Introduction to petroleum seismology - · PDF fileTABLEOFCONTENTS Preface xix Acknowledgments...

INTRODUCTION TO

PETROLEUM SEISMOLOGY

Luc T. Ikelle

Lasse Amundsen

Investigations in Geophysics Series No. 12

Michael R. Cooper, series editor

Anthony F. Gangi, volume editor

Society of Exploration Geophysicists

Tulsa, Oklahoma, USA

TABLE OF CONTENTS

Preface xix

Acknowledgments xxi

About the Authors xxiii

Chapter 1 Introduction 1

The "Bottom Line" of Petroleum Seismology 1

Petroleum Traps 1

How Does Petroleum Seismology Work? I

Challenges of Petroleum Seismology 3

Exploring for Stratigraphic Traps 3

Exploring the Subsalt Stratigraphic Column 3

Exploring the Subbasalt Stratigraphic Column 6

Environmental Challenges of Exploring the Arctic 7

Exploring for Gas Hydrates 8

Petroleum Seismologists in Production of Oil and Gas 10

Technological Advances outside the E&P Industry 11

Technological Advances inside the E&P Industry 11

Instrumented Oil Fields 11

Measurement while Drilling 12

Reservoir Model 12

Box 1.1 Marine Electromagnetic Surveying for Hydrocarbon Detection 13

The Scope of this Book 15

Chapter 2 The Relationship between Propagation of Seismic Waves and Particle Motions

in Isotropic Media 17

An Example of Wave Propagation 17

The Assumption of a Continuous Medium 19

Continuous and Isotropic Media 19

Particle Positions and Coordinate Systems 20

Homogeneous Media and Heterogeneous Media 20

Internal Forces (Stresses) 20

The Stress Tensor 20

Box 2.1 Scalar Product and Vector Product 21

Examples of Stresses 24

Example 1 24

Example 2 24

Example 3 25

Box 2.2 Conventions of Summation 25

Abbreviated Notation of the Stress Tensor 26

vi

Table of Contents vii

Box 2.3 Change of Orthonormal Basis: Vectors 26

Box 2.4 Euler Angles 28

The Stress Field 28

Box 2.5 Changes of Orthonormal Basis (Stress Tensor) 29

Principal Stresses 29

Particle Displacement and Strain 30

Particle Displacement 30

Strain Tensor 31

Abbreviated Notation of the Strain Tensor 32

Examples of Strain Tensors 33

Example 4 33

Example 5 33

Example 6 34

Elastic Moduli 34

Linear Elasticity (Hooke's Law): General Case 34

Hooke's Law with Abbreviated Tensor Notation 35

Box 2.6 Change of Orthonormal Basis (Stiffness Tensor) 36

Linear Elasticity (Hooke's Law): Isotropic Case 36

Physical Interpretation of Elastic Moduli for an Isotropic Medium 38

Equations of Elastodynamic Wave Motion 39

Newton's Equation of Motion,

39

Elastic Waves: P-waves and S-waves 41

Box 2.7 Helmholtz Decomposition of Vector Fields 43

Box 2.8 Plane Waves 44

Parameters of Isotropic, Elastic Rock Formations 45

Relating Elastic Parameters to Petrophysical Parameters 46

Sources of Seismic Waves 49

Definition of Sources in the Context of Petroleum Seismology 49

Equations of Wave Motion and the Generalized Hooke's Law 49

Examples of Seismic-wave Radiation 50

Geometric Spreading 53

Box 2.9 Another Form of the Equations of Wave Motion 54

Box 2.10 Acoustic Equations of Wave Motion 54

Box 2.11 The Equivalence Fluid Model for P-waves in a Solid 55

Isotropy, Anisotropy, Homogeneity, and Heterogeneity 56

Exercises in Problem Solving 60

vi i i Table of Contents

Chapter 3 Partition of Energy at an Interface 63

Introduction 63

Huygens' Principle 63

Fermat's Principle 64

Snell's Law 64

Reflection and Transmission 64

Snell's Law: Fluid-fluid Interface 65

Snell's Law: Solid-solid and Fluid-solid Interfaces 67

Snell's Law: Air-water and Air-solid Interfaces 70

What is a free surface? 70

Snell's law at the free surface 71

Traveltime Equations for a Horizontal Interface 72

Direct Waves 72

Refracted Waves 73

Reflected P-P and S-S Waves 74

P-S Converted Waves 75

Conversion Point Offset 76

Box 3.1 Traveltime in 1D Media 78

Turning Rays 79

Variation of Linear Velocity with Depth 79

Box 3.2 The Notion of rms Velocity for 1D Media 81

Box 3.3 Dix's Formula 84

Boundary Conditions for the Elastodynamic Field 85

Solid-solid Interface 85

Fluid-solid Interface 85

Vacuum-solid Interface 86

Fluid-fluid Interface 86

From elastodynamic to acoustic fields: A brief background 86

Boundary conditions at the interface between two fluids 86

Interface between Vacuum and Fluid 86

Zoeppritz's Equations for a Horizontal Interface 87

Zoeppritz's Equations: Solid-solid Interface 88

Reflection and transmission coefficients for a downward-traveling incident P-wave 89

Special case: i± constant 91

Special case: V$ and p constant 91

Special case: Normal incidence 91

Reflection and transmission coefficients for a downward-traveling incident SV-wave 91

Reflection and transmission coefficients for incident waves from below 92

Zoeppritz's Equations: Fluid-solid Interface 92

Box 3.4 R/T Coefficients in Terms of Slowness: Solid-solid Interface 93

Zoeppritz's Equations: Vacuum-solid Interface 94

Zoeppritz's Equations: Fluid-fluid Interface 94

Table ofContents ix

Box 3.5 R/T Coefficients in Terms of Slowness: Fluid-solid Interface 95

Reflection and Transmission Coefficients for the Energy of Seismic Waves 95

Normal incidence 95

Oblique incidence 96

Examples 96

Example 1 96

Example 2 97

Example 3 98

Example 4 98

Example 5 98

Surface Waves 98

Motivations for Studying Surface Waves 100

Evanescent Plane Waves 101

Phase Velocity of Scholte and Rayleigh Waves 102

Rayleigh waves 104

Scholte waves: A fluid half-space on a solid half-space 105

Scholte waves: A fluid layer above a solid half-space 105

Surface-wave Particle Motion 106

Linearized Zoeppritz's Equations 107

Matrix Form of Zoeppritz's Equations 107

Linearized Versions of Reflection Coefficients 107

Application to AVA Analysis: P-P Reflections Ill

Box 3.6 Some Probable Values of Reflection Coefficients at Normal Incidence 113

Application to AVA Analysis: P-P and P-S Reflections 113

Dipping Interface 114

Traveltime Equation for Refracted Waves 114

Traveltime Equation for Reflected Waves 116

Diffractions 118

An Illustration of Diffractions 119

Traveltime Equation for Refracted Waves 119

Traveltime Equation for Reflected Waves 121


Chapter 4 The Fourier Representation of Seismic Signals 127

Signals and Systems 127

Signals 127

Systems 128

Box 4.1 Periodic and Transient Signals 128

The Cosine Wave: Concept of Frequency 129

Frequency 129

Delays 129

A Useful Form of Cosine Waves 130

The Fourier Series 130

Basis Representation for Signals 131

x Table of Contents

Box 4.2 Orthonormal Basis of the Space of Signals: The Vector Space Analogy 132

The Fourier Series: General Case 132

The Fourier Series: Even and Odd Functions 134

Example 1: The Fourier Series of Sawtooth Waves 135

Example 2: The Fourier Series of Square Waves 136

The Fourier Transform 138

The Fourier Transform of Periodic Functions 139

The Fourier Transform of a Nonperiodic Signal 139

Box 4,3 Fourier Transform and Square Integrable Functions 141

Example 1: The Fourier Transform of the Exponential Function 141

Box 4.4 Nyquist Frequency 142

Example 2: The Fourier Transform of a Symmetrical Rectangular Pulse 142

Properties of the Fourier Transform 143

The Multidimensional Fourier Transform 144

Sampling Theorem and Discrete Fourier Transform 144

Discrete Signal 145

Aliasing 146

Reconstruction of the Continuous Signal from its Discrete Samples 148

The Discrete Fourier Transform 149

Some Properties of the Discrete Fourier Transform 150

Convolution 150

Definition of the Impulse Response of a Linear System 151

Examples of the Impulse Response of a Linear System 151

Example 1 151

Example 2 152

Example 3 152

Convolution Theorem 153

Example 1 154

Example 2 154

Convolution Sum 155

Smoothness 155

Seismic Resolution 156

Filtering 157

Basic Terminology of Filtering 158

Inverse Filtering 159

An Example of Inverse Filters: Multiple Attenuation 159

Classification 161

A Limitation of the Effectiveness of the Fourier Transform Analysis 162

Examples of Nonstationary Signals 162

Example: A signal with impulses 162

Example: A quadratic chirp signal 164

Example: A signal with a shutdown period 164

The Windowed Fourier Transform 165

Example: A signal with impulses 166

Table of Contents xi

Example: A quadratic chirp signal 168

The Wavelet Transform 169

An Example of the Wavelet Transform of Seismic Data 172

Quadratic (Nonlinear) Time-frequency Transforms 173

Box 4.5 The Uncertainty Principle 176


Chapter 5 Characterization of Seismic Signals by Statistical Averages 181

Random Variables 182

Examples of Random Variables 182

Probability Density Functions and Characteristic Functions 182

Probability density functions 182

Characteristic functions 183

Moments and Cumulants 183

Moments 183

Central moments 184

Cumulants 185

Joint Moments and Joint Cumulants 186

Linear Regression: An Application of Joint Moments 188

Statistics of the Optimization Criteria 189

Box 5.1 The Central Limit Theorem 192

Seismic Imaging and Random Variables 192

Stochastic Signals 196

Moments and Cumulants 196

Polyspectra 199

Cross-cumulants and their Spectra 200

Cross-cumulants 200

Cross-cumulant spectra 201

Examples of Calculations of Cumulants, Cross-cumulants, Polyspectra, and Cross-cumulant Spectra 201

Example 1: Quadratic phase coupling 201

Example 2: Non-Gaussian signal applied to a linear system 203

Example 3: Gaussian signal applied to a nonlinear Volterra system 205

Deterministic Signals 208

Moments, Cross-moments, and their Spectra 209

Examples of Calculations of Moments, Cross-moments, and their Spectra 210

Example 4: Time delay 210

Example 5: Minimum-, maximum-, and mixed-phase signals 211

Box 5.2 Similarities between Crosscorrelation and Convolution 214

Application of Autocorrelation to Ghost Identification 214

Ghost Identification 214

A Mathematical Derivation of the Autocorrelation Function 215

Application of Crosscorrelation and Bicoherence Correlation to Moveout Correction 216

Data Set 217

Moveout Correction 217

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Box 5.3 Definition of Bicoherence Correlation 219

Some Differences between Second- and Third-order Cumulants 220

Second-order statistics: Crosscorrelation 220

Second-order statistics: Coherence correlation 220

Third-order techniques: Bispectral correlation 221

Third-order techniques: Bicoherence correlation 221

More Insight into Second- and Third-order Statistics 222

Time Delays 222

Normalized Third-order Cumulants 225

Coherence correlation 225

Bispectral correlation 225

Seismic Resolution 226

Wiener-Hopf Equations and the Quadratic Volterra Model 227

Convolution as a Matrix Equation 228

The Method of Least Squares 228

Quadratic Volterra Model 229

Box 5.4 The Concept of Signal-to-Noise Ratio 230


Chapter 6 The Concepts of Reciprocity and Green's Functions 233

Time-domain Green's Functions in Unbounded Space 233

Acoustic Medium 234

Solving for pressure 234

Analytic solutions for a homogeneous medium 235

Elastic Medium 235

Solving for displacement 236


Frequency-domain Green's Functions in Unbounded Space 236

Acoustic Medium 237

Solving for pressure 237


Elastic Medium 237

Solving for displacement 238

Analytic solution for a homogeneous medium 238

Rayleigh's Reciprocity Theorem 238

General Theory 238

Box 6.1 Divergence Theorem (Gauss's Theorem) 239

Special Cases of Acoustic Reciprocity for Identical Media 239

Box 6.2 Application of Equation (6.68) to Towed-streamer Acquisition 241

Representation Theorem 242

Lippmann-Schwinger Equation 242

Marine-source Radiation-pattern Determination 243

Table of Contents xiii

Betti-Rayleigh's Reciprocity Theorem 244

General Theory 244

Box 6.3 Derivation of Lippmann-Schwinger Equation Using the Perturbation Theory 245

Special Cases of Elastic Reciprocity for Identical Unbounded Media 246

Reciprocity of particle velocity for point forces 247

Reciprocity of strain for stress-point sources 247

Reciprocity of stress for strain-point sources 249

Reciprocity for P-wave source and force 249


Chapter 7 Acquisition Geometries and Seismic Data 255

Seismic Acquisition in Water and in Solids 255

Marine Towed-streamer Seismics 256

Acquisition Geometry 256

Seismic Data 257

Box 7.1 The Superposition Principle 262

Shot and Receiver Gathers 263

Common-midpoint and Common-offset Gathers 264

Out-of-plane Reflections 266

Swell Noise 269

Measurement of Particle Velocity in Towed-streamer Acquisition 270

Box 7.2 Displaying Seismic Data: Amplitude Correction 271

Ocean-bottom Seismics 273

Acquisition Geometry: 4C-OBS Data 273

Ocean-bottom Seismic Data 274

Direct waves 274

Receiver ghosts 275

The dominant converted shear-wave reflections 275

PZdata 276

Brief History of Marine 4C-OBS Experiments 277

Some Benefits of 4C Technology 278

Imaging below gas-invaded sediments 278

Imaging under salt 280

Imaging of reservoirs with low P-wave reflectivity but high PS-wave reflectivity 280

Quantification of amplitude anomalies 281

Quantitative Vp/Vs velocity ratio 283

Overpressured zones 283

Anisotropy: Fractured reservoirs 283

Reservoir monitoring (4D) 284

Imaging of complex structures by multiazimuth, true 3D surveys 284

Box 7.3 4D Seismic Monitoring of a Subsurface CO2 Repository 286

xiv Table of Contents

287Land-surface Seismics

Contrasting Land and Marine Acquisitions 2S7

988Explosive Sources (Dynamite)Vibroseis 288

Land Data 291

Ground roll 292

Statics 294

Transition Zones 294

Box 7.4 Scholte Waves Recorded on the Seafloor 295

Borehole Seismics 29^

VSP Acquisition Geometries and Borehole Seismic Data 296

Check shot 297

Zero-offset VSP 297

Offset VSP 299

Walkaway VSP 299

Walkabove VSP 300

Drill-noise VSP 301

Salt-proximity VSP 301

Shear-wave VSP 301

3D VSPs 302

Through-tubing VSPs 303

Tube Waves 303

Vertical Cables 305

Marine VC 305

Acquisition 305

Data 306

Potential Impact of Land VC 307

VC data 309

Resolution of VC data versus surface data 310


Chapter 8 Wavefield Sampling 315

Plane Waves and the 2D Fourier Transform 315

Apparent Velocity 316

Wavenumber 317

2D Fourier Transform 320

Example 1: 2D Fourier Transform of the Rectangle Function 320

Example 2: 2D Fourier Transform of an Event with Linear Moveout 321

Properties of 2D Fourier Transforms 322

Discrete 2D Fourier Transform 323

Box 8.1 Dispersion, Phase Velocities, and Group Velocities 324

Criteria of Uniform Spatial Sampling 326

Energy Distribution in thef-k Domain 326

Sampling Criteria 326

Spatial Aliasing 327

Table of Contents xv

Dip Filtering 329

An Application of Dip Filtering to Multiple Attenuation 330

An Application of Dip Filtering to Up-down Separation 331

Spatial Resampling Based on a Hardwired Array Summation 332

Definition of Arrays 334

Impulse Responses of Arrays 335

Wavenumber Response of Arrays: General Case 337

Ideal wavenumber response 338

Wavenumber response of an array with an odd number of elements 338

Wavenumber response of an array with an even number of elements 340

Wavenumber Response of Equally Weighted Line Arrays 341

Wavenumber Response of Nonuniformly Weighted Line Arrays 342

Nonuniform line arrays 343

Areal arrays343

Wavenumber Response of a Combination of Source and Receiver Arrays 344

Array System Designed as an Antialiasing Filter 345

Array System Designed as a Surface-noise Suppressor 347

Sensitivity of Array Summation to Sensor Dropouts 349

Spatial Resampling Based on Adaptive Beamforming 350

Single-sensor Recordings 350

What Is Beamforming? 351

A Formulation of Beamforming as a Variant of the Wiener Filter 352

Linearly Constrained Adaptive Beamforming 354

An Example of Swell-noise Attenuation 355

Box 8.2 Crossline Sampling 355

3D Wavefield Sampling 356

The Multisource and Multistreamer Concept 356


Chapter 9 Wavefield Decomposition into P- and S-waves and Upgoing and Downgoing Waves 361

The Concept of Decomposition into P- and S-wave Arrivals (P/S) and Total Upgoing and

Downgoing Waves (U/D) 361

Definitions of P- and S-wave and Upgoing and Downgoing Wave Decomposition 361

The Benefit of Multicomponent Recordings 363

Derivation of P/S and U/D Decomposition 367

The Matrix-vector Differential Equation 367

Decomposition of the Particle-velocity Vertical-traction Vector 368

Box 9.1 The Matrix-vector Differential Equation (9.14) for a Special Case 369

Upgoing and Downgoing P- and S-wave Components 371

Total Upgoing and Downgoing Wave Components 372

Total P- and S-wave Components 375

Box 9.2 Relationship between Vertical-traction and Particle-velocity Vectors for Purely Upgoing or

Purely Downgoing Waves 377

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Application of P/S and U/D Decomposition to 4C OBS Recordings 377

Upgoing and Downgoing P- and S-wave Components 377

Total P- and S-wave Components 378

U/D Decomposition Just below the Seafioor as a Demultiple Process 379

Pressure 379

Horizontal components of the particle velocity 380

Vertical component of the particle velocity 381

Demultiple process as a function of angles 381

Acoustic Wavefield Decomposition 382

Numerical Examples 382

Application of U/D Decomposition to Towed-streamer Data 386

Box 9.3 Reflection and Transmission from a Generalized Interface 387

Box 9.4 The Relationship between Downgoing Field Components below the Seafioor 388

Application of U/D Decomposition to VC Data 389

Application of U/D Decomposition to Snapshots 392


Chapter 10 Multiple Attenuation 395

Multiple Attenuation: Towed-streamer Data 395

The Exercise of Constructing Free-surface Multiples 395

The Representation Theorem and the Kirchhoff Scattering Series 397

The integral relationship between data containing multiples and data without multiples 397

Extrapolation of the vertical component of the particle velocity from the receiver positions to

the sea surface 399

Box 10.1 Formulating the Representation Theorem to Predict Data Containing Multiples 401

Box 10.2 Another Choice for the Surface Integral in the Representation Theorem 402

A Kirchhoff Scattering Series 402

A Physical Interpretation of the Kirchhoff Scattering Series 403

The pressure field and the vertical component of particle velocity without ghosts and without

direct waves 404

The pressure field with ghosts and direct waves, and the vertical component of the

particle velocity without ghosts and without direct waves 405

Both the pressure field and the vertical component of the particle velocity with ghosts and

direct-wave arrivals 407

Box 10.3 Some Basic Taylor Series Expansions 408

Box 10.4 The Two-reflector Problem in Towed-streamer Data 409

Box 10.5 Computing Particle Velocity from Pressure Data 410

Estimation of the Inverse Source Signature 410

Table of Contents xvii

Box 10.6 Extrapolation of Missing Near Traces 415

Barents Sea Example 417

Troll Example 419

Pluto 1.5 Example 421

Multiple Attenuation: OBS and VC Data 423

The Representation Theorem and the Kirchhoff Scattering Series for OBS Data 423

A Physical Interpretation of the Kirchhoff Scattering Series for OBS Data 427

Box 10.7 The Two-reflector Problem in OBS 429

An Optimization of the Kirchhoff Series for the OBS Demultiple Process 429

A Synthetic Example 430

The Demultiple Process for VC Data 435


Chapter 11 An Example of an Inverse Problem: Linearized Seismic Inversion 445

A Multiple-step Inversion Approach 445

Basic Components of an Inverse Problem 445

Nonuniqueness, Instabilities, Convergence, Uncertainties, and Cost 446

A Multiple-step Approach to the Seismic Inverse Problem 448

Key Assumptions of our Example of an Inverse Problem 449

The Born Approximation 449

Solving the forward problem, on the basis of the finite-difference technique 449

Solving the forward problem on the basis of the Bom approximation 450

Smooth-background medium 451

An Illustration of the Limitations of the Born Approximation 452

Straight-ray Approximation: Hyperbolic and Nonhyperbolic Moveouts 456

An Optimal Data Set: The Common-azimuthal-section Example 459

Box 11.1 The Born Scattering Series 463

Box 11.2 The Kirchhoff Approximation 464

An Example of a Linearized Forward Problem 465

Linearization 465

A Physical Interpretation of the Linearized Forward Problem 466

Geometric spreading 466

Traveltimes 467

Amplitude variations with angles (AVA) 467

A Numerical Illustration of Out-of-plane Scattering 468

Box 11.3 Linearized Forward Problem for P-P Scattering 469

Scattered Wavefield 469

AVA Response 471

Box 11.4 Linearized Forward Problem for P-S Scattering 472

An Example of a Linearized Inversion Problem 473

A Compact Notation for the Forward Problem 473

xviii Table of Contents

Data-fitting Approaches 473

Norms and criteria 473

Constrained least squares 478

A Derivation of the Least-squares Solution 480

A Physical Interpretation of the Least-squares Inversion 482

The Hessian Matrix and its Eigenvalues 484

Spatial Resolution 487

Box 1 1.5 Covariance Operator in thef-k Domain 489

Box 11.6 Scalar Products and Norms 490

Linearized Inversion and AVA Inversion 490

Preprocessing by AVA Inversion 491

Linear Regression 491

Migration 492

What Is Migration? 492

Poststack Migration 493

Dip-moveout (DMO) plus stack 493

Normal moveout (NMO) plus stack 495

2D prestackf-k migation plus zero-offset/-^ migration 495

Time Imaging and Depth Imaging 495

Time imaging 495

Box 11.7 f-k Migration and Stolt's Time Stretch 496

Depth imaging 497

Models for Estimating Background Velocity 500

Linking the Imaging Requirements with the Background-velocity Estimation 500

Velocity Spectrum 501

Velocity-migration Analysis 503

Velocity Building 503

Creating an initial-velocity model 503

Iterative process 503

Imaging Receiver Ghosts of Primaries 503

Box 11.8 The Eikonal Equation 507

Box 11.9 Semblance 510


Chapter 12 Anisotropy and Beyond 517

Wave Propagation through 2D Random Media 519

Description of Random Media 519

Setting up the problem 519

Elliptical correlation function 520

Other Evidences of Anisotropy 520

Seismic Coda 525

Table of Contents xix

Seismic Pulse-broadening Effect 525

Scattering Attenuation 526

Box 12.1 Backus' VTI-equivalent Medium 527

Anisotropic Symmetries 529

Isotropic Media 530

Stiffness tensor 530

Small-scale heterogeneity arrangements 530

Transversely Isotropic Media with a Vertical Symmetry Axis 530


Wave propagation in a homogeneous-VTI medium 532

Box 12.2 Quasicompressional and Quasishear Waves 533

Transversely Isotropic Media with a Horizontal Symmetry Axis 533


Wave propagation in a homogeneous-HTI medium 535

Transversely Isotropic Media with a Tilted Symmetry Axis 536

Orthorhombic Media 538


Wave propagation in a homogeneous orthorhombic medium 539

Monoclinic Media 540


Wave propagation in a homogeneous monoclinic medium 540

The Alford Rotation 544

A 2 x 2C Experiment (XX, XY, YX, and YY Experiment) 544

Mathematics of Shear-wave Rotation 545

A Numerical Illustration of the Alford Rotation 547

A Shear Sonic-log Application of the Alford Rotation 549

Box 12.3 A Brief Review of the Principles of Sonic-log Measurements 551

Phase Velocity as a Function of Elastic Moduli 553

Box 12.4 The Christoffel Equation 553

Box 12.5 Phase and Group Velocities 557

Thomsen's Parameterization for VTT 558

qP-wave Velocity 561

qS 1 - and qS2-wave Velocities 563

Box 12.6 Equation of Vertical Slowness 565

Wave Equations 565

Slowness Equations 565

Dispersion Relationships for Anisotropic Media 566

The Dispersion Relationship for qP-waves 567

The Dispersion Relationship for qS-waves 570

xx Table of Contents

Applying the Dispersion Relationship for Phase-shift Migration 571

Common-azimuthal section 571

Phase-shift migration 571

Zoeppritz's Equations for Anisotropic Media 572

Up-down Symmetry 572

Schoenberg and Protazio's Formulation 572

Box 12.7 Linearized Reflection Coefficients for VTI Half-spaces 573

Box 12.8 Vertical Wavenumbers, Polarization Vectors, and Slowness Vectors in Isotropic Media....

575

Amplitude Variations with Offsets and Azimuths (AVO-A) 577

AVO-A Derivation and Analysis for P-P Data 577

Dip and azimuthal angles 577

Decoupling of AVAZ and AVO 578

Heterogeneity versus anisotropy 581

AVO-A analysis for inversion purposes 582

AVO-A Derivation and Analysis for P-SV Data 584



Heterogeneity versus anisotropy 588


AVO-A Derivation and Analysis for P-SH Data 590




AVO-A of a Horizontally Flat Interface 593

Sensitivity of AVO-A to Properties of Fractured Rock Formations 595

Linear Anelasticity 598

Geometric Spreading and the Concept of Attenuation 598

Box 12.9 Relationships of the Phase Velocity and the Quality Factor with Complex Moduli 599

The Maxwell Model 601

The Kelvin-Voigt Model 603

The Standard Linear Solid Model 603

The Constant-Q Model 605

Box 12.10 Models of Linear Attenuation: A 3D Problem 606

Perfect Elasticity 606

Elasticity with Viscosity 606

Standard Linear Solid 606

P-wave and S-wave Drifts: An Interplay of Anisotropy and Anelasticity 606

Anisotropy Effect 608

Intrinsic Attenuation 608

Lateral Inhomogeneity Effects 609

Box 12.11 Intrinsic Attenuation and Scattering Attenuation 611


Table of Contents xxi

Appendix A Some Terminology of Petroleum Geology 615

Appendix B Velocities and Densities of Reservoir Fluids 619

Appendix C A Review of Finite-difference Modeling: Explicit Implementation 623

Basic Equations for Elastodynamic Wave Motion in Elastic Media 623

Discretization in Both Time and Space 624

Staggered-grid Implementation 624

Stability of the Staggered-grid Finite-difference Modeling 625

Grid Dispersion in Finite-difference Modeling 625

Boundary Conditions 626

3D Elastic Finite-difference Modeling 626

Box C. 1 Implicit Finite-difference Modeling 628

Appendix D Definitions of Some of the Integral Transforms Used in Petroleum Seismology 641

The Laplace Transform 641

The Mellin Transform 642

The Hartley Transform 642

The «th-order Hankel Transform 642

The Hilbert Transform 643

Analytic Function and Instantaneous Frequency 643

The Radon Transform in Petroleum Seismology 644

The Abel Transform 645

More on Abel Transform Pairs 645

On the Discrete Fast Radon Transform 646

On the Numerical Implementation of the Triangle Fourier Transform 648

Appendix E 3D-to-2D Transformation and 2D-to-3D Transformation 649

Explosive Point Source in an Acoustic or Elastic Medium 649

Plane-wave Decompositions 649

3D-to-2D Transformation 650


Point Force in an Elastic Medium 651



Appendix F A Derivation of the Linearized Forward Problem 653

Linearization 653

Green's Functions 654

Midpoint and Half-offset Coordinates 655

3D Acquisition Geometry as a Series of 2D Multioffset Profiles 656

References• • 659

Index 669

Introduction to petroleum seismology - · PDF fileTABLEOFCONTENTS Preface xix Acknowledgments...

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Transcript of Introduction to petroleum seismology - · PDF fileTABLEOFCONTENTS Preface xix Acknowledgments...