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# Basic Wave Analysis

Written in three parts, Basic Wave Analysis provides the information required for understanding the fundamental aspects of the elaborate computer processing schemes prevalent in exploration geophysics. Part 1 addresses velocity analysis. The correct determination of velocity is the most important problem in seismic exploration, and an understanding of velocity analysis is a valuable asset for a geophysicist. Part 2 discusses raypath analysis. Raypaths provide a geometrical picture of how waves travel, so that a person can visualize raypaths in their imagination. Geometrical pictures are as important in seismology as they are in optics. Part 3 addresses wavefront analysis. A person cannot easily visualize traveling wavefronts in their imagination; however, a computer can follow their motion and give the geophysicist the final outcome. Knowledge of wavefront analysis helps a geophysicist understand many modern computer methods.

Alongside contemporary technical information, however, this book also serves to remind the readers of our pioneering ancestors of scientific research. Too often, the study of science pays minimal attention to the historical forbearers to whom we owe much. On the contrary, the stories of these important figures provide fascinating insight into the development of ideas which underpin our work today. Basic Wave Analysis was copublished with EAGE.

Preface

Introduction

Part 1: Velocity Analysis

Chapter 1: Survey of Seismic Imaging

Exploration geophysics
Earthquake seismology
Traveling waves
Physical seismology and geometrical seismology
Tomography
Depth points
Prestack reverse-time depth migration
Iterative improvement method
Full waveform inversion

Chapter 2: Time, Distance, and Velocity
Time-distance curve
Alternative expression for the time-distance curve
Single horizontal interface
Critical angle
Interval velocity and average velocity
Oblique and conventional root-mean-square velocity
Snell’s parameter
Linear approximation

Chapter 3: Velocity Estimation
Four important corrections
Multiple coverage
Common midpoint
Coherency
Stacking (NMO) velocity
Dix formula for interval velocity
Approximation of stacking velocity

Part 2: Raypath Analysis

Chapter 4: Traveling Waves
Coordinates
Curvature
Wave motion
Sinusoidal waves

Chapter 5: Basic Properties of Waves
Rays and wavefronts
Velocity
Wave equation
Telegrapher’s equations
Downgoing waves and upgoing waves

Chapter 6: Eikonal Equation and Ray Equation
Principle of least time
Eikonal equation
Ray equation
Diving waves
Fresnel zone
Fermat’s derivation of Snell’s law
Huygens’ derivation of Snell’s law
Comparison of Fermat and Huygens

Chapter 7: Ray Tracing
Classical ray tracing
Determination of wavefront curvature
Propagation and depropagation
Normal moveout velocity
Interval velocities
Verification

Part 3: Waveform Analysis

Chapter 8: Three Prototype Waves
Pioneers of wave study and the invention of radio
Early days of seismic exploration
Seismic waves
Plane waves
Cylindrical waves
Spherical waves

Chapter 9: Singularity Functions
Dirac delta function and Heaviside step function
Fourier transform
Fourier transforms of delta and step functions
Fourier transform of a derivative

Chapter 10: Waves Traveling in Opposite Directions
To see a star
Interference
Matter waves
Diffraction
Seismic convolutional model
Huygens’ principle
One-dimensional wave equation
Initial value problem for the wave equation
Two-dimensional wave equation
Three-dimensional wave equation

Chapter 11: Green’s Function for the Wave Equation
Wavefront cones
Three spatial dimensions
One spatial dimension
Two spatial dimensions
Green’s function for the wave equation

Chapter 12: Initial and Boundary Conditions
The convolution integral
Source signature
Surface convolution
Wave equation with initial conditions
Wave propagation
Conclusion

References

Index

### References

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