The Effects of Errors on the Convergence of an Iterative Deconvolution Method

Abstract

Calculations carried out to investigate the van Cittert iterative deconvolution method and the effects of random noise and truncation errors on its convergence behavior are presented. Gaussian functions are used for "both the true function W and the system, response function A. The model "observed" function S is generated from W and A. Both rms differences between the result of n iterations Wn and the true function, and between Wn*A and S are used to measure convergence. The effects of introducing errors can be measured against standards set by the convergence without such errors. These "no-error" calculations make use of true functions with widths from 2.0 to 0.67 times the response function width. The effects of random noise are investigated by adding noise to W*A before deconvolution. A small amount of random noise initially added builds up rapidly in amplitude during the iterative process and eventually dominates the rms difference calculations. To suppress the effects of random noise build-up; smoothing techniques are applied, the best of which involved smoothing both the noisy observed function, S, and the system response function, A, before deconvolution. The smoothing operation is thus taken as part of the measurement and. the divergence resulting from, the noise build-up is avoided. The results depend strongly upon, the -width of the smoothing function. Uhsymmetric system response functions, similar' to those encountered in soft x-ray appearance potential spectroscopy and in x-ray continuum, isochromat measurements, are used in investigations of truncation errors. Abrupt cut-offs of the model S and A functions before deconvolution result in the build-up of large fluctuations in W . These truncation errors become increasingly localized with continued iterations and make only minor contributions to the errors in Wn in the vicinity of the real peak if the truncations are made sufficiently far from the peak location. Alternatively, the truncation errors can be avoided by analytical continuation.