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Astron. Astrophys. 324, 843-856 (1997)

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4. Data reduction

Because observing conditions are never the same for two successive exposures, three corrections have to be applied to the images before pixel light curves can be extracted:

  1. A pixel light curve makes sense only if a definite pixel always covers the same part of the sky on all successive pictures to a very high degree of precision (within [FORMULA]). This is never the case for raw data to such an accuracy, and we correct for that by software. We call the corresponding correction geometric alignment.
  2. Atmospheric conditions are never the same. In particular, the absorption of light and the sky background change significantly from one exposure to the other (in particular with the moon). The corresponding correction is called photometric alignment.
  3. Seeing changes from night to night and this must also be corrected for. However, when dealing with large enough super-pixels far from bright stars it can be neglected in a first step.

Reference image To apply geometric and photometric alignment, one must choose a reference image. We have chosen images taken on October 26 1994, because observing conditions were good and all fields A to F were available in both colours.

4.1. Star detection and seeing

To find out a maximum of stellar objects on our pictures, we used an adapted version of the program PEIDA, developed by one of us within the EROS collaboration (Ansari 1994), which is optimised to process quickly a large number of images. The main changes we had to implement concern the small number of resolved stars (around 50 per field) and the strong gradient of the background, which compelled us to rethink the star detection.

This treatment left us with 56 stellar objects on the reference image of the A field. Each object plus its background was then fitted by a two-dimensional Gaussian PSF plus a plane (9 parameters altogether). In this way we get the value of the full width at half maximum (FWHM) for each object.

The next step was to distinguish the "real" stars from other types of objects such as globular clusters, which would artificially increase the average seeing of the picture. We did so using the following discriminating method: if on most pictures the FWHM of an object was significantly above the average, it was removed from the average estimate, and the process was iterated. After this treatment, we ended with a total of 32 "real" stars in each of our pictures of the A field.

This procedure allowed us to discard a few bad images, where the [FORMULA] of the PSF fit was poor for most of the 32 stars. We were left with 64 exposures of good quality for the A field, for which the average seeing for the 1994 runs was [FORMULA], and [FORMULA] for the 1995 runs. Fig. 5 shows the evolution with time of the seeing in 1994 and 95, and the distribution of the seeing for both years combined.

[FIGURE] Fig. 5a and b. The seeing for the 1994 and 1995 runs.

Seeing is highly variable and this is a major problem. As mentioned earlier, we cope with these seeing variations by working with super-pixels [FORMULA] wide obtained by replacing each elementary pixel by the square of [FORMULA] elementary pixel centered on it. Far from bright stars, this is sufficient for seeings smaller than [FORMULA], even if we expect to do better in the future.

4.2. Geometric alignment

Geometric alignment involves a two steps procedure:

  1. On the reference and the current images, one detects as many bright stars as possible, one identifies them on the two frames, and one computes the general linear transformation in two dimensions, sometimes called the "Turner tranformation", that corrects for any translation, rotation and scale change between the current and the reference images.
  2. The Turner transformation of the current image to the reference image is implemented by linear interpolation. In general, this can become very complicated as each pixel is not only translated but also scaled and rotated. However, rotations and scale transformations are very small and, although they are important for the position of the transformed pixel, the changes they induce on the pixel orientation, size and shape may be neglected.

This geometric alignment is quite successful as can be seen in Fig. 6. The dispersion of the differences in star positions on two images, after alignment, is of the order of 0.3 pixel, that is [FORMULA]. However this dispersion is dominated by the uncertainty on the determination of the position of each star, therefore the precision of the geometric alignment is better than [FORMULA]

[FIGURE] Fig. 6. Dispersion of the difference of star positions between two images after geometric alignment

4.3. Photometric alignment

In general, photometric alignment is performed assuming that all differences in instrumental absorption between runs have been removed by the correction for flat fields. In this case one may assume the existence of a linear relation (supposing identical seeing) between the intensity in corresponding pixels of the current and reference images:

[EQUATION]

Here a is the ratio of absorptions (due to variations of the atmospheric transmission and/or airmass effects) and b the difference of sky backgrounds (due to moon phases, and/or variations of the atmospheric diffusion) between the reference and the current image.

The usual way to evaluate a is to compare the total intensities of corresponding stars on the two pictures. However, we cannot get in this way a precision better than a few percent on the factor a, because the photometry can be done only on about 50 stars and is difficult on each star, because they are faint and the background is very steep. For this reason, we devised an original global statistical approach to tackle the problem, global in the sense that we take into account all pixels, and not only a few resolved stars. The two methods give equivalent results, but the statistical approach allows to push the precision to about 0.5%.

Statistical approach Assuming relation (20), the variance [FORMULA] and the mean value [FORMULA] of the histograms of pixel intensities on the two images are related by:

[EQUATION]

Relations (21, 22) are valid only when the main cause of variance is the gradient of the surface brightness of M 31. The photon noise and fluctuations due to seeing variations can in principle invalidate Eq. (21). However, in our case, the luminosity gradient of the bulge of M 31 largely supersedes all other causes of variations. The efficiency of this procedure is illustrated in Fig. 7: pixel histograms, for four pictures, that look very different before treatment coincide down to small structures after photometric alignment, using only the two parameters a  and b.

[FIGURE] Fig. 7a and b. The matching of pixel histograms before a and after b photometric alignment

4.4. Filtering out of large spatial scale variations

Reflected light After photometric alignment, there remains a slight gradient in the difference between two images of different runs. This is particularly obvious between runs c and d, when we had to take ISARD down and tune its mirrors. This resulted in a substantial gain of luminosity but introduced a significant gradient between images of runs c and d (Fig. 8).

[FIGURE] Fig. 8. The residual gradient between runs c and d

We think that this residual gradient is due to reflected light for the following reasons. i) It is not cured by the usual debiassing and flat-fielding procedures. ii) Its shape depends on the field but seems constant for each field in a given run. iii) Its intensity seems proportional to the overall luminous intensity.

Median background image To cope with the problem, we construct for each frame a background image where the stars are removed using a median filter. We take a [FORMULA] window for the median filter, that is with a surface much larger than that of the largest seeing disk, therefore all stars but the very brightest completely disappear. We then subtract from each frame its background image and add that of the reference frame.

High spatial passband filter This procedure filters out variations of low spatial frequencies: it insures that, relative to the reference image, all variations on scales larger than 40 pixels are very strongly suppressed whereas variations on scales smaller than 20 pixels are fully preserved. The only remaining differences between images come either from short scale fluctuations (seeing variations around stars and around surface brightness fluctuations, or photon noise) or from varying stellar objects.

Residual gradient and the alignment coefficient a Because of this residual gradient, the sky backgrounds of two images do not stricly satisfy Eq. (20). This introduces a systematic error on a when comparing different runs. This error, however, remains smaller than the error arising from matching resolved stars. As all images have been brought to have the same median background, the error on a only affects the difference of the super-pixel intensity with this background and not the total super-pixel intensity. In other word, the systematic uncertainty on a does not alter our ability to detect variations, but it limits our precision on the time evolution of a variation, once detected.

The pixel stability in time achieved after the processing presented above is described in Sect. 5

4.5. Absolute photometric calibration

Absolute photometric calibration is, strictly speaking, not necessary for microlensing searches which rely solely on the detection of relative luminosity variations in time. Nonetheless, to study the nature of the variable objects we detect, it is necessary to know their absolute magnitude.

We took images of the Palomar-Green PG1657+078 calibration field from Green, Schmidt and Liebert (1986) on 28 July 1995 (calibration day). To determine the flux of reference stars reported in Landolt (1992) UBVRI photoelectric observations, we used the same procedure as for the study of seeing (see Sect. 4.1) except that the fit with a gaussian plus a plane is used only to determine the plane that fits the background, the flux of the star is then obtained by subtracting the estimated background to the observed total flux under the star. The photometry obtained in this way turns out to be much more stable among different images. The colour equations for the Johnson R and B magnitudes, denoted [FORMULA] and [FORMULA], are:

[EQUATION]

where r and b are the instrumental magnitudes with the Gunn r and Johnson B filters:
[FORMULA].
We find, using a [FORMULA] minimisation:

[EQUATION]

We then have to transform our results for [FORMULA] to the reference day where atmospheric absorption was different. The final value is:

[EQUATION]

and the other coefficients are not affected.

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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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