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Applications of the Karhunen-Loève transform in reflection seismology Jones, Ian Frederick


The Karhunen-Loève transform, which optimally extracts coherent information from multichannel data, has been applied to several problems in reflection seismic data processing. The transform is derived by a least-squares construction of an orthogonal set of principal components and eigenvectors, with corresponding eigenvalues. Data are reconstructed as a linear combination of the principal components. The mathematical properties of the Karhunen-Loève transform which render it applicable to problems in seismic data processing are reviewed, and a number of new algorithms developed. Most algorithms are tested on synthetic and real data examples, and 'production-line' industrially viable versions of some of the programs have been developed. A new signal-to-noise ratio enhancement technique, based on reconstruction of stacked seismic sections, has proved to be successful on real data. Reconstruction of less coherent information to emphasize anomalous features in stacked seismic data ("misfit" reconstruction) shows some promise. Diffraction hyperbolae isolated by misfit reconstruction are used to estimate residual migration velocities with some success. And, the ability of the transform to segregate coherent information is used successfully as the basis of a new multiple suppression technique. An anomaly identification scheme, based on cluster analysis of the eigenvectors of the transform, works well on the synthetic data used, and gives promising results when applied to real data. A new velocity analysis method, utilizing a ratio of the eigenvalues, works well for good data at early travel times, and offers a potential for high resolution velocity inversion studies. Use of the eigenvalues in evaluation of a constant phase approximation to dispersion for synthetic data provides promising results, leading to quantification of dispersion in terms of relative phase shifts. As part of this development, an analysis of the effect of dispersion on Vibroseis© data acquisition, which represents an original investigation, is presented.

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