[alma-config]mosaicing and dividing by the PB

[Bryan Butler]

mark,

i disagree (kind of) with your argument that in mosaicing you never divide by
the PB.  i understand that formally this is never done.  instead, what you do 
is
multiply the model by the PB before inverse FTing it and then subtracting from
the measured vis (to get the 'goodness of fit' estimate - or call it the 
'entropy'
if you will).  i would argue that multiplying the model by the PB is equivalent
to dividing the image by the PB, in the sense of SNR.  this is true for each
pointing.  where you gain back is by having many such pointings, of course.

consider the following:

you have mosaiced some section of sky in which there is a source of flux
density F at position X - this is the source you are interested in recovering.
there is another (confusing) source of flux density C at position Z.  ignore
deconvolution for now - consider that we are only interested in the dirty
map (this is the case that leonia has been arguing all along).  in the i'th
pointing, the recovered flux density for your source of interest is:

           PBi(Z) PSFi(Z->X)
    F + C -------------------
                PBi(X)

consider the second term above as an 'imaging noise' term (to distinguish it
from true thermal noise - note that this is not strictly true [that it should
be considered a formal rms], but is probably statistically so).  also, assume
that the PSF in each pointing is exactly the same (again, not strictly true,
but not such a bad assumption if good mosaicing practice is followed).  then,
the effective rms on the final estimation of the flux density of your source
of interest is:

             C PSF(Z->X) sqrt{N}
    s = ------------------------------
         sqrt{sum[PBi^2(X)/PBi^2(Z)]}

this is strictly true in the linear mosaiced dirty image.

now, this is easy to write down for one source of interest, and one confusing
source, but i have a hard time imagining how to extend it to a statistical
argument about sources anywhere in the beam, and my head really starts to
hurt when you include deconvolution.  if you said that you could perfectly
deconvolve, then i think the 'C PSF(Z->X)' term goes away.  if you had perfect
knowledge of your PBs, then the PBi terms go away.  in reality we are in some
odd parameter space where neither of these is true.  it's probably something
more like:

            a C PSF(Z->X) sqrt{N}
    s = --------------------------------
         sqrt{sum[PB'i^2(X)/PB'i^2(Z)]}

where a is some factor which tells us how well we deconvolve (a << 1), and
PB'i(Y) is the _uncertainty_ in the PB at position Y.  if this is indeed
the case, then it's not clear to me how to proceed, since we don't _really_
know how well we will know the PBs, and any guess at the value of a is just
that - a guess (however informed it might be).


	-bryan