Basic Theory in Reflection Seismology, Volume 1: (Handbook of Geophysical Exploration: Seismic Exploration)

Cover......Page 1
Contents......Page 5
List of Figures......Page 9
List of Tables......Page 17
Introduction......Page 19
Acknowledgments......Page 27
Complex Numbers......Page 28
Manipulation of Complex Numbers......Page 29
Real and Complex Exponentials and Trigonometric Functions......Page 32
Powers and Roots of a Complex Number......Page 34
Logarithm of a Complex Number......Page 35
Functions of a Complex Variable......Page 37
Representation of Signals by Phasors......Page 38
Linear Equations......Page 39
Introduction......Page 43
Signal Nomenclature......Page 45
FT—Continuous Time, Continuous Frequency......Page 46
DTFT—Discrete Time, Continuous Frequency......Page 47
DFT—Discrete Time, Discrete Frequency......Page 49
Subroutine FT......Page 51
The Fourier Coefficients......Page 52
Sign Convention......Page 56
Fourier Coefficients from Linear Equations......Page 59
Numerical Example......Page 60
The Average Value of a Function......Page 61
Useful Integrals......Page 62
The Average Values of Sine and Cosine Functions, and Products of Sine and Cosine Functions over a Period......Page 65
Dirichlet Conditions......Page 68
Summary......Page 69
Examples......Page 76
Sign Convention Revisited......Page 78
Seismogram from the Atlantic Coastal Plain......Page 79
Independence of the Fourier coefficients......Page 81
From Fourier Series to Fourier Integrals......Page 87
Complex Forms and Fourier Integral......Page 89
Computer Implementation of the Fourier Series and Fourier Integral......Page 91
Fast Fourier Transform......Page 93
Time-Shifting Theorem......Page 97
Implications......Page 99
Time Differentiation of the Fourier Transform......Page 101
Time Integration of the Fourier Transform......Page 102
Introduction to the Unit-Impulse Function......Page 106
The Sinc Function......Page 111
Impulse Response......Page 113
The Time-Convolution Theorem......Page 115
An Application of the Time-Convolution Theorem......Page 120
Autocorrelation and Crosscorrelation......Page 124
The Frequency-Convolution Theorem......Page 127
The Effect of the Analysis Window Revisited......Page 129
Hilbert Transforms......Page 133
Summary......Page 141
Hilbert Transform of a Sinusoid......Page 142
Fourier Sign Convention......Page 144
The Analytic Signal......Page 145
Mathematical Definition of Hilbert Transformation in the Time and Frequency Domains. The "Quadrature Function."......Page 146
Hilbert Transform of a Seismic Trace......Page 155
Mathematical Derivation of a Hilbert Transform Pair in the Frequency Domain using Continuous Functions......Page 160
z-Transform......Page 167
Factors of a Finite, Discrete Function......Page 169
Phase of Minimum and Maximum Delay Couplets......Page 172
Amplitude and Phase of a z-Transform......Page 173
Introduction to Filters......Page 179
Numerical Example......Page 181
Introduction to the Pole-Zero Design of Digital Filters......Page 182
A Notch Filter Element......Page 183
A Bandpass Filter......Page 184
Example of a Low-Pass Filter—Butterworth......Page 185
Effect of Analysis Window on Fourier Spectrum......Page 189
Aliasing......Page 192
Sampling in the Time Domain - Aliasing in the Frequency Domain......Page 193
Example......Page 194
How to Determine if Aliasing is Present......Page 197
Normal-Incidence Reflection Coefficient......Page 199
Example......Page 203
Values of Reflection Coefficients......Page 205
The Zoeppritz Equations......Page 208
Physical Significance of a Complex Reflection Coefficient......Page 211
AVO and Zoeppritz Equations in T-X Domain......Page 212
Conversion of the Zoeppritz Equations to the Time-Offset Domain......Page 213
Synthetic Seismograms......Page 218
The Reflectivity Function......Page 222
Velocity Functions......Page 223
Step......Page 225
Ramp......Page 227
Wavelet Tuning......Page 235
Summary of Velocity Functions......Page 242
Seismic Trace Attributes......Page 246
Traveltime Curves and Velocity......Page 257
Snell's Law......Page 258
The Ray Parameter p......Page 261
Reflection Traveltime Curves......Page 262
Average Velocity......Page 267
Traveltime Equations using Lagrangian Multipliers......Page 268
The Root-Mean-Square Velocity......Page 270
Determination of Interval Velocities using Dix Equation......Page 275
Effect of Dip on Reflection Traveltime Curves......Page 290
The Normal Moveout Correction......Page 294
Refraction Traveltime Curves......Page 297
Refractions from a Single Horizontal Interface......Page 300
Delay Time......Page 302
Two-Layer Model......Page 306
General Expression for Multilayer Refraction Traveltime Curves from Horizontal Layers......Page 307
Effect of Dip on Refraction Traveltime Curves......Page 309
The Principle of Reciprocity......Page 315
Refraction Traveltime Curves over Various Geologic Models......Page 316
Dipping Plane Interfaces......Page 321
Linear Increase in Velocity with Depth......Page 326
Composite Refraction-Reflection Stacks......Page 329
Refraction Stack......Page 334
Composite Stack Sections......Page 337
Seismic Source Wavelets......Page 342
Vibroseis......Page 346
Marine......Page 356
Mathematical Descriptions of Wavelets......Page 358
Klauder Wavelet......Page 359
Comments......Page 361
Wavelet z—Transform Representation......Page 362
Physical Requirements for Real Wavelets......Page 363
A Simple 2-Point Wavelet......Page 365
Generation of Wavelets......Page 367
Partial Energy of a Wavelet......Page 370
Roots Plotted in the z-Plane......Page 371
Root on the Unit Circle......Page 372
Wavelet Shaping and Deconvolution......Page 374
Inverse Infinite Filters, Finite Input......Page 377
Inverse Filtering of a Minimum-Delay 2-Term Wavelet using z-Transforms......Page 383
Examples......Page 387
Inverse Filtering of a Maximum-Delay 2-Term Wavelet using z-Transforms......Page 388
Examples......Page 392
Inverse Filtering of a Seismic Trace using z-Transforms......Page 396
Exact Filters for Wave Guides......Page 398
z-Transform Notation......Page 406
Design of General Inverse Filters using z-Transforms and Partial Fractions......Page 415
Inverse Filters and Input each of Finite Length......Page 422
General Shaping and Least-Squares Method......Page 424
Wavelet Shaping......Page 429
Figure 6-2 (a) from Robinson and Treitel......Page 432
Figure 6-2 (b) from Robinson and Treitel......Page 438
Figure 6-2 (c) from Robinson and Treitel......Page 439
Figure 6-2 (d) from Robinson and Treitel......Page 441
Maximum-Delay Examples......Page 442
Shaping Mixed-Delay Wavelets......Page 445
Conclusions......Page 449
What about Fourier Theory?......Page 452
Inverse Filtering of a Vibroseis Wavelet......Page 455
Summary......Page 457
Predictive Deconvolution......Page 458
Design of the Predictive Deconvolution Filter......Page 465
Non-Spiking Deconvolution......Page 483
What does Predictive Deconvolution do?......Page 484
Predictive Deconvolution and Mixed-Delay Wavelets......Page 488
Deconvolution of a Seismic Trace......Page 489
Effect of Design Window on Deconvolution......Page 492
What are the Observable Effects of Wavelet Truncation?......Page 493
An Alternative to Spiking Deconvolution?......Page 495
Reverberations......Page 497
Long-Period Multiples......Page 499
Summary Guidelines for Predictive Deconvolution......Page 501
Predictive Deconvolution—Conclusion......Page 503
Spectral Whitening......Page 511
Stretched Automatic Gain Control......Page 513
Discussion......Page 516
Relationship between the Amplitude and Phase Spectrum of a Causal Function......Page 518
Q......Page 522
Introduction......Page 523
Dispersion......Page 525
Comparison of Dispersive Phase Velocity Values of Ecevitoglu and Costain [68] with Futterman [72]......Page 537
Absorption......Page 544
Normalized Dispersion D......Page 546
Comparison of Dispersion D Values with Real Data......Page 547
A Time-Domain Method for Determination of Q......Page 549
Summary......Page 551
References......Page 553
Index......Page 568