nassp masters 5003f - computational astronomy - 2009 lecture 13 further with interferometry –...

NASSP Masters 5003F - Computational Astronomy - 2009

Lecture 13

Further with interferometry –

• Resolution and the field of view;

• Binning in frequency and time, and its effects on the image;

• Noise in cross-correlation;

• Gridding and its pros and cons.


Earth-rotation synthesis

Apply appropriate delays: like measuring Vwith ‘virtual antennas’ in a plane normalto the direction of the phase centre.


Earth-rotation synthesis

Apply appropriate delays: like measuring V

with ‘virtual antennas’ in a plane normal

to the direction of the phase centre.


Field of view and resolution.

Single dish:FOV and resolution are the same.

FOV ~ λ/d(d = dish diameter)

Resolution ~ λ/d



Aperture synthesis array:FOV is much larger than resolution.

FOV ~ λ/d Resolution ~ λ/D(D = longest baseline)

d

D



Phased array:Signals delayed then added.FOV again = resolution.

FOV ~ λ/D Resolution ~ λ/Dd

D

Good for spectroscopy,VLBI.


LOFAR – can see the whole sky at once.


Reconstructing the image.

• The basic relation of aperture synthesis:

where all the (l,m) functions have been bundled into I´. We can easily recover the true brightness distribution from this.

• The inverse relationship is:

• But, we have seen, we don’t know V everywhere.

vmulimlIdmdlvuV 2exp,,

vmulivuVdvdumlI 2exp,,


Sampling function and dirty image• Instead, we have samples of V. Ie V is

multiplied by a sampling function S.

• Since the FT of a product is a convolution,

where the ‘dirty beam’ B is the FT of the sampling function:

ID is called the ‘dirty image’.

vmulivuSvuVdvdumlI 2D e,,,

mlBmlImlI ,,,D

vmulivuSdvdumlB 2e,,


Painting in V as the Earth rotates


But we must ‘bin up’ in ν and t.

This smears out the finer ripples.Fourier theory says: finer ripples come from distant sources.Therefore want small Δν, Δt for wide-field imaging. But: huge files.


We further pretend that these samples are points.


What’s the noise in these measurements?• Theory of noise in a cross-correlation is a little

involved... but if we assume the source flux S is weak compared to sky+system noise, then

• If antennas the same,

• Root 2 smaller SNR from single-dish of combined area (lecture 9).– Because autocorrelations not done information lost.

t

T

A

kS total

erms

2

t

TT

AA

kS total2total1

e2e1

rms

2


Resulting noise in the image:

Spatially uniform – but not ‘white’.

(Note: noise in real and imaginaryparts of the visibility is uncorrelated.)


Transforming to the image plane:

• Can calculate the FT directly, by summing sine and cosine terms.– Computationally expensive - particularly with

lots of samples.• MeerKAT: a day’s observing will generate about

80*79*17000*500=5.4e10 samples.

• FFT:– quicker, but requires data to be on a regular

grid.


How to regrid the samples?

Could simply add samples in each box.


But this can be expressed as a convolution.

Samples convolved with a square box.


Convolution gridding.

• ‘Square box’ convolver is

• Gives

• But the benefit of this formulation is that we are not restricted to a ‘square box’ convolver.– Reasons for selecting the convolver carefully will be

presented shortly.

vvuuGvuVdvduV kjkj ,,,

else. 0 ,5.0,5.0,for 1, vuvuG

vuVdvduV kj ,

5.0,5.0

,


What does this do to the image?

• Fourier theory:– Convolution Multiplication.– Sampling onto a grid ‘aliasing’.


A 1-dimensional example ‘dirty image’ ID:

V I via direct FT:


A 1-dimensional example ‘dirty image’ ID:

Multiplied by the FT ofthe convolver:


A 1-dimensional example:

The aliased resultis in green:

Image boundariesbecome cyclic.


A 1-dimensional example:

Finally, dividingby the FT of theconvolver:


Effect on image noise:

Direct FT Gridded then FFT


Aliasing of sources – none in DT

This is a direct transform. The green box indicatesthe limits of a gridded image.


Aliasing of sources – FFT suffers from this.

The far 2 sources are now wrapped or ‘aliased’into the field – and imperfectly suppressed by thegridding convolver.

nassp masters 5003f - computational astronomy - 2009 lecture 13 further with interferometry –...

Documents

f computational astronomy

nassp masters

d d d good

longest baseline d d

interferometry resolution

earthrotation synthesis

dish diameter resolution

image noise