Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitcl Geophysical References Series No. 15 David V. Fitterman, managing editor Laurence R. Lines, volume editor Society of Exploration Geophysicists The international society of applied geophysics. Tulsa, Oklahoma, U.S.A
Contents Some Perspectives on Svenders xi About the Authors : xiv Preface xvii Acknowledgments : xxi Chapter 1: Wave Motion. 1 Introduction : 1 Wavefronts and raypaths 4 d'alembert's solution 4 One-dimensional waves 5 Sinusoidal waves 7 Phase Velocity,-.'.. 11 Wave pulses 13 Geometric seismology 13 The speed of light 16 Huygens' principle 17 Reflection and refraction 20 Ray theory 23 Fermat's principle 24 Fermat's principle and reflection and refraction 28 Diffraction 30 Analogy 34 Appendix A: Exercise 36 References 37 Chapter 2: Digital Imaging 39 Reflection seismology 39 -Digital processing.' 42 Signal enhancement 43 Migration 45 Interpretation 48 Rays 49 JThe unit tangent vector 50' Traveltime 51
vi Contents The gradient '. 52 The directional derivative...'., 52 The principle of least time.' 54 The eikonal equation 57 Snell's law." 59 Ray equation 60 Ray equation for velocity linear with depth 62 Raypath for velocity linear with depth ' 64 Traveltime for velocity linear with depth 66 Point of maximum depth 67 Wavefront for velocity linear with depth i, 68 Two orthogonal sets of circles, 69 Migration in the case of constant velocity 72 Implementation of migration 73 Appendix B: Exercises...., 76 References...: 77 Chapter 3: Visualization 79 The seismic reflection method 79 Seismic interpretation 80 Porosity : 83 Absorption loss and transmission loss 84 The wave equation 85 Wave velocity ; 89 Velocity analysis 90 Seismic tomography 91 Coherence 93 Appendix C: Exercises 96 References 97 Chapter 4: Sampling 99 Time series 99 The wavelet 100 Digitization 101 Frequency 103..Sinusoidal motion : 103 Aliasing ' 105 The Nyquist frequency 107 Sampling geophysical data 110 Appendix D: Exercises 113 References 114 Chapter 5: Filtering 117 Digital filtering 117 Convolution, 126
Contents vii Feedback filters 131 Inverse filters 133 Minimum delay '. 136 Appendix E: Exercises 5 139 References 140 Chapter 6: Frequency 143 Frequency spectrum 143 Magnitude spectrum and phase spectrum 144 Fourier transform 147 Minimum-phase spectrum 150 Inverse Fourier transform 152 Appendix F: Exercises 154 References 156 Chapter 7: Wavelets 159 Wavelets 159 FourieTtrahsform 161 Z-transform 163 Delay: Minimum, mixed, and maximum 166 Two-length wavelets 166 Illustrations of spectra 169 Delay in general. 172 Energy 176 Autocorrelation " 178 Canonical representation 182 Zero-phase wavelets 182 Symmetric wavelets 183 Ricker wavelet 185 Appendix G: Exercises 187 References :..' 191 Chapter 8: Synthetics 193 Introduction 193 -Polarity 194 Reflection coefficients and transmission coefficients 195 Ghost reflection 197 Layer-cake model 200 Synthetic seismogram without multiples 202,.. Water reverberations 205 Synthetic seismogram with multiples 207 Examples 210 Small and white reflection coefficients 211 Appendix H: Exercises 215 References 215
viii Contents Chapter 9: Wavelet Processing. 217 Wavelets, 217 The shaping filter 220 Spiking filter...'. 223 White convolutional model 225 Wavelet processing 226 All-pass filter 234 Convolutional model 235 Nonminimum-delay wavelet 237 Signature deconvolution 239 Vibroseis \. 242 Dual-sensor wavelet estimation A 243 Deconvolution: Einstein or predictive? 247 Summary : 248 Appendix I: Exercises f. 249 References, 250 Chapter 10: Deconvolution 253 Model used for deconvolution.... : 253 Least-squares prediction and smoothing 255 The prediction-error filter 258 Spiking deconvolution 260 Gap deconvolution 262 Tail shaping and head shaping 263 Seismic deconvolution 267 Piecemeal convolutional model 268 Time-varying convolutional model 269 Random-reflection-coefficient model 272 Implementing deconvolution 273 Canonical representation 275 Appendix J: Exercises 277 References ; 286 Chapter 11: Fine Points 289 Prediction-error filters -...:..,...-.;..-.-..--.:...-.-.-.- :-289 Water reverberations 296 Gap deconvolution of a mixed-delay wavelet 298 Prewhitening 301 Prediction distance...; 303 Model-driven predictive deconvolution 305 Convolutional model in the frequency domain 308 Time-variant spectral whitening 310 Model-based deconvolution 311 Surface-consistent deconvolution 312 Interactive earth-digital processing 314
Contents ix Appendix K: Exercises 318 References 327 Chapter 12: Attributes 329 Interpretive processing 329 Seismic attributes 330 Instantaneous attributes 332. Seismic sequence attribute map (SSAM).. 337 Coherence cube (C3) 338 SSAM and C3 340 Appendix L: Design of Hilbert transforms. 342 Appendix M: Exercises 345 References 347 Chapter 13: Phase 349 Phase rotation 349 Alternate approaches to phase rotation 355 Band-limiting root approximation 357 Appendix N: The energy-delay theorem 359 References 361 Chapter 14: Absorption 363 Attenuation ;... 363 Useful attenuation mechanisms 365 Practical considerations 370 Appendix O: Exercises 371 References : 373 Chapter 15: Input-output Models 375 Introduction 375 Digital linear time-invariant systems 376 Analog linear time-invariant systems 380 Digital transfer functions 381 Analog transfer functions 384 Causality and stability of digital systems 386 Causality and stability of analog systems 390 Frequency response of a digital system 391 Digital prediction 399 Digital prediction error 406 Analog prediction error 412 References.' 416 Index i 417