GEE -- TABLE OF CONTENTS

Introduction (ps.gz 23K) (pdf 30K)
Basic operators and adjoints (ps.gz 1705K) (pdf 1579K) (src 206K)

• 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)
• 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 381K) (pdf 438K) (src 143K)

• HOW TO DIVIDE NOISY SIGNALS
  • Dividing by zero smoothly
  • Damped solution
  • Formal path to the low-cut filter
  • How much damping?
• 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
  • Setting up a generic 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
• 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 848K) (pdf 855K) (src 296K)

• MISSING DATA IN ONE DIMENSION
  • Missing-data program
    • Matrix approach to missing data
    • Operator approach to missing data
• WELLS NOT MATCHING THE SEISMIC MAP
  • Abandoned theory for matching wells and seismograms
• SEARCHING THE SEA OF GALILEE
• INVERSE LINEAR INTERPOLATION
• PREJUDICE, THE CHOICE OF EPSILON, AND CROSS VALIDATION

The helical coordinate (ps.gz 2557K) (pdf 2295K) (src 179K)

• 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
• SPECTRAL 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 211K) (pdf 311K) (src 154K)

• 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
  • Bigsolver
  • Lateral variations
  • Blocky models
• INVERSE LINEAR INTERPOLATION
• EMPTY BINS AND PRECONDITIONING
  • Inverse masking code
• SEABEAM: FILLING THE EMPTY BINS WITH A LAPLACIAN
• UNDERDETERMINED LEAST-SQUARES
• SCALING THE ADJOINT
• ACKNOWLEDGEMENTS

Multidimensional autoregression (ps.gz 1161K) (pdf 1162K) (src 297K)

• 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
• PEF ESTIMATION WITH MISSING DATA
  • Internal boundaries to multidimensional convolution
  • Finding the prediction-error filter
• TWO-STAGE LINEAR LEAST SQUARES
  • Infill of 3-D seismic data from a quarry blast
  • Imposing prior knowledge of symmetry
  • Hexagonal coordinates
• SEABEAM: FILLING THE EMPTY BINS WITH A PEF
• GEOSTATISTICS
• 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
• MULTIVARIATE SPECTRUM
  • What should we optimize?
  • Confusing terminology for data covariance
  • Hermeneutics

Noisy data (ps.gz 579K) (pdf 800K) (src 335K)

• MEANS, MEDIANS, PERCENTILES AND MODES
  • Percentiles and Hoare's algorithm
  • The weighted mean
  • Weighted L.S. conjugate-direction template
  • Multivariate L₁ 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
• ELIMINATING SHIP TRACKS IN GALILEE
  • Using a PEF on the Galilee residual
  • PEFs on both model space and residual space
  • Regridding
  • Treasure hunting at Galilee

Spatial aliasing and scale invariance (ps.gz 52K) (pdf 78K) (src 83K)

• 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 1121K) (pdf 1089K) (src 116K)

• 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
  • Noise removal on Shearer's data
  • The human eye as a dip filter
• SPACE-VARIABLE DECONVOLUTION

Plane waves in three dimensions (ps.gz 648K) (pdf 616K) (src 208K)

• 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 897K) (pdf 783K) (src 2K)

• SOLAR PASSIVE SEISMOLOGY
• GULF OF MEXICO CUBE

Entrance examination (ps.gz 25K) (pdf 46K) (src 13K)

• 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