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.
In order to quantify the internal errors
we produced a one square degree residual map by removing all the sources
detected above
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
and
and
intrinsic angular sizes
(FWHM major axis) between
and
.
Intrinsic sizes were convolved with the synthesized beam
(
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.
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
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 (
)
is much lower than the
-threshold chosen for the catalogue (indicated by the vertical
solid line in Fig. 8): at
we expect flux
over-estimations
.
Some systematic effects appear to be present also for the source size at
:
the major and minor axes tend to be respectively under- and
over-estimated (see Fig. 9).
Such effects disappear at
(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):
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 (
)
the errors are in the range
-
for angular sizes in the range
-
.
For fluxes
the errors are always
.
Deconvolved angular sizes are unreliable for very faint sources
(
), 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
never reaches
:
it goes from
at the lowest fluxes, to
at 1 mJy, to
at the highest
fluxes. We therefore can assume that
is
a critical value for deconvolution at the ATESP resolution,
and that ATESP sources with intrinsic sizes
are to be
considered unresolved.
The positional accuracy for point sources is shown in Fig. 10,
where we plot the difference (
and
)
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):
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 at the lowest flux levels (
)
and gradually disappears going to
higher fluxes (no effect for
).
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
(), the observing frequency (
)
and the synthesized
beam FWHM width (
). 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):
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
pixels,
source flux measures at distances as large as
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
spacing rectangular grid, a source located at the center of phase of one
field (d=0) is measured also
at
in the 4 contiguous E, W, S and N fields
and at
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
,
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
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
,
which represents
the maximum distance to the closest field center for ATESP sources.
This maximum
value gives an upper limit of
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
,
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 (
)
ATESP
sources with
mJy and we have plotted the average values of
the
ratio in different distance intervals
(full dots in Fig. 12). The 2 mJy threshold (
)
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
and do not depend on the actual position of the source in the
mosaics. This result also confirms the
estimate drawn from
Fig. 4.
We finally point out that smearing will affect to some extent also source
sizes and source coordinates.
![]() |
Figure 13: Comparison of NVSS with ATESP flux densities. Dashed lines show the 90% confidence limits in the flux measure |
![]() |
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 (
![]() ![]() ![]() ![]() ![]() |
The NVSS has a poor spatial resolution
(
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).
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
will be only marginally
separated.
To avoid this problem we have restricted the comparison to bright
(
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
(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
).
In Fig. 13 we have plotted the NVSS against
the ATESP flux density for the 443
mJy ATESP sources identified
(sources within the
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
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 (). 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
(
mJy) about 40%
of the sources can be resolved by the ATESP
synthesized beam (intrinsic angular sizes
). This fraction
goes up to
at
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).
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:
and
;
and
.
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
and
(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)
and
.
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.
Copyright The European Southern Observatory (ESO)