67
Calibration Calibration Michael Bietenholz Based on a lecture by George Moellenbrock (NRAO) at the NRAO Synthesis Imaging Workshop
Calibration Calibration
Michael Bietenholz
Based on a lecture by George Moellenbrock (NRAO) at the NRAO Synthesis Imaging Workshop
2SynopsisSynopsis
• 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
3Why Calibration and Editing?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
4Calibration StrategyCalibration 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)
5What Does the Raw Data Look Like? What Does the Raw Data Look Like?
Flux Density Calibrator – e.g., 3C286
Phase calibrator
Program source
Visibility A
mplitude
Time
AIPS Task: UVPLT
Time
Calibration and EditingCalibration 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
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
TerminologyTerminology
7What Does the Raw Data Look Like? What Does the Raw Data Look Like?
Bad data, to be flagged
Flux Density Calibrator – e.g., 3C286
Phase calibrator
Program source
Visibility A
mplitude
Time
AIPS Task: UVPLT
Time
8What Does the Raw Data Look Like? What Does the Raw Data Look Like?
AIPS Task: UVPLT
AIPSAIPS TVFLGTVFLG
Baseline
Time
Color: visibility amplitude in this example. Can be phase or other quantities
Don’t Edit Too MuchDon’t Edit Too MuchRule 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.
11Radio Frequency InterferenceRadio Frequency Interference
• Has always been a problem (Grote Reber, 1944, in total power)!
12Radio Frequency Interference (cont)Radio Frequency Interference (cont)
• Growth of telecom industry threatening radio astronomy!
13Radio Frequency InterferenceRadio 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.
14Radio Frequency InterferenceRadio 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…
15
Calibration: Calibration:
What Is Delivered by a What Is Delivered by a
Synthesis Array?Synthesis Array?
An enormous list of complex numbers (visibility data set)!
E.g., the EVLA:At each timestamp (~1s intervals): 351 baselines (+ 27 auto-
correlations)For each baseline: 1-64 Spectral Windows (“subbands” or “IFs”)For each spectral window: tens to thousands of channelsFor each channel: 1, 2, or 4 complex correlations
RR or LL or (RR,LL), or (RR,RL,LR,LL)
With each correlation, a weight valueMeta-info: Coordinates, antenna, field, frequency label info
Ntotal = Nt x Nbl x Nspw x Nchan x Ncorr visibilitiesEVLA: ~1300000 x Nspw x Nchan x Ncorr vis/hour (10s to 100s of GB
per observation)
MeerKAT: ~8X more baselines than EVLA!
Calibrator SourcesCalibrator Sources
Ideally – they would be very strong, completely point-like 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.
17AIPS Calibration PhilosophyAIPS 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)
18AIPS Calibration TablesAIPS 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
• Formally, we wish to use our interferometer to obtain the visibility function:
• ….which we intend to invert to obtain an image of the sky:
• 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)?
uv
vmuli dudvevuVmlI )(2),(),(
sky
vmuli dldmemlIvuV )(2),(),(
From Idealistic to RealisticFrom 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:
• 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”
ijij
trueijij
tjiijijobsij
vuVJ
txtxvuV
,
, *
From Idealistic to RealisticFrom Idealistic to Realistic
21Practical Calibration ConsiderationsPractical 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
22““Absolute” Astronomical Absolute” Astronomical
CalibrationsCalibrations• 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
A Single Baseline – 3C 286A Single Baseline – 3C 286
3C 286 is one of the strong, stable sources which can be used as a flux density calibrator
105°
120°
Vis. Phase vs freq. (single channel)
Single Baseline, Single Single Baseline, Single Integration Visibility Spectra (4 Integration Visibility Spectra (4
correlations)correlations)
Baseline ea17-ea21 Single integration – typically
1 to 10 seconds
Vis. amp. vs freq. Vis. phase vs freq.
Single Baseline, Single ScanSingle Baseline, Single ScanVisibility Spectra (4 Visibility Spectra (4
correlationscorrelations))
baseline ea17-ea21 Single scan – typically 1 to 30
minutes, 5 to 500 integrations
Vis. amp. vs freq. Vis. phase vs freq.
2626
Single Baseline, Single Scan (time-Single Baseline, Single Scan (time-averaged)averaged)
Visibility Spectra (4 correlations)Visibility Spectra (4 correlations)
baseline ea17-ea21 Single scan – time averaged
Vis. amp. vs freq. Vis. phase vs freq.
29Baseline-based Cross-CalibrationBaseline-based Cross-Calibration
• 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!
trueijij
obsij VJV
30Antenna-Based Cross CalibrationAntenna-Based Cross Calibration
• Measured visibilities are formed from a product of antenna-based 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):
• 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
)()(
)( ),,(),,()(
tnts
tndldmmltsmltJtx
ii
i
sky
iii
Antenna-base Calibration Antenna-base Calibration RationaleRationale
Instru-mental delay 1
Instru-mental delay 2
• 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 frequencyV
output
Atmospheric delay 1
Atmospheric delay 2
32Correlation of Realistic Signals - ICorrelation of Realistic Signals - I
• The correlation of two realistic signals from different antennas:
• Noise signal doesn’t correlate—even if |ni|>> |si|, the correlation process isolates desired signals:
• 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:
tsky
jiji
tsky
jj
sky
ii
tji
jijijiji
tjjiitji
dldmssJJ
dldmsJmdldsJ
ss
nnsnnsss
nsnsxx
**
**
*
****
**
33Correlation of Realistic Signals - IICorrelation 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):
• 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:
• If all J=1, we of course recover the ideal expression:
sky
mvlui
sky
mvluiji
sky
mvlui
tji
tsky
mvluijiij
dldmemlI
dldmemlIJJ
dldmemltsJJ
dldmemltsJJV
ijij
ijij
ijij
ijij
2
2*
22*
22*
,
,
,,
,,
34Aside: Auto-correlations and Single Aside: Auto-correlations and Single
DishesDishes• The auto-correlation of a signal from a single antenna:
• 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
22
222
**
**
, i
sky
i
i
sky
ii
iiii
iiiiii
ndldmmlIJ
ndldmsJ
nnss
nsnsxx
35The Scalar Measurement EquationThe Scalar Measurement Equation
• First, isolate non-direction-dependent effects, and factor them from the integral:
• 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:
• Standard calibration of most existing arrays reduces to solving this last equation for the Ji
trueijji
trueij
visj
visi
obsij
mvlui
sky
visj
visi
mvlui
sky
skyj
skyi
visj
visi
mvlui
sky
jiobsij
VJJVJJV
dldmemlIJJ
dldmemlIJJJJ
dldmemlIJJV
ijij
ijij
ijij
**
2*
2**
2*
,
,
,
36Solving for the Solving for the JJii
• We can write:
• …and define chi-squared:
• …and minimize chi-squared w.r.t. each Ji, yielding (iteration):
• …which we recognize as a weighted average of Ji, itself:
0 *
22
iij
jijj
ijj
ijjtrueij
obsij
i JwJwJ
V
VJ
ij
jij
ijj
ijii wwJJ
0* jitrueij
obsij JJ
V
V
jiji
ijjitrueij
obsij wJJ
V
V
,
2
*2
37Solving for Solving for JJii (cont) (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:
• 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…
1
anttrue
VJ NJV
tobs
i
38Antenna-based Calibration and Antenna-based Calibration and
ClosureClosure• 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):
– Closure amplitude (4 baselines):
• Baseline-based calibration formally violates closure!
trueki
truejk
trueij
iktruekikj
truejkji
trueij
obski
obsjk
obsij
truejl
trueik
truekl
trueij
truejllj
trueikki
truekllk
trueijji
obsjl
obsik
obskl
obsij
VV
VV
VJJVJJ
VJJVJJ
VV
VV
39Simple Scalar Calibration ExampleSimple 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)
40The Calibration ProcessThe Calibration Process
• Solve for antenna-based gain factors for each scan on flux calibrator Ji(fd) (where Vij is known):
Solve also gain factors for phase calibrator(s), Ji(nt)
• Bootstrap flux density scale by enforcing constant mean power response:
• Correct data (interpolate J as needed):
21
2
)(
2
)(
)()(
inti
ifdi
ntinti
J
JJJ
trueijji
obsij VJJV *
obsijj
correctedij VJJV
i
1*1
true
4141
Antenna-Based CalibrationAntenna-Based Calibration
Visibility phase on a several baselines to a common antenna (ea17)
Calibration Effect on ImagingCalibration Effect on Imaging
J1822-0938
(calibrator)
3C391(science
)
How Good is My Calibration?How Good is My Calibration?• Are solutions continuous?
• Noise-like solutions are probably noise! (Beware: calibration of pure noise generates a spurious point source)
• 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
• Any evidence of unsampled variation? Is interpolation of solutions appropriate?• Reduce calibration timescale, if SNR permits
44A prioriA priori Models Required for Models Required for
CalibratorsCalibrators
Point source, but flux density not stable
Stable flux density, but not point sources
45Antenna-based Calibration Image Antenna-based Calibration Image
ResultResult
46Evaluating Calibration PerformanceEvaluating 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)
47Summary of Scalar ExampleSummary 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
48Full-Polarization Formalism Full-Polarization Formalism
(Matrices!)(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):
• 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 dual-polarization matrix description desirable or even required for proper calibration
VI
iUQ
iUQ
VI
V
U
Q
I
i
i
LL
LR
RL
RR
ISI Stokescirccirc
1001
010
010
1001
@
QI
iVU
iVU
QI
V
U
Q
I
i
i
YY
YX
XY
XX
ISI Stokeslinlin
0011
100
100
0011
@
49Full-Polarization Formalism: Signal Full-Polarization Formalism: Signal
DomainDomain
• Substitute:
• The Jones matrix thus corrupts the vector wavefront signal as follows:
qqqp
pqpp
ii
i
q
p
iiJJ
JJJJ
s
sss
@ ,
i
qqqpqp
qpqppp
i
q
p
i
qqqp
pqpp
i
q
p
iii
sJsJ
sJsJ
s
s
JJ
JJ
s
s
sJs
omitted) integral(sky @
50Full-Polarization Formalism: Full-Polarization Formalism:
Correlation - ICorrelation - I• Four correlations are possible from two polarizations. The outer
product (a ‘bookkeeping’ product) represents correlation in the matrix formalism:
• A very useful property of outer products:
qj
qi
pj
qi
qj
pi
pj
pi
j
q
p
i
q
p
jiobsij
ss
ss
ss
ss
s
s
s
sssV
*
*
*
*
*
*
trueijijjijijjiiji
obsij VJssJJsJsJssV
@@@@@ *****
53The Matrix Measurement EquationThe Matrix Measurement Equation
• We can now write down the Measurement Equation in matrix notation:
• …and consider how the Ji are products of many effects.
dldmemlISJJV mvlui
sky
jiobsij
ijij 2* ,@@@
54A Dictionary of Calibration A Dictionary of Calibration
ComponentsComponents• 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• G = electronic gain• B = bandpass response• K = geometric compensation
• 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
iiiiiiiiii FTPXEDGBKJ@@@@@@@@@@
55Ionospheric Effects, Ionospheric Effects, FF• The ionosphere introduces a dispersive phase shift:
• 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
cossin
sincos ;
0
0 iXY
i
iiRL eF
e
eeF
@@
Gin ,cm10in cm,in
deg 15.0
||2-14
||2
Bdsn
dsnB
e
e
60~ cm20 G;1~ ;cm10~ ||-214 BdsnTEC e
(TEC = Total Electron Content)
56Tropospheric Effects, Tropospheric Effects, TT
• 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?
– )
10
01
0
0t
t
tT pq@
57Parallactic Angle, Parallactic Angle, PP
• 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:
• 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!)
cossin
sincos ;
0
0 XY
i
iRL P
e
eP
@@
n declinatio angle,hour )( latitude,
)(cossincoscossin
)(sincosarctan)(
thl
thll
thlt
58Linear Polarization Position Angle, Linear Polarization Position Angle, XX
• 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
cossin
sincos ;
0
0 XY
i
iRL X
e
eX
@@
59Antenna Voltage Pattern, Antenna Voltage Pattern, EE
• 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)
),(0
0),(
mle
mleE
q
ppq
60Polarization Leakage, Polarization Leakage, DD
• 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)
–
1
1q
ppq
d
dD@
61““Electronic” Gain, Electronic” Gain, GG
• 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
q
ppq
g
gG
0
0
62Bandpass Response, Bandpass Response, BB
• 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
)(0
0)(
q
ppq
b
bB
63Geometric Compensation, Geometric Compensation, KK
• 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)
q
ppq
k
kK
0
0
64Baseline-based, Non-closing Effects: Baseline-based, Non-closing Effects:
M, AM, A• 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
65The Full Matrix Measurement The Full Matrix Measurement
EquationEquation
• The total general Measurement Equation has the form:
• S maps the Stokes vector, I, to the polarization basis of the instrument, all calibration terms cast in this basis
• Suppressing the direction-dependence:
• 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
ij
sky
mvluiijijijijijijijijijij AdmdlemlISFTPEDGBKMV ijij
@@@@@@@@@@ , 2
ijtrueijijijijijijijijijij
obsij AVFTPXDGBKMV
@@@@@@@@@
66 Solving the Measurement EquationSolving the Measurement Equation
• Formally, solving for any antenna-based visibility calibration component is always the same non-linear fitting problem:
• Viability of the solution depends on isolation of different effects using proper calibration observations, and appropriate solving strategies
truecorruptedij
solvej
solvei
obscorrectedij VJJV *
67Calibration Heuristics – Spectral LineCalibration Heuristics – Spectral Line
• Spectral Line (B,G):1. Preliminary G solve on B-calibrator:
1. B Solve on B-calibrator:
1. G solve (using B) on G-calibrator:
1. Flux Density scaling:
1. Correct:
1. Image!
obscorrected
fd
trueobs
trueobs
trueobs
VBGV
GGGG
VGVB
VGBV
VGV
@@
@@@@
@@
@@
@
11
2122
1
trueijijij
obsij VGBV
@@
68Calibration Heuristics – Continuum Calibration Heuristics – Continuum
PolarimetryPolarimetry
• Continuum Polarimetry (G,D,X,P):• Preliminary G solve on GD-calibrator (using P):
• D solve on GD-calibrator (using P, G):
• Polarization Position Angle Solve (using P,G,D):
• Flux Density scaling:
• Correct:
• Image!
obscorrected
fd
trueobs
trueobs
trueobs
VGDXPV
GGGG
VPXVGD
VPDVG
VPGV
@@@@
@@@@
@@@@
@@@
@@
1111
2122
11
1
trueijijijijij
obsij VPXDGV
@@@@
Recall:
• P = parallactic angle
• X = linear polarization angle
• D = polarization leakage
• G = electronic gain
• B = bandpass response
69New Calibration Challenges 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?
70
Why Not Just Solve for Generic Why Not Just Solve for Generic JJi i Matrix?Matrix?
• It has been proposed (Hamaker 2000, 2006) that we can self-calibrate the generic Ji matrix, apply “post-calibration” 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…
71SummarySummary
• 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
