Interpolation and extrapolation using a high-resolution discrete Fourier transform

Abstract

We present an iterative nonparametric approach to spectral estimation that is particularly suitable for estimation of line spectra. This approach minimizes a cost function derived from Bayes' theorem. The method is suitable for line spectra since a "long tailed" distribution is used to model the prior distribution of spectral amplitudes. An important aspect of this method is that since the data themselves are used as constraints, phase information can also be recovered and used to extend the data outside the original window. The objective function is formulated in terms of hyperparameters that control the degree of fit and spectral resolution. Noise rejection can also be achieved by truncating the number of iterations. Spectral resolution and extrapolation length are controlled by a single parameter. When this parameter is large compared with the spectral powers, the algorithm leads to zero extrapolation of the data, and the estimated Fourier transform yields the periodogram. When the data are sampled at a constant rate, the algorithm uses one Levinson recursion per iteration. For irregular sampling (unevenly sampled and/or gapped data), the algorithm uses one Cholesky decomposition per iteration. The performance of the algorithm is illustrated with three different problems that frequently arise in geophysical data processing: 1) harmonic retrieval from a time series contaminated with noise; 2) linear event detection from a finite aperture array of receivers [which, in fact, is an extension of 1)], 3) interpolation/extrapolation of gapped data. The performance of the algorithm as a spectral estimator is tested with the Kay and Marple data set. It is shown that the achieved resolution is comparable with parametric methods but with more accurate representation of the relative power in the spectral lines.

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