Spectral Estimation From Irregular Arrays

The problem of spectral estimation from data sampled at irregularly distributed measurement locations is considered from the point of view of an optimal fit of a spectral model when the amount of data available is large but finite. Several methods are shown to be asymptotically equivalent to the optimal one (as the amount of data becomes large), but several others, previously advocated, are shown to be either biased estimations or to have significantly larger estimation uncertainties than the optimal one. This problem is then reconsidered in a Bayesian formulation. The estimation problem is shown to be a combination of the prior prejudices of the investigator and the ability of the data to yield further information beyond the initial prejudices. In particular, the choice of the spectral model is best made in the context of this combination. The problem of an optimal array design is also considered: given a set of prior prejudices and a spectral model to be tested, what is the distribution of measurement points which yields the maximum amount of spectral information per measurement? An extensive set of optimal arrays is presented, with particular focus on red spectra (i.eo, ones monotonically decreasing with wavenumber), Because the optimal estimation procedure can be computationally cumbersome for large arrays, an efficient approximation is developed for red spectral regimes. Finally, a comparison of our estimation procedure with other methods is made, in which it is shown that an analysis of bias, aliasing, and estimation uncertainty can be made in a unified framework for a large class of methods.