Gaussian process regression
A second method for measuring the compound beam (CB) shapes uses a comparison of the continuum images to the public NVSS catalog and Gaussian process regression (GPR) to construct the compound beam shape.
The Apertif images of each individual compound beam (CB) are convolved with a circular PSF of 45″ to match the NVSS resolution. Then the source finding is performed, and the list is cross-matched with the NVSS catalog. After that the distribution of the relation between NVSS and Apertif total flux e=SAPERTIF/SNVSS over a CB field of view is considered. In the absence of any biases it represents the corresponding compound beam shape.
An example of this relation is shown in Figure 1. The top left plot shows the distribution of 7153 sources over Beam 01 observed from August 2019 to June 2020. The size of each symbol corresponds to the fraction e.
All the data for a given CB (40+ epochs, a few thousands cross-matched sources) is considered and the scikit gaussian_process python library is used to construct a gaussian process regression for these data. For that, the mean value is subtracted from the data. The kernel for GPR is chosen as a sum of two squared exponentials and the one representing white noise. The first kernel represents the main CB shape, and the second one represents shorter scale irregularities. After the gaussian process is trained the regression surface is obtained. The surface is then normalized to take values between 0 and 1. An example of the GPR for Beam 01 is shown in the top right panel of Figure 1.
With this method the “average” CB shapes over the span of all observations are obtained. We note, nowever, that the shape of a CB can change in time because of re-measurement of the beamweights, broken/repaired PAF elements, or if a particular antenna is excluded from observation.
In order to address concerns about time-variability, the described method allows one to obtain the CB shape for a given observation, using only measurements obtained within a given beamweights set (usually a two week time span). Typically, around 500 cross-matched sources (5 – 7 observations) are needed to build the GPR accurately.
These all-antenna CB models correspond to the middle frequency of the 150 MHz band and can be scaled further to be used for the HI or polarization cubes.
Figure 1. Top row — the total flux ratio of APERTIF to NVSS and the corresponding GPR. Bottom row — the GPR middle slices along RA and Dec.