Modeling Uncertainty in the Earth Sciences

Jef Caers, Associate Professor of Energy Resources Engineering, Department of Energy Resources Engineering, Stanford University, Stanford, CA.

• Preface xi Acknowledgements xvii1 Introduction 11.1 Example Application 11.1.1 Description 11.1.2 3D Modeling 31.2 Modeling Uncertainty 4Further Reading 82 Review on Statistical Analysis and Probability Theory 92.1 Introduction 92.2 Displaying Data with Graphs 102.2.1 Histograms 102.3 Describing Data with Numbers 132.3.1 Measuring the Center 132.3.2 Measuring the Spread 142.3.3 Standard Deviation and Variance 142.3.4 Properties of the Standard Deviation 152.3.5 Quantiles and the QQ Plot 152.4 Probability 162.4.1 Introduction 162.4.2 Sample Space, Event, Outcomes 172.4.3 Conditional Probability 182.4.4 Bayes’ Rule 192.5 Random Variables 212.5.1 Discrete Random Variables 212.5.2 Continuous Random Variables 212.5.2.1 Probability Density Function (pdf) 212.5.2.2 Cumulative Distribution Function 222.5.3 Expectation and Variance 232.5.3.1 Expectation 232.5.3.2 Population Variance 242.5.4 Examples of Distribution Functions 242.5.4.1 The Gaussian (Normal) Random Variable and Distribution 242.5.4.2 Bernoulli Random Variable 252.5.4.3 Uniform Random Variable 262.5.4.4 A Poisson Random Variable 262.5.4.5 The Lognormal Distribution 272.5.5 The Empirical Distribution Function versus the Distribution Model 282.5.6 Constructing a Distribution Function from Data 292.5.7 Monte Carlo Simulation 302.5.8 Data Transformations 322.6 Bivariate Data Analysis 332.6.1 Introduction 332.6.2 Graphical Methods: Scatter plots 332.6.3 Data Summary: Correlation (Coefficient) 352.6.3.1 Definition 352.6.3.2 Properties of r 37Further Reading 373 Modeling Uncertainty: Concepts and Philosophies 393.1 What is Uncertainty? 393.2 Sources of Uncertainty 403.3 Deterministic Modeling 413.4 Models of Uncertainty 433.5 Model and Data Relationship 443.6 Bayesian View on Uncertainty 453.7 Model Verification and Falsification 483.8 Model Complexity 493.9 Talking about Uncertainty 503.10 Examples 513.10.1 Climate Modeling 513.10.1.1 Description 513.10.1.2 Creating Data Sets Using Models 513.10.1.3 Parameterization of Subgrid Variability 523.10.1.4 Model Complexity 523.10.2 Reservoir Modeling 523.10.2.1 Description 523.10.2.2 Creating Data Sets Using Models 533.10.2.3 Parameterization of Subgrid Variability 533.10.2.4 Model Complexity 54Further Reading 544 Engineering the Earth: Making Decisions Under Uncertainty 554.1 Introduction 554.2 Making Decisions 574.2.1 Example Problem 574.2.2 The Language of Decision Making 594.2.3 Structuring the Decision 604.2.4 Modeling the Decision 614.2.4.1 Payoffs and Value Functions 624.2.4.2 Weighting 634.2.4.3 Trade-Offs 654.2.4.4 Sensitivity Analysis 674.3 Tools for Structuring Decision Problems 704.3.1 Decision Trees 704.3.2 Building Decision Trees 704.3.3 Solving Decision Trees 724.3.4 Sensitivity Analysis 76Further Reading 765 Modeling Spatial Continuity 775.1 Introduction 775.2 The Variogram 795.2.1 Autocorrelation in 1D 795.2.2 Autocorrelation in 2D and 3D 825.2.3 The Variogram and Covariance Function 845.2.4 Variogram Analysis 865.2.4.1 Anisotropy 865.2.4.2 What is the Practical Meaning of a Variogram? 875.2.5 A Word on Variogram Modeling 875.3 The Boolean or Object Model 875.3.1 Motivation 875.3.2 Object Models 895.4 3D Training Image Models 90Further Reading 926 Modeling Spatial Uncertainty 936.1 Introduction 936.2 Object-Based Simulation 946.3 Training Image Methods 966.3.1 Principle of Sequential Simulation 966.3.2 Sequential Simulation Based on Training Images 986.3.3 Example of a 3D Earth Model 996.4 Variogram-Based Methods 1006.4.1 Introduction 1006.4.2 Linear Estimation 1016.4.3 Inverse Square Distance 1026.4.4 Ordinary Kriging 1036.4.5 The Kriging Variance 1046.4.6 Sequential Gaussian Simulation 1046.4.6.1 Kriging to Create a Model of Uncertainty 1046.4.6.2 Using Kriging to Perform (Sequential) Gaussian Simulation 104Further Reading 1067 Constraining Spatial Models of Uncertainty with Data 1077.1 Data Integration 1077.2 Probability-Based Approaches 1087.2.1 Introduction 1087.2.2 Calibration of Information Content 1097.2.3 Integrating Information Content 1107.2.4 Application to Modeling Spatial Uncertainty 1137.3 Variogram-Based Approaches 1147.4 Inverse Modeling Approaches 1167.4.1 Introduction 1167.4.2 The Role of Bayes’ Rule in Inverse Model Solutions 1187.4.3 Sampling Methods 1257.4.3.1 Rejection Sampling 1257.4.3.2 Metropolis Sampler 1287.4.4 Optimization Methods 130Further Reading 1318 Modeling Structural Uncertainty 1338.1 Introduction 1338.2 Data for Structural Modeling in the Subsurface 1358.3 Modeling a Geological Surface 1368.4 Constructing a Structural Model 1388.4.1 Geological Constraints and Consistency 1388.4.2 Building the Structural Model 1408.5 Gridding the Structural Model 1418.5.1 Stratigraphic Grids 1418.5.2 Grid Resolution 1428.6 Modeling Surfaces through Thicknesses 1448.7 Modeling Structural Uncertainty 1448.7.1 Sources of Uncertainty 1468.7.2 Models of Structural Uncertainty 149Further Reading 1519 Visualizing Uncertainty 1539.1 Introduction 1539.2 The Concept of Distance 1549.3 Visualizing Uncertainty 1569.3.1 Distances, Metric Space and Multidimensional Scaling 1569.3.2 Determining the Dimension of Projection 1629.3.3 Kernels and Feature Space 1639.3.4 Visualizing the Data–Model Relationship 166Further Reading 17010 Modeling Response Uncertainty 17110.1 Introduction 17110.2 Surrogate Models and Ranking 17210.3 Experimental Design and Response Surface Analysis 17310.3.1 Introduction 17310.3.2 The Design of Experiments 17310.3.3 Response Surface Designs 17610.3.4 Simple Illustrative Example 17710.3.5 Limitations 17910.4 Distance Methods for Modeling Response Uncertainty 18110.4.1 Introduction 18110.4.2 Earth Model Selection by Clustering 18210.4.2.1 Introduction 18210.4.2.2 k-Means Clustering 18310.4.2.3 Clustering of Earth Models for Response Uncertainty Evaluation 18510.4.3 Oil Reservoir Case Study 18610.4.4 Sensitivity Analysis 18810.4.5 Limitations 191Further Reading 19111 Value of Information 19311.1 Introduction 19311.2 The Value of Information Problem 19411.2.1 Introduction 19411.2.2 Reliability versus Information Content 19511.2.3 Summary of the VOI Methodology 19611.2.3.1 Steps 1 and 2: VOI Decision Tree 19711.2.3.2 Steps 3 and 4: Value of Perfect Information 19811.2.3.3 Step 5: Value of Imperfect Information 20111.2.4 Value of Information for Earth Modeling Problems 20211.2.5 Earth Models 20211.2.6 Value of Information Calculation 20311.2.7 Example Case Study 20811.2.7.1 Introduction 20811.2.7.2 Earth Modeling 20811.2.7.3 Decision Problem 20911.2.7.4 The Possible Data Sources 21011.2.7.5 Data Interpretation 211Further Reading 21312 Example Case Study 21512.1 Introduction 21512.1.1 General Description 21512.1.2 Contaminant Transport 21812.1.3 Costs Involved 21812.2 Solution 21812.2.1 Solving the Decision Problem 21812.2.2 Buying More Data 21912.2.2.1 Buying Geological Information 21912.2.2.2 Buying Geophysical Information 22112.3 Sensitivity Analysis 221Index 225

“This is an outstanding contribution to the current literature, particularly since this book is aimed at an audience of young researchers and modelers that may just be starting their careers.” (Mathematical Geoscience, 29 November 2012)“Overall, I consider this book to be a good addition to a rather limited choice of books for teaching an introductory course on modeling uncertainty in the Earth and environmental sciences. As the author points out in the preface of the book, this is not an encyclopedia on modeling uncertainty, but rather an introduction to the topic that can lead the reader to deeper pursuits on modeling uncertainty.”  (Bulletin of the American Meteorological Society, 1 October 2012) “The book, Modeling Uncertainty in the Earth Sciences, can be of great use for anyone involved with making decisions in Earth sciences. It gives a solid overview on how decisions in Earth Science can be improved by explicit uncertainty modeling.”  (Environmental Earth Science, 1 October 2012)