Calibration and Imaging going deeper than ever before Sarod Yatawatta Sarod Yatawatta – p. 1
Calibration Interferometry, Noise Calibration ˆ θ θ y Sky , Instrument Observation Ideally, find ˆ θ = θ . But in real life ??? Sarod Yatawatta – p. 2
Imaging ⇒ To real images of the sky From irregular sparse uv coverage � Gridding: resample irregular 3D ( u, v, w ) data points onto a 2D regular ( u, v ) grid. � Weighting (sampling density compensation) and FFT. � Take real part and correct final image for gridding (apodization correction). Sarod Yatawatta – p. 3
Sky = Complex Sources , 130 MHz, 1 × 1 sq. deg. NCP Cassiopeia A, 120 MHz Sarod Yatawatta – p. 4
Instrument = Beam Shape Data from 2011: LOFAR beam amplitude 121 MHz 10 deg. FOV Sarod Yatawatta – p. 5
Ionospheric Errors Ionospheric spikes, average and difference between two days Sarod Yatawatta – p. 6
Need for better calibration and imaging � SKA will have better hardware, signal processing, transport, correlation, beamforming etc. � SKA will have many more users. Observing time is precious. � The amount of data that needs to be processed will be greater than ever before. The data processing needs to be done with minimal computing time. � Therefore, better calibration and imaging is required to process more data, giving better quality end products, with minimal computational cost. � Current calibration works at best for about 90% of observed data. So 10% of the data is lost (not due to RFI). � Robustness in calibration is essential. Sarod Yatawatta – p. 7
Formal Description of Calibration For K discrete sources, we observe K � y = s i ( θ ) + n , n ∼ N ( 0 , Π ) i =1 Maximum Likelihood (ML) estimate, under White Gaussian Noise K � � s i ( θ ) � 2 θ = arg min φ ( θ ) = arg min � y − θ θ i =1 Traditional calibration: using Levenberg-Marquardt (LM) algorithm θ k +1 = θ k − ( ∇ θ ∇ T θφ ( θ ) + λ H ) − 1 ∇ θφ ( θ ) | θ k △ = diag( ∇ θ ∇ T where H θφ ( θ )) . Much faster methods are available (Ordered Subsets Acceleration) [Kazemi et al., 2012,2013]. Sarod Yatawatta – p. 8
Expectation Maximization [Dempster, Laird, Rubin, 77] K � y = s i ( θ ) + n i =1 � ML estimate: ˆ log f ( y | θ ) θ ML = arg max θ � Auxiliary random variable x : hidden data, y = F ( x ) � The E Step : compute conditional expectation Q ( θ | θ k ) = E { log f ( x | θ ) | y , θ k } � The M Step : Maximize θ k +1 = arg max Q ( θ | θ k ) θ � Can be simplified for exponential family distributions. � Can be even more simplified for Gaussian distributions. � SAGE: Space Alternating Generalized Expectation Maximization [Fessler and Hero, 94] [Kazemi et al., 2011] gives faster convergence. Sarod Yatawatta – p. 9
SAGECal � Multisource calibration: speed ( 50 × to 100 × faster than anything else), accuracy, convergence, robustness. � Complexity ≈ directions × stations 2 . � Very modest memory usage: (1 million data points, 60 000 parameters, < 6 GB RAM). � Highly parallelized and vectorized. Uses GPU acceleration when available ( > 8 speedup). � Pure C code with only standard libraries used. GPU support using CUDA/CUBLAS/CULA. � Supports all source models: points, Gaussians, disks, rings, (widefield) shapelets (prolate spheroidal wave functions). � Supports non-Gaussian noise models. Sarod Yatawatta – p. 10
Calibration at Work (left) Before (right) After SAGECal Sarod Yatawatta – p. 11
Robustness For K discrete sources (known) K � y = s i ( θ ) + n , n ∼ N ( 0 , Π ) i =1 But in practice, there are many more sources in the sky. With K ′ unknown sources, robust data model is K ′ K � � s i ′ + n , y = s i ( θ ) + n ∼ N ( 0 , Π ) i =1 i ′ =1 The effective noise is K ′ � n ′ = s i ′ + n i ′ =1 which is not necessarily Gaussian. Robust calibration can handle noise deviation from Gaussian model [Kazemi, Yatawatta 2013]. Sarod Yatawatta – p. 12
Conclusions Deepest LOFAR image at 150 MHz, noise 25-30 µ Jy, 6 ′′ PSF � SKA will make better images than above. � Better calibration and imaging are essential to make such images. � Novel algorithm development has to begin now. Sarod Yatawatta – p. 13
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