Effective Beams
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Contents
- 1 Product description
- 1.1 Comparison of the images of compact sources observed by Planck with FEBeCoP products
- 1.2 Histograms of the effective beam parameters
- 1.3 Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian
- 1.4 Statistics of the effective beams computed using FEBeCoP
- 1.5 FEBeCoP deliverables
- 2 Production process
- 3 Inputs
- 4 Related products
- 5 File Names
- 6 Meta data
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 image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map.
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.
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 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
30 GHz (left) and 100GHz (right)[edit]
Histograms of the effective beam parameters[edit]
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[edit]
HFI[edit]
Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian[edit]
LFI[edit]
HFI[edit]
Statistics of the effective beams computed using FEBeCoP[edit]
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.
frequency | mean(fwhm) [arcmin] | sd(fwhm) [arcmin] | mean(e) | sd(e) | mean(psi) [degree] | sd(psi) [degree] | mean(Solid Angle) [arcmin | ]sd(Solid Angle) [arcmin | ]
---|---|---|---|---|---|---|---|---|
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 |
FEBeCoP deliverables[edit]
- 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 | Omega_beam [arcmin | ]Monte Carlo error[arcmin | ^]spatial variation [arcmin | ]bias [arcmin | ]FWHM_eff (Gaussian fit) [arcmin] |
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) |
Beam solid angles for the PCCS[edit]
- 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 | Omega_beam[arcmin | ]spatial variation [arcmin | ]Omega_beam_1 [arcmin | ]spatial variation-1 [arcmin | ]Omega_beam_2 [arcmin | ]spatial variation-2 [arcmin | ]
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]
The methodology for computing effective beams for a scanning CMB experiment like Planck was presented in our [| 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
- 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)[edit]
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[edit]
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[edit]
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.
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[edit]
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.
A list (and brief description to the extent possible) of the input data used to generate this product (down to file names), as well as any external ancillary data sets which were used.
The Focal Plane DataBase (FPDB)[edit]
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ...
- LFI - LFI_RIMO_DX9_PTCOR6.fits
- HFI
The scanning strategy[edit]
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[edit]
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[edit]
N times geometric mean of FWHM of all detectors in a channel, where N
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[edit]
- N'_side_' = 1024 for LFI frequency channels
- N'_side_' = 2048 for HFI frequency channels
Related products[edit]
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[edit]
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[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 \widetilde{C}_l:
- W_l = widetilde{C}_l / C_l
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 C_l, convolve the maps with the pre-computed effective beams, compute the convolved power spectra C^\text{conv}_l, divide by the power spectra of the unconvolved map C^\text{in}_l and average over their ratio. Thus, the estimated window function
- W^{est}_l = < C^{conv}_l / C^{in}_l >
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]
- W'_l_' for Planck mission
LFI (left) and HFI(right)[edit]
LFI (left) and HFI(right)[edit]
File Names[edit]
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[edit]
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.
Cosmic Microwave background
(Hierarchical Equal Area isoLatitude Pixelation of a sphere, <ref name="Template:Gorski2005">HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere, K. M. Górski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, M. Bartelmann, ApJ, 622, 759-771, (2005).
Early Release Compact Source Catalog
(Planck) Low Frequency Instrument
(Planck) High Frequency Instrument
Full-Width-at-Half-Maximum
Data Processing Center
reduced IMO
Operation Day definition is geometric visibility driven as it runs from the start of a DTCP (satellite Acquisition Of Signal) to the start of the next DTCP. Given the different ground stations and spacecraft will takes which station for how long, the OD duration varies but it is basically once a day.