Seismic inversion

Description

Cover Title Page Copyright Page Contents About the Author Preface Acknowledgments Notation Convention Abbreviations Part I: Iterative Optimization Methods Chapter 1: Introduction to Seismic Inversion 1.1. Notation 1.2. Inverse Problem 1.3. Types of Seismic Inversion 1.4. Inverse Crimes 1.5. Summary Appendix 1A: Basics of Exploration Seismology Seismic Sources Nonzero-Offset Seismic Experiment Reflection Amplitudes Seismic Processing Reflection Imaging Key Difficulty with Migration Images Chapter 2: Introduction to Gradient Optimization 2.1. Mathematical Definitions 2.2. Gradient Optimization, Taylor Series, and Newton’s Method 2.3. Geometric Interpretation of the Gradient and Hessian 2.4. Eigenvalues of the Hessian Determine Shape of Contours 2.5. MATLAB Examples of Newton’s Method 2.6. Summary 2.7. Exercises 2.8. Computational Labs Chapter 3: Steepest-Descent Method 3.1. Steepest-Descent Method 3.1.1. Convergence Rate 3.2. Step-Length Calculation 3.2.1. Exact Line Search 3.2.2. Inexact Newton Method and Inexact Line Search 3.2.3. Numerical Line Search 3.2.4. 2D Plane Minimization 3.3. Steepest-Descent Method and Linear Systems of Equations 3.3.1. Regularized Steepest Descent 3.3.2. MATLAB Steepest-Descent Code 3.3.3. Preconditioned Steepest Descent 3.4. Summary 3.5. Exercises 3.6. Computational Labs Appendix 3A: Levenberg-Marquardt Regularization Appendix 3B: Choosing a Value for the Regularization Parameter Chapter 4: Conjugate-Gradient and Quasi-Newton Methods 4.1. Conjugate-Gradient Method 4.1.1. Conjugate-Gradient Algorithm 4.1.2. Conjugate-Gradient Method and Linear Systems of Equations 4.1.3. MATLAB Conjugate-Gradient Code 4.1.4. Convergence Rate 4.1.5. Preconditioned and Regularized Conjugate Gradients 4.2. Quasi-Newton Methods 4.3. Nonlinear Functionals 4.4. What Really Works? 4.5. Summary 4.6. Exercises 4.7. Computational Labs Appendix 4A: Successive Line-Minimization Methods Successive-Eigenvector and Conjugate-Direction Methods Part II: Traveltime Tomography Chapter 5: Raypath Traveltime Tomography 5.1. Perturbed Traveltime Integral 5.2. Raypath Traveltime Tomography 5.2.1. Normal Equations 5.2.2. Poorly Conditioned Equations and Regularization 5.2.3. Reweighted Least Squares 5.3. Iterative Steepest-Descent Solution 5.4. Reflection Tomography 5.5. Field-Data Example 5.6. Summary 5.7. Exercises 5.8. Computational Labs Appendix 5A: Eikonal Equation Derivation Appendix 5B: Lp Misfit Gradient Chapter 6: Traveltime Tomography: Assessing Model Accuracy 6.1. Introduction 6.2. Model Covariance Matrix 6.2.1. Numerical Estimation of Themodel Covariance Matrix 6.3. Model Resolution Matrix 6.4. Analytic Model Covariance Matrices 6.4.1. VSP Transmission Data 6.4.2. VSP Reflection Data 6.4.3. CMP Reflection Data 6.4.4. Uncertainty Principle 6.5. Null Space of LtL for 2D Velocity Models 6.6. Projection-Slice Theorem: Traveltime Tomography 6.7. Summary 6.8. Exercises Appendix 6A: Null-Space Properties of LtL Part III: Numerical Modeling Chapter 7: Traveltime Calculation by Solution of the Eikonal Equation 7.1. Finite-Difference Solution of the Eikonal Equation 7.2. Summary 7.3. Exercises Appendix 7A: Efficient Sorting of Traveltimes Chapter 8: Numerical Solutions to the Wave Equation 8.1. Finite-Difference Method 8.1.1. Finite-Difference Approximation to the wave Equation 8.1.2. Stability and Accuracy Analysis 8.1.3. MATLAB code for Fdsolution of the Acoustic Wave Equation 8.2. Pseudospectral Solution of the Wave Equation 8.2.1. MATLAB Code for Pseudospectral Solution of the Acoustic Wave Equation 8.2.2. Stability and Accuracy Analysis 8.3. Spectral Element Solution of the Wave Equation 8.4. Staggered-Grid FD solution of the Wave Equation 8.4.1. Staggered-Grid FD of the First-Order Acoustic Equations 8.4.2. Staggered-Grid FD of the First-Order Elastodynamic Equations 8.4.3. Matlabcode for Staggered-Grid FD of the First-Order Elastic Equations 8.5. Modeling in the Oil and Gas Industry 8.6. Summary 8.7. Exercises 8.8. Computational Labs Appendix 8A: Absorbing Boundary Conditions 8A.1. Sponge Zone 8A.2. PDE Absorbing Boundary Conditions 8A.3. Hybrid PDE Absorbing Boundary Conditions Chapter 9: The Viscoacoustic Wave Equation 9.1. Introduction to Linear Viscoelasticity 9.2. Viscoacoustic Wave Equation 9.3. Summary 9.4. Exercises 9.5. Computational Labs Appendix 9A: Relaxation Function Appendix 9B: Relation Between Q and T’s Part IV: Reflection Migration Chapter 10: Forward and Adjoint Modeling Using Green’s Functions 10.1. Integral-Equation Forward Modeling 10.1.1. Green’s Functions 10.1.2. (.2 X + K2)-1 Bygreen’s Theorem 10.1.3. Lippmann-Schwinger Solution 10.1.4. Neumann Series Solution 10.1.5. Born Approximation 10.1.6. Matrix Operator Notation 10.2. Integral-Equation Adjoint Modeling 10.2.1. Physical Meaning of the Migration Equation 10.3. Summary 10.4. Exercises 10.5. Computational Labs Appendix 10A: Causal and Acausal Green’s Functions Appendix 10B: Generalized Green’s Theorem Chapter 11: Reverse Time Migration 11.1. Introduction 11.2. General Imaging Algorithm 11.2.1. RTM= Generalized Diffraction-Stack Migration 11.2.2. Reverse Time Migration 11.3. Numerical Examples of RTM 11.4. Practical Implementation of RTM 11.5. Summary 11.6. Exercises 11.7. Computational Labs Appendix 11A: MATLAB RTM Code Chapter 12: Wavepaths 12.1. Traveltime Wavepaths 12.1.1. Computing Traveltime Wavepaths 12.2. Pressure Wavepaths 12.2.1. Computing Pressure Wavepaths 12.3. Summary 12.4. Exercises Chapter 13: Generalized Diffraction-stack Migration and Filtering of Coherent Noise Abstract 13.1. Introduction 13.2. Theory of Generalized Diffraction Migration 13.3. Directional Filtering the Generalized Diffraction-Stack Migration Kernel 13.3.1. Horizontal Reflector Model 13.3.2. Vertical Reflector Model 13.4. Anti-Aliasing Filtering the Generalized Diffraction-Stack Migration Kernel 13.5. Numerical Examples 13.5.1. Directional Filtering Results 13.5.2. Anti-Aliasing Filtering Results 13.6. Conclusions 13.7. Acknowledgements References Appendix A: Migration as a Pattern Matching Operation Appendix B: Computation and Compression of the Migration Kernel Exercises Chapter 14: Resolution Limits for Wave Equation Imaging Abstract 14.1. Introduction 14.1.1. Resolution Limits for Traveltime Tomography 14.1.2. Resolution Limits for Reflection Imaging 14.2. Born Forward and Adjoint Modeling 14.2.1. Born Forward Modeling 14.2.2. Born Adjoint Modeling 14.3. Model Resolution Function and FWI Resolution Limits 14.3.1. Model Resolution Equation: Mmig = L†Lm 14.3.2. Wavelength Imaging at the Diffractor 14.4. Filling in the Model Spectrum with Multiples 14.4.1. Lower Wavenumber Resolution with Prism Waves and Free-Surface Multiples 14.4.2. Intermediate-Wavenumber Resolution with Interbed Multiples 14.5. Discussion and Summary Acknowledgment Appendix A: Resolution Properties of Fresnel Volume in Constant and Layered Media Appendix B: Resolution Limits for Imaging Diving Wave Residuals Appendix C: Determinant of a Jacobian Matrix References Part V: Least-Squares Migration Chapter 15: Iterative Least-Squares Migration 15.1. Least-Squaresmigration Theory 15.2. Overdetermined and Underdetermined Iterative LSM 15.3. Implementation of LSM 15.3.1. Least-Squares Reverse Time Migration 15.3.2. Diffraction-Stack LSM 15.4. Problems with LSM 15.4.1. LSM Sensitivity to Velocity Errors 15.4.2. Computational Cost of LSM 15.5. Numerical Results 15.5.1. 3D Point-Scatterermodel 15.5.2. Partial Compensation for Poor Source and Receiver Sampling 15.5.3. Poststack Migration of Gulf of Mexico Data 15.5.4. Prestack Migration of Gulf of Mexico Data 15.6. LSM with a Crosscorrelation Objective Function 15.7. LSRTM with Internal Multiples 15.8. Trim Statics and LSM 15.9. Artifact Reduction with LSM 15.9.1. Numerical Results 15.10. Summary 15.11. Exercises 15.12. Computational Labs Appendix 15A: Diffraction-Stack LSM MATLAB Codes Implementation of LSM with Regularization Appendix 15B: Multisource LSM with Encoding Chapter 16: Viscoacoustic Least-Squares Migration 16.1. Theory of Viscoacoustic Least-Squares Migration 16.1.1. Viscoacoustic Born Modeling Equations 16.1.2. Viscoacoustic Adjoint Equations 16.1.3. Viscoacoustic Gradient 16.1.4. Algorithm for Q LSRTM 16.2. Numerical Results 16.3. Summary 16.4. Computational Labs Chapter 17: Least-Squares Migration Filtering 17.1. Least-Squaresmigration Filtering 17.2. Numerical Results 17.2.1. LSMF of PS and PP Reflections for a Graben Model 17.2.2. LSMF of Valhall Data 17.3. Problems with LSMF 17.3.1. Encoded Multisource LSMF 17.4. Summary Chapter 18: Migration Deconvolution 18.1. Migration Green’s Function 18.2. Approximations to [L†L]-1 18.2.1. Hessian Inverse by Dij /[L†L]Ij 18.2.2. Hessian Inverse by a Nonstationary Matching Filter 18.2.3. Migration Deconvolution 18.3. Iterative Migration Deconvolution 18.4. Numerical Tests 18.4.1. Point-Scatterer Model 18.4.2. Meandering-Stream Model 18.4.3. Converted-Wave Marine Field Data 18.4.4. 3D Alaska Field Data 18.5. Summary 18.6. Exercises Appendix 18A: Numerical Implementation of MD Part VI: Waveform Inversion Chapter 19: Acoustic Waveform Inversion and its Numerical Implementation Example 19.0.1. Pseudolinear Traveltime Misfit Function with Quasimonotonic Character Example 19.0.2. Highly Nonlinear Waveform Misfit Function Example 19.0.3. Mildly Nonlinear Waveform Misfit Function Example 19.0.4. Finite-Difference Solution 19.1. Numerical Implementation of Waveform Inversion 19.2. Expediting Convergence 19.2.1. Conjugate-Gradient and Quasi-Newton Gradient Methods 19.2.2. Starting Model 19.2.3. Preconditioning 19.2.4. Subspace Decomposition 19.2.5. Adaptive Multiscale FWI 19.2.6. Estimation of the Source Wavelet 19.2.7. Ignoring Amplitudes 19.3. Numerical Tests 19.4. Summary 19.5. Exercises 19.6. Computational Labs Chapter 20: Wave-Equation Inversion of Skeletonized Data 20.1. Implicit Function Theorem 20.2. Examples of Skeletonized Inversion 20.2.1. Wave-Equation Traveltime Tomography 20.2.2. Early-Arrival Waveform Tomography 20.2.3. Wave-Equation Inversion of Surface Waves 20.3. Alternative Objective Functions 20.4. Summary 20.5. Exercises Appendix 20A: Gradient of the Traveltime Misfit Function Appendix 20B: Implementation of WT Forward Modeling Backward Propagation Direction of Updating the Model Calculation of the Step Length Chapter 21: Acoustic Waveform Inversion: Case Histories 21.1. Early-Arrival Waveform Inversion Applied to Land Data 21.1.1. Data Acquisition 21.1.2. Data Processing 21.1.3. Estimating and Correcting for Q 21.1.4. EWT of the Wadi Qudaid Data 21.1.5. Synthetic Data Sanity Test 21.1.6. Key Points 21.2. Acoustic FWI Applied to Gulf of Mexico Marine Data 21.2.1. Hybrid Linear and Nonlinear FWI 21.2.2. Synthetic Two-Box Model Data 21.2.3. Gulf of Mexico Data 21.2.4. Key Points 21.3. Rolling-Offset FWI 21.3.1. Workflow for Rolling-Offset FWI 21.3.2. Rolling-Offset FWI: Synthetic Data 21.3.3. Rolling-Offset FWI: 2D Gulf of Mexico Data 21.3.4. FWI with Macro Windows: 3D Marine Data 21.3.5. Key Points 21.4. Acoustic FWI Applied to Crosswell Data 21.4.1. Synthetic Crosshole Data 21.4.2. Friendswood Crosshole Data 21.4.3. Key Points 21.5. Summary 21.6. Exercises Appendix 21A: Optimal Frequency Bands for EWT Chapter 22: Elastic and Viscoelastic Full-Waveform Inversion 22.1. Elastic FWI 22.2. FWI of Crosswell Hydrophone Records 22.2.1. Acoustic FWI Applied to Acoustic Synthetic Data 22.2.2. Elastic and Viscoelastic FWI Applied to Synthetic Viscoelastic Data 22.3. FWI Applied to McElroy Crosswell Data 22.3.1. Data Processing 22.3.2. Elastic and Viscoelastic Waveform Inversion 22.4. Summary 22.5. Exercises Appendix 22A: Misfit Gradient for λ Appendix 22B: Source-Wavelet Inversion Appendix 22C: Borehole Pressure-Field Simulation Appendix 22D: Estimation of Qp and Qs Appendix 22E: Viscoelastic FWI Gradient Appendix 22F: Elastic FWI Gradient Chapter 23: Vertical Transverse Isotropy FWI 23.1. Theory 23.2. Numerical Results 23.2.1. Synthetic VSP Data 23.2.2. 3D Gulf of Mexico Data 23.3. Extension to TTI Media 23.4. Summary 23.5. Exercises Part VII: Image-Domain Inversion Chapter 24: Classical Migration Velocity Analysis 24.1. Image-Domain Inversion 24.2. Classical Ray-Based MVA 24.3. Angle-Domain CIGs 24.4. Trim Statics MVA 24.5. Ray-Based Tomography 24.6. Summary Chapter 25: Generalized Differential Semblance Optimization 25.1. Introduction 25.2. Theory of Generalized Differential Semblance Optimization 25.2.1. Wave-Equation Traveltime and Waveform Inversion 25.2.2. Differential Semblance Optimization 25.2.3. Generalized Differential Semblance Optimization 25.3. Numerical Examples 25.4. Summary 25.5. Exercises Appendix 25A: Migration Images in the Subsurface Offset Domain Appendix 25B: H-Dsofréchet Derivative and Gradient Chapter 26: Generalized Image-Domain Inversion 26.1. Introduction 26.2. Theory of Generalized Image-Domain Inversion 26.2.1. Interpretation of the Gradient Functions 26.3. Numerical Results 26.4. Summary References Index