Revisiting the radio interferometer measurement equation. III. Addressing direction-dependent effects in 21 cm WSRT observations of 3C 147

1.9 MB·English

by  Oleg M. Smirnov

Astronomy&Astrophysicsmanuscriptno.16435 (cid:13)c ESO2011
January12,2011
Revisiting the radio interferometer measurement equation.
III. Addressing direction-dependent effects in 21 cm WSRT
observations of 3C 147
O.M.Smirnov
NetherlandsInstituteforRadioAstronomy(ASTRON)
P.O.Box2,7990AADwingeloo,TheNetherlands
e-mail:[email protected]
1
1 Received5Nov2010/Accepted5Jan2011
0
2 ABSTRACT

n
Context.PapersIandIIofthisserieshaveextendedtheradiointerferometrymeasurementequation(RIME)formalismtothefull-sky
a case,andprovidedaRIME-baseddescriptionofcalibrationandtheproblemofdirection-dependenteffects(DDEs).
J
Aims.ThispaperaimstoprovideapracticaldemonstrationofaRIME-basedapproachtocalibration,viaanexampleofextremely

1 high-dynamicrangecalibrationofWSRTobservationsof3C147at21cm,withfulltreatmentofDDEs.
1 Methods.AversionoftheRIMEincorporatingdifferentialgainshasbeenimplementedinMeqTrees,andappliedtothe3C147data.
Thiswasusedtoperformregularselfcal,thensolveforinterferometer-basederrorsandfordifferentialgains.

] Results.Theresultingimageofthefieldaround3C147isthermalnoise-limited,hasaveryhighdynamicrange(1.6million),and
M
noneoftheoff-axisartefactsthatplagueregularselfcal.Thedifferentialgainsolutionsshowahighsignal-to-noiseratio,andmaybe
usedtoextractinformationonDDEsanderrorsintheskymodel.
I
. Conclusions.ThedifferentialgainapproachcaneliminateDDE-relatedartefacts,andprovideinformationforiterativeimprovements
h
ofskymodels.Perhapsmostimportantly,sourcesasfaintas2mJyhavebeenshowntoyieldmeaningfuldifferentialgainsolutions,
p
andthuscanbeusedaspotentialcalibrationbeaconsinotherDDE-relatedschemes.
-
o
Key words.Methods: numerical -Methods: analytical-Methods: dataanalysis-Techniques: interferometric-Galaxies:quasars:
r
t individual:3C147
s
a
[
Introduction to reprocessthese data using MeqTrees,to see if DDEs can be

2 eliminatedthroughtheuseofasuitableformoftheRIME.
v The field around the bright radio source 3C 147 is a favourite
8 showcase for dynamic range (DR) demonstrations. 3C 147 it- In the presence of a dominant source (in this case 3C 147
6 self is verybright(22Jy at21cm),which ensuresa highSNR itself), selfcal solutions will tend to subsume all effects in the
7 for selfcal solutions,while the surroundingfield boastsa spec- direction of that source. DDEs will then manifest themselves
1 asartefactsaroundothersources,whichneedto beata certain
tacular collection of mostly point-like fainter sources. The ab-
. distancefromthedominantsourcefortheeffecttobecomeap-
1 sence of significant extended emission has allowed very accu-
0 rate sky modelsto be constructed.It is then not surprisingthat parent. Since the dominant source is usually placed at or near
1 3C147wasthefirstfieldtobreakthe106dynamicrangebarrier thepointingcentre(i.e.on-axis),DDE-relatedartefactsarealso
1 (deBruyn2006;deBruynetal.2011),onasingle12-hoursyn- calledoff-axisartefacts.Theartefactsthemselvesarequiteclear
: (Fig.1,leftinset):thenatureoftheeffectsresponsibleforthem,
v thesis.Thisspectacularresultwasachievedwithregularselfcal
farlessso.Atleastfourpossibilitieshavebeenpostulated:
i implementedin the NEWSTAR package.A majorcontributing
X
factor is the relatively low level of beam-related DDEs at the
r WSRT,asdiscussedinPaperII(Smirnov2011,Sect.2.1)1.An
a
imageofthe3C147fieldat1,600,000:1dynamicrangeisshown – Pointingerrors(PaperII,Smirnov2011,Sect.2.1.4).
inFig.1.ThemakingofthisimageisthesubjectofSection1. – Differentialtroposphericrefraction(ibid.,Sect.2.2.4).
The very same benign properties that allow the WSRT to – Errorsinantennapositions,includingnon-coplanarity.
achieverecordDR also make it a perfectinstrumentfor study- – Othersystematiccoordinateerrors.
ing DDEs. The latter are prominentenoughto clearly show up
in high-DR maps (see e.g. left inset of Fig. 1), but not severe
enoughto hinderthe buildingupof verydeepskymodelsdur- One of the objects of this study was to narrow down these
ingnormalselfcal2.The3C147observationsbydeBruynetal. possibilities. Section 2 therefore analyses the differential gain
(2011) are a perfect example of this. I have therefore decided solutionsobtainedduringthiscalibration,withaviewtocharac-
terizingtheDDEs.
1 Bycontrast,eventhepost-upgradeExpandedVLA(EVLA),with
itsmoresignificantDDEs,hasnotyet(attimeofwriting)exceeded106.
2 Strictlyspeaking,thisisonlytrueincontinuummode.Inspectral Sect.2.1.1)becomesaverytroublesomeDDE.Furtherworkisrequired
linemode,the17MHz“ripple”discussedinPaperII(Smirnov2011, onthissubject.
2 O.M.Smirnov:RevisitingtheRIME.III.AddressingDDEsinWSRTobservationsof3C147.
Fig.1. “Showcase” image of the field around
the bright radio source 3C 147, produced af-
ter reduction with MeqTrees. The image is
noise-limited, andhas adynamicrange of 1.6
million. This DR was already achieved by de
BruynusingregularselfcalinNEWSTAR,but
theresultingimagescontainedartefactsaround
off-axis sources (left inset) due to DDEs. A
MeqTrees reduction incorporating differential
gains,asdescribedinthispaper,hascompletely
eliminated the artefacts (right inset). This
image also appears in Noordam&Smirnov
(2010).
1. Calibrationapproachesandresults X = E2X ,
pq s spq
Xs
1.1.ObservationsandNEWSTARreduction
andconsistsofthefollowingsteps:
TheobservationaldatainquestionwereobtainedbydeBruynin
2003.Asingle12-hoursynthesiswastaken,using8×20MHz
bands(of64channelseach)from1300to1460MHz,with30s 1. FindG˜p thatminimizes|Xpq−Dpq|inaleast-squaressense.
integrationtime. Due to a back-endproblem,one of the cross- Compute“correcteddata”as D′pq = G˜−p1DpqG˜q−1.(Thesolu-
correlationswascorrupted,soonlythe XX andYY correlations tion intervalhere was one timeslot, one frequencychannel.
were usable. De Bruyn then successfully reduced the observa- Only56channelsineachbandwereusable;thesewerefur-
tion using the NEWSTAR package, achieving a world record ther averaged down using Hanning tapering, so in the end
1,600,000:1dynamicrange(deBruynetal.2011).Onlyregular only28frequencypointsperbandwereused.)
selfcal was done and no peeling was attempted, so the result- 2. Find M˜pq that minimizes|Mpq ∗Xpq −D′pq|. Compute “cor-
ingimageshowedDDE-relatedartefactsaroundoff-axissources recteddata”as D′′ = D′ ÷M˜ ,where“÷”iselement-by-
pq pq pq
(Fig. 1, left inset). One result of the reductionwas a very deep elementdivision–theinverseof“∗”.(Thesolutioninterval
NEWSTAR-formatskymodelforthefield,containingabout300 herewasthefull12hours,oneband.)
pointsources.Thisprovidedafantasticplatformfromwhichto 3. Compute“residualdata”asR =D′′ −X .
pq pq pq
begin my DDE study with MeqTrees. I had a ready-made sky
modelthatwas knownto be goodenoughto reach the thermal Thisprocedurewasrepeatedforeachband.Residualvisibil-
noisewiththisparticulardataset,andIhadintermediateimages ities were imaged and summed acrossall 8 bands,then decon-
fromdeBruyn’sreductionthatcouldbeusedascheckpoints. volvedusingHo¨gbomCLEAN.Theskymodelwasthenadded
Thesameobservationwasrepeatedin2006withsomewhat backinusingaGaussianrestoringbeam.
different correlator settings (deBruynetal. 2011, for details,
see),andreducedinasimilarmanner.
1.2.CalibrationinMeqTrees:anoverview
ForhisNEWSTARreduction,deBruynself-calibratedeach
channel independently, rather than explicitly calibrating for a In broad terms, selfcal in MeqTrees also consists of a least-
bandpass (see below). This procedure is described in Paper II squaresfit of an equationsuch as (1) to the data. However,the
(Smirnov2011,Sect.1.1).Itusesthefollowingimplicitformof followingfeaturesaredifferent:
theRIME:
– Thestructureoftheequationisnotfixed:arbitraryformsof
the RIME may be constructed. Crucially for my purposes,
V = G M ∗X GH, (1) thesemayincludeDDEterms.
pq p pq pq q
(cid:16) (cid:17)
O.M.Smirnov:RevisitingtheRIME.III.AddressingDDEsinWSRTobservationsof3C147. 3
– TheelementsoftheJonesmatricesarenotnecessarilysim- 1.3.Bandpassselfcal
ple solvable parameters (though they may be), but can be
A per-channel, per-timeslot solution for G in Eq. (1) has 2N
representedbyarbitraryfunctions.Forexample,ratherthan p
complexunknowns,andN(N−1)complexmeasurements,where
solvefor E-Jones(orZ-Jones)elementsdirectly,MeqTrees
N is the number of stations (14 for the WSRT). While an ac-
can derivethem fromsome modelof the primarybeam (or
ceptable ratio (this is, after all, why selfcal works in the first
ionosphere) and solve for the parameters of the model. An
place!), it does not leave a lot of room for introducing DDE-
exampleofthisisgiveninSect.1.6.
relatedparameters.Ingeneral,wewanttoallowoursolutionsas
– Different solvables may have different solution intervals,
fewDoF’saspossible,andmakingthesolutionintervals(intime
eveninasimultaneoussolution.
and/orfrequency)largerisonewayofensuringthis.
The WSRT has a reasonably stable bandpass, so an obvi-
Because ofthe inherentflexibilityof MeqTrees,calibration
ous way to reduce the parameter count is to separate G into
p
can be pursued in a great variety of ways. During this study,
a bandpasscomponentto capturethe frequencystructure(with
a specific methodology was narrowed down, implemented and
little to none variation in time), and a frequency-independent,
tested.Thisbecamethebasisofthe“Calico”calibrationframe- rapidlyvaryingcomplexgain.Pereachstation/receiver,thisre-
workthatisnowincludedwithMeqTrees.Thesteps(andtermi- placesN ×N parameterswithonlyN +N ofthem.
chan time chan time
nology)ofcalibrationwithCalicoareasfollows:
FortheMeqTreesreduction,Ithereforestartedwiththefol-
lowingRIME:
1. A desired form of the RIME is constructed, by selecting a
sky model, and picking a series of Jones terms (plus, op-
V =M ∗ G B E X EH BHGH (2)
tionally, interferometer-based errors). For example, a form pq pq  p p s spq s  q q
similartoNEWSTAR’simplicitRIME(Eq.1),wouldbe:  Xs  
Here, X , E , G and M have the same meaning as in
spq s p pq
Vpq =Mpq∗(GpXpqGqH) the NEWSTAR equation above (including a similar smearing
correction term in X ). The B term is a second diagonal
spq p
Vpqisusuallycalledthe“corruptedpredict”. Jonesmatrixrepresentingthebandpass.NotethatMeqTreesit-
2. “Corruptedpredict”isfittedto“data”.Thatis,theMeqTrees self makes no special distinction between G and B. Both are
solverisinstructedtofindvaluesofsolvableparametersthat genericdiagonalcomplexJonesmatrices,withsolvablerealand
minimize |Dpq − Vpq| in a least-squares sense. This can be imaginaryparts.Itisonlywhenwespecifythesolutionintervals
(and usually is) done in multiple stages, e.g. G˜ first, fol- thattheseJonestermsacquiretheirintendedmeanings:
p
lowedbyM˜ ,etc.
pq
– TheG solutionintervalis30s,allchannels(thus,oneinde-
3. Outputvisibilitiesarecomputedaseither“correcteddata” p
pendentsolutioneverytimeslot,pertheentireband).
D′ =G˜−1(D ÷M˜ )G˜H−1, – The Bp solutionintervalwasinitiallysetto30minutes,and
pq p pq pq q onechannel(thus, one independentsolution everychannel,
oras“corruptedresiduals”R =D −V ,oras“corrected per 60 timeslots). Note that as the bandpass has significant
pq pq pq
structure,Ididnotattempttofititwithanysmoothfunction,
residuals”
butratherallowedeachchanneltobefittedasanindependent
R′ =G˜−1(R ÷M˜ )G˜H−1. parameter,withatimescaleof30minutes.Thelatterinterval
pq p pq pq q
wasintendedtoaccommodateslowdriftinthebandpass.
SubsequentimagingstepsarenotconsideredpartofCalico Theinitialresultofthiscalibrationwasprofoundlyunsatis-
orMeqTreesper se, since theyapplyequallyto data calibrated factory (Fig. 2, upper left). The residual image was dominated
usinganyothermeans,andareaccomplishedviaexternaltools by spoke-like artefacts centred on 3C 147, at about 10,000:1
suchasthelwimagerprogram(partofthecasarestpackage). level(relativethefluxof3C147itself).Thesespokescorrespond
These steps may also differ from project to project. I routinely to edges of the 30-minutesolution intervals (being an E-W ar-
imagetheper-bandcorrectedresidualsR′pq,sincetheseprovide ray,WSRT hasa one-dimensionalinstantaneousPSF).Theob-
the bestvisualindicatorofthe qualityofa calibration.Forthis viousexplanationfortheerrorisshort-termbandpassinstability.
particular study, I subsequently made 8-band residual images Decreasingthesolutionintervalof B to15and7.5minutesre-
p
in MFS mode, using all data. The images were further decon- duced the artefacts to levels of 50,000:1 to 100,000:1, but did
volved using Cotton-Schwab CLEAN (Schwab 1984), and the noteliminatethem entirely.Smoothingthe 7.5 minutesolution
skymodelwasaddedbackinusingaGaussianrestoringbeam. with a spline (along the time axis, per each channel indepen-
WhileCalicocanproduce“correcteddata”(forpurposesof dently)producedamarginalimprovement(Fig.2,lowerright).
imaging,etc.)atanystepofthereduction,itdoesnotuseitasin- Subtractionartefactsarestillplainlyvisibleinthemap,although
putforsubsequentstageslikeNEWSTARandother2GCpack- atalevelnotsignificantlyabovethermalnoise.
agesdo.Thefittingatstep2isalwaysdoneusingtheoriginal3 AtthisstageIhadtoconcludethattheWSRTbandpassex-
observed data Dpq, and calibration consists of building up the hibitssome verylow-level,butextremelyshort-termjitter,pre-
RIME until the “corrupted predict” fits the observations. Once cludinga separatebandpassselfcalatextremedynamicranges.
DDEs enter the picture, “corrected data” (in the conventional On the other hand, this result also shows that where a single-
understandingof data corrected for instrumentalerrors) do not banddynamicrangeofnomorethan100,000isexpected(asis
reallyexistanymore,sincevisibilitiescanonlybecorrectedfor thecaseformanyotherobservations),bandpassselfcalprovides
the value of a DDE in a particular direction. Hence the Calico perfectly adequate results, and can cut down on the numberof
philosophyistoworkwiththeoriginaldataatalltimes. solvable parameters significantly. In the meantime, for the 3C
147studyIhadtoreverttotheper-channelselfcalapproachof
3 Oratmostpre-averaged. deBruyn.
4 O.M.Smirnov:RevisitingtheRIME.III.AddressingDDEsinWSRTobservationsof3C147.
Fig.3. Single-band residual images produced via per-channel
selfcal. The left image is the result of solving for G . It is
p
dominated by concentric rings centred on 3C 147 (designated
as “A150” here). These are caused by closure errors, and go
awayonceasolutionforinterferometer-basederrorsM isdone
pq
(right image). The remaining artefacts are associated with off-
axissourcesB,CandD,andareduetoDDEs.
20′awayfromcentre.Theartefactsthemselvesareunder50µJy
Fig.2.Single-bandresidualimagesproducedviabandpassself- (thermal noise in one band being ∼ 30µJy), or at a level of
cal with different solution intervals for B : 30 minutes (upper 1:1000relativetotheassociatedsources.ThisiswhydeBruyn
p
left),15minutes(upperright),7.5minutes(lowerleft),andwith (2006)talksaboutan“off-axisdynamicrange”ofa1000:while
the 7.5 minute solution smoothed using splines (lower right). 3C 147 itself (22 Jy) is subtracted without a trace (down to
Even in the best-case image, the dominantsource 3C 147 was the thermal noise), the off-axis sources are only subtracted to
not subtracted out perfectly, leaving behind DR-limiting arte- a precision of about 1000. Some of the artefact structure is
facts. doubtlessly due to slightly under- or overestimated sky model
fluxes (this produces regular rings matching the WSRT PSF),
which can be taken care of during subsequent deconvolution.
1.4.Per-channelselfcal
Most of it, however,is due to DDEs and does not deconvolve,
The RIME for per-channel selfcal is just Eq. (2) without the producing artefacts in the final 8-band images such as the one
bandpass term. It is, in fact, very similar to NEWSTAR’s im- shownintheleftinsetofFig.1(theinsetis, infact,aclose-up
plicitequation(1):4 ofsourceB).
V =M ∗ G E X EH GH (3) 1.5.Interferometer-basederrors
pq pq  p s spq s  q
 Xs   ItisnotclearwhatcausesclosureerrorsattheWSRT.Common
Per-channelselfcalisachievedbysettingthesolutioninter- sensesuggeststheanaloguepartofthesignalchainistoblame,
valofG toonechannelandonetimeslot.Theresultingsingle- butthereisnohardevidenceeitherway.Whatisevidentisthat
p
bandresidualimagesaredominatedbyclosureerrorsatalevel high-DR images exhibit concentric ring-like artefacts such as
of ∼ 100µJy (or 1:200,000 relative to 3C 147 itself); these go those in the left image of Fig. 3, and that these go away once
away after an M solution (Fig. 3). These images are quali- asolutionforaninterferometer-basedmultiplicativeerror–the
pq
tatively very similar to those obtained by de Bruyn during his Mpq termofEq.(3)–isapplied.Asinglesolutionperband,per
NEWSTARcalibration(whichistobeexpected,giventhesim- theentire12hours(percorrelationandinterferometer)issuffi-
ilarityoftheequations). cient.TheMpq solutionsareusuallywithin10−3−10−4ofunity
The remaining artefacts in Fig. 3 are associated with the (as is the case here), but can be much higher in some observa-
three next-brightest sources5 in the field: B (42 mJy), C tions,forreasonsthatremainmysterious.
(52 mJy), and D (22 mJy). The furthest of these (B) is about Thelatterfactsuggestsanintrinsictimevariability,butsolv-
ing for M on short time intervals is very dangerous.Any so-
pq
4 WiththeexceptionofthepositionoftheMpqterm,whichisonthe lutionforMpq willalsotrytocompensateforobservedfluxthat
insideofEq.(1),andontheoutsidehere.Forthisparticulardatasetit is not present in the sky model. Unless the solution interval is
makesnodifference:sinceonlythe XX andYY correlationsareused, sufficientlylong,therewillbeunmodelledsourceswithafringe
allmatricesinEq.(1)arediagonal, andfordiagonal matricesthe“∗”
rateslowenoughthattheirvectoraveragevisibilityovertheso-
operatorisequivalenttomatrixmultiplication,andcommutes.Forthe
lutionintervalwillbesignificantlynon-zero.Thesesourceswill
full-polarizationcase,thetwoapproachesarenotequivalent.
thentendtobeattenuatedbytheM solutions.The3C147ob-
5 The source IDs used here follow the “COPART” (Clustering, pq
servationsprovidea perfectexample(Fig. 4). On the left is an
Order, Position Angle, Radius, Type) convention, as implemented by
theTigger sky model management tool (availablewithMeqTrees). A 8-bandresidualimagewitha12hourMpq solution;ontheright
COPART source ID starts with an alphabetic designator (A, B, ... Z,
aa,ab,...)assignedtosourcesinorderofdecreasingbrightness.Thisis of10◦),andonedigitforradialdistance(inunitsof10′).Optionalsuf-
alreadyauniqueidentifier,andissometimesusedbyitselfforbrevity, fixesindicatesourcetypeandclustering.ThusthefullIDofsourceBis
as in the paragraph above. A full ID also encodes approximate posi- B232;beingslightlyextended,inthisparticularskymodelitisactually
tion relativeto fieldcentre: twodigits for the position angle (inunits representedbyaclusterofsixdeltafunctions:B232,B232a,...B232e.
O.M.Smirnov:RevisitingtheRIME.III.AddressingDDEsinWSRTobservationsof3C147. 5
Fig.4. Source suppression through interferometer-based error Fig.5.Resultsiftheflyswatter.Ontheleftisasingle-bandresid-
solutions. On the left is a deconvolved 8-band residual image ual image after G and M solutions only. On the right is the
p pq
of the centre of the field, with 12 hour solutions for M . On sameimagewithdifferentialgainsolutionsforsourcesB,C,and
pq
the right is the same image with 30 minute solutions. The po- D.
sitions of (subtracted) model sources are indicated by crosses.
Suppression of unmodelled sources is evident in the right im-
4. ThedominantDDEaffectingthisobservationisnotpointing
age.
error.ThiswillbeelaboratedonfurtherinSect.2.
5. Thereissomethingwrongwithmyimplementationofpoint-
isonewitha30minutesolution.Modelsourcesareindicatedby ingselfcal,especiallysincethefiguresinSect.2suggestthat
crosses.Attenuationofunmodelledsourcestowardsphasecen- mispointingsaredetectable.
treisclearlyvisibleintherightimage.
Tryingto geta bettergriponthe problem,I eventuallyset-
Thisimpliesthatclosureerrorscannotbereliablysolvedfor
tled on a “controlled experiment”: locating a field containing
onobservationsshorterthan12hours,unlessa“complete”(i.e.
multiple bright off-axis sources, and observing it with delib-
downtothenoise)skymodelofthecenterofthefieldisavail-
erately exaggerated mispointings, to see if these can be more
able.
readilyrecoveredfromthedata.Thisexperimentbecameknown
as the “QMC Project” (in honour of the long-defunct Quality
1.6.Pointingselfcal MonitoringCommitteeofWSRT),andwassuccessfullycarried
out.Theresultsofthisarestillbeingprocessed,andwillbepre-
SincethemaincauseofartefactsinWSRTimagesiscommonly
sented in a follow-uppaper.In the meantime,I had to look for
considered to be pointing error (see Paper II, Smirnov 2011,
alternativeapproachestoDDEsinthe3C147field.
Sect.2.1.4),Idecidedtoimplementaformofthepointingself-
calalgorithmsuggestedbyBhatnagaretal.(2004).Thisproved
tobeastraightforwardexerciseinMeqTrees,sinceonlyasmall 1.7.Differentialgains:the“flyswatter”
modificationoftheRIMEofEq.(3)wasrequired:
Inthespiritof“phenomenological”equationsdiscussedinPaper
II(Smirnov2011,Sect.1.3),Idecidedtointroduceadifferential
Vpq =Mpq∗Gp EspXspqEHsqGqH (4) gainJones(∆E-Jones)intomyformoftheRIME:
 Xs  
Instead of a per-source beam gain Es = E(ls,ms) (where Vpq =Mpq∗Gp ∆EspEsXspqEHs ∆EHsqGqH (6)
2E0(1l,1m,)Seisct.th2e.1c.1o)s,3wahpipcrhoxIihmadatiboenengiuvseinngbiynPthaepeprreIvIi,oSumseirqnuoav-  Xs  
The ∆E term is meant to subsume all DDEs associated
tions, hereI used a per-antenna,per-sourcebeamgain E , de- sp
sp
withsource sandantenna p(withtheexceptionofthenominal
finedasfollows:
primarybeamgain,whichisalreadyrepresentedbyE ).Solving
s
E = E(l +δl ,m +δm ) (5) forthistermrequiressomecaution,lesttoomanyDoF’sbein-
sp s p s p
troducedintotheequation.Iapproachedthisasfollows:
Per-antenna pointing offsets δl ,δm were then treated as
p p
solvableparameters. – ∆EwasfixedatunityforallsourcesexceptB,C,andD;
Theresultsofthisprovedinconclusive.Eventhoughtheso- – ForB,CandD,∆E wassettoadiagonalmatrixwithsolv-
lutionconvergedtosomephysically-sensibleoffsets(ontheor- ablecomplexelements.
derof arcmin),no tangiblereductionofartefactswas observed – The solution intervals for the ∆E elements were set to one
inresidualimages.Thiscouldbeduetoanumberofreasons: solutionper30minutes,perentireband(andpersource,an-
tenna,receptor).
1. Thecos3 approximationisnotgoodenough–unlikely,asit
has been independently verified at least for the core of the Figure5 showsthe effectof ∆E solutions. All artefacts as-
main lobe, which the sources in question sources are well sociatedwithsourcesB,CandDdisappearedcompletely,tothe
within. point where artefacts aroundfainter sources became just about
2. WithonlyafewsourcessufficientlybrighttoexhibitDDEs, visible. These were later eliminated once ∆E solutions were
we simply don’thaveenoughconstraintsfor a pointingso- enabled for those sources. In fact, any source to which a solv-
lutiononthisfield. able∆Etermwasassignedpromptlyvanishedfromtheresidual
3. The model fluxes/positions for the sources in question are maps, hence my name for the differential gains approach: the
notsufficientlyaccurate. “flyswatter”algorithm.
6 O.M.Smirnov:RevisitingtheRIME.III.AddressingDDEsinWSRTobservationsof3C147.
Theresultingimageiscompletelyartefact-free.Presumably,
60
all other sources in the field are too faint to exhibit any DDE-
ae317 ID position totalflux
40 relatedartefacts.Withadynamicrangeof1,600,000:1,thisim-
F012 B232 20’SW 42.3mJy ageisthedeepestandcleanestsingle-synthesisradiomapinthe
20 K063
C270 4’W 51.8mJy worldtodate.
H283
0 C270 D141 14’SE 21.8mJy
D141 B232 F012 25’N 12.3mJy
(cid:0)20 H283 37’W 9.4mJy 1.9.Flyswatterlimitations
(cid:0)40 K063 31’NE 8.2mJy
ae317 74’NW 2.2mJy Whileitcanhelpproducespectacularimages,theflyswatterhas
(cid:0)60 some serious caveats and drawbacks that need to be explored.
(cid:0)60 (cid:0)40 (cid:0)20 0 20 40 60
First of all, it is a brute-force approach, in the sense that it
Fig.6.Positions(relativetonominalpointingcentre)andaggre- squashes all effects into a single ∆E term. This includes inac-
gate fluxes (apparent) of the seven off-axis source clusters for
curacies in the sky model! Indeed, any missing source flux or
which∆Esolutionswereobtained.Circlesareataradiusof30′
error in source position can be accommodated with a suitable
and1◦.Forreference,theFWHMoftheWSRTvoltagebeamis ∆E.Evenunmodelledsourcestructurecan“leak”intodifferen-
∼50′at1.4GHz. tial gain solutions(Sect. 2.1). Thus, differentialgainsare good
for subtracting sources, but at the cost of mashingup informa-
tiononthesourceperse.(Onedoesnotuseaflyswattertoprobe
Once 8-band residual images were created, the increased afly’sanatomy!)
sensitivitymadeitapparentthatfourmoresourceswereexhibit- Secondly, solvable differential gains can lead to a prolif-
ing a small amountof DDEs. A more in-depth look at the sky eration of DoF’s. Per-channel selfcal has N unknowns per
ant
model also showed that all seven sources were in fact slightly Nant(Nant−1) measurements,or 2 unknownsper measurement;
extended; the NEWSTAR model represented each by a tight 2 Nant−1
differentialgainsadd 2Nsrc unknownspermeasurement,
cluster of point sources. Since all “sources” in such a cluster (Nant−1)NtimeNfreq
where N and N are the sizes of the solution interval for
are subject to the same DDE, it seemed sensible to make sure time freq
thesame∆E termwasappliedtoallofthem.Thiswaseasily ∆E.ThisratioremainsfavourableforsmallNsrc andlargeNtime
sp
done using Tigger, a sky model manager/viewer tool included and/or Nfreq (as is the case for my 3C 147 reduction), but one
mustbecareful.
withMeqTrees.Tiggerautomaticallydetectssourceclustersand
assigns source identifiers appropriately (e.g. B232, B232a, ... Thethirdcaveatisprocessingcost.Whileusuallynotasex-
B232e)ItwasthenasimplemattertotelltheCalicoframework pensive in terms of I/O or CPU as peeling (which, in addition
tousethesame∆E termforallsourcessassociatedwithclus- to the solutions themselves, requires repeated subtraction and
ters . s0p phaseshiftingsteps),theflyswatterisnotfree.Everysourcewith
0
The seven source clusters for which differential gain solu- adifferentialgainsolutionadds4Nantunknowns(assumingadi-
agonal complex ∆E term, hence 4 real values per matrix) to
tions were eventually obtained are summarized in Fig. 6. Two sp
the equations. As the number of unknowns(N = 4N N )
ofthemaresomewhatnoteworthy.SourceC270isverycloseto unk ant src
centre,andthereforeshouldn’tbeaffectedbyDDEsasmuchas grows, inversion of the normal matrix within the least-squares
solver becomes a CPU bottleneck, since it scales as O(N3 ).
the other sources. It is, however,a complicated and highly po- unk
Thismakesitimpracticaltosolvefor∆E’sformorethanahand-
larized source, so perhaps the artefacts it exhibits after regular
ful6ofsourcesatatime.
selfcal are primarily due to sky model inaccuracies, which the
∆E solutionsabsorb(seediscussioninSect.2.1).Sourceae317 One way to mitigate the solver bottleneck is to decompose
isalmosttheopposite:itisveryfaint,butfarenoughoff-axisto the∆Esp’sintonearly-orthogonalsetsofunknowns.Forexam-
beinasidelobeoftheprimarybeam,andsosubjecttoespecially ple, we can treat the set of ∆Esp’s associated with one source
severeDDEs. s as independent from all other sources. The (4NantNsrc)2 nor-
malmatrixinsidethesolverthenbecomesblock-diagonal,com-
posedofN blocksofsize(4N )2.Inversionofthismatrixthen
src ant
1.8.Theshowcaseresult scales as O(N )O(N3 ). Thisschemewas tested in MeqTrees,
src ant
anditwasfoundthatthetrade-offisslowerconvergence,requir-
Theultimateresultofmycalibrationofthe2003observationis
ingmoreiterations.Forlargenumbersofsources,however,this
showninFig.1.Thisimageisatrueshowcaseforthedifferential
becomesveryfavourable.
gainsapproach.Theprecisestepsleadingtothisimagewereas
follows:
2. Analysisofdifferentialgainsolutions
1. Eachofthe8bandswasindependentlycalibratedusingper-
channel selfcal, interferometer-based errors, and ∆E solu- Itistimetoseewhetheranyusefulinformationcanbegleaned
tionsonsevensourceclusters,asdescribedabove.Corrected from the differential gains solutions themselves. As a result of
residualsweregenerated. the reduction, I had obtained: per each source direction (7 of
2. Theresidualsforall8bandswereimagedtogether(inMFS these:seeFig.6),pereachantenna(14),pereachband(8),per
mode) to produce a single residual image. This revealed a 30-minuteinterval(24oftheseina12-hoursynthesis),twocom-
large number of fainter sources not visible in the per-band plexnumbersrepresentingtheapparentdifferentialgainoftheX
maps. andY dipole(“differential”beingrelativetothe gainin thedi-
3. The 8-band image was deconvolved using Cotton-Schwab
CLEAN(Schwab1984). 6 Theprecisemeaningofa“handful”heredependsonadditionalfac-
4. Theskymodelwasaddedbackintothedeconvolvedimage, torssuchasN ,sizeofsolutionintervals,etc.Ineffect,thesefactors
ant
usingaGaussianrestoringbeam. influencetheconstantoftheoverallcubicscalinglaw.
O.M.Smirnov:RevisitingtheRIME.III.AddressingDDEsinWSRTobservationsof3C147. 7
rectionof3C147–almostatthecentreofthefield–whichhad Ifasolvabledifferentialgainisthenassignedtothesource,
beentakencareofbyregularselfcal). themodelcanbemadetofitthedatabyabsorbingtheK (δl)A
p
I then adoptedthe followingapproach.Given the relatively factorintothe∆E solutions.
ps
low fractional bandwidth, I didn’t expect much variation with Even more insidiously, differential gains can absorb some
frequencyin ∆E. I thereforetreated each set of 8 per-bandso- source structure. Consider a source that is slightly extended in
lutionsasindependentsamplesofthesamevariable.Themean onedirection,enoughtoberesolvedonthelongestbaselines.An
of the 8 samples was used as an estimator of variable, and the E-WarrayliketheWSRThasaone-dimensionalinstantaneous
standarddeviationofthe8samplesasanestimatoroftheerror fanbeam.Itwill“see”thesourceasapointsourcewhenthefan
(i.e.theerrorbar). beamisalignedwith thesourceorientation,andstartresolving
Sinceitquicklybecameapparentthatthe∆Esolutionswere it when the fan beam becomes perpendicular to the source. In
exhibitingsomeveryinterestingbehaviour,Iappliedexactlythe otherwords,theapparentfluxofthesourcewillremainconstant
same procedure to the 2006 observations, so that comparisons in time on short baselines, and vary in time on the long base-
couldbemade.Theplotsbelowshowtheresultsfrombothob- linesasthesourceresolves.Ifsuchasourceisrepresentedbya
servations. point source in the sky model, the model flux will be constant
Figure7isasummaryofthedifferentialgain-amplitudesper on all baselines. Now, if some antennas are predominantly in-
source,perantenna.Theprecisequantityplottedherewascom- volvedinlongbaselines(RTCandRTD,inthecaseofWSRT),
puted as follows. First, I computedthe norm of the ∆E matrix ∆E solutionscan compensatefor some of the flux discrepancy
(diagonalbyconstruction)as bychangingthegain-amplitudesoftheseantennas.Iwouldex-
pecttoseeavariationof||∆E||witha12-hourperiod.Sincemost
of the baselines to RTC and RTD are mutually redundant(0-C
a 0
≡ |a|2+|b|2, equals1-D,etc.),theirvariationin||∆E||shouldbeverysimilar.
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 0 b!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p This is exactly what we’re seeing in Fig. 7! The plots very
(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) stronglysuggestthatthetopthreesources(B, CandD)arein-
(cid:12)(cid:12) Ithe(cid:12)n(cid:12) normalized(divided)thisvaluebythemeanvalueper
deedslightlyextended(moresothaninthemodel,thatis).Ifthis
source (that is, the mean across all time intervals, bands, and isthecase,thenthedominantcontributionto||∆E||onantennas
antennas), which was meant to take out the effect of incorrect
RTC and RTD is dueto source structurerather than anyactual
modelfluxes(seebelow).Theresulting“normalizednorm”was
DDE.
thenplottedasafunctionoftime,persource,perantenna.
Figure8isasimilarplotofthedifferentialgain-phases,com-
putedas 2.2.Amplitudebehaviour
IfthebehaviourofdifferentialamplitudeonRTCandRTDisdue
a 0 arga+argb tosourcestructure,thisstillleaveseffectsontheotherantennas
arg ≡ .
0 b! 2 unexplained. First let us consider what a pointing error would
looklike.Withtheexceptionofsourceae317(forwhichtheso-
The moststrikingfeatureofFigs. 7 and8is thehighSNR. lutions are much too noisy anyway), the other six sources are
Theyshowahighdegreeoftemporalcontinuityinthesolutions, wellwithinthemainlobeoftheprimarybeam,wherethebeam
andstatisticallysignificantstructure.Thisstronglysuggeststhat gain can be expected to decrease smoothly with distance from
thesolutionsrepresentrealphysicalornumericaleffects.Asto pointingcentre.We shouldthereforeexpectto see ||∆E || > 1
sp
the nature of these effects, I still do not have satisfactory an- if antenna p mispoints towards source s (the source appears
swers,thoughitishopedthatthe“QMCProject”mentionedear- brighter on antenna p), and ||∆Esp|| < 1 if it mispoints away.
lierwillshedsomemorelightontheissues.Therestofthissec- WSRT dishesareequatoriallymounted,somispointingsdueto
tion discusses some of the more prominentquestions,and pro- mechanicalorelectronicerrorsineitheraxisdrivewouldbesta-
posessomeratherspeculativeexplanations. tionary with respect to the sky, and thus cause constant ||∆E||
offsets. Mispointings due to wind pressure, thermal or gravi-
tational deformation, on the other hand, would be intrinsically
2.1.Absorbingerrorsintheskymodel
time-variable(butwouldperhapscorrelatebetweenadjacentan-
tennas).
Asalreadymentioned,amajorcaveatoftheflyswatterapproach
isthat∆Esolutionswilltendtoabsorbinaccuraciesinthesource QuiteafewplotsinFig.7doshow(mostly)staticoffsets.It
canbeilluminatingtopresent||∆E||inaformatIcallthe“rogues
model. By analogy (and for exactly the same reasons), clas-
gallery”(Figs.9and10).Thisshows,foreachofthe14anten-
sic selfcal alone cannot solve for absolute positions or fluxes.
Indeed,if the true positionof a source l = (l,m) is offset from nas, a 12-houraverage ||∆E|| per source, using circles of vary-
themodelpositionl(mod) = l+δl=(l+δl,m+δm),whilethetrue ing size placed at the position of the source. The magnitudeof
brightnessB differsfromthemodelbrightnessbyamultiplica- (||∆E||−1)isindicatedbycirclesize,andthesignbycolour.A
tive matrix factor: B(mod) = ABAH (the latter being a straight- static mispointingin, e.g., a Northerndirectionwould showup
forwardgeneralizationofascalarfactora2),thenthecoherency asbluecirclesinthetophalfoftheplot(i.e.sourcesappearing
brighter),andredcirclesinthebottomhalf.
termoftheRIMEforthemodelsourcemaybewrittenoutas
Thegalleriesshow exactlysuch a patternforantennasRT5
through RT8 (and perhaps RT4), for both the 2003 and, to a
lesserextent,the2006observations.Itisalittlebitstrangethat
X(mod) = K (l+δl)B(mod)KH(l+δl)
pq p q 4 (oreven5)adjacentantennaswouldso consistentlymispoint
= K (δl)K (l)ABAHKH(l)KH(δl) North,anddothesamethreeyearslater.Perhapsthisisanother,
p p q q
poorlyunderstoodconsequenceofunmodelledsourcestructure.
= [K (δl)A]·[K (l)BKH(l)]·[K (δl)A]H
p p q q SomepointerstothisarethatantennasRT4–8,beinginthemid-
= [K (δl)A]·X ·[K (δl)A]H. dle of the array, form up predominantly shorter baselines, and
p pq q
8 O.M.Smirnov:RevisitingtheRIME.III.AddressingDDEsinWSRTobservationsof3C147.
0 1 2 3 4 5 6 7 8 9 A B C D
1.05
1.00 B232
0.95
11..0005 C270
0.95
1.08
1.02 D141
0.96
1.1
1.0 F012
0.9
1.1
1.0 H283
0.9
1.1
1.0 K063
0.9
1.25
1.00 ae317
0.75
0 1 2 3 4 5 6 7 8 9 A B C D
1.04
1.00 B232
0.96
1.08
1.02 C270
0.96
1.1
1.0 D141
0.9 Fig.7. Differential gain-amplitudes (||∆E||) as
1.1
1.0 F012 a function of time for the 2003 (top) and
0.9
1.1 2006 (bottom) observations. Rowscorrespond
1.0 H283
0.9 to sources, columns to antennas. The vertical
1.2
1.0 K063 plot scaleisfixed withineach row, but differs
0.8
fromrow torow. Horizontal linesindicatethe
1.5
01..50 ae317 ||∆E||=1level.
0 1 2 3 4 5 6 7 8 9 A B C D
2
0 B232
(cid:1)2
2
0 C270
(cid:1)2
2.5
0.0 D141
(cid:1)2.5
4
0 F012
(cid:1)4
10
0 H283
(cid:1)10
8
0 K063
(cid:1)8
20
0 ae317
(cid:1)20
0 1 2 3 4 5 6 7 8 9 A B C D
2.5
0.0 B232
(cid:2)2.5
3
0 C270
(cid:2)3
4
0 D141
(cid:2)4 Fig.8.Differentialgain-phases(arg∆E,inde-
5
0 F012 grees)asafunctionoftimeforthe2003(top)
(cid:2)5
8 and 2006 (bottom) observations. Rows corre-
0 H283
(cid:2)8 spondtosources,columnstoantennas.Thever-
08 K063 ticalplotscaleisfixedwithineachrow,butdif-
(cid:2)8 fersfromrowtorow.Horizontallinesindicate
40
(cid:2)400 ae317 thearg∆E=0level.
that the long-baseline antennas RTC and RTD exhibit the op- tionfromDDE-relatedartefactsinthesourceparameters.Truly
positebehaviour.(Whathinderssuchananalysisistheunfortu- robust methods for disentangling source structure from DDEs
natefactthatthethreebrightestoff-axissourcesallexhibitsome have yet to be developed. It is also clear that an approach that
structure,andallthreelieinthebottomhalfofthefield.)Another parametrizestheDDEsina“global”way,suchaspointingself-
puzzling feature is the consistently low ||∆E|| for sources F, H cal,isthewayforward–whatisnotyetclearishowmuchthe
and K on antenna RTC in 2003 (and to a far lesser extent in globalsolutionitselfcanbeaffectedbyunmodelledstructurein
2006).Ifduetosourcestructure,whydoesitnotrepeatonRTD? thebrightersources,andwhattodoaboutit.
PerhapsRTCismispointingtotheSouth?
Antennas RT0–2, RT9 and RTA, on the other hand, show
2.3.Phasebehaviour
completelydifferentpatterns,withlittletonosimilaritybetween
2003and2006.Some of these are consistentwith a static mis- A rather prominentfeature of the phase plots in Fig. 8 is their
pointing. Some antennas (RT8 and RT9, and RTB especially) continuityfromantennatoantenna,andthefactthatthephases
alsoshowahintoftimevariabilityin||∆E||. at the two ends of the array exhibit opposite temporal trends.
In any case, it is clear that the complicated interaction be- This is suggestive of an evolving phase slope over the array.
tween source structure and differential gain-amplitudes makes Fittingsuchaslope(persource)producessomeverystrikingre-
the latter extremely difficult to interpret. Note also that my sults(Fig.11).Thedominantphaseeffectisclearlyglobalrather
(or rather de Bruyn’s) source model was built by NEWSTAR thanantenna-based,andisextremelyconsistentacrossbothob-
basedonregularselfcal,sothere’sboundtobesomecontamina- servations.
O.M.Smirnov:RevisitingtheRIME.III.AddressingDDEsinWSRTobservationsof3C147. 9
Average ||dE||, RT0 Average ||dE||, RT1 Average ||dE||, RT2 Average ||dE||, RT3 Average ||dE||, RT4
40 40 40 40 40
30 30 30 30 30 60 60
20 20 20 20 20
10 10 10 10 10
0 0 0 0 0 40 0.96" 40 0.91"
(cid:3)10 (cid:4)10 (cid:5)10 (cid:6)10 (cid:7)10
(cid:3)(cid:3)3200 (cid:4)(cid:4)3200 (cid:5)(cid:5)3200 (cid:6)(cid:6)3200 (cid:7)(cid:7)3200 20 0.27" 0.31" 20 0.37" 0.33"
(cid:3)4040 30 20 10 0 (cid:3)10 (cid:3)20 (cid:3)30 (cid:3)40(cid:4)4040 30 20 10 0 (cid:4)10 (cid:4)20 (cid:4)30 (cid:4)40(cid:5)4040 30 20 10 0 (cid:5)10 (cid:5)20 (cid:5)30 (cid:5)40(cid:6)4040 30 20 10 0 (cid:6)10 (cid:6)20 (cid:6)30 (cid:6)40(cid:7)4040 30 20 10 0 (cid:7)10 (cid:7)20 (cid:7)30 (cid:7)40
4300 Average ||dE||, RT5 4300 Average ||dE||, RT6 4300 Average ||dE||, RT7 4300 Average ||dE||, RT8 4300 Average ||dE||, RT9 0 0.18" 0.21" 0.51" 0 0.18" 0.20" 0.49"
20 20 20 20 20 0.20" 0.22"
100 100 100 100 100 !20 "20
(cid:8)10 (cid:9)10 (cid:10)10 (cid:11)10 (cid:12)10
(cid:8)(cid:8)3200 (cid:9)(cid:9)3200 (cid:10)(cid:10)3200 (cid:11)(cid:11)3200 (cid:12)(cid:12)3200 !40 "40
(cid:8)4040 30 20 10 0 (cid:8)10 (cid:8)20 (cid:8)30 (cid:8)40(cid:9)4040 30 20 10 0 (cid:9)10 (cid:9)20 (cid:9)30 (cid:9)40(cid:10)4040 30 20 10 0 (cid:10)10 (cid:10)20 (cid:10)30 (cid:10)40(cid:11)4040 30 20 10 0 (cid:11)10 (cid:11)20 (cid:11)30 (cid:11)40(cid:12)4040 30 20 10 0 (cid:12)10 (cid:12)20 (cid:12)30 (cid:12)40
40 Average ||dE||, RTA 40 Average ||dE||, RTB 40 Average ||dE||, RTC 40 Average ||dE||, RTD !60 "60
30 30 30 30
20 20 20 20 60 40 20 0 !20 !40 !60 60 40 20 0 "20 "40 "60
10 10 10 10
0 0 0 0 Fig.12.Fittedpositionoffsetscorrespondingtothephaseslopes
(cid:13)10 (cid:14)10 (cid:15)10 (cid:16)10
(cid:13)20 (cid:14)20 (cid:15)20 (cid:16)20 of Fig. 11 (2003 observation on the left, 2006 on the right).
(cid:13)30 (cid:14)30 (cid:15)30 (cid:16)30
(cid:13)4040 30 20 10 0 (cid:13)10 (cid:13)20 (cid:13)30 (cid:13)40(cid:14)4040 30 20 10 0 (cid:14)10 (cid:14)20 (cid:14)30 (cid:14)40(cid:15)4040 30 20 10 0 (cid:15)10 (cid:15)20 (cid:15)30 (cid:15)40(cid:16)4040 30 20 10 0 (cid:16)10 (cid:16)20 (cid:16)30 (cid:16)40 Thelengthofthearrowsisexaggeratedbyafactorof1200:the
Fig.9. “Rogues gallery” plot for the 2003 observation. This biggestoffsetisinfactjustunder1′′.
showsthe12-houraverage||∆E||persource,asseenbyeachan-
tenna.Bluecirclescorrespondtovaluesof||∆E||>1,redcircles
tovaluesof||∆E||<1,andareasareproportionalto|||∆E||−1|. observation),is, to first order,consistentwith a rotationof 45′′
Linethicknessindicatesthestatisticalsignificanceof|||∆E||−1|; around the phase centre. The corresponding slope curves are
filledcirclesarefordetectionsofover3σ.Thelargegridcircle plotted in red. While there are some significant differences in
isatradius30′. the brighter sources, it seems clear that the dominant effect is
notaninstrumentalDDEatall,buta systematicrotationofthe
Average ||dE||, RT0 Average ||dE||, RT1 Average ||dE||, RT2 Average ||dE||, RT3 Average ||dE||, RT4 skymodel.ThemodelpositionsarederivedbyNEWSTARfrom
40 40 40 40 40
30 30 30 30 30 directfitstothevisibilities,anddeBruyn(priv.comm.)hasinde-
20 20 20 20 20
10 10 10 10 10 pendentlycross-checkedthepositionsofdistantsourcesagainst
0 0 0 0 0
(cid:17)10 (cid:18)10 (cid:19)10 (cid:20)10 (cid:21)10 theNVSS,whichseemstoprecludearotationinthemodelitself.
(cid:17)(cid:17)3200 (cid:18)(cid:18)3200 (cid:19)(cid:19)3200 (cid:20)(cid:20)3200 (cid:21)(cid:21)3200 Notethata45′′ rotationcanalsobeintroducedbyaclockerror
(cid:17)4040 30 20 10 0 (cid:17)10 (cid:17)20 (cid:17)30 (cid:17)40(cid:18)4040 30 20 10 0 (cid:18)10 (cid:18)20 (cid:18)30 (cid:18)40(cid:19)4040 30 20 10 0 (cid:19)10 (cid:19)20 (cid:19)30 (cid:19)40(cid:20)4040 30 20 10 0 (cid:20)10 (cid:20)20 (cid:20)30 (cid:20)40(cid:21)4040 30 20 10 0 (cid:21)10 (cid:21)20 (cid:21)30 (cid:21)40
Average ||dE||, RT5 Average ||dE||, RT6 Average ||dE||, RT7 Average ||dE||, RT8 Average ||dE||, RT9 ofabout2.9s,oracorrespondingrotationalerrorinconversion
40 40 40 40 40
30 30 30 30 30 of uvw coordinates from apparent to J2000. Since NEWSTAR
20 20 20 20 20
10 10 10 10 10 and MeqTreesuse completely differenttool chains and visibil-
0 0 0 0 0
(cid:22)10 (cid:23)10 (cid:24)10 (cid:25)10 (cid:26)10 itydataformats,Icannotexcludeacoordinateconversionerror
(cid:22)20 (cid:23)20 (cid:24)20 (cid:25)20 (cid:26)20
(cid:22)30 (cid:23)30 (cid:24)30 (cid:25)30 (cid:26)30 somewherealongtheline.Thisneedstobeurgentlyinvestigated.
(cid:22)4040 30 20 10 0 (cid:22)10 (cid:22)20 (cid:22)30 (cid:22)40(cid:23)4040 30 20 10 0 (cid:23)10 (cid:23)20 (cid:23)30 (cid:23)40(cid:24)4040 30 20 10 0 (cid:24)10 (cid:24)20 (cid:24)30 (cid:24)40(cid:25)4040 30 20 10 0 (cid:25)10 (cid:25)20 (cid:25)30 (cid:25)40(cid:26)4040 30 20 10 0 (cid:26)10 (cid:26)20 (cid:26)30 (cid:26)40
Average ||dE||, RTA Average ||dE||, RTB Average ||dE||, RTC Average ||dE||, RTD Ifindeedtheentireskymodelisslightlyrotated,thenperhapsthe
40 40 40 40
30 30 30 30 imageofFig.1canevenbeimprovedupon!
20 20 20 20
10 10 10 10
0 0 0 0
(cid:27)10 (cid:28)10 (cid:29)10 (cid:30)10
(cid:27)(cid:27)3200 (cid:28)(cid:28)3200 (cid:29)(cid:29)3200 (cid:30)(cid:30)3200 2.4.Feedingdifferentialgainsbackintotheskymodel
(cid:27)4040 30 20 10 0 (cid:27)10 (cid:27)20 (cid:27)30 (cid:27)40(cid:28)4040 30 20 10 0 (cid:28)10 (cid:28)20 (cid:28)30 (cid:28)40(cid:29)4040 30 20 10 0 (cid:29)10 (cid:29)20 (cid:29)30 (cid:29)40(cid:30)4040 30 20 10 0 (cid:30)10 (cid:30)20 (cid:30)30 (cid:30)40
TheresultsabovesuggestedthatIcouldimprovemyskymodel
Fig.10. “Rogues gallery” plot for the 2006 observation, using
byfeedingbackinsomeinformationextractedfromthe∆E so-
thesamescaleasFig.9
lutions. In the previous section, I obtained a correction to the
modelpositionsofthesevensources.7Followingthediscussion
B232 C270 D141 F012 H283 K063 ae317 of Sect. 2.1, I could also provide corrections for the I and Q
2 2 3 6 4 12
1 1 2 0.0 3 2 6 fluxesbyapplyingtheper-sourceaverage∆Eamplitudes:
0 0 1 (cid:31)1.5 0 0 0
(cid:31)(cid:31)12 (cid:31)(cid:31)12 (cid:31)01 (cid:31)3.0 (cid:31)(cid:31)63 (cid:31)(cid:31)42 (cid:31)(cid:31)162
B232 C270 D141 F012 H283 K063 ae317
0311....0055 011...055 0311....0055 0311....0055 0522....0055 0633 10868 B(scorr) = |∆|∆Ees|B|s|∆E0s| 1
3.0 3.0 3.0 4.5 5.0 6 16 |∆E | ≡ xs = |∆E (t,ν )|,
Fig.11. Phase slopes over the array as a function of time (in s 0 |∆eys|! NantNtNν Xp,i,j sp i j
deg/km)inthedirectionofthesevensourcesforthe2003(top)
and2006observations(bottom).Thegreenlinesindicatephase where t and ν represent the time and frequency solution
i j
slopescorrespondingto the fitted positionoffsets(Fig. 12), the intervals of ∆E. In terms of the I and Q fluxes, the correction
redlines–phaseslopescorrespondingtoanoverallfieldrotation becomes:
of45′′.
I(corr) =Σ·I+∆·Q, Q(corr) =∆·I+Σ·Q,
Aphaseslopeoverthearraycanbeinterpretedasanappar-
1 2 2 1 2 2
entpositionoffset.ItappearsthattheslopebehaviourinFig.11 Σ= |∆e | +|∆e | , ∆= |∆e | −|∆e | .
xs ys xs ys
can be fitted quite well by constant position offsets. The best- 2(cid:18) (cid:19) 2(cid:18) (cid:19)
fitting position offsets are indicated in Fig. 12, and the corre-
7 Forthemoment,I’veleftasidetheissueofwhetherthepositional
spondingslopecurvesareplottedingreenonFig.11. offsetsareultimatelyduetoaglobalfieldrotation.Improvingtheposi-
Figure12immediatelysuggestsafieldrotation.Andindeed, tionsofsevenofthebrightestoff-axissourcesshouldalreadyproduce
theentirecollectionofphaseslopes(forboththe2003and2006 ansuperiorskymodel.
10 O.M.Smirnov:RevisitingtheRIME.III.AddressingDDEsinWSRTobservationsof3C147.
B232 C270 D141 F012 H283 K063 ae317
0.8 0.8 01..86 01..86 1.5 31..05 63
0.0 0.0 0.0 0.0 0.0 0.0 0
#0.8 #0.8 ##10..68 ##10..68 ##31..05 ##31..05 ##63
B232 C270 D141 F012 H283 K063 ae317
1.6 2 2 3.0 5.0 12
0.8 0.8 1 1 1.5 2.5 6
$$010...068 $00..08 $$012 $$012 $01..05 $02..05 $$1062
Fig.15. Phase slopes over the array as a function of time (in
deg/km),usingimprovedskymodelsforthe2003(top)and2006
observations(bottom).Thegreenlinesindicatephaseslopescor-
responding to the fitted position offsets (Fig. 16). Compare to
Fig.11.
Fig.13. Calibration with an improved sky model. This shows
single-bandresidualimagesafterG andM solutions.Theleft
p pq
imageisfromtheoriginalreduction,therightimageusesasky 60 60
modelimprovedviamy∆E analysis.Crossesindicatetheposi- 40 0.13" 40 0.19"
tionsofsourcesforwhichthemodelwasimproved,plus3C147
0.05"
itself(sourceA). 20 0.04" 0.04" 20 0.04"
0.07" 0.08"
0 0.03" 0 0.04"
0.02" 0.03"
0.03" 0.03"
%20 &20
IthereforeappliedthesecorrectionsforI,Q,andpositionto
my sky models(independentlyforthe 2003and 2006observa- %40 &40
tions),andrepeatedthecalibrationprocedure.Animprovement %60 &60
in single-band residuals was immediately apparent (Fig. 13) –
60 40 20 0 %20 %40 %60 60 40 20 0 &20 &40 &60
afterG andM solutions,thesevenoff-axissourcessubtracted
p pq
noticeably better. Residuals after ∆E solutions, on the other Fig.16.Fittedpositionoffsetscorrespondingtothephaseslopes
sp
of Fig. 15 (2003 observation on the left, 2006 on the right).
hand,lookedprettymuchthesame(thisisnotsurprising,since
differential gains had already taken care of the visible off-axis Thelengthofthearrowsisexaggeratedbyafactorof1200:the
biggestoffsetisinfactjustunder0.2′′.ComparetoFig.12.
errorsintheoriginalreduction),withaveryslightimprovement
around 3C 147 itself, which can be explained by improvedG
p
solutionsduetothemoreaccurateskymodel.
The brighter sources B, C, and (to a lesser extent) D show
clearsecond-orderphaseeffects,bothinthephaseslopes,andin
2.5.PhasebehaviourII the phase solutions themselves. The temporalcontinuity in the
phaseslopescanbeinterpretedasatime-variablepositionoffset.
Presumably, the remaining residual structures in Fig. 13 are Icanspeculativelyoffertwoexplanationsforsuchanoffset:
more representativeof the instrumentalDDEs per se, since in-
accuraciesintheskymodelhavebeensignificantlyreduced.We – Unmodelled source structure (again!) For any given hour
shouldalsoexpectthedifferentialgainsolutionstobemorein- angle, an E-W array only sees an integrated cross-section
dicative of the actual DDEs (apart from the issue of resolved through the source in a given direction. If the source is
sources affecting ||∆E|| on antennas RTC and RTD, which the slightly resolved with an asymmetric “hotspot”, the zero-
improvedskymodelsdonotaddressatall).Ofparticularinterest ordermomentofeachsuchcross-sectionwillbeslightlydif-
istheeffectthattheimprovedmodelhasonthedifferentialgain- ferent.
phases. As for the gain-amplitudes, we would expect them to – Differential tropospheric or large-scale ionospheric refrac-
differbyonlyanoverallper-sourcescalingfactor.Indeed,mak- tion, including perhapsapparentchange of baseline caused
ing the same ||∆E|| plot as in Fig. 7 confirmsthis – it is, to all byrefraction(theAndersoneffect).
intentsandpurposes,identical(andomittedheretosavespace),
sincetheplottedamplitudesarerenormalizedbytheper-source Anotherpuzzlingfeatureofthe arg∆E solutionsin Fig.14
average||∆E||. arethesignificantand(tofirstdegree)constantphaseoffsetsof
The phases, on the other hand, show a marked difference, somesources(e.g.H, K,ae)onsomeantennas.Theoffsetsare
since the formerly dominant effect – that of position offsets – mostly(thoughnotcompletely)consistentbetweenthe2003and
hasbeentakenout.Thearg∆E solutionsthemselvesareshown 2006 observations. None of the explanationsoffered above are
inFig.14.Phaseslopesarestillverymuchinevidence,ascanbe consistentwithaconstantphaseoffset!Couldthisbethephase
seeninFig.15.Somewhatsurprisingly,theseslopesindicatethat componentof the primarybeam? There are too few sources in
someresidualpositionoffsetsremain,atalevelof15%to20% this reduction to infer any sort of directional dependence, but
oftheoriginaloffsets(Fig.16).Thissuggeststhatmyprocedure perhaps the “QMC Project” can provide more insights on this
of fitting phase slopes to arg∆E solutions, followed by fitting effect.
positionoffsetstotheslopes,systematicallyunderestimatesthe
true position offsets. This is possibly an effect of the complex
2.6.Thelurkingerrors
averaging implicit in having one ∆E solution per a relatively
largesolutioninterval(20MHzby30minutes).Ifso,thiscould The two calibrations (with the original and the improved sky
perhapsbe incorporatedas a multiplicativecorrectionfactor in models)describedabovehaveproducedwhatappeartobeiden-
the model update procedure. Further work is required to fully ticalfinalmaps.Thisshowsthatthe“flyswatter”canaccommo-
understandtheeffect. date for significanterrorsin the sky model.On the otherhand,