Imaging the Earth's Interior (IEI)
by Jon Claerbout

TABLE OF CONTENTS

Introduction (ps 46K)
Introduction to imaging (ps 573K) (src 22K)

• EXPLODING REFLECTORS
  • A powerful analogy
  • Huygens secondary point source
  • Migration defined
  • An impulse in the data
  • Hand migration
  • Reflector steepening
  • Limitations of the exploding-reflector concept
  • Plate tectonics example
• SURROUNDING DISCIPLINES
  • Physics
    • Derivation of the acoustic wave equation
    • Reflections and the high-frequency limit
    • Failure of Newtonian viscosity
  • Geophysics
    • Nonobservation of converted shear waves
    • Reliability of reverberation modeling
    • What you can get from reflection seismology
  • Mathematics
    • Philosophy of inverse problems
    • Mathematical inverse problems
  • Geology
    • The clastic section
    • Chrono-stratigraphy

Discrete Fourier transform (ps 126K) (src 18K)

• FT AS AN INVERTIBLE MATRIX
  • The Nyquist frequency
  • Laying out a mesh
• INVERTIBLE SLOW FT PROGRAM
  • The simple FT code
• SETTING UP THE FAST FOURIER TRANSFORM
  • Shifted spectrum
• TWO-DIMENSIONAL FT
  • Basics of two-dimensional Fourier transform
  • Signs in Fourier transforms
  • Examples of 2-D FT
• References

Wide-angle migration methods (ps 425K) (src 146K)

• PLANE-WAVE SUPERPOSITION
  • Migration improves horizontal resolution
  • Rays and fronts
  • Waves in Fourier space
  • The input-output relation
  • Travel-time depth
• SUMMATION AND SUPERPOSITION OF CONICS
  • Hyperbola summation refined into the Kirchhoff method
• THE PHASE-SHIFT METHOD
• THE STOLT METHOD
• INFLUENCES ON MIGRATION QUALITY
  • Spatial aliasing concept
  • Sensitivity of migration to velocity error
  • Subjective comparison and evaluation of methods
  • Evanescence and ground roll
• THE PARAXIAL WAVE EQUATION
  • Why the wave equation is rarely used for migration
  • Fourier derivation of the paraxial wave equation
  • Snell waves
  • Time-shifting equations
  • Fourier decomposition
  • Velocity gradients

Migration by finite differences (ps 197K) (src 131K)

• WAVE-EXTRAPOLATION EQUATIONS
  • Meet the parabolic wave equation
  • Muir square-root expansion
  • Dispersion relations
  • Depth-variable velocity
  • Retardation (frequency domain)
  • Lateral velocity variation
  • Two velocities
  • Splitting diffraction and lens terms
  • Time domain
  • Upcoming waves
• FINITE DIFFERENCING
  • The lens equation
  • First derivatives, explicit method
  • First derivatives, implicit method
  • The explicit heat-flow equation
  • The leapfrog method
  • The Crank-Nicolson method
  • Solving tridiagonal simultaneous equations
  • The xxz derivative
  • Difficulty in higher dimensions
• WAVEMOVIE PROGRAM
  • Earth's surface boundary condition
  • Time-domain analysis
  • Internals of the film-loop program
  • Side-boundary analysis
  • Including lateral velocity variation
  • Phase shift
• MIGRATION IN THE (omega,x)-DOMAIN (KJ)

Splitting and working in the time domain (ps 182K) (src 96K)

• SPLITTING AND FULL SEPARATION
  • The heat-flow equation
  • Splitting
  • Full separation
  • Application to lateral velocity variation
  • Application to 3-d downward continuation
  • Separability of 3-d migration (Jakubowicz)
  • Separability in shot-geophone space
  • Validity of the splitting and full-separation concepts
• RECURSIVE DIP FILTERS
  • Definition of a recursive dip filter
  • Recursive-dip-filter implementation
  • Side boundaries
  • Slicing pies
  • Higher dimensionality
• RETARDED COORDINATES
  • Definition of independent variables
  • Definition of dependent variables
  • The chain rule and the high frequency limit
  • Fourier transforms in retarded coordinates
  • Interpretation of the modulated pressure variable Q
  • Einstein's special relativity theory
• FINITE DIFFERENCING IN (t, x, z)-SPACE
  • Migration in (z, t)-space
  • (t,x,z)-Space, 15 degree Diffraction Program
  • You Can't Time Shift in the Time Domain.
  • (t,x,z)-Space, 45 degree equation
• INTRODUCTION TO STABILITY
  • Stability of the explicit heat-flow equation
  • Explicit 15 degree migration equation
  • Implicit equations
  • Leapfrog equations
  • Tridiagonal equation solver

Offset, another dimension (ps 1695K) (src 65K)

• Stacking diagrams
• What is ``poor quality'' data?
• Texture of horizontal bedding, marine data
• Texture of land data: near-surface problems
• ABSORPTION AND A LITTLE FOCUSING
  • The grand isle gas field: a classic bright spot
  • Kjartansson's model for lateral variation in amplitude
  • Rotten alligators
  • Focusing or absorption?
• INTRODUCTION TO DIP
  • Sections and gathers for planar reflectors
  • The dipping bed
  • The point response
  • Cheops' pyramid
  • Random point scatterers
  • Forward and backward scattering: Larner's streaks
  • Velocity of sideswipe
  • The migration ellipse
• SURVEY SINKING WITH THE DOUBLE-SQUARE-ROOT EQUATION
  • Seismic reciprocity in principle and in practice
  • The survey-sinking concept
  • Review of the paraxial wave equation
  • The DSR equation in shot-geophone space
  • The DSR equation in midpoint-offset space
• THE MEANING OF THE DSR EQUATION
  • Zero-offset migration (H = 0)
  • Zero-dip stacking (Y = 0)
  • Conventional processing-separable approximation
  • Various meanings of H = 0
  • Clayton's cosine corrections
  • Snell-wave stacks and CMP slant stacks
  • Why not downward continue in (S,G)-space?

Velocity and dip moveout (ps 1693K) (src 60K)

• STACKING AND VELOCITY ANALYSIS
  • Normal moveout (NMO)
  • Conventional velocity analysis
  • Mutes and weights
  • NMO equations
  • Linearity allows postponing statistical estimation
  • Lateral interpolation and extrapolation of a CMP gather
  • In and out of velocity space
  • The (z,t)-plane method
  • Decomposing a gather by velocity
  • Reflected head waves on sections
• MIGRATION WITH VELOCITY ESTIMATION
  • Dip moveout-Sherwood's devilish
  • Rocca's smear operator
  • Hale's constant-offset dip moveout
  • Ottolini's radial trace dip moveout
  • Anti-alias characteristic of dip moveout
• LATERAL VELOCITY IN BIGGER DOSES
  • Replacement velocity: freezing the water
  • Lateral shift of the hyperbola top
  • Phantom diffractor
  • Wavefront healing
  • Fault-plane reflection
  • Misuse of v(x) for depth migration
  • First-order effects, the lens term
  • The migrated time section: an industry kludge

Dispersion relations (ps 967K) (src 38K)

• The magic of color
• Survey of migration technique enhancements
• A production pitfall: weak instability from v(x)
• COSMETIC ASPECT OF WAVE EXTRAPOLATION
  • t squared
  • Noise, surface waves and clip
  • Complex velocity in the 5 degree equation
  • Absorption
  • Dispersion
  • False semicircles in migrated data
  • Zapping multiples in dip space
  • Mixed appearance of dip-filtered data
  • Accentuating faults
  • Dip filtering
  • Gain control does dip filtering too
  • Rejection by incoherence or rejection by filtering?
  • Spatial scaling before migration
  • Exponential scaling
  • The substitution operator
• ANISOTROPY DISPERSION IN MIGRATION
  • Rays not perpendicular to fronts
  • Wavefront direction and energy velocity
  • Analyzing errors of migration
  • Derivation of group velocity equation
  • Derivation of energy migration equation
  • Extrapolation equations are not frequency-dispersive.
• FREQUENCY DISPERSION IN WAVE-MIGRATION
  • Spatial aliasing various space axes
  • Second space derivatives
  • The 1/6 trick
  • Time and depth derivatives-the bilinear transform
• ACCURACY-THE CONTRACTOR'S VIEW
  • Lateral derivatives
  • Viscosity and causality
  • Retarded Muir recurrence
  • Stepping in depth
  • Lightning phase shift migration
  • Final appearances
  • Program

The craft of wavefield extrapolation (ps 321K) (src 30K)

• ABSORBING SIDES
  • Data padding
  • Truncation at cable ends and at survey ends
  • Engquist boundaries for the scalar wave equation
  • Engquist sides for the extrapolation equation
  • Size of the reflection coefficient
• TUNING UP FOURIER MIGRATIONS
  • Dips greater than 90 degrees
  • Stopping phase-shift migration wraparound (S. Levin)
  • Controlling Stolt migration wraparound
  • Stolt stretch
  • Gazdag's v(x) method
• IMPEDANCE
  • Beware of infinity!
  • Z - transform
  • Review of impedance filters
  • Causal integration
  • Muir's rules for combining impedances
  • Impedance defined from reflectance
  • Functional analysis
  • Wide-angle wave extrapolation
  • Fractional integration and constant Q
• THE BULLETPROOFING OF MUIR AND GODFREY
  • Stability of the differential equation
  • Stability of the difference equation
  • Application to 45 degree wavefield extrapolation

Slanted waves (ps 557K) (src 26K)

• Interpreting seismic data
• Leaning
• Stepout review
• RADIAL TRACES
  • Moveout-corrected radial traces
  • Snell traces
• SLANT STACK
  • Slant stacking and linear moveout
  • Slant-stack gathers are ellipses
  • Two-layer model
  • Interval velocities from slant stacks
  • Interface velocity from head waves
  • Slant stack and Fourier transform
  • Inverse slant stack
  • Plane-wave superposition
  • Reflection coefficients-spherical versus planar
  • The rho filter
• SNELL WAVES AND SKEWED COORDINATES
  • Snell wave information in field data
  • Muting and data recording
  • Synthesizing the Snell wave experiment
  • What's wrong with Snell waves?
  • Lateral invariance
  • Snell wave coordinates
  • Snell Waves in Fourier space
• INTERVAL VELOCITY BY LINEAR MOVEOUT
  • Graphical method for interval velocity measurement
  • Common-midpoint Snell coordinates
  • Differential equations and Fourier transforms
  • Processing possibilities

Multiple reflections (ps 1491K) (src 28K)

• MULTIPLE REFLECTION, CURRENT PRACTICE
  • Hard sea floor example
  • Deconvolution in routine data processing
  • A vertical seismic profile (VSP)
  • Deep marine multiples, in the polar latitudes
  • Pegleg and intrabed multiple reflections
  • The need to distinguish between types of sections
  • Examples of shallow-water multiples with focusing
  • Why deconvolution fails in deep water
• MULTIPLE REFLECTION-PROSPECTS
  • Transformation to one dimension by slant stack
  • Near-surface inhomogeneity
  • Modeling regimes
  • Subtractive removal of multiple reflections
  • Slanted deconvolution and inversion
  • The split Backus filter
  • Sea-floor consistent multiple suppression
  • Replacement-medium concept of multiple suppression
• PROFILE IMAGING
  • The U/D imaging concept
  • Migration with moveout correction
  • Moveout/radial coordinates in geophone space
  • Historical notes on a mysterious scale factor
• PREDICTIONS FOR THE NEXT DECADE
  • Problems in the database
  • Reuniting optimization theory and wave theory
  • Throw away your paper sections.

Seplib and SEP software (ps 74K) (src 8K)