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6 Source detection and photometry

We used SExtractor (Bertin & Arnouts 1996) with the WEIGHT-IMAGE-option and WEIGHT-TYPE = MAP-WEIGHT for the source detection and extraction on the images. The weight-maps described above were used to account for the spatial dependent noise pattern in the co-added images, and in particular to pass the local noise level of the data to the SExtractor program.


  \begin{figure}
{\hbox{
\psfig{figure=ms2904f2.ps,width=18cm,clip=t} }}
\end{figure} Figure 2: The FDF in I band from FORS observations. The common area of all input frames for a field of view of $6\hbox {$^\prime $ }\times 6\hbox {$^\prime $ }$ is shown here. North is up, east to the left. The total integration time was 6.9 h, mean ${\it FWHM} \sim 0\hbox{$.\!\!^{\prime\prime}$ }53$. The QSO Q 0103-260 is south of the center of the frame and marked with an arrow. This area contains $\sim $6100 galaxies. Note the even distribution of galaxies across the frame, except for the small galaxy concentration in the southwestern corner. The brightest object in the field is the large elliptical galaxy in the southeastern part of the FDF at $z \sim 0.2$ with ${m}_{\rm I} = 16.5$.

To use SExtractor, three parameters have to be set: i) The detection threshold t, which is the minimum signal-to-noise ratio of a pixel to be regarded as a detection, ii) the number n of contiguous pixels exceeding this threshold, iii) the filtering of the data prior to detection (e.g. with a top-hat or a Gaussian filter). We used a Gaussian filter with a width $\theta_F$, for the $\theta_F$ values see below.


  \begin{figure}
{\hbox{
\psfig{figure=ms2904f3.ps,width=12.5cm,clip=t} }}
\end{figure} Figure 3: Pixel-value histograms (in ADU per second) for the (central field) I image at various analysis stages. Upper panel: histogram of the original data (thin line) and after subtracting the low frequency spatial variations due to the non-uniform sky background (thick line). Also included is the difference of the corrected histogram and a Gaussian (shown as thick line in the middle panel) fitted to its negative (ADU/s < 0) wing. This negative wing should not be affected by real objects and therefore should represent the true noise in the image. For clarity the difference has been scaled up by a factor of 10 and the curve has been labeled accordingly. The real objects show up as a positive excess of the pixel values in the corrected data distribution and in the difference function at positive ADU/s. Middle panel: the thick line shows the Gaussian derived by fitting the negative wing of the corrected data distribution as described above. Its difference to the pixel-value distribution derived for those pixels where SExtractor (with optimal parameters but without filtering) finds no objects (or object contributions) is shown as a solid line. The corresponding difference distribution of the inverted image is shown dotted for the negative ADU/s only. The negative excess shows the false detections due to the correlated error. The difference curves are again scaled up by a factor of 10. Lower panel: the thin line shows the histogram of the pixel values of pixels not belonging to objects when SExtractor is run after filtering the corrected data with a (2 pixel FWHM) Gaussian. The dotted line shows the difference between this histogram and the Gaussian fit shown in the middle panel. The number of significant false detections has now dropped to nearly zero.

We varied these parameters to maximize the number of source detections, while minimizing false detections. The following procedure, described here for the I-band data, was used for all filters. We first considered only those pixels in the field where the exposure time equaled the total exposure time (the weight-map took care of the correct scaling of RMS for the full field later on) and called this part of data the "central field".

If there were no objects in the field and if the data reduction resulted in a perfectly flat sky we would expect the histogram of the pixel-values to be a Gaussian, with a width reflecting the photon-noise and the correlated noise of the data reduction and coaddition procedure. The actual histogram of pixel-values of the central-field is shown in Fig. 3 (upper panel, thin line). Even ignoring the wings, the histogram is asymmetric around its center at zero. This stems from the non-uniformities of the sky background, that amount to about 1% (see Sect. 4.1). Therefore, we determined the sky-curvature on large scales and subtracted a 2-dimensional fit to this surface from the original data. The corrected histogram of pixel-values (Fig. 3, upper panel, thick curve) is now symmetric around its center at zero and the left-hand part is well described by a Gaussian (with a width of 0.01295 ADU/s). The right hand part shows an excess above $\approx $0.015 ADU/s, which is due to the objects in the field (see difference curve in Fig. 3, scaled up by a factor 10). We have checked that it does not make any difference for the detection and the photometry of reliable objects whether the procedure is applied to the original or to the corrected data: for each object, the difference between the magnitude estimates of these two cases is smaller than the assigned magnitude RMS-error. This implies that we can carry out the adjustment of optimum SExtractor parameters in the corrected version of the data.

To optimize the pre-detection filtering procedure we made the following numerical experiment. We generated a "negative version'' of an image by multiplying it by -1 and a "randomized version'' by randomly assigning measured pixel values to new positions (the weights of the weight-map are re-localized the same way). With no filtering ( $\theta_F=0$) and using t = 1.7 and n = 3 SExtractor finds about 9000 objects in the original image, 5600 in the negative one and 1100 in the randomized one. The fact that many more objects are detected in the negative image than in the randomized one indicates that correlated noise is present in both the negative and the positive images. Therefore filtering must be used to specifically suppress the small-scale noise. It is possible that large-scale noise is still present, but there is no way to remove such a component. By varying the width $\theta_F$ of a Gaussian filter we found that $\theta_F = 2$ is an optimal choice. With n=3 and t=1.7 the number of objects detected on the negative image dropped to the expected random number, nearly zero. Of course, once $\theta_F$ is fixed, one is still left with the freedom of trading n for t by increasing the number of pixels above the threshold and decreasing the threshold value at the same time. We decided to keep n small, in order to obtain an unbiased detection of faint point sources. This choice allows us to exploit the excellent seeing of the I-band data, where the FWHM is only 2.5 pixels.

Now we illustrate our procedure more quantitatively: we ran SExtractor (for each choice of $\theta_F$, n and t) on the positive, the negative and the randomized images. We registered all pixels which were covered by objects, removed them from the pixel-value statistics and normalized the corresponding pixel-value histogram to the total number of pixels in the central field, and we call that the "background-histogram''. We expect that for good source extraction parameters, the background histograms will look like a Gaussian, more precisely like that Gaussian derived by fitting the negative wing of the corrected data distribution, which we call the "optimum-background-histogram'' below. The difference (magnified by a factor of 10) to that optimum background histogram is shown in the middle panel of Fig. 3 for n=3, t = 1.7, $\theta_F=0$ for detection on the positive (solid) and negative (dotted, for negative ADU/s only) image. The negative excess of these histograms below zero are false detections due to correlated noise. Increasing $\theta_F$ these false detections drop dramatically when $\theta_F = 2$ pixels is reached. Then, n=3 and t=1.7 were fixed by requiring no false detections on the negative image, i.e. no detections due to correlated noise. We finally run SExtractor with this set of parameters on the positive image, obtain the background histogram and show the difference to the optimum background histogram in the lower panel of Fig. 3 (dotted histogram, magnified by a factor of 10). The difference is indeed very small.

Using the above parameters ( $\theta_F = 2$ with a Gaussian convolution, n=3 and t = 1.7), obtained from the optimum pre-detection filtering and the requirement of no-detection on the negative image, we find that the extended wing in the ADU-histogram due to the presence of objects disappears and that the histogram becomes symmetrical and Gaussian (see Fig. 3, bottom panel). This demonstrates that with this choice of parameters we are optimally extracting all objects above the noise level, without getting significant false detections. The adopted parameters give a (total) photometric accuracy better than 5$\sigma$.

The optimum parameters were finally used to run SExtractor on the (positive and negative) images of the total FDF. We found about 6900 objects on the positive and less than a handful of objects on the negative side of the entire I image. All these spurious detections occurred near discontinuities of the S/N level outside the central field and were caused by the non perfectly flat sky, which makes some of the discontinuities more pronounced than they should be according to the photon-noise and the corresponding weight-map.

The same analysis described for the I-band image was carried out for the other filters. We emphasize here that our extraction procedure was optimized to maximize the number of real detections for a reliable photometry and hence reliable photometric redshifts rather than to study galaxy number counts at the faintest limits. For the optical bands, we used the same extraction parameters. For the NIR-data we opted for $\theta_F=3$ pixels to match the pixel size of the original NIR-data, which is roughly 1.5 the pixel size of FORS, and t=2.0 and n=5 for the J band, and t=1.9 and n=5for the Ks band, to take into account the poorer seeing and the different noise level. To illustrate the reliability of our detection procedure we display a detection file returned from SExtractor for a $1\hbox {$^\prime $ }\times 1\hbox {$^\prime $ }$ region of the northern part of the FDF in Fig. 4.

The photometric errors presented in the final catalog are those derived by the SExtractor routine. To make sure that the error calculation was not influenced by correlated noise in the sky background, the results of the SExtractor were verified with aperture photometry with different apertures in areas not covered by objects and by estimating the expected photometric errors from the background variations. In general we found good agreement with the SExtractor derived errors. In particular the SExtractor errors were found to be quite accurate for point sources and for small objects. Only in the case of large extended objects may non-stochastic background variations have resulted in an underestimate of the photometric errors. But the few objects possibly affected are normally bright and have small errors, which should still be correct within the numbers given in the catalog.

Finally, we calculated the 50% completeness levels in each filter band using our extraction parameters and the formula given in Snigula et al. (2002). This approach estimates the completeness limit by calculating the brightness at which the area of pixels brighter than the applied flux limit falls below the size threshold of the detection algorithm (for a given FWHM of a point source). To allow a comparison with other deep fields, the data were corrected for galactic extinction as described in Sect. 7. The results are summarized in Table 3.


  \begin{figure}
\par\psfig{figure=ms2904f4.ps,width=8.8cm,height=9cm,clip=t}\end{figure} Figure 4: Detection file returned from SExtractor for a $1\hbox {$^\prime $ }\times 1\hbox {$^\prime $ }$ region of the northern part of the FDF. It illustrates the reliability of our detection and photometry procedure. The I-band image shown here contains $\sim $160 objects. For some objects the integrated magnitudes are displayed. The detection file shows the elliptical aperture limits used to derive mag_auto. Dashed ellipses denote blended objects.


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