[alma-config]multiplication of synthesized beam by primary beam in optimization

[Steven T. Myers]

I have read over the correspondence that Leonia gave me, and here is my 
take on it (sorry - this will be somewhat rambling): 

The standard "synthesized beam" SB = FT of (weighted) sampling function 
(see imaging lecture in book) is what we normally use as the PSF.  We
wish to use a mock version of this in our optimization of configurations,
e.g. minimize the maximum sidelobe over some area of the sky.  Note that
the sampling used is either a bunch of delta-functions (for a DFT image)
or gridded cells with weights (what we really use in FFT imaging).  I 
assume natural weighting is what is used by the optimizers (but see 
below).

Multiplication by the primary beam PB is effectively a penalty function
on the SB, downweighting the far-out sidelobes with respect to the inner
sidelobes in the optimization.  This makes some sense, in that I would 
probably rather have a 5% sidelobe further out than near the main lobe,
given the choice.  However, using the PB as the penalty function 
introduces some peculiarities --- for example, at the nulls of the PB,
the sidelobes will not count in the optimization, which does not make
alot of sense.

Note, I guess when Dave and Leonia talk about the PB, they use a Gaussian 
approximation to the primary beam (GPB), which has better behavior in this 
regard, but doesn't make much sense physically.  In this case, it is just 
a penalty function (monotonically decreasing with distance) with an inner 
core approximating the inner core of the PB.

Dave argues based on (I have to admit a rather confusing to me) a 
definition of the "image" as reradiation of a perfectly phased voltage
pattern from the feeds back through the antennas and the array, which has
the PB in it.  Personally, I find this description incomplete as it does
not deal with deconvolution (the real issue here, I think).  However,
Dave does also point out that if you use actual delta-functions in the 
uv-plane as the "visibility" sampling the sidelobes go on forever, which
is clearly silly.  A visibility has support over the cross-correlation
between the two aperture (voltage) illumination patterns, so there is
a built-in scale there on the sky (the FT of this, which I think is what
we are calling the PB).  However spreading the sampling function over this
uv kernel isn't exactly right either (in particular, mosaicing changes 
this part), but I would need some more time to look into this.

Leonia (correctly) points out that the effect of the sidelobes on 
reconstruction of emission depends on the location of the emission in the
PB, and thus does not like to use the PB for a general case.  I would say
that using the bare SB is the most "conservative" case in this regard, 
though again I worry a little about the trade of inner for outer 
sidelobes.

I think one compromise we can use if we want a penalty function is the 
auto-correlation of the PB (= PB*PB).  Dave sort of mentions this in
one of his emails.  I think this is what you get if you consider 
mosaicing, for example, dealing with the interaction between different
pointings.  It is sqrt(2) "wider" in the core than the PB 
itself, and has somewhat better behavior (I think). Maybe a Gaussian 
GPB*GPB = sqrt(2) wider than the GPB will work.  This sort of considers 
the mean square effect of sources distributed over the primary beam 
interacting with the sidelobes of each other. 

OK, here is my take on this.  We can use:

1) the "bare" synthesized beam (SB) over some field of view (e.g. N 
primary beamwidths) - this tries to dampen the far-out sidelobes at the
cost of inner sidelobes.  This is probably most appropriate for small
sources which can be all over the sidelobes of the beam.  Note that at 
low frequencies with the EVLA the large number of confusing sources far 
out in the beams would lead me to choose this as a "conservative" choice.  
Might also be ok for ALMA cases where the source is much larger than 
the PB (extreme mosaicing) with lots of bright emission further out, but
I think you would be better off using the correct mosaic primary beam 
here.

2) the SBxPB (or SBxGPB, the Gaussian version), which will downweight the
outer parts of the beam in the optimization, so you will be minimizing the
near-in sidelobe levels.  This would probably be the best choice for 
observing sources smaller than the PB (e.g. EVLA NMA & VLBA).

3) the SBx(PB*PB) or SBx(GPB*GPB), not really much wider than #2, but does
push out a bit more into the beam.  I think this is a better choice when
considering sources around the PB size or a few times larger - and I 
postulate that this approximates the mosaic synthesized beam.  I would say 
this is the normal ALMA case.

In the interest of moving this along, I would adopt #3 for ALMA, I think
in the end the different optimized configs no matter what is chosen will
be more than acceptable, and this at least tries to take the mosaicing
into account.

Note - for an E-config for the EVLA, I would probably also adopt #3, as 
you are probably most interested in using this configuration for 
mosaicing.

Hope this helps and not further confuses...

  -Steve

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|:| Steven T. Myers                      |:|  Scientist                |:|
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