Let's consider the spectral case first.
Let be 
and 
the number of photon counts for each
spectral bin in the source+background and the background spectra
created via the Projection tasks. The two spectra are assumed to have
the same bin structure. The background subtracted spectrum
is then defined as follows:
where 
and 
are the areas of the sky portions from which the
source+background and the background spectra were obtained. The exposure
time for the source+background spectra is assumed by default to be the same
as the exposure time for the background one.
The corresponding statistical error is defined as
This formula is modified consistently for data which have been already corrected in the Projection run in correction mode.
When data were not corrected within the Projection run,
it might be necessary to keep into account the following:
in a pointed observation different sky regions affect
systematically different detector portions. Therefore
when the source+background and the background spectra
are taken from sky regions which are relatively far from each other,
it may be necessary to keep into account the differences between the
corresponding spectral correction vectors. Let be
and 
the effective areas
corresponding to each amplitude channel (ampl) of the two spectra (they
are obtained from the corresponding correction vectors, which are
effective areas as a function of energy). In this specific case
the background subtraction is performed as
All this is not applicable -- because not necessary -- whenever Projection was run in correction mode, since all spectra are processed in such a way that their correction vectors are all identical to the effective area on-axis.
As a last step the result spectra can be optionally rebinned,
either interactively or automatically. The automatic rebin is
performed in order to obtain for each of the new bins
a signal to noise ratio not less than a specified value.
Once the signal to noise ratio is fixed to a value
,
the new binsize of a spectral bin starting from a given amplitude
channel 
is defined as
where 
is a multiple of the original spectra binsize
(i.e. the binsize declared in the Projection command
BIN AMPL which was used in creating the spectra).
The sequence of binsizes in the rebinned spectrum is defined
by the recursive
The results of the spectral
background subtraction are always converted
into counts s 
bin 
.
Let's now briefly consider the light curve background subtraction case. Let be
and 
the two sequences of time bins, and
, 
the corresponding (optional) correction vectors
(they are not produced when Projection is run in correction mode).
The two light curves are assumed to
have the same structure, i.e. the same binsizes, the same gaps and the
same extension in time. The corrected and background subtracted light
curve is then given by
The results of the light curve
background subtraction are always converted
into counts s 
.
All the operations described above are automatically performed either via the Correction commands or (partly) within Projection run in correction mode (applicable only for PSPC data). In the following a detailed description of the correction commands syntax is given: