Characterizing three‐dimensional wavefields with high‐resolution spectra

Abstract

Three‐dimensional spectra (frequency‐x‐wavenumber‐y‐wavenumber or [Formula: see text] spectra) can be used to determine the frequency content, velocity, and direction of waves entering an array of receivers. This information is important in detecting aliasing problems, understanding coherent noise, designing arrays, and determining parameters for coherent noise filters. Because of the limited spatial dimensions of most arrays the discrete Fourier transform produces an estimate of the three‐dimensional (3-D) spectrum with severe wavenumber distortion. We extend a 2-D hybrid spectral estimation method to three dimensions by combining a temporal Fourier transform with a spatial 2-D maximum entropy spectral estimation technique. The method produces [Formula: see text] spectra with higher wavenumber resolution and less spectral distortion than corresponding 3-D Fourier spectra. The 2-D maximum entropy spectral estimation algorithm uses a sequence of Fourier transforms to extrapolate the estimated autocorrelation function of the data. We assume the wavenumber spectrum of the data comprises a sum of a few poles. Field and synthetic data are used to demonstrate how 3-D wavefields can be characterized with this method of spectral analysis. From these results we conclude that the method gives excellent wavenumber resolution but performs poorly in detecting small signals in the presence of high amplitude signals. We feel this limitation is not serious for characterizing strong amplitude coherent energy recorded by an array of receivers.