https://wiki.cosmos.esa.int/planck-legacy-archive/api.php?action=feedcontributions&user=Ajaffe&feedformat=atomPlanck Legacy Archive Wiki - User contributions [en-gb]2022-12-02T05:50:22ZUser contributionsMediaWiki 1.31.6https://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Beams&diff=10614Beams2015-01-29T13:45:16Z<p>Ajaffe: begin to add 2015 information.</p>
<hr />
<div>= The 2015 Data Release = <br />
<br />
Information regarding beams for the 2015 release of ''Planck'' data can be found in the HFI DPC Paper {{PlanckPapers|planck2015-A08}}. <br />
<br />
The following contains information from the 2013 release.<br />
<br />
= Information from 2013 =<br />
<br />
== Scanning Beams ==<br />
<br />
The scanning beams describe the instrument’s instantaneous beam profile. Due to the near constant spin rate of the spacecraft, time domain effects (including residual time response and lowpass filtering) are degenerate with the spatial response due to the optical system. The scanning beam reconstruction recovers both of these effects, aside from residual time domain effects on a longer time scale than can be captured with the extent of the scanning beam model.<br />
<br />
In {{PlanckPapers|planck2013-p03c}} we consider two models of the beam in order to better understand systematics in the reconstruction. Here we describe only the B-Spline beams which are used to compute the delivered effective beam (see next section).<br />
<br />
=== B-Spline Beam construction ===<br />
<br />
We use seasons 1 and 2 of the Mars observation to reconstruct the beam. The data are processed with the bigPlanets TOI processing (see [[Frequency_Maps#HFI_processing | here]] or Section 3.11 of {{PlanckPapers|planck2013-p03}} for more info). We use JPL Horizons ephemerides to determine the pointing of each detector relative to the planet. We subtract the astrophysical background in the time domain using a bicubic interpolation of the Planck maps.<br />
<br />
The time ordered data are used to fit a two dimensional B-Spline surface using a least square minimization and a smoothing criterion to minimize the effects of high spatial frequency variations. We therefore assume the scanning beam to be smooth. The smoothing criterion as well as the locations of the nodes used to compute the B-Spline basis functions are set using GRASP physical optics simulations as inputs which are the best assumptions on the spatial frequency content of the in-flight beams.<br />
<br />
The smoothing criterion is defined as follows:<br />
<br />
<math>\eta = \displaystyle{\sum_{i=1}^{g}\left(b^{k}(\lambda_{i+})-b^{k}(\lambda_{i-})\right)^2}<br />
\label{smoothcrit}</math><br />
<br />
<math>\begin{aligned}<br />
\eta &: \mbox{ Smoothing Criterion}\\<br />
b^k &: \mbox{ $k^{th}$ beam derivative evaluated on the nodes locations}\end{aligned}</math><br />
<br />
And the global inversion criterion :<br />
<br />
<math>\zeta = \eta + p\times \delta</math><br />
<br />
with <math>\delta</math> usual least square estimator and <math>p</math> coefficient giving the relative weight to <math>\delta</math> with respect to the smoothing criterion.<br />
<br />
<math>\delta = \displaystyle{\sum_{r=1}^{m}}\left(y_{r} - b(x_{r})\right)^2\label{estimator}</math><br />
<br />
<math>\begin{aligned}<br />
\delta &: \mbox{ usual least square criterion}\\<br />
r &: \mbox{ indice relative to the m data points, } r \in \{1, \ldots, m\}\\<br />
y_r &: \mbox{ planet data of sample r}\\<br />
x_r &: \mbox{ pointing of sample r}\\<br />
b &: \mbox{ reconstructed beam}\end{aligned}</math><br />
<br />
The B-Spline nodes are located on a regular spaced grid in the detector coordinate framset. At the edge of the reconstructed beam map area, 4 coincident nodes are added to avoid vanishing basis functions.<br />
<br />
Let <math>B_{i, k+1}</math>, <math>k</math> degree B-Spline build using nodes {<math>\lambda_{i}, ..., \lambda_{i+k+1}</math>} (''De Boor &amp; Cox'', 1972) :<br />
<br />
<math>B_{i,1}(x) = \left\{<br />
\begin{array}{l}<br />
1, \mbox{ si } x \in \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}\\<br />
0, \mbox{ si } x \notin \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}<br />
\end{array} \right.</math><br />
<br />
<math>B_{i, l+1}(x) = \displaystyle{\frac{x - \lambda_{i}}{\lambda_{i+l} - \lambda_{i}}} B_{i,l}(x) + \displaystyle{\frac{\lambda_{i+l+1}-x}{\lambda_{i+l+1}-\lambda_{i+1}}} B_{i+1, l}(x)</math><br />
<br />
<math>l=1, \ldots, k</math><br />
<br />
<br />
[[Image:FocalPlane_Map_BSScanningBeams_v53.png|thumb|500px|center|'''Figure 1:''' Focal plane plot of B-Spline scanning beams using in-flight pointing reconstruction. The contours are -3,-10,-20,-30 dB from the peak, and for PSB pairs the "a" bolometer is plotted in black and "b" in blue.]]<br />
<br />
=== Simulations and errors ===<br />
<br />
We estimate the reconstruction bias and noise in the measurements using an ensemble of simulated planet observations for each channel. Further details are discussed in {{PlanckPapers|planck2013-p03c}}. Kept fixed in each simulation are:<br />
<br />
* the input beam assumed: we use a supersampled version of the reconstructed B-Spline beam (or whatever comes out of the current ongoing tests!)<br />
* Astrophyical background is the same as that subtracted from the real data.<br />
* StarTracker pointing (using the ptcor6 [[Detector pointing|pointing model]]).<br />
<br />
The following are varied in each simulation:<br />
<br />
* detector noise realizations obtained by filtering randomly generated white noise with the measured noise PSDs<br />
* random pointing errors with 2" rms, and a spectrum that replicates the real errors.<br />
* simulated glitches and the deglitching procedure<br />
* Mars brightness temperature variability<br />
<br />
400 simulated timelines are generated for each bolometer and for each of the two seasons of Mars observations used in the beam reconstruction. The simulated timelines are made into beam maps, projecting onto the B-Spline basis in the same way as the real data.<br />
<br />
The beam maps are propagated to effective beam window functions using the quickbeam approach (see effective beams below) and used to evaluate the reconstruction bias and to construct error eigenmodes in the effective beam window function.<br />
<br />
<!-- <font color=red>Figure: random pointing error PSD Figures: error envelope plots (or should those go under effective beams?)</font> --><br />
<br />
=== Residuals ===<br />
<br />
There are two known beam effects that are not included in the main beam model and are estimated as a separate bias in flux and angular power spectrum measurement: 1. long tails due to errors in low frequency time response deconvolution, and 2. near sidelobes.<br />
<br />
We stack all five observations of Jupiter to estimate the long time scale residuals due to incomplete deconvolution of the long time scale response.<br />
<br />
<!-- <font color=red>Add some kind of mean tail plot</font><br />
<br />
Near sidelobes are also evaluated using stacked Jupiter (hopefully they will just be part of the v53bis B-Spline beams). The main features in the near sidelobes include a wide beam skirt, and dimpling lobes<br />
<font color=red>Add sidelobe plots and tables</font> --><br />
<br />
== Effect of Mirror Support Print-through ==<br />
<br />
The Planck reflectors suffer from print-through <br />
of the<br />
honeycomb structures that support the carbon-fibre face sheets. <br />
While the size of the deformation has been measured during tests below room temperature to be less than 20 microns, it is the strict periodicity of the deformation that contributes most to the additional beam-shape contribution. A simple grating equation has been shown to describe the angular positions of the resulting near side lobes very well:<br />
<br />
<br />
<math> \displaystyle{ \sin \theta _n = \frac{n \lambda}{Yd} } \label{DimplingLobes}</math><br />
<br />
<br />
* <math>\theta _n </math>: the angular position of the n'th order lobe from the central beam peak<br />
* <math>\lambda </math>: the wavelength of the radiation<br />
* <math>d </math>: the grating spacing of the periodicity<br />
* <math>Y </math>: factor that describes the position of the each reflector along the optical path<br />
* <math>Y=1.00 </math>: for the primary reflector <br />
* <math>Y=1.80 </math>: for the secondary reflector<br />
<br />
<br />
Three possible periodicities (19.6mm, 30mm, 52mm) in the honeycomb array dominate the Planck dimpling pattern for the 857 GHz detectors, though only those for the 52mm periodicity can be seen for the 545 GHz and 353 GHz detectors. For the highest frequency detectors, only the weaker lobes due to the 19.6mm and 30mm periodicities are seen outside the 40 arcminute beam window, but they contribute at most (0.05 ± 0.008)% to the integrated beam. <br />
<br />
A map of Jupiter has been created for each 857 GHz detector, using the first four surveys, with the background subtracted using the same sky area for the survey taken 12 months before in which the planet is not present. The background-subtracted maps for each survey were then stacked to make a single map for each of the four detectors and these were further stacked to create a single 857 GHz band map, using the standard detector weightings proportional to 1/(NET)<sup>2</sup>. All the maps have 96 pixels per side, covering ±1.4º, and are scaled to the RMS noise of the background. Central beam values range from 205,000 to 256,000, depending on the noise of the detector.<br />
<br />
For each of these five maps of Jupiter an elliptical gaussian main beam has been fitted, and the amount of saturation estimated by finding the peak increase that gives the best gaussian fit, typically 10--20%. A circular ruze envelope was also fitted together with a tilted gaussian component slightly offset from the beam in the cross-scan (optical X axis) direction, using data more than 0.13º from the beam, with the known positions of the dimpling lobes masked out. Once these strongest components were removed, a tilted elliptical gaussian was fitted for each visible dimpling lobe in up to 5 different sets of lobes, however the innermost lobe set (52mm periodicity) is generally obscured by the ruze component and for the outermost set (19.6mm periodicity) most lobes are too weak to be fitted for individual detectors. The contributing areas of the lobes were determined by three methods: from the fitted gaussians; from summation within boxed areas using offsets determined from surrounding boxed areas; from summation in the boxed areas using an offset determined from gaussian fitting. <br />
<br />
The uncertainties in the beam component areas given here include the spread in values from these different methods. Similarly, the lobe peaks in decibels are calculated using the raw, fitted and estimated desaturated beam peak values, and the uncertainties reflect the variations produced by the three methods. Averaged over the five map fittings, the dimpling lobes contribute (0.47 ± 0.13)% of the beam area, while the ruze envelope accounts for (9.2 ± 0.7)% of the beam with the main beam making up the remaining (90.3 ± 0.7)%. <br />
<br />
For lower frequencies the dimpling lobes are less visible and will not be discussed here. For example, the noise floor for the 545 GHz band map is about 47 dB below the observed Jupiter peak, so that no dimpling lobes are visible further out than the 52mm lobes at 0.61º. For 353 GHz the noise floor is at about 38 dB so that even the 52mm lobes are not observed.<br />
<br />
[[image:CompleteFitWithPNContoursV53bis.jpg|700px|thumb|center|'''Figure 2:''' Stacked Jupiter map for the 857 GHz detectors. Left: the data with main beam and ruze envelope subtracted. Right: the fitted elliptical gaussian dimpling lobes with GRASP contours plotted on top.]]<br />
<br />
<br />
This figure shows how the dimpling lobes seen for the 857GHz band Jupiter map correspond to the contours calculated by the physical optics GRASP package, produced by TICRA, Denmark. The GRASP simulation was performed for the 857-1 detector and assumed a uniform dimpling distortion of 10 microns. The biggest departure from the simple grating model, is that the lobe pattern is elongated in the cross-scan (vertical) direction---lobes that should lie on a line 30º from the vertical are consistently found around 25º and those on the 60º line cluster around 53º. This is due to the offset geometry of the mirror system, whereby the dimpling print-though is foreshortened along the optical x-axis due to the tilt of the mirrors, producing greater lobe spacing as seen by the incoming photons. The amplitudes of the fitted dimpling lobes vary significantly within each set whereas the GRASP model shows a constant amplitude for lobes within a set, except for the 30mm periodicity. Similar amplitudes are seen in lobes lying directly across the beam centre from each other, indicating that facesheet dimpling has not occurred uniformly in all directions as the GRASP model assumes.<br />
<br />
For the 30mm periodicity the vertical lobes are significantly weaker than those at the sides just as the GRASP model also shows. Generally, the fitted lobes are somewhat stronger than those seen by GRASP, indicating that the dimpling is indeed larger than 10 microns. While the dimpling lobes are measureable at the highest two frequencies, with a strong source (i.e. Jupiter), 90% of the dimpling lobe area at 857GHz, and 10% for all other frequencies, already appears in the beam functions described above and need no further correction. However, the presence of the uniform dimpling lobes provides a useful window by which to investigate the mirror geometry.<br />
<br />
== Effective Beams ==<br />
<br />
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. <br />
<br />
Several methods of effective beams determination have been developped and cross-validated. The main products are produced using FEBeCoP and details of the processing are given in the [[Effective Beams]] products page. See also the equivalent [[Beams_LFI | page discussing the LFI beams]]<br />
<br />
<br />
=== FEBeCoP ===<br />
-------------------<br />
<br />
The full algebra for this method for the calculation of effective beams was presented in {{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>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<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.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \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,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\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) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
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 {{BibCite|mitra2010}}.<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
<br />
=== FICSBell ===<br />
For more details, see {{PlanckPapers|planck2013-p03c}}.<br />
<br />
Since the HFI beams are not azimuthally symmetric, the scanning strategy has to be taken into account in the effective beam response modelling. This is done using the FICSBell method (Hivon et al, in preparation), which generalizes to polarization and to include other sources of systematics the approach used for TT <math>C(l)</math> estimation in WMAP-3yr <br />
{{BibCite|hinshaw2007}} and by Smith et al (2007) {{BibCite|smith2007}}<br />
in the detection of CMB lensing in WMAP maps. The different steps of the method used for this study can be summarized as follows:<br />
<br />
<ol><br />
<!-- 1 --><br />
<li><p>The scanning related information (i.e., statistics of the orientation of each detector within each pixel) is computed first, and only once for a given observation campaign:</p><br />
<p><math><br />
w^d_s({\bf r}_p) = \sum_j e^{i s \psi_j}, \label{eq:e1}<br />
</math></p><br />
<p>where <math>\psi_j</math> is the orientation of the detector with respect to the local<br />
meridian during the measurement $j$ occurring in <br />
the direction ${\bf r}_p$. Note that the $s=0$ moment is simply the<br />
hit count map. The orientation hit moments are computed up to<br />
degree $s=4$. At the same time,<br />
the first two moments of the distribution of samples within each<br />
pixel (i.e., the centre of mass and moments of inertia) are computed and stored on disc.<br />
<br />
<!-- 2 --><br />
<li>The scanning beam map or beam model of each detector <math>d</math> is analyzed into its Spherical Harmonics coefficients</p><br />
<p><math><br />
b^d_{ls} = \int d{\bf r} B_d({\bf r}) Y_{ls}({\bf r})\label{scanningBlm}<br />
</math></p><br />
<p>where <math>B_d(\bf{r})</math> is the beam map centered on the North pole, and <math>Y_{ls}(\bf{r})</math> is the Spherical Harmonics basis function. Higher <math>s</math> indexes describes higher degrees of departure from azimuthal symmetry and, for HFI beams, the coefficients <math>b^d_{ls}</math> are decreasing functions of <math>s</math> at most multipoles considered. <br />
It also appears that, for <math>l<3000</math>, the coefficients with <math>|s| > 4</math> account for much less of <math>1\%</math> of the beam solid angle. <br />
Spot checks where window functions are computed with<br />
$|s|\le 6$ show a difference of less than $10^{-4}$ for<br />
$\ell<2000$ at 100 GHz and for $\ell<3000$ at 143 and 217 GHz. For these reasons, only modes with $|s| \le 4$ are considered in the present analysis.<br />
Armitage-Caplan & Wandelt (2009){{BibCite|armitage-caplan2009}} reached a similar conclusion in their deconvolution of LFI beams.<br />
<br />
<!-- 3 --><br />
<li><br />
For a given CMB sky realization $t$, described by its spherical harmonics coefficients<br />
$a_{\ell m} = \int d{\bf r} t({\bf r}) Y_{\ell m}({\bf r})$, the $b^d_{\ell s}$ coefficients computed above are<br />
used to generate $s$-spin weighted maps, </p><br />
<p><math><br />
m^d_{s}({\bf r}) = \sum_{\ell m} b^d_{\ell s}\ a_{\ell m}\ {}_sY_{\ell m}({\bf r}), \label{eq:e3}<br />
</math></p><br />
<p>as well as the first and second derivatives, using standard HEALPix tools.<br />
<br />
<!-- 4 --><br />
<li>The spin weighted maps and orientation hit moments of the same order <math>s</math> are combined for all detectors involved, to provide an “observed” map</p><br />
<p><math><br />
m({\bf r}) = \left(\sum_d \sum_s w^d_s({\bf r}) m^d_s({\bf r})\right) / \sum_d w^d_0({\bf r}). \label{eq:e4}<br />
</math></p><br />
Similarly the local spatial derivatives are combined with the location hit moments to describe the effect of the non-ideal sampling of each pixel (see [[#Pixelization_Artifacts|Pixelization Artifacts section]]). In this combination, the respective number of hits of each detector in each pixel is considered, as well as the weighting (generally proportional to the inverse noise variance) applied to each detector in order to minimize the final noise.<br />
<br />
<!-- 5 --><br />
<li>The power spectrum of this map can then be computed, and compared to the input CMB power spectrum to estimate the effective beam window function over the whole sky, or over a given region of the sky.<br />
</li></ol><br />
Monte-Carlo (MC) simulations in which the sky realizations are changed can be performed by repeating steps 3, 4 and 5. The impact of beam model uncertainties can be studied by including step 2 into the MC simulations.<br />
<br />
=== QuickBeam ===<br />
For more details, see {{PlanckPapers|planck2013-p03c}}<br />
==== Formalism ====<br />
<br />
Planck observes the sky after convolution with a “scanning beam”, which captures its effective response to the sky as a function of displacement from the nominal pointing direction. Decomposing the scanning beam into harmonic coefficients <math>B_{lm}</math>, each time-ordered data (TOD) sample can be modeled as (neglecting the contribution from instrumental noise, which is independent of beam asymmetry) <br />
<math>%T_i = \sum_{lms} D^{l}_{-m s} (\phi_i, \theta_i, \alpha_i) b_{ls} (-1)^{m) T_{lm} + n_i,<br />
T_i = \sum_{lms} e^{-i s \alpha_i} B_{ls} \tilde{T}_{lm} {}_s Y_{lm}(\theta_i, \phi_i),<br />
\label{eqn:tod_beam}</math><br />
where the TOD samples are indexed by <math>i</math>, and <math>\tilde{T}_{lm}</math> is the underlying sky signal. The spin spherical harmonic <math>{}_s Y_{lm}</math> rotates the scanning beam to the pointing location <math>(\theta, \phi)</math>, while the <math>e^{-i s \alpha_i}</math> factor gives it the correct orientation. Eq. \ref{eqn:tod_beam} may be evaluated with the “TotalConvolver” algorithm of Wandelt and Gorski (2001){{BibCite|wandelt2001}}, accelerated using the “conviqt” recursion relations Prezeau and Reinecke (2010){{BibCite|prezeau2010}} This approach is implemented in LevelS, although because it involves working with a TOD-sized objected it is necessarily slow.<br />
<br />
On the small angular scales comparable to the size of the beam, it is a good approximation to assume that the procedure of mapmaking from TOD samples is essentially a process of binning: <br />
<math>T(p) = \sum_{i \in p} T_i / H(p),\label{eqn:map_beam_full}</math> <br />
where <math>H(p)</math> is the total number of hits in pixel <math>\hat{n}</math>.<br />
<br />
Starting with a normalized, rescaled harmonic transform of the beam <math>B_{lm}</math>, sky multipoles <math>\tilde{T}_{lm}</math> and a scan history object <math>w(\hat{n}, s)</math> given by <math>w(\hat{n}, s) = \sum_{j \in p} e^{i s \alpha_j} / H(\hat{n})</math> where the sum is over all hits <math>j</math> of pixel <math>p</math> at location <math>\hat{n}_p</math>, and <math>\alpha_j</math> is the scan angle for observation <math>j</math>. The harmonic transform of this scan-strategy object is given by<br />
<p><math><br />
{}_{s} w_{L M} = \int d^2 \hat{n} {}_s Y_{LM}^*(\hat{n}) w(\hat{n}, s).<br />
\label{eq:spin_hits}<br />
</math></p><p><br />
The beam-convolved observation is then given by <br />
</p><p><math><br />
\tilde{T}(\hat{n}) = \sum_{slm} w(\hat{n}, -s ) B_{ls} T_{lm} {}_s Y_{lm}(\hat{n}).<br />
\label{eq:tilde_T}<br />
</math></p><p><br />
<br />
Taking the ensemble average of the pseudo-Cl power spectrum of these <math>T_{lm}</math> we find<br />
<br />
</p><p><math><br />
\tilde{C}_{L}^{TT} = \sum_{S S'} \sum_{l_1 l_2} \frac{(2l_1+1)(2l_2+1)}{4\pi}<br />
{}_{(-s -s')}{\cal W}_{l_1} B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}<br />
\left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s\! & -s\! & 0\!<br />
\end{array}<br />
\right) \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s'\! & -s'\! & 0\!<br />
\end{array}<br />
\right)<br />
\label{eq:pseudo_Cl_matrix}<br />
</math></p><p><br />
<br />
where <math>{}_{(s s')}{\cal W}_{L} = \frac{1}{2L+1} \sum_{M} {}_{S} w_{LM} {}_{S'} w_{LM}^*</math> is a cross-power spectrum of scan history objects. Note that the $w(\hat{n}, s )$ which we have used here can also incorporate a position dependent weighting to optimize the pseudo-Cl estimate, such as inverse-noise or a mask– the equations are unchanged. Writing the pseudo-Cl in position space (a la Dvorkin and Smith, 2009{{BibCite|dvorkin2008}}) with Wigner-d matrices we have<br />
<br />
</p><p><math><br />
\tilde{C}_{L}^{TT} = \frac{1}{8\pi} \sum_{S S'} \int_{-1}^{1} dz \ d^{L}_{00}(z)<br />
\left[\sum_{l_1} d^{l_1}_{-s -s'}(z) {}_{(-s -s')}{\cal W}_{l_1} (2l_1+1) \right] <br />
\left[ \sum_{l_2} d^{l_2}_{s s'}(z) B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}(2l_2+1) \right].<br />
\label{eq:pseudo_Cl_corr}<br />
</math></p><p><br />
<br />
This integral can be implemented exactly using Gauss-Legendre quadrature, with a cost of $\cal 0(l_{\rm max}^2 s_{\rm max}^2)$. For simplicity, the equations here are written for the auto-spectrum of a single detector, but the generalization to a map made by adding several detectors with different weighting is straightforward. The cost to compute all of the necessary terms exactly in that case becomes <math>\cal 0(l_{\rm max}^2 s_{\rm max}^2 N_{\rm det}^2)</math>.<br />
<br />
On the flat-sky, beam convolution is multiplication in Fourier space<br />
by a beam rotated onto the scan direction. Multiple hits with<br />
different scan directions are incorporated by averaging (as the scan<br />
history objects above encapsulate).<br />
<br />
A scan strategy which is fairly smooth across the sky is nearly<br />
equivalent to observing many independent flat-sky patches at high<br />
$L$. There is a fairly good approximation to the beam convolved <br />
pseudo-power spectrum which is essentially a flat-sky approximation. In the<br />
limit that $L \gg \ell_1$, with $C_{\ell_2}$ and $B_{\ell_2}$ is a slowly-varying<br />
function in $\ell_2$, and using the equality<br />
<br />
\begin{equation}<br />
\sum_{\ell_2} (2 \ell_2+1) <br />
\left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s\! & -s\! & 0\!<br />
\end{array}<br />
\right)<br />
\left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s'\! & -s'\! & 0\!<br />
\end{array}<br />
\right)<br />
= \delta_{ss'},<br />
\end{equation}<br />
<br />
the pseudo-$C_\ell$ sum above can be approximated as<br />
<br />
<math><br />
{\tilde{C}}_L^{TT} = C_L^{TT} \sum_{M} \left< \left| w(\hat{n}_p, M) \right|^2 \right>_p |B_{L M}|^2,<br />
\label{eq:pseudo_Cl_approx}<br />
</math><br />
where the average $\langle \rangle_p$ is taken over the full sky. It is<br />
illustrative to consider two limits of this equation. Firstly, for a ``raster''<br />
scan strategy in which each pixel is observed with the same direction:<br />
<br />
\begin{equation}<br />
\left< \left| w(\hat{n}, M) \right|^2 \right>_p = 1,<br />
\end{equation}<br />
<br />
and the predicted transfer function is just the power spectrum of the<br />
beam. Secondly, for a ``best-case'' scan strategy, in which each pixel is<br />
observed many times with many different orientation angles, <br />
<br />
\begin{equation}<br />
\left< \left| w(\hat{n}, M) \right|^2 \right>_p = \delta_{M0},<br />
\end{equation}<br />
<br />
and the transfer function is the azimuthally symmetric part of<br />
the beam. Note that this is for a full-sky observation; in the<br />
presence of a mask, the average above produces an $f_{\rm sky}$ factor, as<br />
expected but neglects the coupling between $L$ multipoles (which<br />
can be calculated with the more complete equations above).<br />
<br />
==== Effective beam window functions ====<br />
The effective beam window functions $B(l)$ for HFI, computed using Quickbeam, are available in the [[The RIMO|RIMO]].<br />
They do not contain the pixel window function.<br />
<br />
=== Pixelization Artifacts ===<br />
For more details, see {{PlanckPapers|planck2013-p03c}}<br />
<br />
Planck-HFI maps are produced at HEALPix resolution 11 <math>(N_{\rm side} = 2048)</math>, corresponding to pixels with a typical dimension of <math>1.7'</math>. With the resolution comparable to the spacing between scanning rings there is an uneven distribution of hits within pixels, introducing a complication in the analysis and interpretation of the Planck maps. A sample of the Planck distribution of sample hits within pixels is illustrated in figure below.<br />
<br />
[[Image:pixcoverage.png|thumb|500px|center|Illustration of TOI samples near the Galactic plane (gray<br />
dots), over-plotted on a simulated CMB realization which has been<br />
convolved with a Gaussian 7' FWHM beam and pixelized at $(N_{\rm side} = 2048)$. Associated scanning rings (gray lines) as well as centers of mass for the hit distribution (black arrows) are also plotted.]]<br />
<br />
<br />
The collaboration has produced 3 codes which may be used to simulate the effect of pixelization on the observed sky, LevelS/TotalConvoler/Conviqt{{BibCite|wandelt2001}}{{BibCite|prezeau2010}}{{BibCite|reinecke2006}}, FeBeCoP{{BibCite|mitra2010}}, and [[#FICSBell|FICSBell]].<br />
<br />
For the measurement of CMB fluctuations, the effects of pixelization<br />
may be studied analytically. On the small scales relevant to<br />
pixelization, the observed CMB is smooth, both due to physical damping<br />
as well as the convolution of the instrumental beam. Taylor expanding<br />
the CMB temperature about a pixel center to second order, the typical<br />
gradient amplitude is given by<br />
<math>\langle |\nabla T |^2 \rangle = \frac{1}{4\pi} \sum_{l} l(l+1)(2l+1) C_l^{T} W_l \approx 1\times10^9 \mu K^2 / {\rm rad}^2.</math> <br />
where the approximate value is calculated for a <br />
<math>\Lambda</math> CDM<br />
cosmology with a <br />
<math>7'</math>FWHM Gaussian beam. The typical curvature of the observed temperature, on the other hand is given by <math>\langle |\nabla^2 T |^2 \rangle = \frac{1}{4\pi} \sum_{l} [l(l+1)]^2(2l+1) C_l^{T} W_l \approx 7\times10^{14} \mu K^2 / {\rm rad}^4.</math> On the scales relevant to the maximum displacement from the center of a <math>1.7'</math> pixel, the maximum displacement is <br />
<math>{\cal O}(1' = 3\times10^{-4} {\rm rad})</math>, and so the gradient term tends to dominate, although the curvature term is still non-negligible. For each observation of a pixel, we can denote the displacement from the pixel center as <math>d = d_{\theta} + i d_{\phi}</math>. The average over all hits within a pixel gives an overall deflection vector which we will denote for a pixel center located at <math>\hat{n}</math> as <math>d(\hat{n})</math>. This represents the center of mass of the hit distribution; [[#pixcoverage|Fig. 2]] shows these average deflections using black arrows. The deflection field <math>d(\hat{n})</math> may be decomposed into spin-1 spherical harmonics as <math>d_{lm} = \int_{4\pi} {}_1 Y_{lm}^* d(\hat{n}).</math> With a second order Taylor expansion of the CMB temperature about each pixel center, it is then possible to calculate the average pseudo-Cl power spectrum of the pixelized sky. This is given by<br />
<br />
<math><br />
C_l^{T} = [1-l(l+1)R^d] {C}_l^{T} W_l + <br />
\frac{1}{2} \sum_{l_1 l_2} \frac{l_1(l_1+1)(2l_1+1)(2l_2+1)}{4\pi} <br />
\left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & l\! \\<br />
\! l\! & -l\! & 0\!<br />
\end{array}<br />
\right)^2 C_{l_1}^{T} W_{l_1} \left[ C_{l_2}^{d+} + (-1)^{l + l_1 + l_2} C_{l_2}^{ d-} \right],<br />
\label{eqn:clt_pixelized}<br />
</math><br />
<br />
where <math>R^{d} = \langle |d|^2 \rangle/2</math> is half the mean-squared deflection magnitude (averaged over hits within a pixel, as well as over pixels). <math>C_l^{d+}</math> is the sum of the gradient and curl power spectra of <math>d_{lm}</math>, and <math>C_l^{d-}</math> is the gradient spectrum minus the curl spectrum. The <math>R^{d}</math> term describes a smearing of the observed sky due to pixelization. For uniform pixel coverage of <math>N_{\rm side}=2048</math> pixels <math>\sqrt{ \langle |d|^2 \rangle } = 0.725'</math> while, for the hit distribution of Planck frequency maps, <math>R^{d}</math> is typically within 0.2% of this value for CMB channels, and 0.4% for all channels. This term is therefore accurately described by the HEALPix pixel window function, which is derived under the assumption of uniform pixel coverage, and the resulting relative error on the beam window function is at most <math>4\times 10^{-4}</math> for <math>\ell \le 3000</math>.<br />
<br />
The effect of pixelization is essentially degenerate with that of gravitational lensing of the CMB, with the difference that it (1) acts on the beam-convolved sky, rather than the actual sky and (2) produces a curl-mode deflection field as well as a gradient mode. This is discussed further in the Planck gravitational lensing{{PlanckPapers|planck2013-p12}} paper, where the subpixel deflection field constitutes a potential source of bias for the measured lensing potential. Indeed, Eq. \ref{eqn:clt_pixelized} is just a slightly modified version of the usual first order CMB lensing power spectrum Hu (2000){{BibCite|Hu2000}}, Lewis and Challinor (2006){{BibCite|Lewis2006}} to accommodate curl modes.<br />
<br />
A useful approximation to Eq. \ref{eqn:clt_pixelized} which is derived in the unrealistic limit that the deflection vectors are uncorrelated between pixels, but in practice gives a good description of the power induced by the pixelization, is that the <math>d(\hat{n})</math> couples the CMB gradient into a source of noise with an effective level given by <math>\sigma^{N} \approx \sqrt{ R^T \frac{4\pi}{N_{\rm pix}} \langle | d(\hat{n}) |^2<br />
\rangle }, % (\muKarcmin ),</math><br />
<br />
where the average is taken over all pixels and <math>R^T</math> is half the mean-squared power in the CMB gradient: <math>R^{T} = \frac{1}{8\pi} \sum_{l} l(l+1)(2l+1) \tilde{C}_l^{T}.</math> For frequency-combined maps, <math>\sqrt{ \langle | d(\hat{n}) |^2 \rangle }</math> is typically on the order of <math>0.1'</math>, and so the induced noise is at the level of <math>\sigma^{N} \sim 2 \mu</math>K arcmin. This is small compared to the instrumental contribution, although it does not disappear when taking cross-spectra, depending on how coherent the hit distributions of the two maps in the cross-spectrum are.<br />
<br />
= References =<br />
<References /><br />
<br />
<br />
<br />
[[Category:HFI data processing|003]]</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Detector_pointing&diff=8029Detector pointing2013-10-24T15:51:21Z<p>Ajaffe: </p>
<hr />
<div>== Introduction and Summary ==<br />
<br />
The overall geometry of the Planck focal plane is shown here:<br />
<br />
[[Image:FocalPlane.png| thumb|500px|center| The Planck Focal Plane]]<br />
<br />
In order to take full advantage of the Planck beams, we must know the individual detector positions to a precision of better than several arcseconds, over the course of the entire mission. <br />
<br />
Spacecraft pointing comes from the on-board star tracker at 8 Hz between repointings (Attitude History File, AHF). This is translated via a series of three-dimensional rotations to a pointing for the centre of the focal plane and resampled to the HFI or LFI TOI data rate for convenience after correcting for the wobble angle (see below). We must then further rotate this focal-plane boresight pointing to the individual detector locations. Because neither the rotations from the star tracker to the boresight nor those from the boresight to the individual detectors are known exactly a priori, we must calibrate using flight data.<br />
<br />
Specifically, measurements of HFI and LFI Detector pointing are based largely on observations of the brighter planets, with information from the much more frequent observation of lower-flux galactic and extragalactic high-frequency sources used to monitor and build a model of overall pointing drift. This long-term drift of the spacecraft attitude is due to changes in the moment of inertia of the spacecraft, and includes specific events which may induce sudden changes, essentially random as far as our ability to predict the effects thereof. In this delivery HFI used a model (described below) to follow the pointing drift continuously, while LFI uses two different focal plane description for the two time periods separated by operations perfomed on the instrument that modify the termal behaviour. The two approach are consistent to better than a few arcsec. <br />
<br />
Note that for HFI the resulting pointing model cannot easily be directly compared to a physical/optical model: in particular, it includes a phase shift in the scan direction from the convolution and deconvolution of the detector transfer function, which is complex in the Fourier domain (see <cite>planck2013-p03c</cite> {{P2013|7}} ). This phase shift was not measured during normal operations, but a short campaign during which the spacecraft was spun at a higher rate will be used to determine these offsets in future date releases. Comparison with the initial optical model indicates that the in-scan change due to this phase shift is of the order of 1 arcminute. Note also that aberration is corrected in all observations.<br />
<br />
The final pointing model is measured to be better than 2 arcsecond rms in the co-scan and cross-scan directions averaged over ten-day periods, as shown below. Note that there are larger hourly drifts of up to 10 arcseconds due to interference from the radiometer electronics box assembly (REBA) as discussed more fully in <cite>planck2013-p03</cite> {{P2013|6}}).<br />
<br />
==Wobble Angle ==<br />
<br />
The wobble angle is the angle between the Principal Axis Reference Frame and the Body Reference Frame of Planck, both of which have their origin in the Planck Baricenter (ACMS, AHF-ICD). It is normally decomposed into its three components <math> \psi_1, \psi_2, \psi_3 </math>.<br />
<br />
Pointings are determined by a set of rotations converting coordinates in the STR (Star Tracker) reference frame to ECL (Ecliptic) reference frame, i.e. defining the rotation matrix <math> R_{ecl,str} </math>. The matrix can be decomposend in a sequence of matrix mutiplications:<br />
<br />
:<math> R_{ecl,str} = R_{ecl,A} R_{A,B} R_{B,str} </math><br />
<br />
here we used <math> R_{rfb,rfa} </math> to denote transformation from Rerference Frame RFA to Reference Frame RFB, and <math> R_{rfa,rfb} = R_{rfb,rfa}^{-1} </math>. <br />
<br />
The <math> R_{B,str} </math> converts from STR coordinates to Body Reference Frame coordinates, it is a constant matrix.<br />
<br />
:<math> \begin{bmatrix} \cos \beta & 0 & -\sin \beta \\ 0 & 1 & 0 \\ \sin \beta & 0 & \cos \beta \end{bmatrix} </math><br />
<br />
where <math> \beta = 85 \deg </math> is the STR boresight angle assumed to be constant and aligned with the telescope LOS, but this is not the case. The STR is located on the SVM, at about 1.5 m from the origin of the Body reference frame, a change in its position of 0.15 mm will result in a change of its orientation of about <math> 10^{-4} </math> radiants about 20 arcsec.<br />
<br />
There is no way to measure directly those changes. So the effect is that to have an apparent change in the <math> \psi_1 </math>, <math> \psi_2 </math> (tilt angles as defined in the AHF) and <math> \psi_3 </math> (azimuth angle as defined in the AHF) angles: the reason is apparent immediately when looking at the way a perturbation in STR reference frame orientation propagates.<br />
<br />
AHF provides wobble angle measures at 1 minute (<math> \psi_1 </math>, <math> \psi_2 </math>) and one OD (<math> \psi_3 </math>) rate. Indeed <math> \psi_3 </math> is provided at each pointing period but measures within each given OD are constant.<br />
<br />
Assuming to have quaternions represented by rotation matrix <math> R_{ecl,B}(t) </math> at a time <math> t </math>, and assuming to have representative values of true wobble angles <math> \psi_{1,0} </math>, <math> \psi_{2,0} </math>, <math> \psi_{3,0} </math> and a way to estimate the apparent <math> \delta\psi_1(t) </math>, <math> \delta\psi_2(t) </math>, <math> \delta\psi_3(t) </math> it is possible to remove the apparent effect.<br />
<br />
With the available information it can be done for <math> \psi_1 </math> and <math> \psi_2 </math>. <br />
<br />
The correction algorithm initializes two rotation matrices as references using <math> \psi_1 </math> and <math> \psi_2 </math> from the first pointing period of the nominal mission:<br />
<br />
:<math> R_{psi1} = \begin{bmatrix} \cos \psi_{1,ref} & \sin \psi_{1,ref} & 0 \\ -\sin \psi_{1,ref} & \cos \psi_{1,ref} & 0 \\ 0 & 0 & 1 \end{bmatrix} </math><br />
<br />
:<math> R_{psi2} = \begin{bmatrix} \cos \psi_{2,ref} & 0 & -\sin \psi_{2,ref} \\ 0 & 1 & 0 \\ \sin \psi_{2,ref} & 0 & \cos \psi_{2,ref} \end{bmatrix} </math><br />
<br />
Then, for each pointing period, builds two correction matrices using <math> \psi_1 </math> and <math> \psi_2 </math> as provided by the AHF in the Observation section:<br />
<br />
:<math> R_{psi1}^T = \begin{bmatrix} \cos \psi_{1} & -\sin \psi_{1} & 0 \\ \sin \psi_{1} & \cos \psi_{1} & 0 \\ 0 & 0 & 1 \end{bmatrix} </math><br />
<br />
:<math> R_{psi2}^T = \begin{bmatrix} \cos \psi_{2} & 0 & \sin \psi_{2} \\ 0 & 1 & 0 \\ -\sin \psi_{2} & 0 & \cos \psi_{2} \end{bmatrix} </math><br />
<br />
From these matrices the correction matrix is build:<br />
<br />
:<math> R = R_{psi1}^T R_{psi2}^T R_{psi2} R_{psi1} </math><br />
<br />
Each quaternion of the AHF is finally corrected using <math> R </math>.<br />
<br />
==Stellar Aberration==<br />
<br />
The corrected quaternions are interpolated using Spherical Linear Interpolation algorithm and transformed in cartesian vector, which we call <math> DPT </math>. For each sample the stellar aberration correction is applied:<br />
<br />
:<math> DPT = DPT - {v_{sat} \over c } </math><br />
<br />
where <math> v_{sat} </math> is the satellite velocity and <math> c </math> is the speed of light. After this operation the vector is normalized.<br />
<br />
Finally the cartesian vetor is converted in Ecliptic Coordinates, the detector pointing.<br />
<br />
==Beam Rotation==<br />
<br />
The rotation of the beam with respect to North is the <math> \psi </math> angle and is computed rotating the corrected quaternions <math> Q </math> using:<br />
<br />
:<math> R = R_{\theta} R_{\phi} Q ax2det </math><br />
<br />
The resulting rotation matrix represents the rotation of the beam, the <math> \psi </math> angle is then:<br />
<br />
:<math> \psi = -\arctan (R[0][1],R[0][0]) </math><br />
<br />
== Initial measurements: Mars 1 and other planets ==<br />
<br />
The first observation of Mars, which occured around 180 days from launch, is the baseline against which other objects are compared. Here, we show the relative pointing of Mars 1 to the pre-launch RFFM.<br />
<br />
[[Image:Mars1.png|thumb|500px|center| Individual detector pointing measured from the first Mars crossing, relative to the pre-launch RFFM model.]]<br />
<br />
== Focal plane drift: map-based measurement of detector positions ==<br />
<br />
Prior to modeling of systematic changes in the Planck pointing, we find secular drifts of order one arcminute over the course of the nominal mission. We monitor this by making point-source catalogs based on single Planck surveys, only counting those objects which are observed over the course of less than seven days (this limits us to observations away from the ecliptic poles where Planck's observing strategy is highly cross-linked). We cross-match these catalog positions to the known IRAS point-source catalog <cite>Lingyu2009</cite> and average the deviations in ten-day blocks (the individual measurements are shown as lighter points; error bars assume equal weighting for all points, but the results are not sensitive to any imposed S/N cutoff):<br />
<br />
[[Image:mapbased1.png | thumb|500px|center | Planck pointing before correction, measured by comparing point sources seen at 857 GHz to known IRAS positions. Light-colored points show individual source deviations, points with error bars give ten-day errors. Different colors correspond to the four individual Planck sky surveys. ]]<br />
<br />
An analysis performed directly in the timestream gives similar results, below (evaluated until the end of HFI operations in January 2012).<br />
<br />
[[Image:Pointing_offset_100--217GHz_CPP_better_coords.png | thumb|500px|center | The points give positions of individual objects (planets and bright radio sources) and the green dotted line labeled "PTCOR6" gives the corrected mean pointing, discussed below. (In this plot, "OD" refers to the number of days since launch.)]]<br />
<br />
== A model for Planck Pointing ==<br />
<br />
These analyses allow us to build a model for the pointing drift. The following describes the procedures used for the HFI detectors.<br />
<br />
The correction is based on the measurement of pointing offset directly from the timeline, on a small sample of known and bright radio sources and planets. First the beam shape of each detector is measured by stacking individual timeline observation of bright planets. This process is of course contaminated by pointing error, but this is reduced by allowing each pass on the planet to be slightly displaced in order to correct for the observed location of the planet. This process yields a very clean estimate of the beam of each detector.<br />
<br />
Those beams are then used to determine the offset of the observed location of the planets (Mars, Saturn, and Jupiter) and of a few (~10) bright radio sources. These offsets are estimated using detectors at all frequencies for the planets, and only at 100GHz, 143GHz and 217GHz for the radio sources. <br />
<br />
From the planet observations, one can recover the alignment of each detector in the focal plane, as well as a first pointing correction measured at the times of individual planet crossings. This pointing correction is obtained by fitting splines on the planet observation from each detector. We further allow this correction to have a sharp jump between the second and third survey (the time of the [[LFI_design,_qualification,_and_performance#The_Sorption_Cooler|SCS switchover on OD 455]]).<br />
<br />
This first pointing correction is further refined (at the lower HFI frequencies only) using the point sources observations in order to fill the period between planet observations. The radio source offsets require further treatment than the planet based offsets. First, the brightness of the sources being much smaller than planets, the offset on individual detector observation are very noisy, and we filter out the noise by building median offsets for each point source observation. Second, since the sources can be slightly extended we still observe systematic offset above and below the planet based pointing correction in a regular alternating pattern. This is due to the fact that we are observing those extended objects scanning in alternating directions, and our fit of those extended sources to our beam translate into this systematic alternating pattern. We thus allow for a small, arcsec correction of those offsets: i.e., each object is allowed a single offset displacement of order arc seconds in order to minimize the offset between different observations of the same object at different time. <br />
<br />
The resulting list of point source offsets is then merged with the planet offsets. Again a spline based correction is built fitting all of this data, and allowing rupture of continuity at the time of the SCS switchover and other major onboard event (typically between each surveys).<br />
<br />
Once we assume the full Planck pointing model and re-measure the position of Mars for this observation, we see sub-arcsecond deviations (as expected); this gives an indication of the purely numerical limitations of the method. Further planet observations show the small remaining uncorrected drift present in the pointing model (note that cross-scan positions are measured with considerably less data than in-scan positions due to the scan strategy):<br />
<br />
[[Image:S1vM1_v53.png | thumb|500px|center| Saturn 1 vs Mars 1 ]] [[Image:S2vM1_v53.png| thumb|500px|center| Saturn 2 vs Mars 1]]<br />
<br />
Redoing the single-survey map- and catalog-based computation with the corrected pointing shows that Planck has achieved the arcsecond-scale rms uncertainty of pointing:<br />
<br />
[[Image:mapbased_ptcor6.png | thumb|500px|center| As above, after correction by the "ptcor6" model. This plot shows both 545 GHz and 857 GHz sources.]]<br />
<br />
== References ==<br />
<biblio force=false><br />
#[[References]] <br />
</biblio><br />
<br />
<br />
[[Category:HFI/LFI joint data processing|0000]]</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Detector_pointing&diff=8028Detector pointing2013-10-24T15:34:21Z<p>Ajaffe: Added reference to SCS switchover event</p>
<hr />
<div>== Introduction and Summary ==<br />
<br />
The overall geometry of the Planck focal plane is shown here:<br />
<br />
[[Image:FocalPlane.png| thumb|500px|center| The Planck Focal Plane]]<br />
<br />
In order to take full advantage of the Planck beams, we must know the individual detector positions to a precision of better than several arcseconds, over the course of the entire mission. <br />
<br />
Spacecraft pointing comes from the on-board star tracker at 8 Hz between repointings (Attitude History File, AHF). This is translated via a series of three-dimensional rotations to a pointing for the centre of the focal plane and resampled to the HFI or LFI TOI data rate for convenience after correcting for the wobble angle (see below). We must then further rotate this focal-plane boresight pointing to the individual detector locations. Because neither the rotations from the star tracker to the boresight nor those from the boresight to the individual detectors are known exactly a priori, we must calibrate using flight data.<br />
<br />
Specifically, measurements of HFI and LFI Detector pointing are based largely on observations of the brighter planets, with information from the much more frequent observation of lower-flux galactic and extragalactic high-frequency sources used to monitor and build a model of overall pointing drift. This long-term drift of the spacecraft attitude is due to changes in the moment of inertia of the spacecraft, and includes specific events which may induce sudden changes, essentially random as far as our ability to predict the effects thereof. In this delivery HFI used a model (described below) to follow the pointing drift continuously, while LFI uses two different focal plane description for the two time periods separated by operations perfomed on the instrument that modify the termal behaviour. The two approach are consistent to better than a few arcsec. <br />
<br />
Note that for HFI the resulting pointing model cannot easily be directly compared to a physical/optical model: in particular, it includes a phase shift in the scan direction from the convolution and deconvolution of the detector transfer function, which is complex in the Fourier domain (see <cite>planck2013-p03c</cite> {{P2013|7}} ). This phase shift was not measured during normal operations, but a short campaign during which the spacecraft was spun at a higher rate will be used to determine these offsets in future date releases. Comparison with the initial optical model indicates that the in-scan change due to this phase shift is of the order of 1 arcminute. Note also that aberration is corrected in all observations.<br />
<br />
The final pointing model is measured to be better than 2 arcsecond rms in the co-scan and cross-scan directions averaged over ten-day periods, as shown below. Note that there are larger hourly drifts of up to 10 arcseconds due to interference from the radiometer electronics box assembly (REBA) as discussed more fully in <cite>planck2013-p03</cite> {{P2013|6}}).<br />
<br />
==Wobble Angle ==<br />
<br />
The wobble angle is the angle between the Principal Axis Reference Frame and the Body Reference Frame of Planck, both of which have their origin in the Planck Baricenter (ACMS, AHF-ICD). It is normally decomposed into its three components <math> \psi_1, \psi_2, \psi_3 </math>.<br />
<br />
Pointings are determined by a set of rotations converting coordinates in the STR (Star Tracker) reference frame to ECL (Ecliptic) reference frame, i.e. defining the rotation matrix <math> R_{ecl,str} </math>. The matrix can be decomposend in a sequence of matrix mutiplications:<br />
<br />
:<math> R_{ecl,str} = R_{ecl,A} R_{A,B} R_{B,str} </math><br />
<br />
here we used <math> R_{rfb,rfa} </math> to denote transformation from Rerference Frame RFA to Reference Frame RFB, and <math> R_{rfa,rfb} = R_{rfb,rfa}^{-1} </math>. <br />
<br />
The <math> R_{B,str} </math> converts from STR coordinates to Body Reference Frame coordinates, it is a constant matrix.<br />
<br />
:<math> \begin{bmatrix} \cos \beta & 0 & -\sin \beta \\ 0 & 1 & 0 \\ \sin \beta & 0 & \cos \beta \end{bmatrix} </math><br />
<br />
where <math> \beta = 85 \deg </math> is the STR boresight angle assumed to be constant and aligned with the telescope LOS, but this is not the case. The STR is located on the SVM, at about 1.5 m from the origin of the Body reference frame, a change in its position of 0.15 mm will result in a change of its orientation of about <math> 10^{-4} </math> radiants about 20 arcsec.<br />
<br />
There is no way to measure directly those changes. So the effect is that to have an apparent change in the <math> \psi_1 </math>, <math> \psi_2 </math> (tilt angles as defined in the AHF) and <math> \psi_3 </math> (azimuth angle as defined in the AHF) angles: the reason is apparent immediately when looking at the way a perturbation in STR reference frame orientation propagates.<br />
<br />
AHF provides wobble angle measures at 1 minute (<math> \psi_1 </math>, <math> \psi_2 </math>) and one OD (<math> \psi_3 </math>) rate. Indeed <math> \psi_3 </math> is provided at each pointing period but measures within each given OD are constant.<br />
<br />
Assuming to have quaternions represented by rotation matrix <math> R_{ecl,B}(t) </math> at a time <math> t </math>, and assuming to have representative values of true wobble angles <math> \psi_{1,0} </math>, <math> \psi_{2,0} </math>, <math> \psi_{3,0} </math> and a way to estimate the apparent <math> \delta\psi_1(t) </math>, <math> \delta\psi_2(t) </math>, <math> \delta\psi_3(t) </math> it is possible to remove the apparent effect.<br />
<br />
With the available information it can be done for <math> \psi_1 </math> and <math> \psi_2 </math>. <br />
<br />
The correction algorithm initializes two rotation matrices as references using <math> \psi_1 </math> and <math> \psi_2 </math> from the first pointing period of the nominal mission:<br />
<br />
:<math> R_{psi1} = \begin{bmatrix} \cos \psi_{1,ref} & \sin \psi_{1,ref} & 0 \\ -\sin \psi_{1,ref} & \cos \psi_{1,ref} & 0 \\ 0 & 0 & 1 \end{bmatrix} </math><br />
<br />
:<math> R_{psi2} = \begin{bmatrix} \cos \psi_{2,ref} & 0 & -\sin \psi_{2,ref} \\ 0 & 1 & 0 \\ \sin \psi_{2,ref} & 0 & \cos \psi_{2,ref} \end{bmatrix} </math><br />
<br />
Then, for each pointing period, builds two correction matrices using <math> \psi_1 </math> and <math> \psi_2 </math> as provided by the AHF in the Observation section:<br />
<br />
:<math> R_{psi1}^T = \begin{bmatrix} \cos \psi_{1} & -\sin \psi_{1} & 0 \\ \sin \psi_{1} & \cos \psi_{1} & 0 \\ 0 & 0 & 1 \end{bmatrix} </math><br />
<br />
:<math> R_{psi2}^T = \begin{bmatrix} \cos \psi_{2} & 0 & \sin \psi_{2} \\ 0 & 1 & 0 \\ -\sin \psi_{2} & 0 & \cos \psi_{2} \end{bmatrix} </math><br />
<br />
From these matrices the correction matrix is build:<br />
<br />
:<math> R = R_{psi1}^T R_{psi2}^T R_{psi2} R_{psi1} </math><br />
<br />
Each quaternion of the AHF is finally corrected using <math> R </math>.<br />
<br />
==Stellar Aberration==<br />
<br />
The corrected quaternions are interpolated using Spherical Linear Interpolation algorithm and transformed in cartesian vector, which we call <math> DPT </math>. For each sample the stellar aberration correction is applied:<br />
<br />
:<math> DPT = DPT - {v_{sat} \over c } </math><br />
<br />
where <math> v_{sat} </math> is the satellite velocity and <math> c </math> is the speed of light. After this operation the vector is normalized.<br />
<br />
Finally the cartesian vetor is converted in Ecliptic Coordinates, the detector pointing.<br />
<br />
==Beam Rotation==<br />
<br />
The rotation of the beam with respect to North is the <math> \psi </math> angle and is computed rotating the corrected quaternions <math> Q </math> using:<br />
<br />
:<math> R = R_{\theta} R_{\phi} Q ax2det </math><br />
<br />
The resulting rotation matrix represents the rotation of the beam, the <math> \psi </math> angle is then:<br />
<br />
:<math> \psi = -\arctan (R[0][1],R[0][0]) </math><br />
<br />
== Initial measurements: Mars 1 and other planets ==<br />
<br />
The first observation of Mars, which occured around 180 days from launch, is the baseline against which other objects are compared. Here, we show the relative pointing of Mars 1 to the pre-launch RFFM.<br />
<br />
[[Image:Mars1.png|thumb|500px|center| Individual detector pointing measured from the first Mars crossing, relative to the pre-launch RFFM model.]]<br />
<br />
== Focal plane drift: map-based measurement of detector positions ==<br />
<br />
Prior to modeling of systematic changes in the Planck pointing, we find secular drifts of order one arcminute over the course of the nominal mission. We monitor this by making point-source catalogs based on single Planck surveys, only counting those objects which are observed over the course of less than seven days (this limits us to observations away from the ecliptic poles where Planck's observing strategy is highly cross-linked). We cross-match these catalog positions to the known IRAS point-source catalog <cite>Lingyu2009</cite> and average the deviations in ten-day blocks (the individual measurements are shown as lighter points; error bars assume equal weighting for all points, but the results are not sensitive to any imposed S/N cutoff):<br />
<br />
[[Image:mapbased1.png | thumb|500px|center | Planck pointing before correction, measured by comparing point sources seen at 857 GHz to known IRAS positions. Light-colored points show individual source deviations, points with error bars give ten-day errors. Different colors correspond to the four individual Planck sky surveys. ]]<br />
<br />
An analysis performed directly in the timestream gives similar results, below (evaluated until the end of HFI operations in January 2012).<br />
<br />
[[Image:Pointing_offset_100--217GHz_CPP_better_coords.png | thumb|500px|center | The points give positions of individual objects (planets and bright radio sources) and the green dotted line labeled "PTCOR6" gives the corrected mean pointing, discussed below. (In this plot, "OD" refers to the number of days since launch.)]]<br />
<br />
== A model for Planck Pointing ==<br />
<br />
These analyses allow us to build a model for the pointing drift. The following describes the procedures used for the HFI detectors.<br />
<br />
The correction is based on the measurement of pointing offset directly from the timeline, on a small sample of known and bright radio sources and planets. First the beam shape of each detector is measured by stacking individual timeline observation of bright planets. This process is of course contaminated by pointing error, but this is reduced by allowing each pass on the planet to be slightly displaced in order to correct for the observed location of the planet. This process yields a very clean estimate of the beam of each detector.<br />
<br />
Those beams are then used to determine the offset of the observed location of the planets (Mars, Saturn, and Jupiter) and of a few (~10) bright radio sources. This offset are estimated using detectors at all frequencies for the planets, and only at 100GHz, 143GHz and 217GHz for the radio sources. <br />
<br />
From the planet observations, one can recover the alignment of each detector in the focal plane, as well as a first pointing correction measured at the specific time of planet observations. This pointing correction is obtained by fitting splines on the planet observation from each detector. We further allow this correction to have a sharp jump between the second and third survey (the time of the [[LFI_design,_qualification,_and_performance#The_Sorption_Cooler|SCS switchover on OD 455]]).<br />
<br />
This first pointing correction is further refined (at the lower HFI frequencies only) using the point sources observations in order to fill the period between planet observations. The radio source offsets require further treatment than the planet based offsets. First, the brightness of the sources being much smaller than planets, the offset on individual detector observation are very noisy, and we filter out the noise by building median offsets for each point source observation. Second, since the sources can be slightly extended we still observe systematic offset above and below the planet based pointing correction in a regular alternating pattern. This is due to the fact that we are observing those extended objects scanning in alternating directions, and our fit of those extended sources to our beam translate into this systematic alternating pattern. We thus allow for a small, arcsec correction of those offsets: i.e., each object is allowed a single offset displacement of order arc seconds in order to minimize the offset between different observations of the same object at different time. <br />
<br />
The resulting list of point source offsets is then merged with the planet offsets. Again a spline based correction is built fitting all of this data, and allowing rupture of continuity at the time of the SCS switchover and other major onboard event (typically between each surveys).<br />
<br />
Once we assume the full Planck pointing model and re-measure the position of Mars for this observation, we see sub-arcsecond deviations (as expected); this gives an indication of the purely numerical limitations of the method. Further planet observations show the small remaining uncorrected drift present in the pointing model (note that cross-scan positions are measured with considerably less data than in-scan positions due to the scan strategy):<br />
<br />
[[Image:S1vM1_v53.png | thumb|500px|center| Saturn 1 vs Mars 1 ]] [[Image:S2vM1_v53.png| thumb|500px|center| Saturn 2 vs Mars 1]]<br />
<br />
Redoing the single-survey map- and catalog-based computation with the corrected pointing shows that Planck has achieved the arcsecond-scale rms uncertainty of pointing:<br />
<br />
[[Image:mapbased_ptcor6.png | thumb|500px|center| As above, after correction by the "ptcor6" model. This plot shows both 545 GHz and 857 GHz sources.]]<br />
<br />
== References ==<br />
<biblio force=false><br />
#[[References]] <br />
</biblio><br />
<br />
<br />
[[Category:HFI/LFI joint data processing|0000]]</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Detector_pointing&diff=8027Detector pointing2013-10-24T14:57:22Z<p>Ajaffe: Clarify section 7 -- TBC</p>
<hr />
<div>== Introduction and Summary ==<br />
<br />
The overall geometry of the Planck focal plane is shown here:<br />
<br />
[[Image:FocalPlane.png| thumb|500px|center| The Planck Focal Plane]]<br />
<br />
In order to take full advantage of the Planck beams, we must know the individual detector positions to a precision of better than several arcseconds, over the course of the entire mission. <br />
<br />
Spacecraft pointing comes from the on-board star tracker at 8 Hz between repointings (Attitude History File, AHF). This is translated via a series of three-dimensional rotations to a pointing for the centre of the focal plane and resampled to the HFI or LFI TOI data rate for convenience after correcting for the wobble angle (see below). We must then further rotate this focal-plane boresight pointing to the individual detector locations. Because neither the rotations from the star tracker to the boresight nor those from the boresight to the individual detectors are known exactly a priori, we must calibrate using flight data.<br />
<br />
Specifically, measurements of HFI and LFI Detector pointing are based largely on observations of the brighter planets, with information from the much more frequent observation of lower-flux galactic and extragalactic high-frequency sources used to monitor and build a model of overall pointing drift. This long-term drift of the spacecraft attitude is due to changes in the moment of inertia of the spacecraft, and includes specific events which may induce sudden changes, essentially random as far as our ability to predict the effects thereof. In this delivery HFI used a model (described below) to follow the pointing drift continuously, while LFI uses two different focal plane description for the two time periods separated by operations perfomed on the instrument that modify the termal behaviour. The two approach are consistent to better than a few arcsec. <br />
<br />
Note that for HFI the resulting pointing model cannot easily be directly compared to a physical/optical model: in particular, it includes a phase shift in the scan direction from the convolution and deconvolution of the detector transfer function, which is complex in the Fourier domain (see <cite>planck2013-p03c</cite> {{P2013|7}} ). This phase shift was not measured during normal operations, but a short campaign during which the spacecraft was spun at a higher rate will be used to determine these offsets in future date releases. Comparison with the initial optical model indicates that the in-scan change due to this phase shift is of the order of 1 arcminute. Note also that aberration is corrected in all observations.<br />
<br />
The final pointing model is measured to be better than 2 arcsecond rms in the co-scan and cross-scan directions averaged over ten-day periods, as shown below. Note that there are larger hourly drifts of up to 10 arcseconds due to interference from the radiometer electronics box assembly (REBA) as discussed more fully in <cite>planck2013-p03</cite> {{P2013|6}}).<br />
<br />
==Wobble Angle ==<br />
<br />
The wobble angle is the angle between the Principal Axis Reference Frame and the Body Reference Frame of Planck, both of which have their origin in the Planck Baricenter (ACMS, AHF-ICD). It is normally decomposed into its three components <math> \psi_1, \psi_2, \psi_3 </math>.<br />
<br />
Pointings are determined by a set of rotations converting coordinates in the STR (Star Tracker) reference frame to ECL (Ecliptic) reference frame, i.e. defining the rotation matrix <math> R_{ecl,str} </math>. The matrix can be decomposend in a sequence of matrix mutiplications:<br />
<br />
:<math> R_{ecl,str} = R_{ecl,A} R_{A,B} R_{B,str} </math><br />
<br />
here we used <math> R_{rfb,rfa} </math> to denote transformation from Rerference Frame RFA to Reference Frame RFB, and <math> R_{rfa,rfb} = R_{rfb,rfa}^{-1} </math>. <br />
<br />
The <math> R_{B,str} </math> converts from STR coordinates to Body Reference Frame coordinates, it is a constant matrix.<br />
<br />
:<math> \begin{bmatrix} \cos \beta & 0 & -\sin \beta \\ 0 & 1 & 0 \\ \sin \beta & 0 & \cos \beta \end{bmatrix} </math><br />
<br />
where <math> \beta = 85 \deg </math> is the STR boresight angle assumed to be constant and aligned with the telescope LOS, but this is not the case. The STR is located on the SVM, at about 1.5 m from the origin of the Body reference frame, a change in its position of 0.15 mm will result in a change of its orientation of about <math> 10^{-4} </math> radiants about 20 arcsec.<br />
<br />
There is no way to measure directly those changes. So the effect is that to have an apparent change in the <math> \psi_1 </math>, <math> \psi_2 </math> (tilt angles as defined in the AHF) and <math> \psi_3 </math> (azimuth angle as defined in the AHF) angles: the reason is apparent immediately when looking at the way a perturbation in STR reference frame orientation propagates.<br />
<br />
AHF provides wobble angle measures at 1 minute (<math> \psi_1 </math>, <math> \psi_2 </math>) and one OD (<math> \psi_3 </math>) rate. Indeed <math> \psi_3 </math> is provided at each pointing period but measures within each given OD are constant.<br />
<br />
Assuming to have quaternions represented by rotation matrix <math> R_{ecl,B}(t) </math> at a time <math> t </math>, and assuming to have representative values of true wobble angles <math> \psi_{1,0} </math>, <math> \psi_{2,0} </math>, <math> \psi_{3,0} </math> and a way to estimate the apparent <math> \delta\psi_1(t) </math>, <math> \delta\psi_2(t) </math>, <math> \delta\psi_3(t) </math> it is possible to remove the apparent effect.<br />
<br />
With the available information it can be done for <math> \psi_1 </math> and <math> \psi_2 </math>. <br />
<br />
The correction algorithm initializes two rotation matrices as references using <math> \psi_1 </math> and <math> \psi_2 </math> from the first pointing period of the nominal mission:<br />
<br />
:<math> R_{psi1} = \begin{bmatrix} \cos \psi_{1,ref} & \sin \psi_{1,ref} & 0 \\ -\sin \psi_{1,ref} & \cos \psi_{1,ref} & 0 \\ 0 & 0 & 1 \end{bmatrix} </math><br />
<br />
:<math> R_{psi2} = \begin{bmatrix} \cos \psi_{2,ref} & 0 & -\sin \psi_{2,ref} \\ 0 & 1 & 0 \\ \sin \psi_{2,ref} & 0 & \cos \psi_{2,ref} \end{bmatrix} </math><br />
<br />
Then, for each pointing period, builds two correction matrices using <math> \psi_1 </math> and <math> \psi_2 </math> as provided by the AHF in the Observation section:<br />
<br />
:<math> R_{psi1}^T = \begin{bmatrix} \cos \psi_{1} & -\sin \psi_{1} & 0 \\ \sin \psi_{1} & \cos \psi_{1} & 0 \\ 0 & 0 & 1 \end{bmatrix} </math><br />
<br />
:<math> R_{psi2}^T = \begin{bmatrix} \cos \psi_{2} & 0 & \sin \psi_{2} \\ 0 & 1 & 0 \\ -\sin \psi_{2} & 0 & \cos \psi_{2} \end{bmatrix} </math><br />
<br />
From these matrices the correction matrix is build:<br />
<br />
:<math> R = R_{psi1}^T R_{psi2}^T R_{psi2} R_{psi1} </math><br />
<br />
Each quaternion of the AHF is finally corrected using <math> R </math>.<br />
<br />
==Stellar Aberration==<br />
<br />
The corrected quaternions are interpolated using Spherical Linear Interpolation algorithm and transformed in cartesian vector, which we call <math> DPT </math>. For each sample the stellar aberration correction is applied:<br />
<br />
:<math> DPT = DPT - {v_{sat} \over c } </math><br />
<br />
where <math> v_{sat} </math> is the satellite velocity and <math> c </math> is the speed of light. After this operation the vector is normalized.<br />
<br />
Finally the cartesian vetor is converted in Ecliptic Coordinates, the detector pointing.<br />
<br />
==Beam Rotation==<br />
<br />
The rotation of the beam with respect to North is the <math> \psi </math> angle and is computed rotating the corrected quaternions <math> Q </math> using:<br />
<br />
:<math> R = R_{\theta} R_{\phi} Q ax2det </math><br />
<br />
The resulting rotation matrix represents the rotation of the beam, the <math> \psi </math> angle is then:<br />
<br />
:<math> \psi = -\arctan (R[0][1],R[0][0]) </math><br />
<br />
== Initial measurements: Mars 1 and other planets ==<br />
<br />
The first observation of Mars, which occured around 180 days from launch, is the baseline against which other objects are compared. Here, we show the relative pointing of Mars 1 to the pre-launch RFFM.<br />
<br />
[[Image:Mars1.png|thumb|500px|center| Individual detector pointing measured from the first Mars crossing, relative to the pre-launch RFFM model.]]<br />
<br />
== Focal plane drift: map-based measurement of detector positions ==<br />
<br />
Prior to modeling of systematic changes in the Planck pointing, we find secular drifts of order one arcminute over the course of the nominal mission. We monitor this by making point-source catalogs based on single Planck surveys, only counting those objects which are observed over the course of less than seven days (this limits us to observations away from the ecliptic poles where Planck's observing strategy is highly cross-linked). We cross-match these catalog positions to the known IRAS point-source catalog <cite>Lingyu2009</cite> and average the deviations in ten-day blocks (the individual measurements are shown as lighter points; error bars assume equal weighting for all points, but the results are not sensitive to any imposed S/N cutoff):<br />
<br />
[[Image:mapbased1.png | thumb|500px|center | Planck pointing before correction, measured by comparing point sources seen at 857 GHz to known IRAS positions. Light-colored points show individual source deviations, points with error bars give ten-day errors. Different colors correspond to the four individual Planck sky surveys. ]]<br />
<br />
An analysis performed directly in the timestream gives similar results, below (evaluated until the end of HFI operations in January 2012).<br />
<br />
[[Image:Pointing_offset_100--217GHz_CPP_better_coords.png | thumb|500px|center | The points give positions of individual objects (planets and bright radio sources) and the green dotted line labeled "PTCOR6" gives the corrected mean pointing, discussed below. (In this plot, "OD" refers to the number of days since launch.)]]<br />
<br />
== A model for Planck Pointing ==<br />
<br />
These analyses allow us to build a model for the pointing drift. The following describes the procedures used for the HFI detectors.<br />
<br />
The correction is based on the measurement of pointing offset directly from the timeline, on a small sample of known and bright radio sources and planets. First the beam shape of each detector is measured by stacking individual timeline observation of bright planets. This process is of course contaminated by pointing error, but this is reduced by allowing each pass on the planet to be slightly displaced in order to correct for the observed location of the planet. This process yields a very clean estimate of the beam of each detector.<br />
<br />
Those beams are then used to determine the offset of the observed location of the planets (Mars, Saturn, and Jupiter) and of a few (~10) bright radio sources. This offset are estimated using detectors at all frequencies for the planets, and only at 100GHz, 143GHz and 217GHz for the radio sources. <br />
<br />
From the planet observations, one can recover the alignment of each detector in the focal plane, as well as a first pointing correction measured at the specific time of planet observations. This pointing correction is obtained by fitting splines on the planet observation from each detector. We further allow this correction to have a sharp jump between the second and third survey (the time of the SCS switchover).<br />
<br />
This first pointing correction is further refined (at the lower HFI frequencies only) using the point sources observations in order to fill the period between planet observations. The radio source offsets require further treatment than the planet based offsets. First, the brightness of the sources being much smaller than planets, the offset on individual detector observation are very noisy, and we filter out the noise by building median offsets for each point source observation. Second, since the sources can be slightly extended we still observe systematic offset above and below the planet based pointing correction in a regular alternating pattern. This is due to the fact that we are observing those extended objects scanning in alternating directions, and our fit of those extended sources to our beam translate into this systematic alternating pattern. We thus allow for a small, arcsec correction of those offsets: i.e., each object is allowed a single offset displacement of order arc seconds in order to minimize the offset between different observations of the same object at different time. <br />
<br />
The resulting list of point source offsets is then merged with the planet offsets. Again a spline based correction is built fitting all of this data, and allowing rupture of continuity at the time of the SCS switchover and other major onboard event (typically between each surveys).<br />
<br />
Once we assume the full Planck pointing model and re-measure the position of Mars for this observation, we see sub-arcsecond deviations (as expected); this gives an indication of the purely numerical limitations of the method. Further planet observations show the small remaining uncorrected drift present in the pointing model (note that cross-scan positions are measured with considerably less data than in-scan positions due to the scan strategy):<br />
<br />
[[Image:S1vM1_v53.png | thumb|500px|center| Saturn 1 vs Mars 1 ]] [[Image:S2vM1_v53.png| thumb|500px|center| Saturn 2 vs Mars 1]]<br />
<br />
Redoing the single-survey map- and catalog-based computation with the corrected pointing shows that Planck has achieved the arcsecond-scale rms uncertainty of pointing:<br />
<br />
[[Image:mapbased_ptcor6.png | thumb|500px|center| As above, after correction by the "ptcor6" model. This plot shows both 545 GHz and 857 GHz sources.]]<br />
<br />
== References ==<br />
<biblio force=false><br />
#[[References]] <br />
</biblio><br />
<br />
<br />
[[Category:HFI/LFI joint data processing|0000]]</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Changelog&diff=7560Changelog2013-05-21T20:58:23Z<p>Ajaffe: </p>
<hr />
<div>* 2013-05-21, Jaffe: [[Detector_pointing|HFI Detector Pointing]]: Changes from Reijo Keskitalo and others<br />
* 2013-03-25, Mendes: Added [[Changelog| change log page]].<br />
* 2013-03-26, Moneti: [[Cosmological Parameters]]: added caveat no. 2, improved look of tables, and other small changes to improve overall look.<br />
* 2013-03-26, Lewis: [[Cosmological Parameters]]: write equations in math mode.<br />
* 2013-03-26, Mendes: Added entry on [[Questions and Answers |Q&A page]] explaining broken links in fits file headers.<br />
* 2013-03-27, Moneti/Cardoso: [[CMB and astrophysical component maps]]: update to correct product description, add figures, remove commented text, other cleanup.<br />
* 2013-04-03, Mendes: [[Sky temperature maps]]: fixed incorrect links (Ticket ID DTX-918825).<br />
* 2013-04-03, Mendes: [[CMB and astrophysical component maps]]: Fixed incorrect links to the new nilc and smica CMB maps.<br />
* 2013-04-12, Zacchei: [[Frequency Maps | Sky temperature maps]] : Added LFI 70 GHz 2048 product and caveats on the use of effective beam with those maps.<br />
* 2013-04-12, Zacchei: [[Map-making_LFI | Map-making]] : Added LFI 70 GHz 2048 product.<br />
* 2013-04-13, Rusholme: [[DatesObs]]: Added note on incompatibility with Healpix IDL read_fits_map. <br />
* 2013-04-16: Mendes: [[SREM]]: Fixed link to the SREM files metadata.<br />
* 2013-04-18: Gregorio: [[Map-making LFI|Map-making]]: Fixed Link to the Noise Monte Carlo Simulations page, end of 2.2.1, typo in the title of 3.2.3 and title of 5. "References" (previously "The Bibliograpy") for consistency with the rest of the ES.<br />
* 2013-04-30: Mendes: [[References2]]: Added three missing references.<br />
* 2013-05-07: Mendes: [[Catalogues#Individual_Catalogues | Catalogues]]: Fixed typo (second Y_MIN should be Y_MAX) in table.<br />
* 2013-05-07: Moneti: [[CMB_spectrum_%26_Likelihood_Code | CMB spectrum and Likelihood]]: added structure of file with masks used in Likelihood paper; fixes references, and other minor improvements.<br />
* 2013-05-07: Moneti: [[CMB_and_astrophysical_component_maps | CMB and astroph. comp. maps]] added structure of file with uinon mask.<br />
* 2013-05-08: Mendes: [[References2]]: New references added for use in [[Beams]].<br />
* 2013-05-021: Moneti/Ashdown: [[CMB_and_astrophysical_component_maps | CMB and astroph. comp. maps]] added mask description and fixed last details. <span style="color:Red">Pending CCB approval</span>.</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Changelog&diff=7559Changelog2013-05-21T20:57:34Z<p>Ajaffe: added Detector Pointing entry from Jaffe</p>
<hr />
<div>* 2013-05-21, Jaffe: [[Detector Pointing]]: Changes from Reijo Keskitalo and others<br />
* 2013-03-25, Mendes: Added [[Changelog| change log page]].<br />
* 2013-03-26, Moneti: [[Cosmological Parameters]]: added caveat no. 2, improved look of tables, and other small changes to improve overall look.<br />
* 2013-03-26, Lewis: [[Cosmological Parameters]]: write equations in math mode.<br />
* 2013-03-26, Mendes: Added entry on [[Questions and Answers |Q&A page]] explaining broken links in fits file headers.<br />
* 2013-03-27, Moneti/Cardoso: [[CMB and astrophysical component maps]]: update to correct product description, add figures, remove commented text, other cleanup.<br />
* 2013-04-03, Mendes: [[Sky temperature maps]]: fixed incorrect links (Ticket ID DTX-918825).<br />
* 2013-04-03, Mendes: [[CMB and astrophysical component maps]]: Fixed incorrect links to the new nilc and smica CMB maps.<br />
* 2013-04-12, Zacchei: [[Frequency Maps | Sky temperature maps]] : Added LFI 70 GHz 2048 product and caveats on the use of effective beam with those maps.<br />
* 2013-04-12, Zacchei: [[Map-making_LFI | Map-making]] : Added LFI 70 GHz 2048 product.<br />
* 2013-04-13, Rusholme: [[DatesObs]]: Added note on incompatibility with Healpix IDL read_fits_map. <br />
* 2013-04-16: Mendes: [[SREM]]: Fixed link to the SREM files metadata.<br />
* 2013-04-18: Gregorio: [[Map-making LFI|Map-making]]: Fixed Link to the Noise Monte Carlo Simulations page, end of 2.2.1, typo in the title of 3.2.3 and title of 5. "References" (previously "The Bibliograpy") for consistency with the rest of the ES.<br />
* 2013-04-30: Mendes: [[References2]]: Added three missing references.<br />
* 2013-05-07: Mendes: [[Catalogues#Individual_Catalogues | Catalogues]]: Fixed typo (second Y_MIN should be Y_MAX) in table.<br />
* 2013-05-07: Moneti: [[CMB_spectrum_%26_Likelihood_Code | CMB spectrum and Likelihood]]: added structure of file with masks used in Likelihood paper; fixes references, and other minor improvements.<br />
* 2013-05-07: Moneti: [[CMB_and_astrophysical_component_maps | CMB and astroph. comp. maps]] added structure of file with uinon mask.<br />
* 2013-05-08: Mendes: [[References2]]: New references added for use in [[Beams]].<br />
* 2013-05-021: Moneti/Ashdown: [[CMB_and_astrophysical_component_maps | CMB and astroph. comp. maps]] added mask description and fixed last details. <span style="color:Red">Pending CCB approval</span>.</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Effective_Beams&diff=6872Effective Beams2013-03-15T15:54:50Z<p>Ajaffe: </p>
<hr />
<div><span style="color:red"></span><br />
<br />
==Product description==<br />
----------------------<br />
<br />
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.<br />
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. <br />
<br />
Details of the beam processing are given in the respective pages for [[Beams|HFI]] and [[Beams_LFI|LFI]].<br />
<br />
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.]] <cite>#mitra2010</cite>, and a discussion of its application to Planck data is given in the appropriate LFI <cite>#planck2013-p02d</cite> and HFI <cite>#planck2013-p03c</cite> papers. Relevant details of the processing steps are given in the [[Beams|Effective Beams]] section of this document.<br />
<br />
<!-- Everything from here down to the "Production Process" section should eventually be moved to a new section the Joint Processing pages --><br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions, PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for all LFI and HFI frequency channels, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* 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).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
<br />
<gallery widths=350px heights=350px perrow=3 caption="Comparison images of compact sources and effective beams, PSFs"><br />
File:30.png| '''30GHz'''<br />
File:44.png| '''44GHz'''<br />
File:70.png| '''70GHz'''<br />
File:100.png| '''100GHz'''<br />
File:143.png| '''143GHz'''<br />
File:217.png| '''217GHz'''<br />
File:353.png| '''353GHz'''<br />
File:545.png| '''545GHz'''<br />
File:857.png| '''857GHz'''<br />
</gallery><br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
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.<br />
<br />
Where beam solid angle is estimated according to the definition: '''<math> 4 \pi \sum</math>(effbeam)/max(effbeam)'''<br />
i.e., <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
[[File:ist_BS_Mars12.png | 600px| thumb | center| '''Histograms for HFI effective beam parameters''' ]]<br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
<br />
* 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.<br />
<br />
<br />
<gallery widths=500px heights=500px perrow=2 caption="Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian"><br />
File:e_030_GB.png| '''ellipticity - 30GHz'''<br />
File:solidarc_030_GB.png| '''beam solid angle (relative variations wrt scanning beam - 30GHz'''<br />
File:e_100_BS_Mars12.png| '''ellipticity - 100GHz'''<br />
File:solidarc_100_BS_Mars12.png| '''beam solid angle (relative variations wrt scanning beam - 100GHz'''<br />
</gallery><br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 3072 and 768 directions on the sky for HFI and LFI data respectively. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 and 2048 healpix maps (they include the pixel window function).<br />
* 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.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''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] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|-<br />
| 100 || 9.651 || 0.014 || 1.186 || 0.023 || -0.024 || 55.400 || 105.778 || 0.311 || 9.66 <br />
|-<br />
| 143 || 7.248 || 0.015 || 1.036 || 0.009 || 0.383 || 54.130 || 59.954 || 0.246 || 7.27 <br />
|-<br />
| 217 || 4.990 || 0.025 || 1.177 || 0.030 || 0.836 || 54.999 || 28.447 || 0.271 || 5.01<br />
|-<br />
| 353 || 4.818 || 0.024 || 1.147 || 0.028 || 0.655 || 54.745 || 26.714 || 0.250 || 4.86<br />
|- <br />
| 545 || 4.682 || 0.044 || 1.161 || 0.036 || 0.544 || 54.876 || 26.535 || 0.339 || 4.84 <br />
|-<br />
| 857 || 4.325 || 0.055 || 1.393 || 0.076 || 0.876 || 54.779 || 24.244 || 0.193 || 4.63 <br />
|}<br />
<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
** <math>\Omega_{eff}</math> - 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, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
** from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <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>.<br />
*** 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).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''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>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|-<br />
| 100 || 105.778 || 0.311 || 100.830 || 0.410 || 105.777 || 0.311 <br />
|-<br />
| 143 || 59.954 || 0.246 || 56.811 || 0.419 || 59.952 || 0.246 <br />
|-<br />
| 217 || 28.447 || 0.271 || 26.442 || 0.537 || 28.426 || 0.271 <br />
|-<br />
| 353 || 26.714 || 0.250 || 24.827 || 0.435 || 26.653 || 0.250 <br />
|-<br />
| 545 || 26.535 || 0.339 || 24.287 || 0.455 || 26.302 || 0.337 <br />
|-<br />
| 857 || 24.244 || 0.193 || 22.646 || 0.263 || 23.985 || 0.191 <br />
|}<br />
<br />
<br />
==Production process==<br />
------------------------<br />
<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]], is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* 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)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*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)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
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. <br />
<br />
<br />
===Pixel Ordered Detector Angles (PODA)===<br />
<br />
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.<br />
<br />
===effBeam===<br />
<br />
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.<br />
<br />
<br />
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.<br />
<br />
<br />
===Computational Cost===<br />
<br />
The computation of the effective beams has been performed at the NERSC Supercomputing Center. The table below shows the computation cost for FEBeCoP processing of the nominal mission.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution. Wall-clock time is given as a guide, as found on the NERSC supercomputers.<br />
|-<br />
|Channel ||030 || 044 || 070 || 100 || 143 || 217 || 353 || 545 || 857<br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 || 500 || 500 || 500 || 500 || 500 || 500 <br />
|-<br />
|PODA/Detector Computation time (wall clock hrs) || 7 || 10 || 20 || 20 || 20 || 20 || 20 || 20 || 20<br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 || 2800 || 3800 || 3200 || 3000 || 900 || 1100<br />
|-<br />
|Beam/Channel Computation time (wall clock hrs) || 0.5 || 0.8 || 1 || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5<br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 || 3.6 || 4.8 || 4.0 || 4.1 || 4.1 || 3.7 <br />
|-<br />
|Convolution Computation time (wall clock sec) || 1 || 1 || 1 || 4 || 4 || 4 || 4 || 4 || 4 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 || 187 || 182 || 146 || 132 || 139 || 124<br />
|}<br />
<br />
<br />
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.<br />
<br />
<br />
<br />
==Inputs==<br />
------------<br />
<br />
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. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center| "'Planck Focal Plane''']]<br />
<br />
<br />
===The Focal Plane DataBase (FPDB)===<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ... (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
<br />
{{PLADoc|fileType=rimo|link=The Plank RIMOS}}<br />
<br />
<br />
<br />
===The scanning strategy===<br />
<br />
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.<br />
<br />
===The scanbeam===<br />
<br />
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.<br />
<br />
* 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. <br />
* HFI: [[Beams#Scanning beams|B-Spline, BS]] based on 2 observations of Mars.<br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
[[Beams LFI#Effective beams|LFI effective beams]]<br />
<br />
===Beam cutoff radii===<br />
<br />
N times geometric mean of FWHM of all detectors in a channel, where N<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' || '''fwhm of full beam extent''' <br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|-<br />
|100 || 2.25 || 23.703699<br />
|-<br />
|143 || 3 || 21.057402<br />
|-<br />
|217-353 || 4 || 18.782754<br />
|-<br />
|sub-mm || 4 || 18.327635(545GHz) ; 17.093706(857GHz) <br />
|}<br />
<br />
===Map resolution for the derived beam data object===<br />
<br />
* <math>N_{side} = 1024 </math> for LFI frequency channels<br />
* <math>N_{side} = 2048 </math> for HFI frequency channels<br />
<br />
<br />
==Related products==<br />
----------------------<br />
<br />
===Monte Carlo simulations===<br />
<br />
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:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, C'_\ell_'^out^'}, and 1 sigma errors<br />
<br />
<br />
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#.<br />
For example [[HL-sims#FFP6 data set|FFP6]] <br />
<br />
===Beam Window Functions===<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_l </math> relates the true angular power spectra <math>C_l </math> with the observed angular power spectra <math>\widetilde{C}_l </math>:<br />
<br />
<math><br />
W_l= \widetilde{C}_l / C_l <br />
\label{eqn:wl1}</math> <br />
<br />
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.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_l </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_l </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_l </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_l = < C^{conv}_l / C^{in}_l ><br />
\label{eqn:wl2}</math> <br />
<br />
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.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for Planck mission====<br />
<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, Wl, for LFI channels''']] <br />
[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |'''Beam Window functions, Wl, for HFI channels''']]<br />
<br />
<br />
<br />
<br />
==File Names==<br />
-----------------<br />
<br />
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.<br />
<br />
* beam_index.fits<br />
* beams_run.log<br />
<br />
== Retrieval of effective beam information from the PLA interface ==<br />
<br />
In order to retrieve the effective beam information, the user should first launch the Java interface from this page:<br />
http://www.sciops.esa.int/index.php?project=planck&page=Planck_Legacy_Archive<br />
<br />
One should click on "Sky maps" and then open the "Effective beams" area.<br />
There is the possibility to either retrieve one beam nearest to the input source (name or coordinates), or to retrieve a set of beams in a grid defined by the Nside and the size of the region around a source (name or coordinates).<br />
The resolution of this grid is defined by the Nside parameter.<br />
The size of the region is defined by the "Radius of ROI" parameter.<br />
<br />
Once the user proceeds with querying the beams, the PLA software retrieves the appropriate set of effective beams from the database and delivers it in a FITS file which can be directly downloaded.<br />
<br />
<br />
<br />
==Meta data==<br />
----------------<br />
<br />
The data format of the effective beams is unformatted.<br />
<br />
== References ==<br />
------------------<br />
<br />
<biblio force=false><br />
#[[References]]<br />
</biblio><br />
<br />
<br />
<br />
[[Category:Mission science products|004]]</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Beams&diff=6862Beams2013-03-15T15:47:23Z<p>Ajaffe: </p>
<hr />
<div>== Scanning Beams ==<br />
<br />
The scanning beams describe the instrument’s instantaneous beam profile. Due to the near constant spin rate of the spacecraft, time domain effects (including residual time response and lowpass filtering) are degenerate with the spatial response due to the optical system. The scanning beam reconstruction recovers both of these effects, aside from residual time domain effects on a longer time scale than can be captured with the extent of the scanning beam model.<br />
<br />
In <cite>#planck2013-p03c</cite> we consider two models of the beam in order to better understand systematics in the reconstruction. Here we describe only the B-Spline beams which are used to compute the delivered effective beam (see next section).<br />
<br />
=== B-Spline Beam construction ===<br />
<br />
We use seasons 1 and 2 of the Mars observation to reconstruct the beam. The data are processed with the bigPlanets TOI processing. We use JPL Horizons ephemerides to determine the pointing of each detector relative to the planet. We subtract the astrophysical background in the time domain using a bicubic interpolation of the Planck maps.<br />
<br />
The time ordered data are used to fit a two dimensional B-Spline surface using a least square minimization and a smoothing criterion to minimize the effects of high spatial frequency variations. We therefore assume the scanning beam to be smooth. The smoothing criterion as well as the locations of the nodes used to compute the B-Spline basis functions are set using GRASP physical optics simulations as inputs which are the best assumptions on the spatial frequency content of the in-flight beams.<br />
<br />
The smoothing criterion is defined as follows:<br />
<br />
<math>\eta = \displaystyle{\sum_{i=1}^{g}\left(b^{k}(\lambda_{i+})-b^{k}(\lambda_{i-})\right)^2}<br />
\label{smoothcrit}</math><br />
<br />
<math>\begin{aligned}<br />
\eta &: \mbox{ Smoothing Criterion}\\<br />
b^k &: \mbox{ $k^{th}$ beam derivative evaluated on the nodes locations}\end{aligned}</math><br />
<br />
And the global inversion criterion :<br />
<br />
<math>\zeta = \eta + p\times \delta</math><br />
<br />
with <math>\delta</math> usual least square estimator and <math>p</math> coefficient giving the relative weight to <math>\delta</math> with respect to the smoothing criterion.<br />
<br />
<math>\delta = \displaystyle{\sum_{r=1}^{m}}\left(y_{r} - b(x_{r})\right)^2\label{estimator}</math><br />
<br />
<math>\begin{aligned}<br />
\delta &: \mbox{ usual least square criterion}\\<br />
r &: \mbox{ indice relative to the m data points, } r \in \{1, \ldots, m\}\\<br />
y_r &: \mbox{ planet data of sample r}\\<br />
x_r &: \mbox{ pointing of sample r}\\<br />
b &: \mbox{ reconstructed beam}\end{aligned}</math><br />
<br />
The B-Spline nodes are located on a regular spaced grid in the detector coordinate framset. At the edge of the reconstructed beam map area, 4 coincident nodes are added to avoid vanishing basis functions.<br />
<br />
Let <math>B_{i, k+1}</math>, <math>k</math> degree B-Spline build using nodes {<math>\lambda_{i}, ..., \lambda_{i+k+1}</math>} (''De Boor &amp; Cox'', 1972) :<br />
<br />
<math>B_{i,1}(x) = \left\{<br />
\begin{array}{l}<br />
1, \mbox{ si } x \in \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}\\<br />
0, \mbox{ si } x \notin \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}<br />
\end{array} \right.</math><br />
<br />
<math>B_{i, l+1}(x) = \displaystyle{\frac{x - \lambda_{i}}{\lambda_{i+l} - \lambda_{i}}} B_{i,l}(x) + \displaystyle{\frac{\lambda_{i+l+1}-x}{\lambda_{i+l+1}-\lambda_{i+1}}} B_{i+1, l}(x)</math><br />
<br />
<math>l=1, \ldots, k</math><br />
<br />
<br />
[[Image:FocalPlane_Map_BSScanningBeams_v53.png|800px|frame|center|Focal plane plot of B-Spline scanning beams using in-flight pointing reconstruction. The contours are -3,-10,-20,-30 dB from the peak, and for PSB pairs the "a" bolometer is plotted in black and "b" in blue.]]<br />
<br />
=== Simulations and errors ===<br />
<br />
We estimate the reconstruction bias and noise in the measurements using an ensemble of simulated planet observations for each channel. Further details are discussed in <cite>#planck2013-p03c</cite>. Kept fixed in each simulation are:<br />
<br />
* the input beam assumed: we use a supersampled version of the reconstructed B-Spline beam (or whatever comes out of the current ongoing tests!)<br />
* Astrophyical background is the same as that subtracted from the real data.<br />
* StarTracker pointing (using the ptcor6 pointing model).<br />
<br />
The following are varied in each simulation:<br />
<br />
* detector noise realizations obtained by filtering randomly generated white noise with the measured noise PSDs<br />
* random pointing errors with 2 arcsecond rms, and a spectrum that replicates the real errors.<br />
* simulated glitches and the deglitching procedure<br />
* Mars brightness temperature variability<br />
<br />
400 simulated timelines are generated for each bolometer and for each of the two seasons of Mars observations used in the beam reconstruction. The simulated timelines are made into beam maps, projecting onto the B-Spline basis in the same way as the real data.<br />
<br />
The beam maps are propagated to effective beam window functions using the quickbeam approach (see effective beams below) and used to evaluate the reconstruction bias and to construct error eigenmodes in the effective beam window function.<br />
<br />
<!-- <font color=red>Figure: random pointing error PSD Figures: error envelope plots (or should those go under effective beams?)</font> --><br />
<br />
=== Residuals ===<br />
<br />
There are two known beam effects that are not included in the main beam model and are estimated as a separate bias in flux and angular power spectrum measurement: 1. long tails due to errors in low frequency time response deconvolution, and 2. near sidelobes.<br />
<br />
We stack all five observations of Jupiter to estimate the long time scale residuals due to incomplete deconvolution of the long time scale response.<br />
<br />
<!-- <font color=red>Add some kind of mean tail plot</font><br />
<br />
Near sidelobes are also evaluated using stacked Jupiter (hopefully they will just be part of the v53bis B-Spline beams). The main features in the near sidelobes include a wide beam skirt, and dimpling lobes<br />
<font color=red>Add sidelobe plots and tables</font> --> <br />
<br />
<br />
== Effective Beams ==<br />
<br />
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. <br />
<br />
Several methods of effective beams determination have been developped and cross-validated.<br />
<br />
The main products are produced using FEBeCoP and details of the processing are given in the [[Effective Beams]] products page. See also the equivalent [[Beams_LFI | page discussing the LFI beams]]<br />
<br />
<!-- <font color=red>Need satisfactory comparison plot</font> --><br />
<br />
=== FEBeCoP ===<br />
-------------------<br />
<br />
The full algebra for this method for the calculation of effective beams was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]] <cite>#mitra2010</cite>. 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>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<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.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \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,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\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) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
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}.<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
<br />
<br />
=== FICSBell ===<br />
For more details, see <cite>#planck2013-p03c</cite>.<br />
<br />
Since the HFI beams are not azimuthally symmetric, the scanning strategy has to be taken into account in the effective beam response modelling. This is done using the FICSBell method <font color=red>(Hivon et al, in preparation)</font>, which generalizes to polarization and to include other sources of systematics the approach used for TT <math>C(l)</math> estimation in WMAP-3yr <font color=red>Hinshaw et al (2007)</font> and by <font color=red>Smith et al (2007)</font> in the detection of CMB lensing in WMAP maps. The different steps of the method used for this study can be summarized as follows:<br />
<br />
<ol><br />
<li><p>The scanning related information (i.e., statistics of the orientation of each detector within each pixel) is computed first, and only once for a given observation campaign. Those orientation hit moments are only computed up to degree 4, for reasons described in point 2 below. At the same time, the first two moments of the distribution of samples within each pixel (ie, their center of mass and moments of inertia) are computed and stored on disc.</p></li><br />
<li><p>The scanning beam map or beam model of each detector <math>d</math> is analyzed into its Spherical Harmonics coefficients</p><br />
<p><math>b^d_{ls} = \int d{\bf r} B_d({\bf r}) Y_{ls}({\bf r})\label{scanningBlm}</math></p><br />
<p>where <math>B_d(\bf{r})</math> is the beam map centered on the North pole, and <math>Y_{ls}(\bf{r})</math> is the Spherical Harmonics basis function. Higher <math>s</math> indexes describes higher degrees of departure from azimuthal symmetry and, for HFI beams, the coefficients <math>b^d_{ls}</math> are decreasing functions of <math>s</math> at most <math>l</math> considered. It also appears that, for <math>l<3000</math>, the coefficients with <math>|s| > 4</math> account for <math>1\%</math> or less of the beam throughput. For this reason, only modes with <math>|s| \le 4</math> are considered in the present analysis. <font color=red>Armitage-Caplan and Wandelt (2009)</font> reached a similar conclusion in their deconvolution of Planck-LFI beams.</p></li><li><p>The <math>b^d_{ls}</math> coefficients computed above are used to generate <math>s</math>-spin weighted maps, as well as the first and second order derivatives, for a given CMB sky realization.</p></li><br />
<li><p>The spin weighted maps and orientation hit moments of the same order <math>s</math> are combined for all detectors involved, to provide an “observed” map. Similarly the local spatial derivatives are combined with the location hit moments to describe the effect of the non-ideal sampling of each pixel (see [sec:pixelization]). In this combination, the respective number of hits of each detector in each pixel is considered, as well as the weighting (generally proportional to the inverse noise variance) applied to each detector in order to minimize the final noise.</p></li><li><p>The power spectrum of this map can then be computed, and compared to the input CMB power spectrum to estimate the effective beam window function over the whole sky, or over a given region of the sky.</p></li></ol><br />
Monte-Carlo (MC) simulations in which the sky realisations are changed can be performed by repeating steps 3, 4 and 5. The impact of beam model uncertainties can be studied by including step 2 into the MC simulations.<br />
<br />
=== QuickBeam ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
Planck observes the sky after convolution with a “scanning beam”, which captures its effective response to the sky as a function of displacement from the nominal pointing direction. Decomposing the scanning beam into harmonic coefficients <math>B_{lm}</math>, each time-ordered data (TOD) sample can be modelled as (neglecting the contribution from instrumental noise, which is independent of beam asymmetry) <math>%T_i = \sum_{lms} D^{l}_{-m s} (\phi_i, \theta_i, \alpha_i) b_{ls} (-1)^{m) T_{lm} + n_i,<br />
T_i = \sum_{lms} e^{-i s \alpha_i} B_{ls} \tilde{T}_{lm} {}_s Y_{lm}(\theta_i, \phi_i),<br />
\label{eqn:tod_beam}</math> where the TOD samples are indexed by <math>i</math>, and <math>\tilde{T}_{lm}</math> is the underlying sky signal. The spin spherical harmonic <math>{}_s Y_{lm}</math> rotates the scanning beam to the pointing location <math>(\theta, \phi)</math>, while the <math>e^{-i s \alpha_i}</math> factor gives it the correct orientation. Eq. may be evaluated with the “TotalConvolver” algorithm of <font color=red>Wandelt and Gorski (2001)</font>, accelerated using the “conviqt” recursion relations <font color=red>Prezeau and Reinecke (2010)</font> This approach is implemented in LevelS.<br />
</ref>, although because it involves working with a TOD-sized objected it is necessarily slow.<br />
<br />
On the small angular scales comparable to the size of the beam, it is a good approximation to assume that the procedure of mapmaking from TOD samples is essentially a process of binning: <math>T(p) = \sum_{i \in p} T_i / H(p),<br />
\label{eqn:map_beam_full}</math> where <math>H(p)</math> is the total number of hits in pixel <math>\hat{n}</math>.<br />
<br />
Start with a normalized, rescaled harmonic transform of the beam <math>B_{lm}</math>, sky multipoles <math>\tilde{T}_{lm}</math> and a scan history object <math>w(\hat{n}, s)</math> given by <math>w(\hat{n}, s) = \sum_{j \in p} e^{i s \alpha_j} / H(\hat{n})</math> where the sum is over all hits <math>j</math> of pixel <math>p</math> at location <math>\hat{n}_p</math>, and <math>\alpha_j</math> is the scan angle for observation <math>j</math>. The harmonic transform of this scan-strategy object is given by <math>{}_{s} w_{L M} = \int d^2 \hat{n} {}_s Y_{LM}^*(\hat{n}) w(\hat{n}, s).</math> The beam-convolved observation is then given by <math>\tilde{T}(\hat{n}) = \sum_{slm} w(\hat{n}, -s ) B_{ls} T_{lm} {}_s Y_{lm}(\hat{n}).</math> Taking the ensemble average of the pseudo-Cl power spectrum of these <math>T_{lm}</math> we find<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \sum_{S S'} \sum_{l_1 l_2} \frac{(2l_1+1)(2l_2+1)}{4\pi}<br />
{}_{(-s -s')}{\cal W}_{l_1} B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}<br />
\\ \times\left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s\! & -s\! & 0\!<br />
\end{array}<br />
\right) \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s'\! & -s'\! & 0\!<br />
\end{array}<br />
\right)<br />
<br />
\end{gathered}</math><br />
<br />
where <math>{}_{(s s')}{\cal W}_{L} = \frac{1}{2L+1} \sum_{M} {}_{S} w_{LM} {}_{S'} w_{LM}^*</math> is a cross-power spectrum of scan history objects. Note that the w(n,s) which we have used here can also incorporate a position dependent weighting to optimize the pseudo-Cl estimate, such as inverse-noise or a mask– the equations are unchanged. Writing the pseudo-Cl in position space (a la <font color=red> Dvorkin and Smith (2009)</font>) with Wigner-d matrices we have<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \frac{1}{8\pi} \sum_{S S'} \int_{-1}^{1} dz \ d^{L}_{00}(z)<br />
\\ \times<br />
\left[\sum_{l_1} d^{l_1}_{-s -s'}(z) {}_{(-s -s')}{\cal W}_{l_1} (2l_1+1) \right] <br />
\\ \times<br />
\left[ \sum_{l_2} d^{l_2}_{s s'}(z) B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}(2l_2+1) \right].\end{gathered}</math><br />
<br />
This integral can be implemented exactly using Gauss-Legendre quadrature, with a cost of $\cal 0(l_{\rm max}^2 s_{\rm max}^2)$. For simplicity, we’ve written all the equations here for the auto-spectrum of a single detector, but the generalization to a map made by adding several detectors with different weighting is straightforward. The cost to compute all of the necessary terms exactly in that case becomes <math>\cal 0(l_{\rm max}^2 s_{\rm max}^2 N_{\rm det}^2)</math>.<br />
<br />
Are beams really so difficult? On the flat-sky beam convolution is easy: just multiplication in Fourier space by a beam rotated onto the scan direction. Multiple hits with different scan directions are incorporated by averaging (as the “scan history” objects above encapsulate). Does the sphere really require everything to be so complicated? For a scan strategy which is fairly smooth across the sky, we can pretend that we are observing many independent flat-sky patches at high-L with fairly good accuracy. There is in fact a fairly good approximation to the beam convolved pseudo-Cl power spectrum which is essentially a flat-sky approximation. In the limit that <math>L \gg l_1</math>, with <math>C_{l_2}</math> and <math>B_{l_2}</math> being slowly-varying function in <math>l_2</math> the pseudo-Cl sum above can be approximated as <math>{\tilde{C}}_L^{TT} = C_L^{TT} \sum_{M} \left< \left| w(\hat{n}_p, M) \right|^2 \right>_p |B_{L M}|^2,</math> where the average <math><>_p</math> is taken over the full sky. It’s illustrative to consider three limits of this equation: for a “raster” scan strategy in which each pixel is observed with the same direction, we have <math>\left< \left| w(\hat{n}, M) \right|^2 \right>_p = 1,</math> and the predicted pseudo-Cl is just the power spectrum of the beam. For a &quot;best-case&quot; scan strategy, in which each pixel is observed many times with many different orientation angles, we have &lt; | w(, M) |<sup>2</sup> &gt;<sub>p</sub> = <sub>M0</sub>, and the transfer function is just the azimuthally symmetric part of the beam. Note that this is for a full-sky observation– in the presence of a mask, the average above produces an fsky factor, as expected. It just neglects the coupling between L multipoles (which can be calculated with the more complete equations above).<br />
<br />
==== Effective beam window functions ====<br />
The effective beam window functions $B(l)$ for HFI, computed using Quickbeam, are available in the [[The RIMO|RIMO]].<br />
They do not contain the pixel window function.<br />
<br />
=== Pixelization Artifacts ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
<font color=red><br />
* Several codes available to simulate effects of pixelization.<br />
* Mixes the CMB gradient into a pixelization ``noise'' with a level comparable to that of $2\mu Karcmin$ instrumental noise.<br />
* Quantitative estimate of effect should be included with each released map, but expect not to matter significantly for CMB analysis, as small compared to instrumental noise. <br />
</font><br />
[sec:pixelization]<br />
<br />
Planck maps are produced at resolution 11 <math>(N_{\rm side} = 2048)</math>, corresponding to pixels with a typical dimension of <math>1.7'</math>, comparable to the spacing between scanning rings . This results in an uneven distribution of hits within pixels, which introduces some complications in the analysis and interpretation of the maps. A sample of the hit distribution is illustrated in Fig. [fig:pixcoverage]. Below we discuss the simulation and modeling of this pixelization effect in more detail.<br />
<br />
<br />
[[Image:pixcoverage.png|frame|none|alt=image]]<br />
<br />
[fig:pixcoverage]<br />
<br />
The collaboration has produced 3 codes which may be used to simulate the effect of pixelization on the observed sky, LevelS/TotalConvoler/Conviqt, FeBeCoP, and FICSBell <font color=red> references and further discussion of the three methods<br />
and how they each simulate the pixelization effect.</font>.<br />
<br />
For the measurement of CMB fluctuations, it is also possible to gain intuition for the effects of pixelization analytically. On the small scales relevant to pixelization, the observed CMB is smooth, both due to physical damping as well as the convolution of the instrumental beam. Taylor expanding the CMB temperature about a pixel center to second order, the typical gradient amplitude is given by <math>\langle |\nabla T |^2 \rangle = \frac{1}{4\pi} \sum_{l} l(l+1)(2l+1) C_l^{T} W_l \approx 1\times10^9 \mu K^2 / {\rm rad}^2.</math> where the approximate value is calculated for a <math>\Lambda CDM</math> cosmology with a <math>7'</math>fwhm Gaussian beam. The typical curvature of the observed temperature, on the other hand is given by <math>\langle |\nabla^2 T |^2 \rangle = \frac{1}{4\pi} \sum_{l} [l(l+1)]^2(2l+1) C_l^{T} W_l \approx 7\times10^{14} \mu K^2 / {\rm rad}^4.</math> On the scales relevant to the maximum displacement from the center of a <math>1.7'</math> pixel, the maximum displacement is , and so the gradient term tends to dominate, although the curvature term is still non-negligible. For each observation of a pixel, we can denote the displacement from the pixel center as <math>d = d_{\theta} + i d_{\phi}</math>. The average over all hits within a pixel gives an overall deflection vector which we will denote for a pixel center located at <math>\hat{n}</math> as <math>d(\hat{n})</math>. This represents the center of mass of the hit distribution; in Fig. [fig:pixcoverage] we have plotted these average deflections using black arrows. The deflection field <math>d(\hat{n})</math> may be decomposed into spin-1 spherical harmonics as <math>d_{lm} = \int_{4\pi} {}_1 Y_{lm}^* d(\hat{n}).</math> With a second order Taylor expansion of the CMB temperature about each pixel center, it is then possible to calculate the average pseudo-Cl power spectrum of the pixelized sky. This is given by<br />
<br />
<math>\begin{gathered}<br />
C_l^{T} = [1-l(l+1)R^d] {C}_l^{T} W_l + \\ <br />
\frac{1}{2} \sum_{l_1 l_2} \frac{l_1(l_1+1)(2l_1+1)(2l_2+1)}{4\pi} \\<br />
\times \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & l\! \\<br />
\! l\! & -l\! & 0\!<br />
\end{array}<br />
\right)^2 C_{l_1}^{T} W_{l_1} \left[ C_{l_2}^{d+} + (-1)^{l + l_1 + l_2} C_{l_2}^{ d-} \right],<br />
\label{eqn:clt_pixelized}\end{gathered}</math><br />
<br />
where <math>R^{d} = \langle |d|^2 \rangle/2</math> is half the mean-squared deflection magnitude (averaged over hits within a pixel, as well as over pixels). <math>C_l^{d+}</math> is the sum of the gradient and curl power spectra of <math>d_{lm}</math>, and <math>C_l^{d-}</math> is the gradient spectrum minus the curl spectrum. The <math>R^{d}</math> term describes a smearing of the observed sky due to pixelization. For uniform pixel coverage of <math>N_{\rm side}=2048</math> pixels <math>\sqrt{ \langle |d|^2 \rangle } = 0.725'</math>. For the hit distribution of Planck frequency maps, <math>R^{d}</math> is typically within <font color=red> xxx. calculate for final<br />
maps, looks like will be better than 10%</font>percent of this value, and so this term is accurately described by the pixel window function, which is derived under the assumption of uniform pixel coverage.<br />
<br />
The effect of pixelization is essentially degenerate with that of gravitational lensing of the CMB, with the difference that it (1) acts on the beam-convolved sky, rather than the actual sky and (2) produces a curl-mode deflection field as well as a gradient mode. This is discussed further in the [<cite>#planck2013-p12</cite>|Planck gravitational lensing] paper, where the subpixel deflection field constitutes a potential source of bias for the measured lensing potential. Indeed, Eq. [eqn:clt<sub>p</sub>ixelized] is just a slightly modified version of the usual first order CMB lensing power spectrum (<font color=red>Hu (2000)</font>, <font color=red>Lewis and Challinor (2006)</font>) to accommodate curl modes.<br />
<br />
A useful approximation to Eq. which is derived in the unrealistic limit that the deflection vectors are uncorrelated between pixels, but in practice gives a good description of the power induced by the pixelization, is that the <math>d(\hat{n})</math> couples the CMB gradient into a source of noise with an effective level given by <math>\sigma^{N} \approx \sqrt{ R^T \frac{4\pi}{N_{\rm pix}} \langle | d(\hat{n}) |^2<br />
\rangle }, % (\muKarcmin ),</math><br />
<br />
where the average is taken over all pixels and <math>R^T</math> is half the mean-squared power in the CMB gradient: <math>R^{T} = \frac{1}{8\pi} \sum_{l} l(l+1)(2l+1) \tilde{C}_l^{T}.</math> For frequency-combined maps, <math>\sqrt{ \langle | d(\hat{n}) |^2 \rangle }</math> is typically on the order of <math>0.1'</math>, and so the induced noise is at the level of <math>\sigma^{N} \sim 2 \mu K arcmin</math>. This is small compared to the instrumental contribution, although it does not disappear when taking cross-spectra, depending on how coherent the hit distributions of the two maps in the cross-spectrum are.<br />
<br />
= References =<br />
<biblio force=false><br />
#[[References]] <br />
</biblio></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Beams&diff=6858Beams2013-03-15T15:43:42Z<p>Ajaffe: </p>
<hr />
<div>== Scanning Beams ==<br />
<br />
The scanning beams describe the instrument’s instantaneous beam profile. Due to the near constant spin rate of the spacecraft, time domain effects (including residual time response and lowpass filtering) are degenerate with the spatial response due to the optical system. The scanning beam reconstruction recovers both of these effects, aside from residual time domain effects on a longer time scale than can be captured with the extent of the scanning beam model.<br />
<br />
In <cite>#planck2013-p03c</cite> we consider two models of the beam in order to better understand systematics in the reconstruction. Here we describe only the B-Spline beams which are used to compute the delivered effective beam (see next section).<br />
<br />
=== B-Spline Beam construction ===<br />
<br />
We use seasons 1 and 2 of the Mars observation to reconstruct the beam. The data are processed with the bigPlanets TOI processing. We use JPL Horizons ephemerides to determine the pointing of each detector relative to the planet. We subtract the astrophysical background in the time domain using a bicubic interpolation of the Planck maps.<br />
<br />
The time ordered data are used to fit a two dimensional B-Spline surface using a least square minimization and a smoothing criterion to minimize the effects of high spatial frequency variations. We therefore assume the scanning beam to be smooth. The smoothing criterion as well as the locations of the nodes used to compute the B-Spline basis functions are set using GRASP physical optics simulations as inputs which are the best assumptions on the spatial frequency content of the in-flight beams.<br />
<br />
The smoothing criterion is defined as follows:<br />
<br />
<math>\eta = \displaystyle{\sum_{i=1}^{g}\left(b^{k}(\lambda_{i+})-b^{k}(\lambda_{i-})\right)^2}<br />
\label{smoothcrit}</math><br />
<br />
<math>\begin{aligned}<br />
\eta &: \mbox{ Smoothing Criterion}\\<br />
b^k &: \mbox{ $k^{th}$ beam derivative evaluated on the nodes locations}\end{aligned}</math><br />
<br />
And the global inversion criterion :<br />
<br />
<math>\zeta = \eta + p\times \delta</math><br />
<br />
with <math>\delta</math> usual least square estimator and <math>p</math> coefficient giving the relative weight to <math>\delta</math> with respect to the smoothing criterion.<br />
<br />
<math>\delta = \displaystyle{\sum_{r=1}^{m}}\left(y_{r} - b(x_{r})\right)^2\label{estimator}</math><br />
<br />
<math>\begin{aligned}<br />
\delta &: \mbox{ usual least square criterion}\\<br />
r &: \mbox{ indice relative to the m data points, } r \in \{1, \ldots, m\}\\<br />
y_r &: \mbox{ planet data of sample r}\\<br />
x_r &: \mbox{ pointing of sample r}\\<br />
b &: \mbox{ reconstructed beam}\end{aligned}</math><br />
<br />
The B-Spline nodes are located on a regular spaced grid in the detector coordinate framset. At the edge of the reconstructed beam map area, 4 coincident nodes are added to avoid vanishing basis functions.<br />
<br />
Let <math>B_{i, k+1}</math>, <math>k</math> degree B-Spline build using nodes {<math>\lambda_{i}, ..., \lambda_{i+k+1}</math>} (''De Boor &amp; Cox'', 1972) :<br />
<br />
<math>B_{i,1}(x) = \left\{<br />
\begin{array}{l}<br />
1, \mbox{ si } x \in \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}\\<br />
0, \mbox{ si } x \notin \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}<br />
\end{array} \right.</math><br />
<br />
<math>B_{i, l+1}(x) = \displaystyle{\frac{x - \lambda_{i}}{\lambda_{i+l} - \lambda_{i}}} B_{i,l}(x) + \displaystyle{\frac{\lambda_{i+l+1}-x}{\lambda_{i+l+1}-\lambda_{i+1}}} B_{i+1, l}(x)</math><br />
<br />
<math>l=1, \ldots, k</math><br />
<br />
<br />
[[Image:FocalPlane_Map_BSScanningBeams_v53.png|800px|frame|center|Focal plane plot of B-Spline scanning beams using in-flight pointing reconstruction. The contours are -3,-10,-20,-30 dB from the peak, and for PSB pairs the "a" bolometer is plotted in black and "b" in blue.]]<br />
<br />
=== Simulations and errors ===<br />
<br />
We estimate the reconstruction bias and noise in the measurements using an ensemble of simulated planet observations for each channel. Further details are discussed in <cite>#planck2013-p03c</cite>. Kept fixed in each simulation are:<br />
<br />
* the input beam assumed: we use a supersampled version of the reconstructed B-Spline beam (or whatever comes out of the current ongoing tests!)<br />
* Astrophyical background is the same as that subtracted from the real data.<br />
* StarTracker pointing (using the ptcor6 pointing model).<br />
<br />
The following are varied in each simulation:<br />
<br />
* detector noise realizations obtained by filtering randomly generated white noise with the measured noise PSDs<br />
* random pointing errors with 2 arcsecond rms, and a spectrum that replicates the real errors.<br />
* simulated glitches and the deglitching procedure<br />
* Mars brightness temperature variability<br />
<br />
400 simulated timelines are generated for each bolometer and for each of the two seasons of Mars observations used in the beam reconstruction. The simulated timelines are made into beam maps, projecting onto the B-Spline basis in the same way as the real data.<br />
<br />
The beam maps are propagated to effective beam window functions using the quickbeam approach (see effective beams below) and used to evaluate the reconstruction bias and to construct error eigenmodes in the effective beam window function.<br />
<br />
<!-- <font color=red>Figure: random pointing error PSD Figures: error envelope plots (or should those go under effective beams?)</font> --><br />
<br />
=== Residuals ===<br />
<br />
There are two known beam effects that are not included in the main beam model and are estimated as a separate bias in flux and angular power spectrum measurement: 1. long tails due to errors in low frequency time response deconvolution, and 2. near sidelobes.<br />
<br />
We stack all five observations of Jupiter to estimate the long time scale residuals due to incomplete deconvolution of the long time scale response.<br />
<br />
<!-- <font color=red>Add some kind of mean tail plot</font><br />
<br />
Near sidelobes are also evaluated using stacked Jupiter (hopefully they will just be part of the v53bis B-Spline beams). The main features in the near sidelobes include a wide beam skirt, and dimpling lobes<br />
<font color=red>Add sidelobe plots and tables</font> --> <br />
<br />
<br />
== Effective Beams ==<br />
<br />
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. <br />
<br />
Several methods of effective beams determination have been developped and cross-validated.<br />
<br />
The main products are produced using FEBeCoP and details of the processing are given in the [[Effective Beams]] products page. See also the equivalent [[Beams_LFI | page discussing the LFI beams]]<br />
<br />
<! -- <font color=red>Need satisfactory comparison plot</font> --><br />
<br />
=== FEBeCoP ===<br />
-------------------<br />
<br />
The full algebra for this method for the calculation of effective beams was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]] <cite>#mitra2010</cite>. 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>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<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.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \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,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\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) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
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}.<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
<br />
<br />
=== FICSBell ===<br />
For more details, see <cite>#planck2013-p03c</cite>.<br />
<br />
Since the HFI beams are not azimuthally symmetric, the scanning strategy has to be taken into account in the effective beam response modelling. This is done using the FICSBell method <font color=red>(Hivon et al, in preparation)</font>, which generalizes to polarization and to include other sources of systematics the approach used for TT <math>C(l)</math> estimation in WMAP-3yr <font color=red>Hinshaw et al (2007)</font> and by <font color=red>Smith et al (2007)</font> in the detection of CMB lensing in WMAP maps. The different steps of the method used for this study can be summarized as follows:<br />
<br />
<ol><br />
<li><p>The scanning related information (i.e., statistics of the orientation of each detector within each pixel) is computed first, and only once for a given observation campaign. Those orientation hit moments are only computed up to degree 4, for reasons described in point 2 below. At the same time, the first two moments of the distribution of samples within each pixel (ie, their center of mass and moments of inertia) are computed and stored on disc.</p></li><br />
<li><p>The scanning beam map or beam model of each detector <math>d</math> is analyzed into its Spherical Harmonics coefficients</p><br />
<p><math>b^d_{ls} = \int d{\bf r} B_d({\bf r}) Y_{ls}({\bf r})\label{scanningBlm}</math></p><br />
<p>where <math>B_d(\bf{r})</math> is the beam map centered on the North pole, and <math>Y_{ls}(\bf{r})</math> is the Spherical Harmonics basis function. Higher <math>s</math> indexes describes higher degrees of departure from azimuthal symmetry and, for HFI beams, the coefficients <math>b^d_{ls}</math> are decreasing functions of <math>s</math> at most <math>l</math> considered. It also appears that, for <math>l<3000</math>, the coefficients with <math>|s| > 4</math> account for <math>1\%</math> or less of the beam throughput. For this reason, only modes with <math>|s| \le 4</math> are considered in the present analysis. <font color=red>Armitage-Caplan and Wandelt (2009)</font> reached a similar conclusion in their deconvolution of Planck-LFI beams.</p></li><li><p>The <math>b^d_{ls}</math> coefficients computed above are used to generate <math>s</math>-spin weighted maps, as well as the first and second order derivatives, for a given CMB sky realization.</p></li><br />
<li><p>The spin weighted maps and orientation hit moments of the same order <math>s</math> are combined for all detectors involved, to provide an “observed” map. Similarly the local spatial derivatives are combined with the location hit moments to describe the effect of the non-ideal sampling of each pixel (see [sec:pixelization]). In this combination, the respective number of hits of each detector in each pixel is considered, as well as the weighting (generally proportional to the inverse noise variance) applied to each detector in order to minimize the final noise.</p></li><li><p>The power spectrum of this map can then be computed, and compared to the input CMB power spectrum to estimate the effective beam window function over the whole sky, or over a given region of the sky.</p></li></ol><br />
Monte-Carlo (MC) simulations in which the sky realisations are changed can be performed by repeating steps 3, 4 and 5. The impact of beam model uncertainties can be studied by including step 2 into the MC simulations.<br />
<br />
=== QuickBeam ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
Planck observes the sky after convolution with a “scanning beam”, which captures its effective response to the sky as a function of displacement from the nominal pointing direction. Decomposing the scanning beam into harmonic coefficients <math>B_{lm}</math>, each time-ordered data (TOD) sample can be modelled as (neglecting the contribution from instrumental noise, which is independent of beam asymmetry) <math>%T_i = \sum_{lms} D^{l}_{-m s} (\phi_i, \theta_i, \alpha_i) b_{ls} (-1)^{m) T_{lm} + n_i,<br />
T_i = \sum_{lms} e^{-i s \alpha_i} B_{ls} \tilde{T}_{lm} {}_s Y_{lm}(\theta_i, \phi_i),<br />
\label{eqn:tod_beam}</math> where the TOD samples are indexed by <math>i</math>, and <math>\tilde{T}_{lm}</math> is the underlying sky signal. The spin spherical harmonic <math>{}_s Y_{lm}</math> rotates the scanning beam to the pointing location <math>(\theta, \phi)</math>, while the <math>e^{-i s \alpha_i}</math> factor gives it the correct orientation. Eq. may be evaluated with the “TotalConvolver” algorithm of <font color=red>Wandelt and Gorski (2001)</font>, accelerated using the “conviqt” recursion relations <font color=red>Prezeau and Reinecke (2010)</font> This approach is implemented in LevelS.<br />
</ref>, although because it involves working with a TOD-sized objected it is necessarily slow.<br />
<br />
On the small angular scales comparable to the size of the beam, it is a good approximation to assume that the procedure of mapmaking from TOD samples is essentially a process of binning: <math>T(p) = \sum_{i \in p} T_i / H(p),<br />
\label{eqn:map_beam_full}</math> where <math>H(p)</math> is the total number of hits in pixel <math>\hat{n}</math>.<br />
<br />
Start with a normalized, rescaled harmonic transform of the beam <math>B_{lm}</math>, sky multipoles <math>\tilde{T}_{lm}</math> and a scan history object <math>w(\hat{n}, s)</math> given by <math>w(\hat{n}, s) = \sum_{j \in p} e^{i s \alpha_j} / H(\hat{n})</math> where the sum is over all hits <math>j</math> of pixel <math>p</math> at location <math>\hat{n}_p</math>, and <math>\alpha_j</math> is the scan angle for observation <math>j</math>. The harmonic transform of this scan-strategy object is given by <math>{}_{s} w_{L M} = \int d^2 \hat{n} {}_s Y_{LM}^*(\hat{n}) w(\hat{n}, s).</math> The beam-convolved observation is then given by <math>\tilde{T}(\hat{n}) = \sum_{slm} w(\hat{n}, -s ) B_{ls} T_{lm} {}_s Y_{lm}(\hat{n}).</math> Taking the ensemble average of the pseudo-Cl power spectrum of these <math>T_{lm}</math> we find<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \sum_{S S'} \sum_{l_1 l_2} \frac{(2l_1+1)(2l_2+1)}{4\pi}<br />
{}_{(-s -s')}{\cal W}_{l_1} B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}<br />
\\ \times\left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s\! & -s\! & 0\!<br />
\end{array}<br />
\right) \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s'\! & -s'\! & 0\!<br />
\end{array}<br />
\right)<br />
<br />
\end{gathered}</math><br />
<br />
where <math>{}_{(s s')}{\cal W}_{L} = \frac{1}{2L+1} \sum_{M} {}_{S} w_{LM} {}_{S'} w_{LM}^*</math> is a cross-power spectrum of scan history objects. Note that the w(n,s) which we have used here can also incorporate a position dependent weighting to optimize the pseudo-Cl estimate, such as inverse-noise or a mask– the equations are unchanged. Writing the pseudo-Cl in position space (a la <font color=red> Dvorkin and Smith (2009)</font>) with Wigner-d matrices we have<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \frac{1}{8\pi} \sum_{S S'} \int_{-1}^{1} dz \ d^{L}_{00}(z)<br />
\\ \times<br />
\left[\sum_{l_1} d^{l_1}_{-s -s'}(z) {}_{(-s -s')}{\cal W}_{l_1} (2l_1+1) \right] <br />
\\ \times<br />
\left[ \sum_{l_2} d^{l_2}_{s s'}(z) B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}(2l_2+1) \right].\end{gathered}</math><br />
<br />
This integral can be implemented exactly using Gauss-Legendre quadrature, with a cost of $\cal 0(l_{\rm max}^2 s_{\rm max}^2)$. For simplicity, we’ve written all the equations here for the auto-spectrum of a single detector, but the generalization to a map made by adding several detectors with different weighting is straightforward. The cost to compute all of the necessary terms exactly in that case becomes <math>\cal 0(l_{\rm max}^2 s_{\rm max}^2 N_{\rm det}^2)</math>.<br />
<br />
Are beams really so difficult? On the flat-sky beam convolution is easy: just multiplication in Fourier space by a beam rotated onto the scan direction. Multiple hits with different scan directions are incorporated by averaging (as the “scan history” objects above encapsulate). Does the sphere really require everything to be so complicated? For a scan strategy which is fairly smooth across the sky, we can pretend that we are observing many independent flat-sky patches at high-L with fairly good accuracy. There is in fact a fairly good approximation to the beam convolved pseudo-Cl power spectrum which is essentially a flat-sky approximation. In the limit that <math>L \gg l_1</math>, with <math>C_{l_2}</math> and <math>B_{l_2}</math> being slowly-varying function in <math>l_2</math> the pseudo-Cl sum above can be approximated as <math>{\tilde{C}}_L^{TT} = C_L^{TT} \sum_{M} \left< \left| w(\hat{n}_p, M) \right|^2 \right>_p |B_{L M}|^2,</math> where the average <math><>_p</math> is taken over the full sky. It’s illustrative to consider three limits of this equation: for a “raster” scan strategy in which each pixel is observed with the same direction, we have <math>\left< \left| w(\hat{n}, M) \right|^2 \right>_p = 1,</math> and the predicted pseudo-Cl is just the power spectrum of the beam. For a &quot;best-case&quot; scan strategy, in which each pixel is observed many times with many different orientation angles, we have &lt; | w(, M) |<sup>2</sup> &gt;<sub>p</sub> = <sub>M0</sub>, and the transfer function is just the azimuthally symmetric part of the beam. Note that this is for a full-sky observation– in the presence of a mask, the average above produces an fsky factor, as expected. It just neglects the coupling between L multipoles (which can be calculated with the more complete equations above).<br />
<br />
==== Effective beam window functions ====<br />
The effective beam window functions $B(l)$ for HFI, computed using Quickbeam, are available in the [[The RIMO|RIMO]].<br />
They do not contain the pixel window function.<br />
<br />
=== Pixelization Artifacts ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
<font color=red><br />
* Several codes available to simulate effects of pixelization.<br />
* Mixes the CMB gradient into a pixelization ``noise'' with a level comparable to that of $2\mu Karcmin$ instrumental noise.<br />
* Quantitative estimate of effect should be included with each released map, but expect not to matter significantly for CMB analysis, as small compared to instrumental noise. <br />
</font><br />
[sec:pixelization]<br />
<br />
Planck maps are produced at resolution 11 <math>(N_{\rm side} = 2048)</math>, corresponding to pixels with a typical dimension of <math>1.7'</math>, comparable to the spacing between scanning rings . This results in an uneven distribution of hits within pixels, which introduces some complications in the analysis and interpretation of the maps. A sample of the hit distribution is illustrated in Fig. [fig:pixcoverage]. Below we discuss the simulation and modeling of this pixelization effect in more detail.<br />
<br />
<br />
[[Image:pixcoverage.png|frame|none|alt=image]]<br />
<br />
[fig:pixcoverage]<br />
<br />
The collaboration has produced 3 codes which may be used to simulate the effect of pixelization on the observed sky, LevelS/TotalConvoler/Conviqt, FeBeCoP, and FICSBell <font color=red> references and further discussion of the three methods<br />
and how they each simulate the pixelization effect.</font>.<br />
<br />
For the measurement of CMB fluctuations, it is also possible to gain intuition for the effects of pixelization analytically. On the small scales relevant to pixelization, the observed CMB is smooth, both due to physical damping as well as the convolution of the instrumental beam. Taylor expanding the CMB temperature about a pixel center to second order, the typical gradient amplitude is given by <math>\langle |\nabla T |^2 \rangle = \frac{1}{4\pi} \sum_{l} l(l+1)(2l+1) C_l^{T} W_l \approx 1\times10^9 \mu K^2 / {\rm rad}^2.</math> where the approximate value is calculated for a <math>\Lambda CDM</math> cosmology with a <math>7'</math>fwhm Gaussian beam. The typical curvature of the observed temperature, on the other hand is given by <math>\langle |\nabla^2 T |^2 \rangle = \frac{1}{4\pi} \sum_{l} [l(l+1)]^2(2l+1) C_l^{T} W_l \approx 7\times10^{14} \mu K^2 / {\rm rad}^4.</math> On the scales relevant to the maximum displacement from the center of a <math>1.7'</math> pixel, the maximum displacement is , and so the gradient term tends to dominate, although the curvature term is still non-negligible. For each observation of a pixel, we can denote the displacement from the pixel center as <math>d = d_{\theta} + i d_{\phi}</math>. The average over all hits within a pixel gives an overall deflection vector which we will denote for a pixel center located at <math>\hat{n}</math> as <math>d(\hat{n})</math>. This represents the center of mass of the hit distribution; in Fig. [fig:pixcoverage] we have plotted these average deflections using black arrows. The deflection field <math>d(\hat{n})</math> may be decomposed into spin-1 spherical harmonics as <math>d_{lm} = \int_{4\pi} {}_1 Y_{lm}^* d(\hat{n}).</math> With a second order Taylor expansion of the CMB temperature about each pixel center, it is then possible to calculate the average pseudo-Cl power spectrum of the pixelized sky. This is given by<br />
<br />
<math>\begin{gathered}<br />
C_l^{T} = [1-l(l+1)R^d] {C}_l^{T} W_l + \\ <br />
\frac{1}{2} \sum_{l_1 l_2} \frac{l_1(l_1+1)(2l_1+1)(2l_2+1)}{4\pi} \\<br />
\times \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & l\! \\<br />
\! l\! & -l\! & 0\!<br />
\end{array}<br />
\right)^2 C_{l_1}^{T} W_{l_1} \left[ C_{l_2}^{d+} + (-1)^{l + l_1 + l_2} C_{l_2}^{ d-} \right],<br />
\label{eqn:clt_pixelized}\end{gathered}</math><br />
<br />
where <math>R^{d} = \langle |d|^2 \rangle/2</math> is half the mean-squared deflection magnitude (averaged over hits within a pixel, as well as over pixels). <math>C_l^{d+}</math> is the sum of the gradient and curl power spectra of <math>d_{lm}</math>, and <math>C_l^{d-}</math> is the gradient spectrum minus the curl spectrum. The <math>R^{d}</math> term describes a smearing of the observed sky due to pixelization. For uniform pixel coverage of <math>N_{\rm side}=2048</math> pixels <math>\sqrt{ \langle |d|^2 \rangle } = 0.725'</math>. For the hit distribution of Planck frequency maps, <math>R^{d}</math> is typically within <font color=red> xxx. calculate for final<br />
maps, looks like will be better than 10%</font>percent of this value, and so this term is accurately described by the pixel window function, which is derived under the assumption of uniform pixel coverage.<br />
<br />
The effect of pixelization is essentially degenerate with that of gravitational lensing of the CMB, with the difference that it (1) acts on the beam-convolved sky, rather than the actual sky and (2) produces a curl-mode deflection field as well as a gradient mode. This is discussed further in the [<cite>#planck2013-p12</cite>|Planck gravitational lensing] paper, where the subpixel deflection field constitutes a potential source of bias for the measured lensing potential. Indeed, Eq. [eqn:clt<sub>p</sub>ixelized] is just a slightly modified version of the usual first order CMB lensing power spectrum (<font color=red>Hu (2000)</font>, <font color=red>Lewis and Challinor (2006)</font>) to accommodate curl modes.<br />
<br />
A useful approximation to Eq. which is derived in the unrealistic limit that the deflection vectors are uncorrelated between pixels, but in practice gives a good description of the power induced by the pixelization, is that the <math>d(\hat{n})</math> couples the CMB gradient into a source of noise with an effective level given by <math>\sigma^{N} \approx \sqrt{ R^T \frac{4\pi}{N_{\rm pix}} \langle | d(\hat{n}) |^2<br />
\rangle }, % (\muKarcmin ),</math><br />
<br />
where the average is taken over all pixels and <math>R^T</math> is half the mean-squared power in the CMB gradient: <math>R^{T} = \frac{1}{8\pi} \sum_{l} l(l+1)(2l+1) \tilde{C}_l^{T}.</math> For frequency-combined maps, <math>\sqrt{ \langle | d(\hat{n}) |^2 \rangle }</math> is typically on the order of <math>0.1'</math>, and so the induced noise is at the level of <math>\sigma^{N} \sim 2 \mu K arcmin</math>. This is small compared to the instrumental contribution, although it does not disappear when taking cross-spectra, depending on how coherent the hit distributions of the two maps in the cross-spectrum are.<br />
<br />
= References =<br />
<biblio force=false><br />
#[[References]] <br />
</biblio></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Beams&diff=6857Beams2013-03-15T15:40:12Z<p>Ajaffe: fix ref</p>
<hr />
<div>== Scanning Beams ==<br />
<br />
The scanning beams describe the instrument’s instantaneous beam profile. Due to the near constant spin rate of the spacecraft, time domain effects (including residual time response and lowpass filtering) are degenerate with the spatial response due to the optical system. The scanning beam reconstruction recovers both of these effects, aside from residual time domain effects on a longer time scale than can be captured with the extent of the scanning beam model.<br />
<br />
In <cite>#planck2013-p03c</cite> we consider two models of the beam in order to better understand systematics in the reconstruction. Here we describe only the B-Spline beams which are used to compute the delivered effective beam (see next section).<br />
<br />
=== B-Spline Beam construction ===<br />
<br />
We use seasons 1 and 2 of the Mars observation to reconstruct the beam. The data are processed with the bigPlanets TOI processing. We use JPL Horizons ephemerides to determine the pointing of each detector relative to the planet. We subtract the astrophysical background in the time domain using a bicubic interpolation of the Planck maps.<br />
<br />
The time ordered data are used to fit a two dimensional B-Spline surface using a least square minimization and a smoothing criterion to minimize the effects of high spatial frequency variations. We therefore assume the scanning beam to be smooth. The smoothing criterion as well as the locations of the nodes used to compute the B-Spline basis functions are set using GRASP physical optics simulations as inputs which are the best assumptions on the spatial frequency content of the in-flight beams.<br />
<br />
The smoothing criterion is defined as follows:<br />
<br />
<math>\eta = \displaystyle{\sum_{i=1}^{g}\left(b^{k}(\lambda_{i+})-b^{k}(\lambda_{i-})\right)^2}<br />
\label{smoothcrit}</math><br />
<br />
<math>\begin{aligned}<br />
\eta &: \mbox{ Smoothing Criterion}\\<br />
b^k &: \mbox{ $k^{th}$ beam derivative evaluated on the nodes locations}\end{aligned}</math><br />
<br />
And the global inversion criterion :<br />
<br />
<math>\zeta = \eta + p\times \delta</math><br />
<br />
with <math>\delta</math> usual least square estimator and <math>p</math> coefficient giving the relative weight to <math>\delta</math> with respect to the smoothing criterion.<br />
<br />
<math>\delta = \displaystyle{\sum_{r=1}^{m}}\left(y_{r} - b(x_{r})\right)^2\label{estimator}</math><br />
<br />
<math>\begin{aligned}<br />
\delta &: \mbox{ usual least square criterion}\\<br />
r &: \mbox{ indice relative to the m data points, } r \in \{1, \ldots, m\}\\<br />
y_r &: \mbox{ planet data of sample r}\\<br />
x_r &: \mbox{ pointing of sample r}\\<br />
b &: \mbox{ reconstructed beam}\end{aligned}</math><br />
<br />
The B-Spline nodes are located on a regular spaced grid in the detector coordinate framset. At the edge of the reconstructed beam map area, 4 coincident nodes are added to avoid vanishing basis functions.<br />
<br />
Let <math>B_{i, k+1}</math>, <math>k</math> degree B-Spline build using nodes {<math>\lambda_{i}, ..., \lambda_{i+k+1}</math>} (''De Boor &amp; Cox'', 1972) :<br />
<br />
<math>B_{i,1}(x) = \left\{<br />
\begin{array}{l}<br />
1, \mbox{ si } x \in \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}\\<br />
0, \mbox{ si } x \notin \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}<br />
\end{array} \right.</math><br />
<br />
<math>B_{i, l+1}(x) = \displaystyle{\frac{x - \lambda_{i}}{\lambda_{i+l} - \lambda_{i}}} B_{i,l}(x) + \displaystyle{\frac{\lambda_{i+l+1}-x}{\lambda_{i+l+1}-\lambda_{i+1}}} B_{i+1, l}(x)</math><br />
<br />
<math>l=1, \ldots, k</math><br />
<br />
<br />
[[Image:FocalPlane_Map_BSScanningBeams_v53.png|800px|frame|center|Focal plane plot of B-Spline scanning beams using in-flight pointing reconstruction. The contours are -3,-10,-20,-30 dB from the peak, and for PSB pairs the "a" bolometer is plotted in black and "b" in blue.]]<br />
<br />
=== Simulations and errors ===<br />
<br />
We estimate the reconstruction bias and noise in the measurements using an ensemble of simulated planet observations for each channel. Kept fixed in each simulation are:<br />
<br />
* the input beam assumed: we use a supersampled version of the reconstructed B-Spline beam (or whatever comes out of the current ongoing tests!)<br />
* Astrophyical background is the same as that subtracted from the real data.<br />
* StarTracker pointing (using the ptcor6 pointing model).<br />
<br />
The following are varied in each simulation:<br />
<br />
* detector noise realizations obtained by filtering randomly generated white noise with the measured noise PSDs<br />
* random pointing errors with 2 arcsecond rms, and a spectrum that replicates the real errors.<br />
* simulated glitches and the deglitching procedure<br />
* Mars brightness temperature variability<br />
<br />
400 simulated timelines are generated for each bolometer and for each of the two seasons of Mars observations used in the beam reconstruction. The simulated timelines are made into beam maps, projecting onto the B-Spline basis in the same way as the real data.<br />
<br />
The beam maps are propagated to effective beam window functions using the quickbeam approach (see effective beams below) and used to evaluate the reconstruction bias and to construct error eigenmodes in the effective beam window function.<br />
<br />
<font color=red>Figure: random pointing error PSD Figures: error envelope plots (or should those go under effective beams?)</font><br />
<br />
=== Residuals ===<br />
<br />
There are two known beam effects that are not included in the main beam model and are estimated as a separate bias in flux and angular power spectrum measurement: 1. long tails due to errors in low frequency time response deconvolution, and 2. near sidelobes.<br />
<br />
We stack all five observations of Jupiter to estimate the long time scale residuals due to incomplete deconvolution of the long time scale response.<br />
<br />
<font color=red>Add some kind of mean tail plot</font><br />
<br />
Near sidelobes are also evaluated using stacked Jupiter (hopefully they will just be part of the v53bis B-Spline beams). The main features in the near sidelobes include a wide beam skirt, and dimpling lobes<br />
<font color=red>Add sidelobe plots and tables</font> <br />
<br />
<br />
== Effective Beams ==<br />
<br />
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. <br />
<br />
Several methods of effective beams determination have been developped and cross-validated.<br />
<br />
The main products are produced using FEBeCoP and details of the processing are given in the [[Effective Beams]] products page. See also the equivalent [[Beams_LFI | page discussing the LFI beams]]<br />
<br />
<font color=red>Need satisfactory comparison plot</font><br />
<br />
=== FEBeCoP ===<br />
-------------------<br />
<br />
The full algebra for this method for the calculation of effective beams was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]] <cite>#mitra2010</cite>. 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>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<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.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \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,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\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) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
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}.<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
<br />
<br />
=== FICSBell ===<br />
For more details, see <cite>#planck2013-p03c</cite>.<br />
<br />
Since the HFI beams are not azimuthally symmetric, the scanning strategy has to be taken into account in the effective beam response modelling. This is done using the FICSBell method <font color=red>(Hivon et al, in preparation)</font>, which generalizes to polarization and to include other sources of systematics the approach used for TT <math>C(l)</math> estimation in WMAP-3yr <font color=red>Hinshaw et al (2007)</font> and by <font color=red>Smith et al (2007)</font> in the detection of CMB lensing in WMAP maps. The different steps of the method used for this study can be summarized as follows:<br />
<br />
<ol><br />
<li><p>The scanning related information (i.e., statistics of the orientation of each detector within each pixel) is computed first, and only once for a given observation campaign. Those orientation hit moments are only computed up to degree 4, for reasons described in point 2 below. At the same time, the first two moments of the distribution of samples within each pixel (ie, their center of mass and moments of inertia) are computed and stored on disc.</p></li><br />
<li><p>The scanning beam map or beam model of each detector <math>d</math> is analyzed into its Spherical Harmonics coefficients</p><br />
<p><math>b^d_{ls} = \int d{\bf r} B_d({\bf r}) Y_{ls}({\bf r})\label{scanningBlm}</math></p><br />
<p>where <math>B_d(\bf{r})</math> is the beam map centered on the North pole, and <math>Y_{ls}(\bf{r})</math> is the Spherical Harmonics basis function. Higher <math>s</math> indexes describes higher degrees of departure from azimuthal symmetry and, for HFI beams, the coefficients <math>b^d_{ls}</math> are decreasing functions of <math>s</math> at most <math>l</math> considered. It also appears that, for <math>l<3000</math>, the coefficients with <math>|s| > 4</math> account for <math>1\%</math> or less of the beam throughput. For this reason, only modes with <math>|s| \le 4</math> are considered in the present analysis. <font color=red>Armitage-Caplan and Wandelt (2009)</font> reached a similar conclusion in their deconvolution of Planck-LFI beams.</p></li><li><p>The <math>b^d_{ls}</math> coefficients computed above are used to generate <math>s</math>-spin weighted maps, as well as the first and second order derivatives, for a given CMB sky realization.</p></li><br />
<li><p>The spin weighted maps and orientation hit moments of the same order <math>s</math> are combined for all detectors involved, to provide an “observed” map. Similarly the local spatial derivatives are combined with the location hit moments to describe the effect of the non-ideal sampling of each pixel (see [sec:pixelization]). In this combination, the respective number of hits of each detector in each pixel is considered, as well as the weighting (generally proportional to the inverse noise variance) applied to each detector in order to minimize the final noise.</p></li><li><p>The power spectrum of this map can then be computed, and compared to the input CMB power spectrum to estimate the effective beam window function over the whole sky, or over a given region of the sky.</p></li></ol><br />
Monte-Carlo (MC) simulations in which the sky realisations are changed can be performed by repeating steps 3, 4 and 5. The impact of beam model uncertainties can be studied by including step 2 into the MC simulations.<br />
<br />
=== QuickBeam ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
Planck observes the sky after convolution with a “scanning beam”, which captures its effective response to the sky as a function of displacement from the nominal pointing direction. Decomposing the scanning beam into harmonic coefficients <math>B_{lm}</math>, each time-ordered data (TOD) sample can be modelled as (neglecting the contribution from instrumental noise, which is independent of beam asymmetry) <math>%T_i = \sum_{lms} D^{l}_{-m s} (\phi_i, \theta_i, \alpha_i) b_{ls} (-1)^{m) T_{lm} + n_i,<br />
T_i = \sum_{lms} e^{-i s \alpha_i} B_{ls} \tilde{T}_{lm} {}_s Y_{lm}(\theta_i, \phi_i),<br />
\label{eqn:tod_beam}</math> where the TOD samples are indexed by <math>i</math>, and <math>\tilde{T}_{lm}</math> is the underlying sky signal. The spin spherical harmonic <math>{}_s Y_{lm}</math> rotates the scanning beam to the pointing location <math>(\theta, \phi)</math>, while the <math>e^{-i s \alpha_i}</math> factor gives it the correct orientation. Eq. may be evaluated with the “TotalConvolver” algorithm of <font color=red>Wandelt and Gorski (2001)</font>, accelerated using the “conviqt” recursion relations <font color=red>Prezeau and Reinecke (2010)</font> This approach is implemented in LevelS.<br />
</ref>, although because it involves working with a TOD-sized objected it is necessarily slow.<br />
<br />
On the small angular scales comparable to the size of the beam, it is a good approximation to assume that the procedure of mapmaking from TOD samples is essentially a process of binning: <math>T(p) = \sum_{i \in p} T_i / H(p),<br />
\label{eqn:map_beam_full}</math> where <math>H(p)</math> is the total number of hits in pixel <math>\hat{n}</math>.<br />
<br />
Start with a normalized, rescaled harmonic transform of the beam <math>B_{lm}</math>, sky multipoles <math>\tilde{T}_{lm}</math> and a scan history object <math>w(\hat{n}, s)</math> given by <math>w(\hat{n}, s) = \sum_{j \in p} e^{i s \alpha_j} / H(\hat{n})</math> where the sum is over all hits <math>j</math> of pixel <math>p</math> at location <math>\hat{n}_p</math>, and <math>\alpha_j</math> is the scan angle for observation <math>j</math>. The harmonic transform of this scan-strategy object is given by <math>{}_{s} w_{L M} = \int d^2 \hat{n} {}_s Y_{LM}^*(\hat{n}) w(\hat{n}, s).</math> The beam-convolved observation is then given by <math>\tilde{T}(\hat{n}) = \sum_{slm} w(\hat{n}, -s ) B_{ls} T_{lm} {}_s Y_{lm}(\hat{n}).</math> Taking the ensemble average of the pseudo-Cl power spectrum of these <math>T_{lm}</math> we find<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \sum_{S S'} \sum_{l_1 l_2} \frac{(2l_1+1)(2l_2+1)}{4\pi}<br />
{}_{(-s -s')}{\cal W}_{l_1} B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}<br />
\\ \times\left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s\! & -s\! & 0\!<br />
\end{array}<br />
\right) \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s'\! & -s'\! & 0\!<br />
\end{array}<br />
\right)<br />
<br />
\end{gathered}</math><br />
<br />
where <math>{}_{(s s')}{\cal W}_{L} = \frac{1}{2L+1} \sum_{M} {}_{S} w_{LM} {}_{S'} w_{LM}^*</math> is a cross-power spectrum of scan history objects. Note that the w(n,s) which we have used here can also incorporate a position dependent weighting to optimize the pseudo-Cl estimate, such as inverse-noise or a mask– the equations are unchanged. Writing the pseudo-Cl in position space (a la <font color=red> Dvorkin and Smith (2009)</font>) with Wigner-d matrices we have<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \frac{1}{8\pi} \sum_{S S'} \int_{-1}^{1} dz \ d^{L}_{00}(z)<br />
\\ \times<br />
\left[\sum_{l_1} d^{l_1}_{-s -s'}(z) {}_{(-s -s')}{\cal W}_{l_1} (2l_1+1) \right] <br />
\\ \times<br />
\left[ \sum_{l_2} d^{l_2}_{s s'}(z) B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}(2l_2+1) \right].\end{gathered}</math><br />
<br />
This integral can be implemented exactly using Gauss-Legendre quadrature, with a cost of $\cal 0(l_{\rm max}^2 s_{\rm max}^2)$. For simplicity, we’ve written all the equations here for the auto-spectrum of a single detector, but the generalization to a map made by adding several detectors with different weighting is straightforward. The cost to compute all of the necessary terms exactly in that case becomes <math>\cal 0(l_{\rm max}^2 s_{\rm max}^2 N_{\rm det}^2)</math>.<br />
<br />
Are beams really so difficult? On the flat-sky beam convolution is easy: just multiplication in Fourier space by a beam rotated onto the scan direction. Multiple hits with different scan directions are incorporated by averaging (as the “scan history” objects above encapsulate). Does the sphere really require everything to be so complicated? For a scan strategy which is fairly smooth across the sky, we can pretend that we are observing many independent flat-sky patches at high-L with fairly good accuracy. There is in fact a fairly good approximation to the beam convolved pseudo-Cl power spectrum which is essentially a flat-sky approximation. In the limit that <math>L \gg l_1</math>, with <math>C_{l_2}</math> and <math>B_{l_2}</math> being slowly-varying function in <math>l_2</math> the pseudo-Cl sum above can be approximated as <math>{\tilde{C}}_L^{TT} = C_L^{TT} \sum_{M} \left< \left| w(\hat{n}_p, M) \right|^2 \right>_p |B_{L M}|^2,</math> where the average <math><>_p</math> is taken over the full sky. It’s illustrative to consider three limits of this equation: for a “raster” scan strategy in which each pixel is observed with the same direction, we have <math>\left< \left| w(\hat{n}, M) \right|^2 \right>_p = 1,</math> and the predicted pseudo-Cl is just the power spectrum of the beam. For a &quot;best-case&quot; scan strategy, in which each pixel is observed many times with many different orientation angles, we have &lt; | w(, M) |<sup>2</sup> &gt;<sub>p</sub> = <sub>M0</sub>, and the transfer function is just the azimuthally symmetric part of the beam. Note that this is for a full-sky observation– in the presence of a mask, the average above produces an fsky factor, as expected. It just neglects the coupling between L multipoles (which can be calculated with the more complete equations above).<br />
<br />
==== Effective beam window functions ====<br />
The effective beam window functions $B(l)$ for HFI, computed using Quickbeam, are available in the [[The RIMO|RIMO]].<br />
They do not contain the pixel window function.<br />
<br />
=== Pixelization Artifacts ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
<font color=red><br />
* Several codes available to simulate effects of pixelization.<br />
* Mixes the CMB gradient into a pixelization ``noise'' with a level comparable to that of $2\mu Karcmin$ instrumental noise.<br />
* Quantitative estimate of effect should be included with each released map, but expect not to matter significantly for CMB analysis, as small compared to instrumental noise. <br />
</font><br />
[sec:pixelization]<br />
<br />
Planck maps are produced at resolution 11 <math>(N_{\rm side} = 2048)</math>, corresponding to pixels with a typical dimension of <math>1.7'</math>, comparable to the spacing between scanning rings . This results in an uneven distribution of hits within pixels, which introduces some complications in the analysis and interpretation of the maps. A sample of the hit distribution is illustrated in Fig. [fig:pixcoverage]. Below we discuss the simulation and modeling of this pixelization effect in more detail.<br />
<br />
<br />
[[Image:pixcoverage.png|frame|none|alt=image]]<br />
<br />
[fig:pixcoverage]<br />
<br />
The collaboration has produced 3 codes which may be used to simulate the effect of pixelization on the observed sky, LevelS/TotalConvoler/Conviqt, FeBeCoP, and FICSBell <font color=red> references and further discussion of the three methods<br />
and how they each simulate the pixelization effect.</font>.<br />
<br />
For the measurement of CMB fluctuations, it is also possible to gain intuition for the effects of pixelization analytically. On the small scales relevant to pixelization, the observed CMB is smooth, both due to physical damping as well as the convolution of the instrumental beam. Taylor expanding the CMB temperature about a pixel center to second order, the typical gradient amplitude is given by <math>\langle |\nabla T |^2 \rangle = \frac{1}{4\pi} \sum_{l} l(l+1)(2l+1) C_l^{T} W_l \approx 1\times10^9 \mu K^2 / {\rm rad}^2.</math> where the approximate value is calculated for a <math>\Lambda CDM</math> cosmology with a <math>7'</math>fwhm Gaussian beam. The typical curvature of the observed temperature, on the other hand is given by <math>\langle |\nabla^2 T |^2 \rangle = \frac{1}{4\pi} \sum_{l} [l(l+1)]^2(2l+1) C_l^{T} W_l \approx 7\times10^{14} \mu K^2 / {\rm rad}^4.</math> On the scales relevant to the maximum displacement from the center of a <math>1.7'</math> pixel, the maximum displacement is , and so the gradient term tends to dominate, although the curvature term is still non-negligible. For each observation of a pixel, we can denote the displacement from the pixel center as <math>d = d_{\theta} + i d_{\phi}</math>. The average over all hits within a pixel gives an overall deflection vector which we will denote for a pixel center located at <math>\hat{n}</math> as <math>d(\hat{n})</math>. This represents the center of mass of the hit distribution; in Fig. [fig:pixcoverage] we have plotted these average deflections using black arrows. The deflection field <math>d(\hat{n})</math> may be decomposed into spin-1 spherical harmonics as <math>d_{lm} = \int_{4\pi} {}_1 Y_{lm}^* d(\hat{n}).</math> With a second order Taylor expansion of the CMB temperature about each pixel center, it is then possible to calculate the average pseudo-Cl power spectrum of the pixelized sky. This is given by<br />
<br />
<math>\begin{gathered}<br />
C_l^{T} = [1-l(l+1)R^d] {C}_l^{T} W_l + \\ <br />
\frac{1}{2} \sum_{l_1 l_2} \frac{l_1(l_1+1)(2l_1+1)(2l_2+1)}{4\pi} \\<br />
\times \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & l\! \\<br />
\! l\! & -l\! & 0\!<br />
\end{array}<br />
\right)^2 C_{l_1}^{T} W_{l_1} \left[ C_{l_2}^{d+} + (-1)^{l + l_1 + l_2} C_{l_2}^{ d-} \right],<br />
\label{eqn:clt_pixelized}\end{gathered}</math><br />
<br />
where <math>R^{d} = \langle |d|^2 \rangle/2</math> is half the mean-squared deflection magnitude (averaged over hits within a pixel, as well as over pixels). <math>C_l^{d+}</math> is the sum of the gradient and curl power spectra of <math>d_{lm}</math>, and <math>C_l^{d-}</math> is the gradient spectrum minus the curl spectrum. The <math>R^{d}</math> term describes a smearing of the observed sky due to pixelization. For uniform pixel coverage of <math>N_{\rm side}=2048</math> pixels <math>\sqrt{ \langle |d|^2 \rangle } = 0.725'</math>. For the hit distribution of Planck frequency maps, <math>R^{d}</math> is typically within <font color=red> xxx. calculate for final<br />
maps, looks like will be better than 10%</font>percent of this value, and so this term is accurately described by the pixel window function, which is derived under the assumption of uniform pixel coverage.<br />
<br />
The effect of pixelization is essentially degenerate with that of gravitational lensing of the CMB, with the difference that it (1) acts on the beam-convolved sky, rather than the actual sky and (2) produces a curl-mode deflection field as well as a gradient mode. This is discussed further in the [<cite>#planck2013-p12</cite>|Planck gravitational lensing] paper, where the subpixel deflection field constitutes a potential source of bias for the measured lensing potential. Indeed, Eq. [eqn:clt<sub>p</sub>ixelized] is just a slightly modified version of the usual first order CMB lensing power spectrum (<font color=red>Hu (2000)</font>, <font color=red>Lewis and Challinor (2006)</font>) to accommodate curl modes.<br />
<br />
A useful approximation to Eq. which is derived in the unrealistic limit that the deflection vectors are uncorrelated between pixels, but in practice gives a good description of the power induced by the pixelization, is that the <math>d(\hat{n})</math> couples the CMB gradient into a source of noise with an effective level given by <math>\sigma^{N} \approx \sqrt{ R^T \frac{4\pi}{N_{\rm pix}} \langle | d(\hat{n}) |^2<br />
\rangle }, % (\muKarcmin ),</math><br />
<br />
where the average is taken over all pixels and <math>R^T</math> is half the mean-squared power in the CMB gradient: <math>R^{T} = \frac{1}{8\pi} \sum_{l} l(l+1)(2l+1) \tilde{C}_l^{T}.</math> For frequency-combined maps, <math>\sqrt{ \langle | d(\hat{n}) |^2 \rangle }</math> is typically on the order of <math>0.1'</math>, and so the induced noise is at the level of <math>\sigma^{N} \sim 2 \mu K arcmin</math>. This is small compared to the instrumental contribution, although it does not disappear when taking cross-spectra, depending on how coherent the hit distributions of the two maps in the cross-spectrum are.<br />
<br />
= References =<br />
<biblio force=false><br />
#[[References]] <br />
</biblio></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Beams&diff=6854Beams2013-03-15T15:38:52Z<p>Ajaffe: </p>
<hr />
<div>== Scanning Beams ==<br />
<br />
The scanning beams describe the instrument’s instantaneous beam profile. Due to the near constant spin rate of the spacecraft, time domain effects (including residual time response and lowpass filtering) are degenerate with the spatial response due to the optical system. The scanning beam reconstruction recovers both of these effects, aside from residual time domain effects on a longer time scale than can be captured with the extent of the scanning beam model.<br />
<br />
In the paper <font color=red> cite P03c</font> we consider two models of the beam in order to better understand systematics in the reconstruction. Here we describe only the B-Spline beams which are used to compute the delivered effective beam (see next section).<br />
<br />
=== B-Spline Beam construction ===<br />
<br />
We use seasons 1 and 2 of the Mars observation to reconstruct the beam. The data are processed with the bigPlanets TOI processing. We use JPL Horizons ephemerides to determine the pointing of each detector relative to the planet. We subtract the astrophysical background in the time domain using a bicubic interpolation of the Planck maps.<br />
<br />
The time ordered data are used to fit a two dimensional B-Spline surface using a least square minimization and a smoothing criterion to minimize the effects of high spatial frequency variations. We therefore assume the scanning beam to be smooth. The smoothing criterion as well as the locations of the nodes used to compute the B-Spline basis functions are set using GRASP physical optics simulations as inputs which are the best assumptions on the spatial frequency content of the in-flight beams.<br />
<br />
The smoothing criterion is defined as follows:<br />
<br />
<math>\eta = \displaystyle{\sum_{i=1}^{g}\left(b^{k}(\lambda_{i+})-b^{k}(\lambda_{i-})\right)^2}<br />
\label{smoothcrit}</math><br />
<br />
<math>\begin{aligned}<br />
\eta &: \mbox{ Smoothing Criterion}\\<br />
b^k &: \mbox{ $k^{th}$ beam derivative evaluated on the nodes locations}\end{aligned}</math><br />
<br />
And the global inversion criterion :<br />
<br />
<math>\zeta = \eta + p\times \delta</math><br />
<br />
with <math>\delta</math> usual least square estimator and <math>p</math> coefficient giving the relative weight to <math>\delta</math> with respect to the smoothing criterion.<br />
<br />
<math>\delta = \displaystyle{\sum_{r=1}^{m}}\left(y_{r} - b(x_{r})\right)^2\label{estimator}</math><br />
<br />
<math>\begin{aligned}<br />
\delta &: \mbox{ usual least square criterion}\\<br />
r &: \mbox{ indice relative to the m data points, } r \in \{1, \ldots, m\}\\<br />
y_r &: \mbox{ planet data of sample r}\\<br />
x_r &: \mbox{ pointing of sample r}\\<br />
b &: \mbox{ reconstructed beam}\end{aligned}</math><br />
<br />
The B-Spline nodes are located on a regular spaced grid in the detector coordinate framset. At the edge of the reconstructed beam map area, 4 coincident nodes are added to avoid vanishing basis functions.<br />
<br />
Let <math>B_{i, k+1}</math>, <math>k</math> degree B-Spline build using nodes {<math>\lambda_{i}, ..., \lambda_{i+k+1}</math>} (''De Boor &amp; Cox'', 1972) :<br />
<br />
<math>B_{i,1}(x) = \left\{<br />
\begin{array}{l}<br />
1, \mbox{ si } x \in \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}\\<br />
0, \mbox{ si } x \notin \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}<br />
\end{array} \right.</math><br />
<br />
<math>B_{i, l+1}(x) = \displaystyle{\frac{x - \lambda_{i}}{\lambda_{i+l} - \lambda_{i}}} B_{i,l}(x) + \displaystyle{\frac{\lambda_{i+l+1}-x}{\lambda_{i+l+1}-\lambda_{i+1}}} B_{i+1, l}(x)</math><br />
<br />
<math>l=1, \ldots, k</math><br />
<br />
<br />
[[Image:FocalPlane_Map_BSScanningBeams_v53.png|800px|frame|center|Focal plane plot of B-Spline scanning beams using in-flight pointing reconstruction. The contours are -3,-10,-20,-30 dB from the peak, and for PSB pairs the "a" bolometer is plotted in black and "b" in blue.]]<br />
<br />
=== Simulations and errors ===<br />
<br />
We estimate the reconstruction bias and noise in the measurements using an ensemble of simulated planet observations for each channel. Kept fixed in each simulation are:<br />
<br />
* the input beam assumed: we use a supersampled version of the reconstructed B-Spline beam (or whatever comes out of the current ongoing tests!)<br />
* Astrophyical background is the same as that subtracted from the real data.<br />
* StarTracker pointing (using the ptcor6 pointing model).<br />
<br />
The following are varied in each simulation:<br />
<br />
* detector noise realizations obtained by filtering randomly generated white noise with the measured noise PSDs<br />
* random pointing errors with 2 arcsecond rms, and a spectrum that replicates the real errors.<br />
* simulated glitches and the deglitching procedure<br />
* Mars brightness temperature variability<br />
<br />
400 simulated timelines are generated for each bolometer and for each of the two seasons of Mars observations used in the beam reconstruction. The simulated timelines are made into beam maps, projecting onto the B-Spline basis in the same way as the real data.<br />
<br />
The beam maps are propagated to effective beam window functions using the quickbeam approach (see effective beams below) and used to evaluate the reconstruction bias and to construct error eigenmodes in the effective beam window function.<br />
<br />
<font color=red>Figure: random pointing error PSD Figures: error envelope plots (or should those go under effective beams?)</font><br />
<br />
=== Residuals ===<br />
<br />
There are two known beam effects that are not included in the main beam model and are estimated as a separate bias in flux and angular power spectrum measurement: 1. long tails due to errors in low frequency time response deconvolution, and 2. near sidelobes.<br />
<br />
We stack all five observations of Jupiter to estimate the long time scale residuals due to incomplete deconvolution of the long time scale response.<br />
<br />
<font color=red>Add some kind of mean tail plot</font><br />
<br />
Near sidelobes are also evaluated using stacked Jupiter (hopefully they will just be part of the v53bis B-Spline beams). The main features in the near sidelobes include a wide beam skirt, and dimpling lobes<br />
<font color=red>Add sidelobe plots and tables</font> <br />
<br />
<br />
== Effective Beams ==<br />
<br />
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. <br />
<br />
Several methods of effective beams determination have been developped and cross-validated.<br />
<br />
The main products are produced using FEBeCoP and details of the processing are given in the [[Effective Beams]] products page. See also the equivalent [[Beams_LFI | page discussing the LFI beams]]<br />
<br />
<font color=red>Need satisfactory comparison plot</font><br />
<br />
=== FEBeCoP ===<br />
-------------------<br />
<br />
The full algebra for this method for the calculation of effective beams was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]] <cite>#mitra2010</cite>. 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>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<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.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \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,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\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) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
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}.<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
<br />
<br />
=== FICSBell ===<br />
For more details, see <cite>#planck2013-p03c</cite>.<br />
<br />
Since the HFI beams are not azimuthally symmetric, the scanning strategy has to be taken into account in the effective beam response modelling. This is done using the FICSBell method <font color=red>(Hivon et al, in preparation)</font>, which generalizes to polarization and to include other sources of systematics the approach used for TT <math>C(l)</math> estimation in WMAP-3yr <font color=red>Hinshaw et al (2007)</font> and by <font color=red>Smith et al (2007)</font> in the detection of CMB lensing in WMAP maps. The different steps of the method used for this study can be summarized as follows:<br />
<br />
<ol><br />
<li><p>The scanning related information (i.e., statistics of the orientation of each detector within each pixel) is computed first, and only once for a given observation campaign. Those orientation hit moments are only computed up to degree 4, for reasons described in point 2 below. At the same time, the first two moments of the distribution of samples within each pixel (ie, their center of mass and moments of inertia) are computed and stored on disc.</p></li><br />
<li><p>The scanning beam map or beam model of each detector <math>d</math> is analyzed into its Spherical Harmonics coefficients</p><br />
<p><math>b^d_{ls} = \int d{\bf r} B_d({\bf r}) Y_{ls}({\bf r})\label{scanningBlm}</math></p><br />
<p>where <math>B_d(\bf{r})</math> is the beam map centered on the North pole, and <math>Y_{ls}(\bf{r})</math> is the Spherical Harmonics basis function. Higher <math>s</math> indexes describes higher degrees of departure from azimuthal symmetry and, for HFI beams, the coefficients <math>b^d_{ls}</math> are decreasing functions of <math>s</math> at most <math>l</math> considered. It also appears that, for <math>l<3000</math>, the coefficients with <math>|s| > 4</math> account for <math>1\%</math> or less of the beam throughput. For this reason, only modes with <math>|s| \le 4</math> are considered in the present analysis. <font color=red>Armitage-Caplan and Wandelt (2009)</font> reached a similar conclusion in their deconvolution of Planck-LFI beams.</p></li><li><p>The <math>b^d_{ls}</math> coefficients computed above are used to generate <math>s</math>-spin weighted maps, as well as the first and second order derivatives, for a given CMB sky realization.</p></li><br />
<li><p>The spin weighted maps and orientation hit moments of the same order <math>s</math> are combined for all detectors involved, to provide an “observed” map. Similarly the local spatial derivatives are combined with the location hit moments to describe the effect of the non-ideal sampling of each pixel (see [sec:pixelization]). In this combination, the respective number of hits of each detector in each pixel is considered, as well as the weighting (generally proportional to the inverse noise variance) applied to each detector in order to minimize the final noise.</p></li><li><p>The power spectrum of this map can then be computed, and compared to the input CMB power spectrum to estimate the effective beam window function over the whole sky, or over a given region of the sky.</p></li></ol><br />
Monte-Carlo (MC) simulations in which the sky realisations are changed can be performed by repeating steps 3, 4 and 5. The impact of beam model uncertainties can be studied by including step 2 into the MC simulations.<br />
<br />
=== QuickBeam ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
Planck observes the sky after convolution with a “scanning beam”, which captures its effective response to the sky as a function of displacement from the nominal pointing direction. Decomposing the scanning beam into harmonic coefficients <math>B_{lm}</math>, each time-ordered data (TOD) sample can be modelled as (neglecting the contribution from instrumental noise, which is independent of beam asymmetry) <math>%T_i = \sum_{lms} D^{l}_{-m s} (\phi_i, \theta_i, \alpha_i) b_{ls} (-1)^{m) T_{lm} + n_i,<br />
T_i = \sum_{lms} e^{-i s \alpha_i} B_{ls} \tilde{T}_{lm} {}_s Y_{lm}(\theta_i, \phi_i),<br />
\label{eqn:tod_beam}</math> where the TOD samples are indexed by <math>i</math>, and <math>\tilde{T}_{lm}</math> is the underlying sky signal. The spin spherical harmonic <math>{}_s Y_{lm}</math> rotates the scanning beam to the pointing location <math>(\theta, \phi)</math>, while the <math>e^{-i s \alpha_i}</math> factor gives it the correct orientation. Eq. may be evaluated with the “TotalConvolver” algorithm of <font color=red>Wandelt and Gorski (2001)</font>, accelerated using the “conviqt” recursion relations <font color=red>Prezeau and Reinecke (2010)</font> This approach is implemented in LevelS.<br />
</ref>, although because it involves working with a TOD-sized objected it is necessarily slow.<br />
<br />
On the small angular scales comparable to the size of the beam, it is a good approximation to assume that the procedure of mapmaking from TOD samples is essentially a process of binning: <math>T(p) = \sum_{i \in p} T_i / H(p),<br />
\label{eqn:map_beam_full}</math> where <math>H(p)</math> is the total number of hits in pixel <math>\hat{n}</math>.<br />
<br />
Start with a normalized, rescaled harmonic transform of the beam <math>B_{lm}</math>, sky multipoles <math>\tilde{T}_{lm}</math> and a scan history object <math>w(\hat{n}, s)</math> given by <math>w(\hat{n}, s) = \sum_{j \in p} e^{i s \alpha_j} / H(\hat{n})</math> where the sum is over all hits <math>j</math> of pixel <math>p</math> at location <math>\hat{n}_p</math>, and <math>\alpha_j</math> is the scan angle for observation <math>j</math>. The harmonic transform of this scan-strategy object is given by <math>{}_{s} w_{L M} = \int d^2 \hat{n} {}_s Y_{LM}^*(\hat{n}) w(\hat{n}, s).</math> The beam-convolved observation is then given by <math>\tilde{T}(\hat{n}) = \sum_{slm} w(\hat{n}, -s ) B_{ls} T_{lm} {}_s Y_{lm}(\hat{n}).</math> Taking the ensemble average of the pseudo-Cl power spectrum of these <math>T_{lm}</math> we find<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \sum_{S S'} \sum_{l_1 l_2} \frac{(2l_1+1)(2l_2+1)}{4\pi}<br />
{}_{(-s -s')}{\cal W}_{l_1} B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}<br />
\\ \times\left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s\! & -s\! & 0\!<br />
\end{array}<br />
\right) \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s'\! & -s'\! & 0\!<br />
\end{array}<br />
\right)<br />
<br />
\end{gathered}</math><br />
<br />
where <math>{}_{(s s')}{\cal W}_{L} = \frac{1}{2L+1} \sum_{M} {}_{S} w_{LM} {}_{S'} w_{LM}^*</math> is a cross-power spectrum of scan history objects. Note that the w(n,s) which we have used here can also incorporate a position dependent weighting to optimize the pseudo-Cl estimate, such as inverse-noise or a mask– the equations are unchanged. Writing the pseudo-Cl in position space (a la <font color=red> Dvorkin and Smith (2009)</font>) with Wigner-d matrices we have<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \frac{1}{8\pi} \sum_{S S'} \int_{-1}^{1} dz \ d^{L}_{00}(z)<br />
\\ \times<br />
\left[\sum_{l_1} d^{l_1}_{-s -s'}(z) {}_{(-s -s')}{\cal W}_{l_1} (2l_1+1) \right] <br />
\\ \times<br />
\left[ \sum_{l_2} d^{l_2}_{s s'}(z) B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}(2l_2+1) \right].\end{gathered}</math><br />
<br />
This integral can be implemented exactly using Gauss-Legendre quadrature, with a cost of $\cal 0(l_{\rm max}^2 s_{\rm max}^2)$. For simplicity, we’ve written all the equations here for the auto-spectrum of a single detector, but the generalization to a map made by adding several detectors with different weighting is straightforward. The cost to compute all of the necessary terms exactly in that case becomes <math>\cal 0(l_{\rm max}^2 s_{\rm max}^2 N_{\rm det}^2)</math>.<br />
<br />
Are beams really so difficult? On the flat-sky beam convolution is easy: just multiplication in Fourier space by a beam rotated onto the scan direction. Multiple hits with different scan directions are incorporated by averaging (as the “scan history” objects above encapsulate). Does the sphere really require everything to be so complicated? For a scan strategy which is fairly smooth across the sky, we can pretend that we are observing many independent flat-sky patches at high-L with fairly good accuracy. There is in fact a fairly good approximation to the beam convolved pseudo-Cl power spectrum which is essentially a flat-sky approximation. In the limit that <math>L \gg l_1</math>, with <math>C_{l_2}</math> and <math>B_{l_2}</math> being slowly-varying function in <math>l_2</math> the pseudo-Cl sum above can be approximated as <math>{\tilde{C}}_L^{TT} = C_L^{TT} \sum_{M} \left< \left| w(\hat{n}_p, M) \right|^2 \right>_p |B_{L M}|^2,</math> where the average <math><>_p</math> is taken over the full sky. It’s illustrative to consider three limits of this equation: for a “raster” scan strategy in which each pixel is observed with the same direction, we have <math>\left< \left| w(\hat{n}, M) \right|^2 \right>_p = 1,</math> and the predicted pseudo-Cl is just the power spectrum of the beam. For a &quot;best-case&quot; scan strategy, in which each pixel is observed many times with many different orientation angles, we have &lt; | w(, M) |<sup>2</sup> &gt;<sub>p</sub> = <sub>M0</sub>, and the transfer function is just the azimuthally symmetric part of the beam. Note that this is for a full-sky observation– in the presence of a mask, the average above produces an fsky factor, as expected. It just neglects the coupling between L multipoles (which can be calculated with the more complete equations above).<br />
<br />
==== Effective beam window functions ====<br />
The effective beam window functions $B(l)$ for HFI, computed using Quickbeam, are available in the [[The RIMO|RIMO]].<br />
They do not contain the pixel window function.<br />
<br />
=== Pixelization Artifacts ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
<font color=red><br />
* Several codes available to simulate effects of pixelization.<br />
* Mixes the CMB gradient into a pixelization ``noise'' with a level comparable to that of $2\mu Karcmin$ instrumental noise.<br />
* Quantitative estimate of effect should be included with each released map, but expect not to matter significantly for CMB analysis, as small compared to instrumental noise. <br />
</font><br />
[sec:pixelization]<br />
<br />
Planck maps are produced at resolution 11 <math>(N_{\rm side} = 2048)</math>, corresponding to pixels with a typical dimension of <math>1.7'</math>, comparable to the spacing between scanning rings . This results in an uneven distribution of hits within pixels, which introduces some complications in the analysis and interpretation of the maps. A sample of the hit distribution is illustrated in Fig. [fig:pixcoverage]. Below we discuss the simulation and modeling of this pixelization effect in more detail.<br />
<br />
<br />
[[Image:pixcoverage.png|frame|none|alt=image]]<br />
<br />
[fig:pixcoverage]<br />
<br />
The collaboration has produced 3 codes which may be used to simulate the effect of pixelization on the observed sky, LevelS/TotalConvoler/Conviqt, FeBeCoP, and FICSBell <font color=red> references and further discussion of the three methods<br />
and how they each simulate the pixelization effect.</font>.<br />
<br />
For the measurement of CMB fluctuations, it is also possible to gain intuition for the effects of pixelization analytically. On the small scales relevant to pixelization, the observed CMB is smooth, both due to physical damping as well as the convolution of the instrumental beam. Taylor expanding the CMB temperature about a pixel center to second order, the typical gradient amplitude is given by <math>\langle |\nabla T |^2 \rangle = \frac{1}{4\pi} \sum_{l} l(l+1)(2l+1) C_l^{T} W_l \approx 1\times10^9 \mu K^2 / {\rm rad}^2.</math> where the approximate value is calculated for a <math>\Lambda CDM</math> cosmology with a <math>7'</math>fwhm Gaussian beam. The typical curvature of the observed temperature, on the other hand is given by <math>\langle |\nabla^2 T |^2 \rangle = \frac{1}{4\pi} \sum_{l} [l(l+1)]^2(2l+1) C_l^{T} W_l \approx 7\times10^{14} \mu K^2 / {\rm rad}^4.</math> On the scales relevant to the maximum displacement from the center of a <math>1.7'</math> pixel, the maximum displacement is , and so the gradient term tends to dominate, although the curvature term is still non-negligible. For each observation of a pixel, we can denote the displacement from the pixel center as <math>d = d_{\theta} + i d_{\phi}</math>. The average over all hits within a pixel gives an overall deflection vector which we will denote for a pixel center located at <math>\hat{n}</math> as <math>d(\hat{n})</math>. This represents the center of mass of the hit distribution; in Fig. [fig:pixcoverage] we have plotted these average deflections using black arrows. The deflection field <math>d(\hat{n})</math> may be decomposed into spin-1 spherical harmonics as <math>d_{lm} = \int_{4\pi} {}_1 Y_{lm}^* d(\hat{n}).</math> With a second order Taylor expansion of the CMB temperature about each pixel center, it is then possible to calculate the average pseudo-Cl power spectrum of the pixelized sky. This is given by<br />
<br />
<math>\begin{gathered}<br />
C_l^{T} = [1-l(l+1)R^d] {C}_l^{T} W_l + \\ <br />
\frac{1}{2} \sum_{l_1 l_2} \frac{l_1(l_1+1)(2l_1+1)(2l_2+1)}{4\pi} \\<br />
\times \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & l\! \\<br />
\! l\! & -l\! & 0\!<br />
\end{array}<br />
\right)^2 C_{l_1}^{T} W_{l_1} \left[ C_{l_2}^{d+} + (-1)^{l + l_1 + l_2} C_{l_2}^{ d-} \right],<br />
\label{eqn:clt_pixelized}\end{gathered}</math><br />
<br />
where <math>R^{d} = \langle |d|^2 \rangle/2</math> is half the mean-squared deflection magnitude (averaged over hits within a pixel, as well as over pixels). <math>C_l^{d+}</math> is the sum of the gradient and curl power spectra of <math>d_{lm}</math>, and <math>C_l^{d-}</math> is the gradient spectrum minus the curl spectrum. The <math>R^{d}</math> term describes a smearing of the observed sky due to pixelization. For uniform pixel coverage of <math>N_{\rm side}=2048</math> pixels <math>\sqrt{ \langle |d|^2 \rangle } = 0.725'</math>. For the hit distribution of Planck frequency maps, <math>R^{d}</math> is typically within <font color=red> xxx. calculate for final<br />
maps, looks like will be better than 10%</font>percent of this value, and so this term is accurately described by the pixel window function, which is derived under the assumption of uniform pixel coverage.<br />
<br />
The effect of pixelization is essentially degenerate with that of gravitational lensing of the CMB, with the difference that it (1) acts on the beam-convolved sky, rather than the actual sky and (2) produces a curl-mode deflection field as well as a gradient mode. This is discussed further in the [<cite>#planck2013-p12</cite>|Planck gravitational lensing] paper, where the subpixel deflection field constitutes a potential source of bias for the measured lensing potential. Indeed, Eq. [eqn:clt<sub>p</sub>ixelized] is just a slightly modified version of the usual first order CMB lensing power spectrum (<font color=red>Hu (2000)</font>, <font color=red>Lewis and Challinor (2006)</font>) to accommodate curl modes.<br />
<br />
A useful approximation to Eq. which is derived in the unrealistic limit that the deflection vectors are uncorrelated between pixels, but in practice gives a good description of the power induced by the pixelization, is that the <math>d(\hat{n})</math> couples the CMB gradient into a source of noise with an effective level given by <math>\sigma^{N} \approx \sqrt{ R^T \frac{4\pi}{N_{\rm pix}} \langle | d(\hat{n}) |^2<br />
\rangle }, % (\muKarcmin ),</math><br />
<br />
where the average is taken over all pixels and <math>R^T</math> is half the mean-squared power in the CMB gradient: <math>R^{T} = \frac{1}{8\pi} \sum_{l} l(l+1)(2l+1) \tilde{C}_l^{T}.</math> For frequency-combined maps, <math>\sqrt{ \langle | d(\hat{n}) |^2 \rangle }</math> is typically on the order of <math>0.1'</math>, and so the induced noise is at the level of <math>\sigma^{N} \sim 2 \mu K arcmin</math>. This is small compared to the instrumental contribution, although it does not disappear when taking cross-spectra, depending on how coherent the hit distributions of the two maps in the cross-spectrum are.<br />
<br />
= References =<br />
<biblio force=false><br />
#[[References]] <br />
</biblio></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Effective_Beams&diff=6832Effective Beams2013-03-15T15:17:45Z<p>Ajaffe: all common LFI/HFI result info here</p>
<hr />
<div><span style="color:red"></span><br />
<br />
==Product description==<br />
----------------------<br />
<br />
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.<br />
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. <br />
<br />
Details of the beam processing are given in the respective pages for [[Beams|HFI]] and [[Beams_LFI|LFI]].<br />
<br />
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.]] <cite>#mitra2010</cite>, and a discussion of its application to Planck data is given in the appropriate LFI <cite>#planck2013-p02d</cite> and HFI <cite>#planck2013-p03c</cite> papers. Relevant details of the processing steps are given in the [[Beams|Effective Beams]] section of this document.<br />
<br />
<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for all LFI and HFI frequency channels, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* 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).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
<br />
<gallery widths=350px heights=350px perrow=3 caption="Comparison images of compact sources and effective beams, PSFs"><br />
File:30.png| '''30GHz'''<br />
File:44.png| '''44GHz'''<br />
File:70.png| '''70GHz'''<br />
File:100.png| '''100GHz'''<br />
File:143.png| '''143GHz'''<br />
File:217.png| '''217GHz'''<br />
File:353.png| '''353GHz'''<br />
File:545.png| '''545GHz'''<br />
File:857.png| '''857GHz'''<br />
</gallery><br />
<br />
<br />
===Histograms of the effective beam parameters====<br />
<br />
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.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
[[File:ist_BS_Mars12.png | 600px| thumb | center| '''Histograms for HFI effective beam parameters''' ]]<br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
<br />
* 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.<br />
<br />
<br />
<gallery widths=500px heights=500px perrow=2 caption="Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian"><br />
File:e_030_GB.png| '''ellipticity - 30GHz'''<br />
File:solidarc_030_GB.png| '''beam solid angle (relative variations wrt scanning beam - 30GHz'''<br />
File:e_100_BS_Mars12.png| '''ellipticity - 100GHz'''<br />
File:solidarc_100_BS_Mars12.png| '''beam solid angle (relative variations wrt scanning beam - 100GHz'''<br />
</gallery><br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 3072 and 768 directions on the sky for HFI and LFI data respectively. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 and 2048 healpix maps (they include the pixel window function).<br />
* 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.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''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] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|-<br />
| 100 || 9.651 || 0.014 || 1.186 || 0.023 || -0.024 || 55.400 || 105.778 || 0.311 || 9.66 <br />
|-<br />
| 143 || 7.248 || 0.015 || 1.036 || 0.009 || 0.383 || 54.130 || 59.954 || 0.246 || 7.27 <br />
|-<br />
| 217 || 4.990 || 0.025 || 1.177 || 0.030 || 0.836 || 54.999 || 28.447 || 0.271 || 5.01<br />
|-<br />
| 353 || 4.818 || 0.024 || 1.147 || 0.028 || 0.655 || 54.745 || 26.714 || 0.250 || 4.86<br />
|- <br />
| 545 || 4.682 || 0.044 || 1.161 || 0.036 || 0.544 || 54.876 || 26.535 || 0.339 || 4.84 <br />
|-<br />
| 857 || 4.325 || 0.055 || 1.393 || 0.076 || 0.876 || 54.779 || 24.244 || 0.193 || 4.63 <br />
|}<br />
<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
** <math>\Omega_{eff}</math> - 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, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
** from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <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>.<br />
*** 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).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''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>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|-<br />
| 100 || 105.778 || 0.311 || 100.830 || 0.410 || 105.777 || 0.311 <br />
|-<br />
| 143 || 59.954 || 0.246 || 56.811 || 0.419 || 59.952 || 0.246 <br />
|-<br />
| 217 || 28.447 || 0.271 || 26.442 || 0.537 || 28.426 || 0.271 <br />
|-<br />
| 353 || 26.714 || 0.250 || 24.827 || 0.435 || 26.653 || 0.250 <br />
|-<br />
| 545 || 26.535 || 0.339 || 24.287 || 0.455 || 26.302 || 0.337 <br />
|-<br />
| 857 || 24.244 || 0.193 || 22.646 || 0.263 || 23.985 || 0.191 <br />
|}<br />
<br />
<br />
==Production process==<br />
------------------------<br />
<br />
<br />
FEBeCoP, or Fast Effective Beam Convolution in Pixel space ([[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]]), is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:<br />
* 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)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*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)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
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. <br />
<br />
<br />
===Pixel Ordered Detector Angles (PODA)===<br />
<br />
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.<br />
<br />
===effBeam===<br />
<br />
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.<br />
<br />
<br />
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.<br />
<br />
<br />
===Computational Cost===<br />
<br />
The computation of the effective beams has been performed at the NERSC Supercomputing Center. The table below shows the computation cost for FEBeCoP processing of the nominal mission.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ Computational cost for PODA, Effective Beam and single map convolution. Wall-clock time is given as a guide, as found on the NERSC supercomputers.<br />
|-<br />
|Channel ||030 || 044 || 070 || 100 || 143 || 217 || 353 || 545 || 857<br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 || 500 || 500 || 500 || 500 || 500 || 500 <br />
|-<br />
|PODA/Detector Computation time (wall clock hrs) || 7 || 10 || 20 || 20 || 20 || 20 || 20 || 20 || 20<br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 || 2800 || 3800 || 3200 || 3000 || 900 || 1100<br />
|-<br />
|Beam/Channel Computation time (wall clock hrs) || 0.5 || 0.8 || 1 || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5<br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 || 3.6 || 4.8 || 4.0 || 4.1 || 4.1 || 3.7 <br />
|-<br />
|Convolution Computation time (wall clock sec) || 1 || 1 || 1 || 4 || 4 || 4 || 4 || 4 || 4 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 || 187 || 182 || 146 || 132 || 139 || 124<br />
|}<br />
<br />
<br />
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.<br />
<br />
<br />
<br />
==Inputs==<br />
------------<br />
<br />
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. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center| "'Planck Focal Plane''']]<br />
<br />
<br />
===The Focal Plane DataBase (FPDB)===<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ... (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
<br />
{{PLADoc|fileType=rimo|link=The Plank RIMOS}}<br />
<br />
<br />
<br />
===The scanning strategy===<br />
<br />
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.<br />
<br />
===The scanbeam===<br />
<br />
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.<br />
<br />
* 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. <br />
* HFI: [[Beams#Scanning beams|B-Spline, BS]] based on 2 observations of Mars.<br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
[[Beams LFI#Effective beams|LFI effective beams]]<br />
<br />
===Beam cutoff radii===<br />
<br />
N times geometric mean of FWHM of all detectors in a channel, where N<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' || '''fwhm of full beam extent''' <br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|-<br />
|100 || 2.25 || 23.703699<br />
|-<br />
|143 || 3 || 21.057402<br />
|-<br />
|217-353 || 4 || 18.782754<br />
|-<br />
|sub-mm || 4 || 18.327635(545GHz) ; 17.093706(857GHz) <br />
|}<br />
<br />
===Map resolution for the derived beam data object===<br />
<br />
* <math>N_{side} = 1024 </math> for LFI frequency channels<br />
* <math>N_{side} = 2048 </math> for HFI frequency channels<br />
<br />
<br />
==Related products==<br />
----------------------<br />
<br />
===Monte Carlo simulations===<br />
<br />
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:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, C'_\ell_'^out^'}, and 1 sigma errors<br />
<br />
<br />
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#.<br />
For example [[HL-sims#FFP6 data set|FFP6]] <br />
<br />
===Beam Window Functions===<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_l </math> relates the true angular power spectra <math>C_l </math> with the observed angular power spectra <math>\widetilde{C}_l </math>:<br />
<br />
<math><br />
W_l= \widetilde{C}_l / C_l <br />
\label{eqn:wl1}</math> <br />
<br />
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.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_l </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_l </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_l </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_l = < C^{conv}_l / C^{in}_l ><br />
\label{eqn:wl2}</math> <br />
<br />
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.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for Planck mission====<br />
<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, Wl, for LFI channels''']] <br />
[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |'''Beam Window functions, Wl, for HFI channels''']]<br />
<br />
<br />
<br />
<br />
==File Names==<br />
-----------------<br />
<br />
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.<br />
<br />
* beam_index.fits<br />
* beams_run.log<br />
<br />
== Retrieval of effective beam information from the PLA interface ==<br />
<br />
In order to retrieve the effective beam information, the user should first launch the Java interface from this page:<br />
http://www.sciops.esa.int/index.php?project=planck&page=Planck_Legacy_Archive<br />
<br />
One should click on "Sky maps" and then open the "Effective beams" area.<br />
There is the possibility to either retrieve one beam nearest to the input source (name or coordinates), or to retrieve a set of beams in a grid defined by the Nside and the size of the region around a source (name or coordinates).<br />
The resolution of this grid is defined by the Nside parameter.<br />
The size of the region is defined by the "Radius of ROI" parameter.<br />
<br />
Once the user proceeds with querying the beams, the PLA software retrieves the appropriate set of effective beams from the database and delivers it in a FITS file which can be directly downloaded.<br />
<br />
<br />
<br />
==Meta data==<br />
----------------<br />
<br />
The data format of the effective beams is unformatted.<br />
<br />
== References ==<br />
------------------<br />
<br />
<biblio force=false><br />
#[[References]]<br />
</biblio><br />
<br />
<br />
<br />
[[Category:Mission science products|004]]</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Beams&diff=6825Beams2013-03-15T15:15:43Z<p>Ajaffe: move info to products page, clean up effective beam discussion</p>
<hr />
<div>== Scanning Beams ==<br />
<br />
The scanning beams describe the instrument’s instantaneous beam profile. Due to the near constant spin rate of the spacecraft, time domain effects (including residual time response and lowpass filtering) are degenerate with the spatial response due to the optical system. The scanning beam reconstruction recovers both of these effects, aside from residual time domain effects on a longer time scale than can be captured with the extent of the scanning beam model.<br />
<br />
In the paper <font color=red> cite P03c</font> we consider two models of the beam in order to better understand systematics in the reconstruction. Here we describe only the BSpline beams which are used to compute the delivered effective beam (see next section).<br />
<br />
=== BSpline Beam construction ===<br />
<br />
We use seasons 1 and 2 of the Mars observation to reconstruct the beam. The data are processed with the bigPlanets TOI processing. We use JPL Horizons ephemerides to determine the pointing of each detector relative to the planet. We subtract the astrophysical background in the time domain using a bicubic interpolation of the Planck maps.<br />
<br />
The time ordered data are used to fit a two dimensional BSpline surface using a least square minimization and a smoothing criterion to minimize the effects of high spatial frequency variations. We therefore assume the scanning beam to be smooth. The smoothing criterion as well as the locations of the nodes used to compute the B-Spline basis functions are set using GRASP physical optics simulations as inputs which are the best assumptions on the spatial frequency content of the in-flight beams.<br />
<br />
The smoothing criterion is defined as follows:<br />
<br />
<math>\eta = \displaystyle{\sum_{i=1}^{g}\left(b^{k}(\lambda_{i+})-b^{k}(\lambda_{i-})\right)^2}<br />
\label{smoothcrit}</math><br />
<br />
<math>\begin{aligned}<br />
\eta &: \mbox{ Smoothing Criterion}\\<br />
b^k &: \mbox{ $k^{th}$ beam derivative evaluated on the nodes locations}\end{aligned}</math><br />
<br />
And the global inversion criterion :<br />
<br />
<math>\zeta = \eta + p\times \delta</math><br />
<br />
with <math>\delta</math> usual least square estimator and <math>p</math> coefficient giving the relative weight to <math>\delta</math> with respect to the smoothing criterion.<br />
<br />
<math>\delta = \displaystyle{\sum_{r=1}^{m}}\left(y_{r} - b(x_{r})\right)^2\label{estimator}</math><br />
<br />
<math>\begin{aligned}<br />
\delta &: \mbox{ usual least square criterion}\\<br />
r &: \mbox{ indice relative to the m data points, } r \in \{1, \ldots, m\}\\<br />
y_r &: \mbox{ planet data of sample r}\\<br />
x_r &: \mbox{ pointing of sample r}\\<br />
b &: \mbox{ reconstructed beam}\end{aligned}</math><br />
<br />
The BSpline nodes are located on a regular spaced grid in the detector coordinate framset. At the edge of the reconstructed beam map area, 4 coincident nodes are added to avoid vanishing basis functions.<br />
<br />
Let <math>B_{i, k+1}</math>, <math>k</math> degree B-Spline build using nodes {<math>\lambda_{i}, ..., \lambda_{i+k+1}</math>} (''De Boor &amp; Cox'', 1972) :<br />
<br />
<math>B_{i,1}(x) = \left\{<br />
\begin{array}{l}<br />
1, \mbox{ si } x \in \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}\\<br />
0, \mbox{ si } x \notin \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}<br />
\end{array} \right.</math><br />
<br />
<math>B_{i, l+1}(x) = \displaystyle{\frac{x - \lambda_{i}}{\lambda_{i+l} - \lambda_{i}}} B_{i,l}(x) + \displaystyle{\frac{\lambda_{i+l+1}-x}{\lambda_{i+l+1}-\lambda_{i+1}}} B_{i+1, l}(x)</math><br />
<br />
<math>l=1, \ldots, k</math><br />
<br />
<br />
[[Image:FocalPlane_Map_BSScanningBeams_v53.png|800px|frame|center|Focal plane plot of BSpline scanning beams using in-flight pointing reconstruction. The contours are -3,-10,-20,-30 dB from the peak, and for PSB pairs the "a" bolometer is plotted in black and "b" in blue.]]<br />
<br />
=== Simulations and errors ===<br />
<br />
We estimate the reconstruction bias and noise in the measurements using an ensemble of simulated planet observations for each channel. Kept fixed in each simulation are:<br />
<br />
* the input beam assumed: we use a supersampled version of the reconstructed BSpline beam (or whatever comes out of the current ongoing tests!)<br />
* Astrophyical background is the same as that subtracted from the real data.<br />
* StarTracker pointing (using the ptcor6 pointing model).<br />
<br />
The following are varied in each simulation:<br />
<br />
* detector noise realizations obtained by filtering randomly generated white noise with the measured noise PSDs<br />
* random pointing errors with 2 arcsecond rms, and a spectrum that replicates the real errors.<br />
* simulated glitches and the deglitching procedure<br />
* Mars brightness temperature variability<br />
<br />
400 simulated timelines are generated for each bolometer and for each of the two seasons of Mars observations used in the beam reconstruction. The simulated timelines are made into beam maps, projecting onto the BSpline basis in the same way as the real data.<br />
<br />
The beam maps are propagated to effective beam window functions using the quickbeam approach (see effective beams below) and used to evaluate the reconstruction bias and to construct error eigenmodes in the effective beam window function.<br />
<br />
<font color=red>Figure: random pointing error PSD Figures: error envelope plots (or should those go under effective beams?)</font><br />
<br />
=== Residuals ===<br />
<br />
There are two known beam effects that are not included in the main beam model and are estimated as a separate bias in flux and angular power spectrum measurement: 1. long tails due to errors in low frequency time response deconvolution, and 2. near sidelobes.<br />
<br />
We stack all five observations of Jupiter to estimate the long time scale residuals due to incomplete deconvolution of the long time scale response.<br />
<br />
<font color=red>Add some kind of mean tail plot</font><br />
<br />
Near sidelobes are also evaluated using stacked Jupiter (hopefully they will just be part of the v53bis BSpline beams). The main features in the near sidelobes include a wide beam skirt, and dimpling lobes<br />
<font color=red>Add sidelobe plots and tables</font> <br />
<br />
<br />
== Effective Beams ==<br />
<br />
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. <br />
<br />
Several methods of effective beams determination have been developped and cross-validated.<br />
<br />
The main products are produced using FEBeCoP and details of the processing are given in the [[Effective Beams]] products page. See also the equivalent [[Beams_LFI | page discussing the LFI beams]]<br />
<br />
<font color=red>Need satisfactory comparison plot</font><br />
<br />
=== FEBeCoP ===<br />
-------------------<br />
<br />
The full algebra for this method for the calculation of effective beams was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]] <cite>#mitra2010</cite>. 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>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<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.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \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,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\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) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
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}.<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
<br />
<br />
=== FICSBell ===<br />
For more details, see <cite>#planck2013-p03c</cite>.<br />
<br />
Since the HFI beams are not azimuthally symmetric, the scanning strategy has to be taken into account in the effective beam response modelling. This is done using the FICSBell method <font color=red>(Hivon et al, in preparation)</font>, which generalizes to polarization and to include other sources of systematics the approach used for TT <math>C(l)</math> estimation in WMAP-3yr <font color=red>Hinshaw et al (2007)</font> and by <font color=red>Smith et al (2007)</font> in the detection of CMB lensing in WMAP maps. The different steps of the method used for this study can be summarized as follows:<br />
<br />
<ol><br />
<li><p>The scanning related information (i.e., statistics of the orientation of each detector within each pixel) is computed first, and only once for a given observation campaign. Those orientation hit moments are only computed up to degree 4, for reasons described in point 2 below. At the same time, the first two moments of the distribution of samples within each pixel (ie, their center of mass and moments of inertia) are computed and stored on disc.</p></li><br />
<li><p>The scanning beam map or beam model of each detector <math>d</math> is analyzed into its Spherical Harmonics coefficients</p><br />
<p><math>b^d_{ls} = \int d{\bf r} B_d({\bf r}) Y_{ls}({\bf r})\label{scanningBlm}</math></p><br />
<p>where <math>B_d(\bf{r})</math> is the beam map centered on the North pole, and <math>Y_{ls}(\bf{r})</math> is the Spherical Harmonics basis function. Higher <math>s</math> indexes describes higher degrees of departure from azimuthal symmetry and, for HFI beams, the coefficients <math>b^d_{ls}</math> are decreasing functions of <math>s</math> at most <math>l</math> considered. It also appears that, for <math>l<3000</math>, the coefficients with <math>|s| > 4</math> account for <math>1\%</math> or less of the beam throughput. For this reason, only modes with <math>|s| \le 4</math> are considered in the present analysis. <font color=red>Armitage-Caplan and Wandelt (2009)</font> reached a similar conclusion in their deconvolution of Planck-LFI beams.</p></li><li><p>The <math>b^d_{ls}</math> coefficients computed above are used to generate <math>s</math>-spin weighted maps, as well as the first and second order derivatives, for a given CMB sky realization.</p></li><br />
<li><p>The spin weighted maps and orientation hit moments of the same order <math>s</math> are combined for all detectors involved, to provide an “observed” map. Similarly the local spatial derivatives are combined with the location hit moments to describe the effect of the non-ideal sampling of each pixel (see [sec:pixelization]). In this combination, the respective number of hits of each detector in each pixel is considered, as well as the weighting (generally proportional to the inverse noise variance) applied to each detector in order to minimize the final noise.</p></li><li><p>The power spectrum of this map can then be computed, and compared to the input CMB power spectrum to estimate the effective beam window function over the whole sky, or over a given region of the sky.</p></li></ol><br />
Monte-Carlo (MC) simulations in which the sky realisations are changed can be performed by repeating steps 3, 4 and 5. The impact of beam model uncertainties can be studied by including step 2 into the MC simulations.<br />
<br />
=== QuickBeam ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
Planck observes the sky after convolution with a “scanning beam”, which captures its effective response to the sky as a function of displacement from the nominal pointing direction. Decomposing the scanning beam into harmonic coefficients <math>B_{lm}</math>, each time-ordered data (TOD) sample can be modelled as (neglecting the contribution from instrumental noise, which is independent of beam asymmetry) <math>%T_i = \sum_{lms} D^{l}_{-m s} (\phi_i, \theta_i, \alpha_i) b_{ls} (-1)^{m) T_{lm} + n_i,<br />
T_i = \sum_{lms} e^{-i s \alpha_i} B_{ls} \tilde{T}_{lm} {}_s Y_{lm}(\theta_i, \phi_i),<br />
\label{eqn:tod_beam}</math> where the TOD samples are indexed by <math>i</math>, and <math>\tilde{T}_{lm}</math> is the underlying sky signal. The spin spherical harmonic <math>{}_s Y_{lm}</math> rotates the scanning beam to the pointing location <math>(\theta, \phi)</math>, while the <math>e^{-i s \alpha_i}</math> factor gives it the correct orientation. Eq. may be evaluated with the “TotalConvolver” algorithm of <font color=red>Wandelt and Gorski (2001)</font>, accelerated using the “conviqt” recursion relations <font color=red>Prezeau and Reinecke (2010)</font> This approach is implemented in LevelS.<br />
</ref>, although because it involves working with a TOD-sized objected it is necessarily slow.<br />
<br />
On the small angular scales comparable to the size of the beam, it is a good approximation to assume that the procedure of mapmaking from TOD samples is essentially a process of binning: <math>T(p) = \sum_{i \in p} T_i / H(p),<br />
\label{eqn:map_beam_full}</math> where <math>H(p)</math> is the total number of hits in pixel <math>\hat{n}</math>.<br />
<br />
Start with a normalized, rescaled harmonic transform of the beam <math>B_{lm}</math>, sky multipoles <math>\tilde{T}_{lm}</math> and a scan history object <math>w(\hat{n}, s)</math> given by <math>w(\hat{n}, s) = \sum_{j \in p} e^{i s \alpha_j} / H(\hat{n})</math> where the sum is over all hits <math>j</math> of pixel <math>p</math> at location <math>\hat{n}_p</math>, and <math>\alpha_j</math> is the scan angle for observation <math>j</math>. The harmonic transform of this scan-strategy object is given by <math>{}_{s} w_{L M} = \int d^2 \hat{n} {}_s Y_{LM}^*(\hat{n}) w(\hat{n}, s).</math> The beam-convolved observation is then given by <math>\tilde{T}(\hat{n}) = \sum_{slm} w(\hat{n}, -s ) B_{ls} T_{lm} {}_s Y_{lm}(\hat{n}).</math> Taking the ensemble average of the pseudo-Cl power spectrum of these <math>T_{lm}</math> we find<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \sum_{S S'} \sum_{l_1 l_2} \frac{(2l_1+1)(2l_2+1)}{4\pi}<br />
{}_{(-s -s')}{\cal W}_{l_1} B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}<br />
\\ \times\left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s\! & -s\! & 0\!<br />
\end{array}<br />
\right) \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s'\! & -s'\! & 0\!<br />
\end{array}<br />
\right)<br />
<br />
\end{gathered}</math><br />
<br />
where <math>{}_{(s s')}{\cal W}_{L} = \frac{1}{2L+1} \sum_{M} {}_{S} w_{LM} {}_{S'} w_{LM}^*</math> is a cross-power spectrum of scan history objects. Note that the w(n,s) which we have used here can also incorporate a position dependent weighting to optimize the pseudo-Cl estimate, such as inverse-noise or a mask– the equations are unchanged. Writing the pseudo-Cl in position space (a la <font color=red> Dvorkin and Smith (2009)</font>) with Wigner-d matrices we have<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \frac{1}{8\pi} \sum_{S S'} \int_{-1}^{1} dz \ d^{L}_{00}(z)<br />
\\ \times<br />
\left[\sum_{l_1} d^{l_1}_{-s -s'}(z) {}_{(-s -s')}{\cal W}_{l_1} (2l_1+1) \right] <br />
\\ \times<br />
\left[ \sum_{l_2} d^{l_2}_{s s'}(z) B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}(2l_2+1) \right].\end{gathered}</math><br />
<br />
This integral can be implemented exactly using Gauss-Legendre quadrature, with a cost of $\cal 0(l_{\rm max}^2 s_{\rm max}^2)$. For simplicity, we’ve written all the equations here for the auto-spectrum of a single detector, but the generalization to a map made by adding several detectors with different weighting is straightforward. The cost to compute all of the necessary terms exactly in that case becomes <math>\cal 0(l_{\rm max}^2 s_{\rm max}^2 N_{\rm det}^2)</math>.<br />
<br />
Are beams really so difficult? On the flat-sky beam convolution is easy: just multiplication in Fourier space by a beam rotated onto the scan direction. Multiple hits with different scan directions are incorporated by averaging (as the “scan history” objects above encapsulate). Does the sphere really require everything to be so complicated? For a scan strategy which is fairly smooth across the sky, we can pretend that we are observing many independent flat-sky patches at high-L with fairly good accuracy. There is in fact a fairly good approximation to the beam convolved pseudo-Cl power spectrum which is essentially a flat-sky approximation. In the limit that <math>L \gg l_1</math>, with <math>C_{l_2}</math> and <math>B_{l_2}</math> being slowly-varying function in <math>l_2</math> the pseudo-Cl sum above can be approximated as <math>{\tilde{C}}_L^{TT} = C_L^{TT} \sum_{M} \left< \left| w(\hat{n}_p, M) \right|^2 \right>_p |B_{L M}|^2,</math> where the average <math><>_p</math> is taken over the full sky. It’s illustrative to consider three limits of this equation: for a “raster” scan strategy in which each pixel is observed with the same direction, we have <math>\left< \left| w(\hat{n}, M) \right|^2 \right>_p = 1,</math> and the predicted pseudo-Cl is just the power spectrum of the beam. For a &quot;best-case&quot; scan strategy, in which each pixel is observed many times with many different orientation angles, we have &lt; | w(, M) |<sup>2</sup> &gt;<sub>p</sub> = <sub>M0</sub>, and the transfer function is just the azimuthally symmetric part of the beam. Note that this is for a full-sky observation– in the presence of a mask, the average above produces an fsky factor, as expected. It just neglects the coupling between L multipoles (which can be calculated with the more complete equations above).<br />
<br />
==== Effective beam window functions ====<br />
The effective beam window functions $B(l)$ for HFI, computed using Quickbeam, are available in the [[The RIMO|RIMO]].<br />
They do not contain the pixel window function.<br />
<br />
=== Pixelization Artifacts ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
<font color=red><br />
* Several codes available to simulate effects of pixelization.<br />
* Mixes the CMB gradient into a pixelization ``noise'' with a level comparable to that of $2\mu Karcmin$ instrumental noise.<br />
* Quantitative estimate of effect should be included with each released map, but expect not to matter significantly for CMB analysis, as small compared to instrumental noise. <br />
</font><br />
[sec:pixelization]<br />
<br />
Planck maps are produced at resolution 11 <math>(N_{\rm side} = 2048)</math>, corresponding to pixels with a typical dimension of <math>1.7'</math>, comparable to the spacing between scanning rings . This results in an uneven distribution of hits within pixels, which introduces some complications in the analysis and interpretation of the maps. A sample of the hit distribution is illustrated in Fig. [fig:pixcoverage]. Below we discuss the simulation and modeling of this pixelization effect in more detail.<br />
<br />
<br />
[[Image:pixcoverage.png|frame|none|alt=image]]<br />
<br />
[fig:pixcoverage]<br />
<br />
The collaboration has produced 3 codes which may be used to simulate the effect of pixelization on the observed sky, LevelS/TotalConvoler/Conviqt, FeBeCoP, and FICSBell <font color=red> references and further discussion of the three methods<br />
and how they each simulate the pixelization effect.</font>.<br />
<br />
For the measurement of CMB fluctuations, it is also possible to gain intuition for the effects of pixelization analytically. On the small scales relevant to pixelization, the observed CMB is smooth, both due to physical damping as well as the convolution of the instrumental beam. Taylor expanding the CMB temperature about a pixel center to second order, the typical gradient amplitude is given by <math>\langle |\nabla T |^2 \rangle = \frac{1}{4\pi} \sum_{l} l(l+1)(2l+1) C_l^{T} W_l \approx 1\times10^9 \mu K^2 / {\rm rad}^2.</math> where the approximate value is calculated for a <math>\Lambda CDM</math> cosmology with a <math>7'</math>fwhm Gaussian beam. The typical curvature of the observed temperature, on the other hand is given by <math>\langle |\nabla^2 T |^2 \rangle = \frac{1}{4\pi} \sum_{l} [l(l+1)]^2(2l+1) C_l^{T} W_l \approx 7\times10^{14} \mu K^2 / {\rm rad}^4.</math> On the scales relevant to the maximum displacement from the center of a <math>1.7'</math> pixel, the maximum displacement is , and so the gradient term tends to dominate, although the curvature term is still non-negligible. For each observation of a pixel, we can denote the displacement from the pixel center as <math>d = d_{\theta} + i d_{\phi}</math>. The average over all hits within a pixel gives an overall deflection vector which we will denote for a pixel center located at <math>\hat{n}</math> as <math>d(\hat{n})</math>. This represents the center of mass of the hit distribution; in Fig. [fig:pixcoverage] we have plotted these average deflections using black arrows. The deflection field <math>d(\hat{n})</math> may be decomposed into spin-1 spherical harmonics as <math>d_{lm} = \int_{4\pi} {}_1 Y_{lm}^* d(\hat{n}).</math> With a second order Taylor expansion of the CMB temperature about each pixel center, it is then possible to calculate the average pseudo-Cl power spectrum of the pixelized sky. This is given by<br />
<br />
<math>\begin{gathered}<br />
C_l^{T} = [1-l(l+1)R^d] {C}_l^{T} W_l + \\ <br />
\frac{1}{2} \sum_{l_1 l_2} \frac{l_1(l_1+1)(2l_1+1)(2l_2+1)}{4\pi} \\<br />
\times \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & l\! \\<br />
\! l\! & -l\! & 0\!<br />
\end{array}<br />
\right)^2 C_{l_1}^{T} W_{l_1} \left[ C_{l_2}^{d+} + (-1)^{l + l_1 + l_2} C_{l_2}^{ d-} \right],<br />
\label{eqn:clt_pixelized}\end{gathered}</math><br />
<br />
where <math>R^{d} = \langle |d|^2 \rangle/2</math> is half the mean-squared deflection magnitude (averaged over hits within a pixel, as well as over pixels). <math>C_l^{d+}</math> is the sum of the gradient and curl power spectra of <math>d_{lm}</math>, and <math>C_l^{d-}</math> is the gradient spectrum minus the curl spectrum. The <math>R^{d}</math> term describes a smearing of the observed sky due to pixelization. For uniform pixel coverage of <math>N_{\rm side}=2048</math> pixels <math>\sqrt{ \langle |d|^2 \rangle } = 0.725'</math>. For the hit distribution of Planck frequency maps, <math>R^{d}</math> is typically within <font color=red> xxx. calculate for final<br />
maps, looks like will be better than 10%</font>percent of this value, and so this term is accurately described by the pixel window function, which is derived under the assumption of uniform pixel coverage.<br />
<br />
The effect of pixelization is essentially degenerate with that of gravitational lensing of the CMB, with the difference that it (1) acts on the beam-convolved sky, rather than the actual sky and (2) produces a curl-mode deflection field as well as a gradient mode. This is discussed further in the [<cite>#planck2013-p12</cite>|Planck gravitational lensing] paper, where the subpixel deflection field constitutes a potential source of bias for the measured lensing potential. Indeed, Eq. [eqn:clt<sub>p</sub>ixelized] is just a slightly modified version of the usual first order CMB lensing power spectrum (<font color=red>Hu (2000)</font>, <font color=red>Lewis and Challinor (2006)</font>) to accommodate curl modes.<br />
<br />
A useful approximation to Eq. which is derived in the unrealistic limit that the deflection vectors are uncorrelated between pixels, but in practice gives a good description of the power induced by the pixelization, is that the <math>d(\hat{n})</math> couples the CMB gradient into a source of noise with an effective level given by <math>\sigma^{N} \approx \sqrt{ R^T \frac{4\pi}{N_{\rm pix}} \langle | d(\hat{n}) |^2<br />
\rangle }, % (\muKarcmin ),</math><br />
<br />
where the average is taken over all pixels and <math>R^T</math> is half the mean-squared power in the CMB gradient: <math>R^{T} = \frac{1}{8\pi} \sum_{l} l(l+1)(2l+1) \tilde{C}_l^{T}.</math> For frequency-combined maps, <math>\sqrt{ \langle | d(\hat{n}) |^2 \rangle }</math> is typically on the order of <math>0.1'</math>, and so the induced noise is at the level of <math>\sigma^{N} \sim 2 \mu K arcmin</math>. This is small compared to the instrumental contribution, although it does not disappear when taking cross-spectra, depending on how coherent the hit distributions of the two maps in the cross-spectrum are.<br />
<br />
= References =<br />
<biblio force=false><br />
#[[References]] <br />
</biblio></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Effective_Beams&diff=6651Effective Beams2013-03-15T12:04:26Z<p>Ajaffe: more material moved from DP page</p>
<hr />
<div><span style="color:red"></span><br />
<br />
==Product description==<br />
----------------------<br />
<br />
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.<br />
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. <br />
<br />
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.]] <cite>#mitra2010</cite>, and a discussion of its application to Planck data is given in the appropriate LFI <cite>#planck2013-p02d</cite> and HFI <cite>#planck2013-p03c</cite> papers. Relevant details of the processing steps are given in the [[Beams|Effective Beams]] section of this document.<br />
<br />
<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for all LFI and HFI frequency channels, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* 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).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
<br />
<gallery widths=350px heights=350px perrow=3 caption="Comparison images of compact sources and effective beams, PSFs"><br />
File:30.png| '''30GHz'''<br />
File:44.png| '''44GHz'''<br />
File:70.png| '''70GHz'''<br />
File:100.png| '''100GHz'''<br />
File:143.png| '''143GHz'''<br />
File:217.png| '''217GHz'''<br />
File:353.png| '''353GHz'''<br />
File:545.png| '''545GHz'''<br />
File:857.png| '''857GHz'''<br />
</gallery><br />
<br />
<br />
===Histograms of the effective beam parameters====<br />
<br />
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.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
[[File:ist_BS_Mars12.png | 600px| thumb | center| '''Histograms for HFI effective beam parameters''' ]]<br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
<br />
* 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.<br />
<br />
<br />
<gallery widths=500px heights=500px perrow=2 caption="Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian"><br />
File:e_030_GB.png| '''ellipticity - 30GHz'''<br />
File:solidarc_030_GB.png| '''beam solid angle (relative variations wrt scanning beam - 30GHz'''<br />
File:e_100_BS_Mars12.png| '''ellipticity - 100GHz'''<br />
File:solidarc_100_BS_Mars12.png| '''beam solid angle (relative variations wrt scanning beam - 100GHz'''<br />
</gallery><br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 3072 and 768 directions on the sky for HFI and LFI data respectively. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 and 2048 healpix maps (they include the pixel window function).<br />
* 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.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''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] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|-<br />
| 100 || 9.651 || 0.014 || 1.186 || 0.023 || -0.024 || 55.400 || 105.778 || 0.311 || 9.66 <br />
|-<br />
| 143 || 7.248 || 0.015 || 1.036 || 0.009 || 0.383 || 54.130 || 59.954 || 0.246 || 7.27 <br />
|-<br />
| 217 || 4.990 || 0.025 || 1.177 || 0.030 || 0.836 || 54.999 || 28.447 || 0.271 || 5.01<br />
|-<br />
| 353 || 4.818 || 0.024 || 1.147 || 0.028 || 0.655 || 54.745 || 26.714 || 0.250 || 4.86<br />
|- <br />
| 545 || 4.682 || 0.044 || 1.161 || 0.036 || 0.544 || 54.876 || 26.535 || 0.339 || 4.84 <br />
|-<br />
| 857 || 4.325 || 0.055 || 1.393 || 0.076 || 0.876 || 54.779 || 24.244 || 0.193 || 4.63 <br />
|}<br />
<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
** <math>\Omega_{eff}</math> - 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, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
** from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <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>.<br />
*** 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).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''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>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|-<br />
| 100 || 105.778 || 0.311 || 100.830 || 0.410 || 105.777 || 0.311 <br />
|-<br />
| 143 || 59.954 || 0.246 || 56.811 || 0.419 || 59.952 || 0.246 <br />
|-<br />
| 217 || 28.447 || 0.271 || 26.442 || 0.537 || 28.426 || 0.271 <br />
|-<br />
| 353 || 26.714 || 0.250 || 24.827 || 0.435 || 26.653 || 0.250 <br />
|-<br />
| 545 || 26.535 || 0.339 || 24.287 || 0.455 || 26.302 || 0.337 <br />
|-<br />
| 857 || 24.244 || 0.193 || 22.646 || 0.263 || 23.985 || 0.191 <br />
|}<br />
<br />
<br />
==Production process==<br />
------------------------<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]].<br />
<br />
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:<br />
* 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)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*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)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
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. <br />
<br />
<br />
===Pixel Ordered Detector Angles (PODA)===<br />
<br />
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.<br />
<br />
===effBeam===<br />
<br />
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.<br />
<br />
<br />
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.<br />
<br />
<br />
===Computational Cost===<br />
<br />
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.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ 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<br />
|-<br />
|Channel ||030 || 044 || 070 || 100 || 143 || 217 || 353 || 545 || 857<br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 || 500 || 500 || 500 || 500 || 500 || 500 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 || 20 || 20 || 20 || 20 || 20 || 20<br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 || 2800 || 3800 || 3200 || 3000 || 900 || 1100<br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5<br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 || 3.6 || 4.8 || 4.0 || 4.1 || 4.1 || 3.7 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 || 4 || 4 || 4 || 4 || 4 || 4 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 || 187 || 182 || 146 || 132 || 139 || 124<br />
|}<br />
<br />
<br />
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.<br />
<br />
<br />
<br />
==Inputs==<br />
------------<br />
<br />
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. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center| "'Planck Focal Plane''']]<br />
<br />
<br />
===The Focal Plane DataBase (FPDB)===<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ... (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
<br />
{{PLADoc|fileType=rimo|link=The Plank RIMOS}}<br />
<br />
<br />
<br />
===The scanning strategy===<br />
<br />
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.<br />
<br />
===The scanbeam===<br />
<br />
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.<br />
<br />
* 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. <br />
* HFI: [[Beams#Scanning beams|B-Spline, BS]] based on 2 observations of Mars.<br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
[[Beams LFI#Effective beams|LFI effective beams]]<br />
<br />
===Beam cutoff radii===<br />
<br />
N times geometric mean of FWHM of all detectors in a channel, where N<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' || '''fwhm of full beam extent''' <br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|-<br />
|100 || 2.25 || 23.703699<br />
|-<br />
|143 || 3 || 21.057402<br />
|-<br />
|217-353 || 4 || 18.782754<br />
|-<br />
|sub-mm || 4 || 18.327635(545GHz) ; 17.093706(857GHz) <br />
|}<br />
<br />
===Map resolution for the derived beam data object===<br />
<br />
* <math>N_{side} = 1024 </math> for LFI frequency channels<br />
* <math>N_{side} = 2048 </math> for HFI frequency channels<br />
<br />
<br />
==Related products==<br />
----------------------<br />
<br />
===Monte Carlo simulations===<br />
<br />
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:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, C'_\ell_'^out^'}, and 1 sigma errors<br />
<br />
<br />
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#.<br />
For example [[HL-sims#FFP6 data set|FFP6]] <br />
<br />
<br />
<br />
===Beam Window Functions===<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_l </math> relates the true angular power spectra <math>C_l </math> with the observed angular power spectra <math>\widetilde{C}_l </math>:<br />
<br />
<math><br />
W_l= \widetilde{C}_l / C_l <br />
\label{eqn:wl1}</math> <br />
<br />
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.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_l </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_l </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_l </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_l = < C^{conv}_l / C^{in}_l ><br />
\label{eqn:wl2}</math> <br />
<br />
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.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for Planck mission====<br />
<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, Wl, for LFI channels''']] <br />
[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |'''Beam Window functions, Wl, for HFI channels''']]<br />
<br />
<br />
<br />
<br />
==File Names==<br />
-----------------<br />
<br />
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.<br />
<br />
* beam_index.fits<br />
* beams_run.log<br />
<br />
== Retrieval of effective beam information from the PLA interface ==<br />
<br />
In order to retrieve the effective beam information, the user should first launch the Java interface from this page:<br />
http://www.sciops.esa.int/index.php?project=planck&page=Planck_Legacy_Archive<br />
<br />
One should click on "Sky maps" and then open the "Effective beams" area.<br />
There is the possibility to either retrieve one beam nearest to the input source (name or coordinates), or to retrieve a set of beams in a grid defined by the Nside and the size of the region around a source (name or coordinates).<br />
The resolution of this grid is defined by the Nside parameter.<br />
The size of the region is defined by the "Radius of ROI" parameter.<br />
<br />
Once the user proceeds with querying the beams, the PLA software retrieves the appropriate set of effective beams from the database and delivers it in a FITS file which can be directly downloaded.<br />
<br />
<br />
<br />
==Meta data==<br />
----------------<br />
<br />
The data format of the effective beams is unformatted.<br />
<br />
== References ==<br />
------------------<br />
<br />
<biblio force=false><br />
#[[References]]<br />
</biblio><br />
<br />
<br />
<br />
[[Category:Mission science products|004]]</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Beams&diff=6650Beams2013-03-15T12:04:14Z<p>Ajaffe: More material moved to products page</p>
<hr />
<div>== Scanning Beams ==<br />
<br />
The scanning beams describe the instrument’s instantaneous beam profile. Due to the near constant spin rate of the spacecraft, time domain effects (including residual time response and lowpass filtering) are degenerate with the spatial response due to the optical system. The scanning beam reconstruction recovers both of these effects, aside from residual time domain effects on a longer time scale than can be captured with the extent of the scanning beam model.<br />
<br />
In the paper <font color=red> cite P03c</font> we consider two models of the beam in order to better understand systematics in the reconstruction. Here we describe only the BSpline beams which are used to compute the delivered effective beam (see next section).<br />
<br />
=== BSpline Beam construction ===<br />
<br />
We use seasons 1 and 2 of the Mars observation to reconstruct the beam. The data are processed with the bigPlanets TOI processing. We use JPL Horizons ephemerides to determine the pointing of each detector relative to the planet. We subtract the astrophysical background in the time domain using a bicubic interpolation of the Planck maps.<br />
<br />
The time ordered data are used to fit a two dimensional BSpline surface using a least square minimization and a smoothing criterion to minimize the effects of high spatial frequency variations. We therefore assume the scanning beam to be smooth. The smoothing criterion as well as the locations of the nodes used to compute the B-Spline basis functions are set using GRASP physical optics simulations as inputs which are the best assumptions on the spatial frequency content of the in-flight beams.<br />
<br />
The smoothing criterion is defined as follows:<br />
<br />
<math>\eta = \displaystyle{\sum_{i=1}^{g}\left(b^{k}(\lambda_{i+})-b^{k}(\lambda_{i-})\right)^2}<br />
\label{smoothcrit}</math><br />
<br />
<math>\begin{aligned}<br />
\eta &: \mbox{ Smoothing Criterion}\\<br />
b^k &: \mbox{ $k^{th}$ beam derivative evaluated on the nodes locations}\end{aligned}</math><br />
<br />
And the global inversion criterion :<br />
<br />
<math>\zeta = \eta + p\times \delta</math><br />
<br />
with <math>\delta</math> usual least square estimator and <math>p</math> coefficient giving the relative weight to <math>\delta</math> with respect to the smoothing criterion.<br />
<br />
<math>\delta = \displaystyle{\sum_{r=1}^{m}}\left(y_{r} - b(x_{r})\right)^2\label{estimator}</math><br />
<br />
<math>\begin{aligned}<br />
\delta &: \mbox{ usual least square criterion}\\<br />
r &: \mbox{ indice relative to the m data points, } r \in \{1, \ldots, m\}\\<br />
y_r &: \mbox{ planet data of sample r}\\<br />
x_r &: \mbox{ pointing of sample r}\\<br />
b &: \mbox{ reconstructed beam}\end{aligned}</math><br />
<br />
The BSpline nodes are located on a regular spaced grid in the detector coordinate framset. At the edge of the reconstructed beam map area, 4 coincident nodes are added to avoid vanishing basis functions.<br />
<br />
Let <math>B_{i, k+1}</math>, <math>k</math> degree B-Spline build using nodes {<math>\lambda_{i}, ..., \lambda_{i+k+1}</math>} (''De Boor &amp; Cox'', 1972) :<br />
<br />
<math>B_{i,1}(x) = \left\{<br />
\begin{array}{l}<br />
1, \mbox{ si } x \in \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}\\<br />
0, \mbox{ si } x \notin \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}<br />
\end{array} \right.</math><br />
<br />
<math>B_{i, l+1}(x) = \displaystyle{\frac{x - \lambda_{i}}{\lambda_{i+l} - \lambda_{i}}} B_{i,l}(x) + \displaystyle{\frac{\lambda_{i+l+1}-x}{\lambda_{i+l+1}-\lambda_{i+1}}} B_{i+1, l}(x)</math><br />
<br />
<math>l=1, \ldots, k</math><br />
<br />
<br />
[[Image:FocalPlane_Map_BSScanningBeams_v53.png|800px|frame|center|Focal plane plot of BSpline scanning beams using in-flight pointing reconstruction. The contours are -3,-10,-20,-30 dB from the peak, and for PSB pairs the "a" bolometer is plotted in black and "b" in blue.]]<br />
<br />
=== Simulations and errors ===<br />
<br />
We estimate the reconstruction bias and noise in the measurements using an ensemble of simulated planet observations for each channel. Kept fixed in each simulation are:<br />
<br />
* the input beam assumed: we use a supersampled version of the reconstructed BSpline beam (or whatever comes out of the current ongoing tests!)<br />
* Astrophyical background is the same as that subtracted from the real data.<br />
* StarTracker pointing (using the ptcor6 pointing model).<br />
<br />
The following are varied in each simulation:<br />
<br />
* detector noise realizations obtained by filtering randomly generated white noise with the measured noise PSDs<br />
* random pointing errors with 2 arcsecond rms, and a spectrum that replicates the real errors.<br />
* simulated glitches and the deglitching procedure<br />
* Mars brightness temperature variability<br />
<br />
400 simulated timelines are generated for each bolometer and for each of the two seasons of Mars observations used in the beam reconstruction. The simulated timelines are made into beam maps, projecting onto the BSpline basis in the same way as the real data.<br />
<br />
The beam maps are propagated to effective beam window functions using the quickbeam approach (see effective beams below) and used to evaluate the reconstruction bias and to construct error eigenmodes in the effective beam window function.<br />
<br />
<font color=red>Figure: random pointing error PSD Figures: error envelope plots (or should those go under effective beams?)</font><br />
<br />
=== Residuals ===<br />
<br />
There are two known beam effects that are not included in the main beam model and are estimated as a separate bias in flux and angular power spectrum measurement: 1. long tails due to errors in low frequency time response deconvolution, and 2. near sidelobes.<br />
<br />
We stack all five observations of Jupiter to estimate the long time scale residuals due to incomplete deconvolution of the long time scale response.<br />
<br />
<font color=red>Add some kind of mean tail plot</font><br />
<br />
Near sidelobes are also evaluated using stacked Jupiter (hopefully they will just be part of the v53bis BSpline beams). The main features in the near sidelobes include a wide beam skirt, and dimpling lobes<br />
<font color=red>Add sidelobe plots and tables</font> <br />
== Effective Beams ==<br />
Several methods of effective beams determination have been developped and cross-validated.<br />
<font color=red>Need satisfactory comparison plot</font><br />
<br />
=== FEBeCoP ===<br />
-------------------<br />
<br />
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.<br />
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. <br />
<br />
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.]] <cite>#mitra2010</cite>. 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>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<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.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \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,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\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) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
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}.<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]].<br />
<br />
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:<br />
* 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)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*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)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
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. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
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.<br />
<br />
====effBeam====<br />
<br />
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.<br />
<br />
<br />
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.<br />
<br />
<br />
====Computational Cost====<br />
<br />
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.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ 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<br />
|-<br />
|Channel || 100 || 143 || 217 || 353 || 545 || 857<br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 500 || 500 || 500 || 500 || 500 || 500 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 20 || 20 || 20 || 20 || 20 || 20<br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 2800 || 3800 || 3200 || 3000 || 900 || 1100<br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5<br />
|-<br />
|Convolution Computation time (CPU hr) || 3.6 || 4.8 || 4.0 || 4.1 || 4.1 || 3.7 <br />
|-<br />
|Convolution Computation time (Human sec) || 4 || 4 || 4 || 4 || 4 || 4 <br />
|-<br />
|Effective Beam Size (GB) || 187 || 182 || 146 || 132 || 139 || 124<br />
|}<br />
<br />
<br />
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.<br />
<br />
<br />
<br />
====Inputs====<br />
<br />
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 (top plot in this page). 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. <br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ... (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
<br />
{{PLADoc|fileType=rimo|link=The Plank RIMOS}}<br />
<br />
<br />
<br />
====The scanning strategy====<br />
<br />
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.<br />
<br />
====The scanbeam====<br />
<br />
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.<br />
<br />
* HFI: [[Beams#Scanning beams|B-Spline, BS]] based on 2 observations of Mars.<br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
<br />
<br />
====Beam cutoff radii====<br />
<br />
N times geometric mean of FWHM of all detectors in a channel, where N<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' || '''fwhm of full beam extent''' <br />
|-<br />
|100 || 2.25 || 23.703699<br />
|-<br />
|143 || 3 || 21.057402<br />
|-<br />
|217-353 || 4 || 18.782754<br />
|-<br />
|sub-mm || 4 || 18.327635(545GHz) ; 17.093706(857GHz) <br />
|}<br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 2048 </math> for HFI frequency channels<br />
<br />
<br />
====Statistics of the effective beams computed using FEBeCoP====<br />
<br />
We show figures and tables of various simple statistics of the beam for all LFI and HFI channels on the [[Effective Beams]] product page.<br />
<br />
<br />
====Related products====<br />
<br />
=====Monte Carlo simulations=====<br />
<br />
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:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, C'_\ell_'^out^'}, and 1 sigma errors<br />
<br />
<br />
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#.<br />
For example [[HL-sims#FFP6 data set|FFP6]] <br />
<br />
<br />
<br />
=====Beam Window Functions=====<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_l </math> relates the true angular power spectra <math>C_l </math> with the observed angular power spectra <math>\widetilde{C}_l </math>:<br />
<br />
<math><br />
W_l= \widetilde{C}_l / C_l <br />
\label{eqn:wl1}</math> <br />
<br />
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.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_l </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_l </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_l </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_l = < C^{conv}_l / C^{in}_l ><br />
\label{eqn:wl2}</math> <br />
<br />
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.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for HFI channels====<br />
<br />
[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |'''Beam Window functions, Wl, for HFI channels''']]<br />
<br />
<br />
<br />
<font color=red>To be coordinated with Graca</font>See <font color=red>Mitra et al (2011</font> <br />
<br />
=== FICSBell ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
Since the HFI beams are not azimuthally symmetric, the scanning strategy has to be taken into account in the effective beam response modelling. This is done using the FICSBell method <font color=red>(Hivon et al, in preparation)</font>, which generalizes to polarization and to include other sources of systematics the approach used for TT <math>C(l)</math> estimation in WMAP-3yr <font color=red>Hinshaw et al (2007)</font> and by <font color=red>Smith et al (2007)</font> in the detection of CMB lensing in WMAP maps. The different steps of the method used for this study can be summarized as follows:<br />
<br />
<ol><br />
<li><p>The scanning related information (i.e., statistics of the orientation of each detector within each pixel) is computed first, and only once for a given observation campaign. Those orientation hit moments are only computed up to degree 4, for reasons described in point 2 below. At the same time, the first two moments of the distribution of samples within each pixel (ie, their center of mass and moments of inertia) are computed and stored on disc.</p></li><br />
<li><p>The scanning beam map or beam model of each detector <math>d</math> is analyzed into its Spherical Harmonics coefficients</p><br />
<p><math>b^d_{ls} = \int d{\bf r} B_d({\bf r}) Y_{ls}({\bf r})\label{scanningBlm}</math></p><br />
<p>where <math>B_d(\bf{r})</math> is the beam map centered on the North pole, and <math>Y_{ls}(\bf{r})</math> is the Spherical Harmonics basis function. Higher <math>s</math> indexes describes higher degrees of departure from azimuthal symmetry and, for HFI beams, the coefficients <math>b^d_{ls}</math> are decreasing functions of <math>s</math> at most <math>l</math> considered. It also appears that, for <math>l<3000</math>, the coefficients with <math>|s| > 4</math> account for <math>1\%</math> or less of the beam throughput. For this reason, only modes with <math>|s| \le 4</math> are considered in the present analysis. <font color=red>Armitage-Caplan and Wandelt (2009)</font> reached a similar conclusion in their deconvolution of Planck-LFI beams.</p></li><li><p>The <math>b^d_{ls}</math> coefficients computed above are used to generate <math>s</math>-spin weighted maps, as well as the first and second order derivatives, for a given CMB sky realization.</p></li><br />
<li><p>The spin weighted maps and orientation hit moments of the same order <math>s</math> are combined for all detectors involved, to provide an “observed” map. Similarly the local spatial derivatives are combined with the location hit moments to describe the effect of the non-ideal sampling of each pixel (see [sec:pixelization]). In this combination, the respective number of hits of each detector in each pixel is considered, as well as the weighting (generally proportional to the inverse noise variance) applied to each detector in order to minimize the final noise.</p></li><li><p>The power spectrum of this map can then be computed, and compared to the input CMB power spectrum to estimate the effective beam window function over the whole sky, or over a given region of the sky.</p></li></ol><br />
Monte-Carlo (MC) simulations in which the sky realisations are changed can be performed by repeating steps 3, 4 and 5. The impact of beam model uncertainties can be studied by including step 2 into the MC simulations.<br />
<br />
=== QuickBeam ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
Planck observes the sky after convolution with a “scanning beam”, which captures its effective response to the sky as a function of displacement from the nominal pointing direction. Decomposing the scanning beam into harmonic coefficients <math>B_{lm}</math>, each time-ordered data (TOD) sample can be modelled as (neglecting the contribution from instrumental noise, which is independent of beam asymmetry) <math>%T_i = \sum_{lms} D^{l}_{-m s} (\phi_i, \theta_i, \alpha_i) b_{ls} (-1)^{m) T_{lm} + n_i,<br />
T_i = \sum_{lms} e^{-i s \alpha_i} B_{ls} \tilde{T}_{lm} {}_s Y_{lm}(\theta_i, \phi_i),<br />
\label{eqn:tod_beam}</math> where the TOD samples are indexed by <math>i</math>, and <math>\tilde{T}_{lm}</math> is the underlying sky signal. The spin spherical harmonic <math>{}_s Y_{lm}</math> rotates the scanning beam to the pointing location <math>(\theta, \phi)</math>, while the <math>e^{-i s \alpha_i}</math> factor gives it the correct orientation. Eq. may be evaluated with the “TotalConvolver” algorithm of <font color=red>Wandelt and Gorski (2001)</font>, accelerated using the “conviqt” recursion relations <font color=red>Prezeau and Reinecke (2010)</font> This approach is implemented in LevelS.<br />
</ref>, although because it involves working with a TOD-sized objected it is necessarily slow.<br />
<br />
On the small angular scales comparable to the size of the beam, it is a good approximation to assume that the procedure of mapmaking from TOD samples is essentially a process of binning: <math>T(p) = \sum_{i \in p} T_i / H(p),<br />
\label{eqn:map_beam_full}</math> where <math>H(p)</math> is the total number of hits in pixel <math>\hat{n}</math>.<br />
<br />
Start with a normalized, rescaled harmonic transform of the beam <math>B_{lm}</math>, sky multipoles <math>\tilde{T}_{lm}</math> and a scan history object <math>w(\hat{n}, s)</math> given by <math>w(\hat{n}, s) = \sum_{j \in p} e^{i s \alpha_j} / H(\hat{n})</math> where the sum is over all hits <math>j</math> of pixel <math>p</math> at location <math>\hat{n}_p</math>, and <math>\alpha_j</math> is the scan angle for observation <math>j</math>. The harmonic transform of this scan-strategy object is given by <math>{}_{s} w_{L M} = \int d^2 \hat{n} {}_s Y_{LM}^*(\hat{n}) w(\hat{n}, s).</math> The beam-convolved observation is then given by <math>\tilde{T}(\hat{n}) = \sum_{slm} w(\hat{n}, -s ) B_{ls} T_{lm} {}_s Y_{lm}(\hat{n}).</math> Taking the ensemble average of the pseudo-Cl power spectrum of these <math>T_{lm}</math> we find<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \sum_{S S'} \sum_{l_1 l_2} \frac{(2l_1+1)(2l_2+1)}{4\pi}<br />
{}_{(-s -s')}{\cal W}_{l_1} B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}<br />
\\ \times\left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s\! & -s\! & 0\!<br />
\end{array}<br />
\right) \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s'\! & -s'\! & 0\!<br />
\end{array}<br />
\right)<br />
<br />
\end{gathered}</math><br />
<br />
where <math>{}_{(s s')}{\cal W}_{L} = \frac{1}{2L+1} \sum_{M} {}_{S} w_{LM} {}_{S'} w_{LM}^*</math> is a cross-power spectrum of scan history objects. Note that the w(n,s) which we have used here can also incorporate a position dependent weighting to optimize the pseudo-Cl estimate, such as inverse-noise or a mask– the equations are unchanged. Writing the pseudo-Cl in position space (a la <font color=red> Dvorkin and Smith (2009)</font>) with Wigner-d matrices we have<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \frac{1}{8\pi} \sum_{S S'} \int_{-1}^{1} dz \ d^{L}_{00}(z)<br />
\\ \times<br />
\left[\sum_{l_1} d^{l_1}_{-s -s'}(z) {}_{(-s -s')}{\cal W}_{l_1} (2l_1+1) \right] <br />
\\ \times<br />
\left[ \sum_{l_2} d^{l_2}_{s s'}(z) B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}(2l_2+1) \right].\end{gathered}</math><br />
<br />
This integral can be implemented exactly using Gauss-Legendre quadrature, with a cost of $\cal 0(l_{\rm max}^2 s_{\rm max}^2)$. For simplicity, we’ve written all the equations here for the auto-spectrum of a single detector, but the generalization to a map made by adding several detectors with different weighting is straightforward. The cost to compute all of the necessary terms exactly in that case becomes <math>\cal 0(l_{\rm max}^2 s_{\rm max}^2 N_{\rm det}^2)</math>.<br />
<br />
Are beams really so difficult? On the flat-sky beam convolution is easy: just multiplication in Fourier space by a beam rotated onto the scan direction. Multiple hits with different scan directions are incorporated by averaging (as the “scan history” objects above encapsulate). Does the sphere really require everything to be so complicated? For a scan strategy which is fairly smooth across the sky, we can pretend that we are observing many independent flat-sky patches at high-L with fairly good accuracy. There is in fact a fairly good approximation to the beam convolved pseudo-Cl power spectrum which is essentially a flat-sky approximation. In the limit that <math>L \gg l_1</math>, with <math>C_{l_2}</math> and <math>B_{l_2}</math> being slowly-varying function in <math>l_2</math> the pseudo-Cl sum above can be approximated as <math>{\tilde{C}}_L^{TT} = C_L^{TT} \sum_{M} \left< \left| w(\hat{n}_p, M) \right|^2 \right>_p |B_{L M}|^2,</math> where the average <math><>_p</math> is taken over the full sky. It’s illustrative to consider three limits of this equation: for a “raster” scan strategy in which each pixel is observed with the same direction, we have <math>\left< \left| w(\hat{n}, M) \right|^2 \right>_p = 1,</math> and the predicted pseudo-Cl is just the power spectrum of the beam. For a &quot;best-case&quot; scan strategy, in which each pixel is observed many times with many different orientation angles, we have &lt; | w(, M) |<sup>2</sup> &gt;<sub>p</sub> = <sub>M0</sub>, and the transfer function is just the azimuthally symmetric part of the beam. Note that this is for a full-sky observation– in the presence of a mask, the average above produces an fsky factor, as expected. It just neglects the coupling between L multipoles (which can be calculated with the more complete equations above).<br />
<br />
==== Effective beam window functions ====<br />
The effective beam window functions $B(l)$ for HFI, computed using Quickbeam, are available in the [[The RIMO|RIMO]].<br />
They do not contain the pixel window function.<br />
<br />
=== Pixelization Artifacts ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
<font color=red><br />
* Several codes available to simulate effects of pixelization.<br />
* Mixes the CMB gradient into a pixelization ``noise'' with a level comparable to that of $2\mu Karcmin$ instrumental noise.<br />
* Quantitative estimate of effect should be included with each released map, but expect not to matter significantly for CMB analysis, as small compared to instrumental noise. <br />
</font><br />
[sec:pixelization]<br />
<br />
Planck maps are produced at resolution 11 <math>(N_{\rm side} = 2048)</math>, corresponding to pixels with a typical dimension of <math>1.7'</math>, comparable to the spacing between scanning rings . This results in an uneven distribution of hits within pixels, which introduces some complications in the analysis and interpretation of the maps. A sample of the hit distribution is illustrated in Fig. [fig:pixcoverage]. Below we discuss the simulation and modeling of this pixelization effect in more detail.<br />
<br />
<br />
[[Image:pixcoverage.png|frame|none|alt=image]]<br />
<br />
[fig:pixcoverage]<br />
<br />
The collaboration has produced 3 codes which may be used to simulate the effect of pixelization on the observed sky, LevelS/TotalConvoler/Conviqt, FeBeCoP, and FICSBell <font color=red> references and further discussion of the three methods<br />
and how they each simulate the pixelization effect.</font>.<br />
<br />
For the measurement of CMB fluctuations, it is also possible to gain intuition for the effects of pixelization analytically. On the small scales relevant to pixelization, the observed CMB is smooth, both due to physical damping as well as the convolution of the instrumental beam. Taylor expanding the CMB temperature about a pixel center to second order, the typical gradient amplitude is given by <math>\langle |\nabla T |^2 \rangle = \frac{1}{4\pi} \sum_{l} l(l+1)(2l+1) C_l^{T} W_l \approx 1\times10^9 \mu K^2 / {\rm rad}^2.</math> where the approximate value is calculated for a <math>\Lambda CDM</math> cosmology with a <math>7'</math>fwhm Gaussian beam. The typical curvature of the observed temperature, on the other hand is given by <math>\langle |\nabla^2 T |^2 \rangle = \frac{1}{4\pi} \sum_{l} [l(l+1)]^2(2l+1) C_l^{T} W_l \approx 7\times10^{14} \mu K^2 / {\rm rad}^4.</math> On the scales relevant to the maximum displacement from the center of a <math>1.7'</math> pixel, the maximum displacement is , and so the gradient term tends to dominate, although the curvature term is still non-negligible. For each observation of a pixel, we can denote the displacement from the pixel center as <math>d = d_{\theta} + i d_{\phi}</math>. The average over all hits within a pixel gives an overall deflection vector which we will denote for a pixel center located at <math>\hat{n}</math> as <math>d(\hat{n})</math>. This represents the center of mass of the hit distribution; in Fig. [fig:pixcoverage] we have plotted these average deflections using black arrows. The deflection field <math>d(\hat{n})</math> may be decomposed into spin-1 spherical harmonics as <math>d_{lm} = \int_{4\pi} {}_1 Y_{lm}^* d(\hat{n}).</math> With a second order Taylor expansion of the CMB temperature about each pixel center, it is then possible to calculate the average pseudo-Cl power spectrum of the pixelized sky. This is given by<br />
<br />
<math>\begin{gathered}<br />
C_l^{T} = [1-l(l+1)R^d] {C}_l^{T} W_l + \\ <br />
\frac{1}{2} \sum_{l_1 l_2} \frac{l_1(l_1+1)(2l_1+1)(2l_2+1)}{4\pi} \\<br />
\times \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & l\! \\<br />
\! l\! & -l\! & 0\!<br />
\end{array}<br />
\right)^2 C_{l_1}^{T} W_{l_1} \left[ C_{l_2}^{d+} + (-1)^{l + l_1 + l_2} C_{l_2}^{ d-} \right],<br />
\label{eqn:clt_pixelized}\end{gathered}</math><br />
<br />
where <math>R^{d} = \langle |d|^2 \rangle/2</math> is half the mean-squared deflection magnitude (averaged over hits within a pixel, as well as over pixels). <math>C_l^{d+}</math> is the sum of the gradient and curl power spectra of <math>d_{lm}</math>, and <math>C_l^{d-}</math> is the gradient spectrum minus the curl spectrum. The <math>R^{d}</math> term describes a smearing of the observed sky due to pixelization. For uniform pixel coverage of <math>N_{\rm side}=2048</math> pixels <math>\sqrt{ \langle |d|^2 \rangle } = 0.725'</math>. For the hit distribution of Planck frequency maps, <math>R^{d}</math> is typically within <font color=red> xxx. calculate for final<br />
maps, looks like will be better than 10%</font>percent of this value, and so this term is accurately described by the pixel window function, which is derived under the assumption of uniform pixel coverage.<br />
<br />
The effect of pixelization is essentially degenerate with that of gravitational lensing of the CMB, with the difference that it (1) acts on the beam-convolved sky, rather than the actual sky and (2) produces a curl-mode deflection field as well as a gradient mode. This is discussed further in the [<cite>#planck2013-p12</cite>|Planck gravitational lensing] paper, where the subpixel deflection field constitutes a potential source of bias for the measured lensing potential. Indeed, Eq. [eqn:clt<sub>p</sub>ixelized] is just a slightly modified version of the usual first order CMB lensing power spectrum (<font color=red>Hu (2000)</font>, <font color=red>Lewis and Challinor (2006)</font>) to accommodate curl modes.<br />
<br />
A useful approximation to Eq. which is derived in the unrealistic limit that the deflection vectors are uncorrelated between pixels, but in practice gives a good description of the power induced by the pixelization, is that the <math>d(\hat{n})</math> couples the CMB gradient into a source of noise with an effective level given by <math>\sigma^{N} \approx \sqrt{ R^T \frac{4\pi}{N_{\rm pix}} \langle | d(\hat{n}) |^2<br />
\rangle }, % (\muKarcmin ),</math><br />
<br />
where the average is taken over all pixels and <math>R^T</math> is half the mean-squared power in the CMB gradient: <math>R^{T} = \frac{1}{8\pi} \sum_{l} l(l+1)(2l+1) \tilde{C}_l^{T}.</math> For frequency-combined maps, <math>\sqrt{ \langle | d(\hat{n}) |^2 \rangle }</math> is typically on the order of <math>0.1'</math>, and so the induced noise is at the level of <math>\sigma^{N} \sim 2 \mu K arcmin</math>. This is small compared to the instrumental contribution, although it does not disappear when taking cross-spectra, depending on how coherent the hit distributions of the two maps in the cross-spectrum are.<br />
<br />
= References =<br />
<biblio force=false><br />
#[[References]] <br />
</biblio></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Effective_Beams&diff=6646Effective Beams2013-03-15T12:01:41Z<p>Ajaffe: put some HFI+LFI material back</p>
<hr />
<div><span style="color:red"></span><br />
<br />
==Product description==<br />
----------------------<br />
<br />
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.<br />
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. <br />
<br />
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.]] <cite>#mitra2010</cite>, and a discussion of its application to Planck data is given in the appropriate LFI <cite>#planck2013-p02d</cite> and HFI <cite>#planck2013-p03c</cite> papers. Relevant details of the processing steps are given in the [[Beams|Effective Beams]] section of this document.<br />
<br />
<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products====<br />
<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for all LFI and HFI frequency channels, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* 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).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
<br />
<gallery widths=350px heights=350px perrow=3 caption="Comparison images of compact sources and effective beams, PSFs"><br />
File:30.png| '''30GHz'''<br />
File:44.png| '''44GHz'''<br />
File:70.png| '''70GHz'''<br />
File:100.png| '''100GHz'''<br />
File:143.png| '''143GHz'''<br />
File:217.png| '''217GHz'''<br />
File:353.png| '''353GHz'''<br />
File:545.png| '''545GHz'''<br />
File:857.png| '''857GHz'''<br />
</gallery><br />
<br />
<br />
===Histograms of the effective beam parameters====<br />
<br />
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.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
[[File:ist_BS_Mars12.png | 600px| thumb | center| '''Histograms for HFI effective beam parameters''' ]]<br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian====<br />
<br />
<br />
* 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.<br />
<br />
<br />
<gallery widths=500px heights=500px perrow=2 caption="Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian"><br />
File:e_030_GB.png| '''ellipticity - 30GHz'''<br />
File:solidarc_030_GB.png| '''beam solid angle (relative variations wrt scanning beam - 30GHz'''<br />
File:e_100_BS_Mars12.png| '''ellipticity - 100GHz'''<br />
File:solidarc_100_BS_Mars12.png| '''beam solid angle (relative variations wrt scanning beam - 100GHz'''<br />
</gallery><br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 3072 and 768 directions on the sky for HFI and LFI data respectively. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 and 2048 healpix maps (they include the pixel window function).<br />
* 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.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''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] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|-<br />
| 100 || 9.651 || 0.014 || 1.186 || 0.023 || -0.024 || 55.400 || 105.778 || 0.311 || 9.66 <br />
|-<br />
| 143 || 7.248 || 0.015 || 1.036 || 0.009 || 0.383 || 54.130 || 59.954 || 0.246 || 7.27 <br />
|-<br />
| 217 || 4.990 || 0.025 || 1.177 || 0.030 || 0.836 || 54.999 || 28.447 || 0.271 || 5.01<br />
|-<br />
| 353 || 4.818 || 0.024 || 1.147 || 0.028 || 0.655 || 54.745 || 26.714 || 0.250 || 4.86<br />
|- <br />
| 545 || 4.682 || 0.044 || 1.161 || 0.036 || 0.544 || 54.876 || 26.535 || 0.339 || 4.84 <br />
|-<br />
| 857 || 4.325 || 0.055 || 1.393 || 0.076 || 0.876 || 54.779 || 24.244 || 0.193 || 4.63 <br />
|}<br />
<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
** <math>\Omega_{eff}</math> - 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, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
** from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <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>.<br />
*** 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).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''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>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|-<br />
| 100 || 105.778 || 0.311 || 100.830 || 0.410 || 105.777 || 0.311 <br />
|-<br />
| 143 || 59.954 || 0.246 || 56.811 || 0.419 || 59.952 || 0.246 <br />
|-<br />
| 217 || 28.447 || 0.271 || 26.442 || 0.537 || 28.426 || 0.271 <br />
|-<br />
| 353 || 26.714 || 0.250 || 24.827 || 0.435 || 26.653 || 0.250 <br />
|-<br />
| 545 || 26.535 || 0.339 || 24.287 || 0.455 || 26.302 || 0.337 <br />
|-<br />
| 857 || 24.244 || 0.193 || 22.646 || 0.263 || 23.985 || 0.191 <br />
|}<br />
<br />
<br />
==Production process==<br />
------------------------<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]].<br />
<br />
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:<br />
* 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)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*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)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
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. <br />
<br />
<br />
===Pixel Ordered Detector Angles (PODA)===<br />
<br />
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.<br />
<br />
===effBeam===<br />
<br />
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.<br />
<br />
<br />
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.<br />
<br />
<br />
===Computational Cost===<br />
<br />
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.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ 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<br />
|-<br />
|Channel ||030 || 044 || 070 || 100 || 143 || 217 || 353 || 545 || 857<br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 || 500 || 500 || 500 || 500 || 500 || 500 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 || 20 || 20 || 20 || 20 || 20 || 20<br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 || 2800 || 3800 || 3200 || 3000 || 900 || 1100<br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5<br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 || 3.6 || 4.8 || 4.0 || 4.1 || 4.1 || 3.7 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 || 4 || 4 || 4 || 4 || 4 || 4 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 || 187 || 182 || 146 || 132 || 139 || 124<br />
|}<br />
<br />
<br />
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.<br />
<br />
<br />
<br />
==Inputs==<br />
------------<br />
<br />
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. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center| "'Planck Focal Plane''']]<br />
<br />
<br />
===The Focal Plane DataBase (FPDB)===<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ... (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
<br />
{{PLADoc|fileType=rimo|link=The Plank RIMOS}}<br />
<br />
<br />
<br />
===The scanning strategy===<br />
<br />
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.<br />
<br />
===The scanbeam===<br />
<br />
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.<br />
<br />
* 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. <br />
* HFI: [[Beams#Scanning beams|B-Spline, BS]] based on 2 observations of Mars.<br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
[[Beams LFI#Effective beams|LFI effective beams]]<br />
<br />
===Beam cutoff radii===<br />
<br />
N times geometric mean of FWHM of all detectors in a channel, where N<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' || '''fwhm of full beam extent''' <br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|-<br />
|100 || 2.25 || 23.703699<br />
|-<br />
|143 || 3 || 21.057402<br />
|-<br />
|217-353 || 4 || 18.782754<br />
|-<br />
|sub-mm || 4 || 18.327635(545GHz) ; 17.093706(857GHz) <br />
|}<br />
<br />
===Map resolution for the derived beam data object===<br />
<br />
* <math>N_{side} = 1024 </math> for LFI frequency channels<br />
* <math>N_{side} = 2048 </math> for HFI frequency channels<br />
<br />
<br />
==Related products==<br />
----------------------<br />
<br />
===Monte Carlo simulations===<br />
<br />
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:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, C'_\ell_'^out^'}, and 1 sigma errors<br />
<br />
<br />
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#.<br />
For example [[HL-sims#FFP6 data set|FFP6]] <br />
<br />
<br />
<br />
===Beam Window Functions===<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_l </math> relates the true angular power spectra <math>C_l </math> with the observed angular power spectra <math>\widetilde{C}_l </math>:<br />
<br />
<math><br />
W_l= \widetilde{C}_l / C_l <br />
\label{eqn:wl1}</math> <br />
<br />
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.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_l </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_l </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_l </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_l = < C^{conv}_l / C^{in}_l ><br />
\label{eqn:wl2}</math> <br />
<br />
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.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for Planck mission====<br />
<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, Wl, for LFI channels''']] <br />
[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |'''Beam Window functions, Wl, for HFI channels''']]<br />
<br />
<br />
<br />
<br />
==File Names==<br />
-----------------<br />
<br />
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.<br />
<br />
* beam_index.fits<br />
* beams_run.log<br />
<br />
== Retrieval of effective beam information from the PLA interface ==<br />
<br />
In order to retrieve the effective beam information, the user should first launch the Java interface from this page:<br />
http://www.sciops.esa.int/index.php?project=planck&page=Planck_Legacy_Archive<br />
<br />
One should click on "Sky maps" and then open the "Effective beams" area.<br />
There is the possibility to either retrieve one beam nearest to the input source (name or coordinates), or to retrieve a set of beams in a grid defined by the Nside and the size of the region around a source (name or coordinates).<br />
The resolution of this grid is defined by the Nside parameter.<br />
The size of the region is defined by the "Radius of ROI" parameter.<br />
<br />
Once the user proceeds with querying the beams, the PLA software retrieves the appropriate set of effective beams from the database and delivers it in a FITS file which can be directly downloaded.<br />
<br />
<br />
<br />
==Meta data==<br />
----------------<br />
<br />
The data format of the effective beams is unformatted.<br />
<br />
== References ==<br />
------------------<br />
<br />
<biblio force=false><br />
#[[References]]<br />
</biblio><br />
<br />
<br />
<br />
[[Category:Mission science products|004]]</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Effective_Beams&diff=6640Effective Beams2013-03-15T11:55:56Z<p>Ajaffe: More moves from product page to data processing</p>
<hr />
<div><span style="color:red"></span><br />
<br />
==Product description==<br />
----------------------<br />
<br />
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.<br />
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. <br />
<br />
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.]] <cite>#mitra2010</cite>, and a discussion of its application to Planck data is given in the appropriate LFI <cite>#planck2013-p02d</cite> and HFI <cite>#planck2013-p03c</cite> papers. Relevant details of the processing steps are given in the [[Beams|Effective Beams]] section of this document.<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 3072 and 768 directions on the sky for HFI and LFI data respectively. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 and 2048 healpix maps (they include the pixel window function).<br />
* 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.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''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] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|-<br />
| 100 || 9.651 || 0.014 || 1.186 || 0.023 || -0.024 || 55.400 || 105.778 || 0.311 || 9.66 <br />
|-<br />
| 143 || 7.248 || 0.015 || 1.036 || 0.009 || 0.383 || 54.130 || 59.954 || 0.246 || 7.27 <br />
|-<br />
| 217 || 4.990 || 0.025 || 1.177 || 0.030 || 0.836 || 54.999 || 28.447 || 0.271 || 5.01<br />
|-<br />
| 353 || 4.818 || 0.024 || 1.147 || 0.028 || 0.655 || 54.745 || 26.714 || 0.250 || 4.86<br />
|- <br />
| 545 || 4.682 || 0.044 || 1.161 || 0.036 || 0.544 || 54.876 || 26.535 || 0.339 || 4.84 <br />
|-<br />
| 857 || 4.325 || 0.055 || 1.393 || 0.076 || 0.876 || 54.779 || 24.244 || 0.193 || 4.63 <br />
|}<br />
<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
** <math>\Omega_{eff}</math> - 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, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
** from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <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>.<br />
*** 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).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''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>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|-<br />
| 100 || 105.778 || 0.311 || 100.830 || 0.410 || 105.777 || 0.311 <br />
|-<br />
| 143 || 59.954 || 0.246 || 56.811 || 0.419 || 59.952 || 0.246 <br />
|-<br />
| 217 || 28.447 || 0.271 || 26.442 || 0.537 || 28.426 || 0.271 <br />
|-<br />
| 353 || 26.714 || 0.250 || 24.827 || 0.435 || 26.653 || 0.250 <br />
|-<br />
| 545 || 26.535 || 0.339 || 24.287 || 0.455 || 26.302 || 0.337 <br />
|-<br />
| 857 || 24.244 || 0.193 || 22.646 || 0.263 || 23.985 || 0.191 <br />
|}<br />
<br />
<br />
==Production process==<br />
------------------------<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]].<br />
<br />
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:<br />
* 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)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*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)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
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. <br />
<br />
<br />
===Pixel Ordered Detector Angles (PODA)===<br />
<br />
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.<br />
<br />
===effBeam===<br />
<br />
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.<br />
<br />
<br />
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.<br />
<br />
<br />
===Computational Cost===<br />
<br />
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.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ 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<br />
|-<br />
|Channel ||030 || 044 || 070 || 100 || 143 || 217 || 353 || 545 || 857<br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 || 500 || 500 || 500 || 500 || 500 || 500 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 || 20 || 20 || 20 || 20 || 20 || 20<br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 || 2800 || 3800 || 3200 || 3000 || 900 || 1100<br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5<br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 || 3.6 || 4.8 || 4.0 || 4.1 || 4.1 || 3.7 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 || 4 || 4 || 4 || 4 || 4 || 4 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 || 187 || 182 || 146 || 132 || 139 || 124<br />
|}<br />
<br />
<br />
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.<br />
<br />
<br />
<br />
==Inputs==<br />
------------<br />
<br />
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. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center| "'Planck Focal Plane''']]<br />
<br />
<br />
===The Focal Plane DataBase (FPDB)===<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ... (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
<br />
{{PLADoc|fileType=rimo|link=The Plank RIMOS}}<br />
<br />
<br />
<br />
===The scanning strategy===<br />
<br />
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.<br />
<br />
===The scanbeam===<br />
<br />
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.<br />
<br />
* 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. <br />
* HFI: [[Beams#Scanning beams|B-Spline, BS]] based on 2 observations of Mars.<br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
[[Beams LFI#Effective beams|LFI effective beams]]<br />
<br />
===Beam cutoff radii===<br />
<br />
N times geometric mean of FWHM of all detectors in a channel, where N<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' || '''fwhm of full beam extent''' <br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|-<br />
|100 || 2.25 || 23.703699<br />
|-<br />
|143 || 3 || 21.057402<br />
|-<br />
|217-353 || 4 || 18.782754<br />
|-<br />
|sub-mm || 4 || 18.327635(545GHz) ; 17.093706(857GHz) <br />
|}<br />
<br />
===Map resolution for the derived beam data object===<br />
<br />
* <math>N_{side} = 1024 </math> for LFI frequency channels<br />
* <math>N_{side} = 2048 </math> for HFI frequency channels<br />
<br />
<br />
==Related products==<br />
----------------------<br />
<br />
===Monte Carlo simulations===<br />
<br />
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:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, C'_\ell_'^out^'}, and 1 sigma errors<br />
<br />
<br />
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#.<br />
For example [[HL-sims#FFP6 data set|FFP6]] <br />
<br />
<br />
<br />
===Beam Window Functions===<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_l </math> relates the true angular power spectra <math>C_l </math> with the observed angular power spectra <math>\widetilde{C}_l </math>:<br />
<br />
<math><br />
W_l= \widetilde{C}_l / C_l <br />
\label{eqn:wl1}</math> <br />
<br />
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.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_l </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_l </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_l </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_l = < C^{conv}_l / C^{in}_l ><br />
\label{eqn:wl2}</math> <br />
<br />
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.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for Planck mission====<br />
<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, Wl, for LFI channels''']] <br />
[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |'''Beam Window functions, Wl, for HFI channels''']]<br />
<br />
<br />
<br />
<br />
==File Names==<br />
-----------------<br />
<br />
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.<br />
<br />
* beam_index.fits<br />
* beams_run.log<br />
<br />
== Retrieval of effective beam information from the PLA interface ==<br />
<br />
In order to retrieve the effective beam information, the user should first launch the Java interface from this page:<br />
http://www.sciops.esa.int/index.php?project=planck&page=Planck_Legacy_Archive<br />
<br />
One should click on "Sky maps" and then open the "Effective beams" area.<br />
There is the possibility to either retrieve one beam nearest to the input source (name or coordinates), or to retrieve a set of beams in a grid defined by the Nside and the size of the region around a source (name or coordinates).<br />
The resolution of this grid is defined by the Nside parameter.<br />
The size of the region is defined by the "Radius of ROI" parameter.<br />
<br />
Once the user proceeds with querying the beams, the PLA software retrieves the appropriate set of effective beams from the database and delivers it in a FITS file which can be directly downloaded.<br />
<br />
<br />
<br />
==Meta data==<br />
----------------<br />
<br />
The data format of the effective beams is unformatted.<br />
<br />
== References ==<br />
------------------<br />
<br />
<biblio force=false><br />
#[[References]]<br />
</biblio><br />
<br />
<br />
<br />
[[Category:Mission science products|004]]</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Beams&diff=6639Beams2013-03-15T11:55:37Z<p>Ajaffe: move material from products pages to hear</p>
<hr />
<div>== Scanning Beams ==<br />
<br />
The scanning beams describe the instrument’s instantaneous beam profile. Due to the near constant spin rate of the spacecraft, time domain effects (including residual time response and lowpass filtering) are degenerate with the spatial response due to the optical system. The scanning beam reconstruction recovers both of these effects, aside from residual time domain effects on a longer time scale than can be captured with the extent of the scanning beam model.<br />
<br />
In the paper <font color=red> cite P03c</font> we consider two models of the beam in order to better understand systematics in the reconstruction. Here we describe only the BSpline beams which are used to compute the delivered effective beam (see next section).<br />
<br />
=== BSpline Beam construction ===<br />
<br />
We use seasons 1 and 2 of the Mars observation to reconstruct the beam. The data are processed with the bigPlanets TOI processing. We use JPL Horizons ephemerides to determine the pointing of each detector relative to the planet. We subtract the astrophysical background in the time domain using a bicubic interpolation of the Planck maps.<br />
<br />
The time ordered data are used to fit a two dimensional BSpline surface using a least square minimization and a smoothing criterion to minimize the effects of high spatial frequency variations. We therefore assume the scanning beam to be smooth. The smoothing criterion as well as the locations of the nodes used to compute the B-Spline basis functions are set using GRASP physical optics simulations as inputs which are the best assumptions on the spatial frequency content of the in-flight beams.<br />
<br />
The smoothing criterion is defined as follows:<br />
<br />
<math>\eta = \displaystyle{\sum_{i=1}^{g}\left(b^{k}(\lambda_{i+})-b^{k}(\lambda_{i-})\right)^2}<br />
\label{smoothcrit}</math><br />
<br />
<math>\begin{aligned}<br />
\eta &: \mbox{ Smoothing Criterion}\\<br />
b^k &: \mbox{ $k^{th}$ beam derivative evaluated on the nodes locations}\end{aligned}</math><br />
<br />
And the global inversion criterion :<br />
<br />
<math>\zeta = \eta + p\times \delta</math><br />
<br />
with <math>\delta</math> usual least square estimator and <math>p</math> coefficient giving the relative weight to <math>\delta</math> with respect to the smoothing criterion.<br />
<br />
<math>\delta = \displaystyle{\sum_{r=1}^{m}}\left(y_{r} - b(x_{r})\right)^2\label{estimator}</math><br />
<br />
<math>\begin{aligned}<br />
\delta &: \mbox{ usual least square criterion}\\<br />
r &: \mbox{ indice relative to the m data points, } r \in \{1, \ldots, m\}\\<br />
y_r &: \mbox{ planet data of sample r}\\<br />
x_r &: \mbox{ pointing of sample r}\\<br />
b &: \mbox{ reconstructed beam}\end{aligned}</math><br />
<br />
The BSpline nodes are located on a regular spaced grid in the detector coordinate framset. At the edge of the reconstructed beam map area, 4 coincident nodes are added to avoid vanishing basis functions.<br />
<br />
Let <math>B_{i, k+1}</math>, <math>k</math> degree B-Spline build using nodes {<math>\lambda_{i}, ..., \lambda_{i+k+1}</math>} (''De Boor &amp; Cox'', 1972) :<br />
<br />
<math>B_{i,1}(x) = \left\{<br />
\begin{array}{l}<br />
1, \mbox{ si } x \in \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}\\<br />
0, \mbox{ si } x \notin \mbox{[} \lambda_{i}, \lambda_{i+1} \mbox{[}<br />
\end{array} \right.</math><br />
<br />
<math>B_{i, l+1}(x) = \displaystyle{\frac{x - \lambda_{i}}{\lambda_{i+l} - \lambda_{i}}} B_{i,l}(x) + \displaystyle{\frac{\lambda_{i+l+1}-x}{\lambda_{i+l+1}-\lambda_{i+1}}} B_{i+1, l}(x)</math><br />
<br />
<math>l=1, \ldots, k</math><br />
<br />
<br />
[[Image:FocalPlane_Map_BSScanningBeams_v53.png|800px|frame|center|Focal plane plot of BSpline scanning beams using in-flight pointing reconstruction. The contours are -3,-10,-20,-30 dB from the peak, and for PSB pairs the "a" bolometer is plotted in black and "b" in blue.]]<br />
<br />
=== Simulations and errors ===<br />
<br />
We estimate the reconstruction bias and noise in the measurements using an ensemble of simulated planet observations for each channel. Kept fixed in each simulation are:<br />
<br />
* the input beam assumed: we use a supersampled version of the reconstructed BSpline beam (or whatever comes out of the current ongoing tests!)<br />
* Astrophyical background is the same as that subtracted from the real data.<br />
* StarTracker pointing (using the ptcor6 pointing model).<br />
<br />
The following are varied in each simulation:<br />
<br />
* detector noise realizations obtained by filtering randomly generated white noise with the measured noise PSDs<br />
* random pointing errors with 2 arcsecond rms, and a spectrum that replicates the real errors.<br />
* simulated glitches and the deglitching procedure<br />
* Mars brightness temperature variability<br />
<br />
400 simulated timelines are generated for each bolometer and for each of the two seasons of Mars observations used in the beam reconstruction. The simulated timelines are made into beam maps, projecting onto the BSpline basis in the same way as the real data.<br />
<br />
The beam maps are propagated to effective beam window functions using the quickbeam approach (see effective beams below) and used to evaluate the reconstruction bias and to construct error eigenmodes in the effective beam window function.<br />
<br />
<font color=red>Figure: random pointing error PSD Figures: error envelope plots (or should those go under effective beams?)</font><br />
<br />
=== Residuals ===<br />
<br />
There are two known beam effects that are not included in the main beam model and are estimated as a separate bias in flux and angular power spectrum measurement: 1. long tails due to errors in low frequency time response deconvolution, and 2. near sidelobes.<br />
<br />
We stack all five observations of Jupiter to estimate the long time scale residuals due to incomplete deconvolution of the long time scale response.<br />
<br />
<font color=red>Add some kind of mean tail plot</font><br />
<br />
Near sidelobes are also evaluated using stacked Jupiter (hopefully they will just be part of the v53bis BSpline beams). The main features in the near sidelobes include a wide beam skirt, and dimpling lobes<br />
<font color=red>Add sidelobe plots and tables</font> <br />
== Effective Beams ==<br />
Several methods of effective beams determination have been developped and cross-validated.<br />
<font color=red>Need satisfactory comparison plot</font><br />
<br />
=== FEBeCoP ===<br />
-------------------<br />
<br />
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.<br />
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. <br />
<br />
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.]] <cite>#mitra2010</cite>. 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>:<br />
<br />
<math><br />
\widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T},<br />
\label{eq:a0}<br />
</math><br />
<br />
where<br />
<br />
<math><br />
B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, ,<br />
\label{eq:EBT2}<br />
</math><br />
<br />
<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.<br />
<br />
The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel <math>i</math>, <math>(\widetilde{I}, \widetilde{Q}, \widetilde{U})_i</math>, are related to the true stokes parameters <math>(I, Q, U)_i</math>, by the following relation:<br />
<br />
<math><br />
( \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,<br />
\label{eq:a1}<br />
</math><br />
<br />
where the polarised effective beam matrix<br />
<br />
<math><br />
\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) \, ,<br />
\label{eq:a2}<br />
</math><br />
<br />
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}.<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]].<br />
<br />
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:<br />
* 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)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*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)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
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. <br />
<br />
<br />
====Pixel Ordered Detector Angles (PODA)====<br />
<br />
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.<br />
<br />
====effBeam====<br />
<br />
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.<br />
<br />
<br />
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.<br />
<br />
<br />
====Computational Cost====<br />
<br />
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.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ 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<br />
|-<br />
|Channel || 100 || 143 || 217 || 353 || 545 || 857<br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 500 || 500 || 500 || 500 || 500 || 500 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 20 || 20 || 20 || 20 || 20 || 20<br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 2800 || 3800 || 3200 || 3000 || 900 || 1100<br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5<br />
|-<br />
|Convolution Computation time (CPU hr) || 3.6 || 4.8 || 4.0 || 4.1 || 4.1 || 3.7 <br />
|-<br />
|Convolution Computation time (Human sec) || 4 || 4 || 4 || 4 || 4 || 4 <br />
|-<br />
|Effective Beam Size (GB) || 187 || 182 || 146 || 132 || 139 || 124<br />
|}<br />
<br />
<br />
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.<br />
<br />
<br />
<br />
====Inputs====<br />
<br />
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 (top plot in this page). 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. <br />
<br />
<br />
====The Focal Plane DataBase (FPDB)====<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ... (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
<br />
{{PLADoc|fileType=rimo|link=The Plank RIMOS}}<br />
<br />
<br />
<br />
====The scanning strategy====<br />
<br />
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.<br />
<br />
====The scanbeam====<br />
<br />
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.<br />
<br />
* HFI: [[Beams#Scanning beams|B-Spline, BS]] based on 2 observations of Mars.<br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
<br />
<br />
====Beam cutoff radii====<br />
<br />
N times geometric mean of FWHM of all detectors in a channel, where N<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' || '''fwhm of full beam extent''' <br />
|-<br />
|100 || 2.25 || 23.703699<br />
|-<br />
|143 || 3 || 21.057402<br />
|-<br />
|217-353 || 4 || 18.782754<br />
|-<br />
|sub-mm || 4 || 18.327635(545GHz) ; 17.093706(857GHz) <br />
|}<br />
<br />
====Map resolution for the derived beam data object====<br />
<br />
* <math>N_{side} = 2048 </math> for HFI frequency channels<br />
<br />
<br />
<br />
====Comparison of the images of compact sources observed by Planck with FEBeCoP products====<br />
<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for all LFI and HFI frequency channels, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* 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).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
<br />
<gallery widths=350px heights=350px perrow=3 caption="Comparison images of compact sources and effective beams, PSFs"><br />
File:30.png| '''30GHz'''<br />
File:44.png| '''44GHz'''<br />
File:70.png| '''70GHz'''<br />
File:100.png| '''100GHz'''<br />
File:143.png| '''143GHz'''<br />
File:217.png| '''217GHz'''<br />
File:353.png| '''353GHz'''<br />
File:545.png| '''545GHz'''<br />
File:857.png| '''857GHz'''<br />
</gallery><br />
<br />
<br />
====Histograms of the effective beam parameters====<br />
<br />
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.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
[[File:ist_BS_Mars12.png | 600px| thumb | center| '''Histograms for HFI effective beam parameters''' ]]<br />
<br />
<br />
<br />
====Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian====<br />
<br />
<br />
* 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.<br />
<br />
<br />
<gallery widths=500px heights=500px perrow=2 caption="Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian"><br />
File:e_030_GB.png| '''ellipticity - 30GHz'''<br />
File:solidarc_030_GB.png| '''beam solid angle (relative variations wrt scanning beam - 30GHz'''<br />
File:e_100_BS_Mars12.png| '''ellipticity - 100GHz'''<br />
File:solidarc_100_BS_Mars12.png| '''beam solid angle (relative variations wrt scanning beam - 100GHz'''<br />
</gallery><br />
<br />
<br />
<br />
====Statistics of the effective beams computed using FEBeCoP====<br />
<br />
We tabulate the simple statistics of the beam for all LFI and HFI channels in the [[Effective Beams]] product page.<br />
<br />
<br />
=====Beam solid angles for the PCCS=====<br />
<br />
** <math>\Omega_{eff}</math> - 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, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
** from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <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>.<br />
*** 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).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''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>] <br />
|-<br />
| 100 || 105.778 || 0.311 || 100.830 || 0.410 || 105.777 || 0.311 <br />
|-<br />
| 143 || 59.954 || 0.246 || 56.811 || 0.419 || 59.952 || 0.246 <br />
|-<br />
| 217 || 28.447 || 0.271 || 26.442 || 0.537 || 28.426 || 0.271 <br />
|-<br />
| 353 || 26.714 || 0.250 || 24.827 || 0.435 || 26.653 || 0.250 <br />
|-<br />
| 545 || 26.535 || 0.339 || 24.287 || 0.455 || 26.302 || 0.337 <br />
|-<br />
| 857 || 24.244 || 0.193 || 22.646 || 0.263 || 23.985 || 0.191 <br />
|}<br />
<br />
<br />
====Related products====<br />
<br />
=====Monte Carlo simulations=====<br />
<br />
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:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, C'_\ell_'^out^'}, and 1 sigma errors<br />
<br />
<br />
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#.<br />
For example [[HL-sims#FFP6 data set|FFP6]] <br />
<br />
<br />
<br />
=====Beam Window Functions=====<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_l </math> relates the true angular power spectra <math>C_l </math> with the observed angular power spectra <math>\widetilde{C}_l </math>:<br />
<br />
<math><br />
W_l= \widetilde{C}_l / C_l <br />
\label{eqn:wl1}</math> <br />
<br />
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.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_l </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_l </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_l </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_l = < C^{conv}_l / C^{in}_l ><br />
\label{eqn:wl2}</math> <br />
<br />
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.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for HFI channels====<br />
<br />
[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |'''Beam Window functions, Wl, for HFI channels''']]<br />
<br />
<br />
<br />
<font color=red>To be coordinated with Graca</font>See <font color=red>Mitra et al (2011</font> <br />
<br />
=== FICSBell ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
Since the HFI beams are not azimuthally symmetric, the scanning strategy has to be taken into account in the effective beam response modelling. This is done using the FICSBell method <font color=red>(Hivon et al, in preparation)</font>, which generalizes to polarization and to include other sources of systematics the approach used for TT <math>C(l)</math> estimation in WMAP-3yr <font color=red>Hinshaw et al (2007)</font> and by <font color=red>Smith et al (2007)</font> in the detection of CMB lensing in WMAP maps. The different steps of the method used for this study can be summarized as follows:<br />
<br />
<ol><br />
<li><p>The scanning related information (i.e., statistics of the orientation of each detector within each pixel) is computed first, and only once for a given observation campaign. Those orientation hit moments are only computed up to degree 4, for reasons described in point 2 below. At the same time, the first two moments of the distribution of samples within each pixel (ie, their center of mass and moments of inertia) are computed and stored on disc.</p></li><br />
<li><p>The scanning beam map or beam model of each detector <math>d</math> is analyzed into its Spherical Harmonics coefficients</p><br />
<p><math>b^d_{ls} = \int d{\bf r} B_d({\bf r}) Y_{ls}({\bf r})\label{scanningBlm}</math></p><br />
<p>where <math>B_d(\bf{r})</math> is the beam map centered on the North pole, and <math>Y_{ls}(\bf{r})</math> is the Spherical Harmonics basis function. Higher <math>s</math> indexes describes higher degrees of departure from azimuthal symmetry and, for HFI beams, the coefficients <math>b^d_{ls}</math> are decreasing functions of <math>s</math> at most <math>l</math> considered. It also appears that, for <math>l<3000</math>, the coefficients with <math>|s| > 4</math> account for <math>1\%</math> or less of the beam throughput. For this reason, only modes with <math>|s| \le 4</math> are considered in the present analysis. <font color=red>Armitage-Caplan and Wandelt (2009)</font> reached a similar conclusion in their deconvolution of Planck-LFI beams.</p></li><li><p>The <math>b^d_{ls}</math> coefficients computed above are used to generate <math>s</math>-spin weighted maps, as well as the first and second order derivatives, for a given CMB sky realization.</p></li><br />
<li><p>The spin weighted maps and orientation hit moments of the same order <math>s</math> are combined for all detectors involved, to provide an “observed” map. Similarly the local spatial derivatives are combined with the location hit moments to describe the effect of the non-ideal sampling of each pixel (see [sec:pixelization]). In this combination, the respective number of hits of each detector in each pixel is considered, as well as the weighting (generally proportional to the inverse noise variance) applied to each detector in order to minimize the final noise.</p></li><li><p>The power spectrum of this map can then be computed, and compared to the input CMB power spectrum to estimate the effective beam window function over the whole sky, or over a given region of the sky.</p></li></ol><br />
Monte-Carlo (MC) simulations in which the sky realisations are changed can be performed by repeating steps 3, 4 and 5. The impact of beam model uncertainties can be studied by including step 2 into the MC simulations.<br />
<br />
=== QuickBeam ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
Planck observes the sky after convolution with a “scanning beam”, which captures its effective response to the sky as a function of displacement from the nominal pointing direction. Decomposing the scanning beam into harmonic coefficients <math>B_{lm}</math>, each time-ordered data (TOD) sample can be modelled as (neglecting the contribution from instrumental noise, which is independent of beam asymmetry) <math>%T_i = \sum_{lms} D^{l}_{-m s} (\phi_i, \theta_i, \alpha_i) b_{ls} (-1)^{m) T_{lm} + n_i,<br />
T_i = \sum_{lms} e^{-i s \alpha_i} B_{ls} \tilde{T}_{lm} {}_s Y_{lm}(\theta_i, \phi_i),<br />
\label{eqn:tod_beam}</math> where the TOD samples are indexed by <math>i</math>, and <math>\tilde{T}_{lm}</math> is the underlying sky signal. The spin spherical harmonic <math>{}_s Y_{lm}</math> rotates the scanning beam to the pointing location <math>(\theta, \phi)</math>, while the <math>e^{-i s \alpha_i}</math> factor gives it the correct orientation. Eq. may be evaluated with the “TotalConvolver” algorithm of <font color=red>Wandelt and Gorski (2001)</font>, accelerated using the “conviqt” recursion relations <font color=red>Prezeau and Reinecke (2010)</font> This approach is implemented in LevelS.<br />
</ref>, although because it involves working with a TOD-sized objected it is necessarily slow.<br />
<br />
On the small angular scales comparable to the size of the beam, it is a good approximation to assume that the procedure of mapmaking from TOD samples is essentially a process of binning: <math>T(p) = \sum_{i \in p} T_i / H(p),<br />
\label{eqn:map_beam_full}</math> where <math>H(p)</math> is the total number of hits in pixel <math>\hat{n}</math>.<br />
<br />
Start with a normalized, rescaled harmonic transform of the beam <math>B_{lm}</math>, sky multipoles <math>\tilde{T}_{lm}</math> and a scan history object <math>w(\hat{n}, s)</math> given by <math>w(\hat{n}, s) = \sum_{j \in p} e^{i s \alpha_j} / H(\hat{n})</math> where the sum is over all hits <math>j</math> of pixel <math>p</math> at location <math>\hat{n}_p</math>, and <math>\alpha_j</math> is the scan angle for observation <math>j</math>. The harmonic transform of this scan-strategy object is given by <math>{}_{s} w_{L M} = \int d^2 \hat{n} {}_s Y_{LM}^*(\hat{n}) w(\hat{n}, s).</math> The beam-convolved observation is then given by <math>\tilde{T}(\hat{n}) = \sum_{slm} w(\hat{n}, -s ) B_{ls} T_{lm} {}_s Y_{lm}(\hat{n}).</math> Taking the ensemble average of the pseudo-Cl power spectrum of these <math>T_{lm}</math> we find<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \sum_{S S'} \sum_{l_1 l_2} \frac{(2l_1+1)(2l_2+1)}{4\pi}<br />
{}_{(-s -s')}{\cal W}_{l_1} B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}<br />
\\ \times\left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s\! & -s\! & 0\!<br />
\end{array}<br />
\right) \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & L\! \\<br />
\! s'\! & -s'\! & 0\!<br />
\end{array}<br />
\right)<br />
<br />
\end{gathered}</math><br />
<br />
where <math>{}_{(s s')}{\cal W}_{L} = \frac{1}{2L+1} \sum_{M} {}_{S} w_{LM} {}_{S'} w_{LM}^*</math> is a cross-power spectrum of scan history objects. Note that the w(n,s) which we have used here can also incorporate a position dependent weighting to optimize the pseudo-Cl estimate, such as inverse-noise or a mask– the equations are unchanged. Writing the pseudo-Cl in position space (a la <font color=red> Dvorkin and Smith (2009)</font>) with Wigner-d matrices we have<br />
<br />
<math>\begin{gathered}<br />
\tilde{C}_{L}^{TT} = \frac{1}{8\pi} \sum_{S S'} \int_{-1}^{1} dz \ d^{L}_{00}(z)<br />
\\ \times<br />
\left[\sum_{l_1} d^{l_1}_{-s -s'}(z) {}_{(-s -s')}{\cal W}_{l_1} (2l_1+1) \right] <br />
\\ \times<br />
\left[ \sum_{l_2} d^{l_2}_{s s'}(z) B_{l_2 S} B_{l_2 S'}^* C^{TT}_{l_2}(2l_2+1) \right].\end{gathered}</math><br />
<br />
This integral can be implemented exactly using Gauss-Legendre quadrature, with a cost of $\cal 0(l_{\rm max}^2 s_{\rm max}^2)$. For simplicity, we’ve written all the equations here for the auto-spectrum of a single detector, but the generalization to a map made by adding several detectors with different weighting is straightforward. The cost to compute all of the necessary terms exactly in that case becomes <math>\cal 0(l_{\rm max}^2 s_{\rm max}^2 N_{\rm det}^2)</math>.<br />
<br />
Are beams really so difficult? On the flat-sky beam convolution is easy: just multiplication in Fourier space by a beam rotated onto the scan direction. Multiple hits with different scan directions are incorporated by averaging (as the “scan history” objects above encapsulate). Does the sphere really require everything to be so complicated? For a scan strategy which is fairly smooth across the sky, we can pretend that we are observing many independent flat-sky patches at high-L with fairly good accuracy. There is in fact a fairly good approximation to the beam convolved pseudo-Cl power spectrum which is essentially a flat-sky approximation. In the limit that <math>L \gg l_1</math>, with <math>C_{l_2}</math> and <math>B_{l_2}</math> being slowly-varying function in <math>l_2</math> the pseudo-Cl sum above can be approximated as <math>{\tilde{C}}_L^{TT} = C_L^{TT} \sum_{M} \left< \left| w(\hat{n}_p, M) \right|^2 \right>_p |B_{L M}|^2,</math> where the average <math><>_p</math> is taken over the full sky. It’s illustrative to consider three limits of this equation: for a “raster” scan strategy in which each pixel is observed with the same direction, we have <math>\left< \left| w(\hat{n}, M) \right|^2 \right>_p = 1,</math> and the predicted pseudo-Cl is just the power spectrum of the beam. For a &quot;best-case&quot; scan strategy, in which each pixel is observed many times with many different orientation angles, we have &lt; | w(, M) |<sup>2</sup> &gt;<sub>p</sub> = <sub>M0</sub>, and the transfer function is just the azimuthally symmetric part of the beam. Note that this is for a full-sky observation– in the presence of a mask, the average above produces an fsky factor, as expected. It just neglects the coupling between L multipoles (which can be calculated with the more complete equations above).<br />
<br />
==== Effective beam window functions ====<br />
The effective beam window functions $B(l)$ for HFI, computed using Quickbeam, are available in the [[The RIMO|RIMO]].<br />
They do not contain the pixel window function.<br />
<br />
=== Pixelization Artifacts ===<br />
For more details, see <cite>#planck2013-p03c</cite><br />
<br />
<font color=red><br />
* Several codes available to simulate effects of pixelization.<br />
* Mixes the CMB gradient into a pixelization ``noise'' with a level comparable to that of $2\mu Karcmin$ instrumental noise.<br />
* Quantitative estimate of effect should be included with each released map, but expect not to matter significantly for CMB analysis, as small compared to instrumental noise. <br />
</font><br />
[sec:pixelization]<br />
<br />
Planck maps are produced at resolution 11 <math>(N_{\rm side} = 2048)</math>, corresponding to pixels with a typical dimension of <math>1.7'</math>, comparable to the spacing between scanning rings . This results in an uneven distribution of hits within pixels, which introduces some complications in the analysis and interpretation of the maps. A sample of the hit distribution is illustrated in Fig. [fig:pixcoverage]. Below we discuss the simulation and modeling of this pixelization effect in more detail.<br />
<br />
<br />
[[Image:pixcoverage.png|frame|none|alt=image]]<br />
<br />
[fig:pixcoverage]<br />
<br />
The collaboration has produced 3 codes which may be used to simulate the effect of pixelization on the observed sky, LevelS/TotalConvoler/Conviqt, FeBeCoP, and FICSBell <font color=red> references and further discussion of the three methods<br />
and how they each simulate the pixelization effect.</font>.<br />
<br />
For the measurement of CMB fluctuations, it is also possible to gain intuition for the effects of pixelization analytically. On the small scales relevant to pixelization, the observed CMB is smooth, both due to physical damping as well as the convolution of the instrumental beam. Taylor expanding the CMB temperature about a pixel center to second order, the typical gradient amplitude is given by <math>\langle |\nabla T |^2 \rangle = \frac{1}{4\pi} \sum_{l} l(l+1)(2l+1) C_l^{T} W_l \approx 1\times10^9 \mu K^2 / {\rm rad}^2.</math> where the approximate value is calculated for a <math>\Lambda CDM</math> cosmology with a <math>7'</math>fwhm Gaussian beam. The typical curvature of the observed temperature, on the other hand is given by <math>\langle |\nabla^2 T |^2 \rangle = \frac{1}{4\pi} \sum_{l} [l(l+1)]^2(2l+1) C_l^{T} W_l \approx 7\times10^{14} \mu K^2 / {\rm rad}^4.</math> On the scales relevant to the maximum displacement from the center of a <math>1.7'</math> pixel, the maximum displacement is , and so the gradient term tends to dominate, although the curvature term is still non-negligible. For each observation of a pixel, we can denote the displacement from the pixel center as <math>d = d_{\theta} + i d_{\phi}</math>. The average over all hits within a pixel gives an overall deflection vector which we will denote for a pixel center located at <math>\hat{n}</math> as <math>d(\hat{n})</math>. This represents the center of mass of the hit distribution; in Fig. [fig:pixcoverage] we have plotted these average deflections using black arrows. The deflection field <math>d(\hat{n})</math> may be decomposed into spin-1 spherical harmonics as <math>d_{lm} = \int_{4\pi} {}_1 Y_{lm}^* d(\hat{n}).</math> With a second order Taylor expansion of the CMB temperature about each pixel center, it is then possible to calculate the average pseudo-Cl power spectrum of the pixelized sky. This is given by<br />
<br />
<math>\begin{gathered}<br />
C_l^{T} = [1-l(l+1)R^d] {C}_l^{T} W_l + \\ <br />
\frac{1}{2} \sum_{l_1 l_2} \frac{l_1(l_1+1)(2l_1+1)(2l_2+1)}{4\pi} \\<br />
\times \left(<br />
\begin{array}{ccc}<br />
\! l_1\! & l_2\! & l\! \\<br />
\! l\! & -l\! & 0\!<br />
\end{array}<br />
\right)^2 C_{l_1}^{T} W_{l_1} \left[ C_{l_2}^{d+} + (-1)^{l + l_1 + l_2} C_{l_2}^{ d-} \right],<br />
\label{eqn:clt_pixelized}\end{gathered}</math><br />
<br />
where <math>R^{d} = \langle |d|^2 \rangle/2</math> is half the mean-squared deflection magnitude (averaged over hits within a pixel, as well as over pixels). <math>C_l^{d+}</math> is the sum of the gradient and curl power spectra of <math>d_{lm}</math>, and <math>C_l^{d-}</math> is the gradient spectrum minus the curl spectrum. The <math>R^{d}</math> term describes a smearing of the observed sky due to pixelization. For uniform pixel coverage of <math>N_{\rm side}=2048</math> pixels <math>\sqrt{ \langle |d|^2 \rangle } = 0.725'</math>. For the hit distribution of Planck frequency maps, <math>R^{d}</math> is typically within <font color=red> xxx. calculate for final<br />
maps, looks like will be better than 10%</font>percent of this value, and so this term is accurately described by the pixel window function, which is derived under the assumption of uniform pixel coverage.<br />
<br />
The effect of pixelization is essentially degenerate with that of gravitational lensing of the CMB, with the difference that it (1) acts on the beam-convolved sky, rather than the actual sky and (2) produces a curl-mode deflection field as well as a gradient mode. This is discussed further in the [<cite>#planck2013-p12</cite>|Planck gravitational lensing] paper, where the subpixel deflection field constitutes a potential source of bias for the measured lensing potential. Indeed, Eq. [eqn:clt<sub>p</sub>ixelized] is just a slightly modified version of the usual first order CMB lensing power spectrum (<font color=red>Hu (2000)</font>, <font color=red>Lewis and Challinor (2006)</font>) to accommodate curl modes.<br />
<br />
A useful approximation to Eq. which is derived in the unrealistic limit that the deflection vectors are uncorrelated between pixels, but in practice gives a good description of the power induced by the pixelization, is that the <math>d(\hat{n})</math> couples the CMB gradient into a source of noise with an effective level given by <math>\sigma^{N} \approx \sqrt{ R^T \frac{4\pi}{N_{\rm pix}} \langle | d(\hat{n}) |^2<br />
\rangle }, % (\muKarcmin ),</math><br />
<br />
where the average is taken over all pixels and <math>R^T</math> is half the mean-squared power in the CMB gradient: <math>R^{T} = \frac{1}{8\pi} \sum_{l} l(l+1)(2l+1) \tilde{C}_l^{T}.</math> For frequency-combined maps, <math>\sqrt{ \langle | d(\hat{n}) |^2 \rangle }</math> is typically on the order of <math>0.1'</math>, and so the induced noise is at the level of <math>\sigma^{N} \sim 2 \mu K arcmin</math>. This is small compared to the instrumental contribution, although it does not disappear when taking cross-spectra, depending on how coherent the hit distributions of the two maps in the cross-spectrum are.<br />
<br />
= References =<br />
<biblio force=false><br />
#[[References]] <br />
</biblio></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Effective_Beams&diff=6627Effective Beams2013-03-15T11:45:43Z<p>Ajaffe: start removing material duplicated in data processing section</p>
<hr />
<div><span style="color:red"></span><br />
<br />
==Product description==<br />
----------------------<br />
<br />
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.<br />
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. <br />
<br />
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.]] <cite>#mitra2010</cite>, and a discussion of its application to Planck data is given in the appropriate LFI <cite>#planck2013-p02d</cite> and HFI <cite>#planck2013-p03c</cite> papers. Relevant details of the processing steps are given in the [[Beams|Effective Beams]] section of this document.<br />
<br />
<br />
===Comparison of the images of compact sources observed by Planck with FEBeCoP products===<br />
<br />
We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for all LFI and HFI frequency channels, as an example. We show below a few panels of source images organized as follows:<br />
* Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar<br />
* 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).<br />
* Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line<br />
<br />
<br />
<br />
<gallery widths=350px heights=350px perrow=3 caption="Comparison images of compact sources and effective beams, PSFs"><br />
File:30.png| '''30GHz'''<br />
File:44.png| '''44GHz'''<br />
File:70.png| '''70GHz'''<br />
File:100.png| '''100GHz'''<br />
File:143.png| '''143GHz'''<br />
File:217.png| '''217GHz'''<br />
File:353.png| '''353GHz'''<br />
File:545.png| '''545GHz'''<br />
File:857.png| '''857GHz'''<br />
</gallery><br />
<br />
<br />
<br />
<br />
<br />
===Histograms of the effective beam parameters===<br />
<br />
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.<br />
<br />
Where beam solid angle is estimated according to the definition: '''4pi* sum(effbeam)/max(effbeam)'''<br />
ie <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math><br />
<br />
<br />
[[File:ist_GB.png | 600px| thumb | center| '''Histograms for LFI effective beam parameters''' ]] <br />
[[File:ist_BS_Mars12.png | 600px| thumb | center| '''Histograms for HFI effective beam parameters''' ]]<br />
<br />
<br />
<br />
<br />
===Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian===<br />
<br />
* 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.<br />
<br />
<br />
<gallery widths=500px heights=500px perrow=2 caption="Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian"><br />
File:e_030_GB.png| '''ellipticity - 30GHz'''<br />
File:solidarc_030_GB.png| '''beam solid angle (relative variations wrt scanning beam - 30GHz'''<br />
File:e_100_BS_Mars12.png| '''ellipticity - 100GHz'''<br />
File:solidarc_100_BS_Mars12.png| '''beam solid angle (relative variations wrt scanning beam - 100GHz'''<br />
</gallery><br />
<br />
<br />
<br />
<br />
<br />
===Statistics of the effective beams computed using FEBeCoP===<br />
<br />
We tabulate the simple statistics of FWHM, ellipticity (e), orientation (<math> \psi</math>) and beam solid angle, (<math> \Omega </math>), for a sample of 3072 and 768 directions on the sky for HFI and LFI data respectively. Statistics shown in the Table are derived from the histograms shown above.<br />
<br />
* The derived beam parameters are representative of the DPC NSIDE 1024 and 2048 healpix maps (they include the pixel window function).<br />
* 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.<br />
<br />
<br />
<br />
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ '''Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled'''<br />
|-<br />
! '''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] <br />
|-<br />
| 030 || 32.239 || 0.013 || 1.320 || 0.031 || -0.304 || 55.349 || 1189.513 || 0.842 || 32.34<br />
|-<br />
| 044 || 27.005 || 0.552 || 1.034 || 0.033 || 0.059 || 53.767 || 832.946 || 31.774 || 27.12<br />
|-<br />
| 070 || 13.252 || 0.033 || 1.223 || 0.026 || 0.587 || 55.066 || 200.742 || 1.027 || 13.31 <br />
|-<br />
| 100 || 9.651 || 0.014 || 1.186 || 0.023 || -0.024 || 55.400 || 105.778 || 0.311 || 9.66 <br />
|-<br />
| 143 || 7.248 || 0.015 || 1.036 || 0.009 || 0.383 || 54.130 || 59.954 || 0.246 || 7.27 <br />
|-<br />
| 217 || 4.990 || 0.025 || 1.177 || 0.030 || 0.836 || 54.999 || 28.447 || 0.271 || 5.01<br />
|-<br />
| 353 || 4.818 || 0.024 || 1.147 || 0.028 || 0.655 || 54.745 || 26.714 || 0.250 || 4.86<br />
|- <br />
| 545 || 4.682 || 0.044 || 1.161 || 0.036 || 0.544 || 54.876 || 26.535 || 0.339 || 4.84 <br />
|-<br />
| 857 || 4.325 || 0.055 || 1.393 || 0.076 || 0.876 || 54.779 || 24.244 || 0.193 || 4.63 <br />
|}<br />
<br />
<br />
<br />
<br />
====Beam solid angles for the PCCS====<br />
<br />
** <math>\Omega_{eff}</math> - 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, i.e. <math> 4 \pi \sum(B_{ij}) / max(B_{ij}) </math>.<br />
<br />
** from <math>\Omega_{eff}</math> we estimate the <math>fwhm_{eff}</math>, under a Gaussian approximation - these are tabulated above<br />
** <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>.<br />
*** 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).<br />
<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Band averaged beam solid angles'''<br />
| '''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>] <br />
|-<br />
|30 || 1189.513 || 0.842 || 1116.494 || 2.274 || 1188.945 || 0.847 <br />
|-<br />
| 44 || 832.946 || 31.774 || 758.684 || 29.701 || 832.168 || 31.811 <br />
|-<br />
| 70 || 200.742 || 1.027 || 186.260 || 2.300 || 200.591 || 1.027 <br />
|-<br />
| 100 || 105.778 || 0.311 || 100.830 || 0.410 || 105.777 || 0.311 <br />
|-<br />
| 143 || 59.954 || 0.246 || 56.811 || 0.419 || 59.952 || 0.246 <br />
|-<br />
| 217 || 28.447 || 0.271 || 26.442 || 0.537 || 28.426 || 0.271 <br />
|-<br />
| 353 || 26.714 || 0.250 || 24.827 || 0.435 || 26.653 || 0.250 <br />
|-<br />
| 545 || 26.535 || 0.339 || 24.287 || 0.455 || 26.302 || 0.337 <br />
|-<br />
| 857 || 24.244 || 0.193 || 22.646 || 0.263 || 23.985 || 0.191 <br />
|}<br />
<br />
<br />
==Production process==<br />
------------------------<br />
<br />
The methodology for computing effective beams for a scanning CMB experiment like Planck<br />
was presented in [[http://arxiv.org/pdf/1005.1929| Mitra, Rocha, Gorski et al.]].<br />
<br />
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:<br />
* 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)<br />
* follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position<br />
* project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)<br />
* add up all beams that cross the same pixel and its vicinity over the observing period of interest<br />
*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)<br />
*use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission<br />
<br />
<br />
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. <br />
<br />
<br />
===Pixel Ordered Detector Angles (PODA)===<br />
<br />
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.<br />
<br />
===effBeam===<br />
<br />
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.<br />
<br />
<br />
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.<br />
<br />
<br />
===Computational Cost===<br />
<br />
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.<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+ 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<br />
|-<br />
|Channel ||030 || 044 || 070 || 100 || 143 || 217 || 353 || 545 || 857<br />
|-<br />
|PODA/Detector Computation time (CPU hrs) || 85 || 100 || 250 || 500 || 500 || 500 || 500 || 500 || 500 <br />
|-<br />
|PODA/Detector Computation time (Human minutes) || 7 || 10 || 20 || 20 || 20 || 20 || 20 || 20 || 20<br />
|- <br />
|Beam/Channel Computation time (CPU hrs) || 900 || 2000 || 2300 || 2800 || 3800 || 3200 || 3000 || 900 || 1100<br />
|-<br />
|Beam/Channel Computation time (Human hrs) || 0.5 || 0.8 || 1 || 1.5 || 2 || 1.2 || 1 || 0.5 || 0.5<br />
|-<br />
|Convolution Computation time (CPU hr) || 1 || 1.2 || 1.3 || 3.6 || 4.8 || 4.0 || 4.1 || 4.1 || 3.7 <br />
|-<br />
|Convolution Computation time (Human sec) || 1 || 1 || 1 || 4 || 4 || 4 || 4 || 4 || 4 <br />
|-<br />
|Effective Beam Size (GB) || 173 || 123 || 28 || 187 || 182 || 146 || 132 || 139 || 124<br />
|}<br />
<br />
<br />
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.<br />
<br />
<br />
<br />
==Inputs==<br />
------------<br />
<br />
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. <br />
<br />
<br />
[[File:PlanckFocalPlane.png | 600px| thumb | center| "'Planck Focal Plane''']]<br />
<br />
<br />
===The Focal Plane DataBase (FPDB)===<br />
<br />
The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, ... (see the instrument [[The RIMO|RIMOs]]):<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
<br />
<br />
{{PLADoc|fileType=rimo|link=The Plank RIMOS}}<br />
<br />
<br />
<br />
===The scanning strategy===<br />
<br />
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.<br />
<br />
===The scanbeam===<br />
<br />
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.<br />
<br />
* 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. <br />
* HFI: [[Beams#Scanning beams|B-Spline, BS]] based on 2 observations of Mars.<br />
<br />
(see the instrument [[The RIMO|RIMOs]])<br />
<br />
<br />
<br />
*HFI - LFI_RIMO_DX9_PTCOR6 - {{PLASingleFile|fileType=rimo|name=HFI_RIMO_R1.00.fits|link=The HFI RIMO}}<br />
*LFI - HFI-RIMO-3_16_detilt_t2_ptcor6.fits - {{PLASingleFile|fileType=rimo|name=LFI_RIMO_R1.12.fits|link=The LFI RIMO}}<br />
[[Beams LFI#Effective beams|LFI effective beams]]<br />
<br />
===Beam cutoff radii===<br />
<br />
N times geometric mean of FWHM of all detectors in a channel, where N<br />
<br />
{|border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"<br />
|+'''Beam cut off radius'''<br />
| '''channel''' || '''Cutoff Radii in units of fwhm''' || '''fwhm of full beam extent''' <br />
|-<br />
|30 - 44 - 70 || 2.5 ||<br />
|-<br />
|100 || 2.25 || 23.703699<br />
|-<br />
|143 || 3 || 21.057402<br />
|-<br />
|217-353 || 4 || 18.782754<br />
|-<br />
|sub-mm || 4 || 18.327635(545GHz) ; 17.093706(857GHz) <br />
|}<br />
<br />
===Map resolution for the derived beam data object===<br />
<br />
* <math>N_{side} = 1024 </math> for LFI frequency channels<br />
* <math>N_{side} = 2048 </math> for HFI frequency channels<br />
<br />
<br />
==Related products==<br />
----------------------<br />
<br />
===Monte Carlo simulations===<br />
<br />
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:<br />
<br />
* generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission<br />
* generate 100 realizations of maps from a fiducial CMB power spectrum<br />
* convolve each one of these maps with the effective beams using FEBeCoP<br />
* estimate the average of the Power Spectrum of each convolved realization, C'_\ell_'^out^'}, and 1 sigma errors<br />
<br />
<br />
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#.<br />
For example [[HL-sims#FFP6 data set|FFP6]] <br />
<br />
<br />
<br />
===Beam Window Functions===<br />
<br />
The '''Transfer Function''' or the '''Beam Window Function''' <math> W_l </math> relates the true angular power spectra <math>C_l </math> with the observed angular power spectra <math>\widetilde{C}_l </math>:<br />
<br />
<math><br />
W_l= \widetilde{C}_l / C_l <br />
\label{eqn:wl1}</math> <br />
<br />
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.<br />
<br />
The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial <math> C_l </math>, convolve the maps with the pre-computed effective beams, compute the convolved power spectra <math> C^\text{conv}_l </math>, divide by the power spectra of the unconvolved map <math>C^\text{in}_l </math> and average over their ratio. Thus, the estimated window function<br />
<br />
<math><br />
W^{est}_l = < C^{conv}_l / C^{in}_l ><br />
\label{eqn:wl2}</math> <br />
<br />
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.<br />
<br />
Beam window functions are provided in the [[The RIMO#Beam Window Functions|RIMO]]. <br />
<br />
<br />
====Beam Window functions, Wl, for Planck mission====<br />
<br />
<br />
<br />
[[File:plot_dx9_LFI_GB_pix.png | 600px | thumb | center |'''Beam Window functions, Wl, for LFI channels''']] <br />
[[File:plot_dx9_HFI_BS_M12_CMB.png | 600px | thumb | |center |'''Beam Window functions, Wl, for HFI channels''']]<br />
<br />
<br />
<br />
<br />
==File Names==<br />
-----------------<br />
<br />
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.<br />
<br />
* beam_index.fits<br />
* beams_run.log<br />
<br />
== Retrieval of effective beam information from the PLA interface ==<br />
<br />
In order to retrieve the effective beam information, the user should first launch the Java interface from this page:<br />
http://www.sciops.esa.int/index.php?project=planck&page=Planck_Legacy_Archive<br />
<br />
One should click on "Sky maps" and then open the "Effective beams" area.<br />
There is the possibility to either retrieve one beam nearest to the input source (name or coordinates), or to retrieve a set of beams in a grid defined by the Nside and the size of the region around a source (name or coordinates).<br />
The resolution of this grid is defined by the Nside parameter.<br />
The size of the region is defined by the "Radius of ROI" parameter.<br />
<br />
Once the user proceeds with querying the beams, the PLA software retrieves the appropriate set of effective beams from the database and delivers it in a FITS file which can be directly downloaded.<br />
<br />
<br />
<br />
==Meta data==<br />
----------------<br />
<br />
The data format of the effective beams is unformatted.<br />
<br />
== References ==<br />
------------------<br />
<br />
<biblio force=false><br />
#[[References]]<br />
</biblio><br />
<br />
<br />
<br />
[[Category:Mission science products|004]]</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=Summary_of_HFI_data_characteristics&diff=6597Summary of HFI data characteristics2013-03-15T11:21:39Z<p>Ajaffe: Replace with HFI DPC paper table</p>
<hr />
<div>This page will contains the information from the summary section of the HFI DPC paper (<cite>#planck2013-p03</cite>), which describes the most important characteristics of the HFI instrument and data. <br />
<br />
Note that some of these parameters are available in the RIMO, which is described in detail in [[the RIMO]] section.<br />
<br />
[[File:HFI_Summary.png|1092px|HFI Summary Table]]<br />
<br />
[[Category:Data processing|0048]]</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:HFI_Summary.png&diff=6580File:HFI Summary.png2013-03-15T11:07:04Z<p>Ajaffe: uploaded a new version of "File:HFI Summary.png":&#32;Table of HFI characteristics</p>
<hr />
<div>HFI Characteristics</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:HFI_Summary.png&diff=6578File:HFI Summary.png2013-03-15T11:04:16Z<p>Ajaffe: HFI Characteristics</p>
<hr />
<div>HFI Characteristics</div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:S2vM1_v53.png&diff=4801File:S2vM1 v53.png2013-02-27T14:34:49Z<p>Ajaffe: </p>
<hr />
<div></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:S1vM1_v53.png&diff=4800File:S1vM1 v53.png2013-02-27T14:34:36Z<p>Ajaffe: </p>
<hr />
<div></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:Pointing_offset_100--217GHz_CPP_better_coords.png&diff=4197File:Pointing offset 100--217GHz CPP better coords.png2013-02-14T09:32:51Z<p>Ajaffe: uploaded a new version of "File:Pointing offset 100--217GHz CPP better coords.png"</p>
<hr />
<div></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:Pointing_offset_100--217GHz_CPP_better_coords.png&diff=4179File:Pointing offset 100--217GHz CPP better coords.png2013-02-13T20:15:43Z<p>Ajaffe: </p>
<hr />
<div></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:Mars1.png&diff=4178File:Mars1.png2013-02-13T19:58:24Z<p>Ajaffe: uploaded a new version of "File:Mars1.png"</p>
<hr />
<div></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:Mapbased_ptcor6.png&diff=4177File:Mapbased ptcor6.png2013-02-13T19:55:24Z<p>Ajaffe: uploaded a new version of "File:Mapbased ptcor6.png"</p>
<hr />
<div></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:Mapbased1.png&diff=4169File:Mapbased1.png2013-02-13T16:59:16Z<p>Ajaffe: uploaded a new version of "File:Mapbased1.png"</p>
<hr />
<div></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:Mapbased1.png&diff=2219File:Mapbased1.png2012-10-18T11:53:07Z<p>Ajaffe: uploaded a new version of "File:Mapbased1.png"</p>
<hr />
<div></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:Mapbased_ptcor6.png&diff=2216File:Mapbased ptcor6.png2012-10-18T11:38:41Z<p>Ajaffe: </p>
<hr />
<div></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:Mapbased1.png&diff=2215File:Mapbased1.png2012-10-18T11:35:15Z<p>Ajaffe: </p>
<hr />
<div></div>Ajaffehttps://wiki.cosmos.esa.int/planck-legacy-archive/index.php?title=File:Mars1.png&diff=2213File:Mars1.png2012-10-18T11:34:51Z<p>Ajaffe: </p>
<hr />
<div></div>Ajaffe