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Astron. Astrophys. 363, 476-492 (2000)
5. Expected accuracy on real data
In order to discuss on the expected accuracy and possible
systematic errors when exploring real data, we have performed a
complete set of simulated catalogues, with a more realistic (non
uniform) redshift distribution and
ratio along the SEDs. We have adopted the simple PLE model proposed by
Pozzetti et al. (1996, 1998), with minimal changes, to derive the
redshift distributions and to assign a magnitude to each object in the
different filters. Four galaxy types (exponentially decaying SFR with
characteristic time Gyr and 10 Gyr,
constant SFR with evolution in time and at a fixed age of 0.1 Gyr) and
their corresponding luminosity functions are used to reproduce the
number of galaxies expected at a given redshift and absolute magnitude
. Apparent magnitudes are computed
from the evolved SEDs, with ages depending on the redshift considered,
the formation redshift, set to , and
the cosmological parameters. Photometric errors are scaled to apparent
magnitudes assuming the approximate relation
, where
is the signal to noise ratio, which
is given as a function of the apparent magnitude through
,
being the signal to noise ratio at a given reference magnitude
. For simplicity, the photometric
error is assigned to the apparent magnitude m according to a
Gaussian distribution of fixed
. This relation is set to reproduce
the rapid increase of uncertainties when approaching the limiting
magnitudes. According to these equations, a value of
, corresponding to
, is reached 2.5 magnitudes brighter
than the magnitude corresponding to
. An object with
is non-detected in the involved
filter ( ). An object is included in
the final catalogue if it is detected in the filter I (assuming
that this is the selection filter), and in at least two other filters.
The last requirement is needed to compute
.
The same filter combinations discussed in Sect. 3 have been
used to produce the new simulated catalogues. The simulations in
Sect. 3 represent an ideal case, with an infinite depth and a
fixed photometric error, disregarding the dependence on errors versus
magnitudes. However, the relevant quantities
, l% and g% strongly
depend on the number of objects in each redshift bin and then on the
limiting magnitudes.
To give a qualitative idea of the accuracy expected with different
observational configurations, we consider two representative
cases.
5.1. Deep pencil beam surveys
Firstly, we focus on simulations obtained in the case of a pencil
beam-like survey, i.e. a very deep observation, covering a small area.
From the photometric point of view, the main improvement with respect
to the uniform distributions presented above is that we can introduce,
for each object, a realistic in the
different filters, with different values from filter to filter. We
assume that the detection limit is reached
( ) at magnitudes similar to the
limiting magnitudes of the HDF, as reported in the column
(d) of Table 4 and in the right
part of the same table for the HDF-N filters. To obtain approximately
the same number of galaxies observed in the HDF, a field of
5 arcmin2 has been simulated. The percentages of spectral
types included in the simulated catalogue are
% for E, Sb, Im, and
Im( Gyr) respectively. In order to
reproduce the observed number counts at faint magnitudes (Williams et
al. 1996) we assume an open cosmological model, with
and
. In this case, the peak of the
redshift distribution is at and
very few objects are seen at low-z, in particular at redshifts
between and
. Moreover, the PLE model is known
to overestimate the population of high redshift galaxies.
![[TABLE]](img228.gif)
Table 4. Limiting magnitudes at : (d) deep pencil beam-like survey, (s) shallow ground-based survey.
In Table 5 we display the computed quantities
, l% and g% for the
set of filters of the HDF-N, and for all the other deep survey
combinations considered in Sect. 3 (marked by (d) in the second
column). The table contains the dispersion and the percentages of
spurious and catastrophic objects, computed from a set of 10
independent simulations for each configuration. The interpretation of
data in Table 5 must take into account that the definition of
g% depends on the dispersion
computed using the correctly assigned objects, and this quantity is
quite sensitive to the different filter sets and redshift bins.
Nevertheless, these simulations take properly into account the
observed properties of galaxies in deep surveys, such as the presence
of faint objects with huge photometric errors, and the lack of
detection in some filters leading to an uncertain
estimate (that is, increasing the
probability of misidentifications, enlarging the error bars and the
dispersion around the true value). In particular, this effect is
evident when looking at the trend of the l% values. At higher
redshift we find an increasing number of faint objects that are non
detected in some filters. This leads to an increase of l%.
Because of the depth of limiting magnitudes, we adopt the non
detection law number 1 for optical filters and 2 for the near infrared
ones. In the highest redshift bins, the increase of the dispersion
value tends to mitigate the effect of the
deterioration in the value of
g%.
![[TABLE]](img231.gif)
Table 5. The dispersion and the percentage of catastrophic and spurious objects, l% and g%, with errors, in five redshift bins, computed from 10 realizations of simulated catalogues with a redshift distribution derived from a PLE model. (s) and (d) refer to shallow and deep surveys respectively. The data are replaced by a dash when there are not enough data to compute the statistics. For the three examples in Fig. 9, Fig. 10 and Fig. 11 we also present the quantities mentioned above as a function of the limiting signal to noise ratio considered for the detection.
In the case of HDF-N filter set, we considered also two
subcatalogues built with more restrictive selection criteria,
requiring the detection both in
and in at least two other filters to be
and
. Statistics concerning these
simulations are tabulated in Table 5. Obviously, when considering
objects with increasing , the
accuracy of estimate significantly
improves. In Fig. 9 we show the results obtained on the
comparison between and model
redshifts, and also on the versus
reconstruction. Most of the
discrepancy is due to objects with
.
![[FIGURE]](img250.gif) |
Fig. 9. Top: comparison between and for realistic catalogue HDF-like. See the text for the considered limiting magnitudes. Small dots, crosses and circles correspond to objects brighter than respectively, at least in the I filter and in two other filters. Bottom: redshift distributions for the simulation on the top with . Solid line: . Dashed line: .
|
5.2. Shallow wide field surveys
In the second case, the aim is to reproduce the observational
conditions reached when using 8 m telescopes and a wide field
detector. In particular, we consider the case of a survey in a
arcmin2 field, observed
with all the filter sets considered in Sect. 3. The adopted
limiting magnitudes are shallower and conservative with respect to the
values in the previous simulations. They are shown in the column
(s) of Table 4. The percentages
of the different spectral types for catalogues with these limiting
magnitudes, using the same detection criteria, are similar to the
previous ones, being % for E, Sb,
Im, and Im( Gyr) respectively.
Results for , l% and
g% are presented in Table 5 and marked by a (s). For the
filter sets UBVRI and UBVRIJK we repeated the same
procedure adopted for the HDF-N simulated catalogue, to build two
subcatalogues with higher
thresholds. Fig. 10 presents the results obtained with the five
optical bands only, whereas Fig. 11 displays the equivalent
results with the additional photometry in two near infrared filters.
Fig. 10 and Fig. 11 show the associated input and recovered
redshift distributions.
![[FIGURE]](img265.gif) |
Fig. 10. Top: comparison between and for realistic catalogue for a shallow survey with 5 filters. The symbols are the same as in Fig. 9. Bottom: redshift distributions for the simulation on the top with . Solid line: . Dashed line: .
|
![[FIGURE]](img277.gif) |
Fig. 11. Top: comparison between and for realistic catalogue for a shallow survey with 7 filters. The symbols are the same as in Fig. 9. Bottom: redshift distributions for the simulation on the top with . Solid line: . Dashed line: .
|
The peak of the redshift distribution in this case is at a lower
redshift compared to the HDF simulation. Wide-field surveys allow to
obtain a better sampling of the bright end of the luminosity function
with respect to HDF-like surveys, the later being more suited to
explore the faint luminosity regime. The value of g% and
l% change significantly when considering the same set of
filters, but a different kind of survey. On the contrary,
remains similar.
An interesting feature is the opposite trend displayed by deep
pencil-beam compared to shallow wide-field surveys with respect to the
low and high redshift regimes for a given filter set. At low
redshifts, the value of g% is larger for the deep pencil-beam
survey (type (d) in the Table 5) than for the shallow (s)
wide-field one. Conversely, the accuracy of deep surveys overcome that
of the shallow ones at high redshifts. In this context, the separation
between low and high redshift regimes is marked by the
- 2 bin.
This behaviour could be easily explained when we consider the
different characteristics of the catalogues produced in the two cases.
The deep survey catalogue contains few low redshift galaxies, and most
of them derive from the faint tail of the luminosity function. These
faint galaxies are much more abundant than the bright ones, than they
are present in the catalogue even though the volume covered at low
redshift by this survey is small. The photometric errors for these
intrinsically faint objects are rather large, thus causing a poor
estimate of . On the contrary, the
wide-field survey contains a large quantity of bright galaxies at low
redshift, wich have sufficiently small photometric errors to obtain
accurate s. The faintest objects are
lost in this case because of the shallow detection limits. The
majority of galaxies in the shallow wide survey lies in the low
redshift bins, around the peak of .
When we consider the population of galaxies beyond the peak of the
redshift distribution, the photometric errors in the shallow survey
become important and an increasing fraction of objects is non detected
in various filters. These problems hamper a robust determination of
. On the contrary, the pencil beam
survey take advantage of its depth, allowing to compute
at higher redshifts.
On the basis of these results, we caution that the kind of analysis
presented here is strongly advised when a photometric survey is
undertaken in view of computing s. In
particular, the filter configuration and the photometric depth to
reach in each filter have to be determined accurately in advance, in
order to optimize the survey and to study the feasibility of the
project.
© European Southern Observatory (ESO) 2000
Online publication: December 11, 2000
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