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From Visibilities to Images Tom Muxlow , JBCA

From Visibilities to Images Tom Muxlow , JBCA. Practical guide to basic imaging – mostly within AIPS - RFI excision, delay calibration, phase & gain calibration, band-pass correction, gridding schemes, image de-convolution, non-coplanar baselines and multi-faceted images

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From Visibilities to Images Tom Muxlow , JBCA

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  1. From Visibilities to Images Tom Muxlow, JBCA • Practical guide to basic imaging – mostly within AIPS • - RFI excision, delay calibration, phase & gain calibration, band-pass correction, gridding schemes, image de-convolution, non-coplanar baselines and multi-faceted images • - Wide-field imaging, chromatic aberration, confusion, multi-frequency synthesis , mosiacing, high dynamic range imaging • New wide-bandwidth imaging arrays (ALMA, e-MERLIN, EVLA, LOFAR, etc..) will be utilising both existing AIPS software and new CASA-based routines. CASA will increasingly replace AIPS for future data analysis. MeqTrees will also become increasingly important (software for implementing Measurment Equations)

  2. Visibilities to Images General Introduction to Basic Imaging • Practical description of data processing MERLIN 1658MHz data on DQSO • - historical archival data – shows sequential approach to basic calibration/imaging • - solve for R & L amplitude & phase solutions by antenna • - shows typical image de-convolution schemes • - shows image enhancement through self-calibration • MERLIN is a sparsely-filled array where holes in spatial frequency sampling  point-spread function is severe & raw images need significant de-convolution. • MERLIN has now evolved into e-MERLIN & recent e-MERLIN test data are used to illustrate the initial continuum calibration processes - each 128MHz IF calibrated separately

  3. Visibilities to Images Initial Calibration: RFI Excision – we’re operating well outside our protected bands!! • e-MERLIN test data – Data exported via FITS IDI, to imaging computers & read into AIPS (python scripts) – separate files for each calibration & target source to minimize file sizes since initial spectral flagging is interactive (~500GBytes). • 4x128MHz IFs with 512 channels (250kHz channels), 4 Stokes – LL, RR, LR, RL • 2 IFs illustrated in SPFLG – flag each baseline displaying both LL & RR for all IFs • Display ‘Vector RMS’ • Flag single bad channel • Frequency-dependent RFI is more problematical! • Select a window (or series of windows) enclosing the line, then clip interactively • After flagging raw band-pass response becomes visible  IF 1   IF 2 

  4. Visibilities to Images Initial Calibration: Delay Correction • After spectral flagging data are assembled into a single file and can be averaged up in frequency (e.g. 128 1MHz channels/IF) – performed by python scripting • With very wide-bandwidth data now appearing from optical fibre connected arrays such as the EVLA and e-MERLIN, small changes in delay produce significant phase slopes across each IF. FRING routine calculates delay corrections from calibration data scans • Multi or single IF solutions are possible • Delays vary by a few nano-sec. over hours • Some delay jumps are present • R & L delays are different but track each other closely Multi-IF plots of raw e-MERLIN data – baselines plotted to Pickmere Before After

  5. Visibilities to Images Initial Calibration: Amplitude Calibration • Data scans: Target – Phase calibrator – Point calibrator – Flux density calibrator • Target/Phase calibrator rapid scans (typically – 8min/2min) – within a few degrees of target • Point & Flux calibrator observed at least once per day. • Point cal flux density set by comparison with resolved Flux cal on short spacings • Phase calibrator flux density derived from Point cal comparison • Basic calibration set from gain & phase solutions in ‘CALIB’ on Point and Phase cals. • Final fine adjustments made through band-pass calibration (Point cal only) to flatten the IF responses • Needs automating!! Manually all too slow!!! Passband plots on Phase cal after final gain and bandpass corrections (S~0.7Jy)

  6. Visibilities to Images Initial Calibration: (Semi)-automated processing and telescope-specific calibration (for ALMA) • Tsys measurements for initial amplitude calibration may permit automated RFI flagging • Telescope-specific calibration may be required – Water Vapour Radiometers for ALMA • Fast-switching phase referencing - go to bright source within 2o every 20–300 s • Planets, large moons, asteroids to set flux scales • Array configurable on baselines from 250m to 14.5km • Detailed calibration procedures for longer baselines yet to be determined.....

  7. Visibilities to Images Initial Calibration: Phase Calibration • Phase solutions from CALIB (GAINCAL in CASA) provides basic phase calibration for the Target source, making the instrument phase stable and coherent and allowing initial imaging • First e-MERLIN image – DQSO observed at 6.5-7.0 GHz • highest resolution image of kpc jet yet made • imaged in CASA • - final image quality depends on refining the initial calibration sequence... Passband plots on Phase cal after final gain and bandpass corrections (S~0.7Jy)

  8. Visibilities to Images Conventional Imaging: Phase Calibration - 1 • (e-)MERLIN is phase-stable (all telescope-generated LOs are locked to a Hydrogen maser standard) – but atmospheric delay must be calibrated by phase referencing to a nearby source. • Ideally reference is a point – but if not, can map out the object Phase-referencing cycle 8 mins – 2 mins Atmosphere less stable at end of run – also Cambridge has poor phase over the last six hours

  9. Visibilities to Images Conventional Imaging: Phase Calibration - 2 • Initial phase solutions assume a point – first image of reference source may show structure. • Use image structure to refine phase solutions (self-calibration)

  10. Visibilities to Images Self-Calibration - 1 • Apply initial phase & gain solutions to target • If target is bright enough, use the initial target image to apply further self-calibration refinements to the phase and gain solutions 10

  11. Visibilities to Images Self-Calibration - 2 • If troubled by side-lobes, use windowing to restrict positions of source components. Include components to first negative and restrict u-v range to match flux in model

  12. Visibilities to Images Image Deconvolution • Regions in the 2nu-v data grid that are zero will give rise to severe ripples in the beam response – sampling function • The raw image (dirty map) will usually require significant deconvolution • Conventional AIPS image-plane algorithms (APCLN, SDCLN, VTESS) will require a significant guard band around the source structure to avoid aliasing problems – typically restricted to the inner quarter • Visibility-based algorithms (IMAGR) are able to tolerate such errors and typically allow imaging to within a few pixels of the edge

  13. Visibilities to Images Image Deconvolution Dirty Beam Dirty Map • Conventional cleaning – centre the dirty beam under the peak of the dirty map and subtract the pattern scaled by 0.1 – continue until residual image is ~noise • Visibility-based algorithms (IMAGR) subtract smaller beam patches which are then Fourier transformed back to the data plane and (vector) subtracted before re-gridding and transforming to form a new residual image • In both cases idealised point components smoothed by the fitted beam-shape are restored to the final residual image where each subtraction was performed

  14. Image Deconvolution: – Extended Emission Visibilities to Images • Low surface brightness extended structure, can be subject to fragmentation during deconvolution – especially for arrays with sparse u-v coverage • Can be partially alleviated in conventional cleaning algorithms by setting a low loop gain and using ‘Prussian Hat’ beam modifications • SDCLN (Steer-Dewdney) avoids ripples produced by standard cleaning by selecting all pixels above a certain threshold

  15. Visibilities to Images Image Deconvolution – Extended Emission • Maximum entropy deconvolution will produce smoother images in regions of low signal:noise • The image resolution depends on the local signal:noise • VTESS algorithm will also produce the maximum entropy image convolved with the fitted beam + residuals • VTESS does not deal well with bright points in the image – residual side-lobes are common

  16. Visibilities to Images Image Deconvolution – Extended Emission • Maximum entropy can be used in combination with conventional cleaning to alleviate residual side-lobes from bright points whilst still producing a smoother solution in areas of low surface-brightness • Image is initially cleaned to subtract the bright points and clean components are not restored • VTESS is run on the residual image • RSTOR restores the subtracted clean components to the smoothed maximum entropy image

  17. Visibilities to Images Image Deconvolution (Recent developments) – Multiscale Clean • Multiscale CLEAN as implemented in AIPS and CASA produces superior results to conventional clean when deconvolving images with extended structure in the presence of significant high spatial frequency detail (whisps, sharp edges...) • Much better representation of extended emission with little or no ‘negative bowl’ Residual image and beam response smoothed to a selection of scale sizes (eg 0,2,4,8,16,32... pixels) For each scale find strength & location of peak residual For scale with maximum residual, subtract & add this component to source model Update all residual images and loop around until peak residual flux reaches noise threshold (4σ) Multiscale CLEAN thus finds & detects emission on the largest scales first before moving to finer & finer detail – the opposite of conventional cleaning Can produce very fine images from datasets containing large spatial frequency ranges – needs careful steering for sparsely filled aperture data... Conventional CLEAN VLA Radio image from B,C,D arrays + GBT Dyer et al 2009 Residual images after model subtraction Deconvolved images Multiscale CLEAN

  18. Visibilities to Images Integrations are distributed over a greater number of sampled grid points in the outer u-v plane than in the inner regions Data Gridding - 1 + + + + + + + + + + + • Data are interpolated onto a regular 2n grid with a spheroidal convolution func. • Weights unmodified by local density – ‘Natural’ • Weights divided by local density of points – ‘Uniform’

  19. Visibilities to Images Data Gridding - 2 • Naturally weighted images will give better sensitivity at the expense of angular resolution since low spatial frequencies are weighted up and the data are utilised in an optimum way • Uniformly weighted images will give better angular resolution at the expense of sensitivity since low spatial frequencies are weighted down and the data are not utilised optimally – & may be subject to a deconvolution striping instability

  20. Visibilities to Images Data Gridding - 3 Robust Weighting • Originally derived as a cure for striping – Natural weighting is immune and therefore most ‘robust’ • Varies effective weighting as a function of local u-v weight density • Where weight density is low – effective weighting is natural • Where weight density is high – effective weighting is uniform • Modifies the variations in effective weight found in uniform weighting  more efficient use of data & lower thermal noise • Can produce images close to uniform weighting resolution with noise levels close to natural weighting AIPS IMAGR Robustness Factor ROBUST = – 4 is nearly pure uniform ROBUST = + 4 is nearly pure natural ROBUST = 0 is a good compromise (Contoured) 20

  21. Visibilities to Images Data Weighting By Telescope • Data from mixed-type arrays should be re-weighted by telescope sensitivity in order to minimise thermal noise – for e-MERLIN this can decrease the noise level by ~2 • Diameter • Lovell - 76m • Cambridge - 32m • Others - 25m • Defford - 25m (but mesh surface ~transparent at high frequency)

  22. Visibilities to Images Data Weighting By u-v Distance • Gaussian u-vtaper or u-vrange can smooth the image but at the expense of sensitivity since data are excluded or data usage is non-optimum • For MERLIN/VLBI arrays with already sparse coverage beware compromising image quality by severely restricting the u-v coverage

  23. Visibilities to Images Wide-Field Imaging • Images with large numbers of resolution elements across them • Multiple images distributed across the interferometer primary beam • – M82 MERLIN MFS+VLA 5GHz image >1000 beams wide • Wide-field images are subject to a number of possible distortions: M82 MERLIN MFS + VLA 5GHz Non-coplanar baselines Bandwidth smearing Time-averaging smearing Primary beam response 30 arcseconds Beam 35 mas

  24. Visibilities to Images Wide-Field Imaging • Standard Fourier synthesis assumes planar arrays – only true for E-W interferometers • Errors increase quadratically with offset from phase-centre • Serious errors result if θoffset(radians)x θoffset(beams) > 1 • Need to account for a three-dimensional coherence function V(u, v, w) FTI(l, m, n) image vol. – computationally expensive Non-coplanar baselines M82 MERLIN MFS + VLA 5GHz 1.45x10-4 radians x 850 beams = 0.123 30 arcseconds Beam 35 mas

  25. Visibilities to Images Wide-Field Imaging Non-coplanar baselines • A computational less expensive method of imaging is to adopt a faceted or small field approximation in which the image sphere is approximated by pieces of many smaller tangent planes. The centre of each sub-field is correctly positioned in the three-dimensional image plane. • Within each sub-field fast two-dimensional FFTs may be used. • Errors increase quadratically away from the centre of each sub-field, but these are acceptable if enough small sub-fields are selected. • Facets can be selected so as to cover known sources. • Facets may overlap allowing complete coverage of the primary beam. n l m

  26. w u0,w0 Field of View u1,v1 u0 u Visibilities to Images Wide-Field Imaging W-Projection • Facetted imaging naturally allows spatially dependent correction – separate telescope solutions for each facet • An alternative to multiple facets is being developed by Tim Cornwell in CASA • W-projection allows the projection of each uvw visibility onto a single 2-D uv-plane (w=0) with a phase shift proportional to the distance from the flat plane • Each visibility is mapped to all the uv points lying within a cone whose full angle is equal to the field of view of the required wide-field image – extension to position dependence

  27. Visibilities to Images Wide-Field Imaging Bandwidth smearing (chromatic aberration) • Thus far we have considered monochromatic visibilities. • Finite bandwidth averages the visibility data radially producing a radial smearing in the image plane. • Smearing increases with distance from the pointing centre.

  28. Visibilities to Images Wide-Field Imaging Bandwidth smearing • Bandwidth smearing (chromatic aberration) will produce radial smearing and reduction in source peak • Parameterized by the product of the fractional bandwidth and the source offset in synthesised beamwidths δυ /υ0 x θ/θHPBW • Can be alleviated by observing and imaging in spectral line mode with many narrow frequency channels gridded separately prior to Fourier inversion – reduces δυ • – now practicable with new powerful correlators θ’= 2θ

  29. Visibilities to Images Wide-Field Imaging Bandwidth smearing • MERLIN outlier (confusing source) in the outer part of primary beam • GRS1915 observed ~1000 arcseconds from field centre. • θ= 1000 arcsec • θHPBW= 0.22 arcsec • υ0= 1658MHz • δν = 1MHz •  2.7 Very Smeared! δυ /υ0 x θ/θHPBW

  30. Visibilities to Images Wide-Field Imaging Bandwidth smearing • MERLIN outlier (confusing source) in the outer part of primary beam • GRS1915 observed ~1000 arcseconds from field centre. • θ= 1000 arcsec • θHPBW= 0.22 arcsec • υ0= 1658MHz • δν = 15MHz •  41!!!! δυ /υ0 x θ/θHPBW Same data averaged to 1x15 MHz instead of 15x1 MHz frequency channels 30

  31. Visibilities to Images Wide-Field Imaging Time-averaging smearing • Time-average smearing (de-correlation) will produce tangential smearing • In general cannot be easily parameterized. • However can be alleviated by ensuring that δtint is small enough such that there at least 4 samples per turn of phase: • source offset from pointing centre ~10,000 resolution elements • assume 10,000 turns of phase on longest baselines in 6 hours • require 40,000 samples in 6 hours δtint ~0.5secs • very large datasets will result if you want to image to edge of primary beam – needed for good confusion subtraction...

  32. Visibilities to Images Wide-Field Imaging Primary beam response • The ultimate factor limiting the field of view is the diffraction limit of the individual antennas. • AIPS routines exist to correct for primary beam effects in both homogeneous and mixed arrays (PBCOR) • For mixed arrays, beams for interferometer pairs are simply the voltage polar diagram of one multiplied by that of the second. (Strom 2004, astro-ph/0412687) • The overall correction will also depend on the relative weighting and the data distribution between telescopes. Input parameters for PBCOR can be generated (outside AIPS) from telescope parameters and data weighting schemes.

  33. Visibilities to Images Wide-Field Imaging Confusion • Radio sources on the edge of the primary beam can give rise to ripples in the centre of the field of view – subtract them out • The primary beam size is spectrally dependent, so image subtraction should include such corrections and be performed in full spectral-line mode • Pointing errors will introduce gain and phase changes on the edge of the primary beam. If severe, the apparent source structure may change – attempt multiple snapshot subtraction on timescales comparable with pointing error changes OK for weak confusing sources For stronger confusing sources…… VLA GOODS-N 1.4 GHz

  34. Visibilities to Images Wide-Field Imaging Peeling off Confusion • After phase calibrating the data, perform self-calibration for the brightest confusing source – then subtract it out • Delete phase solutions derived for previous confusing source  • Move to next brightest confusing source, perform self-calibration/imaging cycles – then subtract that source from the dataset  • Perform and  until all confusing sources are subtracted. Delete all self-calibration solutions and image central regions

  35. Visibilities to Images Wide-Field Imaging In-beam self-calibration • After peeling off confusing sources , other sources may lie within the central areas of the primary beam • Provided these are compact and bright enough, they can be used to self-calibrate the target (if they lie within the isoplanatic region of the image). • For non-isoplanatic situations (eg VLA D-array at low frequencies, LOFAR) new routines are under development to solve for direction dependent telescope errors – provided there are enough sources to adequately sample the beam – MERLIN GRS1915 ~1000” from pointing centre used to refine phase solutions from poor initial solutions

  36. Visibilities to Images Wide-Field Imaging In-beam self-calibration • Combination of initial external calibration + subsequent in-beam self-calibration has successfully recovered errors • Target imaging should now be of high quality – however astrometric positioning is dependent on initial phase solutions and is likely to be poor Ionospheric activity produces rapid phase changes on long spacings

  37. Wide-Field Imaging:Mosaicing within the primary beam Visibilities to Images • GMRT observations of cluster Abell 901 at 610 MHz • Close to thermal noise only away from bright sources • 19 facets used – projected to a flat image of a curved sky with correct positions by AIPS routine FLATN • Primary beam correction PBCOR Beswick et al (In preparation)

  38. Visibilities to Images Multi-Frequency Synthesis • New wide-band upgrades (e-MERLIN, EVLA) will require a spectral solution in addition to the radio brightness at each location in the image • Initial data processing will likely be conventional imaging sequentially by sub-band (IF) • MFS imaging across all IFs provides massively increased bandwidths together with improved image fidelity • New algorithms have been developed/are under development to solve for spectral index (or full spectral fitting) in addition to radio brightness pixel by pixel e-MERLIN MFS u-v Coverage 4–8 GHz at Declination 30°

  39. Visibilities to Images Multi-Frequency Synthesis • For ELVA & e-MERLIN MFS at C-& L-Band, fractional bandwidth is substantial: eg 4 – 8 GHz at C-Band. • IMAGR (AIPS) & CLEAN (CASA) now have spectral de-convolution options • The size of the primary beam scales as 1/observing frequency • Full MFS imaging will be restricted to the primary beam at the highest frequency • High dynamic-range confusion subtraction from outer parts of the primary beam is likely to be a challenging problem – will need ‘peeling’ in spectral-line mode, possibly in multi-snapshot mode. Primary beams at 4 GHz and 8 GHz

  40. Visibilities to Images Mosaicing • Ultra-wide fields of view can be built up by mosaicing with multiple pointing centres • Each pointing centre must contain some degree of overlap. • Overlap optimisation depends on desired consistency in sensitivity across the mosaiced image and speed of observation. • For arrays with a single type of element, this is relatively straightforward – a typical compromise is a hexagonal pattern with a beam throw of ~60 % 40

  41. Visibilities to Images Mosaicing • For mixed arrays (eg e-MERLIN including the Lovell telescope), similar principles apply, but now each position will have a final image made from both sensitive central Lovell and less sensitive non-Lovell annuli • Mosaic spacing set by Lovell primary beam • Initial mosaics will be made in the sky-plane, however future software development is likely to permit data-plane combination • Including EVLA will complicate matters further ……

  42. Visibilities to Images High Dynamic Range Imaging Present dynamic range limits (on axis): • Phase calibration – up to 1000  improve with self-calibration • Non-closing data errors – continuum ~20,000 line >100,000 • After non-closing error correction <10,000,000 • Redundant baseline data can help….. • Non-closing errors thought to be dominated by small changes in telescope passbands • Spectral line data configurations are the default for all new wide-band radio telescopes • In order to subtract out confusion we will need to be able to image with these very high dynamic ranges away from the beam centre 3C273 MERLIN 1.7 GHz Image from a single 1 MHz channel – dynamic range ~106

  43. Visibilities to Images High Dynamic Range Imaging Achieving high dynamic range off axis: • It is vitally important to monitor and calibrate your spectral line data for dynamically changing spectral bandpass effects. • You must understand the primary beam response to very high precision well into the near side-lobes • Tests with ATCA data have successfully achieved high dynamic ranges off axis…. ..by establishing an accurate primary beam model over a field of a few degrees and by mosaicing out to the 3rd or 4th sidelobe of the primary beam with the established voltage pattern model.

  44. Visibilities to Images High Dynamic Range Imaging Achieving high dynamic range off axis for SKA: • A challenge !! • Some elements may have a well determined and stable primary beam, but others may change with position on the sky • Dynamic reconfiguration to provide multiple beams may introduce difficulties in achieving the required levels of calibration. As they say – it’s only computing... ...but I feel sorry for the computers tasked to deliver... • The SKA pathfinder projects, LOFAR, and others, will prove invaluable in establishing the optimised configurations that will help to deliver both the desired levels of performance and flexibility.

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