By R. Lynn Kirlin, William J. Done
This quantity is meant to offer the geophysical sign analyst enough fabric to appreciate the usefulness of information covariance matrix research within the processing of geophysical signs. A history of uncomplicated linear algebra, data, and basic random sign research is thought. This reference is exclusive in that the information vector covariance matrix is used all through. instead of facing just one seismic information processing challenge and offering a number of equipment, the focus during this e-book is on just one primary technique - research of the pattern covariance matrix providing many seismic facts difficulties to which the technique applies. This quantity could be of curiosity to many researchers, delivering a style amenable to many specified purposes. It bargains a various sampling and dialogue of the idea and the literature constructed thus far from a typical viewpoint.
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1989, Performance of high resolution frequencies estimation methods compared to the Cramer-Rao bounds: IEEE Trans. , Speech and Sig. , 37, 1703-1720. Kirlin, R. , 1991, A note on the effects of narrowband and stationary signal model assumptions on the covariance matrix of sensor array data vectors: IEEE Trans. Signal Processing, 503-506. Pillai, S. , 1989, Array signal processing: Springer-Verlag Inc. Scharf, L. , 1991, Statistical signal processing: Addison-Wesley Publ. Corp. 50. org/ 50 Chapter 5 Temporal and Spatial Spectral Analysis R.
4, v1 ϭ s. Now note that SVD would have found both eigenstructures. 32) where V are the eigenvectors of XHX(p ϫ p), and U are the eigenvectors of XXH(M ϫ M). Then U 1 ϭ s, v 1 ϭ 1 , and ⌳ 11 ϭ ( E s ϩ n2 ) 1 / 2 ; these are the singular vector u1 of X for the first example, the eigenvector v1 of XHX for the current example, and the first singular value 1 of the SVD of X. The eigenvalues of either XXH or XHX are i ϭ E s ϩ n2 , n2 , … , n2 , to either p or M values, respectively. These two examples are basic to many of the algorithms presented elsewhere in this book.
11a). The factor (XTX)-1XT ϭ X# is the pseudoinverse of X, since X#X ϭ I. Further, T X# ϭ ( X X ) Ϫ1 ⌳ Ϫ1 0 T T U . 23) These are convenient notations for determining the least-squares fit of columns of X to a general vector y. , p )T give a best least-squares fit yˆ x ϭ X to y? 50. org/ 25 Note too that the error y Ϫ yˆ ϭ ( I Ϫ P # )y is orthogonal to the range space. 21)]. 26) where X is M ϫ p, is p ϫ 1, y are M noisy measurements, and M Ͼ p. If p Ͼ M, is known as the minimum-norm solution, since there are many solutions , but only one that minimizes T.
Covariance Analysis for Seismic Signal Processing by R. Lynn Kirlin, William J. Done