Reprise: dirty beam, dirty image'

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Lecture 14

• Reprise dirty beam, dirty image.
• Sensitivity
• Wide-band imaging
• Weighting
• Uniform vs Natural
• Tapering
• De Villiers weighting
• Briggs-like schemes

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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.)

3
Reprise l and m

• Remember that l sin ?. ? is the angle from the
  phase centre.
• For small l, l ? (in radians of course).
• m is similar but for the orthogonal direction.

Direction of phase centre.
Direction of source.
l
?
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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/p 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 lt the minimum
  antenna separation, an interferometer is
  generally poor at imaging large-scale structure.

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Wide-band imaging.
How can we increase UV coverage?we could get
more baselines if we moved the antennas!
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but it is simpler to change the observing
wavelength.
?
eg
?/2
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With many wavelengths
we have many baselines,
and, effectively,
many antennas.
8
Narrow vs broad-band UV coverage
16 x 1 MHz
2000 x 1 MHz
Merlin, d35
eMerlin, d35
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Narrow vs broad-band - without noise
16 x 1 MHz
2000 x 1 MHz
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Narrow vs broad-band - with noise
16 x 1 MHz
2000 x 1 MHz
SNR of each visibility 15.
11
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.

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Weighting

• There are usually far more short than long
  baselines.

The distribution of baselines also nearly always
has a hole in the middle.
Baseline length
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Weighting

• A crude example

This bin has 1 sample.
This bin has 84 samples.
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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,k1) gives the lowest noise in the
  image.
• This is called natural weighting.
• The easiest other thing to do is set Wj,k1/(the
  number of visibilities in the j,kth grid cell).
• This is called uniform weighting.
• Then optionally multiply everything by a
  Gaussian
• Called tapering.

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Natural vs uniform
Natural weighting
Uniform weighting
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The resulting dirty images
Natural weighting
Uniform weighting
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But if we add in some noise...
Natural weighting
Uniform weighting
SNR of each visibility 0.7.
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Tradeoff

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

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Some other recent ideas

• Scheme by Mattieu de Villiers (new, not yet
  published SA work)
• Weight by inverse of density of samples.
• My own contribution
• Iterative optimization. Has the effect of
  rounding the weight distribution to feather out
  sharp edges in the field of weights.
• Havent got the bugs out of it yet.

Ideal smooth weight function (Fourier inverse of
desired PSF)
Densely packed samples are down-weighted.
Isolated samples get weighted higher so that the
average approaches the ideal.
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Weighting schemes
Simulated e-Merlin data. 400 x 5 MHz
channels ?av 6 GHz tint 10 s d 30
Iterative best fit out- side 20-pixel radius
Uniform
Tapered uniform
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Dirty beam images (absolute values).
Iterative best fit out- side 20-pixel radius
Uniform
Tapered uniform
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Comparison slices through the DIs
Natural (narrow-band)
Natural
Uniform
Optimized for rgt10
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More on iterated weights
r 10
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But real data is noisy
SNR of each visibility 5.
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One could think of other feathering schemes.

• Multiply visibilities
• with a vignetting
• function of time and
• frequency, eg

2. Aips task IMAGR parameter UVBOX effectively
smooths the weight function. See also D
Briggs PhD thesis.