Wilkinson Microwave Anisotropy Probe

The low resolution (res 4) maps were used to study the influence of foregrounds on the polarization component of the WMAP data outside the Galactic plane.

A map for each differencing assembly and each year of data was produced by degrading the normal resolution (res9) maps to res 4. The res 9 processing mask was used to block out the Galactic plane, resulting in a set of 85 res 4 pixels that were completely unpopulated. These pixels were repopulated from the res 9 data. The reason for this two-step procedure was to assure that data omitted by the processing mask at res 9 were excluded from the small subset of res 4 pixels that overlap the boundary of the cut. This processing resulted in 50 maps (5 years times 10 D/A's).

The single-D/A maps were combined into one map for each frequency band using a weighted average, resulting in a single map for each frequency band (K, Ka, Q, V, and W). In this case, the res 4 Galactic plane remains unpopulated for the Stokes I and spurious (S) components, because of zero-point effects. This has been done for each single-year set of maps and for the coadded five year maps.

Both the single-year, single-D/A maps and the five-year band-averaged maps are available for download.

The spurious signal (S) is not a Stokes parameter, but rather, an apparent signal generated by differences in the two radiometers that compose a differencing assembly. This component is computed by the map-making process, as discussed in Jarosik, et.al., 2007 and Hinshaw, et.al., 2008.

The Stokes I and spurious (S) components included in the low-resolution map set are not suitable for direct use in analysis because of signal aliasing. Instead, their purpose is to help furnish a complete description of Stokes Q and U.

WMAP maps are stored in FITS binary table extensions. The maps are stored in the first extension in a file; the number of columns in this table depends upon whether polarization maps have been included in the file. Some files also contain the polarization covariance matrices for the maps; if supplied these matrices are stored in a second binary table extension.

As previously stated, the first FITS extension contains the maps. These maps are in a nested HEALPix format, with each row representing a single pixel. The number of columns depends upon which maps are included; these columns may be:

The column names for I, Q, U, and N_Obs maps were selected to be compatible with existing HEALPix software. The TEMPERATURE and N_OBS columns are the only columns that exist in files that do not contain polarization maps. The Q_POLARISATION and U_POLARISATION exist in files containing polarization maps; the names were selected to be compatible with existing HEALPix software. The SPUR_SIGNAL columns exists only in polarization map files that contain an 'S' map.

The WMAP skymaps used the HEALPix nested pixel ordering scheme in Galactic coordinates. For the convenience of users who do not have HEALPix software installed, we provide a series of FITS sky map files that give the sky coordinates of each pixel center for various resolutions.

N^-1 Covariance Matrices

While available with the Res 4 maps, the noise covariance matrices supplied with higher resolution WMAP maps do not fully characterize the pixel-to-pixel noise in the res 4 maps. Therefore a more accurate and much larger inverse covariance matrix is supplied for each map as a separate product. These are supplied in one of two forms:

• A 6144x6144 matrix containing four 3072x3072 blocks representing Q-U covariance with the form:
  

  QQ	QU
  QU	UU

  
  Most of the matrices supplied have this form.
• A 12288x12288 matrix containing sixteen 3072x3072 blocks representing I-Q-U-S covariance with the form:
  

  II	IQ	IU	IS
  IQ	QQ	QU	QS
  IU	QU	UU	US
  IS	QS	US	SS

  
  These matrices are supplied for the single year IQUS maps per differencing assembly.

The elements corresponding to the Galatic plane pixels as defined by the processing mask were set to zero. Therefore, the matrices as supplied are singular. To make them non-singular, the row and the column corresponding to each zero element of the diagonal need to be deleted.

Singular value decomposition was applied to the 12288x12288 matrices for each year and differencing assembly. The form of the decomposition is:
N-1 = u * s * uT
The decomposition was done on N^-1 in Nobs units. Approximate scaling to units of 1/mK2 was done by dividing the original 's' values by the product
σ0(Stokes I) * σ0(Stokes Q,U)
where σ₀ is the noise observation in mK for a res 4 pixel.

This scaling is not exact for the I-I or (Q,U,S)-(Q,U,S) blocks of N^-1, but the error is at most 1.2%. FITS files of the SVD results contain the 12288x12288 'u' matrix, stored in the primary data array, and the diagonal matrix of eigenvalues 's', scaled to 1/mK2 and stored as a vector of length 12288 in the FITS extension.

See Jarosik, et.al., 2007, Hinshaw, et.al., 2008 and Page, et.al., 2007 for more information.

Additional Information

• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Data Processing, Sky Maps, and Basic Results, G. Hinshaw, et.al. (2008)
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Likelihoods and Parameters from WMAP Data, J. Dunkley, et.al (2008).
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Galactic Foreground Emission, B. Gold, et.al. (2008)
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Beam Maps and Window Functions, R. Hill, et.al. (2008)
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation, E. Komatsu, et.al. (2008)
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Angular Power Spectra, M. Nolta, et.al (2008).
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Point Sources, E. Wright, et.al (2008).
• WMAP Five Year Explanatory Supplement

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