COBE Quadrilateralized Spherical Cube

All COBE map data are presented in a quadrilateralized spherical projection, an approximately equal-area projection (to within a few percent) in which the celestial sphere is projected onto an inscribed cube. Unlike the Aitoff, Mollweide, and Global-sinusoidal projections, the "quad-sphere" projection does not have polar singularities. The following Postscript figures depict various aspects of the CSC on an unfolded cube:

Tables that relate pixel numbers to ecliptic, Galactic, and equatorial coordinates according to the two numbering schemes are given in the FITS binary table files DIRBE_SKYMAP_INFO.FITS and DMR_FIRAS_SKYMAP_INFO.FITS. Note that there is only one file for DMR and FIRAS because the pixel size is the same in both cases. These files can be accessed from the Ancilary data product section of the DIRBE Product Table , DMR Product Table or FIRAS Product Table for DMR_FIRAS ancil files.

Note that the skymap_info FITS files do not contain scientific data. The two files differ only in terms of pixel resolution. The DIRBE instrument has a 42' x 42' square instantaneous field of view; a 21' pixel size (quad-sphere resolution index 9) was adopted for presentation of the DIRBE map data. The FIRAS and DMR instruments each have 7 degree beams and the corresponding map data are presented using 2.8 degree pixels (quad-sphere resolution index 6).

A detailed description of the CSC projection is given in the FITS headers. The header from DIRBE_SKYMAP_INFO.FITS is given below for convenient reference:

SIMPLE  =                    T / File conforms to FITS standard                 
BITPIX  =                   32 / Quad-sphere (QS) pixel nos. are 32-bit integers
NAXIS   =                    3 / Data given as a cube of stacked faces          
NAXIS1  =                  256 / x axis of face = number of QS pixels on side   
NAXIS2  =                  256 / y axis of face = number of QS pixels on side   
NAXIS3  =                    6 / Number of faces [faces numbered from 0 to 5]   
EXTEND  =                    T / Binary table extension follows primary array.  
COMMENT                         / The binary table gives the coordinates        
COMMENT                         / for the center of the pixels.                 
ORIGIN  = 'CDAC    '           / NASA Cosmology Data Analysis Center
DATE    = '14/10/94'           / FITS file creation date (dd/mm/yy)             
TELESCOP= 'COBE    '           / COsmic Background Explorer satellite           
INSTRUME= 'DIRBE   '           / COBE instruments for which the Quad-sphere     
COMMENT                         / pixel array diagram in the data array is      
COMMENT                         / appropriate.                                  
OBJECT  = 'ALL-SKY '           / Part of sky given in the array and extensions  
COMMENT    COBE project keywords (non- FITS reserved):                          
PRODUCT = 'PIXINFO9'           / Pixel information file for given resolution    
PROJECTN= 'QUADSPHR'           / Projection used is a Quad-sphere projection    
PIXORDER= 'QUADTREE'           / Pixel ordering used                            
PIX0LOCN= 'BRC     '           / Location of initial pixel on face              
COMMENT                         / BRC = "Bottom Right Corner"                   
COORDSYS= 'E       '           / Pixelization coordinate system                 
COMMENT    Coordinate system is one of E = Ecliptic, G = Galactic,              
COMMENT    C = Celestial, i.e. Equatorial, or S = Special
PIXSIZE =            0.3239008 / Size in deg. of each side of the avg. QS pixel 
PIXINDEX=                    9 / Quad-sphere pixel resolution index             
COMMENT                         / PIXINDEX = 6 for DMR/FIRAS and 9 for DIRBE    
COMMENT    This file is intended to familiarize users of the COBE data sets with
COMMENT    the COBE sky cube representation and to facilitate usage of the data 
COMMENT    sets by providing sky coordinate information. The COBE sky cube is   
COMMENT    based upon the quadrilateralized spherical cube (quad-sphere)        
COMMENT    projection and the quad-tree pixel ordering scheme.                  
COMMENT     The primary data array for this file gives the quad-sphere (QS)  
COMMENT     pixel number as a function of location on the COBE quad-sphere      
COMMENT     cube faces.  The QS pixels are ordered according to the quad-tree   
COMMENT     nearest neighbor pixel numbering scheme, which is hierarchical.     
COMMENT     Because of this, the QS pixel numbers are not sequential along a    
COMMENT     row or column.                                                      
COMMENT     Data in the primary FITS array are ordered such that the first   
COMMENT     row of pixel numbers is at the bottom and the last row of pixel     
COMMENT     numbers on a face is at the top, with the first QS pixel per        
COMMENT     face being in the lower right corner as shown in the diagram below. 
COMMENT     (See the section "Quad-tree pixelization scheme" below for more     
COMMENT     information on how the pixel numbers fall on the sky cube.)         
COMMENT                row  -------------------                                 
COMMENT                 n   |                 |     n = NAXIS1 = NAXIS2         
COMMENT                 ^   |                 |                                 
COMMENT                 |   |                 |                                 
COMMENT                 2   |                 |                                 
COMMENT                 1   |                 |                                 
COMMENT                     -------------------                                 
COMMENT                      n   <---      2 1                                  
COMMENT                           column                                        
COMMENT       This FITS file also contains one binary table extension after     
COMMENT    the primary data array.  There are no other extensions.  The         
COMMENT    binary table gives the coordinates at the QS pixel centers.          
COMMENT    Coordinate information is given in the binary table for various      
COMMENT    reasons.  The primary reason is that the quad-tree ordering of       
COMMENT    the quad-sphere pixels is not linear.  The CD matrix convention      
COMMENT    might handle this, but it is not yet a part of standard FITS.        
COMMENT    Since COBE data will be analyzed on a pixel-by-pixel basis as        
COMMENT    well as by analyzing the full sky, the binary table format seems     
COMMENT    to be the most conceptually clear.  Furthermore, it is useful to     
COMMENT    give the coordinates of the pixel centers in equatorial and          
COMMENT    Galactic coordinates as well as the COBE standard ecliptic 2000,     
COMMENT    and this is readily done via the binary table format.                
COMMENT    COBE Data Sets:                                                      
COMMENT       The FITS files containing COBE data consist either of a           
COMMENT    primary FITS header and primary data array, OR, a primary FITS       
COMMENT    header, a binary table extension header, and a binary table          
COMMENT    extension. The COBE data files, unlike this information file,        
COMMENT    do not have both a primary data array and a binary table extension.  
COMMENT    For those COBE data files that contain binary table extensions,      
COMMENT    there is only one binary table per file, and there are no other      
COMMENT    extensions in these files. In binary tables, the data consists of    
COMMENT    various parameters in fields (table columns) sequenced by quad-tree  
COMMENT    pixel number.                                                        
COMMENT    Quadrilateralized Sky Cube Projection:                               
COMMENT       COBE datasets are lists of records sequenced by pixel numbers     
COMMENT    defined in a quad-tree manner (see "Quad-tree Pixelization Scheme"   
COMMENT    below) on the faces of a cube.  The cube faces are tangent plane     
COMMENT    projections of the celestial sphere. This is the quadrilateralized   
COMMENT    spherical (sky) cube (or quad-sphere) projection which gives         
COMMENT    equally spaced pixels of approximately equal area on the cube,       
COMMENT    which represent approximately equal areas on the sky.                
COMMENT       The COBE quadrilateralized spherical cube (CSC) projection        
COMMENT    conserves area to approximately 1% across the whole cube with        
COMMENT    a maximum discrepancy from equal area of only a few percent at       
COMMENT    the edges of the cube.  Furthermore, the projection distorts         
COMMENT    shapes minimally.  Beam shapes or point response functions will      
COMMENT    differ only slightly depending upon location on the quad-cube face.  
COMMENT       The six faces of the cube, each of dimension NAXIS1 x NAXIS2,     
COMMENT    are the six planes of the primary data array.  The cube faces are    
COMMENT    numbered 0 through 5 corresponding to planes 1 through 6 (NAXIS3)    
COMMENT    in the primary data array, respectively.                             
COMMENT       The dimensions of each face correspond to the COBE sky cube       
COMMENT    resolution index number given by PIXINDEX.  The dimension of a       
COMMENT    side of each face (NAXIS1, which is necessarily the same value as    
COMMENT    NAXIS2) is 2**(PIXINDEX-1).  Thus, the number of pixels in each      
COMMENT    face is NAXIS1 x NAXIS2 or 2**[2*(PIXINDEX-1)].                      
COMMENT    The all-sky, skyward looking, unfolded cube is reassembled by        
COMMENT    arranging the faces in a sideways T with the bar on the right side:  
COMMENT                                      0                                  
COMMENT                          4   3   2   1                                  
COMMENT                                      5                                  
COMMENT    This corresponds to a 4*NAXIS1 by 3*NAXIS2 array as follows:
COMMENT                        -----------------                                
COMMENT                        | b | b | b | 0 |     ^                          
COMMENT                        -----------------     |                          
COMMENT                        | 4 | 3 | 2 | 1 |  3*NAXIS2                      
COMMENT                        -----------------     |                          
COMMENT                        | b | b | b | 5 |     v                          
COMMENT                        -----------------                                
COMMENT                        <-- 4*NAXIS1 --->                                
COMMENT    where b stands for blanks.                                           
COMMENT       COBE data can be displayed on this unfolded cube by arranging     
COMMENT    the pixels in the binary table as they are arranged in the primary   
COMMENT    array of this file.  For example, if the binary table for the COBE   
COMMENT    data set of interest looks like this (this is not necessarily the    
COMMENT    fields that are in the COBE data tables; the fields and data are     
COMMENT    for illustrative purposes only):                                     
COMMENT    QS_Pixel_Num   Signal    Error_in_Signal   Ecl_Long  Ecl_Lat         
COMMENT         0           22           1             315.0      36.7          
COMMENT         1           43           2             317.9      38.1          
COMMENT         2           36           2             312.1      38.1          
COMMENT         3           31           1             315.0      39.7          
COMMENT                                  .                                      
COMMENT                                  .                                      
COMMENT                                  .                                      
COMMENT    then the signal will be mapped where the pixel numbers are in        
COMMENT    face 0 (first plane) of the primary array:                           
COMMENT       QS Pixel Numbers         Signal                                   
COMMENT            3  2 |       --->   31  36 |                                 
COMMENT            1  0 |       --->   43  22 |                                 
COMMENT            ------              --------                                 
COMMENT    This is the quad-tree pixelization scheme described below.           
COMMENT    Note that the tabulated signal values may not necessarily refer      
COMMENT    to measurements made at the pixel centers; thus, the coordinates     
COMMENT    given may not be those of the pixel centers! If the coordinates      
COMMENT    given with the data set differ from those given in the binary        
COMMENT    table below, then they are not for pixel centers, and thus the       
COMMENT    data will be displayed off-center.                                   
COMMENT    Quad-tree Pixelization Scheme:                                       
COMMENT       Pixel numbers begin in the lower right corner of a cube face      
COMMENT    and increase in quad-tree fashion from right to left and bottom      
COMMENT    to top across the face.  This "Z" pattern is followed throughout     
COMMENT    a face as shown by the following pixel numbers in the lower right    
COMMENT    corner of face 0:                                                    
COMMENT                                                y                        
COMMENT                      63 62 59 58 47 46 43 42 | 7                        
COMMENT                      61 60 57 56 45 44 41 40 | 6    "Z" Pattern:        
COMMENT                      55 54 51 50 39 38 35 34 | 5                        
COMMENT                      53 52 49 48 37 36 33 32 | 4       <---             
COMMENT                      31 30 27 26 15 14 11 10 | 3          ^             
COMMENT                      29 28 25 24 13 12  9  8 | 2         /              
COMMENT                      23 22 19 18  7  6  3  2 | 1        /               
COMMENT                      21 20 17 16  5  4  1  0 | 0       <---             
COMMENT                      -----------------------                            
COMMENT                   x   7  6  5  4  3  2  1  0                            
COMMENT       The quad-tree pixel ordering scheme is related to rasterized      
COMMENT    rectangular array positions as follows.  The location within a       
COMMENT    face is specified by the 2*(PIXINDEX-1) least significant bits.      
COMMENT    (As described below, the most significant bits are used to           
COMMENT    identify the cube face.)  The even least significant bits            
COMMENT    (0, 2, . . ., 2*(PIXINDEX-2)) specify the x, or horizontal,          
COMMENT    raster position, and the odd bits (1, 3, . . ., (2*PIXINDEX)-3)      
COMMENT    specify the y position.  Changing the pixel resolution thus          
COMMENT    simply requires adding or deleting pairs of least significant        
COMMENT    bits; each pair represents a resolution change in the x and y        
COMMENT    face coordinates by a factor of 2.                                   
COMMENT    For example (at a PIXINDEX of 4) quad-tree pixel 24 in binary is     
COMMENT                     0 1 1 0 0 0                                         
COMMENT                     y x y x y x                                         
COMMENT    Extracting the even and odd least significant bits gives:            
COMMENT                     y = 010 = 2                                         
COMMENT                     x = 100 = 4                                         
COMMENT       Writing the pixel numbers in binary form clarifies the mapping    
COMMENT    hierarchy:                                                           
COMMENT   Col 7       6        5       4         3       2        1       0  Row
COMMENT    111111  111110   111011  111010 |  101111  101110   101011  101010 7 
COMMENT    111101  111100   111001  111000 |  101101  101100   101001  101000 6 
COMMENT    110111  110110   110011  110010 |  100111  100110   100011  100010 5 
COMMENT    110101  110100   110001  110000 |  100101  100100   100001  100000 4 
COMMENT    --------------------------------+--------------------------------    
COMMENT    011111  011110   011011  011010 |  001111  001110 | 001011  001010 3 
COMMENT    011101  011100   011001  011000 |  001101  001100 | 001001  001000 2 
COMMENT                                    |  ---------------+---------------   
COMMENT    010111  010110   010011  010010 |  000111  000110 | 000011  000010 1 
COMMENT    010101  010100   010001  010000 |  000101  000100 | 000001  000000 0 
COMMENT    Bit numbers      543210  543210    543210  543210   543210  543210   
COMMENT    The lines of dashes show how the hierarchy is built up.              
COMMENT       As illustrated below, a cube face is bisected (PIXINDEX-1)        
COMMENT    times to reach the pixel resolution.  The figure depicts a           
COMMENT    face at a PIXINDEX of 4.                                             
COMMENT                                                        PIXINDEX:        
COMMENT          --------------------------------------------  ^  ^  ^  ^       
COMMENT          |                     |          | 11 | 10 |  4  |  |  |       
COMMENT          |                     |    11    |---10----|  v  3  |  |       
COMMENT          |                     |          | 01 | 00 |     |  |  |       
COMMENT          |         11          |---------10----------     v  2  |       
COMMENT          |                     |          |         |        |  |       
COMMENT          |                     |    01    |   00    |        |  |       
COMMENT          |                     |          |         |        |  |       
COMMENT          |---------------------+---------------------        v  1 (Whole
COMMENT          |                     |                    |           |  Face)
COMMENT          |                     |                    |           |       
COMMENT          |                     |                    |           |       
COMMENT          |         01          |         00         |           |       
COMMENT          |                     |                    |           |       
COMMENT          |                     |                    |           |       
COMMENT          |                     |                    |           |       
COMMENT          --------------------------------------------           v       
COMMENT       The face index is specified in the remaining Most Significant     
COMMENT    Bits, that is, in bits 2*(PIXINDEX-1) through 2*PIXINDEX.  Thus      
COMMENT    for face 0, pixel 24, the binary number would be                     
COMMENT                     0 0 0    0 1 1 0 0 0     ,                          
COMMENT                    | Face | Pixel number |                              
COMMENT    whereas for pixel number 24 on face 5, the binary number would be    
COMMENT                     1 0 1    0 1 1 0 0 0     .                          
COMMENT                    | Face | Pixel number |                              
COMMENT       The pixel numbers run sequentially from 0 on face 0 to (NAXIS1    
COMMENT    x NAXIS2 x NAXIS3) - 1 on face 5.  The last pixel number in a        
COMMENT    face is (face number+1) * 2**[2*(PIXINDEX-1)] - 1.  The first        
COMMENT    pixel number in a face is 0 for face 0, and 1 plus the last pixel    
COMMENT    number in the previous face for the other faces.  Thus, for a        
COMMENT    PIXINDEX of 4 the pixel number illustrated above, pixel 24 on        
COMMENT    face 5, would be pixel number 344 (binary 101011000 = decimal 344)   
COMMENT    with respect to the whole cube.                                      
COMMENT    World Coordinate Information:
COMMENT       World coordinates such as ecliptic, celestial, and Galactic       
COMMENT    are NOT linear across the COBE sky cube.  As mentioned above,        
COMMENT    the coordinates of the center of each pixel are given in the         
COMMENT    binary table extension rather than by a CDELT or CD matrix           
COMMENT    convention.  The following comments are given as a general           
COMMENT    description of how certain astronomical coordinate systems           
COMMENT    would project onto the cube.                                         
COMMENT       The COBE sky cube is defined using the ecliptic J2000 reference   
COMMENT    frame.  With reference to this coordinate system, face 0 is centered 
COMMENT    around the North ecliptic pole (NEP), faces 1 through 4 are along    
COMMENT    the Ecliptic (each spanning 90 degrees of ecliptic longitude), and   
COMMENT    face 5 is centered around the South ecliptic pole (SEP).  Ecliptic   
COMMENT    longitude increases from face 1 to face 4.  The vernal equinox       
COMMENT    (0.0 ecliptic longitude and latitude) lies at the center of face 1.  
COMMENT    (Face 1 spans ecliptic longitudes from -45 (315) to +45 degrees.)    
COMMENT    The anti-vernal equinox point is at the center of face 3.  Faces     
COMMENT    1-4 include the ecliptic latitudes ranging from roughly -45 to +45   
COMMENT    degrees.  Face 0 contains all ecliptic longitudes, and the           
COMMENT    latitudes around the North ecliptic pole, down to approximately      
COMMENT    +45 degrees.  Face 5 contains all ecliptic longitudes, and the       
COMMENT    latitudes around the South ecliptic pole, up to approximately        
COMMENT    -45 degrees.  The latitude ranges in each face are not exact         
COMMENT    because of the curvilinear nature of the coordinates on the          
COMMENT    COBE sky cube.                                                       
COMMENT       The North Galactic Pole is on face 3 at J2000 ecliptic            
COMMENT    longitude 179.32095 and latitude 29.811954 degrees, and the          
COMMENT    Galactic center is on face 4 at J2000 ecliptic longitude             
COMMENT    266.14097 and latitude -5.52967943 degrees.  The South Galactic      
COMMENT    Pole is on face 1, and the Galactic anti-center is on face 2.        
COMMENT    Pixel numbers for these locations can be found in the binary         
COMMENT    extension table of this file.                                        
COMMENT       The ecliptic coordinates of the North Celestial Pole are          
COMMENT    (90.0,66.5607089) degrees, located on face 0.                        
COMMENT    References:                                                          
COMMENT    Chan, F.K. and O'Neill 1975. Feasibility Study of a Quadri-          
COMMENT      lateralized Spherical Cube Earth Data Base, Computer Sciences      
COMMENT      Corp., EPRF Tech. Report 2-75. Prepared for the Environmental      
COMMENT      Prediction Research Facility, Monterey, Calif.                     
COMMENT    Greisen, Eric W. and Mark Calabretta 1993. Representations of        
COMMENT      Celestial Coordinates in FITS. Draft FITS proposal, NRAO.          
COMMENT    O'Neill, I.M. and Laubscher, R.E. 1976. Extended Studies of a        
COMMENT      Quadrilateralized Spherical Cube Earth Data Base, Computer         
COMMENT      Sciences Corp., EPRF Tech. Report 3-76. Prepared for the           
COMMENT      Environmental Prediction Research Facility, Monterey, Calif.       
COMMENT    White, R.A. and Mather, J.C. 1991. Databases from Cosmic             
COMMENT      Background Explorer (COBE) in Databases and On-Line Data in        
COMMENT      Astronomy, ed. M. Albrecht and D. Egret (Kluwer, Dordrecht,        
COMMENT      the Netherlands), pp. 29-34.                                       
COMMENT    White, Richard A. and Stemwedel, Sally W. 1992. "The Quadri-         
COMMENT     lateralized Spherical Cube and Quad-Tree for All Sky Data."         
COMMENT     In Astronomical Data Analysis Software and Systems I, Volume        
COMMENT     25, edited by D.M. Worrall, C. Biemesderfer, and J. Barnes,         
COMMENT     pp. 379-381.                                                        
HISTORY   Created by COBE project software facility UDF routine                 
HISTORY   UDF_CREATE_INFO_FILES using FITSIO version 3.3 routines.              
HISTORY   Primary array created by COBE project software UIDL routine PIX2XY.   
COMMENT   End of creation history.                                              

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