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Subsections

  
4 Errors in the source parameters

Parameter uncertainties are the quadratic sum of two independent terms: the calibration errors, which dominate at high signal-to-noise ratios, and the internal errors, due to the presence of noise in the maps. The latter dominate at low signal-to-noise ratios.

In the following sections we discuss the parameter internal accuracy of our source catalogue. Master equations for total rms error derivation, with estimates of the calibration terms are reported in Appendix A.

  
4.1 Internal accuracy

In order to quantify the internal errors we produced a one square degree residual map by removing all the sources detected above $5\sigma$ in the radio mosaic fld20to25. We performed a set of Montecarlo simulations by injecting Gaussian sources in the residual map at random positions and re-extracting them using the same detection algorithm used for the survey (IMSAD). The Montecarlo simulations were performed by injecting samples of 30 sources at fixed flux and intrinsic angular size. We sampled peak fluxes between $5\sigma$ and $50\sigma$ and intrinsic angular sizes (FWHM major axis) between $4\hbox{$^{\prime\prime}$ }$ and $20\hbox{$^{\prime\prime}$ }$. Intrinsic sizes were convolved with the synthesized beam ( $8.5\hbox{$^{\prime\prime}$ }\times 16.8\hbox{$^{\prime\prime}$ }$ for mosaic fld20to25) before injecting the source in the residual map.

The comparison between the input parameters and the ones provided by IMSAD permitted an estimate of the internal accuracy of the parameters as a function of source flux and intrinsic angular size. In particular we could test the accuracy of flux densities, positions and angular sizes and estimate both the efficiency and the accuracy of the deconvolution algorithm.

  
4.1.1 Flux densities and source sizes


  \begin{figure}
{\includegraphics[width=8.8cm]{H2076f8.ps} }\end{figure} Figure 8: Peak flux density internal accuracy for point sources. Mean and standard deviation for the output/input (IMSAD/injected) ratio, as a function of flux. Expected values (see text) are also plotted for both the mean (dashed line) and the rms (dotted lines). The flux density cut-off chosen for the ATESP catalogue is indicated by the vertical solid line


  \begin{figure}
{\includegraphics[width=8.8cm]{H2076f9.ps} }\end{figure} Figure 9: Fitted FWHM axes internal accuracy for point sources. Top panel: major axis. Bottom panel: minor axis. Mean and standard deviation for the output/input (IMSAD/injected) ratio, as a function of flux. Expected values (see text) are also plotted for both the mean (dashed line) and the rms (dotted lines). The flux density cut-off chosen for the ATESP catalogue is indicated by the vertical solid line

The flux density and fitted angular size errors for point sources are shown in in Figs. 8 and 9 where we plot the ratio of the parameter value found by IMSAD (output) over the injected one (input), as a function of the signal-to-noise ratio. We notice that mean values very far from 1 could indicate the presence of systematic effects in the parameter measure. The presence of such systematic effects is clearly present for peak flux densities in the faintest bins (see Fig. 8). This is the expected effect of the noise on the catalogue completeness at the extraction threshold. Due to its Gaussian distribution whenever an injected source falls on a noise dip, either the source flux is underestimated or the source goes undetected. This produces an incompleteness in the faintest bins. As a consequence, the measured fluxes are biased toward higher values in the incomplete bins, because only sources that fall on noise peaks have been detected and measured. We notice that the mean values found for $S_{\rm output}/S_{\rm input}$ are in good agreement with the ones expected taking into account such an effect (see dashed line). It is worth pointing out that our catalogue is only slightly affected by this effect because the detection threshold ($4.5\sigma$) is much lower than the $6\sigma$-threshold chosen for the catalogue (indicated by the vertical solid line in Fig. 8): at $S_{\rm peak} \geq 6\sigma$ we expect flux over-estimations $\leq 5\%$.

Some systematic effects appear to be present also for the source size at $5\sigma$: the major and minor axes tend to be respectively under- and over-estimated (see Fig. 9). Such effects disappear at $S_{\rm peak} \geq 6\sigma$ (ATESP cut-off). For both the flux densities and the source axes, the rms values measured are in very good agreement with the ones proposed by Condon (1997) for elliptical Gaussian fitting procedures (for details see Appendix A):

   
$\displaystyle \sigma(S_{\rm peak})/S_{\rm peak}$ = $\displaystyle 0.93 \; \left(\frac{S_{\rm peak}}{\sigma}\right)^{-1}$ (3)
$\displaystyle \sigma(\theta_{\rm maj})/\theta_{\rm maj}$ = $\displaystyle 1.24\; \left(\frac{S_{\rm peak}}{\sigma}\right)^{-1}$ (4)
$\displaystyle \sigma(\theta_{\rm min})/\theta_{\rm min}$ = $\displaystyle 0.69\;\left(\frac{S_{\rm peak}}{\sigma}\right)^{-1}$ (5)

where we have applied Eqs. (21) and (41) in Condon (1997) to the case of ATESP point sources (see dotted lines in Figs. 8 and 9).

The fact that a source is extended does not affect the internal accuracy of the fitting algorithm for both the peak flux density and the source axes. In other words the errors quoted for point sources apply to extended sources as well.

However, this is not true for the deconvolution algorithm. The errors for the deconvolved source axes depend on both the source flux and intrinsic angular size. The higher the flux and the larger the source, the smaller the error. In particular, at 1 mJy ( $\sim 12\sigma$) the errors are in the range $35\%$ - $10\%$ for angular sizes in the range $6\hbox{$^{\prime\prime}$ }$ - $20\hbox{$^{\prime\prime}$ }$. For fluxes $>50\sigma$ the errors are always $<10\%$. Deconvolved angular sizes are unreliable for very faint sources ($5-6\sigma$), where only a very small fraction of sources can be deconvolved. The deconvolution efficiency increases with the source flux. In particular, the fraction of deconvolved sources with intrinsic dimension $\leq 4\hbox{$^{\prime\prime}$ }$ never reaches $100\%$: it goes from $3\%$ at the lowest fluxes, to $15\%$ at 1 mJy, to $50\%$ at the highest fluxes. We therefore can assume that $4\hbox{$^{\prime\prime}$ }$ is a critical value for deconvolution at the ATESP resolution, and that ATESP sources with intrinsic sizes $\leq 4\hbox{$^{\prime\prime}$ }$ are to be considered unresolved.

  
4.1.2 Source positions

The positional accuracy for point sources is shown in Fig. 10, where we plot the difference ( $\Delta \alpha $ and $\Delta \delta $) between the position found by IMSAD (output) and the injected one (input), as a function of flux. No systematic effects are present and the rms values are in agreement with the ones expected for point sources (Condon 1998, for details see Appendix A):

  
$\displaystyle \sigma_{\alpha}$ $\textstyle \simeq$ $\displaystyle \frac{b_{\rm min}}{2}\;\left(\frac{S_{\rm peak}}{\sigma}\right)^{-1}$ (6)
$\displaystyle \sigma_{\delta}$ $\textstyle \simeq$ $\displaystyle \frac{b_{\rm maj}}{2}\;\left(\frac{S_{\rm peak}}{\sigma}\right)^{-1}$ (7)

where we have assumed $b_{\rm min}=8\hbox{$^{\prime\prime}$ }$ and $b_{\rm maj}=14\hbox{$^{\prime\prime}$ }$, the average synthesized beam values of the ATESP survey (see dotted lines in Fig. 10). The positional accuracy of ATESP sources is therefore $\sim 1\hbox{$^{\prime\prime}$ }$ at the limit of the survey ($6\sigma$), decreasing to $\sim
0.5\hbox{$^{\prime\prime}$ }$ at $12\sigma$ ($\sim 1$ mJy) and to $\sim 0.1\hbox{$^{\prime\prime}$ }$ at 50$\sigma$.


  \begin{figure}
\resizebox{8.8cm}{!}{\includegraphics{H2076f10.ps}}\par
\end{figure} Figure 10: Position internal accuracy for point sources. Top panel: right ascension. Bottom panel: declination. Mean and standard deviation for the output-input (IMSAD-injected) difference, as a function of flux. Expected values (see text) are also plotted for both the mean (dashed line) and the rms (dotted lines). The flux density cut-off chosen for the ATESP catalogue is indicated by the vertical solid line

  
4.2 Systematic effects


  \begin{figure}
\resizebox{8.8cm}{!}{\includegraphics{H2076f11.ps}}\par
\end{figure} Figure 11: Bandwidth smearing in ATESP single fields calibrated using source ATESP J233758-401025. The measured peak flux density decreases going to larger distances (Top, full circles) while the measured source area increases correspondingly (Bottom, empty circles). As a consequence the integrated flux density remains constant (Top, stars). The curve describing the peak flux density - distance can be used to describe the area - distance relation as well


  \begin{figure}
\resizebox{8.8cm}{!}{\includegraphics{H2076f12.ps}}\par
\end{figure} Figure 12: Bandwidth smearing in ATESP mosaics. Average $S_{\rm peak}/S_{\rm total}$ ratios obtained by summing all the unresolved ATESP sources brighter than 2 mJy in different $d_{\rm min}$ intervals (full dots). $d_{\rm min}$ is defined as the distance to the closest field center. Also displayed are radial smearing expected in ATESP single fields (solid line) and in ATESP mosaics (dotted line)

Two systematic effects are to be taken into account when dealing with ATESP flux densities, the clean bias and the bandwidth smearing effect. Clean bias has been extensively discussed in Paper I of this series (see also Appendix B at the end of this paper). It is responsible for flux density under-estimations of the order of $10-20\%$at the lowest flux levels ( $S<10\sigma$) and gradually disappears going to higher fluxes (no effect for $S\geq 50-100\sigma$).

The effect of bandwidth smearing is well-known. It reduces the peak flux density of a source, correspondingly increasing the source size in radial direction. Integrated flux densities are therefore not affected.

The bandwidth smearing effect increases with the angular distance (d) from the the pointing center of phase and depends on the passband width ($\Delta\nu$), the observing frequency ($\nu$) and the synthesized beam FWHM width ( $\theta_{\rm b}$). The particular functional form that describes the bandwidth smearing is determined by the beam and the passband shapes. It can be demonstrated, though, that the results obtained are not critically dependent on the particular functional form adopted (e.g. Bridle & Schwab 1989).

In the simplest case of Gaussian beam and passband shapes, the bandwidth smearing effect can be described by the equation (see Eq. (12) in Condon et al. 1998):

 \begin{displaymath}
\frac{S_{\rm peak}}{S^0_{\rm peak}} = \frac{1}{\sqrt{1 + \fr...
... \frac{\Delta \nu }{\nu}\frac{d}{\theta_{\rm b}} \right)^2 } }
\end{displaymath} (8)

where the $\frac{S_{\rm peak}}{S^0_{\rm peak}}$ ratio represents the attenuation on peak flux densities by respect to the unsmeared (d=0) source peak value. We have estimated the actual smearing radial attenuation on ATESP single fields, by measuring $S_{\rm peak}$ for a strong source (ATESP J233758-401025) detected in eight contiguous ATESP fields (corrected for the primary beam attenuation) at increasing distance from the field center (full circles in Fig. 11, top panel). The data were then fitted using Eq. (8) by setting $\nu = 1.4$ GHz and $\theta_{\rm b} \simeq 11\hbox{$^{\prime\prime}$ }$ (from ( $b_{\rm maj} + b_{\rm min})/2 =
(14\hbox{$^{\prime\prime}$ }+8\hbox{$^{\prime\prime}$ })/2$), as for the ATESP data. The best fit (dashed line in Fig. 11, top panel) gives $S^0_{\rm peak} = 131$ mJy and an effective bandwidth $\Delta \nu = 9$ MHz (in very good agreement with the nominal channel width $\Delta \nu = 8$ MHz (see Sect. 5.2 in Paper I). As expected, the measured integrated flux density (stars in the same plot) remains constant.

The mosaicing technique consists in a weighted linear combination of all the single fields in a larger mosaiced image (see Eq. (1) in Paper I). This means that, given single fields of size $1800\times 1800$ pixels, source flux measures at distances as large as $\simeq 35\hbox{$^\prime$ }$ from field centers are still used to produce the final mosaic (even if with small weights). As a consequence, the radial dependence of bandwidth smearing tends to cancel out.

For instance, since ATESP pointings are organized in a $20\hbox{$^\prime$ }$ spacing rectangular grid, a source located at the center of phase of one field (d=0) is measured also at $d=20\hbox{$^\prime$ }$ in the 4 contiguous E, W, S and N fields and at $d=20\cdot \sqrt{2} \simeq 28\hbox{$^\prime$ }$ in the other 4 diagonally contiguous fields. Using Eq. (1) of Paper I, we can estimate a 4% smearing attenuation for the mosaic peak flux of that source. In the same way we can estimate indicative values for mosaic smearing attenuations as a function of $d_{\rm min}$, defined as the distance to the closest field center (see dotted line in Fig. 12). We notice that actual attenuations vary from source to source depending on the actual position of the source in the mosaic.

From Fig. 12 we can see that at small $d_{\rm min}$ mosaic smearing is much worse than single field's one (indicated by the solid line). The discrepancy becomes smaller going to larger distances and disappears at $d_{\rm min}\simeq 14\hbox{$^\prime$ }$, which represents the maximum distance to the closest field center for ATESP sources. This maximum $d_{\rm min}$ value gives an upper limit of $\sim 6\%$ to mosaic smearing attenuations.

The expected mosaic attenuations have been compared to the ones obtained directly estimating the smearing from the source catalogue. As already noticed (Sect. 3.1), a ratio $S_{\rm peak}/S_{\rm total}< 1$, is purely determined, in case of point sources and in absence of flux measurement errors, by the bandwidth smearing effect, which systematically attenuates the source peak flux, leaving the integrated flux unchanged. We have then considered all the unresolved ( $\Theta_{\rm maj}=0$) ATESP sources with $S_{\rm peak}>2$ mJy and we have plotted the average values of the $S_{\rm peak}/S_{\rm total}$ ratio in different distance intervals (full dots in Fig. 12). The 2 mJy threshold ( $\sim 25\sigma$) was chosen in order to find a compromise between statistics and flux measure accuracy. The average flux ratios plotted are in very good agreement with the expected ones, especially when considering that the most reliable measures are the intermediate distance ones, where a larger number of sources can be summed. In general we can conclude that on average smearing attenuations are $\sim 5\%$ and do not depend on the actual position of the source in the mosaics. This result also confirms the $5\%$ estimate drawn from Fig. 4. We finally point out that smearing will affect to some extent also source sizes and source coordinates.


  \begin{figure}
\resizebox{8.8cm}{!}{\includegraphics{H2076f13.ps}}\par
\end{figure} Figure 13: Comparison of NVSS with ATESP flux densities. Dashed lines show the 90% confidence limits in the flux measure


  \begin{figure}
\resizebox{8.8cm}{!}{\includegraphics{H2076f14.ps}}\par
\end{figure} Figure 14: Result of the cross-identification between ATESP and NVSS catalogues in the overlapping region. Only isolated ATESP sources have been considered (see Sect. 4.3.1). The position offsets ( $\Delta \alpha $and $\Delta \delta $ are expressed in terms of their rms uncertainties ( $\sigma _{\alpha }$ and $\sigma _{\delta }$). The $3\sigma $ confidence error circle is also shown

  
4.3 Comparison with external data

To determine the quality of ATESP source parameters we have made comparisons with external data. Unfortunately in the region covered by the ATESP, there are not many 1.4 GHz data available. The only existing 1.4 GHz radio survey is the NVSS (Condon et al. 1998), which covers about half of the region covered by the ATESP survey ( $\delta > -40\hbox{$^\circ$ }$) with a point source detection limit $S_{\rm lim}\sim$ 2.5 mJy.

The NVSS has a poor spatial resolution ( $45\hbox{$^{\prime\prime}$ }$ FWHM beam width) compared to ATESP and this introduces large uncertainties in the comparison, especially for astrometry. To test the positional accuracy we have therefore used data at other frequencies as well. In particular we have used VLBI sources extracted from the list of the standard calibrators at the ATCA and the catalogue of PMN compact sources with measured ATCA positions (Wright et al. 1997).

  
4.3.1 Flux densities

In order to estimate the quality of the ATESP flux densities we have compared ATESP with NVSS. To minimize the uncertainties due to the much poorer NVSS resolution we should in principle consider only point-like ATESP sources. Nevertheless, in order to increase the statistics at high fluxes (S>10 mJy), we decided to include extended ATESP sources as well. Another source of uncertainty in the comparison is due to the fact that in the NVSS distinct sources closer together than $50\hbox{$^{\prime\prime}$ }$ will be only marginally separated. To avoid this problem we have restricted the comparison to bright ( $S_{\rm peak}>1$ mJy) isolated ATESP sources: we have discarded all multi-component sources (as defined in Sect. 3.2) and all single-component sources whose nearest neighbor is at a distance $\leq 100\hbox{$^{\prime\prime}$ }$ (as in the comparison between the FIRST and the NVSS by White et al. 1997). We have not used isolated ATESP sources fainter than 1 mJy because we have noticed that there are several cases where NVSS point sources are resolved in two distinct objects in the ATESP images, only one being listed in the ATESP catalogue (i.e. the other one has $S_{\rm peak} <6\sigma$).

In Fig. 13 we have plotted the NVSS against the ATESP flux density for the 443 $S_{\rm peak}>1$ mJy ATESP sources identified (sources within the $3\sigma $ confidence circle in Fig. 14). We have used integrated fluxes for the sources which appear extended at the ATESP resolution (full circles) and peak fluxes for the unresolved ones (dots). Also indicated are the $90\%$ confidence limits (dashed curves), drawn taking into account both the NVSS and the ATESP errors in the flux measure. In drawing the upper line we have also taken into account an average correction for the systematic under-estimation of ATESP fluxes due to clean bias and bandwidth smearing (see Sect. 4.2). The provided NVSS fluxes are already corrected for any systematic effect.

The scatter plot shows that at high fluxes the ATESP flux scale agrees with the NVSS one within a few percents ($\leq 3\%$). This gives an upper limit to calibration errors and/or resolution effects at high fluxes. Going to fainter fluxes the discrepancies between the ATESP and the NVSS fluxes become larger, reaching deviations as high as a factor of 2 at the faintest levels. ATESP fluxes tend to be lower than NVSS fluxes. This could be, at least partly, due to resolution effects. Such effects can be estimated from the comparison between NVSS and ATESP sources itself and from theoretical considerations. Assuming the source integral angular size distribution provided by Windhorst et al. (1990) we have that at the NVSS limit ( $S\simeq 2.5$ mJy) about 40% of the sources can be resolved by the ATESP synthesized beam (intrinsic angular sizes $\geq 4\hbox{$^{\prime\prime}$ }$). This fraction goes up to $50\%$ at $S\simeq 10$ mJy.

On the other hand, we point out that close to the NVSS catalogue cut-off, incompleteness could affect NVSS fluxes. For instance we have noticed that below 5 mJy, there are several cases where the flux given in the NVSS catalogue is overestimated with respect to the one measured in the NVSS image (even taking into account the applied corrections).

  
4.3.2 Astrometry

The region covered by the ATESP survey contains two VLBI sources from the ATCA calibrator catalogue: 2227-399 and 0104-408. These sources were not used to calibrate our data and therefore provide an independent check of our ATCA positions. The offset between ATESP and VLBI positions (ATESP-VLBI) for the first and the second source respectively are: $\Delta\alpha = -0.277\hbox{$^{\prime\prime}$ }$ and $-0.023\hbox{$^{\prime\prime}$ }$; $\Delta\delta =
+0.239\hbox{$^{\prime\prime}$ }$ and $-0.172\hbox{$^{\prime\prime}$ }$. Such offsets indicate that the uncertainty in the astrometry should be within a fraction of arcsec. Obviously, we cannot exclude the presence of systematic effects, but an analysis of the ATESP-NVSS positional offsets gives $\Delta\alpha_{\rm med}=-0.115\hbox{$^{\prime\prime}$ }$ and $\Delta\delta_{\rm med}=-0.8\hbox{$^{\prime\prime}$ }$(see Fig. 14).

A more precise comparison could be obtained by using the PMN sources with ATCA position measurements available. Unfortunately the number of such PMN sources in the region covered by the ATESP survey is very small: we found only 12 identifications. Using the 4.8 GHz positions for the PMN sources, we derived (ATESP-ATPMN) $\Delta\alpha_{\rm med}=-0.115\hbox{$^{\prime\prime}$ }$ and $\Delta\delta_{\rm med}=-0.3\hbox{$^{\prime\prime}$ }$.

We can conclude that all comparisons give consistent results and that the astrometry is accurate within a small fraction of an arcsec. Also systematic offsets, if present, should be very small.

Acknowledgements
We acknowledge R. Fanti for reading and commenting on an earlier version of this manuscript.

The authors aknowledge the Italian Ministry for University and Scientific Research (MURST) for partial financial support (grant Cofin 98-02-32). This project was undertaken under the CSIRO/CNR Collaboration programme. The Australia Telescope is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.


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