The covariance analysis approach to directions of plane-wave arrival or multiple sinusoid frequency estimation have already been detailed in Chapter 5. In Chapter 7 Fu Li and Hui Liu derive optimal subspace estimators of two-way vertical traveltime and rms velocity for single wavefronts. Because hyperbolic wavefronts by definition are not planar and wavelets are not temporally narrowband, direct applications of high-resolution subspace algorithms often do not work well. In this chapter I will give some of the details and trade-offs in the estimation of rms velocity when multiple broadband wavefronts are present in the analysis window. I will also present two enhancements of conventional semblance. The first is a low-rank or signal subspace version of semblance, and the second, the multiple sidelobe canceller, is an adaptive interference cancelling method that presents to the semblance algorithm a best estimate of interference-free wavefronts. These enhancements can substantially improve resolution and coherence estimation
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This reference is intended to give the geophysical signal analyst sufficient material to understand the usefulness of data covariance matrix analysis in the processing of geophysical signals. A background of basic linear algebra, statistics, and fundamental random signal analysis is assumed. This reference is unique in that the data vector covariance matrix is used throughout. Rather than dealing with only one seismic data processing problem and presenting several methods, we will concentrate on only one fundamental methodology—analysis of the sample covariance matrix—and we present many seismic data problems to which the methodology applies.