1. Data Weighting and Imaging Tom Muxlow General introduction Initial data processing Data gridding and weighting schemes De-convolution Wide-field imaging High dynamic-range imaging

2. Data Weighting and Imaging General Introduction Description of data processing MERLIN 1658MHz data on the Double Quasar observed 3 rd May 1993 Shows sequential approach to basic amplitude, phase, and polarization calibration Illustrates basic data gridding, weighting and Fourier inversion to raw instrumental response and ‘dirty map’ Shows typical image de-convolution schemes Shows image enhancement through self-calibration

3. Data Weighting and Imaging Initial Processing Data initially inspected and edited by local software routines (‘d-programmes’) – additional editing performed later in AIPS Amplitudes converted from correlation coefficients to flux density by calibrating with strong point sources (eg OQ208, 0552+398, 2134+004) Flux density of point sources derived from comparison of amplitudes on shortest MERLIN spacings on the resolved flux density calibrator 3C286 Data exported to AIPS via disc FITS multi-source file

4. Data Weighting and Imaging Data Editing - 1 Interactive raw data editing via ‘dplot’

5. Data Weighting and Imaging Data Editing - 2 Subsequent data editing in AIPS ‘IBLED’

6. Data Weighting and Imaging Phase Calibration - 1 MERLIN is phase-stable – but atmospheric delay must be calibrated by phase referencing to a nearby source. Ideally reference is a point – but can map out the object

7. Data Weighting and Imaging Phase Calibration - 2 Initial phase solutions assume a point – first image of reference source may show structure. Use image clean components to refine phase solutions

8. Data Weighting and Imaging Phase Calibration - 3 As image and phase solutions stabilise, subsequent phase corrections become small

9. Data Weighting and Imaging Phase Calibration - 4 With a stable reference source image it is possible to solve for gain fluctuations within the imaging run

10. Data Weighting and Imaging Self-Calibration - 1 If target is bright enough, use the initial target image to apply further refinements to the phase and gain solutions

11. Data Weighting and Imaging Self-Calibration - 2 If troubled by side-lobes, use windowing to restrict positions of clean components. Include components to first negative and restrict u-v range to match flux in model

12. Data Weighting and Imaging Polarization Calibration Telescope leakage terms (typically a few %) are derived from a source of known polarization or from a source with good paralactic angle coverage (phase calibrator usually) Polarization position angle calibrated on 3C286 (+33°) Residual leakage into polarized image for MERLIN (after correction) is typically ~ 0.2%-0.5% of the total intensity image at that position

13. Data Weighting and Imaging Data Gridding - 1 + + + + + Data are interpolated onto a regular 2 n grid with a spheroidal convolution func. Weights unmodified by local density – ‘Natural’ Weights divided by local density of points – ‘Uniform’

14. Data Weighting and Imaging 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 striping instability

15. Data Weighting and Imaging Data Gridding - 3 Robust Weighting Smooths the large 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 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 AIPS IMAGR Robustness Factor ROBUST = – 4 is nearly pure uniform ROBUST = + 4 is nearly pure natural ROBUST = 0 is a good compromise (Contoured)

16. Data Weighting and Imaging Data Gridding - 4 Robust Weighting Can reduce thermal noise of a full track VLA uniformly weighted image by 25%, whilst increasing the fitted beam by only 3% For VLBI arrays thermal noise improvements of ~2 are possible with only ~15% loss in resolution

17. Data Weighting and Imaging Data Weighting By u-v Distance Gaussian u-v taper or u-v range can smooth the image but at the expense of sensitivity since data are excluded or data usage is non-optimum For MERLIN/VLBI arrays beware compromising image quality by severely restricting the u-v coverage

18. Data Weighting and Imaging Data Weighting By Telescope Data from mixed-type arrays should be re-weighted by telescope sensitivity in order to minimise thermal noise – for MERLIN this can decrease the noise level by ~2 Typical WTMOD parameters for 1.7GHz data: Lovell - 30 (76m) Cambridge - 2 (32m) Knockin - 1 (25m) Defford - 0.6 (25m)

19. Data Weighting and Imaging Image De-convolution - 1 The raw image (dirty map) will usually require significant de-convolution Conventional 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, MX) are able to tolerate such errors and typically allow imaging to within a few pixels of the edge Regions in the 2 n u-v data grid that are zero will give rise to severe ripples in the beam response

20. Data Weighting and Imaging Image De-convolution - 2 Extended Emission Low surface brightness extended structure, can be subject to fragmentation during de-convolution – 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

21. Data Weighting and Imaging Image De-convolution - 3 Extended Emission Maximum entropy de-convolution will produce smoother images in regions of low signal:noise VTESS algorithm will produce the maximum entropy image convolved withy the fitted beam + residuals VTESS does not deal well with bright points in the image – residual side-lobes are common

22. Data Weighting and Imaging Image De-convolution - 3 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 Multi-scale clean as developed in AIPS++ may supersede this….

23. Data Weighting and Imaging Image De-convolution - 5 MFS Spatial frequency coverage for continuum imaging may be significantly improved by Multi-Frequency Synthesis observations where data are taken at several frequencies – MERLIN δν ~13–15% Spectral changes across the source structure must be addressed if high dynamic ranges are required Overall spectral index accounted for in IMAGR – but discrepant components (flat-spectrum cores) may need separate subtraction from each frequency dataset – restoring a single central frequency average value MERLIN 3C459 MFS 4546 / 4866 / 5186 MHz

24. Data Weighting and Imaging Image De-convolution - 6 MFS New wide-band upgrades (e-MERLIN, EVLA1) will require a spectral solution in addition to the radio brightness at each location in the image 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 Miriad software package already does this routinely for ATCA datasets e-MERLIN MFS u-v Coverage 4–8 GHz at Declination 30°

25. Data Weighting and Imaging Wide-Field Imaging - 1 Wide-field images are subject to a number of distortions: Non-coplanar baselines Bandwidth smearing Time-averaging smearingPrimary beam response Standard Fourier synthesis assumes planar arrays – only true for E-W interferometers Errors increase quadratically with offset from phase-centre Serious errors result if θ offset (radians) x θ offset (beams) > 1 Effects particularly severe for low-frequency VLA observations Standard imaging software can account for this by making many smaller offset images with the geometry corrected for each image centre M82 MERLIN MFS + VLA 5GHz 1.45x10 -4 radians x 850 beams = 0.123 30 arcseconds Beam 35 mas

26. Data Weighting and Imaging Wide-Field Imaging – 2 Non-coplanar baselines Corner of image 0.00206 radians from phase centre Beam = 2 arcsec – 212 beam offsets 212x0.00206= 0.437 OK! 7 arcmin HDF North VLA 1.4 GHz - but not ok for MERLIN beam 200mas – need to mosaic with many smaller images New software under development will allow projection to a coplanar grid ‘W-projection’ – Tim Cornwell et al

27. Data Weighting and Imaging Wide-Field Imaging - 3 Bandwidth smearing Parameterized by the product of the fractional bandwidth and the source offset in synthesised beamwidths δυ /υ 0 x θ/θ HPBW Bandwidth smearing (chromatic aberration) will produce radial smearing and reduction in source peak Can be alleviated by observing and imaging in spectral line mode with many narrow frequency channels gridded separately prior to Fourier inversion – reduces δυ

28. Data Weighting and Imaging Wide-Field Imaging - 3 Time-averaging smearing In general cannot be easily parameterized. At Declination=+90° a simple case exists where the effects can be parameterized by the equivilent product: ω e δt int x θ/θ HPBW Time-average smearing (decorrelation) will produce tangential smearing For other Declinations the effects are more complicated. However they can be alleviated by ensuring that δt int is small enough such that there at least 4 samples per turn assuming a maximum rate of θ/θ HPBW turns in 6 hours Where ω e is the Earth’s angular rotation rate and δt int is the integration time interval in the dataset

29. Data Weighting and Imaging Wide-Field Imaging – 4 Primary beam response θ HPBW = 1125 / (d ν GHz ) In the limit where one dish is much larger than the other, the beam is determined by the voltage pattern of the large telescope. For a Gaussian shaped beam the FWHM will be  2 larger. (Strom 2004, astro-ph/0412687) – at 1.4 GHz Lovell HPBW beam ~10.4’, but Lovell-Knockin beam ~19’ The overall correction will also depend on the relative weighting and the data distribution between telescopes. Detailed solutions are usually empirical and are derived from measurements on offset sources of known strengths. AIPS routines exist to correct for primary beam effects (PBCOR for existing images and frequency dependent beam corrections within IMAGR when cleaning) For mixed arrays, beams for interferometer pairs are simply the voltage polar diagram of one multiplied by that of the second. The ultimate factor limiting the field of view is the diffraction limit of the individual antennas. For an antenna of diameter d metres an approximate formula for the full width at half power in arcminutes is given by:

30. Data Weighting and Imaging Wide-Field Imaging – 5 Confusion Strong sources on the edge of the primary beam can give rise to ripples in the centre of the field of view VLA HDF North 1.4 GHz 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 – attempt multiple snapshot subtraction on timescale comparable with pointing error changes – or resort to ‘pealing’

31. Data Weighting and Imaging Wide-Field Imaging – 6 Pealing off Confusion After phase calibrating the data, perform self-calibration for the brightest confusing source – then subtract 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

32. Data Weighting and Imaging High Dynamic Range Imaging Phase calibration – up to 1000:1  improve with self-calibration Non-closing data errors – continuum ~20,000:1 line >100,000:1 After baseline calibration (RESOFF, BLCAL) ~1,000,000:1 3C273 MERLIN 1.7 GHz Present dynamic range limits: Non-closing errors thought to be dominated by small changes in telescope pass-bands on short timescales Use line configurations with bandpass corrections or strip and sequentially re- assemble each frequency channel Finally use baseline corrections – but beware these are dangerous since they alter the data to correspond to your image model Image from a 1 MHz single channel – dynamic range ~65:000:1