Robust Calibration of Radio Interferometers in Non-Gaussian Environment V. Ollier, M. N. El Korso, R. Boyer, P. Larzabal, M. Pesavento SATIE - L2S LEME TU-D Universit´ e Paris-Saclay Universit´ e Paris-Ouest Technische Universit¨ at Darmstadt CEA Saclay V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 1 / 34

Outline Introduction to radio astronomy and motivation of this work Data model ◮ Non-structured Jones matrices ◮ Structured Jones matrices (3DC calibration regime) Estimation procedure for non-structured Jones matrices ◮ Principle of the proposed calibration technique ◮ Use of the EM algorithm ◮ Use of the BCD algorithm Estimation procedure for structured Jones matrices (3DC calibration regime) Numerical simulations ◮ Under SIRP noise assumption ◮ Under realistic model ◮ Under 3DC calibration regime Conclusion V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 2 / 34

Introduction Van der Veen et al. (2013) Goal ◮ Measure electromagnetic waves impinging on the Earth thanks to astronomical instruments ◮ Deduce spectral flux density from maps which measure strength of radiation ◮ Study physical phenomena to handle cosmological issues Astronomical instruments ◮ Large telescope dishes: expensive , lack of flexibility ◮ Interferometric array : higher angular resolution ◮ Phased-array system : cheaper dipole antennas, no moving parts (software telescope), huge collecting area V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 3 / 34

Introduction Radio astronomy ◮ Study radio emissions ◮ Case of LOFAR (LOw Frequency ARray): LBA (Low Band Antenna) 30-80 MHz HBA (High Band Antenna) 120-240 MHz V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 4 / 34

Motivation Calibration ◮ Estimation of all perturbations introduced along the signal path ◮ Essential to perform image reconstruction with no distorsions Hamaker et al. (1996) Kazemi and Yatawatta (2013) Non-Gaussianity assumption ◮ Presence of outliers in the data (weak unknown sources, RFI, ...) ◮ Robust calibration in the literature: only Student’s t V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 5 / 34

Data model Non-structured case Data model Non-structured Jones matrices V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 6 / 34

Data model Non-structured case Mathematical model for non-structured Jones matrices Hamaker et al. (1996), Smirnov (2011) ⊲ D sources, M antennas, 2 orthogonal polarization directions ( x , y ) s i = [ s i x , s i y ] T i -th incoming radiation v i p ( θ ) = [ v i px ( θ ) , v i py ( θ )] T generated voltage at p -th antenna ¯ J i p ( θ ) Jones matrix, parametrized by unknown vector θ ⇓ ¯ v i p ( θ ) = J i p ( θ ) s i ⇒ Unknown elements = entries of all Jones matrices = V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 7 / 34

Data model Non-structured case Noise-free case V pq ( θ ) = � D i =1 J i p ( θ ) C i J H i q ( θ ) for p < q , p , q ∈ { 1 , . . . , M } � � v H V pq ( θ ) = E ¯ v p ( θ )¯ q ( θ ) Hamaker et al. (1996) � I i + Q i � U i + jV i C i = E { s i s H i } = intrinsic source coherency matrix U i − jV i I i − Q i V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 8 / 34

Data model Non-structured case Full visibility vector D � J i p ( θ ) C i J H V pq ( θ ) = i q ( θ ) i =1 ⇓ D � � � v pq ( θ ) = vec ˜ V pq ( θ ) = u i pq ( θ ) i =1 � � J ⋆ u i pq ( θ ) = i q ( θ ) ⊗ J i p ( θ ) vec ( C i ) Noisy correlation measurements v pq = ˜ v pq ( θ ) + n pq D ( M − 1) M ] T = x = [ v T 12 , v T 13 , . . . , v T = ⇒ � u i ( θ ) + n i =1 � T � u T i 12 ( θ ) , u T i 13 ( θ ) , . . . , u T u i ( θ ) = i ( M − 1) M ( θ ) � T = � n T 12 , n T 13 , . . . , n T n = ⇒ Gaussian noise & outliers ( M − 1) M V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 9 / 34

Data model Non-structured case Noise modeling Spherically invariant random process (SIRP) n pq = √ τ pq g pq τ pq positive real random variable ( texture ) g pq complex zero-mean Gaussian process ( speckle ) g pq ∼ CN ( 0 , Ω ) s.t. tr { Ω } = 1 ⇒ Remove scaling ambiguities = V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 10 / 34

Data model Structured case Data model Structured Jones matrices V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 11 / 34

Data model Structured case 3DC calibration regime Direction dependent distorsions with compact array Lonsdale (2004) Closely packed group of similar antennas Wide field of view of individual elements V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 12 / 34

Data model Structured case Noordam (1996), Smirnov (2011), Yatawatta (2012) J i p ( θ 3DC ) = G p ( g p ) H i p Z i p ( α i ) F i ( ϑ i ) i p with θ 3DC = [ ϑ i , g T p , α T i ] T i p Ionospheric delay matrix � � Z i p ( α i ) = exp I 2 j ϕ i p where ϕ i p = η i u p + ζ i v p , ◮ α i = [ η i , ζ i ] T source offset ◮ r p = [ u p , v p ] T known antenna position in units of wavelength V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 13 / 34

Data model Structured case Diagonal electronic gain matrix G p ( g p ) = diag { g p } Known matrix H i p ◮ Electromagnetic simulations ◮ A priori knowledge (calibrator sources & antenna positions) Ionospheric Faraday rotation matrix � cos( ϑ i ) − sin( ϑ i ) � F i ( ϑ i ) = sin( ϑ i ) cos( ϑ i ) ◮ Faraday rotation angle ϑ i V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 14 / 34

Estimation procedure Non-structured case Estimation procedure Non-structured Jones matrices V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 15 / 34

Estimation procedure Non-structured case Maximum likelihood method � � 1 − 1 � a H pq ( θ ) Ω − 1 a pq ( θ ) f ( x | θ , τ , Ω ) = | πτ pq Ω | exp τ pq pq τ = [ τ 12 , τ 13 , . . . , τ ( M − 1) M ] T a pq ( θ ) = v pq − ˜ v pq ( θ ) ⇓ log f ( x | θ , τ , Ω ) = 1 � � pq ( θ ) Ω − 1 a pq ( θ ) a H − 4 B log π − 4 log τ pq − B log | Ω | − τ pq pq pq Iterative ML algorithm ⊲ Optimization w.r.t. each unknown parameter: concentrated ML estimator ⊲ Probability density function of all τ pq not specified: relaxed ML estimator V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 16 / 34

Estimation procedure Non-structured case Proposed algorithm 1 Initialize ˆ Ω = Ω init , ˆ τ = τ init 2 Estimation of θ � � − 1 a pq ( θ ) ˆ 1 pq ( θ )ˆ τ pq a H θ = argmin � Ω ˆ pq θ 3 Estimation of Ω a pq (ˆ pq (ˆ θ ) a H θ ) ˆ Ω = 4 ˆ ˆ Ω � Ω = pq (ˆ θ )(ˆ Ω ) − 1 a pq (ˆ tr { ˆ Ω } B a H θ ) pq ‘ 4 Estimation of τ − 1 a pq (ˆ pq (ˆ θ )ˆ τ pq = 1 4 a H ˆ Ω θ ) 5 Repeat steps 2 to 4 until stop criterion reached V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 17 / 34

Estimation procedure Non-structured case Non-structured Jones matrices Partition per source D ] T = [ θ T θ = [ θ T 1 , . . . , θ T 1 1 , . . . , θ T 1 M , . . . , θ T D 1 , . . . , θ T D M ] T J i p ( θ ) parametrized by path from i-th calibrator source to p-th sensor J i p ( θ ) = J i p ( θ i p ) with θ i p ∈ R 8 × 1 V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 18 / 34

Estimation procedure Non-structured case EM algorithm Expectation-Maximization Yatawatta et al. (2009), Kazemi et al. (2011) ⊲ Motivation Decrease computational cost Optimization for single source problems of smaller dimensions ⇒ optimization w.r.t. θ i ∈ C 4 M × 1 instead of θ ∈ C 4 DM × 1 = Ensure convergence Complete data vector w = [ w T 1 , . . . , w T D ] T D w i = u i ( θ i ) + n i s.t. x = � w i i =1 V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 19 / 34

Estimation procedure Non-structured case ⊲ E-step Goal : conditional expectation of complete data ˆ w = E { w | x ; θ , τ , Ω } n = � D i =1 n i ∼ CN ( 0 , Ψ ) n i ∼ CN ( 0 , β i Ψ ) τ 12 Ω 0 . . . D . . ... . . � Ψ = β i = 1 . . i =1 0 · · · τ ( M − 1) M Ω � D � � w i = E { w i | x ; θ , τ , Ω } = u i ( θ i ) + β i ˆ x − u l ( θ l ) l =1 V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 20 / 34

Estimation procedure Non-structured case ⊲ M-step Goal : estimation of θ i D � �� 1 � H � ( β i Ψ ) − 1 � � f (ˆ w | θ , τ , Ω ) = | πβ i Ψ | exp − w i − u i ( θ i ) ˆ w i − u i ( θ i ) ˆ i =1 � H � ( β i Ψ ) − 1 � � w i − u i ( θ i ) w i − u i ( θ i ) φ i ( θ i ) = ˆ ˆ Numerical example: Levenberg-Marquardt (LM) algorithm Analytical method: Block Coordinate Descent (BCD) algorithm V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 21 / 34

Estimation procedure Non-structured case BCD algorithm Block Coordinate Descent Friedman et al. (2007), Hong et al. (2016) Perform optimization of φ i w.r.t. θ i p ∈ C 4 × 1 with fixed θ i q , q � = p M � H � ( β i τ pq Ω ) − 1 � � � φ i ( θ i p ) = w i pq − u i pq ( θ i p ) w i pq − u i pq ( θ i p ) + q =1 q > p M � H � ( β i τ qp Ω ) − 1 � � � w i qp − u i qp ( θ i p ) w i qp − u i qp ( θ i p ) + Cst q =1 q < p V.Ollier (SATIE/L2S) Robust calibration 02/02/2017 22 / 34

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