M.A. Holdaway
National Radio Astronomy Observatory
email: mholdawa@nrao.edu
August 6, 1997
For the purpose of helping to evaluate the various options for collaboration between the LSA and the MMA, we review the optimum point source sensitivity and optimum imaging sensitivity criteria for array design, and also develop a new criteria based on optimizing surface brightness sensitivity in wide field imaging.
We calculate the noises under the various criteria for the various possible LSA/MMA joint arrays. If the array has antennas of two different diameters, cross-correlating all antennas is required to maximize the array's sensitivity. Processing of visibilities from antennas of different diameter is feasible with our current software, although with an array of 8 m and 15 m dishes, full cross-correlation mosaicing will increase the cpu requirements by an order of magnitude. Given that all cross correlations are formed, there is not a great deal of difference between the various sensitivities of the homogeneous array options (50-60 x 12 m antennas) and the heterogeneous array options (40 x 8 m plus 25-35 x 15 m antennas).
On June 25 and 26 of 1997, European representatives of the LSA project met with representatives of the MMA project to discuss the possibility of building a single collaborative array. In addition to a resolution which stated the goal of working towards a single array, we also agreed to study two different array design concepts: a homogeneous array with 50-60 12 m antennas, and a heterogeneous array consisting of both 40 8 m antennas and 25-35 15 m antennas. (The range in the numbers of antennas is due to the uncertainty in the cost of the 12 m and 15 m antennas. We consider both the poor and rich ends of these options.) Here, we investigate the various sensitivities of these different array options and compare them with each other and with the ``default'' arrays of 40 8 m antennas and 35 15 m antennas.
For a heterogeneous array, simple analytical expressions exist for the point sensitivity and the imaging sensitivity. We review these expressions, as well as develop an expression for the imaging surface brightness sensitivity of an array.
The noise 
in a naturally weighted image from interferometric
observation is equal to
where 
is the system temperature, A is the collecting area
of each dish, 
, 
is the integration time, 
is the band width, and 
is the number of baselines
n(n-1)/2, n being the number of antennas. Taking out the items we
are not interested in,
The analogous proportionality for a single dish of area 
, all
other factors such as efficiency, , integration time, and
band width being the same, is
Since 
is very close to n, the noise in an
interferometer image is very nearly inversely proportional to the
total collecting area, and very nearly the same noise as for a large
single dish of the same collecting area. If autocorrelations are
included, then 
, and the interferometer noise is
actually a bit less than for a single dish of the same collecting
area. To optimize the point source sensitivity of an instrument,
we then seek to maximize the total collecting area, for either an
interferometer or a single dish. Hence, we arrive at the
consideration of maximizing
The interferometer offers some advantages: its potential for higher resolution reduces confusion, its ability to image multiple resolution elements simultaneously in the primary beam results in a speed up of its imaging with respect to a single dish of the same collecting area, and its multiple small elements often represent a cost savings, as compared to a single dish of the same collecting area. The single dish also offers one main advantage: it puts a given amount of point source sensitivity (set by the collecting area) into the largest possible beam on the sky, hence optimizing surface brightness sensitivity, at least for single pointing observations.
If point source sensitivity were all that mattered, we would get as much collecting area as we could, as cheaply as we could. However, the universe is not made up of point sources, in spite of the Clean algorithm. Especially at millimeter wavelengths where the primary beams become smaller, there are many sources which will require multiple pointings to image. Since larger dishes result in smaller primary beams, part of the sensitivity gained by increasing the dish size to increase the collecting area is lost as more pointings must be spent to cover a large source and each pointing will receive less integration time, as compared to a smaller dish.
The imaging sensitivity, a measure of the sensitivity in imaging two
dimensional objects which are much large than the primary beam, is
proportional to the point source sensitivity multiplied by the square
root of the amount of time which the array is able to spend on each
pointing, given a fixed amount of time to cover the entire region of
interest. The integration time which each pointing receives will be
inversely proportional to the number of pointings required to cover
the region of interest; the number of pointings is inversely
proportional to the area of the primary beam; so the integration time
which each pointing receives is proportional to the area covered by
the antenna's primary beam B, which is inversely proportional to
. Hence, the imaging sensitivity is given by
For one dimensional objects, the integration time each pointing
receives will be proportional to 
, so the imaging sensitivity
will go like 
. For zero dimensional objects (ie, point
sources, or sources small compared to the beam), the integration time
the pointing receives is independent of the dish size, and is
recovered.
The and nD criteria are well known in array design. We wish
to consider another criteria for the number of elements and the dish
size which has not been considered by others: how does the surface
brightness sensitivity of a packed configuration vary in the case
of wide field imaging?
A naive approach would reason that the brightness sensitivity of an
array is independent of n and D, but only depends upon the filling
factor f of the array (ie, the fraction of the array area which is
filled up with antenna collecting area), which is ultimately limited
by the antenna design and how close it allows the antennas to be
placed. For the time being, we will assume that all antenna diameters
can have designs which permit the same maximum filling factor, which
is something like 0.5. Now, as in imaging sensitivity, the number of
pointings required to span the two dimensional object will be
proportional to , which will augment the sensitivity by .
This suggests that the wide field surface brightness sensitivity
criteria is f/D, or just 1/D for equally filled arrays. How can
this be? In the limit, it seems we should have an unspecified number
of vanishingly small antennas!
In order to compare the wide field surface brightness sensitivity of
different arrays, we really need to hold the resolution constant. A
50% filled array with n large dishes will have higher resolution
than a 50% filled array with n small dishes. To compare the two
arrays, we must taper the array of larger dishes to the natural
resolution of the array of smaller dishes. Some amount of point
source sensitivity is lost from the array of large dishes, but the
point source sensitivity will still be better than for the array of
small dishes. To address how much point source sensitivity is lost to
tapering, we have generated arrays with a range of antenna size and
number, simulated data sets from each array with noise per visibility
proportional to 
, and tapered the simulated visibilities to
produce the same image resolution for each array. The resulting image
noise is proportional to the single pointing surface brightness
sensitivity at the chosen resolution. A power law expression
was fit to the simulated image noise for the
different arrays of various n and D, and the best fit to the point
source noise at constant resolution is 
, though
there is considerable scatter in the relationship, presumably due to
details of the simulated array configurations. The wide field surface
brightness sensitivity at this constant resolution will be one over
the point source noise, divided by D to account for the number of
pointings and time per pointing, or
Hence, for the case of optimally filled arrays, the wide field
brightness sensitivity at constant resolution is approximately
independent of the dish diameter and only depends upon the number of
elements, subject to the implicit constraint that the array is
packed (ie, 
50% filled) at a size corresponding to the
desired resolution.
, nD, and 
Criteria
Following the work of R. Brown, as presented at the June 25 LSA/MMA
meeting, we can optimize the quantities , nD, or
subject to the constraints given by the cost equation and the total
amount of money we think we have to spend. Optimizing will
favor larger antennas (as per the LSA's 15 m antennas), optimizing nD
will favor smaller antenna (as per the MMA's 8 m antennas), and
optimizing will favor small antennas.
While the analytical quantities in the previous sections are approximately correct for homogeneous arrays, computation of the imaging and low resolution surface brightness sensitivities for a heterogeneous array in which all baselines are correlated, such as what we are considering for the joint MMA/LSA array, must be performed in a somewhat more detailed manner. Only the point source sensitivity remains a simply computed quantity.
We consider the various sensitivities of six different array options: 50 12 m antennas, 60 12 m antennas, 40 8 m antennas plus 25 15 m antennas, 40 8 m antennas plus 35 15 m antennas, 40 8 m antennas alone, and 35 15 m antennas alone. We assume here that the 8 m antennas will have 1 arcsecond pointing and 25 micron surfaces, the 12 m antennas will have 1 arcsecond pointing and 25 micron surfaces, and the 15 m antennas will have 1.5 arcsecond pointing and 30 micron surfaces. In addition to the differences in diameter and pointing and surface specifications, there will be other differences in the performance of the antennas. Since the feed legs do not go to the edge of the dish in the LSA design, there will be more blockage and more scatter of stray radiation, so the low frequency antenna efficiency will not be as high as the MMA design, and increased warm spill over will slightly increase the system temperature. Since the efficiency and spill over effects are small, we have not included them in these computations.
In order to maximize the sensitivity of a heterogeneous array, signals from all antenna pairs must be correlated, and all antennas must have a common pointing center at each time. In the case of the heterogeneous choice for the MMA/LSA collaboration in which 40 8 m dishes and between 25 and 35 15 m dishes have been proposed, we will have three different primary beams: the 8 m primary beam, the 15 m primary beam, and the 8x15 m primary beam. If an object is much smaller than the smallest primary beam (as in the case of MERLIN or in ad hoc VLBI arrays), then the different primary beam sizes do not matter. However, if the object is as large or larger than the primary beams, we must correctly treat the effects of the primary beam via a mosaicing algorithm.
Consider an object which is much larger than our primary beams which we wish to image with our heterogeneous array. In order to maintain uniform sensitivity in the mosaic image, the common pointings must be set by the Nyquist sampling rate of the smallest primary beam, in this case, the 15 m. This results in the 8 m data being oversampled by a factor of 15/8, or 3.5 times as many pointings required to sample the same two dimensional region of sky as for an array of only 8 m dishes. However, the oversampling does not result in any loss of speed, as the extra sensitivity from adjacent pointings permits 1/3.5 times less integration time per pointing to obtain the same final noise level in the mosaic.
The non-linear mosaic algorithm (Cornwell, 1988; Cornwell, Holdaway,
and Uson, 1993) is already flexible enough to deal with mosaic data
from a heterogeneous array. To demonstrate this, we have made a toy
simulation with an array of 27 8 m dishes and 13 15 m dishes, with 121
pointings sampled on the sky by 
for D=15m. Baselines
correlating 8x8, 15x15, and 8x15 dishes were treated as different
telescopes, each with their own primary beam. Hence, a total of 363
logical pointings (ie, 121 actual pointings times 3 different primary
beam types) were processed in the mosaic algorithm, as opposed to the
pointings which would have been required if only 8 m dishes
had been used. With 10 times the number of pointings to process, the
cpu time required to image a mosaic of a given size will increase by a
factor of 10. The toy simulation which demonstrates heterogeneous
mosaicing works produced an image with only about 1000:1 dynamic
range. Error-free mosaic simulations with 40 elements typically
produce much higher dynamic range images. We attribute the lower than
expected dynamic range to the fact that the point spread functions for
each logical pointings (ie, one with 
baselines, one
with 
baselines, and one with 
cross
baselines) had higher side lobes than 40 elements in a homogeneous
array (
baselines). A heterogeneous mosaic with 40 8 m
antennas and 25-35 15 m antennas would have higher dynamic range, but
not as high as the same number of elements in a homogeneous array.
However, we expect that pointing errors will be a more serious
limitation to the image quality than the higher side lobes in each
logical pointing, especially for the 15 m dishes which will have large
pointing errors and a smaller beam (a future MMA Memo will address
heterogeneous array mosaicing with pointing errors).
While there is still much work that needs to be done in the subject of heterogeneous mosaicing (ie, examination of pointing errors and total power considerations, optimization, etc), it is reasonable to assume at this time that there are no fundamental software limitations which will prevent us from making heterogeneous mosaic in the future.
The imaging sensitivity of a heterogeneous array mosaic will be somewhat
more complicated than nD: it will be proportional to the quantity S
defined as:
where 
is the imaging sensitivity of the 
primary beam
type, 
is the number of baselines that have the
primary beam type, 
is the effective dish diameter (the
geometric mean in the case of the cross baselines), and 
is the
area on the sky covered by the primary beam. The individual
sensitivities for each different primary beam type (ie, 8x8,
15x15, and 8x15) add in quadrature as they are essentially different
arrays observing the same object.
Because the proposed 8 m, 12 m, and 15 m antennas have different pointing and surface specifications, a proper comparison should look at the array sensitivities over a range of frequencies. To properly do this, we must consider the following frequency dependent effects:
and
otherwise the same analysis as R. Brown's ``Uniform
Assumptions for the Atacama Array Workshop'' analysis,
we adopt the following numbers:
| Freq [GHz] | [K] |
| 230 | 65 |
| 350 | 110 |
| 650 | 280 |
| 850 | 330 |
. The Ruze formula is
actually too pessimistic as 
gets large, due to the
fact that when surface errors cause wave front deformations of
more than 
radians the wave front may actually be more in
phase, and a correction has been made to account for this effect
(Levy, 1996).
| Freq [GHz] | D=8m, PE=1" | D=12m, PE=1" | D=15m, PE=1.5" |
| 230 | 1.00 | 1.00 | 0.98 |
| 350 | 1.00 | 0.99 | 0.96 |
| 650 | 0.99 | 0.96 | 0.87 |
| 850 | 0.97 | 0.94 | 0.81 |
With a 13% sensitivity loss at 650 GHz and a 19% loss at 850 GHz, the 15m dish will sometimes experience extreme pointing deviations during which the visibility amplitude will drop substantially. It will be difficult to correct for the effect of these amplitude fluctuations on imaging, and they will help set the upper frequency at which the 15 m dish will work.
For the six array options mentioned above, we calculate the rms image noise for single pointing observations of 1 minute, 8 Ghz bandwidth, and two polarizations, for frequencies ranging from 230 GHz to 850 GHz. For the two configuration options with both 8 m and 15 m antennas, we assume all possible correlations are performed. Ruze losses and pointing inefficiency are included, but no time losses due to phase calibration or pointing calibration are included. The point source sensitivities are shown in Table 1.
| Freq | 50x12m | 60x12m | 40x8m | 40x8m | 40x8m | 35x15m |
| [GHz] | plus | plus | ||||
| 25x15m | 35x15m | |||||
| 230 | 0.043 | 0.036 | 0.038 | 0.030 | 0.120 | 0.040 |
| 350 | 0.083 | 0.069 | 0.073 | 0.057 | 0.219 | 0.078 |
| 650 | 0.352 | 0.293 | 0.301 | 0.241 | 0.785 | 0.352 |
| 850 | 0.678 | 0.564 | 0.563 | 0.459 | 1.303 | 0.718 |
For the six array options mentioned above, we calculate the rms image noise for observations of a source which is many pointings across in both directions, spending 60 s of time per square arcminute of source, with all other parameters being the same as in the point source sensitivity case. For the two configuration options with both 8 m and 15 m antennas, we assume all possible correlations are performed. The wide field imaging sensitivities are shown in Table 2.
In reviewing the entries in Table 2 for 40 8 m antennas plus 25 or 35 15 m antennas, remember that the 1.5 arcsecond pointing errors and the smaller beam of the 15 m will drastically limit its usefulness in mosaicing at high frequencies. The 15 m dishes will probably not be used at all in the 650 and 850 GHz bands. At 350 GHz, the 15 m dish's pointing errors will set a limit to the image quality which can be achieved. Bright sources which could possibly exceed this image quality limit would be mosaiced with only the 8 m antennas, but weak sources which would be limited by thermal noise could employ both 8 m and 15 m dishes for mosaicing. The image quality of the 12 m mosaics will also degrade more quickly than the 8 m, but not as drastically as the 15 m. However, the 12 m option cannot rely upon smaller dishes for better mosaic image quality at the highest frequencies.
| Freq | 50x12m | 60x12m | 40x8m | 40x8m | 40x8m | 35x15m |
| [GHz] | plus | plus | ||||
| 25x15m | 35x15m | |||||
| 230 | 0.16 | 0.13 | 0.14 | 0.12 | 0.29 | 0.18 |
| 350 | 0.46 | 0.38 | 0.40 | 0.34 | 0.81 | 0.54 |
| 650 | 3.60 | 2.99 | 3.01 | 2.54 | 5.35 | 4.50 |
| 850 | 9.06 | 7.54 | 7.12 | 6.13 | 11.62 | 12.01 |
Using model arrays with approximately 50% filling factor, we have calculated the increase in point source noise which results from tapering to a 3.5" (230 GHz/freq) and a 7.0" (230 GHz/freq) beam, applied this noise increase to the values in Table 2, and converted the noise values into brightness sensitivities. These values for the wide field surface brightness sensitivities of the various array options are presented in Tables 3 and 4
| Freq | 50x12m | 60x12m | 40x8m | 40x8m | 40x8m | 35x15m |
| [GHz] | plus | plus | ||||
| 25x15m | 35x15m | |||||
| 230 | 0.38 | 0.32 | 0.33 | 0.28 | 0.55 | 0.54 |
| 350 | 1.12 | 0.95 | 0.93 | 0.81 | 1.52 | 1.57 |
| 650 | 8.73 | 7.41 | 6.88 | 6.10 | 10.09 | 13.03 |
| 850 | 21.97 | 18.66 | 16.28 | 14.73 | 21.91 | 34.75 |
| Freq | 50x12m | 60x12m | 40x8m | 40x8m | 40x8m | 35x15m |
| [GHz] | plus | plus | ||||
| 25x15m | 35x15m | |||||
| 230 | 0.17 | 0.14 | 0.15 | 0.12 | 0.21 | 0.22 |
| 350 | 0.50 | 0.40 | 0.43 | 0.35 | 0.57 | 0.64 |
| 650 | 3.88 | 3.15 | 3.17 | 2.66 | 3.79 | 5.28 |
| 850 | 9.77 | 7.94 | 7.51 | 6.42 | 8.24 | 14.09 |
From Tables 1 through 4 we see that the homogeneous and heterogeneous arrays actually have very similar sensitivities. The heterogeneous options are a bit better than the homogeneous options (compare the ``poor'' homogeneous array to the ``poor'' heterogeneous array, and the ``rich'' homogeneous array to the ``rich'' heterogeneous array). For the heterogeneous arrays, the addition of the 40 8 m antennas adds a small improvement to the point source sensitivity. The cross-correlations between dishes of different diameter are not so important to point source sensitivity either. However, the 8 m antennas, with their wider beams, make a strong contribution to the imaging and imaging surface brightness sensitivities (as do the inter-dish cross-correlations).
So, it appears that both the homogeneous and the heterogeneous options are acceptable from a sensitivity point of view. A decision between the collaboration options must be made on other issues such as image quality (ie, high frequency mosaicing), scientific flexibility of the instruments, engineering considerations for a single or multiple antenna designs, software considerations, and political factors.
References
Cornwell, 1988, ``Radio-interferometric imaging of very large
objects'', A&A 143, 77.
Cornwell, Holdaway, and Uson, 1993, ``Radio-interferometric imaging of
very large objects: implications for array design'', A&A 271, 697-713.
Levy, Roy, 1996, Structural Engineering of Microwave Antennas,
IEEE Press, New York, p. 288.
.'' GHz, spending 60 s of time per square arcminute.