7 Error estimates for super-pixel fluxes

  As explained in previous sections, it is not straightforward to trace the errors affecting pixel fluxes through the various corrections. Errors are estimated here in a global way for each pixel flux, "global'' meaning that we do not separate the various sources of noise. The images used for the averaging procedure provide a first estimate of these errors. The dispersion of the flux measurements performed over each night allows the computation of an error associated with the averaged pixel flux. We discuss how this estimate deviates from Gaussian behaviour, and which correction can be applied. Gaussian behaviour is of course an ideal case, but it provides a good reference for the different estimates discussed here.

Error estimates on elementary pixel.

When we perform for each night n the averaging of pixel fluxes, we also measure a standard deviation for each pixel ${\sigma_{\phi^p_n}}$. Assuming this dispersion is a good estimate of the error associated to each flux measurement $\phi^p_{n,j}$, and that the errors affecting each measurement are independent, we can deduce an error ${\sigma^p_{n}}$ on $\phi^p_n$ as:  

$${\sigma^p_{n}}^2 = \frac{1}{N^p_n} {\sigma_{\phi^p_n}}^2.$$ (10)

This estimation, however, is uncertain: the number of images per night can be quite small, and Eq. (10) assumes identical weight for all images of the same night.

  

Figure 23: Distributions of ${\chi^p}^2$: on both panels, the dashed line represents the ideal distribution discussed in the text. Panel a) (solid line) displays the ${\chi^p}^2$ distribution with error estimates based on the dispersion of pixel flux measurements over each night. In b), the histogram (solid line) is computed with errors calculated for each pixel flux as the maximum between the photon noise and the errors used in a)

In order to assess our error estimates, we compute the distribution of the ${\chi^p}^2$ values associated with each pixel p light curve.

$${\chi^p}^2 = \sum_n\frac{\left({\phi^p_n - \langle \phi^p \rangle}\right)^2} {{\sigma^p_n}^2}\cdot$$ (11)

Figure 23a displays two $\chi^2$ distributions: the ideal case (dashed line) assumes Gaussian noise and the number of degree of freedom (hereafter NDOF) of the data; the solid line uses actual data with errors computed with Eq. (10): the histogram peaks roughly to the correct NDOF, but exhibits a heavy tail corresponding to non-Gaussian and under-estimated errors. Due to statistical uncertainties on the calculation of the errors ${\sigma^p_{n}}$, it happens that some of them are estimated to be smaller than the corresponding photon noise, in which case the photon noise is adopted as the error. The corresponding $\chi^2$ distribution displayed (solid line) in Fig. 23b has a smaller non-Gaussian tail, but peaks at a smaller NDOF: not surprisingly the errors are now over-estimated.

Correction.

  

Figure 24: Corrected error bars: the upper panel a) displays a zn distribution for a given image n whose errors were over-estimated (histogram). The full line corresponds to the fitted Gaussian distribution, and the dashed line to the normalised Gaussian distribution. The lower panel b) shows the $\chi^2$distribution calculated with the corrected errors (solid line)

  

Figure 25: Example of a stable super-pixel light curve

  

Figure 26: Example of a variable super-pixel light curve in blue (top). The star is unresolved at minimum (bottom left panel) and would even be difficult to detect with classical procedures at maximum (bottom right panel)

The correction described here is intended to account for night to night variations, or important systematic effects altering some images. Although the main variations in the observing conditions have been eliminated by the procedures described above, each night is different and for instance the seeing distribution over one night can differ from the global one. Hence, we weight each error with a coefficient depending on the composite image. The principle is to consider the distribution for each night n of the variable z_n^p given by

$$z_n^p = \frac{{\phi^{p}_{n}} - {\langle{{\phi^{p}}}\rangle}_n}{\sigma^p_n} ,$$ (12)

and to re-normalise it in order to approach a normal Gaussian distribution as well as possible. ${\langle{{\phi^{p}}}\rangle}_n$ is the mean pixel p flux value computed over the whole light curve. The standard deviation $\sigma^\prime_n$ of each of these z_n^p distributions is estimated for each average image n on a central patch of 100$\times$100 pixels. A z_n^p distribution is plotted for each image and is fitted with a Gaussian distribution. This fit is quite good for most of the images and the dispersion of the Gaussian distribution is our estimate of $\sigma^\prime_n$. Figure 24a shows an example of the $\sigma^\prime_n$ estimation. The solid line shows a Gaussian fit to the data. The width is not equal to 1 as it should be, but rather to 0.77, the value of $\sigma^\prime_n$ for this image. For comparison, we show a Gaussian of width 1, with the same normalisation (dashed line).

In the following, the corrected errors

$${{\sigma}^p_n}\vert_{\rm corrected} = \sigma^\prime_n {{\sigma}^p_n} ,$$ (13)

are associated with each pixel flux. ${\sigma^p_{n}}$ is different for each measurement whereas $\sigma^\prime_n$ is a constant for each image n. The resulting $\chi^2$ histogram is displayed in Fig. 24b (full line). The $\chi^2$ distribution peaks at a higher value of $\chi^2$ than before correction (Fig. 23b), which however is still slightly smaller than the NDOF.

From pixel errors to super-pixel errors.

We have seen in Sect. 5 that the use of super-pixel light curves allows us to reduce significantly the flux dispersion along the light curves. The most natural approximation for the computation of super-pixel errors is to assume those on elementary pixels to be independent:

$${\sigma^{\rm sp}_n} = \sqrt{\sum_{p} {{\sigma}^p_n}^2} .$$ (14)

However, errors on neighbouring pixels are not independent, because of the geometrical alignment procedure and of the seeing correction. To take this into account, we correct the error on super-pixels in the same way as above. The factors $\sigma^\prime_n\vert _{\rm sp}$ thus obtained are 20% higher than for elementary pixels.

We have now super-pixel light curves with an error estimate for each flux. Figure 25 displays an example of a typical stable light curve in blue (upper panel) and in red (lower panel), whereas Fig. 26 is an example of a variable light curve.