Appendix A: Temporal method for faint source detection

The principle

As glitches are the main limitations for faint ISOCAM source detection, not the noise, it is clear that an analysis on the final raster image with a standard method (detection at $k\sigma$ the noise level + background) would lead to poor results if the glitches with transients have not been removed. In order to avoid other problems (dark current subtraction, flat field, transient and long drift correction, etc.), a solution is to perform a temporal source detection technique rather than a standard source detection technique on the final raster map. The temporal source detection method is based on the fact that the flux observed by a single detector increases when the detector points toward a source and decreases when the camera is moved to the subsequent phase of a raster observation. This temporal behavior of the flux observed by a detector has the advantage of being dark current and flat field independent. Indeed, the flat field and dark current act as a multiplicative and an additive constant on the total temporal signal, and do not effect the shape of the signal. Thus, the signature of a source can be identified.

Temporal detection

Short glitches (i.e. first type) can be easily removed by masking the position where they appear (Starck et al. 1999). For each pixel (x,y), we indicate the deglitched data as D(x,y,c,r) and the corresponding mask as M(x,y,c,r) (0 if the position is masked, 1 otherwise), where c and r indicate respectively the configuration (raster position) and the readout number in this configuration. Values corresponding to the same sky position and the same configuration are averaged:

$$I(x,y,c) = \frac{ \sum_{r} M(x,y,c,r) D(x,y,c,r)} {\sum_{r} M(x,y,c,r)}\cdot$$ (12)

The temporal noise $\sigma(x,y)$ is estimated for each pixel independently using a k-sigma clipping method, so the noise on the mean value of signal in a configuration is given by:

$$\sigma_I(x,y,c) = \frac{\sigma(x,y)}{ \sqrt{\sum_{r} M(x,y,c,r)}}.$$ (13)

The detection is done by calculating the signal:

$$W(x,y,c)\!=\!I(x,y,c)\!-\!\frac{1}{2}(I(x,y,c\!-\!1)\!+\!I(x,y,c\!+\!1))$$ (14)

and its associated noise:

$$\sigma_W(x,y,c) =$$

$$\sqrt{ ( \sigma_I(x,y,c))^2 \!+ \! \left(\! \frac{1}{2}\sig... ... \!+ \! \left(\! \frac{1}{2}\sigma_I(x,y,c\!+\!1) \!\right)^2}.$$ (15)

Then we consider we have a detection at pixel (x,y) and at the configuration c if:

$$W(x,y,c) \gt k \sigma_W(x,y,c),$$ (16)

where, in general, k is taken equal to 3. If a source is detected at position (x,y,c) we put I(x,y,c)=1, otherwise I(x,y,c)=0. We can therefore coadd the C I(x,y,c) matrixes in order to obtain a matrix of detections with size equal to that of the total image: $Image(\xi,\eta)$ indicates how many times a source has been detected at the sky position $(\xi,\eta)$ during the raster observation. For instance, for a raster observation with half overlapping, $Image(\xi,\eta)$ can take the integer values between 0 and 4.

Constraints for a robust detection

The detection has been made under the assumption of Gaussian noise. Due the large number of glitches, false detections will occur. Two parameters can be adjusted in order to limit the number of false detections:

1.

the detection level (k parameter). By default, the detection is done at $3\sigma$ (it corresponds to a false detection probability of 0.25%). Increasing the detection level eliminates false detections (but also weaker objects).

2.

the number of required redundancy $N_{\rm r}$. For a raster made by overlaping half the array, a source should be detected four times (two times if it is on the border of the raster image). Fixing a minimum of two detections should suppress most of the false detections.

A robust detection is performed by comparing Image(x,y) to the number $N_{\rm r}$ of required redundancy. A high redundancy allows one to increase $N_{\rm r}$ and improve the robustness of the detection. We point out that Image(x,y) is independent of the background level, the flat field, and the dark.

Conclusion

Once (Goldschmidt et al. 1997; Serjeant et al. 1997) the detection is done, sources must be extracted with astrometric and photometric information. This is done using the final calibrated raster image (see Siebenmorgen et al. 1996, for a complete description of each calibration step). For observations with the six arc second lens, the PSF is mainly contained in one single pixel. So PSF fitting does not help, and the flux of an object can be obtained by integrating the flux in a small box around the detected position,using an estimate of the background. The gain variation due to dippers and faders will have an effect on the accuracy of the photometry, because it modifies the background on a individual pixel. To summarize this approach, the advantages are that the detection is relatively robust and independent of the dark current and the flat field, while the drawbacks are:

1.

the photometry is poor;

2.

the temporal detection does not allow the use of correlation between adjacent pixels, which is needed for extended weak sources detection;

3.

data cannot be coadded before detection.

To overcome these problems, the only way is to correct the data from the gain variation due to faders and dippers.