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Interferometric Imaging in CASA An Introduction for ALMA reduction Steven T. Myers National Radio Astronomy Observatory Socorro, NM S. T. Myers NAASC Charlottesville –08 Sep 2011 1 Radio Interferometric Imaging in theory… S. T. Myers NAASC Charlottesville –08 Sep 2011 2 Classic Radio Interferometer • e.g. The EVLA or ALMA – a bunch (27 to 50) antennas connected together – independent elements Earth rotation synthesis NAASC Charlottesville - 08 Sep 2011 3 Interferometer Baselines • Baseline vector B in “aperture plane” – coherent signal applied to interferometer would produce plane-wave interference “fringe” on sky with angular period l/B baseline vector B q=λ/B interferometer naturally decomposes sky into plane waves! NAASC Charlottesville - 08 Sep 2011 4 The Aperture Plane • Correlate wavefronts in plane of apertures (Fourier transform of sky) – dish optics sum aperture plane at focus – visibility is cross-correlation of wavefronts of the 2 apertures visibility contains range of baselines from closest to furthest parts of apertures each point on aperture gets correlated with each autocorrelations point on other aperture measure uv spacings inside D/l interferometer cannot measure “zero-spacing” w/o autocorrelations NAASC Charlottesville - 08 Sep 2011 5 From sky to uv-plane The uv-plane is the Fourier Transform of the tangent plane to the sky aperture xcor width 2D/λ F-1 2.5o baseline u = B/λ F ℓ = 2|u| = 2|B|/λ Sky Plane x = (x,y) Fourier Plane u = (u,v) ~ ~ 2 i vx p V (u) d v A(u v) I ( v) e 2 n NAASC Charlottesville - 08 Sep 2011 6 Example: VLA observes Jupiter • A 6cm VLA observation of Jupiter: Fourier transform of nearly symmetric planetary disk bad data S. T. Myers NAASC Charlottesville –08 Sep 2011 7 Reconstruction of the Sky • Visibilities and the Sky v = A F-1 s + n – A known instrumental response, but is not invertible – instrumental noise n is a random variable • The issues: – unknown random noise n – convolution due to size of A in uv domain – incomplete sampling of uv-plane by visibilities • Maximum Likelihood - Optimal Map mMLE = ( AT N-1 A)-1 AT N-1 v = R-1 d d=Hv No inversion R singular (at best ill-conditioned) for fully sampled s S. T. Myers NAASC Charlottesville –08 Sep 2011 8 The Dirty Map • Grid onto sampled uv-plane d = H v = H s + nd – H should be close to HMLE, e.g. H = AT N-1 : A~A – AT should sample onto suitable grid in uv-plane – reminder: need only be approximate for gridding • Invert onto sky “dirty image” d = F d = R s + nd R = F R F-1 – image is “dirty” as it contains artifacts • convolution by “point spread function” (columns of R) • multiplication by response function (diagonal of R) • noise S. T. Myers NAASC Charlottesville –08 Sep 2011 9 VLA point-spread function (PSF) • The VLA is an example of a sparsely filled array – uv-plane gaps are treated as zeroes, cause “sidelobes” in PSF – many solutions for sky that fit data, “dirty image” is principal solution – must use “deconvolution” techniques to “clean” image snapshot uv-coverage PSF “dirty beam” “dirty” image “clean” image Example: VLA 30s snapshot discovery data for gravitational lens CLASS B1608+656 (Myers et al. 1995, ApJL, 447, L5-L8) S. T. Myers NAASC Charlottesville –08 Sep 2011 10 Image, uv, and Data Spaces • image plane uv-plane visibilities – operators F , H , A handle these transformations – not all operators have inverses (H and A do not) • example: model image m – first transform sky model to uv-plane m = F-1 m – then project onto the visibilities (data space) vm = A m = A F-1 m – form residual dvm = v - vm = A ( s - m ) + n – finding “best model” will involve minimizing this residual S. T. Myers NAASC Charlottesville –08 Sep 2011 11 Classic Deconvolution • uv-plane CLEAN algorithm (“major cycles”) – iterate on residual images removing point models – initial residual data, and model: dv0 = v m0 = 0 – form dirty image: d0 = F H dv0 – locate peak and residual and put fraction f into model dm1 = f M d0 mask M : 1 at max, else 0 – increment model: m1 = m0 + dm1 – form cumulative visibilities and residual v1 = A m1 = A F-1 m1 dv1 = v – v1 – form new dirty residual image: d1 = F H dv1 – and repeat until final residual image df is noise-like S. T. Myers NAASC Charlottesville –08 Sep 2011 12 image-plane CLEAN • CLEAN in the image plane (minor cycles) – initial model: m0 = 0 – form dirty image: d0 = F H dv0 – locate peak and residual and put fraction f into model dm1 = f M d0 mask M : 1 at max, else 0 – increment model: m1 = m0 + dm1 – subtract from dirty image to make dirty residual image d1 = d0 – R dm1 R ~ R (the PSF) – and repeat until final residual image df is noise-like – size of kernel R (Hogbom=full-size, Clark=quarter size) • specified by psfmode in CASA S. T. Myers NAASC Charlottesville –08 Sep 2011 13 CLEAN variations • Cotton-Schwab (CS) CLEAN – break into major (uv-plane) & minor (image-plane) cycles • in CASA initiated by imagermode=‘csclean’ • some options ( ‘mosaic’, ‘mfs’ nterms>1) hardwired cs – clean in minor cycles to a threshold where max residual is some fraction of starting max residual • in CASA will be cyclefactor x max psf sidelobe – more major cycles = more accurate cleaning but slow • poor PSF, simple image structure = lower cyclefactor (<1) • good PSF, complex image structure = raise cyclefactor (>1) – purpose: correct errors from gridding & minor cycles • the transform back from model visibilities is as accurate as we wish to make it S. T. Myers NAASC Charlottesville –08 Sep 2011 14 CLEAN Example • Jupiter 6cm – interactive cleaning in CASA as imaging proceeds, include regions containing emission in mask breaks between interactive cycles are major cycles S. T. Myers NAASC Charlottesville –08 Sep 2011 15 Gridding kernels in CLEAN • This is a choice: d = Hv • Default kernel: H0 = N-1 – includes only noise (inverse variance) weighting • Optimal (mosaic) imaging: H = AT N-1 – uses aperture function (A ~ A) – will grid mosiacs onto same uv-plane – can correct for known pointing errors • Frequency synthesis kernel (MFS) – add kernels corresponding to Taylor expansion terms • G0k= 1 G1k= ln(nk/n0) for each channel k S. T. Myers NAASC Charlottesville –08 Sep 2011 16 Mosaicing in the sky and uv planes offset & add phase gradients S. T. Myers NAASC Charlottesville –08 Sep 2011 17 Mosaicing kernel • the usual equation (aperture and offset term) V (u,v) dx dy Ax x , y y Ix, ye k k 2iux x k vy y k dx dy Bx, y,u,v Ix, y e 2 i ux vy – note that we assume “phased up” at each pointing xk ! • kernel B and its transform (Fourier shift theorem) Bx, y,u,v Ax x k , y y k e2 iux k vyk Bi u,v Au ui,v vi e2iui xk vi yk ˜ ˜ – B = F-1 B F is the “mosaicing” kernel (A-projection) • if offset xk is unknown, then this is a “pointing error” • offset for R and L polarizations is the “squint” term S. T. Myers NAASC Charlottesville –08 Sep 2011 18 Mosaicing considerations • In CASA, imagermode=‘mosaic’ – ftmachine=‘mosaic’ uses A-projection kernel • grids to single uv-plane, optimally weights fields, most efficient • can be some aliasing, keep mosaic to inner quarter of imsize – ftmachine=‘ft’ does standard image-plane shift+add • takes much longer, not recommended – mosaic uses csclean, watch cyclefactor – uses approximate single PSF for entire mosaic • will “correct” in major cycles – uses POINTING table when present (FIELD for phases) • Outputs – standard: .image, .model, .psf – .flux (PB plus weights plus extra PB from A-convolution) – .flux.pbcoverage (just the PB effects) – e.g. to correct to flux on-sky use .image/.flux S. T. Myers NAASC Charlottesville –08 Sep 2011 19 MEM and CLEAN • CLEAN – algorithm: find peak in residual image; add fraction to model; form new residual data & residual image; iterate – performance: good on compact emission, difficult for extended • Maximum Entropy Method (MEM) – algorithm: for pixel values p : maximize entropy -S p ln p ; minimize c2(p) to encourage “smooth” extended emission – performance: complicated, suppresses spiky emission, but fast • CLEAN and MEM use point (pixel) basis – complete basis – unique representation of image S. T. Myers NAASC Charlottesville –08 Sep 2011 20 Sparse Approximation Imaging • Problem: find a model to represent the sky as efficiently as possible, subject to the data constraints and within the noise uncertainty, possibly also subject to prior constraints. – some problems (like ours) cannot be efficiently reconstructed using orthonormal bases (like pixels or Fourier modes) – extensive literature on this! – use non-orthogonal bases: multiscale (e.g. Gaussians) – choose dictionary of model elements (atoms) – efficiency: find a representation that uses the fewest number of atoms S. T. Myers NAASC Charlottesville –08 Sep 2011 21 Example: MEM versus MS-CLEAN Restored Residual Error Maximum Entropy MS Clean S. T. Myers NAASC Charlottesville –08 Sep 2011 22 The Future of Multiscale Methods • Algorithms – mostly iterative, starting from a blank model – “greedy” methods make locally optimal choices at each step • MS-CLEAN is a greedy algorithm in this class! – dictionary of points and Gaussians on different scales – is essentially a “Matching Pursuit” (MP) algorithm (e.g. Tropp 2004) • Key Research Area in CASA – new arrays are designed for high dynamic range & fidelity – we need efficient, robust, and accurate multiscale methods – integrate multiscale (MS) with multispectral (MFS) S. T. Myers NAASC Charlottesville –08 Sep 2011 23 Calibration and Imaging • Some effects corrupt the visibilities – most are on a per-antenna basis, other per-baseline – antenna based effects can be “self-calibrated” out • The Measurement Equation (ME) * * * * Vij J i si J j s j J i J j si s j obs ideal – the Jones matrices J contain the corruptions to V V obs ij M ij Bij Gij Dij E ij Pij Tij Fij V ideal ij Aij – there are different corruption terms to the J • gain G, pol leakage D, ionosphere F, parallactic angle P • can be direction dependent, additive RFI & correlator errors S. T. Myers NAASC Charlottesville –08 Sep 2011 24 Calibration in Image Plane • Calibration errors show up as artifacts in image plane: Before Correction After Correction – given an approximate model for the image we can solve for the errors “self-calibration” through iteration S. T. Myers NAASC Charlottesville –08 Sep 2011 25 Pointing Corrections • Example of direction-dependent errors: – VLA antennas have ~10’’ pointing residual – affects high-dynamic range imaging (>104) – also “squint” between R and L beams • Work by Sanjay Bhatnagar (NRAO) • Simulation of 1.4GHz EVLA observations • Residual images – Before correction: Peak 250Jy, RMS 15Jy – After correction: Peak 5Jy, RMS 1Jy • Can incorporate into standard self-cal • Computational cost ok for now • See EVLA Memos 100 & 84 – Implementing in CASA, testing underway S. T. Myers NAASC Charlottesville –08 Sep 2011 26 Polarization: Vis to Stokes in XY basis • for antennas with parallactic angles fi and fj f v XX cos( i f j ) f cos( i f j ) sin(fi f j ) i sin(fi f j ) I XY v sin(fi f j ) sin(fi f j ) f cos( i f j ) f i cos( i f j ) Q vYX sin(f f ) sin(fi f j ) cos( i f j ) i cos( i f j ) U f f i j vYY cos( f ) cos( f ) sin(f f ) i sin(f f ) V fi j fi j i j i j • and for identical parallactic angles f between antennas: v XX I Q cos 2f U sin 2f Linear Feeds: XY linear polarization v Q sin 2f U cos 2f iV in all hands, circular j vYX fi f Q sin 2f U cos 2f iV f only in cross-hands vYY I Q cos 2f U sin 2f S. T. Myers NAASC Charlottesville –08 Sep 2011 27 Polarization Leakage • Primary on-axis effect is “leakage” of one polarization into the measurement of the other (e.g. X Y) : X X + diX Y – but, direction dependence due to polarization beam! • Calibrate out on-axis leakage and put direction dependence in “beam” – example: expand XY basis with on-axis leakage ˆ XX V XX d X V YX d *X V XY d X d *X V YY Vij ij i ij j ij i j ij ˆ VijXY VijXY diX Vij d *Y VijXX diX d Y Vij YY YX j j – remember XX,YY contain IQU and XY,YX contain QUV – ideally the “d”s are ~1-2% but can be worse, should be stable S. T. Myers NAASC Charlottesville –08 Sep 2011 28 Primary Beam: full field polarization • VLA primary beams – Beam squint due to off-axis system – Instrumental polarization off- axis – AZ-EL telescopes • Instrumental polarization patterns rotate on sky with parallactic angle – Limits polarization imaging – Limits Stokes I dynamic range Green contours: Stokes I 3dB, 6dB, black contours: fractional polarization 1% and up, (via second order terms) vectors: polarization position angle, raster: – must implement during imaging Stokes V S. T. Myers NAASC Charlottesville –08 Sep 2011 29 Simulations on a complex model • VLA simulation of ~ 1 Jy point sources + large I V source with complex polarization (“Hydra A”) • Long integration with full range of parallactic angles • equivalent to weak 1.4GHz source observed with EVLA Q U • Antenna primary beam model by W. Brisken • See EVLA memo 62 S. T. Myers NAASC Charlottesville –08 Sep 2011 30 1-D Symmetric Beam I V Dynamic Range ~200 using symmetric beam model Q U • dynamic range limited by errors from incorrect approximate primary beam S. T. Myers NAASC Charlottesville –08 Sep 2011 31 2-D Polarized Beam I V Dynamic Range ~104 using 2-D beam Q U model • need to use accurate polarized beam to reach high fidelity and dynamic range S. T. Myers NAASC Charlottesville –08 Sep 2011 32 CASA CLEAN Imaging Details • MFS (mode mfs) – nterm=2 compute spectral index, 3 for curvature etc. – needed for bandwidths ~5% or more (S/N dependent) – tt0 average intensity, tt1 alpha*tt0, alpha images output – takes at least nterms longer (image size dependent) • Multiscale (set non-zero multiscale list) – scales are in units of pixels – usually set to be multiples of synth. beam size – e.g. for 3x oversampling of beam: • multiscale = [0,3,6,12,24] – takes at least nscales longer (x nterms?) – can be tricky to get to work right S. T. Myers NAASC Charlottesville –08 Sep 2011 33 CASA CLEAN logistics • clean can be restarted from current state – if imagename used before and files exist and same size – will first recompute residuals from model • user can input some starting conditions – previous mask (e.g. from previous clean) – previous modelimage • key toggles – mode: mfs, channel, velocity, frequency – imagermode: ‘’, csclean, mosaic – psfmode: clark, hogbom – gridmode: advanced stuff, eventually a-projection S. T. Myers NAASC Charlottesville –08 Sep 2011 34 Spectral Cube Considerations • mode: channel, velocity, frequency – sets the grid of planes in output image cube – will apply doppler corrections on the fly – can set frame info, restfreq, etc. • data: taken in sky frequency (terrestrial) frame • velocity: include doppler shifts from: – Earth rotation: few km/s (diurnal) – Earth orbit: 30 km/s (annual) – Earth/Sun motion w.r.t. LSR (e.g. LSRK, LSRD) – maybe galactic rotation to extragalactic frames • these shifts are time dependent – “Doppler Tracking/Setting” adjust sky freq while observing – can shift and regrid data before imaging : cvel S. T. Myers NAASC Charlottesville –08 Sep 2011 35 Continuum Subtraction • If you have a viable continuum model – e.g. via using mode mfs – can use uvsub to remove from data before imaging – BUT: must have really good accurate model (not likely) • In practice, subtract “continuum” in data – use task uvcontsub or uvcontsub2 • particularly if strong short-baseline emission – these fit polynomial to each vis spectrum and subtract • uvcontsub2 can cross spw boundaries – need to know line-free channel ranges • if uvcontsub need to fit in each spw – “continuum” not generally imageable • e.g. no closure !!! S. T. Myers NAASC Charlottesville –08 Sep 2011 36 Odds and Ends • Oversampling PSF – at least 2.5x, I like 3-4x (whichever rounds well), some use 5x, no strong impact (but can make imaging longer) • Boxing – helps guide clean where to pick out real emission, important when psf sidelobes high/complex and/or when image has complicated structure – if initial calibration poor, shallow careful clean with boxing will get better model for selfcal (don’t burn in artifacts) – some cases using csclean (w/cyclefactor) you can “turn it loose”, but be careful before doing this – we are working on autoclean / autoboxing S. T. Myers NAASC Charlottesville –08 Sep 2011 37 Flotsam and Jetsam • What, no clean components? – CASA clean increments the .model image – does not keep track of each iteration component – does not report total cleaned flux until end – does not keep separate multiscale components – might later store cc’s as component lists, but not now – you can mask/edit .model image – you can supply initial “model” using modelimage • this can come from anywhere! e.g. single dish • Stopping clean – when you get residual that are noiselike :) or a mess :( – ideally you should get within 2x thermal • if not, probably calibration errors, bad data, or dyn. range issues S. T. Myers NAASC Charlottesville –08 Sep 2011 38 Fin • Final words – the proof of the quality of the data is in the final imaging, you generally cannot guess this from metrics on the calibrators (particularly when selfcal is possible) • Anything else? • Documentation – inline help <task> – online cookbook (massive but in depth) – casaguides S. T. Myers NAASC Charlottesville –08 Sep 2011 39

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