[alma-config]Comments on Memos 389 and 390 (fwd)

[John Conway]

Hi,

 A slightly edited version of the comments
on memos  389 and 390 that I sent to Dave, 
Leonia and Al a couple of weeks ago.

  John.

---------- Forwarded message ----------
Date: Tue, 18 Sep 2001 15:53:59 +0200 (CEST)
From: John Conway <jconway at oso.chalmers.se>
To: dwoody at ovro.caltech.ed
Cc: lkogan at nrao.edu, jconway at oso.chalmers.se, awooten at nrao.edu
Subject: Memos 389 and 390 


Hi Dave (and Leonia, Al),

  What I was going to write before being interupted by world events
was that I was very impressed by both of Daves new memos - and I think 
they provide a breakthrough in our understanding of what can 
be achieved in terms of the sidelobes as a 
function of radius. As Dave stresses the theory provides 
an important benchmark against which real arrays can then 
be assessed. In addition the criteria suggested for including short 
spacings (that sufficient are available that the inner part
of the dirty beam stays non-negative even with autocorrelations)
 is another breakthrough  in an area which imaging simulations have shown
is  vital but where little attention has previously been paid.


I have though two different comments to make to
the excellent arguments given on the the optimisation 
of single configurations  (1) I think pad
sharing between configurations is an important practical constraint
which is most easily achieved if its built in from the start 
of the design process and (2) I believe there are arrays which 
exist which give close to pseudo-random uv distributions
(little detectable patterns in the uv distributions) even though their pad
distributions
are not derived from a random starting point, some of these
distributions are good from the point of view of pad 
sharing and point (1).


Taking the first point, Dave  suggests in his second memo  that in 
a  multiple configuations optimisation 10%-15% of
pads are chosen randomly to share betwen configurations.
A statement is made which puzzles me a little; it is said  that 
pad sharing should not be too much or else the mumber
of uv point samples achieved is reduced. Presumably 
this is referring to combining multi-configuration  observations.
The currect baseline designs have about 50% pad sharing between
arrays different by a factor of 2 in resolution. The result 
is that 1/4 of the baselines are common between the two 
such subarrays.  The fractional
loss in the number of uv points compared to having arrays
with no common baselines is then (2-0.25)/2 = 0.88. A loss
of 12% of the information seems little compared to having
a significant number of baselines in common to aid
cross calibration. In addition if the purpose of the second
condiguration is 
to fill in the central hole  in the uv plane one would 
probably use as the second array one with more than
a factor of 2 lower resolution, hence even fewer 
common common baselines.  Looking at the resource perspective
however a  50% reuse rate between arrays a factor of 2 apart
in resolution  compared to a 10% reuse rate makes a very big 
difference in the numbers  of pads and number of antenna moves required
(almost a factor of 2).
 
  

 As you know my focus has always been on finding arrays 
which maximised pad sharing while giving good beams (implying
following a smooth radial uv density distribution while being 
locally close to random - the 'pseudo-random' of Daves memo).  
I have concentrated on this because I have felt that practical
issues of minimising resources and moves were likely to be 
very important in the end. I do not believe that there is 
anything 'special' about scale free arrays in terms of uv covergae
compared to ones optimised from a random starting point except that they
naturally  allow a large degree of pad sharing with  a  minimum pattern 
left in the uv coverage (compared to  say other ideas with similar 
degrees of pad sharing such as multiply nested circles).


Others in the configuration group have had a different approach of  
trying  to find  the  best criteria for optimising 
a given configuration (peak sidelobes or number of filled uv cells
out to a maximum radius for instance). Establishing such a criteria and
then letting a final design emerge from a computer  would be very
satisfying (although I believe the practical problem remains 
of minimising the number of pads). As in solving many difficult 
optimisation problems (bus timetables etc) I believe you can get 
to close to a optimum by starting with a good guess based
on general principles (choose out of all arrays which 
are deterministic and self-similar one with a  minimum symmetry). 
However even  even if one  decides some starting generating shape 
with good properties is useful to start 
as a 'backbone' to allow large pad sharing there is still the need
to incorporate a final beam optimisation taking into account terrain and
so Dave is right (as he has  stressed many times in the telecons) 
that a decision on such a criteria  is sorely needed to produce a 
final design. Up to now in opimising my designs I have used
Leonias peaks sidelobe criteria (which have given rise to only small 
changes in the general properties of the array) but for arrays smaller
than  the most compact one the fact that one mimimised within
a specified radius  (30 or 40 beams say) semed very arbitary- 
Daves  analysis and modification to Leonias method  has now
corrected this arbitariness. 


I have also been considering the alternative ideas of
Boone in terms of an optimisation criteria, where one
samples  all cells out to a uv max but allows the overall
density to be bell  shaped to give smaller near-in 
sidelobes (the 'full' uv coverage that Dave also supported originally). 
I and Frederik have had a large amount of 
correspondence in the last month about his ideas.
To  some extent the arguments converges, since optimising 
to an ideal uv coverage or optimising to its FT 
an ideal beam are almost equivalent. I can see 
slight advantages both wasy but from a practical 
point of view I think beam optimisation wins out slighty
for the following regions


advantages of beam optimisation:


1) As Dave notes we only get 'full' covergae at the expense of bell
shaped curve with a cutoff at small radius - giving significant
near in sidelobes.  As I  have always argued I believe it is better to 
undersample to a larger radius (and interpolate) than to 
fully sample to a smaller radius (and have to extrapolate).


2) The 'Cell filling' concept only takes into account the primary
beam in an approximate way. Boones algorithm in the most compact 
array for a snapshot  comes close effectively to minimising the 
dirty beam times  the FT of the square uv cell, it seems more 
sensible to explicitly minimise the sidelobes directly,
since these sidelobes as noted in Dave paper give the cross-coupling
between different points in the image.


3) For the case where the uv cell occucancy is low a Boone type 
optimisation does not gurantee low sidelobes. Imagine a  1D example
where in  a snapshot that the number of uv points is such that the mean
cell  occupancy is 0.5, assume a uniform target density. If the
uv points could all be moved interpendently then the 
smoothest distribution on all scales (which is the one
which would emerge from Boones algorithm) would be 0,2,4,6.. 
(in units of 1/B) [if this is not the smoothest 
possible distribution on all scales according to Boones
algorithm my challenge is to ask what is???] but this would
give 100% sidelobes
within the primary beam. In practice this is prevented
by the fact that the position of uv points cannot be ajusted
interpendently and that we aim for non-uniform radial density
distributions. The fact remains however that there is no explict
optimisation
against these 'resonances' in the algorithm,  [Frederik 
does not agree with at all on the above argument. To be fair  he 
in any case proposes only doing the uv optimisation for long tracks such
that  the cell occupancy approaches one in which case 
the problem above does not arise -  my position is that 
optimising snapshots even for long tracks is useful]


4) From a very practical point of view optimising 
the zenith snapshot (and then the long track looks
after itself because its a superposition of streched and
rotated snapshot beams) seems to be a time efficient approach.
If one wants to optimise the uv coverages of long track over 
a range of declinations, simulations must be done 
for each declination and some form or weighted optimum 
produced. While this has some attractions, given 
the limited time to come to a final design its very 
time consuming.


5) There is a feeling that uv optimistaion is 
more fundamantal because of the sampling theorm.
One can ask about where the sampling theorom comes from. 
in the case of the standard regular sampling  in 
electrical engineering one  derives the Nyquist 
criteria in terms of the smaplin rate such that the FT 
has sidelibes which are zero over the spectrum, the 
uv criteria coresponsd to having zero sidelobes over 
the support of  source; hence reduces to a beam 
criteria.  The question of what mathematical conditions on a
non-regular uv distribution are required to allow 
a linear reconstruction of the true source convolved
with a gaussian (and is well conditioned with respect to noise) 
is an interesting question which 
it would be good to give more exact mathematical 
attention too, I  suspect however it will always be one that has small 
sidelobes (i.e. as is  achieved if we have 
 on average one or more  uv sample per uv
cell).  Fundementally I can't believe that one criteria (beam 
or uv plane) is inherently superior to the 
other since they must be releated by a Fourier 
transform, therefore I believe we should 
choose the one which is  most convenient  to implement.


 

Paradoxically  Boones uv optimisation software is an  
excellent quick way of optimising the beam (snapshot or long track). 
Since it optimises on several scales in the uv plane this coresponds
to minimising the  sidelobes within different radii of
the beam simulataneously. I supsect that starting from 
a random starting point the Boone algorithm  is likely
to be much quicker than the Kogan-Woody one. On the other hand
the Boone method takes only a simple account of the  primary beam 
and does not include for low density uv coverages the effects of
periodicties in the uv coverage in creating sidelobes, and so 
it might be good to do a Woody_Kogan optimisation as a final
step.



I have compared the results of optimising  a spiral and a 
random starting array  with the Kogan or Boone algorithms 
in http://www.oso.chalmers.se/~jconway/ALMA/SIMULATIONS/SIM8/

This was done before Daves two memos came out, but I did
plot the peak sidelobe versus  radial distance. The Boone
optimisation effectively corresponds to optimising the beam on many
different beam radii. What is  interesting is that since the uv covergae
of the spiral was already pretty pseudo-random, then the resulting 
optimised arrays are not much different (note the slight reduction in the
mean square error in applying Boone optimisation is largely due
to reducing the number of short baselines)  In Daves
second memo when staring  with a ring or filled circular shells 
the optimisation broke up the original pattern in the uv plane and then
gave a fairly random pad distribution. The algorithms have not
really done this to the spiral, the final result is close
to the input. Hopefully therefore 
one can have the desirable properties of high pad sharing 
from a self-similar array (and also the option 
for gradually ajustable zoom arrays) AND  have good close 
to pseudo-random uv coverage. It would be intersting to compare the 
results of optimising a truly random distribution of antenna
positions and then one based on a suitable spiral with 
slight perturbations (which had a -uv coverage- with, to the eye, hardly
any discernable pattern). One can then compare the peak sidelobe versus
radius for the two types of beam and compare them with Daves theory.
I suspect that the truly random distribution may have very
slightly smaller sidelobes, but I doubt it would be significant,
and not worth the complexity of finding  some other 
random method of sharing pads between configurations. The 
nice thing is we don't need to argue about any of this(!) we can
simply simulate it and see  how they both perform compared
to the theorectical benchmark!


If I'm right about the above assertion then 
I think now we have all the tools needed to make a great 
ALMA array configuation, incorporating all the ideas that 
have emerged. For the arrays out to 3km starting from a self-similar
backbone  we can optimise  using a Kogan-Woody algorithm
(for thses scales despite the terrrain I can still fit  a good 
slightly offset zoom spiral as a starting point). For larger
arrays the Boone method provides a good way of getting a
good starting point with the desired radial taper for long
track arrays, which can then be optimised.  I was also very 
excited by the modification Dave suggested to incorporate short 
spacings, use the criteria that 
the inner part of the dirty beam does not go negative (even
when not using autocorrelations). This seems a great criteria
which was formerly missing from the debate. The resulting 
uv coverages which emerge, with an inner circle occupied
is similar to what was used in the strawperson array 
at http://www.oso.chalmers.se/~jconway/ALMA/SIMULATIONS/SIM5/
Heepfully the  pads on this circle can be used from the 
outer ring of Leonias Compact ('road first') design.


 John.


P.S. One other thing I wasn't certain about is that I think I 
understand for the larger arrays (with single pointing observations) 
why when optimises the FT of the uv point multiplied by the
autocorrelations 
of the primary beam. However when considering the smaller 
arrays, as we all know,  after mosaicing using mosaicing the effective uv
coverage is the uv points
convolved with the autocorrelation of the circular aperture  of
each antenna. The effetcive beam is then the FT of the uv points
multiplied by the primary beam. So for small arrays, mainly 
used in mosaicing mode, would not minimising the 
sidelobes (times expected radiual dependence) within this
primary beam (rther than the autocorelated one) be 
the correct criteria to employ?.