# Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Series | Geophysical References Series |
---|---|

Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |

Author | Enders A. Robinson and Sven Treitel |

DOI | http://dx.doi.org/10.1190/1.9781560801610 |

ISBN | 9781560801481 |

Store | SEG Online Store |

*Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing* (SEG Geophysical References Series No. 15) covers the basic ideas and methods used in seismic processing, concentrating on the fundamentals of seismic imaging and deconvolution. Most chapters are followed by problem sets. Some exercises supplement textual material; others are meant to stimulate classroom discussions. Text and exercises deal mostly with simple examples that can be solved with nothing more than pencil and paper.

The book covers wave motion; digital imaging; digital filtering; various visualization aspects of the seismic reflection method; sampling theory; the frequency spectrum; synthetic seismograms; wavelets and wavelet processing; deconvolution; the need for continuing interaction between the seismic interpreter and the computer; seismic attributes; phase rotation; and seismic attenuation. The last of the 15 chapters provide a detailed mathematical overview. *Digital Imaging and Deconvolution*, nominated for the Association of Earth Science Editors award for best geoscience publication of 2008–2009, will interest professional geophysicists, graduate students, and upper-level undergraduates in geophysics. The book also will be helpful to scientists and engineers in other disciplines who use digital signal processing to analyze and image wave-motion data in remote-detection applications. The methods described are important in optical imaging, video imaging, medical and biological imaging, acoustical analysis, radar, and sonar.

## Contents

- 1 Chapter 1: Wave motion
- 2 Chapter 2: Digital imaging
- 3 Chapter 3: Visualization
- 4 Chapter 4: Sampling
- 5 Chapter 5: Filtering
- 6 Chapter 6: Frequency
- 7 Chapter 7: Wavelets
- 8 Chapter 8: Synthetics
- 9 Chapter 9: Wavelet processing
- 10 Chapter 10: Deconvolution
- 11 Chapter 11: Fine points
- 12 Chapter 12: Attributes
- 13 Chapter 13: Phase
- 14 Chapter 14: Absorption
- 15 Chapter 15: Input-output models

## Chapter 1: Wave motion

- Introduction
- Wavefronts and raypaths
- d’Alembert’s solution
- One-dimensional waves
- Sinusoidal waves
- Phase velocity
- Wave pulses
- Geometric seismology
- The speed of light
- Huygens’ principle
- Reflection and refraction
- Ray theory
- Fermat’s principle
- Fermat’s principle and reflection and refraction
- Diffraction
- Analogy
- Appendix A: Exercise

## Chapter 2: Digital imaging

- Reflection seismology
- Digital processing
- Signal enhancement
- Migration
- Interpretation
- Rays
- The unit tangent vector
- Traveltime
- The gradient
- The directional derivative
- The principle of least time
- The eikonal equation
- Snell’s law
- Ray equation
- Ray equation for velocity linear with depth
- Raypath for velocity linear with depth
- Traveltime for velocity linear with depth
- Point of maximum depth
- Wavefront for velocity linear with depth
- Two orthogonal sets of circles
- Migration in the case of constant velocity
- Implementation of migration
- Appendix B: Exercises

## Chapter 3: Visualization

- The seismic reflection method
- Seismic interpretation
- Porosity
- Absorption loss and transmission loss
- The wave equation
- Wave velocity
- Velocity analysis
- Seismic tomography
- Coherence
- Appendix C: Exercises

## Chapter 4: Sampling

- Time series
- The wavelet
- Digitization
- Frequency
- Sinusoidal motion
- Aliasing
- The Nyquist frequency
- Sampling geophysical data
- Appendix D: Exercises

## Chapter 5: Filtering

## Chapter 6: Frequency

- Frequency spectrum
- Magnitude spectrum and phase spectrum
- Fourier transform
- Minimum-phase spectrum
- Inverse Fourier transform
- Appendix F: Exercises

## Chapter 7: Wavelets

- Wavelets
- Fourier transform
- Z-transform
- Delay: Minimum, mixed, and maximum
- Two-length wavelets
- Illustrations of spectra
- Delay in general
- Energy
- Autocorrelation
- Canonical representation
- Zero-phase wavelets
- Symmetric wavelets
- Ricker wavelet
- Appendix G: Exercises

## Chapter 8: Synthetics

- Introduction
- Polarity
- Reflection coefficients and transmission coefficients
- Ghost reflection
- Layer-cake model
- Synthetic seismogram without multiples
- Water reverberations
- Synthetic seismogram with multiples
- Examples
- Small and white reflection coefficients
- Appendix H: Exercises

## Chapter 9: Wavelet processing

- Wavelets
- The shaping filter
- Spiking filter
- White convolutional model
- Wavelet processing
- All-pass filter
- Convolutional model
- Nonminimum-delay wavelet
- Signature deconvolution
- Vibroseis
- Dual-sensor wavelet estimation
- Deconvolution: Einstein or predictive?
- Summary
- Appendix I: Exercises

## Chapter 10: Deconvolution

- Model used for deconvolution
- Least-squares prediction and smoothing
- The prediction-error filter
- Spiking deconvolution
- Gap deconvolution
- Tail shaping and head shaping
- Seismic deconvolution
- Piecemeal convolutional model
- Time-varying convolutional model
- Random-reflection-coefficient model
- Implementing deconvolution
- Canonical representation
- Appendix J: Exercises

## Chapter 11: Fine points

- Prediction-error filters
- Water reverberations
- Gap deconvolution of a mixed-delay wavelet
- Prewhitening
- Prediction distance
- Model-driven predictive deconvolution
- Convolutional model in the frequency domain
- Time-variant spectral whitening
- Model-based deconvolution
- Surface-consistent deconvolution
- Interactive earth-digital processing
- Appendix K: Exercises

## Chapter 12: Attributes

- Interpretive processing
- Seismic attributes
- Instantaneous attributes
- Seismic sequence attribute map (SSAM)
- Coherence cube (C3)
- SSAM and C3
- Appendix L: Design of Hilbert transforms
- Appendix M: Exercises

## Chapter 13: Phase

- Phase rotation
- Alternate approaches to phase rotation
- Band-limiting root approximation
- Appendix N: The energy-delay theorem

## Chapter 14: Absorption

## Chapter 15: Input-output models

- Introduction
- Digital linear time-invariant systems
- Analog linear time-invariant systems
- Digital transfer functions
- Analog transfer functions
- Causality and stability of digital systems
- Causality and stability of analog systems
- Frequency response of a digital system
- Digital prediction
- Digital prediction error
- Analog prediction error