[mmaimcal]Re: [alma-config]Re: comparison of the boone's and kogan's compact configurations

Tue Jan 29 15:15:39 EST 2002

Folks,

I think this is a statistical equation.
It will not optimize for any specific case, but it
needs to deal with the ensemble of all possible sources
sprinkled across the primary beam.

	-M

> 
> 
> i should realize that smarter folks have always dealt with things
> before i get to them :)...
> 
> but, seriously, i'm not sure you've got eqns 7 & 8 right.  they are simply
> stated, not proven.
> 
> let me develop a counter-example which i think shows that these equations
> aren't quite right.  consider a point source at the half-power point of
> the PB.  now, consider the sidelobe response to this point source at the
> PB center.  your equation would have that sidelobe reponse downweighted
> by ~0.71 (the value of a PB with sqrt{2} wider FWHM).  but, in fact,
> the sidelobe response here *shouldn't be downweighted at all* by the
> PB response - it's at the center of the PB.  you could argue that it
> should even be 'upweighted' in a strict SNR sense - the signal from
> the point source at the HWHM is only half what it should be, while
> the 'confusing' source at the pointing center is at full value (multiplied
> by the sidelobe level).
> 
> so, i think heuristically you got the argument right - that you take
> the upper envelope of the family of PSF's, and that this then is equivalent
> to taking a modified primary beam - but i am not convinced that you got
> the mathematics right.
> 
> my crude treatment of the problem (which i described in the earlier email)
> takes the modified PB as:
>    PB'[s] = PB[max(0.0,s-FWHM)]
> which is similar to yours, but a bit different (it's flat-topped at 1.0
> out to roughly the FWHM [in radius])...
> 
> this is relatively easy to see and argue about in 1D, but gets
> more complicated in 2D, and, i think most of the argument goes away
> when you start talking about deconvolution.
> 
> 
> 	-bryan
> 
> 
> 
> On 2002.01.29 11:31 David Woody wrote:
> > Bryan
> > I attempted to develop a rigorous solution to this problem
> > in ALMA memo 389 section II.B equ. 7 and 8.
> > Cheers
> > David
> > ----- Original Message -----
> > From: "Bryan Butler" <bbutler at nrao.edu>
> > To: "Leonia Kogan" <lkogan at zia.aoc.NRAO.EDU>
> > Cc: <dwoody at ovro.caltech.edu>; <alma-config at nrao.edu>;
> > <dbock at astron.berkeley.edu>; <mmaimcal at zia.aoc.NRAO.EDU>
> > Sent: Tuesday, January 29, 2002 9:22 AM
> > Subject: Re: [alma-config]Re: comparison of the boone's and kogan's compact
> > configurations
> > 
> > 
> > >
> > >
> > > all,
> > >
> > > leonia and i have been discussing this at some length just now.
> > > i had gone off myself and essentially reproduced all of this
> > > independently, and have now been thinking about it quite a bit
> > > (and leonia has clarified some things for me).
> > >
> > > it seems clear that simply multiplying the synthesized beam by
> > > the primary beam (which is what dave & frederic did) is too
> > > simple.  the problem is that the synthesized beam is convolved
> > > with the sky brightness, and hence 'moves around' (effectively)
> > > in the PB.  so you have to consider multiplying the synthesized
> > > beam by shifted PBs.  the question is how much is the maximum
> > > shift?  a related question is what is the size of the optimization
> > > region on the sky (for leonia's beam optimizations)?
> > >
> > > it also seems clear to me that considering the synthesized beam with
> > > *no* multiplication (what leonia has been doing) is too simple.
> > > this should be considered as the absolute worst case (i.e., a
> > > pathological case where sources on opposites sides of the PB both
> > > happen to fall at locations of peak sidelobes in relation to each
> > > other).  if we were never to deconvolve (always just take the dirty
> > > images), then this would be closer to the right thing to do, but
> > > the fact of the matter is that we will *almost always* deconvolve.
> > >
> > > the exact treatment is not clear to me.  maybe a reasonable
> > > compromise might be to calculate the 'worst case shifted peak
> > > sidelobe' - where, instead of downweighting strictly by the PB,
> > > you take a shifted version of the PB instead (so downweight
> > > by less).  the question, again, is 'shifted by *how much*'?  not
> > > clear to me.  i *would* say that making the shift be as much as
> > > the full width between the nulls in the PB seems too much to me
> > > (this is equivalent to having leonia's optimization go over twice
> > > the width between the nulls).  perhaps make the shift be equal to
> > > the FWHM.  i will have a go at this...
> > >
> > >
> > > -bryan
> > >
> > >
> > > On 2002.01.29 09:07 Leonia Kogan wrote:
> > > > Hi David,
> > > >
> > > > I carried out my calculations at the normalized coordinate:
> > > >
> > > > Radius = number of the lambda/D = number of resolutions;  D = arrray
> > diameter
> > > >
> > > > The two side primary beam for the flat illumination is
> > > > 2.4*lambda/d; d is diameter of the dish
> > > >
> > > > Introduce alpha is a portion of the two side primary beam.
> > > >
> > > > Then Radius = Alpha*2.4*D/d;
> > > >
> > > > Subtitude D= 200; d=12 and get
> > > >
> > > > Radius = alpha*40;
> > > >
> > > > My calaculations  indicate sidelobes of 9.3% at Radius=20 (alpha=0.5).
> > > >
> > > > Of cource the multiplication of the PSF by the primary beam will reduce
> > > > the sidelobes. And this explain our difference. But:
> > > >
> > > > 1. The actual 12 meter dish will have wider primary beam because
> > > >    the illumination will not be flat.
> > > >
> > > > 2. But I do not think that the multiplication of the PSF by the primary
> > beam
> > > >    is the right operation.
> > > >    You wrote:
> > > >
> > > > >You are correct that for
> > > > >mosaicing you need to consider a wider area and Mark said that
> > > > >twice the primary beam should be enough.
> > > >
> > > > I think even for not mosaic observation twice the primary beam should be
> > > > required. Effect of the primary peam on the sidelobes of PSF is not
> > simple
> > > > multiplication ( I think so). The primary beam multiply the sky.
> > > >
> > > > Consider the PSF pointed at the edge of PM. Then the sky at this point
> > > > will be reduced by the PB. But the sidelobes at the oposite edge of the
> > PB
> > > > (remoted by twice distance) will be reduced by the same factor.
> > > > So the ratio of the sinal and the sidelobe at the oposite edge of the PB
> > > > will be determined by the PSF exclusively.
> > > >
> > > > Thanks
> > > >
> > > > Leonia
> > > >
> > > >
> > > > ----- Begin Included Message -----
> > > >
> > > > >From dwoody at ovro.caltech.edu Mon Jan 28 20:21 MST 2002
> > > > Reply-To: "David Woody" <dwoody at ovro.caltech.edu>
> > > > From: "David Woody" <dwoody at ovro.caltech.edu>
> > > > To: "Leonia Kogan" <lkogan at zia.aoc.NRAO.EDU>
> > > > Cc: "douglas bock" <dbock at astron.berkeley.edu>,
> > > >         "Al Wootten" <awootten at cv3.cv.nrao.edu>
> > > > References: <200201271622.JAA00901 at bonito.aoc.nrao.edu>
> > > > Subject: Re: comparison of the boone's and kogan's compact
> > configurations
> > > > Date: Mon, 28 Jan 2002 19:21:58 -0800
> > > > Organization: Caltech
> > > > MIME-Version: 1.0
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> > > >
> > > > Hi Leonia
> > > > My calculations always include the primary beam, i.e. the FT of the
> > > > UV samples multiplied by the primary beam, and hence I do not
> > > > see the far out sidelobes beyond ~15.  You are correct that for
> > > > mosaicing you need to consider a wider area and Mark said that
> > > > twice the primary beam should be enough.  ALMA memo 389
> > > > argues that for a Gaussian primary beam, the correct beam to
> > > > multiple the FT of the UV samples by is only a sqrt(2) larger
> > > > Gaussian beam.  I will run this case soon and send you the
> > > > results for both configurations.  I will also scale Boone's
> > configuration
> > > > down to give the same magnification as your configuration (including
> > > > margin for close packing) so that we are comparing the same capability.
> > > > My guess at this point is that the differences will not be important,
> > > > i.e. very small.
> > > > Cheers
> > > > David
> > > > ----- Original Message -----
> > > > From: "Leonia Kogan" <lkogan at aoc.nrao.edu>
> > > > To: <dwoody at ovro.caltech.edu>; <frederick.boone at obspm.fa>
> > > > Cc: <alma-config at cv3.cv.nrao.edu>
> > > > Sent: Sunday, January 27, 2002 8:22 AM
> > > > Subject: comparison of the boone's and kogan's compact configurations
> > > >
> > > >
> > > >
> > > >
> > > >
> > > > ----- End Included Message -----
> > > >
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> > 
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