Plots of flux repeatability as a function of signal-to-noise ratio, S/N (see, for example, Figures 5-7 of Cutri (2001)), suggest that the quoted PROPHOT uncertainties represent underestimates for low S/N sources. As will be discussed below, the problem is not with the PROPHOT uncertainties, but with the interpretation of those plots; the quoted PROPHOT uncertainties at low S/N are, in fact, good representations of the true error.
Flux repeatability plots were a standard part of the QA
suite for evaluation of the pipeline output. They were derived from
sets of calibration scans in which each source was observed 6 times.
The RMS flux repeatability was calculated
separately for each such set of 6 observations and compared with the
mean value of quoted flux uncertainty, sigma, for the set. Scatter plots
were then made of [RMS-sigma] as
a function of RMS. Scatter plots of this type (made from 1.5 months of
calibration scan data from the northern hemisphere during 10/1/1999 -
11/15/1999 using uncorrected ?_msig values), are shown in
Figure 1a (J-band),
Figure 1b (H-band) and
Figure 1c (K-band). They exhibit similar behavior
to that of Figures 5-7 of
Cutri (2001)).
The above plots show a systematic tendency for [RMS-sigma] to be significantly larger than zero at large RMS, and monotonically increasing with RMS. The interpretation has been that the PROPHOT uncertainties are systematically smaller than the repeatability values for low S/N.
A problem with this interpretation is that the RMS values are not good indicators of the true S/N since they are subject to small-number statistics. Since the RMS values are each determined from only 6 observations, they are somewhat broadly distributed, with a scatter of roughly sigma/sqrt(6) for a given sigma. By contrast, the sigma values themselves are well defined for a given source magnitude. One can then understand the observed trend in terms of the positive correlation between [RMS-sigma] and RMS itself. What happens is that on the right hand side of the plot (large RMS) we are preferentially selecting those cases in the positive statistical tail of the distribution of RMS values corresponding to a given sigma, and in doing so, we have preferentially selected the cases which happen to have a large positive deviation of [RMS-sigma], even if there is no deviation between RMS and sigma on average.
This problem can be overcome by replacing the abscissa of the plot with a different S/N-dependent quantity that does not suffer from small-number statistics, such as magnitude or even sigma itself. The results are shown as a function of sigma in Figure 2a (J-band), Figure 2b (H-band), Figure 2c (K-band), and as a function of magnitude in Figure 3a (J-band), Figure 3b (H-band), Figure 3c (K-band). In the magnitude plots, the open circles represent the mean values, averaged in bins of width 0.25 magnitudes. It is apparent from these plots that the trend of increasing [RMS-sigma] with decreasing S/N is no longer present, confirming that the effect was spurious.
It is interesting, however, that there is a turnover in [RMS-sigma] at the faint-magnitude end of the plot for all three bands. This may be a detection threshold effect, whereby the repeatability of the faintest sources could be artifically raised if they are limited by a constant threshold.
