IMAGE ESTIMATION BY EXAMPLE:
Geophysical Soundings Image Construction (GEE)
by Jon Claerbout

TABLE OF CONTENTS

Introduction (ps.gz 26K) (pdf 83K)
Basic operators and adjoints (ps.gz 1713K) (pdf 664K) (src 211K)

• Programming linear operators
• FAMILIAR OPERATORS
  • Adjoint derivative
  • Transient convolution
  • Internal convolution
  • Zero padding is the transpose of truncation
  • Adjoints of products are reverse-ordered products of adjoints
  • Nearest-neighbor coordinates
  • Data-push binning
  • Linear interpolation
  • Spray and sum : scatter and gather
  • Causal and leaky integration
  • Backsolving, polynomial division and deconvolution
  • The basic low-cut filter
  • Nearest-neighbor normal moveout (NMO)
  • Coding chains and arrays
• ADJOINT DEFINED: DOT-PRODUCT TEST
  • Definition of a vector space
  • Dot-product test for validity of an adjoint
  • The word ``adjoint''
  • Matrix versus operator
  • Inverse operator
  • Automatic adjoints
  
  Model fitting by least squares (ps.gz 385K) (pdf 426K) (src 421K)
  
  • HOW TO DIVIDE NOISY SIGNALS
    • Dividing by zero smoothly
    • Damped solution
    • Formal path to the low-cut filter
  • MULTIVARIATE LEAST SQUARES
    • Inside an abstract vector
    • Normal equations
    • Differentiation by a complex vector
    • From the frequency domain to the time domain
      • Unknown filter
      • Unknown input: deconvolution with a known filter
  • KRYLOV SUBSPACE ITERATIVE METHODS
    • Sign convention
    • Method of random directions and steepest descent
    • Null space and iterative methods
    • Why steepest descent is so slow
    • Conjugate direction
    • Routine for one step of conjugate-direction descent
    • A basic solver program
    • Why Fortran 90 is much better than Fortran 77
    • Test case: solving some simultaneous equations
  • INVERSE NMO STACK
  • VESUVIUS PHASE UNWRAPPING
    • Digression: curl grad as a measure of bad data
    • Estimating the inverse gradient
    • Discontinuity in the solution
    • Fourier solution
    • Integrating time differences
  • THE WORLD OF CONJUGATE GRADIENTS
    • Physical nonlinearity
    • Statistical nonlinearity
    • Coding nonlinear fitting problems
    • Standard methods
    • Understanding CG magic and advanced methods
  • REFERENCES
  
  Empty bins and inverse interpolation (ps.gz 852K) (pdf 625K) (src 325K)
  
  • MISSING DATA IN ONE DIMENSION
    • Missing-data program
      • Matrix approach to missing data
      • Operator approach to missing data
  • WELLS NOT MATCHING THE SEISMIC MAP
  • SEARCHING THE SEA OF GALILEE
  • INVERSE LINEAR INTERPOLATION
    • Abandoned theory for matching wells and seismograms
  • PREJUDICE, BULLHEADEDNESS, AND CROSS VALIDATION
  
  The helical coordinate (ps.gz 3108K) (pdf 0K) (src 208K)
  
  • FILTERING ON A HELIX
    • Review of 1-D recursive filters
    • Multidimensional deconvolution breakthrough
    • Examples of simple 2-D recursive filters
    • Coding multidimensional de/convolution
    • Causality in two-dimensions
  • FINITE DIFFERENCES ON A HELIX
    • Matrix view of the helix
  • CAUSALITY AND SPECTAL FACTORIZATION
    • Cholesky decomposition
    • Toeplitz methods
    • Kolmogoroff spectral factorization
  • WILSON-BURG SPECTRAL FACTORIZATION
    • Wilson-Burg theory
  • HELIX LOW-CUT FILTER
  • THE MULTIDIMENSIONAL HELIX
  • SUBSCRIPTING A MULTIDIMENSIONAL HELIX
  
  Preconditioning (ps.gz 260K) (pdf 357K) (src 659K)
  
  • PRECONDITIONED DATA FITTING
    • Preconditioner with a starting guess
  • PRECONDITIONING THE REGULARIZATION
    • The second miracle of conjugate gradients
    • Importance of scaling
    • Statistical interpretation
    • The preconditioned solver
  • OPPORTUNITIES FOR SMART DIRECTIONS
  • NULL SPACE AND INTERVAL VELOCITY
    • Balancing good data with bad
    • Lateral variations
    • Blocky models
  • INVERSE LINEAR INTERPOLATION
  • EMPTY BINS AND PRECONDITIONING
    • SEABEAM: Filling the empty bins with a laplacian
    • Three codes for inverse masking
  • THEORY OF UNDERDETERMINED LEAST-SQUARES
  • SCALING THE ADJOINT
  • A FORMAL DEFINITION FOR ADJOINTS
  
  Multidimensional autoregression (ps.gz 1561K) (pdf 864K) (src 402K)
  
  • Time domain versus frequency domain
  • SOURCE WAVEFORM, MULTIPLE REFLECTIONS
  • TIME-SERIES AUTOREGRESSION
  • PREDICTION-ERROR FILTER OUTPUT IS WHITE
    • The relationship between spectrum and PEF
    • Undoing convolution in nature
    • Causal with causal inverse
    • PEF output tends to whiteness
    • Spectral estimation
    • Short windows
    • Weathered layer resonance
    • PEF whiteness proof in 1-D
    • Simple dip filters
    • PEF whiteness proof in 2-D
    • Examples of modeling and deconvolving with a 2-D PEF
    • Seismic field data examples
  • PEF ESTIMATION WITH MISSING DATA
    • Internal boundaries to multidimensional convolution
    • Finding the prediction-error filter
  • TWO-STAGE LINEAR LEAST SQUARES
    • Adding noise (Geostat)
    • Inversions with geostat
    • Infill of 3-D seismic data from a quarry blast
    • Imposing prior knowledge of symmetry
    • Hexagonal coordinates
  • BOTH MISSING DATA AND UNKNOWN FILTER
    • Objections to interpolation error
    • Packing both missing data and filter into a vector
  • LEVELED INVERSE INTERPOLATION
    • Test results for leveled inverse interpolation
    • Analysis for leveled inverse interpolation
    • Seabeam: theory to practice
    • Risky ways to do nonlinear optimization
    • The bane of PEF estimation
  • MULTIVARIATE SPECTRUM
    • What should we optimize?
    • Confusing terminology for data covariance
    • Hermeneutics
    
    Noisy data (ps.gz 1978K) (pdf 1957K) (src 6073K)
    
    • MEANS, MEDIANS, PERCENTILES AND MODES
      • Percentiles and Hoare's algorithm
      • The weighted mean
      • Weighted L.S. conjugate-direction template
      • Multivariate $^1$ estimation by iterated reweighting
      • Nonlinear L.S. conjugate-direction template
      • Minimizing the Cauchy function
    • NOISE BURSTS
      • De-spiking with median smoothing
      • De-bursting
    • MEDIAN BINNING
    • ROW NORMALIZED PEF
    • DEBURST
      • Potential seismic applications of two-stage infill
    • TWO 1-D PEFS VERSUS ONE 2-D PEF
    • ALTITUDE OF SEA SURFACE NEAR MADAGASCAR
    • ELIMINATING NOISE AND SHIP TRACKS IN GALILEE
      • Attenuation of noise bursts and glitches
      • Preconditioning for accelerated convergence
      • $^1$ norm
      • Abandoned strategy for attenuating tracks
      • Modeling data acquisition drift
      • Regridding
      
      Spatial aliasing and scale invariance (ps.gz 52K) (pdf 128K) (src 632K)
      
      • INTERPOLATION BEYOND ALIASING
        • Interlacing a filter
      • MULTISCALE, SELF-SIMILAR FITTING
        • Examples of scale-invariant filtering
        • Scale-invariance introduces more fitting equations
        • Coding the multiscale filter operator
      • References
      
      Nonstationarity: patching (ps.gz 1126K) (pdf 0K) (src 138K)
      
      • PATCHING TECHNOLOGY
        • Weighting and reconstructing
        • 2-D filtering in patches
        • Designing a separate filter for each patch
        • Triangular patches
      • STEEP-DIP DECON
        • Dip rejecting known-velocity waves
        • Tests of steep-dip decon on field data
        • Are field arrays really needed?
        • Which coefficients are really needed?
      • INVERSION AND NOISE REMOVAL
      • SIGNAL-NOISE DECOMPOSITION BY DIP
        • Signal/noise decomposition examples
        • Spitz for variable covariances
        • Noise removal on Shearer's data
        • The human eye as a dip filter
      • SPACE-VARIABLE DECONVOLUTION
      
      Plane waves in three dimensions (ps.gz 650K) (pdf 357K) (src 210K)
      
      • THE LEVELER: A VOLUME OR TWO PLANES?
        • PEFs overcome spatial aliasing of difference operators
        • My view of the world
      • WAVE INTERFERENCE AND TRACE SCALING
        • Computing the proper scale factor for a seismogram
      • LOCAL MONOPLANE ANNIHILATOR
        • Mono-plane deconvolution
        • Monoplanes in local windows
        • Crossing dips
        • Tests of 2-D LOMOPLAN on field data
      • GRADIENT ALONG THE BEDDING PLANE
        • Definition of LOMOPLAN in 3-D
        • The quarterdome 3-D synthetic (qdome)
      • 3-D SPECTRAL FACTORIZATION
      
      Some research examples (ps.gz 445K) (pdf 104K) (src 2K)
      
      • GULF OF MEXICO CUBE
      
      Entrance examination (ps.gz 25K) (pdf 118K) (src 16K)
      
      • Changes and backward compatibility
      • Examples
      • Memory allocation in subroutines
      • The main program environment
      • SERGEY'S MAIN PROGRAM DOCS
        • Autocorr - compute autocorrelation for helix filters
        • Bin2 - nearest neighbor binning in 2-D
        • Conv - convolve two helix filters
        • Decon - Deconvolution (N-dimensional)
        • Devector - create one output filter from two input filters
        • Helderiv - Helix derivative filter in 2-D
        • Helicon - Helix convolution and deconvolution (N-dimensional!)
        • Helocut - Helix Lowcut filter in 2-D
        • Hole - Punch ellipsoidal hole in 2-D data
        • Igrad - take gradient on the first axis
        • LPad - Pad and interleave traces
        • LPef - Find PEF on aliased traces
        • Lapfill2 - fill missing data by minimizing the Laplacian
        • LoLPef - Find PEF on aliased traces (with patching)
        • Lomiss - Missing data interpolation with a prescribed helix filter
        • (in local patches)
        • Lopef - Local Prediction-Error Filter (1-D, 2-D, and 3-D)
        • Losignoi - Local signal and noise separation (N-dimensional)
        • MSHelicon - Multi-scale Helix convolution (N-dimensional!)
        • MSMiss - Multiscale missing data interpolation (N-dimensional)
        • MSPef - Multi-scale PEF estimation
        • Make - generate simple 2-D synthetics with crossing plane waves
        • Minphase - create minimum-phase filters
        • Miss - Missing data interpolation with a prescribed helix filter
        • NHelicon - Non-stationary helix convolution and deconvolution
        • NPef - Estimate Non-stationary PEF in N dimensions
        • Nozero - Read (x,y,z) data triples, throw out values of z > thresh, transpose
        • Parcel - Patching illustration
        • Pef - Estimate PEF in N dimensions
        • Sigmoid - generate sigmoid reflectivity model
        • Signoi - Local signal and noise separation (N-dimensional)
        • Tentwt - Tent weight for patching
        • Vrms2int - convert RMS velocity to interval velocity
        • Wilson - Wilson's factorization for helix filters
      • References