The Karhunen-Loeve Transform (KLT) technique is frequently applied to a variety of seismic data processing problems. Initial seismic applications of the KLT (Hemon and Mace, 1978; Jones, 1985) were based on principal components analysis, usually on stacked data. A subset of the principal components obtained from the KLT of a seismic data set is used to reconstruct the data. Using the dominant principal components in the reconstruction emphasizes the lateral coherence which characterizes poststack seismic data. Using subdominant principal components during reconstruction can emphasize detail in the result by eliminating the strong lateral coherency carried by the high-order principal components. The usual approach to principal components analysis is also used to suppress random noise in the final reconstruction by always eliminating the low order principal components from any reconstructions. These low-order principal components contribute to the randomness in the data.
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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.