Lecture 17 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting

Lecture 17 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 I D : I D ( ) ( ) ( ) 2πi( ul+ vm) l, m dudvv u, v Su, v e This is related to the true sky image I by: ( l, m) = I ( l, m) Bl ( m) I, D The dirty beam B is the FT of the sampling function: ( ) ( ) 2πi( ul+ vm), m dudvsu, v e Bl (Can get B by setting all the V to 1, then FT.)

Reprise: l and m Remember that l = sin θ. θ is the angle from the phase centre. Direction of phase centre. l Direction of source. θ For small l, l ~ θ (in radians of course). m is similar but for the orthogonal direction.

Sensitivity Image noise standard deviation (for the weaksource case) is (for natural weighting) σ I = I rms k A ( N 1) t ν N here is the number of antennas. e Note that A e is further decreased by correlator effects for example by 2/π if 1-bit digitization is used. N T 2 total 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.

Wide-band imaging. How can we increase UV coverage? we could get more baselines if we moved the antennas!

but it is simpler to change the observing wavelength. λ eg λ/2

With many wavelengths we have many baselines, and, effectively, many antennas.

A simulated example. The full visibility function V(u,v) (real part only shown). A familiar pattern of sources Red positive; blue negative. (I ve taken some liberties here obviously the stars of the Southern Cross are not strong radio sources I ve also rescaled their angular separations.) 21/43 Talk at Nagoya University IMS Oct 2009

Snapshot sampling of V is poor. Antenna spacings from KAT-7. 22/43 Talk at Nagoya University IMS Oct 2009

Aperture synthesis via the Earth s rotation. For this technique to work perfectly, all sources must be constant over time. Antenna spacings from KAT-7. Dirty image D is the true sky brightness map I, convolved with the dirty beam B. 23/43 Talk at Nagoya University IMS Oct 2009

Frequency synthesis. For this technique to work perfectly, all sources must not only be constant over time, but must also have the same spectra. Antenna spacings from KAT-7. Bandwidth 5 to 6 GHz. The final image is still not as clean as we would like 24/43 Talk at Nagoya University IMS Oct 2009

Narrow vs broad-band: UV coverage 16 x 1 MHz 2000 x 1 MHz Merlin, δ=+35 emerlin, δ=+35

Narrow vs broad-band - without noise: 16 x 1 MHz 2000 x 1 MHz

Narrow vs broad-band - with noise: 16 x 1 MHz 2000 x 1 MHz SNR of each visibility = 15%.

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

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 W j,k =1) gives the lowest noise in the image. This is called natural weighting. The easiest other thing to do is set W j,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 vs uniform: Natural weighting Uniform weighting

The resulting dirty images: Natural weighting Uniform weighting

But if we add in some noise... Natural weighting Uniform weighting SNR of each visibility = 0.7%.

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. Densely packed samples are down-weighted. Ideal smooth weight function (Fourier inverse of desired PSF) Isolated samples get weighted higher so that the average approaches the ideal.

Weighting schemes: Simulated e-merlin data. 400 x 5 MHz channels; ν av = 6 GHz; t int = 10 s; δ = +30 Iterative best fit outside 20-pixel radius Uniform Tapered uniform

Dirty beam images (absolute values). Iterative best fit outside 20-pixel radius 20 Uniform Tapered uniform

Comparison slices through the DIs: Natural (narrow-band) Natural Uniform Uniform Optimized for r>10 Optimized