1/20 Radio interferometric imaging of spatial structure that varies with time and frequency Urvashi Rau ( NRAO ) Wednesday Lunch Talk 29 Aug 2012 Outline : - A multi-scale, multi-frequency, time-variable CLEAN algorithm. - Simulated example : Reconstruction of time-varying ~3D magnetic-field structures in the solar corona ( using VLA uv-coverage ). Points to take away from this talk : --- what these methods can do how far they can be reasonably pushed conditions under which they work best.... Do you have any interesting use-cases ?

2/20 Aperture Synthesis : Earth-rotation + Multi-Frequency Narrow-band 1.5 GHz Broad- band 1 – 2 GHz Standard Imaging equations and reconstruction algorithms apply Only if the sky is invariant with time and frequency. However Partition the data (or) - Model variability Snapshot4 hour observation

3/20 Basic Image Reconstruction - CLEAN Image Reconstruction : Iteratively fit a sky-model to the observed visibilities. Normal Equations : – This describes an image-domain convolution Fit the parameters of via a weighted least-squares optimization : – Minimize ==> Measurement Equation : Let Imaging (Gridding + iFT) Prediction (FT + de-Gridding) Deconvolution Iterative Solution : Hogbom 1974, Schwarz 1978, Clark 1980, Schwab & Cotton 1983

4/20 Basic Image Reconstruction - CLEAN Normal Equations : where – This describes an image-domain convolution Imaging (Gridding + iFT) Prediction (FT + de-Gridding) Deconvolution Iterative Solution : Model the sky as a list of point-sources : Hogbom 1974, Schwarz 1978, Clark 1980, Schwab & Cotton 1983 CLEAN - Calculate dirty image - Normalize by sum of weights - Add location/amp of peak to model - Update residuals

5/20 Multi-Frequency ( or Time-Variable ) CLEAN Model the sky spectrum with polynomial coefficients : (sparse representation of a smooth spectrum) Model the time-variation as another series : ( a polynomial, for smooth time-variation ) Conway et al, 1991, Sault & Wieringa 1994, Rau & Cornwell 2011, Stewart et al, 2011.

6/20 Point-source responses to time and frequency basis functions Flat-spectrum + No time-variability Linear-spectrum + No time-variability Flat-spectrum + Linear time variation. => Minor-cycle does a joint deconvolution => simultaneous solution of coefficients of all 'terms'.

7/20 Extended emission : Multi-Scale CLEAN where is a blob of size 's' and Multi-Scale Sky Model : Linear combination of 'blobs' of different scale sizes Cornwell 2008, Greisen 2008

8/20 Multi-Scale, Multi-Frequency, Time-Variable Image model Multi-scale flux-components whose amplitudes follow dynamic spectra Sparse representation of extended structure : Atom position and shape Sparse representation of dynamic spectrum of each component (or atom): Coefficients of polynomials in time/frequency ( other 2D basis set : Zernicke polynomials ? ) x y Implemented using CASA libraries, as an extension of MS-MFS.

9/20 Solar Coronal Magnetography – 3D structures (Flares + Sunspots) Sunspots are regions of high magnetic fields For a given B-field, radio emission peaks at the gyro-resonance frequency The brightness distribution at a given observing frequency relates to the electron temperature on an isogauss surface. Magnetic field strength (generally) decreases with height from the base of the active region. => Multi-frequency observations (1-20 GHz) trace ~3D structures in the solar corona. + These structures evolve with time The proposed design of FASR is optimized for such structures, with excellent snapshot uv-coverage..... but, what about VLA with newer algorithms ? Lee et. al. 1997, Gary,2001 Model from T.Bastian

10/20 3D Solar Flare Loop : Simulation ( 15 element array, snapshot ) Array : 15 elements, max baseline of 400 m Frequencies : 5 GHz – 15 GHz Angular resolution : 40 arcsec 10 arcsec Snapshot UV-coverage – 10 GHz Wide-Band UV-coverage – 5-15 GHz Loop Model

11/20 3D Solar Flare Loop : Reconstruction Traditional Single-frequency Imaging => Frequency-dependent resolution. Multi-scale Wide-band imaging => Reconstruction at fixed, high resolution - But, a polynomial spectral model may not always be appropriate here. – Use a physical model ? Loop Model Single-channel Imaging Wide-band Imaging Fleishman et all, 2011

12/20 3D Solar Flare Loop : Model + Reconstruction Loop Model Reconstruction - Reconstruction from a wideband (5 – 15 GHz) snapshot observation with a 15-element array - 4 spatial scales [ 0,6,10,20 ] - 4 Taylor-terms to model the spectrum of each component.

13/20 Coronal magnetogram : Simulation ( VLA : 27 elements, 4 hours ) Array : 27 elements, max baseline of 3km Frequencies : 5 GHz – 15 GHz Angular resolution : 4 arcsec – 1 arcsec Largest spatial scale : 6 arcmin – 2 arcmin Model (6, 8, 10,12, 14 GHz) Snapshot + Single-frequency UV-coverage 4-hour, Multi-frequency UV-coverage 1 arcmin Time 1 Time 2

14/20 Coronal magnetogram : Simulation of time-varying 3D structure 6 GHz 10 GHz 14 GHz Time 1 Time 2 Height increases

15/20 Coronal magnetogram : Reconstruction of time-varying 3D structure 6 GHz 10 GHz 14 GHz Time 1 Time 2 Dominant Errors : Frequency dependent structure at the largest spatial scales.

16/20 Coronal magnetogram : Reconstructing time-varying 3D structure Reconstruction from 8 VLA C-config snapshots spread over 4 hours. Bandwidth : 5-15 GHz - 6 spatial scales [0,10,20,40,60,80] - Linear spectral variation + linear time variation = 3 terms in the model. The wideband uv-coverage does not constrain the structure at the largest spatial scales well-enough. Ok for small scales.

17/20 Coronal magnetogram : Imaging the average intensity 1 channel, 1 timestep 20 channels, 8 timesteps Rms : 3.5 mJy Rms : 1.4 mJy

18/20 Coronal magnetogram : Imaging the average intensity 20 channels, 1 timestep 1 channel, 8 timesteps Rms : 0.3 mJy Rms : 0.4 mJy

19/20 Coronal magnetogram : Imaging the average intensity An appropriate model can ---- separate the varying and invariant parts of the signal ---- reconstruct the frequency-structure and time-variability accurately enough for astrophysical interpretation. What is an “ appropriate model “ ? – represents the source structure efficiently ( sparse parameter space ) – the instrument has a significant response to the basis functions. – the instrument can distinguish between the basis functions. 20 channels, 8 timesteps Rms : mJy

20/20 Summary Generalization of 'CLEAN' : Model the sky in a sparse basis, transform to that basis, apply a greedy algorithm. Image model : M ulti-scale flux-components whose amplitudes follow dynamic-spectra parameterized as 2D functions on the time-frequency plane. Example application : 3D time-varying structures in the solar corona – Increase the imaging dynamic-range of the average sky-brightness distribution. – Reconstruct frequency-dependent and time-varying structure ( extended emission ). Limitations : – The basis functions strongly bias the reconstruction : Directly model the Physics ? – Error estimates are tricky with a greedy algorithm : Other (more expensive) approaches ? – Multi-term methods work only with linear-combinations of basis-functions : non-physical ? Work to do : More realistic tests to assess whether this can be scientifically useful, or if it is just an academic exercise. For example.... Can some of the solar coronal magnetography goals planned for FASR be done with VLA uv-coverage ? From this proof-of-concept test, maybe....