[alma-config] Ultracompact array

[Mark Holdaway]

<< Snezana: to bring you up to speed, this e-mail discussion is in
   an ALMA working group charged with designing the ALMA array
   configurations.  We've been talking about both (u,v) metrics
   to judge superior configurations by, and also image plane
   metrics such as the peak sidelobe level and such.  Ed just
   steered us in your direction. >>


Ed Fomalont wrote:

> Hi Mark and others,
> 
>      I have a few questions concerning mosaicing and the inner u-v
> hole on u-v coverage.
> 
>      Since mosaicing may be used alot for the smaller configurations,
> perhaps the u-v coverage should be optimized for more robust mosaicing
> at the expense of slightly poorer beam characteristics.
> 
>      I believe that true mosaicing depends mostly on fringe spacings
> which are between 20% to 50% of the FWHP primary beam size.  It is
> with these spacings that the primary beam shape interacts with the
> more extended objects, so that short spacing information can be
> obtained.  If this is correct, would an array with an excess (probably
> slight) of shorter spacings be more mosaic-friendly?
> 
>      For the u-v and PSF simiulations, what is the status of the u-v
> hole in the middle of the array?  Can it be assumed that somehow the
> visibility at these short spacings will be measured (by total-power
> phased-array imaging around the source or by mosaicing or ???) - and
> thus should be filled in at some level for the simulations?  Or, should
> the hole remain for the simulations?  The reason I am asking is that
> the distortions caused by this u-v hole may actually dominate certain
> types of sidelobe behavior for most reasonable arrays, particularly the
> more compact arrays.
> 

This is a good question.  For example, in the clasical MEM-based
mosaicing, the dirty beams for each pointing are used explicitly to
calculate a "step image" (ie, the negative gradient of chi squared) by
"mosaicing" together the residual images (calculated by multiplying the
model by the primary beam appropriately positioned, convolving that
"single pointing model" by that pointing's PSF, and then subtracting that
off of the dirty image for that pointing).  Total power data could either
be insert at the (0,0) point for each pointing's (u,v) plane if the
pointing centers of the total power and interferometer data were the same
for a homogeneous case, or the total power data could be treated in a
similar way to the interferometer data in adding into the gradient image.  
So, the plain old PSF IS used, but is mitigated in some way by the total
power.

A more explicit use of the single dish beam is found in Stanimirovic's AT
LMC HI work in which a "linear combination" (ie, "lc")  of the dirty
images and psf's is formed:

Dirty_lc = (Dirty_int + a . Dirty_sd) / (1 + a)

PSF_lc   = (PSF_int   + a . PSF_sd)   / (1 + a)

The Dirty_sd and PSF_sd images are just the single dish image and
the single dish beam.  Since the interferometric and single dish
Dirty images and beams obey a convolution relationship with the
true object image, there is also a convolution relationship
between the Dirty_lc, PSF_lc, and truth image:

Dirty_lc = Truth * PSF_lc

(Note: the above completely disregards mosaicing, but it fits nicely
into the mosaicing schemes in AIPS++ which take an approximate
PSF for the whole mosaic, deconvolves down to a certain point,
does an exact subtraction from the data, forms the new residual
mosaic image, and then continues to deconvolve with the single 
approximate PSF.  The approximate PSF could easily be adjusted
by adding the single dish PSF to it, and adding the (residual) single dish
image into the residual mosaic image.)

So, the "lc" beam may be a valid concept to work with.  Huristically, you
are adding a small plateau ("a" is typically smallish, like 0.1, but this
method has not had enough exercise for people to be too clearsighted about
what "a" should be -- see below for discussion), and also have my own) to
the interferometer PSF, and this small plateau has the effect of
anti-negatiing the negative "bowl" in the interferometer PSF.  In the
Fourier plane, just FT the two PSF's, and you are literally filling in the
hole.

>From a huristic point of view, the classical MEM-based mosaicing is
sort of doing the same thing as it builds up its gradient image.

However, we cannot really progress further quantitatvely untill
we have a handle on "a".  I suppose it makes perfect sense that
the design of the compact configuration (and its beam) should depend
upon how much total power you feel you need to add to make the
image.

What should "a" be?  Stanimirovic's argument was based on the ratio of
the beam (ie, PSF_int and PSF_sd) volumes.  I can't remember Helfer's
argument, but it may have been similar.  I argued that the method
is independent of "a", but that "a" should be set based on the ratio
of the SNR.  Helfer claimed that the method only works for a particular
value of "a".  From the current discussion, it seems that "a" could
be chosen in a way such as to optimize the "lc" beam (and perhaps
also to drive the SNR ratio for the interferometer and single dish
observations).

One aspect that we have not addressed explicitly is the "effective
Fourier plane coverage", given by convolving the delta function
coverage with the aperture illumination function.  The FT of the
PSF_int   +   a . PSF_sd  leads to a situation where the single
dish point at (0,00 HAS been convolved with the aperture illumination,
but the interferometer coverage has NOT been convolved with the
illumination, so perhaps I have missed something in this method?

Well, hows THAT for a new can or worms, Ed?


	-Mark