To address the problems described in Sect. 4 we have developed
a method which uses the fact that a position on the sky has been observed
by several ISOCAM pixels at different times (see Fig. 1).
The redundant information is used to separate the contributions of the sky emission
and of instrumental effects to the observed signal.
This approach has already been used for the IRAS mission (e.g. Okumura 1991;
Wheelock et al. 1997)
and may be generalized to every raster type observations, whatever the wavelength of observation.
Formally the processing of astronomical data with spatial redundancy
could be treated as an inversion problem.
We can consider that the data observed O is the result of the convolution
of the real sky S with the instrumental function I plus some additive noise N:
![]() |
(1) |
ISOCAM's response variations are complex and presently we are not able to model it with a reasonable number of parameters which would allow us to address the data processing problem by a global inversion. We have thus adopted a sequential approach where the instrumental effects are treated one at a time. Nevertheless, even if we cannot use such a global method, the idea of minimizing the differences between pixels that have seen the same sky position is the mainstay of our approach. In particular, the ISOCAM long term drift problem is addressed by a least square minimization technique based on the fact that, if the detector is slowly reaching stabilization, the measured flux is also approaching the real observed flux. Here we suppose that every pixels of the array is affected by the same long term drift. The implication of this assumption will be discussed in Sect. 6.
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Figure 4: Data reduction chain illustrating the data processing steps from left to right. The time dependent flat field and bad pixel identification operations can be done iteratively |
Once the LTT is removed, the other response variations (slow glitches, ghosts and residual transient effects) are corrected by comparing the data cube to the sky image. This is done in two steps. First we compute a variable flat field that takes into account pixel-to-pixel response variations. Second, pixels that have a flux that departs significantly from the sky image are flagged (these are called bad pixels in the following). This last operation removes slow glitches and ghosts. These two operations (variable flat field and bad pixel identification) can be done several times to improve the sky image (see Fig. 4).
First of all it is important to determine how the long term transient, the flat field
F and the observed flux
are related to the incident flux
.
We have considered the three following possibilities for the LTT:
The two GRB observations are of great help in identifying the nature of the LTT.
As the two observations were done in exactly the same way, it is possible
to subtract directly the two data cubes.
In Fig. 5A we show the flux history of two pixels of the GRB2 observation.
As one can see there is no long term drift detectable in this observation. The flux difference
between the two selected pixels is due to a 25% flat field difference.
The difference between observation 1 and 2 for these two pixels is shown in
Fig. 5B. Both pixels
have a very similar drift; the 25%
difference between the two pixels flux does not appear in Fig. 5B.
We then conclude that the LTT does not depend on the signal
.
In this context, the only valid description of the LTT is the pure offset
effect:
.
In the following we consider the LTT as a single offset over all the detector;
in other words we suppose that all pixels are affected by the same offset.
For a given pixel at a position (x,y) on the detector array and at a given time t,
the observed flux
is related to the temporally varying flat
field F(x,y,t), the incident flux
and the long term drift
by the following equation:
The offset function
is found using Eq. (2) and the spatial
redundancy inherent to
raster mode observations. We determine
by solving a set of linear equations
obtained by comparing flat field corrected intensities of the same sky positions but obtained
at different times.
The flat field F(x,y,t) is computed using the adaptive flat field
method (see Appendix A.2).
The flat field corrected intensities can be written:
In raster mode the camera does not always point at the same position on the sky.
To compare two pixels that have seen the same position on the sky, we must
put all readouts in a common spatial reference frame. Practically, we project
and F(x,y,t) on the plane of the sky (right ascension (
)
-
declination (
)
reference frame). In this reference frame and following
Eq. (3), the difference
between two pixels that have seen the same sky
at different time (ti and tj) is:
The function of interest
is estimated using a least-square minimization technique
with the following minimization criterion:
The function
which minimizes the value of
is found by solving
the system determined by:
Equation (6) represents a standard set of linear equations
which can be written in a matrix form:
,
with:
![]() |
(7) |
![]() |
(8) |
![]() |
(9) |
Finally
is found from:
.
As the second derivative
of
is always positive, the solution found for
necessarily minimizes
the
criterion.
We add an offset to all values of
to force the correction to be zero
at the end of the observation:
.
This is justified
by the fact that the long term transient tends to stabilize at the end of an observation.
However, there are several observations not stabilized at the end, so that
the absolute brightness may be systematically shifted (generally below a few %).
For the observations of low contrasted clouds on top of a flat large scale emission
(at least the zodiacal emission), this point is not critical as we always remove
the large scale emission at the end of our processing.
Other than detector noise, two additional sources of uncertainties affect the comparison of raster images and thus the LTT correction through the minimization algorithm described in the previous section: slow glitches and flat field variations along the observation. The signal measured on point sources is not identical in the individual readouts, essentially due to the undersampling of the the point spread function for the 6'' pixels. Bright sources are thus an additional source of error but they can easily be discarded. The two other noise sources make the practical implementation of the formalism a non-straightforward procedure.
An extensive use of the LTT inversion method presented here on real data has demonstrated
the extreme importance of an accurate flat field. As the LTT correction is based on the comparison of
the brightness measured by different parts of the array, its result depends on
the accuracy of the flat field.
In particular, when
is not well estimated, oscillations
in phase with the rastering of the observations may appear in the correction found.
To overcome this difficulty we compute an approximate LTT correction
(as oppose to the exact correction which is the solution of Eq. (6)).
The study of low contrast ISOCAM data affected by long term drifts has shown that, in some cases,
the LTT can be approximated by the sum of two exponential functions, one upward and one
downward:
![]() |
(10) |
On the other hand, we have observed cases where this approximation of the LTT does not hold. Sometimes, the LTT shows oscillations that can dominate the emission in low contrasted fields (e.g. cirrus clouds - see Miville-Deschênes et al. 2000). In these cases, an accurate estimation of the flat field and the use of the exact solution for the LTT are mandatory.
Figure 6 presents the LTT corrections found for the GRB1 observation.
A adaptive single flat field (see Appendix A.2) was used and point
sources were discarded to compute the exact and approximated corrections.
The approximated LTT correction is smoother than the exact one, which oscillates
with a
ADU/g/s amplitude and a period of
s.
In this case we have applied the approximated LTT correction.
The sky image computed after that correction has been done is presented in Fig. 8B.
One sees that it is mandatory to apply the LTT correction since its amplitude (3 ADU/g/s) is nearly
three times the amplitude of the emission of interest (1.1 ADU/g/s).
At this stage we have used a single flat field to compute the sky image. It is necessary
to use a variable flat field to correct the artefacts (e.g. periodic patterns) seen in Fig. 8B.
After the LTT has been corrected, we then take into account the pixel-to-pixel
temporal variations of the detector response. These response variations are observed
at various timescales. At short timescales they are due to bad short term transient
correction, particularly on point sources. On longer time scales they are mainly
caused by slow glitches (see Sect. 4).
The use of a single flat field (see Sect. 3)
does not take into account these temporal variations (that represent 1-3% of the average flat field)
preventing the benefit of the optimal ISOCAM sensitivity.
To go further in the data processing, we try to correct these pixel-to-pixel response
variations with a time-dependent flat-field
(called "variable flat field'' in the following).
For LTT corrected data,
the observed flux
at position (x,y) on the array and at time t is
In raster observation mode, many pixels of the data cube have seen the same position
(
)
on the sky. By averaging all these pixels we reduce the noise due to
instrumental effects on the computation of the sky image
.
The estimate of
is made by an inverse projection of
on each readout of the data cube. Then F(x,y,t) is computed by averaging
with a sliding window on the time axis (see Appendix A.1).
Here are the guidelines of this method:
The sky image of the GRB1 observation, obtained with the variable flat field,
is shown in Fig. 8C. The size of the filtering window on the time axis
is 100 (corresponding to 500 s).
As seen in Fig. 8C, the variable flat field removes almost all
periodic patterns due to high-frequency variations of the detector response.
Compared to other methods we have tested (see Appendices A.1 and A.2),
this variable flat field gives by far the best results.
The deglitching process used at the beginning of the reduction chain (see Sect. 3) removes extremely deviant pixels that have been hit by cosmic rays. To go further in the noise minimization process, we again take advantage of the spatial redundancy.
Memory effects often appear after a strong flux step (e.g. after a point source) due to improper short term transient correction; this is what we call ghosts. To identify pixels affected by such effects, we look at the flux history of every pixel for a memory effect after a flux step (see Fig. 7). Again, we use the redundancy information to improve the identification of ghosts. Here is how we proceed:
![]() |
Figure 9: Structure function of the difference between the two GRB final maps (bottom line) and of the difference between the first map of GRB1 and the final map of GRB2 (top line) |
We can go further in the reduction of the noise level by working sky position by sky position instead of working on the time history of every pixel. The idea is to look at pixels in the data cube that have seen the same sky position and discard deviant flux values.
First the sky image is smoothed (median smoothing) with a 10
10 window.
Then, for a given sky position (
,
), we compare the N pixels in the data cube
that have seen that position with the flux of the smoothed sky image.
For most of the sky position, the N pixels are distributed
around the smoothed sky estimate. On the other hand, at point source positions,
the N pixels generally will
be above the smoothed sky estimate. Furthermore, it is also possible to find sky positions where most
of the N pixels have fluxes under the smoothed sky level. Here is how we deal with each case:
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