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Principal and independent component analysis for seismic data Kaplan, Sam T.

Abstract

Principal and Independent component analysis (PCA and ICA) are two ideas which are very much related; both employing a statistical understanding of data to achieve their goals. Whereas PCA exploits statistical correlation, ICA uses statistical independence to glean useful information from data. Seismic data is inherently noisy, and is complicated by the presence of an unknown seismic wavelet. Analysis of the data is aided by, both, noise suppression and blind deconvolution techniques. First, consider the subject of noise suppression. If the data are organized into several sequences where, from one sequence to the next, the signal is correlated while the noise is uncorrelated, then PCA has the ability to separate noise and signal. Here, PCA is analyzed from three points of view, variance maximization, the singular value decomposition and neural networks. The resulting theory is used to filter noise from a set of common midpoint seismic gathers by exploiting correlations which exist from one gather to the next. To further simplify analysis of these data, the Earth is often approximated as a linear system; thus, the seismic trace is subject to the convolutional model. Convolution is a linear operation, and consequently, can be formulated as a linear system of equations. If only the output of the system (the convolved signal) is known, then the problem is blind so that given one equation, two unknowns are sought. This problem is well suited for ICA which has the ability to find some estimate of the two unknowns, and here the blind deconvolution problem is solved using ICA. To facilitate this, several time-lagged versions of the convolved signal are extracted and used to construct realizations of a random vector. For ICA, this random vector is the, so called, mixture vector, created by the matrix-vector multiplication of the two unknowns, the mixing matrix and the source vector. Due to the properties of convolution, the mixing matrix is banded with its nonzero elements containing the convolution's filter. This banded property is incorporated into the ICA algorithm as prior information, giving rise to a banded ICA algorithm (B-ICA) which is, in turn, used in a new blind deconvolution algorithm. This algorithm is considered for both noiseless and noisy data.

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