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Next: THE M.E. FOR A SINGLE POINT SOURCE Up: The MEASUREMENT EQUATION of a generic radio telescope Previous: Contents

INTRODUCTION

The matrix-based Measurement Equation (ME) of a Generic Radio Telescope was developed by Hamaker, Bregman and Sault [2] [3], based on earlier work by Bregman [1]. After discussion by Noordam [5] and Cornwell [6] [7] [8] [9] [10] [11], the M.E. has been adopted as the generic foundation of the uv-data calibration and imaging part of AIPS++. In the not too distant future, an `official' AIPS++ description of the ME will be needed, with agreed conventions and nomenclature (see Appendix A). This note is a step towards that goal.

The heart of the M.E. is formed by the 2 x 2 feed-based `Jones' matrices, which describe the effects of various parts of the observing instrument on the signal. The main section of this document is devoted to describing the basic form of the Jones matrices in linear and circular polarisation coordinates. Another section discusses the conditions under which their order may be modified (matrices do not always commute).

It is expected that the details of the M.E. (and of this note) will be refined during the first few iterations of design and implementation of AIPS++. But the structure of the M.E. formalism as presented here appears to be rich enough to accomodate all existing and planned radio telescopes. This includes `exotic' ones like cylindrical mirrors, phased arrays, and interferometer arrays with very dissimilar antennas. Further refinements should only require the addition of new Jones matrices, or devising new expressions for existing matrix elements.

In order to test this bold assertion, the various institutes might endeavour to model their own telescopes in terms of the precise and common language of the M.E., using this note as a reference. The following `rules' are probably good ones:

In modelling an instrument, stay as close to the actual physical situation as possible. Violations of this principle, for whatever reasons, will lead to problems sooner or later.
It is counterproductive to try and simplify the M.E. to make it `look more tractable'. This practice introduces hidden assumptions, which tend to be forgotten by the programmer, and unknown to the user.
Use the suggested nomenclature and conventions.

It is also good to realise that there are two basic forms of ME, which should not be confused: In the physical form, each instrumental effect is modelled separately by its own matrix. This is useful for simulation purposes. In the mathematical form, effects are `lumped together' if they cannot be solved for separately. Example: the various contributions to the receiver gain, and tropospheric gain.

Acknowledgements: The author has greatly benefited from detailed discussions with Jayaram Chengalur, Jaap Bregman, Johan Hamaker, Tim Cornwell, Wim Brouw and Mark Wieringa.

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