1 Introduction

The way variability is analyzed in astronomical images has been deeply influenced by the development of the microlensing experiments (EROS, OGLE, MACHO, DUO). The enormous quantity of CCD images produced by these experiments has motivated the development of fast and accurate techniques to analyze variability. The first implementations of variability search were based on an analysis of catalogues of stellar objects. The production of the catalogues has been almost exclusively done using the DoPHOT software (Mateo & Schechter). This type of analysis is still used in microlensing experiments, and has proved very successful. However since we are concerned only with differential photometry, such data processing is certainly not optimal. Purely differential methods, like image subtraction, would appear to be better suited for study of variability in astronomical images. However the development of image subtraction techniques has been long delayed by the inherent difficulty and complexity of the method.

The first successful implementation of the method was by Tomaney & Crotts (1996). Further progress has been made by the creation of the fast Optimal Image Subtraction (OIS) method (Alard & Lupton 1998). Due to its ability to solve the full least-square problem, the OIS method proved to produce subtracted images and light curves with an accuracy approaching closely the photon noise limit (Alard 1999). The OIS method is very efficient for crowded stellar fields, like those encountered in microlensing experiments, and could readily be used for massive processing of microlensing data. The current implementation of the OIS works by dividing the field into small sub-areas where constant kernel solutions are derived. In dense crowded fields, sub-areas as small as 128 $\times$ 128 pixels can be used. At this scale kernel variations can usually be ignored to a good approximation. However in case of very bad optics or less dense fields, the constant kernel approximation does not hold any more. For instance, in high latitude fields taken for supernovae search, there are often insufficient bright objects per unit area over which the kernel variations can be ignored. We are then forced to try to make a self consistent fit of the kernel variations, as described in Alard & Lupton (1998). It is important to notice though that even if the spatial variations are fitted to order 1 only, the cost to build the least-square matrix will be about 9 times larger than for a constant kernel solution. Some situations might require order 2 or 3 or even more. Order 3 requires roughly 100 times more calculations than a constant kernel solution. Clearly the problem quickly becomes intractable, and that one of the main advantages of the OIS, the fast computing time, will be completely lost.

Fortunately, it will be demonstrated in this article that the fit of the spatial variations of the kernel can be achieved for little additional computing cost, given a reanalysis of the problem.

Figure 1: Simulated of crowded field images. On the left is the image with constant PSF, and on the right is the image with PSF variations along the Y axis. Note the large amplitude of the PSF variations. A total of 2500 stars has been included in this simulation

Figure 2: On the left is the subtracted image obtained with constant kernel solution. Note the systematic pattern along the Y axis due to the kernel variations. On the right we present the subtracted image obtained by fitting the spatial variations of the kernel to order 2