Basic Earth Imaging (BEI)
by Jon Claerbout
TABLE OF CONTENTS

Introduction (ps.gz 7K) (pdf 6K)
Field recording geometry (ps.gz 324K) (pdf 318K) (src 1715K)

• RECORDING GEOMETRY
  • Fast ship versus slow ship
• TEXTURE
  • Texture of horizontal bedding, marine data
  • Texture of land data: near-surface problems

Adjoint operators (ps.gz 46K) (pdf 69K) (src 88K)

• FAMILIAR OPERATORS
  • Adjoint derivative
  • Zero padding is the transpose of truncation
  • Adjoints of products are reverse-ordered products of adjoints
  • Nearest-neighbor coordinates
  • Data-push binning
  • Linear interpolation
  • Causal integration
• ADJOINTS AND INVERSES
  • Dot product test

Waves in strata (ps.gz 290K) (pdf 336K) (src 576K)

• TRAVEL-TIME DEPTH
  • Vertical exaggeration
• HORIZONTALLY MOVING WAVES
  • Amplitudes
  • LMO by nearest-neighbor interpolation
  • Muting
• DIPPING WAVES
  • Rays and fronts
  • Snell waves
  • Evanescent waves
  • Solution to kinematic equations
• CURVED WAVEFRONTS
  • Root-mean-square velocity
  • Layered media
  • Nonhyperbolic curves
  • Can I abandon the material in this section?
  • Velocity increasing linearly with depth
  • Prior RMS velocity

Moveout, velocity, and stacking (ps.gz 750K) (pdf 818K) (src 169K)

• INTERPOLATION AS A MATRIX
  • Looping over input space
  • Looping over output space
  • Formal inversion
• THE NORMAL MOVEOUT MAPPING
• COMMON-MIDPOINT STACKING
  • Crossing traveltime curves
  • Ideal weighting functions for stacking
  • Gulf of Mexico stack and AGC
• VELOCITY SPECTRA
  • Velocity picking
  • Stabilizing RMS velocity

Zero-offset migration (ps.gz 531K) (pdf 524K) (src 447K)

• MIGRATION DEFINED
  • A dipping reflector
  • Dipping-reflector shifts
  • Hand migration
  • A powerful analogy
  • Limitations of the exploding-reflector concept
• HYPERBOLA PROGRAMMING
  • Tutorial Kirchhoff code
  • Fast Kirchhoff code
  • Kirchhoff artifacts
  • Sampling and aliasing
  • Kirchhoff migration of field data
• STRATIFIED V(z) FANTASY
• MIGRATION VELOCITY ANALYSIS FANTASY

Waves and Fourier sums (ps.gz 129K) (pdf 202K) (src 71K)

• FOURIER TRANSFORM
  • FT as an invertible matrix
  • The Nyquist frequency
  • Laying out a mesh
• INVERTIBLE SLOW FT PROGRAM
  • The simple FT code
• CORRELATION AND SPECTRA
  • Spectra in terms of Z-transforms
  • Two ways to compute a spectrum
  • Common signals
• SETTING UP THE FAST FOURIER TRANSFORM
  • Shifted spectrum
• SETTING UP 2-D FT
  • Basics of two-dimensional Fourier transform
  • Signs in Fourier transforms
  • Simple examples of 2-D FT
• THE HALF-ORDER DERIVATIVE WAVEFORM
  • Hankel tail
• References

Downward continuation (ps.gz 629K) (pdf 641K) (src 109K)

• MIGRATION BY DOWNWARD CONTINUATION
  • Huygens secondary point source
  • Migration derived from downward continuation
• DOWNWARD CONTINUATION
  • Continuation of a dipping plane wave.
  • Downward continuation with Fourier transform
  • Linking Snell waves to Fourier transforms
• PHASE-SHIFT MIGRATION
  • Pseudocode to working code
  • Kirchhoff versus phase-shift migration
  • Damped square root
  • Adjointness and ordinary differential equations
  • Vertical exaggeration example
  • Vertical and horizontal resolution
  • Field data migration

Dip and offset together (ps.gz 1705K) (pdf 1346K) (src 90K)

• PRESTACK MIGRATION
  • Cheops' pyramid
  • Prestack migration ellipse
  • Constant offset migration
• INTRODUCTION TO DIP
  • The response of two points
  • The dipping bed
  • Randomly dipping layers
• TROUBLE WITH DIPPING REFLECTORS
  • Gulf of Mexico example
• SHERWOOD'S DEVILISH
• ROCCA'S SMEAR OPERATOR
  • Push and pull
  • Dip moveout with v(z)
• GARDNER'S SMEAR OPERATOR
  • Restatement of ellipse equations
• DMO IN THE PROCESSING FLOW
  • Residual NMO
  • Results of our DMO program

Finite-difference migration (ps.gz 209K) (pdf 263K) (src 110K)

• THE PARABOLIC EQUATION
• SPLITTING AND SEPARATION
  • The heat-flow equation
  • Splitting
  • Full separation
  • Splitting the parabolic equation
  • Validity of the splitting and full-separation concepts
• FINITE DIFFERENCING IN (omega,x)-SPACE
  • The lens equation
  • First derivatives, explicit method
  • First derivatives, implicit method
  • Explicit heat-flow equation
  • The leapfrog method
  • The Crank-Nicolson method
  • Solving tridiagonal simultaneous equations
  • Finite-differencing in the time domain
• WAVEMOVIE PROGRAM
  • Earth surface boundary condition
  • Frames changing with time
  • Internals of the film-loop program
  • Side-boundary analysis
  • Lateral velocity variation
  • Migration in (omega,x)-space
• HIGHER ANGLE ACCURACY
  • Another way to the parabolic wave equation
  • Muir square-root expansion
  • Dispersion relations
  • The xxz derivative
• END OF CHAPTER FOR NOW
  • Retardation (frequency domain)
  • Lateral velocity variation again
  • Splitting again
  • Time domain
  • Upcoming waves
• SPLITTING AND SEPARATION APPLICATIONS
  • Application to 3-D downward continuation
  • Separability of 3-D migration (Jakubowicz)
  • Separability in shot-geophone space
• THE ACOUSTIC WAVE EQUATION
  • Reflections and the high-frequency limit

Antialiased hyperbolas (ps.gz 300K) (pdf 478K) (src 37K)

• Amplitude pitfall
• MIMICING FIELD ARRAY ANTIALIASING
  • Adjoint of data acquisition
  • NMO and stack with a rectangle footprint
  • Coding a triangle footprint
• MIGRATION WITH ANTIALIASING
  • Use of the antialiasing parameter
  • Orthogonality of crossing plane waves
• ANTIALIASED OPERATIONS ON A CMP GATHER
  • Iterative velocity transform

Imaging in shot-geophone space (ps.gz 911K) (pdf 699K) (src 15K)

• TOMOGRAPY OF REFLECTION DATA
  • The grand isle gas field: a classic bright spot
  • Kjartansson's model for lateral variation in amplitude
  • Rotten alligators
  • Focusing or absorption?
• SEISMIC RECIPROCITY IN PRINCIPLE AND IN PRACTICE
• SURVEY SINKING WITH THE DSR EQUATION
  • The survey-sinking concept
  • Survey sinking with the double-square-root equation
  • The DSR equation in shot-geophone space
  • The DSR equation in midpoint-offset space
• THE MEANING OF THE DSR EQUATION
  • Zero-dip stacking (Y = 0)
  • Giving up on the DSR

Seplib and SEP software (ps.gz 22K) (pdf 37K) (src 9K)