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Subsections

6 Assessment of the method

 


  \begin{figure}
\par\includegraphics[width=17.5cm]{ds8759f10.eps}
\end{figure} Figure 10: Noise maps before (left) and after (right) the processing. For this observation 1 ADU/g/s corresponds to 0.242 mJy/pix

6.1 Comparison of the two GRB observations

 

From the comparison of the final maps of the two GRB observations (Figs. 8D and 8E) we can estimate the reliability of our processing. At first glance we see that the structure of the diffuse emission is very similar in both maps; the LTT correction applied seems to restore properly the large scale structure. Furthermore, almost all point like structures are present in both maps, giving confidence in our bad pixel identification.

The difference of the two final sky images is shown in Fig. 8F. It is dominated by small-scale structure noise but large-scale structures are also apparent. These are probably due to error in the LTT correction. Extra noise is seen at point source positions. This was expected as memory effects are not fully corrected on point sources and as we are undersampling the point spread function (PSF). One also notices that the noise level is higher at the edges of the difference map, due to less redundancy in these regions. The standard deviation of the difference map (Fig. 8F) in the central part of the field is 0.06 ADU/g/s. Therefore, the noise on each GRB final map can be estimated at $0.06/\sqrt{2} = 0.04$ ADU/g/s.

It is clear from the difference map (Fig. 8F) and from the impact of the LTT on the sky image that noise is present at all scales. The processing presented in this paper affects the signal at various scales. To characterize the noise as a function of angular scale we use the structure function of second order:

\begin{displaymath}B^2({l}) \equiv \frac{\sum [I_{\rm sky}({r}) - I_{\rm sky}({r}+{l})]^2}
{N({l})},
\end{displaymath} (12)

where the sum is over all the N(l) pairs of points separated by a distance l. To estimate the noise at each scale in the first map, the one affected by all instrumental effects, we compute the structure function on the difference between the first map of the GRB1 observation and the final one of the GRB2 observation. This difference map, where the sky is removed, is dominated by the noise of the GRB1 observation. The structure function of this difference map rises strongly from small to large scale (see Fig. 9), mainly due to the presence of the LTT. To estimate the noise at the end of the processing, we have computed the structure function on the difference between the final maps of the two GRB observations (see Fig. 8F). This time the structure function is very flat (see Fig. 9). The noise level is reduced by a factor of ten at 8 arcmin scale and a factor of two at the resolution limit. This indicates that the noise level has been lowered at all scales and that it is now uniform at all scales.

6.2 Study of the noise sources

The goal of this section is to show that the high spatial frequency noise of our maps is close to the optimal value obtained with stabilized ground calibration data with no glitches. The data are affected by many sources of noise. First, there are the classical quantum photon noise and the detector readout noise which have been extensively studied in the pre-launch calibration phase (Perault et al. 1994). A conservative value of the readout noise is given in the ISOCAM cookbook: 1.5 ADU/g. Secondly, memory effects (short term transient, long term transient and slow glitches) and fast glitches with small amplitudes substantially increase the noise level. These non-Gaussian events may prevent to reach the optimal sensitivity.

The noise level is measured on the flux history of pixels for which fast glitches have been removed. We have selected pixels not affected by slow glitches. We quantify the noise using the standard deviation of the high frequency component of the pixel flux history. We see in Table 2 that the noise level of the two GRB observations is in total agreement with the readout and photon noise estimated from calibration data (for a 41.5 ADU/g/s flux).

The flux $I_{\rm sky}$ at a given position in the final sky image is the average of N independent flux measurements. The error $\delta$ on $I_{\rm sky}$ can be estimated by

\begin{displaymath}\delta = \frac{\sigma}{\sqrt{N}},
\end{displaymath} (13)

where $\sigma$ is the standard deviation of the N flux measurements used to compute $I_{\rm sky}$. We have computed the error map at each step of the processing. In Fig. 10 we show the error $\delta$ of each sky position for the sky image obtained before the LTT correction (Fig. 10A) and for the final sky image (Fig. 10B) of the GRB1 observation. In the Fig. 10A one sees an enhancement of the error in the southern part of the image, due to the presence of the LTT. Glitches and periodic flat defects appear as noise peaks in this error map. The structure of the final error map (Fig. 10) is dominated by the redundancy effect: the noise is higher on the edges of the map as less flux measurements were obtained in these regions.

For each sky image of Fig. 8, Table 2 lists the median error $\delta$, the median redundancy N and the median standard deviation of the N flux measurements averaged at each sky position. The first thing to notice is that the noise level decreases gradually through the processing. In the final sky image, the median $\sigma$ is only 5% above the noise calibration measurement for the GRB1 observation. The noise in the GRB2 observation is exactly the one obtained with stabilized ground calibration data with no glitches.

The dispersion of the difference between the two final maps (divided by $\sqrt{2}$) is 0.04 ADU/g/s (see Sect. 6.1) which is 35% above the noise level computed on each final map (0.03 ADU/g/s - see Table 2). This 35% difference is partly due to the increased noise on point sources and to the imperfection of the LTT correction. Nevertheless, considering the amplitude of the instrumental effects (3 ADU/g/s for the LTT) we think that the comparison between the two GRB observations is a strong validation of the whole processing.

Finally we conclude from the numbers of Table 2 that the noise level in our final maps are dominated by the readout and photon noise. The other instrumental effects are corrected in the data reduction. Such ISOCAM sensitivity has been obtained in recent point source extraction studies (Desert et al. 1999; Aussel et al. 1999) where the low frequency diffuse emission is removed. It is the first time that such a sensitivity is reached for the emission structure on all scales.


 

 
Table 2: The noise level in the sky image at each step of the processing. In comparison, the noise level measured on the flux history of a single pixel of observation GRB1 and the noise level predicted by ground calibrations (for a mean flux of I=41.5 ADU/g/s) are both 1.15 ADU (which corresponds to 7.26 $\mu $Jy)
  Sky Image $\delta^a$ Nb $\sigma^c$
GRB1 Before LTT correction 0.062 76 2.72
  After LTT correction 0.053 76 2.32
  Variable Flat Field 0.046 76 2.01
  Final map 0.031 58 1.21
GRB2 Final map 0.030 60 1.15

a Median error value of each sky image (in ADU/g/s). b Median redundancy of each sky image (in number of readouts).
c $\sigma$ (in ADU) = Error $\times \sqrt{\mbox{Redundancy}} \times$ Integration time.



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