15.3 Mosaicing in practice

 

Observation and calibration

The observation of a mosaic with the Plateau de Bure interferometer and the calibration of the data do not present any specific difficulties. We just mention here a few practical remarks:

• As shown in the previous paragraph, the optimal spacing between adjacent fields is half the primary beam width. Larger separations can be used (e.g. to map larger field of view in the same amount of time) but the image reconstruction is not optimal in that case. Note that if the two receivers are used simultaneously, the field spacing has to be adapted to one of the frequencies, which results in an over- or undersampling for the other one.

• Even if this is not formally required by the reconstruction and deconvolution algorithm described in the following section, it seems quite important to ensure similar observing conditions for all the pointing centers. Ideally, one would want the same noise level in each field, so that the noise in the final image is uniform, and the same uv coverage, to avoid strong discrepancies (in terms of angular resolution and image artifacts) between the different parts of the mosaic. To handle these constraints in practice, the fields are observed in a loop, each one during a few minutes (similarly to snapshot observations of several sources): hence, atmospheric conditions and uv coverage are similar for all the fields.

• In most cases, a mosaic will not be observed during an amount of time significantly larger than normal projects. As the observing time is shared between the different pointing centers, the sensitivity of each individual field is thus smaller than what would have been achieved with normal single-field observations. Note however that the sensitivity is further increased in the mosaic, thanks to the strong overlap between the adjacent fields (see Fig. 15.1).

• The maximal number of fields it is possible to observe in a mosaic is limited by observational constraints. The fields are observed in a loop, one after the other, and to get a reasonable uv coverage within one transit, only a limited number of fields can be observed. With the Plateau de Bure interferometer, the limit seems to be around 15 fields. Mosaicing even more fields would probably require some other approach (e.g. mosaic of several mosaics). Finally, a potential practical limitation is the disk and memory sizes of the computers, as mosaicing requires to handle very large images.

• The calibration of a mosaic is strictly identical with any other observation performed with the Plateau de Bure interferometer, as only the observations of the calibrators are used. At the end of the calibration process, a uv table and then a dirty map are computed for each pointing center.

Mosaic reconstruction

The point is now to reconstruct a mosaic from the observations of each field, in an optimal way in terms of signal-to-noise ratio. Just forget for the time being the effects of the convolution by the dirty beam. Each field i can thus be written: $F_i = B_i \times I + N_i$, where B_i is the primary beam of the interferometer, centered in a different direction for each observation i, and N_i is the corresponding noise distribution. In practice, the same phase center (i.e. the same coordinate system) is used for all the fields. We are thus in the classical framework of several observations of the same unknown quantity I, each one being affected by a weighting factor B_i. The best estimate of I, in the least-square sense, is thus given by:

$$\widetilde{I} = \frac{\displaystyle \sum\nolimits_i \frac{B_i... ...\,F_i}{\displaystyle \sum\nolimits_i \frac{B_i^2}{\sigma_i^2}}$$ (15.9)

where the sum includes all the observed fields and $\sigma_i$ is the rms of the noise distribution N_i. (Note that in Eq.15.9 as well as in the following equations, $\sigma_i$ is a number while other letters denote two-dimensional distributions).

Linear vs. non-linear mosaics

The problem which remains to be address is the deconvolution of the mosaic. This is actually the main difficulty of mosaic interferometric observations. Two different approaches have been proposed (e.g. [Cornwell 1993]):

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Linear mosaicing: each field is deconvolved using classical technics, and a mosaic is reconstructed afterwards with the clean images, according to equation 15.9.

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Non-linear mosaicing: a joint deconvolution of all the fields is performed, i.e. the reconstruction and the deconvolution of the mosaic are done simultaneously.

The deconvolution algorithms are highly non-linear, and the two methods are therefore not equivalent. The first one is straightforward to implement, but the non-linear mosaicing algorithms give much better results. Indeed, the combination of the adjacent fields in a mosaic allows to estimate visibilities which were not observed (see previous paragraph), it allows to remove sidelobes in the whole mapped area, and it increases the sensitivity in the (large) overlapping regions: these effects make the deconvolution much more efficient.

Non-linear deconvolution methods based on the MEM algorithm have been proposed by [Cornwell 1988]) and [Sault et al 1996]. As CLEAN deconvolutions are usually applied on Plateau de Bure data, a CLEAN-based method adapted to the case of the mosaics has been developed. The initial idea was proposed by F. Viallefond (DEMIRM, Paris) and S. Guilloteau (IRAM), and the algorithm is now implemented in the MAPPING software.