Calibration
Michael Bietenholz
Based on a lecture by George Moellenbrock
(NRAO) at the NRAO Synthesis Imaging
Workshop
Synopsis
•
•
•
•
•
•
•
•
•
•
•
Why calibration and editing?
Editing and RFI
Idealistic formalism → Realistic practice
Practical Calibration
Baseline- and Antenna-based Calibration
Intensity Calibration Example
Full Polarization Generalization
A Dictionary of Calibration Effects
Calibration Heuristics
New Calibration Challenges
Summary
2
Why Calibration and Editing?
• Synthesis radio telescopes, though well-designed, are not
perfect (e.g., surface accuracy, receiver noise, polarization
purity, stability, etc.)
• Need to accommodate deliberate engineering (e.g., frequency
conversion, digital electronics, filter bandpass, etc.)
• Passage of radio signal through the Earth’s atmosphere
• Hardware or control software occasionally fails or behaves
unpredictably
• Scheduling/observation errors sometimes occur (e.g., wrong
source positions)
• Radio Frequency Interference (RFI)
Determining instrumental properties (calibration)
is a prerequisite to
determining radio source properties
3
Calibration Strategy
• Observe calibrator sources in addition to our program
sources
• These are sources with a known location and known properties,
usually point sources (or nearly so)
• Ideally, they are nearby on the sky to our target source
• By examining the visibility measurements for the calibrator
sources, where we know what they should be, we can estimate
our instrumental properties, often called the calibration
• We can then use these estimates of the instrumental properties
to calibrate the visibility data for the program source
• In general the instrumental properties vary with time, with
frequency and with position on the sky
• One usually uses different calibrator sources to obtain different
parts of the calibration (flux density scale, polarization etc, etc),
trying to separate out those aspects which change on different
timescales (generally: instrumental – long timescales;
atmosphere – short timescales)
4
What Does the Raw Data Look Like?
Visibility Amplitude
Flux
Density
Calibrator
– e.g.,
3C286
AIPS Task: UVPLT
Phase
calibrator
Time
Time
Program
source
5
Calibration and Editing
Calibration and editing (flagging) are inter-dependent. If
we derive calibration from visibilities, we want to edit
out corrupted visibilities before obtaining calibration
But: editing data is much easier when its already well
calibrated
Terminology
Integration time – the time interval used to dump the
correlator, typically 1 – 10 secs
Scan – One continuous observation of one source,
typically 1 to 30 minutes
What Does the Raw Data Look Like?
Visibility Amplitude
Flux
Density
Calibrator
– e.g.,
3C286
Bad data, to
be flagged
AIPS Task: UVPLT
Phase
calibrator
Time
Time
Program
source
7
What Does the Raw Data Look Like?
AIPS Task: UVPLT
8
AIPS TVFLG
Color:
visibility
amplitude in
this example.
Can be
phase or
other
quantities
Time

Baseline 
Don’t Edit Too Much
Rule 1) You should examine your data to see if there is anything that needs
to be edited out. If your data is good, there may be nothing to edit out, but you
won’t know till you look!
Rule 2) Try to edit by antenna, not by baseline. The vast majority of problems
are antenna-based, so if baseline ant 1 – ant 2 is bad, try and figure out whether its
ant 1 or ant 2 which has the problem and then flag the antenna. Caveat: RFI is
generally baseline-based.
Rule 3) Don’t edit out data which is just poorly calibrated – fix the calibration
instead.
Rule 4) Don’t be afraid of noise – much of our visibility data, especially on
weak sources, looks very much like pure noise. Don’t throw it out – the signal
you want is buried in that noise.
Rule 5) Don’t edit too much!
– The goal is to remove data which is obviously bad. Generally, if you are
editing out more than 10% of your data, you are probably editing too much.
Rule 6) Remember your program source. If e.g., an antenna is bad for two
calibrator scans, its probably bad for the intervening program source scan, and
should be edited out.
Radio Frequency Interference
• Has always been a problem (Grote Reber, 1944, in total power)!
11
Radio Frequency Interference (cont)
• Growth of telecom industry threatening radio astronomy!
12
Radio Frequency Interference
• RFI originates from man-made signals generated in the antenna
electronics or by external sources (e.g., satellites, cell-phones,
radio and TV stations, automobile ignitions, microwave ovens,
computers and other electronic devices, etc.)
– Adds to total noise power in all observations, thus decreasing the
fraction of desired natural signal passed to the correlator, thereby
reducing sensitivity and possibly driving electronics into non-linear
regimes
– Can correlate between antennas if of common origin and baseline
short enough (insufficient decorrelation via geometry compensation),
thereby obscuring natural emission in spectral line observations
• Some RFI is generated by the instruments themselves (Local
oscillators, high-speed digital electronics, power lines). Careful
design can minimize such internal RFI.
• Least predictable, least controllable threat to a radio astronomy
observation.
13
Radio Frequency Interference
• RFI Mitigation
–
–
–
–
Careful electronics design in antennas, including filters, shielding
High-dynamic range digital sampling
Observatories world-wide lobbying for spectrum management
Choose interference-free frequencies: but try to find 50 MHz (1
GHz) of clean spectrum in the VLA (EVLA) 1.6 GHz band!
– Observe continuum experiments in spectral-line modes so affected
channels can be edited
• Various off-line mitigation techniques under study
– E.g., correlated RFI power that originates in the frame of the array
appears at celestial pole (also stationary in array frame) in image
domain…
14
Calibration:
What Is Delivered by a Synthesis Array?
An enormous list of complex numbers (visibility data set)!
E.g., the EVLA:
At each timestamp (~1s intervals): 351 baselines (+ 27 autocorrelations)
For each baseline: 1-64 Spectral Windows (“subbands” or “IFs”)
For each spectral window: tens to thousands of channels
For each channel: 1, 2, or 4 complex correlations
RR or LL or (RR,LL), or (RR,RL,LR,LL)
With each correlation, a weight value
Meta-info: Coordinates, antenna, field, frequency label info
Ntotal = Nt x Nbl x Nspw x Nchan x Ncorr visibilities
EVLA: ~1300000 x Nspw x Nchan x Ncorr vis/hour (10s to 100s of GB
per observation)
MeerKAT: ~8X more baselines than EVLA!
15
Calibrator Sources
Ideally – they would be very strong, completely pointlike sources which did not vary in time
In practice such sources do not exist. Only a few
sources have reasonably stable flux densities, and
they are usually not very compact.
Most point-like sources, on the other hand, are variable
with time (timescales from days to weeks)
Typical strategy is to use one of the few stable sources
as a flux-density calibrator, observed once or twice in
the observing run, and a point-like source near the
program source as a phase calibrator, which is
observed more frequently.
AIPS Calibration Philosophy
• “Keep the data”
• Original visibility data is
not altered
• Calibration is stored in
tables, which can be
applied to print out or plot
or image the visibilities
• Different steps go into
different tables
• Easy to undo
• Need to store only one copy of the visibility data
set (big file), but can have many versions of the
calibration tables (small files)
17
AIPS Calibration Tables
•
Visibility data file contains the visibility measurements (big file). Associated
with it are various tables which contain other information which might be
needed: here are some of the tables used during calibration:
•
•
•
AN table – Antenna table, lists antenna properties and names
NX table – Index table, start and end times of scans
SU table – Source table, source names and properties (e.g., flux density if
known)
FQ table – frequency structure. Frequencies of different IFs relative to the
header frequency
FG table – flagged (edited) data, marks bad visibilities
SN table – “solution table” , contains solutions for complex gains as a function
of time and antenna
CL table – complex gains as a function of time and antenna interpolated to a
regular grid of times, this is the table that is used to actually calibrate the
visibilities different tables
BP table – bandpass response, complex gain as a function of frequency and
antenna
•
•
•
•
•
18
From Idealistic to Realistic
•
Formally, we wish to use our interferometer to obtain the visibility
function:

i
2

(
ul

vm
)
V
(
u
,
v
)

I
(
l
,
m
)
e
dldm

sky
•
….which we intend to invert to obtain an image of the sky:
i
2

(
ul

vm
)
I
(
l
,
m
)

V
(
u
,
v
)
e
dudv

uv
•
•
V(u,v) set the amplitude and phase of 2D sinusoids that add up to
an image of the sky
How do we measure V(u,v)?
From Idealistic to Realistic
•
In practice, we correlate (multiply & average) the electric field
(voltage) samples, xi & xj, received at pairs of telescopes (i, j ) and
processed through the observing system:
obs
*




V
u
,
v

x
t

x
t
ij
ij ij
i
j

t
true


J
V
u
,v
ijij 
ij
ij
•
•
•
•
xi & xj are delay-compensated for a specific point on the sky
Averaging duration = integration time, is set by the expected timescales
for variation of the correlation result (~seconds)
Jij is an operator characterizing the net effect of the observing
process for baseline (i,j), which we must calibrate
Sometimes Jij corrupts the measurement irrevocably, resulting in data
that must be edited or “flagged”
Practical Calibration Considerations
• A priori “calibrations” (provided by the observatory)
–
–
–
–
–
Antenna positions, earth orientation and rate
Clocks
Antenna pointing, gain, voltage pattern
Calibrator coordinates, flux densities, polarization properties
System Temperature, Tsys, nominal sensitivity
• Absolute engineering calibration?
– Very difficult, requires heroic efforts by observatory scientific and
engineering staff
– Concentrate instead on ensuring instrumental stability on adequate
timescales
• Cross-calibration a better choice
– Observe nearby point sources against which calibration (Jij) can
be solved, and transfer solutions to target observations
– Choose appropriate calibrators; usually strong point sources
because we can easily predict their visibilities
– Choose appropriate timescales for calibration
21
“Absolute” Astronomical Calibrations
• Flux Density Calibration
– Radio astronomy flux density scale set according to several
“constant” radio sources
– Use resolved models where appropriate
• Astrometry
– Most calibrators come from astrometric catalogs; directional
accuracy of target images tied to that of the calibrators
(ICRF = International Celestial Reference Frame)
– Beware of resolved and evolving structures and phase
transfer biases due to troposphere (especially for VLBI)
• Linear Polarization Position Angle
– Usual flux density calibrators also have significant stable
linear polarization position angle for registration
• Relative calibration solutions (and dynamic range)
insensitive to errors in these “scaling” parameters
22
A Single Baseline – 3C 286
Vis. Phase vs freq. (single channel)
120°
105°
3C 286 is one of the strong, stable sources which can be
used as a flux density calibrator
Single Baseline, Single Integration
Visibility Spectra (4 correlations)
Vis. amp. vs freq.
Baseline ea17-ea21
Vis. phase vs freq.
Single integration – typically
1 to 10 seconds
Single Baseline, Single Scan
Visibility Spectra (4 correlations)
Vis. amp. vs freq.
baseline ea17-ea21
Vis. phase vs freq.
Single scan – typically 1 to 30
minutes, 5 to 500 integrations
Single Baseline, Single Scan (time-averaged)
Visibility Spectra (4 correlations)
Vis. amp. vs freq.
baseline ea17-ea21
Vis. phase vs freq.
Single scan – time averaged
26
Baseline-based Cross-Calibration
Vijobs  J ijVijtrue
• Simplest, most-obvious calibration approach: measure complex
response of each baseline on a standard source, and scale
science target visibilities accordingly
– “Baseline-based” Calibration
• Calibration precision same as calibrator visibility sensitivity (on
timescale of calibration solution).
• Calibration accuracy very sensitive to departures of calibrator
from known structure
– Un-modeled calibrator structure transferred (in inverse) to science
target!
29
Antenna-Based Cross Calibration
• Measured visibilities are formed from a product of antennabased signals. Can we take advantage of this fact?
• The net signal delivered by antenna i, xi(t), is a combination of
the desired signal, si(t,l,m), corrupted by a factor Ji(t,l,m) and
integrated over the sky, and diluted by noise, ni(t):
xi (t ) 
 J (t , l , m)s (t , l , m) dldm n (t )
i
i
i
sky
 si(t )  ni (t )
• Ji(t,l,m) is the product of a series of effects encountered by the
incoming signal
• Ji(t,l,m) is an antenna-based complex number
• Usually, |ni |>> |si| - Noise dominated
30
Antenna-base Calibration Rationale
•
•
•
Atmospheric
delay 2
Atmospheric
delay 1
•
•
Instrumental
delay 1
Instrumental
delay 2
V
output
Signals affected by a
number of processes
Due mostly to the
atmosphere and to the
the antenna and the
electronics
The majority of factors
depend on antenna only,
not on baseline
Some factors known a
priori, but most of them
must be estimated from
the data
Factors take the form of
complex numbers, which
may depend on time and
frequency
Correlation of Realistic Signals - I
• The correlation of two realistic signals from different antennas:
xi  x*j
 si  ni   sj  n j 
*
t
t
 si  sj*  si  n*j  ni  sj*  ni  n*j
• Noise signal doesn’t correlate—even if |ni|>> |si|, the correlation
process isolates desired signals:
 si  sj*

t
 J s dl dm   J
i i
sky
* *
j j
s dldm
sky
t
• In the integral, only si(t,l,m), from the same directions correlate
(i.e., when l=l’, m=m’), so order of integration and signal
product can be reversed:

*
*
J
J
s
s
i
j
i
j dldm

sky
t
32
Correlation of Realistic Signals - II
•
The si & sj differ only by the relative arrival phase of signals from different
parts of the sky, yielding the Fourier phase term (to a good approximation):
Vij 
* 2
J
J
i
 j s t , l , me

i 2 uij l  vij m

dldm
sky
•
t
On the timescale of the averaging, the only meaningful average is of the
squared signal itself (direction-dependent), which is just the image of the
source:

*
2
t , l , m 
J
J
s
i
j

sky

*
J
J
 i j I l , me
t
e


i 2 uij l  vij m
i 2 uij l  vij m

dldm
sky
•
If all J=1, we of course recover the ideal expression:

 I l , m e
sky

i 2 uij l  vij m

dldm

dldm
33
Aside: Auto-correlations and Single Dishes
• The auto-correlation of a signal from a single antenna:
*
xi  xi*  si  ni   si  ni 
 si  si*  ni  ni*

 J i si dldm  ni
2
2
2
sky

 J i I l, mdldm  ni
2
2
sky
• This is an integrated power measurement plus noise
• Desired signal not isolated from noise
• Noise usually dominates
• Single dish radio astronomy calibration strategies dominated by
switching schemes to isolate desired signal from the noise
34
35
The Scalar Measurement Equation
obs
ij
V

 Ji J
*
j
 i 2 u l  v m 


dldm
I l, m e
ij
ij
sky
•
First, isolate non-direction-dependent effects, and factor them from the
integral:

*
 J ivis J vis
j
  J
sky
i

*
I l , m e
J sky
j

i 2 uij l  vij m

dldm
sky
•
Here we have included in Jsky only the part of J which varies with position
on the sky. Over small fields of view, J does not vary appreciably, so we
can take Jsky = 1, and then we have a relationship between ideal and
observed Visibilities:
i 2 uij l  vij m 
vis vis*

 Ji J j

  I l, me
dldm
sky

* true
true
*
J
J

V
Vijobs  J ivis J vis
i jVij
ij
j
•
Standard calibration of most existing arrays reduces to solving this last
equation for the Ji
36
Solving for the Ji
• We can write:
Vijobs
true
ij
V


 J i J *j  0
obs
ij
true
ij
V
• …and define chi-squared:  2 

i, j V
i j


2
 J i J *j wij
• …and minimize chi-squared w.r.t. each Ji, yielding (iteration):
 Vijobs


J i   true J j wij 


j  Vij

j i
 J
2
j
wij
j
j i

  2

 *  0 
 J i

• …which we recognize as a weighted average of Ji, itself:
J i   J iwij 
j
j i
 w
ij
j
j i
Solving for Ji (cont)
• For a uniform array (same sensitivity on all baselines, ~same
calibration magnitude on all antennas), it can be shown that the
error in the calibration solution is:
J 
i
V
obs
t 
V true J
N ant  1
• SNR improves with calibrator strength and square-root of Nant
(c.f. baseline-based calibration).
• Other properties of the antenna-based solution:
– Minimal degrees of freedom (Nant factors, Nant(Nant-1)/2
measurements)
– Constraints arise from both antenna-basedness and consistency
with a variety of (baseline-based) visibility measurements in which
each antenna participates
– Net calibration for a baseline involves a phase difference, so
absolute directional information is lost
– Closure…
37
Antenna-based Calibration and Closure
• Success of synthesis telescopes relies on antenna-based calibration
– Fundamentally, any information that can be factored into antenna-based
terms, could be antenna-based effects, and not source visibility
– For Nant > 3, source visibility cannot be entirely obliterated by any
antenna-based calibration
• Observables independent of antenna-based calibration:
– Closure phase (3 baselines):
obs
true
true
true





ijobs   obs
















jk
ki
ij
i
j
jk
j
k
ki   k   i 
true
 ijtrue   true


jk
ki
– Closure amplitude (4 baselines):
VijobsVklobs
obs obs
ik
jl
V V

J i J jVijtrue J k J lVkltrue
true
i k ik
true
j l jl
J J V J JV

VijtrueVkltrue
ViktrueV jltrue
• Baseline-based calibration formally violates closure!
38
Simple Scalar Calibration Example
• Sources:
– Science Target: 3C129
– Near-target calibrator: 0420+417 (5.5 deg from target; unknown
flux density, assumed 1 Jy)
– Flux Density calibrators: 0134+329 (3C48: 5.74 Jy), 0518+165
(3C138: 3.86 Jy), both resolved (use standard model images)
• Signals:
– RR correlation only (total intensity only)
– 4585.1 MHz, 50 MHz bandwidth (single channel)
– (scalar version of a continuum polarimetry observation)
• Array:
– VLA B-configuration (July 1994)
39
The Calibration Process
• Solve for antenna-based gain factors for each scan on flux
true
calibrator Ji(fd) (where Vij is known):


Vijobs  Ji J *j Vijtrue
Solve also gain factors for phase calibrator(s), Ji(nt)
• Bootstrap flux density scale by enforcing constant mean power
response:
 J
 i ( fd )
 J i ( nt ) 
2
 J i ( nt )

2
J i( nt )
12


i


i 
• Correct data (interpolate J as needed):
corrected
ij
V


 J i J  V
1
*1
j
obs
ij
40
Antenna-Based Calibration
Visibility phase on a several
baselines to a common antenna
(ea17)
41
Calibration Effect on Imaging
J1822-0938
(calibrator)
3C391
(science)
How Good is My Calibration?
•
Are solutions continuous?
•
•
•
•
Are calibrator data fully described by antenna-based effects?
•
•
•
Noise-like solutions are probably noise! (Beware: calibration of pure noise
generates a spurious point source)
Discontinuities indicate instrumental glitches
Any additional editing required?
Phase and amplitude closure errors are the baseline-based residuals
Are calibrators sufficiently point-like? If not, self-calibrate: model
calibrator visibilities (by imaging, deconvolving and transforming) and resolve for calibration; iterate to isolate source structure from calibration
components
Any evidence of unsampled variation? Is interpolation of solutions
appropriate?
•
Reduce calibration timescale, if SNR permits
A priori Models Required for Calibrators
Point source,
but flux
density not
stable
Stable flux density, but not point sources
44
Antenna-based Calibration Image Result
45
Evaluating Calibration Performance
• Are solutions continuous?
– Noise-like solutions are just that—noise
– Discontinuities indicate instrumental glitches
– Any additional editing required?
• Are calibrator data fully described by antenna-based effects?
– Phase and amplitude closure errors are the baseline-based
residuals
– Are calibrators sufficiently point-like? If not, self-calibrate: model
calibrator visibilities (by imaging, deconvolving and transforming)
and re-solve for calibration; iterate to isolate source structure from
calibration components
• Mark Claussen’s lecture: “Advanced Calibration” (Wednesday)
• Any evidence of unsampled variation? Is interpolation of
solutions appropriate?
– Reduce calibration timescale, if SNR permits
• Ed Fomalont’s lecture: “Error Recognition” (Wednesday)
46
Summary of Scalar Example
• Dominant calibration effects are antenna-based
•
•
•
•
Minimizes degrees of freedom
More precise
Preserves closure
Permits higher dynamic range safely!
• Point-like calibrators effective
• Flux density bootstrapping
47
Full-Polarization Formalism (Matrices!)
• Need dual-polarization basis (p,q) to fully sample the incoming
EM wave front, where p,q = R,L (circular basis) or p,q = X,Y
(linear basis):
 

I circ  S circ I Stokes
 RR   1
  
 RL   0
 LR    0
  
 LL   1
  
0 0 1  I   I  V 
  

1 i
0  Q   Q  iU 




1 i 0 U
Q  iU 
  





0 0  1 V   I  V 
 

I lin  Slin I Stokes
 XX   1 1

 
XY

 0 0
 YX    0 0

 
 YY   1  1

 
0 0  I   I  Q 
  

1 i  Q  U  iV 




1 i U
U  iV 
  





0 0  V   I  Q 
• Devices can be built to sample these linear or circular basis
states in the signal domain (Stokes Vector is defined in “power”
domain)
• Some components of Ji involve mixing of basis states, so dualpolarization matrix description desirable or even required for
proper calibration
48
Full-Polarization Formalism: Signal Domain
• Substitute:
 sp 
si  si   q  ,
 s i
  J p p
J i  J i   p q
J
J q p 

q q 
J

• The Jones matrix thus corrupts the vector wavefront
signal as follows:
 
si  J i si
 s p   J p  p
 q    p q
 s   J
 i 
(sky integralomitted)
J q p   s p 
 
q q   q 
J
i  s i
 J p p s p  J q p s q 

  p q p
q q q 
s  J s i
J
49
Full-Polarization Formalism: Correlation - I
• Four correlations are possible from two polarizations. The outer
product (a ‘bookkeeping’ product) represents correlation in the
matrix formalism:
*
p
p
 obs






s
s
 *
Vij  si  s j   q    q 
 s  i  s  j







si p  sj* p 

si p  sj*q 

q
*p
si  sj 
siq  sj*q 


• A very useful property of outer products:
     
 obs   *  
* *  *  *   true
Vij  si  s j  J i si  J j s j  J i  J j si  s j  J ijVij

50
The Matrix Measurement Equation
• We can now write down the Measurement Equation in matrix
notation:
 obs
Vij 


  * 
 i 2 uij l  vij m 
J i  J j SI l , m e
dldm
sky
• …and consider how the Ji are products of many effects.
53
A Dictionary of Calibration Components
• Ji contains many components:
•
•
•
•
•
•
•
•
•
F = ionospheric effects
T = tropospheric effects
P = parallactic angle
X = linear polarization position angle
E = antenna voltage pattern
D = polarization leakage
i
G = electronic gain
B = bandpass response
K = geometric compensation

      
J  Ki BiGi Di Ei X i PiTi Fi
• Order of terms follows signal path (right to left)
• Each term has matrix form of Ji with terms embodying its
particular algebra (on- vs. off-diagonal terms, etc.)
• Direction-dependent terms must stay inside FT integral
• Full calibration is traditionally a bootstrapping process wherein
relevant terms are considered in decreasing order of
dominance, relying on approximate orthogonality
54
Ionospheric Effects, F
• The ionosphere introduces a dispersive phase shift:
  0.15 2  B|| ne ds deg
 in cm, ne ds in 1014 cm-2 ,
B|| in G
TEC   ne ds ~ 1014 cm -2 ; B|| ~ 1G;   20cm   ~ 60 
•
•
•
•
(TEC = Total Electron Content)
More important at longer wavelengths (2)
More important at solar maximum and at sunrise/sunset, when
ionosphere is most active and variable
Beware of direction-dependence within field-of-view!
The ionosphere is birefringent; one hand of circular polarization is
delayed w.r.t. the other, thus rotating the linear polarization position
angle
i
 RL

e
F  ei 
0
0   XY
i  cos

; F  e 
i 
e 
 sin 
 sin  

cos 
55
Tropospheric Effects, T
• The troposphere causes polarization-independent amplitude and
phase effects due to emission/opacity and refraction, respectively
•
•
•
•
•
•
•
Typically 2-3m excess path length at zenith compared to vacuum
Higher noise contribution, less signal transmission: Lower SNR
Most important at  > 20 GHz where water vapor and oxygen absorb/emit
More important nearer horizon where tropospheric path length greater
Clouds, weather = variability in phase and opacity; may vary across array
Water vapor radiometry? Phase transfer from low to high frequencies?
Zenith-angle-dependent parameterizations?
– )
 pq  t 0   1 0 
  t 

T  
0 t  0 1
56
57
Parallactic Angle, P
• Visibility phase variation due to changing orientation of sky in
telescope’s field of view
• Constant for equatorial telescopes
• Varies for alt-az-mounted telescopes:

cosl sin h(t ) 

 sin l  cos   cosl sin   cosh(t )  

 (t )  arctan
l  latitude, h(t )  hour angle,   declination
• Rotates the position angle of linearly polarized radiation
• Analytically known, and its variation provides leverage for determining
polarization-dependent effects
• Position angle calibration can be viewed as an offset in 
– Steve Myers’ lecture: “Polarization in Interferometry” (today!)
 RL  ei
P  
0
0   XY  cos 
 ; P  
i 
e 
 sin 
 sin  

cos  
Linear Polarization Position Angle, X
• Configuration of optics and electronics causes a linear
polarization position angle offset
• Same algebraic form as P
• Calibrated by registration with a source of known polarization
position angle
• For linear feeds, this is the orientation of the dipoles in the frame
of the telescope
 RL  ei
X  
 0
0   XY  cos 
 ; X  
i 
e 
 sin 
 sin  

cos  
58
Antenna Voltage Pattern, E
• Antennas of all designs have direction-dependent gain
• Important when region of interest on sky comparable to or larger than
/D
• Important at lower frequencies where radio source surface density is
greater and wide-field imaging techniques required
• Beam squint: Ep and Eq offset, yielding spurious polarization
• For convenience, direction dependence of polarization leakage (D) may
be included in E (off-diagonal terms then non-zero)
– Rick Perley’s lecture: “Wide Field Imaging I” (Thursday)
– Debra Shepherd’s lecture: “Wide Field Imaging II” (Thursday)
E pq
 e p (l , m)

0

 
q



0
e
(
l
,
m
)


59
Polarization Leakage, D
• Antenna & polarizer are not ideal, so orthogonal polarizations
not perfectly isolated
• Well-designed feeds have d ~ a few percent or less
• A geometric property of the optical design, so frequency-dependent
• For R,L systems, total-intensity imaging affected as ~dQ, dU, so only
important at high dynamic range (Q,U,d each ~few %, typically)
• For R,L systems, linear polarization imaging affected as ~dI, so almost
always important
• Best calibrator: Strong, point-like, observed over large range of
parallactic angle (to separate source polarization from D)
–
 pq  1
D   q
d
dp

1 
60
“Electronic” Gain, G
• Catch-all for most amplitude and phase effects introduced by
antenna electronics and other generic effects
• Most commonly treated calibration component
• Dominates other effects for standard VLA observations
• Includes scaling from engineering (correlation coefficient) to radio
astronomy units (Jy), by scaling solution amplitudes according to
observations of a flux density calibrator
• Often also includes ionospheric and tropospheric effects which are
typically difficult to separate unto themselves
• Excludes frequency dependent effects (see B)
• Best calibrator: strong, point-like, near science target; observed
often enough to track expected variations
– Also observe a flux density standard
G pq
gp
 
 0
0

q
g 
61
Bandpass Response, B
• G-like component describing frequency-dependence of antenna
electronics, etc.
•
•
•
•
Filters used to select frequency passband not square
Optical and electronic reflections introduce ripples across band
Often assumed time-independent, but not necessarily so
Typically (but not necessarily) normalized
• Best calibrator: strong, point-like; observed long enough to get
sufficient per-channel SNR, and often enough to track variations
B pq
 b p ( )
0 



q
b ( ) 
 0
62
Geometric Compensation, K
• Must get geometry right for Synthesis Fourier Transform relation
to work in real time; residual errors here require “Fringe-fitting”
•
•
•
•
•
Antenna positions (geodesy)
Source directions (time-dependent in topocenter!) (astrometry)
Clocks
Electronic pathlengths
Longer baselines generally have larger relative geometry errors,
especially if clocks are independent (VLBI)
• Importance scales with frequency
• K is a clock- & geometry-parameterized version of G (see
chapter 5, section 2.1, equation 5-3 & chapters 22, 23)
K
pq
k p
 
0
0

q
k 
63
Baseline-based, Non-closing Effects: M, A
64
• Baseline-based errors which do not decompose into antenna-based
components
– Digital correlators designed to limit such effects to well-understood and
uniform (not dependent on baseline) scaling laws (absorbed in G)
– Simple noise (additive)
– Additional errors can result from averaging in time and frequency over
variation in antenna-based effects and visibilities (practical instruments
are finite!)
– Correlated “noise” (e.g., RFI)
– Difficult to distinguish from source structure (visibility) effects
– Geodetic observers consider determination of radio source structure—a
baseline-based effect—as a required calibration if antenna positions are
to be determined accurately
– Diagonal 4x4 matrices, Mij multiplies, Aij adds
The Full Matrix Measurement Equation
65
• The total general Measurement Equation has the form:


   
     
 i 2 uij l  vij m 
Vij  M ij K ij BijGij  Dij Eij PijTij Fij SI l , m e
dl dm  Aij
sky
• S maps the Stokes vector, I, to the polarization basis of the instrument,
all calibration terms cast in this basis
• Suppressing the direction-dependence:
 obs           true 
Vij  M ij Kij BijGij Dij X ij PijTij FijVij  Aij
• Generally, only a subset of terms (up to 3 or 4) are considered,
though highest-dynamic range observations may require more
• Solve for terms in decreasing order of dominance
Solving the Measurement Equation
• Formally, solving for any antenna-based visibility
calibration component is always the same non-linear
fitting problem:


*
corrupted true
Vijcorrectedobs  Jisolve J solve
V
j
ij
• Viability of the solution depends on isolation of different
effects using proper calibration observations, and
appropriate solving strategies
66
Calibration Heuristics – Spectral Line
•
 obs    true
Spectral Line (B,G): Vij  BijGijVij
1. Preliminary G solve on B-calibrator:
 obs   true
V  GV
1. B Solve on B-calibrator:

 obs    true
V  B GV

1. G solve (using B) on G-calibrator:


  true
 1  obs
B V
 GV
1. Flux Density scaling:
1. Correct:

 
G   G  G fd


 1  1  obs
 corrected
V
 G B V
1. Image!
 2 1 2
G 

2

67
Calibration Heuristics – Continuum
Polarimetry
•
 obs      true
Continuum Polarimetry (G,D,X,P): Vij  Gij Dij X ij PijVij
•
Preliminary G solve on GD-calibrator (using P):

 obs    true
V  G PV
•
D solve on GD-calibrator (using P, G):

 
 1  obs
   true
G V
 D PV

•
•

•
Polarization Position
  Solve (using
 Angle
  P,G,D):
•
Flux Density scaling:
•
Correct:
•
Image!
D
1


G 1V obs  X PV true

 
G  G  G fd



2
Recall:
P = parallactic angle
X = linear polarization
angle
D = polarization leakage
G = electronic gain
B = bandpass response
•
•
•
 2 1 2
G 

 corrected
 1  1  1  1  obs
V
 P X D G V

68
New Calibration Challenges
• Bandpass Calibration
• Parameterized solutions (narrow-bandwidth, high resolution regime)
• Spectrum of calibrators (wide absolute bandwidth regime)
• Phase vs. Frequency (self-) calibration
• Troposphere and Ionosphere introduce time-variable phase effects
which are easily parameterized in frequency and should be (c.f.
sampling the calibration in frequency)
• Frequency-dependent Instrumental Polarization
• Contribution of geometric optics is wavelength-dependent (standing
waves)
• Frequency-dependent Voltage Pattern
• Increased sensitivity: Can implied dynamic range be
reached by conventional calibration and imaging
techniques?
69
Why Not Just Solve for Generic Ji Matrix?
• It has been proposed (Hamaker 2000, 2006) that we
can self-calibrate the generic Ji matrix, apply “postcalibration” constraints to ensure consistency of the
astronomical absolute calibrations, and recover full
polarization measurements of the sky
• Important for low-frequency arrays where isolated
calibrators are unavailable (such arrays see the
whole sky)
• May have a role for MeerKAT (and EVLA & ALMA)
• Currently under study…
70
Summary
• Determining calibration is as important as determining source
structure—can’t have one without the other
• Data examination and editing an important part of calibration
• Beware of RFI! (Please, no cell phones at the VLA site tour!)
• Calibration dominated by antenna-based effects, permits
efficient separation of calibration from astronomical information
(closure)
• Full calibration formalism algebra-rich, but is modular
• Calibration determination is a single standard fitting problem
• Calibration an iterative process, improving various components
in turn, as needed
• Point sources are the best calibrators
• Observe calibrators according requirements of calibration
components
71