Difference between revisions of "CMB and astrophysical component maps"
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Production of the Q and U polarisation maps. | Production of the Q and U polarisation maps. | ||
− | The production of the Q and U maps is similar to the production of the intensity map. The SMICA pipeline uses all the 7 polarized Planck channels. After point source masking and diffusive inpainting, the E and B modes are computed and combined to produce E and B modes of the CMB map. Those combined modes are then used to synthesize the U and Q CMB maps. The E and B parts of the input frequency maps being processed jointly, there are, at each multipole, 2*7=14 coefficients (weights) defined to produce the E modes of the CMB map and as many to produce the B part. The weights are displayed in the figure below. | + | The production of the Q and U maps is similar to the production of the intensity map. The SMICA pipeline uses all the 7 polarized Planck channels. After point source masking and diffusive inpainting, the E and B modes are computed and combined to produce E and B modes of the CMB map. Those combined modes are then used to synthesize the U and Q CMB maps. The E and B parts of the input frequency maps being processed jointly, there are, at each multipole, 2*7=14 coefficients (weights) defined to produce the E modes of the CMB map and as many to produce the B part. The weights are displayed in the figure below. The Q and U maps were originally produced at Nside=2048 with a 5-arc-minute resolution, but were downgraded to Nside=1024 with a 10 arc-minute resolution for this release. |
− | |||
Revision as of 17:04, 30 January 2015
Contents
Overview[edit]
This section describes the maps of astrophysical components produced from the Planck data. These products are derived from some or all of the nine frequency channel maps described above using different techniques and, in some cases, using other constraints from external data sets. Here we give a brief description of the product and how it is obtained, followed by a description of the FITS file containing the data and associated information. All the details can be found in Planck-2013-XII[1].
CMB maps[edit]
CMB maps have been produced by the SMICA SEVEM, NILC and COMMANDER pipelines, which are described in the CMB and foreground separation section and also in Section 3 and Appendices A-D of Planck-2013-XII[1] and references therein.. For each pipeline we provide:
- Full-mission CMB intensity map, confidence mask and beam transfer function.
- Full-mission high-pass filtered CMB polarisation map,
- A confidence mask.
- A beam transfer function.
In addition, and for characterisation purposes, there are six other sets of maps from three data splits: first/second half-ring, odd/even years and first/second half-mission. And for each of these data splits we provide half-sum and half-difference maps. The half-difference maps can be used to provide an approximate noise estimate for the full mission, but they should be used with caution. Each split has caveats in this regard: there are noise correlations between the half-ring maps, and missing pixels in the other splits. The Intensity maps are provided at Nside = 2048, at 5 arcmin resolution, while the Polarisation ones are provided at Nside = 1024 at 10 arcmin resolution. All maps are in units of Kcmb.
These maps can be found in the files
- COM_CMB_IQU-{pipeline}-field-{Int/Pol}_Nside_R2.00.fits.
The Int files have two extensions, for the Intensity maps and the beam transfer function, the Pol files have three extensions, for Q and U maps, and for the beam transfer function. For a complete description of the data structure, see the below; the content of the first extensions is illustrated and commented in the table below.
The gallery below shows the Intensity, noise from half-mission, half-difference, and confidence mask for the four pipelines, in the order SMICA, SEVEM, NILC and COMMANDER, from top to bottom. The Intensity maps scale is [–500.+500] μK, and the noise are between [–25,+25] μK. We do not show the Q and U maps since they have no significant visible structure to contemplate.
Product description[edit]
SMICA[edit]
- Principle
- SMICA produces CMBs map by linearly combining all Planck input channels with multipole-dependent weights. It includes multipoles up to . Temperature and polarization maps are produced independently.
- Resolution (effective beam)
- The SMICA intensity map has an effective beam window function of 5 arc-minutes which is truncated at and is not deconvolved from the pixel window function. Thus the delivered beam window function is the product of a Gaussian beam at 5 arcminutes and the pixel window function for Nside=2048.
- The SMICA Q and U maps are obtained similarly but are produced at Nside=1024 with an effective beam of 10 arc-minutes (to be multiplied by the pixel window function, as for the intensity map).
- Confidence mask
- A confidence mask is provided which excludes some parts of the Galactic plane, some very bright areas and the masked point sources. This mask provides a qualitative (and subjective) indication of the cleanliness of a pixel. See section below detailing the production process.
NILC (done by CB, checks with producers in progress)[edit]
- Principle
- The Needlet-ILC (hereafter NILC) CMB map is constructed both in total intensity as well as polarization, Q and U Stokes parameters. For total intensity, all Planck frequency channels are included. For polarization, all polarization sensitive frequency channels are included, from 30 to 353 GHz. The solution, for T, Q and U is obtained by applying the Internal Linear Combination (ILC) technique in needlet space, that is, with combination weights which are allowed to vary over the sky and over the whole multipole range.
- Resolution (effective beam)
- The spectral analysis, and estimation of the NILC coefficients, is performed up to a maximum FWHM=5 arcminutes. . The effective beam is equivalent of a Gaussian circular beam with
- Confidence mask
- The same procedure is followed by SMICA and NILC for producing confidence masks, though with different parametrizations. A low resolution smoothed version of the NILC map, noise subtracted, is thresholded to 73.5 squared micro-K for T, and 6,75 squared micro-K for Q and U.
SEVEM[edit]
- Principle
- The aim of SEVEM is to produce clean CMB maps at one or several frequencies by using a procedure based on template fitting in real space. 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. In the cleaning process, no assumptions about the foregrounds or noise levels are needed, rendering the technique very robust. The single frequency clean maps are then combined to obtain the final CMB map.
- Resolution
- For intensity the clean CMB map is constructed up to a maximum at Nside=2048 and at the standard resolution of 5 arcminutes (Gaussian beam).
- For polarization the clean CMB map is produced at Nside=1024 with a resolution of 10 arcminutes (Gaussian beam) and a maximum .
- Confidence masks
- The confidence masks cover the most contaminated regions of the sky, leaving approximately 85 per cent of useful sky for intensity and 80 per cent for polarization.
COMMANDER-Ruler[edit]
COMMANDER-Ruler is the Planck software implementing a pixel based parametric component separation. Amplitude of CMB and the main diffuse foregrounds along with the relevant spectral parameters for those (see below in the Astrophysical Foreground Section for the latter) are parametrized and fitted in single MCMC chains conducted at Nside=256 using COMMANDER, implementing a Gibbs Sampling. The CMB amplitude which is obtained in these runs corresponds to the delivered low resolution CMB component from COMMANDER-Ruler which has a FWHM of 40 arcminutes. The sampling of the foreground parameters is applied to the data at full resolution for obtaining the high resolution CMB component from Ruler which is available on the PLA. In the Planck Component Separation paper Planck-2013-XII[1]additional material is discussed, specifically concerning the sky region where the solutions are reliable, in terms of chi2 maps. The products mainly consist of:
- Maps of the Amplitudes of the CMB at low resolution, Nside=256, along with the standard deviations of the outputs, beam profiles derived from the production process.
- Maps of the CMB amplitude, along with the standard deviations, at high resolution, Nside=2048, beam profiles derived from the production process.
- Mask obtained on the basis of the precision in the fitting procedure; the thresholding is evaluated through the COMMANDER-Ruler likelihood analysis and excludes 13% of the sky, see Planck-2013-XII[1].
Production process[edit]
SMICA[edit]
Production of the intensity map.
- 1) Pre-processing
- Before computing spherical harmonic coefficients, all input maps undergo a pre-processing step to deal with regions of very strong emission (such as the Galactic center) and point sources. The point sources with SNR > 5 in the PCCS catalogue are fitted 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 is filled in by a simple diffusive process to ensure a smooth transition and mitigate spectral leakage. The diffusive inpainting process is also applied to some regions of very strong emissions. This is done at all frequencies but 545 and 857 GHz, here all point sources with SNR > 7.5 are masked and filled-in similarly.
- 2) Linear combination
- The nine pre-processed Planck frequency channels from 30 to 857 GHz are harmonically transformed up to and co-added with multipole-dependent weights as shown in the figure.
- 3) Post-processing
- A confidence mask is determined (see the Planck paper) and all regions which have been masked in the pre-processing step are added to it.
Production of the Q and U polarisation maps.
The production of the Q and U maps is similar to the production of the intensity map. The SMICA pipeline uses all the 7 polarized Planck channels. After point source masking and diffusive inpainting, the E and B modes are computed and combined to produce E and B modes of the CMB map. Those combined modes are then used to synthesize the U and Q CMB maps. The E and B parts of the input frequency maps being processed jointly, there are, at each multipole, 2*7=14 coefficients (weights) defined to produce the E modes of the CMB map and as many to produce the B part. The weights are displayed in the figure below. The Q and U maps were originally produced at Nside=2048 with a 5-arc-minute resolution, but were downgraded to Nside=1024 with a 10 arc-minute resolution for this release.
![](/planck-legacy-archive/images/thumb/7/7c/Smica_filterEB_dx11.png/600px-Smica_filterEB_dx11.png)
NILC (done by CB, check by producers in progress)[edit]
- Pre-processing
- All sky frequency maps are deconvolved using the DPC beam transfer function provided, and re-convolved with a 5 arcminutes FWHM circular Gaussian beam. In polarization, prior to the smoothing process, all sky E and B maps are derived from Q and U using standard HEALPix tools from each individual frequency channels
- Linear combination
- Pre-processed input frequency maps are decomposed in needlet coefficients, specified in the Appendix B of the Planck A11 paper, with shape given by Table B.1. Minimum variance coefficients are then obtained, using all channels for T, from 30 to 353 for E and B.
- Post-processing
- E and B maps are re-combined into Q and U products using standard HEALPix tools.
SEVEM[edit]
Usually we construct our templates by subtracting two close Planck frequency channel maps, after first smoothing them to a common resolution to ensure that the CMB signal is properly removed. A linear combination of the templates is then subtracted from (hitherto unused) map d to produce a clean CMB map at that frequency. This is done in real space at each position on the sky: where is the number of templates. The coefficients are obtained by minimising the variance of the clean map outside a given mask. Note that the same expression applies for I, Q and U. 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).
It should be stressed that the method is very fast and permits the generation of thousands of simulations to characterize the statistical properties of the outputs, a critical need for many cosmological applications. The final CMB map retains the angular resolution of the original frequency map.
There are several possible configurations of SEVEM with regard to the number of frequency maps which are cleaned or the number of templates that are used in the fitting. Note that the production of clean maps at different frequencies is of great interest in order to test the robustness of the results. Therefore, to define the best strategy, one needs to find a compromise between the number of maps that can be cleaned independently and the number of templates that can be constructed.
- Intensity
For the CMB intensity map, we have cleaned the 100 GHz, 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), (44-70), (545-353) and 857 as the fourth template. First of all, the six frequency channels which are going to be used to construct templates are inpainted at the point source positions detected using the Mexican Hat Wavelet algorithm (Planck Collaboration A35 2014). The size of the holes to be inpainted is determined taking into account the beam size of the channel as well as the flux of each source. The inpainting algorithm is based on simple diffuse inpainting, which fills one pixel with the mean value of the neighbouring pixels in an iterative way. To avoid inconsistencies when subtracting two channels, each frequency map is inpainted on the sources detected in that map and on the second map (if any) used to construct the template. Then the maps are smoothed to a common resolution (the first channel in the subtraction is smoothed with the beam of the second map and viceversa). For the 857 GHz template, we simply filter this map with the beam from 545 GHz (this is for comparison with the previous pipeline, where the 857 GHz was smoothed at this resolution when using it to construct the 857–545 template).
The coefficients are obtained outside the analysis mask, that covers the 1 per cent brightest emission of the sky as well as point sources detected at all frequency channels. Once the maps are cleaned, each of them is inpainted on the point sources positions detected at that (raw) channel. Then, the MHW algorithm is run again, now on the clean maps. A relatively small number of new sources are found and are also inpainted at each channel. The resolution of the clean map is the same as that of the raw map. Our final CMB map has then been constructed by combining the 143 and 217 GHz 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.
The confidence mask is produced by looking at differences between three different SEVEM CMB reconstructions, leaving a suitable sky fraction of approximately 85 per cent.
- Polarization
To clean the polarization maps, a procedure similar to the one used for intensity data is applied to the Q and U maps independently. In particular, we clean the 70, 100 and 143 GHz using four templates: 30-44 (after being convolved with the beam of each other), 353-217 (smoothed at 10' resolution) and 217-143 and 217-100 (both at 1 degree resolution). Conversely to the intensity case and due to the lower availability of frequency channels, it becomes necessary to use the maps to be cleaned as part of one of the templates: the 100 GHz map is used in the 217-100 template to clean the 143 GHz one and the 143 GHz map is used in the 217-143 template to clean the 100 GHz one, making the clean maps less independent between them than in the intensity case.
The linear coefficients are estimated independently for Q and U outside a mask, that covers point sources and the 3 per cent brightest Galactic emission. Once the maps have been cleaned, inpainting of the point sources detected at each map is carried out. The size of the holes to be inpainted takes into account the additional smoothing of the 100 and 143 GHz maps. The 100 and 143 GHz clean maps are then combined in harmonic space, using E and B decomposition, to produce the final CMB maps for the Q and U components at a resolution of 10′ (Gaussian beam) for a HEALPix parameter nside = 1024. Each map is weighted taking into account its corresponding noise level at each multipole. Finally, before applying the post-processing HPF to the clean polarization data, the region with the brightest Galactic residuals is inpainted (5 per cent of the sky).
The confidence mask includes all the pixels above a given threshold, the CO emission and those pixels more affected by the high-pass filtering, leaving a useful sky fraction os approximately 80 per cent.
COMMANDER-Ruler[edit]
The production process consist in low and high resolution runs according to the description above.
- Low Resolution Runs
- Same as the Astrophysics Foregrounds Section below; The CMB amplitude is fitted along with the other foreground parameters and constitutes the CMB Low Resolution Rendering which is in the PLA.
- Ruler Runs
- the sampling at high resolution is used to infer the probability distribution of spectral parameters which is exploited at full resolution in order to obtain the High Resolution CMB Rendering which is in the PLA.
Inputs[edit]
The input maps are the sky temperature maps described in the Sky temperature maps section. SMICA and SEVEM use all the maps between 30 and 857 GHz; NILC uses the ones between 44 and 857 GHz. Commander-Ruler uses frequency channel maps from 30 to 353 GHz.
File names and structure[edit]
The FITS files corresponding to the three CMB products are the following:
COM_CMB_IQU-{method}-field-{Int,Pol}_Nside_R2.nn.fits
where method is mica, nilc, sevem, or commander, and Int and Pol indicate whether the file contains the temperature (Int) or the polarisation (Pol) maps. For this release the temperature maps are provided at Nside = 2048, and the polarisation maps at Nside = 1024.
The files contain
- a minimal primary extension with no data;
- one or two BINTABLE data extensions with a table of Npix lines by 14 columns in which the first 13 columns is a CMB maps produced from the full or a subset of the data, as described in the table below, and the last column in a confidence mask. There is a single extension for Int files, and two, for Q and U, for Pol files.
- a BINTABLE extension containing the beam window function.
Ext. 1. or 2. EXTNAME = COMP-MAP (BINTABLE) | |||
---|---|---|---|
Column Name | Data Type | Units | Description |
I or Q or U | Real*4 | uK_cmb | I or Q or U map |
HM1 | Real*4 | uK_cmb | Half-miss 1 |
HM2 | Real*4 | uK_cmb | Half-miss 2 |
YR1 | Real*4 | uK_cmb | Year 1 |
YR2 | Real*4 | uK_cmb | Year 2 |
HR1 | Real*4 | uK_cmb | Half-ring 1 |
HR2 | Real*4 | uK_cmb | Half-ring 2 |
HMHS | Real*4 | uK_cmb | Half-miss, half sum |
HMHD | Real*4 | uK_cmb | Half-miss, half diff |
YRHS | Real*4 | uK_cmb | Year, half sum |
YRHD | Real*4 | uK_cmb | Year, half diff |
HRHS | Real*4 | uK_cmb | Half-ring half sum |
HRHD | Real*4 | uK_cmb | Half-ring half diff |
MASK | BYTE | Confidence mask | |
Keyword | Data Type | Value | Description |
AST-COMP | String | CMB | Astrophysical compoment name |
PIXTYPE | String | HEALPIX | |
COORDSYS | String | GALACTIC | Coordinate system |
POLCCONV | String | COSMO | Polarization convention |
ORDERING | String | NESTED | Healpix ordering |
NSIDE | Int | 2048 | Healpix Nside |
METHOD | String | name | Cleaning method (smica/nilc/sevem/commander) |
Keyword | Data Type | Value | Description |
PIXTYPE | String | HEALPIX | |
COORDSYS | String | GALACTIC | Coordinate system |
ORDERING | String | NESTED | Healpix ordering |
NSIDE | Int | 1024 | Healpix Nside |
METHOD | String | name | Cleaning method (SMICA/NILC/SEVEM) |
Ext. 2. or 3. EXTNAME = BEAM_WF (BINTABLE) | |||
Column Name | Data Type | Units | Description |
BEAM_WF | Real*4 | none | The effective beam window function, including the pixel window function. See Note 1. |
Keyword | Data Type | Value | Description |
LMIN | Int | value | First multipole of beam WF |
LMAX | Int | value | Lsst multipole of beam WF |
METHOD | String | name | Cleaning method (SMICA/NILC/SEVEM/COMMANDER-Ruler) |
Notes:
- The beam window function given here includes the pixel window function for the Nside=2048 pixelization. It means that, ideally, .
Astrophysical foregrounds from parametric component separation[edit]
We describe diffuse foreground products for the Planck 2013 release. See Planck Component Separation paper Planck-2013-XII[1] for a detailed description and astrophysical discussion of those.
Product description[edit]
- Low frequency foreground component
- The products below contain the result of the fitting for one foreground component at low frequencies in Planck bands,along with its spectral behavior parametrized by a power law spectral index. Amplitude and spectral indeces are evaluated at N$_\rm{side}$ 256 (see below in the production process), along with standard deviation from sampling and instrumental noise on both. An amplitude solution at N$_\rm{side}$=2048 is also given, along with standard deviation from sampling and instrumental noise as well as solutions on halfrings. The beam profile associated to this component is also provided as a secondary Extension in the N$_\rm{side}$ 2048 product.
- Thermal dust
- The products below contain the result of the fitting for one foreground component at high frequencies in Planck bands, along with its spectral behavior parametrized by temperature and emissivity. Amplitude, temperature and emissivity are evaluated at N$_\rm{side}$ 256 (see below in the production process), along with standard deviation from sampling and instrumental noise on all of them. An amplitude solution at N$_\rm{side}$=2048 is also given, along with standard deviation from sampling and instrumental noise as well as solutions on halfrings. The beam profile associated to this component is provided.
- Sky mask
- The delivered mask is defined as the sky region where the fitting procedure was conducted and the solutions presented here were obtained. It is made by masking a region where the Galactic emission is too intense to perform the fitting, plus the masking of brightest point sources.
Production process[edit]
CODE: COMMANDER-RULER. The code exploits a parametrization of CMB and main diffuse foreground observables. The naive resolution of input frequency channels is reduced to N$_\rm{side}$=256 first. Parameters related to the foreground scaling with frequency are estimated at that resolution by using Markov Chain Monte Carlo analysis using Gibbs sampling. The foreground parameters make the foreground mixing matrix which is applied to the data at full resolution in order to obtain the provided products at N$_\rm{side}$=2048. In the Planck Component Separation paper Planck-2013-XII[1] additional material is discussed, specifically concerning the sky region where the solutions are reliable, in terms of chi2 maps.
Inputs[edit]
Nominal frequency maps at 30, 44, 70, 100, 143, 217, 353 GHz (LFI 30 GHz frequency maps, LFI 44 GHz frequency maps and LFI 70 GHz frequency maps, HFI 100 GHz frequency maps, HFI 143 GHz frequency maps,HFI 217 GHz frequency maps and HFI 353 GHz frequency maps) and their II column corresponding to the noise covariance matrix. Halfrings at the same frequencies. Beam window functions as reported in the LFI and HFI RIMO.
Related products[edit]
None.
File names[edit]
- Low frequency component at N$_\rm{side}$ 256: COM_CompMap_Lfreqfor-commrul_0256_R1.00.fits
- Low frequency component at N$_\rm{side}$ 2048: COM_CompMap_Lfreqfor-commrul_2048_R1.00.fits
- Thermal dust at N$_\rm{side}$ 256: COM_CompMap_dust-commrul_0256_R1.00.fits
- Thermal dust at N$_\rm{side}$ 2048: COM_CompMap_dust-commrul_2048_R1.00.fits
- Mask: COM_CompMap_Mask-rulerminimal_2048_R1.00.fits
Meta Data[edit]
Low frequency foreground component[edit]
Low frequency component at Nside = 256[edit]
File name: COM_CompMap_Lfreqfor-commrul_0256_R1.00.fits
- Name HDU -- COMP-MAP
The Fits extension is composed by the columns described below:
Column Name | Data Type | Units | Description |
---|---|---|---|
I | Real*4 | uKCMB | Intensity |
I_stdev | Real*4 | uKCMB | standard deviation of intensity |
Beta | Real*4 | effective spectral index | |
B_stdev | Real*4 | standard deviation on the effective spectral index |
- Notes
- Comment: The Intensity is normalized at 30 GHz
- Comment: The intensity was estimated during mixing matrix estimation
Low frequency component at Nside = 2048[edit]
- File name: COM_CompMap_Lfreqfor-commrul_2048_R1.00.fits
- Name HDU -- COMP-MAP
The Fits extension is composed by the columns described below:
Column Name | Data Type | Units | Description |
---|---|---|---|
I | Real*8 | uKCMB | Intensity |
I_stdev | Real*8 | uKCMB | standard deviation of intensity |
I_hr1 | Real*8 | uKCMB | Intensity on half ring 1 |
I_hr2 | Real*8 | uKCMB | Intensity on half ring 2 |
- Notes
- Comment: The intensity was computed after mixing matrix application
- Name HDU -- BeamWF
The Fits second extension is composed by the columns described below:
Column Name | Data Type | Units | Description |
---|---|---|---|
BeamWF | Real*4 | beam profile |
- Notes
- Comment: Beam window function used in the Component separation process
Thermal dust[edit]
Thermal dust component at Nside=256[edit]
- File name: COM_CompMap_dust-commrul_0256_R1.00.fits
- Name HDU -- COMP-MAP
The Fits extension is composed by the columns described below:
Column Name | Data Type | Units | Description |
---|---|---|---|
I | Real*4 | MJy/sr | Intensity |
I_stdev | Real*4 | MJy/sr | standard deviation of intensity |
Em | Real*4 | emissivity | |
Em_stdev | Real*4 | standard deviation on emissivity | |
T | Real*4 | uKCMB | temperature |
T_stdev | Real*4 | uKCMB | standard deviation on temerature |
- Notes
- Comment: The intensity is normalized at 353 GHz
Thermal dust component at Nside=2048[edit]
File name: COM_CompMap_dust-commrul_2048_R1.00.fits
- Name HDU -- COMP-MAP
The Fits extension is composed by the columns described below:
Column Name | Data Type | Units | Description |
---|---|---|---|
I | Real*8 | MJy/sr | Intensity |
I_stdev | Real*8 | MJy/sr | standard deviation of intensity |
I_hr1 | Real*8 | MJy/sr | Intensity on half ring 1 |
I_hr2 | Real*8 | MJy/sr | Intensity on half ring 2 |
- Name HDU -- BeamWF
The Fits second extension is composed by the columns described below:
Column Name | Data Type | Units | Description |
---|---|---|---|
BeamWF | Real*4 | beam profile |
- Notes
- Comment: Beam window function used in the Component separation process
Sky mask[edit]
File name: COM_CompMap_Mask-rulerminimal_2048.fits
- Name HDU -- COMP-MASK
The Fits extension is composed by the columns described below:
Column Name | Data Type | Units | Description |
---|---|---|---|
Mask | Real*4 | Mask |
References[edit]
Flexible Image Transfer Specification
Cosmic Microwave background
Full-Width-at-Half-Maximum
Planck Legacy Archive
Data Processing Center
(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).