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MMA Memo 199:
Cost-Benefit Analysis for the Number of MMA Configurations
M.A. Holdaway
National Radio Astronomy Observatory
949 N. Cherry Ave.
Tucson, AZ 85721-0655
email: mholdawa@nrao.edu
February 27, 1998
Abstract:
We perform a new sort of cost/scientific-benefit analysis to determine
the optimal number of configurations the MMA should have. We trade
the costs of building extra configurations, moving antennas among
them, and lost observing time, against the loss of sensitivity which
results from tapering when a specific image resolution different from
the natural resolution of the array is required. With our
assumptions, we find that the optimal number of configurations is
between five and eight, depending upon what fraction of the time tapering
is required. However, four configurations is fairly close to the optimum.
Up until now, we have assumed that the MMA, like the VLA, will have
four different major configurations, with minor modifications of these
configurations to enhance observing to the far north or far south.
However, no justification has been given for four configurations,
rather than three or five or six. With between 32 and 100 antennas,
it would be possible to do many observations with a single
configuration; however, to achieve high brightness sensitivity at low
resolution, one would have to taper the array rather dramatically.
Obviously it is more efficient to build multiple arrays which produced
the desired resolutions. But what resolutions do the astronomers
want? There will clearly be a wide distribution of resolutions, and
we probably cannot settle on just a few required resolutions. If we
build a wide variety of array configurations to cater to everyone's
desired resolution, we also suffer from inefficiencies, as we will
spend an inordinate amount of money to build the vast number of
antenna pads and cabling, and will waste an unfortunate amount of time
moving the antennas among the configurations.
This memo seeks to find that compromise between the few configurations
(which require much tapering) and the many configurations (which require
much construction and moving) which will optimize the integrated sensitivity,
and hence the scientific output, of the MMA.
The cost-benefit approach we take in addressing the issue of the
number of configurations required for the MMA assumes that we can
trade off the extra costs associated with configurations (ie,
pads, cables, antenna moving costs) for scientific benefit in
the form of usable sensitivity (ie, instead of buying configuration,
we buy more antennas).
In order to proceed, we will
need to know a very lot about the Millimeter Array and how it will
be used. Table 1 lays out the symbols for each quantity, what the
quantity is, and our current estimate for that quantity.
We assume that the array configurations are bounded by a most compact
array with approximately filling factor 
of 40%, and by a
most extended array with maximum baselines 
of 
m.
Further, we assume that all configurations except the most compact are
basically ring arrays, and that the resolution of adjacent arrays are
related by a resolution scale factor S. The maximum baseline of the
compact array is given by
However, because the average baseline is shorter in this array than in a ring array,
this array will have the same resolution as a ring array about 70% as large.
Then, S is related to the number of configurations 
and and 
by
or
where R is defined to be the ratio of the largest array and smallest array effective
baselines, 
.
We need to assume the density of required resolutions for the MMA. To
first order, we assume that the required resolutions will be constant
in logrithmic bins. Since the array configurations have been designed
with a constant resolution scaling factor S, this implies that each
configuration will have the same proposal pressure. The VLA, which
also has constant S configurations, finds very similar observing
pressure on each of its four configurations, though the lowest
resolution array has been slightly more oversubscribed than the other
configurations for the last several years. While there are more sources
which can be detected from the compact arrays, it will take longer to
detect the fewer sources in the large arrays, so array use sort of
balances out.
Many observers will simply take the natural resolution of the array
configuration they observed in. An extreme case of this is the observation
of a point source, which can be observed in any configuration large
enough to avoid confusion. So, one component of the density of required resolutions
is a set of delta functions at the full resolution of each array being
considered.
However, sources which are not point sources have a maximum resolution
at which they may be profitably observed. The astronomer is playing
off resolution and brightness sensitivity. The resolution of one
array may not be high enough to see what the astronomer needs to see,
but the brightness sensitivity of the next larger array may not be
high enough to permit the astronomer to see anything. Hence, in
addition to the set of delta functions, there is also a continuous
component of the density of required resolutions. Most observers who
take an array's natural resolution (ie, the delta function crowd) are
actually part of this continuous distribution, but allow themselves to
be lumped into the delta function out of convenience. They might
actually benefit from slightly higher resolution or slightly higher
brightness sensitivity (and lower resolution), but there is no array
configuration which can provide this, so they take the nearest configuration.
And finally, there are people who absolutely need images at
non-standard resolutions. One fundamental analysis method used in
millimeter astronomy is comparing lines of different molecular
species, or different transitions of the same molecular species. In
general, these will be at different frequencies and hence different
resolutions. While the VLA was designed to be scalable for multi-frequency
comparisons (ie, spectral index maps between 15 GHz in C array,
5 GHz in B array and 1.4 GHz in A array), the number of specific frequencies
and resolutions which are important to the MMA is too large to optimize the
configurations for just a few. Rather, we need to accept that astronomers
will be making images of arbitrary resolution, and will have to taper sometimes
to get the resolution they require.
We assume that some fraction of observations, 
, will
require no tapering (the delta function crowd), and the remaining
observations for the 
configuration will have a required
resolution distribution of
The factor 
normalizes 
for
the coordinate convention that 
is the natural resolution of
the array and S is the resolution scaling factor between arrays, or
the most one would ever taper. This expression considers all
resolutions greater than the resolution of the most compact
configuration. The most compact configuration will also require
tapering, but since the configuration is constrained by the close
packing, there are no options open to consideration, and it should not
be included in this optimization attempt.
One complication arises in the case of extreme tapering (ie, by more
than a factor of 
). Imagine we are comparing two different
line maps, and we want to make the resolutions identical. Further
imagine that to get the same resolution in the second line as we have
in the first, we need to observe in the B array and taper almost all
the way back to the C array. From a sensitivity point of view, it is
actually much more advantageous to observe the second line in the C
array and just mildly taper the first line map. Of course, the case
becomes more complicated when more than two lines are included in the
astrophysical analysis and an extreme tapering event (ETE) cannot
always be avoided. However, for calculational purposes, lets assume
that most extreme tapering events can be avoided, and that we only
need to consider tapering up to 
where p is 0.5. Then
we only consider between normalized 
of 1 and for each configuration, and the new normalized
form of is given by
I argued in MMA Memo 156 (Holdaway, 1996) that since tapering would be
so important for the MMA, we should design each configuration such
that it performed optimally with respect to tapering, losing a minimum
of sensitivity. Filled arrays meet this requirement, while ring-like
arrays, with their more uniform Fourier plane coverage, lose more
sensitivity when tapered to a given resolution. Since then, imaging
simulations (Holdaway, unpublished; Morita, in
preparation) indicate that ring arrays provide superior imaging
quality in spite of their large sidelobes. However, the superior
imaging quality is not due to the ring array's ``uniform'' Fourier
plane coverage, but due to the fact that the ring array has much
shorter shortest baselines than the filled array (and quite a lot of
them, too), and the image quality in the simulations is being
dominated by the very short baseline distribution. At this point, I
am ready to move ahead with ring-like arrays, though there are
investigators who still favor filled arrays (Kogan, 1997). Recently,
Kogan (private communication) has produced arrays which are fat rings,
or donut arrays. They produce a partially tapered Fourier plane
distribution, and so will lose less sensitivity upon tapering than the
pure ring-like, uniform coverage arrays. The approach taken in this
memo is more global: to design the entire set of array configurations
to perform optimally with respect to tapering.
For a ring-like array with approximately uniform Fourier plane
coverage, increasing the resolution by a factor a will require
tapering, leaving a fraction of 
of the visibilities. Since
the sensitivity is proportional to the square root of the number of
visibilities, the residual sensitivity after tapering will be
proportional to 
. Hence, we define the sensitivity function,
intended for use between a tapered resolution between 1 and
S:
We now define the normalized observing sensitivity, integrated over all resolutions
between the natural resolution of the largest and the smallest configuration, based on the
above considerations as
If everyone were happy with the natural resolution of the array
configuration, would be 1, and 
would be 1.
Table 1 considers the case were R = 45 (ie, 3000 m/(.7 
95 m) )
(nobody doesn't taper), but with tapers only out to
(ie, p=0.5), for a variety of .
Hence, to get as much as 0.80 of the sensitivity of the MMA when
observers always tapered with a distribution like 
,
you would need 6 different array configurations. Or, with the 4 proposed
MMA configurations, you end up with 0.70 of the sensitivity. We remind
here that the observing efficiency depends strongly on .
For reconfiguring the array, we assume:
-
, the time to move one antenna, is 1 hr (see MMA Memo 147, Holdaway and Owen 1996).
-
, the time available each day for outdoor work, is 10 hr.
-
we will be in each of the configurations twice a year.
-
some of the time spent reconfiguring will permit useful science.
However, sensitivity will be lost while antennas sit during the day,
after they have been moved and before pointing and baseline determinations
have been made at night. Also, we assume that the scientific demand for the oddly
configured hybrid array may not be 100%. Lumping all these factors together,
we assume that a fraction
of the time spent during reconfiguration
will not be useful for scientific purposes.
Then the time lost to the array, in days per year, will be about
and the normalized reconfiguration efficiency 
is given by
In the cost-benefit analysis, we sum the monetary costs of
reconfiguring and trade the money for antennas. We can then ask if it
is better to have a few more antennas and fewer configurations, or
more configurations and somewhat fewer antennas.
The monetary cost to move one antenna is estimated to be
so the cost to move 
antennas through configurations
in a year will be
Meanwhile, the cost of making the configurations will be
Now, since operating expenses and capital costs will come from different sources
for the MMA, we can't really trade one off against the other.
But for the cost-benefit analysis, lets add up the move costs for a 20 year period.
Then the total cost of the configurations plus moves will be
We have calculated the various efficiencies subject to both tapering
and reconfiguration for numbers of configurations ranging from
2 to 12, for 36 10 m antennas, assuming 
(see
Table 2). Even for a very large number of configurations,
the efficiency lost to reconfiguring the array is negligibly small,
and it would seem that the choice would be to make many configurations
We have also calculated the additional costs that extra
configurations impact upon the array. Under our assumptions, the
extra costs are linear with the number of configurations, and
are equivalent to 1.57 antennas per configuration. In order
to compare = 6 on an equal footing with = 4,
we must keep the total cost of the two options equal; in other
words, we must take the $10M which was spent on the
two extra configurations out of the antenna budget, implying
we building 3 fewer antennas and our sensitivity is down.
To reflect this, we correct the total efficiency of the 6 configuration option by
. This corrected total efficiency is reported as
in Table 2.
We plot both the
total efficiency and the corrected total efficiency for the
case in Figure 1, and for the case in
Figure 2. Most remarkably, the results do not come
out too differently from the presumed 
option. In the
case, the optimal is about 5, but 4 and 6 are
also extremely close to the optimal . If a larger
fraction of the observations will require tapering, ie, if ,
the optimal number of configurations will shift upwards to about 
,
but even in this case, the case is still just about 7% below
the optimal .
Holdaway, M.A., 1996, ``What Fourier Plane Coverage is Right for the MMA?'',
MMA Memo 156.
Holdaway, M.A., and Owen, F.N., 1996, ``How Quickly Can the MMA Reconfigure?'',
MMA Memo 147.
Kogan, L., 1997, ``Optimization of an array configuration minimizing side lobes'',
MMA Memo 171.
Figure 1: Total efficiency (dash) and corrected total efficiency (solid) for the
case.
Figure 2: Total efficiency (dash) and corrected total efficiency (solid) for the
case.
.