NoiseCovarMatrices

Introduction

The Planck low resolution frequency maps are provided with a pixel-pixel noise covariance matrix. The matrix is required for maximum likelihood analysis of CMB anisotropies in the maps but also for low resolution component separation.

The first step in calculating the noise matrices is estimating the detector noise. We do this for each detector at roughly one day intervals. The process is

1. time-ordered information (TOI) is cleaned of signal by interpolating the full frequency, full mission map to the sample positions on the sky.
2. Estimate the auto covariance function of the TOI without samples that either have their quality flag raised or that fall within the galactic or point source mask
3. Fourier transform the auto covariance function into a power spectral density (PSD)
4. Reduce realization noise in the PSD by fitting and analytical model

The noise PSDs feed into two noise covariance estimation codes. The LFI noise covariance is estimated using a module integrated in the Madam mapmaking code. It directly estimates the posterior pixel-pixel covariance in the presence of correlated noise residuals after destriping. The HFI noise covariance approach is different because of the longer baseline (offset) used in HFI destriping. We break the HFI residual noise covariance into offset-offset (between rings) and short time scale (within ring) components. The short time scale component is readily available as the noise PSD while the offset-offset covariance requires calculating over all of the offset-offset crossing points and inverting and binning the resulting offset covariance matrix.

Method

The LFI noise covariance follows from the destriping principle, where the correlated noise is approximated by correlated baseline offsets:

[math] d = Pm + Fa + n, [/math]

where the data vector, [math]d[/math], is a linear combination of the sky signal, [math]Pm[/math], baseline offsets, [math]Fa[/math], and white noise, [math]n[/math]. Matrices [math]P[/math] and [math]F[/math] scan the sky map, [math]m[/math] and the baseline offsets, [math]a[/math], into time domain.

The residual pixel-pixel noise covariance under the generalized destriping principle (MADAM) is then

[math] N^{-1} = P^T \mathcal N_w^{-1} P - P^T \mathcal N_w^{-1} F ( \mathcal N_a^{-1} + F^T\mathcal N_w^{-1}F ) F^T N_w^{-1} P^T.[/math]

The short time scale noise correlations are unaffected by the long baseline destriping used in HFI mapmaking. This means that the resulting pixel-pixel covariance can be evaluated simply by binning the their time domain covariance into pixel domain:

[math] N = B \mathcal N B^T [/math],

where [math]B[/math] is the map binning operator, typically constructed from the pointing matrix, [math]P[/math] and the detector noise weights contained in the diagonal white noise covariance matrix, [math]\mathcal N_w[/math]:

[math] B = (P^T \mathcal N_w^{-1} P)^{-1} P^T \mathcal N_w^{-1}[/math].