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4. Photometric analysis

This section briefly describes the photometric software used in the data analysis (SExtractor: Source Extraction software). A complete description is given in Bertin & Arnouts (1996, hereafter BA96).

4.1. Description of photometric package

The photometric analysis is done in four steps:

4.1.1. determination of the sky background

The background map is derived by binning the frame into large meshes (32 tex2html_wrap_inline2463 32 or 64 tex2html_wrap_inline2465 64 pixels), and removing the possible overestimations due to bright objects using a tex2html_wrap_inline2467-sigma clipping algorithm. This sky background map is subtracted from each frame.

4.1.2. Detection of objects

The detection algorithm determines a group of connected pixels above a given threshold. A minimum number of connected pixels is chosen in order to avoid spurious objects (bad pixels, un-removed cosmic rays,...). Convolution with a gaussian with a FWHM close to the seeing improves the detection of objects by decreasing the background noise. The typical threshold used is tex2html_wrap_inline2469 above the background value (tex2html_wrap_inline2471 is measured from the unsmoothed background) corresponding to a surface brightness equal to 27 mag arcsectex2html_wrap_inline2473 in B, 26.5 mag arcsectex2html_wrap_inline2477 in V and 26 mag arcsectex2html_wrap_inline2481 in R. When such a low threshold is used, large spurious faint objects can appear in the wings of objects with shallow profiles. This effect is seen around elliptical galaxies or bright stars where the local background noise increases and can exceed the detection threshold. A cleaning procedure is therefore applied to check if the detected objects are real. For the faint objects in the vicinity of bright objects, a new estimation of the local background is obtained by assuming that the dominant central object has large gaussian wings. If the mean surface brightness of the faint object is lower than the calculated local threshold, the object is rejected.

4.1.3. Deblending of neighbouring objects:

Each detection of a connected set of pixels is processed through a deblending algorithm based on multi-thresholding as described in BA96.

4.1.4. Photometry

Three kinds of magnitudes are used: isophotal, "corrected isophotal''and "adaptive aperture''.

The "corrected isophotal'' retrieves the lost flux in the isophotal magnitude by assuming that the wings of objects outside the limiting isophote are nearly gaussian (see Maddox et al. 1990a). This estimation of the "total'' magnitude could be improved by assuming that the profile of objects follows the more realistic Moffat profile. However, if the value of the threshold is low, the gaussian profile provides an adequate correction as demonstrated by the tests on simulated frames (see Sect. 4.2).

The "adaptive aperture'' magnitude is the best estimation of the "total'' magnitude. The algorithm is similar to the "first-moment'' measure designed by Kron (1980). This magnitude is calculated in two steps:

(1) The object's light distribution above the isophotal threshold is used to measure an isophotal elliptical aperture characterized by the elongation tex2html_wrap_inline2485 and position angle tex2html_wrap_inline2487.

(2) The first moment tex2html_wrap_inline2489 is calculated in an aperture 2 tex2html_wrap_inline2491 larger than the isophotal aperture in order to reach the light profile information below the isophotal threshold.
The first moment is used to define the "adaptive aperture'' of radius tex2html_wrap_inline2493 inside which we measure the "total'' magnitude. The principal axes of each object are defined by tex2html_wrap_inline2495 and tex2html_wrap_inline2497. Kron (1980) uses a circular aperture of radius tex2html_wrap_inline2499 with k=2 which measures 90% of the total flux from the objects. To converge near the "total'' magnitude we can increase the k value, but a compromise must be found between the added measured flux and the increasing noise in larger apertures. As Metcalfe et al. (1991), we choose k=2.5, yielding 94% of the total flux inside the adaptive aperture (this value was calculated using simulated frames with a large variety of galaxy profiles). In contrast to the isophotal magnitude which operates at fixed signal-to-noise, the "adaptive aperture'' magnitude may be determined at very low signal-to-noise. Sometimes for faint objects, tex2html_wrap_inline2507 may converge to erroneously small apertures. We therefore constrain the apertures to a minimum value of tex2html_wrap_inline2509 (where tex2html_wrap_inline2511 is the mean standard deviation of the bivariate gaussian profile defined by the second order moments of the object profile (BA96)).

4.2. Simulations

To test the quality of the photometry for objects with different magnitudes, we use simulated frames provided by E. Bertin (BA96) and generated according to the following scheme. A sky background is defined by a sky surface brightness (tex2html_wrap_inline2513 mag/tex2html_wrap_inline2515), and Poisson noise is added according to a gaussian distribution as observed on real CCD frames. A set of stars with a Moffat profile (Moffat 1969) are generated. Galaxies are produced with a large variety of shapes and sizes. The pixel size and seeing disk are adjusted to resemble as much as possible the real frames. The frames are defined through the R band corresponding to the band used here for the selection of the spectroscopic sample and for the star/galaxy separation. The results of the tests on several simulated images are based on tex2html_wrap_inline2519 6000 galaxies and tex2html_wrap_inline2521 600 stars in total.

4.2.1. Test of photometric accuracy

In Fig.  1 (click here), we compare the mean difference between the different measured magnitudes and the true magnitude for the galaxies. The mean difference is calculated in bins of tex2html_wrap_inline2523 of the true magnitude. The error bars represent the rms scatter around the mean.
As expected, the isophotal magnitude which measures the flux inside the defined isophote looses the flux in the wings outside this isophote. Figure 1 (click here) shows that an increasing fraction of flux is lost for fainter magnitudes. The "corrected isophotal'' magnitude (defined in Sect. 4.1.4) provides a significant improvement.
Figure 1 (click here) also confirms that the "adaptive aperture'' magnitude (Kron magnitude) measures tex2html_wrap_inline2525 of the flux of objects over the entire magnitude range (the systematic offset of tex2html_wrap_inline2527 is indicated by the dashed line in Fig. 1 (click here)). To reach the "total'' magnitude, we then subtract a constant value of tex2html_wrap_inline2529 to all magnitudes. The error bars in Fig. 1 (click here) are tex2html_wrap_inline2531 for galaxies with tex2html_wrap_inline2533, and up to tex2html_wrap_inline2535 at fainter magnitudes. The interest of the "adaptive aperture'' magnitude over the isophotal magnitude originates in it insensitivity to seeing and redshift (Kron 1980).
Because a large fraction of the detected objects in our survey are stars, we also compare in Fig. 2 (click here) the different magnitudes for the simulated stars. Even when a gaussian profile is used to correct the flux lost in the wings of stellar objects with Moffat profile, the "corrected isophotal'' is very close to the total magnitude for tex2html_wrap_inline2537.
Because the fraction of flux measured in a fixed aperture is constant for stars, we check the reliability of the second order moments by comparing the "adaptive'' magnitudes to the "aperture'' magnitudes. The aperture radius is defined by tex2html_wrap_inline2539 where tex2html_wrap_inline2541 is the second order moment along the major axis, estimated by using unsatured stars with high signal-to-noise (typically tex2html_wrap_inline2543). The lost flux is stable in the full magnitude range and is close to tex2html_wrap_inline2545. This value is in good agreement with the "adaptive'' aperture obtained by using a minimum radius defined as tex2html_wrap_inline2547 (where tex2html_wrap_inline2549 is calculated for each object). We can then conclude that the second order moments can be reliably estimated down to very low signal-to-noise ratios and the "adaptive'' magnitude is a good measure of the total magnitude.

Despite the reliability of "adaptive'' magnitude for stars, we use the "corrected isophotal'' magnitude as being the best magnitude to estimate the "total'' magnitude. Because, we make the star/galaxy separation only for tex2html_wrap_inline2551, we use the "corrected isophotal'' magnitude up to this cut-off for the stellar sample and the "adaptive'' magnitude for the galaxy sample. At fainter magnitudes, no classification is done, thus we use the "adaptive'' magnitude for all objects.

Figure 1: Mean difference in tex2html_wrap_inline2553 bins between the measured magnitudes and the true magnitudes for galaxies in simulated R frames. The symbols for the different magnitudes used are given inside the graph. The error bars show the rms scatter around the mean. The dashed line represents the expected position of magnitude difference for the "adaptive aperture'' magnitude corresponding to the 94% enclosed flux

Figure 2: same as Fig. 1 (click here) for stars. The dashed line represents the flux measured inside a radius defined as tex2html_wrap_inline2557, where tex2html_wrap_inline2559 is derived from stars with high signal-to-noise (as described in text)

4.2.2. Conclusion

The results of the tests applied to the simulated images show that:

(1) For galaxies, the "adaptive aperture'' magnitude is a robust estimate of the "total magnitude'' (after correction for an offset of tex2html_wrap_inline2569) down to tex2html_wrap_inline2571. The uncertainty in the measure is close to tex2html_wrap_inline2573 up to tex2html_wrap_inline2575 and increases to tex2html_wrap_inline2577 up to tex2html_wrap_inline2579. However, this is an aperture magnitude and it is sensitive to crowding by close neighbours. Therefore, when an object has a neighbours closer than 2 isophotal radii, we use by default the "corrected isophotal'' magnitude.

(2) For stars, the "corrected isophotal'' magnitude is a reliable estimate of the "total magnitude'' for tex2html_wrap_inline2581.

(3) Because we measure "total'' magnitudes, the colours can be calculated as the difference between the "total'' magnitudes in each passband. The colour error can be estimated as the quadratic sum of the error in each band. For objects brighter than tex2html_wrap_inline2583, the colour error is tex2html_wrap_inline2585 and at fainter magnitudes the colour error is tex2html_wrap_inline2587.

4.3. Star-galaxy separation

To build the spectroscopic catalogue, it is important to exclude the stellar objects. The star/galaxy separation is performed using the neural network included in the photometric software SExtractor. For a complete description of the principle, the training (with using simulated frames containing stars and galaxies), and various tests of this neural network, the reader can refer to BA96. Here, we describe the input and output parameters used for the separation. For each image, the classifier works in a ten-parameter space containing 8 isophotal areas defined by dividing the peak intensity in 8 levels equally spaced in logarithmic scale, the central peak intensity, and one control-parameter which defines the fuzziness of the frame and is chosen to be the seeing (in pixels). This set of input parameters is shown to provide an "optimal'' description of the characteristics of each image (BA96). The neural network provides an output parameter defined as a "stellarity-index''. Because the p.s.f. of a frame is the parameter which determines the quality of the performance of the neural network, SExtractor estimates on each frame the FWHM of the p.s.f. using the unsatured bright stars and uses it for the neural network. The "stellarity-index'' output parameter is a measure of the confidence level in the classification of each image. The value varies between 1 for stars and 0 for galaxies. In Fig. 3 (click here), we show the stellarity index as a function of magnitude for a R frame. For our 50 R frames, a reliable classification can be done for tex2html_wrap_inline2593 with a success rate for the galaxies close to 95% (BA96). A fainter magnitude limit can be reached for several frames with good seeing quality (such a frame is shown in Fig.  3 (click here)). To be sure that galaxies are not mis-classified as stars, we define as stars all objects with a stellarity index greater than 0.8. The corresponding magnitude limit for classification is 1.5 magnitude deeper than our spectroscopic limit of R=20.5, and thus guarantees that our spectroscopic sub-sample is poorly biased by mis-classified objects.
Figure 3 (click here) shows that at faint magnitudes tex2html_wrap_inline2597, the stellarity index approaches 0.5 because no distinction can be made between the two classes when the profiles are dominated by the seeing disk. On the other hand, at bright magnitudes, for the largely saturated stars (tex2html_wrap_inline2599) the stellarity index may drop below 0.8.
When we started the spectroscopic observations, the photometric software by E. Bertin was not avalaible. We thus used INVENTORY, the photometric software developed by A. Kruszweski (West & Kruszewski 1981) and avalaible in the MIDAS environment. INVENTORY provides a reliable star-galaxy separation for tex2html_wrap_inline2601 and was used to generate the entry catalogue for the first half of the spectroscopic observations. The good success rate in the separation is confirmed by only 15 spectra of stars observed out of a total number of 521 reduced spectra (i.e. tex2html_wrap_inline2603) (Bellanger et al. 1995a).

Figure 3: Star/galaxy separation for a science frame observed in the R band with a seeing of 0.9 arcsec. The ordinate shows the output parameter of neural network, the "stellarity index'' which provides an estimate of the confidence level in the classification of each object as a star (CLASS = 1) or a galaxy (CLASS = 0)

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