Synthesis imaging in the presence of image plane effects

S. Bhatnagar (NRAO)

T.J. Cornwell (ATNF, Australia), K. Golap (NRAO)

National Radio Astronomy Observatory

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Basic Interferometry

<eie*j>

Terminology:

Visibility: Vij=<eie*j>, ei is the electric field

Baseline: Length of the

projected separation between the antennas (B).

Only relative separation matters for incoherent emission

Co-ordinates: (uij,vij,wij)

uij=ui-uj

An N-element array instantaneously measures N(N-1)/2 baselines (complex values)

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Basic Imaging

In terms of the sky brightness distribution (Io(l,m))

where S : uv-sampling function, (l,m): direction in the sky

For l2+m2<<1 or small wij, sky is the 2D Fourier transform of the Visibility function (van Cittert-Zernike Theorem)

The Dirty Image The Dirty Beam

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

The Measurement Eq.: where A is the measurement matrix, V, I and N are the Visibility, Image and noise vectors.

A=FS where F is the Fourier Transform operator and S is a diagonal matrix of weights.

Due to finite Fourier plane sampling, A is singular and in general rectangular

Image reconstruction: Solve for Io

B: The Toeplitz Beam Matrix

ATV=Id: The Dirty Image vector

Basic Imaging

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

The image plane

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

The full Measurement Equation:

Data Corruptions Sky

Gij: direction independent corruptions (e.g., multiplicative complex gains, etc.)

Xij: direction dependent corruptions (e.g., Primary beam effects, etc.)

Often Gij, Xij are separable into antenna based quantities as Gij=GiG*j ==> N unknowns; O(N2) measurements

Data corruptions

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Inverse transform strictly does not exist.

Use accurate transform for Image->Data and approximate for Data->Image

Use D to compute approximate update direction to be used in a non-linear iterative scheme to estimate IM (deconvolution).

Use IM to solve for corruptions (calibration):

Use to improve the estimate for IM

Iterate between deconvolution and calibration.

General Imaging approach

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Example: The CLEAN algorithm

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Antenna based, direction dependent errors needs correction in the data from each baseline by a different function:

Such errors need to be corrected during deconvolution.

General structure of deconvolution algorithms:

Major cycle: Compute the update direction:

Minor cycle: Update the model image

Direction dependent errors

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Use FFT for A to compute Vo. Needs data re-sampling.

Use Eij for GCF when predicting model visibilities.

Compute the update direction (IR) as:

Direction dependent error correction

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Non co-planar baselines

W-projection algorithm

Pre-compute GCFs with uniform sampling in such that aliasing effects are less than the required dynamic range.

(Cornwell,Golap,Bhatnagar: EVLA Memo#67;A&A,in press)

Known direction dependent effects

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

e1=e'1(u,v,w) propagated using Fresnel diffraction theory. The above convolution equation is reproduced with

A interferometer is not a device to measure a single Fourier component.

W-projection

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

W-projection

Galactic Plane at P-band – VLA B,C,D (Brogan et al.)

GMRT P-band image of TXS2342+342 (Kanekar et al.)

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Without correction With correction

Scaling laws:

W-projection:

UV-facet:

Ratio:

W-projection scaling laws

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Eij as function of direction is measured a priori

Primary beam effects

Non-identical Primary Beams:

Polarized primary beam: Beam squint

Ei separately measured for the two polarizations

Applied with the known squint

Measured direction dependent effects

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

VLA: The two orthogonal polarization power patterns have a squint w.r.t. to the pointing direction (6% of FWHM).

If not corrected, polarization properties are correct only at the image center.

Polarization Beam Squint: Example

Stokes-V image without squint correction

Stokes-V image with squint correction

Dynamic range improvement: ~15x Fidelity improvement: ~50x

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Eij as a function of direction is not known a priori

Eij is parameterized (pi). Needs a solver

Primary beam effects (Pointing offsets):

Source structure as a function of polarization and frequency: IM is also parameterized

Unknown direction dependent effects

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Separates into constant Eo and antenna dependent terms

Eij (the GCF) is different for each baseline!

Use lookup tables or approximation:

Pointing offset calibration

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Pointing errors for single pointing observation of typical L-Band field limits the RMS noise to ~10microJy.

EVLA L-band sensitivity: 1microJy/beam

Mosacing dynamic range limited by pointing errors.

Significant fraction of ALMA observation will be mosacing observations.

This implies significantly increased computing:

Each iteration involves expensive visibility prediction

Pointing SelfCal

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Model image: 59 sources from NRAO-VLA All Sky Survey(NVSS).

Flux range ~2-200 mJy/beam

Continuous lines: Typical antenna pointing offsets for VLA as a function of time (Mean between +/-20arcsec and RMS of 5arcsec).

Dashed lines: Residual pointing errors. RMS ~ 1arcsec.

Pointing SelfCal: Example

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Residual image without pointing correction:

FT[VObs-VM]

Peak flux ~250 microJy/beam

RMS ~ 15microJy

Residual image after pointing correction:

FT[VObs-Eij*VM]

Peak flux ~5microJy/beam

RMS ~ 1microJy

Ref: Bhatnagar,Cornwell,Golap, EVLA Memo #84

Pointing SelfCal: Example

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

EVLA bandwidth ratio of 2:1

Sky emission, the Primary Beams, etc. become a function of frequency.

Ideas: Apply PB effects during predict. Sky model parameterized in frequency as well.

Combining Scale sensitive + frequency sensitive deconvolution with image plane corrections.

Work in progress: Extension of Multi-Frequency Synthesis and Scale-sensitive Clean.

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Wide-band continuum imaging

Pixel-to-pixel noise in the image is correlated at the scale of the resolution element

The scale of emission fundamentally separates signal (Io) from the noise (IN).

Asp-Clean (Bhatnagar&Cornwell, 2004, A&A)

Explicitly solve for local scale, position and amplitude

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Scale sensitive deconvolution

Asp-Clean (Bhatnagar & Cornwell, A&A, in press)

Large scale emission and asymmetric structures are better reconstructed

Computationally expensive: cost increases with the no. of components

Acceleration: Solve in a sub-space; adaptively determine the sub-space

Asp-Clean acceleration

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

where Kij is the ionospheric, direction dependent phase

The general form for residuals:

where Xij=Wij.Eij.Kij (and other direction dependent terms)

Express ionospheric phase in the antenna beam using ionospheric physics/geometrical effects/primary beam effects (de-focusing, time varying illumination, etc.)

Xij is separable into antenna based quantities X(pi).X*(pj) Solve for pi's

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Ionospheric phase correction

Increase in computing due to more sophisticated parameterization

Deconvolution: Fast evaluation of

Calibration: Fast evaluation of

Cost of computing residual visibilities is dominated by I/O costs for large datasets (~200GB for EVLA)

Deconvolution: Approx. 20 access of the entire dataset

Calibration: Each trial step in the search accesses the entire dataset

Solutions: Analytical approximations, caching, Parallel computing and I/O,...

S.Bhatnagar: Signal Recovery and Synthesis (OSA), Charlotte, June 6, 2005

Computing and I/O costs

Sanjay Bhatnagar

2005-06-04T11:14:14

Sanjay Bhatnagar