High-resolution velocity spectra using eigenstructure methods

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. INTRODUCTION The estimation of stacking velocities is a classic problem in exploration seismology. A related problem is the estimation of the ray parameter of a plane wave. Both problems are particular cases of a more general problem: estimating the parameters that describe the shape of the wavefront (“shape parameters”) for a signal recorded at a linear array of receivers. In the near field, the wavefront is approximately spherical, and therefore the relative time delays between receivers are conveniently parameterized by stacking velocities and zero-offset traveltimes. In the far field, the wavefront is well approximated by plane waves characterized by ray parameters. The standard approach to estimating stacking velocities is to pick the maxima of shape-parameter spectra (Taner and Koehler, 1969). The spectra are computed by the repeated application, for a sweep of shape parameters, of first a time correction that aligns the wavefront in space along the array and then a coherency measure along the spatial direction. The time correction could be normal moveout (NMO) or linear moveout (LMO); the coherency measure could be a simple stacking or the computation of a semblance function. The stacking spectrum, and in particular the stackingvelocity spectrum, has many attractive properties. It yields the maximum-likelihood estimates of the shape parameter when the statistics of the data are Gaussian and there is only one wavefront impinging on the array. Furthermore, the estimate is robust -with respect tom the data’s deviations from the assumed simple propagation model or from Gaussian statistics. When two or more wavefronts impinge on the array, the classical procedure still yields good estimates of their shape parameters, provided that the wavefront shapes are sufficiently different. However, when a pair of wavefronts are too closely interfering, the resulting estimates are biased. Even worse, the spectra may show only one maximum and~indicate only one wavefront~ incident on the array. The main disadvantage of the stacking spectrum is its poor resolution, limited by the aperture of the array and by the Manuscript received by the Editor June 30, 1988; revised manuscript received December 27, 1988. *Geophysics Department, Stanford University, Stanford, CA 943052215.