https://wiki.cosmos.esa.int/planckpla2015/api.php?action=feedcontributions&user=Tvassall&feedformat=atomPlanck PLA 2015 Wiki - User contributions [en-gb]2022-12-08T11:01:56ZUser contributionsMediaWiki 1.31.6https://wiki.cosmos.esa.int/planckpla2015/index.php?title=The_RIMO&diff=11137The RIMO2015-02-04T10:43:17Z<p>Tvassall: /* Beam Window Functions */</p>
<hr />
<div>{{DISPLAYTITLE:The instrument model}}<br />
== Overview ==<br />
<br />
The RIMO, or ''Reduced Instrument Model'' is a FITS file or a set of FITS files containing selected instrument characteristics that are needed by users who work with the released data products. It is described in detail in ''The HFI and LFI RIMO ICD'' (ref). There will be two RIMOs, one for each instrument (the HFI RIMO will consists of several parts), which will follow the same overall structure, but will differ in the details. The type of data in the RIMO can be:<br />
<br />
; Parameter : namely scalars to give properties such as a noise level or a representative beam FWHM<br />
; Vector or Table : to give, e.g., filter transmission profiles or noise power spectra, beam window functions. When possible, namely when they are of equal length, such as the noise power spectra, an effort is made to put them together into a table, otherwise they are given as separate vectors.<br />
; Image : namely 2-D "flat" array, to give, e.g., the beam correlation matrices<br />
<br />
The FITS file begins with primary header that contains some keywords that mainly for internal use and no data. The different types of data are written into different BINTABLE (for parameters and tables) or IMAGE (for 2-D arrays) extensions, as described below. <br />
<br />
The HFI-RIMO separates the beam window functions and associated data from the main set of parameters, and this because the beam window functions are delivered for two cases covering 100% and 75% of the sky, as described in detail below.<br />
<br />
=== File Names ===<br />
<br />
; HFI: {{PLASingleFile|fileType=rimo|name=HFI_RIMO_Rn.mm.fits|link=HFI_RIMO_Rn.mm.fits}}<br />
; LFI: {{PLASingleFile|fileType=rimo|name=LFI_RIMO_Rn.mm.fits|link=LFI_RIMO_Rn.mm.fits}}<br />
<br />
== Detector-level parameter data ==<br />
<br />
<br />
The detector parameter data are given in the form of a table giving the parameter values for each detector. The table columns (whose names are in ''BOLD ITALICS'') are:<br />
<br />
; Bolometer name - ''DETECTOR'' : These are the detector names. For HFI these will be of the form ''217-3'' for SWBs or ''100-3b'' for PSBs, and for LFI they will have the form 27M or 18S. There are 52 HFI detectors and 22 LFI detectors.<br />
<br />
; Focal plane geometry parameters - ''PHI_UV'', ''THETA_UV'', and ''PSI_UV'' : These parameters give the geometry of the focal plane, or the positions of the detectors in the focal plane in the Dxx reference frame. The angles that give the rotation of the beam pattern from a fiducial orientation (forward beam direction (z-axis) pointing along the telescope line of sight, with y-axis aligned with the nominal scan direction) to their positions in the focal plane. The fiducial position is that given by the Star Tracker. All angles are in radians. These parameters are derived from observations of bright planets; see [[Detector_pointing | Detector pointing]] for details.<br />
<br />
; Polarization parameters - ''PSI_POL'', ''EPSILON'' :These are the direction of maximum polarization, defined with the beam in the fiducial orientation described above, that is, before rotation onto the detector position, and the cross-polarization contamination (or leakage). These values are determined from ground-based measurements.<br />
<br />
; Beam parameters - ''FWHM'', ''ELLIPTICITY'', ''POSANG'' : These are the mean FWHM of the scanning beam (in arcmin, the beam ellipticity (no units), and the position angle of the beam major axis. The scanning beam is that recovered from the observation of bright planets; details in [[Beams]] section.<br />
<br />
; Noise parameters - ''NET'' (LFI), ''NET_TOT'' (HFI), ''NET_WHT'' (HFI), ''NET_OOF'' (HFI), ''F_KNEE'', ''ALPHA'', ''F_MIN'' (LFI), ''F_MAX'' (LFI) : Three NETs are given: one determined from the total noise (rms of the noise timeline, excluding glitched data and other non-valid data), one determined from the white noise level of the noise amplitude spectrum, and one determined from fitting a ''1/f'' noise spectrum, described by the function <math>\sigma^2(1+(f_{knee}/f)^\alpha)</math> to the noise '''Power''' spectrum. In the latter, the ''F_KNEE'' and ''ALPHA'' parameters are the frequency where the ''1/f'' component meets the white noise level, and the slope of the former. so that the slope is is about twice the slope of the amplitude spectrum. The NETs are in units of Kcmb or MJy/sr * sqrt(s). <span style="color:red">Not sure what of this applies also to LFI</span> <br />
<br />
; Detector sampling frequency - ''F_SAMP'', which is self-explanatory. <br />
<br />
In the HFI RIMO, this table includes entries for the RTS bolometers (143-8 and 545-3), which are approximate or 0.00 when not evaluated.<br />
<br />
== Map-level parameter data ==<br />
<br />
The map-level data table contains the effective beam solid angle (total and out to different multiples of the beamFWHM) and noise information. It is written into a BINTABLE extension named ''MAP_PARAMS'' whose structure is different for HFI and LFI and is as follows. The noise description below is very simplified; a more accurate rendition can be obtained from the half-ring maps. Regarding the characterization of systematics, the user should use the survey differences.<br />
<br />
=== HFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''217''<br />
; ''OMEGA_F'', ''OMEGA_F_ERR'' (Real*4) : the full beam solid angle and its uncertainty, in armin<sup>2</sup><br />
; ''OMEGA_1'', ''OMEGA_1_DISP'' (Real*4) : the beam solid angle out to 1FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''OMEGA_2'', ''OMEGA_2_DISP'' (Real*4) : the beam solid angle out to 2FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''FWHM'' (Real*4) : FWHM of a Gaussian beam having the same (total) solid angle, in armin<sup>2</sup>. This is the best value for source flux determination<br />
; ''FWHMGAUS'' (Real*4) : FWHM derived from best Gaussian fit to beam maps, in armin<sup>2</sup>. This is the best value for source identification<br />
; ''NOISE'' (Real*4) : This is the typical noise/valid observation sample as derived from the high-''l'' spectra of the half-ring maps, in the units of the corresponding map<br />
<br />
For the Omega columns, the 'DISP' (for ''dispersion'') column gives an estimate of the spatial variation as a function of position on the sky. This is the variation induced by combining the scanning beam determined from the planet observations with the scanning strategy, as described in [[Beams]].<br />
<br />
=== LFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''030, 044, 070''<br />
; ''FWHM'' (Real*8) : FWHM of a Gaussian beam having the same (total) solid angle, in arcmin<br />
; ''NOISE'' (Real*8) : This is the average noise in T<math>\cdot</math>s<sup>1/2</sup> <br />
; ''CENTRALFREQ'' (Real*4) : This is the average central frequency in GHz<br />
; ''FWHM_EFF'', ''FWHM_EFF_SIGMA'' (Real*4) : This is the average FWHM of the effective beam, in arcmin, and its dispersion<br />
; ''ELLIPTICITY_EFF'', ''ELLIPTICITY_EFF_SIGMA'' (Real*4) : This is the average ellipticity and its dispersion<br />
; ''SOLID_ANGLE_EFF'', ''SOLID_ANGLE_EFF_SIGMA'' (Real*4) : This is the average full beam solid angle, in arcmin<sup>2</sup>, and its dispersion<br />
<br />
== Effective band transmission profiles ==<br />
<br />
The effective filter bandpasses are given in different BINTABLE extensions. The extension is named ''BANDPASS_{name}'', where ''name'' specifies the detector or the maps. For the latter, the bandpasses are a weighted average of the bandpasses of the detectors that are used to build the map, using the same weights that are used in the mapmaking. These merged bandpasses are given for the full channel maps (all detectors of the frequency channel) and for the PSBs only in each frequency channel for HFI. For details on the measurements and compilation of the bandpasses see {{PlanckPapers|planck2013-p03d}}. For details on the measurements and compilation of the LFI bandpasses see {{PlanckPapers|planck2014-a05||Planck-2015-A05}}. <br />
<br />
The bandpasses are given as 4-column tables containing:<br />
<br />
=== HFI ===<br />
<br />
; ''WAVENUMBER'' (Real*4) : the wavenumber in cm-1, conversion to GHz is accomplished by multiplying by <math>10^{-7}c</math> [mks].<br />
; ''TRANSMISSION'' (Real*4) : the transmission (normalized to 1 at the max for HFI)<br />
; ''ERROR'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile.<br />
; ''FLAG'' (Integer) : a flag indicating if the data point is an independent frequency data point (nominally the case), or an FTS instrument line shape (ILS)-interpolated data point. The frequency data has been over-sampled by a factor of ~10 to assist in CO component separation efforts {{PlanckPapers|planck2013-p03d}}{{PlanckPapers|planck2013-p03a}}.<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension. Tables with the unit conversion coefficients and color correction factors for the HFI detectors (and LFI in some instances), including uncertainty estimates based on the uncertainty of the HFI detector spectral response are given in [[UC_CC_Tables | this appendix]].<br />
<br />
=== LFI ===<br />
<br />
<span style="color:red"> there was some talk of changing wn to freq ... TBD </span> <br />
<br />
; ''WAVENUMBER'' (Real*8) : the wavenumber in GHz. ??? <br />
; ''TRANSMISSION'' (Real*8) : the transmission (normalized to have an integral of 1 for LFI)<br />
; ''UNCERTAINITY'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile (not provided for LFI)<br />
; ''FLAG'' (Character) : a flag, not used by now by the LFI<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension.<br />
<br />
== Detector noise spectra ==<br />
<br />
; HFI: these are the ring noise amplitude spectra averaged over about 5000 rings in order to give a representative spectrum. The spectra of all 50 valid bolometers are given in a single table. The spectra have a maximum frequency (Nyquist) of 90.18685Hz, also given the the ''F_NYQ'' keyword, and are built over 32768 points, giving a lower frequency of 2.75 mHz.<br />
; LFI : TBW<br />
<br />
The keyword ''F_NYQ'' gives the Nyquist frequency, and can be used together with the number of points in the spectrum to reconstruct the frequency scale. The BINTABLE has Ndetector columns by Npoints rows.<br />
<br />
<br />
== Beam Window Functions ==<br />
<br />
Beam window functions and associated error descriptions are written into a BINTABLE for each ''detection unit'', where ''detection unit'' consists of an auto or a cross product (for HFI only) of one (or two) frequency maps or detset maps used in the likelihood. Here they are: <br />
<br />
; ''For the HFI'':<br />
* the 6 HFI frequency channels + 3 psb-only subsets, producing 45 extensions<br />
** 100, 143, 217, 353, 545, 857, 143p, 217p, 353p<br />
* 26 detsets, producing 351 extensions; the detsets used are, by frequency channel:<br />
** 100-DS1, 100-DS2,<br />
** 143-DS1, 143-DS2, 143-5, 143-6, 143-7,<br />
** 217-DS1, 217-DS2, 217-1, 217-2, 217-3, 217-4, <br />
** 353-DS1, 353-DS2, 353-1, 353-2, 353-7, 353-8,<br />
** 545-1, 545-2, 545-4,<br />
** 857-1, 857-2, 857-3, 857-4<br />
<br />
Note for HFI these (and also the associated covariance matrices) are given in separate files named ''HFI_RIMO-Beams-nnnpc_Rm.nn.fits'', where nnn is 100 or 075 and indicates the percentage (pc) of the sky included (see [[Frequency_Maps#Galactic_Plane_masks | Masks]] section).<br />
<br />
; ''For the LFI'':<br />
* the 3 LFI frequency channels (auto- products only), producing 3 extensions<br />
** 30, 44, 70<br />
* the 3 LFI 70GHz detector pairs (auto- products only), producing 3 extensions<br />
** 18-23, 19-22, 20-21<br />
<br />
The extension names are of the form ''BEAMWF_U1XU2'' where U1 and U2 are one (possibly the same) detection unit from one of the main groups above (i.e. there are no cross products between detsets and frequency channels, or between HFI and LFI). Each extension contains the columns:<br />
; ''NOMINAL'' (HFI, Real*4) : the beam window function proper,<br />
; ''BL'' (LFI, Real*4) : the beam window function proper,<br />
; ''EIGEN_n'' (Real*4, n=1-5 for the HFI, n=1-4 for the LFI): the five/four corresponding error modes.<br />
<br />
and the following keywords give further information, only for the HFI:<br />
; ''NMODES'' (Integer) : the number of EIGEN_* modes,<br />
; ''LMIN'' and ''LMAX'' (Integer) : the starting and ending (both included) multipoles of the vectors NOMINAL and EIGEN_*<br />
; ''LMIN_EM'' and ''LMAX_EM'' (Integer) : that give the range of the valid samples of the EIGEN_* vectors. Here ''LMAX_EM'' is always less than or equal to ''LMAX''. On the range ''LMAX_EM''+1 to ''LMAX'' the values of EIGEN_* are set to NaN, while the values of NOMINAL only are a Gaussian extrapolation of the lower multipole window function, only provided for convenience.<br />
; ''CORRMAT'' (string) : the name of the extension containing the corresponding beam correlation matrix<br />
<br />
And finally see also the ''COMMENTs'' of each header for more specific details.<br />
<br />
== Beam Correlation Matrix ==<br />
<br />
Two beam correlation matrices are given for the HFI, in two ''IMAGE'' extensions:<br />
; ''CORRBEAM_FREQ'' (Real*8) : for the frequency channels (21 units), 105x015 pixel matrix,<br />
; ''CORRBEAM_DSET'' (Real*8) : for the detsets (351 units), 1755x1755 pixel matrix <br />
Each is a symmetric matrix with 1-valued diagonal, made of NBEAMS*NBEAMS blocks, each block being NMODES*NMODES in size. The n$^{th}$ row- (and column-) block entry relates to the B(l) model whose name is indicated in ROWn = BEAMWF_U1XU2 keywords, and the corresponding eigenmodes are stored in a HDU of the same name. <br />
<br />
Each extension contains also the following keywords:<br />
; ''NDETS'' (Integer) : the number of detector units<br />
; ''NBEAMS'' (Integer) : the number of beams = NSETS * (NSETS+1) / 2<br />
; ''NMODES'' (Integer) : here 5<br />
; ''L_PLUS'' (Integer) : Eigenmode > 0 to break degeneracies<br />
; ''BLOCKn'' (string) : for n=1-NBEAMS, gives the name of the extension containing the beam WF and error eigenmodes for the nth block<br />
and some other ones for internal data checking and traceability<br />
<br />
No beam correlation matrices are produced by the LFI by now. And for HFI these, together with the beam window functions, are given in a file separate from the main RIMO (see subsection above).<br />
<br />
==Appendices==<br />
<br />
* [[UC_CC_Tables | Unit correction and color correction tables]]<br />
<br />
<br />
== References ==<br />
<br />
<References /><br />
<br />
<br />
[[Category:Mission products|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=The_RIMO&diff=11136The RIMO2015-02-04T10:38:30Z<p>Tvassall: /* Beam Window Functions */</p>
<hr />
<div>{{DISPLAYTITLE:The instrument model}}<br />
== Overview ==<br />
<br />
The RIMO, or ''Reduced Instrument Model'' is a FITS file or a set of FITS files containing selected instrument characteristics that are needed by users who work with the released data products. It is described in detail in ''The HFI and LFI RIMO ICD'' (ref). There will be two RIMOs, one for each instrument (the HFI RIMO will consists of several parts), which will follow the same overall structure, but will differ in the details. The type of data in the RIMO can be:<br />
<br />
; Parameter : namely scalars to give properties such as a noise level or a representative beam FWHM<br />
; Vector or Table : to give, e.g., filter transmission profiles or noise power spectra, beam window functions. When possible, namely when they are of equal length, such as the noise power spectra, an effort is made to put them together into a table, otherwise they are given as separate vectors.<br />
; Image : namely 2-D "flat" array, to give, e.g., the beam correlation matrices<br />
<br />
The FITS file begins with primary header that contains some keywords that mainly for internal use and no data. The different types of data are written into different BINTABLE (for parameters and tables) or IMAGE (for 2-D arrays) extensions, as described below. <br />
<br />
The HFI-RIMO separates the beam window functions and associated data from the main set of parameters, and this because the beam window functions are delivered for two cases covering 100% and 75% of the sky, as described in detail below.<br />
<br />
=== File Names ===<br />
<br />
; HFI: {{PLASingleFile|fileType=rimo|name=HFI_RIMO_Rn.mm.fits|link=HFI_RIMO_Rn.mm.fits}}<br />
; LFI: {{PLASingleFile|fileType=rimo|name=LFI_RIMO_Rn.mm.fits|link=LFI_RIMO_Rn.mm.fits}}<br />
<br />
== Detector-level parameter data ==<br />
<br />
<br />
The detector parameter data are given in the form of a table giving the parameter values for each detector. The table columns (whose names are in ''BOLD ITALICS'') are:<br />
<br />
; Bolometer name - ''DETECTOR'' : These are the detector names. For HFI these will be of the form ''217-3'' for SWBs or ''100-3b'' for PSBs, and for LFI they will have the form 27M or 18S. There are 52 HFI detectors and 22 LFI detectors.<br />
<br />
; Focal plane geometry parameters - ''PHI_UV'', ''THETA_UV'', and ''PSI_UV'' : These parameters give the geometry of the focal plane, or the positions of the detectors in the focal plane in the Dxx reference frame. The angles that give the rotation of the beam pattern from a fiducial orientation (forward beam direction (z-axis) pointing along the telescope line of sight, with y-axis aligned with the nominal scan direction) to their positions in the focal plane. The fiducial position is that given by the Star Tracker. All angles are in radians. These parameters are derived from observations of bright planets; see [[Detector_pointing | Detector pointing]] for details.<br />
<br />
; Polarization parameters - ''PSI_POL'', ''EPSILON'' :These are the direction of maximum polarization, defined with the beam in the fiducial orientation described above, that is, before rotation onto the detector position, and the cross-polarization contamination (or leakage). These values are determined from ground-based measurements.<br />
<br />
; Beam parameters - ''FWHM'', ''ELLIPTICITY'', ''POSANG'' : These are the mean FWHM of the scanning beam (in arcmin, the beam ellipticity (no units), and the position angle of the beam major axis. The scanning beam is that recovered from the observation of bright planets; details in [[Beams]] section.<br />
<br />
; Noise parameters - ''NET'' (LFI), ''NET_TOT'' (HFI), ''NET_WHT'' (HFI), ''NET_OOF'' (HFI), ''F_KNEE'', ''ALPHA'', ''F_MIN'' (LFI), ''F_MAX'' (LFI) : Three NETs are given: one determined from the total noise (rms of the noise timeline, excluding glitched data and other non-valid data), one determined from the white noise level of the noise amplitude spectrum, and one determined from fitting a ''1/f'' noise spectrum, described by the function <math>\sigma^2(1+(f_{knee}/f)^\alpha)</math> to the noise '''Power''' spectrum. In the latter, the ''F_KNEE'' and ''ALPHA'' parameters are the frequency where the ''1/f'' component meets the white noise level, and the slope of the former. so that the slope is is about twice the slope of the amplitude spectrum. The NETs are in units of Kcmb or MJy/sr * sqrt(s). <span style="color:red">Not sure what of this applies also to LFI</span> <br />
<br />
; Detector sampling frequency - ''F_SAMP'', which is self-explanatory. <br />
<br />
In the HFI RIMO, this table includes entries for the RTS bolometers (143-8 and 545-3), which are approximate or 0.00 when not evaluated.<br />
<br />
== Map-level parameter data ==<br />
<br />
The map-level data table contains the effective beam solid angle (total and out to different multiples of the beamFWHM) and noise information. It is written into a BINTABLE extension named ''MAP_PARAMS'' whose structure is different for HFI and LFI and is as follows. The noise description below is very simplified; a more accurate rendition can be obtained from the half-ring maps. Regarding the characterization of systematics, the user should use the survey differences.<br />
<br />
=== HFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''217''<br />
; ''OMEGA_F'', ''OMEGA_F_ERR'' (Real*4) : the full beam solid angle and its uncertainty, in armin<sup>2</sup><br />
; ''OMEGA_1'', ''OMEGA_1_DISP'' (Real*4) : the beam solid angle out to 1FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''OMEGA_2'', ''OMEGA_2_DISP'' (Real*4) : the beam solid angle out to 2FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''FWHM'' (Real*4) : FWHM of a Gaussian beam having the same (total) solid angle, in armin<sup>2</sup>. This is the best value for source flux determination<br />
; ''FWHMGAUS'' (Real*4) : FWHM derived from best Gaussian fit to beam maps, in armin<sup>2</sup>. This is the best value for source identification<br />
; ''NOISE'' (Real*4) : This is the typical noise/valid observation sample as derived from the high-''l'' spectra of the half-ring maps, in the units of the corresponding map<br />
<br />
For the Omega columns, the 'DISP' (for ''dispersion'') column gives an estimate of the spatial variation as a function of position on the sky. This is the variation induced by combining the scanning beam determined from the planet observations with the scanning strategy, as described in [[Beams]].<br />
<br />
=== LFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''030, 044, 070''<br />
; ''FWHM'' (Real*8) : FWHM of a Gaussian beam having the same (total) solid angle, in arcmin<br />
; ''NOISE'' (Real*8) : This is the average noise in T<math>\cdot</math>s<sup>1/2</sup> <br />
; ''CENTRALFREQ'' (Real*4) : This is the average central frequency in GHz<br />
; ''FWHM_EFF'', ''FWHM_EFF_SIGMA'' (Real*4) : This is the average FWHM of the effective beam, in arcmin, and its dispersion<br />
; ''ELLIPTICITY_EFF'', ''ELLIPTICITY_EFF_SIGMA'' (Real*4) : This is the average ellipticity and its dispersion<br />
; ''SOLID_ANGLE_EFF'', ''SOLID_ANGLE_EFF_SIGMA'' (Real*4) : This is the average full beam solid angle, in arcmin<sup>2</sup>, and its dispersion<br />
<br />
== Effective band transmission profiles ==<br />
<br />
The effective filter bandpasses are given in different BINTABLE extensions. The extension is named ''BANDPASS_{name}'', where ''name'' specifies the detector or the maps. For the latter, the bandpasses are a weighted average of the bandpasses of the detectors that are used to build the map, using the same weights that are used in the mapmaking. These merged bandpasses are given for the full channel maps (all detectors of the frequency channel) and for the PSBs only in each frequency channel for HFI. For details on the measurements and compilation of the bandpasses see {{PlanckPapers|planck2013-p03d}}. For details on the measurements and compilation of the LFI bandpasses see {{PlanckPapers|planck2014-a05||Planck-2015-A05}}. <br />
<br />
The bandpasses are given as 4-column tables containing:<br />
<br />
=== HFI ===<br />
<br />
; ''WAVENUMBER'' (Real*4) : the wavenumber in cm-1, conversion to GHz is accomplished by multiplying by <math>10^{-7}c</math> [mks].<br />
; ''TRANSMISSION'' (Real*4) : the transmission (normalized to 1 at the max for HFI)<br />
; ''ERROR'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile.<br />
; ''FLAG'' (Integer) : a flag indicating if the data point is an independent frequency data point (nominally the case), or an FTS instrument line shape (ILS)-interpolated data point. The frequency data has been over-sampled by a factor of ~10 to assist in CO component separation efforts {{PlanckPapers|planck2013-p03d}}{{PlanckPapers|planck2013-p03a}}.<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension. Tables with the unit conversion coefficients and color correction factors for the HFI detectors (and LFI in some instances), including uncertainty estimates based on the uncertainty of the HFI detector spectral response are given in [[UC_CC_Tables | this appendix]].<br />
<br />
=== LFI ===<br />
<br />
<span style="color:red"> there was some talk of changing wn to freq ... TBD </span> <br />
<br />
; ''WAVENUMBER'' (Real*8) : the wavenumber in GHz. ??? <br />
; ''TRANSMISSION'' (Real*8) : the transmission (normalized to have an integral of 1 for LFI)<br />
; ''UNCERTAINITY'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile (not provided for LFI)<br />
; ''FLAG'' (Character) : a flag, not used by now by the LFI<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension.<br />
<br />
== Detector noise spectra ==<br />
<br />
; HFI: these are the ring noise amplitude spectra averaged over about 5000 rings in order to give a representative spectrum. The spectra of all 50 valid bolometers are given in a single table. The spectra have a maximum frequency (Nyquist) of 90.18685Hz, also given the the ''F_NYQ'' keyword, and are built over 32768 points, giving a lower frequency of 2.75 mHz.<br />
; LFI : TBW<br />
<br />
The keyword ''F_NYQ'' gives the Nyquist frequency, and can be used together with the number of points in the spectrum to reconstruct the frequency scale. The BINTABLE has Ndetector columns by Npoints rows.<br />
<br />
<br />
== Beam Window Functions ==<br />
<br />
Beam window functions and associated error descriptions are written into a BINTABLE for each ''detection unit'', where ''detection unit'' consists of an auto or a cross product (for HFI only) of one (or two) frequency maps or detset maps used in the likelihood. Here they are: <br />
<br />
; ''For the HFI'':<br />
* the 6 HFI frequency channels + 3 psb-only subsets, producing 45 extensions<br />
** 100, 143, 217, 353, 545, 857, 143p, 217p, 353p<br />
* 26 detsets, producing 351 extensions; the detsets used are, by frequency channel:<br />
** 100-DS1, 100-DS2,<br />
** 143-DS1, 143-DS2, 143-5, 143-6, 143-7,<br />
** 217-DS1, 217-DS2, 217-1, 217-2, 217-3, 217-4, <br />
** 353-DS1, 353-DS2, 353-1, 353-2, 353-7, 353-8,<br />
** 545-1, 545-2, 545-4,<br />
** 857-1, 857-2, 857-3, 857-4<br />
<br />
Note for HFI these (and also the associated covariance matrices) are given in separate files named ''HFI_RIMO-Beams-nnnpc_Rm.nn.fits'', where nnn is 100 or 075 and indicates the percentage (pc) of the sky included (see [[Frequency_Maps#Galactic_Plane_masks | Masks]] section).<br />
<br />
; ''For the LFI'':<br />
* the 3 LFI frequency channels (auto- products only), producing 3 extensions<br />
** 30, 44, 70<br />
* the 3 LFI 70GHz detector pairs (auto- products only), producing 3 extensions<br />
** 18-23, 19-22, 20-21<br />
<br />
The extension names are of the form ''BEAMWF_U1XU2'' where U1 and U2 are one (possibly the same) detection unit from one of the main groups above (i.e. there are no cross products between detsets and frequency channels, or between HFI and LFI). Each extension contains the columns:<br />
; ''NOMINAL'' (Real*4) : the beam window function proper,<br />
; ''EIGEN_n'' (Real*4, n=1-5 for the HFI, n=1-4 for the LFI): the five/four corresponding error modes.<br />
<br />
and the following keywords give further information, only for the HFI:<br />
; ''NMODES'' (Integer) : the number of EIGEN_* modes,<br />
; ''LMIN'' and ''LMAX'' (Integer) : the starting and ending (both included) multipoles of the vectors NOMINAL and EIGEN_*<br />
; ''LMIN_EM'' and ''LMAX_EM'' (Integer) : that give the range of the valid samples of the EIGEN_* vectors. Here ''LMAX_EM'' is always less than or equal to ''LMAX''. On the range ''LMAX_EM''+1 to ''LMAX'' the values of EIGEN_* are set to NaN, while the values of NOMINAL only are a Gaussian extrapolation of the lower multipole window function, only provided for convenience.<br />
; ''CORRMAT'' (string) : the name of the extension containing the corresponding beam correlation matrix<br />
<br />
And finally see also the ''COMMENTs'' of each header for more specific details.<br />
<br />
== Beam Correlation Matrix ==<br />
<br />
Two beam correlation matrices are given for the HFI, in two ''IMAGE'' extensions:<br />
; ''CORRBEAM_FREQ'' (Real*8) : for the frequency channels (21 units), 105x015 pixel matrix,<br />
; ''CORRBEAM_DSET'' (Real*8) : for the detsets (351 units), 1755x1755 pixel matrix <br />
Each is a symmetric matrix with 1-valued diagonal, made of NBEAMS*NBEAMS blocks, each block being NMODES*NMODES in size. The n$^{th}$ row- (and column-) block entry relates to the B(l) model whose name is indicated in ROWn = BEAMWF_U1XU2 keywords, and the corresponding eigenmodes are stored in a HDU of the same name. <br />
<br />
Each extension contains also the following keywords:<br />
; ''NDETS'' (Integer) : the number of detector units<br />
; ''NBEAMS'' (Integer) : the number of beams = NSETS * (NSETS+1) / 2<br />
; ''NMODES'' (Integer) : here 5<br />
; ''L_PLUS'' (Integer) : Eigenmode > 0 to break degeneracies<br />
; ''BLOCKn'' (string) : for n=1-NBEAMS, gives the name of the extension containing the beam WF and error eigenmodes for the nth block<br />
and some other ones for internal data checking and traceability<br />
<br />
No beam correlation matrices are produced by the LFI by now. And for HFI these, together with the beam window functions, are given in a file separate from the main RIMO (see subsection above).<br />
<br />
==Appendices==<br />
<br />
* [[UC_CC_Tables | Unit correction and color correction tables]]<br />
<br />
<br />
== References ==<br />
<br />
<References /><br />
<br />
<br />
[[Category:Mission products|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=The_RIMO&diff=11135The RIMO2015-02-04T10:38:10Z<p>Tvassall: /* Beam Window Functions */</p>
<hr />
<div>{{DISPLAYTITLE:The instrument model}}<br />
== Overview ==<br />
<br />
The RIMO, or ''Reduced Instrument Model'' is a FITS file or a set of FITS files containing selected instrument characteristics that are needed by users who work with the released data products. It is described in detail in ''The HFI and LFI RIMO ICD'' (ref). There will be two RIMOs, one for each instrument (the HFI RIMO will consists of several parts), which will follow the same overall structure, but will differ in the details. The type of data in the RIMO can be:<br />
<br />
; Parameter : namely scalars to give properties such as a noise level or a representative beam FWHM<br />
; Vector or Table : to give, e.g., filter transmission profiles or noise power spectra, beam window functions. When possible, namely when they are of equal length, such as the noise power spectra, an effort is made to put them together into a table, otherwise they are given as separate vectors.<br />
; Image : namely 2-D "flat" array, to give, e.g., the beam correlation matrices<br />
<br />
The FITS file begins with primary header that contains some keywords that mainly for internal use and no data. The different types of data are written into different BINTABLE (for parameters and tables) or IMAGE (for 2-D arrays) extensions, as described below. <br />
<br />
The HFI-RIMO separates the beam window functions and associated data from the main set of parameters, and this because the beam window functions are delivered for two cases covering 100% and 75% of the sky, as described in detail below.<br />
<br />
=== File Names ===<br />
<br />
; HFI: {{PLASingleFile|fileType=rimo|name=HFI_RIMO_Rn.mm.fits|link=HFI_RIMO_Rn.mm.fits}}<br />
; LFI: {{PLASingleFile|fileType=rimo|name=LFI_RIMO_Rn.mm.fits|link=LFI_RIMO_Rn.mm.fits}}<br />
<br />
== Detector-level parameter data ==<br />
<br />
<br />
The detector parameter data are given in the form of a table giving the parameter values for each detector. The table columns (whose names are in ''BOLD ITALICS'') are:<br />
<br />
; Bolometer name - ''DETECTOR'' : These are the detector names. For HFI these will be of the form ''217-3'' for SWBs or ''100-3b'' for PSBs, and for LFI they will have the form 27M or 18S. There are 52 HFI detectors and 22 LFI detectors.<br />
<br />
; Focal plane geometry parameters - ''PHI_UV'', ''THETA_UV'', and ''PSI_UV'' : These parameters give the geometry of the focal plane, or the positions of the detectors in the focal plane in the Dxx reference frame. The angles that give the rotation of the beam pattern from a fiducial orientation (forward beam direction (z-axis) pointing along the telescope line of sight, with y-axis aligned with the nominal scan direction) to their positions in the focal plane. The fiducial position is that given by the Star Tracker. All angles are in radians. These parameters are derived from observations of bright planets; see [[Detector_pointing | Detector pointing]] for details.<br />
<br />
; Polarization parameters - ''PSI_POL'', ''EPSILON'' :These are the direction of maximum polarization, defined with the beam in the fiducial orientation described above, that is, before rotation onto the detector position, and the cross-polarization contamination (or leakage). These values are determined from ground-based measurements.<br />
<br />
; Beam parameters - ''FWHM'', ''ELLIPTICITY'', ''POSANG'' : These are the mean FWHM of the scanning beam (in arcmin, the beam ellipticity (no units), and the position angle of the beam major axis. The scanning beam is that recovered from the observation of bright planets; details in [[Beams]] section.<br />
<br />
; Noise parameters - ''NET'' (LFI), ''NET_TOT'' (HFI), ''NET_WHT'' (HFI), ''NET_OOF'' (HFI), ''F_KNEE'', ''ALPHA'', ''F_MIN'' (LFI), ''F_MAX'' (LFI) : Three NETs are given: one determined from the total noise (rms of the noise timeline, excluding glitched data and other non-valid data), one determined from the white noise level of the noise amplitude spectrum, and one determined from fitting a ''1/f'' noise spectrum, described by the function <math>\sigma^2(1+(f_{knee}/f)^\alpha)</math> to the noise '''Power''' spectrum. In the latter, the ''F_KNEE'' and ''ALPHA'' parameters are the frequency where the ''1/f'' component meets the white noise level, and the slope of the former. so that the slope is is about twice the slope of the amplitude spectrum. The NETs are in units of Kcmb or MJy/sr * sqrt(s). <span style="color:red">Not sure what of this applies also to LFI</span> <br />
<br />
; Detector sampling frequency - ''F_SAMP'', which is self-explanatory. <br />
<br />
In the HFI RIMO, this table includes entries for the RTS bolometers (143-8 and 545-3), which are approximate or 0.00 when not evaluated.<br />
<br />
== Map-level parameter data ==<br />
<br />
The map-level data table contains the effective beam solid angle (total and out to different multiples of the beamFWHM) and noise information. It is written into a BINTABLE extension named ''MAP_PARAMS'' whose structure is different for HFI and LFI and is as follows. The noise description below is very simplified; a more accurate rendition can be obtained from the half-ring maps. Regarding the characterization of systematics, the user should use the survey differences.<br />
<br />
=== HFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''217''<br />
; ''OMEGA_F'', ''OMEGA_F_ERR'' (Real*4) : the full beam solid angle and its uncertainty, in armin<sup>2</sup><br />
; ''OMEGA_1'', ''OMEGA_1_DISP'' (Real*4) : the beam solid angle out to 1FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''OMEGA_2'', ''OMEGA_2_DISP'' (Real*4) : the beam solid angle out to 2FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''FWHM'' (Real*4) : FWHM of a Gaussian beam having the same (total) solid angle, in armin<sup>2</sup>. This is the best value for source flux determination<br />
; ''FWHMGAUS'' (Real*4) : FWHM derived from best Gaussian fit to beam maps, in armin<sup>2</sup>. This is the best value for source identification<br />
; ''NOISE'' (Real*4) : This is the typical noise/valid observation sample as derived from the high-''l'' spectra of the half-ring maps, in the units of the corresponding map<br />
<br />
For the Omega columns, the 'DISP' (for ''dispersion'') column gives an estimate of the spatial variation as a function of position on the sky. This is the variation induced by combining the scanning beam determined from the planet observations with the scanning strategy, as described in [[Beams]].<br />
<br />
=== LFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''030, 044, 070''<br />
; ''FWHM'' (Real*8) : FWHM of a Gaussian beam having the same (total) solid angle, in arcmin<br />
; ''NOISE'' (Real*8) : This is the average noise in T<math>\cdot</math>s<sup>1/2</sup> <br />
; ''CENTRALFREQ'' (Real*4) : This is the average central frequency in GHz<br />
; ''FWHM_EFF'', ''FWHM_EFF_SIGMA'' (Real*4) : This is the average FWHM of the effective beam, in arcmin, and its dispersion<br />
; ''ELLIPTICITY_EFF'', ''ELLIPTICITY_EFF_SIGMA'' (Real*4) : This is the average ellipticity and its dispersion<br />
; ''SOLID_ANGLE_EFF'', ''SOLID_ANGLE_EFF_SIGMA'' (Real*4) : This is the average full beam solid angle, in arcmin<sup>2</sup>, and its dispersion<br />
<br />
== Effective band transmission profiles ==<br />
<br />
The effective filter bandpasses are given in different BINTABLE extensions. The extension is named ''BANDPASS_{name}'', where ''name'' specifies the detector or the maps. For the latter, the bandpasses are a weighted average of the bandpasses of the detectors that are used to build the map, using the same weights that are used in the mapmaking. These merged bandpasses are given for the full channel maps (all detectors of the frequency channel) and for the PSBs only in each frequency channel for HFI. For details on the measurements and compilation of the bandpasses see {{PlanckPapers|planck2013-p03d}}. For details on the measurements and compilation of the LFI bandpasses see {{PlanckPapers|planck2014-a05||Planck-2015-A05}}. <br />
<br />
The bandpasses are given as 4-column tables containing:<br />
<br />
=== HFI ===<br />
<br />
; ''WAVENUMBER'' (Real*4) : the wavenumber in cm-1, conversion to GHz is accomplished by multiplying by <math>10^{-7}c</math> [mks].<br />
; ''TRANSMISSION'' (Real*4) : the transmission (normalized to 1 at the max for HFI)<br />
; ''ERROR'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile.<br />
; ''FLAG'' (Integer) : a flag indicating if the data point is an independent frequency data point (nominally the case), or an FTS instrument line shape (ILS)-interpolated data point. The frequency data has been over-sampled by a factor of ~10 to assist in CO component separation efforts {{PlanckPapers|planck2013-p03d}}{{PlanckPapers|planck2013-p03a}}.<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension. Tables with the unit conversion coefficients and color correction factors for the HFI detectors (and LFI in some instances), including uncertainty estimates based on the uncertainty of the HFI detector spectral response are given in [[UC_CC_Tables | this appendix]].<br />
<br />
=== LFI ===<br />
<br />
<span style="color:red"> there was some talk of changing wn to freq ... TBD </span> <br />
<br />
; ''WAVENUMBER'' (Real*8) : the wavenumber in GHz. ??? <br />
; ''TRANSMISSION'' (Real*8) : the transmission (normalized to have an integral of 1 for LFI)<br />
; ''UNCERTAINITY'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile (not provided for LFI)<br />
; ''FLAG'' (Character) : a flag, not used by now by the LFI<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension.<br />
<br />
== Detector noise spectra ==<br />
<br />
; HFI: these are the ring noise amplitude spectra averaged over about 5000 rings in order to give a representative spectrum. The spectra of all 50 valid bolometers are given in a single table. The spectra have a maximum frequency (Nyquist) of 90.18685Hz, also given the the ''F_NYQ'' keyword, and are built over 32768 points, giving a lower frequency of 2.75 mHz.<br />
; LFI : TBW<br />
<br />
The keyword ''F_NYQ'' gives the Nyquist frequency, and can be used together with the number of points in the spectrum to reconstruct the frequency scale. The BINTABLE has Ndetector columns by Npoints rows.<br />
<br />
<br />
== Beam Window Functions ==<br />
<br />
Beam window functions and associated error descriptions are written into a BINTABLE for each ''detection unit'', where ''detection unit'' consists of an auto or a cross product (for HFI only) of one (or two) frequency maps or detset maps used in the likelihood. Here they are: <br />
<br />
; ''For the HFI'':<br />
* the 6 HFI frequency channels + 3 psb-only subsets, producing 45 extensions<br />
** 100, 143, 217, 353, 545, 857, 143p, 217p, 353p<br />
* 26 detsets, producing 351 extensions; the detsets used are, by frequency channel:<br />
** 100-DS1, 100-DS2,<br />
** 143-DS1, 143-DS2, 143-5, 143-6, 143-7,<br />
** 217-DS1, 217-DS2, 217-1, 217-2, 217-3, 217-4, <br />
** 353-DS1, 353-DS2, 353-1, 353-2, 353-7, 353-8,<br />
** 545-1, 545-2, 545-4,<br />
** 857-1, 857-2, 857-3, 857-4<br />
<br />
Note for HFI these (and also the associated covariance matrices) are given in separate files named ''HFI_RIMO-Beams-nnnpc_Rm.nn.fits'', where nnn is 100 or 075 and indicates the percentage (pc) of the sky included (see [[Frequency_Maps#Galactic_Plane_masks | Masks]] section).<br />
<br />
; ''For the LFI'':<br />
* the 3 LFI frequency channels (auto- products only), producing 3 extensions<br />
** 30, 44, 70<br />
* the 3 LFI 70GHz detector pairs (auto- products only), producing 3 extensions<br />
** 18_23, 19_22, 20_21<br />
<br />
The extension names are of the form ''BEAMWF_U1XU2'' where U1 and U2 are one (possibly the same) detection unit from one of the main groups above (i.e. there are no cross products between detsets and frequency channels, or between HFI and LFI). Each extension contains the columns:<br />
; ''NOMINAL'' (Real*4) : the beam window function proper,<br />
; ''EIGEN_n'' (Real*4, n=1-5 for the HFI, n=1-4 for the LFI): the five/four corresponding error modes.<br />
<br />
and the following keywords give further information, only for the HFI:<br />
; ''NMODES'' (Integer) : the number of EIGEN_* modes,<br />
; ''LMIN'' and ''LMAX'' (Integer) : the starting and ending (both included) multipoles of the vectors NOMINAL and EIGEN_*<br />
; ''LMIN_EM'' and ''LMAX_EM'' (Integer) : that give the range of the valid samples of the EIGEN_* vectors. Here ''LMAX_EM'' is always less than or equal to ''LMAX''. On the range ''LMAX_EM''+1 to ''LMAX'' the values of EIGEN_* are set to NaN, while the values of NOMINAL only are a Gaussian extrapolation of the lower multipole window function, only provided for convenience.<br />
; ''CORRMAT'' (string) : the name of the extension containing the corresponding beam correlation matrix<br />
<br />
And finally see also the ''COMMENTs'' of each header for more specific details.<br />
<br />
== Beam Correlation Matrix ==<br />
<br />
Two beam correlation matrices are given for the HFI, in two ''IMAGE'' extensions:<br />
; ''CORRBEAM_FREQ'' (Real*8) : for the frequency channels (21 units), 105x015 pixel matrix,<br />
; ''CORRBEAM_DSET'' (Real*8) : for the detsets (351 units), 1755x1755 pixel matrix <br />
Each is a symmetric matrix with 1-valued diagonal, made of NBEAMS*NBEAMS blocks, each block being NMODES*NMODES in size. The n$^{th}$ row- (and column-) block entry relates to the B(l) model whose name is indicated in ROWn = BEAMWF_U1XU2 keywords, and the corresponding eigenmodes are stored in a HDU of the same name. <br />
<br />
Each extension contains also the following keywords:<br />
; ''NDETS'' (Integer) : the number of detector units<br />
; ''NBEAMS'' (Integer) : the number of beams = NSETS * (NSETS+1) / 2<br />
; ''NMODES'' (Integer) : here 5<br />
; ''L_PLUS'' (Integer) : Eigenmode > 0 to break degeneracies<br />
; ''BLOCKn'' (string) : for n=1-NBEAMS, gives the name of the extension containing the beam WF and error eigenmodes for the nth block<br />
and some other ones for internal data checking and traceability<br />
<br />
No beam correlation matrices are produced by the LFI by now. And for HFI these, together with the beam window functions, are given in a file separate from the main RIMO (see subsection above).<br />
<br />
==Appendices==<br />
<br />
* [[UC_CC_Tables | Unit correction and color correction tables]]<br />
<br />
<br />
== References ==<br />
<br />
<References /><br />
<br />
<br />
[[Category:Mission products|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=The_RIMO&diff=11134The RIMO2015-02-04T10:37:09Z<p>Tvassall: /* Beam Window Functions */</p>
<hr />
<div>{{DISPLAYTITLE:The instrument model}}<br />
== Overview ==<br />
<br />
The RIMO, or ''Reduced Instrument Model'' is a FITS file or a set of FITS files containing selected instrument characteristics that are needed by users who work with the released data products. It is described in detail in ''The HFI and LFI RIMO ICD'' (ref). There will be two RIMOs, one for each instrument (the HFI RIMO will consists of several parts), which will follow the same overall structure, but will differ in the details. The type of data in the RIMO can be:<br />
<br />
; Parameter : namely scalars to give properties such as a noise level or a representative beam FWHM<br />
; Vector or Table : to give, e.g., filter transmission profiles or noise power spectra, beam window functions. When possible, namely when they are of equal length, such as the noise power spectra, an effort is made to put them together into a table, otherwise they are given as separate vectors.<br />
; Image : namely 2-D "flat" array, to give, e.g., the beam correlation matrices<br />
<br />
The FITS file begins with primary header that contains some keywords that mainly for internal use and no data. The different types of data are written into different BINTABLE (for parameters and tables) or IMAGE (for 2-D arrays) extensions, as described below. <br />
<br />
The HFI-RIMO separates the beam window functions and associated data from the main set of parameters, and this because the beam window functions are delivered for two cases covering 100% and 75% of the sky, as described in detail below.<br />
<br />
=== File Names ===<br />
<br />
; HFI: {{PLASingleFile|fileType=rimo|name=HFI_RIMO_Rn.mm.fits|link=HFI_RIMO_Rn.mm.fits}}<br />
; LFI: {{PLASingleFile|fileType=rimo|name=LFI_RIMO_Rn.mm.fits|link=LFI_RIMO_Rn.mm.fits}}<br />
<br />
== Detector-level parameter data ==<br />
<br />
<br />
The detector parameter data are given in the form of a table giving the parameter values for each detector. The table columns (whose names are in ''BOLD ITALICS'') are:<br />
<br />
; Bolometer name - ''DETECTOR'' : These are the detector names. For HFI these will be of the form ''217-3'' for SWBs or ''100-3b'' for PSBs, and for LFI they will have the form 27M or 18S. There are 52 HFI detectors and 22 LFI detectors.<br />
<br />
; Focal plane geometry parameters - ''PHI_UV'', ''THETA_UV'', and ''PSI_UV'' : These parameters give the geometry of the focal plane, or the positions of the detectors in the focal plane in the Dxx reference frame. The angles that give the rotation of the beam pattern from a fiducial orientation (forward beam direction (z-axis) pointing along the telescope line of sight, with y-axis aligned with the nominal scan direction) to their positions in the focal plane. The fiducial position is that given by the Star Tracker. All angles are in radians. These parameters are derived from observations of bright planets; see [[Detector_pointing | Detector pointing]] for details.<br />
<br />
; Polarization parameters - ''PSI_POL'', ''EPSILON'' :These are the direction of maximum polarization, defined with the beam in the fiducial orientation described above, that is, before rotation onto the detector position, and the cross-polarization contamination (or leakage). These values are determined from ground-based measurements.<br />
<br />
; Beam parameters - ''FWHM'', ''ELLIPTICITY'', ''POSANG'' : These are the mean FWHM of the scanning beam (in arcmin, the beam ellipticity (no units), and the position angle of the beam major axis. The scanning beam is that recovered from the observation of bright planets; details in [[Beams]] section.<br />
<br />
; Noise parameters - ''NET'' (LFI), ''NET_TOT'' (HFI), ''NET_WHT'' (HFI), ''NET_OOF'' (HFI), ''F_KNEE'', ''ALPHA'', ''F_MIN'' (LFI), ''F_MAX'' (LFI) : Three NETs are given: one determined from the total noise (rms of the noise timeline, excluding glitched data and other non-valid data), one determined from the white noise level of the noise amplitude spectrum, and one determined from fitting a ''1/f'' noise spectrum, described by the function <math>\sigma^2(1+(f_{knee}/f)^\alpha)</math> to the noise '''Power''' spectrum. In the latter, the ''F_KNEE'' and ''ALPHA'' parameters are the frequency where the ''1/f'' component meets the white noise level, and the slope of the former. so that the slope is is about twice the slope of the amplitude spectrum. The NETs are in units of Kcmb or MJy/sr * sqrt(s). <span style="color:red">Not sure what of this applies also to LFI</span> <br />
<br />
; Detector sampling frequency - ''F_SAMP'', which is self-explanatory. <br />
<br />
In the HFI RIMO, this table includes entries for the RTS bolometers (143-8 and 545-3), which are approximate or 0.00 when not evaluated.<br />
<br />
== Map-level parameter data ==<br />
<br />
The map-level data table contains the effective beam solid angle (total and out to different multiples of the beamFWHM) and noise information. It is written into a BINTABLE extension named ''MAP_PARAMS'' whose structure is different for HFI and LFI and is as follows. The noise description below is very simplified; a more accurate rendition can be obtained from the half-ring maps. Regarding the characterization of systematics, the user should use the survey differences.<br />
<br />
=== HFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''217''<br />
; ''OMEGA_F'', ''OMEGA_F_ERR'' (Real*4) : the full beam solid angle and its uncertainty, in armin<sup>2</sup><br />
; ''OMEGA_1'', ''OMEGA_1_DISP'' (Real*4) : the beam solid angle out to 1FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''OMEGA_2'', ''OMEGA_2_DISP'' (Real*4) : the beam solid angle out to 2FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''FWHM'' (Real*4) : FWHM of a Gaussian beam having the same (total) solid angle, in armin<sup>2</sup>. This is the best value for source flux determination<br />
; ''FWHMGAUS'' (Real*4) : FWHM derived from best Gaussian fit to beam maps, in armin<sup>2</sup>. This is the best value for source identification<br />
; ''NOISE'' (Real*4) : This is the typical noise/valid observation sample as derived from the high-''l'' spectra of the half-ring maps, in the units of the corresponding map<br />
<br />
For the Omega columns, the 'DISP' (for ''dispersion'') column gives an estimate of the spatial variation as a function of position on the sky. This is the variation induced by combining the scanning beam determined from the planet observations with the scanning strategy, as described in [[Beams]].<br />
<br />
=== LFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''030, 044, 070''<br />
; ''FWHM'' (Real*8) : FWHM of a Gaussian beam having the same (total) solid angle, in arcmin<br />
; ''NOISE'' (Real*8) : This is the average noise in T<math>\cdot</math>s<sup>1/2</sup> <br />
; ''CENTRALFREQ'' (Real*4) : This is the average central frequency in GHz<br />
; ''FWHM_EFF'', ''FWHM_EFF_SIGMA'' (Real*4) : This is the average FWHM of the effective beam, in arcmin, and its dispersion<br />
; ''ELLIPTICITY_EFF'', ''ELLIPTICITY_EFF_SIGMA'' (Real*4) : This is the average ellipticity and its dispersion<br />
; ''SOLID_ANGLE_EFF'', ''SOLID_ANGLE_EFF_SIGMA'' (Real*4) : This is the average full beam solid angle, in arcmin<sup>2</sup>, and its dispersion<br />
<br />
== Effective band transmission profiles ==<br />
<br />
The effective filter bandpasses are given in different BINTABLE extensions. The extension is named ''BANDPASS_{name}'', where ''name'' specifies the detector or the maps. For the latter, the bandpasses are a weighted average of the bandpasses of the detectors that are used to build the map, using the same weights that are used in the mapmaking. These merged bandpasses are given for the full channel maps (all detectors of the frequency channel) and for the PSBs only in each frequency channel for HFI. For details on the measurements and compilation of the bandpasses see {{PlanckPapers|planck2013-p03d}}. For details on the measurements and compilation of the LFI bandpasses see {{PlanckPapers|planck2014-a05||Planck-2015-A05}}. <br />
<br />
The bandpasses are given as 4-column tables containing:<br />
<br />
=== HFI ===<br />
<br />
; ''WAVENUMBER'' (Real*4) : the wavenumber in cm-1, conversion to GHz is accomplished by multiplying by <math>10^{-7}c</math> [mks].<br />
; ''TRANSMISSION'' (Real*4) : the transmission (normalized to 1 at the max for HFI)<br />
; ''ERROR'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile.<br />
; ''FLAG'' (Integer) : a flag indicating if the data point is an independent frequency data point (nominally the case), or an FTS instrument line shape (ILS)-interpolated data point. The frequency data has been over-sampled by a factor of ~10 to assist in CO component separation efforts {{PlanckPapers|planck2013-p03d}}{{PlanckPapers|planck2013-p03a}}.<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension. Tables with the unit conversion coefficients and color correction factors for the HFI detectors (and LFI in some instances), including uncertainty estimates based on the uncertainty of the HFI detector spectral response are given in [[UC_CC_Tables | this appendix]].<br />
<br />
=== LFI ===<br />
<br />
<span style="color:red"> there was some talk of changing wn to freq ... TBD </span> <br />
<br />
; ''WAVENUMBER'' (Real*8) : the wavenumber in GHz. ??? <br />
; ''TRANSMISSION'' (Real*8) : the transmission (normalized to have an integral of 1 for LFI)<br />
; ''UNCERTAINITY'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile (not provided for LFI)<br />
; ''FLAG'' (Character) : a flag, not used by now by the LFI<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension.<br />
<br />
== Detector noise spectra ==<br />
<br />
; HFI: these are the ring noise amplitude spectra averaged over about 5000 rings in order to give a representative spectrum. The spectra of all 50 valid bolometers are given in a single table. The spectra have a maximum frequency (Nyquist) of 90.18685Hz, also given the the ''F_NYQ'' keyword, and are built over 32768 points, giving a lower frequency of 2.75 mHz.<br />
; LFI : TBW<br />
<br />
The keyword ''F_NYQ'' gives the Nyquist frequency, and can be used together with the number of points in the spectrum to reconstruct the frequency scale. The BINTABLE has Ndetector columns by Npoints rows.<br />
<br />
<br />
== Beam Window Functions ==<br />
<br />
Beam window functions and associated error descriptions are written into a BINTABLE for each ''detection unit'', where ''detection unit'' consists of an auto or a cross product (for HFI only) of one (or two) frequency maps or detset maps used in the likelihood. Here they are: <br />
<br />
; ''For the HFI'':<br />
* the 6 HFI frequency channels + 3 psb-only subsets, producing 45 extensions<br />
** 100, 143, 217, 353, 545, 857, 143p, 217p, 353p<br />
* 26 detsets, producing 351 extensions; the detsets used are, by frequency channel:<br />
** 100-DS1, 100-DS2,<br />
** 143-DS1, 143-DS2, 143-5, 143-6, 143-7,<br />
** 217-DS1, 217-DS2, 217-1, 217-2, 217-3, 217-4, <br />
** 353-DS1, 353-DS2, 353-1, 353-2, 353-7, 353-8,<br />
** 545-1, 545-2, 545-4,<br />
** 857-1, 857-2, 857-3, 857-4<br />
<br />
Note for HFI these (and also the associated covariance matrices) are given in separate files named ''HFI_RIMO-Beams-nnnpc_Rm.nn.fits'', where nnn is 100 or 075 and indicates the percentage (pc) of the sky included (see [[Frequency_Maps#Galactic_Plane_masks | Masks]] section).<br />
<br />
; ''For the LFI'':<br />
* the 3 LFI frequency channels (auto- products only), producing 3 extensions<br />
** 30, 44, 70<br />
* the 3 LFI 70GHz detector pairs (auto- products only), producing 3 extensions<br />
** 18 & 23, 19 & 22, 20 & 21<br />
<br />
The extension names are of the form ''BEAMWF_U1XU2'' where U1 and U2 are one (possibly the same) detection unit from one of the main groups above (i.e. there are no cross products between detsets and frequency channels, or between HFI and LFI). Each extension contains the columns:<br />
; ''NOMINAL'' (Real*4) : the beam window function proper,<br />
; ''EIGEN_n'' (Real*4, n=1-5 for the HFI, n=1-4 for the LFI): the five/four corresponding error modes.<br />
<br />
and the following keywords give further information, only for the HFI:<br />
; ''NMODES'' (Integer) : the number of EIGEN_* modes,<br />
; ''LMIN'' and ''LMAX'' (Integer) : the starting and ending (both included) multipoles of the vectors NOMINAL and EIGEN_*<br />
; ''LMIN_EM'' and ''LMAX_EM'' (Integer) : that give the range of the valid samples of the EIGEN_* vectors. Here ''LMAX_EM'' is always less than or equal to ''LMAX''. On the range ''LMAX_EM''+1 to ''LMAX'' the values of EIGEN_* are set to NaN, while the values of NOMINAL only are a Gaussian extrapolation of the lower multipole window function, only provided for convenience.<br />
; ''CORRMAT'' (string) : the name of the extension containing the corresponding beam correlation matrix<br />
<br />
And finally see also the ''COMMENTs'' of each header for more specific details.<br />
<br />
== Beam Correlation Matrix ==<br />
<br />
Two beam correlation matrices are given for the HFI, in two ''IMAGE'' extensions:<br />
; ''CORRBEAM_FREQ'' (Real*8) : for the frequency channels (21 units), 105x015 pixel matrix,<br />
; ''CORRBEAM_DSET'' (Real*8) : for the detsets (351 units), 1755x1755 pixel matrix <br />
Each is a symmetric matrix with 1-valued diagonal, made of NBEAMS*NBEAMS blocks, each block being NMODES*NMODES in size. The n$^{th}$ row- (and column-) block entry relates to the B(l) model whose name is indicated in ROWn = BEAMWF_U1XU2 keywords, and the corresponding eigenmodes are stored in a HDU of the same name. <br />
<br />
Each extension contains also the following keywords:<br />
; ''NDETS'' (Integer) : the number of detector units<br />
; ''NBEAMS'' (Integer) : the number of beams = NSETS * (NSETS+1) / 2<br />
; ''NMODES'' (Integer) : here 5<br />
; ''L_PLUS'' (Integer) : Eigenmode > 0 to break degeneracies<br />
; ''BLOCKn'' (string) : for n=1-NBEAMS, gives the name of the extension containing the beam WF and error eigenmodes for the nth block<br />
and some other ones for internal data checking and traceability<br />
<br />
No beam correlation matrices are produced by the LFI by now. And for HFI these, together with the beam window functions, are given in a file separate from the main RIMO (see subsection above).<br />
<br />
==Appendices==<br />
<br />
* [[UC_CC_Tables | Unit correction and color correction tables]]<br />
<br />
<br />
== References ==<br />
<br />
<References /><br />
<br />
<br />
[[Category:Mission products|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=The_RIMO&diff=11133The RIMO2015-02-04T10:36:26Z<p>Tvassall: /* Beam Window Functions */</p>
<hr />
<div>{{DISPLAYTITLE:The instrument model}}<br />
== Overview ==<br />
<br />
The RIMO, or ''Reduced Instrument Model'' is a FITS file or a set of FITS files containing selected instrument characteristics that are needed by users who work with the released data products. It is described in detail in ''The HFI and LFI RIMO ICD'' (ref). There will be two RIMOs, one for each instrument (the HFI RIMO will consists of several parts), which will follow the same overall structure, but will differ in the details. The type of data in the RIMO can be:<br />
<br />
; Parameter : namely scalars to give properties such as a noise level or a representative beam FWHM<br />
; Vector or Table : to give, e.g., filter transmission profiles or noise power spectra, beam window functions. When possible, namely when they are of equal length, such as the noise power spectra, an effort is made to put them together into a table, otherwise they are given as separate vectors.<br />
; Image : namely 2-D "flat" array, to give, e.g., the beam correlation matrices<br />
<br />
The FITS file begins with primary header that contains some keywords that mainly for internal use and no data. The different types of data are written into different BINTABLE (for parameters and tables) or IMAGE (for 2-D arrays) extensions, as described below. <br />
<br />
The HFI-RIMO separates the beam window functions and associated data from the main set of parameters, and this because the beam window functions are delivered for two cases covering 100% and 75% of the sky, as described in detail below.<br />
<br />
=== File Names ===<br />
<br />
; HFI: {{PLASingleFile|fileType=rimo|name=HFI_RIMO_Rn.mm.fits|link=HFI_RIMO_Rn.mm.fits}}<br />
; LFI: {{PLASingleFile|fileType=rimo|name=LFI_RIMO_Rn.mm.fits|link=LFI_RIMO_Rn.mm.fits}}<br />
<br />
== Detector-level parameter data ==<br />
<br />
<br />
The detector parameter data are given in the form of a table giving the parameter values for each detector. The table columns (whose names are in ''BOLD ITALICS'') are:<br />
<br />
; Bolometer name - ''DETECTOR'' : These are the detector names. For HFI these will be of the form ''217-3'' for SWBs or ''100-3b'' for PSBs, and for LFI they will have the form 27M or 18S. There are 52 HFI detectors and 22 LFI detectors.<br />
<br />
; Focal plane geometry parameters - ''PHI_UV'', ''THETA_UV'', and ''PSI_UV'' : These parameters give the geometry of the focal plane, or the positions of the detectors in the focal plane in the Dxx reference frame. The angles that give the rotation of the beam pattern from a fiducial orientation (forward beam direction (z-axis) pointing along the telescope line of sight, with y-axis aligned with the nominal scan direction) to their positions in the focal plane. The fiducial position is that given by the Star Tracker. All angles are in radians. These parameters are derived from observations of bright planets; see [[Detector_pointing | Detector pointing]] for details.<br />
<br />
; Polarization parameters - ''PSI_POL'', ''EPSILON'' :These are the direction of maximum polarization, defined with the beam in the fiducial orientation described above, that is, before rotation onto the detector position, and the cross-polarization contamination (or leakage). These values are determined from ground-based measurements.<br />
<br />
; Beam parameters - ''FWHM'', ''ELLIPTICITY'', ''POSANG'' : These are the mean FWHM of the scanning beam (in arcmin, the beam ellipticity (no units), and the position angle of the beam major axis. The scanning beam is that recovered from the observation of bright planets; details in [[Beams]] section.<br />
<br />
; Noise parameters - ''NET'' (LFI), ''NET_TOT'' (HFI), ''NET_WHT'' (HFI), ''NET_OOF'' (HFI), ''F_KNEE'', ''ALPHA'', ''F_MIN'' (LFI), ''F_MAX'' (LFI) : Three NETs are given: one determined from the total noise (rms of the noise timeline, excluding glitched data and other non-valid data), one determined from the white noise level of the noise amplitude spectrum, and one determined from fitting a ''1/f'' noise spectrum, described by the function <math>\sigma^2(1+(f_{knee}/f)^\alpha)</math> to the noise '''Power''' spectrum. In the latter, the ''F_KNEE'' and ''ALPHA'' parameters are the frequency where the ''1/f'' component meets the white noise level, and the slope of the former. so that the slope is is about twice the slope of the amplitude spectrum. The NETs are in units of Kcmb or MJy/sr * sqrt(s). <span style="color:red">Not sure what of this applies also to LFI</span> <br />
<br />
; Detector sampling frequency - ''F_SAMP'', which is self-explanatory. <br />
<br />
In the HFI RIMO, this table includes entries for the RTS bolometers (143-8 and 545-3), which are approximate or 0.00 when not evaluated.<br />
<br />
== Map-level parameter data ==<br />
<br />
The map-level data table contains the effective beam solid angle (total and out to different multiples of the beamFWHM) and noise information. It is written into a BINTABLE extension named ''MAP_PARAMS'' whose structure is different for HFI and LFI and is as follows. The noise description below is very simplified; a more accurate rendition can be obtained from the half-ring maps. Regarding the characterization of systematics, the user should use the survey differences.<br />
<br />
=== HFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''217''<br />
; ''OMEGA_F'', ''OMEGA_F_ERR'' (Real*4) : the full beam solid angle and its uncertainty, in armin<sup>2</sup><br />
; ''OMEGA_1'', ''OMEGA_1_DISP'' (Real*4) : the beam solid angle out to 1FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''OMEGA_2'', ''OMEGA_2_DISP'' (Real*4) : the beam solid angle out to 2FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''FWHM'' (Real*4) : FWHM of a Gaussian beam having the same (total) solid angle, in armin<sup>2</sup>. This is the best value for source flux determination<br />
; ''FWHMGAUS'' (Real*4) : FWHM derived from best Gaussian fit to beam maps, in armin<sup>2</sup>. This is the best value for source identification<br />
; ''NOISE'' (Real*4) : This is the typical noise/valid observation sample as derived from the high-''l'' spectra of the half-ring maps, in the units of the corresponding map<br />
<br />
For the Omega columns, the 'DISP' (for ''dispersion'') column gives an estimate of the spatial variation as a function of position on the sky. This is the variation induced by combining the scanning beam determined from the planet observations with the scanning strategy, as described in [[Beams]].<br />
<br />
=== LFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''030, 044, 070''<br />
; ''FWHM'' (Real*8) : FWHM of a Gaussian beam having the same (total) solid angle, in arcmin<br />
; ''NOISE'' (Real*8) : This is the average noise in T<math>\cdot</math>s<sup>1/2</sup> <br />
; ''CENTRALFREQ'' (Real*4) : This is the average central frequency in GHz<br />
; ''FWHM_EFF'', ''FWHM_EFF_SIGMA'' (Real*4) : This is the average FWHM of the effective beam, in arcmin, and its dispersion<br />
; ''ELLIPTICITY_EFF'', ''ELLIPTICITY_EFF_SIGMA'' (Real*4) : This is the average ellipticity and its dispersion<br />
; ''SOLID_ANGLE_EFF'', ''SOLID_ANGLE_EFF_SIGMA'' (Real*4) : This is the average full beam solid angle, in arcmin<sup>2</sup>, and its dispersion<br />
<br />
== Effective band transmission profiles ==<br />
<br />
The effective filter bandpasses are given in different BINTABLE extensions. The extension is named ''BANDPASS_{name}'', where ''name'' specifies the detector or the maps. For the latter, the bandpasses are a weighted average of the bandpasses of the detectors that are used to build the map, using the same weights that are used in the mapmaking. These merged bandpasses are given for the full channel maps (all detectors of the frequency channel) and for the PSBs only in each frequency channel for HFI. For details on the measurements and compilation of the bandpasses see {{PlanckPapers|planck2013-p03d}}. For details on the measurements and compilation of the LFI bandpasses see {{PlanckPapers|planck2014-a05||Planck-2015-A05}}. <br />
<br />
The bandpasses are given as 4-column tables containing:<br />
<br />
=== HFI ===<br />
<br />
; ''WAVENUMBER'' (Real*4) : the wavenumber in cm-1, conversion to GHz is accomplished by multiplying by <math>10^{-7}c</math> [mks].<br />
; ''TRANSMISSION'' (Real*4) : the transmission (normalized to 1 at the max for HFI)<br />
; ''ERROR'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile.<br />
; ''FLAG'' (Integer) : a flag indicating if the data point is an independent frequency data point (nominally the case), or an FTS instrument line shape (ILS)-interpolated data point. The frequency data has been over-sampled by a factor of ~10 to assist in CO component separation efforts {{PlanckPapers|planck2013-p03d}}{{PlanckPapers|planck2013-p03a}}.<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension. Tables with the unit conversion coefficients and color correction factors for the HFI detectors (and LFI in some instances), including uncertainty estimates based on the uncertainty of the HFI detector spectral response are given in [[UC_CC_Tables | this appendix]].<br />
<br />
=== LFI ===<br />
<br />
<span style="color:red"> there was some talk of changing wn to freq ... TBD </span> <br />
<br />
; ''WAVENUMBER'' (Real*8) : the wavenumber in GHz. ??? <br />
; ''TRANSMISSION'' (Real*8) : the transmission (normalized to have an integral of 1 for LFI)<br />
; ''UNCERTAINITY'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile (not provided for LFI)<br />
; ''FLAG'' (Character) : a flag, not used by now by the LFI<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension.<br />
<br />
== Detector noise spectra ==<br />
<br />
; HFI: these are the ring noise amplitude spectra averaged over about 5000 rings in order to give a representative spectrum. The spectra of all 50 valid bolometers are given in a single table. The spectra have a maximum frequency (Nyquist) of 90.18685Hz, also given the the ''F_NYQ'' keyword, and are built over 32768 points, giving a lower frequency of 2.75 mHz.<br />
; LFI : TBW<br />
<br />
The keyword ''F_NYQ'' gives the Nyquist frequency, and can be used together with the number of points in the spectrum to reconstruct the frequency scale. The BINTABLE has Ndetector columns by Npoints rows.<br />
<br />
<br />
== Beam Window Functions ==<br />
<br />
Beam window functions and associated error descriptions are written into a BINTABLE for each ''detection unit'', where ''detection unit'' consists of an auto or a cross product (for HFI only) of one (or two) frequency maps or detset maps used in the likelihood. Here they are: <br />
<br />
; ''For the HFI'':<br />
* the 6 HFI frequency channels + 3 psb-only subsets, producing 45 extensions<br />
** 100, 143, 217, 353, 545, 857, 143p, 217p, 353p<br />
* 26 detsets, producing 351 extensions; the detsets used are, by frequency channel:<br />
** 100-DS1, 100-DS2,<br />
** 143-DS1, 143-DS2, 143-5, 143-6, 143-7,<br />
** 217-DS1, 217-DS2, 217-1, 217-2, 217-3, 217-4, <br />
** 353-DS1, 353-DS2, 353-1, 353-2, 353-7, 353-8,<br />
** 545-1, 545-2, 545-4,<br />
** 857-1, 857-2, 857-3, 857-4<br />
<br />
Note for HFI these (and also the associated covariance matrices) are given in separate files named ''HFI_RIMO-Beams-nnnpc_Rm.nn.fits'', where nnn is 100 or 075 and indicates the percentage (pc) of the sky included (see [[Frequency_Maps#Galactic_Plane_masks | Masks]] section).<br />
<br />
; ''For the LFI'':<br />
* the 3 LFI frequency channels (auto- products only), producing 3 extensions<br />
** 30, 44, 70<br />
* the 3 LFI detector pairs (auto- products only), producing 3 extensions<br />
** 18 & 23, 19 & 22, 20 & 21<br />
<br />
The extension names are of the form ''BEAMWF_U1XU2'' where U1 and U2 are one (possibly the same) detection unit from one of the main groups above (i.e. there are no cross products between detsets and frequency channels, or between HFI and LFI). Each extension contains the columns:<br />
; ''NOMINAL'' (Real*4) : the beam window function proper,<br />
; ''EIGEN_n'' (Real*4, n=1-5 for the HFI, n=1-4 for the LFI): the five/four corresponding error modes.<br />
<br />
and the following keywords give further information, only for the HFI:<br />
; ''NMODES'' (Integer) : the number of EIGEN_* modes,<br />
; ''LMIN'' and ''LMAX'' (Integer) : the starting and ending (both included) multipoles of the vectors NOMINAL and EIGEN_*<br />
; ''LMIN_EM'' and ''LMAX_EM'' (Integer) : that give the range of the valid samples of the EIGEN_* vectors. Here ''LMAX_EM'' is always less than or equal to ''LMAX''. On the range ''LMAX_EM''+1 to ''LMAX'' the values of EIGEN_* are set to NaN, while the values of NOMINAL only are a Gaussian extrapolation of the lower multipole window function, only provided for convenience.<br />
; ''CORRMAT'' (string) : the name of the extension containing the corresponding beam correlation matrix<br />
<br />
And finally see also the ''COMMENTs'' of each header for more specific details.<br />
<br />
== Beam Correlation Matrix ==<br />
<br />
Two beam correlation matrices are given for the HFI, in two ''IMAGE'' extensions:<br />
; ''CORRBEAM_FREQ'' (Real*8) : for the frequency channels (21 units), 105x015 pixel matrix,<br />
; ''CORRBEAM_DSET'' (Real*8) : for the detsets (351 units), 1755x1755 pixel matrix <br />
Each is a symmetric matrix with 1-valued diagonal, made of NBEAMS*NBEAMS blocks, each block being NMODES*NMODES in size. The n$^{th}$ row- (and column-) block entry relates to the B(l) model whose name is indicated in ROWn = BEAMWF_U1XU2 keywords, and the corresponding eigenmodes are stored in a HDU of the same name. <br />
<br />
Each extension contains also the following keywords:<br />
; ''NDETS'' (Integer) : the number of detector units<br />
; ''NBEAMS'' (Integer) : the number of beams = NSETS * (NSETS+1) / 2<br />
; ''NMODES'' (Integer) : here 5<br />
; ''L_PLUS'' (Integer) : Eigenmode > 0 to break degeneracies<br />
; ''BLOCKn'' (string) : for n=1-NBEAMS, gives the name of the extension containing the beam WF and error eigenmodes for the nth block<br />
and some other ones for internal data checking and traceability<br />
<br />
No beam correlation matrices are produced by the LFI by now. And for HFI these, together with the beam window functions, are given in a file separate from the main RIMO (see subsection above).<br />
<br />
==Appendices==<br />
<br />
* [[UC_CC_Tables | Unit correction and color correction tables]]<br />
<br />
<br />
== References ==<br />
<br />
<References /><br />
<br />
<br />
[[Category:Mission products|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=The_RIMO&diff=11127The RIMO2015-02-04T10:30:17Z<p>Tvassall: /* Detector-level parameter data */</p>
<hr />
<div>{{DISPLAYTITLE:The instrument model}}<br />
== Overview ==<br />
<br />
The RIMO, or ''Reduced Instrument Model'' is a FITS file or a set of FITS files containing selected instrument characteristics that are needed by users who work with the released data products. It is described in detail in ''The HFI and LFI RIMO ICD'' (ref). There will be two RIMOs, one for each instrument (the HFI RIMO will consists of several parts), which will follow the same overall structure, but will differ in the details. The type of data in the RIMO can be:<br />
<br />
; Parameter : namely scalars to give properties such as a noise level or a representative beam FWHM<br />
; Vector or Table : to give, e.g., filter transmission profiles or noise power spectra, beam window functions. When possible, namely when they are of equal length, such as the noise power spectra, an effort is made to put them together into a table, otherwise they are given as separate vectors.<br />
; Image : namely 2-D "flat" array, to give, e.g., the beam correlation matrices<br />
<br />
The FITS file begins with primary header that contains some keywords that mainly for internal use and no data. The different types of data are written into different BINTABLE (for parameters and tables) or IMAGE (for 2-D arrays) extensions, as described below. <br />
<br />
The HFI-RIMO separates the beam window functions and associated data from the main set of parameters, and this because the beam window functions are delivered for two cases covering 100% and 75% of the sky, as described in detail below.<br />
<br />
=== File Names ===<br />
<br />
; HFI: {{PLASingleFile|fileType=rimo|name=HFI_RIMO_Rn.mm.fits|link=HFI_RIMO_Rn.mm.fits}}<br />
; LFI: {{PLASingleFile|fileType=rimo|name=LFI_RIMO_Rn.mm.fits|link=LFI_RIMO_Rn.mm.fits}}<br />
<br />
== Detector-level parameter data ==<br />
<br />
<br />
The detector parameter data are given in the form of a table giving the parameter values for each detector. The table columns (whose names are in ''BOLD ITALICS'') are:<br />
<br />
; Bolometer name - ''DETECTOR'' : These are the detector names. For HFI these will be of the form ''217-3'' for SWBs or ''100-3b'' for PSBs, and for LFI they will have the form 27M or 18S. There are 52 HFI detectors and 22 LFI detectors.<br />
<br />
; Focal plane geometry parameters - ''PHI_UV'', ''THETA_UV'', and ''PSI_UV'' : These parameters give the geometry of the focal plane, or the positions of the detectors in the focal plane in the Dxx reference frame. The angles that give the rotation of the beam pattern from a fiducial orientation (forward beam direction (z-axis) pointing along the telescope line of sight, with y-axis aligned with the nominal scan direction) to their positions in the focal plane. The fiducial position is that given by the Star Tracker. All angles are in radians. These parameters are derived from observations of bright planets; see [[Detector_pointing | Detector pointing]] for details.<br />
<br />
; Polarization parameters - ''PSI_POL'', ''EPSILON'' :These are the direction of maximum polarization, defined with the beam in the fiducial orientation described above, that is, before rotation onto the detector position, and the cross-polarization contamination (or leakage). These values are determined from ground-based measurements.<br />
<br />
; Beam parameters - ''FWHM'', ''ELLIPTICITY'', ''POSANG'' : These are the mean FWHM of the scanning beam (in arcmin, the beam ellipticity (no units), and the position angle of the beam major axis. The scanning beam is that recovered from the observation of bright planets; details in [[Beams]] section.<br />
<br />
; Noise parameters - ''NET'' (LFI), ''NET_TOT'' (HFI), ''NET_WHT'' (HFI), ''NET_OOF'' (HFI), ''F_KNEE'', ''ALPHA'', ''F_MIN'' (LFI), ''F_MAX'' (LFI) : Three NETs are given: one determined from the total noise (rms of the noise timeline, excluding glitched data and other non-valid data), one determined from the white noise level of the noise amplitude spectrum, and one determined from fitting a ''1/f'' noise spectrum, described by the function <math>\sigma^2(1+(f_{knee}/f)^\alpha)</math> to the noise '''Power''' spectrum. In the latter, the ''F_KNEE'' and ''ALPHA'' parameters are the frequency where the ''1/f'' component meets the white noise level, and the slope of the former. so that the slope is is about twice the slope of the amplitude spectrum. The NETs are in units of Kcmb or MJy/sr * sqrt(s). <span style="color:red">Not sure what of this applies also to LFI</span> <br />
<br />
; Detector sampling frequency - ''F_SAMP'', which is self-explanatory. <br />
<br />
In the HFI RIMO, this table includes entries for the RTS bolometers (143-8 and 545-3), which are approximate or 0.00 when not evaluated.<br />
<br />
== Map-level parameter data ==<br />
<br />
The map-level data table contains the effective beam solid angle (total and out to different multiples of the beamFWHM) and noise information. It is written into a BINTABLE extension named ''MAP_PARAMS'' whose structure is different for HFI and LFI and is as follows. The noise description below is very simplified; a more accurate rendition can be obtained from the half-ring maps. Regarding the characterization of systematics, the user should use the survey differences.<br />
<br />
=== HFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''217''<br />
; ''OMEGA_F'', ''OMEGA_F_ERR'' (Real*4) : the full beam solid angle and its uncertainty, in armin<sup>2</sup><br />
; ''OMEGA_1'', ''OMEGA_1_DISP'' (Real*4) : the beam solid angle out to 1FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''OMEGA_2'', ''OMEGA_2_DISP'' (Real*4) : the beam solid angle out to 2FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''FWHM'' (Real*4) : FWHM of a Gaussian beam having the same (total) solid angle, in armin<sup>2</sup>. This is the best value for source flux determination<br />
; ''FWHMGAUS'' (Real*4) : FWHM derived from best Gaussian fit to beam maps, in armin<sup>2</sup>. This is the best value for source identification<br />
; ''NOISE'' (Real*4) : This is the typical noise/valid observation sample as derived from the high-''l'' spectra of the half-ring maps, in the units of the corresponding map<br />
<br />
For the Omega columns, the 'DISP' (for ''dispersion'') column gives an estimate of the spatial variation as a function of position on the sky. This is the variation induced by combining the scanning beam determined from the planet observations with the scanning strategy, as described in [[Beams]].<br />
<br />
=== LFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''030, 044, 070''<br />
; ''FWHM'' (Real*8) : FWHM of a Gaussian beam having the same (total) solid angle, in arcmin<br />
; ''NOISE'' (Real*8) : This is the average noise in T<math>\cdot</math>s<sup>1/2</sup> <br />
; ''CENTRALFREQ'' (Real*4) : This is the average central frequency in GHz<br />
; ''FWHM_EFF'', ''FWHM_EFF_SIGMA'' (Real*4) : This is the average FWHM of the effective beam, in arcmin, and its dispersion<br />
; ''ELLIPTICITY_EFF'', ''ELLIPTICITY_EFF_SIGMA'' (Real*4) : This is the average ellipticity and its dispersion<br />
; ''SOLID_ANGLE_EFF'', ''SOLID_ANGLE_EFF_SIGMA'' (Real*4) : This is the average full beam solid angle, in arcmin<sup>2</sup>, and its dispersion<br />
<br />
== Effective band transmission profiles ==<br />
<br />
The effective filter bandpasses are given in different BINTABLE extensions. The extension is named ''BANDPASS_{name}'', where ''name'' specifies the detector or the maps. For the latter, the bandpasses are a weighted average of the bandpasses of the detectors that are used to build the map, using the same weights that are used in the mapmaking. These merged bandpasses are given for the full channel maps (all detectors of the frequency channel) and for the PSBs only in each frequency channel for HFI. For details on the measurements and compilation of the bandpasses see {{PlanckPapers|planck2013-p03d}}. For details on the measurements and compilation of the LFI bandpasses see {{PlanckPapers|planck2014-a05||Planck-2015-A05}}. <br />
<br />
The bandpasses are given as 4-column tables containing:<br />
<br />
=== HFI ===<br />
<br />
; ''WAVENUMBER'' (Real*4) : the wavenumber in cm-1, conversion to GHz is accomplished by multiplying by <math>10^{-7}c</math> [mks].<br />
; ''TRANSMISSION'' (Real*4) : the transmission (normalized to 1 at the max for HFI)<br />
; ''ERROR'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile.<br />
; ''FLAG'' (Integer) : a flag indicating if the data point is an independent frequency data point (nominally the case), or an FTS instrument line shape (ILS)-interpolated data point. The frequency data has been over-sampled by a factor of ~10 to assist in CO component separation efforts {{PlanckPapers|planck2013-p03d}}{{PlanckPapers|planck2013-p03a}}.<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension. Tables with the unit conversion coefficients and color correction factors for the HFI detectors (and LFI in some instances), including uncertainty estimates based on the uncertainty of the HFI detector spectral response are given in [[UC_CC_Tables | this appendix]].<br />
<br />
=== LFI ===<br />
<br />
<span style="color:red"> there was some talk of changing wn to freq ... TBD </span> <br />
<br />
; ''WAVENUMBER'' (Real*8) : the wavenumber in GHz. ??? <br />
; ''TRANSMISSION'' (Real*8) : the transmission (normalized to have an integral of 1 for LFI)<br />
; ''UNCERTAINITY'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile (not provided for LFI)<br />
; ''FLAG'' (Character) : a flag, not used by now by the LFI<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension.<br />
<br />
== Detector noise spectra ==<br />
<br />
; HFI: these are the ring noise amplitude spectra averaged over about 5000 rings in order to give a representative spectrum. The spectra of all 50 valid bolometers are given in a single table. The spectra have a maximum frequency (Nyquist) of 90.18685Hz, also given the the ''F_NYQ'' keyword, and are built over 32768 points, giving a lower frequency of 2.75 mHz.<br />
; LFI : TBW<br />
<br />
The keyword ''F_NYQ'' gives the Nyquist frequency, and can be used together with the number of points in the spectrum to reconstruct the frequency scale. The BINTABLE has Ndetector columns by Npoints rows.<br />
<br />
<br />
== Beam Window Functions ==<br />
<br />
Beam window functions and associated error descriptions are written into a BINTABLE for each ''detection unit'', where ''detection unit'' consists of an auto or a cross product (for HFI only) of one (or two) frequency maps or detset maps used in the likelihood. Here they are: <br />
<br />
; ''For the HFI'':<br />
* the 6 HFI frequency channels + 3 psb-only subsets, producing 45 extensions<br />
** 100, 143, 217, 353, 545, 857, 143p, 217p, 353p<br />
* 26 detsets, producing 351 extensions; the detsets used are, by frequency channel:<br />
** 100-DS1, 100-DS2,<br />
** 143-DS1, 143-DS2, 143-5, 143-6, 143-7,<br />
** 217-DS1, 217-DS2, 217-1, 217-2, 217-3, 217-4, <br />
** 353-DS1, 353-DS2, 353-1, 353-2, 353-7, 353-8,<br />
** 545-1, 545-2, 545-4,<br />
** 857-1, 857-2, 857-3, 857-4<br />
<br />
Note for HFI these (and also the associated covariance matrices) are given in separate files named ''HFI_RIMO-Beams-nnnpc_Rm.nn.fits'', where nnn is 100 or 075 and indicates the percentage (pc) of the sky included (see [[Frequency_Maps#Galactic_Plane_masks | Masks]] section).<br />
<br />
; ''For the LFI'':<br />
* the 3 LFI frequency channels (auto- products only), producing 3 extensions<br />
** 30, 44, 70<br />
<br />
The extension names are of the form ''BEAMWF_U1XU2'' where U1 and U2 are one (possibly the same) detection unit from one of the main groups above (i.e. there are no cross products between detsets and frequency channels, or between HFI and LFI). Each extension contains the columns:<br />
; ''NOMINAL'' (Real*4) : the beam window function proper,<br />
; ''EIGEN_n'' (Real*4, n=1-5 for the HFI, n=1-4 for the LFI): the five/four corresponding error modes.<br />
<br />
and the following keywords give further information, only for the HFI:<br />
; ''NMODES'' (Integer) : the number of EIGEN_* modes,<br />
; ''LMIN'' and ''LMAX'' (Integer) : the starting and ending (both included) multipoles of the vectors NOMINAL and EIGEN_*<br />
; ''LMIN_EM'' and ''LMAX_EM'' (Integer) : that give the range of the valid samples of the EIGEN_* vectors. Here ''LMAX_EM'' is always less than or equal to ''LMAX''. On the range ''LMAX_EM''+1 to ''LMAX'' the values of EIGEN_* are set to NaN, while the values of NOMINAL only are a Gaussian extrapolation of the lower multipole window function, only provided for convenience.<br />
; ''CORRMAT'' (string) : the name of the extension containing the corresponding beam correlation matrix<br />
<br />
And finally see also the ''COMMENTs'' of each header for more specific details.<br />
<br />
== Beam Correlation Matrix ==<br />
<br />
Two beam correlation matrices are given for the HFI, in two ''IMAGE'' extensions:<br />
; ''CORRBEAM_FREQ'' (Real*8) : for the frequency channels (21 units), 105x015 pixel matrix,<br />
; ''CORRBEAM_DSET'' (Real*8) : for the detsets (351 units), 1755x1755 pixel matrix <br />
Each is a symmetric matrix with 1-valued diagonal, made of NBEAMS*NBEAMS blocks, each block being NMODES*NMODES in size. The n$^{th}$ row- (and column-) block entry relates to the B(l) model whose name is indicated in ROWn = BEAMWF_U1XU2 keywords, and the corresponding eigenmodes are stored in a HDU of the same name. <br />
<br />
Each extension contains also the following keywords:<br />
; ''NDETS'' (Integer) : the number of detector units<br />
; ''NBEAMS'' (Integer) : the number of beams = NSETS * (NSETS+1) / 2<br />
; ''NMODES'' (Integer) : here 5<br />
; ''L_PLUS'' (Integer) : Eigenmode > 0 to break degeneracies<br />
; ''BLOCKn'' (string) : for n=1-NBEAMS, gives the name of the extension containing the beam WF and error eigenmodes for the nth block<br />
and some other ones for internal data checking and traceability<br />
<br />
No beam correlation matrices are produced by the LFI by now. And for HFI these, together with the beam window functions, are given in a file separate from the main RIMO (see subsection above).<br />
<br />
==Appendices==<br />
<br />
* [[UC_CC_Tables | Unit correction and color correction tables]]<br />
<br />
<br />
== References ==<br />
<br />
<References /><br />
<br />
<br />
[[Category:Mission products|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=The_RIMO&diff=11125The RIMO2015-02-04T10:29:47Z<p>Tvassall: /* Detector-level parameter data */</p>
<hr />
<div>{{DISPLAYTITLE:The instrument model}}<br />
== Overview ==<br />
<br />
The RIMO, or ''Reduced Instrument Model'' is a FITS file or a set of FITS files containing selected instrument characteristics that are needed by users who work with the released data products. It is described in detail in ''The HFI and LFI RIMO ICD'' (ref). There will be two RIMOs, one for each instrument (the HFI RIMO will consists of several parts), which will follow the same overall structure, but will differ in the details. The type of data in the RIMO can be:<br />
<br />
; Parameter : namely scalars to give properties such as a noise level or a representative beam FWHM<br />
; Vector or Table : to give, e.g., filter transmission profiles or noise power spectra, beam window functions. When possible, namely when they are of equal length, such as the noise power spectra, an effort is made to put them together into a table, otherwise they are given as separate vectors.<br />
; Image : namely 2-D "flat" array, to give, e.g., the beam correlation matrices<br />
<br />
The FITS file begins with primary header that contains some keywords that mainly for internal use and no data. The different types of data are written into different BINTABLE (for parameters and tables) or IMAGE (for 2-D arrays) extensions, as described below. <br />
<br />
The HFI-RIMO separates the beam window functions and associated data from the main set of parameters, and this because the beam window functions are delivered for two cases covering 100% and 75% of the sky, as described in detail below.<br />
<br />
=== File Names ===<br />
<br />
; HFI: {{PLASingleFile|fileType=rimo|name=HFI_RIMO_Rn.mm.fits|link=HFI_RIMO_Rn.mm.fits}}<br />
; LFI: {{PLASingleFile|fileType=rimo|name=LFI_RIMO_Rn.mm.fits|link=LFI_RIMO_Rn.mm.fits}}<br />
<br />
== Detector-level parameter data ==<br />
<br />
<br />
The detector parameter data are given in the form of a table giving the parameter values for each detector. The table columns (whose names are in ''BOLD ITALICS'') are:<br />
<br />
; Bolometer name - ''DETECTOR'' : These are the detector names. For HFI these will be of the form ''217-3'' for SWBs or ''100-3b'' for PSBs, and for LFI they will have the form 27M or 18S. There are 52 HFI detectors and 22 LFI detectors.<br />
<br />
; Focal plane geometry parameters - ''PHI_UV'', ''THETA_UV'', and ''PSI_UV'' : These parameters give the geometry of the focal plane, or the positions of the detectors in the focal plane in the Dxx reference frame. The angles that give the rotation of the beam pattern from a fiducial orientation (forward beam direction (z-axis) pointing along the telescope line of sight, with y-axis aligned with the nominal scan direction) to their positions in the focal plane. The fiducial position is that given by the Star Tracker. All angles are in radians. These parameters are derived from observations of bright planets; see [[Detector_pointing | Detector pointing]] for details.<br />
<br />
; Polarization parameters - ''PSI_POL'', ''EPSILON'' :These are the direction of maximum polarization, defined with the beam in the fiducial orientation described above, that is, before rotation onto the detector position, and the cross-polarization contamination (or leakage). These values are determined from ground-based measurements.<br />
<br />
; Beam parameters - ''FWHM'', ''ELLIPTICITY'', ''POSANG'' : These are the mean FWHM of the scanning beam (in arcmin, the beam ellipticity (no units), and the position angle of the beam major axis. The scanning beam is that recovered from the observation of bright planets; details in [[Beams]] section.<br />
<br />
; Noise parameters - ''NET'' (LFI),''NET_TOT'' (HFI), ''NET_WHT'' (HFI), ''NET_OOF'' (HFI), ''F_KNEE'', ''ALPHA'', ''F_MIN'' (LFI), ''F_MAX'' (LFI) : Three NETs are given: one determined from the total noise (rms of the noise timeline, excluding glitched data and other non-valid data), one determined from the white noise level of the noise amplitude spectrum, and one determined from fitting a ''1/f'' noise spectrum, described by the function <math>\sigma^2(1+(f_{knee}/f)^\alpha)</math> to the noise '''Power''' spectrum. In the latter, the ''F_KNEE'' and ''ALPHA'' parameters are the frequency where the ''1/f'' component meets the white noise level, and the slope of the former. so that the slope is is about twice the slope of the amplitude spectrum. The NETs are in units of Kcmb or MJy/sr * sqrt(s). <span style="color:red">Not sure what of this applies also to LFI</span> <br />
<br />
; Detector sampling frequency - ''F_SAMP'', which is self-explanatory. <br />
<br />
In the HFI RIMO, this table includes entries for the RTS bolometers (143-8 and 545-3), which are approximate or 0.00 when not evaluated.<br />
<br />
== Map-level parameter data ==<br />
<br />
The map-level data table contains the effective beam solid angle (total and out to different multiples of the beamFWHM) and noise information. It is written into a BINTABLE extension named ''MAP_PARAMS'' whose structure is different for HFI and LFI and is as follows. The noise description below is very simplified; a more accurate rendition can be obtained from the half-ring maps. Regarding the characterization of systematics, the user should use the survey differences.<br />
<br />
=== HFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''217''<br />
; ''OMEGA_F'', ''OMEGA_F_ERR'' (Real*4) : the full beam solid angle and its uncertainty, in armin<sup>2</sup><br />
; ''OMEGA_1'', ''OMEGA_1_DISP'' (Real*4) : the beam solid angle out to 1FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''OMEGA_2'', ''OMEGA_2_DISP'' (Real*4) : the beam solid angle out to 2FWHM, and its dispersion, in arcmin<sup>2</sup><br />
; ''FWHM'' (Real*4) : FWHM of a Gaussian beam having the same (total) solid angle, in armin<sup>2</sup>. This is the best value for source flux determination<br />
; ''FWHMGAUS'' (Real*4) : FWHM derived from best Gaussian fit to beam maps, in armin<sup>2</sup>. This is the best value for source identification<br />
; ''NOISE'' (Real*4) : This is the typical noise/valid observation sample as derived from the high-''l'' spectra of the half-ring maps, in the units of the corresponding map<br />
<br />
For the Omega columns, the 'DISP' (for ''dispersion'') column gives an estimate of the spatial variation as a function of position on the sky. This is the variation induced by combining the scanning beam determined from the planet observations with the scanning strategy, as described in [[Beams]].<br />
<br />
=== LFI ===<br />
<br />
; ''FREQUENCY'' (String) : a 3-digit string giving the reference frequency in GHz, i.e., of the form ''030, 044, 070''<br />
; ''FWHM'' (Real*8) : FWHM of a Gaussian beam having the same (total) solid angle, in arcmin<br />
; ''NOISE'' (Real*8) : This is the average noise in T<math>\cdot</math>s<sup>1/2</sup> <br />
; ''CENTRALFREQ'' (Real*4) : This is the average central frequency in GHz<br />
; ''FWHM_EFF'', ''FWHM_EFF_SIGMA'' (Real*4) : This is the average FWHM of the effective beam, in arcmin, and its dispersion<br />
; ''ELLIPTICITY_EFF'', ''ELLIPTICITY_EFF_SIGMA'' (Real*4) : This is the average ellipticity and its dispersion<br />
; ''SOLID_ANGLE_EFF'', ''SOLID_ANGLE_EFF_SIGMA'' (Real*4) : This is the average full beam solid angle, in arcmin<sup>2</sup>, and its dispersion<br />
<br />
== Effective band transmission profiles ==<br />
<br />
The effective filter bandpasses are given in different BINTABLE extensions. The extension is named ''BANDPASS_{name}'', where ''name'' specifies the detector or the maps. For the latter, the bandpasses are a weighted average of the bandpasses of the detectors that are used to build the map, using the same weights that are used in the mapmaking. These merged bandpasses are given for the full channel maps (all detectors of the frequency channel) and for the PSBs only in each frequency channel for HFI. For details on the measurements and compilation of the bandpasses see {{PlanckPapers|planck2013-p03d}}. For details on the measurements and compilation of the LFI bandpasses see {{PlanckPapers|planck2014-a05||Planck-2015-A05}}. <br />
<br />
The bandpasses are given as 4-column tables containing:<br />
<br />
=== HFI ===<br />
<br />
; ''WAVENUMBER'' (Real*4) : the wavenumber in cm-1, conversion to GHz is accomplished by multiplying by <math>10^{-7}c</math> [mks].<br />
; ''TRANSMISSION'' (Real*4) : the transmission (normalized to 1 at the max for HFI)<br />
; ''ERROR'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile.<br />
; ''FLAG'' (Integer) : a flag indicating if the data point is an independent frequency data point (nominally the case), or an FTS instrument line shape (ILS)-interpolated data point. The frequency data has been over-sampled by a factor of ~10 to assist in CO component separation efforts {{PlanckPapers|planck2013-p03d}}{{PlanckPapers|planck2013-p03a}}.<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension. Tables with the unit conversion coefficients and color correction factors for the HFI detectors (and LFI in some instances), including uncertainty estimates based on the uncertainty of the HFI detector spectral response are given in [[UC_CC_Tables | this appendix]].<br />
<br />
=== LFI ===<br />
<br />
<span style="color:red"> there was some talk of changing wn to freq ... TBD </span> <br />
<br />
; ''WAVENUMBER'' (Real*8) : the wavenumber in GHz. ??? <br />
; ''TRANSMISSION'' (Real*8) : the transmission (normalized to have an integral of 1 for LFI)<br />
; ''UNCERTAINITY'' (Real*4) : the statistical <math>1\sigma</math> uncertainty for the transmission profile (not provided for LFI)<br />
; ''FLAG'' (Character) : a flag, not used by now by the LFI<br />
<br />
The number of rows will differ among the different extensions, but are the same, by construction, within each extension.<br />
<br />
== Detector noise spectra ==<br />
<br />
; HFI: these are the ring noise amplitude spectra averaged over about 5000 rings in order to give a representative spectrum. The spectra of all 50 valid bolometers are given in a single table. The spectra have a maximum frequency (Nyquist) of 90.18685Hz, also given the the ''F_NYQ'' keyword, and are built over 32768 points, giving a lower frequency of 2.75 mHz.<br />
; LFI : TBW<br />
<br />
The keyword ''F_NYQ'' gives the Nyquist frequency, and can be used together with the number of points in the spectrum to reconstruct the frequency scale. The BINTABLE has Ndetector columns by Npoints rows.<br />
<br />
<br />
== Beam Window Functions ==<br />
<br />
Beam window functions and associated error descriptions are written into a BINTABLE for each ''detection unit'', where ''detection unit'' consists of an auto or a cross product (for HFI only) of one (or two) frequency maps or detset maps used in the likelihood. Here they are: <br />
<br />
; ''For the HFI'':<br />
* the 6 HFI frequency channels + 3 psb-only subsets, producing 45 extensions<br />
** 100, 143, 217, 353, 545, 857, 143p, 217p, 353p<br />
* 26 detsets, producing 351 extensions; the detsets used are, by frequency channel:<br />
** 100-DS1, 100-DS2,<br />
** 143-DS1, 143-DS2, 143-5, 143-6, 143-7,<br />
** 217-DS1, 217-DS2, 217-1, 217-2, 217-3, 217-4, <br />
** 353-DS1, 353-DS2, 353-1, 353-2, 353-7, 353-8,<br />
** 545-1, 545-2, 545-4,<br />
** 857-1, 857-2, 857-3, 857-4<br />
<br />
Note for HFI these (and also the associated covariance matrices) are given in separate files named ''HFI_RIMO-Beams-nnnpc_Rm.nn.fits'', where nnn is 100 or 075 and indicates the percentage (pc) of the sky included (see [[Frequency_Maps#Galactic_Plane_masks | Masks]] section).<br />
<br />
; ''For the LFI'':<br />
* the 3 LFI frequency channels (auto- products only), producing 3 extensions<br />
** 30, 44, 70<br />
<br />
The extension names are of the form ''BEAMWF_U1XU2'' where U1 and U2 are one (possibly the same) detection unit from one of the main groups above (i.e. there are no cross products between detsets and frequency channels, or between HFI and LFI). Each extension contains the columns:<br />
; ''NOMINAL'' (Real*4) : the beam window function proper,<br />
; ''EIGEN_n'' (Real*4, n=1-5 for the HFI, n=1-4 for the LFI): the five/four corresponding error modes.<br />
<br />
and the following keywords give further information, only for the HFI:<br />
; ''NMODES'' (Integer) : the number of EIGEN_* modes,<br />
; ''LMIN'' and ''LMAX'' (Integer) : the starting and ending (both included) multipoles of the vectors NOMINAL and EIGEN_*<br />
; ''LMIN_EM'' and ''LMAX_EM'' (Integer) : that give the range of the valid samples of the EIGEN_* vectors. Here ''LMAX_EM'' is always less than or equal to ''LMAX''. On the range ''LMAX_EM''+1 to ''LMAX'' the values of EIGEN_* are set to NaN, while the values of NOMINAL only are a Gaussian extrapolation of the lower multipole window function, only provided for convenience.<br />
; ''CORRMAT'' (string) : the name of the extension containing the corresponding beam correlation matrix<br />
<br />
And finally see also the ''COMMENTs'' of each header for more specific details.<br />
<br />
== Beam Correlation Matrix ==<br />
<br />
Two beam correlation matrices are given for the HFI, in two ''IMAGE'' extensions:<br />
; ''CORRBEAM_FREQ'' (Real*8) : for the frequency channels (21 units), 105x015 pixel matrix,<br />
; ''CORRBEAM_DSET'' (Real*8) : for the detsets (351 units), 1755x1755 pixel matrix <br />
Each is a symmetric matrix with 1-valued diagonal, made of NBEAMS*NBEAMS blocks, each block being NMODES*NMODES in size. The n$^{th}$ row- (and column-) block entry relates to the B(l) model whose name is indicated in ROWn = BEAMWF_U1XU2 keywords, and the corresponding eigenmodes are stored in a HDU of the same name. <br />
<br />
Each extension contains also the following keywords:<br />
; ''NDETS'' (Integer) : the number of detector units<br />
; ''NBEAMS'' (Integer) : the number of beams = NSETS * (NSETS+1) / 2<br />
; ''NMODES'' (Integer) : here 5<br />
; ''L_PLUS'' (Integer) : Eigenmode > 0 to break degeneracies<br />
; ''BLOCKn'' (string) : for n=1-NBEAMS, gives the name of the extension containing the beam WF and error eigenmodes for the nth block<br />
and some other ones for internal data checking and traceability<br />
<br />
No beam correlation matrices are produced by the LFI by now. And for HFI these, together with the beam window functions, are given in a file separate from the main RIMO (see subsection above).<br />
<br />
==Appendices==<br />
<br />
* [[UC_CC_Tables | Unit correction and color correction tables]]<br />
<br />
<br />
== References ==<br />
<br />
<References /><br />
<br />
<br />
[[Category:Mission products|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Beams_LFI&diff=11121Beams LFI2015-02-04T10:09:53Z<p>Tvassall: /* Polarized Scanning Beams and Focal Plane calibration */</p>
<hr />
<div>{{DISPLAYTITLE:Beams}}<br />
<br />
== Overview ==<br />
<br />
LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.<br />
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]). <br />
<br />
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completeness and they are not representative of flight beams.''']]<br />
<br />
Details are given in {{PlanckPapers|planck2014-a05||Planck-2015-A05}}.<br />
<br />
<!--<br />
<br />
For the beam we consider these three regions:<br />
<br />
<br />
; main beam: is the portion of the pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.<br />
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called <b>intermediate beam</b>).<br />
; far sidelobes: is the pattern at angular regions more than 5 degrees from the beam center.<br />
--><br />
<br />
== Polarized Scanning Beams and Focal Plane calibration ==<br />
<br />
As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurements using Jupiter transits. The parameters that characterise the beam pointing are the following:<br />
<br />
* THETA_UV (<math>\theta_{uv}</math>)<br />
* PHI_UV (<math>\phi_{uv}</math>)<br />
<br />
They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as <br />
<br />
<math>\theta_{uv} = \arcsin(u^2+v^2)</math><br />
<br />
<math>\phi_{uv} = \arctan(v/u)</math><br />
<br />
Two additional angles are used to characterize the beams in the RIMO: <br />
<br />
* PSI_UV (<math>\psi_{uv}</math>)<br />
* PSI_POL (<math>\psi_{pol}</math>)<br />
<br />
<math>\psi_{uv}</math> and <math>\psi_{pol}</math> are '''not''' derived from measurements but they are estimated from '''optical simulations'''. They are the quantities that represent the polarization direction of each beam, in the following approximation: '''the M- and S- beams of the same RCA point at the same direction on the sky'''.<br />
<br />
The polarized scanning beams have been evaluated from optical simulations using GRASP Physical Optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM) {{PlanckPapers|tauber2010b}}. <br />
<br />
The Radio Frequency Tuned Model, called RFTM, was implemented to fit the in-flight beam measurements with the electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross polar beam in flight, we have relied on simulations validated by accurate beam measurements.<br />
<br />
The model beams were monochromatic and were computed throughout a 6 GHz band around the Optical Center Frequency (OCF) with non-regular step (denser sampling where the band-pass was higher). For the RFTM model the OCF were at <math>28.0, \, 44.0, \, 70.0</math> GHz. <br />
<br />
For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the [[The_RIMO#LFI_2|RIMO]] transmission function.<br />
<br />
The delivered [[Scanning_Beams|products]] include the in-band averaged Stokes scanning maps of Main Beams, Intermediate Beams and Sidelobes.<br />
<br />
== Effective beams ==<br />
<br />
The '''effective beam''' is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky.<br />
It captures the complete information about the difference between the true and observed image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map. <br />
<br />
The full algebra involving the effective beams for temperature and polarisation was presented in {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature sky <math>\widetilde{\mathbf{T}} </math> is a convolution of the true sky <math>\mathbf{T} </math> and the effective beam <math>\mathbf{B}</math>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<math>t</math> is time samples, <math>A_{ti}</math> is <math>1</math> if the pointing direction falls in pixel number <math>i</math>, else it is <math>0</math>, <math>\mathbf{p}_t</math> represents the exact pointing direction (not approximated by the pixel centre location), and <math>\hat{\mathbf{r}}_j</math> is the centre of the pixel number <math>j</math>, where the scanbeam <math>b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)</math> is being evaluated (if the pointing direction falls within the cut-off radius of <math>\sim 2.5 \times</math> FWHM.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^T \ = \ \Delta\Omega \sum_j \mathbf{B}_{ij} \cdot (I \quad Q \quad U)_j^T,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\mathbf{B}_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^T_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^T(\hat{\mathbf{n}}_j,\hat{\mathbf{p}}_t) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
and <math>\mathbf{w}_t </math>and <math>\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) </math> are the the polarisation weight vectors, as defined in \cite{mitra2010}.<br />
<br />
The task is to compute <math>B_{ij}</math> for temperature only beams and the <math>3 \times 3</math> matrices <math>\mathbf{B}_{ij}</math> for each pixel <math>i</math>, at every neighbouring pixel <math>j</math> that fall within the cut-off radius around the the center of the <math>i^\text{th}</math> pixel.<br />
<br />
<br />
<br />
The effective beam is computed by stacking within a small field around each pixel of the HEALPix sky map. Due to the particular features of Planck scanning strategy coupled to the beam asymmetries in the focal plane, and data processing of the bolometer and radiometer TOIs, the resulting Planck effective beams vary over the sky. <br />
<br />
FEBeCoP, given information on Planck scanning beams and detector pointing during a mission period of interest, provides the pixelized stamps of both the Effective Beam, EB, and the Point Spread Function, PSF, at all positions of the HEALPix-formatted map pixel centres.<br />
<br />
<br />
===Production process===<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in {{BibCite|mitra2010}}.<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scanbeam models and different lengths of datasets. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixels---the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.<br />
<br />
====effBeam====<br />
<br />
The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.<br />
<br />
<br />
One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.<br />
<br />
<br />
====Computational Cost====<br />
<br />
The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below it isn displayed the computation cost on NERSC for nominal mission both in terms of CPU hrs and in Human time.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution.The cost in Human time is computed using an arbitrary number of nodes/core on Carver or Hopper NERSC Supercomputers<br />
|-<br />
|Channel ||030 || 044 || 070 <br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 <br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 <br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 <br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 <br />
|}<br />
<br />
<br />
The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.<br />
<br />
===Inputs===<br />
<br />
<br />
In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane''']]<br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====The scanning strategy====<br />
<br />
The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day (OD) 91 to OD 563 covering 3 surveys and half.<br />
<br />
====The scanbeam====<br />
<br />
The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.<br />
<br />
* LFI: [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beam]] - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. <br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====Beam cutoff radii====<br />
<br />
* N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.<br />
<!--<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' ||<br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|}<br />
--><br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 1024 </math> for all LFI frequency channels.<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for 30GHz frequency channel, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for LFI and B-Spline beams,BS, Mars12 apodized for the CMB channels and the BS Mars12 for the sub-mm channels, for HFI (see section Inputs below).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
[[File:30.png| 600px| thumb | center| '''30GHz''']]<br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
Here we present histograms of the three fit parameters - beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The shy is sampled (pretty sparsely) at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
* The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.<br />
<br />
<br />
[[File:e_030_GB.png| 600px| thumb | center| '''ellipticity - 30GHz''']]<br />
[[File:solidarc_030_GB.png| 600px| thumb | center| '''beam solid angle (relative variations wrt scanning beam - 30GHz''']]<br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 healpix maps (they include the pixel window function).<br />
* The reported FWHM_eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''frequency''' || '''mean(fwhm)''' [arcmin] || '''sd(fwhm)''' [arcmin] || '''mean(e)''' || '''sd(e)''' || '''mean(<math> \psi</math>)''' [degree] || '''sd(<math> \psi</math>)''' [degree] || '''mean(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''sd(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''FWHM_eff''' [arcmin] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|}<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
* <math>\Omega_{eff}</math> - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: <math> 4 \pi*sum(effective_{beam})/max(effective_{beam})</math> , i.e. as an integral over the full extent of the effective beam, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <math>\Omega^{(1)}_{eff}</math> is the beam solid angle estimated up to a radius equal to one <math>fwhm_{eff}</math> and <math>\Omega^{(2)}_{eff}</math> up to a radius equal to twice the <math>fwhm_{eff}</math>.<br />
*** These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''Band''' || '''<math>\Omega_{eff}</math>'''[arcmin<math>^{2}</math>] || '''spatial variation''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(1)}_{eff}</math>''' [arcmin<math>^{2}</math>]|| '''spatial variation-1''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(2)}_{eff}</math>''' [arcmin<math>^{2}</math>] || '''spatial variation-2''' [arcmin<math>^{2}</math>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|}<br />
<br />
===Related products===<br />
<br />
===Monte Carlo simulations===<br />
<br />
FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, <math>C_\ell^{out}</math>, and 1 sigma errors<br />
<br />
<br />
As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be CMB or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations.<br />
<!--, FFP#,<br />
for example [[HL-sims#FFP6 data set|FFP6]].<br />
--><br />
<br />
== Window Functions ==<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_\ell </math> relates the true angular power spectra <math>C_\ell </math> with the observed angular power spectra <math>\widetilde{C}_\ell </math>:<br />
<br />
<math><br />
W_\ell= \widetilde{C}_\ell / C_\ell<br />
\label{eqn:wl1}</math> <br />
<br />
Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_\ell </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_\ell </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_\ell </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_\ell = < C^{conv}_\ell / C^{in}_\ell ><br />
\label{eqn:wl2}</math> <br />
<br />
For subtle reasons, we perform a more rigorous estimation of the window function by comparing <math> C^{conv}_\ell</math> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, <math>W_\ell </math>, for LFI channels====<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, <math>W_\ell </math>, for LFI channels''']]<br />
<br />
== Sidelobes ==<br />
<br />
There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD.<br />
We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi.<br />
The beams have been computed in the same frames used for the main beams.<br />
The intermediate beam region (theta < 5 degrees) has been replaced with null values.<br />
<br />
*In the computation we considered:<br />
**the direct field from the feed<br />
**the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr<br />
**the 2nd order contributions SrPd and SdPd <br />
<br />
where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box. <br />
and where d=diffraction, r=reflection.<br />
For example Br, means that we considered in the calculation the reflection on the telescope baffle system. <br />
<br />
A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.<br />
<br />
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.''']]<br />
<br />
== References ==<br />
<br />
<References /> <br />
<br />
<br />
[[Category:LFI data processing|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Beams_LFI&diff=11120Beams LFI2015-02-04T10:08:24Z<p>Tvassall: /* Polarized Scanning Beams and Focal Plane calibration */</p>
<hr />
<div>{{DISPLAYTITLE:Beams}}<br />
<br />
== Overview ==<br />
<br />
LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.<br />
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]). <br />
<br />
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completeness and they are not representative of flight beams.''']]<br />
<br />
Details are given in {{PlanckPapers|planck2014-a05||Planck-2015-A05}}.<br />
<br />
<!--<br />
<br />
For the beam we consider these three regions:<br />
<br />
<br />
; main beam: is the portion of the pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.<br />
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called <b>intermediate beam</b>).<br />
; far sidelobes: is the pattern at angular regions more than 5 degrees from the beam center.<br />
--><br />
<br />
== Polarized Scanning Beams and Focal Plane calibration ==<br />
<br />
As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurements using Jupiter transits. The parameters that characterise the beam pointing are the following:<br />
<br />
* THETA_UV (<math>\theta_{uv}</math>)<br />
* PHI_UV (<math>\phi_{uv}</math>)<br />
<br />
They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as <br />
<br />
<math>\theta_{uv} = \arcsin(u^2+v^2)</math><br />
<br />
<math>\phi_{uv} = \arctan(v/u)</math><br />
<br />
Two additional angles are used to characterize the beams in the RIMO: <br />
<br />
* PSI_UV (<math>\psi_{uv}</math>)<br />
* PSI_POL (<math>\psi_{pol}</math>)<br />
<br />
<math>\psi_{uv}</math> and <math>\psi_{pol}</math> are '''not''' derived from measurements but they are estimated from '''optical simulations'''. They are the quantities that represent the polarization direction of each beam, in the following approximation: '''the M- and S- beams of the same RCA point at the same direction on the sky'''.<br />
<br />
The polarized scanning beams have been evaluated from optical simulations using GRASP Physical Optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM) {{PlanckPapers|tauber2010b}}. <br />
<br />
The Radio Frequency Tuned Model, called RFTM, was implemented to fit the in-flight beam measurements with the electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross polar beam in flight, we have relied on simulations validated by accurate beam measurements.<br />
<br />
The model beams where monochromatic and were computed throughout a 6 GHz band around the Optical Center Frequency (OCF) with non-regular step (denser sampling where the band-pass was higher). For the RFTM model the OCF were at <math>28.0, \, 44.0, \, 70.0</math> GHz. <br />
<br />
For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the [[The_RIMO#LFI_2|RIMO]] transmission function.<br />
<br />
The delivered [[Scanning_Beams|products]] include the in-band averaged Stokes scanning maps of Main Beams, Intermediate Beams and Sidelobes.<br />
<br />
== Effective beams ==<br />
<br />
The '''effective beam''' is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky.<br />
It captures the complete information about the difference between the true and observed image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map. <br />
<br />
The full algebra involving the effective beams for temperature and polarisation was presented in {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature sky <math>\widetilde{\mathbf{T}} </math> is a convolution of the true sky <math>\mathbf{T} </math> and the effective beam <math>\mathbf{B}</math>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<math>t</math> is time samples, <math>A_{ti}</math> is <math>1</math> if the pointing direction falls in pixel number <math>i</math>, else it is <math>0</math>, <math>\mathbf{p}_t</math> represents the exact pointing direction (not approximated by the pixel centre location), and <math>\hat{\mathbf{r}}_j</math> is the centre of the pixel number <math>j</math>, where the scanbeam <math>b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)</math> is being evaluated (if the pointing direction falls within the cut-off radius of <math>\sim 2.5 \times</math> FWHM.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^T \ = \ \Delta\Omega \sum_j \mathbf{B}_{ij} \cdot (I \quad Q \quad U)_j^T,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\mathbf{B}_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^T_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^T(\hat{\mathbf{n}}_j,\hat{\mathbf{p}}_t) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
and <math>\mathbf{w}_t </math>and <math>\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) </math> are the the polarisation weight vectors, as defined in \cite{mitra2010}.<br />
<br />
The task is to compute <math>B_{ij}</math> for temperature only beams and the <math>3 \times 3</math> matrices <math>\mathbf{B}_{ij}</math> for each pixel <math>i</math>, at every neighbouring pixel <math>j</math> that fall within the cut-off radius around the the center of the <math>i^\text{th}</math> pixel.<br />
<br />
<br />
<br />
The effective beam is computed by stacking within a small field around each pixel of the HEALPix sky map. Due to the particular features of Planck scanning strategy coupled to the beam asymmetries in the focal plane, and data processing of the bolometer and radiometer TOIs, the resulting Planck effective beams vary over the sky. <br />
<br />
FEBeCoP, given information on Planck scanning beams and detector pointing during a mission period of interest, provides the pixelized stamps of both the Effective Beam, EB, and the Point Spread Function, PSF, at all positions of the HEALPix-formatted map pixel centres.<br />
<br />
<br />
===Production process===<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in {{BibCite|mitra2010}}.<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scanbeam models and different lengths of datasets. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixels---the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.<br />
<br />
====effBeam====<br />
<br />
The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.<br />
<br />
<br />
One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.<br />
<br />
<br />
====Computational Cost====<br />
<br />
The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below it isn displayed the computation cost on NERSC for nominal mission both in terms of CPU hrs and in Human time.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution.The cost in Human time is computed using an arbitrary number of nodes/core on Carver or Hopper NERSC Supercomputers<br />
|-<br />
|Channel ||030 || 044 || 070 <br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 <br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 <br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 <br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 <br />
|}<br />
<br />
<br />
The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.<br />
<br />
===Inputs===<br />
<br />
<br />
In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane''']]<br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====The scanning strategy====<br />
<br />
The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day (OD) 91 to OD 563 covering 3 surveys and half.<br />
<br />
====The scanbeam====<br />
<br />
The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.<br />
<br />
* LFI: [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beam]] - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. <br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====Beam cutoff radii====<br />
<br />
* N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.<br />
<!--<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' ||<br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|}<br />
--><br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 1024 </math> for all LFI frequency channels.<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for 30GHz frequency channel, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for LFI and B-Spline beams,BS, Mars12 apodized for the CMB channels and the BS Mars12 for the sub-mm channels, for HFI (see section Inputs below).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
[[File:30.png| 600px| thumb | center| '''30GHz''']]<br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
Here we present histograms of the three fit parameters - beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The shy is sampled (pretty sparsely) at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
* The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.<br />
<br />
<br />
[[File:e_030_GB.png| 600px| thumb | center| '''ellipticity - 30GHz''']]<br />
[[File:solidarc_030_GB.png| 600px| thumb | center| '''beam solid angle (relative variations wrt scanning beam - 30GHz''']]<br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 healpix maps (they include the pixel window function).<br />
* The reported FWHM_eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''frequency''' || '''mean(fwhm)''' [arcmin] || '''sd(fwhm)''' [arcmin] || '''mean(e)''' || '''sd(e)''' || '''mean(<math> \psi</math>)''' [degree] || '''sd(<math> \psi</math>)''' [degree] || '''mean(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''sd(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''FWHM_eff''' [arcmin] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|}<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
* <math>\Omega_{eff}</math> - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: <math> 4 \pi*sum(effective_{beam})/max(effective_{beam})</math> , i.e. as an integral over the full extent of the effective beam, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <math>\Omega^{(1)}_{eff}</math> is the beam solid angle estimated up to a radius equal to one <math>fwhm_{eff}</math> and <math>\Omega^{(2)}_{eff}</math> up to a radius equal to twice the <math>fwhm_{eff}</math>.<br />
*** These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''Band''' || '''<math>\Omega_{eff}</math>'''[arcmin<math>^{2}</math>] || '''spatial variation''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(1)}_{eff}</math>''' [arcmin<math>^{2}</math>]|| '''spatial variation-1''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(2)}_{eff}</math>''' [arcmin<math>^{2}</math>] || '''spatial variation-2''' [arcmin<math>^{2}</math>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|}<br />
<br />
===Related products===<br />
<br />
===Monte Carlo simulations===<br />
<br />
FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, <math>C_\ell^{out}</math>, and 1 sigma errors<br />
<br />
<br />
As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be CMB or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations.<br />
<!--, FFP#,<br />
for example [[HL-sims#FFP6 data set|FFP6]].<br />
--><br />
<br />
== Window Functions ==<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_\ell </math> relates the true angular power spectra <math>C_\ell </math> with the observed angular power spectra <math>\widetilde{C}_\ell </math>:<br />
<br />
<math><br />
W_\ell= \widetilde{C}_\ell / C_\ell<br />
\label{eqn:wl1}</math> <br />
<br />
Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_\ell </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_\ell </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_\ell </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_\ell = < C^{conv}_\ell / C^{in}_\ell ><br />
\label{eqn:wl2}</math> <br />
<br />
For subtle reasons, we perform a more rigorous estimation of the window function by comparing <math> C^{conv}_\ell</math> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, <math>W_\ell </math>, for LFI channels====<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, <math>W_\ell </math>, for LFI channels''']]<br />
<br />
== Sidelobes ==<br />
<br />
There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD.<br />
We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi.<br />
The beams have been computed in the same frames used for the main beams.<br />
The intermediate beam region (theta < 5 degrees) has been replaced with null values.<br />
<br />
*In the computation we considered:<br />
**the direct field from the feed<br />
**the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr<br />
**the 2nd order contributions SrPd and SdPd <br />
<br />
where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box. <br />
and where d=diffraction, r=reflection.<br />
For example Br, means that we considered in the calculation the reflection on the telescope baffle system. <br />
<br />
A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.<br />
<br />
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.''']]<br />
<br />
== References ==<br />
<br />
<References /> <br />
<br />
<br />
[[Category:LFI data processing|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10918Scanning Beams2015-02-02T12:54:57Z<p>Tvassall: /* FITS file structure */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged [[Beams LFI#Polarized_Scanning_Beams_and_Focal_Plane_calibration|scanning beams]]. The averages are performed using the [[The_RIMO#LFI_2|RIMO]] bandpass, assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region expressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || Comment<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns of the BINDATA extension contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead"<br />
! Column Name || Data Type || Units || Description ||Comment<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number || colatitude<br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number || longitude<br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] ||<br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10917Scanning Beams2015-02-02T12:51:37Z<p>Tvassall: /* FITS file structure */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged [[Beams LFI#Polarized_Scanning_Beams_and_Focal_Plane_calibration|scanning beams]]. The averages are performed using the [[The_RIMO#LFI_2|RIMO]] bandpass, assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region expressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || Comment<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead"<br />
! Column Name || Data Type || Units || Description ||Comment<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number || colatitude<br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number || longitude<br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] ||<br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10916Scanning Beams2015-02-02T12:27:08Z<p>Tvassall: /* FITS file structure */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged [[Beams LFI#Polarized_Scanning_Beams_and_Focal_Plane_calibration|scanning beams]]. The averages are performed using the [[The_RIMO#LFI_2|RIMO]] bandpass, assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region expressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead"<br />
! Column Name || Data Type || Units || Description ||<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number || colatitude<br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number || longitude<br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] ||<br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10915Scanning Beams2015-02-02T12:25:34Z<p>Tvassall: </p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged [[Beams LFI#Polarized_Scanning_Beams_and_Focal_Plane_calibration|scanning beams]]. The averages are performed using the [[The_RIMO#LFI_2|RIMO]] bandpass, assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead"<br />
! Column Name || Data Type || Units || Description ||<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number || colatitude<br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number || longitude<br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] ||<br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Beams_LFI&diff=10914Beams LFI2015-02-02T12:21:00Z<p>Tvassall: /* Polarized Scanning Beams and Focal Plane calibration */</p>
<hr />
<div>{{DISPLAYTITLE:Beams}}<br />
<br />
== Overview ==<br />
<br />
LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.<br />
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]). <br />
<br />
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completness and they are not representative of flight beams.''']]<br />
<br />
<br />
<br />
<!--<br />
<br />
For the beam we consider these three regions:<br />
<br />
<br />
; main beam: is the portion of the pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.<br />
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called <b>intermediate beam</b>).<br />
; far sidelobes: is the pattern at angular regions more than 5 degrees from the beam center.<br />
--><br />
<br />
== Polarized Scanning Beams and Focal Plane calibration ==<br />
<br />
As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurements using Jupiter transits. The parameters that characterise the beam pointing are the following:<br />
<br />
* THETA_UV (<math>\theta_{uv}</math>)<br />
* PHI_UV (<math>\phi_{uv}</math>)<br />
<br />
They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as <br />
<br />
<math>\theta_{uv} = \arcsin(u^2+v^2)</math><br />
<br />
<math>\phi_{uv} = \arctan(v/u)</math><br />
<br />
Two additional angles are used to characterize the beams in the RIMO: <br />
<br />
* PSI_UV (<math>\psi_{uv}</math>)<br />
* PSI_POL (<math>\psi_{pol}</math>)<br />
<br />
<math>\psi_{uv}</math> and <math>\psi_{pol}</math> are '''not''' derived from measurements but they are estimated from '''optical simulations'''. They are the quantities that represent the polarization direction of each beam, in the following approximation: '''the M- and S- beams of the same RCA point at the same direction on the sky'''.<br />
<br />
The polarized scanning beams have been evaluated from optical simulations using GRASP Physical Optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM) {{PlanckPapers|tauber2010b}}. <br />
<br />
The Radio Frequency Tuned Model, called RFTM, was implemented to fit the in-flight beam measurements with the electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross polar beam in flight, we have relied on simulations validated by accurate beam measurements.<br />
<br />
The simulated beams where monochromatic and were computed throughout a 6 GHz band around the Optical Center Frequency (OCF). For the RFTM model the OCF were at <math>28.0, \, 44.0, \, 70.0</math> GHz. <br />
For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the [[The_RIMO#LFI_2|RIMO]] transmission function.<br />
<br />
The delivered [[Scanning_Beams|products]] include the in-band averaged Stokes scanning maps of Main Beams, Intermediate Beams and Sidelobes.<br />
<br />
== Effective beams ==<br />
<br />
The '''effective beam''' is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky.<br />
It captures the complete information about the difference between the true and observed image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map. <br />
<br />
The full algebra involving the effective beams for temperature and polarisation was presented in {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature sky <math>\widetilde{\mathbf{T}} </math> is a convolution of the true sky <math>\mathbf{T} </math> and the effective beam <math>\mathbf{B}</math>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<math>t</math> is time samples, <math>A_{ti}</math> is <math>1</math> if the pointing direction falls in pixel number <math>i</math>, else it is <math>0</math>, <math>\mathbf{p}_t</math> represents the exact pointing direction (not approximated by the pixel centre location), and <math>\hat{\mathbf{r}}_j</math> is the centre of the pixel number <math>j</math>, where the scanbeam <math>b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)</math> is being evaluated (if the pointing direction falls within the cut-off radius of <math>\sim 2.5 \times</math> FWHM.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^T \ = \ \Delta\Omega \sum_j \mathbf{B}_{ij} \cdot (I \quad Q \quad U)_j^T,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\mathbf{B}_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^T_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^T(\hat{\mathbf{n}}_j,\hat{\mathbf{p}}_t) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
and <math>\mathbf{w}_t </math>and <math>\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) </math> are the the polarisation weight vectors, as defined in \cite{mitra2010}.<br />
<br />
The task is to compute <math>B_{ij}</math> for temperature only beams and the <math>3 \times 3</math> matrices <math>\mathbf{B}_{ij}</math> for each pixel <math>i</math>, at every neighbouring pixel <math>j</math> that fall within the cut-off radius around the the center of the <math>i^\text{th}</math> pixel.<br />
<br />
<br />
<br />
The effective beam is computed by stacking within a small field around each pixel of the HEALPix sky map. Due to the particular features of Planck scanning strategy coupled to the beam asymmetries in the focal plane, and data processing of the bolometer and radiometer TOIs, the resulting Planck effective beams vary over the sky. <br />
<br />
FEBeCoP, given information on Planck scanning beams and detector pointing during a mission period of interest, provides the pixelized stamps of both the Effective Beam, EB, and the Point Spread Function, PSF, at all positions of the HEALPix-formatted map pixel centres.<br />
<br />
<br />
===Production process===<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in {{BibCite|mitra2010}}.<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scanbeam models and different lengths of datasets. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixels---the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.<br />
<br />
====effBeam====<br />
<br />
The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.<br />
<br />
<br />
One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.<br />
<br />
<br />
====Computational Cost====<br />
<br />
The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below it isn displayed the computation cost on NERSC for nominal mission both in terms of CPU hrs and in Human time.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution.The cost in Human time is computed using an arbitrary number of nodes/core on Carver or Hopper NERSC Supercomputers<br />
|-<br />
|Channel ||030 || 044 || 070 <br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 <br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 <br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 <br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 <br />
|}<br />
<br />
<br />
The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.<br />
<br />
===Inputs===<br />
<br />
<br />
In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane''']]<br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====The scanning strategy====<br />
<br />
The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day (OD) 91 to OD 563 covering 3 surveys and half.<br />
<br />
====The scanbeam====<br />
<br />
The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.<br />
<br />
* LFI: [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beam]] - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. <br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====Beam cutoff radii====<br />
<br />
* N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.<br />
<!--<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' ||<br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|}<br />
--><br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 1024 </math> for all LFI frequency channels.<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for 30GHz frequency channel, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for LFI and B-Spline beams,BS, Mars12 apodized for the CMB channels and the BS Mars12 for the sub-mm channels, for HFI (see section Inputs below).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
[[File:30.png| 600px| thumb | center| '''30GHz''']]<br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
Here we present histograms of the three fit parameters - beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The shy is sampled (pretty sparsely) at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
* The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.<br />
<br />
<br />
[[File:e_030_GB.png| 600px| thumb | center| '''ellipticity - 30GHz''']]<br />
[[File:solidarc_030_GB.png| 600px| thumb | center| '''beam solid angle (relative variations wrt scanning beam - 30GHz''']]<br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 healpix maps (they include the pixel window function).<br />
* The reported FWHM_eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''frequency''' || '''mean(fwhm)''' [arcmin] || '''sd(fwhm)''' [arcmin] || '''mean(e)''' || '''sd(e)''' || '''mean(<math> \psi</math>)''' [degree] || '''sd(<math> \psi</math>)''' [degree] || '''mean(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''sd(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''FWHM_eff''' [arcmin] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|}<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
* <math>\Omega_{eff}</math> - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: <math> 4 \pi*sum(effective_{beam})/max(effective_{beam})</math> , i.e. as an integral over the full extent of the effective beam, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <math>\Omega^{(1)}_{eff}</math> is the beam solid angle estimated up to a radius equal to one <math>fwhm_{eff}</math> and <math>\Omega^{(2)}_{eff}</math> up to a radius equal to twice the <math>fwhm_{eff}</math>.<br />
*** These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''Band''' || '''<math>\Omega_{eff}</math>'''[arcmin<math>^{2}</math>] || '''spatial variation''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(1)}_{eff}</math>''' [arcmin<math>^{2}</math>]|| '''spatial variation-1''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(2)}_{eff}</math>''' [arcmin<math>^{2}</math>] || '''spatial variation-2''' [arcmin<math>^{2}</math>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|}<br />
<br />
===Related products===<br />
<br />
===Monte Carlo simulations===<br />
<br />
FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, <math>C_\ell^{out}</math>, and 1 sigma errors<br />
<br />
<br />
As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be CMB or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations.<br />
<!--, FFP#,<br />
for example [[HL-sims#FFP6 data set|FFP6]].<br />
--><br />
<br />
== Window Functions ==<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_\ell </math> relates the true angular power spectra <math>C_\ell </math> with the observed angular power spectra <math>\widetilde{C}_\ell </math>:<br />
<br />
<math><br />
W_\ell= \widetilde{C}_\ell / C_\ell<br />
\label{eqn:wl1}</math> <br />
<br />
Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_\ell </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_\ell </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_\ell </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_\ell = < C^{conv}_\ell / C^{in}_\ell ><br />
\label{eqn:wl2}</math> <br />
<br />
For subtle reasons, we perform a more rigorous estimation of the window function by comparing <math> C^{conv}_\ell</math> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for LFI channels====<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, <math>W_\ell </math>, for LFI channels''']]<br />
<br />
== Sidelobes ==<br />
<br />
There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD.<br />
We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi.<br />
The beams have been computed in the same frames used for the main beams.<br />
The intermediate beam region (theta < 5 degrees) has been replaced with null values.<br />
<br />
*In the computation we considered:<br />
**the direct field from the feed<br />
**the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr<br />
**the 2nd order contributions SrPd and SdPd <br />
<br />
where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box. <br />
and where d=diffraction, r=reflection.<br />
For example Br, means that we considered in the calculation the reflection on the telescope baffle system. <br />
<br />
A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.<br />
<br />
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.''']]<br />
<br />
== References ==<br />
<br />
<References /> <br />
<br />
<br />
[[Category:LFI data processing|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10911Scanning Beams2015-02-02T11:58:36Z<p>Tvassall: </p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged [[Beams LFI#Polarized_Scanning_Beams_and_Focal_Plane_calibration|scanning beams]]. The averages are performed using the RIMO bandpass, assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead"<br />
! Column Name || Data Type || Units || Description ||<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number || colatitude<br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number || longitude<br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] ||<br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10909Scanning Beams2015-02-02T11:57:41Z<p>Tvassall: </p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged [[Beams#Polarized_Scanning_Beams_and_Focal_Plane_calibration|scanning beams]]. The averages are performed using the RIMO bandpass, assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead"<br />
! Column Name || Data Type || Units || Description ||<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number || colatitude<br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number || longitude<br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] ||<br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10908Scanning Beams2015-02-02T11:56:56Z<p>Tvassall: </p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged [[Beams#Polarized_Scanning_Beams_and_Focal_Plane_calibration][scanning beams]]. The averages are performed using the RIMO bandpass, assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead"<br />
! Column Name || Data Type || Units || Description ||<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number || colatitude<br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number || longitude<br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] ||<br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10906Scanning Beams2015-02-02T11:55:14Z<p>Tvassall: /* FITS file structure */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The averages are performed using the RIMO bandpass , assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead"<br />
! Column Name || Data Type || Units || Description ||<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number || colatitude<br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number || longitude<br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] ||<br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10905Scanning Beams2015-02-02T11:53:52Z<p>Tvassall: /* FITS file structure */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The averages are performed using the RIMO bandpass , assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead"<br />
! Column Name || Data Type || Units || Description ||<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description ||<br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number ||<br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number ||<br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] ||<br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10904Scanning Beams2015-02-02T11:52:23Z<p>Tvassall: /* FITS file structure */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The averages are performed using the RIMO bandpass , assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description ||<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description ||<br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number ||<br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number ||<br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] ||<br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10903Scanning Beams2015-02-02T11:52:04Z<p>Tvassall: /* FITS file structure */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The averages are performed using the RIMO bandpass , assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description ||<br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number ||<br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number ||<br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] ||<br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10902Scanning Beams2015-02-02T11:50:56Z<p>Tvassall: /* FITS file structure */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The averages are performed using the RIMO bandpass , assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || <br />
|-<br />
|Ntheta || Int || || <math>\theta</math> samples number || <br />
|-<br />
|Nphi || Int || || <math>\phi</math> samples number || <br />
|-<br />
|Mintheta || Float || || Minimum value of <math>\theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>\theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] || <br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10901Scanning Beams2015-02-02T11:49:42Z<p>Tvassall: /* FITS file structure */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The averages are performed using the RIMO bandpass , assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || <br />
|-<br />
|Ntheta || Int || || <math>theta</math> samples number || <br />
|-<br />
|Nphi || Int || || <math>phi</math> samples number || <br />
|-<br />
|Mintheta || Float || || Minimum value of <math>theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] || <br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10900Scanning Beams2015-02-02T11:49:16Z<p>Tvassall: /* FITS file structure */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The averages are performed using the RIMO bandpass , assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || <br />
|-<br />
|Ntheta || Int || || <math>theta</math> samples number || <br />
|-<br />
|Nphi || Int || || <math>phi</phi> samples number || <br />
|-<br />
|Mintheta || Float || || Minimum value of <math>theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] || <br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10899Scanning Beams2015-02-02T11:47:03Z<p>Tvassall: </p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The averages are performed using the RIMO bandpass , assuming a flat spectrum of the incoming radiation. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Float || || X step in radians || <br />
|-<br />
|Ydelta || Float || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Intermediate Beam and Sidelobes file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Ntheta || Int || || <math>theta</math> samples number || <br />
|-<br />
|Nphi || Int || || <math>phi</phi> samples number || <br />
|-<br />
|Mintheta || Float || 0.0 || Minimum value of <math>theta</math> ||<br />
|-<br />
|Maxtheta || Float || || Maximum value of <math>theta</math> ||<br />
|-<br />
|angularCut || Float || || Angular cut [deg] || <br />
<br />
|}</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10898Scanning Beams2015-02-02T11:33:35Z<p>Tvassall: </p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. <br />
Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || <br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|Nx || Int || 601 || X axis samples number || X axis aligned with the direction of the S arm<br />
|-<br />
|Ny || Int || 601 || Y axis samples number || Y axis aligned with the direction of the M arm<br />
|-<br />
|Xcentre || Int || 301 || X coordinate of the beam centre ||<br />
|-<br />
|Ycentre || Int || 301 || Y coordinate of the beam centre || <br />
|-<br />
|Xdelta || Real*4 || || X step in radians || <br />
|-<br />
|Ydelta || Real*4 || || Y step in radians || <br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
The stokes map files consist in four columns containing the Stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math>.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10897Scanning Beams2015-02-02T11:11:07Z<p>Tvassall: </p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
<br />
==FITS file structure==<br />
The FITS files contain a primary extension with no data and a minimal header, and one BINTABLE extension with data and with a description of the data in the header keywords. <br />
The BINTABLE extension consist in four columns, each containing the array of one Stokes parameter. <br />
The columns are called:<br />
* ''Beamdata'' , containing <math>I</math><br />
* ''BeamdataQ'', containing <math>Q</math><br />
* ''BeamdataU'', containing <math>U</math><br />
* ''BeamdataV'', containing <math>V</math>.<br />
<br />
Main Beam, Intermediate Beams and Sidelobes are saved with a different data format: <br />
<br />
* Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The grid include an angular region gexpressed by the keyword ''angularCut'', and its dimensions are given by the keywords ''Nx'' and ''Ny'', representing the number of columns and rows. <br />
Each column of the BINDATA extension contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords ''Xcentre'' and ''Ycentre'' express the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:left" width=800px<br />
|+ '''Main Beam file structure''' <br />
<br />
|- bgcolor="ffdead" <br />
! Column Name || Data Type || Units || Description || Comment<br />
|-<br />
|BEAMDATA || Real*4 || none || Stokes parameter I || <br />
|-<br />
|BEAMDATAQ || Real*4 || none || Stokes parameter Q ||<br />
|-<br />
|BEAMDATAU || Real*4 || none || Stokes parameter U ||<br />
|-<br />
|BEAMDATAV || Real*4 || none || Stokes parameter V ||<br />
<br />
|- bgcolor="ffdead" <br />
<br />
! Keyword || Data Type || Value || Description || Comment<br />
|-<br />
|UNIT || String || || Units of signal || Only HFI, LFI always K<sub>cmb</sub><br />
|-<br />
|DESTRIPE || 1/0 || || whether timeline is destriped || Only HFI<br />
|-<br />
|OD || Int || || OD covered (as in filename) ||<br />
|-<br />
|BEGIDX || Int || || first sample index of given OD || Only HFI<br />
|-<br />
|ENDIDX || Int || || last sample index of given OD || Only HFI<br />
|-<br />
|BEGRING || Int || || first ring in given OD || Only HFI<br />
|-<br />
|ENDRING || Int || || last ring in given OD || Only HFI<br />
|}<br />
<br />
<br />
<br />
<br />
* Intermediate Beams and Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.<br />
The stokes map files consist in four columns containing the Stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math>.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10887Scanning Beams2015-02-02T09:51:28Z<p>Tvassall: </p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
==File Names==<br />
<br />
The file names are of the form:<br />
<br />
''LFI_ScanBeam-{mb,ib,sl}_{fff}-{rrr}_R2.{nn}.stokes''<br />
<br />
where<br />
* ''fff'' denotes the frequency<br />
* mb denotes the Main Beams<br />
* ib denotes the Intermediate Beam <br />
* sl denotes the Sidelobes<br />
* ''rrr'' denotes the radiometer<br />
* R2.nn is the version.<br />
<br />
<br />
At the present time, HFI is not releasing the Scanning Beams.<br />
<br />
==File format==<br />
<br />
The stokes map files consist in four columns containing the Stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math>.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows. <br />
Each of the four columns contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Beams_LFI&diff=10515Beams LFI2015-01-22T10:32:03Z<p>Tvassall: /* Polarized Scanning Beams and Focal Plane calibration */</p>
<hr />
<div>{{DISPLAYTITLE:Beams}}<br />
<br />
== Overview ==<br />
<br />
LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.<br />
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]). <br />
<br />
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completness and they are not representative of flight beams.''']]<br />
<br />
<br />
<br />
<!--<br />
<br />
For the beam we consider these three regions:<br />
<br />
<br />
; main beam: is the portion of the pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.<br />
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called <b>intermediate beam</b>).<br />
; far sidelobes: is the pattern at angular regions more than 5 degrees from the beam center.<br />
--><br />
<br />
== Polarized Scanning Beams and Focal Plane calibration ==<br />
<br />
As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurements using Jupiter transits. the parameters that characterise the beam pointing are the following:<br />
<br />
* THETA_UV (<math>\theta_{uv}</math>)<br />
* PHI_UV (<math>\phi_{uv}</math>)<br />
<br />
They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as <br />
<br />
<math>\theta_{uv} = \arcsin(u^2+v^2)</math><br />
<br />
<math>\phi_{uv} = \arctan(v/u)</math><br />
<br />
Two additional angles are used to characterize the beams in the RIMO: <br />
<br />
* PSI_UV (<math>\psi_{uv}</math>)<br />
* PSI_POL (<math>\psi_{pol}</math>)<br />
<br />
<math>\psi_{uv}</math> and <math>\psi_{pol}</math> are '''not''' derived from measurements but they are estimated from '''optical simulations'''. They are the quantities that represent the polarization direction of each beam, in the following approximation: '''the M- and S- beams of the same RCA point at the same direction on the sky'''.<br />
<br />
The polarized scanning beams have been evaluated from optical simulations using GRASP Physical Optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM) {{PlanckPapers|tauber2010b}}. <br />
<br />
The Radio Frequency Tuned Model, called RFTM, was implemented to fit the in-flight beam measurements with the electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross polar beam in flight, we have relied on simulations validated by accurate beam measurements.<br />
<br />
The simulated beams where monochromatic and were computed throughout a 6 GHz band around the Optical Center Frequency (OCF). For the RFTM model the OCF were at <math>28.0, \, 44.0, \, 70.0</math> GHz. <br />
For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the [[The_RIMO#LFI_2|RIMO]] transmission function.<br />
<br />
The delivered [[Scanning_Beams|products]] include the in-band averaged Stokes scanning maps of Main Beams, Intermediate Beams and Sidelobes.<br />
<br />
== Effective beams ==<br />
<br />
The '''effective beam''' is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky.<br />
It captures the complete information about the difference between the true and observed image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map. <br />
<br />
The full algebra involving the effective beams for temperature and polarisation was presented in {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature sky <math>\widetilde{\mathbf{T}} </math> is a convolution of the true sky <math>\mathbf{T} </math> and the effective beam <math>\mathbf{B}</math>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<math>t</math> is time samples, <math>A_{ti}</math> is <math>1</math> if the pointing direction falls in pixel number <math>i</math>, else it is <math>0</math>, <math>\mathbf{p}_t</math> represents the exact pointing direction (not approximated by the pixel centre location), and <math>\hat{\mathbf{r}}_j</math> is the centre of the pixel number <math>j</math>, where the scanbeam <math>b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)</math> is being evaluated (if the pointing direction falls within the cut-off radius of <math>\sim 2.5 \times</math> FWHM.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^T \ = \ \Delta\Omega \sum_j \mathbf{B}_{ij} \cdot (I \quad Q \quad U)_j^T,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\mathbf{B}_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^T_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^T(\hat{\mathbf{n}}_j,\hat{\mathbf{p}}_t) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
and <math>\mathbf{w}_t </math>and <math>\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) </math> are the the polarisation weight vectors, as defined in \cite{mitra2010}.<br />
<br />
The task is to compute <math>B_{ij}</math> for temperature only beams and the <math>3 \times 3</math> matrices <math>\mathbf{B}_{ij}</math> for each pixel <math>i</math>, at every neighbouring pixel <math>j</math> that fall within the cut-off radius around the the center of the <math>i^\text{th}</math> pixel.<br />
<br />
<br />
<br />
The effective beam is computed by stacking within a small field around each pixel of the HEALPix sky map. Due to the particular features of Planck scanning strategy coupled to the beam asymmetries in the focal plane, and data processing of the bolometer and radiometer TOIs, the resulting Planck effective beams vary over the sky. <br />
<br />
FEBeCoP, given information on Planck scanning beams and detector pointing during a mission period of interest, provides the pixelized stamps of both the Effective Beam, EB, and the Point Spread Function, PSF, at all positions of the HEALPix-formatted map pixel centres.<br />
<br />
<br />
===Production process===<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in {{BibCite|mitra2010}}.<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scanbeam models and different lengths of datasets. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixels---the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.<br />
<br />
====effBeam====<br />
<br />
The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.<br />
<br />
<br />
One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.<br />
<br />
<br />
====Computational Cost====<br />
<br />
The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below it isn displayed the computation cost on NERSC for nominal mission both in terms of CPU hrs and in Human time.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution.The cost in Human time is computed using an arbitrary number of nodes/core on Carver or Hopper NERSC Supercomputers<br />
|-<br />
|Channel ||030 || 044 || 070 <br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 <br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 <br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 <br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 <br />
|}<br />
<br />
<br />
The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.<br />
<br />
===Inputs===<br />
<br />
<br />
In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane''']]<br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====The scanning strategy====<br />
<br />
The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day (OD) 91 to OD 563 covering 3 surveys and half.<br />
<br />
====The scanbeam====<br />
<br />
The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.<br />
<br />
* LFI: [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beam]] - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. <br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====Beam cutoff radii====<br />
<br />
* N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.<br />
<!--<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' ||<br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|}<br />
--><br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 1024 </math> for all LFI frequency channels.<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for 30GHz frequency channel, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for LFI and B-Spline beams,BS, Mars12 apodized for the CMB channels and the BS Mars12 for the sub-mm channels, for HFI (see section Inputs below).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
[[File:30.png| 600px| thumb | center| '''30GHz''']]<br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
Here we present histograms of the three fit parameters - beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The shy is sampled (pretty sparsely) at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
* The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.<br />
<br />
<br />
[[File:e_030_GB.png| 600px| thumb | center| '''ellipticity - 30GHz''']]<br />
[[File:solidarc_030_GB.png| 600px| thumb | center| '''beam solid angle (relative variations wrt scanning beam - 30GHz''']]<br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 healpix maps (they include the pixel window function).<br />
* The reported FWHM_eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''frequency''' || '''mean(fwhm)''' [arcmin] || '''sd(fwhm)''' [arcmin] || '''mean(e)''' || '''sd(e)''' || '''mean(<math> \psi</math>)''' [degree] || '''sd(<math> \psi</math>)''' [degree] || '''mean(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''sd(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''FWHM_eff''' [arcmin] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|}<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
* <math>\Omega_{eff}</math> - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: <math> 4 \pi*sum(effective_{beam})/max(effective_{beam})</math> , i.e. as an integral over the full extent of the effective beam, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <math>\Omega^{(1)}_{eff}</math> is the beam solid angle estimated up to a radius equal to one <math>fwhm_{eff}</math> and <math>\Omega^{(2)}_{eff}</math> up to a radius equal to twice the <math>fwhm_{eff}</math>.<br />
*** These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''Band''' || '''<math>\Omega_{eff}</math>'''[arcmin<math>^{2}</math>] || '''spatial variation''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(1)}_{eff}</math>''' [arcmin<math>^{2}</math>]|| '''spatial variation-1''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(2)}_{eff}</math>''' [arcmin<math>^{2}</math>] || '''spatial variation-2''' [arcmin<math>^{2}</math>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|}<br />
<br />
===Related products===<br />
<br />
===Monte Carlo simulations===<br />
<br />
FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, <math>C_\ell^{out}</math>, and 1 sigma errors<br />
<br />
<br />
As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be CMB or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations.<br />
<!--, FFP#,<br />
for example [[HL-sims#FFP6 data set|FFP6]].<br />
--><br />
<br />
== Window Functions ==<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_\ell </math> relates the true angular power spectra <math>C_\ell </math> with the observed angular power spectra <math>\widetilde{C}_\ell </math>:<br />
<br />
<math><br />
W_\ell= \widetilde{C}_\ell / C_\ell<br />
\label{eqn:wl1}</math> <br />
<br />
Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_\ell </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_\ell </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_\ell </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_\ell = < C^{conv}_\ell / C^{in}_\ell ><br />
\label{eqn:wl2}</math> <br />
<br />
For subtle reasons, we perform a more rigorous estimation of the window function by comparing <math> C^{conv}_\ell</math> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for LFI channels====<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, <math>W_\ell </math>, for LFI channels''']]<br />
<br />
== Sidelobes ==<br />
<br />
There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD.<br />
We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi.<br />
The beams have been computed in the same frames used for the main beams.<br />
The intermediate beam region (theta < 5 degrees) has been replaced with null values.<br />
<br />
*In the computation we considered:<br />
**the direct field from the feed<br />
**the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr<br />
**the 2nd order contributions SrPd and SdPd <br />
<br />
where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box. <br />
and where d=diffraction, r=reflection.<br />
For example Br, means that we considered in the calculation the reflection on the telescope baffle system. <br />
<br />
A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.<br />
<br />
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.''']]<br />
<br />
== References ==<br />
<br />
<References /> <br />
<br />
<br />
[[Category:LFI data processing|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10272Scanning Beams2014-12-17T13:39:14Z<p>Tvassall: /* Note on the file format */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the Stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math>.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows. <br />
Each of the four columns contain the sequence of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10271Scanning Beams2014-12-17T13:38:53Z<p>Tvassall: /* Note on the file format */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the Stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math>.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows. <br />
Each of the four columns contain the succession of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the sequence of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10270Scanning Beams2014-12-17T13:37:26Z<p>Tvassall: /* Note on the file format */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the Stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math>.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows. <br />
Each of the four columns contain the succession of the <math>Nx \times Ny</math> samples of the map in row major order. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10269Scanning Beams2014-12-17T13:34:04Z<p>Tvassall: /* Note on the file format */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the Stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math>.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows. <br />
Each of the four columns contain the succession of the <math>Nx \times Ny</math> samples of the map in major order. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum (i.e. where the sphere intersect the tangent plane).<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10268Scanning Beams2014-12-17T13:33:32Z<p>Tvassall: /* Note on the file format */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the Stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math>.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows. <br />
Each of the four columns contain the succession of the <math>Nx \times Ny</math> samples of the map in major order. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum (i.e where the sphere intersect the tangent plane).<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10267Scanning Beams2014-12-17T13:32:15Z<p>Tvassall: /* Note on the file format */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the Stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math>.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows. <br />
Each of the four stokes columns contain the succession of the <math>Nx \times Ny</math> samples in major order. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum (i.e where the sphere intersect the tangent plane).<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10266Scanning Beams2014-12-17T13:31:47Z<p>Tvassall: /* Note on the file format */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the Stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively the <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math> Stokes parameters.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows. <br />
Each of the four stokes columns contain the succession of the <math>Nx \times Ny</math> samples in major order. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum (i.e where the sphere intersect the tangent plane).<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10265Scanning Beams2014-12-17T13:31:26Z<p>Tvassall: </p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information about the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively the <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math> Stokes parameters.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows. <br />
Each of the four stokes columns contain the succession of the <math>Nx \times Ny</math> samples in major order. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum (i.e where the sphere intersect the tangent plane).<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10264Scanning Beams2014-12-17T13:30:45Z<p>Tvassall: /* Note on the file format */</p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information on the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively the <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math> Stokes parameters.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are projected on the tangent plane to the sphere, and sampled on a grid. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows. <br />
Each of the four stokes columns contain the succession of the <math>Nx \times Ny</math> samples in major order. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum (i.e where the sphere intersect the tangent plane).<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10263Scanning Beams2014-12-17T13:14:24Z<p>Tvassall: </p>
<hr />
<div><br />
We released maps of the Stokes parameters of the band-averaged scanning beams. The Stokes parameters maps contain the complete information on the field intensity and polarization properties. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the stokes parameters for each sample. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively the <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math> Stokes parameters.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are sampled with a grid map on the tangent space. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows of the grid map. <br />
The four stokes columns contain the succession of the <math>Nx \times Ny</math> samples of the map. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum.<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10262Scanning Beams2014-12-17T13:08:58Z<p>Tvassall: </p>
<hr />
<div><br />
The polarized scanning beams are released as maps of the Stokes polarization parameters of the electromagnetic field. Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the stokes parameters for each sample, encoding all the information on the field intensity and polarization properties. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively the <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math> Stokes parameters.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are sampled with a grid map on the tangent space. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows of the grid map. <br />
The four stokes columns contain the succession of the <math>Nx \times Ny</math> samples of the map. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum.<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Beams_LFI&diff=10255Beams LFI2014-12-17T11:32:04Z<p>Tvassall: /* Polarized Scanning Beams and Focal Plane calibration */</p>
<hr />
<div>{{DISPLAYTITLE:Beams}}<br />
<br />
== Overview ==<br />
<br />
LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.<br />
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]). <br />
<br />
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completness and they are not representative of flight beams.''']]<br />
<br />
<br />
<br />
<!--<br />
<br />
For the beam we consider these three regions:<br />
<br />
<br />
; main beam: is the portion of the pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.<br />
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called <b>intermediate beam</b>).<br />
; far sidelobes: is the pattern at angular regions more than 5 degrees from the beam center.<br />
--><br />
<br />
== Polarized Scanning Beams and Focal Plane calibration ==<br />
<br />
As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurements using Jupiter transits. the parameters that characterise the beam pointing are the following:<br />
<br />
* THETA_UV (<math>\theta_{uv}</math>)<br />
* PHI_UV (<math>\phi_{uv}</math>)<br />
<br />
They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as <br />
<br />
<math>\theta_{uv} = \arcsin(u^2+v^2)</math><br />
<br />
<math>\phi_{uv} = \arctan(v/u)</math><br />
<br />
Two additional angles are used to characterize the beams in the RIMO: <br />
<br />
* PSI_UV (<math>\psi_{uv}</math>)<br />
* PSI_POL (<math>\psi_{pol}</math>)<br />
<br />
<math>\psi_{uv}</math> and <math>\psi_{pol}</math> are '''not''' derived from measurements but they are estimated from '''optical simulations'''. They are the quantities that represent the polarization direction of each beam, in the following approximation: '''the M- and S- beams of the same RCA point at the same direction on the sky'''.<br />
<br />
The polarized scanning beams have been evaluated from optical simulations using GRASP Physical Optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM) {{PlanckPapers|tauber2010b}}. <br />
<br />
The Radio Frequency Tuned Model, called RFTM, was implemented to fit the in-flight beam measurements with the electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross polar beam in flight, we have relied on simulations validated by accurate beam measurements [[PlancPapers4]].<br />
<br />
The simulated beams where monochromatic and were computed throughout a 6 GHz band around the Optical Center Frequency (OCF). For the RFTM model the OCF were at <math>28.0, \, 44.0, \, 70.0</math> GHz. <br />
For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the [[The_RIMO#LFI_2|RIMO]] transmission function.<br />
<br />
The delivered [[Scanning_Beams|products]] include the in-band averaged Stokes scanning maps of Main Beams, Intermediate Beams and Sidelobes.<br />
<br />
== Effective beams ==<br />
<br />
The '''effective beam''' is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky.<br />
It captures the complete information about the difference between the true and observed image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map. <br />
<br />
The full algebra involving the effective beams for temperature and polarisation was presented in {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature sky <math>\widetilde{\mathbf{T}} </math> is a convolution of the true sky <math>\mathbf{T} </math> and the effective beam <math>\mathbf{B}</math>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<math>t</math> is time samples, <math>A_{ti}</math> is <math>1</math> if the pointing direction falls in pixel number <math>i</math>, else it is <math>0</math>, <math>\mathbf{p}_t</math> represents the exact pointing direction (not approximated by the pixel centre location), and <math>\hat{\mathbf{r}}_j</math> is the centre of the pixel number <math>j</math>, where the scanbeam <math>b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)</math> is being evaluated (if the pointing direction falls within the cut-off radius of <math>\sim 2.5 \times</math> FWHM.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^T \ = \ \Delta\Omega \sum_j \mathbf{B}_{ij} \cdot (I \quad Q \quad U)_j^T,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\mathbf{B}_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^T_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^T(\hat{\mathbf{n}}_j,\hat{\mathbf{p}}_t) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
and <math>\mathbf{w}_t </math>and <math>\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) </math> are the the polarisation weight vectors, as defined in \cite{mitra2010}.<br />
<br />
The task is to compute <math>B_{ij}</math> for temperature only beams and the <math>3 \times 3</math> matrices <math>\mathbf{B}_{ij}</math> for each pixel <math>i</math>, at every neighbouring pixel <math>j</math> that fall within the cut-off radius around the the center of the <math>i^\text{th}</math> pixel.<br />
<br />
<br />
<br />
The effective beam is computed by stacking within a small field around each pixel of the HEALPix sky map. Due to the particular features of Planck scanning strategy coupled to the beam asymmetries in the focal plane, and data processing of the bolometer and radiometer TOIs, the resulting Planck effective beams vary over the sky. <br />
<br />
FEBeCoP, given information on Planck scanning beams and detector pointing during a mission period of interest, provides the pixelized stamps of both the Effective Beam, EB, and the Point Spread Function, PSF, at all positions of the HEALPix-formatted map pixel centres.<br />
<br />
<br />
===Production process===<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in {{BibCite|mitra2010}}.<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scanbeam models and different lengths of datasets. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixels---the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.<br />
<br />
====effBeam====<br />
<br />
The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.<br />
<br />
<br />
One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.<br />
<br />
<br />
====Computational Cost====<br />
<br />
The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below it isn displayed the computation cost on NERSC for nominal mission both in terms of CPU hrs and in Human time.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution.The cost in Human time is computed using an arbitrary number of nodes/core on Carver or Hopper NERSC Supercomputers<br />
|-<br />
|Channel ||030 || 044 || 070 <br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 <br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 <br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 <br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 <br />
|}<br />
<br />
<br />
The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.<br />
<br />
===Inputs===<br />
<br />
<br />
In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane''']]<br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====The scanning strategy====<br />
<br />
The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day (OD) 91 to OD 563 covering 3 surveys and half.<br />
<br />
====The scanbeam====<br />
<br />
The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.<br />
<br />
* LFI: [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beam]] - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. <br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====Beam cutoff radii====<br />
<br />
* N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.<br />
<!--<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' ||<br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|}<br />
--><br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 1024 </math> for all LFI frequency channels.<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for 30GHz frequency channel, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for LFI and B-Spline beams,BS, Mars12 apodized for the CMB channels and the BS Mars12 for the sub-mm channels, for HFI (see section Inputs below).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
[[File:30.png| 600px| thumb | center| '''30GHz''']]<br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
Here we present histograms of the three fit parameters - beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The shy is sampled (pretty sparsely) at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
* The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.<br />
<br />
<br />
[[File:e_030_GB.png| 600px| thumb | center| '''ellipticity - 30GHz''']]<br />
[[File:solidarc_030_GB.png| 600px| thumb | center| '''beam solid angle (relative variations wrt scanning beam - 30GHz''']]<br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 healpix maps (they include the pixel window function).<br />
* The reported FWHM_eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''frequency''' || '''mean(fwhm)''' [arcmin] || '''sd(fwhm)''' [arcmin] || '''mean(e)''' || '''sd(e)''' || '''mean(<math> \psi</math>)''' [degree] || '''sd(<math> \psi</math>)''' [degree] || '''mean(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''sd(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''FWHM_eff''' [arcmin] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|}<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
* <math>\Omega_{eff}</math> - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: <math> 4 \pi*sum(effective_{beam})/max(effective_{beam})</math> , i.e. as an integral over the full extent of the effective beam, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <math>\Omega^{(1)}_{eff}</math> is the beam solid angle estimated up to a radius equal to one <math>fwhm_{eff}</math> and <math>\Omega^{(2)}_{eff}</math> up to a radius equal to twice the <math>fwhm_{eff}</math>.<br />
*** These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''Band''' || '''<math>\Omega_{eff}</math>'''[arcmin<math>^{2}</math>] || '''spatial variation''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(1)}_{eff}</math>''' [arcmin<math>^{2}</math>]|| '''spatial variation-1''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(2)}_{eff}</math>''' [arcmin<math>^{2}</math>] || '''spatial variation-2''' [arcmin<math>^{2}</math>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|}<br />
<br />
===Related products===<br />
<br />
===Monte Carlo simulations===<br />
<br />
FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, <math>C_\ell^{out}</math>, and 1 sigma errors<br />
<br />
<br />
As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be CMB or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations.<br />
<!--, FFP#,<br />
for example [[HL-sims#FFP6 data set|FFP6]].<br />
--><br />
<br />
== Window Functions ==<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_\ell </math> relates the true angular power spectra <math>C_\ell </math> with the observed angular power spectra <math>\widetilde{C}_\ell </math>:<br />
<br />
<math><br />
W_\ell= \widetilde{C}_\ell / C_\ell<br />
\label{eqn:wl1}</math> <br />
<br />
Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_\ell </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_\ell </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_\ell </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_\ell = < C^{conv}_\ell / C^{in}_\ell ><br />
\label{eqn:wl2}</math> <br />
<br />
For subtle reasons, we perform a more rigorous estimation of the window function by comparing <math> C^{conv}_\ell</math> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for LFI channels====<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, <math>W_\ell </math>, for LFI channels''']]<br />
<br />
== Sidelobes ==<br />
<br />
There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD.<br />
We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi.<br />
The beams have been computed in the same frames used for the main beams.<br />
The intermediate beam region (theta < 5 degrees) has been replaced with null values.<br />
<br />
*In the computation we considered:<br />
**the direct field from the feed<br />
**the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr<br />
**the 2nd order contributions SrPd and SdPd <br />
<br />
where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box. <br />
and where d=diffraction, r=reflection.<br />
For example Br, means that we considered in the calculation the reflection on the telescope baffle system. <br />
<br />
A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.<br />
<br />
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.''']]<br />
<br />
== References ==<br />
<br />
<References /> <br />
<br />
<br />
[[Category:LFI data processing|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Beams_LFI&diff=10254Beams LFI2014-12-17T11:29:49Z<p>Tvassall: /* Polarized Scanning Beams and Focal Plane calibration */</p>
<hr />
<div>{{DISPLAYTITLE:Beams}}<br />
<br />
== Overview ==<br />
<br />
LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.<br />
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]). <br />
<br />
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completness and they are not representative of flight beams.''']]<br />
<br />
<br />
<br />
<!--<br />
<br />
For the beam we consider these three regions:<br />
<br />
<br />
; main beam: is the portion of the pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.<br />
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called <b>intermediate beam</b>).<br />
; far sidelobes: is the pattern at angular regions more than 5 degrees from the beam center.<br />
--><br />
<br />
== Polarized Scanning Beams and Focal Plane calibration ==<br />
<br />
As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurements using Jupiter transits. the parameters that characterise the beam pointing are the following:<br />
<br />
* THETA_UV (<math>\theta_{uv}</math>)<br />
* PHI_UV (<math>\phi_{uv}</math>)<br />
<br />
They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as <br />
<br />
<math>\theta_{uv} = \arcsin(u^2+v^2)</math><br />
<br />
<math>\phi_{uv} = \arctan(v/u)</math><br />
<br />
Two additional angles are used to characterize the beams in the RIMO: <br />
<br />
* PSI_UV (<math>\psi_{uv}</math>)<br />
* PSI_POL (<math>\psi_{pol}</math>)<br />
<br />
<math>\psi_{uv}</math> and <math>\psi_{pol}</math> are '''not''' derived from measurements but they are estimated from '''optical simulations'''. They are the quantities that represent the polarization direction of each beam, in the following approximation: '''the M- and S- beams of the same RCA point at the same direction on the sky'''.<br />
<br />
The polarized scanning beams have been evaluated from optical simulations using GRASP Physical Optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM) {{PlanckPapers|tauber2010b}}. <br />
<br />
The Radio Frequency Tuned Model, called RFTM, was implemented to fit the in-flight beam measurements with the electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross polar beam in flight, we have relied on simulations validated by accurate beam measurements [[PlancPapers4]].<br />
<br />
The simulated beams where monochromatic and were computed throughout a 6 GHz band around the Optical Center Frequency (OCF). For the RFTM model the OCF were at <math>28.0, \, 44.0, \, 70.0</math> GHz. <br />
For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the [[The_RIMO|RIMO]] R2.2 band-pass function [[]].<br />
<br />
The delivered [[Scanning_Beams|products]] include the in-band averaged Stokes scanning maps of Main Beams, Intermediate Beams and Sidelobes.<br />
<br />
== Effective beams ==<br />
<br />
The '''effective beam''' is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky.<br />
It captures the complete information about the difference between the true and observed image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map. <br />
<br />
The full algebra involving the effective beams for temperature and polarisation was presented in {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature sky <math>\widetilde{\mathbf{T}} </math> is a convolution of the true sky <math>\mathbf{T} </math> and the effective beam <math>\mathbf{B}</math>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<math>t</math> is time samples, <math>A_{ti}</math> is <math>1</math> if the pointing direction falls in pixel number <math>i</math>, else it is <math>0</math>, <math>\mathbf{p}_t</math> represents the exact pointing direction (not approximated by the pixel centre location), and <math>\hat{\mathbf{r}}_j</math> is the centre of the pixel number <math>j</math>, where the scanbeam <math>b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)</math> is being evaluated (if the pointing direction falls within the cut-off radius of <math>\sim 2.5 \times</math> FWHM.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^T \ = \ \Delta\Omega \sum_j \mathbf{B}_{ij} \cdot (I \quad Q \quad U)_j^T,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\mathbf{B}_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^T_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^T(\hat{\mathbf{n}}_j,\hat{\mathbf{p}}_t) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
and <math>\mathbf{w}_t </math>and <math>\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) </math> are the the polarisation weight vectors, as defined in \cite{mitra2010}.<br />
<br />
The task is to compute <math>B_{ij}</math> for temperature only beams and the <math>3 \times 3</math> matrices <math>\mathbf{B}_{ij}</math> for each pixel <math>i</math>, at every neighbouring pixel <math>j</math> that fall within the cut-off radius around the the center of the <math>i^\text{th}</math> pixel.<br />
<br />
<br />
<br />
The effective beam is computed by stacking within a small field around each pixel of the HEALPix sky map. Due to the particular features of Planck scanning strategy coupled to the beam asymmetries in the focal plane, and data processing of the bolometer and radiometer TOIs, the resulting Planck effective beams vary over the sky. <br />
<br />
FEBeCoP, given information on Planck scanning beams and detector pointing during a mission period of interest, provides the pixelized stamps of both the Effective Beam, EB, and the Point Spread Function, PSF, at all positions of the HEALPix-formatted map pixel centres.<br />
<br />
<br />
===Production process===<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in {{BibCite|mitra2010}}.<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scanbeam models and different lengths of datasets. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixels---the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.<br />
<br />
====effBeam====<br />
<br />
The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.<br />
<br />
<br />
One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.<br />
<br />
<br />
====Computational Cost====<br />
<br />
The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below it isn displayed the computation cost on NERSC for nominal mission both in terms of CPU hrs and in Human time.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution.The cost in Human time is computed using an arbitrary number of nodes/core on Carver or Hopper NERSC Supercomputers<br />
|-<br />
|Channel ||030 || 044 || 070 <br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 <br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 <br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 <br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 <br />
|}<br />
<br />
<br />
The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.<br />
<br />
===Inputs===<br />
<br />
<br />
In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane''']]<br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====The scanning strategy====<br />
<br />
The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day (OD) 91 to OD 563 covering 3 surveys and half.<br />
<br />
====The scanbeam====<br />
<br />
The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.<br />
<br />
* LFI: [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beam]] - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. <br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====Beam cutoff radii====<br />
<br />
* N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.<br />
<!--<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' ||<br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|}<br />
--><br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 1024 </math> for all LFI frequency channels.<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for 30GHz frequency channel, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for LFI and B-Spline beams,BS, Mars12 apodized for the CMB channels and the BS Mars12 for the sub-mm channels, for HFI (see section Inputs below).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
[[File:30.png| 600px| thumb | center| '''30GHz''']]<br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
Here we present histograms of the three fit parameters - beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The shy is sampled (pretty sparsely) at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
* The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.<br />
<br />
<br />
[[File:e_030_GB.png| 600px| thumb | center| '''ellipticity - 30GHz''']]<br />
[[File:solidarc_030_GB.png| 600px| thumb | center| '''beam solid angle (relative variations wrt scanning beam - 30GHz''']]<br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 healpix maps (they include the pixel window function).<br />
* The reported FWHM_eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''frequency''' || '''mean(fwhm)''' [arcmin] || '''sd(fwhm)''' [arcmin] || '''mean(e)''' || '''sd(e)''' || '''mean(<math> \psi</math>)''' [degree] || '''sd(<math> \psi</math>)''' [degree] || '''mean(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''sd(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''FWHM_eff''' [arcmin] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|}<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
* <math>\Omega_{eff}</math> - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: <math> 4 \pi*sum(effective_{beam})/max(effective_{beam})</math> , i.e. as an integral over the full extent of the effective beam, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <math>\Omega^{(1)}_{eff}</math> is the beam solid angle estimated up to a radius equal to one <math>fwhm_{eff}</math> and <math>\Omega^{(2)}_{eff}</math> up to a radius equal to twice the <math>fwhm_{eff}</math>.<br />
*** These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''Band''' || '''<math>\Omega_{eff}</math>'''[arcmin<math>^{2}</math>] || '''spatial variation''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(1)}_{eff}</math>''' [arcmin<math>^{2}</math>]|| '''spatial variation-1''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(2)}_{eff}</math>''' [arcmin<math>^{2}</math>] || '''spatial variation-2''' [arcmin<math>^{2}</math>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|}<br />
<br />
===Related products===<br />
<br />
===Monte Carlo simulations===<br />
<br />
FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, <math>C_\ell^{out}</math>, and 1 sigma errors<br />
<br />
<br />
As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be CMB or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations.<br />
<!--, FFP#,<br />
for example [[HL-sims#FFP6 data set|FFP6]].<br />
--><br />
<br />
== Window Functions ==<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_\ell </math> relates the true angular power spectra <math>C_\ell </math> with the observed angular power spectra <math>\widetilde{C}_\ell </math>:<br />
<br />
<math><br />
W_\ell= \widetilde{C}_\ell / C_\ell<br />
\label{eqn:wl1}</math> <br />
<br />
Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_\ell </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_\ell </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_\ell </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_\ell = < C^{conv}_\ell / C^{in}_\ell ><br />
\label{eqn:wl2}</math> <br />
<br />
For subtle reasons, we perform a more rigorous estimation of the window function by comparing <math> C^{conv}_\ell</math> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for LFI channels====<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, <math>W_\ell </math>, for LFI channels''']]<br />
<br />
== Sidelobes ==<br />
<br />
There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD.<br />
We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi.<br />
The beams have been computed in the same frames used for the main beams.<br />
The intermediate beam region (theta < 5 degrees) has been replaced with null values.<br />
<br />
*In the computation we considered:<br />
**the direct field from the feed<br />
**the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr<br />
**the 2nd order contributions SrPd and SdPd <br />
<br />
where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box. <br />
and where d=diffraction, r=reflection.<br />
For example Br, means that we considered in the calculation the reflection on the telescope baffle system. <br />
<br />
A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.<br />
<br />
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.''']]<br />
<br />
== References ==<br />
<br />
<References /> <br />
<br />
<br />
[[Category:LFI data processing|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Beams_LFI&diff=10253Beams LFI2014-12-17T11:27:49Z<p>Tvassall: /* Polarized Scanning Beams and Focal Plane calibration */</p>
<hr />
<div>{{DISPLAYTITLE:Beams}}<br />
<br />
== Overview ==<br />
<br />
LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.<br />
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]). <br />
<br />
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completness and they are not representative of flight beams.''']]<br />
<br />
<br />
<br />
<!--<br />
<br />
For the beam we consider these three regions:<br />
<br />
<br />
; main beam: is the portion of the pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.<br />
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called <b>intermediate beam</b>).<br />
; far sidelobes: is the pattern at angular regions more than 5 degrees from the beam center.<br />
--><br />
<br />
== Polarized Scanning Beams and Focal Plane calibration ==<br />
<br />
As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurements using Jupiter transits. the parameters that characterise the beam pointing are the following:<br />
<br />
* THETA_UV (<math>\theta_{uv}</math>)<br />
* PHI_UV (<math>\phi_{uv}</math>)<br />
<br />
They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as <br />
<br />
<math>\theta_{uv} = \arcsin(u^2+v^2)</math><br />
<br />
<math>\phi_{uv} = \arctan(v/u)</math><br />
<br />
Two additional angles are used to characterize the beams in the RIMO: <br />
<br />
* PSI_UV (<math>\psi_{uv}</math>)<br />
* PSI_POL (<math>\psi_{pol}</math>)<br />
<br />
<math>\psi_{uv}</math> and <math>\psi_{pol}</math> are '''not''' derived from measurements but they are estimated from '''optical simulations'''. They are the quantities that represent the polarization direction of each beam, in the following approximation: '''the M- and S- beams of the same RCA point at the same direction on the sky'''.<br />
<br />
The polarized scanning beams have been evaluated from optical simulations using GRASP Physical Optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM) {{PlanckPapers|tauber2010b}}. <br />
<br />
The Radio Frequency Tuned Model, called RFTM, was implemented to fit the in-flight beam measurements with the electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross polar beam in flight, we have relied on simulations validated by accurate beam measurements [[PlancPapers4]].<br />
<br />
The simulated beams where monochromatic and were computed throughout a 6 GHz band around the Optical Center Frequency (OCF). For the RFTM model the OCF were at <math>28.0, \, 44.0, \, 70.0</math> GHz. <br />
For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the RIMO R2.2 band-pass function [[]].<br />
<br />
The delivered [[Scanning_Beams|products]] include the in-band averaged Stokes scanning maps of Main Beams, Intermediate Beams and Sidelobes.<br />
<br />
== Effective beams ==<br />
<br />
The '''effective beam''' is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky.<br />
It captures the complete information about the difference between the true and observed image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map. <br />
<br />
The full algebra involving the effective beams for temperature and polarisation was presented in {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature sky <math>\widetilde{\mathbf{T}} </math> is a convolution of the true sky <math>\mathbf{T} </math> and the effective beam <math>\mathbf{B}</math>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<math>t</math> is time samples, <math>A_{ti}</math> is <math>1</math> if the pointing direction falls in pixel number <math>i</math>, else it is <math>0</math>, <math>\mathbf{p}_t</math> represents the exact pointing direction (not approximated by the pixel centre location), and <math>\hat{\mathbf{r}}_j</math> is the centre of the pixel number <math>j</math>, where the scanbeam <math>b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)</math> is being evaluated (if the pointing direction falls within the cut-off radius of <math>\sim 2.5 \times</math> FWHM.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^T \ = \ \Delta\Omega \sum_j \mathbf{B}_{ij} \cdot (I \quad Q \quad U)_j^T,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\mathbf{B}_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^T_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^T(\hat{\mathbf{n}}_j,\hat{\mathbf{p}}_t) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
and <math>\mathbf{w}_t </math>and <math>\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) </math> are the the polarisation weight vectors, as defined in \cite{mitra2010}.<br />
<br />
The task is to compute <math>B_{ij}</math> for temperature only beams and the <math>3 \times 3</math> matrices <math>\mathbf{B}_{ij}</math> for each pixel <math>i</math>, at every neighbouring pixel <math>j</math> that fall within the cut-off radius around the the center of the <math>i^\text{th}</math> pixel.<br />
<br />
<br />
<br />
The effective beam is computed by stacking within a small field around each pixel of the HEALPix sky map. Due to the particular features of Planck scanning strategy coupled to the beam asymmetries in the focal plane, and data processing of the bolometer and radiometer TOIs, the resulting Planck effective beams vary over the sky. <br />
<br />
FEBeCoP, given information on Planck scanning beams and detector pointing during a mission period of interest, provides the pixelized stamps of both the Effective Beam, EB, and the Point Spread Function, PSF, at all positions of the HEALPix-formatted map pixel centres.<br />
<br />
<br />
===Production process===<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in {{BibCite|mitra2010}}.<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scanbeam models and different lengths of datasets. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixels---the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.<br />
<br />
====effBeam====<br />
<br />
The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.<br />
<br />
<br />
One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.<br />
<br />
<br />
====Computational Cost====<br />
<br />
The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below it isn displayed the computation cost on NERSC for nominal mission both in terms of CPU hrs and in Human time.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution.The cost in Human time is computed using an arbitrary number of nodes/core on Carver or Hopper NERSC Supercomputers<br />
|-<br />
|Channel ||030 || 044 || 070 <br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 <br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 <br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 <br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 <br />
|}<br />
<br />
<br />
The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.<br />
<br />
===Inputs===<br />
<br />
<br />
In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane''']]<br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====The scanning strategy====<br />
<br />
The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day (OD) 91 to OD 563 covering 3 surveys and half.<br />
<br />
====The scanbeam====<br />
<br />
The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.<br />
<br />
* LFI: [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beam]] - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. <br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====Beam cutoff radii====<br />
<br />
* N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.<br />
<!--<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' ||<br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|}<br />
--><br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 1024 </math> for all LFI frequency channels.<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for 30GHz frequency channel, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for LFI and B-Spline beams,BS, Mars12 apodized for the CMB channels and the BS Mars12 for the sub-mm channels, for HFI (see section Inputs below).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
[[File:30.png| 600px| thumb | center| '''30GHz''']]<br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
Here we present histograms of the three fit parameters - beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The shy is sampled (pretty sparsely) at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
* The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.<br />
<br />
<br />
[[File:e_030_GB.png| 600px| thumb | center| '''ellipticity - 30GHz''']]<br />
[[File:solidarc_030_GB.png| 600px| thumb | center| '''beam solid angle (relative variations wrt scanning beam - 30GHz''']]<br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 healpix maps (they include the pixel window function).<br />
* The reported FWHM_eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''frequency''' || '''mean(fwhm)''' [arcmin] || '''sd(fwhm)''' [arcmin] || '''mean(e)''' || '''sd(e)''' || '''mean(<math> \psi</math>)''' [degree] || '''sd(<math> \psi</math>)''' [degree] || '''mean(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''sd(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''FWHM_eff''' [arcmin] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|}<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
* <math>\Omega_{eff}</math> - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: <math> 4 \pi*sum(effective_{beam})/max(effective_{beam})</math> , i.e. as an integral over the full extent of the effective beam, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <math>\Omega^{(1)}_{eff}</math> is the beam solid angle estimated up to a radius equal to one <math>fwhm_{eff}</math> and <math>\Omega^{(2)}_{eff}</math> up to a radius equal to twice the <math>fwhm_{eff}</math>.<br />
*** These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''Band''' || '''<math>\Omega_{eff}</math>'''[arcmin<math>^{2}</math>] || '''spatial variation''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(1)}_{eff}</math>''' [arcmin<math>^{2}</math>]|| '''spatial variation-1''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(2)}_{eff}</math>''' [arcmin<math>^{2}</math>] || '''spatial variation-2''' [arcmin<math>^{2}</math>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|}<br />
<br />
===Related products===<br />
<br />
===Monte Carlo simulations===<br />
<br />
FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, <math>C_\ell^{out}</math>, and 1 sigma errors<br />
<br />
<br />
As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be CMB or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations.<br />
<!--, FFP#,<br />
for example [[HL-sims#FFP6 data set|FFP6]].<br />
--><br />
<br />
== Window Functions ==<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_\ell </math> relates the true angular power spectra <math>C_\ell </math> with the observed angular power spectra <math>\widetilde{C}_\ell </math>:<br />
<br />
<math><br />
W_\ell= \widetilde{C}_\ell / C_\ell<br />
\label{eqn:wl1}</math> <br />
<br />
Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_\ell </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_\ell </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_\ell </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_\ell = < C^{conv}_\ell / C^{in}_\ell ><br />
\label{eqn:wl2}</math> <br />
<br />
For subtle reasons, we perform a more rigorous estimation of the window function by comparing <math> C^{conv}_\ell</math> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for LFI channels====<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, <math>W_\ell </math>, for LFI channels''']]<br />
<br />
== Sidelobes ==<br />
<br />
There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD.<br />
We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi.<br />
The beams have been computed in the same frames used for the main beams.<br />
The intermediate beam region (theta < 5 degrees) has been replaced with null values.<br />
<br />
*In the computation we considered:<br />
**the direct field from the feed<br />
**the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr<br />
**the 2nd order contributions SrPd and SdPd <br />
<br />
where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box. <br />
and where d=diffraction, r=reflection.<br />
For example Br, means that we considered in the calculation the reflection on the telescope baffle system. <br />
<br />
A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.<br />
<br />
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.''']]<br />
<br />
== References ==<br />
<br />
<References /> <br />
<br />
<br />
[[Category:LFI data processing|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Beams_LFI&diff=10252Beams LFI2014-12-17T11:26:55Z<p>Tvassall: /* Polarized Scanning Beams and Focal Plane calibration */</p>
<hr />
<div>{{DISPLAYTITLE:Beams}}<br />
<br />
== Overview ==<br />
<br />
LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.<br />
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]). <br />
<br />
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completness and they are not representative of flight beams.''']]<br />
<br />
<br />
<br />
<!--<br />
<br />
For the beam we consider these three regions:<br />
<br />
<br />
; main beam: is the portion of the pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.<br />
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called <b>intermediate beam</b>).<br />
; far sidelobes: is the pattern at angular regions more than 5 degrees from the beam center.<br />
--><br />
<br />
== Polarized Scanning Beams and Focal Plane calibration ==<br />
<br />
As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurements using Jupiter transits. the parameters that characterise the beam pointing are the following:<br />
<br />
* THETA_UV (<math>\theta_{uv}</math>)<br />
* PHI_UV (<math>\phi_{uv}</math>)<br />
<br />
They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as <br />
<br />
<math>\theta_{uv} = \arcsin(u^2+v^2)</math><br />
<br />
<math>\phi_{uv} = \arctan(v/u)</math><br />
<br />
Two additional angles are used to characterize the beams in the RIMO: <br />
<br />
* PSI_UV (<math>\psi_{uv}</math>)<br />
* PSI_POL (<math>\psi_{pol}</math>)<br />
<br />
<math>\psi_{uv}</math> and <math>\psi_{pol}</math> are '''not''' derived from measurements but they are estimated from '''optical simulations'''. They are the quantities that represent the polarization direction of each beam, in the following approximation: '''the M- and S- beams of the same RCA point at the same direction on the sky'''.<br />
<br />
The polarized scanning beams have been evaluated from optical simulations using GRASP Physical Optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM) {{PlanckPapers|tauber2010b}}. <br />
<br />
The Radio Frequency Tuned Model, called RFTM, was implemented to fit the in-flight beam measurements with the electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross polar beam in flight, we have relied on simulations validated by accurate beam measurements [[PlancPapers4]].<br />
<br />
The simulated beams where monochromatic and were computed throughout a 6 GHz band around the Optical Center Frequency (OCF). For the RFTM model the OCF were at <math>28.0, \, 44.0, \, 70.0</math> GHz. <br />
For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the RIMO R2.2 band-pass function [[]].<br />
<br />
The delivered products include the in-band averaged Stokes scanning maps of Main Beams, Intermediate Beams and Sidelobes[[Scanning_Beams]].<br />
<br />
== Effective beams ==<br />
<br />
The '''effective beam''' is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky.<br />
It captures the complete information about the difference between the true and observed image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map. <br />
<br />
The full algebra involving the effective beams for temperature and polarisation was presented in {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature sky <math>\widetilde{\mathbf{T}} </math> is a convolution of the true sky <math>\mathbf{T} </math> and the effective beam <math>\mathbf{B}</math>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<math>t</math> is time samples, <math>A_{ti}</math> is <math>1</math> if the pointing direction falls in pixel number <math>i</math>, else it is <math>0</math>, <math>\mathbf{p}_t</math> represents the exact pointing direction (not approximated by the pixel centre location), and <math>\hat{\mathbf{r}}_j</math> is the centre of the pixel number <math>j</math>, where the scanbeam <math>b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)</math> is being evaluated (if the pointing direction falls within the cut-off radius of <math>\sim 2.5 \times</math> FWHM.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^T \ = \ \Delta\Omega \sum_j \mathbf{B}_{ij} \cdot (I \quad Q \quad U)_j^T,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\mathbf{B}_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^T_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^T(\hat{\mathbf{n}}_j,\hat{\mathbf{p}}_t) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
and <math>\mathbf{w}_t </math>and <math>\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) </math> are the the polarisation weight vectors, as defined in \cite{mitra2010}.<br />
<br />
The task is to compute <math>B_{ij}</math> for temperature only beams and the <math>3 \times 3</math> matrices <math>\mathbf{B}_{ij}</math> for each pixel <math>i</math>, at every neighbouring pixel <math>j</math> that fall within the cut-off radius around the the center of the <math>i^\text{th}</math> pixel.<br />
<br />
<br />
<br />
The effective beam is computed by stacking within a small field around each pixel of the HEALPix sky map. Due to the particular features of Planck scanning strategy coupled to the beam asymmetries in the focal plane, and data processing of the bolometer and radiometer TOIs, the resulting Planck effective beams vary over the sky. <br />
<br />
FEBeCoP, given information on Planck scanning beams and detector pointing during a mission period of interest, provides the pixelized stamps of both the Effective Beam, EB, and the Point Spread Function, PSF, at all positions of the HEALPix-formatted map pixel centres.<br />
<br />
<br />
===Production process===<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in {{BibCite|mitra2010}}.<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scanbeam models and different lengths of datasets. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixels---the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.<br />
<br />
====effBeam====<br />
<br />
The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.<br />
<br />
<br />
One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.<br />
<br />
<br />
====Computational Cost====<br />
<br />
The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below it isn displayed the computation cost on NERSC for nominal mission both in terms of CPU hrs and in Human time.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution.The cost in Human time is computed using an arbitrary number of nodes/core on Carver or Hopper NERSC Supercomputers<br />
|-<br />
|Channel ||030 || 044 || 070 <br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 <br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 <br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 <br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 <br />
|}<br />
<br />
<br />
The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.<br />
<br />
===Inputs===<br />
<br />
<br />
In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane''']]<br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====The scanning strategy====<br />
<br />
The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day (OD) 91 to OD 563 covering 3 surveys and half.<br />
<br />
====The scanbeam====<br />
<br />
The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.<br />
<br />
* LFI: [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beam]] - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. <br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====Beam cutoff radii====<br />
<br />
* N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.<br />
<!--<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' ||<br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|}<br />
--><br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 1024 </math> for all LFI frequency channels.<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for 30GHz frequency channel, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for LFI and B-Spline beams,BS, Mars12 apodized for the CMB channels and the BS Mars12 for the sub-mm channels, for HFI (see section Inputs below).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
[[File:30.png| 600px| thumb | center| '''30GHz''']]<br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
Here we present histograms of the three fit parameters - beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The shy is sampled (pretty sparsely) at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
* The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.<br />
<br />
<br />
[[File:e_030_GB.png| 600px| thumb | center| '''ellipticity - 30GHz''']]<br />
[[File:solidarc_030_GB.png| 600px| thumb | center| '''beam solid angle (relative variations wrt scanning beam - 30GHz''']]<br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 healpix maps (they include the pixel window function).<br />
* The reported FWHM_eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''frequency''' || '''mean(fwhm)''' [arcmin] || '''sd(fwhm)''' [arcmin] || '''mean(e)''' || '''sd(e)''' || '''mean(<math> \psi</math>)''' [degree] || '''sd(<math> \psi</math>)''' [degree] || '''mean(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''sd(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''FWHM_eff''' [arcmin] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|}<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
* <math>\Omega_{eff}</math> - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: <math> 4 \pi*sum(effective_{beam})/max(effective_{beam})</math> , i.e. as an integral over the full extent of the effective beam, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <math>\Omega^{(1)}_{eff}</math> is the beam solid angle estimated up to a radius equal to one <math>fwhm_{eff}</math> and <math>\Omega^{(2)}_{eff}</math> up to a radius equal to twice the <math>fwhm_{eff}</math>.<br />
*** These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''Band''' || '''<math>\Omega_{eff}</math>'''[arcmin<math>^{2}</math>] || '''spatial variation''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(1)}_{eff}</math>''' [arcmin<math>^{2}</math>]|| '''spatial variation-1''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(2)}_{eff}</math>''' [arcmin<math>^{2}</math>] || '''spatial variation-2''' [arcmin<math>^{2}</math>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|}<br />
<br />
===Related products===<br />
<br />
===Monte Carlo simulations===<br />
<br />
FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, <math>C_\ell^{out}</math>, and 1 sigma errors<br />
<br />
<br />
As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be CMB or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations.<br />
<!--, FFP#,<br />
for example [[HL-sims#FFP6 data set|FFP6]].<br />
--><br />
<br />
== Window Functions ==<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_\ell </math> relates the true angular power spectra <math>C_\ell </math> with the observed angular power spectra <math>\widetilde{C}_\ell </math>:<br />
<br />
<math><br />
W_\ell= \widetilde{C}_\ell / C_\ell<br />
\label{eqn:wl1}</math> <br />
<br />
Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_\ell </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_\ell </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_\ell </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_\ell = < C^{conv}_\ell / C^{in}_\ell ><br />
\label{eqn:wl2}</math> <br />
<br />
For subtle reasons, we perform a more rigorous estimation of the window function by comparing <math> C^{conv}_\ell</math> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for LFI channels====<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, <math>W_\ell </math>, for LFI channels''']]<br />
<br />
== Sidelobes ==<br />
<br />
There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD.<br />
We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi.<br />
The beams have been computed in the same frames used for the main beams.<br />
The intermediate beam region (theta < 5 degrees) has been replaced with null values.<br />
<br />
*In the computation we considered:<br />
**the direct field from the feed<br />
**the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr<br />
**the 2nd order contributions SrPd and SdPd <br />
<br />
where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box. <br />
and where d=diffraction, r=reflection.<br />
For example Br, means that we considered in the calculation the reflection on the telescope baffle system. <br />
<br />
A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.<br />
<br />
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.''']]<br />
<br />
== References ==<br />
<br />
<References /> <br />
<br />
<br />
[[Category:LFI data processing|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10251Scanning Beams2014-12-17T10:46:37Z<p>Tvassall: </p>
<hr />
<div><br />
The polarized scanning beams are released as maps of the Stokes polarization parameters of the electromagnetic field. The maps of Main Beam, Intermediate Beam, and Sidelobes are released separately.<br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the stokes parameters for each sample, encoding all the information on the field intensity and polarization properties. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively the <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math> Stokes parameters.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
<br />
Main Beams are sampled with a grid map on the tangent space. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows of the grid map. <br />
The four stokes columns contain the succession of the <math>Nx \times Ny</math> samples of the map. The keywords "Xcentre" and "Ycentre" contain the coordinates of the beam maximum.<br />
<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Beams_LFI&diff=10248Beams LFI2014-12-17T10:01:03Z<p>Tvassall: /* Polarized Scanning Beams and Focal Plane calibration */</p>
<hr />
<div>{{DISPLAYTITLE:Beams}}<br />
<br />
== Overview ==<br />
<br />
LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.<br />
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]). <br />
<br />
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completness and they are not representative of flight beams.''']]<br />
<br />
<br />
<br />
<!--<br />
<br />
For the beam we consider these three regions:<br />
<br />
<br />
; main beam: is the portion of the pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.<br />
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called <b>intermediate beam</b>).<br />
; far sidelobes: is the pattern at angular regions more than 5 degrees from the beam center.<br />
--><br />
<br />
== Polarized Scanning Beams and Focal Plane calibration ==<br />
<br />
As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurements using Jupiter transits. the parameters that characterise the beam pointing are the following:<br />
<br />
* THETA_UV (<math>\theta_{uv}</math>)<br />
* PHI_UV (<math>\phi_{uv}</math>)<br />
<br />
They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as <br />
<br />
<math>\theta_{uv} = \arcsin(u^2+v^2)</math><br />
<br />
<math>\phi_{uv} = \arctan(v/u)</math><br />
<br />
Two additional angles are used to characterize the beams in the RIMO: <br />
<br />
* PSI_UV (<math>\psi_{uv}</math>)<br />
* PSI_POL (<math>\psi_{pol}</math>)<br />
<br />
<math>\psi_{uv}</math> and <math>\psi_{pol}</math> are '''not''' derived from measurements but they are estimated from '''optical simulations'''. They are the quantities that represent the polarization direction of each beam, in the following approximation: '''the M- and S- beams of the same RCA point at the same direction on the sky'''.<br />
<br />
The polarized scanning beams have been evaluated from optical simulations using GRASP Physical Optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM) {{PlanckPapers|tauber2010b}}. <br />
<br />
The Radio Frequency Tuned Model, called RFTM, was implemented to fit the in-flight beam measurements with the electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross polar beam in flight, we have relied on simulations validated by accurate beam measurements [[PlancPapers4]].<br />
<br />
The simulated beams where monochromatic and were computed throughout a 6 GHz band around the Optical Center Frequency (OCF). For the RFTM model the OCF were at <math>28.0, \, 44.0, \, 70.0</math> GHz. <br />
For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the RIMO R2.2 band-pass function [[]].<br />
<br />
The delivered products include the in-band averaged Stokes scanning maps of Main Beams, Intermediate Beams and Sidelobes[[]].<br />
<br />
== Effective beams ==<br />
<br />
The '''effective beam''' is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky.<br />
It captures the complete information about the difference between the true and observed image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map. <br />
<br />
The full algebra involving the effective beams for temperature and polarisation was presented in {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature sky <math>\widetilde{\mathbf{T}} </math> is a convolution of the true sky <math>\mathbf{T} </math> and the effective beam <math>\mathbf{B}</math>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<math>t</math> is time samples, <math>A_{ti}</math> is <math>1</math> if the pointing direction falls in pixel number <math>i</math>, else it is <math>0</math>, <math>\mathbf{p}_t</math> represents the exact pointing direction (not approximated by the pixel centre location), and <math>\hat{\mathbf{r}}_j</math> is the centre of the pixel number <math>j</math>, where the scanbeam <math>b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)</math> is being evaluated (if the pointing direction falls within the cut-off radius of <math>\sim 2.5 \times</math> FWHM.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^T \ = \ \Delta\Omega \sum_j \mathbf{B}_{ij} \cdot (I \quad Q \quad U)_j^T,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\mathbf{B}_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^T_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^T(\hat{\mathbf{n}}_j,\hat{\mathbf{p}}_t) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
and <math>\mathbf{w}_t </math>and <math>\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) </math> are the the polarisation weight vectors, as defined in \cite{mitra2010}.<br />
<br />
The task is to compute <math>B_{ij}</math> for temperature only beams and the <math>3 \times 3</math> matrices <math>\mathbf{B}_{ij}</math> for each pixel <math>i</math>, at every neighbouring pixel <math>j</math> that fall within the cut-off radius around the the center of the <math>i^\text{th}</math> pixel.<br />
<br />
<br />
<br />
The effective beam is computed by stacking within a small field around each pixel of the HEALPix sky map. Due to the particular features of Planck scanning strategy coupled to the beam asymmetries in the focal plane, and data processing of the bolometer and radiometer TOIs, the resulting Planck effective beams vary over the sky. <br />
<br />
FEBeCoP, given information on Planck scanning beams and detector pointing during a mission period of interest, provides the pixelized stamps of both the Effective Beam, EB, and the Point Spread Function, PSF, at all positions of the HEALPix-formatted map pixel centres.<br />
<br />
<br />
===Production process===<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in {{BibCite|mitra2010}}.<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scanbeam models and different lengths of datasets. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixels---the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.<br />
<br />
====effBeam====<br />
<br />
The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.<br />
<br />
<br />
One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.<br />
<br />
<br />
====Computational Cost====<br />
<br />
The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below it isn displayed the computation cost on NERSC for nominal mission both in terms of CPU hrs and in Human time.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution.The cost in Human time is computed using an arbitrary number of nodes/core on Carver or Hopper NERSC Supercomputers<br />
|-<br />
|Channel ||030 || 044 || 070 <br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 <br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 <br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 <br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 <br />
|}<br />
<br />
<br />
The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.<br />
<br />
===Inputs===<br />
<br />
<br />
In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane''']]<br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====The scanning strategy====<br />
<br />
The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day (OD) 91 to OD 563 covering 3 surveys and half.<br />
<br />
====The scanbeam====<br />
<br />
The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.<br />
<br />
* LFI: [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beam]] - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. <br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====Beam cutoff radii====<br />
<br />
* N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.<br />
<!--<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' ||<br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|}<br />
--><br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 1024 </math> for all LFI frequency channels.<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for 30GHz frequency channel, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for LFI and B-Spline beams,BS, Mars12 apodized for the CMB channels and the BS Mars12 for the sub-mm channels, for HFI (see section Inputs below).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
[[File:30.png| 600px| thumb | center| '''30GHz''']]<br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
Here we present histograms of the three fit parameters - beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The shy is sampled (pretty sparsely) at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
* The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.<br />
<br />
<br />
[[File:e_030_GB.png| 600px| thumb | center| '''ellipticity - 30GHz''']]<br />
[[File:solidarc_030_GB.png| 600px| thumb | center| '''beam solid angle (relative variations wrt scanning beam - 30GHz''']]<br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 healpix maps (they include the pixel window function).<br />
* The reported FWHM_eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''frequency''' || '''mean(fwhm)''' [arcmin] || '''sd(fwhm)''' [arcmin] || '''mean(e)''' || '''sd(e)''' || '''mean(<math> \psi</math>)''' [degree] || '''sd(<math> \psi</math>)''' [degree] || '''mean(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''sd(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''FWHM_eff''' [arcmin] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|}<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
* <math>\Omega_{eff}</math> - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: <math> 4 \pi*sum(effective_{beam})/max(effective_{beam})</math> , i.e. as an integral over the full extent of the effective beam, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <math>\Omega^{(1)}_{eff}</math> is the beam solid angle estimated up to a radius equal to one <math>fwhm_{eff}</math> and <math>\Omega^{(2)}_{eff}</math> up to a radius equal to twice the <math>fwhm_{eff}</math>.<br />
*** These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''Band''' || '''<math>\Omega_{eff}</math>'''[arcmin<math>^{2}</math>] || '''spatial variation''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(1)}_{eff}</math>''' [arcmin<math>^{2}</math>]|| '''spatial variation-1''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(2)}_{eff}</math>''' [arcmin<math>^{2}</math>] || '''spatial variation-2''' [arcmin<math>^{2}</math>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|}<br />
<br />
===Related products===<br />
<br />
===Monte Carlo simulations===<br />
<br />
FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, <math>C_\ell^{out}</math>, and 1 sigma errors<br />
<br />
<br />
As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be CMB or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations.<br />
<!--, FFP#,<br />
for example [[HL-sims#FFP6 data set|FFP6]].<br />
--><br />
<br />
== Window Functions ==<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_\ell </math> relates the true angular power spectra <math>C_\ell </math> with the observed angular power spectra <math>\widetilde{C}_\ell </math>:<br />
<br />
<math><br />
W_\ell= \widetilde{C}_\ell / C_\ell<br />
\label{eqn:wl1}</math> <br />
<br />
Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_\ell </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_\ell </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_\ell </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_\ell = < C^{conv}_\ell / C^{in}_\ell ><br />
\label{eqn:wl2}</math> <br />
<br />
For subtle reasons, we perform a more rigorous estimation of the window function by comparing <math> C^{conv}_\ell</math> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for LFI channels====<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, <math>W_\ell </math>, for LFI channels''']]<br />
<br />
== Sidelobes ==<br />
<br />
There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD.<br />
We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi.<br />
The beams have been computed in the same frames used for the main beams.<br />
The intermediate beam region (theta < 5 degrees) has been replaced with null values.<br />
<br />
*In the computation we considered:<br />
**the direct field from the feed<br />
**the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr<br />
**the 2nd order contributions SrPd and SdPd <br />
<br />
where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box. <br />
and where d=diffraction, r=reflection.<br />
For example Br, means that we considered in the calculation the reflection on the telescope baffle system. <br />
<br />
A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.<br />
<br />
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.''']]<br />
<br />
== References ==<br />
<br />
<References /> <br />
<br />
<br />
[[Category:LFI data processing|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Beams_LFI&diff=10247Beams LFI2014-12-17T09:57:32Z<p>Tvassall: /* Polarized Scanning Beams and Focal Plane calibration */</p>
<hr />
<div>{{DISPLAYTITLE:Beams}}<br />
<br />
== Overview ==<br />
<br />
LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.<br />
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]). <br />
<br />
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completness and they are not representative of flight beams.''']]<br />
<br />
<br />
<br />
<!--<br />
<br />
For the beam we consider these three regions:<br />
<br />
<br />
; main beam: is the portion of the pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.<br />
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called <b>intermediate beam</b>).<br />
; far sidelobes: is the pattern at angular regions more than 5 degrees from the beam center.<br />
--><br />
<br />
== Polarized Scanning Beams and Focal Plane calibration ==<br />
<br />
As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurements using Jupiter transits. the parameters that characterise the beam pointing are the following:<br />
<br />
* THETA_UV (<math>\theta_{uv}</math>)<br />
* PHI_UV (<math>\phi_{uv}</math>)<br />
<br />
They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as <br />
<br />
<math>\theta_{uv} = \arcsin(u^2+v^2)</math><br />
<br />
<math>\phi_{uv} = \arctan(v/u)</math><br />
<br />
Two additional angles are used to characterize the beams in the RIMO: <br />
<br />
* PSI_UV (<math>\psi_{uv}</math>)<br />
* PSI_POL (<math>\psi_{pol}</math>)<br />
<br />
<math>\psi_{uv}</math> and <math>\psi_{pol}</math> are '''not''' derived from measurements but they are estimated from '''optical simulations'''. They are the quantities that represent the polarization direction of each beam, in the following approximation: '''the M- and S- beams of the same RCA point at the same direction on the sky'''.<br />
<br />
The polarized scanning beams have been evaluated from optical simulations using GRASP Physical Optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM) {{PlanckPapers|tauber2010b}}. <br />
<br />
The Radio Frequency Tuned Model, called RFTM, was implemented to fit the in-flight beam measurements with the electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross polar beam in flight, we have relied on simulations validated by accurate beam measurements [[PlancPapers4]].<br />
<br />
The simulated beams where monochromatic and were computed throughout a 6 GHz band around the Optical Center Frequency (OCF). For the RFTM model the OCF were at <math>28.0, \, 44.0, \, 70.0</math> GHz. <br />
For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the RIMO R2.2 band-pass function [[]].<br />
<br />
The delivered products include the in-band averaged Stokes maps of Main Beams, Intermediate Beams and Sidelobes[[]].<br />
<br />
== Effective beams ==<br />
<br />
The '''effective beam''' is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky.<br />
It captures the complete information about the difference between the true and observed image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map. <br />
<br />
The full algebra involving the effective beams for temperature and polarisation was presented in {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature sky <math>\widetilde{\mathbf{T}} </math> is a convolution of the true sky <math>\mathbf{T} </math> and the effective beam <math>\mathbf{B}</math>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<math>t</math> is time samples, <math>A_{ti}</math> is <math>1</math> if the pointing direction falls in pixel number <math>i</math>, else it is <math>0</math>, <math>\mathbf{p}_t</math> represents the exact pointing direction (not approximated by the pixel centre location), and <math>\hat{\mathbf{r}}_j</math> is the centre of the pixel number <math>j</math>, where the scanbeam <math>b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)</math> is being evaluated (if the pointing direction falls within the cut-off radius of <math>\sim 2.5 \times</math> FWHM.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^T \ = \ \Delta\Omega \sum_j \mathbf{B}_{ij} \cdot (I \quad Q \quad U)_j^T,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\mathbf{B}_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^T_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^T(\hat{\mathbf{n}}_j,\hat{\mathbf{p}}_t) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
and <math>\mathbf{w}_t </math>and <math>\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) </math> are the the polarisation weight vectors, as defined in \cite{mitra2010}.<br />
<br />
The task is to compute <math>B_{ij}</math> for temperature only beams and the <math>3 \times 3</math> matrices <math>\mathbf{B}_{ij}</math> for each pixel <math>i</math>, at every neighbouring pixel <math>j</math> that fall within the cut-off radius around the the center of the <math>i^\text{th}</math> pixel.<br />
<br />
<br />
<br />
The effective beam is computed by stacking within a small field around each pixel of the HEALPix sky map. Due to the particular features of Planck scanning strategy coupled to the beam asymmetries in the focal plane, and data processing of the bolometer and radiometer TOIs, the resulting Planck effective beams vary over the sky. <br />
<br />
FEBeCoP, given information on Planck scanning beams and detector pointing during a mission period of interest, provides the pixelized stamps of both the Effective Beam, EB, and the Point Spread Function, PSF, at all positions of the HEALPix-formatted map pixel centres.<br />
<br />
<br />
===Production process===<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in {{BibCite|mitra2010}}.<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scanbeam models and different lengths of datasets. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixels---the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.<br />
<br />
====effBeam====<br />
<br />
The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.<br />
<br />
<br />
One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.<br />
<br />
<br />
====Computational Cost====<br />
<br />
The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below it isn displayed the computation cost on NERSC for nominal mission both in terms of CPU hrs and in Human time.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution.The cost in Human time is computed using an arbitrary number of nodes/core on Carver or Hopper NERSC Supercomputers<br />
|-<br />
|Channel ||030 || 044 || 070 <br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 <br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 <br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 <br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 <br />
|}<br />
<br />
<br />
The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.<br />
<br />
===Inputs===<br />
<br />
<br />
In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane''']]<br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====The scanning strategy====<br />
<br />
The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day (OD) 91 to OD 563 covering 3 surveys and half.<br />
<br />
====The scanbeam====<br />
<br />
The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.<br />
<br />
* LFI: [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beam]] - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. <br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
====Beam cutoff radii====<br />
<br />
* N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.<br />
<!--<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' ||<br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|}<br />
--><br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 1024 </math> for all LFI frequency channels.<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for 30GHz frequency channel, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for LFI and B-Spline beams,BS, Mars12 apodized for the CMB channels and the BS Mars12 for the sub-mm channels, for HFI (see section Inputs below).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
[[File:30.png| 600px| thumb | center| '''30GHz''']]<br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
Here we present histograms of the three fit parameters - beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The shy is sampled (pretty sparsely) at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
* The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.<br />
<br />
<br />
[[File:e_030_GB.png| 600px| thumb | center| '''ellipticity - 30GHz''']]<br />
[[File:solidarc_030_GB.png| 600px| thumb | center| '''beam solid angle (relative variations wrt scanning beam - 30GHz''']]<br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 healpix maps (they include the pixel window function).<br />
* The reported FWHM_eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''frequency''' || '''mean(fwhm)''' [arcmin] || '''sd(fwhm)''' [arcmin] || '''mean(e)''' || '''sd(e)''' || '''mean(<math> \psi</math>)''' [degree] || '''sd(<math> \psi</math>)''' [degree] || '''mean(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''sd(<math> \Omega </math>)''' [arcmin<math>^{2}</math>] || '''FWHM_eff''' [arcmin] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|}<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
* <math>\Omega_{eff}</math> - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: <math> 4 \pi*sum(effective_{beam})/max(effective_{beam})</math> , i.e. as an integral over the full extent of the effective beam, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <math>\Omega^{(1)}_{eff}</math> is the beam solid angle estimated up to a radius equal to one <math>fwhm_{eff}</math> and <math>\Omega^{(2)}_{eff}</math> up to a radius equal to twice the <math>fwhm_{eff}</math>.<br />
*** These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''Band''' || '''<math>\Omega_{eff}</math>'''[arcmin<math>^{2}</math>] || '''spatial variation''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(1)}_{eff}</math>''' [arcmin<math>^{2}</math>]|| '''spatial variation-1''' [arcmin<math>^{2}</math>] || '''<math>\Omega^{(2)}_{eff}</math>''' [arcmin<math>^{2}</math>] || '''spatial variation-2''' [arcmin<math>^{2}</math>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|}<br />
<br />
===Related products===<br />
<br />
===Monte Carlo simulations===<br />
<br />
FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, <math>C_\ell^{out}</math>, and 1 sigma errors<br />
<br />
<br />
As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be CMB or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations.<br />
<!--, FFP#,<br />
for example [[HL-sims#FFP6 data set|FFP6]].<br />
--><br />
<br />
== Window Functions ==<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_\ell </math> relates the true angular power spectra <math>C_\ell </math> with the observed angular power spectra <math>\widetilde{C}_\ell </math>:<br />
<br />
<math><br />
W_\ell= \widetilde{C}_\ell / C_\ell<br />
\label{eqn:wl1}</math> <br />
<br />
Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_\ell </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_\ell </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_\ell </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_\ell = < C^{conv}_\ell / C^{in}_\ell ><br />
\label{eqn:wl2}</math> <br />
<br />
For subtle reasons, we perform a more rigorous estimation of the window function by comparing <math> C^{conv}_\ell</math> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for LFI channels====<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, <math>W_\ell </math>, for LFI channels''']]<br />
<br />
== Sidelobes ==<br />
<br />
There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD.<br />
We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi.<br />
The beams have been computed in the same frames used for the main beams.<br />
The intermediate beam region (theta < 5 degrees) has been replaced with null values.<br />
<br />
*In the computation we considered:<br />
**the direct field from the feed<br />
**the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr<br />
**the 2nd order contributions SrPd and SdPd <br />
<br />
where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box. <br />
and where d=diffraction, r=reflection.<br />
For example Br, means that we considered in the calculation the reflection on the telescope baffle system. <br />
<br />
A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.<br />
<br />
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.''']]<br />
<br />
== References ==<br />
<br />
<References /> <br />
<br />
<br />
[[Category:LFI data processing|003]]</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10245Scanning Beams2014-12-16T18:26:34Z<p>Tvassall: /* Note on the file format */</p>
<hr />
<div><br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the stokes parameters for each sample, encoding all the information on the field intensity and polarization properties. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively the <math>I</math>, <math>Q</math>, <math>U</math> and <math>V</math> Stokes parameters.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
Main Beams are sampled with a grid map on the tangent space. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows of the grid map . <br />
The four stokes columns contain the succession of the <math>Nx \times Ny</math> samples of the map.<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10244Scanning Beams2014-12-16T18:25:57Z<p>Tvassall: /* Note on the file format */</p>
<hr />
<div><br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the stokes parameters for each sample, encoding all the information on the field intensity and polarization properties. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively the <math>I</math> <math>Q</math>,<math>U</math> and <math>V</math> Stokes parameters.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
Main Beams are sampled with a grid map on the tangent space. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows of the grid map . <br />
The four stokes columns contain the succession of the <math>Nx \times Ny</math> samples of the map.<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of Nphi Stokes parameters for each <math>\theta</math>.</div>Tvassallhttps://wiki.cosmos.esa.int/planckpla2015/index.php?title=Scanning_Beams&diff=10243Scanning Beams2014-12-16T18:25:36Z<p>Tvassall: /* Note on the file format */</p>
<hr />
<div><br />
<br />
===Note on the file format===<br />
<br />
The stokes map files consist in four columns containing the stokes parameters for each sample, encoding all the information on the field intensity and polarization properties. <br />
The columns are called "Beamdata" , "BeamdataQ", "BeamdataU" and "BeamdataV", containing respectively the <math>I</math> <math>Q</math>,<math>U</math> and <math>V</math> Stokes parameters.<br />
There are differences in the data format between the Main Beam and the near and far Sidelobes.<br />
Main Beams are sampled with a grid map on the tangent space. The dimension of the grid is given by the keywords "Nx" and "Ny", representing the number of columns and rows of the grid map . <br />
The four stokes columns contain the succession of the <math>Nx \times Ny</math> samples of the map.<br />
Sidelobes are sampled on the sphere. The resolution in <math>\theta</math> and <math>\phi</math> is given by the keywords "Ntheta" and "Nphi". The columns contain the succession of <math>Nphi</math> Stokes parameters for each <math>\theta</math>.</div>Tvassall