Map Making with Non-Phase Stable Interferometers

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.