Modeling Uncertainty in the Earth Sciences highlights the various issues, techniques and practical modeling tools available for modeling the uncertainty of complex Earth systems and the impact that it has on practical situations. The aim of the book is to provide an introductory overview which covers a broad range of tried-and-tested tools. Descriptions of concepts, philosophies, challenges, methodologies and workflows give the reader an understanding of the best way to make decisions under uncertainty for Earth Science problems.

The book covers key issues such as: Spatial and time aspect; large complexity and dimensionality; computation power; costs of 'engineering' the Earth; uncertainty in the modeling and decision process. Focusing on reliable and practical methods this book provides an invaluable primer for the complex area of decision making with uncertainty in the Earth Sciences.

Acknowledgements xvii

1 Introduction 1

1.1 Example Application 1

1.1.1 Description 1

1.1.2 3D Modeling 3

1.2 Modeling Uncertainty 4

Further Reading 8

2 Review on Statistical Analysis and Probability Theory 9

2.1 Introduction 9

2.2 Displaying Data with Graphs 10

2.2.1 Histograms 10

2.3 Describing Data with Numbers 13

2.3.1 Measuring the Center 13

2.3.2 Measuring the Spread 14

2.3.3 Standard Deviation and Variance 14

2.3.4 Properties of the Standard Deviation 15

2.3.5 Quantiles and the QQ Plot 15

2.4 Probability 16

2.4.1 Introduction 16

2.4.2 Sample Space, Event, Outcomes 17

2.4.3 Conditional Probability 18

2.4.4 Bayes' Rule 19

2.5 Random Variables 21

2.5.1 Discrete Random Variables 21

2.5.2 Continuous Random Variables 21

2.5.2.1 Probability Density Function (pdf) 21

2.5.2.2 Cumulative Distribution Function 22

2.5.3 Expectation and Variance 23

2.5.3.1 Expectation 23

2.5.3.2 Population Variance 24

2.5.4 Examples of Distribution Functions 24

2.5.4.1 The Gaussian (Normal) Random Variable and Distribution 24

2.5.4.2 Bernoulli Random Variable 25

2.5.4.3 Uniform Random Variable 26

2.5.4.4 A Poisson Random Variable 26

2.5.4.5 The Lognormal Distribution 27

2.5.5 The Empirical Distribution Function versus the Distribution Model 28

2.5.6 Constructing a Distribution Function from Data 29

2.5.7 Monte Carlo Simulation 30

2.5.8 Data Transformations 32

2.6 Bivariate Data Analysis 33

2.6.1 Introduction 33

2.6.2 Graphical Methods: Scatter plots 33

2.6.3 Data Summary: Correlation (Coefficient) 35

2.6.3.1 Definition 35

2.6.3.2 Properties of r 37

Further Reading 37

3 Modeling Uncertainty: Concepts and Philosophies 39

3.1 What is Uncertainty? 39

3.2 Sources of Uncertainty 40

3.3 Deterministic Modeling 41

3.4 Models of Uncertainty 43

3.5 Model and Data Relationship 44

3.6 Bayesian View on Uncertainty 45

3.7 Model Verification and Falsification 48

3.8 Model Complexity 49

3.9 Talking about Uncertainty 50

3.10 Examples 51

3.10.1 Climate Modeling 51

3.10.1.1 Description 51

3.10.1.2 Creating Data Sets Using Models 51

3.10.1.3 Parameterization of Subgrid Variability 52

3.10.1.4 Model Complexity 52

3.10.2 Reservoir Modeling 52

3.10.2.1 Description 52

3.10.2.2 Creating Data Sets Using Models 53

3.10.2.3 Parameterization of Subgrid Variability 53

3.10.2.4 Model Complexity 54

Further Reading 54

4 Engineering the Earth: Making Decisions Under Uncertainty 55

4.1 Introduction 55

4.2 Making Decisions 57

4.2.1 Example Problem 57

4.2.2 The Language of Decision Making 59

4.2.3 Structuring the Decision 60

4.2.4 Modeling the Decision 61

4.2.4.1 Payoffs and Value Functions 62

4.2.4.2 Weighting 63

4.2.4.3 Trade-Offs 65

4.2.4.4 Sensitivity Analysis 67

4.3 Tools for Structuring Decision Problems 70

4.3.1 Decision Trees 70

4.3.2 Building Decision Trees 70

4.3.3 Solving Decision Trees 72

4.3.4 Sensitivity Analysis 76

Further Reading 76

5 Modeling Spatial Continuity 77

5.1 Introduction 77

5.2 The Variogram 79

5.2.1 Autocorrelation in 1D 79

5.2.2 Autocorrelation in 2D and 3D 82

5.2.3 The Variogram and Covariance Function 84

5.2.4 Variogram Analysis 86

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