Modeling Uncertainty In The Earth Sciences

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.

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

Preface.

Acknowledgements.

1 Introduction.

1.1 Example Application.

1.1.1 Description.

1.1.2 3D Modeling.

1.2 Modeling Uncertainty.

2 Review on Statistical Analysis and Probability Theory.

2.1 Introduction.

2.2 Displaying Data with Graphs.

2.2.1 Histograms.

2.3 Describing Data with Numbers.

2.3.1 Measuring the Center.

2.3.2 Measuring the Spread.

2.3.3 Standard Deviation and Variance.

2.3.4 Properties of the Standard Deviation.

2.3.5 Quantiles and the QQ Plot.

2.4 Probability.

2.4.1 Introduction.

2.4.2 Sample Space, Event, Outcomes.

2.4.3 Conditional Probability.

2.4.4 Bayes' Rule.

2.5 Random Variables.

2.5.1 Discrete Random Variables.

2.5.2 Continuous Random Variables.

2.5.2.1 Probability Density Function (pdf).

2.5.2.2 Cumulative Distribution Function.

2.5.3 Expectation and Variance.

2.5.3.1 Expectation.

2.5.3.2 Population Variance.

2.5.4 Examples of Distribution Functions.

2.5.4.1 The Gaussian (Normal) Random Variable and Distribution.

2.5.4.2 Bernoulli Random Variable.

2.5.4.3 Uniform Random Variable.

2.5.4.4 A Poisson Random Variable.

2.5.4.5 The Lognormal Distribution.

2.5.5 The Empirical Distribution Function versus the Distribution Model.

2.5.6 Constructing a Distribution Function from Data.

2.5.7 Monte Carlo Simulation.

2.5.8 Data Transformations.

2.6 Bivariate Data Analysis.

2.6.1 Introduction.

2.6.2 Graphical Methods: Scatter plots.

2.6.3 Data Summary: Correlation (Coefficient).

2.6.3.1 Definition.

2.6.3.2 Properties of r.

3 Modeling Uncertainty: Concepts and Philosophies.

3.1 What is Uncertainty?

3.2 Sources of Uncertainty.

3.3 Deterministic Modeling.

3.4 Models of Uncertainty.

3.5 Model and Data Relationship.

3.6 Bayesian View on Uncertainty.

3.7 Model Verification and Falsification.

3.8 Model Complexity.

3.9 Talking about Uncertainty.

3.10 Examples.

3.10.1 Climate Modeling.

3.10.1.1 Description.

3.10.1.2 Creating Data Sets Using Models.

3.10.1.3 Parameterization of Subgrid Variability.

3.10.1.4 Model Complexity.

3.10.2 Reservoir Modeling.

3.10.2.1 Description.

3.10.2.2 Creating Data Sets Using Models.

3.10.2.3 Parameterization of Subgrid Variability.

3.10.2.4 Model Complexity.

4 Engineering the Earth: Making Decisions Under Uncertainty.

4.1 Introduction.

4.2 Making Decisions.

4.2.1 Example Problem.

4.2.2 The Language of Decision Making.

4.2.3 Structuring the Decision.

4.2.4 Modeling the Decision.

4.2.4.1 Payoffs and Value Functions.

4.2.4.2 Weighting.

4.2.4.3 Trade-Offs.

4.2.4.4 Sensitivity Analysis.

4.3 Tools for Structuring Decision Problems.

4.3.1 Decision Trees.

4.3.2 Building Decision Trees.

4.3.3 Solving Decision Trees.

4.3.4 Sensitivity Analysis.

5 Modeling Spatial Continuity.

5.1 Introduction.

5.2 The Variogram.

5.2.1 Autocorrelation in 1D.

5.2.2 Autocorrelation in 2D and 3D.

5.2.3 The Variogram and Covariance Function.

5.2.4 Variogram Analysis.

5.2.4.1 Anisotropy.

5.2.4.2 What is the Practical Meaning of a Variogram?

5.2.5 A Word on Variogram Modeling.

5.3 The Boolean or Object Model.

5.3.1 Motivation.

5.3.2 Object Models.

5.4 3D Training Image Models.

6 Modeling Spatial Uncertainty.

6.1 Introduction.

6.2 Object-Based Simulation.

6.3 Training Image Methods.

6.3.1 Principle of Sequential Simulation.

6.3.2 Sequential Simulation Based on Training Images.

6.3.3 Example of a 3D Earth Model.

6.4 Variogram-Based Methods.

6.4.1 Introduction.

6.4.2 Linear Estimation.

6.4.3 Inverse Square Distance.

6.4.4 Ordinary Kriging.

6.4.5 The Kriging Variance.

6.4.6 Sequential Gaussian Simulation.

6.4.6.1 Kriging to Create a Model of Uncertainty.

6.4.6.2 Using Kriging to Perform (Sequential) Gaussian Simulation.

7 Constraining Spatial Models of Uncertainty with Data.

7.1 Data Integration.

7.2 Probability-Based Approaches.

7.2.1 Introduction.

7.2.2 Calibration of Information Content.

7.2.3 Integrating Information Content.

7.2.4 Application to Modeling Spatial Uncertainty.

7.3 Variogram-Based Approaches.

7.4 Inverse Modeling Approaches.

7.4.1 Introduction.

7.4.2 The Role of Bayes’ Rule in Inverse Model Solutions.

7.4.3 Sampling Methods.

7.4.3.1 Rejection Sampling.

7.4.3.2 Metropolis Sampler.

7.4.4 Optimization Methods.

8 Modeling Structural Uncertainty.

8.1 Introduction.

8.2 Data for Structural Modeling in the Subsurface.

8.3 Modeling a Geological Surface.

8.4 Constructing a Structural Model.

8.4.1 Geological Constraints and Consistency.

8.4.2 Building the Structural Model.

8.5 Gridding the Structural Model.

8.5.1 Stratigraphic Grids.

8.5.2 Grid Resolution.

8.6 Modeling Surfaces through Thicknesses.

8.7 Modeling Structural Uncertainty.

8.7.1 Sources of Uncertainty.

8.7.2 Models of Structural Uncertainty.

9 Visualizing Uncertainty.

9.1 Introduction.

9.2 The Concept of Distance.

9.3 Visualizing Uncertainty.

9.3.1 Distances, Metric Space and Multidimensional Scaling.

9.3.2 Determining the Dimension of Projection.

9.3.3 Kernels and Feature Space.

9.3.4 Visualizing the Data–Model Relationship.

10 Modeling Response Uncertainty.

10.1 Introduction.

10.2 Surrogate Models and Ranking.

10.3 Experimental Design and Response Surface Analysis.

10.3.1 Introduction.

10.3.2 The Design of Experiments.

10.3.3 Response Surface Designs.

10.3.4 Simple Illustrative Example.

10.3.5 Limitations.

10.4 Distance Methods for Modeling Response Uncertainty.

10.4.1 Introduction.

10.4.2 Earth Model Selection by Clustering.

10.4.2.1 Introduction.

10.4.2.2 k-Means Clustering.

10.4.2.3 Clustering of Earth Models for Response Uncertainty Evaluation.

10.4.3 Oil Reservoir Case Study.

10.4.4 Sensitivity Analysis.

10.4.5 Limitations.

11 Value of Information.

11.1 Introduction.

11.2 The Value of Information Problem.

11.2.1 Introduction.

11.2.2 Reliability versus Information Content.

11.2.3 Summary of the VOI Methodology.

11.2.3.1 Steps 1 and 2: VOI Decision Tree.

11.2.3.2 Steps 3 and 4: Value of Perfect Information.

11.2.3.3 Step 5: Value of Imperfect Information.

11.2.4 Value of Information for Earth Modeling Problems.

11.2.5 Earth Models.

11.2.6 Value of Information Calculation.

11.2.7 Example Case Study.

11.2.7.1 Introduction.

11.2.7.2 Earth Modeling.

11.2.7.3 Decision Problem.

11.2.7.4 The Possible Data Sources.

11.2.7.5 Data Interpretation.

12 Example Case Study.

12.1 Introduction.

12.1.1 General Description.

12.1.2 Contaminant Transport.

12.1.3 Costs Involved.

12.2 Solution.

12.2.1 Solving the Decision Problem.

12.2.2 Buying More Data.

12.2.2.1 Buying Geological Information.

12.2.2.2 Buying Geophysical Information.

12.3 Sensitivity Analysis.

Index.