1. Advanced Calibration Techniques
Mark Claussen
NRAO/Socorro
(based on previous self-calibration
lectures)
Ok, in this talk I will speak about advanced calibration techniques. The major part of this talk will be on the technique called self-calibration. This technique can be used to reduce the noise in an image to the thermal level.
Calibrating a synthesis array is one of the most difficult parts of its operation. Small quasi-random errors in the amplitude and phase calibration of the visibility data scatter power all around the image and so produce an increased level of ``rumble'' in the weaker regions of the image, and other systematic errors can lead to a variety of artifacts in the image.

2. Self-calibration of a VLA snapshot
Final image
Initial image
Original Image
An example of using self-calibration is this snapshot image of a radio galaxy (NGC 5322) which has a bright core, jet and lobe emission. We will revisit this image and the steps in self-calibration later in the talk. The image on the left was made after doing the “normal” steps of flagging, calibration, and imaging. The image on the right is the same image after self calibration.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

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Calibration equation
Fundamental calibration equation
The ordinary calibration procedure which we have heard about from George Moellenbrock yesterday relies on frequent observations of radio source of known structure, strength, and position in order to determine empirical corrections for time-variable instrumental and environmental factors that cannot be measured or monitored directly.
A simplified version of the calibration equation which relates the true visibility and the observed visibility at time t on baseline i-j is given here. We have neglected non-factorable parts of the gain (i.e. Baseline-based gain and noise offset terms). Epsilon (ij) is a pure, zero-mean noise term representing the thermal noise.
For simplicity we neglect the effects of time averaging and finite bandwidth --- these have relatively little impact here. The element or antenna gain describes the properties of each of the antennas relative to some reference. The gain for any one antenna has two components: first, a slowly varying instrumental part, and second, a more rapidly varying part due to the troposphere or the ionosphere. Variations in the phase part of the atmospheric component nearly always dominate the overall variation of the antenna gains.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

4. Standard calibration using a point source calibrator
Calibration equation becomes
These are calibrator visibilities; given a point source can solve for the complex gains
Works well - lots of redundancy
N-1 baselines contribute to gain estimate for any given antenna
Given a calibration source near the region to be imaged, one can solve for the antenna gains as a
function of time. Interpolations of the solutions then provides approximate vales for use in
correcting the source visibility data. If the equations are over-determined then a least-squares
technique can be utilized to good effect in overcoming the random errors embodied in epsilon(ij, t).
So for a point source the calibration equation becomes what we see here, where S is the brightness
or strength of the point source.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

5. Why is a priori calibration insufficient?
Initial calibration based on calibrator observed before/after target
Gains were derived at a different time
Troposphere and ionosphere are variable
Electronics may be variable
Gains were derived for a different direction
Troposphere and ionosphere are not uniform
Observation might have been scheduled poorly for the existing conditions
Calibrator may have structure and may not be as strong as expected
The main drawback to ordinary calibration arises from temporal and spatial variations
in the atmosphere (tropo and iono) through which the wavefront passes before reaching
the antennas. We observe a calibrator before and/or after the target so the time and
direction are not exactly the same for the target and calibrator. Values for the gain of a
given antenna inferred from observations of a calibration source may not apply to a
source observed at a different time and in a different part of the sky. Hence the effects
of the antenna gains cannot be removed completely, and residual errors remain.
The level of error varies a lot with the frequency at which the observations are made
and with the lengths of the baselines involved, but on a source of appreciable strength
this atmospheric error nearly always overwhelms the error due to the receiver noise
term.
Other obstacles to ordinary calibration are the strength of the calibrators, and any resolved structure they may contain. In some circumstances one may not be able to find a sufficiently strong unresolved calibration source anywhere near the source of interest.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

6. What is the troposphere doing?
Neutral atmosphere contains water vapor
Index of refraction differs from “dry” air
Variety of moving spatial structures
As an example of an experiment in which self-calibration can be effective, consider the
effects of the neutral atmosphere. The troposphere contains cells of enhanced water
vapor, and so the index of refraction for these cells is different from “dry” air. This
causes trophospheric delay differences over two antennas for a given baseline. Thus,
since self-calibration uses antenna-based solutions, this kind of phase difference (error)
can be effectively reduced by self-calibration.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

7. Movie of point source at 22GHz
Here is a sequence of images, taken with the VLA at 22 GHz of a source. The images are
made every ten seconds, over a time frame of about 30 minutes. One can see that because
of phase errors the source moves about, the structure changes, etc.
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Self-calibration
Use target visibilities and allow the antenna gains to be free parameters.
If all baselines correlated, there are N complex gain errors corrupting the N*(N – 1) / 2 complex visibility measurements for a given time.
Therefore there are N * (N – 1) / 2 - N complex numbers that can be used to constrain the true sky brightness distribution.
Even after adding the degrees of freedom from the antenna gains, the estimation of an adequate model of the target brightness is still overdetermined.
Even after adding the degrees of freedom introduced by the antenna gains, the estimation of an adequate model is still overdeterimined. Self-calibration is really just another method (like CLEAN) which is used to interpret the target visibility data by introducing some plausible assumptions about the source structure.
The aim is to produce a model of the sky brightness distribution, the Fourier transform of which, when corrected by some complex gain factors, reproduces the observed visibilities to within the noise level.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

9. Self-Calibration using a model of a complex source
So begin by using a model of the source visibilities:
One can form a sum of squares of residuals between the observed visibilities and the product of gains and model visibilities and do some kind of minimization by adjusting the gains.
So one begins with a model of the source visibilities, and solves for the complex
gains, in the “standard” fashion of “normal” calibration.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

10. Relationship to point source calibration
We can relate this to using a point source for calibration
Made a fake point source by dividing by model visibilities
Another way to think about this is to relate this to point-source calibration by making a
pseudo-point source by dividing by the model visibilities. Except for the modified noise
term, then this equation looks like normal point-source calibration. The overdetermined
solutions suggest a possible line of attack in which the model is iteratively refined.
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How to self-calibrate
Create an initial source model, typically from an initial image (or else a point source)
Use full resolution information from the clean components or MEM image NOT the restored image
Find antenna gains
Using “least squares” (L1 or L2) fit to visibility data
Apply gains to correct the observed data
Create a new model from the corrected data
Using for example Clean or Maximum Entropy
Go to (2),unless current model is satisfactory
shorter solution interval, different uv limits/weighting
phase  amplitude & phase
Here is a list of iterative steps to go through for self calibration. As we go through these steps,
We are refining the complex gains of the antennas as we make the model better and better.
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Closure phase
self-calibration preserves the Closure Phase which is a good observable even in the presence of antenna- based phase errors
As a bit of an aside, self-calibration preserves the closure phase that George talked about
Yesterday --- if one sums the phases around a triangle of antennas, you can see that the
Closure phase is a sum of the true phases around the triangle --- the instrumental/atmospheric
Phases cancel out. In the next slide we see a dramatic demonstration of this.
Does self-cal also preserve closure amplitude ? Yes. Hybrid imaging in
Earlier days of VLBI.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

13. SMA closure phase measurements at 682GHz
Here is a demo of the closure phase. This is a plot, from the submillimeter array
On mauna kea, of the phases on a triangle of baselines, observed at 682 GHz.
You can see that the phases for each individual baseline varies widely in the one
Hour of data. The closure phase is represented by the black dots, which is clearly
A good observeable --- flat over that hour.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

14. Advantages and disadvantages of self-calibration
Gains derived for correct time --- no interpolation
Gains derived for correct position --- no atmospheric assumptions
Solution is fairly robust if there are many baselines
More time on-source
Disadvantages
Requires a sufficiently bright source
Introduces more degrees of freedom into the imaging: results might not be robust and stable
Absolute position information lost
Advantages and disadvantages of self-calibration
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

15. When to and when not to self-calibrate
Calibration errors may be present if one or both of the following are true:
The background noise is considerably higher than expected
There are convolutional artifacts around objects, especially point sources
Don’t bother self-calibrating if these signatures are not present
Don’t confuse calibration errors with effects of poor Fourier plane sampling such as:
Low spatial frequency errors (woofly blobs) due to lack of short spacings
Multiplicative fringes (due to deconvolution errors)
Deconvolution errors around moderately resolved sources
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16. Choices in self-calibration
Initial model?
Point source often works well
Simple fit (e.g., Gaussian) for barely-resolved sources
Clean components from initial image
Don’t go too deep!
Simple model-fitting in (u,v) plane
Self-calibrate phases or amplitudes?
Usually phases first
Phase errors cause anti-symmetric structures in images
For VLA and VLBA, amplitude errors tend to be relatively unimportant at dynamic ranges < 1000 or so
Choices in self calibration --- choose the initial model – perhaps a point source,
Or a simple fit for barely resolved sources – or first few clean components of the
Image.
Self-calibrate the phases first. Be careful as phase errors cause anti-symmetric
Structures in images. Ed Fomalont will talk about error recognition later this
Afternoon. Amplitude errors tend to be relatively unimportant at dynamic ranges
Less than 1000 or so.
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More choices….
Which baselines?
For a simple source, all baselines can be used
For a complex source, with structure on various scales, start with a model that includes the most compact components, and use only the longer baselines
What solution interval should be used?
Generally speaking, use the shortest solution interval that gives “sufficient” signal/noise ratio (SNR)
If solution interval is too long, data will lose coherence
Solutions will not track the atmosphere optimally
More choices:
Limit baselines … choose solution interval ? Time scale depends upon the timescale for
Gain changes and upon the source strength… question is also one of atmospheric time
Scales as the phase errors are random … solution interval too long, lose coherence.
Time scale for gain changes should be much greater than the time taken for the noise
Per antenna to equal the source flux density.
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Sensitivity limit
Can self-calibrate if SNR on most baselines is greater than one
For a point source, the error in the gain solution is
Sensitivity limit. Can self-cal if SNR on most baselines is greater than unity. Once
The error n the gain is reduced to less than one, then the noise in the final image
Will be close to theoretical.
If error in gain is much less than 1, then the noise in the final image will be close to theoretical
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

19. You can self-calibrate on weak sources!
For the VLA at 8 GHz, the noise in 10 seconds for a single 50 MHz IF is about 13 mJy on 1 baseline
Average 4 IFs (2 RR and 2 LL) for 60 seconds to decrease this by (4 * 60/10)1/2 to 2.7 mJy
If you have a source of flux density about 5 mJy, you can get a very good self-cal solution if you set the SNR threshold to For 5 min, 1.2 mJy gives SNR = 1
For the EVLA at 8 GHz and up, the noise in 10 seconds for an 8 GHz baseband will be about 1 mJy on 1 baseline!
Weak source self-cal ! Self-cal on a 5 mJy source over 60s by averaging 4 Ifs
In the VLA. EVLA will be able to do 10 times better, so self-cal on 2 mJy source
With 10s averaging --- with 60s averaging self-cal on a 0.5 mJy source.
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20. Hard example: VLA Snapshot, 8 GHz, B Array
LINER galaxy NGC 5322
Data taken in October 1995
Poorly designed observation
One calibrator in 15 minutes
Can self-cal help?
15 minutes of data, one calibrator observation (see solution plots)
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Initial NGC 5322 Imaging
Cleaned Image
Synthesized Beam
Image after best editing, normal calibration on left. Dirty beam (psf) on right.
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First pass
Used 4 (merged) clean components in model
10-sec solutions, no averaging, SNR > 5
CALIB1: Found good solutions
CALIB1: Failed on solutions
CALIB1: 2473 solutions had insufficient data
30-sec solutions, no averaging, SNR > 5
CALIB1: Found good solutions
CALIB1: Failed on solutions
CALIB1: 125 solutions had insufficient data
30-sec solutions, average all IFs, SNR > 2
CALIB1: Found good solutions
Three different calib tries for first pass … note keeping snr > 2. Lot of good solns.
Investigating on how long to average. Pick 3.
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23. Phase Solutions from 1st Self-Cal
Reference antenna has zero phase correction
 No absolute position info.
Corrections up to 150° in 14 minutes
Typical coherence time is a few minutes
Phase solution of first self-cal. Corrections are large. Note time interval
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

24. Image after first pass Original Image Self-Calibrated Image
Same contour levels. Power has been collected into the center model.
Note noise has gone down dramatically.
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25. Phase Solutions from 2nd Self-Cal
Used 3 components
Corrections are reduced to 40° in 14 minutes
Observation now quasi-coherent
Next: shorten solution interval to follow troposphere even better
Solutions from second phase cal. Used 3 (merged) components.
Still 30s integration. Corrections are reduced to 40 degrees in 14 minutes.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

26. Image after 2nd Self-Calibration
Original Contour Level
Deeper Contouring
Image after 2nd self calibration. RHS has changed contour levels to see the fainter structure.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

27. Result after second self-calibration
Image noise is now 47 microJy/beam
Theoretical noise in 10 minutes is 45 microJy/beam for natural weighting
For 14 minutes, reduce by (1.4)1/2 to 38 microJy/beam
For robust=0, increase by 1.19, back to 45 microJy/beam
Image residuals look “noise-like”
Expect little improvement from further self-calibration
Dynamic range is 14.1/0.047 = 300
Amplitude errors typically come in at dynamic range ~ 1000
Concern: Source “jet” is in direction of sidelobes
Close to theoretical noise, after second phase-cal, don’t expect much improvement.
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

28. Phase Solutions from 3rd Self-Cal
11-component model used
10-second solution intervals
Corrections look noise-dominated
Expect little improvement in resulting image
Use 11 component model, 10 second solutions. Corrections are now noise-dominated.
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Image Comparison
2nd Self-Calibration
3rd Self-Calibration
Same contouring. Second on left, third on right.
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When Self-cal Fails
Astrometry (actually it just doesn’t help)
Variable sources
Incorrect model
barely-resolved sources
self-cal can embed mistakes in the data
Bad data
Images dominated by deconvolution errors
poor boxing
insufficient uv-coverage
Not enough flux density
fast-changing atmosphere
Errors which are not antenna-based & uniform across the image
baseline-based (closure) errors (e.g., bandpass mismatches)
imaging over areas larger than the isoplanatic patch
antenna pointing and primary beam errors
Self-cal can fail, but not too often. Main point is number of free parameters vs number of
Visibilities, i.e. when N is small. If there are deconvolution errors, then there is a possible
Problem of using these errors as part of the model. Once in the model, the gains are
Adjusted so that such errors will always remain. This is one of most important things about
Self-cal.
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How well it works
Can be unstable for complex sources and poor Fourier plane coverage
VLA snapshots, sparse arrays (VLBA, MERLIN)
Basic requirement is that the total number of degrees of freedom (number of gains plus the number of free parameters in the model) should not be greater than the number of independent vis. measurements.
Quite stable for well sampled VLA observations and appropriately complex sources
Standard step in most non-detection experiments
Bad idea for detection experiments
Will manufacture source from noise
Use in-beam calibration for detection experiments
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Recommendations
Flag your data carefully before self-cal
Expect to self-calibrate most experiments (other than detection checks)
For VLA observations, expect convergence in iterations
Monitor off-source noise, peak brightness, “unbelievable” features to determine convergence
Few antennas (VLBI) or poor (u,v) coverage can require many more iterations of self-cal
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Recommendations
Be careful with the initial model
Don’t go too deep into your clean components!
don’t embed junk in your calibration
False symmetrization in phase self-cal (using, e.g., a point source model)
If it’s important, leave it out: is this feature required by the data?
If desperate, try a model from a different configuration or a different band
Experiment with tradeoffs on solution interval
Average IFs
Shorter intervals follow the atmosphere better
Don’t be too afraid to accept low SNRs
Eleventh Synthesis Imaging Workshop, June 10-17, 2008

34. More Calibration Techniques
Water vapor radiometers
22 GHz (EVLA) and 180 GHz (ALMA)
Ionospheric measurements (TEC)
Dual-frequency observations – calibration transfer
Use self-cal to transfer phase solutions from narrow-band to broad-band (strong-line, weak continuum)
Baseline-based calibration (removal of closure errors)
antenna delay errors, antenna IF bandpasses
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Finis
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Easy example
8.4GHz observations of Cygnus A
VLA C configuration
Deconvolved using AIPS++ multi-scale clean
Calibration using AIPS++ calibrater tool
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37. Image without self-calibration
Phase calibration using nearby source observed every 20 minutes
Peak ~ 22Jy
Display shows Jy to 0.5Jy
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38. After 1 phase-only self-calibration
Phase solution every 10s
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39. After 1 amplitude and phase calibrations
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40. After 2 amplitude and phase calibrations
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41. After 3 amplitude and phase calibrations
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42. After 4 amplitude and phase calibrations
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43. Summary of Cygnus A example
~ Factor of three reduction in off source error levels
Peak increases slightly as array phases up
Off source noise is less structured
Still not noise limited - we don’t know why
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44. Final image showing all emission > 3 sigma
Eleventh Synthesis Imaging Workshop, June 10-17, 2008