Seismic reflections from an interface are relatively short in time; they are transient. Because of this fact, estimation of the covariance matrix is accomplished by the use of either time-slice vectors (across traces) or Fourier transform value vectors. Although we might have a dozen or so time-slice vectors, this is generally not adequate, (a few hundred would be nice, and appropriate for temporally stationary signals). In the frequency domain, the short reflection only affords one Fourier transform. Thus, the multiplicity of vector samples must be found another way if possible.
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Covariance Analysis for Seismic Signal Processing
This reference is intended to give the geophysical signal analyst sufficient material to understand the usefulness of data covariance matrix analysis in the processing of geophysical signals. A background of basic linear algebra, statistics, and fundamental random signal analysis is assumed. This reference is unique in that the data vector covariance matrix is used throughout. Rather than dealing with only one seismic data processing problem and presenting several methods, we will concentrate on only one fundamental methodology—analysis of the sample covariance matrix—and we present many seismic data problems to which the methodology applies.