We investigate robust estimation of two-dimensional power spectra of signals which are adequately represented by Gaussian random field models but for which we have imperfect observations. Two situations of particular interest occur when the contaminating noise is additive and when the contaminating noise appears in the innovations. In these cases, the observed data are not Gaussian and conventional procedures are no longer efficient.
To estimate the parameters of the signal model from the contaminated data, we describe two new procedures which were originally proposed for estimation of scale and location from independent data and adapted to one-dimensional autoregression parameter estimation by previous researchers. The first algorithm is a robustification of least squares and equivalent to an iterated weighted least squares problem where the weights are data dependent. Known as the generalized maximum likelihood (GM) estimator, its analysis is accomplished by the use of a so-called “influence function” or directional derivative of the estimator in the direction of the contamination. We compute expressions for relative efficiency of the estimator using the influence function and specify criteria for selection of the estimator’s robustifying functions.
The second algorithm is an iterative procedure known as a filter cleaner. This procedure is shown to be approximately equivalent to an optimal minimization problem.