nassp masters 5003f - computational astronomy - 2009 lecture 14 reprise: dirty beam, dirty image....

NASSP Masters 5003F - Computational Astronomy - 2009

Lecture 14

• Reprise: dirty beam, dirty image.

• Sensitivity

• Wide-band imaging

• Weighting– Uniform vs Natural– Tapering– De Villiers weighting– Briggs-like schemes


Reprise: dirty beam, dirty image.• Fourier inversion of V times the sampling

function S gives the dirty image ID:

• This is related to the ‘true’ sky image I´ by:

• The dirty beam B is the FT of the sampling function:

• (Can get B by setting all the V to 1, then FT.)

vmulivuSvuVdvdumlI 2D e,,,

mlBmlImlI ,,,D

vmulivuSdvdumlB 2e,,


Reprise: l and m• Remember that l = sin θ. θ is the angle from

the phase centre.

• For small l, l ~ θ (in radians of course).• m is similar but for the orthogonal direction.

Direction of phase centre.

Direction ofsource.

l

θ


Sensitivity• Image noise standard deviation (for the weak-

source case) is (for natural weighting)

• N here is the number of antennas.

• Note that Ae is further decreased by correlator effects – for example by 2/π if 1-bit digitization is used.

• Actual sensitivity (minimum detectable source flux) is different for different sizes of source.– Due to the absence of baselines < the minimum

antenna separation, an interferometer is generally poor at imaging large-scale structure.

tNN

T

A

kII

1

2 total

erms


How can we increase UV coverage?…we could get more baselines if we moved the antennas!

Wide-band imaging.


…but it is simpler to change the observing wavelength.

eg

λ

λ/2


…we have many baselines,

and, effectively,

many antennas.

With many wavelengths…


16 x 1 MHz 2000 x 1 MHz

Merlin, δ=+35° eMerlin, δ=+35°

Narrow vs broad-band: UV coverage


16 x 1 MHz 2000 x 1 MHz

Narrow vs broad-band - without noise:


SNR of each visibility = 15%.

16 x 1 MHz 2000 x 1 MHz

Narrow vs broad-band - with noise:


Weighting: or how to shape the dirty beam.

• Why should we weight the visibilities before transforming to the sky plane?– Because the uneven distribution of samples of V means that the dirty beam has lots of ripples or sidelobes, which can extend a long way out.

• These can hide fainter sources.

– Even if we can subtract the brighter sources, there are always errors in our knowledge of the dirty beam shape.

• If there must be some residual, the smoother and lower it is, the better.


Weighting

• There are usually far more short than long baselines.

Baseline length

The distribution of baselinesalso nearly always hasa ‘hole’ in the middle.


Weighting• A crude example:

This bin has 1 sample.

This bin has 84 samples.


Weighting• What do we get if we leave the visibilities alone?

– The resulting dirty beam will be broad ( low resolution), because there are so many more visibility samples at small (u,v) than large (u,v).

– BUT, if the uncertainties are the same for every visibility, leaving them unweighted (ie, all weights Wj,k=1) gives the lowest noise in the image.

– This is called natural weighting.

• The easiest other thing to do is set Wj,k=1/(the number of visibilities in the j,kth grid cell).– This is called uniform weighting.

• Then optionally multiply everything by a Gaussian:– Called tapering.


Natural weighting Uniform weighting

Natural vs uniform:



The resulting dirty images:


SNR of each visibility = 0.7%.


But if we add in some noise...


Tradeoff

• This sort of tradeoff, between increasing resolution on the one hand and sensitivity on the other, is unfortunately typical in interferometry.


Some other recent ideas:1. Scheme by Mattieu

de Villiers (new, not yet published SA work):– Weight by inverse of

‘density’ of samples.

2. My own contribution:– Iterative optimization.

Has the effect of rounding the weight distribution to ‘feather out’ sharp edges in the field of weights.

– Haven’t got the bugs out of it yet.

Ideal smooth weight function(Fourier inverse of desired PSF)

Isolated samplesget weighted higherso that the averageapproaches the ideal.

Densely packedsamples aredown-weighted.


Uniform

Tapered uniform

Iterative best fit out-side 20-pixel radius

Simulated e-Merlin data.400 x 5 MHz channels;νav = 6 GHz;tint = 10 s;δ = +30°

Weighting schemes:


‘Dirty beam’ images (absolute values).

20

Iterative best fit out-side 20-pixel radius

Tapered uniform

Uniform


Natural

Uniform

Optimized

Natural (narrow-band)

Natural

Uniform

Optimized for r>10

Comparison – slices through the DIs:


r = 10

More on iterated weights:


SNR of each visibility = 5.

But real data is noisy…


1. Multiply visibilitieswith a vignettingfunction of time andfrequency, eg

2. Aips task IMAGRparameter UVBOX:effectively smoothsthe weight function.See also D Briggs’PhD thesis.

One could think of other ‘feathering’ schemes.

nassp masters 5003f - computational astronomy - 2009 lecture 14 reprise: dirty beam, dirty image....

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