GEM -- TABLE OF CONTENTS

Introduction (ps 0K)
Basic operators and adjoints (ps 1755K) (src 254K)

• 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
  • Polarity-preserving 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

Model fitting by least squares (ps 441K) (src 189K)

• 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
• ITERATIVE METHODS
  • Sign convention
  • Method of random directions and steepest descent
  • Null space and gradient methods
  • Why steepest descent is so slow
  • Conjugate direction
  • Routine for one step of conjugate-direction descent
  • Setting up a generic solver program
  • 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 307K) (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
  • Abandoned theory for matching wells and seismograms
• SEARCHING THE SEA OF GALILEE
• INVERSE LINEAR INTERPOLATION

The helical coordinate (ps 1775K) (src 148K)

• FILTERING ON A HELIX
  • Review of 1-D recursive filters
  • Examples of simple 2-D recursive filters
  • Coding multidimensional de/convolution
  • Causality in two-dimensions
• FACTORING THE LAPLACIAN ON A HELIX
• SPECTRAL FACTORIZATION CODE AND THEORY
  • The exponential of a causal is causal.
  • Finding a causal wavelet from a prescribed spectrum
  • Why the causal wavelet can be a divisor
• WILSON'S SPECTRAL FACTORIZATION
• HELIX LOW-CUT FILTER

Preconditioning (ps 481K) (src 134K)

• PRECONDITIONED DATA FITTING
  • Preconditioner conflicts with starting guess
• PRECONDITIONING THE REGULARIZATION
  • The second miracle of conjugate gradients
  • Importance of scaling
  • Statistical interpretation
• PRECONDITIONING AND INTERVAL VELOCITY
• NULL SPACE AND INTERVAL VELOCITY
  • Balancing good data with bad
  • Blocky models
  • Lateral variations
• INVERSE LINEAR INTERPOLATION
• EMPTY BINS AND PRECONDITIONING
• VIRTUAL-RESIDUAL PRECONDITIONING
  • The classic underdetermined least-squares problem
  • The virtual-residual preconditioning algorithm
• SCALING THE ADJOINT
• ACKNOWLEDGEMENTS

Multidimensional autoregression (ps 313K) (src 269K)

• 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
  • Simple dip filters
  • PEF whiteness proof in 1-D
  • PEF whiteness proof in 2-D
  • The shape of the 3-D prediction-error filter
  • Infill of 3-D seismic data from a quarry blast
• WEIGHTED PEF ESTIMATION
• TWO-STAGE LINEAR LEAST SQUARES
• OPTIMUM FILLING OF EMPTY BINS
  • Two-dimensional convolutions, transient and internal
  • Finding the prediction-error filter
  • Imposing prior knowledge of symmetry
• SEABEAM: FILLING THE EMPTY BINS
  • Using linear interpolation instead of binning
• 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
  • Risky ways to do nonlinear optimization
• MULTIVARIATE SPECTRUM
  • What should we optimize?
  • Confusing terminology for data covariance
• TIME-VARIABLE DECONVOLUTION

Nasty noise and robust estimation (ps 163K) (src 90K)

• 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

Noisy data (ps 581K) (src 179K)

• ROW NORMALIZED PEF
• DEBURST
  • Potential seismic applications of two-stage infill
• TWO 1-D PEFS VERSUS ONE 2-D PEF
• SPARSE TRACKS IN SATELLITE ALTIMETRY
  • A deeper model than anisotropy
  • Distribution of sparsity
  • Seasat modeling and solution formulation
  • Along-track noise and crooked tracks
  • Seasat optimization
• 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 87K) (src 63K)

• 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 1385K) (src 85K)

• 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?
• SIGNAL ENHANCEMENT BY PREDICTION
  • Parameters for signal enhancement by prediction
• INVERSION AND NOISE REMOVAL
• SIGNAL-NOISE DECOMPOSITION BY DIP
  • 2-D examples of multidip filtering
  • The human eye as a dip filter
  • Theory for noise along with missing data

Plane waves in three dimensions (ps 704K) (src 62K)

• 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)

Entrance examination (ps 58K) (src 11K)

• Changes and backward compatibility
• Examples
• Memory allocation in subroutines
• The main program environment
• References