,
S. Guilloteau
and
F.
Viallefond
e-mail: gueth@iram.fr
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F. Gueth,
S. Guilloteau and
F.
Viallefond
e-mail: gueth@iram.fr
Institut de Radio Astronomie Millimétrique, 300, rue de
la Piscine, F-38406 St Martin d'Hères, FRANCE
DEMIRM, Observatoire de Paris, 61, avenue de l'Observatoire, F-75014
Paris, FRANCE
Simulations show that this method gives correct clean maps and recovers most of the flux of the sources. The introduction of the short-spacing visibilities in the data set is strongly required. Their absence actually introduces artificial lack of structures on the corresponding scale in the mosaic images. The formation of ``stripes'' in clean maps may also occur, but this phenomenon can be significantly reduced by using the Steer-Dewdney-Ito algorithm (Steer, Dewdney & Ito (1984)) to identify the CLEAN components. Typical IRAM interferometer pointing errors do not have a significant effect on the reconstructed images.
The field of view of aperture synthesis observations is limited by the size of
the primary beams of the antennas. Whereas this is not critical in centimeter
interferometry (field of view 
), it becomes a very strong constraint in
the millimeter range. In the case of the
IRAM millimeter
interferometer of Plateau de
Bure (Hautes Alpes, France), the 15-m dishes
limit the field of view to about 50 arcsec at 
mm. To map
larger fields, one has to perform a mosaic, in which several overlapping
fields are observed in a sequence which guarantees some homogeneity in terms of
coverage and noise level (see, for example, Cornwell (1994)).
One of the most important step in the analysis of interferometric data is the deconvolution of the image, necessary to reduce the effects of the very strong sidelobes of the observed ``dirty beam'' and thus to obtain a map which allows a reliable interpretation. Two methods are used for the deconvolution of radio synthesis images: the CLEAN algorithm (Högbom (1974),Clark (1980)) and the Maximum Entropy Method (see, for example, Narayan & Nityananda (1986)).
So far, only MEM techniques have been used to deconvolve mosaics (Cornwell (1988)). We present here a CLEAN-based method for the mosaic deconvolution.
The maps produced by an interferometer are given by the following measurement equation:
where 
is the map, 
the dirty beam, 
the primary beam, 
the sky
brightness distribution, 
the noise distribution, and 
indicates a
convolution product.
The CLEAN algorithm attempts to replace directly 
by a clean gaussian beam.
Note that it is possible to correct for the primary beam attenuation after the
deconvolution by a simple division.
To build a mosaic, one has to make the sum of the different observed fields and to correct for the primary beam attenuation, which is inhomogeneous over the whole region covered by the observations. For optimum results from the signal to noise point of view, this operation is done by a weighted sum:
where the subscript 
corresponds to the field number 
, and 
is
the noise level: 
. 
is here
directly homogeneous to a sky brightness distribution.
The mosaic reconstruction has to be done before the deconvolution, because of the non-linearity of the CLEAN or MEM algorithms. The joint deconvolution actually gives better results than a mosaic of individually deconvolved fields, because the adjacent pointings reinforce each other in the estimation of the missing spacings (see Ekers & Rots (1979),Cornwell (1988)), and of course because of the improvement in the signal-to-noise ratio due to the redundancy of the observations.
But the equation 2.2 is not a simple convolution equation, and does not allow to directly and properly replace the dirty beam by a clean beam. This is the reason why a specific CLEAN method had to be developed for the mosaic deconvolution.
In practice, we reconstruct the mosaic in a slightly different way. To avoid contaminating field centers by (small, but possibly significant) errors coming from poor knowledge of the far-off center beam pattern from nearby fields, we use the following expression:
where 
is a truncated version of the primary beam, down to some
arbitrary level, typically 10 to 30 %.
The noise level in 
is not constant:
At the edges of the mosaic, where the 
are low, the noise is thus very
high. This is a basic problem in mosaic deconvolution: if we intend to apply
CLEAN on 
, the risk of taking a noise peak as a CLEAN component is too
important. Instead, we construct the following expression:
In the limit of identical noise 
on all fields, one obtains:
is essentially the signal to noise ratio at each point. Thus, locating
CLEAN components from 
is a safer procedure than from 
.
The proposed algorithm is then, for each iteration 
:
from the maximum of 
(or 
) and find the corresponding value 
of 
.
, i.e. subtract from 
the contribution of a supposed point
source at 
:
where 
is the loop gain. 
can be calculated in the same way.
Note that it is possible to use the different enhancements of CLEAN (e.g.
Clark or Steer-Dewdney-Ito variants) in the process, the basic idea being
always to find the position of the components on 
and to correct 
.
However, the Multi Resolution Clean (Wakker & Schwarz (1988)) seems not to be easily
adaptable, since it requires a convolution equation.
Figure 1: Left: Model of a sky brightness distribution. Right:
Simulation of a ten fields mosaic observed with the IRAM Plateau de Bure
interferometer. The separation between the adjacent pointings is half the
primary beam. The clean beam is 
.
To check and test this algorithm, various simulations have been performed. The flow-chart of these simulations is the following:
Figure 1 shows one of the model and the corresponding map obtained with
a mosaic of ten fields. The algorithm reconstructs correctly the images, and
recovers most of the flux of the sources (
% in this example). These
results are better than those obtained in the same conditions with MEM
techniques, which seem to need a better 
coverage than the one available
with the Plateau de Bure interferometer.
Figure 2: Left: Model of a sky brightness distribution. Middle:
Simulation of a ten field mosaic observed with the PdBI. Right: The
same with the short spacing information: the continuous structure is
reconstructed.
The problem of the short-spacing information is well-known in aperture synthesis: due to the physical size of the dishes, the shortest baselines cannot be observed with an interferometer. This usually limits the maximal size of a structure it is possible to map in a single-field observation. But is appears to be even more important in the case of mosaics, where the absence of the short spacings introduces artificial lack of structure on intermediate scale. Figure 2 shows, on a particular example, how a continuous large source can be split into several structures, which are only reconstruction artefacts.
The correct way to get rid of this problem is to observe the same object with a single-dish antenna that is larger than the shortest baseline available, or with a smaller interferometer, and to combine these observations with the interferometric data. The third image of Figure 2 corresponds to such a simulation: the continuous structure is now reconstructed in a correct way.
Figure 3: Left: Mosaic deconvolved with the Clark algorithm, and a very
high gain loop, so as to more clearly show the stripes formation. Right:
The same simulation, deconvolved with the SDI-Clean and the same parameters.
Another phenomenon that occurs in CLEANed maps is the so-called ``stripes formation'' (Schwarz (1984)). This can however be significantly reduced by using the Steer, Dewdney & Ito (1984) enhancement of CLEAN. Figure 3 shows the improvements in the image quality when using SDI-Clean instead of the Clark algorithm.
Simulations including pointing errors have also been performed, each field of
the mosaic being shifted randomly. This global error on the position of the
whole field maximizes the effect. Rms shifts of 1/10 of the primary beam (i.e.
the typical pointing error at Plateau de Bure) do not change significantly the
images obtained after deconvolution. Important modifications of the
reconstructed structures occur with pointing errors of 
of the primary
beam.
A new CLEAN-based algorithm for mosaic deconvolution has been developed.
Simulations of Plateau de Bure observations and image reconstruction show the
efficiency of the method, which recovers in a correct way both the structure
and the flux of the sources. It actually seems to give better results than MEM
techniques in the case of the ``sparse'' 
coverages obtained with the
Plateau de Bure interferometer.
The most important artefacts in the final maps are due to the missing short spacing information. Its introduction in the data set (single-dish observation) is then strongly required, especially for observations of very large and continuous structures.
This deconvolution method will allow now the use of the Plateau de Bure interferometer for mapping large objects (e.g. galaxies, molecular outflows).
