Estimation of the discrete Fourier transform, a linear inversion approach

Spatio‐temporal analysis of seismic records is of particular relevance in many geophysical applications, e.g., vertical seismic profiles, plane‐wave slowness estimation in seismographic array processing and in sonar array processing. The goal is to estimate from a limited number of receivers the 2-D spectral signature of a group of events that are recorded on a linear array of receivers. When the spatial coverage of the array is small, conventional f-k analysis based on Fourier transform leads to f-k panels that are dominated by sidelobes. An algorithm that uses a Bayesian approach to design an artifacts‐reduced Fourier transform has been developed to overcome this shortcoming. A by‐product of the method is a high‐resolution periodogram. This extrapolation gives the periodogram that would have been recorded with a longer array of receivers if the data were a limited superposition of monochromatic planes waves. The technique is useful in array processing for two reasons. First, it provides spatial extrapolation of the array (subject to the above data assumption) and second, missing receivers within and outside the aperture are treated as unknowns rather than as zeros. The performance of the technique is illustrated with synthetic examples for both broad‐band and narrow‐band data. Finally, the applicability of the procedure is assessed analyzing the f-k spectral signature of a vertical seismic profile (VSP).