Noise characterization for 3D RM-synth / RM-clean outputs

[AstroRipples]

Some tools for characterizing the noise in the RM cubes (those produced by RM-synthesis and/or RM-clean) would be useful for a number of purposes, e.g. source-finding, combating Ricean bias.

[PaddyLeahy]

Emma Alexander and I have noticed that the "noise" in RM cubes has very nasty properties, e.g. apparently randomly varying from pixel to pixel despite the maps being oversampled, and often with higher values than on-source. That is, sources of genuine polarized emission appear as islands of low polarized intensity surrounded by random spikes of much higher polarized intensity in off-source pixels. We attribute this to deriving spectra from (Q/I) and (U/I) where I is a non-significant quantity that frequently passes through zero. The solution to this would be to divide by a model power-law fitted to the I values rather than the raw values. We think this would also stabilise the RM-fitting where there is genuine signal of relatively low significance. A model power law would by definition not change sign. In regions of noise the amplitude will often be negative but that does not matter since there is actually no signal and we are only interested in the noise. On-source, a power-law fit would nearly always be an adequate approximation for data taken in a single band, covering a few hundred MHz. For fitting over larger spectral ranges the best approach would be a low-order polynomial in log-log space: the order of polynomial could be one of the input parameters. Obviously fitting a power law would require a non-linear least-squares fit, but this can be done efficiently by initially doing a linear least-squares fit in log-log space and using the results as the starting estimate for the non-linear fit.