3 Image alignments

 The alignments described in this section are needed in order to build pixel light curves from images that are never taken under the same observing conditions. Firstly, the telescope never points exactly twice in the same direction so that the geometric alignment must ensure that the same area of the LMC contributes to the same pixel flux, through the entire period of observation. Secondly, photometric conditions, atmospheric absorption and sky background light change from one frame to another. The photometric alignment corrects for these global variations.

Errors affecting pixel fluxes after these corrections are a key issue as discussed through this section. It is not obvious how to disentangle the various sources of error introduced at each step, in particular after the geometric alignment. Global errors for each pixel flux, including all sources of noise, will be evaluated in Sect. 7.

3.1 Geometric alignment

 Between exposures, images are shifted by as much as 40 pixels and this displacement has to be corrected, in order to ensure that each pixel always covers the same area of the LMC. As emphasised below, errors affect the pixel flux after the geometric alignment and two components can be distinguished. The first one, resulting from the uncertainty in the parameters of displacement, turns out to be negligible, whereas the second one, introduced by the linear interpolation, is a more important source of noise. In this sub-section, we give a qualitative overview of these sources of errors. This study, based on synthetic images, allows us to disentangle errors due to the geometrical alignment from other effects present on real images, because the position and content of unaligned synthetic frames are known by construction.

Displacement parameters.

The parameters of displacement are determined with the PEIDA algorithm (Ansari 1994), based on the matching of star positions. Beside translation, rotation and dilatation are also taken into account as far as their amplitude remains small (otherwise the corresponding images are removed from further consideration).

A series of mock images synthesised with the parameters of real images (geometric displacement, absorption, sky background and seeing) allows us to estimate the mean error on the pixel position to be $0.011~\pm~0.005$ pixel. Similar estimates have been obtained by the EROS group (Ansari, private communication) on real data.

This introduces a small mismatch between pixel fluxes: in first approximation, the error on the flux is proportional to the pixel area corresponding to the difference between the true and the computed pixel position.

Linear interpolation.

 Once the parameters of displacement are estimated, pixel fluxes are corrected with a linear interpolation. This interpolation is necessary in order to monitor pixel fluxes, and to build pixel light curves.

  

Figure 1: Error due to linear interpolation estimated with two sets of synthetic images: $\sigma$ is the dispersion measured on the flux difference between pixels on the "reference'' image and corrected images, while v is a function of the displacement, as discussed in the text (Eq. 5)

We use synthetic images to understand qualitatively the residual errors. Two sets of blue images are simulated with the identical fluxes (new moon condition) and seeings (2.5 arcsec) but shifted with respect to one of them (the "reference'' image). A linear interpolation is applied to each of these images in order to match the position of the reference. In case of pure translation, the corrected flux is computed with the flux of the 4 pixels overlapping the pixel p on the reference frame: the areas of these intersections with this pixel p are used to weight each pixel flux. The square of the variable v, depending upon $\delta x$ and $\delta y$, the displacement in the x and y directions,  

$$v = \sqrt{\left(\delta x^2+{\left(1-\delta x\right)}^2\right) \left(\delta y^2+{\left(1-\delta y\right)}^2\right)} ,$$ (5)

is the sum of the square of these overlapping surfaces. It characterises the mixing of pixel fluxes produced by this interpolation: the smaller v is, the more pixels are mixed by the interpolation.

Figure 1 displays an estimate of the residual errors affecting pixel fluxes for different displacement parameters, and shows a correlation of the errors with the variable v. The first set of images, simulated without photon noise, shows errors on pixel fluxes due to linear interpolation smaller than 5.5 ADU (about 4.5% of the mean flux). The second set of images, simulated with photon noise, allows us to check that the photon noise adds quadratically with the "interpolation'' noise and that residual errors are smaller than 7 ADU. The correlation observed on this figure between the error $\sigma$ and the variable v can be understood as follows: when v decreases, the interpolated image gets more and more degraded, and the interpolation noise increases while the poisson noise is smeared out.

This residual error is strongly seeing dependent. If the above operation is performed on an image with a seeing of 2 arcsec, the residual errors are as large as 10% of the mean flux: the larger the seeing difference, the larger the residual error. As the seeing of raw images varies between 1.6 and 3.6 arcsec, this makes a detailed error tracing very difficult. The PSF is also slightly widened due to the re-sampling, but this effect remains small compared to other sources of PSF variability, and is largely averaged out when summing over the images of a night (see Sect. 4).

3.2 Photometric alignment

 

  

Figure 2: Photometric alignment: a) absorption and b) sky background flux (in ADU) estimated in each blue image

Changes in observational conditions (atmospheric absorption and background flux) are taken into account with a global correction relative to the reference image. We assume that a linear correction is sufficient:  

$$\phi_{\rm corrected}=a\phi_{\rm raw}+b,$$ (6)

where $\phi_{\rm corrected}$ and $\phi_{\rm raw}$ are the pixel fluxes after and before correction respectively. The absorption factor a is estimated for each image with a PEIDA procedure, based on the comparison of star fluxes between this image and the reference frame (Ansari 1994). A sky background excess is supposed to affect pixel fluxes by an additional term b which differs from one image to another.

In Fig. 2, we plot the absorption factor (top) and the sky background (bottom) estimated for each image with respect to the reference image as a function of time. The absorption can vary by as much as a factor 2 within a single night. During full moon periods, the background flux can be up to 20 times higher than during moonless nights, increasing the statistical fluctuations by a factor up to 4.5. However, this high level of noise concerns very few images (see Fig. 2), and only about 20% of the images more than double their statistical fluctuations. Despite their large noise, full moon images improve the time sampling, and at the end of the whole treatment, the error bars associated with these points are not significantly larger than those corresponding to new moon periods, except for a few nights.

3.3 Residual large-scale variations and their correction

We note the presence of a variable spatial pattern particularly important during full moon periods. This residual effect, probably due to reflected light, can be eliminated with a procedure similar to that applied to the AGAPE data, as described by Ansari et al. (1997). We calculate a median image with a sliding window of $9\times 9$ pixels on the difference between each image and the reference image. It is important to work on the difference in order to eliminate the disturbing contributions of stars, and to get a median that retains only large-scale spatial variations. We then subtract the corresponding median from each image, to filter out large-scale spatial variations. In Fig. 3, we show a light curve before and after this correction. Above, the pixel light curve presents important systematic effects during full moon periods, effects which have disappeared below, after correcting for these large-scale variations.

  

Figure 3: Pixel light curve before (above) and after (below) filtering out the large-scale spatial variations. Fluxes are given in ADU

3.4 Image selection

After these alignments, we eliminate images whose parameters lie in extreme ranges. We keep images which have no obvious defects and parameters in the following range:

• Absorption factor:
  0.6 < a^R < 1.5; 0.6 < a^B < 1.5.
• Mean flux (ADU):
  $100.0 < \phi^{R} < 2000.0$
  $70.0 < \phi^{B} < 1500.0$.
• Seeing (arcsec):
  S^R<3.6 ; S^B < 3.6.

This procedure rejects about 33% of the data.

3.5 Stability of elementary pixels after alignment

 

  

Figure 4: Relative flux stability of about 1000 flux measurements in the blue band, spread over 120 days for pixels within a $50~\times~50$ patch of CCD 3

We are now able to build pixel light curves, made of about 1000 measurements spread over 120 days. The stability can be expressed in term of the relative dispersion ${\sigma}/{\phi}$ measured for each light curve, where $\phi$ stands for the mean flux and $\sigma$ for the dispersion of the light curve. This dispersion gives us a global estimate of the errors introduced by the alignments, combined with all other sources of noise (photon noise, read-out noise...). In Fig. 4, we present the histogram of this dispersion for one $50\times 50$ patch of one CCD field, which shows a mean dispersion of 9.1%. We estimate the contribution of the photon noise alone to be as high as 7%. With such a noise level, dominated at this stage by photon counting and flux interpolation errors, one does not expect a good sensitivity to luminosity variations. Fortunately, various improvements described in the following (namely the averaging of the images of each night, the super-pixels and the seeing correction) will further reduce this dispersion by a factor of 5.