Parallel multi-wavelength calibration algorithm for radio astronomical arrays

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Parallel multi-wavelength calibration algorithm for radio astronomical arrays

Martin Brossard, Mohammed Nabil El Korso, Marius Pesavento, Rémy Boyer, Pascal Larzabal, Stefan J. Wijnholds

To cite this version:

Martin Brossard, Mohammed Nabil El Korso, Marius Pesavento, Rémy Boyer, Pascal Larzabal, et

al.. Parallel multi-wavelength calibration algorithm for radio astronomical arrays. Signal Processing,

Elsevier, 2018, 145, pp.258-271. �10.1016/j.sigpro.2017.12.014�. �hal-01675545�

Parallel multi-wavelength calibration algorithm for radio astronomical arrays ^R

Martin Brossard

^a,∗

, Mohammed Nabil El Korso

^d

, Marius Pesavento

^c

, Rémy Boyer

^e

, Pascal Larzabal

^b

, Stefan J. Wijnholds

^f

aSATIE, UMR 8029, École Normale Supérieure de Cachan, Cachan, France

bCAOR, Mines Paristech, 60 boulevard Saint-Michel, 75006 Paris,

cCommunication Systems Group, Technische Universität, Darmstadt, Germany

dIUT de Ville d’Avray, University of Paris Ouest Nanterre La Défense, Lemé EA 4416, France

eLaboratoire des Signaux et Systèmes (L2S), University of Paris-Sud, Gif-sur-Yvette, France

fNetherlands Institute for Radio Astronomy (ASTRON), P.O. Box 2, Dwingeloo NL-7990 AA, The Netherlands

1. Introduction

Advanced radio astronomicalarrays, such asthe existing LOw FrequencyARray(LOFAR)[1]andthefutureSquare KilometreAr- ray (SKA) [2], form large sensor arrays, which are constituted of manysmallantennaelements.Asan example,theLOFAR consists of50stations, mainlylocated inThe Netherlands.Eachstation is a closed packed sensor array,composed ofat least 96 low-band antennas(LBA,30–90MHz)and48high-bandantenna(HBA,109–

240MHz)tiles, each tilebeinga 4-by-4uniformregular array of HBAs.Suchinterferometersoffer alargeaperture sizeanddeliver largeamountsofdatainordertoreachhighperformanceinterms of resolution, sensitivity and survey speed [2]. Nevertheless, to

R This work was supported by MAGELLAN (ANR-14-CE23-0 0 04-01) ASTR-2017- MARGARITA and ON FIRE project (Jeunes Chercheurs GDR-ISIS). This work is also funded by IBM, ASTRON, the Dutch Ministry of Economic Affairs and the Province of Drenthe.

∗ Corresponding author at: CAOR, Mines Paristech, 60 boulevard Saint-Michel, 75006 Paris, France.

E-mail addresses: martin.brossard@mines-paristech.fr (M. Brossard),

achievethetheoreticaloptimalperformancebounds,aplethoraof signalprocessingchallengesmustbetreated[3,4].Thiscoverscali- bration,imagesynthesisanddatareduction.Inthispaper,wefocus oncalibrationissuesby designingacomputationally efficientpar- allelalgorithm.Calibrationproceduresdevisedforsuchradiointer- ferometersmust estimate:(i) the gainresponse andnoise power ofeachantenna[5–8];and(ii)thepropagationdisturbances,espe- ciallythephasedelayscausedbytheionosphere,whichscalewith wavelength[9,10].

Specifically, in this paper, we focus on the regime where the linesofsightfromeachantennatowarda sourceintheskycross the sameionosphericlayer andwherethe thicknessofthe iono- sphere can be direction dependent [11], which is represented in Fig.1andwell adaptedforthecalibrationofaLOFAR stationand thefutureSKAstationsaswell asthecoreofthesearrays.Conse- quently,inthisregime,theionosphericphase delaysareaddedto the geometric delays andintroduce angular-shifts for the source directions [7,12], which are direction and wavelength dependent [9,13].Byestimating calibratorshifts (i.e.,thedifference between thetruecalibratordirections,knownfromtables[14–17],andtheir estimatedapparentdirections),interpolation methodscanbeeffi- cientlyapplied inordertoobtainaphase screenmodelthatcap- m.elkorso@u-paris10.fr (M.N. El Korso), mpesa@nt.tu-darmstadt.de (M. Pesavento),

remy.boyer@l2s.centralesupelec.fr (R. Boyer), pascal.larzabal@satie.ens-cachan.fr (P.

Larzabal), wijnholds@astron.nl (S.J. Wijnholds).

Fig. 1. The so-called regime 3, which is considered in this paper, assumes that V S and A S . This leads to ionospheric perturbations which are direction dependent (after [8,11] ).

turestheionosphericdelaysovertheentireField-of-View[12].We emphasizethatinadditiontothephasescreenreconstructionstep, thecalibration usuallyinvolves theestimationofthecomplex di- rectionindependent(DI)gainsoftheantennas,their directionde- pendent(DD)gainstowardeachcalibratorandtheirnoisepowers [6],forthewholeavailablerangeofwavelengths.

The characteristics of the calibration sources, i.e., their true/nominal directions and their powers without the effects of theionosphereorantennaimperfections,areaprioriknownfrom tables which is required to solve such calibration problems [7]. Based on this knowledge, state-of-the-art calibration algorithms operatemostlyinan iterativemannerinamono-wavelengthsce- nario [5–7,18–22]. For instance, the (Weighted) Alternating Least Squares approachhas beenadapted for LOFAR stationcalibration [5,6],inwhichclosed-formexpressionshavebeenobtainedforan- tennagainandsensornoisepowerparameters.Nevertheless,such algorithms presentthree major limitations: (i) suboptimalitydue totheconsiderationofonlyonewavelengthatatime;(ii)theas- sumption of a centralized processor, i.e., a single compute agent simultaneous accessingalldata;and(iii)theinefficiency withre- specttotheDirection-of-Arrival(DoA)estimationperformancesin thesevereradioastronomicalcontexts.

Concerning limitation(i), mostcurrent approachescalibrate a single wavelength at a time [5–7,18–20] assuming that solutions can be combined afterwards. This is usually a suboptimal ap- proach. Forthe real-time systemof the MurchisonWidefield Ar- ray(MWA)[23]severalfilteringtechniquesarecombinedtoisolate theresponseacrossfrequencysuchthationosphericrefractionand instrumental gainscanbe fittedacrossfrequencyona per-source bases [24].The MWAreal-time system is highly optimizedfor a specific case,whilewe aimtoprovide amoregeneralframework

thatcanbeadjustedasneeded.Amoregeneralapproachtomulti- wavelengthcalibration in thecontext of large radio astronomical arrays is the procedure presented in [25]. That procedure treats calibrationasa single,mathematicallydefinedoptimizationprob- lemthatissolvedusingageneralsolvingstrategy.Unlike[25],we considera structured parametric modelbased onthe sampleco- variancematrixinthearray processingframeworkbasedonsome a prioriknowledge about thephysics of the systemasdescribed inthedatamodelsection.Asaconsequence,theproblemsaredif- ferent,sincewedonotestimatetheelementoftheJonesmatrices asin[25],butwe aimto estimatethe apparent directionsofthe calibrationsources,thedirectiondependentanddirectionindepen- dentcomplexgainsofthearray elementsandtheir noisepowers usingphysicalconstraintsgiveninSection2.

Furthermore,regarding the limitation (ii),the aforementioned state-of-the-art methods typically are designed for a centralized hardwarearchitecture, whereas, takingthe LOFAR stations asex- ample,processingall512frequencybandssimultaneouslyatasin- gle location, iffeasible,is challenging. As a solution,parallel and consensus algorithms, mostly based on the Alternating Direction ofMultipleMultipliers(ADMM)[26],haverecentlybeenmassively investigated in parametric estimation frameworks [27–34]. These consensusschemescanoperateinvariousnetworktopologies.We will consider a group of compute agents, where each agent ac- cessesdataacrossasmallbandwidthandcanonlycommunicates witha fusion centerthrough low data ratechannels. Thisarchi- tecture models correctly the situation for radio interferometers, where data forthe full observing bandwidth is typically divided intochannelsandchannelsaregroupedintosubbands.

Finally, regarding the limitation(iii), classical subspace meth- ods, such as MUSIC [35], have been commonly applied in radio astronomical calibration [6]. However, these techniques are inef- ficient in low Signal-to-Noise-Ratio (SNR) scenarios and require knowledgeoftheexactnumberofsourcesinthescene.Asanal- ternative,recentapproaches,basedonsparsereconstructionmeth- ods, came into focus of DoA estimation for fully calibrated ar- rays [36–38] aswell as forpartially calibratedarrays [39]. These approachesexhibit the super-resolution property, robustness and computational efficiency, without the aforementioned limitations of subspace-based methods [36]. However, most methods based onthecompressivesensingframeworkaredesignedforacentral- izedhardwarearchitecture andare appliedinthesignaltime do- main[20,40].Thesemethods becomecomputationally impractical withhuge numbers of observations, making them unsuitable for calibrationintheradio interferometercontextforwhichwecom- monly accessonly the sample covariancematrix rather than the time signal itself [7]. This issue was also recognized in recently in[22,41],in whichthe authors propose a techniquefor,respec- tively,fullcalibrationandblind calibrationofthe DIgainsforin- dividualfrequencychannelsby assuming thattheobserved scene issparse.Westressthatmanyworksbasedofcompressedsensing hasbeendevelopedforimage reconstruction intheradio astron- omycommunity[42–45]asaresultofthesparsenatureofthein- terferometricsampling.Suchsparseimplementationsproducebet- ter results on large extended objects with high angular resolu- tion than other classical deconvolution methods. However, these worksassumeusuallyanalreadycalibratedarrayandtheproblem- aticofreconstructedanimagestronglydifferfromthecalibration.

In image reconstruction, compressed sensing studies are based on sparse analysis and/or sparse synthesis using, e.g., wavelets, unionofwaveletbases orsynthesis IUWT(IsotropicUndecimated WaveletTransform).Inourcalibrationproblem, compressedsens- ing techniquesare used inorder toestimate the phase shiftdue tothe ionosphereusing aDistributed IterativeHardThresholding methodinwhichthegridisconstructedbasedonarrayprocessing modeling.

Fig. 2. Operation flow and signaling exchange between a local processor Az and the central processor. The first third arrows are performed repetitively during Algorithm 2 and the two middle arrows during Algorithm 4 .

Fig. 3. LOFAR’s initial test station antenna locations [73] .

In this paper, we propose an iterative algorithm, namely the ParallelMulti-wavelengthCalibration Algorithm(PMCA),that per- formsthecalibrationofaradioastronomicalarray usingitsarray covariance matrix (usually referred to as matrix of visibilities in radioastronomy), involvingdisturbances duetoindividual anten- nasandpropagationeffects.Weassumethat thesensorarray has an arbitrary geometry, identical elements and is simultaneously excited by inaccurately known calibration sources and unknown weaknon-calibrationsources. Theproposed PMCAovercomesthe aforementioned limitations, by: (i) reformulating the parametric modelinthemulti-wavelengthscenarioinordertoexploitwave- lengthdiversity;(ii)relying onparallelandconsensus algorithms;

and(iii) adapting the sparse reconstruction methods to the cali- brationofradio astronomicalarrays. Fromthe parallelcalibration perspective,thePMCAsuccessivelyestimatestheDIantennagains along with the DD and noise parameters for multiple subbands, wherewe enforcethecoherence overthewavelengthoftheesti- matesbasedonphysicalandastronomicalphenomena[8,9,13,46].

The rest of the paper is organized as follows: in Section 2, we formulate the data model and its associated parallel multi- wavelength calibration problem. In Section 3, we present the overviewoftheproposedschemeandthen describeitstwomain alternating steps.The constrainedCramér–Rao bound ofthe data modelisderivedinSection4.Numericalsimulationsandrealdata tests,inSection5,showthefeasibilityandsuperiorityofthepro- posedscheme compared to mono-wavelengthcalibration. Finally, wegiveourconclusionsinSection6.

Inthefollowing,(.)^∗,(.)^T,(.)^H,(.)†,(.)^α,(.),(.)and[.]n de- note,respectively,conjugation,transposition,Hermitian transposi-

tion, pseudo-inverse, element-wiseraising to

α

^{, real part,imagi-}

narypart andthe nthelement ofavector. The expectation oper- ator is E

{

.

}

, denotes the Kronecker product, exp(.) and rep- resent the element-wise exponential function and multiplication (Hadamardproduct), respectively. The operatordiag(.)^{converts a} vector to a diagonalmatrix withthe vector alignedon the main diagonal, blkdiag(.) ^{is the} block-diagonal operator for matrices, whereas vecdiag(.) ^{produces a vector fromthe main diagonal of} its matrix argument andvec(.) ^{converts a matrixto a vector by} stackingthecolumnsofitsentry.Theoperators

^.

0,

^.

2and

.

F

referto thel₀ norm,i.e., thenumberofnon-zero elementsofits entry, the l₂ andFrobenius norms, respectively. x0 means that eachelementinxisnon-negative.

2. Datamodelandproblemstatement 2.1. Covariancematrixmodel

Consider an array comprisedofP elements, withknown loca- tions, each referred by its Cartesian coordinates

ξ

p=[xp,yp,zp]^T forp=1,...,P,thatwestackin=

ξ

1,...,

ξ

P

T

∈R^P×3.Thisar- ray isexposedto Qknownstrongcalibration sourcesandQ^U un- knownweaknon-calibrationsources.LetD^K=

d^K₁,...,d^K_Q

∈R^3×Q andD^U=

d^U₁,...,d^U

Q^U

∈R^3×QU denote theknown (true/nominal) calibrator direction cosines and unknown non-calibrator direc- tion cosines, respectively, in which each source direction d= [d_l,dm,dn]^Tcan beuniquely describedbya couple(d_l,dm), since dn=

1−d²_l −d_m² [6,9]. The ionosphere introduces an unknown angular-shiftforeach sourcedirection[3,12,13],dependingonthe wavelength

λ

^{,which is related to the frequency f}=c/

λ

, withc denotingthespeedoflight.Consequently,wedistinguishbetween theunknownapparentdirectionsw.r.t.thecalibrators,denotedby D_λ=

d_λ,1,. . .,d_λ,Q

, that depend on

λ

^{since the shift is wave-}

lengthdependent,andtheirtrue/nominalknowndirectionsD^K,i.e., withoutthepropagationdisturbances.

Inthefollowing,wedescribethesignalforonewavelengthbin

λ

^{.Underthenarrowband}assumption,thesteeringvectora_λ(d)to-

wardthedirectiond(atwavelength

λ

^)isgivenby a_λ

(

^dl,dm

)

= 1

√Pexp

−j2

π λ

^d

, (1)

thatwegatherformultipledirectionsinthesteeringmatrix

AD_λ= 1

√Pexp

−j2

π λ

^Dλ

. (2)

Asin[6],weassumethatallantennashaveidenticaldirectional responses. Their DD gain responses (and propagation losses) are modeled by two diagonal matrices, _λ∈C^Q×Q and^U_λ∈C^QU×QU towardthecalibrationandnon-calibrationsources,respectively.

Thereceivedsignalsfromeachantennaarepartitionedintonar- rowsubbands. The measurement ofthe sensorarray collectedin thesubbandwithwavelength

λ

^{arestackedtothereceivedsignal}

vector x_λ

(

ⁿ

)

=G_λ

AD_λ

λs_λ

(

ⁿ

)

+A_D^U

λ

^U_λ^sU_λ

(

ⁿ

)

+n_λ

(

ⁿ

)

^{, (3)}

forthenthobservation,wherewestressthateach elementofthe right part of (3)depend on

λ

^andAU_λ ^{is the steering matrix for}

unknownsources,with[x_λ(n)]p denotingthesignalcorresponding tothepthantenna,whereG_λ=diag(^g_λ)∈C^P×P modelstheDIan- tenna gains,with [g_λ]p the DI antenna gain forthe pth antenna.

s_λ(ⁿ)∈C^Q ands^U_λ(ⁿ)∈C^QU represent, respectively, the i.i.d.cali- brator andnon-calibrator signals, with[s_λ(n)]q and[s^U_λ(ⁿ)^]q, re- spectively, thesignal correspondingto theqthcalibrator andqth non-calibrator,whereasn_λ(ⁿ)∼CN(⁰,ⁿ_λ)^{denotesthei.i.d.noise} vector, with [n_λ(n)]p the thermal noise for the pth antenna [7]. Let _λ=diag(

σ

λ)∈R^Q×Q, ^U_λ=diag

σ

^U_λ

andⁿ_λ=diag

σ

ⁿ_λ

∈ R^P×P bethediagonalcovariancematricesforthecalibrators,non- calibration sources and sensor noises, respectively, and assume thatthesourcesarestatisticallyindependentfromeachother.Con- sequently, s_λ(ⁿ)∼CN(⁰,_λ), s^U_λ(ⁿ)∼CN(⁰,^U_λ) ^{andthecovari-} ance matrix R_λ=E x_λx^H_λ

of the observations corresponding to model(3)isgivenby

R_λ=ED_λM_λE^H_Dλ+R^U_λ+

ⁿ_λ^{, (4)}

inwhich

ED_λ=G_λAD_λ

_λ^{12, (5)}

M_λ=

_λ

^H_λ=diag

(

^mλ

)

^{, (6)}

andwherewehavedefinedtheunknowncovariancematrixforthe non-calibrationsourcesas

R^U_λ=G_λA_D^U

λ

^U_λ

^U_λ

G_λA_D^U

λ

^U_λ

H

. (7)

Inradioastronomy,sources(includingthecalibrationsources),are typicallymuchweakerthantheantennanoise[18],sothecovari- ance matrix isusually approximated by R_λ≈ⁿ_λ^{.Since the array} consistsofidenticalelementsandmutualcouplingcanusually be ignored,itiscommonlyassumedthat

ⁿ_λ=diag

σ

ⁿ_λ

≈

σ

_λ^{nI. (8)}

Suchnomutualcouplingiscurrentlyassumedduetoantennasep- aration(thoughthismaynotexactlybetrueinastationarray)and carefulhardwareimplementationsasinLOFAR[1].

Inordertoovercomethescalingambiguitiesintheobservation model(4),weconsiderthefollowingcommonlyusedassumptions inradioastronomy[6–8]:(i)toresolvethephaseambiguityofg_λ, we take its first element as the phase reference [47,48]; (ii) the phaseinformationof_λ^drops,consequently,weonlyestimateM_λ, i.e., itsabsolutepart;(iii)m_λ sharesacommonscalarfactorwith g_λandconsequently,weassumethatthedirectionalgaintowards the firstcalibration sourceis known/fixed;and(iv)when solving

forthecalibratordirections,acommonrotationofallsteeringvec- torscanbecompensatedbytheundirectionalgainphasesolution.

Wethereforefixthedirectionofthefirstcalibrationsource atits knownposition[5].

2.2.Modeleffectsofthewavelengthonantennagains,source directionshiftsandsourcepowers

Intheradio astronomicalcontext, theantennaandsourcepa- rametersofthecovariancematrixarecommonlyassumedaswave- length dependent [7,8]. Consequently, we assume smooth and/or known variations of the parameters g_λ, _λ^, _λ^{, U}_λ ^{and n}_λ ⁱⁿ (4)over

λ

^{,ascommonlyusedinrecentworksonarray}calibration inradioastronomy[25,49].Wesummarizetheparticularbehavior oftheunderlyingparametersasfollows:

• The DIgains,g_λ,varysmoothly over

λ

^{.Hence, inordertoim-}

pose coherence along subbands (not along different sensors), we define a set ofsmooth basis functions b_k,λ asa function ofthewavelength

λ

^,fork=1,...,K.Thesebasisfunctionsde- fineourcoherencemodel,inwhich,forthepthsensor,itsgain [g_λ]_pisafunctionofwavelength

λ

^andrepresentedasalinear combinationofbasisfunctions,hence:

[g_λ]p= K

k=1

b_k,λ

α

k,p,

∀ λ

^∈

,p=1,...,P, (9)

where

α

k,p represents the linear coefficient corresponding to the kthbasis functionforthepthsensor.Common modelsfor characterizing thisbehavior consist ofclassical polynomialsof power law over

λ

^{[8,25], e.g., we can consider theset of Kth}

order basis functions centeredaround a referencewavelength

λ

0=c/f₀ thatis defined byb_k,λ=

λ−λ0

λ0

1−k

[25]that impose smoothness w.r.t. frequency (keeping in mind that b_k,λ

0=1).

Letusdenote b_λ=

b_1,λ,...,b_K,λ

T

∈R^K, (10)

representingallpolynomialtermsandrewrite(9)as g_λ=

b^T_λI

α

=B_λ

α

,

∀ λ

∈

^{, (11)}

whereB_λ=

b^T_λI

∈R^P×PKand

α

^{isthustheaugmentedvec-}

torofhiddenvariablesdefinedby

α

=[

α

1,1,...,

α

1,P,

α

2,1,...,

α

K,P]^T∈C^PK. (12)

• TheDDgains,_λ^{,areinversely}proportionalto

λ

^,i.e.,_λ∝

λ

⁻¹, asobservedinpractice[46].Notethattheproposed algorithm can be straightforwardly adapted withanothergivenbehavior (including the simplestcaseofa constantbehavior across the wavelengthrange).

• As a consequence of the ionospheric delays, the directional shiftsareproportionalto

λ

^{2 [9,13,46].}Furthermore,weassume that the calibrationsources are wellseparated,which iscom- moninradioastronomy[6,7],andconsiderintheremainderof thispaperthatforeverywavelength:

(A1)Eachapparentcalibrationsourceliesinanuncertaintysec- torwithapredefinedangularspreadarounditsnominallo- cation.

(A2) The displacement sectorsof different calibration sources arenotoverlapping.

• The sourcepowers,_λ ^andU_λ,varycommonlywitha power lawwithdifferentspectralindexes.Weconsider thecalibrator powers,_λ^{,tobeknownfromtables,e.g.,[14–17].}

• The antenna noise covariance matrices, ⁿ_λ, do not exhibit a smooth behavior w.r.t.

λ

^{andnoise isassumedtobe indepen-}

dentoverwavelength.Nevertheless,ifanyparticularcoherence

modelforthenoisecovariancesisavailable,thisknowledgecan beincorporatedintheproposedalgorithminastraightforward manner.

2.3.Jointparameterestimationproblem

Inthissection,weformulatethecalibrationproblemasthees- timationoftheparametervectorofinterest,p,definedas

p=

p^T_λ

1,...,p^T_λ

J

^T

, (13)

under the constraints given in Section 2.2, in which p_λ=

g^T_λ,d^T_λ,1,...,d^T_λ,Q,m^T_λ,

σ

ⁿ_λ^T

T

,fromJsamplecovariancematrices

Rˆ_λ= 1 N

N

n=1

x_λ

(

ⁿ

)

^xH_λ

(

ⁿ

)

λ∈, (14)

where=

λ

1,. . .,

λ

J

representsthe setof the Javailablewave- lengthsforthewholenetworkandassumethatJ≥K,i.e.,accessing todataforatleast Kwavelengths, whichisassumedto be satis- fiedsince, e.g., forthe LOFAR, thesignal is typically divided into 512subbandswhileusuallyalowpolynomialorderissufficientto representthe variationsacrosswavelength(3–5 inour numerical simulations).

Data parallelismacrosswavelengthisarecentfieldofresearch inradioastronomicalobservations,inwhichdataare recordedas multiplechannelsat differentwavelengths [1], thus,leading toa data which is not centralized but paralleled across the network.

Thisnetworkconsistsof:(i)onefusioncenter,thatdoesnotaccess data;and(ii)Zcomputeagents.Thezthagent,Az,canonlyaccess dataforasubset z⊂^ofJz≤Jsubbands, andforeachavailable wavelength,its associated samplecovariance matrix isaccessible forexactlyoneagent.Moreover,theagentscannotexchangeinfor- mationamongthemselves.Operationflowandsignalingexchange betweenlocalprocessorsandthecentralprocessorisgiveninthe endofthealgorithmdescription(c.f.Fig.2).

Note that the estimation of the unknown matrices R^U_λ repre- sentsthe imaging step which is beyond the scope of the paper [6,7,9,50].Imagesynthesis[51–55]isusuallyperformedasasepa- rateprocedure afterthecalibrationandcanbe complementedby theproposedcalibration approach.Themain reasonforthistwo- step procedure is that the calibration step is usually carried out basedon a point source model (unlikethe imaging step) witha knownnumberofstrongcalibrators,whereas,theeffectofanun- knownnumberoftheweakest(non-calibration)sourcescanbeas- sumedabsorbedbythestrongestcalibrationsources.Thiscausesa biasintheDIgainsolutionsthatresultsinimagingartefactsknow asghosts[56,57].Thiseffectcanbemitigatedbyimprovingthesky model.Consequently,intheLOFARpipeline,analternatingscheme betweencalibrationandimagingisperformed[1,3].

3. Proposedparallelmulti-wavelengthcalibrationalgorithm

3.1.Overviewoftheproposedparallelmulti-wavelengthcalibration algorithm

In this section, we define the main steps of our proposed al- gorithm,then,inthefollowingsections,we describeeachstep in detail.

It iswell established thata statisticallyefficientestimatorcan be obtained via the Maximum Likehood method. However, from acomputational viewpoint, itsexact evaluationappearsto be in- tractableintheradio astronomicalcontext[6].Withalargenum- berofsamples,statisticallyefficientestimatorscanbedevisedus- ingtheWeightingLeastSquaresapproach.Inthiscontext,wede- fine,foreach

λ

∈^{,thelocalcostfunction[5]tobeminimizedas}

κ

λ

(

^pλ

)

=

^W−λ¹²

R_λ

(

^pλ

)

−Rˆ_λ

W⁻_λ¹²

²

F, (15)

inwhich

R_λ

(

^pλ

)

^=EDλM_λE^H_D

λ+

ⁿ_λ ⁽¹⁶⁾

denotesthecovariancematrixwhenthecontributionofthe(weak- est)non-calibratorsisabsorbedinthenoise,andW_λistheweight- ingmatrix.TheoptimalweightingmatrixforGaussiannoiseisthe inverseofthecovarianceofthe residuals[58],whichisgenerally unknown.Justifiedbyphysicalreasonsand(8),weconsiderW_λ=I in our alternating algorithm as an initialization and refine it as W_λ=ⁿ_λ ^{once we obtain an estimate ofn}_λ^{. Since n}_λ ^{is diago-} nal,we rewritethe localcost function(15),i.e., thecost function associatedwiththewavelength

λ

^,as

κ

λ

(

^pλ

)

⁼

R_λ

(

^pλ

)

^−Rˆλ

λ

²

F,with (17)

λ=

σ

ⁿ_λ

σ

ⁿ_λ^T

−¹2

. (18)

Finally,with (13),we define the costfunction forthe entirenet- workas

κ (

^p

)

= λ∈

κ

λ

(

^pλ

)

^{. (19)}

Our aim is to estimate p by minimizing

κ

^{(p) in an alternat-}

ing and parallel manner. Note that the overall problem is non- convex,sowecannotclaimfindingtheglobalminimumwhenthe algorithmconverge(nevertheless,inpracticewemighthavegood initialization that converges to the globalminimum as shownin the realdata simulation). We first estimate locally ineach agent the parametervector {g_λ}_λ∈,with theremaining parameters in p fixed as described in Section 3.2, by reformulating the prob- lem asa consensus problem, in which the updatesof

α

^provide

coherence among wavelength.In a second step,we estimate the variables m_λ,d_λ,1,...,d_λ,Q,

σ

ⁿ_λ

λ∈ forfixed{g_λ}_λ∈,byusinga sparserepresentationapproachasdescribedinSection3.3.Finally, we update theweighting matrices {_λ^}_λ∈. Duringtheseproce- dures,theamountofinformationthatneeds tobeexchanged be- tweenthefusioncenterandthecomputeagentsismuchlessthan the volume of data being calibrated, making this scheme com- putationally feasible. The overall procedure, referred to as Paral- lel Multi-wavelength Calibration Algorithm (PMCA), is presented in Algorithm 1. The algorithm is carefully initialized with the

Algorithm1:ParallelMulti-wavelengthCalibrationAlgorithm Input: Rˆ_λ,m^[0]_λ

λ∈,D^K,,

η

^p;

Init:seti=0, g_λ=g^[0]_λ

λ∈, D_λ=D^K,m_λ=m^[0]_λ ,_λ=1_P×P

λ∈; repeat

1 i=i+1;

2 Estimateparalelelly g^[_λ^i]

λ∈ withAlgorithm1.2;

3 Estimateparalelelly D^[_λ^i],m^[_λ^i],

σ

^n[_λ^i]

λ∈ withAlgorithm 1.3;

4 Updatelocally ^[_λ^i]=

σ

^n[_λ^i]

σ

^n[_λ^i]T

−¹₂

λ∈; until

p^[i−1]−p^[i]

2≤

p^[i]

2

η

^p;

Output:pˆ=

p^[_λ^i]T

1 ,...,p^[_λ^i]T

J

T

;

true/nominalcalibratorparametersandaninitialguessforthean- tenna gains,e.g. from precedent calibration,or by defaultby the

unit sensorgain,g^[0]_λ =1.Thestoppingcriterion

η

phastobesuf- ficientlysmalltoassureconvergence.Inthefollowingsections,we detailthetwomajoralternatingoptimizationstepsoftheproposed PMCA.

3.2. Directionindependentantennagainestimation(Algorithm2)

In this section, we describe Algorithm 2 of the PMCA. As shown in Algorithm 2, this optimizationstep is performedw.r.t.

the DIgain parameters {g_λ}_λ∈,while theremaining parameters m_λ,d_λ,1,...,d_λ,Q,

σ

ⁿ_λ

λ∈ ofp are fixed. During this step, each agent calibrates the sensor gains g_λ using the data available lo- cally andAlgorithm 3.Then, theagentstransfers their parameter estimatestothecentralizedlocation.Atthefusioncenter,smooth- ness of the parameters across wavelength is enforced (line 3 of Algorithm 2), using the coherence parameter vector

α

^{, that is}

Algorithm2:Estimationof{g_λ}_λ∈ Input: Rˆ_λ

λ∈,p^[i−1],

η

^g;

Init:sett=0, g^[_λ^t]=g^[_λ^i−1]

λ∈, R^K_λ=A_D[i−1]

λ M^[_λ^i−1]A^H

D^[i−1]

λ

λ∈; repeat

1 t=t+1;

2 Estimatelocally g^[_λ^t]

λ∈withAlgorithm3;

3 Estimate

α

^{[t]atthefusioncenterwith(36);}

4 UpdatelocallytheLagrangemultipliers y^[_λ^t]

λ∈with (24);

until

λ∈

^g[_λ^t−1]−g[_λ^t]

₂^≤λ∈

^g[_λ^t]

₂

η

g;

5Deducelocally gˆ_λ

λ∈from

α

^[t]with(11);

Output: gˆ_λ

λ∈;

passed to each agent to provide coherent processing across the whole wavelengthrange,andthusimprovingthelocalcalibration procedure.

At thispoint, we distinguish betweencentralized andparallel basedestimationof

α

^.Specifically:

• Jointcalibrationleadstoadirectestimationschemeof

α

^from

thedata,inwhichwesubstituteintheminimizationof(19)the sensorgains{g_λ}_λ∈ by

α

^{accordingto(11).However,thisre-}

quiresaccess toall data by minimizing (19)w.r.t.

α

^{, whichis}

computationallyunfeasibleduetotherestrictionresultingfrom thelargedatavolumes.

• LetusrecallthatZcomputationalagentsaredisposedonanet- work(see,Section 2.3), wherethezthagent,Az,accessesdata forwavelengths

λ

∈z⊂^{.In this} perspective, we propose a parallel calibration scheme, in which the sensor gains corre- spondingtoeach wavelength

λ

^{areestimatedlocallyandcon-}

sensusisenforcedamongagentsbyimposingconstraint(11). Withthisnetwork setup,we formulatethe parallelcalibration procedureas

minimize α,{^gλ}λ∈

λ∈

κ

˜_λ^[i]

(

^gλ

)

subjectto g_λ=B_λ

α

^,

∀ λ

^∈

^{, (20)}

inwhich

κ

˜_λ^[i](^g_λ)=

κ

_λ(^p_λ

|

^m[_λ^i],D^[_λ^i],

σ

^n[_λ^i]),iisthe[i]thiterationof Algorithm 1, and where the cost function consists of a sum of independent cost functions, one foreach subband, that are cou- pled through the coherence constraintswhich howeverare inde- pendentacross sensors.Acommonlywaytosolve(20)isto con- sider the problemasa consensus optimizationproblem [26] and

consequentlytousetheaugmentedLagrangian,givenby[59]

L^[i]

{

^gλ

}

λ∈,

α

,

{

^yλ

}

λ∈

= λ∈

κ

˜_λ^[i]

(

^gλ

)

^{+ yH}_λ

(

^gλ−B_λ

α )

+

ρ

2

^gλ−B_λ

α

²2

=

λ∈

L^[_λ^i]

(

^gλ,

α

^,yλ

)

, (21)

where{y_λ}_λ∈ aretheJLagrangemultipliersand

ρ

^{istheregular-}

izationterm,chosenbythepractitioner.Inordertosolve(20),we resorttotheconsensusADMM[26].Lettdenotethelocaliteration counterofAlgorithm2,theupdatesforthe[t]thiterationaregiven by

g^[_λ^t]=argmin

g_λ

L^[_λ^i]

g_λ,

α

^[t−1],y^[_λ^t−1]

,

λ

∈

^{, (22)}

α

^[t]=argmin

α

λ∈

L^[_λ^i]

g^[_λ^t],

α

^,y[_λ^t−1]

, (23)

y^[_λ^t]=y^[_λ^t−1]+

ρ

g^[_λ^t]−B_λ

α

^[t]

,

λ

∈

^{, (24)}

that correspond to, respectively, the sensor gain update, the smoothnessparameterupdateandtheLagrangemultiplierupdate.

The minimization of (22) is the computationally most expensive stepandisperformedlocallybyeachagent.Similarly,(24)issolve locally,whereas(23)issolved atthefusioncenter.Proceduresfor obtaining(22)and(23)aredetailedinthefollowing.

3.2.1. Minimizationof(22)

Towardthe minimization of (22),we followan iterativeopti- mization approach based on [18,30], that we adapt to our par- allel minimization. Let us assume that g_λ and g^∗_λ are two in- dependent variables. We then regard g^∗_λ as fixed and minimize L^[_λ^i]

g_λ,g^∗_λ,

α

^[t],y^[_λ^t]

w.r.t.g_λonly,andwithoutconsideringthedi- agonalelementsinthecostfunctionsin(20)that containtheun- knownnoisevariances

σ

ⁿ_λ^{.Inthiscase,thelocalcostfunctionbe-}

comesseparablew.r.t.theelementsofg_λ,hence,

κ

˜_λ^[i]

(

^gλ

)

= P

p=1

κ

˜_λ^p[i]

(

^[gλ]p

)

^{, (25)}

where

κ

˜_λ^p[i](^[g_λ^]p)correspondstothecostfunctionforthepthrow ofR_λ, whichdependsonlyon[g_λ]p sincetheremaining parame- tersareconsideredasfixedinthisstep.Letusdefinetheoperator Sp(.), that convertsthe pth rowof a matrixto a vector andre- movesthe pthelement ofthisselectedvector. Further,define the vectorˆr_λ^p=Sp

_ˆ

R_λ

andtheweightingvector

ω

=Sp

^[_λ^i]

.Wecan thuswrite

κ

˜_λ^p[i](^[g_λ^]p)^in(25)as

κ

˜_λ^p[i]

(

^[g_λ^]p

)

=

ˆr_λ^p−z[g_λ]p

ω

²

2, (26)

inwhichz=Sp

R^K_λG^∗_λ

andwhere

R^K_λ=AD_λM_λA^H_Dλ (27)

represents the estimated calibratorsky model. Then, we decom- posetheaugmentedLagrangianin(22)w.r.t.theelementsofg_λas L^[_λ^i]

g_λ,

α

^[t],y^[_λ^t]

= P

p=1

L^p[_λ^i]

[g_λ]p,

α

^[t],y^[_λ^t]

, (28)