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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 vx 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 Ax  x , y  y Ix, ye
                                                   k              k
                                                                                   2iux x k vy y k 



                        dx dy Bx, y,u,v Ix, y e                  2 i ux vy


              – note that we assume “phased up” at each pointing xk !
  • kernel B and its transform (Fourier shift theorem)
           Bx, y,u,v   Ax  x k , y  y k e2 iux k vyk 

                          Bi u,v  Au  ui,v  vi e2iui 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 250Jy, RMS 15Jy
      – After correction: Peak 5Jy, RMS 1Jy
• 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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