Event Horizon Telescope
Memo Series

EHT Memo 2019-CE-03

Calibration & Error Analysis WG
Global calibration of instrumental polarimetric phase gains
S. Steel1,2,∗, M. Wielgus1,2,†, L. Blackburn1,2 , S. Issaoun3 , M. D. Johnson1,2
Jul 9, 2019 – Version 1.0

Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA
Black Hole Initiative at Harvard University, 20 Garden St., Cambridge, MA 02138, USA
3
Department of Astrophysics/IMAPP, Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

12AbstractThis memo summarizes formalism describing the polarimetric calibration, focusing on measuring LCP-RCP
phase offsets for each station from differences between phases in LL and RR visibilities on baselines to a reference
station. We introduce a new calibration procedure utilizing multiple sources and globally fitting to the phaseoffsets in order to prevent overfitting to the intrinsic source polarimetric properties. In this procedure we calculatea family of polynomial and piecewise linear fits and select the optimal one using a proper information criterion.

∗ sangersteel@gmail.com† maciek.wielgus@gmail.com

1

Polarimetric calibration formalism

1.1

Principles of interferometry

The quantity measured in a radiointerferometric observation is the mutual coherence Γ of the incoming radiation, between
two observing points r1 and r2 . We denote the electromagnetic field measured by each point (each telescope) as E1 and E2
respectively. The mutual coherence function between these two observation points can be written as
1T →∞ 2T

Γ12 (τ, r1 , r2 ) = lim

Z T−T

E1 (t) E2∗ (t − τ) dt(1)for τ representing the time offset between the arrival of signal at the two telescopes. T is the averaging time [1, 2]. Once theoffset τ is accounted for [3, 4], we can represent the coherence function as the value of correlation between the electromagnetic
fields E1 and E2 ,
Γ12 = hE1 , E2∗ i .
(2)
We refer to this correlated product as a visibility function V. By the Van Cittert-Zernike theorem the visibility function isequivalent to the Fourier transform of the brightness in the sky I, that is"V(u, v) =I(l, m) e−2πi(ul+vm) dl dm .
(3)
Coordinates (u, v) represent the projection of baseline r2 − r1 onto the plane of observations, in the units of wavelength λ. The
coordinates l and m are direction cosines to the source measured with respect to the u and v axes. The overarching goal ofvery long baseline interferometry (VLBI, [1]) as is concerned by the Event Horizon Telescope (EHT) is then to characterize
the spatial distribution of the source brightness I(l, m) based on the sparsely sampled Fourier domain data V(u, v).

1.2

Polarimetric calibration

Polarized emission from the astronomical source may encode important characterization of the magnetic field geometry [5].
A full polarized state of the radiation field can be measured in a VLBI observation by using two receivers, susceptible toorthogonally polarized components. In case of the circular polarization basis measurements, a following relation with theStokes components [6, 7] is found˜ ∗∗ hE1,R E2,R i + hE1,L E2,L i  I1,2  VRR + VLL ∗∗Q̃1,2  1  hE1,L E2,Ri + hE1,R E2,Li  1  VLR + VRL  = .
= ∗∗
Ũ1,2  2 i (hE1,L E2,R i − hE1,R E2,L i) 2 i (VLR − VRL )

∗
∗
Ṽ1,2
VRR − VLL
hE1,R E2,Ri − hE1,L E2,Li(4)

in which the left hand side of the equation 4 represents Stokes parameters, and the right hand side represents the measured
circular polarization components, with indices ”R” and ”L” denoting right and left circular polarization, respectively. The
majority of the EHT array telescopes measure the electric field in a circular basis (with the exception of ALMA, which
performs measurement in a linear basis, subsequently converted to a circular basis at the correlation stage [8]). In equation 4,I˜1,2 denotes the Fourier transform of the total intensity of the light I, and the remaining parameters can be thought as degrees
to which the light is polarized linearly and circularly. Particularly, Stokes V, is related to the amount of circular polarization.
If Ṽ = 0 then VRR = VLL , so there is no phase difference between the two. This constitutes an approximation that is useful forthe data calibration and in many cases sufficiently accurate, as based on physical motivations we would expect I˜ > Q̃ ≈ Ũ > Ṽ.
However, it is scientifically interesting to be able to detect Ṽ, or at least to put constraints on its magnitude. For that reason inthis memo we attempt to relax the V = 0 calibration assumption.

2

The mathematical framework for the polarimetric calibration of the measured quantities E0R and E0L can be described in termsof Jones matrices J [5]!
!
!
!
!
!
ER0
GR 0
1 DR e−iψ 0 ERER==J,(5)
E L00 G L DL 10eiψ E L
EL
where ER and EL model the true underlying values. Different components of the equation 5 correspond to:
1. field rotation angle ψ, geometric effect related to rotation of polarized component on the sky, known a priori,
2. leakage terms D, parametrizing susceptibility of polarimetric feeds to the perpendicular polarization component, often
poorly characterized a priori,
3. gains G, representing scaling of the signal energy related to the sensitivity of telescopes, for which a priori characterization is present, [9], atmospheric variability effects stabilized at the fringe fitting calibration stage [4], and instrumental
polarimetric phase offsets, that are relevant for this memo.
From equation 5, combined for a pair of telescopes, we can derive the visibility calibration formula!
!
00
VRR VRL
VRL0†
0−1 †−10−1 †−1 VRR= V12 = J−1V(J)=J(E(J−1 )† ,⊗E)
(J)
=J1
12211212
VLR VLL
V0LR V0LL 2(6)describing the relation between measured visibilities V0 and the true underlying values V.

1.3

Calibration of polarimetric phase gains

The aim of this memo is to describe the calibration of polarimetric phase gains, corresponding to the phase offset between twopolarized feeds measured by the telescope. To this end, we assume that field rotation angle ψ, leakage terms D and amplitude
components of gains G have been accounted for. In that case equation 5 simplifies to!
!
!
!
ER0
10ERER== Jϑ.
(7)
E L00 eiϑRL E L
ELIf additionally we assume absolute calibration of one station (treated here are the first one in the pair), that is Jϑ,1 is an identity
matrix, then on baselines to this station we find
VRR
VLR!
VRL
1=VLL
00
1

!−10
VRR
V0LR!
0 1VRL
0VLL  0!−1 †0 = VRRiϑRL,2eV0LR0!
0VRL
1V0LL 00eiϑRL,2!
.

(8)

Under these assumptions the problem reduces to finding a family of phase offsets ϑRL,i for all telescopes apart from theabsolutely calibrated reference station. We also see that!
0 !
VRRVRRε ≡ arg= arg− ϑRL,2 = ϑ0RL,2 − ϑRL,2(9)VLLV0LLand finallyϑ0RL,2 = ϑRL,2 + ε.

(10)Equation 10 gives a relation between measured phase differences between RR and LL visibilities on baselines to an absolutelycalibrated station ϑ0RL,2 , instrumental phase offset between the right and left circular polarization receivers on non-calibratedstation ϑRL,2 and unknown true phase difference between RR and LL visibilities ε, expected to be small, ε ≈ 0. The Stokes
V = 0 calibration is tantamount to assuming ϑ0RL,2 = ϑRL,2 . In this work we attempt to disentangle source dependent ε fromthe instrumental effect ϑRL .
3

measured R-L phase offset ϑ0RL

60

true instrumental offset ϑRL

50

ϑ0RL,A = ϑRL + εA (source A)
ϑ0RL,B = ϑRL + εB (source B)

40
30
20
100−100246810timeFigure 1: Illustration of the concept of a global in time polarimetric phase gains fit across multiple sources. When using a single source,
there is a risk of overfitting to source intrinsic properties (thin dashed lines). This risk can be reduced by using multiple sources.

2

Calibrating polarimetric phase gains

Polarimetric calibration of the EHT data is performed in the framework of the eat library [10]. A standard approach topolarimetric phase gains would be to individually treat observations of different sources, calibrating them locally in time to amodel of equal mean RR and LL visibility phases. This poses a risk of wiping out signatures of Stokes V by overfitting tosource-specific phases ε.
The approach we investigate here is different in two ways:
1. global fits in time are utilized rather than local in time calibration. We can often expect that the instrumental offsets ϑRLshould either be constant in time, or exhibit linear trends,
2. utilizing multiple sources for the calibration to prevent the local overfitting.
A schematic description of this approach is shown in Figure 1. We assume perfect calibration of the ALMA array, andestimate station gain phases on baselines to ALMA within the framework described in Section 1.3. We utilize aggregated data
representing multiple observed sources (e.g., the science target source and its calibrators), potentially representing multiple
days of an observational campaign. We then fit multiple different models to the data, that is, to the measured phase offsets
0between RR and LL visibilities on ALMA baselines, ϑ0RL = arg(VRR/V0LL ). The models are fitted utilizing the data thermaluncertainties σth,i and estimated systematic uncertainties σsys , with weights

2
2 −1
wi = σ−1.
(11)i = σsys + σth,iFor the presented test we have used σsys = 2◦ . Models that are fitted to the data fall into two categories:
1. piecewise regression, linear on subdomains, specified number of segments, boundaries between segments are solved for,number of degrees of freedom = 3 × number of segments. Fitted using mlinsights.mlmodel.PiecewiseRegressor
within the scikit-learn framework [11],42. global polynomials, specified order, number of degrees of freedom = polynomial order +1. Fitted using numpy.polifit.
Finally, the optimal model is selected based on an information criterion and added to a calibration text file, utilized by theHOPS-EHT data reduction pipeline [4, 10].

2.1

Information criteria

The two popular methods for optimal model selection are Akaike Information Criterion (AIC, [12]) and Bayesian Information
Criterion (BIC, [13]). Formulae governing the criteria are
AIC = 2k − 2 ln (L̂) ,(12)BIC = ln (N) k − 2 ln (L̂) ,(13)in which k represents the number of parameters (degrees of freedom) in the model, N represents the number of datapoints, andL̂ is the maximum of the likelihood function for the model. Approximating distributions associated with measurements withnormal distributions of known (but different for each datapoint) standard deviation σi given by equation 11, we findN

1 X xi − miln L̂ ≈ −2 i=1
σi!21+ C = − χ2 + C.
2(14)Here xi is an i-th measurement with uncertainty σi , mi is model prediction for the i-th measurement and C is a modelindependent constant. Then, neglecting the constant, the criteria become simply
AIC = 2k + χ2 ,(15)BIC = ln (N) k + χ .
2(16)The χ2 square term is reduced when more degrees of freedom are added to the model. This behavior is counteracted by theterm proportional to the number of degrees of freedom k. The optimal model in this framework is simply the one resulting inthe smallest value of the information criterion.
We illustrate the model evaluation with the Global Millimeter VLBI Array data from 2017 observation campaign on Galactic
Center [14] in Figures 1-3. Baseline connecting ALMA (AA, Chile) and Green Bank Telescope (GB, West Wirginia, USA) isshown.
AIC: 659.70, BIC: 671.62−201749+096
NRAO530
SGRA
1921-293−60−80−100−120

1749+096
NRAO530
SGRA
1921-293−40
R-L phase (deg)R-L phase (deg)−40

AIC: 429.44, BIC: 453.27−20−60−80−1000.20

0.25
0.30
0.35
AA-GB: MJD - 57846.15−1200.40

50.20

0.25
0.30
0.35
AA-GB: MJD - 57846.150.40

AIC: 346.41, BIC: 382.15−201749+096
NRAO530SGRA
1921-293−60−80−100−120

1749+096
NRAO530
SGRA
1921-293−40
R-L phase (deg)R-L phase (deg)−40

AIC: 350.28, BIC: 397.94−20−60−80−1000.20

0.25
0.30
0.35
AA-GB: MJD - 57846.15−1200.40

0.20

0.25
0.30
0.35
AA-GB: MJD - 57846.15

0.40

Figure 1: Examples of models with piecewise linear fit with 1,2,3 and 4 segments used. Both criteria favor model with 3 segments(bottom left).

AIC: 667.80, BIC: 675.74−201749+096
NRAO530
SGRA
1921-293−60−80−1000.20−80−1200.400.20−80−1000.40

1749+096
NRAO530
SGRA
1921-293−40−600.25
0.30
0.35
AA-GB: MJD - 57846.15
AIC: 411.87, BIC: 431.73−201749+096
NRAO530
SGRA
1921-293−40
R-L phase (deg)

0.25
0.30
0.35
AA-GB: MJD - 57846.15
AIC: 409.93, BIC: 425.81−20−120−60−100

R-L phase (deg)−1201749+096
NRAO530
SGRA
1921-293−40
R-L phase (deg)R-L phase (deg)−40

AIC: 472.26, BIC: 484.17−20−60−80−1000.20

0.25
0.30
0.35
AA-GB: MJD - 57846.15−1200.40

0.20

0.25
0.30
0.35
AA-GB: MJD - 57846.15

0.40

Figure 2: Examples of models with polynomial fit of order 1, 2, 3, 4. Both criteria select third order polynomial (bottom left) as an
optimal polynomial degree choice. However, both also point at 3-segments piecewise model as better than any polynomial one.

650

R-L phase (deg)251749+096
NRAO530

SGRA
1921-293

0−25
−50
−75
−100
−125

0.2

0.3

0.4

0.2

0.3

0.4

AA-GB: MJD - 57846.15
Figure 3: Comparison of R-L phase offsets before and after application of the described calibration procedure.

2.2Discussion

The method for calibrating instrumental polarimetric phase gains using global in time, multi-source modeling was presented.
The method constitutes an alternative to a more standard approach of local in time aligning of right and left circular polarization
components phases, and offers a chance to relax the assumptions of the alternative procedure. Because equality between R
and L phases is not assumed, data calibrated with this method can be used to obtain more reliable upper limits on the Stokes V
component value.
The choice of a particular information criterion is not entirely clear. Both BIC and AIC are the option in the current
implementation, and they are often consistent, as in the presented example.
R-L phase offsets could be measured in the future using a tone injection into the recorded data stream. In this way the offsets
could be measured, rather than modeled.

7

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