In radio astronomy, the Fourier transform of the sampled visibilities with no phase errors yields the dirty image
where 
is the visibility function and 
is the sampling
function. By the convolution theorem, multiplication by the sampling
function 
leads to a convolution of the true image by a point
spread function which is given by the Fourier transform of 
.
In the presence of antenna based phase errors 
,
where 
represents the combined effects of the phases in the
Fourier plane. Hence, the dirty image will be the true image
convolved with a point spread function given by
The problem with this formalism is that we do not know 
.
There is a nice analog to this approach in optical astronomical
imaging, in which the resolution is limited by phase fluctuations in
the atmosphere which are generally too fast to correct. A typical
optical field of view contains several bright stars, and the profiles
of these stars can be used to derive an effective point spread
function representing the statistical effects of the atmosphere. The
phase errors are occurring so quickly that we have thousands of
independent instantiations of the phase errors, and even though the
phase errors are not known in any detail, the form of 
can be
determined. Some degree of superresolution can then be achieved by
deconvolving the effects of this point spread function from the entire
image.
The situation at millimeter frequencies is different from the optical situation in two respects: we will have tens of crossing times of the turbulence over our aperture instead of thousands, and it will be rare to encounter bright point sources in the field of interest (Holdaway, Owen, and Rupen, 1994). Holdaway and Owen (1995) have recently analized the residual phase errors which result from imperfect atmospheric cancellation when switching between the target source and a nearby calibrator. Phase fluctuations which occur faster than the switching time scale cannot be corrected, and will result in significant decorrelation if the residual phase errors are about 30 degrees or larger. However, it is possible to use the statistics of the phase errors as measured on a calibrator to simulate an effective point spread function which would include the effects of both the incomplete Fourier sampling and the phase jitter. We can determine the phase structure function from the calibrator phase time series, which allows us to construct a model phase screen which will have the same statistical properties as the actual atmosphere, but which will not have the correct detailed phases. Hence, deconvolving with the point spread function which includes phase errors from this model atmosphere can correct for the decorrelation, as can the amplitude correction scheme. As in the amplitude correction scheme, details of the phase time series derived from the model atmosphere will be wrong, so errors will be made. However, on average, the model phases will affect the point spread function in a manner which is representative of how the actual target source phase errors scatter flux in the target source. We found that superior results were achieved when we calculated the effective point spread function, including phase errors, from several (ten) different model atmospheres, averaged the different model point spread functions, and then deconvolved the dirty image with this effective point spread function.
This method, or the statistical deconvolution of phase errors, is quite similar to correcting the decorrelated amplitudes. After averaging several effective PSF's, the phases will be small, so the effective PSF will be dominated by the Fourier sampling and the amplitude decorrelation. Consider the case of perfect Fourier sampling, so the effective PSF is due entirely to the amplitude decorrelation. Deconvolving by this function is equivalent to dividing the Fourier transform of the image by the Fourier transform of the effective PSF, or boosting up the amplitudes of the outer visibilities as performed when correcting the decorrelated amplitudes.
In both the amplitude correction and the statistical deconvolution of phase errors schemes, the results are dependent upon the post observation averaging time in a complicated manner which has not yet been fully explored.