The calibration of interferometric visibilities obtained with single-mode optical interferometers. Computation of error bars and correlations

10 Pages·0.27 MB·English

by  G. Perrin

Astronomy & Astrophysics manuscript no. February 2, 2008
(DOI: will be inserted by hand later)
The calibration of interferometric visibilities obtained with
single-mode optical interferometers
Computation of error bars and correlations
1
G. Perrin
3
0 LESIA,FRE 2461, Observatoire de Paris, section de Meudon, 5, place Jules Janssen 92190 Meudon, France
0 e-mail: [email protected]
2

n Received 4 September2002 / Accepted 19 December2002
a
J
Abstract. I present in this paper a method to calibrate data obtained from optical and infrared interferometers.

8 I show that correlated noises and errors need to be taken into account for a very good estimate of individual

error bars butalso when model fittingthedatato derivemeaningful model parameters whose accuracies are not
1
overestimated. It is also shown that under conditions of high correlated noise, faint structures of the source can
v
be detected. This point is important to define strategies of calibration for difficult programs such as exoplanet
0
detection. The limits of validity of theassumptions on the noise statistics are discussed.
4
1
1 Key words.techniques:interferometric – methods: data reduction
0
3
0 1. Introduction 2. Principles of data reduction and calibration
/
h
p With optical-infrared interferometry becoming more ma- In this section the general scheme of data reduction is
-
o ture, the quality of visibility measurements have become reviewedto introduce the vocabularyand notations. Two
r anissue.Single-modeinterferometers(seeSect.2.3)allow mainstepsaretobeconsidered.Inthefirstone(Sect.2.1),
t
s onetoeliminatenon-stationaryeffectsbyfilteringoutthe thefringeprocessing,fringecontrastsarederivedfromraw
a
spatial modes of turbulence. The response of interferom- signals.Becauseofcontrastlosses,fringecontrastsarecal-
:
v eters is therefore very stable and the issue of estimating ibrated in a second step (Sect. 2.2) to provide the visibil-
i
X the accuracies of non-biased data is raised.The final visi- ities directly linked to the spatialintensity distribution of
bility estimate is a complex quantity as it is a non-linear the source.
r
a mix of noisy measurements and of parameter estimates
with their own uncertainties. Estimating the stability of
the instrument, a crucial point for calibration,and the fi- 2.1. Fringe contrasts estimates
nalerroronvisibilitiesisthereforenon-trivialandmustbe
considered with caution. Moreover, data analysis mainly Inthefollowingwewilldistinguishbetweenthefringecon-
consists of model fitting the final visibilities and the mat- trastobtainedonasourceandthevisibilityofthissource.
ter of their potential correlations becomes important, es- The fringe contrast measured from a single exposure or
pecially if some very faint structures are looked for, as is scanis calledthe coherencefactor andis notedµwhereas
the case in extra-solarplanet detection. the visibility is noted V.
InthispaperIproposeamethodtomeetthesechallenges. Whatever the beamcombining technique, µ being the
Themethodhasbeentestedandelaboratedalongwiththe modulus of a complex number, unbiased estimators are
FLUORinterferometer,the firstsingle-modeinterferome- only obtainedforsquaredquantities fromwichbiasesdue
ter. This method was firstpublished in Perrin (1996)and to additive noise can be subtracted. In the future, phase
used in Perrin et al. (1998). It is updated and improved referencing techniques may allow one to directly measure
in this paper by accounting for correlations. complex visibilities (real and imaginary parts) but this is
not the case yet and I will only consider measurements of
fringe contrast moduli. The result of the processing of a
series of scans on a single source is a series of realizations
Send offprint requests to: G. Perrin of the µ2 estimator or is an average value of the realiza-
2 G. Perrin: The calibration of interferometric visibilities obtained with single-mode optical interferometers
30 30 35
71 UMa d Sge V636 Her
25 <m 2> = 0.8022 25 <m 2> = 0.9842 30 <m 2> = 0.7268
rms = 0.1474 rms = 0.1301 rms = 0.0975
20 r = 0.67 20 r = 0.57 25 r = 0.27
20
Count 15 Count 15 Count
15
10 10
10
5 5 5
0 0 0
0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1
m2 m2 m2
Fig.1.ExamplesofsquaredcoherencefactorhistogramsobtainedwithFLUORinoneoftheinterferometricchannels.
About100interferogramshavebeenrecordedforeachobject.Themeanandrmsofindividualmeasurementsaregiven
forthischannel.Thecorrelationfactorr measuresthenoisecorrelationbetweenthetwointerferometricchannels.The
amount of atmospheric piston is decreasing from the left to the right.
tions with a 1σ error bar if their statistical distribution aperfectsingle-modeinterferometerinwhichsingle-mode
can be trusted to be Gaussian. fibersareusedtofilterthephaseaberrationsproducedby
atmosphericturbulence(exceptforthepistonmode)these
non-stationaryeffectsareeliminated.Inordertoavoidin-
2.2. Necessity for a calibration
stabilities dueto the pistonmode,the fringesarescanned
Theaverageµ2 isnotdirectlyanestimatorofthesquared at a frequency far above the characteristic frequency of
modulusofthevisibilityofthesourcebecausesomephysi- piston. In interferometers where the piston is stabilized
calphenomenadegradethecoherencefactor.Amongthese by a fringe tracking servo loop, this issue is solved. The
phenomena, polarization mismatches between the inter- remaining main sources of variation of the transfer func-
ferometric beams are the most common. Without perfect tion are basically temperature drifts and differential po-
beamcleaningby afiber,atmosphericturbulence alsode- larisation effects due to the change of beam inclination
grades the fringe contrast. It is necessary to estimate the onthe firstmirrorswithchangingpositions ofthe sources
lossofcoherenceonacalibratorsourceforwhichthe visi- in the sky. In both cases the transfer function drifts are
bilityis known.AtransferfunctionT isobtainedbycom- veryslowandagoodestimateofthetransferfunctioncan
puting the ratio of the measured coherence factor µ to beobtainedbyinterpolatingtwoestimatesbracketingthe
c
the expected visibility V . With squared quantities this sourceto be calibrated.This has been demonstratedwith
exp
yields: the FLUOR beamcombiner, as will be shown is Sect. 7.2.
In the following we will therefore consider that the effi-
µ2
T2 = c . (1) ciency of the interferometer is continuously assessed by
Ve2xp observingcalibratorsbefore andafter science sources.We
will not consider the case where the transfer function is
If the instrument is stable enough then the estimate of
derived by averaging individual transfer functions on a
T2 obtained on the calibrator can be used to derive an
large temporal scale, as this is not required for a single-
estimated visibility for the source from the measured co-
mode interferometer. This technique does not allow one
herence factor:
to assess the quality of the calibration in detail.
µ2
V2 = =coT2µ2. (2)
T2
whereI callcoT2 the squaredco-transferfunction. Its use
will be detailed in Sect. 4.
3. Estimating fringe contrasts
2.3. Assumptions in the case of a single-mode
interferometer
This section focuses on estimating the statistical prop-
The calibration process may fail if the assumption that erties of fringe contrasts. I will not describe the method
the transfer function is stable is wrong.This usually hap- to compute coherence factors from single exposures and
pens if non-stationary processes like atmospheric phase I will refer the reader to appropriate articles in the next
turbulence play a role in the fringe formation process. In paragraphs.
G. Perrin: The calibration of interferometric visibilities obtained with single-mode optical interferometers 3
May 15, 2000
1,8
1,6
2T 1,4
o
c
1,2
1
6:20 6:40 7:00 7:20 7:40 8:00 8:20 8:40 9:00
UT Time (hh:mm)
May 22, 2000
1,8
1,6
2T 1,4
o
c
1,2
1
5:50 6:40 7:30 8:20 9:10 10:00 10:50 11:40 12:30
UT Time (hh:mm)
Fig.2. Examples of squared co-transfer functions measured with FLUOR. The two curves for each night correspond
to the two interferometric channels of the coaxialinterferometer.The full circles are the squaredco-transferfunctions
measuredoncalibratorswhereastheopencirclesarethe valuesinterpolatedatthe time whenthe sciencetargetswere
observed. 1σ error bars are displayed.
3.1. Single-channel spatial modulation interferometer The estimate of the variance of the coherence factor esti-
mator µ2 is then:
Inamultiaxialinterferometer,distantparallelbeamsfeed S2
Var(µ2)= . (4)
a focusing optic. The beams are recombined in the focal
N
plane where they overlapat the focus locus.The modula-
tionis spatialasthefringe phasevariesacrossthediffrac-
3.2. Two-channel temporal modulation interferometer
tionpattern.Amethodtoderivefringecontrastshasbeen
published by Mourard et al. (1994) in the case of GI2T. Inacoaxialinterferometer,beamsaresuperimposedinpo-
The method has been adapted to AMBER which is a sition and in direction.This can be realizedwith a beam-
single-modemultiaxialinterferometer(Chelli et al. 2000). splitter or with a fiber coupler. A relative phase between
Thankstothefilteringofthenon-stationarymodesoftur- the beams is introduced by setting an optical path differ-
bulence, the statistics of µ2 can be well approximated by ence. This is achieved with a moving mirror in one of the
a Gaussiandistribution.This will be demonstratedinthe twobeams,hencethetemporalmodulationofthephase.A
case of the data obtained with FLUOR in Sect. 7.1. The method to compute fringe contrastsfor this type of inter-
estimate of the squared coherence factor is therefore the ferometer is described in Coud´e du Foresto et al. (1997).
mean of the distribution of the realizations denoted µ2. Amorerecentmethodbasedonwaveletsanalysishasbeen
An unbiased estimate of the variance of individual mea- proposed by Segransan et al. (1999). The method to ob-
surements is: tain an estimate of the coherence factor without the pho-
ton noise bias is explained in Perrin (2002). A prototype
instrument for this kind of interferometer is the FLUOR
beamcombiner.
ThedifferencewiththepreviousinterferometerofSect.3.1
N−1
S2 = 1 (µ2 −µ2). (3) is that it produces two interferometric beams and there-
N −1 n foretwosetsofcoherencefactorsestimates.Thestatistics
n=0
X
4 G. Perrin: The calibration of interferometric visibilities obtained with single-mode optical interferometers
1
Without correlations
f =17,05±0,13 mas
0,8 UD
0,75 c 2=0,92
0,7
With correlations
0,65
0,6 f =17,27±0,31 mas
y 0,6 UD
t
li 0,55 c 2=13,30
i
b
i 0,5
s 30 32 34 36 38 40
i
V 0,4 0,75
0,7
0,65
0,2 0,6
0,55
0,5
30 32 34 36 38 40
0
0 10 20 30 40 50 60
Spatial frequency (cycles/arcsec)
Fig.3. Example of visibility fit. The source is SW Vir and the model is a uniform disk. Errors are 1σ errors. Open
circlesanddashedlinearethevisibilitiesandmodelfitcomputedwithouttakingcorrelationsintoaccount.Fullcircles
andcontinuouslinearetheequivalentwiththemethoddescribedinthispaper.Thetwocasesareseparatelypresented
in the little windows.
ofeachsetcanbe wellapproximatedbya gaussianstatis- 4. Estimating the transfer function
tics as will be shown in Sect. 7. The photon noises of the
twointerferometricsignalsareuncorrelated.Theread-out Itisassumedthatthetransferfunctionisaslowlyvarying
noisesaregenerallyconsidereduncorrelatedbutsomecor- function which is rapidly sampled. This property will be
relationmayoccurasdifferentpixelssharethesameread- illustrated with real data in Sect. 7. It is then legitimate
outelectronics.In additionthe two beams suffer fromthe tolinearlyinterpolatethesquaredtransferfunctionatthe
same turbulence effects (residual piston and photometric time when the science source was observed. Because the
beamfluctuations)whichgeneratesomenoiseinthemea- variances of products of random variables are more eas-
surements. Part of the noise is therefore common to the ily calculated than those of ratios, the reciprocal of the
two signals and the coherence factors estimates are corre- squared transfer function, the squared co-transfer func-
lated. The correlation factor r is directly estimated from tion, is interpolated instead of the squared transfer func-
the µ2 distributions: tion.The use ofone orthe otheris equivalent.Inorderto
begeneral,twointerferometricoutputsarealwaysconsid-
ered. The particular case of the multiaxial interferometer
willbeconsideredindiscussions.Theexpressionofthein-
hµ2−µ2ihµ2−µ2i terpolated co-transfer functions in the two outputs of the
r = 1 1 2 2 , (5) instrument is :
Var(µ2)Var(µ2)
1 2
p coT2 = xVa2 +(1−x)Vb2
where the subscripts describe the two interferometric coT12 = xVµa1a22 +(1−x)Vµb1b22 (6)
channels.  2 µa22 µb22

G. Perrin: The calibration of interferometric visibilities obtained with single-mode optical interferometers 5
X Her G Her
1 1
f =12,38±0,04 mas f =16,32±0,08 mas
UD UD
0,8 c 2=1,48 0,8 c 2=1,03
y 0,6 y 0,6
bilit bilit
Visi 0,4 Visi 0,4
0,2 0,2
0 0
0 20 40 60 80 100 0 20 40 60 80 100
Spatial frequency (cycles/arcsec) Spatial Frequency (cycles/arcsec)
BK Vir R Hya
1 1
f =11,49±0,40 mas f =23,75±0,11 mas
UD UD
0,8 c 2=0,23 0,8 c 2=3,99
y 0,6 y 0,6
bilit bilit
Visi 0,4 Visi 0,4
0,2 0,2
0 0
0 20 40 60 80 100 0 20 40 60 80 100
Spatial Frequency (cycles/arcsec) Spatial Frequency (cycles/arcsec)
Fig.4. Examples of fits of visibility data with a uniform disk model. All correlations are taken into account. Errors
are 1σ errors.The visibility point around 85 cycles/arcsec in the G Her panel is not taken into account in the fit.
Va and Vb are the expected visibilities of calibrators A the confidence interval.The upper andlower limits of the
and B. Coefficient x is the relative time distance between confidence interval at 68.3% for a ratio of two Gaussian
theobservationofthesciencetargetandtheobservationof random variables α and β are given by (Pelat (1992)):
calibrator A. The expected visibilities are supposed to be
α± β/σβ σ
Gthaeuvsasriaiannrcaensdoofmthevasrqiuaabrleesd.cDor-otrpapnisnfgertfhuencchtiaonnnsealriendeqicueasl, L± = ((α/σα)2+(β/σβ)2−1)1/2 α (9)
β∓ α/σα σ
to: ((α/σα)2+(β/σβ)2−1)1/2 β
Var(coT2)=x2Var(coTa2)+(1−x)2Var(coTb2) (7) This interval is not symmetric. I choose as error bar the
larger distance of the mean to the limits, thus slightly
when the two calibrators are different. When the two cal-
overestimating the error. The probability that the true
ibrators are the same then the variance is equal to:
valueisinthisintervalaroundthemeanisthereforelarger
Var(coT2)= x2Var(coTa2)+(1−x)2Var(coTb2) (8) than 68.3%. In the following I will consider that ratios
x(1−x) of Gaussian variables are Gaussian variables. This is not
+2 Var(Va2) rigourously true but this allows one to derive expressions
µa2µb2
otherwisedifficulttohandle.TheGaussianandLorentzian
The squared co-transfer functions estimated on the cali- functions are both bell curves, the wings of the latter be-
bratorsare ratios of Gaussianestimators.These new ran- ing more extended than that of the Gaussian. The ap-
dom distributions are not Gaussian. They are Cauchy proximation amounts to giving more weight to the center
distributions (the density probability of which is a of the lorentzian. The other possibility would be to apply
Lorentzian) with no mean and no variance. By analogy Monte-Carlo or bootstrap methods to all error bar com-
with the standard deviation of a Gaussian law, an esti- putations, which would make data reduction a very long
mate of the uncertainty can be derived from the width of process for a verylimited gain. Nevertheless,this method
6 G. Perrin: The calibration of interferometric visibilities obtained with single-mode optical interferometers
willhavetobeappliedoncetocomputethecorrelationbe- expected visibilities in the two interferometric channels.
tween the visibilities of a two-channel coaxial interferom- This can be illustrated by the equation below when the
eter (Sect. 5). Consistency of error bars will be addressed first and second calibrators are different:
in Sect. 7. This final consistency is the justification of the
approximations performed. x2 µ21 µ22 Var(Va2)+(1−x)2 µ21 µ22 Var(Vb2)
ρ = µa12µa22 µb12µb22 (14)
12
Var(V2)Var(V2)
1 2
5. Estimating visibilities
Whenthetwocalibrpatorsarethesamethecorrelationfac-
5.1. Mean and variance of channel visibilities
torhastheparticularexpression(forsakeofsimplicitythe
twoestimatesofthevisibilityatslighltydifferentbaselines
The single-channelsquaredvisibility in the case ofa mul-
are supposed to be the same):
tiaxial interferometer or the two-channel squared visibili-
ties of a coaxial interferometer are simply the product of
the science target squared coherence factors and squared x µ21 +(1−x) µ21 x µ22 +(1−x) µ22 Var(Va2)
co-transfer functions of Eq. (6) yielding: ρ =(cid:20) µa12 µb12(cid:21)(cid:20) µa22 µb22(cid:21) (15)
12
Var(V2)Var(V2)
1 2
V12 = xµµa1212Va2+(1−x)µµb1212Vb2 (10) ItisnoweasytoseepfromEq.(10)thatifthe noiseonthe
V22 = xµµa2222Va2+(1−x)µµb2222Vb2 mtieesasounrethmeenextspeisctneedglviigsiibblielitwieitshofretshpeeccatltibortahteorusntcheerntatihne-
correlation tends towards 1. In this case, the second in-
The variances of the two-channel squared visibilities are
terferometric channel brings no extra information except
calculated with the variances of the squared co-transfer
a consistency check and the precision on the visibility of
functions and of the squared coherence factors with the
the science targetis directly proportionalto the precision
following formula :
on the expected visibilities of the calibrators.
Var(AB)= Var(A)Var(B)+A¯2Var(B) (11)
+B¯2Var(A) 5.3. Comments on visibility variances
We therefore have: It is also interesting to analyze the propagation of noises
inthevisibilityestimates.Forexample,ifthenoiseonthe
Var(V2)= Var(µ2)Var(coT2)+µ2Var(coT2) (12)
1 1 1 1 1 measurements is negligible, it is possible to evaluate the
+coT2Var(µ2) amountof variance due to the uncertainty onthe calibra-
1 1
Var(V2)= Var(µ2)Var(coT2)+µ2Var(coT2) torsvisibilities (ordiameters).Droppingchannelindices I
2 2 2 2 2
obtain:
+coT2Var(µ2)
2 2
2
µ2
Var(V2)| = x2 Var(Va2) (16)
µ,µa,µb
5.2. Correlation between two channel visibilities "µa2#
2
Theaboveequationsdefinethe uncertaintiesonthe chan- +(1−x)2 µ2 Var(Vb2)
nelestimates ofthe visibilities.In the case ofa multiaxial
"µb2#
interferometer, this is the final estimate of the visibility.
Inthecaseofcoaxialinterferometers,thetwoestimatesof Ifthe uncertaintiesonthe calibrators´squaredvisibilities
the visibility need to be averaged at this stage. For this, are equal to 1% and if the coherence factors are all equal
itis necessaryto assessthe correlationfactorbetweenthe to 1 and observations are equally spaced in time then the
twoestimates.Bydefinition,thecorrelationfactorisequal uncertainty on the measured squared visibility is equal
to: to 0.7%. This equation also shows that the smaller the
visibility of the calibrator, the more amplified the noise
h(V2−V2)(V2−V2)i
ρ = 1 1 2 2 (13) onitsexpectedvisibilityis.Symmetrically,thesmallerthe
12
Var(V2)Var(V2)
1 2 visibilityofthesciencetarget,thesmallerthecontribution
This qupantity is defined by sums, ratios and products of to the noise of the calibrators´ expected visibilities.
ten random variables. The correlation factor has to be
computedwithaMonte-Carlomethodbysimulatingeach
5.4. Final estimate of the visibility in a two-channel
random variable from its mean and variance (assuming
interferometer
it has Gaussian statistics) and by correlating the series
of V2 and V2. In the special case when the correlations The final squared visibility V2 is estimated from the two
1 2
betweenmeasuredquantitiescanbeneglectedbecausethe squared visibilities obtained from each output of the in-
correlatednoise levelis farbelow that ofthe uncorrelated terferometerV2 andV2 andtheirrespectivevariances(or
1 2
noise, it is to be noticed that there is still a correlation equivalently uncertainties σ and σ ). I define the final
1 2
due to the common estimated values of the calibrators estimate V2 as being the least squares fit estimator of
G. Perrin: The calibration of interferometric visibilities obtained with single-mode optical interferometers 7
the squared visibility as this is an optimal estimator for matrix. The correlations may be as high as the correla-
Gaussian random variables. In this fit, the model is lin- tionsbetweenthetwochannelsofacoaxialinterferometer
ear and has only one parameter: V2. Let us call C the asallcalibratorsarecommontoallbaselines.Themethod
covariance matrix of V2 and V2: used in Sect. 5.2 should be applied. A correlation matrix
1 2
for the µ2 should be computed first. The final correlation
σ2 ρ σ σ
C = 1 12 1 2 . (17) factorsforthefinalvisibilityestimatesarethencomputed
ρ σ σ σ2
(cid:20) 12 1 2 2 (cid:21) with a Monte-Carlo method.
The quantityto minimize inthe leastsquaresfitcanthen
be written: 6. Correlations between non-simultaneous
S(V2)= V12−V2 tC−1 V12−V2 =YtC−1Y. (18) visibilities
(cid:20)V22−V2(cid:21) (cid:20)V22−V2(cid:21) Visibilities obtained on different baselines or on different
daysareusuallyconsideredindependent.Inthe lastpara-
It can be shown that the minimum is reached for:
graph, we focused on the possible correlations of visibili-
V2 =(XtC−1X)−1XtC−1 V12 . (19) ties recorded simultaneously on baselines with telescopes
V2 in common. In this section, we will consider the correla-
(cid:20) 2 (cid:21)
tion due to common uncertainties in the calibration pro-
with
cess for independent baselines or for visibilities measured
1 atdifferenttimes.The calibrationofthe transferfunction
X = , (20)
1 mayhaverequiredustousethesamecalibratorshencethe
(cid:20) (cid:21)
samediameterestimates.Theerrorsonthevisibilitiesare
the uncertainty on V2 being: therefore not independent. It is the purpose of this para-
σ2 =(XtC−1X)−1XtC−1. (21) graph to establish a method to compute this correlation
V2 and, more important, to be able to trace it to compute it
The above equations yield the final visibility estimate: a posteriori long after taking the data at the telescopes.
LetS andS betwospatialfrequenciesatwhichsquared
1 2
V2 = V12(σ22−ρ12σ1σ2)+V12(σ12−ρ12σ1σ2) (22) visibilities V2(S1) and V2(S2) have been measured. The
σ12σ22−2ρ12σ1σ2 visibilityestimatesofEqs.(10)and(22)cantaketheform:
and the associated error: V2(S ) = α Va2(S )+β Vb2(S )
1 1 1 1 1 (25)
σ2 = (1−ρ212)σ12σ22 (23) (V2(S2) = α2Vc2(S2)+β2Vd2(S2)
V2 σ2+σ2−2ρ σ σ
1 2 12 1 2
wherethecalibratorsareA,B,CandD.Tosaveroom,the
If ρ12 = 1 then the two single-output squared visibilities calibratorvisibilities arereplacedbythe capitalletters.It
V2 and V2 are fully correlated and the above expression can be shown that the correlationfactor between the two
1 2
does not apply. In this case V is equal to one of the two squared visibilities is:
single-output visibilities with its associated error bar.
The quality of the fit is expressed by the χ2: ρ(V2(S ),V2(S )) = α1α2 Var(A)Var(B)ρ(A,C)(26)
1 2
Var(V2(S ))Var(V2(S ))
(V2−V2)2 p 1 2
χ2 = σ12+σ122−2ρ212σ1σ2 (24) × αp1β2 Var(A)Var(D)ρ(A,D)
Var(V2(S ))Var(V2(S ))
p 1 2
This parameteris importantbecause itallowsus to check
the consistency of the instrument and of the method to βp1α2 Var(B)Var(C)ρ(B,C)
×
measure the visibilities and the error bars. If all assump- Var(V2(S ))Var(V2(S ))
p 1 2
tionsarecorrectthentheχ2 shouldbeequalto1onaver-
βp1β2 Var(B)Var(D)ρ(B,D)
age.IntheFLUORsoftwareweusethisnumberasadata ×
Var(V2(S ))Var(V2(S ))
quality parameter. Data with χ2 greater than 3 should p 1 2
be examined in detail and rejected for science programs Whenallcalibratorsarepdifferent,allcorrelationsarezero.
requiring a very good quality of calibration as the proba- The correlation is maximum when a single calibrator has
bility to get a value larger than 3 is only of 8.33%. systematicallybeenused.Measurementnoiseispresentat
the denominator only and the correlation is of course all
the larger as the measurement noise is smaller.
5.5. Correlations of multiple baseline interferometer
The correlationbetween two expected squared visibilities
simultaneous visibilities
at two different baselines is not easy to evaluate analyti-
Coherence factors recorded simultaneously on different cally. Besides, it may depend upon the model of the cali-
baselines with telescopes in common may also be corre- brator.Acomputationcanbeperformedwhichshowsthat
lated. This correlation should be taken into account and the correlation is indeed equal to 1 with an excellent ac-
saved with the reduced data in the form of a correlation curacy as long as no baseline is equal to 0. This can also
8 G. Perrin: The calibration of interferometric visibilities obtained with single-mode optical interferometers
be shown by expanding the visibility function. Thus, the ble and may experience variations. In some cases like on
expected visibilities derived froma uniform disk model of May 15, 2000 at 8:07, an error of calibration may have
a same calibrator at two different baselines are fully cor- happened as the co-transfer functions jump by a few per-
relatedto the firstorder.This holds aslong asthe second cent. Yet, in most cases, the transfer functions variations
derivative of the model is small (which in the case of the are slow on time scales of a few hours and variations can
uniformdisk modelis trueexceptcloseto the zerosofthe be wellapproximatedto the firstorder.Data collectedon
model) and as a condition, none of the baselines is very May22,2000showthatthis isstillthe casewhenthe cal-
close to zero.In practice, the error on the diameter being ibrator diameters are known with a very good precision.
usually small (less than 5%), the first order approxima-
tion is valid and the two expected squared visibilities can
7.3. Discussion of model fitting and examples
thereforebeconsideredfully correlated.Thisistruedown
to very short baselines as for example for a diameter of
7.3.1. Amount of correlation
10±0.5mas the correlation starts to decrease for a base-
line below 5cm. Beforepresentingexamplesletussummarizethe different
For practical use, Eq. (26) can be simplified as the corre- levels of correlations we have encountered so far:
lationsbetweenexpectedvisibilitiesareeither0or1when
thecalibratorsarerespectivelydifferentoralike.Theonly 1. correlationof coherence factors (coaxial beamcombin-
requirement to compute this correlation is therefore that ers)
the variances of the expected visibilities and the coeffi- 2. correlation of interferometric channels (coaxial beam-
cientsαandβ besavedwiththereduceddata.Thesecor- combiners)
relations will have to be computed to model fit the data. 3. correlationof simultaneous baselines
The generalization of the Levenberg-Marquardt method 4. correlationof non-simultaneous baselines
with correlated data is given in the Appendix at the end
of the paper. The firstlevel(r) wasaddressedin Sect.7.1.The amount
of correlation between interferometric channels (ρ ) for
12
a coaxial beamcombiner like FLUOR varies from a few
7. Validation of the method
percent for faint sources calibrated by very well-known
Examples of data reduction results and calibrations are calibrators to almost 100% for bright sources calibrated
presented. The quantities introduced in the previous sec- by sources whose diameters are known with an accuracy
tionsarediscussedinpracticalsituationsandgeneralcom- of a few percent. The two channels are therefore not fully
ments on observing strategies are expressed. independent in this case and it is important to check the
χ2 defined by Eq. (24). A large χ2 may indicate that ei-
thertheassumptionsonGaussianstatisticswerewrongfor
7.1. Squared coherence factors statistics
these particular data or that the transfer function varia-
IhaveplottedinFig.1threeexamplesofµ2 distributions. tion is not well measured. In either case, data should be
In the case of V636 Her, the fringe speed puts the fringe examined in detail to decide whether the visibility value
frequency far above the turbulence piston spectrum. The canbe usedornot.Ablind methodis to rejectvisibilities
pistonisalmostfrozenduringeachscanandtheamountof withaχ2 aboveacertainlevelthatcanbeof3fordifficult
correlatednoiseissmall.Inthecaseof71UMa,thefringe programsorrelaxedto a largervalue for easierprograms.
speed is lower and the measurements are more sensitive It is important to note that if the transfer function has
to piston hence the higher correlated noise. δ Sge is an varied accordingly in both channels at the time the sci-
intermediate case.Inallthree examples,the distributions ence target was observed by an amount larger than the
of µ2 are compatible with Gaussian distributions hence error bars then this χ2 test will fail to detect it. It can
validatingthebasicassumptiononthestatisticsoftheµ2. only be detected if the variations are opposite in the two
An important fact is that the amount of correlated noise channels. This is certainly a weakness.
isnotnegligibleandmustbetakenintoaccount.However, It will be interesting to assess the level of correlations of
a test on distributions is performed to detect deviations visibilities measured with multiple beam interferometers.
from Gaussianstatistics. Deviations are not common and Itcanbeanticipatedthatitwillnotbenegligibleandwill
are always due to instrumental problems. In such cases, be of the same level as ρ .
12
depending on the requiredlevelof data quality, data may The importance of correlation between visibilities
be eliminated. recordedseparatelyis illustrated in Fig. 3. The data have
been reduced in two different ways. Data plotted with
opencirclesandfittedbyadashed-lineuniformdiskmodel
7.2. Examples of transfer functions
arereducedwithouttakingcorrelationsintoaccount.Data
Figure 2 presents two examples of squared co-transfer plotted with full circles and fitted by the continuous line
functions. Full circles are measurements on calibrators were reduced with the method of this paper. In the first
whereasopencirclesareinterpolationsforsciencetargets. case,thefitisofverygoodqualitywithaχ2 smallerthan
Itisvisiblethattheco-transferfunctionisnotalwayssta- 1. Yet, all visibilities have been calibrated with the same
G. Perrin: The calibration of interferometric visibilities obtained with single-mode optical interferometers 9
source,henceastrongcorrelationbetweenvisibilityvalues accuracies better than 0.001 have to be obtained on visi-
as the 3×3 correlation matrix shows: bilitiesthenitisverylikelythatnocalibratorcanbeused
twice, unless the error on the expected visibility of this
1 0.96 0.96
calibrator is less than the level of accuracy required.This
C = 0.96 1 0.97 (27)
  would suppose that the visibility model of the calibrator
0.96 0.97 1
bemeasuredfirst.Anotherpossibilityisrelativedetection.
 
It is to be noticed that the ρ12 correlationfactor is larger As illustrated by the example of SW Vir, if the same cal-
than 90% for all three visibilities, a large fraction of this ibrator is systematically used, the fit is sensitive to very
correlation being due to the common calibrator. If corre- low levels as the correlated noise does not contribute to
lations are ignored then noise is considered independent the value ofthe χ2.Inthis example,adeparturefromthe
fromonevisibilitytotheotherandthisiswhythefirstχ2 uniform disk model or a calibration error may have been
is smaller, as a large global noise is now interpreted as a detected to a level much lower than the error bars. For
largefluctuatingnoisefromonevisibilitytotheother.On very faint detail detection, this can work if the visibility
the contrary, when correlations are used, a tiny fraction curve of the calibrator is smooth and without wiggles of
ofnoise(4%atmost)canbe considereda fluctuationgiv- similar amplitude as the ones searched for on the science
ing degrees of freedom for the adjustement of the model. target.
This is equivalent to reducing error bars on visibilities by In any case, the observing strategy should be prepared in
96% in the fit. The common noise due to the uncertainty advance andshould take the problem of data correlations
on the calibrator is then a simple common bias on the into account.
visibilities but does not contribute to the noise in the fit,
hencethemuchlargerχ2.Inthezoomedpartofthissame
figure, one can see that the fit now conforms to only one
8. Conclusion
of the visibility data as the correlation matrix is close to
being non-invertible (see the Appendix for the use of the Ihaveproposedinthispaperamethodtocalibratevisibil-
correlationmatrixinthefittingprocess).Inmorephysical ity data obtained with single mode interferometers. The
terms,thecorrelationsbeingveryclosetoone,alldataare single mode character is required to make valid the as-
equivalent andthe fit can be derivedfrom one of the visi- sumption that the statistics of coherence factors data are
bility data. If all data points were compatible despite the Gaussianandstationary.Itispossibletoderivereliableer-
largecorrelationsthenthebestfitcurvewouldgothrough rorbarsifallcorrelationsareconsideredinthe derivation
the error bars. It is not the case here and this is why the of all estimators. Correlations also need to be taken into
χ2 is large. account when fitting the data by models. The validity of
the method has been demonstrated on real interferomet-
ric data recorded with FLUOR. An important conclusion
7.3.2. Examples of visibility accuracies
of this work is that the strategy of calibration has to be
Other examples of model fitting are presented in Fig. 4. adapted for specific programs requiring high standards of
TheBKVirdatawerecalibratedwiththesamecalibrator calibration.
(the correlation matrix is similar to the matrix above) as
SW Vir but the visibilities are very consistent with each
others. Data for the three other sources are either totally
References
independentorslightlycorrelated.Onlythefirstlobedata
were used for the fit of G Her. These four examples show Chelli, A., 2000, AMBER report AMB-IGR-017
very good fits and consistency of data. In particular, this Coud´e du Foresto, V., Ridgway, S.T., Mariotti, J.-M., 1997,
sets the best absolute accuracy of the calibration of visi- A&ASS,121, 379
bilities with FLUOR to 0.004 (equivalent to an accuracy Mourard, D., Tallon-Bosc, I., Rigal, F., et al. 1994, A&A288,
of 0.004 on V2 with V = 0.5 as σV2 =2σV.V to the first 675
order). Pelat, D., 1992, Cours “Bruits et Signaux”, E´cole Doctorale
d’ˆIle deFrance, Astronomie-Astrophysique
Perrin, G., 1996, PhD thesis, Universit´e Paris VII
7.3.3. Calibration strategies
Perrin, G., Coud´e du Foresto, V. Ridgway, S.T., et al., 1998,
It is important to adapt the strategyof calibrationto the A&A331, 619
type of astrophysicalstudies addressedwith opticalinter- Perrin, G., 2002, A&A,in press
ferometers.Formoststudies where visibility accuraciesof
Press,W.H.,Flannery,B.P.,Teukolsky,S.A.,Vetterling,W.T.,
a few percent are acceptable, the repeated use of a single 1988, ”Numerical Recipes in C”, Cambridge University
or of a few calibrators is possible. For difficult programs Press
like exoplanet detection, a very high level of accuracy is Segransan, D., Forveille, T., Millan-Gabet, R., Perrier, C.,
requiredand the strategy needs to be well prepared. Two Traub,W.A.,1999,“WorkingontheFringe:OpticalandIR
cases may arise depending on whether the required cal- Interferometry from Ground and Space”, ASP Conference
ibration of visibilities is absolute or relative. If absolute Vol. 194, p.290, S. Unwin and R. Stachnik Eds.
10 G. Perrin: The calibration of interferometric visibilities obtained with single-mode optical interferometers
Appendix A: Practical implementation of model ∂2S ∂M t ∂M ∂2M
= C−1 − C−1 V2−M
fitting with correlated visibilities ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ
k l (cid:20) k(cid:21) l (cid:20) k l(cid:21)
k,l =1,...,p (cid:2) (cid:3)
Algorithms for model fitting are well known. One of the
most commonly used is the Levenberg-Marquardt algo- IncaseS isequaltoitssecondorderexpansionthenθˆcan
rithms. An example of practical implementation is given
be guessed directly through:
in Press et al. (1988) in the case where data are not cor-
related. Here, I give a generalization of this algorithm to ∂2S −1 ∂S
θˆ=θ− (A.7)
the case of correlated data. ∂θ2 ∂θ
(cid:20) (cid:21) (cid:20) (cid:21)
IftheexpansionisapoorapproximationofS thenasteep-
A.1. Definitions
est descent method has to be applied with:
Let (Vi2)i=1,...,N be a series of squaredvisibility measure- ∂S
ments of an astronomical source. Let C be the matrix of θnext =θ−constant× ∂θ (A.8)
variances-covariancesof these measurements: (cid:20) (cid:21)
This the basis of the Levenberg-Marquardt method. It is
σ2 ρ σ σ ··· ρ σ σ
1 1,2 1 2 1,N 1 N classical to use the following notations:
.
C = ρ1,2σ1σ2 σ22 ··· ..  (A.1) ∂S
 ... ... ... ...  βk = −∂θ ,k =1,...,p (A.9)
  k
Let (Mρi(1θ,N))σi=1σ1,N...,N b·e··the m··o·del tσoN2fit the data with θ a αkl = (cid:20)∂∂Mθk(cid:21)tC−1∂∂Mθl ,k,l=1,...,p
vectorofpparameters.Itisassumedthatthenoiseonthe
where ∂M is the kth column of the matrix:
squared visibilities is Gaussian. ∂θk
∂M ∂M(x ,θ)
i
= (A.10)
A.2. Method ∂θ ∂θ
(cid:20) k (cid:21)i,k
The best estimates of the parameters in the sense of the The matrixequivalentofthe abovedefinitionistherefore:
highest likelihood are obtained by maximizing the likeli-
hood function of the Y =[V2−M(θ)] vector which leads β = ∂M tC−1 V2−M (A.11)
to minimizing the following functional: ∂θ
(cid:20) (cid:21)
S(θ)= 1YtC−1Y (A.2) α = ∂M tC−1∂(cid:2)M (cid:3)
2 ∂θ ∂θ
(cid:20) (cid:21)
The optimum θˆis obtained when: Atoptimumthevariance-covariancematrixoftheparam-
eters is:
∂S = 0
∂∂2θS > 0 (A.3) Var(θˆ)=[α]−1 (A.12)
(cid:26) ∂θ2
Besides, the covariance matrix of the estimated parame- With these definitions, 2S(θ) is a χ2 with N −p degrees
ters is asymptotically (N →+∞) equal to: offreedom.Theincrementinparameterspaceistherefore
the solution of the linear system:
∂2S −1
Var(θˆ)= (A.4) p
∂θ2
(cid:20) (cid:12)θ=θˆ(cid:21) α δθ =β (A.13)
(cid:12) kl l k
Thesearchforth(cid:12)eoptimumcanbeperformedwithagen- l=1
(cid:12) X
eralized Levenberg-Marquardtalgorithm.
when the quadratic form is a good approximation to S
otherwise the increment is given by:
A.3. Generalized Levenberg-Marquardt algorithm
δθ =constant×β (A.14)
l l
Closetominimum,S canbeexpandedtothesecondorder
The readerisreferredtoPress et al. (1988)forthe imple-
in θ:
mentation of this modified algorithm.
∂S t 1 t ∂2S
S(θ)=S(θˆ)+ θ−θˆ + θ−θˆ θ−θˆ (A.5)
∂θ 2 ∂θ2
(cid:20) (cid:21) h i h i (cid:20) (cid:21)h i
ThefirstandsecondderivativeofSarerespectivelyavec-
tor and a matrix whose elements are:
∂S ∂M t
= − C−1 V2−M ,k =1,...,p (A.6)
∂θ ∂θ
k (cid:20) k(cid:21)
(cid:2) (cid:3)