SEISMIC WAVE THEORY

EDWARD S . KREBES University ofCalgary

UCAMBRIDGE V UNIVERSITY PRESS

Contents

Preface page xiii List of Symbols XV

1 Vectors, Tensors, and Fourier Transforms 1 1.1 Basic Definitions 1 1.2 Directional Derivatives 7 1.3 Curl and Rotation 8 1.4 Divergence and Flux 9 1.5 Line Integrals 10 1.6 Curvilinear Coordinates 12 1.7 Cylindrical Coordinates 16 1.8 Spherical Coordinates 18 1.9 Laplacian of a Vector 19 1.10 Rotation of Coordinates 20 1.11 Invariance 26 1.12 Rotation of Vectors 28 1.13 Translation of Coordinates 28 1.14 Cartesian Tensors 29 1.15 General Tensors 33 1.16 Fourier Transforms 35 1.17 Appendix lA: General Rotation with Euler Angles 41 1.18 Appendix lB: The Convolution of Continuous Functions 42

1.18 .1 Reversing and Time Shifting of Signals 43 1.18.2 Convolution of Continuous Signals 43

Exercises 45

2 Stress, Strain, and Seismic Waves 51 2.1 Stress 51 2.2 Strain 54

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2.3 Dilatation 57 2.4 Equation of Motion 57 2.5 Conservation of Energy 60 2.6 Strain Energy Density 63 2.7 Stress-Strain Relation 63 2.8 More on Strain Energy and Energy Conservation 65 2.9 Isotropy 66 2.10 Hydrostatic Stress 67 2.11 Other Elastic Constants 69 2.12 The Equation of Motion and the Wave Equations 70 2.13 Elastic Constants in Terms of a and ß 72 2.14 The lD Wave Equation 72 2.15 Harmonie Waves 74 2.16 3D Plane Wave Solutions of the Equation of Motion 76 2.17 Displacement Potentials 79 2.18 Helrnholtz Equations 82 2.19 Near-Field and Far-Field Waves 83 2.20 Mean Values of Energy-Related Quantities 84 2.21 Inhomogeneous Media 87 2.22 Other Types of Media 87 2.23 The Acoustic Wave Equation 88 2.24 Seismic Waves Generated by a Buried Explosive Charge 88 2.25 Seismic Waves Generated by a Directed Point Force 91 2.26 Appendix 2A: Apparent Velocities 102 Exercises 104

3 Reflection and Transmission of Plane Waves 112 3.1 A Normally Incident P Wave Pulse 112 3.2 More on Polarity Reversals 118 3.3 Reflection and Transmission of Harmonie Plane Waves 120 3.4 Reflection and Transmission of Plane Waves at Nonnormal

Incidence 123 3.5 Reflection Off a Free Surface 132 3.6 Critical Angles 133 3.7 Energy Reflection and Transmission Coefficients 139 3.8 A Liquid-Liquid Interface 144 3.9 A Liquid-Solid Interface 144 3.10 A Rigid Boundary 144 3.11 Approximations 144 Exercises 146

Contents ix

4 Surface Waves, Head Waves, and Normal Modes 152 4.1 Surface Waves 152 4.2 Surface Waves for a Homogeneous Half-Space 152 4.3 Interface Waves 157 4.4 Huygens' Principle and Applications 158 4.5 Phase Velocity, Group Velocity, and Dispersion 164 4.6 Normal Modes in a Surface Layer over a Rigid Half-Space 167 4.7 Love Waves 172 4.8 The Airy Phase 178 4.9 Other Cases of Normal Modes 179 4.10 Measurement of Phase and Group Velocities 180 4.11 Air-Coupled Ground Roll 183 Exercises 184

5 Waves in Heterogeneous Media 187 5.1 Waves in a Vertically Inhomogeneous Medium 187 5.2 Applications to Well Surveys and Logs 195 5.3 The Wiechert-Herglotz Method 197 5.4 Wave Propagation in a Slowly Varying Vertically

Inhomogeneous Medium 201 5.5 Tue Eikonal Equation 205 5.6 Zero-Offset Ray Tracing 207 5.7 Seismic Diffractions 212 5.8 Diffraction Moveout on a Shot Record and a Stack Section 213 5.9 Diffraction Amplitudes and Phases 214 5.10 A Terminating Reftector 215 5.11 A Flat Continuous Reftector with a Laterally Changing

Reftection Coefficient 216 5.12 A Bent Reftector 217 5.13 Appendix 5A: Acoustic Wave Equations for Heterogeneous

Media 218 5 .13 .1 Equation for the Potential for Constant Density 218 5.13.2 Equation for the Pressure for Constant Density 219 5.13.3 Equation for the Pressure for Nonconstant Density 219 5.13.4 A Point Source 220

5.14 Appendix 5B: Eikonal, Transport, and Ray Equations 220 5.14.1 Eikonal and Transport Equations 220 5.14.2 Ray Equations 222 5.14.3 Another Form of the Ray Equations 223

Exercises 224 5.15 Formulas That May Prove Useful in Doing Exercises 2, 4, and 5 229

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6 Data Transformations 230 6.1 The Frequency Domain 230 6.2 The Frequency-Wavenumber Domain 232 6.3 The r-p Domain 235 Exercises 240

7 Synthetic Seismograms 242 7.1 The Subsurface Model and Synthetic Trace 242 7.2 Reftection Losses 244 7.3 Transmission Losses 244 7.4 Geometrical Spreading Losses 246 7.5 Absorption Losses 247 7.6 Tue General Case 251 7.7 Tue Reftectivity Function 254 7.8 Interlerence Effects 255 7.9 Appendix 7 A: Synthetic Shot Records and Wavefronts 260 Exercises 267

8 Seismic Migration 270 8.1 A Point Reftector 270 8.2 A Dipping Reflector 271 8.3 Diffraction-Summation Migration 274 8.4 Wavefront Migration 275 8.5 Wave Equation Migration 276 8.6 Frequency-Wavenumber Migration 278 8.7 Finite Difference Migration 285 8.8 Kirchoff Migration 291 8.9 Depth Migration 291 Exercises 293

9 Plane Waves in Anisotropie Media 294 9.1 Anisotropy 294 9.2 3D Plane Wave Solutions 294 9.3 Weak VTI-Anisotropy 301 9.4 Additional Comments 302 9.5 Appendix 9A: Velocities, Elliptical Anisotropy, Slowness 305

9.5.1 Velocity Formulas 305 9.5.2 Elliptical Anisotropy 306 9.5.3 Slowness Surface and Ray (Group) Velocity 306 9.5.4 Slowness Diagrams 307

Exercises 310

Contents

10 Plane Waves in Anelastic Media 10.1 Absorption of Seismic Waves 10.2 Waveforms and Dispersion 10.3 Anelasticity Models and Stress-Strain Relations 10.4 General Linear Viscoelasticity 10.5 Complex Modulus and Q 10.6 Generalized Standard Linear Solid 10.7 The lD Anelastic Equation of Motion 10.8 The 3D Anelastic Equation of Motion 10.9 Reftection and Transmission of General Plane Waves 10.10 An Alternative Form of the Stress-Strain Relation 10.11 Measuring Absorption 10.12 Exactly Constant Q Exercises

Answers to Selected Exercises References Index

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312 312 313 316 318 319 321 323 324 326 329 330 331 332

334 338 344