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[Min Yun]

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Date: Tue, 14 Mar 2000 18:22:26 -0500 (EST)
From: Eric Keto <keto at dogstar.harvard.edu>
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Min,
Paul gave me some ALMA memos, #283,291,292, by John Conway 
on possible array configurations. Although you might already
know this, Paul urged me to send you an e-mail pointing out
that there are some problems
with the methodology and the conclusions in these memos. 
In particular, memo #291, "First Simulations of Imaging
Performance ..." concludes that the lower sidelobe level
of the centrally condensed spiral array results in better
imaging than achievable with the sharp sidelobes
of the ring or triangular arrays. 

First, as you no doubt
know, the spiral array must be twice as big to achieve
the same angular resolution. The longer baselines will
be noisier because of atmospheric decorrelation, and this
means the image will be noisier. A better method would be
a comparison including this effect. This is one reason
why we did not chose the spiral for the SMA. The angular
resolution is not as good for the same maximum baseline.

Second, the IMAGR implementation of the clean algorithm 
is one which has been developed over 20 years to clean the 
centrally condensed VLA configuration. But this algorithm 
is not going to do a good job on the uniform distribution
of the ring or triangular arrays. The very high angular
resolution of the Reuleaux triangle does come with the
price of higher sidelobe levels if not treated carefully 
and these can result in ringing in the image. IMAGR has
never had to deal with this situation and as a result
has little capability for effectively deconvolving such
images. For example, it is well known in the optical world
where apertures are always quite sharp edged, that an
easy and effective tapering function for rounding the 
edge in Fourier space is a so called super-gaussian 
exp(-(x/s)^n) where n is some number greater than 2, 
maybe 4 or 8. This function is much flatter than a regular 
gaussian, then cuts off quickly at (s) with a rounded edge.
AIPS implements a gaussian taper which is not going to be
effective with the sharp edge in Fourier space of the triangular
configuration. For example as a very simple experiment,
one could first taper the UV data from the ring array 
with a super-gaussian to eliminate the ringing. 
There are more clever ways to deconvolve these images which
result in less loss of signal-to-noise.
For example, the best edge weighting function is
given by the eigenvalues of the prolate spheroidal wave
function, but the super-gaussian is easier to implement.
I would imagine that one could design a clean-style processing 
algorithm with no tapering that would eliminate the sidelobes
in the image, but I don't have one ready myself. The point 
is one should be careful about designing the 
ALMA array based on the limitations of one image 
deconvolution algorithm.

While there is certainly a trade-off between the resolution
of a configuration and the sidelobe level, the higher
sidelobe levels are due to the sharp edge in
UV space created by some ring arrays; however, the
high angular resolution of the ring or triangular array
is more the result of the flat coverage in UV space
than the sharp edge. It is possible to  design a triangular
array with a softer edge than the ring array, but which
still preserves much of the angular resolution. One
example is in my paper, figure 7 (1997 ApJ 475, 834)
The point of this configuration is that the density of
UV points is flat over much of the UV plane than falls
off more gradually near the edge. I put this configuration
in the paper as an example of how one could fulfill
a different set of imaging criteria. But the best configuration
depends on the criteria. Those for the SMA were quite 
simple, maximum signal-to-noise at the highest possible 
angular resolution.

Although we never published the less desirable configurations
we did go through the analyses of both spirals and concentric
circles for the SMA. So we are quite familiar with the
imaging characteristics of many configurations besides the
Reuleaux triangle. If I can be of some help, let me know.

-Eric Keto



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