[mmaimcal]Re: [alma-config]Re: comparison of the boone's and kogan's compact configurations
Bryan Butler
bbutler at cv3.cv.nrao.edu
Tue Jan 29 15:19:22 EST 2002
if it's statistical in nature then i think there is no reason
to optimize over 2*(width between nulls), since the argument
for going that far out is based on specific examples. going
to 1*(width between nulls) seems plenty far to me, and i suspect
satisfies the statistical argument.
-bryan
On 2002.01.29 13:15 Mark Holdaway wrote:
>
> 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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> > > > > X-Priority: 3
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> > > > > X-Mailer: Microsoft Outlook Express 5.50.4522.1200
> > > > > X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4522.1200
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> > > > > Status: RO
> > > > >
> > > > > 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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