2003 Distinguished Instructor Short Course Distinguished Instructor Series, No. 6 sponsored by the Society of Exploration Geophysicists European Association of Geoscientists & Engineers SUB Gottingen 7 212 189 654 iiiiiiiiiwi Geostatistics for Seismic Data Integration in Earth Models presented by Olivier Dubrule EUROPEAN ASSOCIATION OF GEOSCIENTISTS & ENGINEERS
Table of Contents 1 Introduction 1-1 1.1 Historical Perspective 1-1 1.2 The Role of Geostatistics at Different Steps of the Earth Modeling Workflow 1-3 1.3 The Goal of This Course 1-10 1.4 Basics of Univariate Statistics 1-10 1.4.1 Random variables 1-10 1.4.2 Mean, variance, standard deviation, and support effect 1-13 1.4.3 Measures of central tendency and quantiles 1-17 1.4.4 Two important distributions 1-20 1.4.5 Two important theorems 1-22 1.5 Basics of Bivariate Statistics 1-22 1.5.1 Covariance and correlation coefficient 1-23 1.5.2 Fitting a regression line 1-23 1.6 The Multivariate Normal Distribution 1-26 1.7 Trend Surface Analysis 1-29 2 The Covariance and the Variogram 2-1 2.1 Stationarity Versus Nonstationarity 2-1 2.2 The Stationary Model 2-4 2.3 Calculation of a Variogram 2-7 2.4 Stationary Variogram Models 2-16 2.5 Examples of Anisotropic Experimental Models 2-17 2.6 Unbounded Variogram Models and Their Relationship with Fractals..2-19 2.7 First Uses of the Variogram: Predicting the Support Effect 2-23. 2.8 Cross-covariance and the Variogram 2-24 2.9 Practical Considerations about the Variogram 2-26 Covariance, Fractals, and Spectral Density 2-31 3 Interpolation: Kriging, Cokriging, Factorial Kriging, and Splines 3-1 3.1 Introduction 3-1 3.2 Kriging, an Interpolation Technique 3-1 3.2.1 Introduction 3-1 Distinguished Instructor Short Course v
Geostatistics for Seismic Data Integration in Earth Models 3.2.2 Universal kriging 3-1 3.2.3 Generalized covariances of order k 3-14 3.2.4 Kriging considered as an interpolating function 3-14 3.2.5 Cross-validation 3-16 3.2.6 Conclusion on kriging 3-20 3.3 Error Cokriging and Factorial Kriging to Distinguish Noise from Signal 3-21 3.3.1 Error cokriging for V sta ck interpolation 3-21 3.3.2 Factorial kriging 3-28 3.4 Kriging with an External Drift 3-39 3.4.1 The external-drift model 3-39 3.4.2 Examples of time-to-depth conversion using the external-drift approach 3-40 3.5 Bayesian Kriging, a Generalization of Kriging with an External Drift..3-44 3.6 Cokriging and Collocated Cokriging 3-47 3.6.1 Introduction 3-47 3.6.2 A cokriging example 3-47 3.6.3 Collocated cokriging 3-49 A Few Words on Bayesian Statistics 3-53 3.6.4 Revisiting collocated cokriging 3-56 3.6.5 Collocated cokriging versus external drift 3-58 3.6.6 Factorial cokriging 3-63 3.7 Some Relationships between the Kriging Techniques 3-63 3.8 Kriging Versus Other Interpolation Techniques 3-63 3.8.1 Introduction 3-63 3.8.2 Splines 3-63 3.8.3 Comparison among kriging, splines, and radial-basis functions... 3-69 Why this relationship between splines and kriging? 3-74 4 Conditional Simulation for Heterogeneity Modeling and Uncertainty Quantification 4-1 4.1 Introduction 4-1 4.2 A Few Reminders on Monte-Carlo Simulation 4-1 4.3 Conditional Simulations for Continuous Parameters 4-7 4.3.1 Example 4-7 4.3.2 Algorithms 4-14 4.4 Cosimulation 4-22 vi Society of Exploration Geophysicists / European Association of Geoscientists & Engineers
4.4.1 Collocated simulation with seismic data as a secondary variable.. 4-22 4.4.2 Joint simulation of two parameters 4-24 4.4.3 Cascade or parallel conditional simulation of several parameters?. 4-25 4.5 Conditional Simulation for Geological Facies Modeling 4-25 4.5.1 Introduction 4-25 4.5.2 Pixel-based models 4-29 4.5.3 Object-based models 4-37 4.5.4 Facies models constrained by seismic information 4-39 4.5.5 Hierarchical modeling of geology and petrophysical parameters 4-46 Geostatistics, Inverse of the Covariance, and Filtering 4-52 5 Geostatistical Inversion 5-1 5.1 Introduction 5-1 5.2 Basics of Geostatistical Inversion 5-2 5.3 Accounting for Faults 5-10 5.4 A Variety of Methods 5-10 5.4.1 A different sampling algorithm 5-10 5.4.2 Geostatistical inversion based on fractals 5-15 5.4.3 Analytical approach 5-16 5.4.4 Emerging techniques 5-16 5.5 A Generalized Downscaling Approach 5-18 5.6 Going Further with Geostatistical Inversion Results 5-19 5.6.1 Two-step approach: from seismic to impedance, from impedance to other properties 5-19 5.6.2 Predicting porosity during the acoustic-impedance inversion process 5-21 5.6.3 Predicting facies during the acoustic-impedance inversion process 5-21 6 Stochastic Earth Modeling That Integrates All Subsurface Uncertainties... 6-1 6.1 Introduction 6-1 6.2 Geometrical Uncertainties 6-4 6.2.1 Two-step approach 6-4 6.2.2 Bayesian kriging approach 6-10 6.2.3 How many realizations? 6-11 6.3 Static and Dynamic Model Uncertainties 6-13
Geostatistics for Seismic Data Integration in Earth Models 6.3.1 Static model uncertainties 6-13 6.3.2 Link with dynamic flow simulation 6-15 6.4 Multirealization Uncertainty-quantification Approach: A Panacea?....6-18 6.4.1 Approaches by scenarios 6-19 6.4.2 Combining scenarios and geostatistical realizations 6-20 6.5 Conclusion on Uncertainties: A Word of Caution 6-24 7 Conclusions 7-1 7.1 What We Have Learned 7-1 7.2 Future Topics 7-2 7.3 Websites and Software 7-4 7.4 The Role of Geostatistics in Geophysics 7-1 8 Exercises 8-1 8.1 Exercise 1: Fitting a Variogram Model 8-1 8.2 Exercise 2: Understanding the Kriging System 8-2 8.3 Exercise 3: Generating a Nonconditional Simulation in ID 8-3 9 Notation 9-1 10 Acknowledgments 10-1 11 References * 11-1 viii Society of Exploration Geophysicists / European Association of Geoscientists & Engineers