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Astron. Astrophys. 342, 313-336 (1999)

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5. Simulations

The PRETI method has been developed to detect faint sources in all ISOCAM data (if they are taken in fields where the background is more or less constant), but has to be tuned in order to work very efficiently, as shown in the previous section. This fine tuning has been done through various tests and simulations, that have proven to be very useful to ascertain detections, study the completeness, the number of false detections and to calibrate the source photometry.

In order to test the PRETI method, we have tried to build simulated data. These should be as near as possible of real ISOCAM data, that is, contain glitches, transients, and sources. A special attention has to be payed to the `fader' and `dipper' long term effects of glitches since they are the most limiting factor for the faint source detection. Unfortunately, no available model is able to reproduce accurately these behaviours of the LW detector. The only solution is to use real ISOCAM data to build simulations.

5.1. Cosmic rays in simulated data

Cosmic ray effects are dependent of three parameters

  1. the integration time because it is directly related to the mean number of cosmic rays that will hit the detector per readout.

  2. the background emission, because in the long term effect a cosmic ray may affect the gain of the pixels (though this is not yet clearly established).

  3. the period of the observation since the cosmic ray density varies along the orbit of the satellite as it crosses the radiation belts. We have noticed variations of 40% in the number of `faders' and `dippers' between various observations, early part of the orbit being most affected.

For these reasons, data for the simulation are taken with the same integration time, same gain, same filter and zodiacal background as the observation that is to be simulated. Moreover, in order to avoid any modification of the time history of pixels, these data have to be taken on periods at least as long as the observation. For example, data taken on a shorter period and duplicated would cause discontinuities in the time history and produce spurious sources.

5.2. Suppression of real sources in the data set used for simulations

Since we look for faint sources, it is very likely that the observations selected to be used for simulations will contain such sources.

The ideal case is when these are `staring' observations, i.e. when the satellite points during a very long time at the same position. Since the flat-field is built directly from the observation, any source present in the field is cleaned during the operation.

One less ideal case is when the raster parameters of the observation are different from the simulated data. the sources in the simulation data set are blurred when the raster map is reconstructed with different steps. If there are bright sources in the simulated data, which would despite the blurring process still have a high S/N level, these can be masked out.

5.3. Addition of sources

Point sources of various fluxes are added using the PSF model of M. Pérault and K. Okumura, that match well the measurements (Okumura et al. 1997). Each source flux is converted in ADU by using the ISOCAM cookbook. Object are randomly distributed on the raster map, at a precision of a tenth of a pixel, then each position on the detector array is recomputed, without taking the field distortion into account.

Each raster image is then translated into a data cube by using the transient model from Abergel et al. (1996), where the detector response D is a function of the incoming flux I with:

[EQUATION]

It is clear from this formula that the response depends on the intensity of the source and on the background, because the exponential time constant is a function of I. For this reason, we add to the simulated source images the median of the deglitched cube that is to be simulated before running the transient simulator. We subtract this image later, in order to have a simulated cube on a zero background, and add it to the raw data selected to perform the simulation.

5.4. ISO-HDF simulations

We have searched the ISOCAM calibration database for observations matching the ISO-HDF configurations. For the LW3 observation, we found a set of flat-field observations, called `FLAT16' that were done with the same filter, the same lens and same integration time. Seven bright sources present in the field were masked out. Fainter sources have been blurred since this observation is a raster with half an array of raster step (16 pixels). We checked the PRETI method by running it completely on this set. We set our deglitching parameters in order to avoid any detection in that field (since their should be no source), except the ones that could be related to a source present in the original dataset. We also checked that the `fader' and `dipper' rate is about the same in both observations. This set of data is a perfect tool to build complete simulations. Since it is an empty field, we are able to study very carefully the level at which PRETI begins to give us false detections.

For the LW2 ISO-HDF, we could not find any satisfactory set of data: only a few calibration have been performed with an integration time of 10 s, and they are always short ones. We eliminated also data from our IDSPCO survey program that were matching all the configuration requirements, because we found a rate of long term effect cosmic rays about 20% higher: this causes severe differences in the sensitivity that can be reached by an observation. Since our first reductions show us that the source density is not very high in the ISO-HDF LW2 observation, we decided to use these data themselves for the simulations. The drawback is that this prevents us to fully study the confidence level and the false detections.

We performed two kinds of simulations. First, 10 sources of a given flux were randomly added, and the operation was repeated from 1 mJy to 100 µJy by steps of 100 µJy, and from 100 to 10 µJy by steps of 10 µJy. This allowed us to investigate our detection limits and to calibrate our source photometry. Finally, we performed simulations containing a wide range of fluxes, more or less matching number count models, in order to test the completeness of the method.

5.5. Simulation results on the ISO-HDF

Fig. 6 presents the results of our counts on simulated data for LW2 (a) and LW3 (b). For the LW2 filter, all added sources in the field are detected down to 60 µJy for a 5 [FORMULA] detection, showing that we are complete at this level. However, this result does not take into account confusion. A faint source is not detected if is is too close to a bright one ([FORMULA] 9" in LW3, depending on the relative brightness of the two sources). This effect has to be taken into account when deriving number counts. For the LW3 observation, the 99.9% completion is obtained above 200 µJy, for a 5[FORMULA] detection and stays above 85% at 100 µJy. 7[FORMULA] detection, that are more reliable in terms of avoiding false detections, are only 99.9% complete above 1 mJy, but stay of the order of 90% down to 220 µJy. We could not measure the number of false detections in the LW2 band since we used the data set itself to do the simulations. Since PRETI is run with the same arguments on both sets, we assume that the difference should be small between the two filters and tune our criteria according to LW3.

[FIGURE] Fig. 6a and b. Percentage of detected sources in simulations of the ISO-HDF, as a function of the input flux of the sources, for both filters. a  LW2, 5[FORMULA] detection level. b  LW3. Solid: 5[FORMULA] detection level. Dashed: 7[FORMULA] detection level.

Figs. 7 and 8 show the calibrations curves deduced from simulations, i.e. the relation between the measured flux to the real flux that was simulated, for two measurement methods: (1) an aperture photometry taking a radius of 3" for LW2 and 6" for LW3 (two pixels in each case), (2) the integrated flux of the objects reconstructed by our wavelet based detection program (Starck et al. 1998). The two methods display very different behaviors for the faint end of the plots.

[FIGURE] Fig. 7a-d. Photometry results of LW2 simulations. Comparison of photometric methods and errors. a  Measured flux as a function of input flux for 2 photometry methods. Solid: 2 pixel aperture photometry (3"). Dot-dashed: wavelet photometry. Dashed: the identity curve. b  Ratio of measured flux over real flux as a function of the real flux. Solid: 2 pixel aperture photometry (3"). Dot-dashed: 3 pixel aperture photometry (4.5"). Dashed: wavelet photometry. c  Measured flux as a function of input flux for the wavelet method. Solid: measure. Dash-dotted: 1[FORMULA] error on measure. Dotted: Identity d  Measured flux as a function of input flux for 2 pixel (3") aperture photometry. Solid: measure. Dash-dotted: 1[FORMULA] error on measure. Dotted: Identity.

[FIGURE] Fig. 8a-d. Photometry results of LW3 simulations. Comparison of photometric methods and errors. a  Measured flux as a function of input flux for 2 photometry methods. Solid: 2 pixels aperture photometry (3"). Dot-dashed: wavelet photometry. Dashed: the identity curve. b  Ratio of measured flux over real flux as a function of the real flux. Solid: 2 pixel aperture photometry (3"). Dot-dashed: wavelet photometry. c  Measured flux as a function of input flux for the wavelet method. Solid: measure. Dash-dotted: 1[FORMULA] error on measure. Dotted: Identity. d  Measured flux as a function of input flux for 2 pixels (3") aperture photometry. Solid: measure. Dash-dotted: 1[FORMULA] error on measure. Dotted: Identity.

Photometry on wavelet reconstructed objects (panels (c)) shows a non-linearity effect for input fluxes below 100 µJy in LW2 (resp. 200 µJy in LW3). Above these levels, the ratio of the measured flux over the input flux is roughly constant ([FORMULA] for LW2). We interpret this factor in terms of the effect of transients (factor of 0.6), and by the fact that only the central core of the PSF is reconstructed by the algorithm (factor of 0.8), thus giving a theoretical correction of 0.48. Below these levels, the non-linearity is explained by the fact that the wavelet analysis program reconstructs objects with extended low levels wings. This effect might be due to the fact that we use a b-spline wavelet transform of our map that does not match closely enough the ISOCAM PSF. We note that the non-linearity is more marked for LW3 than for LW2 where the PSF is less extended, and is therefore closer to the wavelet shape. For the last filter, the measured level stays constant at 30 µJy while it rises in the first one for decreasing fluxes. The reason for this difference is not clear, but might be explained by the confusion of the the observations in the LW3 map, or by residual glitches 1. This non-linearity prevents to use the "wavelet" photometry for low levels objects, thus this method is not appropriate for the HDF data set.

Aperture photometry (panels (d)) does not display such non-linearity. In LW2, the ratio of the measured flux over the input flux decreases as the input flux decreases. The explanation for this behaviour is the following: when the flux decreases, the outer pixels of the central core of the PSF are more and more dominated by the noise, and the summed flux contains therefore negative terms. Indeed, due to the baseline substraction, a few readouts are slightly negative before and after the source, (the background level is overestimated). Therefore, the average level on the final map around the source is slightly negative. This effect is illustrated in Fig. 7 (b) were a 3 pixel radius aperture photometry has also been drawn (dash-dotted curve). For this method, the ratio at high flux is larger than for a 2 pixel radius aperture (more of the central core is taken into account), but decreases more abruptly toward faint fluxes, because more noise and negative values are integrated in the aperture. We note that at high flux, the ratio between the measured flux in the 3 pixels and 2 pixels apertures is [FORMULA], in good agreement with the PSF model that predicts 1.209. Again for this photometric method, the behaviour of the LW3 filter is slightly different, because the slope of the calibration curve is less steep toward faint fluxes. The change of slope occurs at the same level (200 µJy) where the "wavelet" method begin to be non-linear. This indicates that the same process is at work for both methods, but that aperture photometry is less affected. For these reasons, we have used a two pixel aperture photometry for the HDF.

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© European Southern Observatory (ESO) 1999

Online publication: February 22, 1999
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