Difference between revisions of "Effective Beams"

Latest revision as of 02:01, 22 March 2018

Product description[edit]

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. It captures the complete information about the difference between the true and observed images of the sky. The effective beams are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map.

Details of the beam processing are given in the respective sections for HFI and LFI.

The full algebra involving the effective beams for temperature and polarization was presented in ^[1], and a discussion of its application to Planck data is given in the appropriate LFI (Planck-2013-IV^[2], Planck-2015-A05^[3]) and HFI (Planck-2013-VII^[4]) papers. Relevant details of the processing steps are given in the Effective Beams section of this document.

Comparison of the images of compact sources observed by Planck with FEBeCoP products[edit]

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 for a few example compact sources observed by Planck, for all LFI and HFI frequency channels. We show below a few panels of source images organized as follows:

Row #1 - DX9 images of four ERCSC objects with their Galactic (l,b) coordinates shown under the colour bar;
Row #2 - linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams (GBs) for LFI and B-Spline beams (BSs) of Mars12 apodized for the HFI CMB channels and Mars12 for the sub-mm channels (see section "Inputs" below);
Row #3 - log scale of #2, PSF iso-contours being shown as solid lines, elliptical Gaussian fit iso-contours shown as broken lines.

Comparison images of compact sources, effective beams, and PSFs.
30GHz
44GHz
70GHz
100GHz
143GHz
217GHz
353GHz
545GHz
857GHz

Histograms of the effective beam parameters[edit]

Here we present histograms of the three fit parameters, i.e., beam FWHM, ellipticity, and orientation with respect to the local meridian, and of the beam solid angle. The sky is uniformly sampled (although fairly sparsely) in 3072 directions that were chosen as HEALPix N_side=16 pixel centres for HFI and in 768 directions that were chosen as HEALPix N_side=8 pixel centres for LFI.

Here the beam solid angle is estimated according to the definition: "4π Σ(effbeam)/max(effbeam)" i.e., 4π Σ(B_ij) / max(B_ij).

Histograms for LFI effective beam parameters.

Histograms for HFI effective beam parameters.

Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian[edit]

The discontinuities at the HEALPix domain edges in the maps are a visual artefact due to the interplay of the discretized effective beam and the HEALPix pixel grid.

Sky variation of ellipticity of the best-fit Gaussian for the effective beams.
Ellipticity - 30GHz.
Ellipticity - 44GHz.
Ellipticity - 70GHz.
Ellipticity - 100GHz.
Ellipticity - 143GHz.
Ellipticity - 217GHz.
Ellipticity - 353GHz.
Ellipticity - 545GHz.
Ellipticity - 857GHz.

Sky variation of the relative solid angle of the best-fit Gaussian for the effective beams.
Relative beam solid angle variations w.r.t. scanning beam - 30GHz.
Relative beam solid angle variations w.r.t. scanning beam - 44GHz.
Relative beam solid angle variations w.r.t. scanning beam - 70GHz.
Relative beam solid angle variations w.r.t. scanning beam - 100GHz.
Relative beam solid angle variations w.r.t. scanning beam - 143GHz.
Relative beam solid angle variations w.r.t. scanning beam - 217GHz.
Relative beam solid angle variations w.r.t. scanning beam - 353GHz.
Relative beam solid angle variations w.r.t. scanning beam - 545GHz.
Relative beam solid angle variations w.r.t. scanning beam - 857GHz.

Sky variation of the relative FWHM of the best-fit Gaussian for theeffective beams.
Relative FWHM variations w.r.t. scanning beam - 30GHz.
Relative FWHM variations w.r.t. scanning beam - 44GHz.
Relative FWHM variations w.r.t. scanning beam - 70GHz.
Relative FWHM variations w.r.t. scanning beam - 100GHz.
Relative FWHM variations w.r.t. scanning beam - 143GHz.
Relative FWHM variations w.r.t. scanning beam - 217GHz.
Relative FWHM variations w.r.t. scanning beam - 353GHz.
Relative FWHM variations w.r.t. scanning beam - 545GHz.
Relative FWHM variations w.r.t. scanning beam - 857GHz.

Sky variation of effective beams <math
ψ - 30GHz.
ψ - 44GHz.
ψ - 70GHz.
ψ - 100GHz.
ψ - 143GHz.
ψ - 217GHz.
ψ - 353GHz.
ψ - 545GHz.
ψ - 857GHz.

Statistics of the effective beams computed using FEBeCoP[edit]

Below we tabulate the simple statistics of FWHM, ellipticity (e), orientation (ψ) and beam solid angle, (Ω), for a sample of 3072 and 768 directions on the sky for HFI and LFI data, respectively. Statistics shown in the Table are derived from the histograms shown above.

The derived beam parameters are representative of the DPC N_side 1024 and 2048 HEALPix maps (including the pixel window function).
The reported FWHM_eff values are derived from the beam solid angles, under a Gaussian approximation. These are appropriate for flux determination, while the the Gaussian fits to the effective beam maps are more suited for source identification.

**Statistics of the FEBeCoP effective beams computed with the BS Mars12 data apodized for the CMB channels and oversampled.**
Frequency [GHz]	Mean(FWHM) [arcmin]	σ(FWHM) [arcmin]	Mean(e)	σ(e)	Mean(ψ) [deg]	σ(ψ) [deg]	Mean(Ω) [arcmin²]	σ(Ω) [arcmin²]	FWHM_eff [arcmin]
30	32.239	0.013	1.320	0.031	-0.304	55.349	1189.513	0.842	32.34
44	27.005	0.552	1.034	0.033	0.059	53.767	832.946	31.774	27.12
070	13.252	0.033	1.223	0.026	0.587	55.066	200.742	1.027	13.31
100	9.651	0.014	1.186	0.023	-0.024	55.400	105.778	0.311	9.66
143	7.248	0.015	1.036	0.009	0.383	54.130	59.954	0.246	7.27
217	4.990	0.025	1.177	0.030	0.836	54.999	28.447	0.271	5.01
353	4.818	0.024	1.147	0.028	0.655	54.745	26.714	0.250	4.86
545	4.682	0.044	1.161	0.036	0.544	54.876	26.535	0.339	4.84
857	4.325	0.055	1.393	0.076	0.876	54.779	24.244	0.193	4.63

Beam solid angles for the PCCS[edit]

For the Planck Catalogue of Compact Sources we also need the following definitions.

Ω_eff - is the mean beam solid angle of the effective beam, where the solid angle is estimated according to the definition "4 π Σ(effbeam)/max(effbeam)", i.e., as an integral over the full extent of the effective beam, 4π Σ (B_ij)/max(B_ij).

From Ω_eff we estimate FWHM_eff, under a Gaussian approximation - these are tabulated above.

Ω⁽¹⁾_eff is the beam solid angle estimated up to a radius equal to 1 FWHM_eff and Ω⁽²⁾_eff up to a radius equal to twice the FWHM_eff.

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).

**Band averaged beam solid angles**
Band [GHz]	Ω_eff[arcmin²]	Spatial variation [arcmin²]	Ω⁽¹⁾_eff [arcmin²]	Spatial variation-1 [arcmin²]	Ω⁽²⁾_eff [arcmin²]	Spatial variation-2 [arcmin²]
30	1189.513	0.842	1116.494	2.274	1188.945	0.847
44	832.946	31.774	758.684	29.701	832.168	31.811
70	200.742	1.027	186.260	2.300	200.591	1.027
100	105.778	0.311	100.830	0.410	105.777	0.311
143	59.954	0.246	56.811	0.419	59.952	0.246
217	28.447	0.271	26.442	0.537	28.426	0.271
353	26.714	0.250	24.827	0.435	26.653	0.250
545	26.535	0.339	24.287	0.455	26.302	0.337
857	24.244	0.193	22.646	0.263	23.985	0.191

Production process[edit]

FEBeCoP, the Fast Effective Beam Convolution in Pixel space^[1], is an approach for representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:

identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from the observations of planets, but in general, an arbitrary mathematical representation of the beam can be used on input);
exactly follow the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position;
project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM in diameter);
add up all beams that cross the same pixel and its vicinity over the observing period of interest,
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 times a few hundred);
use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission.

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, so that several estimations of the effective beams were possible for all Planck detectors for different scan and beam models, as well as different lengths of data set.

Pixel-ordered detector angles (PODA)[edit]

The main challenge in computing the effective beams is to go through the roughly trillion samples, which becomes severely limited by I/O. In the first stage, for a given data set, ordered lists of pointing angles for each pixel (the pixel-ordered detector angles, PODA) are made. This is an one-time process for each data set. We used computers with large memory and tedious memory management and bookkeeping to make this step efficient.

Effective beam[edit]

The "effBeam" part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we have tried to made sure that no repetition occurs in evaluating any trigonometric quantity.

One important reason for separating the two steps is that they use different parallel computing schemes. The PODA part requires parallelization over time-ordered data samples, while the effBeam part requires distribution of pixels among different computers.

Computational Cost[edit]

The computation of the effective beams has been performed at the NERSC Supercomputing Center. The table below shows the computational cost for FEBeCoP processing of the nominal mission.

**Computational cost for PODA, effective beam and single map convolution. Wall-clock time is given as a guide, as found on the NERSC supercomputers.**
Channel [GHz]	030	044	070	100	143	217	353	545	857
PODA/detector computation time [CPU hrs]	85	100	250	500	500	500	500	500	500
PODA/detector computation time [wall clock hrs]	7	10	20	20	20	20	20	20	20
Beam/channel computation time [CPU hrs]	900	2000	2300	2800	3800	3200	3000	900	1100
Beam/channel computation time [wall clock hrs]	0.5	0.8	1	1.5	2	1.2	1	0.5	0.5
convolution computation time [CPU hrs]	1	1.2	1.3	3.6	4.8	4.0	4.1	4.1	3.7
convolution computation time [wall clock sec]	1	1	1	4	4	4	4	4	4
Effective beam size [GB]	173	123	28	187	182	146	132	139	124

The computation cost, especially for PODA and convolution, is heavily limited by the I/O capacity of the disk and so it depends on the overall usage of the cluster by other users.

Inputs[edit]

In order to describe the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by incoming CMB photons. The scan direction is marked, and the centre of the focal plane is at an angle of 85° with respect to the spin axis, pointing upward in the picture.

Planck focal plane.

The focal plane database (FPDB)[edit]

The FPDB contains information on each detector, e.g., the orientation of the polarization axis, and different weight factors. Details are provided in the instrument RIMOs):

for HFI - the HFI RIMO;
for LFI - the LFI RIMO.

The scanning strategy[edit]

The scanning strategy description consists of the three pointing angles for each detector for each sample. Detector pointings for the nominal mission cover about 15 months of observation from Operational Day (OD) 91 to OD 563, covering three and a half surveys.

The scanning beam[edit]

The "scanbeam" is modelled for each detector through the observation of planets. This was assumed to be constant over the whole mission, although FEBeCoP could be used for a few sets of scanbeams too.

For LFI the GRASP scanning beams are used; these are based on the radio-frequency tuned model (RFTM), smeared to simulate the in-flight optical response.
For HFI the B-Spline, BS beams are used based on two observations of Mars.

Beam cutoff radii[edit]

The cutoffs are set to N times the geometric mean of the FWHM of all detectors in a channel, where N is given below.

**Beam cut off radius.**
Channel [GHz]	Cutoff radii in units of FWHM	FWHM of full beam extent [arcmin]
30, 44, 70	2.5
100	2.25	23.703699
143	3	21.057402
217, 353	4	18.782754
545	4	18.327635
857	4	17.093706

Map resolution for the derived beam data object[edit]

The resolution is

N_side = 1024 for LFI frequency channels, and
N_side = 2048 for HFI frequency channels.

Related products[edit]

Monte Carlo simulations[edit]

FEBeCoP software enables fast, full-sky convolutions of the sky signals with the effective beams in the pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps can be easily generated, which are convolved with realistically rendered, spatially varying, asymmetric Planck beams. We performed the following steps:

generate the effective beams with FEBeCoP for all frequencies for "dDX9" data and the Nominal Mission;
generate 100 realizations of maps from a fiducial CMB power spectrum;
convolve each one of these maps with the effective beams using FEBeCoP;
estimate the average of the power spectrum of each convolved realization, and 1 σ errors.

Since FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations can be generated. These Monte Carlo simulations of the sky signal (which might be CMB or a foreground, e.g., dust) along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission and tagged as full focal-plane simulations, FFP# (for example the FFP6).

Beam Window Functions[edit]

The "Transfer Function" or the "Beam Window Function", W_ℓ relates the true angular power spectra C_ℓ with the observed angular power spectra Ĉ_ℓ:

Note that, the window function can contain a pixel window function (depending on the definition) and it is not the angular power spectra of the scanbeams, although in principle one may be able to connect them via some complicated algebra.

The window functions are estimated by performing Monte Carlo simulations. We generate several random realizations of the CMB sky, starting from a given fiducial C_ℓ, convolve the maps with the pre-computed effective beams, compute the convolved power spectra C_ℓ^conv, divide by the power spectra of the unconvolved map C_ℓⁱⁿ and average over their ratio. Thus, the estimated window function is

For subtle reasons, we perform a more rigorous estimation of the window function by comparing C_ℓ^conv with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but not necessarily exactly the same) size and then scaling the estimated window function accordingly.

Beam window functions are provided in the RIMO.

Beam window functions, W_ℓ, for the Planck mission.[edit]

Beam window functions, W_ℓ, for LFI channels.

Beam window functions, W_ℓ, for HFI channels.

File Names[edit]

The effective beams are provided in the PLA as FITS files containg HEALPix maps of the beams. For the file names the following convention is used.

Single beam query, "beams_FFF_PixelNumber.fits" :
- FFF is the channel frequency (one of "30", "44", "70", "100", "143", "217", "353", "545", "857");
- "PixelNumber" is the number of the pixel to which the beam corresponds, 0–12 × N_side² - 1, with N_side=1024 for the LFI and N_side=2048 for the HFI.
Multiple beam query, "beams_FFF_FirstPixelNumber-LastPixelNumber.zip" .
- The compressed files contains a set of files with the beams for the pixels covereing the selected region, with "FFF" as for sinle beam query and the naming convention for the beam files contained in the .zip file being the same as for single beam queries;
- FirstPixelNumber is the lowest pixel number for the area covered by the request;
- LastPixelNumber is the highest pixel number for the area covered by the request.

File format[edit]

The FITS files provided in the PLA contain HEALPix maps of the beams.

Previous Releases: (2015) and (2013) effective beams[edit]

2015 Release of effective beams

Product description 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. It captures the complete information about the difference between the true and observed image of the sky. The effective beams are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map.

Details of the beam processing are given in the respective pages for HFI and LFI.

The full algebra involving the effective beams for temperature and polarisation was presented in ^[1], and a discussion of its application to Planck data is given in the appropriate LFI Planck-2013-IV^[2], Planck-2015-A05^[3] and HFI Planck-2013-VII^[4] papers. Relevant details of the processing steps are given in the Effective Beams section of this document.

Comparison of the images of compact sources observed by Planck with FEBeCoP products 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 all LFI and HFI frequency channels, as an example. We show below a few panels of source images organized as follows:

Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar
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).
Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line

Comparison images of compact sources and effective beams, PSFs
30GHz
44GHz
70GHz
100GHz
143GHz
217GHz
353GHz
545GHz
857GHz

Histograms of the effective beam parameters 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 sky is sampled (pretty sparsely) at 3072 directions which were chosen as HEALpix nside=16 pixel centers for HFI and at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI. These uniformly sample the sky.

Where beam solid angle is estimated according to the definition: [math] 4 \pi \sum[/math](effbeam)/max(effbeam) i.e.,

Histograms for LFI effective beam parameters

Histograms for HFI effective beam parameters

Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian

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.

Sky variation of effective beams ellipticity of the best-fit Gaussian
ellipticity - 30GHz
ellipticity - 44GHz
ellipticity - 70GHz
ellipticity - 100GHz
ellipticity - 143GHz
ellipticity - 217GHz
ellipticity - 353GHz
ellipticity - 545GHz
ellipticity - 857GHz

Sky (relative) variation of effective beams solid angle of the best-fit Gaussian
beam solid angle (relative) variations wrt scanning beam - 30GHz
beam solid angle (relative) variations wrt scanning beam - 44GHz
beam solid angle (relative) variations wrt scanning beam - 70GHz
beam solid angle (relative) variations wrt scanning beam - 100GHz
beam solid angle (relative) variations wrt scanning beam - 143GHz
beam solid angle (relative) variations wrt scanning beam - 217GHz
beam solid angle (relative) variations wrt scanning beam - 353GHz
beam solid angle (relative) variations wrt scanning beam - 545GHz
beam solid angle (relative) variations wrt scanning beam - 857GHz

Sky (relative) variation of effective beams fwhm of the best-fit Gaussian
fwhm (relative) variations wrt scanning beam - 30GHz
fwhm (relative) variations wrt scanning beam - 44GHz
fwhm (relative) variations wrt scanning beam - 70GHz
fwhm (relative) variations wrt scanning beam - 100GHz
fwhm (relative) variations wrt scanning beam - 143GHz
fwhm (relative) variations wrt scanning beam - 217GHz
fwhm (relative) variations wrt scanning beam - 353GHz
fwhm (relative) variations wrt scanning beam - 545GHz
fwhm (relative) variations wrt scanning beam - 857GHz

Sky variation of effective beams <math
[math]\psi[/math] - 30GHz
[math]\psi[/math] - 44GHz
[math]\psi[/math] - 70GHz
[math]\psi[/math] - 100GHz
[math]\psi[/math] - 143GHz
[math]\psi[/math] - 217GHz
[math]\psi[/math] - 353GHz
[math]\psi[/math] - 545GHz
[math]\psi[/math] - 857GHz

Statistics of the effective beams computed using FEBeCoP

We tabulate the simple statistics of FWHM, ellipticity (e), orientation ([math] \psi[/math]) and beam solid angle, ([math] \Omega [/math]), for a sample of 3072 and 768 directions on the sky for HFI and LFI data respectively. Statistics shown in the Table are derived from the histograms shown above.

The derived beam parameters are representative of the DPC NSIDE 1024 and 2048 healpix maps (they include the pixel window function).
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.

**Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled**
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]
030	32.239	0.013	1.320	0.031	-0.304	55.349	1189.513	0.842	32.34
044	27.005	0.552	1.034	0.033	0.059	53.767	832.946	31.774	27.12
070	13.252	0.033	1.223	0.026	0.587	55.066	200.742	1.027	13.31
100	9.651	0.014	1.186	0.023	-0.024	55.400	105.778	0.311	9.66
143	7.248	0.015	1.036	0.009	0.383	54.130	59.954	0.246	7.27
217	4.990	0.025	1.177	0.030	0.836	54.999	28.447	0.271	5.01
353	4.818	0.024	1.147	0.028	0.655	54.745	26.714	0.250	4.86
545	4.682	0.044	1.161	0.036	0.544	54.876	26.535	0.339	4.84
857	4.325	0.055	1.393	0.076	0.876	54.779	24.244	0.193	4.63

Beam solid angles for the PCCS

[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 [/math](effbeam)/max(effbeam), i.e. as an integral over the full extent of the effective beam, i.e. .

from [math]\Omega_{eff}[/math] we estimate the [math]fwhm_{eff}[/math], under a Gaussian approximation - these are tabulated above
- [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].
  - 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).

**Band averaged beam solid angles**
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]]
30	1189.513	0.842	1116.494	2.274	1188.945	0.847
44	832.946	31.774	758.684	29.701	832.168	31.811
70	200.742	1.027	186.260	2.300	200.591	1.027
100	105.778	0.311	100.830	0.410	105.777	0.311
143	59.954	0.246	56.811	0.419	59.952	0.246
217	28.447	0.271	26.442	0.537	28.426	0.271
353	26.714	0.250	24.827	0.435	26.653	0.250
545	26.535	0.339	24.287	0.455	26.302	0.337
857	24.244	0.193	22.646	0.263	23.985	0.191

Production process FEBeCoP, or Fast Effective Beam Convolution in Pixel space^[1], is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:

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)
follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position
project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)
add up all beams that cross the same pixel and its vicinity over the observing period of interest
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)
use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission

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 scan and beam models, as well as different lengths of datasets.

Pixel Ordered Detector Angles (PODA) 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 pixel - 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.

effBeam 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.

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.

Computational Cost The computation of the effective beams has been performed at the NERSC Supercomputing Center. The table below shows the computation cost for FEBeCoP processing of the nominal mission.

**Computational cost for PODA, Effective Beam and single map convolution. Wall-clock time is given as a guide, as found on the NERSC supercomputers.**
Channel	030	044	070	100	143	217	353	545	857
PODA/Detector Computation time (CPU hrs)	85	100	250	500	500	500	500	500	500
PODA/Detector Computation time (wall clock hrs)	7	10	20	20	20	20	20	20	20
Beam/Channel Computation time (CPU hrs)	900	2000	2300	2800	3800	3200	3000	900	1100
Beam/Channel Computation time (wall clock hrs)	0.5	0.8	1	1.5	2	1.2	1	0.5	0.5
Convolution Computation time (CPU hr)	1	1.2	1.3	3.6	4.8	4.0	4.1	4.1	3.7
Convolution Computation time (wall clock sec)	1	1	1	4	4	4	4	4	4
Effective Beam Size (GB)	173	123	28	187	182	146	132	139	124

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.

Inputs 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.

Planck Focal Plane

The Focal Plane DataBase (FPDB)

The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument RIMOs):

HFI - The HFI RIMO
LFI - The LFI RIMO

The scanning strategy 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.

The scanbeam 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.

LFI: GRASP scanning beam - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response.
HFI: B-Spline, BS based on 2 observations of Mars.

(see the instrument RIMOs).

Beam cutoff radii N times geometric mean of FWHM of all detectors in a channel, where N

**Beam cut off radius**
channel	Cutoff Radii in units of fwhm	fwhm of full beam extent
30 - 44 - 70	2.5
100	2.25	23.703699
143	3	21.057402
217-353	4	18.782754
sub-mm	4	18.327635(545GHz) ; 17.093706(857GHz)

Map resolution for the derived beam data object

[math]N_{side} = 1024 [/math] for LFI frequency channels
[math]N_{side} = 2048 [/math] for HFI frequency channels

Related products Monte Carlo simulations

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:

generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission
generate 100 realizations of maps from a fiducial CMB power spectrum
convolve each one of these maps with the effective beams using FEBeCoP
estimate the average of the Power Spectrum of each convolved realization, and 1 [math]\sigma[/math] errors

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, FFP#. For example FFP6

Beam Window Functions The Transfer Function or the Beam Window Function [math] W_l [/math] relates the true angular power spectra [math]C_l [/math] with the observed angular power spectra [math]\widetilde{C}_l [/math]:

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.

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_l [/math], convolve the maps with the pre-computed effective beams, compute the convolved power spectra [math] C^\text{conv}_l [/math], divide by the power spectra of the unconvolved map [math]C^\text{in}_l [/math] and average over their ratio. Thus, the estimated window function

For subtle reasons, we perform a more rigorous estimation of the window function by comparing C^{conv}_l 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.

Beam window functions are provided in the RIMO.

Beam Window functions, Wl, for Planck mission

Beam Window functions, Wl, for LFI channels

Beam Window functions, Wl, for HFI channels

File Names The effective beams are provided by the PLA as FITS files containg HEALPix maps of the beams. For the file names the following convention is used:

Single beam query: beams_FFF_PixelNumber.fits
- FFF is the channel frequency (one of 30, 44, 70, 100, 143, 217, 353, 545, 857);
- PixelNumber is the number of the pixel to which the beam corresponds ([math]0[/math] - ). For the LFI [math]N_{\rm side}=1024[/math], for the HFI [math]N_{\rm side}=2048[/math];
Multiple beam query: beams_FFF_FirstPixelNumber-LastPixelNumber.zip
The compressed files contains a set of files with the beams for the pixels covereing the selected region. FFF as for sinle beam query. The naming convention for the beam files contained in the .zip file is the same as for single beam queries.
- FirstPixelNumber is the lowest pixel number for the area covered by the request;
- LastPixelNumber is the highest pixel number for the area covered by the request;

File format The FITS files provided by the PLA contain HEALPix maps of the beams.

2013 Release of effective beams

Product description 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. It captures the complete information about the difference between the true and observed image of the sky. The effective beams are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map.

Details of the beam processing are given in the respective pages for HFI and LFI.

The full algebra involving the effective beams for temperature and polarisation was presented in ^[1], and a discussion of its application to Planck data is given in the appropriate LFI Planck-2013-IV^[2], Planck-2015-A05^[3] and HFI Planck-2013-VII^[4] papers. Relevant details of the processing steps are given in the Effective Beams section of this document.

Comparison of the images of compact sources observed by Planck with FEBeCoP products 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 all LFI and HFI frequency channels, as an example. We show below a few panels of source images organized as follows:

Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar
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).
Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line

Comparison images of compact sources and effective beams, PSFs
30GHz
44GHz
70GHz
100GHz
143GHz
217GHz
353GHz
545GHz
857GHz

Histograms of the effective beam parameters 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 sky is sampled (pretty sparsely) at 3072 directions which were chosen as HEALpix nside=16 pixel centers for HFI and at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI. These uniformly sample the sky.

Where beam solid angle is estimated according to the definition: [math] 4 \pi \sum[/math](effbeam)/max(effbeam) i.e.,

Histograms for LFI effective beam parameters

Histograms for HFI effective beam parameters

Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian

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.

Sky variation of effective beams ellipticity of the best-fit Gaussian
ellipticity - 30GHz
ellipticity - 44GHz
ellipticity - 70GHz
ellipticity - 100GHz
ellipticity - 143GHz
ellipticity - 217GHz
ellipticity - 353GHz
ellipticity - 545GHz
ellipticity - 857GHz

Sky (relative) variation of effective beams solid angle of the best-fit Gaussian
beam solid angle (relative) variations wrt scanning beam - 30GHz
beam solid angle (relative) variations wrt scanning beam - 44GHz
beam solid angle (relative) variations wrt scanning beam - 70GHz
beam solid angle (relative) variations wrt scanning beam - 100GHz
beam solid angle (relative) variations wrt scanning beam - 143GHz
beam solid angle (relative) variations wrt scanning beam - 217GHz
beam solid angle (relative) variations wrt scanning beam - 353GHz
beam solid angle (relative) variations wrt scanning beam - 545GHz
beam solid angle (relative) variations wrt scanning beam - 857GHz

Sky (relative) variation of effective beams fwhm of the best-fit Gaussian
fwhm (relative) variations wrt scanning beam - 30GHz
fwhm (relative) variations wrt scanning beam - 44GHz
fwhm (relative) variations wrt scanning beam - 70GHz
fwhm (relative) variations wrt scanning beam - 100GHz
fwhm (relative) variations wrt scanning beam - 143GHz
fwhm (relative) variations wrt scanning beam - 217GHz
fwhm (relative) variations wrt scanning beam - 353GHz
fwhm (relative) variations wrt scanning beam - 545GHz
fwhm (relative) variations wrt scanning beam - 857GHz

Sky variation of effective beams <math
[math]\psi[/math] - 30GHz
[math]\psi[/math] - 44GHz
[math]\psi[/math] - 70GHz
[math]\psi[/math] - 100GHz
[math]\psi[/math] - 143GHz
[math]\psi[/math] - 217GHz
[math]\psi[/math] - 353GHz
[math]\psi[/math] - 545GHz
[math]\psi[/math] - 857GHz

Statistics of the effective beams computed using FEBeCoP

We tabulate the simple statistics of FWHM, ellipticity (e), orientation ([math] \psi[/math]) and beam solid angle, ([math] \Omega [/math]), for a sample of 3072 and 768 directions on the sky for HFI and LFI data respectively. Statistics shown in the Table are derived from the histograms shown above.

The derived beam parameters are representative of the DPC NSIDE 1024 and 2048 healpix maps (they include the pixel window function).
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.

**Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled**
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]
030	32.239	0.013	1.320	0.031	-0.304	55.349	1189.513	0.842	32.34
044	27.005	0.552	1.034	0.033	0.059	53.767	832.946	31.774	27.12
070	13.252	0.033	1.223	0.026	0.587	55.066	200.742	1.027	13.31
100	9.651	0.014	1.186	0.023	-0.024	55.400	105.778	0.311	9.66
143	7.248	0.015	1.036	0.009	0.383	54.130	59.954	0.246	7.27
217	4.990	0.025	1.177	0.030	0.836	54.999	28.447	0.271	5.01
353	4.818	0.024	1.147	0.028	0.655	54.745	26.714	0.250	4.86
545	4.682	0.044	1.161	0.036	0.544	54.876	26.535	0.339	4.84
857	4.325	0.055	1.393	0.076	0.876	54.779	24.244	0.193	4.63

Beam solid angles for the PCCS

[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 [/math](effbeam)/max(effbeam), i.e. as an integral over the full extent of the effective beam, i.e. .

from [math]\Omega_{eff}[/math] we estimate the [math]fwhm_{eff}[/math], under a Gaussian approximation - these are tabulated above
- [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].
  - 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).

**Band averaged beam solid angles**
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]]
30	1189.513	0.842	1116.494	2.274	1188.945	0.847
44	832.946	31.774	758.684	29.701	832.168	31.811
70	200.742	1.027	186.260	2.300	200.591	1.027
100	105.778	0.311	100.830	0.410	105.777	0.311
143	59.954	0.246	56.811	0.419	59.952	0.246
217	28.447	0.271	26.442	0.537	28.426	0.271
353	26.714	0.250	24.827	0.435	26.653	0.250
545	26.535	0.339	24.287	0.455	26.302	0.337
857	24.244	0.193	22.646	0.263	23.985	0.191

Production process FEBeCoP, or Fast Effective Beam Convolution in Pixel space^[1], is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:

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)
follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position
project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)
add up all beams that cross the same pixel and its vicinity over the observing period of interest
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)
use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission

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 scan and beam models, as well as different lengths of datasets.

Pixel Ordered Detector Angles (PODA) 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 pixel - 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.

effBeam 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.

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.

Computational Cost The computation of the effective beams has been performed at the NERSC Supercomputing Center. The table below shows the computation cost for FEBeCoP processing of the nominal mission.

**Computational cost for PODA, Effective Beam and single map convolution. Wall-clock time is given as a guide, as found on the NERSC supercomputers.**
Channel	030	044	070	100	143	217	353	545	857
PODA/Detector Computation time (CPU hrs)	85	100	250	500	500	500	500	500	500
PODA/Detector Computation time (wall clock hrs)	7	10	20	20	20	20	20	20	20
Beam/Channel Computation time (CPU hrs)	900	2000	2300	2800	3800	3200	3000	900	1100
Beam/Channel Computation time (wall clock hrs)	0.5	0.8	1	1.5	2	1.2	1	0.5	0.5
Convolution Computation time (CPU hr)	1	1.2	1.3	3.6	4.8	4.0	4.1	4.1	3.7
Convolution Computation time (wall clock sec)	1	1	1	4	4	4	4	4	4
Effective Beam Size (GB)	173	123	28	187	182	146	132	139	124

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.

Inputs 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.

Planck Focal Plane

The Focal Plane DataBase (FPDB)

The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument RIMOs):

HFI - The HFI RIMO
LFI - The LFI RIMO

The scanning strategy 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.

The scanbeam 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.

LFI: GRASP scanning beam - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response.
HFI: B-Spline, BS based on 2 observations of Mars.

(see the instrument RIMOs).

Beam cutoff radii N times geometric mean of FWHM of all detectors in a channel, where N

**Beam cut off radius**
channel	Cutoff Radii in units of fwhm	fwhm of full beam extent
30 - 44 - 70	2.5
100	2.25	23.703699
143	3	21.057402
217-353	4	18.782754
sub-mm	4	18.327635(545GHz) ; 17.093706(857GHz)

Map resolution for the derived beam data object

[math]N_{side} = 1024 [/math] for LFI frequency channels
[math]N_{side} = 2048 [/math] for HFI frequency channels

Related products Monte Carlo simulations

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:

generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission
generate 100 realizations of maps from a fiducial CMB power spectrum
convolve each one of these maps with the effective beams using FEBeCoP
estimate the average of the Power Spectrum of each convolved realization, and 1 [math]\sigma[/math] errors

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, FFP#. For example FFP6

Beam Window Functions The Transfer Function or the Beam Window Function [math] W_l [/math] relates the true angular power spectra [math]C_l [/math] with the observed angular power spectra [math]\widetilde{C}_l [/math]:

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.

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_l [/math], convolve the maps with the pre-computed effective beams, compute the convolved power spectra [math] C^\text{conv}_l [/math], divide by the power spectra of the unconvolved map [math]C^\text{in}_l [/math] and average over their ratio. Thus, the estimated window function

For subtle reasons, we perform a more rigorous estimation of the window function by comparing C^{conv}_l 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.

Beam window functions are provided in the RIMO.

Beam Window functions, Wl, for Planck mission

Beam Window functions, Wl, for LFI channels

Beam Window functions, Wl, for HFI channels

File Names The effective beams are provided by the PLA as FITS files containg HEALPix maps of the beams. For the file names the following convention is used:

Single beam query: beams_FFF_PixelNumber.fits
- FFF is the channel frequency (one of 30, 44, 70, 100, 143, 217, 353, 545, 857);
- PixelNumber is the number of the pixel to which the beam corresponds ([math]0[/math] - ). For the LFI [math]N_{\rm side}=1024[/math], for the HFI [math]N_{\rm side}=2048[/math];
Multiple beam query: beams_FFF_FirstPixelNumber-LastPixelNumber.zip
The compressed files contains a set of files with the beams for the pixels covereing the selected region. FFF as for sinle beam query. The naming convention for the beam files contained in the .zip file is the same as for single beam queries.
- FirstPixelNumber is the lowest pixel number for the area covered by the request;
- LastPixelNumber is the highest pixel number for the area covered by the request;

File format The FITS files provided by the PLA contain HEALPix maps of the beams.

References[edit]

↑ ^1.0^1.1^1.2^1.3^1.4^1.5 Fast Pixel Space Convolution for Cosmic Microwave Background Surveys with Asymmetric Beams and Complex Scan Strategies: FEBeCoP, S. Mitra, G. Rocha, K. M. Górski, K. M. Huffenberger, H. K. Eriksen, M. A. J. Ashdown, C. R. Lawrence, ApJS, 193, 5-+, (2011).
↑ ^2.0^2.1^2.2 Planck 2013 results. IV. Low Frequency Instrument beams and window functions, Planck Collaboration, 2014, A&A, 571, A4.
↑ ^3.0^3.1^3.2 Planck 2015 results. IV. LFI beams and window functions, Planck Collaboration, 2016, A&A, 594, A4.
↑ ^4.0^4.1^4.2 Planck 2013 results. VII. HFI time response and beams, Planck Collaboration, 2014, A&A, 571, A7.

[.5B.5B:Template:Mitra2010.5D.5D-1] 1.0^1.1^1.2^1.3^1.4^1.5 Fast Pixel Space Convolution for Cosmic Microwave Background Surveys with Asymmetric Beams and Complex Scan Strategies: FEBeCoP, S. Mitra, G. Rocha, K. M. Górski, K. M. Huffenberger, H. K. Eriksen, M. A. J. Ashdown, C. R. Lawrence, ApJS, 193, 5-+, (2011).

[planck2013-p02d-2] 2.0^2.1^2.2 Planck 2013 results. IV. Low Frequency Instrument beams and window functions, Planck Collaboration, 2014, A&A, 571, A4.

[planck2014-a05-3] 3.0^3.1^3.2 Planck 2015 results. IV. LFI beams and window functions, Planck Collaboration, 2016, A&A, 594, A4.

[planck2013-p03c-4] 4.0^4.1^4.2 Planck 2013 results. VII. HFI time response and beams, Planck Collaboration, 2014, A&A, 571, A7.

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
-<span style="color:red"></span>
+{{DISPLAYTITLE:Effective Beams}}
 ==Product description==
-----------------------
+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.
+It captures the complete information about the difference between the true and observed images of the sky. The effective beams are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map.
+Details of the beam processing are given in the respective sections for [[Beams|HFI]] and [[Beams_LFI|LFI]].
+The full algebra involving the effective beams for temperature and polarization was presented in {{BibCite|mitra2010}}, and a discussion of its application to Planck data is given in the appropriate LFI ({{PlanckPapers|planck2013-p02d}}, {{PlanckPapers|planck2014-a05||Planck-2015-A05}}) and HFI ({{PlanckPapers|planck2013-p03c}}) papers. Relevant details of the processing steps are given in the [[Beams|Effective Beams]] section of this document.
+<!-- Everything from here down to the "Production Process" section should eventually be moved to a new section the Joint Processing pages -->
+===Comparison of the images of compact sources observed by Planck with FEBeCoP products===
+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 for a few example compact sources observed by Planck, for all LFI and HFI  frequency channels.  We show below a few panels of source images organized as follows:
+* Row #1 - DX9 images of four ERCSC objects with their Galactic (<i>l</i>,<i>b</i>) coordinates shown under the colour bar;
+* Row #2 - linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams (GBs) for LFI and B-Spline beams (BSs) of Mars12 apodized for the HFI CMB channels and Mars12 for the sub-mm channels (see section "Inputs" below);
+* Row #3 - log scale of #2, PSF iso-contours being shown as solid lines, elliptical Gaussian fit iso-contours shown as broken lines.
+<center>
+<gallery widths=400px heights=400px perrow=2 caption="Comparison images of compact sources, effective beams, and PSFs.">
+File:30.png| '''30GHz'''
+File:44.png| '''44GHz'''
+File:70.png| '''70GHz'''
+File:100.png| '''100GHz'''
+File:143.png| '''143GHz'''
+File:217.png| '''217GHz'''
+File:353.png| '''353GHz'''
+File:545.png| '''545GHz'''
+File:857.png| '''857GHz'''
+</gallery>
+</center>
+===Histograms of the effective beam parameters===
+Here we present histograms of the three fit parameters, i.e., beam FWHM, ellipticity, and orientation with respect to the local meridian, and of the beam solid angle. The sky is uniformly sampled (although fairly sparsely) in 3072 directions that were chosen as HEALPix <i>N</i><sub>side</sub>=16 pixel centres for HFI and in 768 directions that were chosen as HEALPix <i>N</i><sub>side</sub>=8 pixel centres for LFI.
-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.
+Here the beam solid angle is estimated according to the definition:  "4&pi; &Sigma;(effbeam)/max(effbeam)"
-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.
+i.e., 4&pi; &Sigma;(<i>B<sub>ij</sub></i>) /  max(<i>B<sub>ij</sub></i>).
+[[File:ist_GB.png | 800px| thumb | center| '''Histograms for LFI effective beam parameters.''' ]]
+[[File:ist_BS_Mars12.png | 800px| thumb | center| '''Histograms for HFI effective beam parameters.''' ]]
+===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===
+* The discontinuities at the HEALPix domain edges in the maps are a visual artefact due to the interplay of the discretized effective beam and the HEALPix pixel grid.
+<gallery widths=300px heights=220px perrow=3 caption="Sky variation of ellipticity of the best-fit Gaussian for the effective beams.">
+File:e_030_GB.png| '''Ellipticity - 30GHz.'''
+File:e_044_GB.png| '''Ellipticity - 44GHz.'''
+File:e_070_GB.png| '''Ellipticity - 70GHz.'''
+File:e_100_BS_Mars12.png| '''Ellipticity - 100GHz.'''
+File:e_143_BS_Mars12.png| '''Ellipticity - 143GHz.'''
+File:e_217_BS_Mars12.png| '''Ellipticity - 217GHz.'''
+File:e_353_BS_Mars12.png| '''Ellipticity - 353GHz.'''
+File:e_545_BS_Mars12.png| '''Ellipticity - 545GHz.'''
+File:e_857_BS_Mars12.png| '''Ellipticity - 857GHz.'''
+</gallery>
+<gallery widths=300px heights=220px perrow=3 caption="Sky variation of the relative solid angle of the best-fit Gaussian for the effective beams.">
+File:solidarc_030_GB.png| '''Relative beam solid angle variations w.r.t. scanning beam - 30GHz.'''
+File:solidarc_044_GB.png| '''Relative beam solid angle variations w.r.t. scanning beam - 44GHz.'''
+File:solidarc_070_GB.png| '''Relative  beam solid angle variations w.r.t. scanning beam - 70GHz.'''
+File:solidarc_100_BS_Mars12.png| '''Relative beam solid angle variations w.r.t. scanning beam - 100GHz.'''
+File:solidarc_143_BS_Mars12.png| '''Relative beam solid angle variations w.r.t. scanning beam - 143GHz.'''
+File:solidarc_217_BS_Mars12.png| '''Relative beam solid angle variations w.r.t. scanning beam - 217GHz.'''
+File:solidarc_353_BS_Mars12.png| '''Relative beam solid angle variations w.r.t. scanning beam - 353GHz.'''
+File:solidarc_545_BS_Mars12.png| '''Relative beam solid angle variations w.r.t. scanning beam - 545GHz.'''
+File:solidarc_857_BS_Mars12.png| '''Relative beam solid angle variations w.r.t. scanning beam - 857GHz.'''
+</gallery>
+<gallery widths=300px heights=220px perrow=3 caption="Sky variation of the relative FWHM of the best-fit Gaussian for theeffective beams.">
+File:fwhm_030_GB.png| '''Relative FWHM variations w.r.t. scanning beam - 30GHz.'''
+File:fwhm_044_GB.png| '''Relative FWHM variations w.r.t. scanning beam - 44GHz.'''
+File:fwhm_070_GB.png| '''Relative FWHM variations w.r.t. scanning beam - 70GHz.'''
+File:fwhm_100_BS_Mars12.png| '''Relative FWHM variations w.r.t. scanning beam - 100GHz.'''
+File:fwhm_143_BS_Mars12.png| '''Relative FWHM variations w.r.t. scanning beam - 143GHz.'''
+File:fwhm_217_BS_Mars12.png| '''Relative FWHM variations w.r.t. scanning beam - 217GHz.'''
+File:fwhm_353_BS_Mars12.png| '''Relative FWHM variations w.r.t. scanning beam - 353GHz.'''
+File:fwhm_545_BS_Mars12.png| '''Relative FWHM variations w.r.t. scanning beam - 545GHz.'''
+File:fwhm_857_BS_Mars12.png| '''Relative FWHM variations w.r.t. scanning beam - 857GHz.'''
+</gallery>
+<gallery widths=300px heights=220px perrow=3 caption="Sky variation of effective beams <math>\psi</math> angle of the best-fit Gaussian">
+File:psi_030_GB.png| '''&psi; - 30GHz.'''
+File:psi_044_GB.png| '''&psi; - 44GHz.'''
+File:psi_070_GB.png| '''&psi; - 70GHz.'''
+File:psi_100_BS_Mars12.png| '''&psi; - 100GHz.'''
+File:psi_143_BS_Mars12.png| '''&psi; - 143GHz.'''
+File:psi_217_BS_Mars12.png| '''&psi; - 217GHz.'''
+File:psi_353_BS_Mars12.png| '''&psi; - 353GHz.'''
+File:psi_545_BS_Mars12.png| '''&psi; - 545GHz.'''
+File:psi_857_BS_Mars12.png| '''&psi; - 857GHz.'''
+</gallery>
+<!--
+<gallery widths=500px heights=500px perrow=2 caption="Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian">
+File:e_030_GB.png| '''ellipticity - 30GHz'''
+File:solidarc_030_GB.png| '''beam solid angle (relative variations wrt scanning beam - 30GHz'''
+File:e_100_BS_Mars12.png| '''ellipticity - 100GHz'''
+File:solidarc_100_BS_Mars12.png| '''beam solid angle (relative variations wrt scanning beam - 100GHz'''
+</gallery>
+-->
+===Statistics of the effective beams computed using FEBeCoP===
+Below we tabulate the simple statistics of FWHM, ellipticity (<i>e</i>), orientation (&psi;) and beam solid angle, (&Omega;), for a sample of 3072 and 768 directions on the sky for HFI and LFI data, respectively. Statistics shown in the Table are derived from the histograms shown above.
+* The derived beam parameters are representative of the DPC <i>N</i><sub>side</sub> 1024 and 2048 HEALPix maps (including the pixel window function).
+* The reported FWHM<sub>eff</sub> values are derived from the beam solid angles, under a Gaussian approximation. These are appropriate for flux determination, while the  the Gaussian fits to the effective beam maps are more suited for source identification.
+{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
+|+ '''Statistics of the FEBeCoP effective beams computed with the BS Mars12 data apodized for the CMB channels and oversampled.'''
+|-
+!  '''Frequency''' [GHz] ||  '''Mean(FWHM)''' [arcmin]  ||  '''&sigma;(FWHM)''' [arcmin]  || '''Mean(<i>e</i>)'''  ||  '''&sigma;(<i>e</i>)'''  ||  '''Mean(&psi;)''' [deg]  ||  '''&sigma;(&psi;)''' [deg]  || '''Mean(&Omega;)''' [arcmin<sup>2</sup>]  ||  '''&sigma;(&Omega;)''' [arcmin<sup>2</sup>] || '''FWHM<sub>eff</sub>''' [arcmin]
+|-
+| 30 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34
+|-
+| 44 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774  || 27.12
+|-
+| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027  || 13.31
+|-
+| 100 || 9.651 || 0.014 || 1.186 || 0.023 || -0.024 || 55.400 || 105.778 || 0.311 || 9.66
+|-
+| 143 || 7.248 || 0.015 || 1.036 || 0.009 || 0.383   || 54.130 || 59.954 || 0.246  || 7.27
+|-
+| 217 || 4.990 || 0.025 || 1.177 || 0.030 || 0.836   || 54.999 || 28.447 || 0.271  || 5.01
+|-
+| 353 || 4.818 || 0.024 || 1.147 || 0.028 || 0.655   || 54.745 || 26.714 || 0.250  || 4.86
+|-
+| 545 || 4.682 || 0.044 || 1.161 || 0.036 || 0.544   || 54.876 || 26.535 || 0.339  || 4.84
+|-
+| 857 || 4.325 || 0.055 || 1.393 || 0.076 || 0.876   || 54.779 || 24.244 || 0.193  || 4.63
+|}
+====Beam solid angles for the PCCS====
+For the Planck Catalogue of Compact Sources we also need the following definitions.
+*  &Omega;<sub>eff</sub> - is the mean beam solid angle of the effective beam, where the solid angle is estimated according to the definition "4 &pi; &Sigma;(effbeam)/max(effbeam)", i.e., as an integral over the full extent of the effective beam, 4&pi; &Sigma; (<i>B<sub>ij</sub></i>)/max(<i>B<sub>ij</sub></i>).
+* From &Omega;<sub>eff</sub> we estimate FWHM<sub>eff</sub>, under a Gaussian approximation - these are tabulated above.
+*&Omega;<sup>(1)</sup><sub>eff</sub> is the beam solid angle estimated up to a radius equal to 1 FWHM<sub>eff</sub> and &Omega;<sup>(2)</sup><sub>eff</sub> up to a radius equal to twice the FWHM<sub>eff</sub>.
+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).
+{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
+|+'''Band averaged beam solid angles'''
+|  '''Band [GHz]'''  ||  '''&Omega;<sub>eff</sub>'''[arcmin<sup>2</sup>] || '''Spatial variation''' [arcmin<sup>2</sup>] ||  '''&Omega;<sup>(1)</sup><sub>eff</sub>''' [arcmin<sup>2</sup>]|| '''Spatial variation-1''' [arcmin<sup>2</sup>] ||  '''&Omega;<sup>(2)</sup><sub>eff</sub>''' [arcmin<sup>2</sup>] || '''Spatial variation-2''' [arcmin<sup>2</sup>]
+|-
+| 30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847
+|-
+| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811
+|-
+| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027
+|-
+| 100 || 105.778 || 0.311 || 100.830 || 0.410 || 105.777 || 0.311
+|-
+| 143 || 59.954 || 0.246 || 56.811 || 0.419 || 59.952 || 0.246
+|-
+| 217 || 28.447 || 0.271 || 26.442 || 0.537 || 28.426 || 0.271
+|-
+| 353 || 26.714 || 0.250 || 24.827 || 0.435 || 26.653 || 0.250
+|-
+| 545 || 26.535 || 0.339 || 24.287 || 0.455 || 26.302 || 0.337
+|-
+| 857 || 24.244 || 0.193 || 22.646 || 0.263 || 23.985 || 0.191
+|}
+==Production process==
+FEBeCoP, the Fast Effective Beam Convolution in Pixel space{{BibCite|mitra2010}}, is an approach for representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:
+* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from the observations of planets, but in general, an arbitrary mathematical representation of the beam can be used on input);
+* exactly follow the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position;
+* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM in diameter);
+* add up all beams that cross the same pixel and its vicinity over the observing period of interest,
+*create a data object of all beams pointed at all <i>N</i><sub>pix</sub> directions of pixels in the map at a resolution at which this precomputation was executed (dimension <i>N</i><sub>pix</sub> times a few hundred);
+*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission.
+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, so that several estimations of the effective beams were possible for all Planck detectors for different scan and beam models, as well as different lengths of data set.
+===Pixel-ordered detector angles (PODA)===
+The main challenge in computing the effective beams is to go through the roughly trillion samples, which becomes severely limited by I/O. In the first stage, for a given data set, ordered lists of pointing angles for each pixel (the pixel-ordered detector angles, PODA) are made. This is an one-time process for each data set. We used computers with large memory and tedious memory management and bookkeeping to make this step efficient.
+===Effective beam===
+The "effBeam" part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we have tried to made sure that no repetition occurs in evaluating any trigonometric quantity.
+One important reason for separating the two steps is that they use different parallel computing schemes. The PODA part requires parallelization over time-ordered data samples, while the  effBeam part requires distribution of pixels among different computers.
+===Computational Cost===
+The computation of the effective beams has been performed at the NERSC Supercomputing Center. The table below shows the computational cost for FEBeCoP processing of the nominal mission.
+{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
+|+ '''Computational cost for PODA, effective beam and  single map convolution. Wall-clock time is given as a guide, as found on the NERSC supercomputers.'''
+|-
+|Channel [GHz] ||030 || 044 || 070 || 100 || 143 || 217 || 353 || 545 || 857
+|-
+|PODA/detector computation time [CPU hrs] || 85 || 100  || 250 || 500 || 500 || 500 || 500 || 500 || 500
+|-
+|PODA/detector computation time [wall clock hrs] || 7 || 10  || 20 || 20 || 20 || 20 || 20 || 20 || 20
+|-
+|Beam/channel computation time [CPU hrs] || 900 || 2000 || 2300 || 2800 ||  3800 || 3200 || 3000 || 900 || 1100
+|-
+|Beam/channel computation time [wall clock hrs] || 0.5 || 0.8  || 1 || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5
+|-
+|convolution computation time [CPU hrs] || 1 || 1.2 || 1.3 || 3.6 || 4.8 || 4.0 || 4.1 || 4.1 || 3.7
+|-
+|convolution computation time [wall clock sec] || 1 || 1 || 1 || 4 || 4 || 4 || 4 || 4 || 4
+|-
+|Effective beam size [GB] || 173 || 123 || 28 || 187 || 182 || 146 || 132 || 139 || 124
+|}
+The computation cost, especially for PODA and convolution, is heavily limited by the I/O capacity of the disk and so it depends on the overall usage of the cluster by other users.
+==Inputs==
+In order to describe the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by incoming CMB photons. The scan direction is marked, and the centre of the focal plane is at an angle of 85&deg; with respect to the spin axis, pointing upward in the picture.
+[[File:PlanckFocalPlane.png | 500px| thumb | center|'''Planck focal plane.''']]
+===The focal plane database (FPDB)===
+The FPDB contains information on each detector, e.g., the orientation of the polarization axis, and different weight factors. Details are provided in the instrument [[The RIMO|RIMOs]]):
+*for HFI - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=the HFI RIMO}};
+*for LFI - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=the LFI RIMO}}.
+<!--
+*HFI - LFI_RIMO_DX9_PTCOR6 -  {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}
+*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}
+{{PLADoc|fileType=rimo|link=The Plank RIMOS}}
+-->
+===The scanning strategy===
+The scanning strategy description consists of the three pointing angles for each detector for each sample.  Detector pointings for the nominal mission cover about 15 months of observation from Operational Day (OD) 91 to OD 563, covering three and a half surveys.
+===The scanning beam===
+The "scanbeam" is modelled for each detector through the observation of planets. This was assumed to be constant over the whole mission, although FEBeCoP could be used for a few sets of scanbeams too.
+* For LFI the [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beams]] are used; these are based on the radio-frequency tuned model (RFTM), smeared to simulate the in-flight optical response.
+* For HFI the [[Beams#Scanning beams|B-Spline, BS]] beams are used based on two observations of Mars.
+<!--
+*HFI - LFI_RIMO_DX9_PTCOR6 -  {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}
+*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}
+[[Beams LFI#Effective beams|LFI effective beams]]
+-->
+===Beam cutoff radii===
+The cutoffs are set to <i>N</i> times the geometric mean of the FWHM of all detectors in a channel, where <i>N</i> is given below.
+{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
+|+'''Beam cut off radius.'''
+| '''Channel [GHz]''' || '''Cutoff radii in units of FWHM''' || '''FWHM of full beam extent [arcmin]'''
+|-
+|30, 44, 70 || 2.5 ||
+|-
+|100 || 2.25 || 23.703699
+|-
+|143 || 3 || 21.057402
+|-
+|217, 353 || 4 || 18.782754
+|-
+|545 || 4 || 18.327635
+|-
+|857 || 4 || 17.093706
+|}
+===Map resolution for the derived beam data object===
+The resolution is
+* <i>N</i><sub>side</sub> = 1024 for LFI frequency channels, and
+* <i>N</i><sub>side</sub> = 2048 for HFI frequency channels.
+==Related products==
+===Monte Carlo simulations===
+FEBeCoP software enables fast, full-sky convolutions of the sky signals with the effective beams in the pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps can be easily generated, which are convolved with realistically rendered, spatially varying, asymmetric Planck beams. We performed the following steps:
+* generate the effective beams with FEBeCoP for all frequencies for "dDX9" data and the Nominal Mission;
+* generate 100 realizations of maps from a fiducial CMB power spectrum;
+* convolve each one of these maps with the effective beams using FEBeCoP;
+* estimate the average of the power spectrum of each convolved realization, and 1 &sigma; errors.
+Since FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations can be generated. These Monte Carlo simulations of the sky signal (which might be CMB or a foreground, e.g., dust) along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission and tagged as full focal-plane simulations, FFP#
+(for example the [[HL-sims#FFP6 data set|FFP6]]).
+===Beam Window Functions===
+The "Transfer Function" or the "Beam Window Function", <i>W</i><sub>&#8467;</sub> relates the true angular power spectra <i>C</i><sub>&#8467;</sub> with the observed angular power spectra &#264;<sub>&#8467;</sub>:
-The full algebra involving the effective beams for temperature and polarisation was presented in~\citet{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>,
-%
 <math>
-\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},
+W_l= \tilde{C}_\ell / C_\ell .
-</math>
+\label{eqn:wl1}</math>
-%
-where
+Note that, the window function can contain a pixel window function (depending on the definition) and it is <i>not the angular power spectra of the scanbeams</i>, although in principle one may be able to connect them via some complicated algebra.
-%
+The window functions are estimated by performing Monte Carlo simulations. We generate several random realizations of the CMB sky, starting from a given fiducial <i>C</i><sub>&#8467;</sub>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <i>C</i><sub>&#8467;</sub><sup>conv</sup>, divide by the power spectra of the unconvolved map <i>C</i><sub>&#8467;</sub><sup>in</sup> and average over their ratio. Thus, the estimated window function is
 <math>
-B_{ij} \ = \ \frac{\sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)}{\sum_t A_{ti}} \, ,
+W^{\rm est}_\ell  = < C^{\rm conv}_\ell /  C^{\rm in}_\ell >.
-\label{eq:EBT}</math>
+\label{eqn:wl2}</math>
-%
-<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.
+For subtle reasons, we perform a more rigorous estimation of the window function by comparing <i>C</i><sub>&#8467;</sub><sup>conv</sup> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but not necessarily exactly the same) size and then scaling the estimated window function accordingly.
+Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]].
+====Beam window functions, <i>W</i><sub>&#8467;</sub>, for the Planck mission.====
+[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam window functions, <i>W</i><sub>&#8467;</sub>, for LFI channels.''']]
+[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |'''Beam window functions, <i>W</i><sub>&#8467;</sub>, for HFI channels.''']]
-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:
+==File Names==
-%
+The effective beams are provided in the PLA as FITS files containg HEALPix maps of the beams. For the file names the following convention is used.
-<math>
+*Single beam query, "beams_FFF_''PixelNumber''.fits" :
-( \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,
+** FFF is the channel frequency (one of "30", "44", "70", "100", "143", "217", "353", "545", "857");
-</math>
+** "PixelNumber" is the number of the pixel to which the beam corresponds, 0&ndash;12 &times; <i>N</i><sub>side</sub><sup>2</sup> - 1, with <i>N</i><sub>side</sub>=1024 for the LFI and <i>N</i><sub>side</sub>=2048 for the HFI.
-%
+*Multiple beam query, "beams_FFF_''FirstPixelNumber''-''LastPixelNumber''.zip" .
-where the polarised effective beam matrix
+** The compressed files contains a set of files with the beams for the pixels covereing the selected region, with "FFF" as for sinle beam query and the naming convention for the beam files contained in the ''.zip'' file being the same as for single beam queries;
-%
+** ''FirstPixelNumber'' is the lowest pixel number for the area covered by the request;
-<math>
+** ''LastPixelNumber'' is the highest pixel number for the area covered by the request.
-\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) \, ,
-</math>
-%
-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}.
-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.
+==File format==
+The FITS files provided in the PLA contain HEALPix maps of the beams.
+== Previous Releases: (2015) and (2013) effective beams ==
+<div class="toccolours mw-collapsible mw-collapsed" style="background-color: #FFDAB9;width:80%">
+'''2015 Release of effective beams'''
+<div class="mw-collapsible-content">
-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.
+'''Product description'''
+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.
+It captures the complete information about the difference between the true and observed image of the sky. The effective beams are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map.
-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.
+Details of the beam processing are given in the respective pages for [[Beams|HFI]] and [[Beams_LFI|LFI]].
-----
+The full algebra involving the effective beams for temperature and polarisation was presented in  {{BibCite|mitra2010}}, and a discussion of its application to Planck data is given in the appropriate LFI {{PlanckPapers|planck2013-p02d}}, {{PlanckPapers|planck2014-a05||Planck-2015-A05}} and HFI {{PlanckPapers|planck2013-p03c}} papers. Relevant details of the processing steps are given in the [[Beams|Effective Beams]] section of this document.
-===Comparison of the images of compact sources observed by Planck with FEBeCoP products===
+<!-- Everything from here down to the "Production Process" section should eventually be moved to a new section the Joint Processing pages -->
-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 and 100GHz frequency channels. We show below a few panels of source images organized as follows:
+'''Comparison of the images of compact sources observed by Planck with FEBeCoP products'''
+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 all LFI and HFI  frequency channels, as an example.  We show below a few panels of source images organized as follows:
 * Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar
 * 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).
 * Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line
+<center>
+<gallery widths=400px heights=400px perrow=2 caption="Comparison  images of compact sources and effective beams, PSFs">
+File:30.png| '''30GHz'''
+File:44.png| '''44GHz'''
+File:70.png| '''70GHz'''
+File:100.png| '''100GHz'''
+File:143.png| '''143GHz'''
+File:217.png| '''217GHz'''
+File:353.png| '''353GHz'''
+File:545.png| '''545GHz'''
+File:857.png| '''857GHz'''
+</gallery>
+</center>
-====30 GHz (left) and 100GHz (right) ====
+'''Histograms of the effective beam parameters'''
+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 sky is sampled (pretty sparsely) at 3072 directions which were chosen as HEALpix nside=16 pixel centers for HFI and at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI.  These uniformly sample the sky.
-[[File:30.png | 500px| thumb | left | 30GHz ]] [[File:100.png | 500px| thumb | right | 100GHz ]]
+Where beam solid angle is estimated according to the definition:  '''<math> 4 \pi \sum</math>(effbeam)/max(effbeam)'''
+i.e., <math> 4 \pi \sum(B_{ij}) /  max(B_{ij}) </math>
+[[File:ist_GB.png | 800px| thumb | center| '''Histograms for LFI effective beam parameters''' ]]
+[[File:ist_BS_Mars12.png | 800px| thumb | center| '''Histograms for HFI effective beam parameters''' ]]
-----
+'''Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian'''
+* 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.
-===Histograms of the effective beam parameters===
-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 3072 directions which were chosen as HEALpix nside=16 pixel centers for HFI and at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.
+<gallery widths=300px heights=220px perrow=3 caption="Sky variation of effective beams ellipticity of the best-fit Gaussian">
+File:e_030_GB.png| '''ellipticity - 30GHz'''
+File:e_044_GB.png| '''ellipticity - 44GHz'''
+File:e_070_GB.png| '''ellipticity - 70GHz'''
+File:e_100_BS_Mars12.png| '''ellipticity - 100GHz'''
+File:e_143_BS_Mars12.png| '''ellipticity - 143GHz'''
+File:e_217_BS_Mars12.png| '''ellipticity - 217GHz'''
+File:e_353_BS_Mars12.png| '''ellipticity - 353GHz'''
+File:e_545_BS_Mars12.png| '''ellipticity - 545GHz'''
+File:e_857_BS_Mars12.png| '''ellipticity - 857GHz'''
+</gallery>
-Where beam solid angle is estimated according to the definition:  '''4pi* sum(effbeam)/max(effbeam)'''
-====LFI====
+<gallery widths=300px heights=220px perrow=3 caption="Sky (relative) variation of effective beams solid angle of the best-fit Gaussian">
+File:solidarc_030_GB.png| '''beam solid angle (relative) variations wrt scanning beam - 30GHz'''
+File:solidarc_044_GB.png| '''beam solid angle (relative) variations wrt scanning beam - 44GHz'''
+File:solidarc_070_GB.png| '''beam solid angle (relative) variations wrt scanning beam - 70GHz'''
+File:solidarc_100_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 100GHz'''
+File:solidarc_143_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 143GHz'''
+File:solidarc_217_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 217GHz'''
+File:solidarc_353_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 353GHz'''
+File:solidarc_545_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 545GHz'''
+File:solidarc_857_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 857GHz'''
+</gallery>
-[[File:ist_GB.png | 500px| thumb | center| Histograms for LFI effective beam parameters ]]
-====HFI====
+<gallery widths=300px heights=220px perrow=3 caption="Sky (relative) variation of effective beams fwhm of the best-fit Gaussian">
+File:fwhm_030_GB.png| '''fwhm (relative) variations wrt scanning beam - 30GHz'''
+File:fwhm_044_GB.png| '''fwhm (relative) variations wrt scanning beam - 44GHz'''
+File:fwhm_070_GB.png| '''fwhm (relative) variations wrt scanning beam - 70GHz'''
+File:fwhm_100_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 100GHz'''
+File:fwhm_143_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 143GHz'''
+File:fwhm_217_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 217GHz'''
+File:fwhm_353_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 353GHz'''
+File:fwhm_545_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 545GHz'''
+File:fwhm_857_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 857GHz'''
+</gallery>
-[[File:ist_BS_Mars12.png | 500px| thumb | center| Histograms for HFI effective beam parameters ]]
+<gallery widths=300px heights=220px perrow=3 caption="Sky variation of effective beams <math>\psi</math> angle of the best-fit Gaussian">
+File:psi_030_GB.png| '''<math>\psi</math> - 30GHz'''
+File:psi_044_GB.png| '''<math>\psi</math> - 44GHz'''
+File:psi_070_GB.png| '''<math>\psi</math> - 70GHz'''
+File:psi_100_BS_Mars12.png| '''<math>\psi</math> - 100GHz'''
+File:psi_143_BS_Mars12.png| '''<math>\psi</math> - 143GHz'''
+File:psi_217_BS_Mars12.png| '''<math>\psi</math> - 217GHz'''
+File:psi_353_BS_Mars12.png| '''<math>\psi</math> - 353GHz'''
+File:psi_545_BS_Mars12.png| '''<math>\psi</math> - 545GHz'''
+File:psi_857_BS_Mars12.png| '''<math>\psi</math> - 857GHz'''
+</gallery>
+<!--
+<gallery widths=500px heights=500px perrow=2 caption="Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian">
+File:e_030_GB.png| '''ellipticity - 30GHz'''
+File:solidarc_030_GB.png| '''beam solid angle (relative variations wrt scanning beam - 30GHz'''
+File:e_100_BS_Mars12.png| '''ellipticity - 100GHz'''
+File:solidarc_100_BS_Mars12.png| '''beam solid angle (relative variations wrt scanning beam - 100GHz'''
+</gallery>
+-->
-----
+'''Statistics of the effective beams computed using FEBeCoP'''
-===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian - 30GHz (top) and 100GHz (bottom)===
+We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 3072 and 768 directions on the sky for HFI and LFI data respectively. Statistics shown in the Table are derived from the histograms shown above.
-[[File:e_030_GB.png | 520px| thumb | left | Sky variation of  effective beams ellipticity of best-fit Gaussian for 30GHz]]
+* The derived beam parameters are representative of the DPC NSIDE 1024 and 2048 healpix maps (they include the pixel window function).
-[[File:solidarc_030_GB.png | 520px| thumb | right | Sky variation of  effective beams solid angle of best-fit Gaussian for 30GHz relative to scanning beam solid angle]]
+* 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.
-[[File:e_100_BS_Mars12.png | 520px| thumb | left | Sky variation of  effective beams ellipticity of best-fit Gaussian for 100GHz]]
-[[File:solidarc_100_BS_Mars12.png | 520px| thumb | right | Sky variation of  effective beams solid angle of best-fit Gaussian for 100GHz relative to scanning beam solid angle]]
+{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
+|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''
+|-
+!  '''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]
+|-
+| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34
+|-
+| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774  || 27.12
+|-
+| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027  || 13.31
+|-
+| 100 || 9.651 || 0.014 || 1.186 || 0.023 || -0.024 || 55.400 || 105.778 || 0.311 || 9.66
+|-
+| 143 || 7.248 || 0.015 || 1.036 || 0.009 || 0.383   || 54.130 || 59.954 || 0.246  || 7.27
+|-
+| 217 || 4.990 || 0.025 || 1.177 || 0.030 || 0.836   || 54.999 || 28.447 || 0.271  || 5.01
+|-
+| 353 || 4.818 || 0.024 || 1.147 || 0.028 || 0.655   || 54.745 || 26.714 || 0.250  || 4.86
+|-
+| 545 || 4.682 || 0.044 || 1.161 || 0.036 || 0.544   || 54.876 || 26.535 || 0.339  || 4.84
+|-
+| 857 || 4.325 || 0.055 || 1.393 || 0.076 || 0.876   || 54.779 || 24.244 || 0.193  || 4.63
+|}
+'''Beam solid angles for the PCCS'''
+*  <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 </math>(effbeam)/max(effbeam)''', 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>.
-===Statistics of the effective beams computed using FEBeCoP===
+* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above
+** <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>.
+*** 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).
-We tabulate the simple statistics of FWHM, ellipticity, orientation and beam solid angle for a sample of 3072 and 768 directions on the sky for HFI and LFI data respectively. Statistics shown in the Table are derived from the histograms shown above.
+{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
-{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
+|+'''Band averaged beam solid angles'''
-|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''
+|  '''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>]
+|-
+|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847
+|-
+| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811
+|-
+| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027
+|-
+| 100 || 105.778 || 0.311 || 100.830 || 0.410 || 105.777 || 0.311
 |-
-!  '''frequency'''  ||  '''mean(fwhm)''' [arcmin]  ||  '''sd(fwhm)''' [arcmin]  || '''mean(e)'''  ||  '''sd(e)'''  ||  '''mean(psi)''' [degree]  ||  '''sd(psi)''' [degree]  || '''mean(Solid Angle)''' [arcmin<math>^{2}</math>]  ||  '''sd(Solid Angle)''' [arcmin<math>^{2}</math>]
+| 143 || 59.954 || 0.246 || 56.811 || 0.419 || 59.952 || 0.246
 |-
-| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842
+| 217 || 28.447 || 0.271 || 26.442 || 0.537 || 28.426 || 0.271
 |-
-| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774
+| 353 || 26.714 || 0.250 || 24.827 || 0.435 || 26.653 || 0.250
 |-
-| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027
+| 545 || 26.535 || 0.339 || 24.287 || 0.455 || 26.302 || 0.337
 |-
-| 100 || 9.651 || 0.014 || 1.186 || 0.023 || -0.024 || 55.400 || 105.778 || 0.311
+| 857 || 24.244 || 0.193 || 22.646 || 0.263 || 23.985 || 0.191
+|}
+'''Production process'''
+FEBeCoP, or Fast Effective Beam Convolution in Pixel space{{BibCite|mitra2010}}, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:
+* 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)
+* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position
+* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)
+* add up all beams that cross the same pixel and its vicinity over the observing period of interest
+*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)
+*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission
+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 scan and beam models, as well as different lengths of datasets.
+'''Pixel Ordered Detector Angles (PODA)'''
+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 pixel - 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.
+'''effBeam'''
+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.
+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.
+'''Computational Cost'''
+The computation of the effective beams has been performed at the NERSC Supercomputing Center. The table below shows the computation cost for FEBeCoP processing of the nominal mission.
+{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
+|+ '''Computational cost for PODA, Effective Beam and  single map convolution. Wall-clock time is given as a guide, as found on the NERSC supercomputers.'''
 |-
-| 143 || 7.248 || 0.015 || 1.036 || 0.009 || 0.383   || 54.130 || 59.954 || 0.246
+|Channel ||030 || 044 || 070 || 100 || 143 || 217 || 353 || 545 || 857
 |-
-| 217 || 4.990 || 0.025 || 1.177 || 0.030 || 0.836   || 54.999 || 28.447 || 0.271
+|PODA/Detector Computation time (CPU hrs) || 85 || 100  || 250 || 500 || 500 || 500 || 500 || 500 || 500
 |-
-| 353 || 4.818 || 0.024 || 1.147 || 0.028 || 0.655   || 54.745 || 26.714 || 0.250
+|PODA/Detector Computation time (wall clock hrs) || 7 || 10  || 20 || 20 || 20 || 20 || 20 || 20 || 20
 |-
-| 545 || 4.682 || 0.044 || 1.161 || 0.036 || 0.544   || 54.876 || 26.535 || 0.339
+|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 || 2800 ||  3800 || 3200 || 3000 || 900 || 1100
+|-
+|Beam/Channel Computation time (wall clock hrs) || 0.5 || 0.8  || 1 || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5
+|-
+|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 || 3.6 || 4.8 || 4.0 || 4.1 || 4.1 || 3.7
 |-
-| 857 || 4.325 || 0.055 || 1.393 || 0.076 || 0.876   || 54.779 || 24.244 || 0.193
+|Convolution Computation time (wall clock sec) || 1 || 1 || 1 || 4 || 4 || 4 || 4 || 4 || 4
+|-
+|Effective Beam Size (GB) || 173 || 123 || 28 || 187 || 182 || 146 || 132 || 139 || 124
 |}
-----
-===FEBeCoP deliverables===
+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.
+'''Inputs'''
+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.
+[[File:PlanckFocalPlane.png | 500px| thumb | center|'''Planck Focal Plane''']]
+'''The Focal Plane DataBase (FPDB)'''
+ The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):
+*HFI - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}
+*LFI - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}
+<!--
+*HFI - LFI_RIMO_DX9_PTCOR6 -  {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}
+*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}
+{{PLADoc|fileType=rimo|link=The Plank RIMOS}}
+-->
+'''The scanning strategy'''
+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.
+'''The scanbeam'''
+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.
+* 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.
+* HFI: [[Beams#Scanning beams|B-Spline, BS]] based on 2 observations of Mars.
+(see the instrument [[The RIMO|RIMOs]]).
+<!--
+*HFI - LFI_RIMO_DX9_PTCOR6 -  {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}
+*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}
+[[Beams LFI#Effective beams|LFI effective beams]]
+-->
-* The derived beam parameters are representative of the DPC NSIDE 2048 healpix maps (they include the pixel window function).
+'''Beam cutoff radii'''
-* The reported FHWM are derived from the solid angles, under a Gaussian approximation.  The value in the values in parenthesis are the Gaussian fits to the effective beam maps tabulated above.  The former is best used for flux determination, the latter for source identification.
+N times geometric mean of FWHM of all detectors in a channel, where N
 {|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
-|+ '''Band averaged beam solid angles'''
+|+'''Beam cut off radius'''
+| '''channel''' || '''Cutoff Radii in units of fwhm''' || '''fwhm of full beam extent'''
+|-
+|30 - 44 - 70 || 2.5 ||
+|-
+|100 || 2.25 || 23.703699
+|-
+|143 || 3 || 21.057402
+|-
+|217-353 || 4 || 18.782754
 |-
-| '''Band'''  ||  '''Omega_beam''' [arcmin<math>^{2}</math>]  ||  '''Monte Carlo error'''[arcmin<math>^{2}</math>^]  || '''spatial variation''' [arcmin<math>^{2}</math>]  ||  '''bias''' [arcmin<math>^{2}</math>]  || '''FWHM_eff''' (Gaussian fit) [arcmin]
+|sub-mm || 4 || 18.327635(545GHz) ;  17.093706(857GHz)
+|}
+'''Map resolution for the derived beam data object'''
+* <math>N_{side} = 1024 </math> for LFI frequency channels
+* <math>N_{side} = 2048 </math> for HFI frequency channels
+'''Related products'''
+'''Monte Carlo simulations'''
+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:
+* generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission
+* generate 100 realizations of maps from a fiducial CMB power spectrum
+* convolve each one of these maps with the effective beams using FEBeCoP
+* estimate the average of the Power Spectrum of each convolved realization, and 1 <math>\sigma</math> errors
+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, FFP#.
+For example [[HL-sims#FFP6 data set|FFP6]]
+'''Beam Window Functions'''
+The '''Transfer Function''' or the '''Beam Window Function'''  <math> W_l </math> relates the true angular power spectra <math>C_l </math> with the observed angular power spectra <math>\widetilde{C}_l </math>:
+<math>
+W_l= \widetilde{C}_l / C_l
+\label{eqn:wl1}</math>
+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.
+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_l </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_l </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_l </math> and average over their ratio. Thus, the estimated window function
+<math>
+W^{est}_l  = < C^{conv}_l /  C^{in}_l >
+\label{eqn:wl2}</math>
+For subtle reasons, we perform a more rigorous estimation of the window function by comparing C^{conv}_l 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.
+Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]].
+'''Beam Window functions, Wl, for Planck mission'''
+[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, Wl, for LFI channels''']]
+[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |'''Beam Window functions, Wl, for HFI channels''']]
+'''File Names'''
+The effective beams are provided by the PLA as FITS files containg HEALPix maps of the beams. For the file names the following convention is used:
+*Single beam query: '''beams_FFF_''PixelNumber''.fits'''
+**FFF is the channel frequency (one of '''30''', '''44''', '''70''', '''100''', '''143''', '''217''', '''353''', '''545''', '''857''');
+** ''PixelNumber'' is the number of the pixel to which the beam corresponds (<math>0</math> - <math>12 \times N_{\rm side}^2 - 1</math>). For the LFI <math>N_{\rm side}=1024</math>, for the HFI <math>N_{\rm side}=2048</math>;
+*Multiple beam query: '''beams_FFF_''FirstPixelNumber''-''LastPixelNumber''.zip''' <br /> The compressed files contains a set of files with the beams for the pixels covereing the selected region. FFF as for sinle beam query. The naming convention for the beam files contained in the ''.zip'' file is the same as for single beam queries.
+** ''FirstPixelNumber'' is the lowest pixel number for the area covered by the request;
+** ''LastPixelNumber'' is the highest pixel number for the area covered by the request;
+'''File format'''
+The FITS files provided by the PLA contain HEALPix maps of the beams.
+</div>
+</div>
+<div class="toccolours mw-collapsible mw-collapsed" style="background-color: #EEE8AA;width:80%">
+'''2013 Release of effective beams'''
+<div class="mw-collapsible-content">
+'''Product description'''
+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.
+It captures the complete information about the difference between the true and observed image of the sky. The effective beams are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map.
+Details of the beam processing are given in the respective pages for [[Beams|HFI]] and [[Beams_LFI|LFI]].
+The full algebra involving the effective beams for temperature and polarisation was presented in  {{BibCite|mitra2010}}, and a discussion of its application to Planck data is given in the appropriate LFI {{PlanckPapers|planck2013-p02d}}, {{PlanckPapers|planck2014-a05||Planck-2015-A05}} and HFI {{PlanckPapers|planck2013-p03c}} papers. Relevant details of the processing steps are given in the [[Beams|Effective Beams]] section of this document.
+<!-- Everything from here down to the "Production Process" section should eventually be moved to a new section the Joint Processing pages -->
+'''Comparison of the images of compact sources observed by Planck with FEBeCoP products'''
+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 all LFI and HFI  frequency channels, as an example.  We show below a few panels of source images organized as follows:
+* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar
+* 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).
+* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line
+<center>
+<gallery widths=400px heights=400px perrow=2 caption="Comparison  images of compact sources and effective beams, PSFs">
+File:30.png| '''30GHz'''
+File:44.png| '''44GHz'''
+File:70.png| '''70GHz'''
+File:100.png| '''100GHz'''
+File:143.png| '''143GHz'''
+File:217.png| '''217GHz'''
+File:353.png| '''353GHz'''
+File:545.png| '''545GHz'''
+File:857.png| '''857GHz'''
+</gallery>
+</center>
+'''Histograms of the effective beam parameters'''
+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 sky is sampled (pretty sparsely) at 3072 directions which were chosen as HEALpix nside=16 pixel centers for HFI and at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI.  These uniformly sample the sky.
+Where beam solid angle is estimated according to the definition:  '''<math> 4 \pi \sum</math>(effbeam)/max(effbeam)'''
+i.e., <math> 4 \pi \sum(B_{ij}) /  max(B_{ij}) </math>
+[[File:ist_GB.png | 800px| thumb | center| '''Histograms for LFI effective beam parameters''' ]]
+[[File:ist_BS_Mars12.png | 800px| thumb | center| '''Histograms for HFI effective beam parameters''' ]]
+'''Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian'''
+* 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.
+<gallery widths=300px heights=220px perrow=3 caption="Sky variation of effective beams ellipticity of the best-fit Gaussian">
+File:e_030_GB.png| '''ellipticity - 30GHz'''
+File:e_044_GB.png| '''ellipticity - 44GHz'''
+File:e_070_GB.png| '''ellipticity - 70GHz'''
+File:e_100_BS_Mars12.png| '''ellipticity - 100GHz'''
+File:e_143_BS_Mars12.png| '''ellipticity - 143GHz'''
+File:e_217_BS_Mars12.png| '''ellipticity - 217GHz'''
+File:e_353_BS_Mars12.png| '''ellipticity - 353GHz'''
+File:e_545_BS_Mars12.png| '''ellipticity - 545GHz'''
+File:e_857_BS_Mars12.png| '''ellipticity - 857GHz'''
+</gallery>
+<gallery widths=300px heights=220px perrow=3 caption="Sky (relative) variation of effective beams solid angle of the best-fit Gaussian">
+File:solidarc_030_GB.png| '''beam solid angle (relative) variations wrt scanning beam - 30GHz'''
+File:solidarc_044_GB.png| '''beam solid angle (relative) variations wrt scanning beam - 44GHz'''
+File:solidarc_070_GB.png| '''beam solid angle (relative) variations wrt scanning beam - 70GHz'''
+File:solidarc_100_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 100GHz'''
+File:solidarc_143_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 143GHz'''
+File:solidarc_217_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 217GHz'''
+File:solidarc_353_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 353GHz'''
+File:solidarc_545_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 545GHz'''
+File:solidarc_857_BS_Mars12.png| '''beam solid angle (relative) variations wrt scanning beam - 857GHz'''
+</gallery>
+<gallery widths=300px heights=220px perrow=3 caption="Sky (relative) variation of effective beams fwhm of the best-fit Gaussian">
+File:fwhm_030_GB.png| '''fwhm (relative) variations wrt scanning beam - 30GHz'''
+File:fwhm_044_GB.png| '''fwhm (relative) variations wrt scanning beam - 44GHz'''
+File:fwhm_070_GB.png| '''fwhm (relative) variations wrt scanning beam - 70GHz'''
+File:fwhm_100_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 100GHz'''
+File:fwhm_143_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 143GHz'''
+File:fwhm_217_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 217GHz'''
+File:fwhm_353_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 353GHz'''
+File:fwhm_545_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 545GHz'''
+File:fwhm_857_BS_Mars12.png| '''fwhm (relative) variations wrt scanning beam - 857GHz'''
+</gallery>
+<gallery widths=300px heights=220px perrow=3 caption="Sky variation of effective beams <math>\psi</math> angle of the best-fit Gaussian">
+File:psi_030_GB.png| '''<math>\psi</math> - 30GHz'''
+File:psi_044_GB.png| '''<math>\psi</math> - 44GHz'''
+File:psi_070_GB.png| '''<math>\psi</math> - 70GHz'''
+File:psi_100_BS_Mars12.png| '''<math>\psi</math> - 100GHz'''
+File:psi_143_BS_Mars12.png| '''<math>\psi</math> - 143GHz'''
+File:psi_217_BS_Mars12.png| '''<math>\psi</math> - 217GHz'''
+File:psi_353_BS_Mars12.png| '''<math>\psi</math> - 353GHz'''
+File:psi_545_BS_Mars12.png| '''<math>\psi</math> - 545GHz'''
+File:psi_857_BS_Mars12.png| '''<math>\psi</math> - 857GHz'''
+</gallery>
+<!--
+<gallery widths=500px heights=500px perrow=2 caption="Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian">
+File:e_030_GB.png| '''ellipticity - 30GHz'''
+File:solidarc_030_GB.png| '''beam solid angle (relative variations wrt scanning beam - 30GHz'''
+File:e_100_BS_Mars12.png| '''ellipticity - 100GHz'''
+File:solidarc_100_BS_Mars12.png| '''beam solid angle (relative variations wrt scanning beam - 100GHz'''
+</gallery>
+-->
+'''Statistics of the effective beams computed using FEBeCoP'''
+We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 3072 and 768 directions on the sky for HFI and LFI data respectively. Statistics shown in the Table are derived from the histograms shown above.
+* The derived beam parameters are representative of the DPC NSIDE 1024 and 2048 healpix maps (they include the pixel window function).
+* 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.
+{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
+|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''
 |-
-| 030 || 1188.513 || --- || 0.842 || --- || 32.34 (32.24)
+!  '''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]
 |-
-| 044 || 832.946 || --- || 31.774 || --- || 27.12 (27.01)
+| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34
 |-
-| 070 || 200.742 || --- || 1.027 || --- || 13.31 (13.25)
+| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774  || 27.12
 |-
-| 100 || 105.78 || --- || 0.31 || --- || 9.66 (9.65)
+| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027  || 13.31
 |-
-| 143 || 59.95 || --- || 0.25 || --- || 7.27 (7.25)
+| 100 || 9.651 || 0.014 || 1.186 || 0.023 || -0.024 || 55.400 || 105.778 || 0.311 || 9.66
 |-
-| 217 || 28.45 || --- || 0.27 || --- || 5.01 (4.99)
+| 143 || 7.248 || 0.015 || 1.036 || 0.009 || 0.383   || 54.130 || 59.954 || 0.246  || 7.27
 |-
-| 353 || 26.71 || --- || 0.25 || --- || 4.86 (4.82)
+| 217 || 4.990 || 0.025 || 1.177 || 0.030 || 0.836   || 54.999 || 28.447 || 0.271  || 5.01
 |-
-| 545 || 26.54 || --- || 0.34 || --- || 4.84 (4.68)
+| 353 || 4.818 || 0.024 || 1.147 || 0.028 || 0.655   || 54.745 || 26.714 || 0.250  || 4.86
+|-
+| 545 || 4.682 || 0.044 || 1.161 || 0.036 || 0.544   || 54.876 || 26.535 || 0.339  || 4.84
 |-
-| 857 || 24.24 || --- || 0.19 || --- || 4.63 (4.33)
+| 857 || 4.325 || 0.055 || 1.393 || 0.076 || 0.876   || 54.779 || 24.244 || 0.193  || 4.63
 |}
-----
+'''Beam solid angles for the PCCS'''
-====Beam solid angles for the PCCS====
+*  <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 </math>(effbeam)/max(effbeam)''', 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>.
-**  Omega_beam - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition:  4pi/sum(effbeam)/max(effbeam), i.e. as an integral over the full extent of the effective beam
+* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above
-** from Omega_beam we estimate the fwhm_eff, under a Gaussian approximation - these are tabulated above
+** <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>.
-** Omega_beam_1 is the beam solid angle estimated up to a radius equal to 1xfwhm_eff and Omega_beam_2 up to a radius equal to 2xfwhm_eff
 *** 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).
@@ Line 165: / Line 831: @@
 {|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
 |+'''Band averaged beam solid angles'''
-|  '''Band'''  ||  '''Omega_beam'''[arcmin<math>^{2}</math>] || '''spatial variation''' [arcmin<math>^{2}</math>] ||  '''Omega_beam_1''' [arcmin<math>^{2}</math>]|| '''spatial variation-1''' [arcmin<math>^{2}</math>] ||  '''Omega_beam_2''' [arcmin<math>^{2}</math>] || '''spatial variation-2''' [arcmin<math>^{2}</math>]
+|  '''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>]
 |-
 |30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847
@@ Line 186: / Line 852: @@
 |}
+'''Production process'''
+FEBeCoP, or Fast Effective Beam Convolution in Pixel space{{BibCite|mitra2010}}, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:
-==Production process==
-------------------------
-The methodology for computing effective beams for a scanning CMB experiment like Planck
-was presented in our [[http://arxiv.org/pdf/1005.1929 | paper]].
-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:
 * 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)
 * follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position
@@ Line 206: / Line 862: @@
-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.
+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 scan and beam models, as well as different lengths of datasets.
-===Pixel Ordered Detector Angles (PODA)===
-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.
+'''Pixel Ordered Detector Angles (PODA)'''
+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 pixel - 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.
-===effBeam===
+'''effBeam'''
 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.
@@ Line 221: / Line 875: @@
-===Computational Cost===
+'''Computational Cost'''
+The computation of the effective beams has been performed at the NERSC Supercomputing Center. The table below shows the computation cost for FEBeCoP processing of the nominal mission.
-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.
 {|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
-|+ 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
+|+ '''Computational cost for PODA, Effective Beam and  single map convolution. Wall-clock time is given as a guide, as found on the NERSC supercomputers.'''
 |-
 |Channel ||030 || 044 || 070 || 100 || 143 || 217 || 353 || 545 || 857
@@ Line 232: / Line 885: @@
 |PODA/Detector Computation time (CPU hrs) || 85 || 100  || 250 || 500 || 500 || 500 || 500 || 500 || 500
 |-
-|PODA/Detector Computation time (Human minutes) || 7 || 10  || 20 || 20 || 20 || 20 || 20 || 20 || 20
+|PODA/Detector Computation time (wall clock hrs) || 7 || 10  || 20 || 20 || 20 || 20 || 20 || 20 || 20
 |-
 |Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 || 2800 ||  3800 || 3200 || 3000 || 900 || 1100
 |-
-|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8  || 1 || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5
+|Beam/Channel Computation time (wall clock hrs) || 0.5 || 0.8  || 1 || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5
 |-
 |Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 || 3.6 || 4.8 || 4.0 || 4.1 || 4.1 || 3.7
 |-
-|Convolution Computation time (Human sec) || 1 || 1 || 1 || 4 || 4 || 4 || 4 || 4 || 4
+|Convolution Computation time (wall clock sec) || 1 || 1 || 1 || 4 || 4 || 4 || 4 || 4 || 4
 |-
 |Effective Beam Size (GB) || 173 || 123 || 28 || 187 || 182 || 146 || 132 || 139 || 124
@@ Line 248: / Line 901: @@
 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.
+'''Inputs'''
+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.
-==Inputs==
+[[File:PlanckFocalPlane.png | 500px| thumb | center|'''Planck Focal Plane''']]
-------------
-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.
-[[File:PlanckFocalPlane.png | 500px| thumb | center|  "'Planck Focal Plane''']]
+'''The Focal Plane DataBase (FPDB)'''
+ The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument [[The RIMO|RIMOs]]):
-===The Focal Plane DataBase (FPDB)===
+*HFI - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}
+*LFI - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}
-The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ...
-*LFI - LFI_RIMO_DX9_PTCOR6
-*HFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits
-[<span style="color:Red"> reference-link to wikipage with RIMO </span>]
+<!--
+*HFI - LFI_RIMO_DX9_PTCOR6 -  {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}
+*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}
-===The scanning strategy===
+{{PLADoc|fileType=rimo|link=The Plank RIMOS}}
+-->
+'''The scanning strategy'''
 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.
-===The scanbeam===
+'''The scanbeam'''
+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.
+* 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.
+* HFI: [[Beams#Scanning beams|B-Spline, BS]] based on 2 observations of Mars.
-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.
+(see the instrument [[The RIMO|RIMOs]]).
-* LFI: GRASP scanning beam - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response.
+<!--
-* HFI: B-spline based on 2 observations of Mars
-===Beam cutoff radii===
+*HFI - LFI_RIMO_DX9_PTCOR6 -  {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}
+*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}
+[[Beams LFI#Effective beams|LFI effective beams]]
+-->
+'''Beam cutoff radii'''
 N times geometric mean of FWHM of all detectors in a channel, where N
@@ Line 287: / Line 947: @@
 | '''channel''' || '''Cutoff Radii in units of fwhm''' || '''fwhm of full beam extent'''
 |-
-|30 - 44 - 70 || 2.5
+|30 - 44 - 70 || 2.5 ||
 |-
 |100 || 2.25 || 23.703699
@@ Line 298: / Line 958: @@
 |}
-===Map resolution for the derived beam data object===
+'''Map resolution for the derived beam data object'''
+* <math>N_{side} = 1024 </math> for LFI frequency channels
-* N'_side_' = 1024 for LFI frequency channels
+* <math>N_{side} = 2048 </math> for HFI frequency channels
-* N'_side_' = 2048 for HFI frequency channels
-==Related products==
-----------------------
-<span style="color:Red">A description of other products that are related and share some commonalities with the product being described here. E.g. if the description is of a generic product (e.g. frequency maps), all the products falling into that type should be listed and referenced.</span>
-===Monte Carlo simulations===
+'''Related products'''
+'''Monte Carlo simulations'''
 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:
@@ Line 317: / Line 970: @@
 * generate 100 realizations of maps from a fiducial CMB power spectrum
 * convolve each one of these maps with the effective beams using FEBeCoP
-* estimate the average of the Power Spectrum of each convolved realization, C'_\ell_'^out^'}, and 1 sigma errors
+* estimate the average of the Power Spectrum of each convolved realization, and 1 <math>\sigma</math> errors
 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, FFP#.
-For example FFP6 [<span style="color:Red"> reference-link to wikipage on FFP6 simulations </span>]
+For example [[HL-sims#FFP6 data set|FFP6]]
-===Beam Window Functions===
+'''Beam Window Functions'''
 The '''Transfer Function''' or the '''Beam Window Function'''  <math> W_l </math> relates the true angular power spectra <math>C_l </math> with the observed angular power spectra <math>\widetilde{C}_l </math>:
@@ Line 342: / Line 993: @@
 For subtle reasons, we perform a more rigorous estimation of the window function by comparing C^{conv}_l 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.
-Beam window functions are provided in the RIMO [<span style="color:Red"> reference-link to RIMO</span>]
+Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]].
-====Wl for Planck mission: LFI (left) and HFI (right)====
-[[File:plot_dx9_LFI_GB_pix.png | 450px | thumb | left |Beam Window functions, Wl, for LFI channels]]
+'''Beam Window functions, Wl, for Planck mission'''
+[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, Wl, for LFI channels''']]
+[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |'''Beam Window functions, Wl, for HFI channels''']]
-[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |Beam Window functions, Wl, for HFI channels]]
+'''File Names'''
+The effective beams are provided by the PLA as FITS files containg HEALPix maps of the beams. For the file names the following convention is used:
+*Single beam query: '''beams_FFF_''PixelNumber''.fits'''
+**FFF is the channel frequency (one of '''30''', '''44''', '''70''', '''100''', '''143''', '''217''', '''353''', '''545''', '''857''');
+** ''PixelNumber'' is the number of the pixel to which the beam corresponds (<math>0</math> - <math>12 \times N_{\rm side}^2 - 1</math>). For the LFI <math>N_{\rm side}=1024</math>, for the HFI <math>N_{\rm side}=2048</math>;
+*Multiple beam query: '''beams_FFF_''FirstPixelNumber''-''LastPixelNumber''.zip''' <br /> The compressed files contains a set of files with the beams for the pixels covereing the selected region. FFF as for sinle beam query. The naming convention for the beam files contained in the ''.zip'' file is the same as for single beam queries.
+** ''FirstPixelNumber'' is the lowest pixel number for the area covered by the request;
+** ''LastPixelNumber'' is the highest pixel number for the area covered by the request;
+'''File format'''
+The FITS files provided by the PLA contain HEALPix maps of the beams.
+</div>
+</div>
-----
-==File Names==
-The effective beams are stored as unformatted files in directories with the frequency channel's name, e.g., 100GHz, each subdirectory contains N unformatted files with names beams_###.unf, a beam_index.fits and a beams_run.log. For 100GHz and 143GHz: N=160, for 30, 44, 70 217 and 353GHz: N=128; for 545GHz: N=40; and 857GHz: N=32.
-* beam_index.fits
-* beams_run.log
+== References ==
+<References />
-==Meta data==
-<span style="color:Red">A detailed description of the data format of each file, including header keywords for fits files, extension names, column names, formats….</span>
-The data format of the effective beams is unformatted.
-[[Category:Mission science products|004]]
+[[Category:Mission products|004]]

Difference between revisions of "Effective Beams"

Latest revision as of 02:01, 22 March 2018

Contents

Product description[edit]

Comparison of the images of compact sources observed by Planck with FEBeCoP products[edit]

Histograms of the effective beam parameters[edit]

Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian[edit]

Statistics of the effective beams computed using FEBeCoP[edit]

Beam solid angles for the PCCS[edit]

Production process[edit]

Pixel-ordered detector angles (PODA)[edit]

Effective beam[edit]

Computational Cost[edit]

Inputs[edit]

The focal plane database (FPDB)[edit]

The scanning strategy[edit]

The scanning beam[edit]

Beam cutoff radii[edit]

Map resolution for the derived beam data object[edit]

Related products[edit]

Monte Carlo simulations[edit]

Beam Window Functions[edit]

Beam window functions, Wℓ, for the Planck mission.[edit]

File Names[edit]

File format[edit]

Previous Releases: (2015) and (2013) effective beams[edit]

References[edit]

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Beam window functions, W_ℓ, for the Planck mission.[edit]