# Difference between revisions of "Astrophysical component separation"

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The aim of Sevem is to produce clean CMB maps at one or several frequencies by using a procedure based on template fitting. The templates are internal, i.e., they are constructed from Planck data, avoiding the need for external data sets, which usually complicates the analyses and may introduce inconsistencies. The method has been successfully applied to Planck simulations Leach et al., 2008 and to WMAP polarisation data Fernandez-Cobos et al., 2012. In the cleaning process, no assumptions about the foregrounds or noise levels are needed, rendering the technique very robust. | The aim of Sevem is to produce clean CMB maps at one or several frequencies by using a procedure based on template fitting. The templates are internal, i.e., they are constructed from Planck data, avoiding the need for external data sets, which usually complicates the analyses and may introduce inconsistencies. The method has been successfully applied to Planck simulations Leach et al., 2008 and to WMAP polarisation data Fernandez-Cobos et al., 2012. In the cleaning process, no assumptions about the foregrounds or noise levels are needed, rendering the technique very robust. | ||

− | The input maps used are all the Planck frequency channels. In particular, we have cleaned the 100, 143 GHz and 217 GHz maps using | + | The input maps used are all the Planck frequency channels. In particular for intensity, we have cleaned the 100, 143 GHz and 217 GHz maps using three templates constructed as the difference of the following Planck channels (smoothed to a common resolution): (30-44) GHz, (44-70) GHz, (545-353) GHz and 857 GHz (smoothed at the resolution of the 545 GHz channel). For polarization the clean maps are at 70, 100 and 143 GHz and the templates are: (30-44) GHz smoothed to a common resolution, (353-217) GHz smoothed at 10', (217-143) GHz and (217-100) GHz smoothed at 1 degree. Before the subtraction we perform in painting to reduce the point source contamination. |

− | + | A linear combination of the templates is then subtracted from the Planck sky map at the frequency to be cleaned, in order to produce the clean CMB. The coefficients of the linear combination are obtained by minimising the variance of the clean map outside a given mask. Although we exclude very contaminated regions during the minimization, the subtraction is performed for all pixels and, therefore, the cleaned maps cover the full-sky (although we expect that foreground residuals are present in the excluded areas). | |

− | + | The final CMB intensity map has then been constructed by combining the 143 and 217 GHz cleaned maps by weighting the maps in harmonic space taking into account the noise level, the resolution and a rough estimation of the foreground residuals of each map (obtained from realistic simulations). This final map has a resolution corresponding to a Gaussian beam of FWHM =5 arcminutes at nside=2048. The final CMB polarization map has been obtained by combining the 100 and 143 GHz clean maps with nside=1024 and resolution of 10 arc minutes. | |

− | |||

− | The final CMB map has then been constructed by combining the 143 and 217 GHz cleaned maps by weighting the maps in harmonic space taking into account the noise level, the resolution and a rough estimation of the foreground residuals of each map (obtained from realistic simulations). This final map has a resolution corresponding to a Gaussian beam of FWHM =5 arcminutes. | ||

===SMICA=== | ===SMICA=== |

## Revision as of 12:01, 29 January 2015

## Contents

## CMB and foreground separation[edit]

See the Component Separation paper Planck-2013-XII^{[1]} for details.

### NILC[edit]

NILC is a linear method for combining the input channels. It implements an ILC with weighting coefficients varying over the sky and over the multipole range up to

and it does so using 'needlets' which are spherical wavelets. A special procedure is used for processing the coarsest needlet scale which contains the large scale multipoles.### SEVEM[edit]

The aim of Sevem is to produce clean CMB maps at one or several frequencies by using a procedure based on template fitting. The templates are internal, i.e., they are constructed from Planck data, avoiding the need for external data sets, which usually complicates the analyses and may introduce inconsistencies. The method has been successfully applied to Planck simulations Leach et al., 2008 and to WMAP polarisation data Fernandez-Cobos et al., 2012. In the cleaning process, no assumptions about the foregrounds or noise levels are needed, rendering the technique very robust.

The input maps used are all the Planck frequency channels. In particular for intensity, we have cleaned the 100, 143 GHz and 217 GHz maps using three templates constructed as the difference of the following Planck channels (smoothed to a common resolution): (30-44) GHz, (44-70) GHz, (545-353) GHz and 857 GHz (smoothed at the resolution of the 545 GHz channel). For polarization the clean maps are at 70, 100 and 143 GHz and the templates are: (30-44) GHz smoothed to a common resolution, (353-217) GHz smoothed at 10', (217-143) GHz and (217-100) GHz smoothed at 1 degree. Before the subtraction we perform in painting to reduce the point source contamination.

A linear combination of the templates is then subtracted from the Planck sky map at the frequency to be cleaned, in order to produce the clean CMB. The coefficients of the linear combination are obtained by minimising the variance of the clean map outside a given mask. Although we exclude very contaminated regions during the minimization, the subtraction is performed for all pixels and, therefore, the cleaned maps cover the full-sky (although we expect that foreground residuals are present in the excluded areas).

The final CMB intensity map has then been constructed by combining the 143 and 217 GHz cleaned maps by weighting the maps in harmonic space taking into account the noise level, the resolution and a rough estimation of the foreground residuals of each map (obtained from realistic simulations). This final map has a resolution corresponding to a Gaussian beam of FWHM =5 arcminutes at nside=2048. The final CMB polarization map has been obtained by combining the 100 and 143 GHz clean maps with nside=1024 and resolution of 10 arc minutes.

### SMICA[edit]

A linear method, SMICA reconstructs a CMB map as a linear combination in the harmonic domain of input frequency maps with weights that depend on multipole . Given the vector of spherical harmonic coefficients for the input maps, it computes coefficients for the CMB map as

where the CMB with minimum variance. This is achieved with

vector which contains the multipole-dependent weights is built to offer unit gain to thewhere vector CMB evaluated at each channel (allowing for possible inter-channel recalibration factors) and is the spectral covariance matrix of . Taking in Eq. \ref{eq:smica:w} to be the sample spectral covariance matrix of the observations:

is the emission spectrum of thewould implement a simple harmonic-domain ILC. This is not what SMICA does. As discussed below, we instead use a model

and determine the covariance matrix to be used in Eq. \ref{eq:smica:w} by fitting to . This is done in the maximum likelihood sense for stationary Gaussian fields, yielding the best fit model parameters θ as

SMICA models the data is a superposition of CMB, noise and
foregrounds. The latter are not parametrically modelled; instead, we
represent the total foreground emission by templates
with arbitrary frequency spectra, angular spectra and correlations:

where CMB, is a matrix, is a positive matrix, and is a diagonal matrix representing the noise power spectrum. The parameter vector contains all or part of the quantities in Eq. (5).

is the angular power spectrum of the

The above equations summarize the founding principles of SMICA; its
actual operation depends on a choice for the spectral model
and on several
implementation-specific details.

The actual implementation of SMICA includes the following steps:

- Inputs
- All nine Planck frequency channels from 30 to 857 GHz, harmonically transformed up to .
- Fit
- In practice, the SMICA fit,i.e.,the minimization of Eq. (4), is conducted in three successive steps: We first estimate the CMB spectral law by fitting all model parameters over a clean fraction of sky in the range and retaining the best fit value for vector . In the second step, we estimate the foreground emissivity by fixing a to its value from the previous step and fitting all the other parameters over a large fraction of sky in the range and retaining the best fit values for the matrix . In the last step, we fit all power spectrum parameters; that is, we fix and to their previously found values and fit for each and at each .
- Beams
- The discussion thus far assumes that all input maps have the same resolution and effective beam. Since the observed maps actually vary in resolution, we process the input maps in the following way. To the HEALPix grid with , the CMB sky multipole actually contributes , where is the pixel window function for the grid at . Seeking a final CMB map at 5-arcmin resolution, the highest resolution of Planck, we work with input spherical harmonics re-beamed to 5 arcmins, ; that is, SMICA operates on vectors with entries , where is a 5 arcmin Gaussian beam function. By construction, SMICA then produces an CMB map with an effective Gaussian beam of 5 arcmin (without the pixel window function). -th input map with effective beam and sampled on an
- Pre-processing
- We start by fitting point sources with SNR > 5 in the PCCS catalogue in each input map. If the fit is successful, the fitted point source is removed from the map; otherwise it is masked and the hole in-painted. This is done at all frequencies but 545 and 857 GHz, where all point sources with SNR > 7.5 are masked and in-painted.
- Masking and in-painting
- In practice, SMICA uses a small Galactic mask leaving 97% of the sky. We deliver a full-sky CMB map in which the masked pixels (Galactic and point-source) are replaced by a constrained Gaussian realization.
- Binning
- In our implementation, we use binned spectra.
- High
- Since there is little point trying to model the spectral covariance at high multipoles, because the sample estimate is sufficient, SMICA implements a simple harmonic ILC at ; that is, it applies the filter (Eq. 2) with .

Viewed as a filter, SMICA can be summarized by the weights

applied to each input map as a function of multipole. In this sense, SMICA is strictly equivalent to co-adding the input maps after convolution by specific axi-symmetric kernels directly related to the corresponding entry of . The SMICA weights used here are shown in figure below for input maps in units of K . They show, in particular, the (expected) progressive attenuation of the lowest resolution channels with increasing multipole.### Commander-Ruler[edit]

The Commander-Ruler (C-R) approach implements Bayesian component separation in pixel space, fitting a parametric model to the data by sampling the posterior distribution for the model parameters. For computational reasons, the fit is performed in a two-step procedure: First, both foreground amplitudes and spectral parameters are found at low-resolution using MCMC/Gibbs sampling algorithms (Jewell et al. 2004; Wandelt et al. 2004; Eriksen et al. 2004, 2007, 2008). Second, the amplitudes are recalculated at high resolution by solving the generalized least squares system (GLSS) per pixel with the spectral parameters fixed to the their values from the low-resolution run. For the CMB-oriented analysis presented in this paper, we only use the seven lowest Planck frequencies, i.e., from 30 to 353 GHz. We first downgrade each frequency map from its native angular resolution to a common resolution of 40 arcminutes and re-pixelize at HEALPix N = 256. Second, we set the monopoles and dipoles for each frequency band using a method that locally conserves spectral indices (Wehus et al. 2013, in preparation). We approximate the effective instrumental noise as white with an RMS per pixel given by the Planck scanning pattern and an amplitude calibrated by smoothing simulations of the instrumental noise including correlations to the same resolution. For the high-resolution analysis, the important pre-processing step is the upgrading of the effective low-resolution mixing matrices to full Planck resolution: this is done by repixelizing from N = 256 to 2048 in harmonic space, ensuring that potential pixelization effects from the low-resolution map do not introduce sharp boundaries in the high-resolution map.

## References[edit]

- ↑
**Planck 2013 results. XI. Component separation**, Planck Collaboration, 2014, A&A, 571, A11

Cosmic Microwave background

Full-Width-at-Half-Maximum

(Hierarchical Equal Area isoLatitude Pixelation of a sphere, <ref name="Template:Gorski2005">**HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere**, K. M. Górski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, M. Bartelmann, ApJ, **622**, 759-771, (2005).