[alma-config] [Fwd: Re: Memo 538 draft]

[John Conway]

Hi,

 I was planning to write a detailed reply to all of Frederic's
comments tomorrow (when I have a gap from teaching), but since debate
on the Nyquist issue has started again I have decided to reply on this point
straight away

Several years ago we debated endlessley the issue of the appropriate
value of the edge taper 10db  20db etc and the related issue of the minimum
Nyquist sampling to allow at that edge. There was no universal agreement.
At the Socorro delta-PDR meeting I suggested that we could circumvent this
problem by considering as a practical matter what the impact on performace
atually is. This argument I summarise in point 1 below. Once we establish that we
are talking about small factors, we can relax and hopefully then not get
hot under the collar when we discuss what the actual answer is, which I
discuss in 2)

1) Frederic points out that my 3.5km array has points seperated by
4*nyquist at the edge  when an array with 1*nyqusit could be build instead
but this would better fit the design goals. Now I don't have the actual numbers handy (I'm
writing this between meetings) but the overall density distribution in my
design is close to  gauusian  with a cutoff  at the edge of say about 15dB
and Frederick  suggests we now redesign it instead to have a cutoff of say
10db; because those sub-Nyquist  sampled points  cannot be used
(especially  for Mosaicing). Whether or not  this is correct  is the
subject of point 2 below. But let us assume he is  right, what happens?,
we must then delete the uv points between 10dB and 15dB, but because
a Gaussian fall off very fat beyond teh 10dB point there
are  very few points in this region (!). I beleive the result is only an
increase in the noise at the percent level and  a reduction in resolution
of a  similar amount (If I get time tomorrow I  will look at the exact
values).

Still even if the savings are small why not built the array to avoid these (assumed) useless
points?. Such a design would argue for a less centrally condensed array, but there are costs
involved in this both in terms of imaging and money. While all this attention is given to
the outer uv edge and the exact value of its sampling we are in danger of forgetting
what I think is a much more critical area for imaging quality, the short spacing coverage
and specifically the shortest baseline for a given array resolution. A less centrally
condensed  usually has a larger shortest baseline. Likewise a less centrally  condensed array
requires more pads to span a given range of resolutions. The project is very short of money,
one of my goals in the redesign going to 50 antennas was to reduce the number of pads as much as
possible, the consequence was a highly concentrated array. If we have less of an edge
taper we will need more pads, since each including roads each pad costs at least $150,000, the
small amount of sensitivity we recover by having more pads and a better edge taper we may
lose again because we cannot afford an antenna (which then loses us 2% in sensitivity).
A final point is  that as designed the pad distribution has a 1/r^2 density distribution
which then allows continuous zoom configuration with  all the operational
and observational advantages that gives. With such a constraint on the pad density
distribution the range  of occupied radii which gives close to gaussian uv density is
constrained, and this value of occupied raddi leads to the particular edge
taper we have. If we decide  to keep a 1/r^2 pad density distribution to
allow zooming and then adjust the range of occupied radii in any  configuration to
give say 10dB edge  taper, then as well as needing more pads the  uv
density function will be  far from  gaussian and we will  lose sensitity
from this. I can of course relax  the 1/r^2 pad desity requirement but
then I lose the zoom capability,



2) Now we can consider the problem of whether these uv point at the edge of the
uv coverage which are sub-Nyquist sampled really are useless or not. First I try
to characterise the nature of the problem then give my views.

Consider a thought experiment where I sample at greater than Nyquist
within a boundary in the uv plane. I am now given a few extra points to place beyond
this boundary. Clearly  their sampling will be highly sub-Nyquist.
If I am an optimist I see the glass as half full - and argue that at least
these points give me SOME information about the region beyond the boundary
to constrain extrapolation and that 'some information is always better than none'.
If I am a pessimist I say the glass is half-empty. I admit that there is extra
information in these points but I argue that there is no way of making use of this
information. If  I do try to make use of these points the disaster results, with huge
increases in noise because I am trying to interpolate between samples which are too
far apart.

To decide which of the two points is correct I can first
look at linear methods of interpolation (and I define a linear
operation F rigoursly such that F(u1 + u2) = F(u1) + F(u2), while for non-linear operations I have an
inequality,  where u1 and u2 are subsets  of the uv  data). One can look at an infinite sequence
of interleaved points with mean seperations Nyquist but with half the seperations larger
than Nyquist and half smaller and show that the noise increases significantly when I
have gaps which are much larger than Nyquist. Frederic has shown something
similar but more general  in one of his  papers. I am not sure how
applicable these ideas are to the problem at hand however, where
we have rapid decrease in density to the uv edge rather than an infinite sequence and where
although  we will get larger errors towards the uv edge they will have less impact because
of the restoration with a CLEAN beam.

To try to see whether highly sub-Nyquist samples caused a serious problem or not
for linear deconvolution schemes I once simulated doing a pseudo-inverse restoration of a 1D
uv coverage where the samples at the edge were highly sub-Nyquist, see

http://www.oso.chalmers.se/~jconway/ALMA/SIMULATIONS/SIM12/

I got good reconstructions
and no large amplifications of the noise. Of course in a sense a pseudo-inverse method is
a sophisicated version of throwing away data, but in this case these points at the edge of
the uv coverage make some contributions to the eigenvectors which are eventually included in
the final image, wheras if one just deletes points beyond a given radius they have no impact
at all.


In practice we will almost certainly continue to use non-linear deconolution methods
to fit our visibility data, and we can ask whether these can make use of the sub-Nyquist
sampled points.  Frederik in his paper characterises the impact of such algorithms as
improving on Nyquist by some fixed factor, so they would allow the use of 2x Nyquist
spacing points instead of Nyquist. I am sure that he would be first to admit (and I think its
said in his papers) that this is just a crude approximation. In particular
performance for non-linear
methods will depend on the source structure. If on the longest baselines all the extended emission
is resolved out except for one bright unresolved point the low denisty uv samples certainly
are useful for constraining the position and flux density of this point [I anticipate
an objection here from Frederic that I am yet again mixing up imaging and model-fitting, I will
explain my position more fully  on this issue in a subsequent email, briefly I consider imaging
just as model fitting using a model  representation which is a bed of delta functions and the
model  parameters to be adjusted the fluxes of each delta function]. In the above case on the long
baselines the size of the source which is influencing the visibilities is much smaller than the
primary beam. The effective size of the source theta is much less than the primary beam so the
'bandwidth' of the uv data is smaller than the primary beam, and the Nyquist sampling interval
in the uv plane correspondingly larger.Hence low density uv points can become in effect more than
Nyquist sampled and I can readily use  their information. Non-linear algorithms like CLEAN
(and more advanced versions like Multi-resolution or wavelet CLEAN) I believe can identify regions
of signal above the noise and then adjust the value of the pixles  within
these regions to fit the data. So  even if there was not one but several
disconnected compact regions contributing to the uv data on the longest baseline outlier uv points
even a handful of such outlier uv points would be  sifficient to constrain
the structures of the compact regions.

In conclusion I believe (A) Robust deconvolution schemes exist and can be built for which
can deal with data with sub-Nyquist sampled uv points, such that they don't in any way casue disaster
 and high noise in the final  image (this is clearly true since CLEAN and MEM daily work with VLA
data with sub-Nyquist  sampling and get final map noises very close to the noise in Jy/beam
for  the dirty map). (B) Algorithms exist  where in all cases the
outlier points contribute -some- useful information to the final image without dramtically
increasing the noise. (C) the degree  to which outlier points are useful
for imaging is very source structure dependant. In the case that the longest baselines only sees
structure above the noise from a small faction of the primary beam they can very useful. Non-linear
algorithms can be effective in seperating signal from noise in the image domain and concentrating
on adjusting the parameters of the pixel
delta functions to the uv data.

I am sure there will be a lot of disagreement  with what is written above!
However if we are to have any further discussions on item 2 please let us also keep in mind
my item 1 so we keep a sense of perspective about what we are taking about in practice!

    John