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Is this a useful abstraction of calibration? Imaging Model is useful principally because we know that we will call different types of Imaging Model from various types of Imagers. Similarly, we will call Telescope Model from self-calibrating Imagers so having a concise description of calibration helps considerably. Perhaps more important however is the conceptualization of calibration as a non-linear machine. Solving for calibration changes the internal state of the machine, and applying calibration invokes the machine on some data.

I argued above that calibration could be split into linear, mildly non-linear and fully non-linear cases. This means that it would be a good idea to develop a facility in AIPS++ for these forms of optimization. It would be best to do this now, ahead of any significant work in implementing calibration.

To summarize the net effect of Imaging Model and Telescope Model, I can now describe the general structure of the deconvolution and self-calibration procedures. For deconvolution alone, algorithms look like this:

1.

Select an initial model I for the sky,

2.

Use the residual service of Imaging Model to find a residual image A^Tw(AI - D),

3.

Use the residual image in some algorithm-dependent way to update the current estimate of I,

4.

Return to step 2 unless convergence is obtained.

For deconvolution/self-calibration, algorithms will follow the following general pattern:

1.

Select an initial model I for the sky,

2.

Use the predict service of Imaging Model to predict the data $\widehat{D}$,

3.

Use the solve service of Telescope Model to find the parameters P,

4.

Use the apply service of Telescope Model to correct the data: $\widehat{D}$ = C^-1(D, P),

5.

Use the residual service of Imaging Model to find a residual image A^Tw(AI - $\widehat{D}$),

6.

Use the residual image in some algorithm-dependent way to update the current estimate of I,

7.

Return to step 2 unless convergence is obtained.

Variants on this exist. For example, the initial model may be generated using an approximately calibrated dataset. Furthermore, steps 3 and 4 may involve many cycles of a deconvolution procedure.

In the terminology introduced by Holdaway and Bhatnagar, both deconvolution and deconvolution/self-calibration algorithms are of the class of Imagers.

If AIPS++ decides to use this formulation of Telescope Model (which is more or less that selected at Green Bank: Cornwell and Shone, 1992) the next concern is to build the internals for many different types of Telescope Models. The interesting questions then become ones of sharing class designs and concepts amongst different Telescopes. It is this step which has consumed much time and threatens to lead one into the old mistake of trying to design a very general model which can then be specialized to particular cases. I think that the answer to this temptation is just to design fairly simple Telescope Models specialized to particular Telescopes. Too much agonizing about a putative common calibration scheme will lead to paralysis.

Finally, I would like to discuss how this can be applied to very difficult imaging problems in which the calibration intimately affects the way the imaging is performed. An example of such a problem is non-isoplanatic selfcalibration/deconvolution. In such cases we can assign the Imaging Model calibratable parameters P so that we have the matrix equation:

D = A(P)I (8)

The Telescope Model is therefore responsible for creating an Imaging Model with estimates for some parameters. In the case of non-isoplanatic imaging, these parameters would be the phases of various parts of the sky at various times. The imaging proceeds thus:

1.

Select an initial model I for the sky,

2.

Select an initial Imaging Model A(P),

3.

Use the predict service of Imaging Model A(P) to predict the data $\widehat{D}$,

4.

Use the solve service of Telescope Model to find the parameters P,

5.

Generate a new Imaging Model A(P),

6.

Use the apply service of Telescope Model to correct the data: $\widehat{D}$ = C^-1(D, P),

7.

Use the residual service of Imaging Model A(P) to find a residual image A(P)^Tw(A(P)I - $\widehat{D}$),

8.

Use the residual image in some algorithm-dependent way to update the current estimate of I,

9.

Return to step 2 unless convergence is obtained.

In this approach calibration yields two types of parameters, those independent of imaging (e.g. flux scale, overall positions, etc.) and those involved in imaging (phases of patches of the sky as a function of time).