1. Self-calibration Steven Tingay Swinburne University of Technology
2006 ATNF synthesis imaging school

2. Outline Brief recap; What, why……;
Minimal post-conference-dinner mathematics;
Closure quantities;
The general implementation of self-calibration;
Related topics:
Redundant calibration;
(u,v) crossing points;
Explicit use of closure quantities;
Practical considerations;
ATCA example;
VLBI example;
Final comments.

3. Brief recap……
target
Calibrator
Correlator measures visibility…….(talks on first couple of days)
Instrumental calibration…..(Reynolds talk)
“Primary” and “secondary” calibration……(Reynolds/Chapman talks)
- Self-calibration is often the final step in an overall calibration strategy;
- Self-calibration uses essentially the same calculation as used for calibration using a reference source (primary and secondary calibration at the ATCA);
- Usually undertake self-calibration as part of an iterative scheme during imaging;

4. Target
scans
Calibration
scans

5. What is self-calibration?
Consider the antenna-based complex gains as free parameters:
One can seek a set of complex gains and a model for the source structure that are self-consistent with the observed visibilities - all parameters (gains and source structure parameters) are derived from the same dataset.
Estimation of the gains using such a method is known as “self-calibration”. The technique is most readily recognised by users as part of an iterative imaging/deconvolution/self-calibration strategy.

6. Literature…… Read, ask, discuss, re-read, do, re-read………….
“ The white book ” : Cornwell and Fomalont, Chapter 9 (1994 edition);
“ The blue book ” : Thompson, Moran, and Swenson;
Pearson and Readhead, 1984, Ann.Rev.Astron.Astrophys., 22, 97
Ekers, 1984, “Serendipitous Discoveries in Radio Astronomy”, pp 154
Read, ask, discuss, re-read, do, re-read………….

7. Why use self-calibration?
ATCA; NGC 4945 (starburst galaxy):
21 GHz, 375m + 6 km, reference calibration every 5 minutes

8. The “economy model” calibration equation
…in order to solve for these…
Measure this…
…and determine this.
Basic assumption is that the complex gains can be factored
into antenna-based quantities.

9. The gi contain residual errors after instrumental calibration and any calibration transferred from a reference source.
For an array of n antennas, we need to derive improvements to n complex gains.
At any point in time, one has
n(n-1)/2 measurements of Vij to work with i.e. for the ATCA, 15 measurements.
After solving for the gains,
n(n-1)/2 - n “good” complex numbers are available with which to determine the structure of the source (9 for the ATCA)
# antennas
# complex gains
# complex visibilities
# “good” complex numbers 3 4 6 2 5 10
(ATCA) 6 15 9
(VLA) 27 27
351
324
(SKA) 125
125
7750
7625
Table 1: increasing constraints
on source structure as a function
of number of antennas

10. Exercise
For
students
Closure quantities
Closure phase is the sum of measured phases around a closed loop of antennas. The antenna-based phase errors cancel and a quantity purely reflective of the true source structure remains.
There are n(n-1)/2 - (n-1) closure phases. These are half of the “good” numbers in the visibility data - in Table 1. j
An amplitude closure quantity exists as well - an exercise to the student to look this up and derive it. i k

11. General implementation
Observed visibilities
Complex gains
Model visibilities
The complex gains are treated as free parameters that need to be solved.
A consequence of this is that certain important astronomical information can be absorbed into the free parameters, and lost, Most notably the absolute celestial coordinates and the strength of the source.
In general, one can imagine a scheme whereby the parameters of the source structure and the complex gains can be varied in order to minimise the following:
A modified version of
this least-squares method can be used,
which mimics the derivation of the
antenna gains for a point source:

12. Self-calibration in imaging: an iterative approach
Fit the data to the model visibilities by adjusting the complex gains
Make a model using whatever constraints are available
Good model?
Use the data (partly corrected by the estimated gains) to derive a new model for the source
Fourier transform the model to give model visibilities No
Yes
Finished

13. Amplitude part of gain can be held constant - phase-only self-calibration;
Phase part of gain can be held constant - amplitude-only self-calibration;
Amplitude and phase parts of gain can be varied simultaneously.

14. Redundant calibration
Redundant calibration: Array designed so that multiple baselines measure same Fourier component of brightness distribution.
N(N-1)/2 measurements, (N-1) true visibilities to
determine + N gains (maximally redundant array);
Over-determined system, use least-squares
methods to suppress noise and estimate true
visibilities + gains.
V12=V12t g1g2*
V23=V23t g2g3*
V12t=V23t
V12/V23=g1g2*/g2g3*
Exercise
For
students 3 2 1

15. Crossing points
Different baselines make different tracks through the (u,v) plane as a function of time - these can cross (generally not at the same point in time).
This leads to similar (but not identical) constraints to those given by redundant baselines.
VAB(t1)=VABt(t1)gA(t1)gB(t1)*
VCD(t2)=VCDt(t2)gC(t2)gB(t2)*
VABt(t1)=VCDt(t2)
VAB(t1)/VCD(t2)=gA(t1)gB(t1)*/gC(t2)gD(t2)* CD AB

16. Explicit use of closure quantities
Estimate from model
Closure phase is a robust quantity;
Alternative to a global minimisation of gain parameters:
Use model of source to determine two phases around the closure loop;
Use the closure phase to calculate the remaining phase;
Use these improved gains to correct the data and form a new model;
Possible to use closure amplitude in a similar way;
Equivalent to a global minimisation scheme;
Somewhat problematic in the implementation.
Calculate this from the closure

17. When is self-calibration most useful?
ATCA (6 km baselines, 22 GHz band, summer - see example);
VLBI (1000 km baseline, relative clock drifts, possible antenna/source position errors, vastly different weather at different sites - see example);
SKA (~10 km, ~300 MHz, ionospheric gradients and variations);
a priori (instrumental) and/or primary/secondary calibration not very good:
Poorly designed experiment;
Failure of instrument;
Phase reference calibrator has structure;
Phase calibrator is variable in time;

20. When is self-calibration least useful?
When your data are already very well calibrated:
Image noise close to theoretical;
No significant artifacts apparent in image;
Primary/secondary/instrumental calibration does the job.
Experiments for the detection of very weak sources (not enough signal to noise to self-calibrate). Need to relay on a priori and secondary calibration.

21. Some practical considerations
Signal to noise requirements:
Need good SNR to undertake meaningful
least-squares fit;
Solution interval:
Choose to give good SNR well within the
coherence time of the data (see Norris talk);
Quality of initial model:
Most sources structures only require a rudimentary initial model. Do you have a pathological source (well-separated equal double);
Is a single set of solutions valid over the entire field of view?
Introduce “peeling” (see Annie Hughes poster) or similar techniques into the imaging process. 
V()
G() (S=0.1 Jy; n=6)
10 s
0.010 Jy
0.050 Jy
30 s
0.005 Jy
0.025 Jy
60 s
0.003 Jy
0.015 Jy
G2()= V2()/(n-2)S2

22. An ATCA example NGC 4945, southern starburst galaxy: Starburst region
Use ATCA at 17, 19, 21, and 23 GHz to measure flux densities and spectral indices for supernova remnants in disk of galaxy;
Compare high frequency ATCA data to low frequency VLBI data to examine spectral turnover in supernova remnant spectra and infer properties of the free-free absorbing ISM in NGC 4945
Starburst region
In NGC 4945

23. ATCA example (cont) 21 GHz is close to the water line at 22 GHz;
Observation date is March Very hot in Narrabri at this time of year. Atmosphere can hold a lot of water. Thunderstorms, turbulent atmosphere;
Significant short-timescale phase variations
due to varying path length through the wet
atmosphere;
Two configurations that both use the 6 km antenna on different days. 6 km baseline required for best resolution but most susceptible to phase excursions;
Phase self-calibration will likely be very important in imaging.

24. ATCA example (cont) 375m configuration (with 6 km)
+ 6 km configuration;
21 GHz;
128 MHz bandwidth;
Nearby phase calibrator every
5 minutes;
Iterative deconvolution and
self-calibration;

25. VLBI example Centaurus A ( = -44 deg.); 8.4 GHz;
VLBA (elevation ~ 20o);
VLBI fringe-fitting performs self-calibration in order to determine delays and delay rates, leaving residual phase errors in the data.
Amp.
Phase

26. Advice for rookies
Does your science require self-calibration? Yes/No?;
Don’t attempt amplitude self-calibration until you have accounted for all of the flux on the shortest baselines. Try phase-only self-calibration initially. Phase errors are usually larger and have more effect than amplitude errors;
Think carefully about SNR and self-calibration solution interval;
Learn to recognise amplitude and phase errors in both the (u,v) plane and the image plane. Use the knowledge to guide your choice of self-calibration (see Ekers lecture);
Make improvements to your model slowly and inspect the model often - look at the residual image. What is the residual RMS? Is the model sensible?
Inspect your self-calibration solutions as a reality check if you can. Are the self-calibration solutions sensible?
Are you trying to get too much out of your data?

27. Summary
Benefits:
When you have enough signal to noise and enough antennas, self-calibration vastly improves your images in many cases.
Many projects with the ATCA will benefit from self-calibration.
Warnings:
Careful evaluation of the data is required and an understanding of when self-calibration should and should not be attempted;
Self-calibration can be pushed too far under inappropriate conditions and can lead to images that look good but are wrong (see Ekers talk - image of AP Librae)