[mmaimcal] draft results of holography sensitivity calculations

[Mark Holdaway]

DRAFT!

The basic question: is it worthwhile to set up a correlator at the
OSF site to do two element holography on astronomical sources?


I am scaling from a base sensitivity for a two element interferometer
operating at 90~GHz with 8~GHz bandwidth.  I had calculated the
1-sigma noise for a 30~s integration, averaging two polarizations,
would be 3.5~mJy.  If this is in error, we need to scale the results I
present here.  Furthermore, I use my canonical ``quiescent'' 3C273
spectrum, which pegs the non-flaring 90~GHz flux of 3C273 at 15 Jy.
Planets cannot be used for interferometric holography, and 3C273 will
be among the brightest of compact sources that could be used at 90
GHz.

I further assume that we need to perform a complete holography scan in
1 hour so we can track surface changes with elevation.  This could be
somewhat relaxed, ie, we could spend two hours doing holography, and
the sensitivity should be scaled by sqrt(2).


OK, heres Table 1:

t_int		sigma		Peak		NxN
[s]		[mJy]		SNR		in 1 hour

30.0		 3.5		>4000		11 x 11  (useless?)
 3.0		11.1		1300		34 x 34
 0.3		35.0		>400		110 x 110 (could set 
panels?)


This table is simple to create.  The problem is now: what does the
peak SNR mean?  Darrel Emerson made a hand-waving argument that
translates the peak SNR in the image plane to the sensitivity to
surface errors in the aperture plane, and it is probably correct to within
a factor of 2-4, depending on how we slice it.

I've made a simple holography simulation package in AIPS++/glish
(this software package is really great for things like this, I must
say;  it is such a pity that AIPS++/glish is so underappreciated
and underutilized).  The package performs the following steps:

* We select an aperture-plane cell-size (ie, 0.20~m), a holography 
  observation size (ie, 128x128), and a taper level at the edge
  of the dish (ie, 0.25 in voltage).  A 128x128 pattern with 0.20~m
  aperture-plane cells will lead to a factor of 1.7 oversampled in the
  sky plane.  From these input parameters, we generate the
  amplitude of the aperture-plane voltage pattern.

* We can optionally simulate surface errors, but this doesn't quite
  work yet, so we assume zero surface errors and evaluate the
  success at reconstructing the surface errors by the rms deviation
  from zero in the reconstructed surface pattern later.  Surface
  errors, measured as a fraction of a wavelength, would contribute
  twice (ie, once pre-reflction, once post-reflection) toward the
  phase of the aperture-plane voltage pattern.
 

* We Fourier transform the complex aperture-plane voltage pattern to 
  obtain the complex sky-plane voltage pattern.  This is a simulation of 
  what we would obtain if one antenna tracked 3C273 and the other antenna
  performed an NxN raster scan about 3C273.  In the sky-plane, we
  can verify the oversampling.

* The complex sky-plane voltage pattern is normalized wrong for our
  purposes, so we scale the peak to the brightness of 3C273 (15 Jy).
  We also add independent complex thermal noise at each
  pixel.  For a 128x128 raster, we added 0.05 Jy (this obviously
  doesn't account for any move time between observations).  
  For a 64x64 raster, we can spend 4 times as much time integrating
  at each point, so we added 0.025 Jy to each pixel.

* We then perform another complex-to-complex Fourier transform back 
  into the aperture-plane to obtain an estimate of the phase errors
  across the aperture.  We convert these phase distribution into
  a surface error estimate by scaling by wave_length /(4 pi) (the
  extra factor of 2 being again due to the coming-and-going nature
  of phase errors due to surface errors.

* Basically, we just transformed thermal noise distributed over the
  sky-plane holography observation into errors in our surface
  determination.  As we started with zero surface errors, any
  ``surface errors'' we think we see are actually due to thermal noise.
  We evaluate our ability to measure surface errors by taking the
  RMS in 1~m wide aunnuli on the dish.


Here are the results:

For a 128x128 holography observation, oversampled, with 0.20 m pixels
in the aperture plane, and 0.05~Jy noise per sky-plane pixel:

Radius Range	RMS Error in Surface
[m]			[micron]
0-1			7.9
1-2			8.9
2-3			9.8
3-4			12.7
4-5			16.3
5-6			22.0

We get essentially the same results from a ``just about'' Nyquist-sampled
64x64 holography observation with 0.2m pixels and 0.025 Jy noise.
I posit that the noise limitation to surface error detection in the 
aperture plane for a given amount of total integration time is
a function only of the aperture-plane cell-size, and not of the number of
points observed in the holography raster.

A cell-size of 0.2 m is sort of the largest cell-size which would
permit us to make panel adjustments, but we don't have the sensitivity
at the outer edge of the dish to detect the expected 25 micron surface
errors.  If sensitivity were not an issue, we would probably prefer
0.1 m cell sizes so we could get the slope and curvature
of the panel settings right and do a really nice job of it.

For a 64x64 holography observation, oversampled, with 0.40 m pixels
in the aperture plane, and 0.025~Jy noise per sky-plane pixel:

Radius Range	RMS Error in Surface
[m]			[micron]
0-1			2.4
1-2			2.3
2-3			2.2
3-4			3.1
4-5			4.1
5-6			5.8

Now, this is the sort of accuracy we WANT to set the panels, but
we don't have the resolution we need to set the pannels.


Basically, our accuracy in the surface measurement will be
proportional to 1/cell**2, where cell is the aperture plane
cell size.  Making the cell a bit smaller will make the
error in the surface determination a lot larger.  So, it is
anticipated that with a two element interferometer doing 
holography on 3C273, we will hit a hard wall at around 0.3
m cell sizes, and it will be very hard to get the desired
accuracy the with smaller cell-sizes that are required for
accurate panel settings.  On the other hand, if we relax to 0.4 m
cell sizes, which are too large to set the panels, we will
be able to do a basic verification of the surface accuracy
of a dish using two element interferometric holography.




Summary:

Using two-element interferometric holography and the brightest compact
celestial radio sources available, we will have enough sensitivity
to accurately set the pannels near the center of the dish, and
not at the edge of the dish.  Alternatively, using a larger cell
size (0.4 m) which won't permit panel setting, we can very accurately
confirm the surface accuracy of the dishes.