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HEALPix -- a Framework for High Resolution Discretization, and Fast Analysis of Data Distributed on the Sphere

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abstract

HEALPix -- the Hierarchical Equal Area iso-Latitude Pixelization -- is a versatile data structure with an associated library of computational algorithms and visualization software that supports fast scientific applications executable directly on very large volumes of astronomical data and large area surveys in the form of discretized spherical maps. Originally developed to address the data processing and analysis needs of the present generation of cosmic microwave background (CMB) experiments (e.g. BOOMERanG, WMAP), HEALPix can be expanded to meet many of the profound challenges that will arise in confrontation with the observational output of future missions and experiments, including e.g. Planck, Herschel, SAFIR, and the Beyond Einstein CMB polarization probe. In this paper we consider the requirements and constraints to be met in order to implement a sufficient framework for the efficient discretization and fast analysis/synthesis of functions defined on the sphere, and summarise how they are satisfied by HEALPix.

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  • abstract HEALPix -- the Hierarchical Equal Area iso-Latitude Pixelization -- is a versatile data structure with an associated library of computational algorithms and visualization software that supports fast scientific applications executable directly on very large volumes of astronomical data and large area surveys in the form of discretized spherical maps. Originally developed to address the data processing and analysis needs of the present generation of cosmic microwave background (CMB) experiments (e.g. BOOMERanG, WMAP), HEALPix can be expanded to meet many of the profound challenges that will aris
  • method 0 scenario [31] with time-varying greenhouse gas and ozone concentrations. ICON started 1 January 2020 at approximately 10 km atmospheric and 5 km oceanic resolution; IFS-FESOM started 20 January 2020 at approximately 9 km atmospheric and 5 km oceanic resolution. Output was post-processed onto a Hierarchical Equal Area IsoLatitude Pixelation (HEALPix) grid [32] at∼12.7 km horizontal resolution. 2.2 K¨ oppen-Geiger climate classification We use the K¨ oppen-Geiger classification as described in [
  • background 08022]. [45] H. Li, S.-Y. Li, Y. Liu, Y.-P. Li, Y. Cai, M. Li et al.,Probing Primordial Gravitational Waves: Ali CMB Polarization Telescope,Natl. Sci. Rev.6(2019) 145 [1710.03047]. [46] M. Hazumi et al.,LiteBIRD: A Satellite for the Studies of B-Mode Polarization and Inflation from Cosmic Background Radiation Detection,J. Low Temp. Phys.194(2019) 443. [47] C. Chiocchetta, A. Gruppuso, M. Lattanzi, P. Natoli and L. Pagano,Lack-of-correlation anomaly in CMB large scale polarisation maps,JCAP08(202
  • method 500 snapshots are stored, linearly spaced in scale factor 15. halos are also identified and used in the baryonification process, which we detail in Section 3.3. We build past lightcones by discretizing the volume around an observer into concentric shells, computing the projected density of particles in each of those shells, and storing them on Healpixmaps [65, 66] 16 with anNsideof 8192. This resolution corresponds to a physical scale of 0.43 arcmin or 26 arcsec. The lightcone construction proce
  • method parameters, the marginal widths ∆λand ∆βalone are not sufficient to fully capture the two-dimensional poste- rior structure, especially when the sky posterior exhibits clear multimodality. In such cases, the sky area provides a more direct measure of the overall geometric extent of the localization region. In this work, the sky area is com- puted using a HEALPix-based pixelization scheme [65], with implementation details given in Appendix B. The corresponding parameter-precision, divergence, and
  • background 8 ˚A for all fibers and exposures, which simplifies the co-addition (or averaging) of spectra from the same target obtained in different exposures. The data pro- cessing pipeline provides 2 sets of co-added spectra, one per spectrograph and tile, combining data of several exposures and nights for the same pointing and fiber allocation, and one per HEALPix pixel [42] on the sky where the full co-added spectra (across exposures and tiles) of all the targets in a HEALPix pixel are saved. We use thi
  • background E[d] =WE[a] +E[ϵ] =Wµ a,(19) Cov(d) = Cov(Wa) + Cov(ϵ) =WΣ a W T + Σd.(20) Hence, p(d|θ) =N  d Wµ a, WΣ a W T + Σd  ,(21) so that marginalization over the high-dimensional sur- face map is available in closed form without explicitly computing any integrals. The posterior distribution of the nonlinear parameters θfollows from Bayes' theorem: p(θ|d)∝p(d|θ)p(θ),(22) wherep(d|θ) is the marginal likelihood in Eq. (21), andp(θ) denotes the prior on the nonlinear parameters. Bayesian Doppler Imaging5

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