Fit for the offset position and flux of a point source.

Arguments:

Double x_hpbw

The full-width at half maximum of the beam along the X (cross-elevation) axis (degrees).

The half-power full-width of the beam along the Y (elevation) axis (degrees).

listof PointSourceSample samples

The received power of the source at a list of offset positions.

Context:

This function is designed to be a building block within pointing procedures. Given a list of received powers or fluxes from a number of positions around the expected position of a point source, it attempts to fit for the actual position and flux of the point source, under the assumption that the received power measurements sample a specified beam profile, convolved with the point source.

The beam profile is assumed to be reasonablly well approximated be an elliptical gaussian of the dimensions specified by the x_hpbw and y_hpbw arguments. Thus a point source sampled by this beam, should look like a gaussian of the same size, centered on the position of the source, scaled by the flux of the source, and offset by the background flux level of the sky. The fitting function thus has to fit for the flux of the source, the 2D offset of the source from the expected position, and the background flux level. It does this using the Levenberg Marquardt non-linear least-squares algorithm.

The input samples.

The offset positions of the received power measurements from the expected position of the source, must be specified as sky X,Y offsets (ie. X=cross-elevation, and Y=elevation). These offsets can be commanded using the x, and y arguments of the offset command, The fitted position of the source is also returned in this form.

The chosen sample positions must provide good coverage in both X and Y, within a beam width of the source in all directions, in order that the X and Y offsets of the source can be determined. They must also include at least one sample that is a few beam-widths away from the expected position of the point source, to aid in fitting the background flux level.

In principle, since there are 4 free parameters to be fit, at least 4 distinct positions need to be sampled. However this small number of points is unlikely to give a reasonable fit, especially given the need for one background sample, far off source. At the other extreme, the use of a large number of samples may also result in a poor fit, because the sky background flux may vary significantly during the time needed to measure a large number of samples. The optimal number is probably between 10 and 20 samples, depending on the accuracy needed and the time available.

The return value.

The return value of the fit_point_source function is a PointSourceFit group value. This contains the fitted X, Y offset of the source, suitable for passing to the x and y arguments of the offset command, the flux of the source, the background flux, the reduced chi-squared of the fit, and a quantity that is refered to as the normalized deviation of the fit (see below).

In practice, the reduced chi-squared of the fit is generally not useful, because it depends on the correct estimation of the uncertainties of the flux measurements in the input samples. These uncertainties are unknowable, since they depend on atmospheric variations, such as clouds passing in front of the telescope, and how well the telescope tracks the intended offset position. It also depends on the assumed gaussian being a good approximation of the actual beam profile.

A better measure of how well the measured fluxes fit the scaled and offset guassian beam profile, is returned in the ndev member of the returned group value. This, normalized deviation, is the RMS difference between the measured and model fluxes, normalized by the fitted gaussian flux. A maximum value of 0.2 appears to be a reasonable level above which to reject poor fits, along with a lower flux cutoff and a maximum correction cutoff. Note that the fitting algorithm aborts the fitting procedure if the fitted X or Y offsets exceed one beam width.

Error estimates.

The fit_point_source function returns error estimates along with each of the fitted parameters. They are Asymptotic standard errors, meaning that they are computed from the square root of the diagonal of the covariance matrix of the fit, as though the fitted parameters were all linear. In practice this turns out to be a good approximation for the pointing fitting algorithm. To test this, confidence interval searches were performed on the pointing samples of a number of test sources, to determine more robust estimates of the errors, and the results were essentially the same as the asymptotic standard error estimates. Since the error estimates of the input data tend to significantly underestimate the errors of the sample fluxes, all of the calculated uncertainties are scaled by the square root of the reduced chi-squared. This is a commonly used trick, which works reasonably well when there are large numbers of data points. For the small number of samples that go into pointing fits, it should work on average, but may produce large variations in individual fits.

Occassionally, when there is a lot of noise, such as during bad weather, it isn't possible to compute the covariance matrix. In such cases, the returned uncertainties are set to zero.