Restoration in the Presence of Errors

The observed antenna temperature due to a celestial source distribution differs from the true distribution in being smoother; various methods are known for operating on the observed distribution to gain better agreement with the true distribution. Current practice includes restoration by successive substitutions, by the chord construction, etc., but also shows a trend towards complete omission of any restoration, which is, no doubt, partly due to awareness that the true distribution is not fully determined by the observations. The isolation of a principal solution has, however, removed uncertainty as to the significance of nonuniqueness. The remaining question as to the degree of restoration which can be tolerated is solved here in terms of statistical properties of the errors. The solution shows that any proposed method of restoration must include mention of the character of the true distribution and of the errors; otherwise cases could be constructed where the proposed method led to deterioration. In the simplest case of independent random errors and a poorly resolved distribution, the rms level of the errors relative to the observed temperature is the relevant parameter (represented by (c/b)1/2 in Fig. 3); e.g., with a random error level as great as 15 per cent, agreement with the true distribution improves up to the third stage of restoration by successive substitutions (Table I) or on application of the chord construction.