Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion

Transcription

1 N. Bleistein J.K. Cohen J.W. Stockwell, Jr. Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion With 71 Illustrations Springer

2 Contents Preface List of Figures vii xxiii 1 Multidimensional Seismic Inversion Inverse Problems and Imaging The Nonlinearity of the Seismic Inverse Problem High Frequency Migration Versus Inversion Source-Receiver Configurations Band and Aperture Limiting of Data Dimensions: 2D Versus 2.5D Versus 3D Acoustic Versus Elastic Inversion A Mathematical Perspective on the Geometry of Migration 22 2 The One-Dimensional Inverse Problem Problem Formulation in One Spatial Dimension The 1D Model in a Geophysical Context The 1D Model as a Mathematical Testground Mathematical Tools for Forward Modeling The Governing Equation and Radiation Condition Fourier Transform Conventions Green's Functions 32

3 xvi Contents Green's Theorem The Forward Scattering Problem The Forward Scattering Problem in 1D The Born Approximation and Its Consequences The Inverse Scattering Integral Equation Constant-Background, Zero-Offset Inversion Constant-Background, Single-Layer More Layers, Accumulated Error A Numerical Example Summary Inversion in a Variable-Background Medium Modern Mathematical Issues Summary Implementation of the Variable-Wavespeed Theory Summary Reevaluation of the Small-Perturbation Assumption Computer Implementation Sampling Variable Density 81 3 Inversion in Higher Dimensions The Scattering Problem in Unbounded Media The Born Approximation The Born Approximation and High Frequency The Constant-Background Zero-Offset Equation One Experiment, One Degree of Freedom in a Zero-Offset Constant-Background Inversion in 3D Restrictions on the Choice of k^ High Frequency, Again Reflection from a Single Tilted Plane The Reflectivity Function Alternative Representations of the Reflectivity Function Two-and-One-Half Dimensions Zero-Offset, Two-and-One-Half Dimensional Inversion Kirchhoff Inversion Stationary Phase Computations Two-and-One-Half-Dimensional Kirchhoff Inversion D Modeling and Inversion Testing the Inversion Formula with Kirchhoff Data The Kirchhoff Approximation Asymptotic Inversion of Kirchhoff Data 146

4 Contents xvii Summary Reverse-Time Wave-Equation Migration Deduced from the Kirchhoff Approximation Large-Wavenumber Fourier Imaging The Concept of Aperture The Relationship Between Aperture and Survey Parameters Rays, Fourier Transforms, and Apertures Aperture and Migration Dip Migration Dip and Apertures Summary Examples of Aperture-Limited Fourier Inversion Aperture-Limited Inversion of a Dirac Delta Function (A Point Scatterer) Aperture-Limited Inversion of a Singular Function (a Reflecting Plane) Generalization to Singular Functions of Other Types of Surfaces Asymptotic Evaluation Relevance to Inverse Scattering Aperture-Limited Fourier Inversion of Smoother Functions Aperture-Limited Fourier Inversion of Steplike Functions Aperture-Limited Fourier Inversion of a Ramplike Function Aperture-Limited Inversion of an Infinitely Differentiable Function Summary Aperture-Limited Fourier Identity Operators The Significance of the Boundary Values in D y > Stationary Phase Analysis for Jo The Near-Surface Condition Extracting Information About / on S y > Processing for a Scaled Singular Function of the Boundary Surface S y < The Normal Direction Integrands with Other Types of Singularities Summary Modern Mathematical Issues Inversion in Heterogeneous Media Asymptotic Inversion of the Born-Approximate Integral Equation General Results Recording Geometries 217

5 xviii Contents Formulation of the 3D, Variable-Background, Inverse-Scattering Problem Inversion for a Reflectivity Function Summary of Asymptotic Verification Inversion in Two Dimensions General Inversion Results, Stationary Triples, and cos 9 S An Alternative Derivation: Removing the Small-Perturbation Restriction at the Reflector Discussion The Beylkin Determinant h, and Special Cases of 3D Inversion General Properties of the Beylkin Determinant Common-Shot Inversion Common-Offset Inversion Zero-Offset Inversion Beylkin Determinants and Ray Jacobians in the Common-Shot and Common-Receiver Configurations Asymptotic Inversion of Kirchhoff Data for a Single Reflector Stationary Phase Analysis of the Inversion of Kirchhoff Data Determination of cos 0 S and c Finding Stationary Points Determination of the Matrix Signature The Quotient/i/l det^h 1 / Verification Based on the Fourier Imaging Principle Variable Density Variable-Density Reflectivity Inversion Formulas The Meaning of the Variable-Density Reflectivity Formulas Discussion of Results and Limitations Summary Two-and-One-Half-Dimensional Inversion D Ray Theory and Modeling Two-and-One-Half-Dimensional Ray Theory D Inversion and Ray Theory The 2.5D Beylkin Determinant The General 2.5D Inversion Formulas for Reflectivity The Beylkin Determinant H and Special Cases of 2.5D Inversion General Properties of the Beylkin Determinant Common-Shot Inversion 299

6 Contents xix A Numerical Example Extraction of Reflectivity from a Common-Shot Inversion Constant-Background Propagation Speed Vertical Seismic Profiling Well-to-Well Inversion Invert for What? Common-Offset Inversion A Numerical Example Extraction of the Reflection Coefficient and cosö s from a Common-Offset Inversion A Numerical Example Imaging a Syncline with Common-offset Inversion Constant Background Inversion Zero-Offset Inversion The General Theory of Data Mapping Introduction to Data Mapping Kirchhoff Data Mapping (KDM) Amplitude Preservation A Rough Sketch of the Formulation of the KDM Platform Possible Kirchhoff Data Mappings Derivation of a 3D Kirchhoff Data Mapping Formula Spatial Structure of the KDM Operator Frequency Structure of the Operator and Asymptotic Preliminaries Determination of Incidence Angle D Kirchhoff Data Mapping Determination of Incidence Angle Application of KDM to Kirchhoff Data in 2.5D Asymptotic Analysis of 2.5D KDM Stationary Phase Analysis in Validity of the Stationary Phase Analysis Common-Shot Downward Continuation of Receivers (or Sources) Time-Domain Data Mapping for Other Implementations Stationary Phase in tj D Transformation to Zero-Offset (TZO) TZO in the Frequency Domain A Haie-Type TZO Gardner/Forel-Type TZO On the Simplification of the Second Derivatives of the Phase D Data Mapping 374

7 xx Contents Stationary Phase in Discussion of the Second Derivatives of the Phase D Constant-Background TZO The 72 Integral As a Bandlimited Delta Function Space/Frequency TZO in Constant Background A Haie-Type 3D TZO Summary and Conclusions 387 A Distribution Theory 389 A.l Introduction 389 A.2 Localization via Dirac Delta functions 390 A.3 Fourier Transforms of Distributions 397 A.4 Rapidly Decreasing Functions 399 A.5 Temperate Distributions 400 A.6 The Support of Distributions 401 A.7 Step Functions 402 A.7.1 Hubert Transforms 404 A.8 Bandlimited Distributions 405 B The Fourier Transform of Causal Functions 409 B.l Introduction 409 B.2 Example: the 1D Free-Space Green's Function 415 C Dimensional Versus Dimensionless Variables 418 Ol The Wave Equation 419 C.l.l Mathematical Dimensional Analysis 419 C.1.2 Physical Dimensional Analysis 421 C.2 The Helmholtz Equation 422 C.3 Inversion Formulas 425 D An Example of Ill-Posedness 430 D.l Ill-posedness in Inversion 431 E An Elementary Introduction to Ray Theory and the Kirchhoff Approximation 435 E.l The Eikonal and Transport Equations 436 E.2 Solving the Eikonal Equation by the Method of Characteristics 438 E.2.1 Characteristic Equations for the Eikonal Equation 442 E.2.2 Choosing A = : a as the Running Parameter E.2.3 Choosing A = c 2 /2: r, Traveltime, as the Running Parameter 444 E.2.4 Choosing A = c( x)/2: s, Arclength, as the Running Parameter 444

8 Contents xxi E.3 Ray Amplitude Theory 446 E.3.1 The ODE Form of the Transport Equation E.3.2 Differentiation of a Determinant 449 E.3.3 Verification of ( E.3.12) 452 E.3.4 Higher-Order Transport Equations 453 E.4 Determining Initial Data for the Ray Equations 453 E.4.1 Initial Data for the 3D Green's Function 454 E.4.2 Initial Data for the 2D Green's Function 457 E.4.3 Initial Data for Refiected and Transmitted Rays. 459 E D Ray Theory 463 E D Ray Equations 464 E D Amplitudes 465 E.5.3 The 2.5D Transport Equation 465 E.6 Raytracing in Variable-Density Media 467 E.6.1 Ray Amplitude Theory in Variable-Density Media 468 E.6.2 Refiected and Transmitted Rays in Variable-Density Media 469 E.7 Dynamic Raytracing 470 E.7.1 A Simple Example, Raytracing in Constant-Wavespeed Media 473 E.7.2 Dynamic Raytracing in er 474 E.7.3 Dynamic Raytracing in r 475 E.7.4 Two Dimensions 475 E.7.5 Conclusions 475 E.8 The Kirchhoff Approximation 476 E.8.1 Problem Formulation 478 E.8.2 Green's Theorem and the Wavefield Representation 479 E.8.3 The Kirchhoff Approximation 483 E D 486 E.8.5 Summary 487 References 489 Author Index 499 Subject Index 503