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Next: THE CASE OF A CENTRAL POINT SOURCE Up: Some practical aspects of the matrix-based MEASUREMENT EQUATION Previous: Contents

INTRODUCTION

Over the last few years, various people have suggested that the Measurement Equation of a radio telescope (which includes interferometers and single dishes) can best be described in the form of matrices. In 1993, Bregman [Bregman93] and later Hjellming [Hjellming93] suggested that such a `universal' formalism is essential for AIPS++, which is supposed to deal with all the Consortium telescopes, both by themselves and in combination with each other (e.g. VLBI). Thus, a good formalism should be adopted from the start. They provided an initial formulation based on 2 x 2 instrumental response matrices, and a 2 x 2 coherency matrix. Later that year, Hamaker realised that the `direct matrix product' provided an elegant way to link a 4-dimensional coherence response matrix of an interferometer with the 2 x 2 response matrices of a single antenna. This also opened the way to a uniform treatment of radio and optical polarimetry, where Stokes and coherence 4-vectors are related by such interferometer matrices. For a full account, see the two papers by Hamaker et al [Hamaker95], and Sault et al [Sault95].

Since the work mentioned above limits itself to uv-domain effects, while ignoring image-plane effects, it is strictly speaking only valid for the case of a point source in the centre of the field. However, it is a good approximation for the many actual cases where a compact source dominates the field. It correctly describes most of the important differences between the various Consortium telescopes, like alt-az and equatorial mounts, and linearly and circularly polarised feeds. Therefore, this uv-domain Measurement Equation is used in section 3 to demonstrate the generic nature of the formalism by giving a `catalog' of practical applications to existing telescopes.

The full Measurement Equation, which includes image-plane effects, was developed by Bregman during an AIPS++ workshop in Dwingeloo in June 1995. Section 4 gives a first description of it for the benefit of AIPS++ developers. It will be published in the open literature next year [Bregman96].

It should be emphasised that some of the matrix elements in this paper may not have reached their final form (particularly in the description of the antenna beam). But the overall structure of the formalism seems solid enough for AIPS++ to start building on. Some implementation issues are discussed. There is also an outline of a `universal' Solver for parameters of the Measurement Model and Sky Model.

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