N. Bleistein J.K. Cohen J.W. Stockwell, Jr.

Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion

With 71 Illustrations

Springer

Contents

Preface vii

List of Figures xxiii

1 Multidimensional Seismic Inversion 1 1.1 Inverse Problems and Imaging 2 1.2 The Nonlinearity of the Seismic Inverse Problem 4 1.3 High Frequency 5 1.4 Migration Versus Inversion 7 1.5 Source-Receiver Configurations 12 1.6 Band and Aperture Limiting of Data 18 1.7 Dimensions: 2D Versus 2.5D Versus 3D 20 1.8 Acoustic Versus Elastic Inversion 20 1.9 A Mathematical Perspective on the Geometry of

Migration 22

2 The One-Dimensional Inverse Problem 24 2.1 Problem Formulation in One Spatial Dimension 25

2.1.1 The 1D Model in a Geophysical Context 25 2.1.2 The 1D Model as a Mathematical Testground . . 27

2.2 Mathematical Tools for Forward Modeling 28 2.2.1 The Governing Equation and Radiation

Condition 28 2.2.2 Fourier Transform Conventions 29 2.2.3 Green's Functions 32

xvi Contents

2.2.4 Green's Theorem 33 2.3 The Forward Scattering Problem 35

2.3.1 The Forward Scattering Problem in 1D 36 2.3.2 The Born Approximation and Its Consequences . 40 2.3.3 The Inverse Scattering Integral Equation 41

2.4 Constant-Background, Zero-Offset Inversion 42 2.4.1 Constant-Background, Single-Layer 43 2.4.2 More Layers, Accumulated Error 56 2.4.3 A Numerical Example 58 2.4.4 Summary 59

2.5 Inversion in a Variable-Background Medium 60 2.5.1 Modern Mathematical Issues 64 2.5.2 Summary 66 2.5.3 Implementation of the Variable-Wavespeed

Theory 70 2.5.4 Summary 76

2.6 Reevaluation of the Small-Perturbation Assumption . . . 77 2.7 Computer Implementation 78

2.7.1 Sampling 79 2.8 Variable Density 81

3 Inversion in Higher Dimensions 88 3.1 The Scattering Problem in Unbounded Media 90 3.2 The Born Approximation 94

3.2.1 The Born Approximation and High Frequency . . 99 3.2.2 The Constant-Background Zero-Offset Equation . 103 3.2.3 One Experiment, One Degree of Freedom in a . . 103

3.3 Zero-Offset Constant-Background Inversion in 3D . . . . 106 3.3.1 Restrictions on the Choice of k^ 111

3.4 High Frequency, Again 113 3.4.1 Reflection from a Single Tilted Plane 117 3.4.2 The Reflectivity Function 119 3.4.3 Alternative Representations of the Reflectivity

Function 121 3.5 Two-and-One-Half Dimensions 123

3.5.1 Zero-Offset, Two-and-One-Half Dimensional Inversion 125

3.6 Kirchhoff Inversion 127 3.6.1 Stationary Phase Computations 127 3.6.2 Two-and-One-Half-Dimensional Kirchhoff

Inversion 136 3.6.3 2D Modeling and Inversion 138

3.7 Testing the Inversion Formula with Kirchhoff Data . . . 144 3.7.1 The Kirchhoff Approximation 144 3.7.2 Asymptotic Inversion of Kirchhoff Data 146

Contents xvii

3.7.3 Summary 152 3.8 Reverse-Time Wave-Equation Migration Deduced from

the Kirchhoff Approximation 156

4 Large-Wavenumber Fourier Imaging 161 4.1 The Concept of Aperture 163 4.2 The Relationship Between Aperture and Survey

Parameters 164 4.2.1 Rays, Fourier Transforms, and Apertures 165 4.2.2 Aperture and Migration Dip 166 4.2.3 Migration Dip and Apertures 168 4.2.4 Summary 176

4.3 Examples of Aperture-Limited Fourier Inversion 176 4.3.1 Aperture-Limited Inversion of a Dirac Delta

Function (A Point Scatterer) 177 4.3.2 Aperture-Limited Inversion of a Singular

Function (a Reflecting Plane) 179 4.3.3 Generalization to Singular Functions of Other

Types of Surfaces—Asymptotic Evaluation . . . . 184 4.3.4 Relevance to Inverse Scattering 189 4.3.5 Aperture-Limited Fourier Inversion of Smoother

Functions 189 4.3.6 Aperture-Limited Fourier Inversion of Steplike

Functions 189 4.3.7 Aperture-Limited Fourier Inversion of a

Ramplike Function 192 4.3.8 Aperture-Limited Inversion of an Infinitely

Differentiable Function 194 4.3.9 Summary 196

4.4 Aperture-Limited Fourier Identity Operators 196 4.4.1 The Significance of the Boundary Values in Dy> . 198 4.4.2 Stationary Phase Analysis for Jo 201 4.4.3 The Near-Surface Condition 208 4.4.4 Extracting Information About / on Sy> 208 4.4.5 Processing for a Scaled Singular Function of the

Boundary Surface Sy< 209 4.4.6 The Normal Direction 211 4.4.7 Integrands with Other Types of Singularities . . . 212 4.4.8 Summary 214 4.4.9 Modern Mathematical Issues 215

5 Inversion in Heterogeneous Media 216 5.1 Asymptotic Inversion of the Born-Approximate Integral

Equation—General Results 217 5.1.1 Recording Geometries 217

xviii Contents

5.1.2 Formulation of the 3D, Variable-Background, Inverse-Scattering Problem 220

5.1.3 Inversion for a Reflectivity Function 225 5.1.4 Summary of Asymptotic Verification 227 5.1.5 Inversion in Two Dimensions 227 5.1.6 General Inversion Results, Stationary Triples,

and cos 9S 232 5.1.7 An Alternative Derivation: Removing the

Small-Perturbation Restriction at the Reflector . 238 5.1.8 Discussion 241

5.2 The Beylkin Determinant h, and Special Cases of 3D Inversion 242 5.2.1 General Properties of the Beylkin Determinant . 243 5.2.2 Common-Shot Inversion 245 5.2.3 Common-Offset Inversion 247 5.2.4 Zero-Offset Inversion 249

5.3 Beylkin Determinants and Ray Jacobians in the Common-Shot and Common-Receiver Configurations . . 250

5.4 Asymptotic Inversion of Kirchhoff Data for a Single Reflector 257 5.4.1 Stationary Phase Analysis of the Inversion of

Kirchhoff Data 258 5.4.2 Determination of cos 0S and c+ 263 5.4.3 Finding Stationary Points 265 5.4.4 Determination of the Matrix Signature 268 5.4.5 The Quotient/i/l d e t ^ H 1 / 2 269

5.5 Verification Based on the Fourier Imaging Principle . . . 271 5.6 Variable Density 276

5.6.1 Variable-Density Reflectivity Inversion Formulas . 277 5.6.2 The Meaning of the Variable-Density Reflectivity

Formulas 278 5.7 Discussion of Results and Limitations 279

5.7.1 Summary 281

6 Two-and-One-Half-Dimensional Inversion 282 6.1 2.5D Ray Theory and Modeling 283

6.1.1 Two-and-One-Half-Dimensional Ray Theory . . . 283 6.2 2.5D Inversion and Ray Theory 290

6.2.1 The 2.5D Beylkin Determinant 292 6.2.2 The General 2.5D Inversion Formulas for

Reflectivity 293 6.3 The Beylkin Determinant H and Special Cases of 2.5D

Inversion 297 6.3.1 General Properties of the Beylkin Determinant . . 297 6.3.2 Common-Shot Inversion 299

Contents xix

6.3.3 A Numerical Example—Extraction of Reflectivity from a Common-Shot Inversion 299

6.3.4 Constant-Background Propagation Speed 301 6.3.5 Vertical Seismic Profiling 302 6.3.6 Well-to-Well Inversion 304 6.3.7 Invert for What? 304 6.3.8 Common-Offset Inversion 305 6.3.9 A Numerical Example—Extraction of the

Reflection Coefficient and cosös from a Common-Offset Inversion 307

6.3.10 A Numerical Example—Imaging a Syncline with Common-offset Inversion 307

6.3.11 Constant Background Inversion 309 6.3.12 Zero-Offset Inversion 309

7 The General Theory of Data Mapping 311 7.1 Introduction to Data Mapping 312

7.1.1 Kirchhoff Data Mapping (KDM) 314 7.1.2 Amplitude Preservation 315 7.1.3 A Rough Sketch of the Formulation of the KDM

Platform 315 7.1.4 Possible Kirchhoff Data Mappings 317

7.2 Derivation of a 3D Kirchhoff Data Mapping Formula . . 319 7.2.1 Spatial Structure of the KDM Operator 322 7.2.2 Frequency Structure of the Operator and

Asymptotic Preliminaries 323 7.2.3 Determination of Incidence Angle 327

7.3 2.5D Kirchhoff Data Mapping 328 7.3.1 Determination of Incidence Angle 330

7.4 Application of KDM to Kirchhoff Data in 2.5D 331 7.4.1 Asymptotic Analysis of 2.5D KDM 339 7.4.2 Stationary Phase Analysis in 7 341 7.4.3 Validity of the Stationary Phase Analysis . . . . 344

7.5 Common-Shot Downward Continuation of Receivers (or Sources) 347 7.5.1 Time-Domain Data Mapping for Other

Implementations 349 7.5.2 Stationary Phase in tj 351

7.6 2.5D Transformation to Zero-Offset (TZO) 354 7.6.1 TZO in the Frequency Domain 355 7.6.2 A Haie-Type TZO 361 7.6.3 Gardner/Forel-Type TZO 363 7.6.4 On the Simplification of the Second Derivatives

of the Phase 365 7.7 3D Data Mapping 374

xx Contents

7.7.1 Stationary Phase in 7 375 7.7.2 Discussion of the Second Derivatives of the Phase . 377 7.7.3 3D Constant-Background TZO 380 7.7.4 The 72 Integral As a Bandlimited Delta Function . 381 7.7.5 Space/Frequency TZO in Constant Background . 384 7.7.6 A Haie-Type 3D TZO 386

7.8 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 O l 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 . . 443 E.2.3 Choosing A = c2/2: r , Traveltime, as the

Running Parameter 444 E.2.4 Choosing A = c( x)/2: s, Arclength, as the

Running Parameter 444

Contents xxi

E.3 Ray Amplitude Theory 446 E.3.1 The ODE Form of the Transport Equation . . . . 448 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.5 2.5 D Ray Theory 463 E.5.1 2.5D Ray Equations 464 E.5.2 2.5D 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.8.4 2.5D 486 E.8.5 Summary 487

References 489

Author Index 499

Subject Index 503