ALMA CSV: Holography Discussion
Table of Contents:
Two-Element Sensitivity Using Continuum Sources
Email communication from Mark Holdaway regarding interferometric holography sensitivity.
Mark Holdaway: 01 Dec 2004 (DRAFT)
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.
Discussion:
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.
Continuum Versus Spectral Line Sources
Email discussion where Mark Holdaway spells-out the sensitivity argument which favours continuum sources over spectral line (SiO maser) sources.
> >>> > 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.
>
> the SiO masers (mostly from the envelopes of stars) will be better.
> couple of hundred Jy, if memory serves. very compact (for the purposes
> of doing two-element interferometry, where you want the dishes close
> together). variable (factor of 2 or so), but so what, since you're just
> doing holography.
Hey! The calculation:
Quoting from some great past unknown E-mail god
(probably Rick Perley:)
>
> For a line source, it is
>
> Smean x sqrt(Linewidth).
>
> where Smean is the mean flux density in the line over the
> Linewidth.
>
> Wright et al. (AJ, Vol 99, p1299, 1990) give calibrated profiles
> of SiO transitions from numerous sources. The beefiest of these is
> W Hya, although Orion-IRc2 is close. For both, the mean intensity
> is about 500 Jy, and the equivalent linewidth is 10 km/sec = 2.9 MHz.
> Thus, in 'astronomers' units', the equivalent SNR in a line
> is 750 Jy.sqrt(MHz).
>
3C273: 15 Jy * sqrt(8000MHz) = 1340 Jy.sqrt(MHz)
ie, 3C273 WINS
> >>> > 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).
>
> i made a similar simulation package in IDL, which i am happy to send to
> anybody if they want it. it is described in VLBA test memos 57 (the
> theory) and 62 (describing the simulations). i also implemented it in
> good old FORTRAN, which is significantly faster and doesn't need an IDL
> license, but doesn't give you a nice graphical display... it allows for
> investigations of sensitivity to raster size, oversampling factor, SNR,
> phase rms, amplitude rms (gain fluctuations), pointing errors (both
> fixed offset and rms for both the fixed and rastering antennas), and
> type of transform...
Sounds like I reinvented your wheel.
> >>> > 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.
>
> you don't have to hand wave (and i'm sure darrel can calculate this
> properly, he's an expert in these things...). the errors look like:
> e_{max} ~ l N / (pi SNR)
> e_{rms} ~ l N / (5 pi SNR)
> for wavelength l, and raster size N. again, see the above two VLBA
> memos for the derivation, theoretically, and the simulations...
>
> -bryan
>
Response from Bryan
hmmm, i thought i had once calculated that the line sources win. i must
have miscalculated or used different numbers. planets don't help much -
uranus is only around 8 Jy. mars might be OK if the baseline is short
enough and the geometry is right (so that it's bigger than a few asec,
but not at opposition [20 asec-ish]). if you thought you could do
holography at 1mm, then uranus would start to be a good option, though
it starts to get resolved unless the baseline is short...
-bryan
Email from Mark Holdaway regarding sensitivity and resolution.
> e_{max} ~ l N / (pi SNR)
> e_{rms} ~ l N / (5 pi SNR)
> for wavelength l, and raster size N. again, see the above two VLBA
> memos for the derivation, theoretically, and the simulations...
>
>
Bryan,
I think for the VLBA case, with the restricted bandwidth, SiO masers must
come out ahead, but for the ALMA case with super-duper 8 GHz bandwidth,
3C273 comes out ahead.
BTW: I went around the block a few times with Darrel, and we agree that
the RESOLUTION will be about TWO PIXELS -- I quoted the pixel size in my
earlier e-mail, but not the resolution. SO, we can get plenty good
SNR on the surface (5.8 microns at the edge) with 32x32 pointings, just
Nyquist sampling, and 0.4 m pixels, or 0.8 m resolution, which means
that surface setting is not really possible. THOUGH, as Jeff Mangum
points out, that is not their goal. This should be fine for verifying
that the surface is OK.
BTW: Bryan's above expression, e_{rms} ~ l N / (5 pi SNR), for
the 32x32 case, SNR = 1300, l = 3333 microns (90 GHz), yields
e_{rms} = 5.22 microns.
In my simulation for Nyquist-sampled 32x32 holography, I get 5.8 microns
at the EDGE of the dish, as low as 2.3 microns near the center. OK, I'm
willing to call it a day on this topic.
Note that in Bryan's expression, there is a hidden N dependence in the
SNR; assuming the setup time is neglible (which will not really be the
case) and that we have the same total integration time, the SNR ~ 1/N,
and then the error will be proportional to N^2.
Take care,
-Mark
Delay Tracking Required for Interferometric Holography
Email from Darrel Emerson regarding the level of delay tracking necessary for interferometric holography.
Just to spell out something which may or may not be obvious:
There's a huge difference between full interferometric OTF mapping, and
astronomical OTF interferometric holography. In the holography case,
you want to keep the phase and delay center tracking the point source -
you're not interested in the sky around the point source, but you want
to know the complex response of the antenna in different off-boresight
directions. From the phase & delay tracking point of view, OTF
holography is the same as just making a regular interferometric observation
on any single field point source.
For the antenna OTF motion that happens at the same time, it's just
the same as has been done for single dish OTF mapping for years.
Yes, you need to take into account the boxcar smoothing that happens
as you slew the antenna in the OTF motion, but that's more or less a
question of making sure the data acquisition is sufficiently fast that
the boxcar smoothing on the sky is sufficiently small, and then offline using
a proper gridding function (sinc, spheroidal or whatever) of this highly
oversampled data onto the appropriate grid before doing the FFT. If
we're careful we can even apply a correction for the boxcar smoothing
at this point. I don't think true interferometric holography this way
is any different from single dish OTF holography that's been done for
years. It should be fairly straightforward.
However, if you're making a true large-scale astronomical map, the phase & delay
tracking center has to move. It doesn't work if you just literally make the
phase center following the instantaneous single dish pointing position,
because for reasonable OTF mapping speeds you end up with, effectively,
huge artificial fringe rates that are faster than the ALMA hardware can
cope with. It doesn't work if you just leave the phase center at the
center of the map, because you end up with large offset fringe rates at
the edge of the mosaiced field. A possible solution is to track the fringe
frequency associated with each instantaneous pointing, which then only changes
relatively slowly, rather than tracking the actual phase center. There was some
discussion about this a couple of years ago involving Barry Clark, Larry D'Addario,
Robert Lucas Brian Glendenning and myself, and I believe Robert Lucas, working
with the ALMA computing group, volunteered to take on the task of
cracking this particular algorithm in a way that's consistent with
the hardware. However, this is a messy thing that I think may turn
out to be difficult to get right on the first few iterations, and all
manner of subtle difficulties are likely to turn up.
That's why I think that true OTF large scale astronomical interferometric
mapping is a horrendous problem to solve, and we shouldn't expect it to be
working until fairly late on. The simpler case however, which is
what we'll want for OTF interferometric holography, is relatively
trivial.
Just a couple of cents.
Cheers,
Darrel.
--
JeffMangum - 13 Dec 2004
Concerning The Need for OTF Holography and Full ALMA Holographic Sensitivity
Formerly the Appendix in Mark's email
OTF Holography: why do we need it?
32 x 32 holography gives 0.8m resolution, not fine enough
for panel setting. If we try to do it in 1 hour, that is
3.5 s per pointing, including setup time. OK, we could
actually do this one in POINT & SHOOT mode, taking about 1.5 s
to move, and spending 2 s to integrate.
If we want to set panels, we need 64x64 or 128x128:
64x64 holography gives 0.4m resolution (marginal for panel settings?).
That would be 0.9 s per point -- we clearly need OTF, or
we must relax to 2.7 hours for a point and shoot observation
of the same sensitivity.
And what IS that sensitivity?
1 baseline gave about 22 microns at the edge of the dish
(observing 3C273 at 90 GHz).
If we had 10 antennas, that would be more like 7 microns at the
dish edge. SO, if we want to do panel settings from astronomical
holography at the AOS, we will have to wait until we have
10 antennas.
128x128 holography gives 0.2m resolution (excellent for panel settings).
This is only 0.22s per point, we ABSOLUTELY NEED OTF.
And sensitivity? Things are 4 times worse than 64x64 --
so we need 16 times as many antennas -- or 160! to reach the
same sensitivity level.
SO: if we want to do 128x128 astronomical holography, we PROBABLY
will wait until we have all 64(?) antennas, and we will
either integrate LONGER (ie, 6 times longer to reach the same
sensitivity level) or LIVE WITH poor sensitivity at the dish edge.
--
MarkHoldaway - 13 Dec 2004