Difference between revisions of "Effective Beams"

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|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''
|+ '''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>']
!  '''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  
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842  
Line 219: Line 219:
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ...
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ...

Revision as of 06:17, 9 March 2013

The second part of this section is mostly unreadable. Please use proper style in terms of sectioning (i.e. pseudo headings that are underlined), lists (bullet or numbered) - see other pages for examples. Info regarding M3 (and NERCS) is probably irrelevant to external users)

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.

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[edit]


100 GHz[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)


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]


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[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.

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

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 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]
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 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]]
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.


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.

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.


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

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

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[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.

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….

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


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.