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8. Matching CCD overlaps

Because our images were taken over a 6-year period using different telescopes (3.60 m and NTT), instruments (EFOSC2 and EMMI) and CCDs (different pixel size and color coefficients), it is crucial to adjust the zero-points of our magnitude scales over the whole survey. This is done by comparing the measured magnitudes of objects located in the overlapping edges of the CCD images (1/10 of the CCD size, i.e. tex2html_wrap_inline2829 arcmin). This adjustment guarantees an internally homogeneous photometry, and corrects for the systematic offsets in the photometric calibrations from one observing night/run to another, as well as other possible variations not accounted for like the systematic variations in the extinction curves due to the eruption of the Pinatubo volcano in 1991 (Burki et al. 1995).
The input images have the standard magnitudes derived in Sect. 7, and the equatorial coordinates (tex2html_wrap_inline2831, tex2html_wrap_inline2833) described in Sect. 5. Because the overlaps contain few objects, we use all stars and galaxies with R < 22 mag. The magnitudes can then be directly compared in the overlaps of each CCD, and we show below how the mean offsets provide the correction of zero-point to apply to each CCD.

8.1. Method

Because each CCD frame has 2 to 4 overlaps, the adjustment of zero-points must be done by a global least-squares fit. We use the method described by Maddox et al. (1990a) and originally proposed by Seldner et al. (1977). Seldner designed this method for adjustment of galaxy counts in the overlapping cells on neighbouring plates to correct the systematic variations in the magnitude limits from plate to plate. Instead of comparing the counts, Maddox formalism allows to compare the differences of magnitude for common objects on overlapping photographic plates. For a complete description of the technique the reader can refer to Maddox et al. (1990a). Here we briefly review the principle of the method.
For each galaxy k on CCD i, we define a correction for the zero point tex2html_wrap_inline2841 by
where tex2html_wrap_inline2843 is the "true magnitude'' and tex2html_wrap_inline2845 is the measured magnitude with an error tex2html_wrap_inline2847. On each overlapping CCD pair (i,j), we obtain
We define a mean offset tex2html_wrap_inline2849 so that
here, we assume that tex2html_wrap_inline2851. Because the CCD provides a linear response in a large magnitude range, tex2html_wrap_inline2853 is independent of the magnitude (tex2html_wrap_inline2855)/2. We thus measure tex2html_wrap_inline2857 by an offset in the zerotex2html_wrap_inline2859 order of (tex2html_wrap_inline2861) versus (tex2html_wrap_inline2863)/2 (for the photographic plates, Maddox et al. (1990a) define tex2html_wrap_inline2865 by a third order adjustment). Figure 8 (click here) shows an example of the differences of magnitudes for all objects in one overlap (i, j). We require that the overlaps taken into account in the equation system have more than 5 objects in order to guarantee a reliable measure of the mean offset value.
For CCD i, we obtain several correction factors corresponding to each of its neighbours j which we denote as
where tex2html_wrap_inline2875 is the correction factor estimated with neighbouring CCD j, and tex2html_wrap_inline2879 denotes all the neighbours j of CCD i.

The goal is to determine the unique value tex2html_wrap_inline2885 for CCD i which globally minimizes the scatter in the tex2html_wrap_inline2889. These coefficients are calculated by minimization of the function F defined by:
where tex2html_wrap_inline2893 are the weighting factors which favor the overlaps with a small variance.
With tex2html_wrap_inline2895 = 0, we obtain the following set of equations:
which are solved by iteration. To increase the convergence speed and the stability of this iterative method, we adopt the technique from Maddox of adding the previous value tex2html_wrap_inline2897 into Eq. (10):
where tex2html_wrap_inline2899 is the mean value of all tex2html_wrap_inline2901 of each CCD i

Figure 8: Difference of magnitudes for common objects in two overlapping CCDs j and i. The solid line shows the linear fit for objects brighter than 22 mag which measures tex2html_wrap_inline2909

Figure 9: The upper graph shows the estimated magnitude correction for the R filter to be applied to each CCD zero-point. The first 30 CCDs are fields observed at the 3.60 m and the remaining come from NTT observations. CCDs from 47 to 50 are used as reference frames with no correction factors which provide the constraints to the other frames The lower graph shows the histogram of tex2html_wrap_inline2913 before correction of the zero-point (empty histogram) and after correction (hashed histogram)

8.2. Results

To constrain the set of correction factors, Maddox used tex2html_wrap_inline2919. In our case, we have re-observed several fields across the whole survey during the last observing run: 4 fields in B with TEK#31, 4 in V with TEK#36 and 5 fields in R with LOR#34. The seeing conditions were good (tex2html_wrap_inline2927, tex2html_wrap_inline2929, tex2html_wrap_inline2931 in B, V, R respectively) and the photometric calibrations were performed with great caution. We use these fields as fixed references for the adjustments of the zero-points of the other frames. During this process, we detected systematic offsets in the initial V and R zero-points of the 3.6 m frames derived from calibration sequences (Sect. 6) when compared with the NTT reference frames: tex2html_wrap_inline2943 in the R band, tex2html_wrap_inline2947 in the V band. These shifts cannot be explained by a problem in the colour correction because the offsets are independent of the colour of the objects. Because of the absence of a similar shift in the B band, this effect cannot be attributed to the variation of the extinction coefficients due to the eruption of the Pinatubo volcano in 1991, which occurred when the observations switched from the 3.60 m telescope to the NTT telescope. To prevent the minimization method from introducing a gradient in the set of correction factors around the boundary between NTT and 3.60 m frames the equation systems in each band for the NTT and 3.60 m are processed separately.
First, we use tex2html_wrap_inline2953 as a constraint for both systems. The solutions of the two systems provide the internal zero-point corrections to be applied to each CCD. Then, we use the reference fields to derive the global magnitude shift of each ensemble of zero-points (3.60 m and NTT). These various steps allow us to adjust the two systems to the same zero-point. The resulting correction factors for the R band are shown in Fig. 9 (click here)a. Each point defines one CCD and corresponds (with some gaps) to a sequence number (see figure captions). The iteration of Eq. (11) is stopped when the difference between the previous and new estimation (tex2html_wrap_inline2957) is smaller than 0.005tex2html_wrap_inline2959 for all CCDs i. The typical number of required iterations is 10.
In Fig. 9 (click here)b, we plot the histogram of initial and corrected tex2html_wrap_inline2963, tex2html_wrap_inline2965 defined as:
In these histograms, we exclude the symmetric terms tex2html_wrap_inline2967.
The rms scatter in the initial tex2html_wrap_inline2969 in B, V, R bands are respectively tex2html_wrap_inline2977, tex2html_wrap_inline2979, tex2html_wrap_inline2981. After correction by tex2html_wrap_inline2983, we reduce the scatter to tex2html_wrap_inline2985 in all 3 bands.

8.3. Conclusion

These results demonstrate the efficiency of the method to reduce the dispersion between the CCD frames over the whole survey. They also allow us to estimate the dispersion in our magnitude system. In fact, if there was no error in the measures of magnitudes, tex2html_wrap_inline2993 would be equal to zero. The dispersion in the tex2html_wrap_inline2995 after adjustments gives a tex2html_wrap_inline2997 error in our measured magnitudes in all 3 bands and is in good agreement with the tex2html_wrap_inline2999 in the adaptive aperture magnitude uncertainty given by the simulations (see Sect. 4.2). To check the consistency of zero-point calibrations between the NTT and 3.60 m fields, we have applied a K-S two-sample test to the B-R and B-V colour distributions obtained as described in last section. The probability that the 2 distributions are drawn from the same parent distribution is 0.45 after correction of the systematic zero-point offsets between 3.60 m  and NTT frames, it is 0.01 when the correction is not applied. This confirms that correction of the offsets in zero-point for the 3.60 m  in R and V bands is necessary. Finally, there could be a systematic shift in our overall zero-point, but it would be within the error bars estimated from our individual zero-points (i.e. tex2html_wrap_inline3009).

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