The data model. Panos Labropoulos. Kapteyn Instituut

Transcription

1 The data model Panos Labropoulos "

2 Challenges The actual 21-cm signal is several times weaker than the foregrounds and sits below the telescope noise Foregrounds: galactic and extragalactic (polarization?) / RFI RFI Ionosphere Large data sets Interferometry Intrinsic differencing (fringe pattern) Filtered images Equivalent to masking parts of a telescope aperture Resolution at the cost of surface brightness sensitivity Works in spatial Fourier domain Element gain spread in image plain 4/17/08 2

3 The Van Cittert Zernike relation The antenna size smears out the coherence function response Lose ability to localize wavefront direction (FOV): Small antenna diameter wide field The visibility function in the image and uv planes 4/17/08 3

4 uv plane 4/17/08 4

5 uv plane 4/17/08 5

6 uv plane On short baselines a visibility can correlate with other visibilities as well as its conjugate 4/17/08 6

7 Polarized feeds Radio astronomers had to deal early on with polarization as the complex feeds/dipoles of the antennas led to a polarized response 4/17/08 7

8 TECV Movie Vertical 2D TEC as a function of time Nearly Kolmogorov turbulence with a cut off at certain scales + Large scale waves 4/17/08 8

9 The Sky Jelic & Zaroubi 4/17/08 9

10 Jones Vector Instead of using the Jones vector one can form the coherency matrix: C = D e (t) e (t) E = e1 (t) e 1 (t) e 1 (t) e 2 (t) e 2 (t) e 1 (t) e 2 (t) e 2 (t) This matrix encodes all the statistical properties. It is a correlation matrix providing the second moments of the signal. «4/17/08 10

11 Effects of media 4/17/08 11

12 Comparison Problems involving polarizers & retarders Jones Calculus Yes Mueller Calculus Yes Scatters No Yes Fully polarized light Yes Yes Partially polarized light No Yes Directly related to exptl. measurements Light vector specified by Less generally applicable No Amplitude & phase of the electric field vector More generally applicable Yes Intensities Phase info Retained Lost Number Complex Real Computational effort Less Vector: 2x1; Matrix: 2x2 More Vector: 4x1; Matrix: 4x4 4/17/08 12

13 The Measurement equation Polarization effects in the signal chain appear as error terms in the Measurement Equation F = ionospheric Faraday rotation (T = tropospheric effects) P = parallactic angle E = antenna voltage pattern D = polarization leakage G = electronic gain B = bandpass response A i = B i G i D i E i P i T i F i A * j = F * j T * j P * j E * j D * j G * * j B j I + Q U + iv V = B i G i D i E i P i T i F i U iv I Q l,m,n V = A j CA j * + n F * j T * j P * j E * j D * j G * j B * j + diag{ σ } ij 4/17/08 13

14 Using the identity AXB = C The equation can be rewritten in the form v =Au for each interferometer. The covariance is then: vv * = A uu * A * + N Conjugates should also be considered. The ML solution over n is: A = (A t N -1 A) -1 A t N -1 4/17/08 14

15 For every interferometer A is a 4x4 complex matrix where each element is a function of the parameters that enter the MEQ through the Jones matrices. We assume that we know those parameters to a reasonable precision and their errors are Gaussian and independent. For an element of A the errors in the linear regime would propagate as: δa ij ( x) = n A ij x n 2 Fisher information matrix - CR bounds MC Monte Carlo simulations / explore likelihood surface SVD of the covariance matrix 4/17/08 15

16 The inversion is O(N^3) process: Ideally we would like to have O(N) The large dataset needs to be reduced. Solution: convolve with a matched filter kernel Lossy but fast and generally safe!! Does not include the application of the primary calibration. Complicates covariance calculation. 4/17/08 16

17 Terms for known effects instrumental noise (?) residual source foreground (?) incorporate as noise matrices with known prefactors Terms for unknown effects e.g. foreground sources with known positions known structure in C incorporate as noise matrices with large prefactors equivalent to downweighting contaminated modes in data 4/17/08 17

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19 Computational cost Forward 25 hrs per frequency 4hrs-10s Trivial parallelization along the frequency 5.6 TB of data for the full cube 300GB for a selection of frequencies and longer averaging Inverse TFLOP / frequency 16 hrs per frequency (Assuming access to a 10 TFLOP system ) 170 days for all frequencies Extensive I/O 4/17/08 19

20 GPUs GPUs are getting faster 350 GFLOPS Multicore: Scale cores instead of clock frequencies 100s cores, 1000s threads, non cc NUMA access NVIDA G80: 128 cores, ATI R cores It is not about hacking the GPU, real programming languages are developed Cell processor 4/17/08 20

21 GPUs 4/17/08 21

22 GPUs 4/17/08 22

23 GPUs 4/17/08 23

24 GPUs 4/17/08 24

25 GPUs 4/17/08 25

26 GPUs Maximize independent parallelism Maximize math intensity (math operations/thread) Better to recompute than to cache, GPUS spend transistors on ALUs, not memory Do as much computations on the GPU as possible to avoid I/O Global vs local memory = order of magnitude Local memory can be access 100x times faster and by ALL threads 4/17/08 26

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28 GPUs 4/17/08 28

29 GPUs 4/17/08 29

30 GPUs 4/17/08 30

31 GPUs 4/17/08 31

32 Sanity checks Let P = Q 2 + U 2 + V 2 I 2 be the degree of polarization and π=(q,u,v). One can decompose the stokes vector in several ways i.e. polarized and non polarized states.» 1+P 1 s = s P» «1 2 π 2 π Energy conservation: The input to output intensity ration must be less or equal to one (energy conservation) 4/17/08 32

33 Sanity checks The density matrix describes the statistical state of a system. It is similar to the Liouville density (quantum phase space equivalent). The extension of classical entropy concepts in the field of QM is the von Neumann entropy. For the polarization states we can define it as: S = tr Ĉ ln Ĉ nsity. It can be expressed ` as a func S = ˆ( P ) ln 1+P of information betw We can see that it is decreasing with P and that it is bound. It gets its maximum value ln2 for unpolarized light and its minimum value 0 for totally polarized light. It a very handy criterion of quality of the data i.e. check for depolarizing effects. 4/17/08 33

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35 Faraday Rotation The ionosphere shows a birefringent material behavior The linear polarization rotation angle is rotated as Δφ 0.15 λ 2 B n e ds deg λ in cm, n e ds in cm -2, B in G Important at low frequencies Severe during solar maxima and sunrise/sunset 4/17/08 35