ECP pixelization      

GLESP(-like) pixelization:

• The S2HAT pixelization structure, pixelization, of a type s2hat_pixeltype, describing this scheme is set by calling
  
  
  set_pixelization( PIXCHOICE_GLESP, pixpars, pixelization),
  
  where PIXCHOICE_GLESP is a predefined S2HAT parameter (a 4-byte integer set to 1) and:
  
  the field par1 of the pixpars variable (of type s2hat_pixparameters) is equal to nrings, i.e., a number of all isolatitude rings;
  the field par2 of the pixpars variable (of type s2hat_pixparameters) is equal to nphi, i.e., a maximal number of pixels per isolatitude ring (and equal to a number of pixels at the equator).
  
  Check here or here to find more about the set_pixelization routine in Fortran 90 and C, respectively.
  
  Note that the GLESP pixelization implemented here is consistent with the description contained in Doroshkievich et al paper (see below) but does not reimplement directly routines from the GLESP package and therefore its consistency with the latter may need to be checked on the case-by-case basis. Nevertheless the GLESP pixelization as defined in S2HAT is internally consistent, tested and viable pixelization scheme.
  
  Basic description:
  
  
  In the GLESP pixelization the constant iso-latitude rings of pixels are located at the polar angles which correspond to the zeros of the Legendre polynomial of the order nrings. We denote those hereafter as, theta(i). In the particular implementation considered in S2HAT the number of pixels in the equatorial ring of pixels, np(eq) is given by an input parameter, nphi, while the number of the last ring of the Norther hemisphere (plus the equatorial ring if present) ring, nr(eq), is,
  
  

  nr(eq) = (nrings+1)/2

  
  The (approximate) overall pixel area, Sapprox, is then given by,
  
  

  Sapprox = 2 pi / np(eq) (cos( theta[nr(eq)] - dtheta(eq)) - cos( theta[nr(eq)] + dtheta(eq))),

  
  where
  
  

  dtheta(eq) = (theta(nr(eq)+1) - theta(nr(eq)-1))/2,

  and
  
  

  theta(nr(eq)) = pi/2,

  
  if there is an equatorial ring of pixels (i.e., nrings is an odd number).
  A number of pixels in any other ring of pixels is then set to be,
  
  

  np(nr) = round( 2 pi (cos( theta(nr) - dtheta(nr)) - cos( theta(nr) + dtheta(nr))/Sapprox)

  
  where
  
  

  dtheta(nr) = (theta(nr+1) - theta(nr-1))/2.

  
  That makes the pixel area somewhat ring dependent and given by,
  
  

  Spixel(nr) = 2 pi / np(nr) (cos( theta(nr) - dtheta(nr)) - cos( theta(nr) + dtheta(nr)),

  
  for a ring nr.
  In addition the first pixel in each isolatitudinal ring is always centered on meridian, 0. (Possibly there are better, more symmetric, ways to do so though ?!)
  
  Global pixel numbers start from pixel 0 at the North Pole and follow ring-by-ring left-to-right until npixsall-1, where npixsall is defined by the field with same name of the respective s2hat_pixeltype structure.
  
  For more details see:

  • GLESP home page;
  • GLESP paper
    by A.G. Doroshkevich, P.D. Naselsky, O. V. Verkhodanov, D.I. Novikov, V.I. Turchaninov, I.D. Novikov, P.R. Christensen, and, L-Y. Chiang.