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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 offer the geophysical sign analyst enough fabric to appreciate the usefulness of knowledge covariance matrix research within the processing of geophysical indications. A historical past of easy linear algebra, statistics, and basic random sign research is believed. This reference is exclusive in that the information vector covariance matrix is used all through. instead of facing just one seismic facts processing challenge and offering numerous tools, the focus during this booklet is on just one basic method - research of the pattern covariance matrix providing many seismic facts difficulties to which the technique applies. This quantity will be of curiosity to many researchers, offering a style amenable to many special purposes. It bargains a various sampling and dialogue of the idea and the literature built up to now from a standard viewpoint.

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50. 33) ϭ T ( As ϩ w ) ϭ TAs ϩ Tw . The covariance matrix of z is H H 2 R z ϭ TAPs A T ϩ ␴ n I . 34) As long as THT ϭ I (or orthonormal transformation vectors are employed), the eigenvalues of Rz are identical to those of Rx. To demonstrate, observe that if l and u are an eigenvalue-eigenvector pair for Rz, then H R z u ϭ TR x T u ϭ lu , H H H T TR x T u ϭ lT u , H H Rx ( T u ) ϭ l ( T u ) . 35) This shows that l is an eigenvalue of both Rx and Rz. The eigenvector v ϭ THu of Rx is simply a rotation of u because the magnitude is the same: H H 2 H H H H 2 T u ϭ u TT u ϭ u T Tu ϭ u u ϭ u .

Sometimes the approximate signal dimensionality is known. In many cases, it is possible to either know or estimate what the quantitative trade is statistically. The following presentation is based on Scharf (1991, Chapter 9). Suppose that the data matrix X, M traces of length LT is the sum of a signal matrix S plus an independent white zero-mean noise matrix N. 33) where we are indicating estimates of ULVH ϭ S. We have seen in the previous sections that a flattened event, if it is the only event, will cause u1 ϭ s and v 1H ϭ ( 1 1 … 1 ) .

The above four eigenstructure properties are explained as follows. First, we note that As has r degrees of freedom, therefore AE{ssH}AH has rank r. Further, APs AH must have r positive eigenvalues, the last M Ϫ r equaling zero. Next, 2 we observe that if an eigenvalue of APs AH is ␭, then ␭ ϩ ␴ n is an eigenvalue H 2 of APs A ϩ ␴ n I ; because if v is the eigenvector associated with ␭, then H 2 2 Rv ϭ ( APs A ϩ ␴ I )v ϭ ␭v ϩ ␴ n v . 2 ϭ ( ␭ ϩ ␴ n )v . Then by definition, ␭ ϩ ␴2 must be an eigenvalue of R.

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