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MMA Memo 157: Exploring the Clustered Array Concept for the Atacama Array
M.A. Holdaway
National Radio Astronomy Observatory
949 N. Cherry Ave.
Tucson, AZ 85721-0655
email: mholdawa@nrao.edu
P.J. Napier
National Radio Astronomy Observatory
Socorro, NM 87801
email: pnapier@nrao.edu
F. Owen
National Radio Astronomy Observatory
Socorro, NM 87801
email: fowen@nrao.edu
July 9, 1996
Abstract:
One mode of collaboration between the LMSA and the MMA is to form a
10 km array using all 90 antennas to gain the sensitivity required for
such high resolution observations. This array has been called the
``Atacama Array''. One possible configuration of the 90 element
Atacama Array is to arrange the antennas in clusters of 2-6 antennas,
scattered in a ring-like shape. The signals from the antennas in a
cluster would be phased up, and the phased cluster signals would be
correlated, some by the LMSA correlator and some by the MMA
correlator. The advantage of this scheme is that the number of
correlations which must be performed is greatly reduced, permitting
much wider bandwidth per visibility, and hence, continuum sensitivity
which is a factor of 2-5 higher (depending upon the number of
clusters) than using the two arrays separately. With 90 antennas, we
can afford to trade Fourier plane coverage for bandwidth, but this
trick would not work very well for the arrays individually. One
antenna from each cluster could also be used for phase calibration. The
disadvantages of this configuration include a reduced field of view, a
reduced number of baselines resulting in poorer image quality, the
possibility of atmospheric phase fluctuations across the cluster
dephasing the signal, and greatly increased correlator complexity.
The field of view problem can best be overcome by correlating all
antennas (with a decrease in bandwidth), but most astronomical targets
of the Atacama Array are very compact objects and would not be
troubled by the field of view restrictions. The long track Fourier
coverage of an array with 20-30 clusters is still quite good
(comparable to the VLA, which produces excellent images), and we
expect most astronomical targets of the Atacama Array will be
sensitivity limited rather than limited by the quality of the Fourier
plane coverage. The atmosphere at Chajnantor is good enough to permit
the clusters to phase up at 950 GHz with less than 15% loss in
sensitivity about 50% of the time. In fact, the phasing works much
better than fast switching phase connection between the clusters. If
radiometric phase correction worked better than fast switching, it
would also work well enough to improve the phasing of the clusters.
In order to correlate signals from both the LMSA and the MMA in the
guise of the Atacama Array, the two instruments must interface
cleanly, which will require extensive design work. We feel that this
early design work, drawing upon the expertise of both the NRO and the
NRAO, should result in strengthening the designs of both instruments.
The clustered array should be considered as a viable way to get the
full collecting area of the Atacama Array while gaining, rather than
losing, continuum bandwidth.
While there is a fantastic amount of new science which the proposed
MMA could perform, there are several interesting problems which are
just at the sensitivity threshold of the instrument. This is
especially true at the highest resolutions where the small beam
results in rather poor brightness sensitivities in feasible
integration times. We are currently discussing possibilities for
combining the LMSA and MMA antennas into a 90 element, 10 km
configuration, called the Atacama Array, to improve the sensitivity at
very high resolution. If the LMSA and MMA are used as separate
arrays, and signals from the two arrays are not cross correlated, the
sensitivity is improved by something like 
. But if all the
elements are correlated, we gain by a factor of about 2 (assuming
equal collecting areas for the two arrays, which is not quite true).
Hence, cross correlating all elements results in an array which is
twice as fast as the two arrays observing separately. However, since
the number of baselines is approximately proportional to the number of
antennas squared, cross-correlating all antennas requires a correlator
which is about twice as large as the LMSA and MMA correlators
combined.
It was pointed out early on that if we only correlated the total
intensity visibilities and threw away the polarization, we could fit
all (or almost all) baselines into the correlator at full bandwidth.
However, the current LMSA and MMA antenna designs rotate differently
on the sky, and hence all four polarization correlations must be
performed to obtain total intensity.
Another way to overcome this limitation is to reduce the bandwidth of
the observations to fit the increased number of baselines into the
correlator, but this also loses sensitivity, cancelling the increase
in sensitivity gained by cross-correlating the two instruments in the
first place. In other words, the continuum sensitivity of the Atacama
Array, made by naively combining the MMA and the LMSA, is limited by
the correlator and not by the collecting area. Since the brightness
sensitivity of the Atacama Array precludes observation of most thermal
emission lines, continuum observations will likely dominate the
Atacama Array.
We can more than recover this lost bandwidth with a clustered array,
in which we configure the antennas into several very tight clusters
and correlate the phased signals from each cluster, resulting in fewer
baselines to be correlated, which can, depending upon correlator
design, be translated into increased bandwidth and increased
sensitivity. We explore the advantages and disadvantages of this
scheme with respect to sensitivity, phase stability, field of view,
and imaging.
The improvement in continuum sensitivity is
the primary reason for a clustered array configuration.
The image noise from one array will be proportional to
where 
is the number of antennas in array 1 and 
is the
dish diameter of array 1's antennas.
If two different arrays observe the same object, and the data is added
after correlation, then the image noise will be
but if all the elements from the two arrays are cross-correlated, the
image noise will be
However, if we are correlator limited, we will need to decrease
our bandwidth by a factor of
For 
, 
, and 
= 8m, the
bandwidth reduction is 
, then
(In this memo, we will normalize all quoted sensitivities to a 90
element array with no change in the bandwidth.) However, if =
10m and 
= 8m, then
If both arrays have antennas of the same diameter, cross correlating
all antennas is quite attractive if we can keep all of the bandwidth.
The = 10m, = 8m case still results in a significant
sensitivity improvement. In either case, if we must sacrifice half of
our bandwidth in order to perform all of the correlations, we gain
nothing over adding the visibilities from the two arrays separately.
The clustered array concept allows us to get the sensitivity of the
full correlated array without losing any bandwidth. In fact, we can
gain an appreciable amount of bandwidth through using a
clustered array since the number of baselines being correlated could
be smaller than either correlator normally correlates. The full
bandwidth increase might not be realizable due to limitations imposed
by the atmospheric windows or by the correlators flexibility, but the
potential bandwidth increase is given by the factor
where 
is the number of antenna clusters. Table 1
shows the potential bandwidth increase for , ,
and a variety of possible .
So, the cases with = 45, 30, and 22 look particularly interesting,
potentially increasing the bandwidth by almost an order of magnitude.
A natural observing mode which follows from the clustered array is
paired antenna calibration (Holdaway, 1992; Asaki et al., 1996).
One of the antennas in each cluster can be used to observe a
calibrator source while the other antenna(s) observe the target
source, and the calibrator phase is applied to the target source. We
do not perform any detailed analysis of paired antenna calibration in
a 90 element clustered array, but it works about as well as fast
switching, and can be used even if the LMSA or MMA antennas cannot
switch quickly. This mode will require the
correlation of two subarrays of the same size, or twice as many
correlations as the regular clustered array. The optimal division of
the bandwidth between the target array and the calibration array is
not clear, but if we assume equal bandwidths for the two arrays, the
bandwidth increases in Table 1 should be reduced by 2.0
and the noises increased by .
There are many arguments against the clustered array approach, including
complications due to the cluster phasing hardware and the ultra-flexible
correlator, the stability of the atmosphere over each cluster, limitations
of the field of view, and decrease in the maximum potential dynamic range
achievable due to fewer baselines. We speak to each of these concerns,
except for the hardware complications, which may very well prevent this
idea from becoming part of the MMA design.
It has been argued that the clustered array is potentially risky
because it may be difficult to keep each cluster in phase, and if the
cluster antennas are out of phase, they will decorrelate and all
baselines to the affected cluster will be down by 
in
amplitude (the effective collecting area of a randomly phased array is
the collecting area of a single antenna). There will be electronic,
atmospheric, and antenna deformation contributions to the relative
antenna phases. We assume the electronics will hold its phase over
time scales of at least 10 minutes, and the electronic phasing can be
accomplished by observing a bright calibrator for a fraction of a
minute. Since the atmospheric phase errors are a function of the
baseline length, we will need to investigate cluster sizes.
To eliminate shadowing above elevation angle 
for all
azimuths, antennas must be separated by a baseline of 
, where D is the antenna dish diameter. To observe
down to 30 degrees elevation in all directions requires 2D minimum
antenna separations, down to 20 degrees requires 2.9D separations.
It is beneficial to place the antennas as close as possible to
maximize the field of view and to minimize the phase errors, but we
must be able to see a reasonable fraction of the sky. While
30 degrees elevation seems too high for an elevation limit, this limit
applies only in certain directions, and we can certainly arrange the
cluster antennas so they are not shadowed until a lower elevation when
looking in an important direction such as north. Hence, we will adopt
2D as the minimum distance between antennas in a cluster for the
calculation here.
The number of antennas per cluster 
and the geometry of the cluster
will also affect cluster size. For this calculation, we adopt the
cluster geometries given in Figure 1, with 2D as the minimum
spacing. From these geometries, the maximum cluster size (from tip to
tip of the extreme antennas) and the effective cluster size (ie,
the decorrelation of the cluster is the same as if all baselines in
it were of this length) are given in Table 2.
Since these cluster sizes are generally smaller than the effective
calibration baseline length for fast switching, whenever the
atmosphere prevents the cluster from phasing up, phase calibration of
the whole array will probably be impossible.
The decorrelation in the voltage produced by summing the signals in a
given cluster subject to antenna based phase errors 
will
be 
, and the decorrelation in the
visibilities formed from two decorrelated clusters will be
, or 
for
per baseline phase errors such as atmospheric phase errors.
There are several sources of these phase errors, including atmosphere,
path length changes due to thermal or wind induced changes in the
shape or position of the antenna, and electronic phase drifts, mainly due to
thermal drifts in the electronics. It might seem that the two step
decorrelation in a clustered array, which first adds several antenna
voltages, and then correlates different cluster voltages with
wandering phase, might be different from the decorrelation in a
generic interferometer. However, after some thought and numerical
simulations of the decorrelation in the clustered array scenario, we
are confident that the decorrelation is not fundamentally different
from the case of a generic interferometer. The only difference is
that decorrelation suffered prior to multiplying the summed cluster signals is
irreversible, while a generic interferometer could conceivably recover
from this decorrelation if data were taken with very short integration
times and self-calibration were possible.
First of all, self-calibration will probably seldom be able to correct
for these phase fluctuations on time scales of a few seconds; sources
bright enough to permit this will seldom be targets of the Atacama
array. Hence, the clustered array is not at a disadvantage with
respect to a generic interferometer. Second, we will also show that
the decorrelation is not too extreme by examining the magnitude of the
various phase errors expected:
-
Phase errors due to antenna deformations
will not only occur slowly enough to permit their calibration by phasing
up the array every 10 minutes, but since the antennas in a given
cluster will be subject to nearly identical thermal and wind
environments, antenna phase errors will be nearly identical for
all antennas in a cluster. Antennas in different clusters will have
different environments, and hence different phases, and will decorrelate
when the final cross-correlation is performed among cluster voltages,
but this is the same situation as for a generic interferometer.
-
Electronic phase errors will be primarilly due to temperature
changes in the electronics, part of which will be correlated, and part
of which will be uncorrelated between antennas. However, any variations on time scales
longer than 10 minutes will be removed as each cluster is phased up
every 10 minutes. Our electronic phase stability specifications are
7 microns (best) to 16 microns (median) per antenna, (Woody et
al.). At 850 GHz, 7 microns will reduce the visibility amplitude to
0.98%, while 16 microns will reduce the visibility amplitude to 0.92%.
The prime conditions for high frequency observing with the Atacama
array will occur at night, and the relatively constant temperatures
may result in somewhat better electronic stability. Hence, we adopt
the value of 12 microns rms for the electronic phase specification for
high frequency observing in the Atacama array. At lower frequencies,
much larger electronic and atmospheric phase variations can be
tolerated, and day time observations would not be a problem.
-
Atmospheric phase errors on the effective baselines quoted in
Table 2 will largely be benign on the Chile site. The
site test interferometer (Radford et al., 1996) provides the
distribution of phase structure functions for the Chajnantor site in
Chile (Holdaway et al., 1995), which can be used to determine
the atmospheric contribution to the cluster voltage decorrelation.
It is possible that the atmospheric phase errors could be reduced
by highly effective water vapor monitoring instrumentation on each
antenna, but we do not need to rely upon this.
Following the above reasoning, the decorrelation in
adding the antenna voltages in each cluster will result in
power sensitivity which is lowered by a factor of
where 
is the rms path length due to fluctuations in
the electronics (assumed to be 12 microns), and
is the rms atmospheric path length error
predicted by the phase structure function on a baseline of
, the effective baseline length of the cluster. Now we
ask the question: how often will 
be greater than 0.85
(ie, when will the electronic and atmospheric phase result in an
irreversible cluster decorrelation sensitivity loss of less than
15%)? We answer this question using one year of site testing data to
evaluate what the phase errors are on the effective cluster baselines
for several different observing frequencies.
Figure 2 indicates that the atmosphere will often
permit the array to phase up with very little sensitivity loss, even
at the highest frequencies. In fact, the atmosphere will cause more
problems with connecting the phases of the different clusters via fast
switching, say, than phasing up the clusters.
As addressed above, it may not be possible, nor is it necessary, to
track the atmospheric phase across a cluster, as these phase errors
will be very fast and usually small. Neither is it necessary to track
the antenna deformation phase in a cluster, as each antenna in a
cluster will suffer very similar deformations. However, we must be
able to track the electronic phase of the antennas, at least on
intermediate time scales like 10 minutes. Since we are not concerned
with the other phase contributions, we are not constrained to phase up
on a source which is very close to the target source. We can easily
afford the luxury of phasing up the electronics on a bright 1 Jy
calibrator at 90 GHz which might typically be 15 degrees away (15 s
round trip slew time). Then, even for the worst case of 850 GHz
observations which requires a very accurate phase determination at
90 GHz, a mini-correlator with 2 GHz bandwidth could phase a three
element cluster to 1% decorrelation in about 20 s. The entire
phasing operation could take place in about 40 s. If phasing were
performed every 10 minutes, the lost time amounts to a 3% increase in
noise. This is a very small penalty in time and sensitivity.
When we phase up each cluster and add the signals, we are synthesizing
an aperture the size of the cluster, and this aperture will have a smaller
primary beam on the sky. Another way to look at the problem is that we
are taking data which belongs in several different (u,v) cells and
placing the data in the same cell. Hence, it does no good to have a cell
size finer than the cluster size, again limiting our field of view in
the image plane. In order to recover the full field of view, we must
correlate all antennas, which eliminates the bandwidth advantage of
the clustered array, or separately observe each sub-field phasing the
clusters to point to that sub-field, which results in an even larger
loss in sensitivity if the full primary beam must be observed.
Figure 3 illustrates how the field of view decreases as
the number of antennas per cluster increases. However the key targets
of the Atacama Array will be very small objects such as protostars.
It is quite possible that we could live with a limited field of view.
In investigating the limitations in image quality due to the reduced
number of baselines in the clustered array, it is tempting to make
arguments about the number of filled cells in the Fourier plane.
Since the Fourier plane cell size increases as the cluster size
increases, but the maximum baseline remains constant, the number of
cells to fill decreases even as the number of baselines decreases.
However, such arguments are artificial: we could always get a much
higher number of filled cells from an unclustered array simply by
restricting the field of view (ie, using the same large cells in the
Fourier plane) and taking advantage of the superior number of
baselines. Also, the fraction of filled cells in the Fourier plane
may be an attractive relative measure to help compare different
arrays, but we cannot at this time translate the fraction of
filled cells into an image quality. Hence, we must resort to image
simulations.
In the absence of a better working model image, we have used the same
old MMA simulation model source, generated from an H 
image of
an HII region in M31. This source is much more complicated than most
potential targets of the Atacama array. The resolution of the
uniformly weighted 10 km Atacama Array is about 14 mas at 300 GHz. We
have chosen to use a model cell size of 7 mas, and have tapered to a
resolution of 18 mas to reduce the size of the inner sidelobes and
improve image quality. The model source is about 630 mas across, or
about 35 resolution elements in one dimension.
We have simulated 2 hour observations with 60 s integrations for
arrays of between 15 and 90 clusters. This object is large enough
that it requires some short spacing information, which we add in the
form of 10 minutes of simulated observations from a 40 element 3 km
array. No noise or other errors were added to the data as we would
like to determine the dynamic range limitations caused by the
incomplete Fourier plane coverage. The arrays were made by placing
the clusters psuedo-randomly along a ring, but no effort was made to
optimize the arrays for good Fourier plane coverage or low sidelobes.
The uniformly weighted data (appropriate weighting for high dynamic
range objects) were gridded and FFT'ed, and the resulting dirty maps
were Clark cleaned with a reasonably tight clean box. Atacama Arrays
with 15 and 18 clusters required 3E+4 clean components before the
image quality saturated, arrays with 22 and 30 clusters required 1E+5
clean components before the image quality saturated, and arrays with
45 and 90 clusters required about 2E+5 clean components. When the
simulated maps generated from a configuration with fewer antenna
clusters were cleaned to the same number of clean components as
required by simulated maps made with a configuration with more
clusters, image quality began to decrease due to over-cleaning, so we
have carefully monitored the cleaning processes and stopped the clean
when the dynamic range saturates.
The quality of the resulting images is judged by the image fidelity
(which gauges the level of errors on source) and the dynamic range
(which gauges the level of errors scattered off source). These
quantities are plotted against the number of clusters on log-log
plots in Figure 4.
Now, each astronomical object will be imaged differently by a given
array, so our simulations are only a rough benchmark. However, our
simulation object is much more complex than most potential targets of
the Atacama Array. At this point we need to estimate the maximum
signal to noise we expect to get on our targets based on thermal
noise. If we can only obtain 200:1 dynamic range on our brightest
target due to thermal noise limitations, it certainly doesn't make
sense to keep an array that is capable of a deconvolution limited
dynamic range of 50,000:1. Even if we opted for the 22 cluster array,
we appear to be able to make 10,000:1 dynamic range images of very
complex sources, more image quality than we could ever take advantage
of.
We have explored some of the advantages and disadvantages of the
clustered array concept for the Atacama Array. This array design
stands to make a very large gain in continuum sensitivity over using
the MMA or the LMSA in a 10 km configuration. The atmosphere will not
prevent the phasing of the clusters unless it is so bad that fast
switching phase calibration will not work anyway. In the event that a
radiometric phase correction technique works for phase calibration
under these conditions, it will also work to keep the clusters in
phase. In fact, the phasing of the clusters appears to be little
enough of a problem that we should consider placing the antennas in
each cluster further apart to allow observations to lower elevation
angles and to prevent solar shadowing and wind blockage in the cluser,
which would promote a more homogeneous thermal environment for the
cluster antennas. The limited field of view cannot be overcome, but
we expect that this will not limit most Atacama Array observations.
Very high dynamic range images can be achieved with modest numbers of
antenna clusters (22 clusters results in 10,000:1 dynamic range on a
very complex source).
A serious discussion of a clustered Atacama Array design would require
more scientific input on the field of view limitation and the maximum
required dynamic range of a clustered array. Furthermore, is the
assumption that the 10 km Atacama Array will be dominated by continuum
observations correct? If the Atacama Array ends up performing mostly
spectral line observations, there is little utility and much expense
in the clustered array concept. The correlator designers must also give
some serious thought to the cluster phasing hardware and the added
complications of the correlator before we can say that the clustered
array concept is feasible. Also, the systems engineers will
probably let us know how the maximum transmittable bandwidth will
limit our clustered array bandwidth, and hence our sensitivity.
Asaki, it et al., 1996, ``Phase Compensation Experiments with
the Paired Antenna Method'', submitted to Radio Science.
Holdaway, 1992, ``Paired Antenna Phase Calibration: Residual Phase
Errors and Configuration Study'', MMA Memo 88
Holdaway, Radford, Owen, and Foster, 1995, ``Data Processing for
Site Test Interferometers'', MMA Memo 129
Radford, Reiland, and Shillue, 1996, ``Site Test Interferometer'',
PASP 108, 441-445.
Woody, David, et al., ``MDC Phase Calibration Working Group Report'',
MMA Memo 144.
Figure 1: Assumed geometry of the antenna clusters. The minimum
separation between antenna centers is 2D.
Figure 2: What fraction of the time will the atmosphere permit
clusters of various sizes to phase up at various frequencies?
Figure 3: Linear size of the field of view as a fraction of the
primary beam size, as a function of the number of antennas in each cluster.
Figure 4: Dynamic range and image fidelity as a function of
the number of clusters in the Atacama array for our simulations.
, 
) 
), and the weighted cluster size.