#### Single-dish vs interferometer weight

In all cases involving short spacings, the relative weight of the single dish data to interferometer data is critical. Within the restrictions imposed by the noise level, this relative weight is a free parameter. It is all the more important that the Fourier transform of the uv plane density of weights is the dirty beam, a key parameter of the deconvolution. The general goal is to have a dirty beam as close as possible to a Gaussian. As the Fourier transform of a Gaussian is a Gaussian, we search for obtaining a uv plane density of weights as close as possible to a Gaussian. In general, the short spacing frequencies are small compared to the largest spatial frequency measured by an interferometer. This implies we can use the linear approximation of a Gaussian, i.e. a constant, in the range of frequencies used for the short spacing processing. We thus end up with the need to match (as far as possible) the Single-Dish and interferometric densities of weights in the uv plane. In practice, we compute the density of weights from the single-dish in a uv circle of radius 1.25 d and we match it to the averaged density of weights from the interferometer in a uv ring between 1.25 and 2.5 d. Experience shows that this gives the right order of magnitude for the relative weight and that a large range of relative weight around this value gives very similar final results.

When processing IRAM-30m data to combine to NOEMA data, this criterion implies a large down-weighting of the IRAM-30m data which may make think that too much observing time was used at the IRAM-30m data. However, just using the above criterion to determine the observing time needed at the IRAM-30m would in general lead to very low signal-to-noise ratio of the single-dish map. In such a case, it is very difficult to detect problems which would translate in strong artifacts in the IRAM-30m + NOEMA combination. We recommend to ask for enough IRAM-30m time to get a median signal-to-noise ratio of 5 on the single-dish map. This ratio should be achieved for all velocity channels of interest (which may include the line wings).