Two weeks in a class on WAVES
by Jon Claerbout

TABLE OF CONTENTS

Introduction (ps.gz 10K) (pdf 9K)
Convolution and Spectra (ps.gz 114K) (pdf 222K) (src 69K)

• SAMPLED DATA AND Z-TRANSFORMS
  • Linear superposition
  • Convolution with Z-transform
  • 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
• FT AS AN INVERTIBLE MATRIX
  • The Nyquist frequency
  • Convolution in one domain...
  • Inverse Z-transform
• SYMMETRIES
  • Laying out a mesh
• CORRELATION AND SPECTRA
  • Spectra in terms of Z-transforms
  • Two ways to compute a spectrum
  • The comb function
  • Undersampled field data
  • Common signals
  • Spectral transfer function

Two-dimensional Fourier transform (ps.gz 67K) (pdf 90K) (src 18K)

• SLOW PROGRAM FOR INVERTIBLE FT
  • The slow FT code
• TWO-DIMENSIONAL FT
  • Signs in Fourier transforms
  • Examples of 2-D FT

Kirchhoff migration (ps.gz 184K) (pdf 194K) (src 5K)

• THE EXPLODING REFLECTOR CONCEPT
• THE KIRCHHOFF IMAGING METHOD
  • Tutorial Kirchhoff code
  • Kirchhoff artifacts
• SAMPLING AND ALIASING

Downward continuation of waves (ps.gz 75K) (pdf 83K) (src 52K)

• DIPPING WAVES
  • Snell waves
  • Evanescent waves
• DOWNWARD CONTINUATION
  • Continuation of a dipping plane wave.
  • Downward continuation with Fourier transform
  • Linking Snell waves to Fourier transforms
• DOWNWARD CONTINUATION AND IMAGING

Waves in layered media (ps.gz 66K) (pdf 68K) (src 89K)

• REFLECTION AND TRANSMISSION COEFFICIENTS
• THE LAYER MATRIX
• ENERGY FLUX
• GETTING THE WAVES FROM THE REFLECTION COEFFICIENTS
  • Layers between two halfspaces
  • Reflection seismology geometry
  • Getting reflection coefficients from the waves

Multidimensional deconvolution examples (ps.gz 673K) (pdf 620K) (src 4K)

• 2-D MODELING AND DECON
• DATA RESTORATION

Resolution and random signals (ps.gz 103K) (pdf 208K) (src 59K)

• 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 form of the uncertainty principle
    • Proof
• 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