Aerial Smoothing in Radio Astronomy

Abstract

SummaryWhen an aerial is used to survey the distribution of radio brightness over the sky, the observed distribution is smoother than the true distribution; the broader the beam of the aerial, the greater the smoothing. It is shown that the aerial does not register those spatial Fourier components of the true distribution having frequencies beyond a cut-off determined by the aerial aperture. Components of lower frequency are registered but their relative strengths are altered.Two important consequences follow. (i) There are invisible distributions which produce no response when scanned by the aerial. Consequently there is not a unique solution to the problem of correcting for aerial smoothing. The established method of correcting by successive smoothing, leads to the principal 8olution, in which Fourier components accepted by the aerial have been restored to their full values, but the components rejected by the aerial are still not represented. (ii) In conducting a survey it is sufficient to observe at discrete intervals. The measuring points must be closer together than half the period of the Fouri,er component at cut-off. For an aperture of width w, this peculiar interval is equal to fA/w (radians).

A multi-scale multi-frequency deconvolution algorithm for synthesis imaging in radio interferometry

“…In the case of perfect sampling (where the Hessian blocks are truly 7 Even if [H] were invertible, it is impractical to evaluate and invert the full Hessian (each row of each Hessian block represents an image). 8 The principal solution (as defined in Bracewell & Roberts (1954) and used in Cornwell et al (1999)) is a term specific to radio interferometry and represents the dirty image normalized by the sum of weights. It is the image formed purely from the measured data, with no contribution from the invisible distribution of images (unmeasured spatialfrequencies).…”

Brightness Distribution in Discrete Radio Sources. I. Observations with an East-West Interferometer.

“…The recovery of the source brightness distribution is therefore not complete, since it lacks the fine detail contributed by the unknown Fourier components of high spatial frequency. The smoothing of a source distribution by omission of its highfrequency Fourie~ components has been discussed in detail by Bracewell and Roberts (1954). The resolution obtainable with finite apertures of different shapes has been discussed in the literature (see, for example, Covington and Harvey 1959;Ryle and Hewish 1960;Bracewell 1961 (15) integrating T'(x,y) over y,…”

Chord Construction for Correcting Aerial Smoothing

“…where the bars indicate Fourier transforms, and therefore for ]'ourier components such that 0 <s < tb-I, where b-l is the cut-off value of s (see Bracewell and Roberts 1954), and partial correction may be expected for components of higher spatial frequency up to b-l . As shown by Bracewell and Roberts Ta(CP) will not contain Fourier components with more than b-1 waves per unit of cpo For this case, then, Wf~ find h,=-t,…”

“…The general problem of aerial smoothing in radio astronomy has been discussed by Bracewell and Roberts (1954) with a view to clarifying the situation. In that paper current practice as regards the correction of smoothing was critically examined.…”