Robust deconvolution by modified Wiener filtering

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Abstract
Deconvolution in the presence of additive noise is a well known problem for which there exists a Wiener filter which spectrally whitens while also suppressing the noise. A simple variant of this standard Wiener filter incorporates a parameter p which is intended to allow further weight to be given to noise suppression. This filter is often called a modified Wiener filter. To design such a filter, one must know the frequency characteristics of the wavelet precisely, plus the spectra of the input and additive noise. Typically, some appropriate estimate of the frequency function of the wavelet is taken, and the modified Wiener filter is designed from that estimate. A more realistic practical viewpoint is to think of the estimated wavelet response as one of a set of possible frequency response functions. By using statistical information obtained during wavelet estimation, a value of p can be chosen which gives a modified Wiener filter equivalent to a statistically robust deconvolution filter. Here “robust” means that the error criterion which defines the deconvolution filter allows for the set of possible wavelet frequency functions. Two different error criteria are considered: (1) the minimization of the average mean‐squared error, and (2) the minimization of the maximum mean‐squared error. Deconvolution using an estimated wavelet can thus be made robust to wavelet uncertainties in an easily followed technique.

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