1. 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

2. NASSP Masters 5003F - Computational Astronomy - 2009 Reprise: dirty beam, dirty image. Fourier inversion of V times the sampling function S gives the dirty image I D : 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. NASSP Masters 5003F - Computational Astronomy - 2009 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. NASSP Masters 5003F - Computational Astronomy - 2009 Sensitivity Image noise standard deviation (for the weak- source case) is (for natural weighting) N here is the number of antennas. Note that A e 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. NASSP Masters 5003F - Computational Astronomy - 2009 How can we increase UV coverage? …we could get more baselines if we moved the antennas! Wide-band imaging.

6. NASSP Masters 5003F - Computational Astronomy - 2009 …but it is simpler to change the observing wavelength. eg λ λ/2

7. NASSP Masters 5003F - Computational Astronomy - 2009 …we have many baselines, and, effectively, many antennas. With many wavelengths…

8. NASSP Masters 5003F - Computational Astronomy - 2009 16 x 1 MHz2000 x 1 MHz Merlin, δ =+35°eMerlin, δ =+35° Narrow vs broad-band: UV coverage

9. NASSP Masters 5003F - Computational Astronomy - 2009 16 x 1 MHz2000 x 1 MHz Narrow vs broad-band - without noise:

10. NASSP Masters 5003F - Computational Astronomy - 2009 SNR of each visibility = 15%. 16 x 1 MHz2000 x 1 MHz Narrow vs broad-band - with noise:

11. NASSP Masters 5003F - Computational Astronomy - 2009 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.

12. NASSP Masters 5003F - Computational Astronomy - 2009 Weighting There are usually far more short than long baselines. Baseline length The distribution of baselines also nearly always has a ‘hole’ in the middle.

13. NASSP Masters 5003F - Computational Astronomy - 2009 Weighting A crude example: This bin has 1 sample. This bin has 84 samples.

14. NASSP Masters 5003F - Computational Astronomy - 2009 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.

15. NASSP Masters 5003F - Computational Astronomy - 2009 Natural weightingUniform weighting Natural vs uniform:

16. NASSP Masters 5003F - Computational Astronomy - 2009 Natural weightingUniform weighting The resulting dirty images:

17. NASSP Masters 5003F - Computational Astronomy - 2009 SNR of each visibility = 0.7%. Natural weightingUniform weighting But if we add in some noise...

18. NASSP Masters 5003F - Computational Astronomy - 2009 Tradeoff This sort of tradeoff, between increasing resolution on the one hand and sensitivity on the other, is unfortunately typical in interferometry.

19. NASSP Masters 5003F - Computational Astronomy - 2009 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 samples get weighted higher so that the average approaches the ideal. Densely packed samples are down-weighted.

20. NASSP Masters 5003F - Computational Astronomy - 2009 Uniform Tapered uniform Iterative best fit out- side 20-pixel radius Simulated e-Merlin data. 400 x 5 MHz channels; ν av = 6 GHz; t int = 10 s; δ = +30° Weighting schemes:

21. NASSP Masters 5003F - Computational Astronomy - 2009 ‘Dirty beam’ images (absolute values). 20 Iterative best fit out- side 20-pixel radius Tapered uniform Uniform

22. NASSP Masters 5003F - Computational Astronomy - 2009 Natural Uniform Optimized Natural (narrow-band) Natural Uniform Optimized for r>10 Comparison – slices through the DIs:

23. NASSP Masters 5003F - Computational Astronomy - 2009 r = 10 More on iterated weights:

24. NASSP Masters 5003F - Computational Astronomy - 2009 SNR of each visibility = 5. But real data is noisy…

25. NASSP Masters 5003F - Computational Astronomy - 2009 1.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. One could think of other ‘feathering’ schemes.