18.3 Short Spacings

Extended structure are missed, attenuated or distorted in interferometric maps by lack of short $uv$ spacing information. While this effect may be negligible for some astronomical problems, it could also be essential in a proper analysis. Deconvolution recovers some of them, but under-estimate the total flux because the integral of the dirty beam is zero (the integral of the dirty beam is the weight of the (0,0) $uv$ cell in the $uv$ data set).

Constructing a beam with a non zero integral can help deconvolution. This can be done by incorporating the Zero spacing flux or spectrum.

Short spacings provides even more information, because they give information on the spatial distribution of this flux on scales between half the primary beam and the primary beam itself. Short spacings can be provided by a smaller interferometer (e.g. BIMA) or a large single dish (e.g. 30-m). In theory, short spacings can also be provided by the interferometer antennas used in single-dish mode. However, because most interferometer have not been designed with total power stability as a goal, this has not been practiced so far.

Incorporating short spacings into interferometer data is a two step process. Task UV_SINGLE extracts short spacing information from single dish data (spectra) and creates a $uv$ table. Task UV_MERGE merges the single-dish and interferometer tables. Coordinates system should be consistent and checked before (coordinates are always J2000.0 at Plateau de Bure, often B1950.0 at the other observatories...).

18.3.1 UV_SINGLE

Incorporating short spacings from the 30-m into Plateau de Bure data is a 3 step process for the user.

• Creation of a table of spectra
  First, one should resample (in frequency) all spectra to same frequency grid than interferometer data, using command RESAMPLE in CLASS. Then a table of spectra is produced using command GRID (with no options) in CLASS.
• Image creation
  The next step is the creation of ``well behaved'' map from the table of spectra. It starts with resampling (in space) on a regular grid by a convolution kernel (interpolation techniques such as the GREG command RANDOM_MAP are inappropriate). Weights are also resampled. Then, we follow by extrapolation to zero outside the convex hull of the mapped region, using a kernel twice broader than the single-dish beam, to avoid introducing spurious structure.

  Because of noise, the map still contains spurious high spatial frequencies. These are removed during the $uv$ table creation. The algorithm steps are

  • Fourier transform of map and weight images
  • Division by Fourier Transform of the single-dish beam
  • Gridding correction (division by Fourier Transform of the gridding function)
  • Truncation to some maximum $uv$ distance ($<$ dish diameter)
  • Inverse Fourier Transform back to image plane
  • Multiplication by primary beam of the interferometer
  • Fourier Transform to $uv$ plane
  • Normalization of the weights so that the sum of weights is the weight of the total flux (derived from integration time, bandwidth and system temperature).
  • Optional application of an amplitude scaling factor.
  • Optional application of a weight scaling factor.
  This produces a $uv$ table with optimal weights in terms of signal to noise ratio for the total flux, and with effective tapering following the single-dish illumination pattern, except for the truncation at some $uv$ distance.

• Merging with interferometer data
  The final step is to merge the resulting $uv$ table with the interferometer $uv$ table. Re-weighting and re-scaling is again possible at this stage. Note that the choice of weighting function is arbitrary. It may result in poorly behaved synthesized beams when combined with the interferometer $uv$ data. Weights can be lowered by any arbitrary factor (increasing the weights is only allowed if signal to noise is not an issue). A good choice is to adjust the weights so that there is almost no negative sidelobe.