By R. Lynn Kirlin, William J. Done
This quantity is meant to offer the geophysical sign analyst adequate fabric to appreciate the usefulness of knowledge covariance matrix research within the processing of geophysical indications. A historical past of simple linear algebra, information, and basic random sign research is thought. This reference is exclusive in that the knowledge vector covariance matrix is used all through. instead of facing just one seismic information processing challenge and proposing a number of equipment, the focus during this e-book is on just one primary technique - research of the pattern covariance matrix featuring many seismic information difficulties to which the method applies. This quantity might be of curiosity to many researchers, delivering a style amenable to many special purposes. It bargains a various sampling and dialogue of the speculation and the literature constructed to this point from a standard viewpoint.
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Extra resources for Covariance Analysis for Seismic Signal Processing
Org/ 42 ␦v i֊ ϭ P B ␦v i . 19) It is shown in Clergeot et al. 20b) where q is the usual time index on the subarray snapshots xk(q) of length m and k is the index on subarrays used for spatial smoothing (see Chapter 8) giving K ϭ M Ϫ m ϩ 1. 6), except that we have indexed time samples and subarrays. 20a) indicates no spatial smoothing and x(q) has length M. Ak and Bk contain appropriate transformation matrices to yield coherence of signals in subarray k with signals at the reference subarray (see Chapter 8).
This spacing in frequency is also the resolution. The Nyquist frequency or folding frequency is half the sampling frequency. FFT coefficients between the folding frequency fs / 2 and the sampling frequency fs are identical to those between –fs / 2 and 0, due to the periodicity of the coefficients in the frequency domain. We assume the reader is also familiar with the z-transform, which is essentially the Fourier transform’s equivalent with application to equispaced samples of either time or spatial signals.
The eigenstructure properties discussed for the examples above still hold under certain restrictions. Basically, the wavefronts to be analyzed must be fairly flat and mostly encompassed by the window of analysis. However, because an unflattened wavefront has nonzero relative delays and the seismic wavefronts are broadband, each frequency component would have its unique phase rotation at each sensor. Thus, straightforward application of the above will yield eigenstructures with an unclear demarcation of the signal subspace, because there will be more than one “larger” eigenvalue per complex wavefront present.