Acknowledgments
Contents
1 Introduction, Objectives
2 Overview of Uncertainty Propagation Methods
2.1 Approaches to Uncertainty
2.2 Monte Carlo Simulations (with or Without a Metamodel)
2.2.1 Direct Monte Carlo Simulations
2.2.2 Indirect Monte Carlo Simulations via a Metamodel
2.2.3 Summary on Direct Versus Indirect Monte Carlo Simulations
2.3 Joint Sets of Parameters, Cross-Correlations, Design of Experiments
2.3.1 Sampling and Uncertainty Propagation for Each Input Parameter (One-By-One)
2.3.2 Joint Uncertainty Propagation for a Parameter Set {P1,P2,…,PK}
2.4 Applications to Risk Assessment in Field Pollution
2.5 Causes and Sources of Uncertainty
2.6 Uncertainty Propagation: Output “Criteria”
References
3 Review of Probabilistic Versus Fuzzy Approaches to Uncertainty Propagation in Geosciences
3.1 Probabilistic Characterization of Uncertainty
3.1.1 Univariate Probabilistic Characterization (Single Parameter)
3.1.2 Multivariate Probabilistic Characterization (Two or More Uncertain Parameters)
3.1.3 Estimating a Covariance or Correlation from a Data Sample
References
4 Fuzzy Set Characterization of Uncertainty (Fuzzy Variables)
4.1 Fuzzy Sets, Fuzzy Numbers, Fuzzy Logic and Arithmetic
4.1.1 Fuzzy Sets, Fuzzy Numbers, Membership Functions, and α-Cuts
4.1.2 Fuzzy Logic and Fuzzy Arithmetic Rules in Models
4.2 From Fuzzyness to “Possibility” (Fuzzy + Probabilistic)
4.2.1 Fuzzy Probability Bounds: Possibility (Upper CDF), Necessity (Lower CDF)
4.2.2 Intermediate CDF’s via α-Cuts (Confidence Levels and Intervals)
4.2.3 Random + Fuzzy Uncertainty: IRS Algorithm for Propagating Possibilistic CDFs
4.2.4 Comparative Review: Softwares and Uncertainty Propagation
References
5 Applications of Uncertainty Analyses on Simplified Models
5.1 Introduction: Models with Uncertain Parameters (Classification)
5.2 Decay Kinetics with Uncertain Decay Parameter
5.2.1 Introduction to the First Order Decay Kinetics Model
5.2.2 Exact Explicit Analytical Expression of C(t)’s Probability Law
5.2.3 The Mean Concentration Curve C(t)
5.3 Taylor Expansion Around the Mean: MVFO Method and Application to Corrosion Pits
5.4 Coupled 1D Model of Contaminant Migration and Source Dissolution
5.4.1 Monte Carlo Analysis of 1D Dissolution/Transport via a Metamodel
5.4.2 Possibilistic Analysis of 1D Contaminant Migration with IRS
References
6 Applications of Uncertainty Analysis to 3D Subsurface Contamination Problems
6.1 Application of Uncertainty Analysis to 3D Contaminant Advection–Dispersion from a Decaying Source (Quasi-Analytical ESPER-1 Model)
6.1.1 Model Equations: PDE, Geometry, Boundary and Initial Conditions
6.1.2 The Quasi-analytical Solution C(x, y, z, t) and M(t)
6.1.3 Visualizations of the Solution C(x, y, z, t) and M(t)
6.1.4 Monte Carlo Analyses of 3D Analytical Model (ESPER1 Package)
6.2 Uncertainty Analysis of 3D Solute Migration and Source Attenuation in a Polluted Site (MODFLOW-SURFACT)
6.2.1 The Modflow-Surfact Code and Model Equations
6.2.2 Main Hydrogeologic Characteristics of the Polluted Test Site
6.2.3 Uncertainty Analysis: Metamodeling Procedure for the Field Site
6.2.4 Results: Probabilistic Characterization of Output Criterion (Critical Time)
References
7 Discussion and Conclusions