## Covariance Analysis for Seismic Signal Processing by R. Lynn Kirlin, William J. Done

By R. Lynn Kirlin, William J. Done

This quantity is meant to provide the geophysical sign analyst enough fabric to appreciate the usefulness of information covariance matrix research within the processing of geophysical indications. A history of simple linear algebra, facts, and basic random sign research is believed. This reference is exclusive in that the knowledge vector covariance matrix is used all through. instead of facing just one seismic info processing challenge and offering numerous equipment, the focus during this booklet is on just one primary method - research of the pattern covariance matrix offering many seismic information difficulties to which the technique applies. This quantity will be of curiosity to many researchers, supplying a mode amenable to many detailed functions. It deals a various sampling and dialogue of the idea and the literature constructed thus far from a standard viewpoint.

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**Covariance Analysis for Seismic Signal Processing**

This quantity is meant to provide the geophysical sign analyst enough fabric to appreciate the usefulness of knowledge covariance matrix research within the processing of geophysical indications. A history of easy linear algebra, statistics, and primary random sign research is thought. This reference is exclusive in that the knowledge vector covariance matrix is used all through.

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Jw1 9⌬ e 1 e e Ϫ jw 2 ⌬ Ϫ jw 2 2⌬ . . 23b) 2 This covariance matrix will have two eigenvalues greater than n and 2 eight equal to n . The two eigenvectors associated with the larger eigenvalues span the same signal subspace as do h1 and h2, the columns of H. Both col2 umns of H are orthogonal to the eigenvector associated with 3 ϭ n . We note that in no case will either eigenvector in ⑀s equal either h1 or h2, but will always be some combination of both h1 and h2. However, with only one sig2 2 nal, the rank 1 case, v 1 ϭ h 1 ր 10 and 1 ϭ 10 s1 ϩ n .

The classical reference by Oppenheim and Schafer (1975) discusses the z-transform, the discrete Fourier transform (DFT), and fast Fourier transform (FFT). Much information can be gained from the frequency domain representation of signals. Just as the FFT can apply to temporal signal samples, it may equally apply to spatial signal samples. Quite commonly, two-dimensional (2D) FFTs are applied to 2-D seismic data, where one dimension is temporal and the other is spatial. 50. org/ 51 sinusoids that may be present in the data, (5) it does not result in a polynomial ratio form, and (6) the spectral resolution is limited to T –1.

2 Null Space and the Minimum Norm Solution Clearly for p Ͻ M there are vectors y of length M which are not LC of the p independent vectors xi in X. Such vectors lie in the “null space” of X; they are not LC of either the p vectors xi or of the p vectors in U1, the submatrix of U ϭ (U1U2) associated with the p nonzero singular values of U. Rather they are LC of the M Ϫ p vectors in U2. 50. org/ 24 the p independent columns in X. Let the two components of y be denoted yx and yൿ, respectively, in the range and null space: y ϭ yx ϩ y֊ .