Effective Beams

Product description

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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. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map.

The full algebra involving the effective beams for temperature and polarisation was presented in [Mitra, Rocha, Gorski et al.]. 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}, \label{eq:a0} [/math]

where

[math] B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, , \label{eq:EBT2} [/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.

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:

[math] ( \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, \label{eq:a1} [/math]

where the polarised effective beam matrix

[math] \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) \, , \label{eq:a2} [/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.

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.

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.

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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 30GHz and 100GHz 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 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

30GHz

30GHz

100GHz

100GHz

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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.

Where beam solid angle is estimated according to the definition: 4pi* sum(effbeam)/max(effbeam)

LFI

Histograms for LFI effective beam parameters

HFI

Histograms for HFI effective beam parameters

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Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian - 30GHz (top) and 100GHz (bottom)

Sky variation of effective beams ellipticity of best-fit Gaussian for 30GHz

Sky variation of effective beams solid angle of best-fit Gaussian for 30GHz relative to scanning beam solid angle

Sky variation of effective beams ellipticity of best-fit Gaussian for 100GHz

Sky variation of effective beams solid angle of best-fit Gaussian for 100GHz relative to scanning beam solid angle

Statistics of the effective beams computed using FEBeCoP

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.

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(psi) [degree]	sd(psi) [degree]	mean(Solid Angle) [arcmin[math]^{2}[/math]]	sd(Solid Angle) [arcmin[math]^{2}[/math]]
030	32.239	0.013	1.320	0.031	-0.304	55.349	1189.513	0.842
044	27.005	0.552	1.034	0.033	0.059	53.767	832.946	31.774
070	13.252	0.033	1.223	0.026	0.587	55.066	200.742	1.027
100	9.651	0.014	1.186	0.023	-0.024	55.400	105.778	0.311
143	7.248	0.015	1.036	0.009	0.383	54.130	59.954	0.246
217	4.990	0.025	1.177	0.030	0.836	54.999	28.447	0.271
353	4.818	0.024	1.147	0.028	0.655	54.745	26.714	0.250
545	4.682	0.044	1.161	0.036	0.544	54.876	26.535	0.339
857	4.325	0.055	1.393	0.076	0.876	54.779	24.244	0.193

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FEBeCoP deliverables

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

Band averaged beam solid angles

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]
030	1188.513	---	0.842	---	32.34 (32.24)
044	832.946	---	31.774	---	27.12 (27.01)
070	200.742	---	1.027	---	13.31 (13.25)
100	105.78	---	0.31	---	9.66 (9.65)
143	59.95	---	0.25	---	7.27 (7.25)
217	28.45	---	0.27	---	5.01 (4.99)
353	26.71	---	0.25	---	4.86 (4.82)
545	26.54	---	0.34	---	4.84 (4.68)
857	24.24	---	0.19	---	4.63 (4.33)

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Beam solid angles for the PCCS

• • 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 Omega_beam we estimate the fwhm_eff, under a Gaussian approximation - these are tabulated above
  • 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).

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]]
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

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The methodology for computing effective beams for a scanning CMB experiment like Planck was presented in [Mitra, Rocha, Gorski et al.].

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
• 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 scanbeam models and 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.

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 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.

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

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 (Human minutes)	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
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
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

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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):

• LFI - LFI_RIMO_DX9_PTCOR6
• HFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits

[ reference-link to wikipage with 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 based on 2 observations of Mars

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

• N'_side_' = 1024 for LFI frequency channels
• N'_side_' = 2048 for HFI frequency channels

Related products

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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.

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, C'_\ell_'^out^'}, and 1 sigma 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 [ reference-link to wikipage on FFP6 simulations ]

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 = \lt C^{conv}_l / C^{in}_l \gt \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 RIMO. [ reference-link to RIMO]

Wl for Planck mission: LFI (left) and HFI (right)

Beam Window functions, Wl, for LFI channels

Beam Window functions, Wl, for HFI channels

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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

Meta data

A detailed description of the data format of each file, including header keywords for fits files, extension names, column names, formats….

The data format of the effective beams is unformatted.

References

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1. References

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