Issue |
A&A
Volume 517, July 2010
|
|
---|---|---|
Article Number | A70 | |
Number of page(s) | 9 | |
Section | Astronomical instrumentation | |
DOI | https://doi.org/10.1051/0004-6361/200913803 | |
Published online | 10 August 2010 |
Atmospheric turbulence in phase-referenced and wide-field interferometric images
Application to the Square Kilometre Array
I. Martí-Vidal1,2 - J. C. Guirado2 - S. Jiménez-Monferrer2 - J. M. Marcaide2
1 - Max-Planck-Institut für Radiastronomie, Auf dem Hügel 69, 53121 Bonn, Germany
2 -
Dpt. Astronomia i Astrofísica, Universitat de València, Dr. Moliner 50,
46100 Burjassot, Spain
Received 3 December 2009 / Accepted 2 April 2010
Abstract
Phase referencing is a standard calibration procedure in radio
interferometry. It allows us to detect weak sources by using
quasi-simultaneous observations of closeby sources acting as
calibrators. However, atmospheric turbulence may introduce strong
differences in the optical paths of the signals of the target and
calibrator and affect, or even waste, phase referencing in cases
of relatively large calibrator-to-target separations and/or bad
weather. The situation is similar in wide-field interferometric
observations. We present the results of a Monte Carlo study
of the astrometric precision and sensitivity of an interferometric
array (a realization of the Square Kilometre Array, SKA) in
phase-referenced and wide-field observations. These simulations can be
extrapolated to other arrays by applying the corresponding corrections.
We consider several effects from the turbulent atmosphere
(i.e., ionosphere and wet component of the troposphere) and also
from the antenna receivers. We study the changes in dynamic range and
astrometric precision as a function of observing frequency, source
separation, and strength of the turbulence. We find that, for
frequencies between 1 and 10 GHz, it is possible to
obtain images with high fidelity, although the atmosphere strongly
limits the sensitivity of the instrument compared to the case with no
atmosphere. Outside this frequency window, the dynamic range of the
images and the accuracy of the source positions decrease. We also find
that, even if a good model of the atmospheric turbulence (with an
accuracy of 99%) is used in the imaging, residual effects from the
turbulence can still limit the dynamic ranges of deep,
high-contrast (
105-106), images.
Key words: atmospheric effects - techniques: high angular resolution - techniques: interferometric - telescopes
1 Introduction
It is well-known that ground-based astronomical observations are affected by the atmosphere. Changes in the atmospheric opacity produce a bias in the source flux density, while changes in the refraction index distort the shape of the electromagnetic frontwave of the source. This distortion translates into a deformation of the observed source structure and/or a variation in the relative positions of all sources observed in a given field. For astronomical devices based on interferometry, atmospheric effects can be accurately modelled if the atmosphere above each element of the interferometer (hereafter, station) remains unchanged over the whole portion of the sky being observed. In these cases, the observed visibilities can be calibrated using station-based algorithms, which are relatively simple and computationally inexpensive (e.g. Readhead & Wilkinson 1978).
![]() |
Figure 1: Array distribution used in our simulations (empty and filled squares). Axes are relative longitude (horizontal axis) and latitude (vertical axis) in km. Left, the whole array. Center, a zoom into the core. Right, a zoom into the inner core. The stations marked with empty squares were removed from the array in a second run of our simulations, to check the dependence of the results on different array distributions (see text). |
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However, when the spatial variations in the atmosphere are significant within the observed portion of the sky, as happens if there is atmospheric turbulence, the opacity and dispersive effects cannot be modelled as a single time-dependent station-based complex gain over the field of view. Unless more complicated calibration algorithms are used (e.g., van der Tol 2007), the effects of these errors on the image are difficult to correct. In this paper, we study the effects that a turbulent atmosphere may introduce in interferometric observations. We focus our study on the effects produced by turbulence in the dynamic range and astrometric accuracy after a phase-referenced calibration between a strong (calibrator) source and a weak source, located a few degrees away. This study is numerically equivalent to the study of the deformation of a wide-field interferometric image at any point located at a given distance from the center of the field (i.e., the phase center of the image). In both cases, the phases introduced by the atmosphere in the signal of each antenna for the different pointing directions are the same, so the effects of the atmosphere in Fourier space (and therefore on the sky plane) will also be the same.
The results reported here are an extended version of those previously reported in the SKA memo by Martí-Vidal et al. (2009). In the next section, we describe the details of the array distribution used, as well as the characteristics of the simulated observations. In Sect. 3, we describe how the noise from the atmosphere and the receivers was added to the visibilities and in Sect. 4 describe the procedures followed in our Monte Carlo analysis. In Sect. 5, we present the main results obtained; in Sect. 6, we summarize our conclusions.
2 Array geometry and sensitivity
We simulated an interferometric array similar to the planned station distribution of the Square Kilometre Array (SKA). We simulated a total of 200 stations distributed in the following way: 50% are randomly distributed within a circle of 5 km radius (inner core); 25% are distributed outside this circle out to a distance of 150 km (core), following 5 equiangular spiral arms; the remaining antennae are distributed following the same spiral arms, but out to a distance of 3000 km from the inner core. This array distribution is similar to that used in Vir Lal, Lobanov & Jiménez-Monferrer (2009). The curvature of the Earth surface was taken into account in our simulations. We show the resulting array distribution in Fig. 1. We also repeated all the simulations reported here, but after subtracting a subset of 100 (randomly selected) stations from the array, to check the sensitivity of the main conclusions of this paper to different array distributions (see Appendix A). In Fig. 1, we also show the modified array after subtracting the 100 stations.
2.1 Sensitivity and bandwidth
We simulated interferometric observations using 16 different frequencies, spanning logarithmic bins from 150 MHz to 24 GHz. This is the theoretical frequency window of the SKA. According to Jones (2004), the maximum observing bandwidth of the SKA will be around 25% of the central observing frequency (up to a maximum bandwidth of 4 GHz for all frequencies above 16 GHz). This (maximum) frequency-dependent bandwidth translates in our simulations into a changing sensitivity of the SKA as a function of frequency.
The sensitivities of the simulated stations were also chosen to be similar to those of the SKA, which were taken from Jones (2004). These values are set for an elevation of 45 degrees and differ from those given in Schilizzi et al. (2007), but the use of the values given in Schilizzi et al. (2007), instead, does not affect the main conclusions of this paper. We interpolated the sensitivities given in Table 1 of Jones (2004) to the frequencies used in our simulations. In Fig. 2, we show the station sensitivities used.
![]() |
Figure 2: Station sensitivities (i.e., effective areas over system temperatures) used in our simulations. |
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2.2 Source position
We set the target source coordinates to be those of the zenith of the array center and the calibrator to an hour angle of 0 degrees, also with respect to the array center. This positioning of the sources minimizes the optical paths of the signals through the atmosphere (since both, source and calibrator, are at maximum elevations), thus enhancing the quality of the phase-referenced observations. The results given in this paper should be interpreted by taking this into account.
If the source were located far from the zenith, the mapping function of the tropospheric delay and the finite width of the ionosphere would increase the effect of turbulence on the phase-referenced visibilities of the target. In addition, the uv coverage of the interferometer would have shorter projected baselines in declination, thus decreasing the synthesized resolution in declination. Therefore, placing the sources at maximum elevations is a key limiting factor of the simulations reported here, especially if they are to be compared to real observations.
3 Noise model
We simulated phase-referencing observations in the following way: we assumed that the calibrator source is sufficiently strong to allow for a perfect antenna-gain calibration at its location; we then determined the image of the target source by computing the differential antenna-gain errors expected at the target location. Therefore, under the effect of atmospheric turbulence, these results depend on the calibrator-to-target separation.
We implemented two kinds of atmospheric turbulence. The first
turbulence was associated with the ionosphere (the free electron
content, which introduces dispersion in the radiation) and the other
turbulence was associated with the wet troposphere (the water
vapour, close to the Earth surface, which is not in a state of
thermodynamic equilibrium). The effect of ionospheric turbulence on the
signal phase varies as ,
affecting the low-frequency observations; the effect of the wet troposphere on the phase varies as
,
affecting the high-frequency observations. The dry troposphere (which
is more homogeneously distributed over each station than the wet
troposphere) was not considered in our simulations, since it can be
easily modelled and removed from the data to a level lower than the
effects coming from the water vapour and the ionosphere. Models of the
turbulence from the ionosphere and troposphere can be found in many
publications (e.g., Thomson et al. 1991).
Here, suffices to say that this turbulence follows a Kolmogorov
distribution. This distribution has a phase structure function
given by
![]() |
(1) |
where




The global factors for both distributions (ionosphere and troposphere)
were computed according to the typical values of ionospheric and
tropospheric conditions. For the ionosphere, the Fried length
(i.e., distance in the ionosphere for which the structure function
rises to 1 rad2) was set to be 3 km at 100 MHz. For the wet troposphere, we set the parameter C2n L (i.e., the integral of the profile of C2n along the zenith direction) to be 10-11 m1/3 (see Eq. (13.100) and Table 13.2 of Thomson et al. 1991); this value translates into a Fried length of 3 km for a frequency of GHz.
Since the Kolmogorov distribution is self-similar, it is possible
to adapt the results here reported to any other atmospheric conditions
(see Sect. 5.3),
by scaling accordingly the source separation to the Fried length
of the ionosphere (for low-frequency observations) or the wet
troposphere (for high-frequency observations). We note that the
self-similarity of the tropospheric turbulence does not apply on very
large scales (the typical baseline lengths in VLBI observations),
since there is a saturation in the power spectrum of the distribution
(see, e.g., Thomson et al. 1991).
However, this is not important to our analysis, since we did not use
the absolute phase of the signal coming from a given direction in the
sky, but computed the differential effects at each station from two
different (closeby) directions, which depend on short-scale turbulence.
Therefore, the saturation of tropospheric turbulence on large
scales does not affect our results.
![]() |
Figure 3: Example of a turbulent phase screen with Kolmogorov statistics. The grey scale shows variations in optical-path phases, normalized between -1 and 1 radians. The final values of the phases depend (given the self-similarity of the distribution) on a global factor related to the observing frequency and the ionospheric and/or wet tropospheric conditions. |
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We computed the differential effects from the turbulent atmosphere in
two ways. For the antennas of the core (within the central
300 km), we generated synthetic phase screens for the ionosphere
and troposphere. We show an example of one such screen in Fig. 3.
We notice that this figure could represent either ionospheric or
tropospheric turbulence in our modelling, just by scaling the screen by
the corresponding factor. Two different screens were generated in each
Monte Carlo simulation. The screen for simulating the ionosphere
was placed at at a height of 300 km and the screen for simulating
the troposphere at a height of 5 km. For the antennas out of the
core, we computed the term
separately.
We proceeded in this way (i.e., we generated a phase screen only
for the core antennas, thus without generating a much larger screen for
the whole array), because the distances between stations outside the
core are large enough to ensure that the cross-correlation of
turbulence above different stations is negligible compared to the
correlation between turbulence on the calibrator and turbulence on the
target source at the same station. This numerical strategy also
accelerated our simulations.
We note that we did not introduce any time evolution in the turbulent phase screens in our simulations. Any evolution of the turbulence could dramatically affect the observations if the acquisition times were longer than the coherence time of the signal, which depends on the evolution of the turbulence and the observing frequency. However, for snapshot-like observations, of the order of a fraction of a minute or so, we could consider, as a good first approximation, a constant turbulence phase screen.
Noise from the receivers was added to our model by generating a
random Gaussian noise in both the real and imaginary parts of the
visibilities. The mean deviation, ,
of the Gaussian noise added to the visibilities was (e.g., Thomson et al. 1991, Eq. (6.43)):
![]() |
(2) |
where k is the Boltzmann constant,




4 Estimate of dynamic range and astrometric precision
We simulated different sets of phase-referenced observations. In all cases, the observations were snapshots of duration t0 = 60 s. Longer observing times, t, would, in principle, increase the dynamic ranges and astrometric precisions shown in all the following sections as
,
as long as the changing atmosphere (and, therefore, the changing
source positions and shapes) did not introduce important smearing
effects in the images after the combination of all visibilities.
In our first series of simulations, we generated visibilities of targets with flux densities of 0.1, 1, and 10 Jy
for a separation of 5 degrees between target and calibrator.
A total of 1500 simulations were performed for each flux
density and frequency. We used such a large separation between
calibrator and target, because these simulations of phase-referenced
observations can also be applied to the study of deformations of
wide-field images affected by a turbulent atmosphere.
In a second series of simulations, we studied the effects of the
atmosphere as a function of calibrator-to-target separation. We
simulated 1500 observations at 1420 MHz (i.e., the
hydrogen line) of a source with 1 Jy for different separations from the calibrator (2-6 degrees).
In a third run of simulations, we used only one Kolmogorov screen (which can represent either ionospheric or tropospheric turbulence, depending on the observing frequency) with different Fried lengths, to study the scalability of the simulations for different source separations and/or atmospheric conditions.
In all these simulations, we added the noise from both the atmosphere
and the receivers. For each simulated phase-reference image,
obtained by applying uniform weighting to the visibilities, the
brightness peak was found and the corresponding point source was
subtracted from the visibilities. To subtract the point source,
the brightness peak was shifted to the phase center of the image
by multiplying the visibilities by the corresponding plane-wave factor
in Fourier space. The flux density of the point source was then
estimated to be the average of the real part of the resulting
visibilities, and the resulting point-source model was subtracted from
the data. Afterwards, a Fourier inversion of the new visibilities
produced an image of the residuals, from which the root-mean-square
(rms) of all the pixels was computed. On the one hand, the deviation of
the brightness peak from the image center was assumed to be the
astrometry error in that image. On the other hand, the source peak
divided by the rms of the residuals was taken to be the dynamic range.
In Fig. 4, we show the distribution of astrometric deviations and dynamic ranges for the case of a target source of 1 Jy observed at 1420 MHz (which corresponds to an interferometric beam of
13 mas) located at 5 degrees from the calibrator. When the distributions such as those shown in Fig. 4
were obtained, we computed the standard deviation in the astrometric
corrections and the mean value of dynamic ranges for each source flux
density, frequency, and separation. The first quantity was our estimate
of the astrometric uncertainty, and the second quantity was an estimate
of the achievable dynamic range.
![]() |
Figure 4:
Distribution of right ascension shifts a) and dynamic ranges b) of 1500 snapshot phase-referenced images, simulated at 1420 MHz, for a 1 |
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5 Results
5.1 Observing frequency and signal decoherence
If the atmospheric turbulence is not taken into account and only the noise from the receivers
is added to the visibilities, our simulations reproduce the dynamic ranges given by Eq. (6.53) of Thomson et al. (1991),
as expected. In addition, the noise from the receivers
does not introduce considerable changes in the source position of the
phase-referenced images (changes of the order of 10 as or lower).
When the turbulent ionosphere and wet troposphere are added to the
simulations, the dynamic range of the images is notably affected,
especially at low (
)
and high (
)
frequencies. In Fig. 5, we show phase-referenced images of a 1
Jy
source, located 5 deg. from its calibrator, observed
at 0.5, 5, and 15 GHz. It can be readily seen that
the inclusion of effects from the atmospheric turbulence produces an
important extra noise in the images at 0.5 and 15 GHz, but
not at 5 GHz.
![]() |
Figure 5:
Simulated phase-reference images of a 1 |
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Following the algorithm described in Sect. 4, we obtained the astrometric uncertainties and dynamic ranges shown in Fig. 6. At very low frequencies (below
500 MHz),
the ionosphere prevents a clear and precise detection of all
sources, regardless of their flux densities. At higher frequencies, the
astrometric uncertainty decreases notably (mainly because of the
dependence of ionospheric effects on
)
and is limited only by diffraction and sensitivity between 1 and
10 GHz (this frequency window depends slightly on the source flux
density, as can be seen in the figure). For higher
frequencies, the wet troposphere begins to affect the astrometric
uncertainty, which increases to around 10 mas at the highest
frequencies. We find that the highest astrometric accuracy,
at least for reasonably well-detected sources, is achieved
for frequencies around 4 GHz. This is where the ionospheric and
(wet) tropospheric components are roughly equal.
![]() |
Figure 6:
Astrometric accuracy a) and dynamic range b)
as a function of frequency, in the case of atmospheric turbulence
and a separation of 5 degrees between calibrator and target. The
calibrator source is located at an hour angle of 0 and the target
source is located at the zenith of the array center. Different lines
correspond to different target source flux densities (10 |
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The dynamic range of the phase-referenced images is limited
considerably by the atmosphere. When the atmosphere adds noise to the
visibility phases, an additional rms is added to the residual images,
which depends on the visibility amplitudes, thus limiting the
achievable dynamic range regardless of the flux density of the source.
That is, when the source flux density is higher, the noise of
the image is also higher. This limitation is, of course, more
important for the brightest sources. In our case, the brightest
source has a flux density of 10 Jy. For this source, the maximum dynamic range achieved is only 110, which is
times
smaller than the dynamic range that would be obtained without the
atmosphere. This situation can also be understood in another way. The
rms of the final image can be divided into two components, which are
added in quadrature. One component,
,
comes from the receiver noise and is independent of the source flux density. The other component,
,
comes from the atmospheric refraction and is equal to a percentage of the source flux density (
,
where S is the source flux density and
depends on the atmospheric refraction). Hence, the dynamic range, D, is
For high flux densities (




5.2 Angular separation and signal decoherence
The results shown in the previous subsection correspond to a
separation of 5 degrees between source and calibrator. These
results change when the angular separation changes. We computed
astrometric uncertainties and dynamic ranges for a source with a flux
densitity of 1 Jy
located at 2-6 degrees from its calibrator. Noise from the
atmosphere and the receivers was taken into account in these
simulations. We used an observing frequency of 1420 MHz
(the hydrogen line) which is inside the frequency window where the
atmospheric effects are minimised. Therefore, all the astrometric
errors derived were small (of the order of a few mas), allowing us
to use image sizes small enough to sample the beam with more pixels (
30 pixels) using a grid of 1024
1024 pixels. This fine gridding of the beam allowed us to determine the location of the image
peak more accurately and, therefore, obtain a more precise estimate of the astrometric error. The
results obtained are shown in Fig. 7.
In that figure, we also plot two analytical (phenomenological) models
for the estimate of the increase in astrometric uncertainty and the
loss of dynamic range (i.e., degree of signal decoherence) as a
function of angular separation. On the one hand, the phenomenological
model proposed for the estimate of loss of dynamic range is
where D is the dynamic range,




As can be seen, this model agrees well with the simulations. We obtain
10-5 deg
and
0.06.
![]() |
Figure 7:
Effects of target-to-calibrator separation in the quality of the
phase-reference images. Dots: simulated astrometric accuracy a) and dynamic range b) for observations at 1420 MHz and a target flux density of 1 |
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On the other hand, the proposed phenomenological model for the increase in the astrometric uncertainty is
where









We note that when we change D0 by D in Eq. (5), the newly fitted k' and
are 1.03
0.09 and 0.07
0.06, respectively. This new value of
is compatible with zero. In other words, the diffraction limit
divided by the dynamic range of the image is an excellent estimator of
the astrometric uncertainty, at least for the range of simulated
calibrator-to-target separations at 1.4 GHz (which falls within
the frequency window where the atmospheric effects are minimized).
For calibrator-to-target separations larger than 6 degrees,
the situation changes. We simulated phase-referenced images for
calibrator-to-target separations of up to 12 degrees, and found
that the model of dynamic range given by Eq. (4) remains valid, but the astrometric uncertainty increases more rapidly, with
= 2.38
0.08 (
,
if we change D0 by D in Eq. (4)). This last
reproduces the astrometric uncertainties well for large source
separations, but the fit is poorer for separations smaller than
5-6 degrees.
5.3 Scalability of the results and use of turbulence models in the data calibration
In the previous subsections, we have reported the effects of
atmospheric turbulence in phase-referenced (and wide-field)
interferometric images using fixed values for the Fried lengths of the
Kolmogorov distributions of the ionosphere and wet troposphere. Since
the Kolmogorov distribution is self-similar, the results presented
can be scaled and adapted to other atmospheric conditions. These
simulations can also be used to estimate the limiting dynamic range and
astrometric uncertainty if an a priori model of the tropospheric
and/or ionospheric turbulence is used in the imaging. In these
cases, the effective Fried length,
,
to compare to our simulations can be estimated to be
![]() |
(6) |
where r0 is the Fried length of the real turbulence and the other factor is related to the fractional precision of the turbulence model,







![]() |
Figure 8:
Maximum dynamic range as a function of |
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where h is the height of the phase screen and
is the calibrator-to-target separation
(or half the size of the wide-field image). Equation (7) can be used with Fig. 8
to compute the maximum achievable dynamic range for many different
combinations of source separations, atmospheric conditions, and
observing frequencies (
for the ionosphere and
for the trosposphere). We note, however, that Fig. 8
has been generated using only one Kolmogorov screen, so it is
applicable to ionospheric dispersion (for low frequencies) or
tropospheric dispersion (for high frequencies), but not to a situation
where ionospheric and tropospheric effects are similar. In these cases,
and as a first approximation, we could set
![]() |
(8) |
where




where D1 = 48.38










![]() |
Figure 9:
Peak flux density of the phase-referenced image, relative to the real
peak flux density of the source, as a function of |
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For a more comprehensive representation of our results, we show in Fig. 10 how the achievable dynamic range (computed from Eq. (9))
depends on the uncertainty in the model of atmospheric turbulence used
in the data calibration. We show this relationship for different
observing frequencies and calibrator-to-target separations. For
instance, a dynamic range 104in observations at
100 MHz for a calibrator-to-target separation of 1 deg
(i.e., the same for a wide-field image of 2
2 deg) would require a turbulence model with an accuracy of
%
for an observing time of 60 s. This requirement would relax
to an accuracy of 97-98% for an observing time of 6000 s
(provided the dynamic range increases as the square root of the
observing time).
![]() |
Figure 10: Maximum dynamic range as a function of the uncertainty in the model of the atmospheric turbulence for a 1-scan snapshot of 60 s. |
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For completeness, we also computed the loss in the recovered flux
density of the source caused by the turbulent atmosphere.
In Fig. 9,
we show the ratio of peak flux densities of the phase-referenced images
to those computed without accounting for the effects of atmospheric
turbulence. For dynamic ranges of 40-50, the loss of flux density
can be as large as 25%. The phenomenological model (also shown in
the figure) used to fit the data is
where D2 = 0.32



The applicability of Eqs. (9) and (10) is not restricted to the array used in the simulations reported here. The exponents
and
depend only on the structure of the atmospheric turbulence and are thus
independent of the interferometer used in the observations. However,
the parameters D1 and D2 also depend on the stations of the interferometer. Therefore, Eqs. (9) and (10) can be adapted to any other interferometer by finding the correct values of D1 and D2. As an example of this generalization of Eqs. (9) and (10), Martí-Vidal et al. (2010)
studied the achievable dynamic range in phase-referenced observations
preformed by the Very Long Baseline Array (VLBA) at 8.4 GHz and
15 GHz. This study was based on quasi-simultaneous observations of
13 sources located at separations ranging from 1.5 to
20.5 degrees. These authors were able to model the dynamic ranges
obtained in the phase-referenced images and the loss of recovered flux
densities (by comparing phase-referenced images to images obtained
from self-calibrated visibilities) using Eqs. (9) and (10) with values for D1 and D2 different to those reported here, but using the same values reported here for the exponents
and
.
6 Conclusions
We have presented Monte Carlo estimates of the sensitivity and
astrometric precision of an interferometric array, with a station
distribution similar to that of the planned SKA, as a function of
observing frequency, flux density, and source separation. These results
can also be applied to other array distributions by taking into account
the corresponding correction factors. Our estimates are based on
simulations of snapshot phase-referenced observations, in which we take
into account several effects including those of the turbulent
atmosphere and the finite temperature of the receivers. We have found
that the astrometric uncertainty strongly depends on the observing
frequency and smoothly increases as the source separation increases.
For frequencies below 1 GHz, ionospheric effects dominate and the astrometry uncertainties (when the source is detectable) can be as large as
1 as.
For frequencies between 1 and 10 GHz (these values slightly
depend on the source flux density), atmospheric effects are minimal and
we roughly reach the theoretical astrometric precision of the
interferometer. Above these frequencies, the wet troposphere begins to
have an important effect and the astrometric uncertainty increases to
10 mas
for the highest simulated frequency (25 GHz). The dynamic range of
the images is strongly limited by atmospheric turbulence at all
frequencies and for all flux densities (it can decrease, in the
worst cases, by several orders of magnitude).
We have proposed analytical models for the loss of dynamic range, astrometric accuracy, and recoverable flus density as a function of distance between calibrator and target source. These expressions may also be used to estimate the deformations and local dynamic ranges of wide-field images as a function of distance to the image phase center (i.e., the point in the sky where the data correlation is centered).
We thank Ed Fomalont for his very useful comments and suggestions. I.M.V. is a fellow of the Alexander von Humboldt Foundation. This work has been supported by the European Community Framework Programme 6, Square Kilometre Array Design Studies (SKADS), contract number 011938. This work has also been partially founded by grants Prometeo 2009/104 of the GVA and AYA2009-13036-CO2-2, AYA2006-14986-CO2-01, and AYA2005-08561-C03 of the Spanish DGICYT.
Appendix A: Complementary simulations: different number of stations and array sensitivities
Our simulations are based on a given realization of the SKA. However, the main structure of the array distribution used in our simulations is not exclusive to the SKA. Other interferometric arrays, such as ALMA or LOFAR, are being built with similar station distributions, consisting of a compact core and several extensions along the shape of spiral arms. Hence, our study can be extended to those arrays by taking into account the difference between the number of stations and the station sensitivities.
![]() |
Figure A.1:
Ratios of dynamic ranges obtained with our original array and those obtained with a subarray of 100 stations a) and an array with half the sensitivity of the original array b). Different lines correspond to different target source flux densities (10 |
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The amount of noise added to the data is proportional to the number of stations, since each
station receives the signal through the turbulent atmosphere. However, in cases of clear source
detections (D > 20-30), the dynamic range does not depend (or the dependence is weak)
on the thermal noise of the receivers (see Sect. 5.1).
Therefore, to estimate the achievable dynamic range for an array with a
different number of stations, the results shown in Fig. 6 should be divided by N/N', where N is the number of stations used in our simulations (N= 200) and N' is the number of stations of the other array. This is true for detections with a relatively large dynamic range (D>20-30). For weak sources, the noise
from the receivers may also contribute to the rms of the residual images, so the factor to apply in
these cases should be
,
where
is the thermal noise of the stations
used in our simulations and
is that of the other array.
We repeated the simulations described in Sect. 5.1 using different arrays to compare the results with those obtained with the original array. On the one hand, we created a smaller array by subtracting 100 stations (those marked with empty squares in Fig. 1) from the original array. On the other hand, we created another array with all the 200 stations, but decreasing their sensitivity by a factor of 2. We show the results obtained in Fig. A.1. Special care must be taken interpretating these figures, given that the computed ratios of dynamic ranges are only meaningful when the detections of the sources are clear (i.e., when no spurious noise peaks appear stronger than the source). This is true for D > 20-30, which corresponds approximately to frequencies between 1 and 10 GHz (although it depends slightly on the source flux density, see Fig. 6).
By taking these considerations into account, we find that the ratio of
dynamic ranges for an array with 100 stations falls
between 0.7 and 0.5 compared to the array with
200 stations. The expected value is 0.5 (since
N' = 0.5 N and
).
Other factors, such as the different coverages of Fourier space by
both arrays, may affect the dynamic range of the images, thus
increasing the ratio in some cases. For an array with lower
station sensitivities (but the same number of stations), the ratio
of dynamic ranges falls between 0.75 and 1 for the strongest
sources (as expected, since N' = N and the noise from
the receivers is muchsmaller than the noise from the atmosphere), but
is close to 0.5 for the weakest source (also as expected, since
the thermal noise from the receivers begins to dominate in
this case).
References
- Eckers, R. D. 1999, in Synthesis Imaging in Radio Astronomy II, ed. Taylor, Carilli, & Perley, ASP Conf. Ser., 180 [Google Scholar]
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All Figures
![]() |
Figure 1: Array distribution used in our simulations (empty and filled squares). Axes are relative longitude (horizontal axis) and latitude (vertical axis) in km. Left, the whole array. Center, a zoom into the core. Right, a zoom into the inner core. The stations marked with empty squares were removed from the array in a second run of our simulations, to check the dependence of the results on different array distributions (see text). |
Open with DEXTER | |
In the text |
![]() |
Figure 2: Station sensitivities (i.e., effective areas over system temperatures) used in our simulations. |
Open with DEXTER | |
In the text |
![]() |
Figure 3: Example of a turbulent phase screen with Kolmogorov statistics. The grey scale shows variations in optical-path phases, normalized between -1 and 1 radians. The final values of the phases depend (given the self-similarity of the distribution) on a global factor related to the observing frequency and the ionospheric and/or wet tropospheric conditions. |
Open with DEXTER | |
In the text |
![]() |
Figure 4:
Distribution of right ascension shifts a) and dynamic ranges b) of 1500 snapshot phase-referenced images, simulated at 1420 MHz, for a 1 |
Open with DEXTER | |
In the text |
![]() |
Figure 5:
Simulated phase-reference images of a 1 |
Open with DEXTER | |
In the text |
![]() |
Figure 6:
Astrometric accuracy a) and dynamic range b)
as a function of frequency, in the case of atmospheric turbulence
and a separation of 5 degrees between calibrator and target. The
calibrator source is located at an hour angle of 0 and the target
source is located at the zenith of the array center. Different lines
correspond to different target source flux densities (10 |
Open with DEXTER | |
In the text |
![]() |
Figure 7:
Effects of target-to-calibrator separation in the quality of the
phase-reference images. Dots: simulated astrometric accuracy a) and dynamic range b) for observations at 1420 MHz and a target flux density of 1 |
Open with DEXTER | |
In the text |
![]() |
Figure 8:
Maximum dynamic range as a function of |
Open with DEXTER | |
In the text |
![]() |
Figure 9:
Peak flux density of the phase-referenced image, relative to the real
peak flux density of the source, as a function of |
Open with DEXTER | |
In the text |
![]() |
Figure 10: Maximum dynamic range as a function of the uncertainty in the model of the atmospheric turbulence for a 1-scan snapshot of 60 s. |
Open with DEXTER | |
In the text |
![]() |
Figure A.1:
Ratios of dynamic ranges obtained with our original array and those obtained with a subarray of 100 stations a) and an array with half the sensitivity of the original array b). Different lines correspond to different target source flux densities (10 |
Open with DEXTER | |
In the text |
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