evolution of an atmospheric plume . . . . . Schematic depiction of the entrainment and detrainment steps in the RPM 126 118 6. . 110 5. . . . . . . . . . . Evaluation of ECM: uncertainty in downwind ozone concentrations . . . . Dependence of plume average O3 concentration on horizontal resolution of the RPM-IV in a VOC dominant regime . . 5. . . . Evaluation of SRSM-ADIFOR: uncertainty in downwind ozone concentrations 101 5. .4. 132 6. . Evaluation of SRSM-ADIFOR: uncertainty in CML . . . . . . . . . 130 6. . 5. . . . . . . 5. . . . Probability distribution of the daily maximum 8 hr running average ozone concentration . . . . . . . . . .12. . . . . . . . 109 5. 100 5. .17.5. . . . . . RPM-IV) . . . . . . . .1. . . . 136 6. . . 134 6. . . Dependence of plume average O3 concentration on vertical resolution of the RPM-3D in a NOx dominant regime . . . 137 6. .7. . .r. . . . . . . . . . . . . . . . . . 134 6. . . . . Vertical profile of the O3 concentration at the plume centerline (w. .8. . . . . .6. . . . . . RPM-3D) . . . . . . . Dependence of plume average O3 concentration on horizontal resolution of the RPM-IV in a NOx dominant regime . . . . . . . . horizontal) in a VOC limited (NOx dominant) regime (4 horizontal cells and 6 vertical cells. . 135 6. . . . .10. . . . . . . . . . . . . 137 C. . . Horizontal profile of the O3 concentration in the plume in a NOx dominant regime (6 horizontal cells.9. . . . . . . . .r. RPM-IV) . . . . horizontal) in a VOC dominant regime (4 horizontal cells and 6 vertical cells. . . . . . . . . .11. . .6. . . . . . . . . . . Horizontal profile of the O3 concentration in the plume in a VOC dominant regime (6 horizontal cells. . . . . . . .t. . . . . . . RPM-3D) . . .t. . . Major species in the CB-IV and their hierarchical relationship . 183 xvii . . Vertical profile of the O3 concentration at the plume centerline (w. . . . . . . . . for example. The purpose is not to somehow “reduce” uncertainty – reduction in uncertainty can only come from gathering additional information and filling “data gaps”. entails uncertainties in the estimates of toxicant emissions.1 Uncertainty Analysis Mechanistic modeling of physical systems is often complicated by the presence of uncertainties. The purpose of quantitative uncertainty analysis is to use currently available information in order for quantifying the degree of confidence in the existing data and models. with respect to the selection of a strategy for the control of pollutant levels.1 Chapter 1 INTRODUCTION 1. In fact. transformation and transport parameters. as well as the sources of uncertainty that are not important with respect to a given response. it can lead to the identification of the key sources of uncertainty (such as data gaps) which merit further research. Further. for example. The implications of these uncertainties are particularly important in the assessment of several potential regulatory options. etc. it is precisely for problems where data are limited and where simplifying assumptions . this could potentially result in providing useful information that can aid in decision making. that impact the estimates of related health risks. understanding the judgments associated with the modeling process is more valuable than side-stepping the uncertainty analysis. Environmental and biological modeling. A systematic uncertainty analysis provides insight into the level of confidence in model estimates. Even though the applicability of a model is limited by the model assumptions and the uncertainties in the evaluation data.. Even though significant effort may be needed to incorporate uncertainties into the modeling process. and can aid in assessing how various possible model estimates should be weighed. and on a mass-balance of chemical or radioactive species in a control volume.2 Transport-Transformation Models A wide range of environmental and biological models involve the fate and transport of a chemical species that originates from a source. often described by ordinary or partial differential equations. 1. they are categorized together because the underlying physical and chemical processes. The following stages are involved in the uncertainty analysis of a model: (a) estimation of uncertainties in model inputs and parameter (characterization of input uncertainties). These models are similar in structure. (c) characterization of uncertainties associated with different model structures and model formulations (characterization of model uncertainty). the term “transport-transformation models” refers to a wide range of biological. In this thesis. The term “transport” refers to the movement of material through the surroundings as a result of associated flows and diffusive processes. are examples of transport. and engineering problems. The flow of gaseous pollutants along with the wind. Even though the complexity of these models varies significantly. or specifically an organ. Additionally. and (d) characterization of the uncertainties in model predictions resulting from uncertainties in the evaluation data. or the absorption of pollutants by the layers of skin. the movement of toxic substances along with the blood flow. (b) estimation of the uncertainty in model outputs resulting from the uncertainty in model inputs and model parameters (uncertainty propagation). and the mathematical equations describing these systems. to help identify how robust the conclusions about model results are. and eventually comes in contact with a receptor (e. These models are based on the continuity equation. undergoes changes due to chemical or radioactive processes (“transformation”). environmental.. the diffusion of chemical .2 have been used that a quantitative uncertainty analysis can provide an illuminating role. and to help target data gathering efforts [72]. are similar. The inhalation of chemicals. a human. travels through a medium (“transport”). or a tissue).g. and the flow of contaminants through and along with surface and ground water are also examples of transport processes. The term “transformation” refers to the change of a species (physical. In the atmosphere.the enzymatic reactions account for transformations. transformation can result from a series of nonlinear photochemical reactions. or biological). In groundwater systems. In biological systems. Uncertainty occurs in transport-transformation models due to a number of factors. such as the reaction rates.3 Table 1. and S is the net rate of injection of the species into the control volume (also called as source/sink term). errors in the estimates of transport properties of a species in a medium. chemical. transformation can sometimes result from radioactive processes. and sometimes due to chemical reactions. R is the net where rate of production of the species due to chemical or radioactive processes. contribute to the overall uncertainty in these models. Table 1. Many transport-transformation models can be represented in a simplified form by the following equation: dc =F +R+S dt dc is the rate of change of a species concentration in a control volume under dt consideration. The randomness inherent in natural systems. .1 presents examples of the components of some transport-transformation models of environmental and biological systems. The main sources of uncertainty in transport-transformation systems are presented in more detail in Chapter 2. F is the net inflow of the species into the control volume.1: Examples of transport-transformation components in biological and environmental models Component Model Biological Air Quality Water Quality Flow Blood Wind Surfacewater/groundwater Reaction Enzymatic Source/Sink Absorption into tissues Photochemical Emissions/Deposition Radioactive Leachate species across a concentration gradient is also a transport process. and inaccurate estimates of transformation properties. have to be developed.e.3 Limitations in Performing Uncertainty Analysis The main limitation in performing comprehensive uncertainty analyses of transporttransformation models is the associated cost and effort. as described in more detail in Chapter 2. Finally. The model results corresponding to all the possible combinations give an estimate of the range of the associated model uncertainty. each corresponding to a different formulation of the mathematical problem corresponding to the original physical system.4 1. the data needs for characterizing input uncertainties are often substantial. The computer resources required for uncertainty propagation using conventional methods (to be discussed in the following paragraph) can sometimes be prohibitively expensive. such as the standard Monte Carlo and Latin Hypercube Sampling methods. This is addressed from the perspective of (a) computational requirements of the methods. 1. in order to estimate the uncertainties associated with model formulation. On the other hand. Thus.4 Objective of the Thesis The primary objective of this thesis is the development of computationally efficient methods for uncertainty propagation. Further.. Development and application of several alternative computational models can require substantial time and effort. i. Conventional methods for uncertainty propagation typically require several model runs that sample various combinations of input values. of the order of many thousands. The number of model runs can sometimes be very large. and (c) ease of use of the methods. Conventional methods for uncertainty propagation. (b) applicability of the methods to a wide range of models. several different models. The development of alternative methods that are applicable to a wide range of transport-transformation models could . the incorporation of uncertainty associated with structural and formulation aspects of the models requires significant effort. resulting in substantial computational demands. the costs associated with uncertainty analysis may sometimes be prohibitively high. necessitating a large number of model simulations and/or the development of several alternative models. may be computationally prohibitively expensive. These objectives are accomplished via the development of the Stochastic Response Surface Method (SRSM). which fall beyond the scope of input uncertainty propagation. In fact. the SRSM is coupled with an existing sensitivity analysis method. Additionally. and the development of auxiliary tools that facilitate easy use of these methods. computationally demanding models where the application of traditional methods may not be feasible due to computational and time limitations. Finally. such methods could facilitate the uncertainty analysis of complex. The extra effort required in understanding and applying a new method can sometimes offset other advantages that the method has to offer.5 substantially reduce the computational costs associated with uncertainty propagation (in terms of both the time and the resources required). Further.” Another objective of this thesis is to study the uncertainties associated with model structure and formulation. Furthermore. uncertainties . the ease of use of any method is an important factor in its applicability. Thus. there is a need for an extensive evaluation of the alternative methods with realistic case studies. attention must be paid to the development of auxiliary tools that facilitate the easy use of the new methods. Additionally. it can be used to identify the relative magnitudes of uncertainties associated with model inputs and model formulation. Such an evaluation addresses the issues associated with the wider applicability of the methods. in conjunction with uncertainty propagation. an easy to use. Further. and its evaluation for a range of transport-transformation models. the primary objective of this thesis can be stated as “the development of computationally efficient alternative methods for uncertainty propagation that are applicable to a wide range of transporttransformation models. This provides information as to where available resources should be focused: filling data gaps versus refining the model used for the problem. Such a study provides help in the selection of the appropriate level of model detail. in order to further improve the computational efficiency of the SRSM. web-based interface is developed to facilitate black-box use of this method. Hence. ADIFOR (Automatic DIfferentiation of FORtran). Interval Mathematics. classified here as sensitivity testing methods. and Probabilistic Analysis (discussed further in Appendix A).6 associated with model structure and formulation are addressed in the framework of photochemical air quality modeling. Individual sections elaborate on the steps involved in the application . sampling based methods. such as. analytical methods. model. Chapter 2 presents the relevant background information in the following order: • classification of uncertainty into natural.1). The merits and limitations of each method are discussed. The formulation. and into reducible and irreducible uncertainties. and the importance of uncertainty analysis in the context of transport-transformation models. and the associated limitations. 1. It has also discussed the steps involved in a systematic uncertainty analysis. • approaches for representing parametric/data uncertainty. which is developed here as a computationally efficient alternative method for uncertainty analysis of numerical models. Chapter 3 presents the Stochastic Response Surface Method (SRSM). Fuzzy Theory. and an argument is made for alternative methods that can be used for combined sensitivity and uncertainty analysis. and parametric/data uncertainties. and application of the SRSM are discussed in detail. and computer algebra based methods. and from the perspective of obtaining sensitivity information. Additionally. This argument is made both from the perspective of the computational cost reduction.5 Outline of the Thesis Chapter 1 has presented a brief introduction to the concepts of uncertainty. Additionally. Chapter 1 summarizes the objectives and presents a brief overview of this thesis (Figure 1. and • conventional methods for sensitivity and sensitivity/uncertainty analysis. Chapter 2 presents several applications of the conventional methods to the uncertainty analysis of transport-transformation models. implementation. and (e) evaluation of the approximation.g. The SRSM is evaluated for all the four models. coefficients in a series expansion). as follows: I: A zero-dimensional physiologically-based pharmacokinetic model. Chapter 3 presents a web-interface for the application of the SRSM through a browser. ADIFOR (Automatic DIfferentiation of FORtran). ADIFOR is a computer algebra based method that reads the Fortran source code of a model and constructs the code that calculates first order partial derivatives of model outputs with respect to model inputs or parameters. whereas the SRSM-ADIFOR is evaluated for the first two models.g.. II: A two-dimensional reactive atmospheric plume model. without requiring additional effort in studying the mathematical details of the method. SRSM-ADIFOR. III: A three-dimensional urban/regional scale photochemical air quality model. series expansions). Additionally. (d) calculation of the statistical properties of outputs. covering a wide range of transport-transformation models. the coupling of the SRSM with ADIFOR is presented.7 of the SRSM: (a) representation of random inputs in a “standardized manner”. Chapter 4 presents the coupling of the SRSM with the Automated Differentiation method. first the rationale for coupling the SRSM with ADIFOR is presented. (b) expression of model outputs in a parametric form (e. case studies for the evaluation of the SRSM are presented in Chapter 5.. is illustrated with an example problem. Each case study description contains the information about the . This evaluation consists of four case studies. Chapter 3 also presents an example problem that illustrates the application of the SRSM. In Chapter 4. IV: A one-dimensional ground water model with discontinuous probability density functions for inputs and parameters. (c) estimation of the parameters of output approximation (e. explaining all the steps in detail. Chapter 5 presents a comprehensive evaluation of both the SRSM and the SRSMADIFOR methods. and is followed by a description of ADIFOR. and the application of the coupled method. Then. . and recommendations for future work. Chapter 6 addresses the issues involved with model uncertainty. This is followed by bibliography. Then. Subsequently. This chapter describes how such uncertainty can be addressed in the framework of photochemical air quality modeling. and the application of different uncertainty propagation methods is presented. The development and implementation of the RPM-3D. the uncertainties associated with the model are described. is presented in this Chapter. which arises due to uncertainties in model formulation and model structure. a three-dimensional version of the RPM. Chapter 7 presents the conclusions of this thesis. A study is presented involving a regulatory atmospheric photochemical plume model. Appendix E lists the programs that are included with the accompanying CDROM. Appendix A presents background information on probabilistic analysis that is useful in understanding the SRSM and the SRSM-ADIFOR. case studies are presented to compare the estimates of RPM-3D and RPM. Appendix B. along with a brief discussion at the end. in order to characterize uncertainties associated with model resolution. The estimates from the uncertainty propagation methods are then presented graphically. and Appendix D present information on the models used in the case studies in Chapter 5. Appendix C. in the form of probability density functions (pdf s).8 model and its application in exposure/risk assessment. the Reactive Plume Model (RPM). 2) Novel Methods SRSM (ch. 2) Conventional Probabilistic Methods: Monte Carlo (MC) and Latin Hypercube Sampling (LHS) (ch. 6) Stochastic Response Surface Method (SRSM) (ch. 1) Background: Uncertainties and Classification (ch. 5.9 Introduction (ch. 5. 3) Implementation of MC and LHS (ch.2) Model Uncertainty (ch.3) Groundwater Model (Leachate) (ch. 5. 2. The shaded areas indicate the contributions of this work . 4) Addressing Uncertainties Associated with Model Formulation and Resolution (ch.3) Coupling of SRSM and ADIFOR (SRSM-ADIFOR) (ch. a 3D Version of RPM-IV (ch.1) Air Quality Model (local scale) Photochemical Plume (ch. 4) Automatic Differentiation (ADIFOR) (ch.4) Implementation of SRSM (ch. 3) Implementation of SRSM-ADIFOR (ch. ch.1: Schematic depiction of the outline of this thesis.4) Figure 1.2) Non-Probabilistic Methods: Fuzzy Theory Interval Methods (ch. 3) SRSM-ADIFOR (ch.2) Input Uncertainty and its Representation (ch. 5. 6) Air Quality Model (urban scale) Airshed Model (ch.2) Development and Application of RPM-3D. 4) Biological Model (Human PBPK) (ch. depending on their origins and on how they can be addressed.1 Natural Uncertainty and Variability Environmental and biological systems are inherently stochastic due to unavoidable unpredictability (randomness).1.10 Chapter 2 BACKGROUND 2. the modeling of emissions from a source are sometimes modeled via a mean value and a “random error”. For example. On the other hand. in air pollution systems. For example.1 Types and Origins of Uncertainty in Transport-Transformation Models Uncertainties in transport-transformation models can be classified as Natural. Some quantities are random even in principle. some quantities vary over time. and is typically characterized through a frequency distribution. and the uncertainty can be characterized through ensemble averages. It is possible to interpret variability as uncertainty under certain conditions. Variability is the heterogeneity between individual members of a population of some type. Model. and Data uncertainties. over space. or across individuals in a population. this is termed “variability”. since it is impractical to continuously monitor the emissions. the turbulent atmosphere and unpredictable emission-related activities contribute to “natural uncertainty”. a precise estimation of system properties is not possible. Further. the implications of the differences in uncertainty and variability are relevant in decision making. since both can be addressed in terms of “frequency” distributions. However. In such cases. 2. the knowledge of the frequency distribution for . They are briefly described in the following. while some quantities that are precisely measurable are modeled as “random” quantities as a practical matter (due to cost and effort involved with continuous and precise measurement). In this context. Hence.11 Evaluation Data Uncertainty Natural Uncertainty Model Uncertainty Input Data Uncertainty Figure 2. the limited spatial or temporal resolution (e. Furthermore.1. The optimal mechanistic model will provide the greatest simplifications while providing an adequately accurate representation of the processes affecting the phenomena of interest. sometimes competing models may be available. In addition to the significant approximations often inherent in modeling.g. In contrast. 2. numerical grid cell size) of many models is also a type of approximation that introduces uncertainty into model results.2 Model Uncertainty Mathematical models are necessarily simplified representations of the phenomena being studied and a key aspect of the modeling process is the judicious choice of model assumptions.. the knowledge of uncertainty can aid in determining areas where additional research or alternative measurement techniques are needed to reduce uncertainty.1: Types of uncertainty present in transport-transformation modeling applications and their interrelationships (adapted from [78]) variability can guide the identification of significant subpopulations which merit more focused study. different sources of model uncertainties . the structure of mathematical models employed to represent transport-transformation systems is often a key source of uncertainty. there is a trade-off between the computation time (hence cost) and prediction accuracy. a dose-response model based on high-dose. then one can be confident that the decision is robust in the face of uncertainty.1) is valid for grid cell sizes between 2 km and 20 km [137]. On the other hand. selection of a spatial and/or temporal grid size often involves uncertainty.12 are summarized in the following: Model Structure: Uncertainty arises when there are alternative sets of scientific or technical assumptions for developing a model. the atmospheric diffusion equation (ADE) (see Equation 6. . In such cases. models that are evaluated only for base case emission levels. a coarse-grid model ignores significant “sub-grid” detail. These include assumptions to convert a complex nonlinear model to a simpler linear model in a parameter space of interest. Very often. and may even produce incorrect results. inclusive models. however. Model Resolution: In the application of numerical models. may introduce uncertainties when they are employed in the study of future scenarios that involve significantly different emissions levels (also known as ∆E caveat). in atmospheric modeling. alternative model formulations lead to different conclusions. For example. as the governing equation may not be appropriate. Model Detail: Often. long duration human exposures. In this case. and a fine-grid model would require more computer resources. further model evaluation might be required. short duration animal tests may incur significant errors when applied to study low-dose. in photochemical grid modeling. a finer grid resolution need not necessarily result in more accurate predictions. models are simplified for purposes of tractability. if the results from competing models result in similar conclusions. Extrapolation: Models that are validated for one portion of input space may be completely inappropriate for making predictions in other regions of the parameter space. Uncertainty in the predictions of simplified models can sometimes be characterized by comparison of their predictions to those of more detailed. there is a trade-off between resolution and the validity of the governing equations of the model at such scales. On one hand. Similarly. For example. If. a coarse grid resolution introduces approximations and uncertainties into model results. Sometimes. or by comparing results based on different grid sizes. 2. Within the boundary. Even though many parameters could be measurable up to any desired precision. types of pathways. at least in principle. . the model may be an accurate representation.1. The selection of a model boundary may be a type of simplification.1: Examples of the sources of uncertainty in the formulation and application of transport-transformation models Uncertainty in Model Formulation (Structural Uncertainty) Simplifications in Conceptual Formulation Simplifications in Mathematical Formulation Ergodic-type Hypotheses Idealizations in Constitutive Relations Independence Hypotheses Spatial Averaging Temporal Averaging Process Decoupling Lumping of Parameters Discretization Numerical Algorithm/Operator Splitting Approximations in Computer Coding Uncertainty in Model Application (Data/Parametric Uncertainty) Constitutive Parameter Selection Design/Structural Parameter Selection Input Data Development/Selection Source Information Meteorology (air quality models) Hydrology (water quality models) Initial and Boundary Conditions Operational Model Evaluation Uncertainty in model estimates Uncertainty in observations Nonexistence of observations Response Interpretation ∆E caveat and other implicit hypotheses This type of uncertainty is sometimes dealt with through an appropriate selection of model domain parameter values. Model Boundaries: Any model may have limited boundaries in terms of time. but other overlooked phenomenon not included in the model may play a role in the scenario being modeled. number of chemical species.3 Parametric/Data Uncertainty Uncertainties in model parameter estimates stem from a variety of sources.13 Table 2. space. and so on. or (c) inaccuracies in the assumptions used to infer the actual quantity of interest from the observed readings of a “surrogate” or “proxy” variable. the averaging processes involved in model formulation unavoidably “lump” together natural and modeling uncertainty and only . the distinction between reducible and irreducible model and data uncertainties is to a great extent a matter of convention since it may not be feasible to eliminate the presence of an error (reducible uncertainty) in measurement or modeling beyond a certain level. improved instrumentation. better than “artificial” reducible uncertainty. and via mechanistic modeling. Other potential sources of uncertainties in estimates of parameters include misclassification. Further. improvements in model formulation. and estimation of parameters through non-representative samples. (b) systematic biases that occur due to imprecise calibration. uncertainty in model application arises from uncertainties associated with measurement data used for the model evaluation. Natural uncertainty is “inherent” or irreducible. e..14 there are often significant uncertainties associated with their estimates. these in turn can involve (a) random errors in analytic devices (e. The irreducible uncertainty in data and models is generally a result of the presence of natural uncertainty.2 Reducible and Irreducible Uncertainty Uncertainty associated with model formulation and application can also be classified as “reducible” and “irreducible”. 2.g.. whereas data and model uncertainty contain both reducible and irreducible components.g. Some uncertainties arise from measurement errors. by better inventorying methods. Irreducible modeling uncertainty reflects the “current” model formulation and may actually change when improved theories describing the phenomena under consideration become available. Nevertheless. Table 2. Furthermore. the imprecision of continuous monitors that measure stack emissions).1 lists some of the sources of model and parametric uncertainty associated with the formulation and the application of transport-transformation models. Also. what is perceived as irreducible natural uncertainty may be quantified in a statistical sense. etc. Reducible uncertainty can be lowered. estimation of parameters through a small sample. • Fuzzy set theory: Fuzzy sets. A detailed description of the probability theory is presented in Appendix A. each consisting of exactly one alternative. retrodictions. the uncertainty arises from the conflict among likelihood claims associated with the subsets of the universal set. Large sets result in less specific predictions. than smaller sets. nonzero probabilities assigned to two or more events conflict with one another since only one of them can be the desired one. Since these alternatives are mutually exclusive. and their interrelationships. This includes diagnostic. Figure 2. are capable of expressing nonspecificity. . they are also capable of expressing vagueness. etc. expresses the likelihood that the desired unique alternative is in this subset. in particular definitions of linguistic terms. In fuzzy sets. • Probability theory: Uncertainty is expressed in terms of a measure on subsets of a universal set of alternatives (events). Here. called probability of the subset. The uncertainty measure is a function that. according to the situation. 2. In addition. Full specificity is obtained only when one alternative is possible. Vagueness is different from nonspecificity in the sense that vagueness emerges from imprecision of definitions. the membership is not a matter of affirmation or denial. and are briefly summarized in the following: • Classical set theory: Uncertainty is expressed by sets of mutually exclusive alternatives in situations where one alternative is desired. similar to classical sets. but rather a matter of degree.15 a quantification of this lumped uncertainty may be possible or desirable. Here. assigns a number between 0 and 1 to each subset of the universal set. predictive and retrodictive uncertainties.1 depicts schematically the types of uncertainties present in transport-transformation models..3 Approaches for Representation of Uncertainty Various approaches for representing uncertainty in the context of different domains of applicability are presented by Klir [125]. the uncertainty arises from the nonspecificity inherent in each set. This number. each of which is characterized by a special property. and probabilistic analysis. it may not be possible to obtain the probabilities of different values of imprecision in data. fuzzy theory. every uncertain parameter is described by an interval. 24]. and each of them has upper and lower limits without a probability structure [184]. These approaches are presented in the following sections. This is especially true in case of conflicting theories for the estimation of model parameters.3. but the information about an element is insufficient to determine whether it satisfies those conditions. In the interval mathematics approach. interval mathematics can be used for uncertainty estimation. in the sense that “probabilities” cannot be assigned to the validity of one theory over another.16 • Fuzzy measure theory: This theory considers a number of special classes of measures. If a parameter xi of a model is known to be . In such cases. uncertain parameters are assumed to be “unknown but bounded”. Some of the widely used uncertainty representation approaches used in transporttransformation modeling include interval mathematics. 2. in some cases only error bounds can be obtained. the approximations could themselves be imprecise or fuzzy. and the classical probability measures. as this method does not require information about the type of uncertainty in the parameters [5. Some of the measures used in this theory are plausibility and belief measures. The objective of interval analysis is to estimate the bounds on various model outputs based on the bounds of the model inputs and parameters. In many cases. • Rough set theory: A rough set is an imprecise representation of a crisp set in terms of two subsets. or theories to estimate model parameters. Fuzzy measure theory and fuzzy set theory differ significantly: in the fuzzy set theory. the conditions are precise. whereas in the fuzzy measure theory. and (b) due to the existence of several alternative methods.1 Interval Mathematics Interval mathematics is used to address data uncertainty that arises (a) due to imprecise measurements. Further. techniques. the conditions for the membership of an element into a set are vague. a lower approximation and upper approximation. a range of computational tools exist for performing interval arithmetic on arbitrary functions of variables that are specified by intervals [122. 152. the interval representation of xi is given by [xi − i . b u bl Furthermore. au − bl ] a·b a/b (2. 151. For example. simple arithmetic operations are given by the following: a + b = [al + bl . Therefore. 120]. where al ≤ au and bl ≤ bu . Interval analysis has also been applied to estimate the uncertainty in displacements in structures due to the uncertainties in external loads [129]. in the cost benefit analysis of power distribution [184]. In addition. and in decision evaluation [24]. bu ]. in principle.1) = [min(al bl . Applications in transport-transformation modeling include uncertainty analysis of groundwater flow models [143. if two variables.17 between xi − i and xi + i . au bu )] 1 1 = [al . the ranges of uncertainty in any analytical or numerical model (FORTRAN model) can be analyzed by existing tools. presented a methodology for the propagation of uncertainties using intervals [52]. Hurme et al. xi + i ]. extensions to FORTRAN language with libraries for interval arithmetic are also available [121. max(al bl . au + bu ] a − b = [al − bu . Various applications of interval analysis in the literature include the treatment of uncertainty in the optimal design of chemical plants [69]. 132]. the model estimates would also belong to another interval. Special arithmetic procedures for calculation of functions of intervals used in this method are described in the literature [5. presented a review on semi qualitative reasoning based on interval mathematics in relation to chemical and safety engineering [104]. 155]. 0 ∈ [bl . Symbolic computation packages such as Mathematica [10] and Maple [94] support interval arithmetic. au bl . au bu ). al bu . au ] · . bu ] / . . au bl . au ] and [bl . Correspondingly. The primary advantage of interval mathematics is that it can address problems of uncertainty analysis that cannot be studied through probabilistic analysis. 53]. a and b are given by [al . It may be useful for cases in which the probability distributions of the inputs are not known. al bu . Dong et al. This is based on a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth – truth values between “completely true” and “completely false”. fuzzy theory allows for a continuous value of µA between 0 and 1. Fuzzy theory uses the process of “fuzzification” as a methodology to generalize any specific theory from a crisp (discrete) to a continuous (fuzzy) form. However. This can be illustrated as follows: In the classical set theory. certain attributes of systems cannot be ascribed to one set or another. the truth value of a statement can be given by the membership function µA (x). as   1 iff µA (x) =  0 iff x∈A x∈A / (2. It was introduced by Zadeh as a means to model the uncertainty of natural language [222].2 Fuzzy Theory Fuzzy theory is a method that facilitates uncertainty analysis of systems where uncertainty arises due to vagueness or “fuzziness” rather than due to randomness alone [65]. fuzzy theory provides allows for a gradual degree of membership. On the other hand.3. For example. Especially when the probability structure of inputs is known. as all the uncertainties are forced into one arithmetic interval [134].3) . an attribute of a system can be specified as either “low” or “high”. as µA (x) =            1 iff 0 iff p. uncertainty arises out of vagueness involved in the definition of that attribute.18 However. 0 < p < 1 if x∈A x∈A / x partially belongs toA (2.2) On the other hand. and hence is not recommended. In such a case. Classical set theory has a “crisp” definition as to whether an element is a member of a set or not. 2. Classical set theory allows for either one or the other value. this method does not provide adequate information about the nature of output uncertainty. the application of interval analysis would in fact ignore the available information. in sludge application land selection [45]. Fuzzy theory appears to be more suitable for qualitative reasoning. Specifically.0 − µA (x) µA∩B (x) = min(µA (x). studied structural reliability assessment by using fuzzy theory to study the uncertainty associated with ambiguity [9]. 174]. [166. based on the grade of membership of a parameter of interest in a set. and by Puri et al. statements are described in terms of membership functions. For example. Kraslawski et al. the grade of membership of a model output in another set can be calculated.19 In fuzzy theory. demonstrated the applicability of fuzzy theory in the modeling and analysis of non-random uncertainties in geotechnical engineering [115]. drawbacks in its applicability to uncertainty analysis has been noted by some researchers [193. It must be noted that if the membership grades are restricted to only 0 and 1. that are continuous and have a range [0.1]. A vast amount of literature is available on fuzzy theory. fuzzy logic is defined by the following the set relationships: µA (x) = 1. the membership function gives the “degree of truth” that the parameter is “high” or “low”.. in quality control [167]. Fuzzy arithmetic and its applications are described by Klir et al.143].212]. However. it has been applied to characterize uncertainty in engineering design calculations [219. and in solute transport modeling [53. 191]. Further. by Kauffmann et al. µB (x)) µA∪B (x) = max(µA (x). 119. the fuzzy theory simplifies to classical set theory. Ayyub et al. Fuzzy theory can be considered to be a generalization of the classical set theory.54. including extensive introductions to the concepts involved by Bezdek and by Smithson [15. applied fuzzy set theory to study uncertainty associated with incomplete and subjective information in process engineering [130]. than for quantitative estimation of uncertainty.131].. given the measured value of a parameter. and classification of elements into a fuzzy set. 126]. Further literature on industrial applications of fuzzy theory is available for the interested reader [221. (2. µB (x)) Using fuzzy arithmetic.4) . Juang et al. especially when estimates of the probability distributions of uncertain parameters are available.3 Probabilistic Analysis In the probabilistic approach.103]. a uniform distribution is selected if only a range of possible values for an input is available. In this approach. For example. The probability of an event can be interpreted in terms of the frequency of occurrence of that event. Similarly. In cases where limited data are available. Techniques for estimating probability distributions from data are presented in the literature [37. emissions. and the the objective is to estimate the output probability distributions.20 2.” This is accomplished by using either statistical estimation techniques or expert judgments. This process comprises of two stages: • Probability encoding of inputs: This process involves the determination of the probabilistic distribution of the input parameters. variability conditions. the uncertainties associated with model inputs are described by probability distributions. but no information about which values are more likely to occur is available. Statistical estimation techniques involve estimating probability distributions from available data or by collection of a large number or representative samples. under identical conditions. normal distribution is typically . For example. the number of times the value of c lies between c1 and c2 is roughly equal to the fraction p of the total number of samples.3. an expert judgment provides the information about the input probability distribution. uncertainties are characterized by the probabilities associated with events. Probabilistic analysis is the most widely used method for characterizing uncertainty in physical systems. and risk considerations. meteorology) and “errors. This approach can describe uncertainty arising from stochastic disturbances. When a large number of samples or experiments are considered.. the probability of an event is defined as the ratio of the number of times the event occurs to the total number of samples or experiments. and incorporation of random variations due to both natural variability (from. e.g. the statement that the probability that a pollutant concentration c lies between c1 and c2 equals p means that from a large number of independent measurements of the concentration c. 2: A schematic depiction of the propagation of uncertainties in transporttransformation models used to describe unbiased measurement errors.2 depicts schematically the concept of uncertainty propagation: each point of the response surface (i.e. 201]. • Propagation of uncertainty through models: Figure 2. The techniques for uncertainty propagation are discussed in the following. each calculated output value) of the model to changes in inputs “1” and “2” will be characterized by a probability density function (pdf ) that will depend on the pdf s of the inputs.. 75. A number of excellent texts on probabilistic analysis are available in the literature for the interested reader [160.2 presents a list of some commonly used probability distributions in the uncertainty analysis of transporttransformation models. Appendix A presents some background information on probabilistic analysis and random variables. Table 2. .21 response 1-2 probability density of response 1-2 input 2 input 1 probability density of input 2 probability density of input 1 Figure 2. σ > 0 xσ 2π 1 √ − e (log(x) − µ)2 2σ 2 2 Mean = e(µ + σ /2) 2 2 Var = (eσ − 1)e(2µ + σ ) 2 Mode = e(µ − σ ) Gamma a. a ≥ 1 a Exponential λ. σ . λ > 0 λe−λx .2: Selected probability density functions useful in representing uncertainties in environmental and biological model inputs Distribution Parameters & Conditions Probability density function 1 b−a a+b Mean = 2 (b − a)2 Var = 12 Moments Uniform a. σ . σ > 0 − 1 √ e σ 2π (x − µ)2 2σ 2 Mean = µ Var = σ 2 Mode = µ Lognormal µ. b. b > 0 x − 1 a−1 x e b x>0 Γ(a)ba Mean = ab Var = ab2 Mode = (a − 1)b 1 Mean = λ 1 Var = 2 λ Mode = 0 1 Mean = Γ(1 + ) a 2 1 Var = Γ(1 + ) − Γ(1 + ) a a 1 1 Mode = (1 − ) a . a > 0. b Normal µ.22 Table 2. x > 0 Weibull a a axa − 1 e−x (x ≥ 0) Extreme Value −x e−x − e Mean = 0 Var = 1 Mode = 0 . and (c) understanding the behavior of the system being modeled. This sensitivity . normalized response and extrema. • Local sensitivity analysis: Here. Sensitivity measures that are appropriate for this type of analysis include the response from arbitrary parameter variation. the sensitivity involves the study of the system behavior over the entire range of parameter variation. Of these measures.23 2. the extreme values are often of critical importance in environmental applications. The choice of a sensitivity analysis method depends to a great extent on (a) the sensitivity measure employed. often taking the uncertainty in the parameter estimates into account.5) Based on the choice of a sensitivity metric and the variation in the model parameters.3 presents some of the sensitivity measures that are often employed in the sensitivity analysis of a mathematical model of the form F(u. sensitivity analysis methods can be broadly classified into the following categories: • Variation of parameters or model formulation: In this approach. (b) the desired accuracy in the estimates of the sensitivity measure. the focus is on estimates of model sensitivity to input and parameter variation in the vicinity of a sample point. the meaning of the term “sensitivity analysis” depends greatly on the sensitivity measure that is used. the model is run at a set of sample points (different combinations of parameters of concern) or with straightforward changes in model structure (e. in model resolution).. In general. Such knowledge is important for (a) evaluating the applicability of the model. and (c) the computational cost involved.g.4 Sensitivity and Sensitivity/Uncertainty Analysis The aim of sensitivity analysis is to estimate the rate of change in the output of a model with respect to changes in model inputs. k) = 0 where k is a set of m parameters. (b) determining parameters for which it is important to have more accurate values. and u is a vector of n output variables. • Domain-wide sensitivity analysis: Here. Table 2. (2. . ui (kP (k)dk k 2 2 δi (k) = ui (k)2 − ui (k) max [ui (k)]. Sensitivity testing involves studying model response for a set of changes in model formulation. . Sij = k j ∂ui ui (k) ∂kj ∂ui ∂kj Normalized Gradient n Sij = is often characterized through gradients or partial derivatives at the sample point. dk k Expected Value Variance Extrema Local Gradient Approximation ui (k) = . Analytical methods involve either the differentiation of model equations and subsequent solution of a set . . . and for a selected model parameter combinations. adapted from McRae et al. and (d) computer algebra based methods. .3: A summary of sensitivity measures employed in sensitivity analysis. 1982 [148] Sensitivity Measure Definition Response from arbitrary parameter variation Normalized Response u = u(k + δk) − u(k) Di = δui ui (k) Average Response ui (k) = . . 2. (b) analytical methods.5 Conventional Sensitivity/Uncertainty Analysis Methods Conventional methods for sensitivity analysis and uncertainty propagation can be broadly classified into four categories: (a) “sensitivity testing”..24 Table 2. (c) sampling based methods. min [ui (k)] δu ≈ [S]δk . ui (k)dk k . the main disadvantage of this approach is that detailed information about the uncertainties is difficult to obtain using this approach. .g. These methods are elaborated in the following. [207] and Vanderperk [206] have used a similar approach in water quality modeling. the sampling based methods involve running the original model for a set of input/parameter combinations (sample points) and estimating the sensitivity/uncertainty using the model outputs at those points. the sensitivity information obtained depends to a great extent on the choice of the sample points. or the reformulation of original model using stochastic algebraic/differential equations. and it has been widely employed in conjunction with transport-transformation modeling. [4] used this approach for the sensitivity analysis of urban air quality models.5. The application of this approach is straightforward. On the other hand. Vieux et al. 2. Yet another sensitivity analysis method is based on direct manipulation of the computer code of the model.1 Sensitivity Testing Methods In this approach. Further. the model is run for a set of sample points for the parameters of concern or with straightforward changes in model structure (e. in model resolution). by testing whether the model response changes significantly in relation to changes in model parameters and structural formulation of the model. and is termed automatic differentiation. This approach is often used to evaluate the robustness of the model. The primary advantage of this approach is that it accommodates both qualitative and quantitative information regarding variation in the model.. Roselle [173] used this approach to study the effect of biogenic emission uncertainties by performing model simulations at three emission biogenic estimate levels.25 of auxiliary sensitivity equations. especially when only a small number of simulations can be performed. Sistla et al. However. [189] and Al-Wali et al. 198]. 140]. Spectral Based Stochastic Finite Element Method This method relies on the use of representing stochastic processes in terms of a series expansion.5. 160]. and (d) coupled and decoupled direct methods. The Neumann expansion method involves finding the inverse of the model operator through the expansion of the model equations. 209]. This approach has been applied to the sensitivity analysis of atmospheric chemistry models [106.26 2. 224]. The sensitivity equations are then solved by constructing an auxiliary set of Green’s functions. Differential Analysis Methods Differential analysis methods include the Neumann expansion [3. and hence has limitations on the type of model equations it can address. specifically the Karhunen-Loeve expansion [83. This method minimizes the number of differential equations that are solved for sensitivity. The perturbation method involves expansion of model outputs as a series in terms of small random perturbations in model parameters. 2. as the model equations are often mathematically intractable. and the subsequent solution of the series coefficients. and the perturbation method [3. 118. (c) spectral based stochastic finite element method. (b) Green’s function method.2 Analytical Methods Some of the widely used analytical methods for sensitivity/uncertainty are: (a) differential analysis methods. The Neumann expansion and perturbation based methods have been applied in uncertainty analysis of ground water models [1. Further. and in the design of structures [26. For finite element . Green’s Function Method In the Green’s function method. The main limitation of these methods is the requirement that the perturbation terms be small. these methods are in general difficult to apply in conjunction with the modeling of complex. and replaces them with integrals that can be easily evaluated [55. 197]. the sensitivity equations of a model are obtained by differentiating the model equations. 56]. nonlinear systems. 2. The analytical methods require access to the governing model equations. and establishing a relationship between inputs and outputs using the model results at the sample points. Coupled/Decoupled Direct Method The direct method involves the differentiation of model equations and the subsequent solution of the sensitivity equations. which may be impractical and sometimes impossible. such as CHEMKIN [41]. (b) Fourier . One of the main features of this method is the representation of random parameters in terms of orthogonal functions of a set of standardized random variables. The decoupled method is also reported to be more efficient than the Green’s function method [60].5. Some of the widely used sampling based sensitivity/uncertainty analysis methods are: (a) Monte Carlo and Latin Hypercube Sampling methods. 165].3 Sampling Based Methods Sampling based methods do not require access to model equations or even the model code. or separately (Decoupled Direct Method) [60. and may involve writing additional computer code for the solution of the auxiliary equations. This method has evolved into a standard module in conjunction with commercial sensitivity analysis codes for chemical kinetics. The matrix equations are solved either using operator expansions or by using the Galerkin’s method [208]. this approach results in a set of linear matrix equations with deterministic matrices multiplied by random vectors. and forms the basis for the development of the “Stochastic Response Surface Method” (SRSM). The polynomial chaos expansion and the SRSM are discussed in detail in Chapter 3. the expansion is also known as “polynomial chaos expansion”. reformulating an existing computational model developed by others could require prohibitive amounts of resources. The sensitivity equations are then solved along with the original model equations (Coupled Direct Method) [200]. and GCKP86 [168]. The decoupled method is advantageous both in terms of computational efficiency and stability of the solution.27 method problems. For example. These methods involve running a set of model at a set of sample points. Problems such as optimization and the simulation of movement of fluid molecules are often addressed through Monte Carlo simulations [75]. the range of probable values for For example. their applicability is sometimes limited to simple models. tools.28 Amplitude Sensitivity Test (FAST) (c) reliability based methods. such as uncertainty propagation in models or solution of stochastic equations. For the interested reader. which requires about six hours of CPU time for the simulation of air quality near the vicinity of New Jersey (see the example in Section 5. 71. and the applicability of the Monte Carlo methods is available in the literature [116. 177. These methods involve random sampling from the distribution of inputs and successive model runs until a statistically significant distribution of outputs is obtained. Since these methods require a large number of samples (or model runs). Monte Carlo methods are also used in the solution of problems that can be modeled by the sequence of a set of random steps that eventually converge to a desired solution.75]. 51. with applications ranging from aerospace engineering [12] to zoology [28]. so that the number of necessary solutions compared to the simple Monte Carlo method is significantly reduced. such as finding the area under a curve [49. and (d) response surface methods. 194] is one such widely used variant of the standard Monte Carlo method. Monte Carlo and Latin Hypercube Sampling Methods Monte Carlo (MC) methods are the most widely used means for uncertainty analysis.∗ . or can be used to solve non-probabilistic problems. in order to perform the uncertainty analysis of a grid-based photochemical model. 192]. In this method. In case of computationally intensive models. a thousand samples would require more than eight months of computer time ∗ .3). The Latin Hypercube Sampling [107. the time and resources required by these methods could be prohibitively expensive. 146. They can be used to solve problems with physical probabilistic structures. A degree of computational efficiency is accomplished by the use of Modified Monte Carlo (MMC) methods that sample from the input distribution in an efficient manner. such as the Urban Airshed Model (UAM-IV). a wide range of literature describing the methodology. In the present work. Guidelines for the application of Monte Carlo and Latin Hypercube Sampling methods in risk assessment are provided by the U. consisting of all the uncertain parameters. and is freely available to the research community. Commercial plugins based on spreadsheet software include Crystal ball [46]. 89. 172.EPA [204. 187]. 203]. This list is only representative of the available software tools. Monte Carlo methods and Latin Hypercube Sampling have been applied in probabilistic assessment of risk in physiologically based pharmacokinetic models by considering uncertainty in the estimates of the model parameters [44. The EPA approved FORTRAN package for Latin Hypercube Sampling by Iman et al. A detailed description of the implementation of Latin Hypercube Sampling is presented in Section 3. 67. . is partitioned into cells having equal probability. 190. and in the reliability assessment of structures [8]. 157. These methods have also been applied in groundwater contamination models [14. Thus. For a model with m model parameters. in air pollution modeling [48. 74. @RISK [218]. 84]. 179. 215]. Several computer packages containing routines for Monte Carlo and Latin Hypercube Sampling methods are reported in the literature [112. 34. The advantage of this approach is that the random samples are generated from all the ranges of possible values.1. in process engineering [169. [108] is available through the EPA’s Exposure Models Library [202]. thus giving insight into the extremes of the probability distributions of the outputs. 143]. the the whole parameter space. thus reducing the a multidimensional model into a single dimensional one. 108].29 each uncertain input parameter is divided into ordered segments of equal probability. a Latin Hypercube Sampling utility in FORTRAN has been developed as a part of the implementation of the Stochastic Response Surface Method.S. RISKMAN [162]. and MonteCarlo [164]. and they are sampled in an “efficient” manner such that each parameter is sampled once from each of its possible segments.8. Fourier Amplitude Sensitivity Test (FAST) Fourier Amplitude Sensitivity Test (FAST) is a method based on Fourier transformation of uncertain model parameters into a frequency domain. k1 (s). The details of the transformation of model parameters into the frequency domain. the FAST method involves the transformation of the parameters into a frequency domain spanned by a scalar s. . . . and the subsequent sampling are explained by Koda et al. . For example. . . . which corresponds to the sampling in the multidimensional model parameter space. k1 (s). Reliability Based Methods (FORM and SORM) First. [127]. k2 (s). in conjunction with land surface processes in global climate models models [145. FAST has been applied in the sensitivity/uncertainty analysis of atmospheric photochemical mechanisms [66]. 40]. . n. . . u1 . . k2 (s). A review article by Helton [96] reports that Monte Carlo methods are more widely applicable than FAST. km (s))ds − u2 i i These integrals are evaluated by repeatedly sampling the parameter space of s. .92. . The outputs are then approximated as: 1 2π 1 2π π l = 1. .6) u2 (t. In addition. . km ). such that ui = fi (t. . m ui (t) = 2 σi (t) = ui (t.117]. k2 . k2 . . . and in the disposal of radioactive waste [124].30 k1 . 2. . i = 1. and a computational implementation of FAST is presented by McRae et al. km (s))ds −π π −π (2. . at no additional computational effort [147. u2 . . the probability of failure). 2. um .and second-order reliability methods (FORM and SORM. km . respectively) are approximation methods that estimate the probability of an event under consideration (typically termed “failure”). k1 . [149]. and n outputs. . . . . as follows: kl = Gl (sin wl s). these methods provide the contribution to the probability of failure from each input random variable. . . these methods can provide the probability that a contaminant concentration exceeds a target level at a location (or. . . . Xn ). X2 . . the computation time grows as mn [97]. as X(U ) and G(U ) respectively. Xn ) < 0 the objective of the reliability based approach is to estimate the probability of failure. and a failure condition g(X1 . through a series of the following simple steps. where CR is a pre-specified maximum permissible concentration at a location of interest. since the value of the integrand is small. . If the joint probability density function for the set X is given by fX . then the probability of failure is given by the n-fold integral: PF = P {g(X) < 0} = P {CR < C(X)} = fX dX g(X) < 0 where the integration is carried out over the failure domain.31 These methods are useful in uncertainty analysis of models with a single failure criterion. In addition. . as illustrated by Bjerager [20]: • mapping the basic random variables X and the failure function g(X). in fact if m is the number of function calls of the integrand per dimension. FORM and SORM use analytical schemes to approximate the probability integral. into a vector of standardized and uncorrelated normal variates U . the numerical inaccuracies can be considerably magnified when integrated over a multi-dimensional space [23]. . . The evaluation of this integral becomes computationally demanding as the number of random variables (the dimension of the integration) increases. • approximating the function G(U ) by a tangent (FORM) or a paraboloid (SORM) at a failure point u∗ closest to the origin. and . X2 . In case of pollutant contamination exceedance. the failure condition can be defined as g(X) = CR − C(X) < 0. . and n is the dimension. For a model with random parameters X = (X1 . . involve significant computational effort for nonlinear black box numerical models [82]. [22. 92. but computationally more intensive. In addition. and with structural reliability problems [27]. 21] describe the adaptive nature of these methods. Further. 123]. . Helton [96] studied the applicability of response surface methods in performance assessment of radioactive waste disposal. This fitted responsesurface is then used as a replacement or proxy for the computer model. since it involves a higher order approximation. This approach is sometimes termed as a “secondary model technique” [68]. These methods are reported to be computationally very efficient compared to Monte Carlo methods.32 • calculating the probability of failure as a simple function of u∗ . with vegetative plant growth modeling [141]. Box et al. SORM is more accurate than FORM. 147]. 90] and to problems in safety assessment [39. and the methodology for identifying the sampling points and for subsequent analysis is presented in detail in the literature [59. 91. Furthermore. these methods impose some conditions on the joint distributions of the random parameters [92]. It has also been used in conjunction with soil erosion modeling [33]. (ii) making multiple runs of the computer model using specific values and pairings of these input parameters. and the subsequent minimization of the function. 70. and (iii) fitting a general polynomial model to the model data (using the method of least squares). especially for scenarios corresponding to low probabilities of failure [92]. Response Surface Methods The response surface methods consist of (i) screening to determine a subset of important model input parameters. thus limiting their applicability. These methods have been applied to problems of ground-water flow and contaminant transport [111. simultaneous evaluation of probabilities corresponding to multiple failure criteria would involve significant additional effort. The main drawbacks of FORM and SORM are that the mapping of the failure function on to a standardized set. and all inferences related to sensitivity/uncertainty analysis for the original model are derived from this fitted model. [32] Horwedel [100] 1989 1996 1996 1994 Fortran C/C++ Fortran 90 Fortran Fortran Solution of ODEs [32] Radionuclide transport [99. pattern-matching algorithms. 101] ADGEN ADOL-C ADOL-F ATOMFT GRESS Worley et al. 161. to generate a “derivative code” based on the model code. 156] Odyssee PADRE2 PCOMP SMS Rostaing et al. [171] 1993 1992 Fortran Matlab Optimization of structures [158] Process engineering [150] Estimates rounding errors Radionuclide modeling [98] Groundwater flow [42] Computer algebra based methods involve the direct manipulation of the computer code. typically available in the form of a high level language code (such as C or FORTRAN). 110. [19] Bischof et al. These methods do not require information about the model structure or the model equations. One of the main computer algebra based methods is the automatic differentiation. and estimation of the sensitivity and uncertainty of model outputs with respect to model inputs. [175] Kubota [133] Dobmann et al. [158] Rich et al. [17] Year 1997 1996 Language C Fortran Applications/Features 3-D CFD grid generator [19] A wide range of applications [105.4: A representative list of packages available for automatic differentiation Package ADIC ADIFOR Authors Bischof et al.4 Computer Algebra Based Methods Table 2. Automatic differentiation involves direct manipulation of a model code to generate . which is sometimes also termed automated differentiation. [220] Griewank [87] Shiriaev et al. [186] Change et al.5. [50] Korelc [128] 1993 1991 1995 1997 Fortran Fortran Fortran Mathematica Automatic derivation of finite element formulae [128] TEXPANDER Unnamed Ozaki et al.33 2. 29. and use mechanical. and this approach results in substantial computer time savings [113. In addition.4 lists a majority of the automatic differentiation tools currently available. The main advantages of this approach are that partial derivatives can be calculated with the accuracy of the machine precision [17]. can sometimes limit its applicability [16]. along with some of their applications reported in the literature. This interpretation overhead associated with the storage. This approach is essentially a “black-box” approach. process engineering and numerical analysis can be found in the literature [13. 109].34 a corresponding “derivative calculating code”. Table 2. 35. The differentiation is carried out for each statement and the results are stored numerically at each stage of computation. various reviews of automatic differentiation in the fields of engineering design. as it does not require access to the model formulation or model equations. AIRDOS [98. The main limitation of GRESS is that the particulars of every operation performed in the code are stored in the memory at each step. 152]. 86]. and can be executed to calculate derivatives in the model.161]. 105]. the automatic differentiation tools can generate derivative information without further user intervention. and the information about what the dependent and independent variables of interest are. and analyzes the functional dependence of independent and dependent variables. Given the source code. and rewrites the original code. It has also been applied a model describing the atmospheric dispersion of radionuclides. along with the . An overview of the methodology and applications of automatic differentiation is presented by Griewank [85. ADIFOR (Automatic DIfferentiation of FORtran) [17] applies the rules of automatic differentiation. Two of the notable automatic differentiation tools that have been applied in conjunction with transport-transformation models are GRESS and ADIFOR. GRESS (GRadient Enhanced Software System) [100] is a Fortran preprocessor system that searches the model equations based on the appearance of the “=” symbol. The generated code can then be compiled just like a normal program. This method has been applied to models describing transport of radionuclides through groundwater [156. A large number of packages for automatic differentiation of model codes written in various programming languages are currently available. and the potentially great memory demands. inserting statements for computing the first-order derivatives. analytical methods require the information about the mathematical equations of a model. In the area of photochemical air quality modeling. They calculated the sensitivity of concentration to a global perturbation of wind velocity in advection models and comparing that with other methods. require a substantial number of model runs to obtain a good approximation of the output pdf s. which is a computationally efficient uncertainty analysis method developed as part of this work. . Hwang et al. and that the CPU requirements were lower for the ADIFOR method. This approach avoids the limitation of GRESS. On the other hand.6 Need for Alternative Sensitivity/Uncertainty Analysis Methods Traditional sampling methods for sensitivity and uncertainty analysis. and in aeronautics [101]. 2. especially for cases involving a several inputs. The details of the ADIFOR approach and its application are presented in Chapter 4 where the coupling of the Stochastic Response Surface Method and ADIFOR is described. Carmichael et al. and often are restricted in their applicability to cases where the uncertainties are small. in weather modeling [18]. such as groundwater modeling [214]. in biostatistics [110]. with respect to the interpretation overhead. [29] applied the ADIFOR method to calculate the sensitivity of ozone levels to the initial concentrations (84 species) and all reaction rate constants (178 chemical reactions) for six different chemical regimes.35 model outputs. The following chapter describes the development of the Stochastic Response Surface Method (SRSM). Therefore there is a need for a computationally efficient method for uncertainty propagation that is robust and also applicable to a wide range of complex models. Further. such as the Monte Carlo and Latin Hypercube Sampling. ADIFOR has also been applied in several other research areas. Their results indicate that ADIFOR-generated code can produce exact sensitivity information up to the machine epsilon. [105] studied the applicability of automatic differentiation techniques in numerical advection schemes in air quality models. Random variables with normal distributions. so this method is applicable to a wide variety of of transporttransformation models. N(0. 49. and (3) the parameters of the functional approximation are determined.21. the propagation of input uncertainty through a model using the SRSM consists of the following steps: (1) input uncertainties are expressed in terms of a set of standard random variables. 198]. The SRSM was developed as a part of this work with the objective of reducing the number of model simulations required for adequate estimation of uncertainty. can be viewed as an extension to the classical deterministic Response Surface Method (RSM). as compared to conventional methods. 83. The SRSM is founded on the principle that random variables with well-behaved (square-integrable) pdf s can be represented as functions of independent random variables [25. the series expansions of the outputs contain unknown coefficients which can be calculated from the results of a limited number of model simulations. the inputs are deterministic variables. where as in the latter.70. as the name suggests. This is accomplished by approximating both inputs and outputs of the uncertain system through series expansions of standard random variables. The above integrability requirement is usually satisfied by uncertain quantities of interest.123]. (2) a functional form is assumed for selected outputs or output metrics. The main difference between the SRSM and the RSM is that in the former the inputs are random variables. Conceptually.1 Overview of the Method The Stochastic Response Surface Method (SRSM).36 Chapter 3 THE STOCHASTIC RESPONSE SURFACE METHOD: DEVELOPMENT AND IMPLEMENTATION 3. on which extensive literature is available [22. are often .1). a). • Models consisting of nonlinear operators with mathematically manipulable equations: a functional form for outputs. Since model outputs are deterministic functions of model inputs. Then. Once the inputs are expressed as functions of the selected srvs. First. In the present work these random variables are referred to as “Standard Random Variables” (srvs). and minimizing an appropriate norm of the resultant error in approximation. a). 160]. Galerkin’s method used in the Deterministic Equivalent Modeling Method [198. such as y = f (ξ. where the set of random inputs is represented by the vector x. The specific method employed in the estimation of parameters of approximation depends on the complexity of the model. is assumed. where ξ denotes the vector of the selected srvs. of the form y = f (ξ. and a set of selected random outputs or output metrics is represented by the vector y. they have at most the same number of degrees of freedom in uncertainty. as y = F −1 (x) = g(ξ). the output metrics can also be represented as functions of the same set of srvs. even a specific functional form for y need not be assumed a priori. where a denotes a parameter vector. Consider a model y = F (x). The minimum number of srvs needed to represent the inputs is defined as the “number of degrees of freedom” in input uncertainty. • Black-box type models (possibly reflecting the numerical implementation of a . For many cases. where a denotes the parameters of approximation. which can be broadly categorized into the following: • Models consisting of simple linear algebraic operators: the outputs can be directly obtained as analytical functions of srvs. The parameters are then calculated by substituting the transformed inputs and outputs into the model equations. the vector of input random variables is expressed as a function of the form x = h(ξ).37 selected to represent input uncertainties due to the mathematical tractability of functions of these random variables [49. 208] is an example of one such case. is assumed. a functional representation for outputs. 38 complex mechanistic model that one may decide to treat as a “black-box” inputoutput system): a functional representation for the output is assumed. identically distributed (iid) normal random variables. IV calculation of statistical properties of the outputs. 83. where the direct solution requires a significant amount of algebraic manipulation. The following sections describe the SRSM in more detail.2 Step I: Representation of Stochastic Inputs The first step in the application of the SRSM is the representation of all the model inputs in terms of a set of “standardized random variables”. 49. ξi . 160]. II representation of model outputs. is expressed directly as a function of the ith srv. This is analogous to normalization process used in transforming deterministic variables. III estimation of approximation parameters. where n is i=1 the number of independent inputs. and elaborate on the following steps that are involved in the application of the SRSM: I representation of uncertain inputs. normal random variables are selected as srvs as they have been extensively studied and their functions are typically well behaved [25. a transformation of Xi to ξi . In this work.1 further offers a schematic illustration of the application of the SRSM for various scenarios. 3. and model outputs for several realizations of the srvs are used as “data” for the estimation of parameters of the functional approximation. Figure 3.e. the srvs are selected from a set of independent.. Sometimes this approach may be preferable in the case of complex nonlinear models. the uncertainty in the ith model input Xi . and V evaluation of the approximation of model outputs. and each ξi has zero mean and unit variance. i. Here. When the input random variables are independent. This approach can be used for cases where it is not possible to directly substitute the input representations into the model. as described above. {ξi }n . Y) = 0 direct solution Y = G (F (ξ) ) assume a form for Y (e. each of which could have very different distribution properties.39 inputs (X) computational model outputs (Y) X = F (ξ) (exact.g..α) Figure 3. if possible) express inputs as functions of srvs X ≅ F (ξ) (for complex cases) explicit systems Y = G (X) implicit systems G (X.g. These techniques are described in detail in the following sections. and (b) series approximations in srvs.. Devroye [49] presents transformation techniques and approximations for a wide variety of random variables. . Galerkin’s method) estimate α using: collocation methods (e. One of two approaches may be taken to represent the uncertain inputs in terms of the selected srvs: (a) direct transformation of inputs in terms of the srvs.. ECM or orthogonal collocation) regression methods (if collocation methods don’t converge) Y evaluate = h (ξ. series expansion) Y = h (ξ. Such transformations are useful in the standardized representation of the random inputs. Input random variables whose transformations are not found in literature can be approximated either by algebraic manipulation or by series expansion techniques.1: Schematic depiction of the Stochastic Response Surface Method is employed.α) black-box models or complex numerical codes mathematically tractable equations estimate α by standard error minimization methods (e.g. Although a large number of transformations for distributions exist in the literature. and fX (x).1) and y is exponential (1) distributed 3.1].1 Direct Transformations Direct transformations for expressing a given random variable as a function of another random variable are typically derived by employing the following relation [160]: If Y = h(X). . and if X is normal. then − log X is λ exponential (λ) distributed. . . xn . For example. .40 Table 3. . Transformations are available in the literature 1 for some common distributions. b) Normal (µ. where X is a random variable with the pdf fX (x). given fY (y). the X 2 is gamma (1. . are the real roots of the equation y = h(x).1: Representation of common univariate distributions as functionals of normal random variables Distribution Type Uniform (a.b) Exponential (λ) Weibull (a) Extreme Value a Transformationa a + (b − a) √ 1 1 + erf(ξ/ 2) 2 2 µ + σξ exp(µ + σξ) 1 1 ab ξ +1− 9a 9a √ 1 1 1 − log + erf(ξ/ 2) λ 2 2 ya − log(y) 1 3 ξ is normal (0. the transformation Y = h(X) is identified by means of trial and error.σ) Gamma (a. they are not directly applicable for the case of normal random variables because of the following reasons: (a) these transformations are optimized for computational speed as .σ) Lognormal (µ.1) where x1 . is given by fY (y) = fX (x1 ) fX (xn ) + ··· + + ··· |h (x1 )| |h (xn )| (3. then the pdf of Y . Using the above equation.1) distributed. fY (y). if X is uniform [0. .2. thus resulting in a set of equations in the unknown coefficients.. and (b) a majority of these transformations are in terms of the uniform random numbers. where .2. measurements or the given non-standard or empirical pdf ).2. In addition. and are difficult to derive. the approximation of the random variable can be obtained as follows: • the moments of the random variable are computed based on the given information (e.g. 3..41 they are used as random number generators. In some cases the uncertainty in a model input x is specified by an empirical cumulative distribution function (cdf ).g. some transformations were derived during the course of the development of the SRSM. through least squares). • the moments of the corresponding series expansion are derived as functions of unknown coefficients. 3. However. • a series expansion in terms of an srv is assumed to represent the random variable. and • the process is repeated using a higher order approximation. some transformations in terms of the standard normal random variables are presented by Devroye [49]. in the form Fx (x) = g(x). In such cases.2 Transformation via Series Approximation When a model input follows a non-standard distribution. • the error between the derived moments and the calculated moments is minimized (e. a sample of random data points.3 Transformation of Empirical Distributions In the case of empirical distributions. until the desired accuracy is obtained. Table 3.e. or a frequency histogram). direct transformation in terms of standard random variables is often difficult to obtain. the probability distribution function of an input is specified in terms of a set of measurements (i.1 presents a list of transformations for some probability distributions commonly employed in transport-transformation modeling. This may be true for some distributions for which direct transformations are not available in the literature.. respectively. and is described briefly here: . the interdependence of a set of n correlated random variables is often described by a covariance matrix. and similar distribution functions. The transformation process for n correlated random variables with zero mean. obtain the value of the input x. and g(x) is the cdf of x 2 2 (3. where z = √ 1 1 + erf(ξ/ 2) . is presented by Devroye [49].2. In this process. where ηxi and ηxj are the means of xi and xj . x is sampled uniformly from the all its percentiles. In such cases. x can be approximated in terms of an srv ξ. is a symmetric matrix with the following elements: Σi. or (b) a look-up table. as described in Appendix A. for which Fx (x) equals the sample value of z • Repeat the above steps for additional samples. or (c) a numerical subroutine.4 Transformation of Correlated Inputs For cases in which the random inputs are correlated. The covariance matrix. Σ.1)) • For the calculated value of z. and hence the samples represent the true distribution of x.42 g(x) can be one of the following: (a) an algebraic function. as follows: • Generate a sample of srv ξ • Generate a uniform random variate z from ξ as z= (z corresponds to Uniform(0. These steps can be represented by the following transformation equation: x = g −1 (z).2) √ 1 1 + erf(ξ/ 2) 2 2 3.j = E{(xi − ηxi )(xj − ηxj )}. with n components. using xi = µi + σi z i . The above method is generalized here for the case of transformation of an arbitrary random vector X with a covariance matrix Σ. the following conditions are satisfied: E{BB T } = 1 E{A} = HE{B} = 0 E{AAT } = HE{BB T }H T = HH T = C The problem reduces to finding a nonsingular matrix H such that HH T = C.1 • obtain the required vector X.j /(σi σj ) i. If the ith random variable. we start with a vector B comprising of n iid unit random variables. with mean 0 and covariance C. xi . • calculating the vector Y through yi = gi (ξi ). Since covariance matrices are positive semidefinite. assuming a linear transformation A = HB. • calculating the vector Z through Z = HY . the steps involved in the transformation into functions of srvs are as follows: • construct a new vector Y such that y i = (xi − µi )/σi • construct a new covariance matrix Σ∗ such that Σ∗ = Σi. and standard deviation σi . ξi ’s. given a random vector X with n elements and a covariance matrix Σ.43 To generate a random vector A. . Then.j • solve for a nonsingular matrix H such that HH T = Σ∗ • obtain transformations gi (ξi ) for the ith random variable y i . as described in Table 3. where Z = HY Thus. mean µi . has a probability density function fi (x). using transformation methods for univariate random variables. the sampling from X involves the following steps: • sampling of n iid normal random variables. one can always find a nonsingular H. and • calculating the vector X through xi = µi + σi z i . a) =    a 1 x 2 −1 e−x/2 . mole fractions vary independently. X1 + X2 + . as follows: E[Xi ] = xi. Hence. This distribution can also be used to represent uncertainty in compartmental mass fractions in physiological modeling. the expansion is called “polynomial chaos expansion” [83]. When normal random variables are used as srvs. the output metrics can also i=1 be represented in terms of the same set. for example. should take into account uncertainties in all inputs. Each component of a Dirichlet distribution is specified by a χ2 distribution.3 Step II: Functional Approximation of Outputs An output of a model may be influenced by any number of model inputs. as the uncertainty in the outputs is solely due to the uncertainty of the inputs [198]. + Xn 3. x > 0 a/2 Γ(a/2)2   0.m Θ The ith component can now be directly obtained as: xi = Xi where n is the number of components in the Dirichlet distribution. an output can be approximated . independent of each other. any general functional representation of uncertainty in model outputs. yet obeying the condition that their sum remains fixed. in terms of Hermite polynomials. . and the Dirichlet distribution parameter Θ. This work addresses one specific form of representation. the series expansion of normal random variables. but their sum always equals unity. . xi. if the inputs are represented in terms of the set {ξi }n .44 Specific Example: The Dirichlet Distribution The Dirichlet distribution provides a means of expressing quantities that vary randomly. it can be used to represent uncertainty in chemical composition. For a deterministic model with random inputs. given by f (x. In environmental modeling. and is related to the mean of the ith component. elsewhere The mean of the χ2 distributions is a.m . . .5) U3 = a0. It is known that the set of multi-dimensional Hermite polynomials forms an orthogonal basis for the space of square-integrable pdf s. .3) where. ξi3 ) + . given by Γp (ξi1 .2 (ξi − 1) + i=1 n i=1 j>i n 2 aii. ∂ξip p (3. an uncertain model output U .3 ξi ξj ξk (3.3 ξi ξj + aijk.2 ξi ξj (3. ∂ξi1 . (3. .. . ξi2 ) + i1 =1 i2 =1 i3 =1 ai1 i2 i3 Γ3 (ξi1 . Hermite polynomials on {ξi }n are random variables. .2 ξi + ai. The order of the expansion can be selected to reflect accuracy needs and computational constraints.3 ξi + i=1 n aij. the accuracy of the approximation increases as the order of the polynomial chaos expansion increases. .45 by a polynomial chaos expansion on the set {ξi }n . ’s are deterministic constants to be estimated. that are used to repk=1 resent input uncertainty. ξip ) are multi-dimensional Hermite polynomials of degree p. .3 + n−1 + i=1 j>i aij. can be expressed as second and third order Hermite polynomial approximations. U2 and U3 as follows: n n n−1 2 aii.4) where ξ is the vector of p iid normal random variables {ξik }p . E[Γp Γq ] = 0 iff Γp = Γq . since i=1 they are functions of the random variables {ξi }n . given by: i=1 n n i1 y = a0 + i1 =1 n i1 ai1 Γ1 (ξi1 ) + i1 =1 i2 =1 i2 ai1 i2 Γ2 (ξi1 .3 (ξi − 3ξi ) n U2 = a0. ξi2 . the Hermite polynoi=1 mials defined on {ξi }n are orthogonal with respect to an inner product defined as the i=1 expectation of the product of two random variables [83]. . For example. . . . and the Γp (ξi1 .3 (ξi − 1) + i=1 n i=1 n 2 aijj. y is any output metric (or random output) of the model. ξip ) = (−1) e ξ ξ p 1 T 2 1 T ∂ e− 2 ξ ξ .2 + i=1 n ai.6) . and that the polynomial chaos expansion is convergent in the mean-square sense [83].3 (ξi ξj − ξi ) + i=1 j=1 i=1 j>i k>j n−2 n−1 n 3 aiii. In general. . the ai1 .. Furthermore. Thus. the unknown coefficients in the polynomial chaos expansion can be determined by minimizing an appropriate norm of the residual.. • For nonlinear operators with mathematically manipulable equations. output metrics can be directly calculated as explicit functions of the srvs (ξi s). aij.. ’s) can be estimated by one of the following methods.m2 . • For cases in which the model equations are not easy to manipulate. the unknown coefficients can be obtained by a collocation method [198.7) (3. depending on the complexity of the model: • For cases in which the model is invertible.m3 . respectively. The unknown coefficients are estimated by equating model outputs and the corresponding polynomial chaos . and aijk. thus making the residual at those points equal to zero. ai.m2 .46 where n is the number of srvs used to represent the uncertainty in the model inputs. In such simple cases. are: N2 = 1 + 2n + N3 n(n − 1) 2 3n(n − 1) n(n − 1)(n − 2) = 1 + 3n + + . it can be seen that the number of unknowns to be determined for second and third order polynomial chaos expansions of dimension n. or when the model is of “black-box” type. after substituting the transformed inputs into the model equations. and the coefficients ai.4 Step III: Estimation of Parameters in the Functional Approximation The unknown coefficients in the polynomial chaos expansion (aik.8) 3. From the above equation. there is no need to assume a functional form for the output metrics.m3 . 208]. This method imposes the requirement that the estimates of model outputs are exact at a set of selected collocation points. the expressions for input variables in terms of ξi s are substituted into the inverted model to obtain explicit functions for the output metrics in terms of ξi s. Gelarkin’s method [208] is commonly used with weight function corresponding to the expectation of the random variables [198]. denoted by N2 and N3 . 2 6 (3.m3 are the coefficients to be estimated. aij. are approximately equal.1 Deterministic Equivalent Modeling Method/Probabilistic Collocation Method (DEMM/PCM) The Deterministic Equivalent Modeling Method (DEMM) [198] incorporates the polynomial chaos expansion technique. In DEMM/PCM the collocation points are selected following the orthogonal collocation method suggested by Villadsen and Michelsen [208]. described here. 159]. for each output metric. 3. This method uses the Galerkin’s method to relate the input and output approximations for models with tractable equations. to provide a prototype language for automated uncertainty analysis. a set of linear equations results with the coefficients as the unknowns. in order to solve for a two dimensional second order polynomial chaos expansion.47 expansion.− 3 and 0 are used.4. at a set of collocation points in the parameter space. and hence is regarded as an “optimal method”. The collocation points correspond to the roots of the polynomial of one degree higher than the order of the polynomial chaos expansion. For one-dimensional problems. the expansion is assumed to have converged. If the estimates of the pdf s of the output metrics of concern. is used for applying DEMM in the case of black-box models. the roots of the √ √ third order Hermite polynomial. the higher order approximation is used to estimate the pdf s of output metrics. these equations can be readily solved using standard linear solvers. DEMM uses either the Galerkin’s method or the Probabilistic Collocation Method (PCM). along with symbolic computation methods and compiler technology. the next order polynomial chaos expansion is employed. 3. Thus. This approach is adapted for multi-dimensional cases in DEMM [198]. For example. the Probabilistic Collocation Method (PCM) [199. hence the possible collocation . Depending on the complexity of the model. As a part of DEMM. Then. The coefficients of the polynomial chaos expansion are obtained using the model outputs at selected collocation points. and the solution process is repeated. this method gives the same results as Galerkin’s method [208]. the number of collocation points should be equal to the number of unknown coefficients to be found. for the case of n = 6. this poses the problem of the optimal selection of collocation points. and model results at these points cannot be obtained. There are nine possible collocation points. and similarly 46 for a third order expansion.6 with n=2 and taking terms only up to Γ2 . 3). which introduces a problem of selecting the appropriate collocation points. using Equation 3. Since the collocation points are selected from the combinations of the roots of one order higher Hermite polynomial.− 3). for higher dimension systems and higher order approximations. (− 3. the collocation points are typically selected at random from the set of available points.6 and 3. the number of collocation points required for second and third order expansions are 28 and 84 respectively. the selection of the required collocation points in an efficient manner from the large number of possible points is not addressed in the present implementation of DEMM. hence the number of possible collocation points is 36 . respectively . any choice of collocation points from the available ones. 3). ( 3. However.48 √ √ √ √ √ √ √ √ √ √ points are (0. the number of available collocation points increases exponentially.3. the number of collocation points available for second and third order expansions are 36 = 729 and 46 = 4096. In the absence of selection criteria at present. (− 3. 3) √ √ and ( 3. and there are six variables.0).− 3). the number of available collocation points increases exponentially. (0. but from Equations 3. some of the collocation points could be outside the range of the algorithmic or numerical applicability of the model. as the number of degrees of freedom increases. Furthermore.0). These issues are addressed in the following .0). ( 3. in a computational setting. (0. Similarly. In fact. should give adequate estimates of the polynomial chaos expansion coefficients.there are three roots for a third order Hermite polynomial (used for obtaining collocation points for a second order expansion).− 3). (− 3. there are only six unknowns. Limitations of DEMM/PCM As noted above. For example. the number of available collocation points is always greater than the number of collocation points needed. As the number of inputs and the order of expansion increase. different combinations of collocation points may result in substantially different estimates of the pdf s of output metrics. In principle. For example.2 Efficient Collocation Method (ECM) This method addresses some limitations of DEMM/PCM. For terms involving only one variable. and − √ 3 − 6 are used as the collocation points. the roots √ √ √ of the fourth order Hermite polynomial. 208]. all the standard normal variables (ξ’s) are set to value zero. − 3 + 6. Further.4. However. i. a “corresponding” collocation point is selected.4.1. Here. The limitations of DEMM/PCM are shown in the context of a case study involving a human Physiologically Based Pharmacokinetic (PBPK) Model. For example. 3. Hence. the exclusion of the origin as a collocation point could potentially lead to a poor approximation.. the collocation point corresponding to the constant is the origin. 3 − 6. The points are selected so that each standard normal random variable ξi takes the values of either zero or of one of the roots of the higher order Hermite-polynomial. and the results are presented in Section 5. and by letting the corresponding variable take values as the roots of the higher order Hermite polynomial. A simple heuristic technique is used to select the required number of points from the large number of potential candidates: for each term of the series expansion. the collocation points for the estimation of the coefficients of the polynomial chaos expansion are selected based on a modification of the standard orthogonal collocation method of [198. where an algorithm for collocation point selection method is described. For terms involving two or more random variables. for a third order approximation. the points which are closer to the origin are preferred. the origin corresponds to the region of highest probability for a normal random variable. as they fall in regions of higher probability.e.49 section. and so on. the collocation points are selected by setting all other ξ’s to zero value. Another shortcoming of the standard collocation technique suggested in DEMM is that sometimes the roots of the Hermite polynomials do not correspond to high probability regions and these regions are therefore not adequately represented in the polynomial chaos expansion. the values of the corresponding variables are set to the values of the roots of the higher order polynomial. when there is still an . If more points “corresponding” to a set of terms are available than needed. namely 3 + 6. 2. the Reactive Plume Model. (RPM-IV). Furthermore.4. This method resulted in accurate estimates of the output pdf s while requiring much fewer model simulations as compared to standard Monte Carlo methods. which is presented in Section 5.4. while following a technique similar to the “orthogonal collocation” approach. For that case study. version 4. This is attributed to the inherent instability of the collocation based approaches.4– 5. 3. The advantage of this method is that the behavior of the model is captured reasonably well at points corresponding to regions of high probability.1. the pdf s of the output metrics are likely to be approximated better than by a random choice of collocation points. especially with polynomial approximations of high orders. as described by Atkinson [6]. but robust . The ECM method was applied to a more complex model.50 unresolved choice. the collocation point is selected randomly. Applicability and Limitations of ECM The ECM method was applied to a case study involving a human Physiologically Based Pharmacokinetic (PBPK) model. as the collocation points are selected to correspond to the terms of the polynomial chaos expansion. the collocation points are selected such that the overall distribution of the collocation points is more symmetric with respect to the origin.7 in Section 5. Thus. the ECM method failed to converge for a third order polynomial chaos approximation. more points are available. singularities can be avoided in the resultant linear equations for the unknown coefficients. the two-dimensional atmospheric photochemical plume model. Thus any one collocation point in the model space could significantly alter the behavior of the polynomial [6]. regression-based methods provide slightly more computationally expensive. as described in Section 5. as shown in Figures 5. In this context.4.1.3 Regression Based SRSM Collocation methods are inherently unstable. If still.3. since the requirement that the polynomial (curve or a surface) has to pass through all the collocation points. described in Section 5. the ECM method is denoted as “ECM” and the regression based method is denoted simply as “SRSM”. as described earlier. is offset by the robustness this method provides for many cases. joint . The model outputs at the selected sample points are equated with the estimates from the series approximation. 3. Hence this method is recommended for use in general. an atmospheric photochemical plume model. such as the density functions. resulting in a set of linear equations with more equations than unknowns. and is the approach used in the case studies presented in this work. The higher computational burden.3. The number of sample points selected must be higher than the number of unknown coefficients to be estimated. compared to the ECM. the statistical properties of the outputs. This extension to the collocation method uses regression in conjunction with an improved input sampling scheme to estimate the unknown coefficients in the polynomial expansion. This system of equations is then solved using the singular value decomposition method [163]. and the influence of each sample point is moderated by all other sample points. because model results at more points are used in this approach. Unless otherwise specified. but the method requires a higher number of model simulations to obtain estimates of output uncertainty.5 Step IV: Estimation of the Statistics of the Output Metrics Once the coefficients used in the series expansion of model outputs are estimated. In the regression based method. selecting a number of points equaling twice the number of coefficients is recommended for obtaining robust estimates. Throughout the rest of this document. The advantages of this method are demonstrated by comparing the results of this method with those of ECM for a the case study involving RPM-IV.2.51 means of estimation of coefficients of the functional approximation. The regression method has the advantage that it is more robust than the collocation method. the regression based method is found to be robust and the method resulted in good approximations of model outputs. moments. the term SRSM in the case studies refers to a regression based SRSM. Based on preliminary case studies (presented in Chapter 5). a set of sample points is selected in the same manner as the ECM method. joint moments. then the following steps are involved in the estimation of the statistics of the inputs and outputs. or between an output and an input. This results in a large number of samples of inputs and outputs.j j=1 N 1 N −1 1 3 N σyi 1 4 N σyi (yi. and hence substantial savings in the computer time are accomplished. These samples can then be statistically analyzed using standard methods.j − ηyi ) j=1 N (yi.9..j − ηyi )4 j=1 Further. and does not involve model runs. if the inputs xi ’s are represented as xi = Fi (ξi ). the moments of the distribution of an output y i can be calculated as follows: ηyi 2 σ yi = Mean(y i ) = = = = Var(y i ) Skew(y i ) Kurt(y i ) = = = E{y i } E{(y i − ηyi )2 } = = 1 N N yi. From a set of N samples. . One way of accomplishing this is through the generation of a large number of realizations of the srvs. • calculation of the values of input and output random variables from the samples. ξ2 . and the calculation of the values of inputs and outputs from the transformation equations. . As an example.yi γ2. correlation between two outputs. and if the outputs y j ’s are estimated as xi = Gi (ξ1 . • generation of a large number of samples of (ξ1 ξ2 . ξn. can be readily calculated. etc. . • calculation of the moments using Equation 3. ξn ).yi 1 E{(yi − ηyi )3 } = 3 σ yi 1 E{(yi − ηyi )4 } = 4 σ yi (yi.9) 3 γ1. .j − ηyi )2 j=1 N (3. It must be noted that the calculation of model inputs and outputs involves evaluation of simple algebraic expressions.52 densities.i ). and • calculation of density functions and joint density functions using the sample values. the correlation coefficient of an input xk and an output y i can be calculated . j − ηxk )(yi. and two outputs Y1 and Y2 .7 An Illustration of the Application of the SRSM As an illustration. the expansion is assumed to have converged. consider a computational model with three independent random inputs X1 .6 Step V: Evaluation of Convergence. is used.j − ηyi ) j=1 (3. and the higher order approximation is used to calculate the pdf s of output metrics. where the input random variables . the convergence of the approximation is determined through comparison with the results from a higher order approximation. This verification is performed to ensure that the approximation converges to the true distribution. yet another series approximation. of the next order. X2 . The efficiency of the SRSM is further evaluated by comparing the number of simulations required for the SRSM method. The next order polynomial chaos expansion is used.yi E{(xk − ηxk )(y i − ηyi )} 1 = = σ xk σ y i N σ xk σ yi N (xk. Accuracy and Efficiency of Approximation Once the coefficients of the polynomial chaos expansion are obtained using one of the methods described above. and the entire process is repeated until convergence is reached. and the process for the estimation of unknown coefficients is repeated. or the correlation between two outputs.10) Similarly. the accuracy of the approximation is evaluated by comparing the results from SRSM with the results obtained from Monte Carlo analyses. If the estimates differ significantly. with the number required for a Latin Hypercube sampling method. In the present work. or between an input and an output can be directly calculated. If the estimates of pdf s of output metrics agree closely. and X3 . 3. 3.53 using the following: rxk . higher moments of outputs. . . (ξ1. These sample points correspond to the original model input samples. in the context of the ECM method. . (x1.4.2 ξ3..1 ) . ξ3 is given by 2 2 2 Y1 = a0 + a1 ξ1 + a2 ξ2 + a3 ξ3 + a4 (ξ1 − 1) + a5 (ξ2 − 1) + a6 (ξ3 − 1) +a7 ξ1 ξ2 + a8 ξ2 ξ3 + a9 ξ1 ξ3 2 2 2 Y2 = b0 + b1 ξ1 + b2 ξ2 + b3 ξ3 + b4 (ξ1 − 1) + b5 (ξ2 − 1) + b6 (ξ3 − 1) (3. .54 X1 .N ξ2. ξ2 .N ) These sample points can be readily generated using a simple algorithm developed as a part of the SRSM implementation. ξ2 . ξ3 . q3 ) The input random variables can be represented by three srvs. ξ2 .1 ).13) +b7 ξ1 ξ2 + b8 ξ2 ξ3 + b9 ξ1 ξ3 In order to estimate the 10 unknown coefficients (for each output) from the above equation.1 ξ3. .2 ). A second order polynomial chaos approximation for Y1 and Y2 in terms of ξ1 .e.2 ξ2.1 x2. q1 ) X2 = Lognormal(p2 . in the form (ξ1.1 ξ2. using Table 3.1 x3.N ξ3. The selection of the sample points is discussed in detail in Section 3. q2 ) X3 = Gamma(p3 .2. the selection of a set of N sample points (equaling about twice the number of sample points.1) random variables. . 20 in this case) is recommended for regression based SRSM.1 as follows: X1 = p1 + (q1 − p1 ) X2 = exp(p2 + q2 ξ2 ) X3 = p3 q3 ξ3 2 2 +1− 9p3 9p3 3 (3. (ξ1.11) √ 1 1 + erf(ξ1 / 2) 2 2 (3. i. ξ3 are iid N(0.12) Where ξ1 . ξ1 . X2 and X3 are given by X1 = Uniform(p1 . .   y2.1 x3.N x3..15)  2 2 2 ξ1.2 − 1 ξ2.1 x2.3 ξ3.N     ξ2.N x2.1 − 1 ξ3. Z can be calculated from the values of ξ’s at each sample point.2 ξ3. b10 by solving the following linear equations through singular value decomposition.i and y2...i )    3  2 2 +1− p3 q3 ξ3.i   x1.i are the corresponding model outputs. .N . .2 ξ1.3 − 1 . 2 2 2 ξ2.1 ξ1.2 ξ2.  .N − 1     2 ξ3.1 ξ3.3 . .i          →  x2. ..1 1 ξ1. .3 ξ3.N  a0 b0    a1 b1    y1.. . the only . 1   ξ1. Then the outputs at these sample points.. .1 ξ2. . .         T Z ×          1               Z=              ξ1..N (3.3 .3 ξ2.3 ξ1.2 1 ξ1.N and y2. . .2 − 1 ξ3. . . After obtaining the original model input sample points.N ).    a8 b8  y1.1 ξ2.. whereas.2 ξ2.i      =       p1 + (q1 − p1 ) 1 + 1 erf(ξ1. . Thus.. . N . the model simulation is performed at the points given by (x1. y1.1 ξ3.   .3 − 1 . .1 ξ1.  .14) for i = 1 . . (3.N x2. 2 2 2 ξ3.. .N    ξ2.1     y a2 b2   =  1. .1   y2.N    ξ3. .N − 1    ξ1. y2. .i     ξ3.3 ξ2.i / 2)  2 2     exp(p2 + q2 ξ2.N ξ3.3 ξ3.1 ξ3. y1.i 9p3 9p3 √            (3..3 − 1 .N  a9 b9  y2. .   .2 − 1 ξ1.N ξ2.N x3. (x1. where .i         x3.2   ..N − 1    2 ξ2. .2  .N ).1 .1 − 1 ξ1..1 ) .16) ξ1. .3 ξ2.55 (x1. In the above equations.  .N    ξ1. .N    2 ξ1. as follows:      ξ1. y1.N ξ3.2 ξ1.i      ξ2.1 − 1 ξ2.1 ξ2.2 ξ3. . a10 and b1 . .2 ξ2.. are used to calculate the coefficients a1 . .2 ξ3.1 . the distributions of Y1 and Y2 are fully described by the polynomial chaos expansions as shown in Equation 3.8. . joint moments. Once the coefficients are estimated.14. calculating the values of model inputs and outputs for those samples.56 Input Distributions Model Output Distributions Generate a set of random inputs Figure 3.i ).2. the coefficients ai ’s and bi ’s.1 Computational Implementation Implementation of the Conventional Uncertainty Propagation Methods Modules for the application of conventional Monte Carlo and the Latin Hypercube Sampling methods were developed as part of this work. 3. such as the individual moments. they can be used in isolation. correlations between the outputs and correlations between inputs and outputs. (b) performing model simulations for the combination of the sampled inputs.15.i ξ3.2: Schematic depiction of the steps involved in the application of conventional Monte Carlo method unknowns in Equation 3. Though these modules are mainly used in this work for the evaluation of alternative methods. and (c) statistically analyzing the model outputs.5. the conventional Monte Carlo method involves the following steps: (a) obtaining random samples from the probability distributions of the inputs. can be readily estimated. The next step involves determination of the statistical properties of the outputs. As shown in Figure 3.8 3. This process is done by numerically generating a large number of random samples of (ξ1. and computing the statistical properties from the values of inputs and outputs as described in Section 3.i ξ2. and applying the transformation shown in Table 3. Further. The probability density is then plotted against the range of values that the output can belong to.1. in the study involving a physiologically based pharmacokinetic model. The ratio of this probability to the interval size gives the probability density at the mid-point of the interval. these were based on equations presented in Section 3. R-GAUSS. the normal random numbers were generated through the function GASDEV provided with the mathematical routines of the Numerical Recipes [163]. [108].1. for the PCs have LHS routines embedded in the modeling platform (typically spreadsheet based). FORTRAN routines to calculate the mean. For all the cases. and the Monte Carlo method involved sampling from random numbers generated by the pseudo-gaussian random number generator function in SimuSolv. In the case of Dirichlet distributions. the original numerical models were modified accordingly to accept the uncertain model inputs and parameters as inputs from a file. In all the case studies described in this work. and model simulations were performed at the sampled values of the inputs. described in Section 5. the model was implemented on the SimuSolv modeling platform [58]. mainly because of the lack of availability of a standardized routine readily usable on Unix operating system. and the corresponding ratios were calculated using Equation 3. and higher moments were also implemented as part of the SRSM. The outputs of the simulations were then analyzed by “binning” them into a set of intervals. from all the model simulations performed. as presented in the case studies. Gamma random variables were generated by first generating a normal random number. gives the probability that the output belongs to that interval. a bin number associated with an interval contains the number of outputs realizations that belong to that interval. the independent random numbers were generated first. developed by Iman et al. In one specific case. and other commercial software such as Crystal ball [46] and @RISK [218]. The Latin Hypercube Sampling (LHS) method was developed and implemented from basic principles.57 The random numbers for sampling the input distributions were generated depending on the case study considered. The ratio of the bin number to the total number of samples. The EPA approved routine for the LHS.4.5.2. is developed for the VAX/VMS operating system. variance. and . The cell number indicates the segment number the sample belongs to. a cell number (2. till Ntot samples are generated. The routine generates a set of M sample points in each iteration. This is accomplished as follows: first.e.58 thus cannot be used in conjunction with stand-alone numerical codes. segment 1 with respect to second parameter. At each successive step. and is accepted only if it does not agree with any previous sample on any of the segment numbers. with respect to each of the parameters. but correspond to extreme values). Thus. For example. Further. the parameter space is divided into 4 × 4 × 4 cells. and (c) the input probability distributions. In order to generate samples from input distributions. is partitioned into M N cells. The main aspects of the LHS implementation developed here are presented in the following. with each cell corresponding to equal probability regions from the input sampling space. and its cell number is calculated. A much higher value results in too small a range for each sampled input. For example. the whole parameter space. The LHS FORTRAN subroutine developed here requires the following inputs: (a) the total number of samples to be generated. the number of ranges the parameter range of each input should be divided (M ) is also required as an input. the range of probable values for each uncertain input parameter is divided into M segments of equal probability. The next step involves the generation of M samples from M N cells. and the distribution parameters (dependent on the distribution type). for every M samples. regions that have very low probabilities of occurrence. each parameter is sampled from each of its M subranges. (b) the number of random inputs. Ntot . thus providing samples from the tails of the probability distributions (i. requiring a large number of trial samples before an acceptable sample is generated.. Thus. for the case of 3 input parameters and 4 segments. . and segment 2 with respect to third parameter. consisting of N parameters. no two samples have any input corresponding to the same segment. a random sample is generated. Further. The advantage of this approach is that the random samples are generated from all the ranges of possible values. whereas much smaller values do not adequately represent all ranges of possible values. A typical value of M used in the case studies is about 50. consisting of the distribution type (a number).2) indicates that the sample lies in the segment 2 with respect to first parameter. N . a random sample is generated.1. 2 Implementation of the Stochastic Response Surface Method The steps involved in the application of the SRSM are presented in Figure 3. and to calculate the numerical values of the polynomial chaos expansions. the third order expansion was the highest required. to generate polynomial chaos expansions of arbitrary order. • A FORTRAN subroutine to generate polynomial chaos expansions of up to fifth order for an arbitrary number of parameters.3: Schematic depiction of the steps involved in the application of the Stochastic Response Surface Method especially the tails of the distributions. . The routine. The extension to higher orders is straightforward.59 Input Distributions Model Output Distributions Select a set of srvs and transform inputs in terms of srvs Express outputs as series in srvs with unknown coefficients Generate a set of regression points Estimate the coefficients of output approximation Figure 3. The implementation of the SRSM is based on the following routines: • A symbolic calculation routine in Maple [94].8. • A set of FORTRAN utilities to sample distributions in the form of the srvs. This routine is developed using the results from the Maple routine. but not pursued because in all the case studies considered. to calculate the corresponding transformed random variables. • A FORTRAN subroutine to generate the sample points based on the input distributions and the order of the polynomial chaos expansion selected.3. 3. consisting of an arbitrary number of parameters. along with all other routines is included in the CDROM accompanying this document. the application of the SRSM method for a simple black-box model is presented.60 • A FORTRAN subroutine to perform collocation or regression (based on the scenario) on a set of sample points and the corresponding model outputs. These are used to estimate the coefficients of a polynomial chaos expansion. In order to use the SRSM. as shown in Figure 3. at the Environmental and Occupational Sciences Institute. and displays the probability density functions for each of the outputs. 3.4. Additionally. and other statistical properties of both the inputs and outputs.edu/srsm. Here. The program responds with the recommended sample points. that could be used for uncertainty propagation with black-box models. This has also been implemented as a “Web Based” tool.rutgers. have to be specified. a joint project of University of Medicine and Dentistry. New Jersey.9 Implementation of a Web Interface to the SRSM All numerical routines associated with the development of the SRSM have been incorporated into a a stand-alone application. and the order of polynomial chaos expansion desired. An operational version can be accessed through the location http://www.html The web server is located in the Computational Chemodynamics Laboratory.5. as shown in Figure 3.ccl. The State University of New Jersey. The interface was implemented by using the Common Gateway Interface (CGI) programming using the Perl language. their probability distributions. by obtaining a large number of samples of srvs.6. first the number of inputs. • A FORTRAN subroutine to generate the probability density functions from the coefficients of a polynomial chaos expansion in srvs. The program uses the model outputs at these sample points and calculates. a range for plotting the output probability density functions is also required as input. the number of outputs. percentiles. and calculating the statistics of the corresponding outputs. as shown in Figure 3. . and Rutgers. • FORTRAN subroutines to calculate the moments. and the model outputs at these sample points have to be input. that can be run from remote web browsers. 61 Figure 3.4: Application of the web based SRSM tool: specification of input uncertainties . 62 Figure 3.5: Application of the web based SRSM tool: recommended sample points and the corresponding model outputs . 6: Application of the web based SRSM tool: estimates of the pdf s of the outputs .63 Figure 3. 64 Chapter 4 COUPLING OF THE SRSM WITH SENSITIVITY ANALYSIS METHODS: DEVELOPMENT AND IMPLEMENTATION OF SRSM-ADIFOR The objective of this effort is to couple the Stochastic Response Surface Method (SRSM) with sensitivity analysis methods, to obtain savings in the computer resources required for uncertainty propagation. In the construction of an approximate model, sensitivity analysis methods can provide information that can help reduce the number of model simulations required to estimate the approximation parameters. Consequently, the use of sensitivity information in conjunction with uncertainty propagation methods can reduce the model simulations required to estimate uncertainties in model outputs. The rationale for the coupling of the SRSM with a sensitivity analysis method is illustrated in Figure 4.1 for the simple case of approximation of a curve in a 2-D Euclidean space using a set of sample points. Figure 4.1 (A) represents the model response, which can be approximated accurately using the five sample points, as shown in Figure 4.1 (B). In contrast, using three sample points, only a poor approximation as shown in Figure 4.1 (C) can be obtained. However, using the derivative information at the three sample points (tangents at these points), a good approximation, as shown in Figure 4.1 (D) can be obtained. As an extension of this reasoning to multi-dimensional systems, the partial derivatives of model outputs with respect to model inputs (i.e., tangents in the multi-dimensional space) can be used to reduce the number of sample points required to construct an adequately approximate model. An overview of various sensitivity metrics is presented in Section 2.4, and the metrics are listed in Table 2.3. Of these sensitivity metrics, the derivative information can be 65 B E A C (A) Model Response (B) Approximation with sufficient data D E C A A C E (C) Approximation with fewer data (D) Approximation with fewer data and derivative information Figure 4.1: Rationale for coupling of the SRSM with a sensitivity analysis method directly utilized in constructing an approximate model. For example, consider a model z = f (x, y) for which the output z is to be approximated as a function of inputs x and y through the expression: z = a0 + a1 x + a2 y + a3 x2 + a4 y 2 + a5 xy In order to estimate the coefficients ai ’s, the model outputs for z are needed at six sample points, requiring six model runs. This results in a set of six equations with the left hand side consisting of the model outputs and the right hand side consisting of linear equations in the coefficients. For the same case, if all the first order partial derivatives are available at each sample point, the number of model runs can be reduced to two, and the derivative information 66 at two points, (x1 , y1 ) and (x2 , y2 ) can be used to construct the following closed system of equations: z|(x1 ,y1 ) = a0 + a1 x1 + a2 y1 + a3 x1 2 + a4 y1 2 + a5 x1 y1 ∂z ∂x ∂z ∂y ∂z ∂x ∂z ∂y = a1 + 2a3 x1 + a5 y1 (x1 ,y1 ) = a2 + 2a4 y1 + a5 x1 (x1 ,y1 ) z|(x2 ,y2 ) = a0 + a1 x2 + a2 y2 + a3 x2 2 + a4 y2 2 + a5 x2 y2 = a1 + 2a3 x2 + a5 y2 (x2 ,y2 ) = a2 + 2a4 y2 + a5 x2 (x2 ,y2 ) Similarly, for a model with M parameters, the number of model runs required to estimate the unknown coefficients decreases by a factor of M + 1, since the first order partial derivatives with respect to M parameters result in M additional equations at each sample point, whereas calculation of the model output at a given point results in only one equation. 4.1 Methods for Estimation of Partial Derivatives The following approaches are commonly used to estimate derivatives for a model [16, 29]: (a) Variational Equations: When the model equations are available, one can, in principle, compute the derivatives from these equations by directly differentiating the equations and performing the necessary algebraic manipulations. This approach results in an additional set of equations that can be solved either in a coupled manner [200], or in a decoupled manner [60]. For lower dimensional, linear models, these equations can be solved directly along with the model equations to estimate the derivatives. However, for nonlinear functions the derivatives are generally more complicated than the function itself. Further, if the original model has n outputs and m model parameters, in order to estimate the partial derivatives of all the outputs with respect to all the parameters, 67 the number of equations required would be m × n. Hence, this approach requires a considerable amount of effort for generating all the necessary model equations. Further, this approach requires access to the original model formulation equations. (b) “Brute-force” approach: The two commonly used techniques in this approach are: • Central differences: For the model represented by Equation 2.5, the partial derivative of ui with respect to a parameter kj at a point (k1,0 , . . . , kj,0 , . . . , km,0 ) is estimated by solving the model at two points (k1,0 , . . . , kj,0 − h, . . . , km,0 ) and (k1,0 , . . . , kj,0 + h, . . . , km,0 ), and the respective outputs are obtained as ui,l and ui,r . The partial derivative is then given by ui,r − ui,l ∂ui ≈ ; h→0 ∂kj 2h • One-sided differences: Here, the model is solved at a point (k1,0 , . . . , kj,0 , . . . , km,0 ), resulting in output ui,0 and either at the point (k1,0 , . . . , kj,0 − h, . . . , km,0 ) or at the point (k1,0 , . . . , kj,0 + h, . . . , km,0 ). The left-side derivative approximation is given by ui,0 − ui,l ∂ui ≈ ; h→0 ∂kj h and the right-side derivative approximation is given by ui,r − ui,0 ∂ui ; h→0 ≈ ∂kj h In order to estimate the partial derivatives of outputs with respect to m parameters, the central difference method requires solution of the model equations 2m times (i.e., 2m “model runs”), that is two times for each parameter considered. On the other hand, the one-sided difference method requires m + 1 model runs, one for each parameter considered, and one run at the point under consideration. The primary disadvantage of these methods is the computational cost associated with a large number of model runs. Another disadvantage with these methods involves the selection of a small enough value of h; a small h results in “roundoff errors”, resulting from subtracting two almost equal numbers, and a large value results in truncation as the omitted higher order terms become significant. By applying the chain rule ∂ f (g(t)) ∂t = t=t0 ∂ f (s) ∂s s=g(t0 ) ∂ g(t) ∂t t=t0 over and over again to the composition of those elementary operations. that can be considered as “black-box” structures. the derivative expression in essence doubles.68 errors. . one can compute. (d) Automated Differentiation: Automated differentiation. another advantage is that this method can be used in conjunction with complex numerical models. leading to a combinatorial explosion [16]. unable to deal with constructs such as branches.4. relies on the fact that every function is executed on a computer as a sequence of elementary operations such as additions. The main advantage of this method is that the partial derivatives can be estimated using a single model run. for every operation. also known as automatic differentiation. This approach essentially consists of using a computational tool that can “understand” the model code and subsequently produce the code for estimating partial derivatives. in a completely mechanical fashion. (b) the symbolic manipulators are in general. The following sections describe the ADIFOR methodology and the coupling with the SRSM. symbolic manipulators such as Maple [94] and MACSYMA [95] can be used for manipulating the model equations and obtaining expressions for the partial derivatives. because.5. multiplications. and (c) the derivative code expressions could be very complex. derivatives of f that are correct up to the machine precision [16]. loops or subroutines that are inherent in computer codes. ADIFOR (Automatic DIfferentiation of FORtran) is used in this work for coupling with the SRSM. The automatic differentiation method has been widely applied in sensitivity analysis of transport/transformation modeling. (c) Symbolic Differentiation: When the model equations are mathematically tractable. and elementary functions. The practical limitations are numerous: (a) many computational models do not have mathematically tractable equations. as mentioned in Section 2. the if and the do statements in Figure 4. Figure 4.3 (a))..g.3 (a) shows a segment of a sample Fortran program for which derivatives are desired.3 (b) shows the derivative code produced through the application of ADIFOR. The dependent variable is y and the independent variables are x(1) through x(10).2: Schematic illustration of the Automatic Differentiation method (adapted from Bischof et al. Figure 4. ADIFOR) DERIVATIVE CODE Further Analysis Derivative Computing Code Compile and Link Figure 4. Using the resultant code. and ADIFOR can calculate derivatives for .2 presents a schematic depiction of the automatic differentiation procedure using ADIFOR.g.2 Automatic Differentiation using ADIFOR ADIFOR (Automatic DIfferentiation of FORtran) [17] transforms the model source code based on the specification of the dependent and independent variables. the partial derivatives of y with respect to independent variable x(i) can be obtained through the corresponding element dy(i) from the array dy.69 COMPUTER MODEL (To Be Analyzed) Automatic Derivative Calculator (e. 1994 [16]) 4. Figure 4.. This example illustrates some key advantages of using ADIFOR over other methods: ADIFOR steps through the conditional execution loops and iterative loops (e.3 presents an example of the transformation of source code using ADIFOR. and produces source code that calculates the derivatives of the dependent variables with respect to the independent variables. Figure 4. 1) where. . . as described in Section 3. this method can be easily extended to compute derivatives for computer code of arbitrary length and complexity. the iterative loops and conditional loops are common in computer codes. . the ar. the user has to only (a) specify the information about the dependent and independent variables. . • an approximation for the model outputs is assumed in the same form as presented in Section 3.g. Since ADIFOR methodically follows the model code. Further. The coupled method. ξi2 . . the statement y =x(i) * y *y).i1 i2 i3 Γ3 (ξi1 . .2) .. . whereas they can be readily expressed in computer code.i1 . . and (b) add statements to the derivative code to output the derivatives. It must be noted that the value of the variable y depends to a great extent the “sign” of the various x(i)’s.i1 i2 Γ2 (ξi1 . denoted by the set {ξi }n . 4. . all the complex logic of the model code is followed exactly by the derivative code. ’s are deterministic constants to be estimated. . In principle.2. ∂ξip p (4. ξip ) = (−1)p e 2 ξ ξ 1 T 1 T ∂ e− 2 ξ ξ . ξi2 ) + i1 =1 i2 =1 i3 =1 ar. given by Γp (ξi1 . and the Γp (ξi1 .0 + i1 =1 n i1 ar. while they do not appear in algebraic or differential equations.i1 Γ1 (ξi1 ) + i1 =1 i2 =1 i2 ar. The following steps are involved in the i=1 application of the SRSM-ADIFOR: • the model inputs are expressed as functions of selected srvs. in the application of ADIFOR. .3 Coupling of SRSM and ADIFOR The coupling of SRSM and ADIFOR follows the same steps as the SRSM with respect to input and output transformations. such situations are difficult to express in algebraic forms. . yr is the rth output metric (or random output) of the model. Furthermore. SRSM approximates uncertain model outputs in terms of a set of “standard random variables” (srvs).. (4. ξip ) are multidimensional Hermite polynomials of degree p. specifically of x(n). SRSM-ADIFOR. ξi3 ) + .70 assignment and iteration statements (e.3: n n i1 yr = ar.. ∂ξi1 . ∂ξj • the model derivative code is generated using the original model code and ADIFOR and modified so that it can calculate the first order partial derivatives of model outputs with respect to the srvs. the first order partial derivatives of the rth model output with ∂yr respect to jth srvξj .10) do j=1.0) then y = x(i) * y * y endif enddo dimension x(10) dimension dy(10). n if (x(i) .i) + c*dy(j) enddo y = a*y endif enddo (a) (b) Figure 4.3) + i1 =1 i2 =1 i3 =1 ar.0 do i = 1.gt.3: FORTRAN derivative code generation using ADIFOR: (a) Sample code fragment and (b) ADIFOR generated derivative code • correspondingly. ξi2 ) ∂ξj (4. dx(10. ξi2 . 10 dy(j) = 0.n if (x(i) .0 enddo do j = 1. 0. given by are expressed as ∂ξj ∂yr ∂ξj n = i1 =1 n ∂Γ1 (ξi1 ) ar..0 enddo y = 1.0) then a = x(i)*y b = y*y c = a + y*x(i) do j = 1.i1 i2 i1 =1 i2 =1 ∂Γ2 (ξi1 .gt.0 do i = 1.i1 + ∂ξj i1 i2 n i1 ar..i) = 1.10 dx(i.i1 i2 i3 ∂Γ3 (ξi1 .71 dimension x(10) y = 1. 10 dy(j) = b*dx(j. ξi3 ) + . . 0. Here.3.1 is used to calculate the number of coefficients to be estimated (say K). Since for each sample point.4. to obtain estimates of ai ’s. The execution of the model derivative code at one sample point gives the model calculations for the outputs and M first order partial derivatives for each output. the recommended number of equations is about twice the number of coefficients to be estimated.1 Selection of Sample Points In the application of SRSM-ADIFOR to the uncertainty analysis of a model with M inputs. • outputs at these points are equated with the polynomial chaos approximation (Equation 4.3. as illustrated in Section 4. Equation 4.3 in conjunction with these calculations result in M + 1 linear equations at each sample point.4) the number is approximate because 2K may not always be exactly divisible by M + 1 . the number of sample points required for SRSM-ADIFOR. is approximately given by∗ : N≈ ∗ 2K M +1 (4. N . the number of recommended sample points is based on the rationale behind the regression based SRSM: the number of resulting equations should be higher than the number of coefficients estimated in order to obtain robust estimates of the coefficients. equations 4. 4. as shown in Equation 4. • The resultant equations are solved using singular value decomposition. Thus. the number of sample points selected is smaller than the number of coefficients to be estimated.4.4. resulting in a set of linear equations in ai ’s. and partial derivatives at these points are equated with the corresponding approximation given by Equation 4. Thus. Here. the number of resultant equations is M + 1. K coefficients need to be estimated for each output. • the model outputs and the partial derivatives with respect to the srvs are calculated at these sample points. for a given order of expansion.1.2. P outputs.1 and 4.72 • a set of sample points is selected using the sampling algorithm presented in Section 3. are given by Equation 3. X2 . compared to 20 sample points required in the application of the SRSM.73 4. N . Thus. the number of sample points.4.4 An Illustration of the Application of the SRSM-ADIFOR This section presents an illustration of the SRSM-ADIFOR method by revisiting the example presented in Section 3. required to estimate the 10 unknown coefficients equals 5. and the two model outputs. The probability distributions of the three model inputs.7 involving the application of the SRSM. the partial derivatives of Y1 and Y2 with respect to the srvs are given by: ∂Y1 ∂ξ1 ∂Y1 ∂ξ2 ∂Y1 ∂ξ3 ∂Y2 ∂ξ1 ∂Y2 ∂ξ2 ∂Y2 ∂ξ3 = a1 + 2a4 ξ1 + a7 ξ2 + a9 ξ3 = a2 + 2a5 ξ2 + a7 ξ1 + a8 ξ3 = a3 + 2a6 ξ3 + a8 ξ2 + a9 ξ1 (4.12. the uncertainty in the model outputs can be estimated by obtaining the model outputs at N = 5 sample points. The model equations corresponding to the estimation of unknown coefficients. X1 . and X3 . Y1 and Y2 .6) = = = b1 + 2b4 ξ1 + b7 ξ2 + b9 ξ3 b2 + 2b5 ξ2 + b7 ξ1 + b8 ξ3 b3 + 2b6 ξ3 + b8 ξ2 + b9 ξ1 Based on Equation 4. are approximated by polynomial chaos expansions 2 2 2 Y1 = a0 + a1 ξ1 + a2 ξ2 + a3 ξ3 + a4 (ξ1 − 1) + a5 (ξ2 − 1) + a6 (ξ3 − 1) +a7 ξ1 ξ2 + a8 ξ2 ξ3 + a9 ξ1 ξ3 2 2 2 Y2 = b0 + b1 ξ1 + b2 ξ2 + b3 ξ3 + b4 (ξ1 − 1) + b5 (ξ2 − 1) + b6 (ξ3 − 1) (4.5) +b7 ξ1 ξ2 + b8 ξ2 ξ3 + b9 ξ1 ξ3 Clearly. are given by the following equations: . using SRSM-ADIFOR. N                 (4. ξ1.N 0 0 0 1 0 1 2ξ3.1 .N ∂y2 ∂ξ1.1 ξ2.N ξ3. .1 ξ3.1 ξ3.N ∂y1 ∂ξ2.1 0 0 0 1 0 1 . .1 − 1 2 .N 2 . .1 ξ2.N ∂y1 ∂ξ1.1 ξ1.N 0 ξ1. . . . . .N y2. .1 0 1 0 0 0 0 1 0 0 2ξ2.1 ∂y2 ∂ξ2.1 − 1 2ξ1.1 ∂y1 ∂ξ1. . . . .N 4.1 .N − 1 0 0 ξ2. . .. y2.N ξ2.1 .1 ∂y2 ∂ξ1.. and then making changes in the resultant code so that the derivatives with respect to the srvs can be calculated.1 0 ξ1... ξ2.N ξ1.N − 1 ξ1. a8 a9 (4.1 ξ2. ξ1.1 ξ1. .N 1 ξ1.N .1 2 ξ2.N                       . .. ξ3.. .. . .1 ξ3. . by first processing the code using ADIFOR. .1 0 0 ξ3.   . . . where                               T Z ×            a0 a1 a2 .. .. ..         b8          b9         y1.N 0 0 ξ3.N ∂y1 ∂ξ3. .1 ∂y2 ∂ξ3. ξ2.5 Implementation of the SRSM-ADIFOR The implementation of the SRSM-ADIFOR involves the modification of the model source code.. ξ1.1 ξ2. .N − 1 2ξ1. .1 ∂y1 ∂ξ2. .8)              2 ξ1. . .N 0 2 2ξ3. . .. .N ξ3.1 .N ξ2. .. . .1 0 0 0 ξ2. . .. . . .N ∂y2 ∂ξ3. . .N ∂y2 ∂ξ2.N ξ3. 0 1 0 0 0 0 1 0 0 2ξ2.N . . .7)  1               Z=              ξ1.  . .. .N ξ3.1 − 1 2 ξ3.1 ∂y1 ∂ξ3.74             b0         b1          b2    =    . y1..N ξ2. .1 ξ3. .7. In order to accomplish this.4: Schematic depiction of the steps involved in the application of the SRSMADIFOR Subsequently the derivative code has to be coupled with a main routine that produces the derivatives of concern as program outputs. Figure 4. and to generate a matrix similar to the illustration shown in Equation 4. a set of routines is developed to generate expressions for the derivatives of a polynomial chaos expansion.4 presents the steps involved in the application of the SRSM-ADIFOR. The subsequent procedure is similar to the application of the SRSM.75 Input Distributions Model Output Distributions Select a set of srvs and transform inputs in terms of srvs ADIFOR application Estimate the coefficients of output approximation Express outputs as series in srvs with unknown coefficients Derivative Code Generate a set of regression points Figure 4. The calculated derivatives are then coupled with the SRSM to estimate the coefficients of the polynomial chaos approximations of the outputs. The main objective of these case studies is to evaluate the applicability of the SRSM and the SRSM-ADIFOR to a wide range of transport-transformation models. but is presented at the end. III: A three-dimensional urban/regional scale air quality model. both from the perspective of model application (such as biological.76 Chapter 5 CASE STUDIES FOR THE EVALUATION OF THE SRSM AND THE SRSM-ADIFOR This chapter presents four case studies to evaluate the SRSM and the SRSM-ADIFOR. through a few seconds (case II). as well as to the associated complexity. and potentially to numerical models used in other areas. and groundwater) and from the perspective of model complexity (ranging from a zero dimensional model to a three-dimensional urban/regional scale photochemical model). The computational time for each model simulation in the first three cases is of the order of a fraction of a second (case I). II: A two-dimensional atmospheric plume model. IV: A one-dimensional ground water model with discontinuous probability density functions. . air quality. The case studies are presented here in the following order: I: A zero-dimensional pharmacokinetic model. The order corresponds to both the chronological evaluation of the SRSM method. to approximately 6 hours (case III). since the complexity with respect to the specification of uncertainties in inputs is high. The groundwater model required about a few seconds per model simulation. The execution times of the models also vary from a fraction of a second to a few hours per model run. These models cover a range of applications. the uncertainties associated with the model are described. CML is considered to be an alternative indicator of risk than simply the total uptake of PERC. is of concern since PERC is a potential carcinogen [61]. Further.1 Description of the Case Study There is often significant uncertainty associated with human Physiologically Based PharmacoKinetic (PBPK) model parameters since human in vivo experimental data are not usually available for toxic chemicals. since the health risks are thought to be associated with the metabolism of . (b) area under venous blood from liver (AUCL). 5. along with a brief discussion at the end. a PBPK model for perchloroethylene (PERC) [61] is considered for uncertainty analysis. and the subsequent metabolism of PERC in the body. Here. The estimates from the uncertainty propagation methods are then presented graphically. The uncertainty in human PBPK parameters includes a significant amount of natural variability. in the form of probability density functions (pdf s). It is desirable to estimate the effects that uncertainties (including variability) associated with model inputs and parameters have on output metrics such as internal and biologically effective doses. Then. PERC is used as a cleaning solvent in the dry cleaning industry. Thus many of the parameters in human PBPK models are generally estimated by in vitro experimentation and by inter-species scale-up of animal PBPK model parameters. reflecting the interindividual variability inherent in human populations. and hence human exposures to it are very common. and the application of different uncertainty propagation methods is presented. and (c) the cumulative amount metabolized in the liver (CML).1 Case Study I: A Zero Dimensional Physiological System Uncertainty Analysis of a Physiologically Based PharmacoKinetic Model for Perchloroethylene 5. Human exposure to PERC.77 Each case study description contains the information about the model and its application in exposure/risk assessment. Three dose surrogates for PERC inhalation are considered in this study: (a) area under the arterial blood concentration (AUCA).1. In the second stage. • Stochastic Response Surface Method with Efficient Collocation (ECM). the DEMM approach and the ECM are evaluated through comparison with the results from a standard Monte Carlo simulation. the ECM. the human body is considered to consist of a set of compartments corresponding to different organs and tissues. also referred to simply as SRSM. respectively. For the second stage (11 parameters). • Latin Hypercube Sampling (LHS). AUCA and AUCL are indicators of the amount of time PERC resides in the arterial blood and liver. In the first stage (6 parameters).1).78 PERC in the body. The effect of uncertainty in the parameters of the PERC PBPK on the three dose surrogates is analyzed using the following: • Standard Monte Carlo approach. • Deterministic Equivalent Modeling Method/Probabilistic Collocation Method (DEMM/PCM). and additional details can be found in the literature [176. • Regression based SRSM. These five additional parameters are described by a set of mutually dependent Dirichlet distributed random variables.1 presents the schematic of the PERC PBPK model used in this case study. the effect of uncertainty in the metabolic constants (Km and Vmax ) and the partition coefficients are considered (see Table 5. Figure 5. and • Combined Application of the Stochastic Response Surface Method and Automatic Differentiation Method (SRSM-ADIFOR). The uncertainty in these six parameters are described by independent log-normally distributed random variables. and thus are also considered as alternative indicators of risk. 78]. The uncertainty analysis consisted of two stages: In the first stage. the uncertainty in compartmental volume proportions are included in the analysis. Appendix B presents the formulation of a PBPK model. Regression Based . These are modeled as a series of interconnected continuous stirred tank reactors (CSTRs). The preferred values for all parameters other than metabolic constants. The preferred values of metabolic constants are given by the geometric mean of values suggested by Hattis et al [93]. The PBPK model parameter uncertainties used in the present work are adopted from their work.1: Schematic representation of a PBPK model for PERC SRSM and SRSM-ADIFOR are evaluated by comparing the results from these methods with the results from the standard Monte Carlo and LHS methods. and are specified by the preferred values (PVs) and uncertainty factors (UFs) given in Table 5. where XP is the preferred value of parameter X. [67] examined the effect of PBPK model parameter uncertainty on the three dose surrogates.2 Specification of Parameter Uncertainty Partition coefficients and metabolic constants are assumed to be lognormally distributed. Farrar et al. and AUCL. The uncertainty factor for a parameter is defined such that the interval from XP /UF to XP UF is the 95% confidence interval. and by Reitz and Nolan [170].1. 5. according to Farrar et al [67]. In an earlier study. The pdf for a given lognormally distributed parameter is specified by assuming that the preferred value of the parameter corresponds Venous Blood Slowly Perfused . AUCA.79 Qrp Crp Qsp Csp Qfat Cfat Qliv Cliv Qca Cinh Rapidly Perfused Qrp Qsp Arterial Blood Fat Qfat Qliv metabolism Qca Liver Lungs Qalv Cexh Qalv Figure 5. correspond to the nominal values in the PERC PBPK model [61]. updated as suggested by Reitz and Nolan [170].1. CML. 25 1.37 Value 70 UF — .0% 2.3 2.50 1.4 87.1: Deterministic and uncertain parameters used in the uncertainty analysis of the PERC PBPK model Preferred Symbol BW Description body weight [Kg] Partition Coefficients Pblood/air Pfat/blood Psp/blood Prp/blood Pliv/blood blood:air partition coefficient fat:blood partition coefficient slowly perfused tissue:blood partition coefficient rapidly perfused tissue:blood partition coefficient liver:blood partition coefficient Blood Flows Qc Qp Qfat Qsp Qrp Qliv cardiac output [liters/hr] alveolar ventilation [liters/hr] blood flow to fat [liters/hr] blood flow to slowly perfused tissue [liters/hr] blood flow to rapidly perfused tissue [liters/hr] blood flow to liver [liters/hr] Compartment Mass Fractions Vfat Vsp Vrp Vliv mass of fat compartment [Kg] mass of slowly perfused tissue compartment [Kg] mass of rapidly perfused tissue compartment [Kg] mass of liver compartment [Kg] Metabolic Constants Km V max.04 1.69 9.6 1.84 23.47 0.6% 1.33 12.66 5.0 153 90.0 5.7 2.09 1.04 1.0% 5.09 1.25 1.15 11.0% 62.c Michaelis-Menten constant for metabolism [mg/liter] maximum rate of metabolism [mg/hr] 1.35 348 288 17.35 12 102 2.05 1.05 5.12 1.80 Table 5. Compartment blood flows are assumed to be specified as deterministic functions of compartment volumes and preferred values of compartmental blood flows. the blood flow to compartment j is given by: Qj = Qj.1) The mean of the χ2 distributions is a.96 respectively. a) =   0. where qc is a lognormally distributed random variable. the pdf of which is given by:  a 1   x 2 −1 e−x/2 .P Vj . The total cardiac output is given by Qc = Qliv + Qfat + Qsp + Qrp . the parameters of the pdf s of the compartment volumes can be obtained. The form of the expression for Qcw . Vj.P − Qc. since the mean and standard deviation that specify the corresponding normal distribution are then given by ln XP and ln(UF)/1. x > 0 a/2 Γ(a/2)2 f (x. as follows: E[Xi ] = xip Θ where Xi is the ith compartment volume.P )qc . and the total cardiac output in waking humans is given by: Qcw = Qc + (Qcw. This is convenient. From the values of Θ and the preferred values for each compartment volume fraction.81 to the median of the distribution.1. E[Xi ] is its mean. elsewhere (5.P where the notation XP for the preferred value of the parameter X is employed. the preferred value and uncertainty factor for which are given in Table 5.1. This implies that each compartment volume is specified by a χ2 distribution [201. and xip is the preferred value of the fraction Xi . In sleeping humans. 114]. as given in Table 5. and is related to the preferred value and the Dirichlet distribution parameter Θ. Compartment volume fractions are assumed to be specified by a Dirichlet distribution [67]. 3 Implementation of the PERC PBPK Model The PERC PBPK model has been implemented in SimuSolv [58]. can be represented in terms of a standard normal random variable ξ = N(0. RGAUSS. Alveolar ventilation rate is a deterministic function of the cardiac output. is accounted for by an increase in blood flow to the slowly perfused compartment. CML. which describe the dynamic variation in the amount of PERC in each of the compartments shown in Figure 5. as follows: Qspw = Qsp.P + (Qcw. 1).1.P − Qc.P )qc . and an uncertainty factor UFX . which is a typical high concentration in urban/industrial areas [102]. The increase in cardiac output in waking humans. is described here.P Qc Qc w . a software environment for numerical modeling and simulation.1. with median µX .P Qcw. The representation of pdf s of uncertain parameters in terms of standard gaussian random variables.2) A χ2 distributed random variable with parameter a. can be approximated in terms . AUCA.P 5. distribution. as follows [201]: X = exp ln(µX ) + ln(UFX ) ξ 1. Daily values of dose surrogates. metabolism and excretion processes.82 ensures that Qcw > Qc always. The pdf for qc is constructed in a manner analogous to that of the other lognormally distributed parameters as described above. resulting from uptake.P Qc. each with zero mean and unit variance. and the preferred values for ventilation are given in Table 5. and Qpw = Qpw. and AUCL.1. are computed using the PBPK model to simulate a 24 hour inhalation exposure to 10 ppb of PERC. A lognormally distributed random variable X.96 (5. having zero mean and unit variance. The PBPK model consisted of a set of coupled ordinary differential equations. Alveolar ventilation rates in sleeping and waking humans are given by: Qp = Qp. The random parameters in the model have been sampled for the simulations using the pseudo-gaussian random number generator function in SimuSolv. 10.08 Cumulative Amount Metabolized [mg] (normalized by mass of liver) 0. The number of model simulations used for DEMM/PCM and the ECM are equal.1.4 Results for the PBPK case study Stage I (Six Uncertain Parameters): Evaluation of DEMM/PCM and ECM The effect of six uncertain PBPK model parameters on the dose surrogate pdf s is studied in the first stage of uncertainty analysis.3) 5. resulting from 6 uncertain parameters of a normal random variable ξ as follows [217]: Y =a ξ 2 2 +1− 9a 9a 3 (5. respectively.05 0. for 2nd and 3rd order approximations.02 0.01 0. 10. Additionally.000 Monte Carlo simulations are used in the first stage for evaluation of the DEMM/PCM and ECM.03 0.04 0. as both these methods differ only with respect to the selection of collocation points.07 0.000 runs Probability Density [1/mg] 40 30 20 10 0 0 0. . The number of simulations required for both these methods for the case of six uncertain inputs are 28 and 84.06 0. Second and third order approximations are used for both DEMM/PCM and the ECM. over a period of one day.83 50 DEMM Trial 1 DEMM Trial 2 DEMM Trial 3 DEMM Trial 4 Monte Carlo .2: Evaluation of DEMM/PCM: uncertainty in the cumulative amount of PERC metabolized.09 Figure 5. 10.005 0.02 0.01 0. resulting from 6 uncertain parameters .03 (a) 20 DEMM Trial 1 DEMM Trial 2 DEMM Trial 3 DEMM Trial 4 Monte Carlo .hr/cm3] 0.05 0.015 0.hr] 15 10 5 0 0 0.hr/cm3] 0.1 0.10.hr] 200 DEMM Trial 1 DEMM Trial 2 DEMM Trial 3 DEMM Trial 4 Monte Carlo .84 250 Probability Density [cm3/µg.2 0.3 (b) Figure 5.000 runs Probability Density [cm3/µg.025 Area Under the Arterial Concentration Curve [µg.25 Area Under Liver Concentration Curve [µg. over a period of one day.15 0.3: Evaluation of DEMM/PCM: uncertainty in the area under the PERC concentration-time curve in (a) arterial blood (AUCA) and in (b) liver (AUCL).000 runs 150 100 50 0 0 0. . Since the DEMM/PCM method failed to produce consistent results for a simple model with independent input probability distributions. 2. AUCL and CML.3 present the pdf s estimated from four different sets of collocation points selected from the same set of original 4096 points (DEMM Trial 1.85 Figures 5. respectively. From these figures. which involved more complex models or input distributions. it can be seen that the DEMM/PCM approach cannot consistently guarantee convergence of the approximation to the true probability density. for the dose surrogates AUCA. 3 and 4). They also present the pdf s estimated by standard Monte Carlo approach.2 and 5. this method was not used in other case studies. 3rd (84 runs) Probability Density [cm3/µg.hr] 14 12 10 8 6 4 2 0 0 0.10.015 0.2nd (28 runs) ECM .86 200 Monte Carlo .000 runs ECM .45 0.10.5 Monte Carlo .005 0.025 (a) 18 16 Probability Density [cm3/µg.3rd (84 runs) (b) Figure 5. over a period of one day.2nd (28 runs) ECM .2 0.25 0.02 Area Under the Arterial Concentration Curve [µg.hr] 150 100 50 0 0 0.4: Evaluation of ECM: uncertainty in the area under the PERC concentrationtime curve in (a) arterial blood (AUCA) and in (b) liver (AUCL).1 0. resulting from 6 uncertain parameters .000 runs ECM .35 0.05 0.15 0.hr/cm3] 0.3 0.4 Area Under Liver Concentration Curve [µg.hr/cm3] 0.01 0. AUCL.3rd (84 runs) 0.2 Cumulative Amount Metabolized [mg] (normalized by mass of liver) 0. the shapes of the dose surrogate pdf s vary considerably.15 0.05 0.25 Figure 5.5: Evaluation of ECM: uncertainty in the cumulative amount of PERC metabolized.87 45 40 Probability Density [1/mg] 35 30 25 20 15 10 5 0 0 Monte Carlo . and CML dose surrogates.4 and 5. obtained using 2nd and 3rd order ECM and Monte Carlo method. resulting from 6 uncertain parameters A comparison of the pdf s of the AUCL. In all the cases. and uncertainty analysis has not been performed with higher order approximations. the ECM method produced results very close to those of the Monte Carlo method. over a period of one day. but required substantially fewer model runs.10. The pdf s estimated by both 2nd and 3rd order DEM/ECM appear to agree well with each other and with the pdf s estimated using Monte Carlo simulation.000 runs ECM . however. for the six uncertain parameter case is shown in Figures 5. .1 0. from the near symmetrical pdf for AUCA to the highly skewed pdf s for AUCL and CML. Although the pdf s for all six input parameters considered in the first stage of the uncertainty analysis are log-normally distributed.2nd (28 runs) ECM .5. Since the differences between the 2nd order and 3rd order approximations do not appear to be significant. the approximation is judged to have converged. resulting in eleven uncertain parameters.1 0. Regression Based SRSM and LHS In the second stage. and the AUCL dose surrogates.6 and 5.000 runs ECM . Figures 5. These five additional parameters are described by a set of mutually dependent Dirichlet distributed random variables. over a period of one day.25 Figure 5.100.6: Evaluation of ECM: uncertainty in the cumulative amount of PERC metabolized. compared to 100.15 0. Again.88 40 35 Probability Density [1/mg] 30 25 20 15 10 5 0 Monte Carlo . the uncertainty in compartmental volume proportions are included in the analysis.2 Cumulative Amount Metabolized [mg] (normalized by mass of liver) 0. resulting from 11 uncertain parameters Stage II (Eleven Uncertain Parameters): Evaluation of ECM. The pdf s estimated by both 2nd and 3rd order ECM appear to agree well with each other and with the pdf s estimated using Monte Carlo simulation. .3rd (84 runs) 0 0. and 364.05 0. The number of simulations required for the 2nd and 3rd order DEM/ECM are 78.2nd (28 runs) ECM .7 show the comparison of the pdf s of the CML.000 Monte Carlo simulations used to generate the dose surrogate pdf s. AUCA. 89 180 160 Probability Density [cm3/µg.hr] 140 120 100 80 60 40 20 0 0 Monte Carlo - 100,000 runs ECM - 2nd (28 runs) ECM - 3rd (84 runs) 0.005 0.01 0.015 0.02 Area Under the Arterial Concentration Curve [µg.hr/cm3] 0.025 (a) 18 16 Probability Density [cm3/µg.hr] 14 12 10 8 6 4 2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Area Under Liver Concentration Curve [µg.hr/cm3] 0.45 0.5 Monte Carlo - 100,000 runs ECM - 2nd (28 runs) ECM - 3rd (84 runs) (b) Figure 5.7: Evaluation of ECM: uncertainty in the area under the PERC concentrationtime curve in (a) arterial blood (AUCA) and in (b) liver (AUCL), over a period of one day, resulting from 11 uncertain parameters 90 40 35 Probability Density [1/mg] 30 25 20 15 10 5 0 Monte Carlo - 100,000 runs LHS - 1,000 runs SRSM - 2nd (130 runs) SRSM - 3rd (600 runs) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Cumulative Amount Metabolized [mg] (normalized by mass of liver) Figure 5.8: Evaluation of SRSM (regression based) and LHS: uncertainty in the the CML resulting from 11 uncertain parameters The number of simulations used for the 2nd and 3rd order SRSM (regression based SRSM) are 130 and 600, respectively. On the other hand, 1,000 Latin Hypercube samples, and 100,000 Monte Carlo simulations used to generate the dose surrogate pdf s. For this case study, both the ECM and the regression based methods resulted in close approximations, indicating that for simple systems, the ECM method is preferable to the regression based SRSM. The pdf s estimated by both 2nd and 3rd order SRSM appear to agree well with each other and with the pdf s estimated using Monte Carlo simulation. Further, the SRSM approximations have a closer agreement with the Monte Carlo simulation results than the results from the Latin Hypercube sampling method. The results indicate that the SRSM is computationally significantly more efficient than the standard Monte Carlo and more accurate than the Latin Hypercube sampling method. 91 180 160 Probability Density [cm3/µg.hr] 140 120 100 80 60 40 20 0 0 Monte Carlo - 100,000 runs LHS - 1,000 runs SRSM - 2nd (130 runs) SRSM - 3rd (600 runs) 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 Area Under the Arterial Concentration Curve [µg.hr/cm3] 0.018 (a) 18 16 14 Probability Density (1/µg) 12 10 8 6 4 2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Area Under Liver Concentration Curve (µg) 0.35 0.4 Monte Carlo - 100,000 runs LHS - 1,000 runs SRSM - 2nd (130 runs) SRSM - 3rd (600 runs) (b) Figure 5.9: Evaluation of SRSM (regression based) and LHS: uncertainty in (a) AUCL and in (b) AUCA (Figure b) resulting from 11 uncertain parameters 92 50 45 40 Probability Density [1/mg] 35 30 25 20 15 10 5 0 0 Monte Carlo - 100,000 runs SRSM - 3rd Order (600 runs) LHS - 1,000 runs SRSM-ADIFOR - 3rd Order (70 runs) 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Cumulative Amount Metabolized [mg] (normalized by mass of liver) Figure 5.10: Evaluation of SRSM-ADIFOR: uncertainty in the cumulative amount of PERC metabolized, over a period of one day, resulting from 11 uncertain parameters Evaluation of SRSM-ADIFOR In order to apply the SRSM-ADIFOR, the PERC PBPK model has been reimplemented in standard FORTRAN and that model was used in conjunction with the ADIFOR system to obtain the derivative code. The derivative model was then run at a set of selected sample points for the inputs. The model outputs and derivatives obtained at those points were then used to approximate the pdf s of the three dose surrogates. The number of simulations used for the SRSM-ADIFOR approach for the third order was 70, compared to 600 simulations for the regression based SRSM. 3rd Order (600 runs) LHS .006 0.4 Area Under Liver Concentration Curve [µg.35 0.hr/cm3] 0.15 0.000 runs SRSM .1.000 runs SRSM .45 0.05 0.2 0. resulting from 11 uncertain parameters .008 0.000 runs SRSM-ADIFOR .3rd Order (70 runs) Probability Density [cm3/µg.hr] 15 10 5 0 0 0.01 0.3 0.014 Area Under the Arterial Concentration Curve [µg.93 250 Probability Density [cm3/µg.002 0.3rd Order (600 runs) LHS .3rd Order (70 runs) 150 100 50 0 0.11: Evaluation of SRSM-ADIFOR: uncertainty in the area under the PERC concentration-time curve in (a) arterial blood (AUCA) and in (b) liver (AUCL).100.012 0.016 (a) 20 Monte Carlo .hr] 200 Monte Carlo .100.1 0.hr/cm3] 0.25 0.5 (b) Figure 5.004 0.1.000 runs SRSM-ADIFOR . over a period of one day. 94 5.2 Case Study II: A Two-Dimensional Photochemical Air Quality Model Uncertainty Analysis of the Reactive Plume Model (RPM-IV) The Reactive Plume Model, version IV (RPM-IV), is a standard regulatory model used for calculating pollutant concentrations and establishing causal relationships between ambient pollutant concentrations and the emissions from point sources such as industrial stacks [153, 195]. It uses either point source emission estimates or initial plume concentrations as inputs, and calculates downwind concentrations, as the plume expands. RPM-IV is often applied for regulatory purposes to calculate the ozone (O3 ) concentrations at locations downwind of industrial point sources, since high ozone concentrations in the ambient environment lead to adverse health effects. Ozone is primarily formed in the atmosphere through a series of complex chemical reactions involving oxides of nitrogen (NOx ) and volatile organic compounds (VOCs) in the presence of sunlight. Some of the major point sources of NOx and VOCs are industrial units, such as the power plants (NOx sources) and refineries (VOC sources). The application of RPM-IV helps in establishing a quantitative causal relationship between emissions and ambient pollutant concentrations, which is useful in assessing various control strategies for emission reductions. However, there are significant uncertainties in developing estimates of the emissions from industrial sources. These uncertainties occur with respect to the amounts of emissions (e.g., total amounts of VOCs and NOx ), and with respect to their chemical compositions (or speciations, i.e., the fractions of various chemicals within these groups of compounds). These uncertainties arise due to a variety of reasons: for example, emission estimates are typically derived from hourly averages projected from annual or seasonal averages [38]. Since there could be a significant variation in the load and operating conditions of an industrial unit, emission estimates and chemical compositions for specific days under consideration may differ significantly from the averages, and thus result in significant uncertainties. Hence, an uncertainty analysis that takes into account the emission estimate uncertainties is useful for a better understanding of the 95 effects of control strategies for emission reductions. 5.2.1 Description of RPM-IV A brief description of the RPM-IV model is presented here; additional details are presented in Chapter 6, which deals with the model uncertainty associated with RPM-IV, and which presents an improved, three-dimensional version of the model. RPM-IV simulates mechanistically the complex nonlinear photochemistry and dispersion processes occurring in an expanding plume. The nonlinear atmospheric gas phase photochemistry is described by the Carbon Bond IV (CB-IV) mechanism [216], which consists of a set of 95 chemical reactions among 35 surrogate chemical species corresponding to organic bonds and functional groups. Details on the CB-IV mechanism are presented in Appendix C. This model follows the trajectory of an expanding, moving plume and simulates its evolution. In this model, the pollutant mass is initially divided into cells containing equal amounts of pollutants. As the plume expands, the individual cells expand in volume and pollutant mass is transferred across cell boundaries in two phases: (a) an “entrainment” phase, where the expanding cell boundaries entrain the pollutants from other cells, and (b) a “detrainment” phase, where the pollutants diffuse across cell boundaries, due to concentration gradients. Further, the pollutants in each cell undergo chemical transformation governed by the Carbon Bond-IV mechanism. The equation describing the concentration changes within a cell is given by: dci j = dt dci j dt − chem 1 dwj · wj ds uci − j 1 dhj · hj ds uci + Fji j (5.4) where ci denotes the concentration of chemical species i in cell j. Fji is the net j i i i flux into the cell, and is given by Fji = Ej − Dj , where Ej is the entrainment as the i cell expands, and Dj is the detrainment due to the diffusion of the species across the boundaries of the cell. Expressions for entrainment and detrainment are given by the following equations: i Ej = 1 yj − yj−1 dyj dt ci − j+1 dyj−1 dt ci j (5.5) 96 i Dj = ∂ ∂y K ∂c ∂y = 2 yj − yj−1 Kj ci − ci j+1 j yj+1 − yj−1 − Kj−1 ci − ci j j−1 yj − yj−2 (5.6) where yj is the distance from the plume centerline to the far most side of the cell j, in the horizontal direction. An additional condition is imposed on the expansion of the cells: the cells are allowed to expand in a way such that the amount of an inert species remains constant within each cell, i.e., no net transfer of an inert species occurs from a cell. This condition results in the following equation for expansion of the cells, as the plume travels downwind: I I Ej − Dj = 0 RPM-IV was selected for evaluation of the SRSM and SRSM-ADIFOR because it is sufficiently complex to represent a wide range of environmental models, and at the same time it is computationally feasible to perform a large number of Monte Carlo simulations with this model. Further, the complex nonlinear photochemistry of this model is employed by a number of photochemical models that are computationally very demanding. Thus, evaluation of the SRSM and SRSM-ADIFOR with RPM-IV could potentially serve as a preliminary test of applicability of this method to a wide range of complex photochemical models. 5.2.2 Uncertainty Analysis of RPM-IV The present work studies the effect of uncertainty in emission estimates on predicted downwind secondary pollutant concentrations; here, the specific focus is on the uncertainties in the calculated ozone concentrations resulting from the uncertainties in amounts and the chemical composition of the emissions. This study uses emission estimates and field data measured near Marathon oil refinery at Robinson, Illinois, during June and July, 1977 by Sexton et al. [183]. Since there was no characterization of uncertainty, and since the focus of this work is to evaluate the applicability of SRSMADIFOR, only representative probability distributions to describe the uncertainties in emissions were assumed. 97 The following distributions for the emissions of VOCs and NOx are used: (a) the amounts of VOCs and NOx released are assumed to have normal distributions with a standard deviation of 20% of the mean value, and (b) the chemical compositions of VOCs and NOx are assumed to follow a Dirichlet distribution [201]. The Dirichlet distribution satisfies the condition that the sum of mole fractions is unity. According to this distribution the mole fraction of the ith compound, yi , is given by: yi = xi n , (5.7) xi i=1 where xi is an independent random variable, representing the the amount (in moles) of the ith compound; here a normal distribution is assumed for all xi s, with a nominal standard deviation of 40% of mean value. In this case study, the VOC group consists of five subgroups (paraffins, ethylene, olefins, toluene, and xylene), and the NOx group consists of NO and NO2 . Thus, the total number of uncertain parameters for the model is nine (including two parameters representing the total amounts of VOCs and NOx ). Thus, a total of 9 uncertain parameters with 7 degrees of freedom were used in this case study; these were represented by 9 srvs. The output metrics considered are the average ozone concentration in the plume for selected distances downwind from the source. The original RPM-IV model was implemented in Fortran and obtained from the EPA Exposure Models Library [202]. The derivative code was obtained using the ADIFOR system on RPM-IV code. For Monte Carlo, LHS, and the SRSM, the model was run at selected sample points, whereas for the SRSM-ADIFOR, the derivative model was run. 5.2.3 Results and Discussion From equations 3.6 and 3.6 for the number of unknown coefficients for a second and third order polynomial chaos expansions, for n=9, the number of coefficients to be determined is 78 and 364 for second and third order approximations, respectively. For a regression based method, 300 model simulations were selected for a second order approximation, and 600 model simulations were selected for a third order approximation, 095 0.1 0.07 0.2nd Order (78 runs) Average Plume O3 Concentration (ppm) at 20 km downwind from the source (b) Figure 5.14 0.085 0.3rd Order (364 runs) ECM .12 0.98 Evaluation of The ECM Method 160 140 Probability Density (1/ppm) 120 100 80 60 40 20 0 0.12: Evaluation of ECM: uncertainty in the predicted ozone concentrations at (a) 2 km and (b) 20 km downwind from the source .065 0.15 0.13 0.075 0.1 Monte Carlo Method (10.000 runs) ECM .000 runs) ECM .2nd Order (78 runs) Average Plume O3 Concentration (ppm) at 2 km downwind from the source (a) Evaluation of The ECM Method 160 140 Probability Density (1/ppm) 120 100 80 60 40 20 0 0.09 0.08 0.16 Monte Carlo Method (10.3rd Order (364 runs) ECM .11 0. The results indicate that. compared to the collocation methods. representative of near-source transport and transformation. SRSM/ADIFOR gave closer estimates with only 80 model runs. representative of larger scale transport and transformation. and (b) ozone concentration at a downwind distance of 20 km. The output pdf s were obtained first using second and third order approximations of the SRSM/ECM.14 shows the uncertainty estimates obtained. For the evaluation of the SRSM-ADIFOR method with RPM-IV. On the other hand. a regression based method resulted in similar estimates for both second and third order approximations and was consistent for all ranges of downwind distances. their robustness makes them a more viable tool for uncertainty analysis of complex environmental models. Two representative output metrics were considered here: (a) ozone concentration at a downwind distance of 2 km. As shown in Figures 5. For the case of the SRSM-ADIFOR. traditional Monte Carlo and Latin Hypercube Sampling. indicating the lack of robustness in the collocation method.000 Monte Carlo simulations and 10. while 10000 Latin Hypercube samples were . This behavior is more prominent for large downwind distance. the the predictions of ECM become inaccurate as the order of approximation increases.13. as estimated by the regression based SRSM. The estimates of pdf s for these cases were evaluated with the results obtained from 10.000 Latin Hypercube sampling methods. 80 simulations were used for a third order approximation.12 shows the pdf s of ozone concentration at a downwind distance of 2 km and of 20 km.13 shows the pdf s of the predicted ozone concentrations at downwind distances of 2 km and 20 km. as estimated by the ECM. the results agree very closely with the Monte Carlo results. Figure 5. Figure 5.12 and 5. indicating that the collocation method may not converge when used with highly nonlinear models (the near-source behavior is expected to be less nonlinear than far-source behavior). On the other hand. and the conventional Monte Carlo method. although the regression methods require a higher number of model simulations for uncertainty propagation. As shown in the figure. although the regression based-method required significantly fewer runs than the Monte Carlo method.99 in order to facilitate regression with an adequate number of model outputs. Figure 5. 075 0.2nd order (91 runs) SRSM .12 0.13 0.135 0.2nd order (91 runs) SRSM .10.3rd order (600 runs) 0.3rd order (600 runs) 0.125 0.115 0.145 Average Plume O3 Concentration (ppm) 20 km (left) from the Source (b) Figure 5.095 Average Plume O3 Concentration (ppm) 2 km (left) from the Source (a) 160 140 Probability Density (1/ppm) 120 100 80 60 40 20 0 0.10.07 Monte Carlo : 10.09 0.14 0.000 runs SRSM .000 runs SRSM .13: Evaluation of SRSM (regression based): uncertainty in the predicted ozone concentrations at (a) 2 km and (b) 20 km downwind from the source .000 runs LHS .000 runs LHS .08 0.11 Monte Carlo : 10.100 160 140 Probability Density (1/ppm) 120 100 80 60 40 20 0 0.085 0. 13 0.135 0.80 runs) 0.075 0.10.12 0.125 0.000 runs SRSM-ADIFOR (3rd .10.14: Evaluation of SRSM-ADIFOR: uncertainty in the predicted ozone concentrations at (a) 2 km and (b) 20 km downwind from the source .095 Average Plume O3 Concentration (ppm) 2 km from the Source (a) 160 140 Probability Density (1/ppm) 120 100 80 60 40 20 0 0.000 runs SRSM .000 runs SRSM .08 0.14 Average Plume O3 Concentration (ppm) 20 km from the Source (b) Figure 5.115 Monte Carlo : 10.101 160 140 Probability Density (1/ppm) 120 100 80 60 40 20 0 0.09 0.000 runs SRSM-ADIFOR (3rd .80 runs) 0.085 0.3rd order (600 runs) LHS .07 Monte Carlo : 10.3rd order (600 runs) LHS . However.102 not sufficient to achieve agreement with the results from the Monte Carlo methods. The modeling process is further complicated by the presence of natural. It is one of the most abundant phytogenic chemical species found in the ambient air [73]. model. Uncertainties in the input data and parameters (such as emission estimates and reaction rates) become especially significant when photochemical models are used to evaluate control strategies for the reduction of ozone levels [43].3 Case Study III: A Three-Dimensional Urban/Regional Scale Photochemical Air Quality Model Uncertainty Analysis of the Urban Airshed Model (UAM-IV) for the New Jersey/Philadelphia Domain Photochemical modeling of ozone (O3 ) levels in the atmosphere is complicated by the fact that ozone is a secondary pollutant. increases in biogenic emissions decrease O3 concentrations [173]. and volatile organic compounds (VOCs). and affects the chemistry of the troposphere because its oxidation products are precursors for the photochemical production of ozone [11]. biogenic emission estimates are laden with large uncertainties [173]. modeling studies suggest that in some extremely NOx -limited areas. it is formed through complex chemical reactions involving oxides of nitrogen (NO2 and NO). depending on the availability of NOx . Biogenic emissions have a significant impact on atmospheric ozone concentrations. In addition. 88]. The focus of the . Earlier studies on the effects of biogenic emissions on O3 concentrations indicate that typically O3 concentrations increase in response to increases in biogenic hydrocarbons.3. relative to anthropogenic hydrocarbons (AHCs) [142]. Isoprene forms a main component of biogenic emissions. 5. such as uncertainty in the chemical reaction rates and uncertainties arising from the lumping of chemical species [7]. as shown in Figure 5. Sensitivity tests in a case study show that local biogenic emissions are an important contributor to local O3 production. there are significant uncertainties associated with the chemical mechanisms employed in photochemical modeling. and accounts for a substantial portion of the atmospheric hydrocarbon load [185. However. and data uncertainties. 3. Such uncertainty analysis could provide information that could be useful in identifying the area to focus resources: studying the reaction mechanisms in more detail versus improving the inventorying methods. and there is a wide range of plant species in an urban/regional scale modeling area. the aim is to compare the effects of uncertainties associated with the emission rates of isoprene with the effects of uncertainties in the rates of reactions involving isoprene. These uncertainties occur mainly due to the following factors: • variability in the vegetation: the amounts of biogenic emissions differ significantly from one plant species to other. .1 Uncertainties Associated with Biogenic Emission Estimates There are significant uncertainties associated with biogenic emission estimates.103 Model Uncertainty Atmospheric Turbulence Variabilty Simplifications in Model Formulation Spatial and Temporal Averaging Lumping in Chemical Mechanisms Natural Uncertainty Data Uncertainty Emission Estimate Uncertainty Errors in Measurement Insufficient Data Figure 5. with respect to isoprene (2-methyl-1. 5. More specifically. and specifically with isoprene emission estimates.15: Origins and types of uncertainty present in Photochemical Air Quality Simulation Models present work is on the comparative evaluation of the effects of uncertainty in the biogenic emission estimates and uncertainties in the reaction rate coefficients on the estimated ozone levels. since it is one of the major components of biogenic emissions.3-butadiene). S. Some studies report the uncertainty in the biogenic emission estimates of an inventory for the United States to be of the order of 300% [136. nitrogen and water availability. Due to the lack of computationally efficient uncertainty analysis methods. 5. temperature. For simple models. grid-based photochemical models are typically very demanding computationally. it is important to study how these uncertainties propagate through the model and affect the predicted O3 concentrations. Hence significant uncertainties occur in the emission estimates. Recently reported studies show that different photochemical mechanisms with different treatments of isoprene chemistry give significantly different estimates of ozone levels [225].2 Uncertainty Analysis Considering the importance of biogenic emissions. and (b) insufficient data to estimate reaction rate parameters associated with biogenic compounds. a number . In another study.104 • variability in emission rates: the emission rates depend on factors such as sunlight. emission estimates of biogenics are reported to range from 22 to 50 Tg/yr depending upon the formulation of different emission rate factors [135]. It is reported in the literature that some reaction rate parameters associated with isoprene chemistry are uncertain up to a range of 150% [7]. However. 173]. The total U. In addition. uncertainty propagation can be accomplished through conventional methods such as Monte Carlo methods. Currently employed models and associated inventories for estimating biogenic emissions typically cannot treat the above variability in adequate detail. and are associated with significant spatial and temporal variability. the uncertainties in the emission estimates of isoprene in Europe are reported to be of the order of 500% [188]. there are significant uncertainties associated with the description of isoprene chemistry.3. The main sources of uncertainties in existing chemical mechanisms are: (a) lumping of various biogenic compounds. and the uncertainties involved in their estimates and reaction rates. thus making conventional Monte Carlo methods prohibitively expensive. However. so that resources could be focused to reduce uncertainties in those factors. and are then propagated through the Urban Airshed Model (UAM-IV) to study urban/regional scale effects of these uncertainties. More specifically.3 Estimation of Uncertainties The present work involves the study of the effects of uncertainty in the emission and reaction rates of isoprene.3. 5. resulting from the above mentioned input uncertainties. Another goal is to obtain an approximate estimate of the range of uncertainty in the predicted ozone concentration. and the rate constants for reactions involving isoprene in the CB-IV chemical mechanism. The results . • approximate estimates of uncertainty in the parameters thus identifies are obtained from the literature. one goal of this approach is to characterize individual contributions of uncertainties in chemistry and emissions on the ambient ozone levels. The parameters considered include the the initial concentration of isoprene. on the ozone levels in the atmosphere. The main advantage of this approach is that it is simple and requires few model runs. This is pursued as follows: • a single cell (“box”) model is used as a screening tool to identify parameters that could potentially affect the results of the grid-based PAQSMs. this approach (“sensitivity/uncertainty” testing) does not provide insight into the distribution of the model results. such an estimate could potentially assist in answering some regulatory questions. Here. and thus obtaining some estimates on the “bounds” of uncertainty in model results.105 of studies reported in the literature with respect to uncertainty in photochemical models involved running the model for selected sets of parameter or input values. a photochemical box model is used to perform preliminary screening based on the effects of uncertainty in isoprene reaction rate constants and initial concentrations of isoprene on the time-dependent concentrations of ozone. As mentioned earlier. and to identify important contributors to uncertainty. the regression based SRSM is applied to estimate the probability distributions of the estimated ozone concentrations. 106 from this exercise are utilized in identifying key factors contributing to uncertainty in the application of a grid-based PAQSM. The biogenic emissions were obtained from the EPA’s Biogenic Emissions Inventory System (BEIS) model [31]. since there is no transfer of material occurring from outside the cell. This model requires meteorological inputs and the emission information for both anthropogenic and biogenic emissions. The CB-IV mechanism (see Appendix C) consists of a set of 95 chemical reactions among 35 surrogate chemical species corresponding to organic bonds and functional groups. Europe and the Far East [78. specifically the UAM-IV. and the evolution of chemical species depends only on time dependent photochemical reaction rates. the anthropogenic emissions were obtained from State-provided inventories. 63]. 1991. Atmospheric chemistry is described by the CB-IV mechanism. The Urban Airshed Model. version IV. Further details of the UAM-IV model are presented in Appendix C. The domain and the modeling grid (at a horizontal resolution of 5 km×5 km) for the case study are shown in Figure 5. is employed to describe the chemistry in both RPM-IV and UAM-IV. A single cell trajectory model. In the present work.16. The meteorological inputs were developed using the Diagnostic Wind Model (DWM) [57]. a single cell option of the Reactive Plume Model (RPM-IV) is used for the screening runs. [154] is a photochemical grid model that has been applied to several urban areas in the United States. processed using the Emission Processing System (EPS) [31]. A box-model consists of a single well mixed reactor cell. los et al. the Carbon-Bond IV (CB-IV) mechanism [216]. when a severe ozone episode occurred in this region. is equivalent to a box model. [78] The procedure used for uncertainty analysis consisted of the following steps: • Performing systematic screening tests to study the sensitivity of the results of the These inputs were obtained from the input data from Georgopou- . The RPM-IV was used here since the same chemical mechanism. This model solves the atmospheric diffusion equation (ADE) [181] on a three dimensional grid covering the airshed of interest. This case study considers the modeling of the Philadelphia-New Jersey region for July 19 and 20. 107 Philadelphia/New Jersey Modeling Domain County Locations UTM ZONE 18 (SW −350. and initial concentration of isoprene • Identifying the factors that have clear effects of the predicted ozone concentration .4575 km) UTM North (km) UTM East (km) Figure 5.4285 km.16: The Philadelphia/New Jersey modeling domain used for uncertainty analysis box-model with respect to selected reaction rate constants of isoprene. NE −610. 3. Of these.16 0. and the rates of seven reactions of isoprene with the O3 .19 Figure 5. namely the initial concentration of isoprene.17 Maximum Daily O3 Concentration (ppm) 0.108 1400 July 19 July 20 1200 Probability Density (ppm ) -1 1000 800 600 400 200 0 0. . and studying the contributions of these uncertainties to the uncertainty in the ozone concentrations predicted by the UAM-IV • Characterizing the relative importance of these parameters.17: Probability distribution of the daily maximum ozone concentration • Obtaining approximate uncertainty estimates for these factors from the literature [7] • Performing uncertainty analysis using the Stochastic Response Surface Method.18 0. NO3 . and the objective was to identify parameters that have a significant effect on the O3 concentration predictions. 5. and the overall uncertainty in predicted ozone concentrations for specific case studies. only three factors. The box model was used to to calculate the O3 concentration.15 0. O and OH radicals.14 0. The nine parameters included initial concentrations of isoprene and monoterpenes. nine parameters were considered. and the reaction rates of the reactions of isoprene with NO3 and with the OH radical were found to have an appreciable effect on the calculated O3 concentrations.4 Results of the Case Study In the first phase (screening phase). Here. The third order approximations were used to estimate the . The application of SRSMs.06 0.07 0.5 is used as a truncation factor to specify the probability distribution uniquely. Truncated lognormal probability distributions were used to represent uncertainties in these parameters. The approximation converged for the third order .08 Daily Average O3 Concentration (ppm) 0. were all assumed to be represented by a truncated lognormal distributions with uncertainty factors of 1. and a variance log(f )/2.there was an almost negligible difference between the estimates of the second and third order approximations. 2. and 20 simulations for a third order approximation. could lie between xm /f and xm f . required 10 simulations for a second order approximation method.05 0. the uncertainties in isoprene emission estimates. representative estimates of uncertainties in these parameters were used to study their effects on urban scale O3 levels. and the rate constants for the reactions of isoprene with NO3 and with the OH radical.09 -1 July 19 July 20 0. this is a rough estimate of the uncertainties based on the articles by Roselle [173] and by Atkinson [7].5. and to limit the range of values x could have. More specifically.18: Probability distribution of the daily average ozone concentration In the second phase. where xm is the reported value and f is an uncertainty factor that is estimated a priori.5. x. for this case study involving 3 uncertain parameters. This means that log(x) is normally distributed with a mean log(xm ).109 5000 4500 4000 Probability Density (ppm ) 3500 3000 2500 2000 1500 1000 500 0 0. The value of a given parameter.1 Figure 5. 008 0. on four representative metrics of photochemical air quality O3 levels through the study domain was studied.110 2000 1800 1600 Probability Density (ppm ) 1400 1200 1000 800 600 400 200 0 0. (b) the daily average O3 concentration.16 Figure 5. (c) the daily maximum eight hour running average (an important .002 0 0 500 1000 1500 2000 2500 3000 Pervasiveness (no units) 3500 4000 4500 5000 Figure 5.01 0.20: Probability distribution of the pervasiveness of the ozone episode uncertainties in the output metrics considered.012 Probability Density (no units) 0.004 0. The four metrics were: (a) the daily maximum of hourly average O3 concentration.13 0.12 0.14 Maximum 8hr O3 Running Average (ppm) 0.014 July 19 July 20 0.15 -1 July 19 July 20 0.19: Probability distribution of the daily maximum 8 hr running average ozone concentration 0. The effect of uncertainty in these parameters.006 0.11 0. Table 5.18 and 5. The coefficients in the table indicate the uncertainty contributed to a metric. an alternative metric. • The variance in pervasiveness was of the order of 5%-10% of the mean value. and the daily maximum eight hour running average concentration. • The daily maximum eight hour running average was affected mildly by the input uncertainties. they indicate that the uncertainty in the isoprene emission rate contributes . In summary. Figures 5.19 describe the uncertainty in the daily average concentration. whereas Figure 5. • The variance in the concentration metrics for all cases was less than 1% of the mean value. and (d) the “pervasiveness”.17 shows the probability distributions describing the uncertainty in the daily maximum of hourly average concentration for the episode days of July 19 and 20. the following conclusions were reached: • The daily average O3 concentration was not affected significantly by the input uncertainties considered.20 shows the uncertainty in the pervasiveness. 1991. Figure 5.2 summarizes the uncertainty in the uncertainty estimates resulting from this analysis. defined as the number of grid cell hours that have ozone concentrations in excess of the hourly maximum O3 standard of 120ppb [81.111 metric in relation to the new ozone regulatory standard). for the same case. • The daily maximum hourly average was also affected mildly. 78]. • The probability distributions of output metrics differed significantly for the two episode days considered. implying that the uncertainty analysis method needs to be applied on a case by case basis. • The pervasiveness was affected significantly by the uncertainty in the inputs considered. The coefficients indicate that the uncertainty contribution is very small relative to the mean value of the metric. for the Philadelphia-New Jersey domain. Further. due to the uncertainty in the individual parameter. Since the exposure estimates are utilized in .71 OH 0.0001 0.178 0.0003 -0.0 OH Jul 20 NO3 -0. This case study also demonstrates how the SRSM.2: Contribution of uncertainties in the individual parameters to the overall uncertainty in the UAM-IV. about five days (20 model runs × 6 hours per model run) is much less when compared with a Monte Carlo analysis involving.30 significantly to the overall uncertainty. 5.0000 -9.0003 31.143 0.0000 -0.0000 0.0000 0.0000 6. and due to incomplete understanding of the processes involved in transport of contaminants.0000 0.112 Table 5.072 0.0009 0.0005 47.0002 0.0003 0. The execution time.4 Case Study IV: A Ground Water Model with Discontinuous Probability Distributions Uncertainty Analysis of Tritium Contamination in a Landfill using the EPACMTP Model Characterization of exposure due to leakage of hazardous contaminants from a land disposal unit involves significant uncertainty due to inherent variability in the hydrogeologic properties of the site.0003 0. the uncertainty in the OH-isoprene reaction constant contributes to an extent comparable to the isoprene emission uncertainty.0001 -0. the above analysis indicates that the isoprene emission rates and the reaction rate constant of the isoprene-OH reaction should be studied in more detail.53 Mean 0. as estimated by the SRSM Jul 19 Mean Max (ppm) Ave (ppm) 8hr-Ave (ppm) Pervasiveness 0. requiring 125 days of computer time.0001 0.22 ISOP 0.0001 -1. However.32 NO3 0. can be used to estimate the uncertainties in the outputs of a complex model such as UAM-IV. and that the uncertainty in the reaction rate of the NO3 -isoprene reaction has an almost negligible effect. 500 runs. for example.149 4075.062 0. So.0001 157.14 0. while requiring a small number of simulations.17 ISOP 0.121 341. a case study is presented in relation to characterization of uncertainty in estimated tritium exposure at a receptor well. . it is important that any exposure characterization should take into account the variability and uncertainty involved with the physical system. and predicts the associated groundwater exposure in a domestic drinking water receptor well. Environmental metrics considered are the estimated maximum concentration of tritium in a “receptor” well and the estimated time of occurrence of the maximum concentration. unidirectional groundwater flow. The main source for the contamination is a hypothetical landfill unit in the southern United States.4. (iii) finite source as well as continuous source scenarios. This model is an improvement over the EPA’s Composite Model for Landfills (EPACML) [62]. In addition. and (iv) metals transport.1 EPACMTP Model The EPA’s Composite Model for leachate Migration and Transformation Products (EPACMTP) provides estimates of potential human exposure to hazardous chemicals leaching from land disposal facilities [64]. EPACMTP can take into consideration: (i) chain decay reactions and transport of daughter and grand-daughter products. (ii) effects of water-table mounding on groundwater flow and contaminant migration.113 making policy decisions. EPACMTP simulates the subsurface fate and transport of contaminants released from land disposal sites. EPACML can describe only uniform. On the other hand. Here. The variability and uncertainty associated with the hydrogeologic parameters and with the physical properties of the landfill unit are considered to characterize the uncertainty in the calculated concentrations of tritium. EPACML accounts for the first-order decay and sorption of chemicals. and with the modeling process. In the following sections. but disregards the formation and transport of transformation products. a brief description of the model used. The fate and transport model used in this work is the EPA’s Composite Model for leachate Migration and Transformation Products (EPACMTP). 5. the estimates of the model parameters and the uncertainty analysis methods is given. and simulations from the HELP model [180]. . FECTUZ is a one-dimensional model that simulates vertically downward steady-state flow and contaminant transport through the unsaturated zone above an unconfined aquifer.4. infiltration rates. • the infiltration and recharge analysis performed by OSW. • the USGS state-by-state inventory of groundwater resources. the data sets from the National Oceanic and Atmospheric Administration. and • the Hydrogeologic Database for Modeling (HGDB). the volume and area of the landfills. Appendix D provides detailed description of the formulation and implementation of FECTUZ and CANSAZ-3D modules. and a saturated zone module called Combined Analytical-Numerical SAturated Zone in 3-Dimensions (CANSAZ-3D). The data for the hypothetical scenario in this case study were adapted from a data set provided by Hydrogeologic Inc. These datasets were used in conjunction with the soil mapping database provided by the Soil Conservation Service (SCS).2 Data Sources for the Application of EPACMTP Data for the site characteristics. 5. In this review. Four of these sets were selected to derive the regional characteristics of important parameters for each sampled site: • the OPPI survey of waste management units (EPA.S. and the modules are optimized for computational efficiency. EPACMTP currently uses a simplified 2-D version of the CANSAZ-3D. CANSAZ-3D simulates 3-D steady-state groundwater flow and transient or steady state contaminant transport.114 EPACMTP consists of two modules: an unsaturated zone module called Finite Element and semi-analytical Contaminant Transport in the Unsaturated Zone (FECTUZ). developed from a survey of hazardous waste field sites in the U. a number of different sources were used for the development of this site-based approach.1986). and the probability distributions for the hydrogeologic parameters are obtained from a review conducted by the Hydrogeologic Inc. [64]. with a leaching duration of 20 years. 5. The various steps involved in the application of a Monte Carlo simulation are: • selection of representative cumulative probability distribution functions for the relevant input variables. a probability distribution (or a frequency distribution) be provided. and a recharge rate of 0. The landfill is a finite source. of 0. representing a possible set of values (a realization) for the input variables.35 km2 area. These responses are then statistically analyzed to yield the probability distribution of the model output. . • application of the model to compute the derived inputs and outputs.4. • generation of a pseudo-random number from the selected distributions. and • statistical analysis of the series of the output values generated. The method involves the repeated generation of pseudo-random values of the uncertain input variables (drawn from the known distribution and within the range of any imposed bounds) and the application of the model using these values to generate a set of model responses or outputs (for example. CRW ).115 5. The receptor well is located at a distance of 1635 m from the landfill.4.3 Implementation of the Monte Carlo Method in EPACMTP The Monte Carlo method requires that for each input parameter that has associated uncertainty or variability. the receptor well concentration.381 m/yr.4 Uncertainty Analysis This case study consists of a site based landfill modeling for tritium contamination at a groundwater well resulting from a hypothetical landfill unit in the souther United States. The EPACMTP model is used to simulate the radioactive decay and transport of tritium through the saturated and unsaturated zone underlying the landfill. • checking whether the realization satisfies the constraints or bounds on the distribution (and resampling till a reasonable realization is obtained). • repeated application of the above steps for a specified number of iterations. The data used here is adapted from data provided by Hydrogeologic Inc. as a result of the leaching from the landfill unit.000432. σ = 5. σ = 0.4.Johnson SB.4. and SB .343. σ = 1.0456 η = 800. min .5.0 units cm/hr m/yr cm−1 cm m m m Description Saturated hydraulic conductivity Hydraulic conductivity Moisture retention parameter Moisture retention parameter Residual water content Percentage of organic matter Particle diameter Longitudinal dispersivity Aquifer pH Fractional organic carbon content Radial distance of observation well Angle off-center Aquifer thickness a b Distribution types : LN .Normal. and max . σ = 5.88 η = 0.0.75 η = 0. σ = 0.629 η = 0.012 η = 1.a LN U LN SB SB SB LN LN LN SB N U U Distribution Parametersb η = 0.minimum value.mean.019. σ = 0. σ = 4.Lognormal. max = 2114 η = 0.409.989 min = 1500. max = 90◦ min = 30.0 η = 6. N .3.Uniform.standard deviation.3: Probability distributions used for the EPACMTP model parameters for the uncertainty analysis Param Ks K α β θr %OM d αL pH foc m θ B Dist. σ . U .09. Distribution parameters: η .116 Table 5.21 shows the uncertainty associated with the estimation of maximum tritium concentration in the ground water. σ = 150 min = 0◦ .maximum value The probability distributions of the site-specific parameters for the land fill at the Old Burial Ground are given in Table 5. σ = 0. and the values of the constant model parameters for EPACMTP are given in Table 5. 5.5 Results and Discussion Figure 5.071 η = 0.0. σ = 0. max = 35.105.068. .0 η = 10. 0 0.381 17.67 0. there are certain other advantages in using the SRSM.22 shows the uncertainty associated with the estimated time of occurrence of the maximum tritium concentration in the ground water at the well.0 1. while requiring much fewer number of runs.65 Units m2 m m m/yr ◦ Description Area of the source Length of the source Width of the source Areal recharge rate Aquifer temperature Radioactive decay coefficient Source leaching duration Transverse dispersivity Vertical dispersivity Saturated water content Depth to water table Hydraulic gradient Retardation factor Bulk density C 1/yr yr m m m g/cm3 Figure 5.0 0. The figures show the pdf s estimated by the Monte Carlo simulations and by the SRSM.4: Deterministic Parameters in the EPACMTP model for the case study Param Aw Xw Yw IR T λ ts αT αv θs Dw S Rs ρb Value 3. that are focused in the ongoing research: • The relationships between inputs and outputs can be easily estimated. Further.0 160. Results of the SRSM with 350 model runs are compared with those of Monte Carlo simulations with 1000. since both .0086 1. 3000 and 5000 model runs.0564 20 8.117 Table 5.8 1143.45 10. The results indicate that the SRSM shows close agreement with the Monte Carlo results.5 0.5×105 304. 08 0.21: Uncertainty in estimated maximum tritium concentration in a receptor well 0.07 0.1 Probability Density 0.06 0.05 0.12 Uncertainty in Time of Occurrence of Max.09 Predicted Maximum Concentration of Tritium At Well N1 (Normalized) Figure 5.04 0.118 Uncertainty in Max.08 0.22: Uncertainty in estimated time of occurrence of maximum tritium concentration in a receptor well the input and output pdf s are expressed as a series expansion in srvs.04 0. Tritium Concentration SRSM: 350 Runs 1000 Runs 5000 Runs 0.02 0. Tritium Concentration 50 SRSM : 350 Runs 1000 Runs 5000 Runs 40 Probability Density 30 20 10 0 0.03 0.06 0. This implies that SRSM provides insight into the relative contribution of each uncertain .02 0 15 20 25 30 35 40 45 50 Predicted Time of Occurrence of Maximum Concentration of Tritium At Well N1 (yrs) Figure 5. indicating that they represent only an approximate estimate. Monte Carlo methods require significantly large number of model simulations to obtain a reasonably good approximation of the correlation between outputs. that is a focus of the ongoing research. this method assumes that the probability distributions are continuous. is as follows: the current implementation of the SRSM does not take into account constraints on the input probability distributions that make the distributions discontinuous. • Correlation between outputs can be easily estimated since all outputs are expressed in terms of srvs.21 and 5. and the results shown in Figures 5. One of the limitations of the Stochastic Response Surface Method. On the other hand. thus giving extensive sensitivity and uncertainty information. such constraints are modeled as truncated probability distributions. Current work in progress involves the refinement of the SRSM so that all types of constraints can be propagated. • The algebraic expressions for the outputs in terms of the srvs are smooth and continuous and could efficiently model the tails of the probability distributions of the outputs (which typically correspond to either high risk or worst case scenarios). In short. and discontinuities in probability distributions can be fully addressed. Conventional Monte Carlo methods do not provide such information.119 input each output. . In the present work.22 for the SRSM are obtained in such a manner. 1. The approach followed here for characterizing model uncertainty here is based on the development of models corresponding to different model formulations subsequent comparison of the model results. ranging from simplified to more detailed.. including the following: (a) when alternative sets of scientific or technical assumptions for developing a model exist (Model Structure). Such a comparison is useful in focusing resources where it is most appropriate (e. When a model application involves both model and data uncertainties. but also due to uncertainty in model formulation (i.120 Chapter 6 CHARACTERIZATION AND REDUCTION OF MODEL UNCERTAINTY: AN ATMOSPHERIC PLUME MODEL STUDY 6. The availability of the alternative models. data gaps versus model refinement). Some of the major sources of model uncertainty in transport-transformation modeling are listed in Table 2.. and (c) when a coarse numerical discretization is used to reduce the computation demands of the model (Model Resolution). then the low resolution models are preferable. model uncertainty). since they typically .g. (b) when models are simplified for purposes of tractability (Model Detail). it is important to identify the relative magnitudes of the uncertainties associated with data and model formulation..e.e. As mentioned in Chapter 1.1 Introduction and Background Uncertainty in the application of transport-transformation models arises not only due to uncertainty in model inputs or parameters (i. model uncertainty arises under several conditions. parametric or data uncertainty). aids in evaluating the applicability of the low resolution models – if the results of the low resolution models agree closely with those of the high resolution models. two aspects of the model are studied: (a) the uncertainty associated with the assumption of uniform vertical concentration profile. As mentioned in Section 5. The relevant background information on photochemical modeling. termed RPM-3D.3. Then a “three-dimensional version” of the RPM. RPM-IV. the RPM describes the evolution of a photochemical plume from “point sources” of emissions. Furthermore. For example. such as the stacks of refineries and of power plants. as the model assumptions include uniform mixing in the vertical direction. A case study is presented involving an EPA regulatory atmospheric photochemical trajectory model. is presented here.2. is developed by incorporating vertical resolution into the RPM-IV.2. Here. Specifically. In this process. . the results of the RPM at varying horizontal resolutions are studied first. the lack of resolution is highlighted in Figure 6. the Reactive Reactive Plume Model (RPM) [195]. required for a detailed understanding of the RPM. The results from the RPM and the RPM-3D are subsequently compared with each other in order to characterize the uncertainty associated with the vertical resolution in the RPM. which is briefly described in Section 5. This chapter focuses on the model uncertainties associated with photochemical air quality modeling applications. if the bounds are narrow the uncertainty associated with the model formulation can be considered to be relatively small. The currently used version of RPM. the uncertainties associated with model resolution and assumptions of “well mixedness” are addressed here. and is described in more detail in the following sections. and (b) the uncertainty associated with the choice of horizontal resolution. That is followed by a description of the formulation of RPM-IV and RPM-3D. lacks vertical resolution.121 require fewer computational resources and lesser input data. the availability of the alternative models aids in the characterization of the model uncertainty associated with the application of those models – the bounds on model calculations corresponding to different model formulations provide an estimate of the uncertainty associated with the model formulation. 2 Photochemical Air Pollution Modeling The main objective of Photochemical Air Quality Simulation Models (PAQSMs) is to establish causal relations among emission levels. to urban and regional scales (e. The PAQSMs are employed to study the effectiveness of various alternative strategies for reductions in emissions with respect to reductions in ambient ozone levels. plume models). airshed models). In the application of PAQSMS. 6.. given by ∂ ci ∂ ci ∂ + uj = ∂t ∂xj ∂xj Kjj ∂ ci ∂xj + Ri ( c1 .2 and Appendix C. The atmospheric diffusion equation. that is based on the fundamental physiochemical principles governing air pollution.1 Mathematical and Numerical Formulation of PAQSMs Many PAQSMs are based on the fundamental description of atmospheric transport and chemical processes. meteorological conditions and ambient air quality.1) is typically employed to describe the pollutant concentration in a selected control volume.. t) (6. . cN ) + Si (x. oxides of nitrogen (NOx ) and volatile organic compounds (VOCs) that are emitted by a wide variety of both anthropogenic and natural (biogenic) emissions sources. . and as via diagnostic modeling. These are numerical models that simulate the transport and chemical transformation of pollutants in the atmosphere. Ozone is a “secondary pollutant” formed through nonlinear chemical reactions between precursor species. 78]. This equation expresses the conservation of mass of each pollutant in a turbulent . that is statistical description of observed data. leading to substantial health risks [182. . one of the main pollutants of concern is ozone.122 6.g. The applications of PAQSMs range from simulation of transport and transformation of chemicals over a few kilometers over few hours (local scale) to over a few thousand kilometers over a few weeks (regional scale). as mentioned in Chapter 5. Certain meteorological conditions can significantly enhance the ambient concentrations of ozone in specific locales. Air quality can be modeled via prognostic modeling. . These models have been applied to local scale (e.2.g. (b) negligence of molecular diffusion compared to turbulent dispersion. Grid models as currently designed are generally not capable of resolving pollutant . or box. Under these conditions. and (d) approximation of ensemble average of reaction rates with the reaction rates for ensemble averages. Horizontal dimensions of each cell usually measure on the order of kilometers. Since these models lack spatial resolution. · represents the ensemble average.or three-dimensional array of uniform grid cells. In the above equation. The region to be modeled is treated as a single cell. they cannot be used in situations where the meteorological or emissions patterns vary significantly across the modeling region. Air quality models can be broadly categorized into the following types: Box models: These are the simplest of the numerical models for air quality modeling. Kjj is the turbulent dispersion coefficient (obtained by approximating turbulent transport with a gradient diffusion scheme). the pollutant concentrations can be described by a balance among the rates they are transported in and out of the volume. A more detailed description of the atmospheric diffusion is given in Appendix C (Equation C. Ri is the rate of formation of the ith species. turbulent fluxes are approximated by a gradient driven fluxes). The Einstein summation convention is used here with respect to the subscript j. and some upper limit to the mixing on the top. and Si is the rate of emissions of the ith species into the control volume. Here. and the east-west and north-south boundaries on the sides.1). (c) incompressibility of the atmosphere. and the rates at which pollutants react chemically or decay. The assumptions leading to the atmospheric diffusion equation above include: (a) first order closure approximation (i. bounded by the ground at the bottom.123 fluid in which chemical reactions occur. briefly described in Section 5. Grid Models: Grid models employ a fixed Cartesian reference system and divide the modeling region into a two. One example of the grid models is the Urban Airshed Model (UAM). a “box”.. the pollutant concentrations in a volume of air. while vertical dimensions can vary from a few meters to a few hundred meters. ci is the concentration. their rates of emissions.3 and in Appendix C.e. are spatially homogeneous and instantaneously mixed. There are practical and theoretical limitations to the minimum grid cell size. such as the wind flows. Further. Here. Plume models are trajectory models that describe short term behavior of the plume of emissions from a point source. most applications employ grid cell sizes of a few kilometers. such as high pollutant concentrations in the immediate vicinity of point sources. and emissions. important local effects. In order to describe the sub-grid phenomena. tend to become obscured by the averaging of pollutant concentrations over the grid cell volume. Thus. In addition. at scales smaller than the size of a grid cell. with length scales . Since the numerical grids used in grid models are considerably large in size (on the length scales of a few kilometers). which is assumed to travel solely with the horizontal wind. choice of a grid cell implies that the input data. the assumptions involved in the formulation of the atmospheric diffusion equation make it applicable for gird models with horizontal resolution coarser than 2 km [137]. However. and do not permit construction of the spatial and temporal variation of concentrations over the entire region. Increasing the number of cells increases computing and data acquisition effort and costs. all the local scale processes are lumped into one average quantity. plume models are employed. considering the domain-wide effects of emissions reductions. are resolved to that scale. turbulence. an understanding of the “sub-grid” effects is important in order to assess local scale effects. As these models assume uniform conditions within each grid cell. with specific emphasis on establishing causal relationships between emissions from point sources (such as industrial stacks) and the pollutant concentrations in the neighborhood of the point sources. Trajectory Models: The trajectory models use a moving coordinate approach to describe pollutant transport. their utility is in the calculation of extreme concentrations that are typically found in the plumes from point sources. the atmospheric diffusion equation is solved in an air parcel of interest. In essence. These models are used to assess the effectiveness of emissions reductions at a urban/regional scale. In practice. The typical time scales range from 1 hour to 24 hours.124 concentrations at the microscale – that is. These models give results only on the path of traversed by the air parcel described. . The formulation of the grid models and of the plume models is discussed extensively in the literature [80. 223]. The expansion of the boundaries. This model considers a control volume that moves downwind along the plume trajectory.3 Formulation of The Reactive Plume Model (RPM) The Reactive Plume Model. Further. The transfer of pollutant mass across cell boundaries is modeled in two phases: (a) an “entrainment” phase. 78. the cells are modeled to expand in a manner such that the amount of an inert species remains constant within each cell. version IV (RPM-IV).125 ranging from a few hundred meters to a few kilometers. resulting from the emissions from point sources such as industrial stacks [153. 195]. 181.1). but with dispersion parameters that evolve with time. As the plume expands. . and (b) a “detrainment” phase.e. by assuming a Gaussian distribution of the pollutant mass along the plume centerline. 6. 144. as the plume expands. and solves the diffusion/reaction equation by considering the nonlinear photochemistry and diffusion of chemical species from one cell into another cell. where the pollutants diffuse across cell boundaries due to concentration gradients. It uses either point source emission estimates or initial plume concentrations as inputs. the individual cells expand in volume correspondingly. Such detail cannot be achieved by increasing the resolution of grid-based models. since the model formulation of the grid-based models is not valid for finer resolutions (i. is a standard regulatory model used for estimating pollutant concentrations in the atmosphere. In this model. and calculates downwind concentrations. the plume is modeled by a set of “cells” consisting of equal masses of pollutants. where the expanding cell boundaries entrain the pollutants from other cells. and the equations governing equations for pollutant concentrations within each cell are as described in the following section. RPM-IV makes use of a diffusion/reaction equation similar in form to the atmospheric diffusion equation (Equation 6. less than 2 km length scales). 2: Schematic depiction of entrainment and detrainment steps simulated in the RPM (adapted from [195]) .126 Touch Down L = Depth of mixing layer Z = Effective stack height L Z Source L0 Downwind Distance X TD Figure 6.1: Schematic depiction of the evolution of an atmospheric plume (adapted from [195]) Plume Centerline t = t0 Entrainment t = t0+ dt j = -2 j = -1 j=1 j=2 Detrainment t = t0+ dt j = -2 j = -1 j=1 j=2 Figure 6. as functions of standard normal quantiles.0 implies that 95. wj . as follows: W (s) = fx σx (s) and H(s) = fy σz (s).2 RPM-IV Model Equations Based on mass balance. with only one cell in the vertical. 6. term (B) the dilution as the jth cell expands in the horizontal. . (6.4) Here. and fz are selected as cutoff values of the plume boundaries.2) where s is the total distance traveled by the plume up to time t under consideration. at the start of the simulation each cell contains equal amount of pollutant mass.3) where wj is the width of the jth cell [195]. For a plume described by M cells in the horizontal direction. (6.3. and fx . A typical value of fx = fy = 4. Based on the Gaussian distribution of initial plume mass [181]. the term (A) describes the change in pollutant concentrations due to chemical transformation.127 6. and term (D) the net flux of the ith contaminant into the cell.4% of the total pollutant mass is included within the plume boundaries. can be calculated using the recursive equation 2 = erf M wj √ 2σy − erf wj−1 √ 2σy .3. term (C) the dilution as the jth cell expands in the vertical. Since the pollutant mass is assumed to follow a Gaussian distribution. fx and fz indicate the amount of pollutant mass that is included in the model boundaries. the governing equation of the Reactive Plume Model is given by: B dci j = dt dci j dt A D − chem 1 dwj · wj ds uci j − 1 dhj · hj ds C uci j + Fji (6.1 Initial Physical Dimensions of the Plume The total width W (s) and the total depth H(s) of the photochemical plume can be estimated as functions of the horizontal and vertical dispersion parameters. the width of the cell j. as the plume travels downwind. in the horizontal. 6.2 describes the process of entrainment and detrainment in detail. In RPM-IV. and wj and hj denote the width and height of the jth. Once Kj s are computed. where Ej is the entrainment as the cell expands. respectively.6 can be used to compute the concentration in each cell at different times. Expressions for entrainment and detrainment are given by the following equations: i Ej = 1 yj − yj−1 dyj dt ci − j+1 dyj−1 dt ci j (6. Additionally.e. Further. for the expansion of the cells. Figure 6. Fji is the net flux into the cell. CB-IV describes the complex non-linear .7) I where Cj is the concentration of an inert species I in the jth cell. This I I condition results in the equation Ej − Dj = 0. the nonlinear plume photochemistry is modeled through the carbonbond mechanism. Equations 6. and i i i i is given by Fji = Ej − Dj . This condition leads to a recursive relationship to compute Kj . i. the condition of zero concentration gradient at the plume centerline results in the equation: I C2 y2 − y0 I I 2 C2 − C1 K1 = dy1 dt (6.4. and Dj is the detrainment due to the diffusion of the species across the boundaries of the cell.6) where yj is the distance from the plume centerline to the far most side of the cell j. ci denotes the concentration of chemical species i in cell j. and 6..5.8) The values of Kj can then be recursively obtained using the above two equations. u j denotes the speed at which the cells travel downwind. no net transfer of an inert species occurs from a cell. An additional condition is imposed on the expansion of the cells: the cells are allowed to expand in a way such that the amount of an inert species remains constant within each cell.5) i Dj = ∂ ∂y K ∂c ∂y = 2 yj − yj−1 Kj ci − ci j+1 j yj+1 − yj−1 − Kj−1 ci − ci j j−1 yj − yj−2 (6. as follows: dyj dt I Cj+1 − dyj−1 dt I Cj = 2 Kj I I Cj+1 − Cj yj+1 − yj−1 − Kj−1 I I Cj − Cj−1 yj − yj−2 (6. version IV (CB-IV) [216].128 In this equation. 129 gas-phase atmospheric photochemistry through a set of 95 reactions among 35 surrogate chemical species corresponding to organic bonds/functional groups [216]. This mechanism lumps similarly bonded carbon atoms, resulting in a condensed mechanism of that is used widely in the regulatory photochemical models. A brief description of the CB-IV mechanism, and the chemical reactions considered, is presented in Appendix C. 6.3.3 Limitations of the RPM-IV Figure 6.1 presents a schematic depiction of the physical structure of a plume as it evolves from a stack, and travels downwind. The assumption of uniform vertical concentration in the RPM (hence, only one “well mixed” cell in the vertical) is valid only at distances much greater than XT D from the point source. This poses a limitation, since the typical touchdown distance could vary between a few hundred meters to a few kilometers, depending on the meteorological conditions, the height of the stack, and the exit velocity at the plume source. When the value of XT D is large, the well-mixedness assumption is not valid at the vicinity of the plume, thus lessening the main advantage of using a plume model to study the “sub-grid” scale, local phenomena. In such cases, if the RPM is used with the uniform vertical mixing assumption, significant “errors” could result in the model calculations. In order to address this limitation of the RPM-IV, a corresponding “threedimensional” version of the RPM, called RPM-3D, is developed as part of this work. This RPM-3D simulates the evolution of a plume by dividing the plume cross-section into rectangular regions consisting of equal initial pollutant mass; the plume cross section is divided into columns of cells, as in the case of RPM-IV, and each column is further subdivided into rows. Figure 6.3 illustrates the physical structure of the plume cross section in the RPMIV and RPM-3D. In the figure, R1 is the region containing the cell closest to the plume centerline, R2 contains the ground-level cell along the plume centerline in the horizontal, and R3 contains the centerline cells with respect to both the horizontal and the vertical. Here, R1 corresponds to the representation used by RPM-IV, whereas R2 and 130 R1 R3 Z’ Y’ X’ X Y Z B C C’ B’ A’ A B C C’ B’ R2 RPM-IV Figure 6.3: Discretization of the plume cross section by RPM-IV (left) and RPM-3D (right) R3 correspond to representations used by RPM-3D. In typical cases, the extreme pollutant concentration occurs along the plume centerline, and the pollutant concentration decreases as the distance of the cells from the centerline increases. Hence, the extreme pollutant concentration (e.g., concentration of ozone) calculated by the RPM-IV is an average over the region R1. This provides a concentration for the estimation of exposure to ozone. However, as shown in the figure, the region R2 corresponds to the most likely ground level maximum ozone concentration (typically covering the region from the ground to up to less than a hundred meters), whereas the region R3 corresponds to the maximum ozone level concentration at the plume centerline. Clearly, the RPM-3D provides estimates of pollutant concentrations at a detailed spatial resolution that is appropriate for estimating human exposure. 6.4 Formulation of the RPM-3D The equations describing the transport of chemical species in RPM-IV are modified accordingly to accommodate the three dimensional description of the plume. Thus, Equation 6.4 is modified as follows: dci j,k dt = dci j,k dt chem − dwj,k 1 · wj,k ds uci − j,k 1 dhj,k · hj,k ds ¡¡ ¤ ¤ ¡¡ ¡¡ ¡¡ ¡¡¤ ¡¡£ £¤£¤£ £¤£¤£ ¡¡¤ ¡¡¤£ ¡¡£ ¡¡ ¡¡¤ ¤££¤£¤ ¤££¤£¤ ¡¡£¤£¤£ ¢¡¢ ¡ ¡ ¡ ¡  ¢ ¢ ¡ ¢ ¢  ¡ ¡ ¡ ¡ ¢ ¢ ¢¡¢ ¢ ¢ ¡ ¡ ¡ ¢ ¢ ¡¢ ¢  ¡ ¡ ¡ ¢ ¢ ¡¢ ¢  ¡ ¡ ¡ ¡  ¢¢  ¡ ¢¢   ¡ ¡ ¡ ¡ ¢ ¢ ¢¡¢ ¢ ¢ ¡ ¡ ¡ ¢ ¢ ¡¢ ¢  ¡ ¡ ¡ ¡  ¢ ¢ ¡ ¢ ¢  A’ A RPM-3D i i uci +FY,j,k +FZ,j,k (6.9) j,k 131 where ci is the concentration of the ith species in the cell (j, k); j denotes the cell j,k i number in the horizontal direction, and k denotes the cell number in the vertical. FY,j,k i and FZ,j,k denote the fluxes along the horizontal and vertical directions respectively. The equations for the calculation of the fluxes, are similar to Equations 6.5 and 6.6, and are as follows: i i i FY,j,k = EY,j,k − DY,j,k i i i FZ,j,k = EZ,j,k − DZ,j,k i EY,j,k = i DY,j,k 1 yj,k − yj−1,k 2 yj,k − yj−1,k 1 zj,k − zj,k−1 2 zj,k − zj,k−1 dyj,k dt KY,j,k dzj,k dt KZ,j,k ci j+1,k − dyj−1,k dt ci j,k ci − ci j−1,k j,k yj,k − yj−2,k (6.10) = i ci j+1,k − cj,k yj+1,k − yj−1,k ci j,k+1 − − KY,j−1,k ci j,k i EZ,j,k = i DZ,j,k = dzj,k−1 dt i ci j,k+1 − cj,k zj,k+1 − zj,k−1 − KZ,j,k−1 ci − ci j,k j,k−1 zj,k − zj,k−2 where yj,k , and zj,k denote the distances from the plume centerlines to the far most sides, in the horizontal and vertical directions respectively. KZ,j,k and KY,j,k are solved using recursive relationships similar to Equation 6.7: the relationships are obtained I I from the following conditions: (a) FY,j,k and FZ,j,k are equal to zero for an inert species I, (b) there is a zero concentration gradient at the plume centerline in both horizontal and vertical directions. The RPM-IV uses the Gear algorithm [76] to solve the stiff set of equations describing the chemical reactions. In this method, the step size for each successive calculation is adjusted according to the local truncation errors. The same approach is followed in the RPM-3D. 6.5 Case Studies The RPM-IV and RPM-3D are both applied to two significantly different case studies. An attempt is made to understand the extent of the uncertainty associated with the model assumptions and model resolution, by comparing the predictions of these two models. 132 0.13 0.125 Average Plume Concentration (ppm) 0.12 0.115 0.11 0.105 0.1 0.095 0.09 0.085 0.08 0.075 0 20 40 60 80 Plume Evolution Time (min) 2 Cells 4 Cells 6 Cells 8 Cells 10 Cells 100 120 Figure 6.4: Dependence of plume average O3 concentration on horizontal resolution of the RPM-IV in a VOC dominant regime The first case study uses emission estimates and field data measured near Marathon Oil Refinery at Robinson, Illinois, during June and July, 1977 [183]. This case study involves a primarily VOC dominant source (i.e., the volatile organic compounds emissions are significantly higher than the NO/NO2 emissions). The meteorological data and initial plume concentrations were obtained from the work of Georgopoulos and Roy [79], which was based on the measurements reported by Sexton et al. [183]. Figure 6.4 shows the average ozone concentrations in the plume, calculated by RPMIV, as a function of the plume evolution time for different horizontal resolutions, ranging from two cells to ten cells in the horizontal. Similarly, Figure 6.5 shows the average plume concentrations calculated by RPM-3D for different vertical resolutions with four cells in the horizontal. The calculations from RPM-IV, which has no vertical resolution, and RPM-3D with vertical resolution corresponding to two cells, four cells, and six cells, are shown in the figure. The results indicate that the different vertical and horizontal resolutions produce similar estimates of the average plume concentrations. Hence, it is sufficient to use a low-resolution two-dimensional model, for studying the plume average concentrations in such cases. Figure 6. this would clearly produce an under-estimation of the ground level ozone levels. it is clear that in a NOx dominant regime. The cells are illustrated in Figure 6.3 for RPM-IV). this would clearly produce an over-estimation of the ground level ozone levels. The horizontal centerline cell is given by “A”.5: Dependence of plume average O3 concentration on vertical resolution of the RPM-3D in a VOC dominant regime (4 horizontal cells) Figure 6. and “AX” (centerline).7 shows the vertical profile of the ozone concentrations calculated by RPM-3D for the centerline horizontal cell when the plume is divided into four horizontal cells and each cell is in turn divided into six vertical cells. Examining the results. and the corresponding vertical slices are “AZ” (outermost). “AY” (middle).09 0. Examining the results. The plots indicate the average of the vertical cells along the horizontal centerline.6 shows the ozone concentrations within each horizontal cell calculated by RPM-IV when the plume cross section is divided into six cells.3.13 Average Plume Concentration (ppm) 0.133 0.1 0.08 0. . The plots indicate the average ozone concentrations in the different horizontal cells (indicated by “A”. “B”. in a VOC dominant regime. the RPM-IV approximates the highest possible ground level concentration by the average in the vertical.07 RPM-IV 2 Cells 4 Cells 6 Cells 0 20 40 60 80 Plume Evolution Time (min) 100 120 Figure 6. Hence.12 0. and individual vertical cells at the horizontal line. Similarly. and “C” in Figure 6. it is clear that by ignoring the vertical structure of a plume.11 0. 13 0.14 0. RPM-IV) 0.134 0. horizontal) in a VOC dominant regime (4 horizontal cells and 6 vertical cells.16 0.06 Centerline Cell (Horizontal) Outermost Vertical Cell Middle Vertical Cell Centerline Cell 0 20 40 60 80 Plume Evolution Time (min) 100 120 Figure 6.14 0.12 0.08 0.18 0.08 0. RPM-3D) Concentration (ppm) Concentration (ppm) .7: Vertical profile of the O3 concentration at the plume centerline (w.2 0.1 0.1 0.12 0.09 0.11 0.07 Plume Average Outermost Horizontal Cell Middle Horizontal Cell Centerline Cell 0 20 40 60 80 Plume Evolution Time (min) 100 120 Figure 6.r.6: Horizontal profile of the O3 concentration in the plume in a VOC dominant regime (6 horizontal cells.t. New Jersey.02 0. described by Georgopoulos et al. Figure 6. 1988.04 0. The calculations from RPM-IV. [81]. The simulation is performed for the period of July 7. ranging from two cells to ten cells in the horizontal. This case study involves a NO/NO2 dominant regime. which has no vertical resolution. 8:00 am to 10:00 am.08 Average Plume Concentration (ppm) 0. Similarly. The model results are analyzed in the same manner as described in the earlier case study involving a VOC dominant regime. . calculated by RPMIV.8 shows the average ozone concentrations in the plume.05 0.06 0.9 shows the average plume concentrations calculated by RPM-3D for different vertical resolutions with four cells in the horizontal. The meteorological inputs were obtained from the model inputs of the UAM-IV simulation for the modeling domain [81].135 0. corresponding to the occurrence of a severe ozone episode in the region.07 0.03 0. as a function of the plume evolution time for different horizontal resolutions. 1988. Ambient pollutant concentrations were obtained from the results of an Urban Airshed Model (UAM-IV) simulation for the Philadelphia/New Jersey modeling domain for the time period of July 6 to July 8.8: Dependence of plume average O3 concentration on horizontal resolution of the RPM-IV in a NOx dominant regime The second case study considers a single point source in Mercer County. Figure 6.01 0 0 20 40 60 Plume Evolution Time (min) 2 Cells 4 Cells 6 Cells 8 Cells 10 Cells 80 100 Figure 6. 06 0. this would clearly produce an under-estimation of the ground level ozone levels.01 0 0 20 40 60 Plume Evolution Time (min) RPM-IV 2 Cells 4 Cells 6 Cells 80 100 Figure 6.04 0. Figure 6.136 0. it is clear that in a NOx dominant regime.05 0.10 shows the ozone concentrations within each horizontal cell calculated by RPM-IV when the plume cross section is divided into six cells. for studying the plume average concentrations in such cases.9: Dependence of plume average O3 concentration on vertical resolution of the RPM-3D in a NOx dominant regime and RPM-3D with vertical resolution corresponding to two cells. Hence. are shown in the figure. .08 Average Plume Concentration (ppm) 0.07 0. it is sufficient to use a low-resolution two-dimensional model. Examining the results.02 0. four cells. The results indicate that the different vertical and horizontal resolutions produce similar estimates of the average plume concentrations.03 0.11 shows the vertical profile of the ozone concentrations calculated by RPM-3D for the centerline horizontal cell when the plume is divided into four horizontal cells and each cell is in turn divided into six vertical cells. Figure 6. and six cells. Similarly. 137 0. horizontal) in a VOC limited (NOx dominant) regime (4 horizontal cells and 6 vertical cells.01 0 Centerline Cell (Horizontal) Outermost Vertical Cell Middle Vertical Cell Centerline Cell 0 20 40 60 Plume Evolution Time (min) 80 100 Figure 6.r.08 0.05 0.04 0. RPM-3D) .05 0.10: Horizontal profile of the O3 concentration in the plume in a NOx dominant regime (6 horizontal cells. RPM-IV) 0.04 0.02 0.03 0.03 0.11: Vertical profile of the O3 concentration at the plume centerline (w.02 0.06 Concentration (ppm) 0.08 0.06 Concentration (ppm) 0.t.07 0.01 0 Plume Average Outermost Horizontal Cell Middle Horizontal Cell Centerline Cell 0 20 40 60 Plume Evolution Time (min) 80 100 Figure 6.07 0. Further. the three dimensional version is to be preferred. Further. construction of a hierarchy of models with increasing detail.138 6. . results from models with varying detail provide an estimate of the range of model calculations. one that produces outputs similar to more detailed models. since both the two. However. thus helping in characterizing the uncertainty associated with model formulation and detail. the ground level ozone concentrations are required when calculating human exposure to ozone). The two case studies presented show that the two-dimensional version is preferable to the three-dimensional version when the average plume concentrations are calculated. the uncertainty associated with the calculation of average plume concentrations is relatively small. the approach suggested here. The comparison of the model results can provide insight into what level of detail is sufficient to produce results similar to more detailed models. when the output metrics require a higher resolution (for example.and three-dimensional versions give close estimates.and three-dimensional versions of the RPM indicate that the choice of model resolution is dependent on the output metric(s) considered. Such knowledge is very useful in building an “optimal model”. is recommended. but requires much less detailed inputs and much fewer computational resources. When evaluating the uncertainties associated with model formulation.6 Discussion The comparison of calculations of the two. the series expansions of the outputs contain unknown coefficients which are calculated by a method that uses the results of a limited number of model simulations. the main limitations in performing comprehensive uncertainty analyses of transport-transformation models are the associated cost and effort. there is a need for methods that can reduce the cost and effort associated with uncertainty analysis of transport-transformation models. The SRSM has been implemented as a modular and readily portable stand-alone tool that can be used by researchers as a black-box tool. without requiring the mathematical details of the SRSM. The SRSM approximates the model inputs and the outputs through a series of “well behaved” standard random variables. 7. with respect to control strategy selection for decreasing pollutant levels. In fact. for example. the development of the Stochastic Response Surface Method (SRSM) (Chapter 3). it is important to address these uncertainties. and the increased data needs for characterizing uncertainties in inputs. Hence. The SRSM is based on the Stochastic Finite Element Method [83] and the Deterministic Equivalent Modeling Method [198]. However.139 Chapter 7 CONCLUSIONS AND DISCUSSION The application of transport-transformation models involves significant uncertainties that may have implications on the confidence in model estimates and on their use in the assessment of possible alternative scenarios in a regulatory framework.1 Development and Application of the SRSM One of the main contributions of this research is the development of computationally efficient methods for uncertainty propagation. the SRSM has been implemented as a web-based . specifically. Clearly. the Reactive Plume Model (RPM). (b) a two-dimensional atmospheric photochemical plume model that calculates pollutant concentrations downwind of point sources. and for the estimation of correlations between two model outputs. The main impact of this research is the facilitation of fast uncertainty analyses of complex. the performance of the SRSM was significantly superior to both the Monte Carlo and the Latin Hypercube Sampling methods . and between a model input and a model output. mathematical formulae have been developed and presented for the extension of this method for dealing with arbitrary empirical distributions (with correlated and uncorrelated inputs). the EPA’s Composite Model for leachate Migration and Transformation Products (EPACMTP). a lumped system described by a set of stiff ODEs) describing the uptake and metabolism of perchloroethylene in humans. (d) the estimation of the unknown coefficients in the approximations. and (d) a one-dimensional ground water model with discontinuous probability density functions. and the estimation of the relative contribution of each model input to the uncertainties in the outputs. and also the need for identification of inputs that contribute to the output uncertainties the most.e. (c) a three-dimensional urban/regional scale air quality model. the Urban Airshed Model (UAM). (b) the construction of polynomial chaos approximations for the model outputs.the SRSM application . The SRSM has been evaluated for case studies involving the following models: (a) a biological model (a Physiologically Based PharmacoKinetic (PBPK) model. In the first two cases. Further. (c) the identification of sample points for model runs. version IV. that can be accessed through a web browser. as shown towards the end of Chapter 3. and (e) the subsequent calculation of the statistical properties of the model outputs. The development and the application of the SRSM meets the need for fast and accurate estimates of the uncertainties associated with the model outputs. computationally intensive models.140 tool.. This implementation includes modules for (a) the transformation of non-standard distributions into functions of the srvs. i. 2 Development and Application of the SRSM-ADIFOR In order to further improve the computational efficiency of the SRSM. it was impractical to perform hundreds of Monte Carlo simulations.141 required up to an order of magnitude fewer simulations for estimating the uncertainties in the model outputs. The case studies indicate substantial computer time savings. Hence. ADIFOR (Automatic DIfferentiation of FORtran) [17]. this method was coupled with a state of the art automated sensitivity analysis method. developing a “black-box tool” for using the combined SRSM-ADIFOR approach (as was done for the “stand-alone” SRSM) is not possible. The application of the SRSM-ADIFOR involves processing the original model code using ADIFOR. provides the potential to substantially reduce the computer demands. involving the Urban Airshed Model. For this reason. the SRSM. ADIFOR produces fast and accurate estimates of first order partial derivatives of model outputs with respect to model inputs or any intermediate quantities in the model code. SRSM-ADIFOR. required significantly fewer model runs compared to the Monte Carlo method. The coupled method. EPACMTP. up to two orders of magnitude reductions in the required number of model simulations compared to “conventional” methods were achieved. and subsequent use of the derivative code for uncertainty analysis. the SRSM required just twenty (20) simulations to estimate the output uncertainties: this demonstrates the potential of the SRSM to address problems that are beyond the scope of conventional methods (in terms of computational and time limitations). 7. In the third case study. . as before. In the last case study involving the ground water model. where the series expansions of the SRSM are used in conjunction with outputs and partial derivatives from the ADIFOR generated code. and the Reactive Plume Model (RPM). whereas. ADIFOR accomplishes this by rewriting the model code and producing code that calculates first order partial derivatives. This demonstrates the advantages of combining sensitivity analysis methods with uncertainty propagation methods. was applied to two of the case studies used for evaluating SRSM: a human PBPK model. the coupling of the SRSM and the ADIFOR. The case study stresses the need for comprehensive hierarchical models. moments of the output metrics. In order to extend this list.1 Directions for Future Work Improvements to the Stochastic Response Surface Method In the present form. filling data gaps versus refining a model). several other statistical metrics. However.4. When a model application involves both model and data uncertainties. This can be accomplished by following the techniques outlined in . can be calculated.2. Chapter 6) examines the issues associated with the characterization of uncertainties associated with model formulation. and percentile estimates. the methods outlined for transforming random variables that follow empirical distributions (Section 3. Lack of sufficient model resolution could in fact result in model predictions that may either significantly over-estimate or under-estimate the quantities that are important for decision making.4 7.4) can be incorporated into the SRSM tool. the SRSM includes modules for the transformation of probability distributions commonly used to represent uncertainties in the inputs of environmental and biological models. Such a comparison is useful in focusing resources where it is most appropriate (e. The SRSM calculates probability density functions (pdf s) to quantify the uncertainties in model outputs. since model inputs can follow one of several other distributions. Model uncertainty arises mainly when competing models.. The case study involving the development of a three dimensional version of the Reactive Plume Model (RPM-3D. this list is not extensive. Additionally. there are often uncertainties associated with model formulation and structure.2.3 Consideration of Uncertainties Beyond Parametric/Data Uncertainty In addition to the uncertainties associated with model inputs. exist.3) and for transforming random variables that are correlated (Section 3.142 7. it is important to identify the relative magnitudes of the uncertainties associated with data and model formulation. The transformations are listed in Table 3.1. for the systematic evaluation of the simplified models.g. ranging from simplified to more detailed models. such as the cumulative density function. 7. Transport-transformation models sometimes have constraints on the values the model inputs can assume. percentiles and correlation coefficients. This would facilitate the identification of individual contributions of model inputs to the uncertainties in model outputs. the diameter and porosity of a particle may not assume certain combinations in the EPACMTP model (Chapter 5). in a more rigorous manner than that presented in relation to the case study in Chapter 5 (Table 5. While many constraints are defined in terms of joint pdf s. that would be based not only on the pdf s of model outputs. application of the SRSM and the SRSMADIFOR to more complex models could also aid in the identification of areas where these methods could be further improved. For instance. the constraints could follow a discontinuous pattern. Additionally. often. statistically based evaluation of the SRSM and the SRSM-ADIFOR. Finally. Furthermore. can also be incorporated into the SRSM tool. but also on the moments. the sample value of random input x may affect the range from which another random input y is sampled. 7. cumulative densities. 7. These improvements could further enhance the already wide range of applicability of the SRSM. the methods for calculating correlations between outputs and between an input and an output. in some cases.4. and even in the wed-based SRSM tool. The Monte Carlo method can address such constraints by following the rules listed below: . For example.2).5.143 Section 3.4. would aid in the wider acceptance of this method.5.3 Addressing uncertainty propagation under constraints The case study for the evaluation of the SRSM with a groundwater model showed certain limitations in the application of the SRSM to models with discontinuous probability distributions.2 Further evaluation of the SRSM and the SRSM-ADIFOR Further rigorous. the constraints are based on the values of other model inputs. presented in Section 3. the same improvements translate into the improvements in the SRSM-ADIFOR method. • approximate the outputs using the polynomial chaos expansions. One possible approach that could be followed is suggested here:∗ • express inputs in terms of srvs. It must be noted that samples are drawn from the given pdf s. In the SRSM. and its applicability is still unknown. This means that the inputs have pdf s that are continuous and well behaved (as opposed to the actual pdf s which are discontinuous). • repeat the above procedure till required number of sample points are obtained. and • statistically analyze the model outputs corresponding to all the sample points.144 • generate sample point by randomly sampling all inputs from their respective distributions ignoring constraints. because the inputs are represented as algebraic functions of the srvs. and • generate the sample points for the srvs without considering the constraints. One potential focus of the future efforts could be on the evaluation of this approach. • estimate the unknown coefficients in the series expansion through regression on the model outputs at these sample points. • run the model at the accepted sample point. and repeat the above step till an acceptable sample point is obtained. in addition to the identification of other techniques to address uncertainty propagation under constraints. . and • calculate the statistical properties of the model outputs from the srvs using the methods listed in Chapter 3. such an approach does not appear to be readily applicable. and they also obey the constraints. ∗ This approach has not been tested. • ignore the sample point if any constraints are not satisfied.5 and in the process rejecting the combinations of srvs that do not obey the constraints. ignoring constraints. That is an advantage of using Monte Carlo methods in a brute-force manner. at a given time. ζi ). Further research could focus on extending the SRSM so that it can address uncertainty propagation involving model inputs defined by random processes and random fields. random fields are random functions of spatial coordinates and time.. y reduces to a random variable. ζ) is a stochastic process. The relationship between random processes. and emissions from area sources.5 Uncertainties Associated with Evaluation Data Characterization of uncertainty associated with the evaluation data is an important component of a comprehensive uncertainty analysis. and • x(ti . ζ) where t denotes the time dimension. • x(t. Similarly. in its present form addresses only random variables. The uncertainties in the evaluation .4 Random processes and random fields The SRSM.4. ti . Random processes and random fields are common in environmental and biological systems: for example. the random variables are analogous to points in a multi-dimensional space. random quantities that do not vary with time or space. and deterministic variables can be summarized as follows [160]: • x(t.145 7. or a family of time functions. and ζ is used to denote the randomness. deterministic functions. Here.4. i.ζ) is a random variable. the flow rates or emissions from a point source are random processes. Further. a particular realization of y can be considered as a deterministic function y(t. 7. Random processes and random fields are generalizations of random variables to multi-dimensional spaces.ζi ) is a single number. where ζi denotes one realization of the various possible functions that the random process y can assume. as y(t. 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Gardiner [75] presents the applications of probabilistic analysis in modeling. There are a number of text books that describe the concepts and application of probabilistic analysis in detail. Additionally.165 Appendix A PROBABILISTIC APPROACH FOR UNCERTAINTY ANALYSIS Probabilistic approach is the most widely used technique for uncertainty analysis of mathematical models. Johnson [114] explains the multivariate random variables. is an event. and Papoulis [160] presents an excellent description on probability and random variables from a mathematical view point. such as the tossing of a coin. the situation where the contaminant concentration exceeds a regulatory level can be an event. Tsokos [201] presents excellent introductory material for probabilistic analysis. uncertainties are characterized by the probabilities associated with events. In the probabilistic approach. experimental measurement of contaminant concentration. or any of the possible predictions of a model describing the system. Similarly. A probability of 0 for an event means that the event will never occur. In short. This appendix attempts to merely summarize some basic information on probability and random variables. the probability of an event is defined as the ratio of the number of times the event occurs to the total number of samples or experiments.as and mathematical modeling of a physical system. the situation where the model outputs fall in a certain range. This means that if the coin is tossed a large number of times. Examples of experiments or samples include repeated coin tossing. or each sub-range of values. the probability of heads turning up is 0. The following examples illustrate the concept of probability based on a large number of samples: (a) if a coin is tossed. a large number of independent measurements of contaminant concentration. Examples include contaminant concentrations in the environment. In these cases. Based on the nature of the values that random variables can assume. (b) the statement that the probability that a pollutant concentration c lies between c1 and c2 equals p means the following: from a large number of independent measurements of the concentration c. heads will turn up roughly half the time and tails half the time. one cannot define the probability that a random variable x is exactly equal to a value x0 . and there is an associated probability for each value (or interval) that x takes.5.166 of 1 indicates that the event will always occur. For example. they can be classified into three types: (a) continuous random variables: these variables can assume continuous values from an interval. emissions from an industrial source. under identical conditions.5 and that of the tails is 0. the number of times the value of c lies between c1 and c2 is roughly equal to the fraction p of the total number of samples. what is the probability that c takes a value lower than c0 .1 Random Variables Random variables are mathematical quantities that are used to represent probabilistic uncertainty. since . They can describe the probabilities associated with each value. A. and physical parameters in an exposure model. what is the probability that c takes values between c1 and c2 ? Mathematically. an uncertain quantity can take. the following questions can be answered by analyzing the random variable c: Given a concentration c0 . or a large number of simulations using a mathematical model with randomly varying parameters. or given two numbers c1 and c2 . if a random variable c represents the uncertainty in the concentration of a pollutant. such as body weight or respiration rate. The random variable x can assume values from −∞ to +∞. a random variable x maps a probability space Ω onto the real line. Continuous and discrete random variables can be contrasted as follows: one can define the probability that the number of air quality exceedences to be exactly equal to a given number n. probabilities are defined on intervals (e. A major part of uncertainty analysis in environmental modeling and risk characterization involves uncertainties in continuous quantities.1 Continuous Random Variables Continuous random variables are characterized through the following functions: . and the answer for each point would be zero.167 there are uncountably infinite number of possible values. a probability density can be defined at each point in the interval. and the number of defective cars in a production line. (c) mixed random variables: these variables can assume continuous as well as discrete values. where as one cannot define the probability that the atmospheric concentration is exactly equal to a given concentration c. Further.this can assume integral values between 0 and 365. (b) discrete random variables: these variables can assume discrete values from a set.. The present work focuses mainly on uncertainties described by continuous random variables. and may have properties of discrete random variables in others. the sum of a discrete and a continuous random variable results in a mixed random variable. in such cases. the probability density at a point is representative of the probability in the vicinity of that point. the number of days an air quality standard is violated in a year .g. where the outcome can have only integer values between 1 and 6. These variables have probabilities associated with a countable number of values they can assume. the probability that x lies between x1 and x2 ). Examples include the uncertainties in measured or predicted concentrations. Hence. and the uncertainties in the estimated time of exposure to a contaminant. A. Examples include the following: rolling of dice.1. They may have the properties of continuous random variables in certain ranges. For example. . Fx (x) is monotonous 0 ≤ Fx (x) ≤ 1. fx (x). This denotes the probability that the random variable x has a value less than or equal to x. the probability that x takes on values between x1 and x2 for x2 ≥ x1 . The probability that x has a value greater than x is given by 1-Fx (x).e. This is the derivative of the cumulative density function. fx (x) = dFx (x) dx The main properties of the probability density function are: fx (x) ≥ 0. This function is also known as the cumulative distribution. for − ∞ < x < ∞ fx (−∞) = fx (∞) = 0 x Fx (x) = −∞ fu (u)du x2 Pr{x1 ≤ x ≤ x2 } = Fx (x2 ) − Fx (x1 ) = x1 fx (x)dx (c) Expected value of a function. for − ∞ < x < ∞ Fx (−∞) = 0 and Fx (∞) = 1 In population risk characterization. Fx (x).168 (a) Cumulative density function. This function is also known as the probability distribution. Further. is given by: Pr{x1 ≤ x ≤ x2 } = Fx (x2 ) − Fx (x1 ) The important characteristics of the cumulative density function are: Fx (x2 ) ≥ Fx (x1 ) iff x2 ≥ x1 i. of a random variable. (b) Probability density function. This is defined as ∞ E{g(x)} = −∞ g(x)fx (x)dx . g(x). the corresponding cumulative density function can be considered analogous to the fraction of the population that is at risk with respect to a given risk criterion. the probability density function fx (x) and the random variable x are interchangeably used. (a) The expected value or mean of a random variable: this denotes the average value for the distribution of the random variable. which is not a possible outcome in the coin tossing problem. Mathematically. the terms “distribution” and “random variable” are also interchangeably used. especially when the distribution functions consist of a complex algebraic expressions. and a higher value indicates that the distribution is spread out over a wider range of possible values. . the mean η of a random variable x is given by ∞ η = E{x} = −∞ xfx (x)dx It should be noted that the mean does not necessarily represent a realistic value from the distribution. A lower value of variance indicates that the distribution is concentrated close to the mean value. the expected value of the distribution is equal to the arithmetic mean of the sample values. If a large number of samples from the distribution are considered. they do not provide information on uncertainty at a quick glance. (b) The variance or dispersion of a distribution: this indicates the spread of the distribution with respect to the mean value. and if heads is assigned a value 1. as they both represent the same. In such cases. and tails a value 0. if a coin is tossed a large number of times. A. Further.2 Moments of a Random Variable The moments of a random variable provide concise information about a random variable. For example. The following are the moments that describe a random variable.1. Even though the density functions of a random variable provide all the information about that random variable.169 A probability density function or a cumulative density function fully characterizes a random variable. the mean is 1/2. In the following sections. the moments of a random variable serve as useful metrics that provide a significant amount of information about a distribution. (d) The kurtosis of a distribution indicates the flatness of the distribution with respect to the normal distribution. m1 = η. It is given by γ2 = 1 1 E{(x − η)4 } = 4 4 σ σ ∞ (x − η)4 fx (x)dx −∞ A value of kurtosis higher than 3 indicates that the distribution is flatter compared to the normal distribution. µ0 = 1. (e) Higher order moments: The higher order moments of a random variable are defined as follows: The kth moment mk of x is given by ∞ mk = E{xk } = −∞ xk fx (x)dx The kth central moment µk of a random variable x is defined as ∞ µk = E{(x − η)k } = −∞ (x − η)k fx (x)dx Clearly. characterizing the shape of the distribution. It is given by γ1 = 1 1 E{(x − η)3 } = 3 3 σ σ ∞ (x − η)3 fx (x)dx −∞ A positive value of skewness indicates that the distribution is skewed towards values greater than the mean (i..170 The variance σ 2 of x is given by: ∞ σ 2 = E{(x − η)2 } = −∞ (x − η)2 fx (x)dx The square root of variance is called the standard deviation (σ). Further. µ1 = 0.e. the central moments µk ’s and the moments mk ’s are related as follows: k µk = r=0 k k Cr (−1)r η r mk−r Cr η r µk−r and mk = r=0 k . and a smaller value indicates a higher peak (relative to the normal distribution) around the mean value. skewed towards the right side) and a negative value indicates that the distribution is skewed towards the left side. m0 = 1. (c) The skewness of a distribution indicates the asymmetry of the distribution around its mean. and µ2 = σ 2 . (k − 1) .171 where k Cr = k. .5). the probability that a random variable takes a value below the median is 0. LN(µ. the median is eµ . the 95th percentile indicate the values above which 5% of the samples occur. (k − r − 2)(k − r − 1) k! = 1. For example. σ). and the mode is e(µ−σ ) . Additionally. the mean is e(µ+σ 2 2 /2) . and mode are equal to µ. whereas for the lognormal distribution. .1. . The mode of a distribution is the value at which the probability density is the highest. Mode and Percentiles of a distribution In addition to the information provided by the moments of a distribution. the mean.. For example.3 Median. for the normal distribution N(µ.2. . The median of a distribution is the value for the 50th percentile of the distribution (i.e. some other metrics such as the median and the mode provide useful information. other percentiles of the distribution may sometimes be desired. . σ).r (k − r)! r! A. median. with zero mean. x2 ] and [y1 . Similarly. Fxy (∞. y) = ∂ 2 Fxy (x.y2 ] is given by: x2 y2 Pr{x1 ≤ x ≤ x2 and y1 ≤ y ≤ y2 } = x1 y1 fxy (x. y ≤ y} The joint probability density function fxy (x. ∞) = Fx (x) . the relationships among these random inputs need to be considered for uncertainty analysis. Fxy (−∞. To illustrate the concept of probability analysis involving many random variables. Fxy (x. This is especially useful in situations where the knowledge of one variable provides information regarding other variables. y)dy = fx (x) and −∞ −∞ fxy (x. y) = Fy (y) Fxy (∞. Such relationships can be described by joint probability distributions. y) = 0 .172 A. the joint distribution function can be defined as follows: Fxy (x. y) = Pr{x ≤ x. The following relationships hold for the joint distribution function: Fxy (x. y)dx = fy (y) Further. some of which may be interdependent. y) is defined as: fxy (x. −∞) = 0 ∞ ∞ fxy (x. considering the event that x ≤ x and y ≤ y. and Fy (y) is the marginal distribution of y. fx (x) is called the marginal density function of x. thus restricting the range of possible values the other variables can assume. ∞) = 1 . y)dydx . and the results are extended for the general case involving a set of random variables. Given two random variables x and y. the probability that the random variables x and y lie in intervals given by [x1 .2 Jointly Distributed Random Variables When the behavior of a system depends on more than one random input. and fy (y) is the marginal density function of y. the case for two random variables is presented first. the probabilities Pr{x ≤ x} = Fx (x) and Pr{y ≤ y} = Fy (y) can be defined. Further. y) ∂x∂y Fx (x) is called the marginal distribution function of x. . . . xn in } = ∞ −∞ ∞ ∞ −∞ . . ∞) The marginal density function of xi is given by: ∞ ∞ ∞ −∞ fxi (xi ) = . xn )dx1 dx2 · · · dxi−1 dxi+1 · · · dxn Further. .1 Moments of Jointly Distributed Random Variables For two jointly distributed random variables x and y. y)dxdy Similarly. . . . . . . . .xn . x2 . xn ) is given by: E{g(x1 . . . x2 . . . dxn The covariance of two random variables. . xn )fx1 . . . xn (x1 . xn (x1 . is defined as µ1. . the i..x2 . xn )dx1 dx2 . xn ) The probability density function is given by: fx1 . . x2 . . . x2 . . . xn ≤ xn } = Fx1 . . . .173 In general. . jth joint moment is defined as follows: ∞ ∞ mi. . the i1 .j = E{xi y j } = −∞ −∞ xi y j fxy (x. . x2 . −∞ x1 i1 x2 i2 .. . . xn in fx1 . . . . dxn A. . ∂xn The marginal distribution function of random variable xi is given by: Fxi (xi ) = Fx1 . . . . i2 . xn (x1 . . . x2 . . x2 . . . xn (x1 .. x2 . . . . . . . for the case of n random variables. x2 . x2 . . . x2 . the joint distribution function of random variables x1 . .. . . . . . . xn (∞. . xn )} = ∞ −∞ ∞ ∞ −∞ ...1 = E{(x − ηx )(y − ηy )} . x2 . . x2 ≤ x2 ... . . . . . ∞. . . . . . when n random variables are considered. −∞ g(x1 . . xn )dx1 dx2 . xn ) = ∂ n Fx1 . . . −∞ −∞ fx1 . . . . . . . ∞. . . . . x2 . widely used in probabilistic analysis.in = E{x1 i1 x2 i2 .i2 . . xn (x1 . . xi . .2. . . xn is given by: Pr{x1 ≤ x1 . . xn (x1 . in th moment is defined as: mi1 . x2 . the expected value of a function g(x1 . x1 . . x2 . . . . x2 ... . . xn ) ∂x1 ∂x2 . x2 . x2 . are uncorrelated if E{xi xj } = E{xi }E{xj } for all i = j They are orthogonal if E{xi xj } = 0 for all i = j and they are independent if fx1 . . σy are means and standard deviations of x and y respectively. . 114.1 . n random variables. ηy . . x1 . The covariance matrix.2. . . . . . 75]. xn ) = fx1 (x1 )fx2 (x2 ) . A.174 The correlation coefficient of x and y is defined as rxy = µ1.2 Dependence of Random Variables Two random variables x and y are uncorrelated if E{xy} = E{x}E{y} They are orthogonal if E{xy} = 0 and they are independent if fxy (x.Σ. . 201.j = E{(xi − ηxi )(xj − ηxj )} The uncertainty in model parameters are very often provided in terms of the covariance matrix. fxn (xn ) The interested reader can find additional material on probability concepts from the excellent texts on this subject [160. xn (x1 . . σx . . .1 σx σy where ηx . x2 . x2 . . . y) = fx (x)fy (y) Similarly. xn . of n random variables is defined as a symmetric matrix with the following elements: Σi. The mass balance on all compartments j in the PBPK model. The set of differential equations of the PBPK model are derived through the mass balance across various compartmens (see Figure 5. Qcardiac + Qalveolar /Pblood/air (B. cvenous = 1 Qcardiac n Qcardiac cvenous + Qalveolar Cair(inhaled) . Rj . or continuous stirred tank reactors (CSTRs). Each compartment can describe either an organ or a tissue.j Kmj + cv. as follows.j (B. (B.2) and Qj and cj are volumetric blood flow and concentration with respect to compartment j.175 Appendix B BASIC PBPK MODEL EQUATIONS A Physiologically Based PharmacoKinetic (PBPK) model for humans descibes the body as a set of interconnected compartments.1) Qj cj j=1 (B.1.3) The rate of metabolism. . Roy [176] presents a thorough description of PBPK models for humans. is given by: Vj dcj cj = Qj carterial − dt Pj/blood − Rj . on Page 79). in any given compartment is given by: Rj = Vmaxj cv. A mass balance around the equilibrium lung compartment results in: carterial = where. other than the viable skin and stratum corneum compartments.4) where Vmaxj and Kmj are Michaelis-Menten constants. 176 Appendix C URBAN AIRSHED MODEL (UAM-IV) AND CARBON BOND MECHANISM (CB-IV)∗ C. and removal processes are expressed as follows: ∂Ci ∂(U Ci ) ∂(V Ci ) ∂(W Ci ) + + + ∂t ∂x ∂y ∂z ∂ ∂x ∂Ci ∂x ∂ ∂y ∂Ci ∂y ∂ ∂z ∂Ci ∂z = where. KH + KH + KV + Ri + S + Li (C.1 C. A detailed description of the UAM can be found in the UAM-IV Users Guide by Morris and Myers [154] ∗ .1. chemical reactions. This equation represents a mass balance in which all of the relevant emissions. transport.1 The Urban Airshed Model – UAM-IV Conceptual Overview The Urban Airshed Model is a three-dimensional photochemical grid model designed to calculate the concentrations of both inert and chemically reactive pollutants by simulating the physical and chemical processes in the atmosphere that affect pollutant concentrations [178]. The basis for the UAM is the atmospheric diffusion equation (ADE).1) adapted with permission from Wang [213] and from Walia [211]. diffusion. .. . w KH . t. z. Pp ) net rate of removal of pollutant i by surface uptake processes f (x. y.177 i Ci u. . . i. . The major factors that affect photochemical air quality include [223]: • The spatial and temporal distribution of emissions of NOx and VOC (both anthropogenic and biogenic). . UAM is mainly used to study the photochemical air quality. • The dynamics of the atmospheric boundary layer. . . y.hour period to model air quality “episodes” – periods during which adverse meteorological conditions result in elevated pollutant concentrations of the chemical species of interest. . . • The spatial and temporal variations in the wind fields. . including stability and the level of mixing. Ozone is primarily formed in the atmosphere through a complex chemical mechanism involving oxides of nitrogen (NOx ) and volatile organic compounds (VOCs) in the presence of sunlight. . z. t) Li = = This model is typically applied to an 8. which pertains to air quality with respect to ambient ozone concentrations. CN ) emission rate of all precursors P1 . High ozone concentrations in the ambient environment lead to adverse health effects [78. .181]. . z. and other important species. . . y.to 72. 2. • The composition of the emitted VOC and NOx . . . Pi . z. S(x. N ) pollutant concentration of species i horizontal and vertical wind speed components = f (x.2 is implemented in the UAM). . NOx . .. Ci . (the Carbon Bond Mechanism-IV described in detail in Table C. v. y. P2 . P1 . . Pp . • The diurnal variations of solar radiation and temperature. t) rate of formation of i by chemical reactions = f (C1 . K V Ri S = = = = = = = chemical species i (where i = 1. . . . . . t) horizontal and vertical turbulent diffusion coefficients = f (x. • The chemical reactions involving VOC. P2 . C2 . u and v wind components in each grid cell) for each vertical layer. C.e. In each advection time step.and y-directions. The diffusion break corresponds to the top of a mixed layer. Proper specification of the hourly and three-dimensional . The UAM is typically run with a horizontal resolution of 2-5 km and a vertical resolution of 4-6 layers up to about 2 km above ground level. Vertical layer thicknesses are defined by the user based on the diffusion break. and other species at the immediate outside of the boundary of the region of study. either an unstable convective layer during the day (i. that is by the mean or bulk motion of the air. The vertical wind velocity in the UAM terrain-following coordinate system can then be calculated from the conservation of mass equation.1. The advection time step must be divided into multiple chemistry and vertical diffusion time steps to maintain numerical stability. The region to be simulated is divided up into a three-dimensional grid covering the region of interest.g. the mixing height) or a shallow mechanically mixed layer at night.178 • The loss of ozone and ozone precursors by dry deposition. is typically on the order of 5 min. and • The ambient background of VOC. The master or advection time step. advection. The UAM solves the atmospheric diffusion equation (ADE) using the method of fractional steps. determined by numerical stability criteria (grid size vs domain-maximum wind speed). NOx . The region top is usually defined at or slightly above the maximum daily diffusion break. the number of layers below and above the diffusion break and the minimum layer thickness. the terms in the equation that represent the different atmospheric processes (e. Advection in the UAM is treated by specifying horizontal wind fields (i. chemistry or vertical diffusion) are solved separately using the most efficient numerical integration technique for the given process. and above the region of study.2 Treatment of Physical/Chemical Processes Advective pollutant transport Pollutants are transported primarily by advection. Horizontal grid cells are rectangular with constant lengths in the x. the top of the region.e. the equations contain wide variations in time (reaction rate) constants. The transport resistance is estimated from theoretical considerations of turbulent transfer in .179 varying winds is one of the key steps to successful application of the UAM. ozone. Surface Removal Processes Many types of pollutants. dry deposition is assumed to occur in a two-step process: the transfer of pollutants through the atmosphere to the surface and the uptake of the pollutants by vegetation and other materials at the surface. In the UAM. carbon monoxide and sulfur compounds. Atmospheric Chemistry The UAM employs the Carbon Bond Mechanism (CBM-IV) for solving chemical kinetics of atmospheric chemistry. are removed from the surface layer by such features as vegetation through the process of dry deposition. 138]. Turbulent diffusion Turbulent diffusion (dispersion) is assumed to be proportional to the rate of change of concentration in space (i. Control theory techniques are employed in conjunction with the results of a planetary boundary layer model to generate optimal diffusivity coefficients in the UAM [139. advected downwind and diluted. the CBM-IV contains 86 reactions and 35 species. the solution of the CBM-IV in the UAM uses quasi-steady-state assumptions (QSSA) for the low-mass fast-reacting species (i. The winds influence how different emissions are mixed together. The differential equations that describe the CBM-IV are a “stiff” system. such as the one developed by Gear [77]. The proportionality factor is termed the eddy diffusivity coefficient (Kx .e. Because it has been difficult to obtain precise measurements of the eddy diffusivity coefficients. would result in prohibitively expensive computer time. As implemented in the UAM. the concentration gradient).e. Ky and Kz ) in the ADE). that is. Thus. This process involves a resistance to mass transport and a resistance to surface removal. Solving these equations with a “stiff” numerical integration scheme. including nitrogen oxides. theoretical estimates have been used. the stiff species) and the more computationally efficient Crank-Nicholson algorithm for the remainder of the state species. and air quality data for a particular day or days. emission inventory development. Data collection.1.180 the atmospheric boundary layer. Once the model results have been evaluated and determined to perform within prescribed levels. Philadelphia/New Jersey. the same meteorological inputs and a projected emission inventory can be used to simulate possible future emission scenarios. Louis UAM applications and an extensive series of UAM sensitivity studies designed to provide guidance concerning the types and amounts of data required to support the UAM application. were collected in Tulsa. model performance evaluation and application were major elements of this nation-wide program. In the late 1970s. Baltimore/Washington and . The UAM was used to evaluate whether various transportation plans and programs were consistent with the SIP and to evaluate the effects on Denver’s air quality of urban growth and development that might result from the construction of proposed wastewater treatment facilities. The model is then applied with these inputs and the results are evaluated to determine its performance. supported by OAQPS. data for an application of the UAM. Model inputs are prepared from observed meteorological. The first regulatory use and practical applications of the UAM were carried out for the Denver area on behalf of the Colorado Division of Highways and EPA’s Region VII in 1978. it is ideally suited for evaluating the effects of emission control scenarios on urban air quality. emission. The surface resistance is obtained from experimental data on the uptake of pollutants by various surface features. C. Building off the St. the model will calculate hourly ozone patterns likely to occur under the same meteorological conditions as the base case. EPA’s OAQPS initiated a program to examine the applicability and practicality of the UAM in routine ozone attainment demonstrations required by the SIP process. That is.3 Applications of the UAM Because the UAM accounts for spatial and temporal variations as well as differences in the reactivity/speciation of emissions. This is accomplished by first replicating a historical ozone episode to establish a base case simulation. molecular lumping. These mechanisms are lumped. is treated explicitly because it reacts much slower than other alkenes. Isoprene is an Alkene but it is treated explicitly also because its reactions with free radicals are much faster. This mechanism is a hybrid of explicit chemistry. the California Air Resources Board. The lumping method of carbon-bonds is used mainly for paraffins and olefins. and the Reactive Plume Model (RPM) [195. Molecular surrogates Toluene and Xylene are used for higher . British Leyland. the UAM has been or is being used by the EPA. Arizona Department of Health. and the cities of Taipei and Kaohsiung in Taiwan. Since its first use for air quality planning in 1978. Routine use of the UAM is an emerging practice. the Exxon Corporation. the South Coast (California) Air Quality Management District. by one of several strategies: mathematical lumping. 153]. and it forms a prominent part of biogenic emissions in rural areas. due to high computational overhead involved with their solution at each time-step. and that it yields a high fraction of Formaldehyde. Explicit chemistry is used for the inorganic and Carbonyl species and the chemistries of Ethene. or approximated by a smaller number of reactions. and Formaldehyde. or structural lumping. Isoprene. the TNO in the Netherlands.2 Carbon Bond Mechanism (CB-IV) Chemistry mechanisms included in most Photochemical Air Quality Simulation Models (PAQSMS) use reduced versions of known chemical mechanisms. The Carbon Bond-IV Mechanism was developed mainly for urban smog and regional atmospheric modeling [154]. the New York Department of Environmental Protection.181 New York. Most chemical mechanisms are too big to be implemented efficiently in PAQSMS. The Ethene chemistry. as is. Many peroxy radicals have been lumped into a single XO2 universal Peroxy radical. This photochemistry mechanism is implemented in the Urban Airshed Model (UAM) [154]. C. and lumped or generalized chemistry designed to simulate the features of urban smog chemistry. for example. it is a large fraction of common hydrocarbon emissions. the New Jersey Department of Environmental Protection. Southern California Edison Company. Pacific Gas and Electric. surrogate approximations. are placed at the highest levels. . in terms of their oxidation products (paraffins. some of the rates and the stoichiometries of some incorporated reactions depend upon the atmospheric composition of reacting hydrocarbons.2 presents the chemical reactions that constitute the CB-IV mechanism. OLE. The hierarchical relationship between the major species in the carbon bond mechanism is depicted in Figure C.182 aromatic compounds.1. and PAR surrogates. olefins. The most complex species. Also. The simplest species. isoprene and aromatic hydrocarbons). The chemical species and surrogates that are explicitly represented in CBM-IV are listed in Table C. a complex molecule with both aromatic and alkene structures might be represented with a combination of TOL.1. in terms of molecular complexity ( NOx. CO and formaldehyde). are placed at the lowest levels. HOx. For instance. Table C. HOx Figure C.183 PAR OLE ISOP TOL. XYL KETONES ALD2 DICARBONYLS PAN METHANE ETH FORM CO NOx.1: Major species in the CBM and their hierarchical relationship (adapted from Wang [213]) . 1: Chemical Species in the CBM-IV mechanism Species Name Nitric oxide Nitrogen dioxide Nitrogen trioxide (nitrate radical) Dinitrogen pentoxide Nitrous acid Nitric acid Peroxy-nitric acid (HO2 NO2 ) Oxygen atom (singlet) Oxygen atom (triplet) Hydroxyl radical Water Ozone Hydroperoxy radical Hydrogen peroxide Carbon monoxide Formaldehyde (CH2 =O) High molecular weight aldehydes Peroxyacyl radical (CH3 C(O)OO·) Peroxyacyl nitrate (CH3 C(O)OONO2 ) Paraffin carbon bond (C-C) Secondary organic oxy radical Olefinic carbon bond Ethene (CH2 =CH2 ) Toluene (C6 H5 -CH3 ) Cresol and higher molecular weight phenols Toluene-hydroxyl radical adduct Methylphenoxy radical High molecular weight aromatic oxidation ring fragment Xylene (C6 H4 -(CH3 )2 ) Methylglyoxal (CH3 C(O)C(O)H) Isoprene NO-to-NO2 operation NO-to-nitrate operation Total representation NO NO2 NO3 N2O5 HONO HNO3 PNA O1D O OH H2O O3 HO2 H2O2 CO FORM ALD2 C2O3 PAN PAR ROR OLE ETH TOL CRES TO2 CRO OPEN XYL MGLY ISOP XO2 XO2N 33 .184 Table C. 185 Table C. R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24 R25 R26 R27 R28 R29 1 Reaction NO2 + hν O O3 + NO O + NO2 O + NO2 O + NO NO2 + O3 O3 + hν O3 + hν O1 D O D + H2 O O3 + OH O3 + HO2 NO3 + hν NO3 + NO NO3 + NO2 NO3 + NO2 N 2 O5 + H 2 O N 2 O5 NO + NO NO + NO2 H2 O NO + OH HNO2 + hν OH + HNO2 HNO2 + HNO2 NO2 + OH OH + HNO3 HO2 + NO HO2 + NO2 −→ O2 .2: Chemical Reactions in the Carbon Bond IV Mechanism Reaction No.89 NO2 + 0.89 O + 0.11 NO 2 NO2 NO + NO2 N 2 O5 2 HNO3 NO3 + NO2 2 NO2 2 HNO2 HNO2 NO + OH NO2 NO + NO2 HNO3 NO3 OH + NO2 PNA −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ M M M M O2 M M M M M CB-IV reactions – continued on next page .M NO + O O3 NO2 NO NO3 NO2 NO3 O O1 D O 2 OH HO2 OH 0. 94 HO2 + 0.1 OH CB-IV reactions – continued on next page .13 XO2N + 0.22 XO2 + 0.5 ALD2 + 0.79 XO2 + 0.38 HO2 + 0.2 FORM + 0.76 ROR .87 XO2 + 0.28 XO2 + 0.02 XO2N + 0.96 XO2 + 1.0.11 PAR R53 ROR −→ 0.79 HO2 + 0.79 FORM + 0.186 CB-IV reactions – continued from previous page Reaction No.22 PAR + 0.11 HO2 + 0.1 PAR R54 R55 R56 ROR ROR + NO2 O + OLE −→ −→ −→ HO2 0.74 FORM + 0.2 OH R57 R58 OH + OLE O3 + OLE −→ −→ FORM + ALD2 .04 XO2N + 0.79 OH FORM + XO2 + HO2 0.11 ALD2 + 0.63 ALD2 + 0.1 ALD2 + 0.02 ROR .2.3 CO + 0.PAR + XO2 + HO2 0. R30 R31 R32 R33 R34 R35 R36 R37 R38 R39 R40 R41 R42 R43 R44 R45 R46 R47 R48 R49 R50 R51 R52 PNA OH + PNA HO2 + HO2 HO2 + HO2 H2 O H2 O2 + hν OH + H2 O2 OH + CO FORM + OH FORM + hν FORM + hν FORM + O FORM + NO3 ALD2 + O ALD2 + OH ALD2 + NO3 ALD2 + hν C2 O3 + NO C2 O3 + NO2 PAN C2 O3 + C2 O3 C2 O3 + HO2 OH PAR + OH Reaction −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ O2 O2 O2 O2 2O2 O2 O2 M HO2 + NO2 NO2 H 2 O2 H 2 O2 2 OH HO2 HO2 HO2 + CO 2 HO2 + CO CO OH + HO2 + CO HNO3 + HO2 + CO C2 O3 + OH C2 O3 C2 O3 + HNO3 FORM + 2 HO2 + CO + XO2 FORM + NO2 + HO2 + XO2 PAN C2 O3 + NO2 2 FORM + 2 XO2 + 2 HO2 0. 67 HO2 + 0.03 ALD2 + 0.6 XO2 + 0.13 XO2N + ETH + 0.7 HO2 + CO + 0.03 XO2 + 0.36 CRES + 0.62 C2 O3 + 0.56 TO2 0.1 OH R78 R79 R80 R81 R82 NO3 + ISOP XO2 + NO XO2 + XO2 XO2N + NO XO2 + HO2 −→ −→ −→ −→ −→ XO2N NO2 . R59 NO3 + OLE Reaction + 0.56 FORM + 0.PAR R60 R61 R62 R63 R64 R65 R66 R67 R68 R69 O + ETH OH + ETH O3 + ETH TOL + OH TO2 + NO TO2 OH + CRES CRES + NO3 CRO + NO2 OH + XYL −→ −→ −→ −→ −→ −→ −→ −→ −→ −→ FORM + 1.7 HO2 + XO2 + 0.6 HO2 + 0.44 HO2 + 0.4 ALD2 + 0.7 FORM + 0.2 C2 O3 + 0.3 OH XO2 + 1.PAR −→ 0.69 CO + 0.2 ALD2 R77 O3 + ISOP −→ FORM + 0.1 PAR + 0.1 PAR + 0.9 PAR R76 OH + ISOP −→ XO2 + FORM + 0.7 XO2 + 0.187 CB-IV reactions – continued from previous page Reaction No.9 NO2 + 0.76 HO2 + 0.12 HO2 0.08 XO2 + 0.9 OPEN CRES + HO2 0.91 XO2 + FORM + 0.5 CO + 0.44 HO2 .22 ALD2 + HO2 FORM + 0.06 CO + 0.2 CRES + 0.45 ETH + 0.3 OPEN CRO + HNO3 0.4 MGLY + 0.2 MGLY + 0.8 MGLY + 1.4 CRO + 0.55 ETH + 0.6 HO2 + 0.3 TO2 R70 R71 R72 OPEN + OH OPEN + hν OPEN + O3 −→ −→ −→ XO2 + 2 CO + 2 HO2 + C2 O3 + FORM C2 O3 + HO2 + CO 0.09 XO2N + ALD2 + NO2 .8 ALD2 + 0.44 HO2 + 0.2 MGLY R73 R74 R75 OH + MGLY MGLY + hν O + ISOP −→ −→ −→ XO2 + C2 O3 C2 O3 + HO2 + CO 0.9 HO2 + 0.08 OH + 0.55 OLE + XO2 + 0.42 CO + 0.33 CO + 0. EPACMTP consists of two modules: an unsaturated zone module called Finite Element and semi-analytical Contaminant Transport in the Unsaturated Zone (FECTUZ). EPACMTP currently uses a simplified 2-D version of the CANSAZ-3D. and (iv) metals transport. but disregards the formation and transport of transformation products. . In addition.1 Introduction The EPA’s Composite Model for leachate Migration and Transformation Products (EPACMTP) provides estimates of potential human exposure to hazardous chemicals leaching from land disposal facilities [64]. FECTUZ is a one-dimensional model that simulates vertically downward steady-state flow and contaminant transport through the unsaturated zone above an unconfined aquifer. and a saturated zone module called Combined Analytical-Numerical SAturated Zone in 3-Dimensions (CANSAZ-3D). (ii) effects of water-table mounding on groundwater flow and contaminant migration. and predicts the associated groundwater exposure in a domestic drinking water receptor well. EPACML accounts for the first-order decay and sorption of chemicals. EPACMTP simulates the subsurface fate and transport of contaminants released from land disposal sites. and the modules are optimized for computational efficiency. (iii) finite source as well as continuous source scenarios. unidirectional groundwater flow. EPACML can describe only uniform.188 Appendix D THE EPACMTP MODEL D. CANSAZ-3D simulates 3-D steady-state groundwater flow and transient or steady state contaminant transport. On the other hand. EPACMTP can take into consideration: (i) chain decay reactions and transport of daughter and grand-daughter products. This model is an improvement over the EPA’s Composite Model for Landfills (EPACML) [62]. FECTUZ consists of two modules: a flow module and a solute transport module. The important assumptions incorporated in the transport module are as follows: • advection and dispersion are one-dimensional. . incompressible soil layers. • transport in the porous medium is governed by Fick’s law. which contains more detailed descriptions of the modules and the methods described here.2 Description of the FECTUZ Module The model for the Finite Element and semi-analytical Contaminant Transport in the Unsaturated Zone (FECTUZ) is a one-dimensional flow and transport code. which is assumed to occur at a constant rate. and is not affected by the presence of dissolved chemicals. in the downward vertical direction. typically a landfill or surface impoundment. individually uniform. to an unconfined aquifer. through the unsaturated zone underlying the disposal unit. and • the effects of hysteresis in soil constitutive relations are negligible. • the soil profile consists of one or more. The material presented here is summarized from the EPACMTP background document [64]. This model describes the migration of contaminants downward from a disposal unit. This model can simulate cases where the flow and transport are one-dimensional. and there are certain assumptions incorporated into both the unsaturated flow module and the unsaturated transport module. D. The important assumptions in the flow module are as follows: • the flow of the fluid phase is one-dimensional.189 The following sections describe in detail the formulation and implementation of FECTUZ and CANSAZ-3D. isothermal. Flow is assumed to be always at steady state. the flow and transport are driven by the seepage from a landfill or a surface impoundment. while either transient or steady-state solute transport simulations can be performed. that simulates vertical water flow and solute transport through the unsaturated zone above an unconfined aquifer. steady and governed by Darcy’s law. the fluid is slightly compressible and homogeneous. 1 involves the following steps: .1) where ψ(L) is the pressure head. • the effects of bio-chemical decay can be described by first-order degradation and zero-order production reactions. parameters β and γ are related through the equation γ = 1 − 1/β. Descriptive statistical values for α and β have been determined by Carsel and Parrish [30] for 12 soil classifications. krw is the relative permeability.2. ψl = 0 (D. Additional details on the model formulation and the solution method can be found in the EPACMTP background document [64]. and γ are soil specific shape parameters. ψ<0 . ψ≥0 (D. A brief description of the mathematical formulation of the flow module and of the transport module is presented here. The permeability-water content relation is given assumed to follow the Mualem-van Genuchten model [205].2) where   1 + (−αψ)β Se =  1 −γ .1 Steady State Flow Module Formulation The governing equation for the steady state flow module of FECTUZ is given by Darcy’s Law: −Ks krw dψ −1 =I dz . Solution Method for Flow Module The solution of Equation D. The symbols inside the parentheses denote the dimensions of the physical quantities. Ks (L/T) is the saturated hydraulic conductivity.190 • sorption reactions can be described by Freundlich equilibrium isotherm. I(L/T) is the infiltration rate. z(L) is the depth coordinate which is taken positive downward.3) where the parameters α. and is given by 1/2 1/γ krw = Se 1 − 1 − Se γ 2 (D. D. β. Further. and • fluid properties are independent of concentrations of contaminants. and l(L) is the thickness of the unsaturated zone. R is the retardation factor. The coefficient ξim is a constant related to the decay reaction stoichiometry. V is the Darcy velocity (L/T) obtained from the solution of the flow equation. and • solution of the resultant equation using a combined Newton-Raphson and bisection method. • replacement of the derivative in Equation D. The surface impoundment boundary condition is solved by using the Darcy’s Law.2 and D. where M is the total number of parents. R signifies the effect of equilibrium sorption. In this method. λi is the first order decay constant (1/T).1.4) where ci is the concentration of the ith species (M/L3 ).5) where αL is the longitudinal dispersivity (L) and D∗ is the effective molecular diffusion coefficient (L2 /T).1 with a backward finite difference approximation.191 • substitution of Equations D.2 Solute Transport Module Formulation The one-dimensional transport of solute species is modeled in FECTUZ using the following advection-dispersion equation: ∂ ∂ci ∂ci ∂ci D −V = θRi + θQi λi Ci − ∂z ∂z ∂z ∂t M θξim Qm λm cm m=1 (D.2. and the summation on the righthand side of Equation D. the unsaturated zone is discretized into a number of one-dimensional segments.4 represents the production due to the decay of parent species. D is the apparent dispersion coefficient (L2 /T). The dispersion coefficient D is defined as D = αL V + θD∗ (D. and iteratively solving for the infiltration rate through the surface impoundment liner. Q is a coefficient to incorporate decay in the sorbed phase. whereas Q signifies the . representing the fraction of mth species that decays to the ith daughter species.3 into Equation D. D. with that a high resolution close to the water table and also close to layer interfaces for layered soils. simulates the three-dimensional steady state groundwater flow and contaminant transport. three solution methods are employed based on the complexity of the problem: (a) for single species. (D. c0 (t) is the leachate conceni i tration emanating from the disposal facility. a semi-analytical method is used. a numerical finite element solution is used. The initial and boundary conditions of the one-dimensional transport problem can be given by one of the following equations: ci (z. t) = c0 (t) i ∂ci (l. t) = 0 ∂t where cin is the initial concentration of the ith species. t) = v(c0 (t) − ci ) i ∂z cI (0. and (c) for cases involving nonlinear sorption. 0) = cin i −D ∂ci (0. CANSAZ3D describes advective-dispersive transport in an aquifer with a steady state flow field and a patch source of contaminants at the water table. (b) for transient or steadystate decay chain simulations with linear sorption.192 effect of decay. an analytical solution is used. Combined Analytical-Numerical SAturated Zone 3-D (CANSAZ-3D).6) D.7) (D.3 Description of the CANSAZ-3D Module The three-dimensional model. This model consists of two . and l is the depth of the water table. and forms the saturated zone module of the EPACMTP. z is the downward vertical coordinate. Solution Method for Solute Transport Module In the FECTUZ solute transport model. steady-state simulations involving a linear adsorption isotherm. They are given by ρb k1 ηcη−1 θ ρ Q = 1 + b k1 cη−1 θ R = 1+ where k1 and η are nonlinear Freundlich parameters. and governed by Darcy’s Law. . and the saturated aquifer thickness is large relative to the head difference that establishes the regional gradient. zero-order production. • the contribution of recharge from the unsaturated zone is small relative to the regional flow in the aquifer. with an optional two-dimensional simulation. • the aquifer is homogeneous and initially contaminant free. The flow module considers three-dimensional steady state flow. • the mass flux of contaminants through the source is constant. and estimates the contaminant concentrations at a receptor well. and • the principal directions of the hydraulic conductivity tensors are aligned with the Cartesian coordinate system.193 modules: (a) a module for the ground water flow in the saturated zone. • adsorption of contaminants onto solid phase is described by the Freundlich equilibrium isotherm. The important assumptions in the solute transport module are as follows: • the flow field is at a steady state. The important assumptions in the groundwater flow module are as follows: • the aquifer is homogeneous • the groundwater flow is steady-state. • the chemical is dilute and is present in the solution or adsorbed onto the soil matrix. • chemical/bio-chemical degradation of the contaminant can be described by a firstorder process. and (b) a module for the contaminant migration through the saturated zone. whereas the transport module simulates the transport of dissolved contaminants. the transport module considers advection. hydrodynamic dispersion. isothermal. equilibrium sorption. • the fluid is slightly compressible and homogeneous. On the other hand. and first-order decay. The flow module estimates the hydraulic head and the flow velocities in an aquifer of constant thickness. z) = H2 ∂H (x. D. z) = 0 ∂y 2 ∂H (x. by numerically discretizing the aquifer region of interest into threedimensional elements. y or z (D. 0) = 0. xd . Ky . y. 0. A brief description of the groundwater flow module and of the transport module follows. ± L .194 • hydrodynamic dispersion can be described as a Fickian process. and are given by Vl = −Kl dc . yL and B are length. Additional details can be found in the EPACMTP background document [64]. The Vi ’s are the Darcy velocities obtained from the FECTUZ model. z) = H1 H(xL . and Kx . ∂z and (D.9)    Ixu ≤ x ≤ xd . xu and yD are the upstream and downstream coordinates and the width of the source. − yD ≤ y ≤ yD ∂H 2 2 −KZ (x.10) These equations can be solved by either finite difference technique or finite element methods. and Kz are hydraulic conductivities (L/T). B) =  ∂z  Ir elsewhere where xL . z) = 0 ∂y ∂H y (x. The boundary conditions are given by H(0. and • the saturated zone is a chemically buffered system with constant and uniform geochemical characteristics. width and thickness of the aquifer system. y.1 Formulation of the Groundwater Flow Module The governing equation for steady state flow in three dimensions is Kx ∂ 2H ∂ 2H ∂ 2H + K y 2 + Kz 2 = 0 ∂2x ∂ y ∂ z (D. I is the infiltration rate through the rectangular surface patch source.8) where H is the hydraulic head (L).3. In addition. CANSAZ-3D can also simulate a 2-D groundwater . y. y. dl where l corresponds to x. and Ir is the recharge rate at the water table outside the patch area. y and z directions respectively. Here i. j and k denote the x. respectively. nc i. λp (1/T) and Rp are the first order decay and retardation coefficients. The dispersion coefficients Dij are given by Dii = αL Dij Vj2 V2 Vi2 + αT + αV k + φD∗ |V | |V | |V | (D.2 Formulation of the Saturated Zone Contaminant Migration Module The three dimensional transport of contaminants in an aquifer can be described by ∂ ∂xi Dij ∂cp ∂xj − Vi ∂cp ∂xi M ∂cp −φ = φQp λp cp + φRp ∂t ξpm Qm λm cm . and φ is the aquifer effective porosity.11) where cp (M/L3 ) is the concentration of the pth component species in the nc member decay chain.2. The present implementation of the EPACMTP model uses the finite difference technique for a 2-D solution.12) = (αL − αT )Vi Vj /|V | where αL . D.6 in Appendix D. horizontal transverse and vertical dispersivity (L) respectively. and D∗ is the effective molecular diffusion coefficient (L2 /T). The details on the implementation of the finite element and finite difference techniques in CANSAZ-3D are presented in background document [64]. The retardation factor R and the decay coefficient Q are given as in Equation D. The factors ξpm .2.195 flow in the x−y plane. 2.3. j = 1. m=1 p = 1.2. 3 (D. αT and αV are the longitudinal.2. Qp and Qm are correction factors to account for sorbed phase decay of species p and parent m. and the Einstein summation convention is used to simplify the notation. Qm and λm are the same as described in Appendix D. . y> D 2 2 top boundary: xu ≤ x ≤ xd . z. 0) = c0 (t) p ∂cp + Vz cp ∂z x = 0. − left boundary: 0 ≤ x ≤ xL .196 The initial and boundary conditions for the problem are as follows: upstream boundary: cp = 0 ∂cp =0 ∂x ∂cp =0 ∂y ∂cp =0 ∂y ∂cp =0 ∂z cp (x. y. y = − right boundary: 0 ≤ x ≤ xL . 0 ≤ z ≤ B 2 2 yL . 0≤z≤B 2 bottom boundary: xu ≤ x ≤ xd . z =0 2 2 y yD ≤y≤ D 2 2 yD y . x > xd . . − y yD ≤y ≤ D. − and −Dzz =0 z=B x < xu . 0 ≤ z ≤ B 2 2 yL y ≤ y ≤ L. y < Solution Method for the Transport Module For steady state cases. CANSAZ-3D has options for either a finite difference or a finite element solution. y = yL . For time dependent simulations. the governing equation is solved by using the Laplace Transform Galerkin (GLT) technique. A detailed description of the solution method can be found in the background document [64]. developed by Sudicky [196]. 0≤z≤B 2 downstream boundary: x = xL . − yL y ≤ y ≤ L. the concentration in a receptor well is estimated by applying the de Hoog algorithm [47]. .. Numerical Recipes is a commercial package. the Web Based SRSM tool. the code included here is easily configurable to work on any system that has plotting packages and a web server. the SRSMADIFOR.g.197 Appendix E PROGRAMS INCLUDED IN THE CDROM A list of the programs that are required for the application of the SRSM. In addition. are presented here. Please note that the application of the SRSM requires some system dependent files and auxiliary software such as plotting utilities. and the RPM-3D. but requires permission for use). the SRSM and SRSM-ADIFOR require some standard numerical modules that are commercially available for a low cost such as Numerical Recipes [163]..g. the plotting utilities used here are system dependent). However. whereas ADIFOR is free. or they are system dependent (e. whereas the application of the SRSM-ADIFOR requires the ADIFOR package. These are not included here either because they cannot be redistributed (e. LUDCMP. RAN2.html srsm srsm-form.map Configuration variables for the SRSM Make file for the SRSM program Contains Maple code for the generation of polynomial chaos expansions and auxiliary modules requires a plotting program and a web server Table E.cgi Calculates the coefficients of polynomial chaos expansion and plots the output pdf s srsm-form-errors.198 Table E. SVBKSB. SVDCMP) srsmcfg. PYTHAG.for Description Main program for the SRSM Auxiliary subroutines required for the calculation of polynomial chaos expansions.cgi Description A web page for the SRSM CGI-program for the SRSM web interface Generates the collocation points based on input/output specification srsm-post-proc.for srsmsub. and the calculation of unknown coefficients collsub. LUBKSB.for Contains subroutines for collocation point generation From Numerical Recipes [163] (not included here due to copyright considerations) subroutines required are: GASDEV.2: Programs required for the web based SRSM tool Web Based SRSM – Directory: srsmweb File srsm.for recipes.h makefile chaos.1: Programs required for the implementation of the SRSM Stochastic Response Surface Method (SRSM) – Directory: srsm File srsmmain.pl srsm-post-proc.pl Error handler for input/output specification Auxiliary Perl program for post processing . in RPM-3D input file for the Mercer County case study RPM-3D input file for the Marathon Refinery case study .in marathon.3: Programs required for the implementation of the SRSM-ADIFOR SRSM-ADIFOR – Directory: srsmadi File senscoll.199 Table E.for recipes.for polygen.for makefile Main program for the SRSM-ADIFOR files from Numerical Recipes Make file for the SRSM-ADIFOR program requires the ADIFOR package Table E.in Description RPM-IV code RPM-IV input file for the Mercer County case study RPM-IV input file for the Marathon Refinery case study RPM-3D – Directory: rpm3d makefile Makefile containing the list of files (45 files.4: Other Programs and Input Files RPM – Directory: rpm4 File rpmgear.in marathon. all included in the folder) mercer.for polysens.for Description Collocation generation module Polynomial chaos expansion routines Modules for derivatives of polynomial chaos expansions sensmain.for mercer. Rutgers. (Chemical Engineering). Environmental and Occupational Health Sciences Institute. The State University of New Jersey.D. Research Assistant. Department of Chemical Engineering. 1997-1998 1999 . Rutgers. The State University of New Jersey. Indian Institute of Technology. Madras.200 Vita Sastry S. Teaching Assistant. Research Teaching Specialist.Tech. The State University of New Jersey. The State University of New Jersey. (Chemical Engineering). (Chemical Engineering). Rutgers. Environmental and Occupational Health Sciences Institute. a Joint Project of UMDNJ and Rutgers. The State University of New Jersey. Isukapalli 1993 1993-1994 1995 1994-1997 B. INDIA. M. a Joint Project of UMDNJ and Rutgers. Ph.S.