Data Weighting and Imaging Tom Muxlow General introduction Initial data processing Data gridding and...

Data Weighting and ImagingData Weighting and ImagingTom MuxlowTom Muxlow

•General introduction

•Initial data processing

•Data gridding and weighting schemes

•De-convolution

•Wide-field imaging

•High dynamic-range imaging

Data Weighting and ImagingData Weighting and Imaging

General Introduction

•Description of data processing MERLIN 1658MHz data on the Double Quasar observed 3rd 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


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


Data Editing - 1

•Interactive raw data editing via ‘dplot’


Data Editing - 2

•Subsequent data editing in AIPS ‘IBLED’


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



•Initial phase solutions assume a point – first image of reference source may show structure.

•Use image clean components to refine phase solutions



•As image and phase solutions stabilise, subsequent phase corrections become small



•With a stable reference source image it is possible to solve for gain fluctuations within the imaging run


Self-Calibration - 1

•If target is bright enough, use the initial target image to apply further refinements to the phase and gain solutions


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


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


Data Gridding - 1

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•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’


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


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)


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


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


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)


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 2n u-v data grid that are zero will give rise to severe ripples in the beam response


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



•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



•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….


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


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°


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 θoffset(radians) x θoffset(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


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


Wide-Field Imaging - 3 Bandwidth smearing

•Parameterized by the product of the fractional bandwidth and the source offset in synthesised beamwidths

Chromatic Aberration

0.00

20.00

40.00

60.00

80.00

100.00

120.00

0.00 0.50 1.00 1.50 2.00 2.50

Fractional Bandwidth x Source Offset

Pe

rce

nta

ge

Smearing

Peak

δυ /υ0 x θ/θHPBW

•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


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:

ωeδtint x θ/θHPBW

•Time-average smearing (decorrelation) will produce tangential smearing

•For other Declinations the effects are more complicated. However they can be alleviated by ensuring that δtint is small enough such that there at least 4 samples per turn assuming a maximum rate of θ/θHPBW turns in 6 hours

Where ωe is the Earth’s angular rotation rate and δtint is the integration time interval in the dataset


Wide-Field Imaging – 4 Primary beam response

θHPBW = 1125 / (d νGHz )

•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:


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’


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


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

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Data Weighting and Imaging Tom Muxlow General introduction Initial data processing Data gridding and...

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