Investigate high ell in output GNILC-based dust maps

[zonca]

However, we still want to better understand the impact of the logpoltens transform on the map, it is useful to have a caveat about this in the paper and in the documentation. It is possible that the logpoltens anti-transform makes the map not band limited, so there is residual power at high ell that could impact analysis. Will open a separate issue to track this.

Originally posted by @zonca in #116 (comment)

[zonca]

Using presmooth functionality in #119, trying to understand if the GNILC output maps show worse behavior due to the non-linear logpoltens transform making the map not band-limited.

In this notebook I compare dust maps at 353 GHz that have been either presmoothed (i.e. small scales have been generated already smoothed) and postsmoothed (i.e. as usual, run modelling with a lot of power at high ell and smooth the results).

I try to assess that by comparing spectra of the maps to spectra after another trip to spherical harmonics space. I also compare norm in map space.

https://gist.github.com/c8b42c12e0eec32882c57c47d5ecdab5

It seems actually that doing presmoothing does not help at all. I think the problem is that the nonlinear transform transfers power to higher ells, with post-smooth, whatever happens is filtered partially out. Instead with presmooth, even if the templates are better behaved, there is anyway transfer of power to high ell and the outputs are very similar, actually a bit better with post-smooth.

I'll think about this a bit more, but we can probably proceed with what we have, the effect seems small and similar to the residuals that are intrinsic to spherical harmonics transforms between ell=2nside and 3nside.