Image Estimation by Example – Geophysical Soundings Image Construction
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TOC
1 Basic operators and adjoints
1.0.1 Programming linear operators
1.1 FAMILIAR OPERATORS
1.1.1 Adjoint derivative
1.1.2 Transient convolution
1.1.3 Internal convolution
1.1.4 Zero padding is the transpose of truncation
1.1.5 Adjoints of products are reverse-ordered products of adjoints
1.1.6 Nearest-neighbor coordinates
1.1.7 Data-push binning
1.1.8 Linear interpolation
1.1.9 Spray and sum : scatter and gather
1.1.10 Causal and leaky integration
1.1.11 Backsolving, polynomial division and deconvolution
1.1.12 The basic low-cut filter
1.1.13 Nearest-neighbor normal moveout (NMO)
1.1.14 Coding chains and arrays
1.2 ADJOINT DEFINED: DOT-PRODUCT TEST
2 Model fitting by least squares
2.1 HOW TO DIVIDE NOISY SIGNALS
2.1.1 Dividing by zero smoothly
2.1.2 Damped solution
2.1.3 Smoothing the denominator spectrum
2.1.4 Imaging
2.1.5 Formal path to the low-cut filter
2.2 MULTIVARIATE LEAST SQUARES
2.2.1 Inside an abstract vector
2.2.2 Normal equations
2.2.3 Differentiation by a complex vector
2.2.4 From the frequency domain to the time domain
2.3 KRYLOV SUBSPACE ITERATIVE METHODS
2.4 INVERSE NMO STACK
2.5 THE WORLD OF CONJUGATE GRADIENTS
3 Empty bins and inverse interpolation
3.1 MISSING DATA IN ONE DIMENSION
3.1.1 Missing-data program
3.2 WELLS NOT MATCHING THE SEISMIC MAP
3.3 SEARCHING THE SEA OF GALILEE
3.4 INVERSE LINEAR INTERPOLATION
3.4.1 Abandoned theory for matching wells and seismograms
3.5 PREJUDICE, BULLHEADEDNESS, AND CROSS VALIDATION
4 The helical coordinate
4.1 FILTERING ON A HELIX
4.1.1 Review of 1-D recursive filters
4.1.2 Multidimensional deconvolution breakthrough
4.1.3 Examples of simple 2-D recursive filters
4.1.4 Coding multidimensional de/convolution
4.1.5 Causality in two-dimensions
4.2 FINITE DIFFERENCES ON A HELIX
4.2.1 Matrix view of the helix
4.3 CAUSALITY AND SPECTAL FACTORIZATION
4.3.1 The spectral factorization concept
4.3.2 Cholesky decomposition
4.3.3 Toeplitz methods
4.3.4 Kolmogoroff spectral factorization
4.4 WILSON-BURG SPECTRAL FACTORIZATION
4.4.1 Wilson-Burg theory
4.5 HELIX LOW-CUT FILTER
4.6 THE MULTIDIMENSIONAL HELIX
4.7 SUBSCRIPTING A MULTIDIMENSIONAL HELIX
5 Preconditioning
5.1 PRECONDITIONED DATA FITTING
5.1.1 Preconditioner with a starting guess
5.2 PRECONDITIONING THE REGULARIZATION
5.2.1 The second miracle of conjugate gradients
5.2.2 Importance of scaling
5.2.3 Statistical interpretation
5.2.4 The preconditioned solver
5.3 OPPORTUNITIES FOR SMART DIRECTIONS
5.4 NULL SPACE AND INTERVAL VELOCITY
5.4.1 Balancing good data with bad
5.4.2 Lateral variations
5.4.3 Blocky models
5.5 INVERSE LINEAR INTERPOLATION
5.6 EMPTY BINS AND PRECONDITIONING
5.6.1 SEABEAM: Filling the empty bins with a laplacian
5.6.2 Three codes for inverse masking
5.7 THEORY OF UNDERDETERMINED LEAST-SQUARES
5.8 SCALING THE ADJOINT
5.9 A FORMAL DEFINITION FOR ADJOINTS
6 Multidimensional autoregression
6.0.1 Time domain versus frequency domain
6.1 SOURCE WAVEFORM, MULTIPLE REFLECTIONS
6.2 TIME-SERIES AUTOREGRESSION
6.3 PREDICTION-ERROR FILTER OUTPUT IS WHITE
6.3.1 PEF whiteness proof in 1-D
6.3.2 Simple dip filters
6.3.3 PEF whiteness proof in 2-D
6.3.4 Examples of modeling and deconvolving with a 2-D PEF
6.3.5 Seismic field data examples
6.4 PEF ESTIMATION WITH MISSING DATA
6.4.1 Internal boundaries to multidimensional convolution
6.4.2 Finding the prediction-error filter
6.5 TWO-STAGE LINEAR LEAST SQUARES
6.5.1 Adding noise (Geostat)
6.5.2 Inversions with geostat
6.5.3 Infill of 3-D seismic data from a quarry blast
6.5.4 Imposing prior knowledge of symmetry
6.5.5 Hexagonal coordinates
6.6 BOTH MISSING DATA AND UNKNOWN FILTER
6.6.1 Objections to interpolation error
6.6.2 Packing both missing data and filter into a vector
6.7 LEVELED INVERSE INTERPOLATION
6.8 MULTIVARIATE SPECTRUM
7 Spatial aliasing and scale invariance
7.1 INTERPOLATION BEYOND ALIASING
7.1.1 Interlacing a filter
7.2 MULTISCALE, SELF-SIMILAR FITTING
7.2.1 Examples of scale-invariant filtering
7.2.2 Scale-invariance introduces more fitting equations
7.2.3 Coding the multiscale filter operator
8 Nonstationarity: patching
8.1 PATCHING TECHNOLOGY
8.1.1 Weighting and reconstructing
8.1.2 2-D filtering in patches
8.1.3 Designing a separate filter for each patch
8.1.4 Triangular patches
8.2 STEEP-DIP DECON
8.2.1 Dip rejecting known-velocity waves
8.2.2 Tests of steep-dip decon on field data
8.2.3 Are field arrays really needed?
8.2.4 Which coefficients are really needed?
8.3 INVERSION AND NOISE REMOVAL
8.4 SIGNAL-NOISE DECOMPOSITION BY DIP
8.4.1 Signal/noise decomposition examples
8.4.2 Spitz for variable covariances
8.4.3 Noise removal on Shearer’s data
8.4.4 The human eye as a dip filter
8.5 SPACE-VARIABLE DECONVOLUTION
9 Plane waves in three dimensions
9.1 THE LEVELER: A VOLUME OR TWO PLANES?
9.1.1 PEFs overcome spatial aliasing of difference operators
9.1.2 My view of the world
9.2 WAVE INTERFERENCE AND TRACE SCALING
9.2.1 Computing the proper scale factor for a seismogram
9.3 LOCAL MONOPLANE ANNIHILATOR
9.3.3 Crossing dips
9.3.4 Tests of 2-D LOMOPLAN on field data
9.4 GRADIENT ALONG THE BEDDING PLANE
9.4.1 Definition of LOMOPLAN in 3-D
9.4.2 The quarterdome 3-D synthetic (qdome)
9.5 3-D SPECTRAL FACTORIZATION
10 Some research examples
10.1 GULF OF MEXICO CUBE
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