Aperture synthesis is the method used by astronomers to determine the accurate brightness distribution of the radio sky with a resolution much better than that possible with a single large antenna. The technique, now over a decade old, utilizes a large number of connected radio antennas, some of them physically moveable, to follow a region of sky for many hours or days in order to sample the spatial coherence function of the radiation field over a sufficiently large area and with a reasonable filling factor. Landmark references for aperture synthesis are McCready et al. (1947), Stanier (1950), Christiansen and Warburton (1955), Lequeux et al. (1962), Read (1961) and Ryle and Hewish (1960).

A convolution technique for the reduction of three dimensional interferometer measurements to two dimensions is described. With the addition of relatively simple hardware to a general purpose computer the technique allows fast, efficient processing of three dimensional data.

The effect of errors in an aperture synthesis map is intimately related to the geometry of the synthesis instrument and the properties of the two-dimensional Fourier transformation. This paper concentrates on E-W rotational synthesis, the simple geometry of which allows a systematic treatment. After some preliminary remarks about error statistics and tolerances, the relevant properties of the Fourier transformation in polar coordinates are first reviewed. A classification of error types as “global”, “circular”, “radial” or “local” is introduced, based on the geometry of the area in the aperture over which they are coherent. A few special errors not fitting into this classification are also discussed. The characteristics of the corresponding patterns in the synthesized beam are described and illustrated with computer simulations and real-life examples.

Often, particular unnatural-looking patterns appear in a synthesis map. An experienced observer can recognize the type of error that produced them and may even be able to apply the corrections necessary to get a better-looking map. A systematic approach is described in this paper: A model of the errors most likely to occur is fitted to the patterns in the map, and the fitted parameters are used to apply corrections. The presence of a point source that can serve as a reference is required. This requirement limits the applicability of the method, but is met by quite a few objects of interest. Where interference from other sources would be too strong, their influence may be reduced by CLEANing the map prior to model fitting. One example of the results for atmospheric position scintillation and one for baseline calibration errors will be shown.

For a short description of the problem posed by missing information at short interferometer spacings in aperture synthesis we would like to refer to the introduction of the previous paper by Ekers and Rots. The solution that is suggested in this paper is not a true interpolation but somewhat intermediate between interpolation and Clean.

The effects of the missing short spacings in correlation interferometers measurements on the synthesized antenna pattern and on the maps produced, are analysed. A method for correcting those effects, based on the obtention of the missing components from single dish observations, is described. The procedure has been developed for the WSRT observations of extended objects and it has been applied on several cases. Some results are shown for M31 and NGC4258.

Aperture synthesis instruments providing a generally highly uniform sampling of the visibility function often leave an unsampled hole near the origin of the (u,v)-plane. One of the common configurations of the Westerbork Synthesis Radio Telescope (WSRT) provides sampling in concentric rings with radii of n × 18 m. However, the samples with n=0 and n=1 cannot be obtained directly with the instrument because of rather obvious physical constraints. This missing short spacing information causes a central depression of the synthesized beam pattern, resulting in large negative, bowl shaped areas around (especially) extended sources. Very large sources will only partially be present in the maps. Annoying as this can be for continuum observations, in the case of spectral line data it causes serious distortions of the individual line profiles.

Much ingenuity has been devoted to the determination of mutual coherence functions at radio wavelengths. Measurements of the amplitude and phase of the interference fringes with uncertainties of only 1 percent are now attainable. Unfortunately the atmosphere intervenes between the measured quantities and the mutual coherence function itself and the phase uncertainties introduced by the ionospheric and tropospheric irregularities are often the factor limiting the quality of radioastronomical maps.

The ability to make reliable maps of radio sources at any available frequency and on all feasible angular scales is obviously a fundamental goal of observational radio astronomy. Unfortunately, until very recently, our map-making capability has been extremely restricted, limited in fact to those physical baselines and frequencies where phase-stable interferometers could be operated. However it is now becoming increasingly clear that phase stability is not an essential prerequisite for reliable mapping. The basic point is that whereas in principle there are an infinite number of brightness distributions which could give rise to the observed amplitudes of a set of Fourier components whose phases are unknown, in practice the a priori information that the required distribution is real and positive severely constrains the range of possibilities. Both Bates and his co-workers (e.g. Bates and Napier 1974) and Ross et al. (1978) have pointed out that the brightness distributions we are seeking are examples of ‘entire’ functions. In particular it is known that the real and imaginary parts of such functions are not independent (see references in Ross et al.) and that by studying the positions of their complex zeroes one may well be able to deduce the brightness distribution from knowing only the modulus of its Fourier transform. Unfortunately such a rigorous approach appears to be rather difficult to implement in practice. However the results of Napier and Bates (1974) have confirmed that in two dimensions reliable structure determinations can be made without phase information.

A two step procedure is developed for the determination of the angular structure of the compact components of radio sources from observations made with very long baseline interferometry (VLBI). The first step is to fit a model composed of circularly symmetric Gaussian components to the observations. Then this, or any other, model and closure phase information is used to predict visibility phases which, when paired with the corresponding visibility amplitude observation, are combined in a direct synthesis; the resulting map is deconvolved from the point source response by means of the “CLEAN” procedure. If necessary, the “cleaned” components can be used as a starting model to predict visibility phases and the process iterated. The initial model in many cases may be as simple as a point source. Convergence of this process appears to be quite rapid, requiring only a few iterations when the source is simple or the starting model is good. These two steps for source mapping can be used either together or separately. Examples are given using data generated from test models.

The use of entropy as a basis for object/image reconstruction procedures is not new, but with the appearance of new, faster algorithms the actual use of these algorithms for the reconstruction of objects from ‘real’ data is likely to increase.

The purpose of this contribution is not to discourage such applications, but to illustrate that, under certain circumstances, there is a need for caution in interpreting the results obtained from such algorithms. Specifically, we shall show that the application of statistical methods to problems of object reconstruction, in situations where only the modulus of the object Fourier transform is known, could lead to wholly false conclusions. Indeed, we shall primarily be concerned here with situations for which there is no ‘correct’ solution. In such situations it is pointless to speak of ‘safe’ object reconstruction algorithms. The important point here is that the user of a statistically based ‘object reconstruction algorithm’ may be totally ignorant of whether or not he is working in this régime.

In radio mapping, one of the problems encountered is the random bias in the visibility estimate. The bias can be divided into two parts: (a) the positive bias due to the common sky background seen by all elements of the interferometer, and (b) the negative bias due to phase noise present in the system. The first kind of bias can be easily removed by subtracting the correlation between the signals at two interferometer sites when the source is not in the antenna beams from that measured with the source in the antenna beams. This bias will therefore not be considered here. In contrast, the second kind of bias is more difficult to remove. When the signal-to-noise ratio of the interferometer system is high, incoherent averaging techniques can be utilized in the fringe frequency or in the time domain (Clark et al., 1969; Moran, 1973).

The rapid growth, over the last two decades, of the amount of digital arithmetic capability available for a given cost is obvious to all, and is apparently not yet at an end. Therefore, any discussion of the best equipment and even of the best algorithms with which to attack a given problem will soon become obsolete as decreasing costs bring things previously considered inconceivable within economic feasibility. The general trend is that, as the cost of hardware decreases, the discussions of algorithms and procedures becomes simpler, as fewer approximations have to be made, and the mathematically simple correct formulae may be directly implemented.

Digital techniques for the direct transformation of rotational synthesis data are discussed with particular emphasis on current cost and technological limits. A prototype system designed to test various approaches will be described and its application to “clean” and maximum entropy reconstructions considered.

In aperture synthesis the formation of an image involves the two steps of spatial correlation across an aperture and transformation to the image. This is closely related to conventional imaging with a lens (Cole, 1977a), which Abbé interpreted as two successive transformations at the surfaces of the lens. With the simple lens the image is the light intensity in the output plane (Figure 1(a)). In aperture synthesis (Figure 1(b)) the image is the transform of the correlation but no detection takes place. The image corresponds to the ‘light’ amplitude rather than intensity.

Many of the image restoration algorithms discussed during these past days work best on fields which are virtually empty except for a few discrete sources. But the sky also has faint regions of extended emission; these regions invariably turn out to be of great astrophysical interest and must be represented as accurately as possible in the maps. The methods chosen to present the data in pictorial form can have an important effect on the speed and efficacity with which the astronomer can extract the useful information. I would be afraid of boring all of you with this discussion of display methods if I did not know that, besides being well versed in subjects of probability, statistics, and applied mathematics (and in some cases philosophies and polemics), you are also astronomers with a strong motivation to sift out of your data the useful information on the physics of cosmic radio sources, and to do it in the most efficient way possible.

We describe the processing and display of aperture synthesis maps of the H I absorption against the radio source Cas A. Then data were obtained with the NRAO interferometer and constitute 384 narrow band maps of Cas A combined to produce a picture of the optical depth and mean velocity of the H I absorption due to the Perseus arm.

Both digital and analogue data techniques have been used to Fourier transform, combine and display the data and the merits and usefulness of each are described.

The original design of the Culgoora radioheliograph (Wild, 1967; Sheridan et al., 1973) incorporated a number of features, such as high time resolution and limited declination coverage, which were particularly suited to its primary function of observing the Sun. Although some of these features are disadvantageous for cosmic work, the instrument has nevertheless proved useful for a variety of non-solar studies as well.