Calibration and Imaging going deeper than ever before Sarod - - PowerPoint PPT Presentation

Calibration and Imaging going deeper than ever before

Sarod Yatawatta

Sarod Yatawatta – p. 1

Calibration

Sky, Instrument Observation

θ y ˆ θ

Calibration Interferometry, Noise Ideally, find ˆ θ = θ. But in real life ???

Sarod Yatawatta – p. 2

Imaging

From irregular sparse uv coverage ⇒ To real images of the sky 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

Cassiopeia A, 120 MHz NCP , 130 MHz, 1×1 sq. deg.

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 y =

K

• i=1

si(θ) + n, n ∼ N(0, Π) Maximum Likelihood (ML) estimate, under White Gaussian Noise

• θ = arg min

θ φ(θ) = arg min θ y −

K

• i=1

si(θ)2 Traditional calibration: using Levenberg-Marquardt (LM) algorithm θk+1 = θk − (∇θ∇T θφ(θ) + λH)−1∇θφ(θ)|θ

k

where H

△

= diag(∇θ∇T 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] y =

K

• i=1

si(θ) + n ML estimate: ˆ θML = arg max θ log f(y|θ) Auxiliary random variable x: hidden data, y = F(x) The E Step: compute conditional expectation Q(θ|θk) = E{log f(x|θ)|y, θk} The M Step: Maximize θk+1 = arg max θ Q(θ|θk) 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 × stations2. 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) y =

K

• i=1

si(θ) + n, n ∼ N(0, Π) But in practice, there are many more sources in the sky. With K′ unknown sources, robust data model is y =

K

• i=1

si(θ) +

K′

• i′=1

si′ + n, n ∼ N(0, Π) The effective noise is n′ =

K′

• i′=1

si′ + n 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 µJy, 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