17.3 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 half-power 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. Since observations are performed with dual-channel receivers (operating at 1.3 and 3 mm), the field spacing has to be chosen for one of the frequencies. Consequently, the mosaic observed at the other frequency is either under- or oversampled.

• 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 wants 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. In practice, the fields are observed in a track-sharing mode, i.e. in a loop with a few minutes integration time per pointing direction: hence, atmospheric conditions and $uv$-coverage are similar for all the fields.

• In most cases, a mosaic is not 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 below, Fig. 17.1).

• The number of fields, and therefore the size of the mosaic, is limited by the requirement to get good enough sensitivity and $uv$-coverage for all the fields in a reasonable amount of observing time. The current observing mode used at the Plateau de Bure limits the maximum number of fields to about 20. Observing more fields is in principle possible, but would require (much) more observing time and/or an other approach (e.g. mosaic of several mosaics). Note that in any case, the $uv$-coverage obtained for each field is sparse as compared to normal synthesis observations. Finally, a potential practical limitation is the disk and memory sizes of the computers, as mosaicing requires to handle very large data cubes.

• The calibration of the data, including the atmospheric phase correction, is strictly identical with any other observation performed with the Plateau de Bure interferometer, as only the observations of the calibrators (quasars) 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. For the time being, let's forget the effects of the convolution by the dirty beam. Each field $i$ can then 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.

Hence, the mosaic observations can be described as several measurements of the same unknown quantity $I$, each one being affected by a weighting factor $B_i$. This is a classical mathematical problem: the best estimate $J$ of $I$, in the least-square sense, is given by:

$$J = \frac{\displaystyle \sum\nolimits_i \frac{B_i}{\sigma_i^2} F_i}{\displaystyle \sum\nolimits_i \frac{B_i^2}{\sigma_i^2}}$$ (17.9)

where the sum includes all the observed fields and $\sigma_i$ is the rms of the noise distribution $N_i$. (Note that in Eq. 17.9 as well as in the following equations, $\sigma_i$ is a number while other letters denote two-dimensional distributions).

Linear vs. non-linear mosaicing

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 techniques, and a mosaic is reconstructed afterwards with the clean images, using Eq. 17.9.

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

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 Maximum Entropy Method (MEM) 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.