Difference between revisions of "Beams LFI"

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== Overview ==
 
== Overview ==
---------------
 
  
LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.
+
LFI observed the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers.
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]).   
+
Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is labelled as LFIXXM or LFIXXS. Here "XX" is the RCA number ranging from 18 to 28, while "M" and "S" are the two polarizations, namely the main-arm and side-arm of the orthomode transducers {{BibCite|darcangelo2009b}} (see also [[LFI design, qualification, and performance#Naming Convention|LFI naming convention]]).   
 
 
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completness and they are not representative of flight beams.''']]
 
  
 +
[[File:fieldofview.png|500px|thumb|centre|'''Figure 1. A sketch of the Planck LFI field of view in the (<i>u</i>,<i>v</i>) plane is shown. The polarizations direction on the sky are highlighted by the coloured arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completeness and they are not representative of flight beams.''']]
  
 +
Details are given in {{PlanckPapers|planck2014-a05}}. Please note that many figures below refer to the Planck release {{PlanckPapers|planck2013-p02d}}, since they have not changed significantly.
  
 
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; main beam: is the portion of the pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.
+
; the main beam: is the portion of the beam pattern that extends up to 1.9, 1.3, and 0.9 degrees from the beam center at 30, 44, and 70 GHz, respectively.
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called <b>intermediate beam</b>).
+
; near sidelobes: is the pattern contained between the main beam angular limit and 5 degrees from the beam center (this is often called the <b>intermediate beam</b>).
; far sidelobes: is the pattern at angular regions more than 5 degrees from the beam center.
+
; far sidelobes: is the pattern at angular distance more than 5 degrees from the beam center.
 
-->
 
-->
  
== Main Beams and Focal Plane calibration ==
+
== Beam normalization ==
  
As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurments using Jupiter transits. the parametes that characterise the beam pointing are the following:
+
Following the same procedure applyed during the 2015 release {{PlanckPapers|planck2014-a05}} we didn't normalize the beam since up to 1% of the solid angle of the LFI beams falls into the sidelobes, unevenly distributed and concentrated mainly in two areas, namely the main part and sub-spillover.  
  
* THETA_UV ($\theta_{uv}$)
+
We employ full 4&pi; beams. Important to note is that roughly 1% of the signal found in the sidelobes is missing from the vicinity of the main beam, so the main beam efficiency &eta; &asymp; 99%, which must be accounted for in any analysis of the maps. In particular, the window function used to correct the power spectra extracted from the maps allows for this efficiency.
* PHI_UV ($\phi_{uv}$)
 
  
They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as
+
Details are given in {{PlanckPapers|planck2014-a05}}.
  
$\theta_{uv} = \arcsin(u^2+v^2)$
+
== Polarized scanning beams and focal plane calibration ==
  
$\phi_{uv} = \arctan(v/u)$
+
Focal plane calibration is based on the determination of the beam pointing parameters in the nominal line of sight (LOS) frame derived from measurements during Jupiter transits. The parameters that characterize the beam pointing are:
 +
 
 +
* THETA_UV (&theta;<sub>uv</sub>);
 +
* PHI_UV (&phi;<sub>uv</sub>).
 +
 
 +
They are calculated starting from <i>u</i>,<i>v</i> coordinates derived form the beam reconstruction algorithm:
 +
 
 +
<math>\theta_{uv} = \arcsin(u^2+v^2)</math>;
 +
 
 +
<math>\phi_{uv} = \arctan(v/u)</math>.
  
 
Two additional angles are used to characterize the beams in the RIMO:  
 
Two additional angles are used to characterize the beams in the RIMO:  
  
* PSI_UV ($\psi_{uv}$)
+
* PSI_UV (&psi;<sub>uv</sub>);
* PSI_POL ($\psi_{pol}$)
+
* PSI_POL (&psi;<sub>pol</sub>).
 
   
 
   
$\psi_{uv}$ and $\psi_{pol}$ are '''not''' derived from measurements but they are extimated form '''optical simulations'''. They are the quantities that represent the polarization direction of each beam, in the following approximation: '''the M- and  S- beams of the same RCA point at the same direction on the sky'''.
+
The angles &psi;<sub>uv</sub> and &psi;<sub>pol</sub> are <i>not</i> derived from measurements, but rather are estimated from optical simulations. They are the quantities that represent the polarization direction of each beam, assuming that the M- and  S-beams of the same RCA point at the same direction on the sky.
  
The main beams are characterised by 2 method:
+
The polarized scanning beams were evaluated from optical simulations using the GRASP physical optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM, {{PlanckPapers|tauber2010b}}).
  
* elliptical (or bivariate) gaussian fit as in {{BibCite|[planck2011-1-6]}} with modification explained in {{BibCite|[planck2013-p02d]}} {{P2013|4}}. This method is used to determine
+
The Radio Frequency Tuned Model (RFTM) was implemented to fit the in-flight beam measurements with an electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross-polar beam in flight, we have relied on simulations validated by accurate beam measurements.
**the beam centre
 
**the average full width half maximum defined as $\sqrt{FWHM_{max}\cdot FWHM_{min}}$
 
**the beam ellipticity defined as $FWHM_{max}\over{FWHM_{min}}$
 
**the beam tilting, $\psi_{ell}$, with respect the u-axis.  
 
  
* Electromagnetic simulation (using GRASP Physical Optics code) by appropriately tuning the Radio Frequency Flight Model (RFFM) {{BibCite|tauber2010b}}. The Radio Frequency Tuned Model, called RFTM, was implemented to fit the beam data with electromagnetic model. It is derived as follow:
+
The model beams were monochromatic and were computed at various frequencies across a 6-GHz band around the optical centre frequency (OCF) with non-uniform steps (denser sampling where the bandpass was higher). For the RFTM model the OCFs were set at 28.0, 44.0, and 70.0 GHz.  
**the Focal plane unit electromagnetic model has shifted by 3.5mm toward the secondary mirror;
+
 
** All the simulated beams where monochromatic, i.e calculated at a single frequency called Optical Center Frequency (OCF). For the RFTM model the OCF has been chosen at $28.0$, $44.0$, $70.0$. In fact the optical and radiometer bandshapes as reported in {{BibCite|zonca2009}} demonstrates that for the 30 GHz channel, the radiometer responses are better described by a central frequency closer to 28 GHz with respect to the nominal one, whereas for the other two frequency channels the OCF is close to the nominal one.
+
For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the [[The_RIMO#LFI_2|RIMO]] transmission function.
**each feed horn phase centre has been moved alogn horn axis to optimize the match between simulations and data. The optimization was obtained by minimizing the variance according to the following definition: If $B_s[u,v]$ is the peak-normalized scanning beam matrix (for semplicity we use here $(u,v)$ coordinates also as indexes of the beam matrix) and  $B_o[u,v]$ is the smeared peak-normalized simulated GRASP beam, the variance, $\sigma$, can be evaluated for each beam:
+
 
\begin{eqnarray}
+
The delivered [[Scanning_Beams|products]] includes the in-band averaged Stokes parameter scanning maps of main beams, intermediate beams, and sidelobes.
\label{eqsigma}
 
\sigma &=& {\sum_{u,v}{(f[u,v] - \overline{f})^2}}\cdot {1\over N} \\
 
f[u,v]&=&w[u,v] \cdot (B^{dB}_s(u,v)- B^{dB}_o(u,v)) \\
 
w[u,v]&=&\sqrt{T[u,v]}
 
\end{eqnarray} where also $T[u,v]$ is the temperature or the scanning beam not normalized to peak, and $N$ is the number of points considered in the comparison so that the number of point in the (u,v) plane. The parameter $\sigma$, as already said, is the variance of the difference between two beams weighted by the measured beam itself. For each beam the variance has computed computed down to $-15$dB from beam peak to avoid bias due to noise and background. The comparison between the simulated RFTM beams and the data are reported in {{BibCite|planck2013-p02d}} {{P2013|4}}.
 
  
 
== Effective beams ==
 
== Effective beams ==
-----------------------
 
  
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.
+
The 2018 effective beam estimation procedure didn't change with respect 2015 release, new summary values are perfectly in line with the once obtained in the 2015, see {{PlanckPapers|planck2016-l02}}.
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 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. These beams are, by definition, the objects whose convolution with the true CMB sky produces the observed sky map.  
  
The full algebra involving the effective beams for temperature and polarisation was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]] {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature sky <math>\widetilde{\mathbf{T}} </math> is a convolution of the true sky <math>\mathbf{T} </math> and the effective beam <math>\mathbf{B}</math>:
+
The full algebra involving the effective beams for temperature and polarisation was presented in {{BibCite|mitra2010}}. Here we summarise the main results. The observed temperature on the 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>
 
<math>
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},
+
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \ast \mathbf{T},
 
\label{eq:a0}
 
\label{eq:a0}
 
</math>
 
</math>
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<math>
 
<math>
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,
+
B_{ij} \ = \frac{\sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)} {\sum_t A_{ti}}.
 
\label{eq:EBT2}
 
\label{eq:EBT2}
 
</math>
 
</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.
+
Here <i>t</i> labels the time samples, <i>A<sub>ti</sub></i> is 1 if the pointing direction falls in pixel number <i>i</i>, else it is 0, <math>\hat{\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 <i>j</i>, where the scanning beam <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 2.5&times;FWHM).
  
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at 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:
+
The algebra is a bit more involved for polarized detectors. The observed stokes parameters at 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>
 
<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,
+
( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^{\rm T} \ = \ \Delta\Omega \sum_j B_{ij} \ast (I \quad Q \quad U)_j^{\rm T},
 
\label{eq:a1}
 
\label{eq:a1}
 
</math>
 
</math>
  
where the polarised effective beam matrix
+
where the polarized effective beam matrix
  
 
<math>
 
<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) \, ,
+
B_{ij} \ = \  \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^{\rm T}_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t  \mathbf{W}^{\rm T}(\hat{\mathbf{r}}_j,\hat{\mathbf{p}}_t) \, ,
 
\label{eq:a2}
 
\label{eq:a2}
 
</math>
 
</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}.
+
and <math>\mathbf{w}_t </math>and <math>\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) </math> are the the polarization weight vectors, as defined in Mitra et al. (2010).{{BibCite|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 task is to compute <math>B_{ij}</math> for temperature-only beams and the 3 &times; 3 matrices <math>\mathbf{B}_{ij}</math> for each pixel <i>i</i>, at every neighbouring pixel <i>j</i> that falls within the cut-off radius around the the centre of the <i>i</i>th 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 the 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.
  
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 particular period, provides the pixelized cut-outs of both the effective beam (EB) and the point spread function (PSF), at all pixel-centre positions of the HEALPix-formatted 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.
 
  
  
 
===Production process===
 
===Production process===
 
  
 
The methodology for computing effective beams for a scanning CMB experiment like Planck
 
The methodology for computing effective beams for a scanning CMB experiment like Planck
was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]].
+
was presented in Mitra et al. (2010).{{BibCite|mitra2010}}
 
 
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
 
  
 +
FEBeCoP, or the Fast Effective Beam Convolution in Pixel space, is an approach for representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that consists of the following steps:
 +
* identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets, but 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 during the 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 the same resolution for 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, 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 the implementation of which enabled us to compute the pixel-based effective beams using moderate computational resources. The algorithm uses different mathematical and computational techniques to bring down the computation cost to a practical level.  That allowed several estimations of the effective beams for all Planck detectors for different scan and beam models as well as data sets of different length.  
  
  
====Pixel Ordered Detector Angles (PODA)====
+
====Pixel-ordered detector angles====
  
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 main challenge in computing the effective beams is to go through the trillion samples, which becomes severely limited by I/O. In the first stage, for a given data set, ordered lists of pointing angles are made for each pixel &ndash; the Pixel-Ordered Detector Angles (PODA). This is an one-time process for each dataset. We used computers with large memory and used careful 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.
+
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.
+
One important reason for separating the two steps is that they use different schemes for parallel computing. The PODA part requires parallelization over time-ordered data samples, while the  effBeam part requires distribution of pixels among different computers.
  
  
====Computational Cost====
+
====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.
+
The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below we display the computation cost at NERSC for the nominal mission both in terms of CPU hours and in human time.
  
 
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
 
{|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
+
|+ <small>'''Computational cost for PODA, effective beam and single map convolution. The cost in human time is computed using an arbitrary number of nodes/cores on the Carver or Hopper NERSC supercomputers.'''</small>
 
|-
 
|-
 
|Channel ||030 || 044 || 070  
 
|Channel ||030 || 044 || 070  
 
|-
 
|-
|PODA/Detector Computation time (CPU hrs) || 85 || 100  || 250  
+
|PODA/Detector computation time (CPU hours) || 85 || 100  || 250  
 
|-
 
|-
|PODA/Detector Computation time (Human minutes) || 7 || 10  || 20  
+
|PODA/Detector computation time (human minutes) || 7 || 10  || 20  
 
|-  
 
|-  
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300  
+
|Beam/Channel computation time (CPU hours) || 900 || 2000 || 2300  
 
|-
 
|-
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8  || 1  
+
|Beam/Channel computation time (human hours) || 0.5 || 0.8  || 1  
 
|-
 
|-
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3  
+
|Convolution computation time (CPU hours) || 1 || 1.2 || 1.3  
 
|-
 
|-
|Convolution Computation time (Human sec) || 1 || 1 || 1  
+
|Convolution computation time (human seconds) || 1 || 1 || 1  
 
|-
 
|-
|Effective Beam Size (GB) || 173 || 123 || 28  
+
|Effective beam size (GB) || 173 || 123 || 28
 
|}
 
|}
  
 
+
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.
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===
 
===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.  
+
In order to fix the convention for presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photons. The scan direction is marked, and the direction toward the centre of the focal plane is at 85&deg; with respect to the spin axis, pointing upward in the picture.  
  
  
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane''']]
+
[[File:PlanckFocalPlane.png | 600px| thumb | center|'''Planck Focal Plane.''']]
  
  
====The Focal Plane DataBase (FPDB)====
+
====Focal plane database:====
 
   
 
   
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) contains information on each detector, e.g., the orientation of the polarization axis, different weight factors, etc. See the instrument [[The RIMO|RIMOs]]:
 
 
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=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 scanning strategy:====
  
====The scanbeam====
+
The scanning strategy, specifically includes 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 3&frac12; surveys.
  
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.
+
====Scanning beam:====
  
* 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.  
+
The "scanning beam" was modelled for each detector using observations of planets. This was assumed to be constant over the whole mission, although FEBeCoP could be used for a few sets of scanning beams.  
  
(see the instrument [[The RIMO|RIMOs]])
+
* LFI: [[Beams LFI#Main beams and Focalplane calibration|GRASP scanning beam]] - the scanning beams used are based on the Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. A description of the GRASP beams file formar (.grd) can be found in the "Technical Note PL-LFI-PST-TN-044: LFI Beams Delivery: Format Specifications": [[File: PST-TN-044_1-0.pdf|'''PST-TN-044_1-0.pdf''']].
  
* {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}
+
See the instrument [[The RIMO|RIMOs]]:
 +
{{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}.
  
====Beam cutoff radii====
+
====Beam cutoff radii:====
  
* N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.
+
* <i>N</i> times the geometric mean of the FWHM values of all detectors in a channel, where <i>N</i>=2.5 for all LFI frequency channels.
 
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{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
 
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
 
|+'''Beam cut off radius'''
 
|+'''Beam cut off radius'''
| '''channel''' || '''Cutoff Radii in units of fwhm''' ||
+
| '''channel''' || '''Cutoff Radii in units of FWHM.''' ||
 
|-
 
|-
 
|30 - 44 - 70 || 2.5 ||
 
|30 - 44 - 70 || 2.5 ||
Line 208: Line 195:
 
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====Map resolution for the derived beam data object====
+
====Map resolution for the derived beam data object:====
  
* <math>N_{side} = 1024 </math> for all LFI frequency channels.
+
* <i>N</i><sub>side</sub>=1024 for all LFI frequency channels.
  
 
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===
 
===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 frequency channel, as an example. We show below a few panels of source images organized as follows:
+
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.  The 30GHz frequency channel images are used 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 1 &ndash; 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, 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 2 &ndash; linear scale FEBeCoP PSFs computed using input scanning beams, GRASP beams (GB) for LFI, B-spline (BS) Mars12 beams apodized for the HFI CMB channels, and the BS Mars12 beams for the HFI sub-mm channels (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
+
* Row 3 &ndash; log scale of Row 2, PSF iso-contours are shown as solid lines, elliptical Gaussian fit iso-contours are shown as broken lines.
 
 
 
 
[[File:30.png| 600px| thumb | center| '''30GHz''']]
 
  
 +
[[File:30.png| 600px| thumb | center| '''30GHz.''']]
  
 
===Histograms of the effective beam parameters===
 
===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 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.
+
Here we present histograms of the three fit parameters &ndash; beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The sky is sampled (quite sparsely) in 768 directions, which were chosen as HEALpix <i>N</i><sub>side</sub>=8 pixel centres for LFI in order to uniformly sample the sky.
  
Where beam solid angle is estimated according to the definition:  '''4pi* sum(effbeam)/max(effbeam)'''
+
Here beam solid angle is estimated according to the definition:  '''4&pi;&times;&Sigma;(effbeam)/max(effbeam)''',
ie <math> 4 \pi \sum(B_{ij}) /  max(B_{ij}) </math>
+
i.e.,
  
 +
<math> 4 \pi \sum(B_{ij}) /  {\rm max}(B_{ij}) </math>.
  
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]]  
+
[[File:DX12_effective_beam_histogram.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters.''' ]]
  
 +
===Sky variation of effective beam solid angle and ellipticity of the best-fit Gaussian===
  
 +
Note that 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 effective beams solid angle and ellipticity of the best-fit Gaussian===
+
[[File:ellipticity_ns8_030_beams_DX12.png| 600px| thumb | center| '''Ellipticity &ndash; 30GHz.''']]
 
 
* 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.
 
 
 
 
 
[[File:e_030_GB.png| 600px| thumb | center| '''ellipticity - 30GHz''']]
 
[[File:solidarc_030_GB.png| 600px| thumb | center| '''beam solid angle (relative variations wrt scanning beam - 30GHz''']]
 
 
 
 
 
  
 
===Statistics of the effective beams computed using FEBeCoP===
 
===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 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.
+
We tabulate the simple statistics of FWHM, ellipticity (<i>e</i>), orientation (&psi;), and beam solid angle, (&Omega;), for a sample of 768 directions on the sky for LFI data. Statistics presented in the table are derived from the histograms shown above:
 
 
* The derived beam parameters are representative of the DPC NSIDE 1024 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.
 
  
 +
* the derived beam parameters are representative of the DPC <i>N</i><sub>side</sub>=1024 HEALPix maps (they include the pixel window function);
 +
* the reported FWHM<sub>eff</sub> values are derived from the beam solid angles, under a Gaussian approximation &ndash; these are best used for flux determination, while 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"
 
{| 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'''
+
|+ '''Statistics of the FEBeCoP effective beams computed with the BS Mars12 beams, 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]  
+
!  '''Frequency'''  ||  '''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]  
 
|-
 
|-
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34
+
| 030 || 32.288 || 0.021 || 1.315 || 0.031 || 1.347 || 54.074 || 1190.111 || 0.705 || 32.41
 
|-
 
|-
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12
+
| 044 || 26.997 || 0.583 || 1.190 || 0.030 || 2.088 || 53.905 || 831.611 || 35.041 || 27.09
 
|-
 
|-
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31  
+
| 070 || 13.218 || 0.031 || 1.223 || 0.037 || 2.154 || 54.412 || 200.803 || 0.991 || 13.31  
 
|}
 
|}
  
  
 +
====Beam solid angles for the PCCS====
  
====Beam solid angles for the PCCS====
+
&Omega;<sub>eff</sub> is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: 4&pi;&Sigma;(effective<sub>beam</sub>)/max(effective<sub>beam</sub>), i.e., as an integral over the full extent of the effective beam:
  
*  <math>\Omega_{eff}</math> - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: <math> 4 \pi*sum(effective_{beam})/max(effective_{beam})</math> , i.e. as an integral over the full extent of the effective beam, i.e. <math> 4 \pi \sum(B_{ij}) /  max(B_{ij}) </math>.
+
<math> 4 \pi \sum(B_{ij}) /  {\rm max}(B_{ij}) </math>.
  
* from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above
+
From &Omega;<sub>eff</sub> we estimate the FWHM<sub>eff</sub> value, under a Gaussian approximation and 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>.
+
The quantity &Omega;<sup>(1)</sup><sub>eff</sub> is the beam solid angle estimated up to a radius equal to 1&times;FWHM<sub>eff</sub> and &Omega;<sup>(2)</sup><sub>eff</sub> is the value up to a radius equal to 2&times;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).
+
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"
 
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
|+'''Band averaged beam solid angles'''
+
|+'''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>]  
+
|  '''Band'''  ||  '''&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
+
| 30 || 1190.111 || 0.705 || 1117.876 || 1.968 || 1188.991 || 0.713
 
|-
 
|-
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811
+
| 44 || 831.611 || 35.041 || 757.891 || 32.782 || 830.767 || 35.074
 
|-
 
|-
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027
+
| 70 || 200.803 || 0.991 || 186.055 || 1.883 || 200.497 || 0.991
 
|}
 
|}
  
===Related products===
+
<!-- ===Related products=== -->
  
 
===Monte Carlo simulations===
 
===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:
+
FEBeCoP software enables fast, full-sky convolutions of the sky signals with the effective beams in the pixel domain. This enables a large number of Monte Carlo simulations of the sky signal maps to be easily generated, convolved with realistically rendered, spatially varying, asymmetric, Planck beams. We performed the following steps:
  
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data
+
* generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data;
* generate 100 realizations of maps from a fiducial CMB power spectrum
+
* generate 100 realizations of maps from a fiducial CMB power spectrum;
* convolve each one of these maps with the effective beams using FEBeCoP
+
* convolve each one of these maps with the effective beams using FEBeCoP;
* estimate the average of the Power Spectrum of each convolved realization, <math>C_\ell^{out}</math>, and 1 sigma errors
+
* estimate the average of the power spectrum of each convolved realization, <i>C</i><sub>&#8467;</sub><sup>out</sup>, and 1&sigma; errors.
  
 +
Since FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the sky signal (that might be CMB or a foreground, such as dust emission), 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.
  
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#,
 
<!--, FFP#,
 
for example [[HL-sims#FFP6 data set|FFP6]].
 
for example [[HL-sims#FFP6 data set|FFP6]].
 
-->
 
-->
  
== Window Functions ==
+
== Window functions ==
-----------------------
 
  
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_\ell </math> relates the true angular power spectra <math>C_\ell </math> with the observed angular power spectra <math>\widetilde{C}_\ell </math>:
+
The "Transfer Function" or the "Beam Window Function" <i>B</i><sub>&#8467;</sub> relates the true angular power spectra <i>C</i><sub>&#8467;</sub> to the observed angular power spectra <i>&#264;</i><sub>&#8467;</sub>. In the current release, we deliver both TT and EE window functions defined as
  
 
<math>
 
<math>
W_\ell= \widetilde{C}_\ell / C_\ell
+
B_\ell^{TT,EE}= \widetilde{C}_\ell^{TT,EE} / C_\ell^{TT,EE}.
 
\label{eqn:wl1}</math>  
 
\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.
+
Note that the window function can contain a pixel window function (depending on the definition) and it is <i>not</i> the angular power spectra of the scanning beams, although, in principle, one may be able to connect them algebraically.
 
 
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_\ell </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_\ell </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_\ell </math> and average over their ratio. Thus, the estimated window function
 
 
 
<math>
 
W^{est}_\ell  = < C^{conv}_\ell /  C^{in}_\ell >
 
\label{eqn:wl2}</math>
 
 
 
For subtle reasons, we perform a more rigorous estimation of the window function by comparing <math> C^{conv}_\ell</math> with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.
 
 
 
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]].
 
 
 
 
 
====Beam Window functions, Wl, for LFI channels====
 
  
 +
The window function computed for the 2018 release has been calculated via simulated timelines. This procedure allow us to include the sidelobes. Procedure and consitency with the once computed with the convolution of teh effective beam is discussed in to {{PlanckPapers|planck2014-a05}}.
  
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, <math>W_\ell </math>, for LFI channels''']]
+
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]].
  
 
== Sidelobes ==
 
== Sidelobes ==
---------------
 
  
There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD.
+
There are no direct measurements of sidelobes for LFI. The sidelobe patterns for LFI were simulated using GRASP9 multi-reflector GTD.
We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi.
+
We used the RFTM electromagnetic model. Seven beams for each radiometer were computed in spherical polar cuts with a step of 0.5&deg; in both &theta; and &phi;.
The beams have been computed in the same frames used for the main beams.
+
The beams were computed in the same frames used for the main beams.
The intermediate beam region (theta < 5 degrees) has been replaced with null values.
+
The intermediate beam region (&theta; < 5&deg;) was replaced with null values.
  
*In the computation we considered:
+
In the computation we considered:
**the direct field from the feed
+
*the direct field from the feed;
**the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr
+
*the 1st-order contributions, Bd, Br, Pd, Pr, Sd, Sr, and Fr;
**the 2nd order contributions SrPd and SdPd  
+
*the 2nd-order contributions, SrPd, and SdPd.
  
where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box.
+
Here B = baffle', P = primary reflector, S = secondary reflector, F = focal plane unit box, d = diffraction, and r = reflection.
and  where d=diffraction, r=reflection.
+
For example "Br" means that we considered in the calculation reflections on the telescope baffle system.  
For example Br, means that we considered in the calculation the reflection on the telescope baffle system.  
 
  
A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.
+
A refinement of the sidelobe model will be considered in a future release, taking into account more contributions, together with physical optics models.
  
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.''']]
+
[[File:slb_lfi_30_27_y_tricromia.png|500px|thumb|centre|'''Image of the LFI27-M sidelobes presented as an RGB picture, where the red channel is 27 GHz (f0), the green channel is 30 GHz (f3), and the blue channel is 33 GHz (f6). Because the combined map does not show any wide white region, the sidelobe pattern changes with frequency, as expected.''']]
  
 
== References ==
 
== References ==
------------------
 
  
 
<References />  
 
<References />  

Latest revision as of 15:15, 9 November 2018


Overview[edit]

LFI observed the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers. Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is labelled as LFIXXM or LFIXXS. Here "XX" is the RCA number ranging from 18 to 28, while "M" and "S" are the two polarizations, namely the main-arm and side-arm of the orthomode transducers [1] (see also LFI naming convention).

Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarizations direction on the sky are highlighted by the coloured arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completeness and they are not representative of flight beams.

Details are given in Planck-2015-A04[2]. Please note that many figures below refer to the Planck release Planck-2013-IV[3], since they have not changed significantly.


Beam normalization[edit]

Following the same procedure applyed during the 2015 release Planck-2015-A04[2] we didn't normalize the beam since up to 1% of the solid angle of the LFI beams falls into the sidelobes, unevenly distributed and concentrated mainly in two areas, namely the main part and sub-spillover.

We employ full 4π beams. Important to note is that roughly 1% of the signal found in the sidelobes is missing from the vicinity of the main beam, so the main beam efficiency η ≈ 99%, which must be accounted for in any analysis of the maps. In particular, the window function used to correct the power spectra extracted from the maps allows for this efficiency.

Details are given in Planck-2015-A04[2].

Polarized scanning beams and focal plane calibration[edit]

Focal plane calibration is based on the determination of the beam pointing parameters in the nominal line of sight (LOS) frame derived from measurements during Jupiter transits. The parameters that characterize the beam pointing are:

  • THETA_UV (θuv);
  • PHI_UV (φuv).

They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm:

[math]\theta_{uv} = \arcsin(u^2+v^2)[/math];

[math]\phi_{uv} = \arctan(v/u)[/math].

Two additional angles are used to characterize the beams in the RIMO:

  • PSI_UV (ψuv);
  • PSI_POL (ψpol).

The angles ψuv and ψpol are not derived from measurements, but rather are estimated from optical simulations. They are the quantities that represent the polarization direction of each beam, assuming that the M- and S-beams of the same RCA point at the same direction on the sky.

The polarized scanning beams were evaluated from optical simulations using the GRASP physical optics code, by appropriately tuning the Radio Frequency Flight Model (RFFM, Planck-PreLaunch-II[4]).

The Radio Frequency Tuned Model (RFTM) was implemented to fit the in-flight beam measurements with an electromagnetic model. The LFI main beams can be considered linearly polarized, but the non-null cross-polarization has an impact on the polarization measurements. Since we are not able to measure the cross-polar beam in flight, we have relied on simulations validated by accurate beam measurements.

The model beams were monochromatic and were computed at various frequencies across a 6-GHz band around the optical centre frequency (OCF) with non-uniform steps (denser sampling where the bandpass was higher). For the RFTM model the OCFs were set at 28.0, 44.0, and 70.0 GHz.

For each simulated beam we created a map of the Stokes polarization parameters. On those maps we performed a weighted in-band average to recover our best estimation of the polarized beam shape. The weighting function was the RIMO transmission function.

The delivered products includes the in-band averaged Stokes parameter scanning maps of main beams, intermediate beams, and sidelobes.

Effective beams[edit]

The 2018 effective beam estimation procedure didn't change with respect 2015 release, new summary values are perfectly in line with the once obtained in the 2015, see Planck-2020-A2[5]. 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. These beams are, by definition, the objects whose convolution with the true CMB sky produces the observed sky map.

The full algebra involving the effective beams for temperature and polarisation was presented in [6]. Here we summarise the main results. The observed temperature on the 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} \ast \mathbf{T}, \label{eq:a0} [/math]

where

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

Here t labels the time samples, Ati is 1 if the pointing direction falls in pixel number i, else it is 0, [math]\hat{\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 j, where the scanning beam [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 2.5×FWHM).

The algebra is a bit more involved for polarized detectors. The observed stokes parameters at 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^{\rm T} \ = \ \Delta\Omega \sum_j B_{ij} \ast (I \quad Q \quad U)_j^{\rm T}, \label{eq:a1} [/math]

where the polarized effective beam matrix

[math] B_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^{\rm T}_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^{\rm T}(\hat{\mathbf{r}}_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 polarization weight vectors, as defined in Mitra et al. (2010).[6]

The task is to compute [math]B_{ij}[/math] for temperature-only beams and the 3 × 3 matrices [math]\mathbf{B}_{ij}[/math] for each pixel i, at every neighbouring pixel j that falls within the cut-off radius around the the centre of the ith 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 the 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 particular period, provides the pixelized cut-outs of both the effective beam (EB) and the point spread function (PSF), at all pixel-centre positions of the HEALPix-formatted map.


Production process[edit]

The methodology for computing effective beams for a scanning CMB experiment like Planck was presented in Mitra et al. (2010).[6]

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

  • identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets, but 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 during the period of interest;
  • create a data object of all beams pointed at all Npix directions of pixels in the map at the same resolution for which this precomputation was executed (dimension Npix × 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 the implementation of which enabled us to compute the pixel-based effective beams using moderate computational resources. The algorithm uses different mathematical and computational techniques to bring down the computation cost to a practical level. That allowed several estimations of the effective beams for all Planck detectors for different scan and beam models as well as data sets of different length.


Pixel-ordered detector angles[edit]

The main challenge in computing the effective beams is to go through the trillion samples, which becomes severely limited by I/O. In the first stage, for a given data set, ordered lists of pointing angles are made for each pixel – the Pixel-Ordered Detector Angles (PODA). This is an one-time process for each dataset. We used computers with large memory and used careful 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 for parallel computing. 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 whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below we display the computation cost at NERSC for the nominal mission both in terms of CPU hours 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/cores on the Carver or Hopper NERSC supercomputers.
Channel 030 044 070
PODA/Detector computation time (CPU hours) 85 100 250
PODA/Detector computation time (human minutes) 7 10 20
Beam/Channel computation time (CPU hours) 900 2000 2300
Beam/Channel computation time (human hours) 0.5 0.8 1
Convolution computation time (CPU hours) 1 1.2 1.3
Convolution computation time (human seconds) 1 1 1
Effective beam size (GB) 173 123 28

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.

Inputs[edit]

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


Planck Focal Plane.


Focal plane database:[edit]

The Focal Plane DataBase (FPDB) contains information on each detector, e.g., the orientation of the polarization axis, different weight factors, etc. See the instrument RIMOs:

The scanning strategy:[edit]

The scanning strategy, specifically includes 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 3½ surveys.

Scanning beam:[edit]

The "scanning beam" was modelled for each detector using observations of planets. This was assumed to be constant over the whole mission, although FEBeCoP could be used for a few sets of scanning beams.

  • LFI: GRASP scanning beam - the scanning beams used are based on the Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response. A description of the GRASP beams file formar (.grd) can be found in the "Technical Note PL-LFI-PST-TN-044: LFI Beams Delivery: Format Specifications": File:PST-TN-044 1-0.pdf.

See the instrument RIMOs: The LFI RIMO.

Beam cutoff radii:[edit]

  • N times the geometric mean of the FWHM values of all detectors in a channel, where N=2.5 for all LFI frequency channels.

Map resolution for the derived beam data object:[edit]

  • Nside=1024 for all LFI frequency channels.

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. The 30GHz frequency channel images are used 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 colour bar;
  • Row 2 – linear scale FEBeCoP PSFs computed using input scanning beams, GRASP beams (GB) for LFI, B-spline (BS) Mars12 beams apodized for the HFI CMB channels, and the BS Mars12 beams for the HFI sub-mm channels (see section Inputs below);
  • Row 3 – log scale of Row 2, PSF iso-contours are shown as solid lines, elliptical Gaussian fit iso-contours are shown as broken lines.
30GHz.

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 sky is sampled (quite sparsely) in 768 directions, which were chosen as HEALpix Nside=8 pixel centres for LFI in order to uniformly sample the sky.

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

[math] 4 \pi \sum(B_{ij}) / {\rm max}(B_{ij}) [/math].

Histograms for LFI effective beam parameters.

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

Note that 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.

Ellipticity – 30GHz.

Statistics of the effective beams computed using FEBeCoP[edit]

We tabulate the simple statistics of FWHM, ellipticity (e), orientation (ψ), and beam solid angle, (Ω), for a sample of 768 directions on the sky for LFI data. Statistics presented in the table are derived from the histograms shown above:

  • the derived beam parameters are representative of the DPC Nside=1024 HEALPix maps (they include the pixel window function);
  • the reported FWHMeff values are derived from the beam solid angles, under a Gaussian approximation – these are best used for flux determination, while 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 beams, apodized for the CMB channels and oversampled.
Frequency Mean (FWHM) [arcmin] σ (FWHM) [arcmin] Mean (e) σ (e) Mean (ψ) [deg] σ (ψ) [deg] Mean (Ω) [arcmin2] σ (Ω) [arcmin2] FWHMeff [arcmin]
030 32.288 0.021 1.315 0.031 1.347 54.074 1190.111 0.705 32.41
044 26.997 0.583 1.190 0.030 2.088 53.905 831.611 35.041 27.09
070 13.218 0.031 1.223 0.037 2.154 54.412 200.803 0.991 13.31


Beam solid angles for the PCCS[edit]

Ωeff is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: 4πΣ(effectivebeam)/max(effectivebeam), i.e., as an integral over the full extent of the effective beam:

[math] 4 \pi \sum(B_{ij}) / {\rm max}(B_{ij}) [/math].

From Ωeff we estimate the FWHMeff value, under a Gaussian approximation and these are tabulated above. The quantity Ω(1)eff is the beam solid angle estimated up to a radius equal to 1×FWHMeff and Ω(2)eff is the value up to a radius equal to 2×FWHMeff. 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 Ωeff [arcmin2] Spatial variation [arcmin2] Ω(1)eff [arcmin2] Spatial variation-1 [arcmin2] Ω(2)eff [arcmin2] Spatial variation-2 [arcmin2]
30 1190.111 0.705 1117.876 1.968 1188.991 0.713
44 831.611 35.041 757.891 32.782 830.767 35.074
70 200.803 0.991 186.055 1.883 200.497 0.991


Monte Carlo simulations[edit]

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

  • generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data;
  • 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, Cout, and 1σ errors.

Since FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the sky signal (that might be CMB or a foreground, such as dust emission), 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.


Window functions[edit]

The "Transfer Function" or the "Beam Window Function" B relates the true angular power spectra C to the observed angular power spectra Ĉ. In the current release, we deliver both TT and EE window functions defined as

[math] B_\ell^{TT,EE}= \widetilde{C}_\ell^{TT,EE} / C_\ell^{TT,EE}. \label{eqn:wl1}[/math]

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 scanning beams, although, in principle, one may be able to connect them algebraically.

The window function computed for the 2018 release has been calculated via simulated timelines. This procedure allow us to include the sidelobes. Procedure and consitency with the once computed with the convolution of teh effective beam is discussed in to Planck-2015-A04[2].

Beam window functions are provided in the RIMO.

Sidelobes[edit]

There are no direct measurements of sidelobes for LFI. The sidelobe patterns for LFI were simulated using GRASP9 multi-reflector GTD. We used the RFTM electromagnetic model. Seven beams for each radiometer were computed in spherical polar cuts with a step of 0.5° in both θ and φ. The beams were computed in the same frames used for the main beams. The intermediate beam region (θ < 5°) was replaced with null values.

In the computation we considered:

  • the direct field from the feed;
  • the 1st-order contributions, Bd, Br, Pd, Pr, Sd, Sr, and Fr;
  • the 2nd-order contributions, SrPd, and SdPd.

Here B = baffle', P = primary reflector, S = secondary reflector, F = focal plane unit box, d = diffraction, and r = reflection. For example "Br" means that we considered in the calculation reflections on the telescope baffle system.

A refinement of the sidelobe model will be considered in a future release, taking into account more contributions, together with physical optics models.

Image of the LFI27-M sidelobes presented as an RGB picture, where the red channel is 27 GHz (f0), the green channel is 30 GHz (f3), and the blue channel is 33 GHz (f6). Because the combined map does not show any wide white region, the sidelobe pattern changes with frequency, as expected.

References[edit]

  1. The Planck-LFI flight model ortho-mode transducers, O. D'Arcangelo, A. Simonetto, L. Figini, E. Pagana, F. Villa, M. Pecora, P. Battaglia, M. Bersanelli, R. C. Butler, S. Garavaglia, P. Guzzi, N. Mandolesi, C. Sozzi, Journal of Instrumentation, 4, 2005-+, (2009).
  2. 2.02.12.22.3 Planck 2015 results. IV. LFI beams and window functions, Planck Collaboration, 2016, A&A, 594, A4.
  3. Planck 2013 results. IV. Low Frequency Instrument beams and window functions, Planck Collaboration, 2014, A&A, 571, A4.
  4. Planck pre-launch status: The optical system, J. A. Tauber, H. U. Nørgaard-Nielsen, P. A. R. Ade, et al. , A&A, 520, A2+, (2010).
  5. Planck 2018 results. II. Low Frequency Instrument data processing, Planck Collaboration, 2020, A&A, 641, A2.
  6. 6.06.16.2 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).

(Planck) Low Frequency Instrument

LFI Radiometer Chain Assembly

Line Of Sight

reduced IMO

Cosmic Microwave background

Full-Width-at-Half-Maximum

(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).

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.

Early Release Compact Source Catalog

(Planck) High Frequency Instrument

Data Processing Center