No subject

[Mark Holdaway]

> The imortant question is:
>
> what is better to have very low nearest side lobes, but rather big far
sidelob$
> or
> to have medium sidelobes at the total primary beam?
>
> We need to have an answer carrying out a simulation or?
>
> Thanks
>
> Leonia
>

I would like to repeat my suggestion which I made in 1997 and 1998:

1) For the compact array, we need to consider a region which is
TWICE the entire primary beam: for example, lets say the primary beam
is 30" from center to null, and we've got a point source at 29".
That point source will scatter its sidelobes accross the whole field:   
ie, to the pointing center, and all the way to 30" to the OTHER side.
SO: the first order answer is that for this primary beam which is 60"
null to null, we must be concerned with optimizing the PSF sidelobes
on a 120" by 120" image.


2) Now, obviously a point source at 29" out is going to be modulated by
the primary beam, so we don't need to worry about it so much.  
For example, if the PB knocks its amplitude down to 0.1, then we can
actually tolerate 10 x large sidelobes due to this source than due to a
comparable source at the pointing center.  (This assumes we have perfect
knowledge of the primary beam, however; factoring in our imperfect
knowledge of the primary beam results in a tighter constraint; I'll   
see if I can do that analysis later in the e-mail).
In addition, in the mosaicing process, we make the dirty image for
this pointing (mathematically, (sky brightness x PB) * PSF) and multiply
THAT by the PB, so the sidelobes at the other end of the beam are
now down by another factor of the PB.  (The first factor of the
PB is due to the physical measurement, the second is due to the
software processing.)

<Complete answer deferred to section 4.>

3) As I have argued in the past, I believe the correct procedure which
should be followed is to NOT optimize the PSF, but to optimize the PSF
multiplied by SOME FUNCTION of the PB, over the double-sized region. In
such a case, sidelobes in the inner part of the PB would be well
optimized, and larger sidelobes further out would be acceptable; however,
rather than having a catastrophic cutoff (as in the current algorithm),
the increase in sidelobes as you go out would be gradual.

4) What is the correct FUNCTION to multiply the PSF by prior to
optimization?
   Again, assuming perfect PB knowledge:

The way I am looking at the problem right now, there is actually a
different function to multiply the PSF by in optimization for each
possible location of a point source in the observed field.  ie, if you
only have a source at the field center, you only go out to the 60" x 60"
PSF, and just multiply the PSF by the PB prior to optimization.  If you
have a source elsewhere, you go out further with a different function.

Consider a pointing centered on xp, a point source at position x1,
and a primary beam function PB where PB(0) = 1.  delta = xp - x1.
Consider further the local coordinate x which is with respect to the
PSF center.  Then the FUNCTION F1(x) to multiply the PSF by is

F1(x) = PB(delta) . PB( x - delta ).

The first factor represents the physical damping by the PB, the second
factor represents the software damping by the PB.  For the case of a
source at the field center, xp = x1 and delta = 0, the first term is
unity, the second term becomes PB(x).  If the Kogan algorithm
resulted in homogeneous peak sidelobes over the region of
optimization, application of this function prior to PSF sidelobe
minimization will result in equal sidelobe level in the FINAL MOSAIC
IMAGE.

For mosaics, you generally have sources everywhere in the field, so the 
correct F(x) is given by the average of all such specifc F1(x) for
point sources at all locations x1 accross the PB.


5) What should we really do?  Using the average of all

F1(x) = PB(delta) . PB( x - delta )  

permits VERY LARGE sidelobes far out.  I think in actuality, we WON'T
get such large sidelobes very far out (ie, if we have 0.1 peak sidelobes
in the center, and F(x) = 0.1 at some point far out, we probably AREN'T
going to get 1.0 sidelobes at that point.  We certainly WON'T get 2.0
sidelobes at a point where F(x) = 0.05!

If you have uncertainties in the PB, you would want to damp out the
PSF sidelobes (ie, you DON'T want equal peak sidelobes accross the
mosaic after PB application) more at the edges (ie, an F(x) which does
not fall off so much).  However, given the situation that having a low
value of F(X) doesn't mandate high sidelobes out there, I see no
reason to loosen this function to account for effects such as
imperfect knowledge of the PB.  We should implement and run some cases
to see ho wthey come out, as I might be incorrect.

So, I think we should try to implement the PSF sidelobe optimization with
the
PSF pre multipying function.

The ensemble average of F1(x) = PB(delta) . PB( x - delta ) over all
possible delta will be a scaled version of the autocorrelation of the PB,
so we should just use the autocorrelation of the PB.



	-Mark