EARTH SOUNDINGS ANALYSIS:
Processing versus Inversion (PVI)
by Jon Claerbout

TABLE OF CONTENTS

Preface (ps 161K)
Introduction (ps 295K)
Convolution and Spectra (ps 471K) (src 199K)

• SAMPLED DATA AND Z-TRANSFORMS
  • Linear superposition
  • Convolution with Z-transform
  • Dissecting systems by factoring
  • Convolution equation and program
  • Negative time
• FOURIER SUMS
  • Superposition of sinusoids
  • Sampled time and Nyquist frequency
  • Fourier sum
• FOURIER AND Z-TRANSFORM
  • Unit circle
  • Differentiator
  • Gaussian examples
  • Complex roots
  • Inverse Z-transform
• CORRELATION AND SPECTRA
  • Spectra in terms of Z-transforms
  • Two ways to compute a spectrum
  • Common signals
  • Spectra of complex-valued signals
  • Time-domain conjugate
  • Spectral transfer function
  • Crosscorrelation
  • Matched filtering

Discrete Fourier transform (ps 529K) (src 114K)

• FT AS AN INVERTIBLE MATRIX
  • The Nyquist frequency
  • Laying out a mesh
  • The comb function
  • Undersampled field data
• INVERTIBLE SLOW FT PROGRAM
  • The slow FT code
  • Truncation problems
  • FT by Z-transform
• SYMMETRIES
  • Plot interpretation
  • Convolution in the frequency domain
• SETTING UP THE FAST FOURIER TRANSFORM
  • Shifted spectra
• TWO-DIMENSIONAL FT
  • Basics of two-dimensional Fourier transform
  • Signs in Fourier transforms
  • Examples of 2-D FT
• HOW FAST FOURIER TRANSFORM WORKS
• References

Z-plane, causality, and feedback (ps 567K) (src 350K)

• LEAKY INTEGRATION
  • Plots
  • Two poles
• SMOOTHING WITH BOX AND TRIANGLE
  • Smoothing with a rectangle
  • Smoothing with a triangle
• CAUSAL INTEGRATION FILTER
  • The accuracy of causal integration
  • Examples of causal integration
  • Symmetrical double integral
  • Nonuniqueness of the integration operator
• DAMPED OSCILLATION
  • Narrow-band filters
  • Polynomial division
  • Spectrum of a pole
  • Rational filters
• INSTABILITY
  • Anticausality
  • Inverse filters
  • The unit circle
  • The mapping between Z and complex frequency
  • The meaning of divergence
  • Boundedness
  • Causality and the unit circle
• MINIMUM-PHASE FILTERS
  • Mechanical interpretation
  • Laurent expansion
• INTRODUCTION TO ALL-PASS FILTERS
  • Notch filter
• PRECISION EXHAUSTION
• MY FAVORITE WAVELET
• IMPEDANCE FILTERS

Univariate problems (ps 640K) (src 100K)

• INSIDE AN ABSTRACT VECTOR
• SEGREGATING P AND S CROSSTALK
  • Two univariate problems
  • The physics of crosstalk
  • Failure of straightforward methods
    • Failure of independence assumption
  • Solution by weighting functions
    • A nonlinear-estimation method
    • Clarity of nonlinear picture
    • Nonuniqueness and instability
    • Estimating the noise variance
    • Colored noise
  • Noise as strong as signal
  • Spectral weighting function
  • Flame out
• References
• HOW TO DIVIDE NOISY SIGNALS
  • Dividing by zero smoothly
  • Damped solution
  • Example of deconvolution with a known wavelet
  • Deconvolution with an unknown filter
  • Explicit model for noise
  • A self-fulfilling prophecy?
• NONSTATIONARITY
• DIP PICKING WITHOUT DIP SCANNING
  • The plane-wave destructor
  • Moving windows for nonstationarity

Adjoint operators (ps 569K) (src 65K)

• FAMILIAR OPERATORS
  • Transient convolution
  • Zero padding is the transpose of truncation.
  • Product of operators
  • Convolution end effects
  • Kirchhoff modeling and migration
  • Migration defined
• ADJOINT DEFINED: DOT-PRODUCT TEST
  • What is an adjoint operator?
• NORMAL MOVEOUT AND OTHER MAPPINGS
  • Nearest-neighbor interpolation
  • A family of nearest-neighbor interpolations
    • Looping over input space
    • Looping over output space
  • Formal inversion
  • Nearest-neighbor NMO
  • Stack
  • Pseudoinverse to nearest-neighbor NMO
  • Null space and inconsistency
  • NMO with linear interpolation
• DERIVATIVE AND INTEGRAL
  • Adjoint derivative
• CAUSAL INTEGRATION RECURSION
  • Readers' guide
• UNITARY OPERATORS
  • Meaning of B'B
  • Unitary and pseudounitary transformation
  • Pseudounitary NMO with linear interpolation
• VELOCITY SPECTRA
• INTRODUCTION TO TOMOGRAPHY
  • Units
• STOLT MIGRATION
• References

Model fitting by least squares (ps 475K) (src 82K)

• MULTIVARIATE LEAST SQUARES
  • Inverse filter example
  • Normal equations
  • Differentiation by a complex vector
  • Time domain versus frequency domain
    • Unknown filter
    • Unknown input: deconvolution with a known filter
• ITERATIVE METHODS
  • Method of random directions and steepest descent
  • Conditioning the gradient
  • Why steepest descent is so slow
  • Conjugate gradient
  • Magic
  • Conjugate-gradient theory for programmers
  • First conjugate-gradient program
  • Preconditioning
• INVERSE NMO STACK
• MARINE DEGHOSTING
  • Synthetics
• CG METHODOLOGY
  • Programming languages and this book
• References

Time-series analysis (ps 782K) (src 66K)

• SHAPING FILTER
  • Source waveform and multiple reflections
  • Shaping a ghost to a spike
• SYNTHETIC DATA FROM FILTERED NOISE
  • Gaussian signals versus sparse signals
  • Random numbers into a filter
  • Random numbers into the seismic spectral band
• THE ERROR FILTER FAMILY
  • Prediction-error filters on synthetic data
  • PE filters on field data
  • Prediction-error filter output is white.
  • Proof that PE filter output is white
  • Nonwhiteness of gapped PE-filter output
  • Postcoloring versus prewhitening
• BLIND DECONVOLUTION
• WEIGHTED ERROR FILTERS
  • Automatic gain control
  • Gain before or after convolution
  • Meet the Toeplitz matrix
  • Setting up any weighted CG program
• CALCULATING ERROR FILTERS
  • Stabilizing technique
• INTERPOLATION ERROR
  • Blind all-pass deconvolution

Missing-data restoration (ps 579K) (src 142K)

• INTRODUCTION TO ALIASING
  • Relation of missing data to inversion
  • My model of the world
• MISSING DATA IN ONE DIMENSION
  • Missing-data program
• MISSING DATA AND UNKNOWN FILTER
  • Objections to interpolation error
  • Packing both missing data and filter into a CG vector
  • Spectral preference and training data
  • Summary of 1-D missing-data restoration
  • 2-D interpolation before aliasing
• 2-D INTERPOLATION BEYOND ALIASING
  • Interpolation with spatial predictors
  • Refining both t and x with a spatial predictor
  • The prediction form of a two-dip filter
  • The regression codes
  • Zapping the null space with envelope scaling
  • Narrow-band data
• A FULLY TWO-DIMENSIONAL PE FILTER
  • The hope method
  • An alternative principle for 2-D interpolation
• TOMOGRAPHY AND OTHER APPLICATIONS
  • Clash in philosophies
  • An aside on theory-of-constraint equations
• References

Hyperbola tricks (ps 620K) (src 39K)

• PIXEL-PRECISE VELOCITY SCANNING
  • Smoothing in velocity
  • Rho filter
• GEOMETRY-BASED DECON
  • A model with both signature and reverberation
    • Reverberation
    • Signature
  • Regressing simultaneously before and after NMO
  • A model for convolution both before and after NMO
  • Heavy artillery
• References

Spectrum and phase (ps 694K) (src 144K)

• HILBERT TRANSFORM
  • A Z-transform view of Hilbert transformation
  • The quadrature filter
  • The analytic signal
  • Instantaneous envelope
  • Instantaneous frequency
• SPECTRAL FACTORIZATION
  • The exponential of a causal is causal.
  • Finding a causal wavelet from a prescribed spectrum
  • Why the causal wavelet is minimum-phase
  • Pathological examples
  • Relation of amplitude to phase
• A BUTTERWORTH-FILTER COOKBOOK
  • Butterworth-filter finding program
  • Examples of Butterworth filters
• PHASE DELAY AND GROUP DELAY
  • Phase delay
  • Group delay
  • Group delay as a function of the FT
  • Observation of dispersive waves
  • Group delay of all-pass filters
• PHASE OF A MINIMUM-PHASE FILTER
  • Phase of a single root
  • Phase of a rational filter
• ROBINSON'S ENERGY-DELAY THEOREM
• FILTERS IN PARALLEL

Resolution and random signals (ps 655K) (src 45K)

• TIME-FREQUENCY RESOLUTION
  • A misinterpretation of the uncertainty principle
  • Measuring the time-bandwidth product
  • The uncertainty principle in physics
  • Gabor's proof of the uncertainty principle
  • My rise-time proof of the uncertainty principle
    • Proof by way of the dual problem
• FT OF RANDOM NUMBERS
  • Bandlimited noise
• TIME-STATISTICAL RESOLUTION
  • Ensemble
  • Expectation and variance
  • Probability and independence
  • Sample mean
  • Variance of the sample mean
• SPECTRAL FLUCTUATIONS
  • Paradox: large n vs. the ensemble average
  • An example of the bandwidth/reliability tradeoff
  • Spectral estimation
• CROSSCORRELATION AND COHERENCY
  • Correlation
  • Coherency
  • The covariance matrix of multiple signals
  • Bispectrum
• SMOOTHING IN TWO DIMENSIONS
  • Tent smoothing
  • Gaussian mounds
  • Speed of 2-D Gaussian smoothing
• PROBABILITY AND CONVOLUTION
• THE CENTRAL-LIMIT THEOREM

Entropy and Jensen inequality (ps 328K) (src 6K)

• THE JENSEN INEQUALITY
  • Examples of Jensen inequalities
• RELATED CONCEPTS
  • Prior and posterior distributions
  • Jensen average
  • Additivity of envelope entropy to spectral entropy

Seplib and SEP software (ps 345K) (src 9K)
The Zplane program (ps 293K) (src 10K)

• OPERATORS
• SCALARS
• FILTERS, SIGNALS, AND THEIR TRANSFORMS
• MATRICES AND VECTORS
• CHANGES FROM FGDP