Problems with PS2HAT estimator at low ell
December 7 2018, Clem Pryke
Recently we have been running map based sims using three masks Sims_with_nominal_Chile_and_Pole_masks. The sigma(r) results perhaps look a little unexpected - the Pole mask delivers higher sigma(r) than the circular even at low A_L despite having smaller area. It was noted that in Sims_with_nominal_Chile_and_Pole_masks_II the noise spectra of the Pole mask rises above that for the circular at low ell. This seems odd and might increase sigma(r). This was backed up by some toy sims.
In the plot below I take a look at the raw per-ell spectra from S2HAT interface code which Justin wrote.
This is the mean over the first 10 realizations and shows C_ell (no ell^2 scaling). In the lower right we see again the result that Pole crosses over circular (red crosses over blue) at low ell. Recall that in these sims the effect of timestream filtering is approximated by cutting off the signal and noise input spectra for ell<30 - in most of the spectra we see this clearly. However, the PS2HAT estimator gives a lot of spurious low ell response and in the lower right this makes the Pole noise spectrum (red) look like a continuous 1/f rise to low ell. We certainly need the pureB estimator as the dashed red shows - without it the spurious low ell response goes away but in the lower left panel we see severe E to B leakage at intermediate ell.
Looking at the PS2HAT paper arxiv/0903.2350 (e.g. fig9) it becomes clear that this behavior is expected. For those with access to the BK logbook Justin made some preliminary posts  and  where we can already see this.
I want to note that this low ell leakage is not a big deal. It should be correctly handled by the bandpower window functions which Ben plots here in the sense that it should not produce bias. (I think it appears as the increased high side shoulder in the BPWF curves red versus blue/green.) This effect can however, produce enhanced fluctuation of lower bandpowers and hence increased sigma(r).
Probably we can reduce it by increased mask smoothing or other manipulation. But any further deviation from nhits masking also degrades the statistical power. The right solution is probably a different estimator - e.g. the matrix purification used for BICEP/Keck.