The nominal output 
of a CCD--element to a quantum of light 
can be given as
where the additive contribution 
is caused by the dark current,
by pre-flashing, by charge that may skimmed from columns having a deferred
charge (skim) and by bias added to the output electronically to avoid
problems with digitizing values near to zero. Quantum and transfer efficiency
of the optical system enter into the multiplicative term M. The term I
consist of various components: object, sky and the photons emitted from
the telescope structure. It is known that the response of the CCD can show
non-linear effects that can be as larger as 5-10%. These effects are
represented by the term 
.
In the following we ignore the pre-flash and skim term, and hence only take
the bias and dark frames into account. The objective in reducing CCD frames
is to determine the relative intensity 
of a science data frame. In
order to do this, at least two more frames are required in addition to the
science frame, namely:
, and
.
The dark current dark is measured in absence of any external input signal. By considering a number of dark exposures a medium <dark> can be determined:
The method to correct the frame for multiplicative spatial systematics is
know as flat fielding. Flat fields are made by illuminating the CCD with
a uniformly emitting source. The flat field then describes the sensitivity
over the CCD which is not uniform. A mean flat field frame with a higher
S/N ratio can be obtained buy combining a number of flat exposures. The
mean flat field and the science frame can be described by
Equation (
):
where 
represents the intensity frame, and ../icons a brightness
distribution from a uniform source. If set to the average signal of the dark
corrected flat frame or a subimage thereof:
then the reduced intensity frame intens will have similar data values as the original science frame science.
Combining Eqs.(), () and () we isolate:
Here ../icons can be any number, and term 
now
denotes a dark frame obtained by e.g. applying a local median over a
stack of single dark frames. The subscript in 
denotes that this dark exposures by not necessarily be the same frame
used to subtract the additive spatial systematics from the raw science frame.
The mean absolute error of 
yields with ../icons = 1 (only the
first letter is used for abbreviations):
Computing the partial derivatives we get
A small error in 
is obtained if 
, 
and
are kept small. This is achieved by averaging Dark, Flat and
Science frames. 
is further reduced if S=F, then
Equation () simplifies to
This equation holds only at levels near the sky--background and is relevant for detection of low--brightness emission. In practice however it is difficult to get a similar exposure level for the flatfrm and science since the flats are usually measured inside the dome. From this point of view it is desirable to measure the empty sky (adjacent to the object) just before or after the object observations. In the case of infrared observations this is certainly due to the variation of the sky.
