rms noise, confusion and dynamic range in radio images

Question

I have been trying to understand imaging in radio astronomy. Below are some of my questions related to it and my understanding of their answers. I am not very confident about my understanding of them and it needs to be taken with a grain of a salt. If someone could validate my understanding of it and add corrections and elaborate explanations of them for my better understanding, that would be great:

1. What is RMS noise (mJy/beam) or image rms and how it relates to flux density and images? What are its causes? Why is it measured per beam? How is it related to sensitivity? My Understanding - Root mean square (RMS) noise (fluctuation level) is the total(average) noise level in the image which could be caused by antenna electronics as well as confusion. It seems to be a noise(fluctuations) with gaussian distribution and so RMS noise seems to be synonimous to the standard deviation of that distribution. Lower noise means better interferometer sensitivity because it can detect fainter sources with lower noise. Not sure why is it measured per beam though.

2. There are also quantities like 5σ or 7σ related to RMS noise, peak residual (mJy/beam), etc. What do they mean? My Understanding - 5σ or 7σ, etc. seems to be the Signal-to-Noise ratio of any source within the abeam/fov. Higher the brightness of a source within the beam, the less likely that it is due to the noise (random fluctuations) and more confident we are that it is a real source. I am not quite sure what peak residual is.

3. What is dynamic range? Why its greater value is better? My Understanding - The ratio of the brightest source in the field to the off-source rms in the image. The brighter the source, higher could be the dynamic range. Reduction in rms noise (probably with longer integration times) could also increase dynamic range. I am not sure why higher value of dynamic range is better. Is it better because it gives us better σ-levels (confidence) on the image content and thus makes image more accurate and reliable? What are other benefits of higher dynamic range and other ways to improve it? Do gaps in the uv coverage or beam size, resolution affect it?

4. What is confusion limit and its significance? My Understanding - Confusion is actually not the noise, it's noise like distribution of faint sky objects around the source within the beam/fov which obscure the target source. This noise cannot be removed with long integration times as it is real distribution of sky and is correlated. So, rms noise level of the image cannot go below confusion limit even with longer integration times.

Answer 1

1. RMS noise: Your understanding is mostly correct. RMS noise is the root mean square of fluctuations. Its square equals the sum of square of mean fluctuation and square of standard deviation of fluctuation. The mean is important because it typically encodes the background level (offset from zero). Why Jy/beam?
   • First, the beam: radio astronomers use beam, beam size, beam FWHM, and resolution interchangeably. Note that beams are typically assumed to be Gaussian, and the pixel size is always a few times smaller than the beam; resolution is NOT the same as the pixel size.
   • Now, Jy: If a detector receives one Watt per meter square of detector area per Hz, then the flux density of the source being observed is = 1 Jansky. If the source appears smaller than the beam, its flux density is just the peak value 1 Jansky. But what if it is resolved, ie it covers several beams? Then we need to integrate over its (angular) area to measure its flux density.
   • This is the reason why radio maps are almost always in Jy/beam: the final units are Jy and you obtain it by integrating over an area, hence the integrand must be Jy per solid angle, the relavent solid angle here being the beam area. (I guess that a more appropriate name would be "Jansky per beam area"). Noise, then, which must also be in the same units as the image units, is also given in Jy/beam.
   • Brightness temperatures are equivalent, and are actually easier to handle in some cases. The conversion between Jy/beam and brightness temperature depends on the frequency and the beam size naturally; it is a straight-forward application of Rayleigh-Jeans approximation. See this astropy page for example.
2. $n\sigma$: Yes, the higher the better. Peak residual is quite technical: it is the brightest feature that has not been deconvolved during the radio interferometric deconvolution process. A beginner's way to use this concept is the following: it is the brightest feature that may not be real, ie, the maximum level of noise in the image.
3. Dynamic range: You're right, $DR=\frac{\mathrm{peak}}{\mathrm{RMS}}$. Consider a field of just point sources. The peak of the brightest source does not change. So, to improve DR, we need to reduce RMS noise. Better $uv$-coverage means that we sampled the Fourier space better, so the image reconstruction is better, and hence, we're likely to see DR improvements. Resolution plays no role if all other things are kept constant (play around with VLA exposure calculator to see what I mean).
4. Confusion: Yep, that's correct. The number of sources usually has a power-law distribution with brightness if you're looking at a relatively empty field. At a given resolution, there are always many faint sources along the line of sight, which will all contribute to the noise. The only way to get around this limit is to use higher resolution.

Almost one year late, but I hope this helps at least one person :)