What to do about huge LIGO/ Virgo sky maps Leo Singer, NASA/GSFC LIGO-G1800186-v3 !1 The problem We’re getting better at pinpointing gravitational-wave sources as more detectors come online and existing detectors become more sensitive. Unfortunately, as position accuracy improves, the size of the sky maps that we send to observing partners is going to blow up. This started being a minor inconvenience in O2 with GW170817. It will get slowly worse as we approach design sensitivity. It’s already a major pain if you are studying future detector networks with simulations. !2 HEALPix primer Górski+ 2005 HEALPix 763 radians measured eastward. Pixel centers on the northern hemi- sphere are given by the following equations: North polar cap.—Forph ( p 1)=2, the ring index 1 ¼ þ Hierarchical Equal Area isoLatitude Pixelization: data i < Nside, and the pixel-in-ring index 1 j 4i,where structure for storing data that lives on the unit sphere i I ph Iph 1; 2 ¼ À ðÞþ ð Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j p 1 p2ffiffiffiffiffiffiffiffiffiffiffii(i 1); 3 ¼ þ À À ð Þ • Hierarchical: each tile contains four higher-resolution i2 z 1 2 ; 4 ¼ À 3Nside ð Þ daughter tiles, forming a tree structure s j ; and s 1: 5 ¼ 2i À 2 ¼ ð Þ North equatorial belt.— For p0 p 2Nside(Nside 1), Nside ¼ À À • Equal Area: all tiles of a given resolution have the i 2Nside,and1 j 4Nside,where same area and approximately the same shape i Ip0=4 Nside Nside; 6 ¼ ðÞþ ð Þ j p0 mod4 Nside 1; 7 ¼ ðÞþ ð Þ 4 2i • isoLatitude: tiles are arranged on rings of constant z ; 8 ¼ 3 À 3Nside ð Þ No.Fig. 2, 20053.—Orthographic view of theHEALPix HEALPix partition of the sphere. The761 s latitude to permit fast convolution using spherical overplot of equator and meridians illustrates the octahedral symmetry of j ; and s (i Nside 1) mod 2; 9 ¼ 2 Nside À 2 ¼ À þ ð Þ HEALPix. Light gray shading shows one of the 8 (4 north and 4 south) identical harmonics polar base-resolution pixels. Dark gray shading shows one of the 4 identical equatorial base-resolution pixels. Moving clockwise from the top left panel, the where the auxiliary index s describes the phase shifts along the grid is hierarchically subdivided with the grid resolution parameter equal to rings and I(x)isthelargestintegernumbersmallerthanx. Nside 1, 2, 4, 8, and the corresponding total number of pixels equal to Pixel center positions in the southern hemisphere are obtained by ¼ 2 Npix 12 ; N 12, 48, 192, 768. All pixel centers are located on Nring ¼ side ¼ ¼ the mirror symmetry of the grid with respect to the equator (z 0). • Pixelization: good at storing images 4Nside 1 rings of constant latitude. Within each panel the areas of all pixels are ¼ identical.À One can check that the discretized area element ÁzÁ pix is a constant by defining Áz and Á as the variationjj of z and¼ when i and j,respectively,areincreasedbyunity. Fig. 1.—Quadrilateral tree pixel numbering scheme. The coarsely pixelized coordinate patch on the left consists of 4 pixels. Two bits suffice to label the pixels. !3 To increase the resolution, every pixel splits into 4 daughter pixels, shown on the right. These daughters inherit the pixel index of their parent (boxed) and acquire shouldtwo new bits to form be the new retained pixel index. Several suchfor curvilinearly reasons mapped coordinate related patches (12 in to the case the of HEALPix, fast and 6 inharmonic the case of the COBE QuadCube) are joined at the boundaries to cover the sphere. All pixel indices carry a prefix (here omitted for clarity) that identifies which base-resolution pixel they transform.belong to. 4.2. Pixel Indexing This preferred implementation, which is referred to as satisfies desiderata 1 and 3 but by construction fails with desid- rilateral tree, which admits an elegant binary indexation (il- Specific geometrical properties allow HEALPix to support HEALPix,eratum 2. This is a nuisance is afrom geometrically the point of view of application constructed,lustrated in Fig. 1) self-similar, previously employed in the refinable construction of to full-sky survey data as a result of wasteful oversampling near the QuadCube spherical pixelization. two different numbering schemes for the pixels, as illustrated in quadrilateralthe poles of the map. While mesh the angular on resolution the of sphere the mea- asNext, shown let us consider in the Figure base-level spherical 3. The tessellation. base An surements is fixed by the instrument and does not vary over the entire class of such tessellations can be constructed as illus- Figure 4. resolutionsky, the map resolution, comprises or pixel size, depends 12 on the pixels distance intrated three in Figure rings 2. These constructions around are characterized the poles by two from the poles. This must also be accounted for in work related to parameters: N—the number of base-resolution pixel layers First, in the ring scheme, one can simply count the pixels mov- the integration of data or discretized functions over the sphere. between the north and south poles and N—the multiplicity of and(c) Hexagonal equator. sampling Thegrids with resolution icosahedral symmetry per- of thethe meridional grid is cuts, expressed or the number of equatorial by or the circumpolar pa- form superbly in those applications where near uniformity of base-resolution pixels. Obviously, the total number of base- ing down from the north to the south pole along each isolatitude rametersampling on the sphereNside is essential,whichdefinesthenumberofdivisionsalongthe (Saff & Kuijlaars 1997), and resolution pixels is equal to Nbase pix NN, and the area of À ¼ they can be devised to meet desideratum 2 (Tegmark 1996). each one of them is equal to base pix 4=(NN). One should ring. It is in this scheme that Fourier transforms with spherical sideHowever, of by construction a base-resolution they fail both desiderata 1 and pixel 3. thatalso isnotice needed that each tessellation to reachÀ includes¼ two a single desired layers of (d) Igloo-type constructions are devised to satisfy11 desidera- polar cap pixels (with or without an azimuthal twist in their harmonics are easy to implement. Second, one can replicate the high-resolutiontum 3 (E. L. Wright 1997, private partition. communication; CrittendenAll pixelrespective centers positions on the are sphere placed for odd or even on values rings of N, & Turok 1998). Desideratum 2 can be satisfied to reasonable ac- respectively) and (N 2) layers of equatorial zone pixels, tree structure of pixel numbering used, e.g., with the QuadCube. ofcuracy constant if quite a large number latitude, of base-resolution and pixels are is used equidistantwhich form a regular in rhomboidalÀ azimuth grid in the (on cylindrical each pro- (which, however, precludes the efficient construction of simple jection of the sphere. Since the cylindrical projection is an area- This can easily be implemented since, because of the simple ring).wavelet transforms). All isolatitude Conversely, a tree structure rings seeded located with a preserving between mapping, the this property upper immediately and illustrates lower that small number of base-resolution pixels forces significant var- the areas of equatorial zone pixels are all equal, and to meet description of pixel boundaries, the analytical mapping of the iations in both the area and shape of the pixels. our requirement of a fully equal area partition of the sphere,2 we corners of the equatorial base-resolution pixels (i.e., 3 < (e) The GLESP2 construction (Doroshkevich et al. 2005) ex- need to demonstrate that our constructions render identical HEALPix base-resolution elements (curvilinear quadrilaterals) cosplicitly implements< ), the or Gauss-Legendre in the quadrature equatorial scheme to zone,areas for are the polar divided pixels as well. into Indeed, thisthe allowsÀ same one to render high accuracy3 in numerical integrations with respect to formulate a constraint on the colatitude * at which the lateral latitude but allows irregular variations in the pixel area and is vertices of both polar and equatorial pixels meet: into a 0; 1 ; 0; 1 square exists. This tree structure, aka nested numbernot hierarchical—in of fact, pixels: it offers noN relationeq between4N theside tes- . The remaining rings are located ½ ½ sellations derived at different resolutions. ¼ 2 N base pix N 1 scheme, allows one to implement efficiently all applications in- within the polar cap regions ( cos 2 1>cos)andcontainavarying À ;hencecos À : ðÞ¼À 3 à 2 à ¼ N 4. MEETING THE REQUIREMENTS: jj volving nearest-neighbor searches (Wandelt et al. 1998), and number of pixels, increasing from ring to ring, with increasing1 THE HEALPix SOLUTION ð Þ also allows for an immediate construction of the fast Haar wave- All the requirements introduced in 3 are satisfied by the class The curvilinear quadrilateral pixels of this tessellation class distance from thex poles, by one pixel within each quadrant. A of spherical tessellations structured as follows (Go´rski et al. 1999). 2 retain equal areas but vary in shape, depending on their posi- let transform on HEALPix. HEALPixFirst, let us assume map that the has sphereN is partitioned12 intoN a tionspixels on the sphere. of the We have same chosen the areaN 3, N 4grid number of curvilinear quadrilaterals, whichpix constitute the base- side(Fig.2,middle row, right column) as the definition¼ ofpix our¼ digital level tessellation.2 If there exists a mapping of each¼ element of full-sky map data standard. This choice was based on three driv-¼ The base-resolution pixel index number f runs in 0; N N partition=(3N ontoside a square). 0; 1 ; 0; 1 ,thenanestedn; n sub- ing requirements: that there should be no more than 4 pixels at À division of the square into½ ever diminishing½ subelements is ob- the poles to avoid acute angles, that the elongation of equa- 1 0; 11 .Introducingtherowindex tained trivially, and a hierarchical tree structure for the resulting torial pixels should be simultaneously minimized, and that g ¼ f g È database follows.