High‐resolution velocity spectra using eigenstructure methods

Abstract

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.

High-resolution imaging of diffractions — A window-steered MUSIC approach

“…This smoothing operation is more important in the case of real data, where the optimal window size and the operator length of the smoothing filter need to be tested for. Note that the spatial smoothing operation could alternatively be replaced by a spatial smoothing of the covariance matrix (Biondi and Kostov, 1989;Kirlin, 1992). Now, the question is, what kind of response will be obtained in case the image point considered does not represent the location of a true scatterer?…”

“…The pseudospectrum in equation 16 is expressed by a nil-space projection. Alternatively, that pseudospectrum can be constructed using its counterpart signal-space projection (Biondi and Kostov, 1989;Asgedom et al, 2011a). After spatial smoothing, the steered data window will contain essentially one aligned event in case of a true diffractor.…”

Statistical Stability and Time-Reversal Imaging in Random Media

“…Some of the most popular ones in recent years continue to be matched-field processing (Bucker, 1976;Jensen et al, 1994), MUSIC (MUltiple SIgnal Classification) (Schmidt, 1979;Johnson, 1982;Schmidt, 1986;Biondi and Kostov, 1989), and other linear subspace methods (Johnson, 1982;Johnson and DeGraaf, 1982;Cheney, 2001). When the targets are imbedded in heterogeneous media so that significant multiple scattering occurs in the background medium during wave propagation between array and target, the randomness has a different character than that usually envisioned in these traditional analyses.…”

Phase and amplitude tracking for seismic event separation

“…There exist at least two high-quality methods for this task: the matrix pencil method of Hua and Sarkar (1990) and the MUSIC algorithm (Schmidt, 1986;Biondi and Kostov, 1989;Kirlin and Done, 1999). We choose the latter for its simplicity and robustness.…”