Standard reduction consisting of bias subtraction and flat fielding was performed on the images using the ESO-Midas package and our own software called "Mana'' (Magnier 1996). The INT data required special care in the bias subtraction as there were significant variations in the bias across the chip. The images were corrected using a template of the bias plus an offset determined from the overscan for each image.
The INT images had substantial variations in the point spread function (PSF) as a function of position on the image, possibly due to a mis-alignment of the CCD in the optical plane. We split each of these images into 9 subimages to minimize this problem. This problem was less severe for the MDM data, so that the entire images could be kept intact, but variations in the PSF with position for both telescopes continued to plague the analysis of all images during relative photometry, and forced some variation in the standard analysis, as discussed below.
Photometry was performed with the program DoPHOT (Mateo & Schechter 1989). This is a PSF fitting program which is designed to be easy to use automatically with a large number of images; DoPHOT needs a relatively small amount of information to start running, and it runs without any interaction with the user. A complete discussion of DoPHOT can be found in Mateo & Schechter (1989). Freedman (1989) also discusses DoPHOT in comparison with other commonly used photometry routines.
DoPHOT models the objects on an image using an elliptical
Gaussian with 7 free parameters: a zero level (sky), the central
intensity (I0), the centroid (X and Y), the semi-major and
semi-minor axes of the ellipse (
and 
), and
the angle between the semi-major axis of the ellipse and the pixel
coordinate system (
). For a particular image, DoPHOT uses a
single set of values for the 3 "shape'' parameters (,
, and ) to represent the PSF. DoPHOT
distinguishes between stars (which are well fit with the PSF) and
extended objects (which are well fit by a profile significantly more
extended than the PSF). DoPHOT reports instrumental magnitudes and
positions in pixel coordinates for all objects as determined from
these Gaussian fits.
Astrometry was performed on each image. Two databases were
used as a reference. For most images, the MIT/Amsterdam CCD survey of M 31
(Magnier et al. 1992; Haiman et al. 1993) was
used as an astrometric reference. For images without sufficient overlap
with this survey, the survey by Berkhuijsen et al. (1988) was
used, after correction for the systematic error reported in the astrometry
(Magnier et al.\
1993a). Linear astrometric conversions were used; i.e.\
translations, scaling, and rotations were included in the conversion between
pixel coordinates (X, Y) and sky coordinates (, 
). The
astrometry for certain images was determined from other program
observations of the same field, for which the above astrometric
parameters had been succesfully determined. Typically between 30 and
100 stars from either of the reference catalogs were identified on
each CCD frame. The residuals of the fit were also measured to
provide an estimation of the astrometric error, which we determined to
be typically 
1.0 arcseconds, dominated by the errors in the
original catalogs.
Relative photometry was performed on the images to convert the instrumental magnitudes to a common system. This allows us to make corrections for a variety of effects which may alter the throughput for a given image, in particular small amounts of clouds.
For every star on every image, there exists a relationship between the
observed instrumental magnitude m,
and the apparent magnitude in a common system 
:
The subscripts i and j refer to a particular image and a
particular star, respectively. 
is a correction for the
throughput for a given image, and may incorporate effects such as
cloud level. The goal of relative photometry is to determine
for each image, then use this to find
for all stars from Eq. (2 (click here)). Once one has
for each star, one can then convert them to a standard system,
such as the Johnson system, using appropriate color corrections. This last
step is in general more inaccurate for a variety of reasons, particularly
because of the undersampling of existing photometry bands (see Young
1992). For the purpose of identifying variable stars, however, it
is more important to have an accurate relative magnitude in an
ill-defined system than well-calibrated magnitudes in a commonly used
system.
We performed relative photometry according to the scheme outlined
above, using an iterative method to minimize the 
, defined as
where 
is the error in the measurement mi,j. For
the present dataset, some modifications were necessary. First, we
measured the for each star independently and removed those
stars with unusually high values. This is needed to remove
both the true variable stars from the calibration, as well as those
stars which have a single or a few extreme outlying points, due to,
e.g., cosmic ray hits or the star falling on a bad column. We also
found that the residuals for a given image were a clear function of
the position on the image. We traced this problem to the variation of
the PSF across the images, for both telescopes. Since the model PSF
is kept fixed for a given image (or portion of an image in the case of
the INT images), stars which fit the model less well than other stars
on the same image will have their flux poorly measured. Thus, a trend
across the image in the size of the PSF is translated to a trend in
the effective magnitude of a star measured at that position. To
compensate for this effect, we modified Eq. (2 (click here)) to
incorporate a trend across the image:
where Ai through Fi are kept fixed for each image and x and
y are the position of a measured star on an image (subscripts
dropped for clarity). We solved the system for Ai through Fi
along with the and terms. In fact, this last
correction was not crucial; the actual magnitude of the correction
introduced by the terms Ai through Fi was not very large
compared to the variability of interest: typically only about 
. The fact that every frame had an arbitrary zero point allows a
simple connection of the data from the INT 2.5 m and the data from the
MDM 1.3 m. In practice, we made initial guesses at the relative zero
points of the solutions by correcting for the relative areas of the
two telescopes. All data could then be processed simultaneously. The
remaining scatter observed for non-variable stars which were bright
enough (
) that photon noise was not significant was 
. The relative photometry was converted to the Johnson system
by calibrating each field relative to the MIT/Amsterdam CCD survey of M 31
(Magnier et al. 1992; Haiman et al. 1993).