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Anna Scaife - Wide Field Imaging (W-projection (The higher the number of w…
Anna Scaife - Wide Field Imaging
Multiple effects become important when making very wide field images
Non- coplanarity
Primary beam rotation
Direction-dependent phase calibration
Non coplanarity (w-effect)
Overview
Solutions
Worked examples
uvw is the vector between the two interferometer elements expressed in units of wavelength
Maths!!
fourier transforms
w-term is a phase term, but phase is position dependent
If we ignore the w-term and do a 2d transform we see image distortions, and further from the centre the sources are more distorte
(l',m') is a warped version of (l,m)
The w-phase screen
Effectively, every visability with a different w-value sees a different sky
The w-value changes as a function of time, so the measured sky is time variable
An array with North-south extensions will have non-zero w-terms
On the 2d image plane, source motions follow conic sections
(l',m') scales quadratically with offset from the centre, i.e. sources that are further away move more
How to fix?
3D fourier transform? - recover F(l,m,n) where the only non zero values lie on a 2d surface
Would end up with a cube where the n-direction was almost completely 0-values
Also, no complete sampling in uvw, so Flmn will be convolved with a dirty beam in 3d
Convolution function is known as the w-kernal
Mathematically this means that the visibility for non-zero w can be calculated from the visibility for w=0
For making an image from visibilities we need to go the other way - from visibility data to images
To do this convolve Vuvw with the inverse of Guvw which is just Guv-w
w-projection
Probably using w-planes we use a w-kernel in the convolutional gridding step of the imaging process
w-snapshots
We image each time step seperately, re-project the warped images individually and then stack theresults
Faciting
Break up the sky into pieces and image them separately
W-projection
For making an image from visibilities we need to go the other way - from visibility data to images
Therefore need to project V(u,v,w) --> V(Uu,v,w=0) to use the 2d fourier transform
To do this convolve V(u,v,w) with the inverse of G(u,v,w) which conveniently is just G(u,v,-w)
In practice most w-projection implenentations do not calculate a w-kernal for every individual visibility
Typically the visibility data is ordered in increasing w-value and then divided into w-planes
A Kernel is then created for each plane, with a w-value that represents the mean w-position within that plane
The higher the number of w-planes the more accurate the imaging will be - but also the more computatinally expensive it will be
As w increases the size of the w-kernal also increases
but performing the convolution operation gets more expensive as the kernal gets larger
CASA will look at the available memory on your computer and limit the size (support) of your w-kernals
Example: East-West interferometers
In E-W don't need to worry about coplanarity
Antenna separations remain constant in amplitude as a function of time when viewed from the north pole
lie in a w=0 plane
This is the same as saying that they sit on a w=0 plane in uvw space in a co-ordnate frame with the w-axis oriented towards the NCP