Stacking spectra provide maximum-likelihood estimates for the stacking velocity, or for the ray parameter, of well separated reflections in additive white noise. However, the resolution of stacking spectra is limited by the aperture of the array and the frequency of the data. Despite these limitations, parametric spectral estimation methods achieve better resolution than does stacking. To improve resolution, the parametric methods introduce a parsimonious model for the spectrum of the data. In particular, when the data are modeled as the superposition of wavefronts, the properties of the eigenstructure of the data covariance matrix can be used to obtain high-resolution spectra. The traditional stacking spectra can also be expressed as a function of the data covariance matrix and directly compared to the eigenstructure spectra. The superiority of the latter in separating closely interfering reflections is then apparent from a simple geometric interpretation. Eigenstructure methods were originally developed for use with narrow-band signals. while seismic reflections are wide-band and transient in time Taking advantage of the full bandwidth of seismic data, we average spectra from several frequency bands. We choose each frequency band wide enough, so that we can average over time estimates of the covariance matrix. Thus, we obtain a robust estimate of the covariance matrix from short data sequences. A field-data example shows that the high-resolution estimators are particularly attractive for use in the estimation of local spectra in which short arrays are considered. Several realistic synthetic examples of stacking-velocity spectra illustrate the improved performance of the new methods in comparison with conventional processing.

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