The recorded AO image I of a science object is described by the following convolution equation if we ignore anisoplanatism:
where O is the science object intensity distribution.
The point spread function knowledge is required to a-posteriori
deconvolve AO images (or respectively do model-fitting) and
reconstruct the science object intensity distribution.
Section 3 (click here) has shown that
only a calibration during the observations
is able to get a good psf estimation.
Since seeing statistics are not stationary and the psf is directly
affected by the seeing changes, the psfs is a non-stationary
variable. Consequently, the psf calibration should be done
simultaneously (variable t) to the science object observations and in the
same viewing direction (variable 
).
Such a technique is not yet available but under testing;
Véran et al.
(1995) have estimated from simulations this new calibration technique
to be relatively accurate to a few per cent. However, this technique has
also some drawbacks (Tessier 1995).
The current technique is to calibrate the on-axis psf on a point source (referred to here as the calibrator source) in conditions as similar as possible to the astronomical target source one but sequentially. It is referred to as the sequential calibration technique. In AO, this means to get an equivalent correction on the point source. This will be fulfilled essentially if the WFS noise level is the same. Fortunately, this could be technically done. E.g. the AO system Come-on+ used here is well equipped for this purpose (Beuzit et al. 1994). However, the shape and the color of the source may still induce some differences.
When those precautions are taken, Tessier (1995) has shown on real data that the mismatch in the psf calibration mainly comes from the seeing differences between the observations of the astronomical target and its calibrator point source. The psf charactetistics shown in Sect. 3 (click here) demonstrate that the sequential psf calibration is similar to the calibration of the speckle transfer function. Both are sensitive to any temporal variations from the seeing (the so-called seeing effects) or the instrument. We also recall the seeing dependence with the airmass and so the elevation of the source. Some observing procedures are known to improve the calibration in speckle (Perrier 1988), e.g., selecting a calibrator close in the sky, observing both sources within a time as short as possible (a few minutes), and repeating this coming and going between the source and its calibrator a few times if possible. The lattest point is poorly compatible with CCD-like single very long-exposure shot, however, in the infrared, because of the background variations, exposure times are limited in general to five minutes. We have shown that this procedure identically improves the calibration efficiency in adaptive optics (Tessier 1995; Tessier & Perrier 1996). At last, for a good quality psf, the total integration time must be equal for both science object and its calibrator in order to get a comparable smoothness.
Thus the calibration quality depends on the observational procedure but still depends on the seeing stability which varies with the site and the turbulence conditions. From COP data, I have estimated that the calibration rms error is in relative quite constant over the psf extension and is typically between 5 and 10% of the signal when one applies the described procedure. On the other hand, if the calibration is not done properly, some bias (stronger halo, wider psf) up to 50% will show up. To summarize, for a good calibration, it is needed to integrate enough to have a good signal to noise for the halo and to sample well the seeing statistics to overcome the bad averaging capability of the central part of the psf. These points could be related to the peculiar psf statistics shown in Sect. 3 (click here). Roddier et al. (1995) have noticed for AO observations carried out at the Canada-France-Hawaii telescope that it was more difficult to calibrate the internal part of the psfs than the external part of the halo.
Blind deconvolution (e.g. Christou 1995) intends to reconstruct the science object intensity distribution with the associated synthetic psf directly from the AO image. However, these algorithms are not able to retrieve the true psf in general. This is not surprising since Sect. 3 (click here) has shown how complex the psf is. The problem is ill-conditioned and needs additional constraints. Thiébaut & Conan (1995) assume the circular symmetry for the psf, this allows them to reconstruct synthetic psfs. This assumption is not quite satisfactory in the sense that the deviation of the psf to the circular symmetry is currently larger than the uncertainties of the sequential calibration technique. But it could be very useful in the bad case when the psf is unknown, e.g. in the lack of any calibration data.
The sequential calibration technique is not perfect and its
accuracy is limited by the seeing statistics to about 5-10%
in relative. However, it is still a very valuable
information on how the psf looks like. Blind deconvolution cannot
replace this calibration, on the contrary, this information is really
needed to constrain near-sighted
deconvolution algorithms
which should be more powerful than standard deconvolution ones
(see Sect. 6 (click here)).
