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. 
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 (
, 
) 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.
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
by
where 
is the "true magnitude'' and 
is the measured magnitude
with an error 
.
On each overlapping CCD pair (i,j), we obtain
We define a mean offset 
so that
here, we assume that 
.
Because the CCD provides a linear response in a large magnitude range,
is independent of the magnitude (
)/2.
We thus measure 
by an offset in the zero
order
of (
) versus (
)/2 (for the photographic plates,
Maddox et al. (1990a)
define 
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 
is the correction factor estimated with neighbouring
CCD j, and 
denotes all the neighbours j of CCD i.
The goal is to determine the unique value 
for CCD i which globally
minimizes the scatter in the 
. These coefficients are calculated
by minimization of the function F defined by:
where 
are the weighting factors
which favor the overlaps with a small variance.
With 
= 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 
into Eq. (10):
where 
is the mean value of all 
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 
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 
before correction of
the zero-point (empty histogram) and after correction (hashed histogram)
To constrain the set of correction factors, Maddox used
. 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 (
, 
,
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:
in
the R band, 
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 
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 (
) is
smaller than 0.005
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 
, 
defined as:
In these histograms, we exclude the symmetric terms 
.
The rms scatter in the initial 
in B, V, R bands are
respectively 
, 
, 
. After correction by 
, we
reduce the scatter to 
in all 3 bands.
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, 
would be equal to zero.
The dispersion in the 
after adjustments gives a 
error in our measured magnitudes in all 3 bands and is in good agreement with
the 
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. 
).