Transcript Lecture 1
Lecture 19
• MEM • Calibration • Self-calibration • VLBI • Spectroscopic interferometry – Finishing with a couple of nice movies.
NASSP Masters 5003F - Computational Astronomy - 2009
MEM – the Maximum Entropy Method.
• References: (there are many more) – Cornwell T J, chapter 7, NRAO 1985 Synthesis Imaging Summer School.
– Cornwell T J & Evans K F, Astron. Astrophys. 143, 77 (1985) • Notation: –
I j
are the pixel values of the ‘true’ sky image.
–
D j
represent the pixel values of the data (ie the dirty image).
– –
M j
– an image representing
a priori
information.
R j
are the pixel values of the solution image.
• (2-dimensional images are here represented by a single pixel index
j
just for brevity.) NASSP Masters 5003F - Computational Astronomy - 2009
MEM – the Maximum Entropy Method.
Data
D j
Solution
R j
Prior information
M j
?
?
…We need ways to measure the ‘distance’ between these various images.
NASSP Masters 5003F - Computational Astronomy - 2009
MEM – the Maximum Entropy Method.
Data
D j
Solution
R j
2 Chi squared:
j
D j
j
2
R j
2 …but this is not really valid because the adjacent pixels of the dirty image are not independent. We would need to include covariances
σ jk
.
?
Prior information
M j
NASSP Masters 5003F - Computational Astronomy - 2009
MEM – the Maximum Entropy Method.
Data
D j
Solution
R j
Chi squared: Better: 2 1 2
M
k V k
V k
2 where
V k V k
are the measured visibilities
n k
j I j
exp 2
i
u
k
and
V k
j R j
exp 2
i
u
k
x
j
x
j
?
Prior information
M j
NASSP Masters 5003F - Computational Astronomy - 2009
MEM – the Maximum Entropy Method.
Data
D j
2 Chi squared: 1 2
M
k V k
Solution
R j
V k
2 Prior information
M j
‘Entropy’:
H
j R j
ln
R j M j
(Other choices are possible.) NASSP Masters 5003F - Computational Astronomy - 2009
MEM – the Maximum Entropy Method.
Data
D j
Solution
R j
Prior information
M j
Chi squared: ‘Entropy’: 2 1 2
M
k V k
V k
2
H
j R j
ln
R j M j
(Other choices are possible.) •The actual algorithm employed in interferometry maximizes
H
subject to the constraint that χ 2 equals the expected value.
•Constrained optimization can be done via the technique of Lagrange Multipliers (too advanced for this course).
•Additional constraints (eg on the total image flux Σ
R j
) can easily be added.
NASSP Masters 5003F - Computational Astronomy - 2009
MEM – the Maximum Entropy Method.
• Choice of
M
?
– Usually flat.
•
M
can be adjusted during the procedure – as can other parameters.
• MEM performs well on extended sources – complements CLEAN which is better for compact sources.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration
• So far we have been assuming that geometry is the only source of phase differences.
• But phase (and amplitude) may also vary because of – Imperfections and/or instabilities in the electronics; – Changes in gain originating in properties of the primary beam; – Changes in refraction and path length through • the atmosphere (troposphere); Most important • the ionosphere.
for an observer.
• All these need to be calibrated out .
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration - the troposphere
• Phase delay due to troposphere varies with zenith angle α (approx as 1/cos α ).
– Although for α -> 90 °, curvature of the earth stops this going to infinity.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration - the troposphere
• Components: – ‘Dry air’ component: • Varies slowly.
• Range in variation is small.
• Directly measurable (via pressure).
– ‘Wet’ component (ie, water vapour): • Can vary quickly.
• Can vary over a large range.
• Unpredictable...
– although H 2 O radiometers can help.
• Typical addition to ‘optical path length’ is over 2 metres.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration - the ionosphere
• Variation with zenith angle is very similar to that for the troposphere.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration - the ionosphere
• Excess path length varies with frequency as
L
40
N
2 e • where
N
e is the electron column density in m Note that Δ
L
is negative -2 . because the refractive index is negative (yes it is weird).
• A typical value is about 1 m.
• Varies by a factor of 5 between day and night; • Also large variation during ‘solar storms’.
• The variation with ν means one can calculate it by simultaneous measurements at different frequencies.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration
• Methods of calibration include: – Internal monitoring of antenna parameters (eg pointing, gain), and corrections calculated from these.
– Infrequent observation (usually by observatory support staff) of calibration sources to calibrate a host of antenna parameters which vary only slowly with time.
– Once-per-observing-session observation of a ‘primary flux calibrator’.
– Frequent during-observation peeks at a ‘phase calibrator’.
• The last two are the most important for an observer to know about.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration – some maths.
• A measurement of visibility
V
~
j,k j
and
k
between antennas is related to the ‘true’ visibility
V j,k
by ~
V j
,
k
G j
,
k V j
,
k
j
,
k
j
,
k
(Note: all these are complex numbers.) • • where
G j,k η j,k
is noise.
is the complex gain,
ε j,k G j,k
can be resolved into a product complex-valued antenna gains
g j
is an offset and
g j g*
and
g k k
of , which depend purely on the antennas
j
and
k
respectively, times a so called ‘closure error’
g j,k
.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration – some maths.
• Since the correlation is a digital operation, and thus not subject to analog-type errors, most errors occur upstream of the correlation, and are therefore antenna specific.
• Hence, the closure error
g j,k
is usually close to 1+i0, and can be ignored, at least at a first pass..
• Similarly, the offset
ε j,k
ignorable.
is usually small and • Noise
η j,k
can be minimized through integration over time.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration – some maths.
• Thus we can write, to first order approximation, ~
V j
,
k
V j
,
k a j a k e i
j
k
• where
a j
and
a k
are (real-valued) amplitude errors and
φ j
-
φ k
is the (real valued) phase-difference error.
• Note that both
a
s and
φ
s vary with time an unpredictable fashion. (
φ
much more in than
a
.) • Most of this variation is due to random changes in the atmosphere + ionosphere.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration – some maths.
• Note that there are only 2
N
calculate (1
a
and 1
φ
quantities to per antenna) but we have
N
(
N
-1) ~
N
2 measurements (a real and an imaginary value from each of
N
(
N
1)/2 correlations). This is a happy situation whenever
N
>4 (as it is for all modern arrays of any importance).
• This means that we don’t need measurements of
V
~ from all baselines – which turns out to be useful for reasons shortly to be revealed.
NASSP Masters 5003F - Computational Astronomy - 2009
• • •
The ideal calibrator
Position and structure: 1. point-like; 2. not confused; 3. of exactly known position; 4. near the object of interest.
Flux: 5. strong; 6.
doesn’t vary with time; 7. smooth, flat spectrum.
Different situations impose differing relative emphasis.
NASSP Masters 5003F - Computational Astronomy - 2009
The real calibrator
• Of course (just like with people) we have to make do with calibrators which never meet all these desirables at once, and sometimes don’t meet any of them!
• Some trade-offs: – To find a calibrator near our object, we might have to accept one which is not as strong as we’d like (
n
~
S
-5/2 rule).
– A compact calibrator is usually time-variable, sometimes on timescales as short as a day.
– We may have to discard data from short baselines (because the calibrators are confused) and also from long ones (because they resolve the calibrator).
NASSP Masters 5003F - Computational Astronomy - 2009
‘Goldilocks’ baselines
What short baselines see: confused.
What long baselines see: resolved.
What just-right BL see: isolated points.
NASSP Masters 5003F - Computational Astronomy - 2009
A simulated observation –
V
(
t
) from 1 baseline:
Amplitudes shown here.
Telescopes are moving between the target (long cycle) and the calibrator (short cycle).
NASSP Masters 5003F - Computational Astronomy - 2009
A simulated observation –
V
(
t
) from 1 baseline:
Phases shown here – all cycles.
Telescopes are moving between the target (long cycle) and the calibrator (short cycle).
NASSP Masters 5003F - Computational Astronomy - 2009
A simulated observation –
V
(
t
) from 1 baseline:
Phases shown here cal cycles only.
Telescopes are moving between the target (long cycle) and the calibrator (short cycle).
NASSP Masters 5003F - Computational Astronomy - 2009
Special calibrations:
• Spectral-line: – requires band-pass calibration.
• Polarization: – ‘leakage terms’; • Most modern feeds have 2 detectors, of opposite polarisation; • but their mutual isolation is not perfect.
– variation in polarization across the beam; • A kind of polarized spherical aberration.
– ionospheric Faraday rotation.
NASSP Masters 5003F - Computational Astronomy - 2009
Self-calibration
• Ordinary calibration using an offset source relies on: – interpolation between peeks at the cal; – the assumption that the phase shift in the direction of the cal is the same as towards the target.
• Variations due to imperfection of these assumptions is usually >> thermal noise.
– Effect of this depends on size of array: • Small array: may degrade image a little.
• VLBI: completely impossible to calibrate this way!
• So - can we do better...?
NASSP Masters 5003F - Computational Astronomy - 2009
Self-calibration
• Suppose we knew
I
(
l,m
), the brightness distribution of the target.
• Fourier transform this to get a continuous model visibility function
V
^ (
u,v
).
• Divide the measured visibility samples
V
~ j,k (
u,v,t
) for baseline
j-k
• Is not the result the
V
by this model we’d get from a
V
^ .
point source, multiplied by the product of the antenna ‘gains’
g j g* k
(slide 8)?
• Then just perform normal calibration of
g j
and
g k
.
NASSP Masters 5003F - Computational Astronomy - 2009
Bootstrap
• But...
I
is what we are trying to find out!
• However – an iterative approach works well, provided: – the target is fairly bright; – a starting model can be obtained.
• A quite separate approach utilizes redundant spacings – 2 or more visibility samples which lie very close to each other in the
u-v
plane.
• Westerbork was specifically designed to facilitate this – lots of antennas are the same distance apart.
NASSP Masters 5003F - Computational Astronomy - 2009
Very Long Baseline Interferometry (VLBI)
• Maths is the same as standard interferometry – it is the practical details which tend to be different.
• Antennas are thousands rather than tens of km apart. Hence: – Resolution can be 1 milliarcsecond (mas) or even lower.
– It isn’t practical to lock all antennas to the same frequency standard (LO). Each has its own...
• thus the LOs must be very stable (eg H maser). - $$$ – Data are stored on tape and correlated later.
– With the longer baselines, the array is proportionately more sensitive to phase errors...
• so special calibration techniques are needed.
NASSP Masters 5003F - Computational Astronomy - 2009
VLBI – a typical image:
CSO 1943+546 Polatidis et al NewAR 43, 657 (1999) Courtesy the EVN archive.
NASSP Masters 5003F - Computational Astronomy - 2009
Spectroscopy
• What physical environment is associated with spectral lines?
– Isolated atoms (may be partially ionized).
• Spectral lines can be observed both in emission and absorption.
• Information coming out of spectroscopic interferometry: – composition, column densities, temperature, velocity (both ‘bulk’ or nett, and velocity dispersion).
• Processing: – each channel imaged and CLEANed separately.
NASSP Masters 5003F - Computational Astronomy - 2009
Spectroscopy
• We are now measuring not just
x
also a 3 rd and
y
axis, ν . Hence we deal not in images but in data cubes .
– A challenge to handle • Can be huge: 2048 2 x 1024 pixels = 16 GB – A challenge to visualize • Channel maps and movies • Moment maps: intensity, velocity, width – 3D rendering and visualization software… – A challenge to analyse… • Model for structure & kinematics • Optical depth effects • Excitation: collisions vs radiation but NASSP Masters 5003F - Computational Astronomy - 2009
M82 cube movie
Shows HI absorption – different velocities (therefore different Doppler-shifted frequency) at different places.
Many thanks to Rob Beswick.
Continuum plus line
M82 HI (Wills, Pedlar et al) NASSP Masters 5003F - Computational Astronomy - 2009
HI imaging
• Valuable for cosmology: – Traces galaxies large-scale structure.
– Also shows kinematics inside galaxies.
• But, it’s hard to get high resolution...
– because the brightness temperature of HI is limited to about 100 K; – bandwidth is limited; – wavelength is relatively long; – and, for an interferometer,
A
e is true for a single dish.
A
e property of the antennas).
is not
λ
2 / Ω A , as stays fixed (it’s a • Thus smaller beam (ie Ω A ) gets less flux. For HI, practical limit to Ω A (and thus resolution) is ~1 arcsec.
NASSP Masters 5003F - Computational Astronomy - 2009
Another M82 movie, showing OH emission and absorption, with continuum subtracted.
Many thanks to Rob Beswick.
M82 OH – 1665 & 1667 lines – masers & absorption VLA A-array data (Argo et al 2007) Two lines in band Absorption in black Masers in blue Note ringing about the brightest 1667MHz maser.
NASSP Masters 5003F - Computational Astronomy - 2009