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

[David Woody]

To add more to the discussion.  In ALMA memo 389 I came up with the
same PB down weighting by considering a measurement vector description.
That is each UV sample is an element in a vector.  Then the PSF gotten from
the down weighting by the PB is the same as the dot product of two
measurement
vectors as a function of where they are in the sky.  This can handle the PSF
centered
anywhere, not just at the center of the PB.
***********************************
This version of the PSF is a measure of how well the instrument can
distinguish between two unit strength sources in the sky.
***********************************
It is independent of imaging algorithms and
is not intended to reflect any particular imaging process
and particularly not the CLEAN or Mosaicing algorithms.
It also is not used directly in reconstructing an image from
the measured visibilities.

You have to truncate the SB somewhere to limit how far out you are going to
deal
with the sidelobes and image.  An arbitrary brick wall truncation is not
very physical
and then we would have to argue about where to put the brick wall since the
largest
sidelobes will be just before the truncation.  The above
measurement vector approach, or equivalently the reradiated beam, accounting
for centering the PSF anywhere produces something like the PB*PB
(*=convolved).
described by Steve.

Cheers
David

----- Original Message -----
From: "Steven T. Myers" <smyers at aoc.nrao.edu>
To: <alma-config at cv3.cv.nrao.edu>
Cc: "Leonia Kogan" <lkogan at zia.aoc.NRAO.EDU>
Sent: Monday, July 08, 2002 2:49 PM
Subject: Re: [alma-config]multiplication of synthesized beam by primary beam
in optimization


>
> 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
>
> -------------------------------------------------------------------------
> |:| Steven T. Myers                      |:|  Scientist                |:|
> |:| National Radio Astronomy Observatory |:|                           |:|
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