Lecture 17

1. Lecture 17 • Reprise: dirty beam, dirty image. • Sensitivity • Wide-band imaging • Weighting • Uniform vs Natural • Tapering • De Villiers weighting • Briggs-like schemes

2. 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 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 of source. l θ

4. 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.

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

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

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

8. 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

9. ‘Snapshot’ sampling of V is poor. Antenna spacings from KAT-7. 22/43

10. 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

11. 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

12. Narrow vs broad-band: UV coverage 16 x 1 MHz 2000 x 1 MHz Merlin, δ=+35° eMerlin, δ=+35°

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

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

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.

16. 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

17. Weighting • A crude example: This bin has 1 sample. This bin has 84 samples.

18. 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.

19. Natural vs uniform: Natural weighting Uniform weighting

20. The resulting dirty images: Natural weighting Uniform weighting

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

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

23. 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. • Haven’t 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.

24. Weighting schemes: Simulated e-Merlin data. 400 x 5 MHz channels; νav = 6 GHz; tint = 10 s; δ = +30° Iterative best fit out- side 20-pixel radius Uniform Tapered uniform

25. ‘Dirty beam’ images (absolute values). 20 Iterative best fit out- side 20-pixel radius Uniform Tapered uniform

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

27. More on iterated weights: r = 10

28. But real data is noisy… SNR of each visibility = 5.

29. 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.

30. MeerKAT tapering schemes