PLANE WAVES AT SMALL ARRAYS

Abstract

The resolving power of a seismic array is defined in terms of the array response function and via the classical uncertainty principle. Using the theory of maximum likelihood wavenumber spectra (Capon, 1969), we show for the case of two correlated plane waves that arbitrarily high resolution is achievable in the limit as the background white noise tends to zero. This extends Barnard’s (1969) result to the case of correlated plane waves. The increased resolution arises from the additional assumption that the data are plane waves over all space, and not zero off the array as the classical result assumes. It is found that a sample rate (in time) large compared to the Nyquist rate, is needed in the case of a short time gate at a small array. Cross‐power spectral matrices are estimated at 4 hz from 1 sec of computer generated data consisting of two correlated plane waves in white noise. These spectral matrices are then used to generate maximum likelihood wavenumber spectra. The two plane waves are resolved at various signal‐to‐noise ratios and at correlations up to ρ=0.8. The need for using a high sampling rate is demonstrated. Results are compared with conventional wavenumber spectra, where the classical resolution results hold. The use of a 1‐sec window provides improved resolution of the wavenumber structure as it changes in time, resulting in better separation of any time‐overlapping phases and multipathed waves that arise from one event.