Multidimensional autoregression

  The many applications of least squares to the one-dimensional convolution operator constitute the subject known as ``time-series analysis.'' The autoregression filter, also known as the prediction-error filter (PEF), gathers statistics for us, not the autocorrelation or the spectrum directly but it gathers them indirectly as the inverse of the amplitude spectrum of its input. The PEF plays the role of the so-called ``inverse-covariance matrix'' in statistical estimation theory. Given the PEF, we use it to find missing portions of signals.
 

• 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
  • Seismic field data examples
• PEF ESTIMATION WITH MISSING DATA
  • Internal boundaries to multidimensional convolution
  • Finding the prediction-error filter
• TWO-STAGE LINEAR LEAST SQUARES
  • Adding noise (Geostat)
  • Inversions with geostat
  • Infill of 3-D seismic data from a quarry blast
  • Imposing prior knowledge of symmetry
  • Hexagonal coordinates
• 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
  • The bane of PEF estimation
• MULTIVARIATE SPECTRUM
  • What should we optimize?
  • Confusing terminology for data covariance
  • Hermeneutics