Averaging and Propagation of Uncertainties

The fitting routines used in the pipeline outputs the centre, width, and height of the fitted Gaussian, as well as the standard errors of these parameters. If a user sets num_chunks to a value greater than 2, the visibility or gain amplitudes are generated for the specified number of frequency chunks, and must be combined to produce the final result.

The pipeline computes the weighted-average of the visibility or gain amplitudes for each frequency chunk within each scan of the pointing observation as the input to the fitter. This averaging is either carried out in time only, or in time and polarisation. Averaging only in time allows Fitting the Beams to the Parallel-hands Gain Amplitudes. Averaging over both time and polarisation allows Fitting the Beams to the Stokes I Amplitudes. It is worth noting that the pipeline currently does not support fitting the primary beams on the parallel-hands polarisation visibility amplitudes.

Averaging the Visibility and Gain Amplitudes

When computing the weighted-average of the gain amplitudes, the weights on the gains are extracted from the gain solver used by the pipeline. For the weighted-average of the visibility amplitudes, the weights are extracted from the WEIGHT column of the MAIN table of the Measurement Sets. The weights are used as a measure of the reliability of each sample.

The 2D Gaussian fitter also uses the standard deviation for each averaged visibility or gain amplitude (see scikits.fitting). The pipeline calculates the standard deviation for each point from the unbiased weighted-variance of the visibility or gain amplitudes. The fitter then determines the best fit parameters of interest (i.e the fitted Gaussian centre, width, and height) for each frequency chunk, along with the standard error on each fitted parameter.

Averaging the Fitted Parameters

The pipeline data product contains a single set of fitted parameters for each dish, and so if more than one frequency chunk are used, the results must be combined. This is carried out using a weighted-average for each fitted parameter (ignoring invalid fits - see Validity of the Gaussian Fits for more information on how the fits are declared valid). The weights used in the average for each fit parameter are calculated from the inverse square of their standard errors. The final weighted-standard error is calculated from the square root of the unbiased weighted variance adjusted by the effective sample size.