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module eq_regions_mod |
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! |
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! Purpose. |
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! -------- |
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! eq_regions_mod provides the code to perform a high level |
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! partitioning of the surface of a sphere into regions of |
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! equal area and small diameter. |
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! the type. |
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! |
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! Background. |
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! ----------- |
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! This Fortran version of eq_regions is a much cut down version of the |
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! "Recursive Zonal Equal Area (EQ) Sphere Partitioning Toolbox" of the |
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! same name developed by Paul Leopardi at the University of New South Wales. |
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! This version has been coded specifically for the case of partitioning the |
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! surface of a sphere or S^dim (where dim=2) as denoted in the original code. |
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! Only a subset of the original eq_regions package has been coded to determine |
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! the high level distribution of regions on a sphere, as the detailed |
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! distribution of grid points to each region is left to IFS software. |
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! This is required to take into account the spatial distribution of grid |
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! points in an IFS gaussian grid and provide an optimal (i.e. exact) |
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! distribution of grid points over regions. |
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! |
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! The following copyright notice for the eq_regions package is included from |
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! the original MatLab release. |
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! |
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! +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
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! + Release 1.10 2005-06-26 + |
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! + + |
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! + Copyright (c) 2004, 2005, University of New South Wales + |
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! + + |
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! + Permission is hereby granted, free of charge, to any person obtaining + |
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! + a copy of this software and associated documentation files (the + |
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! + "Software"), to deal in the Software without restriction, including + |
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! + without limitation the rights to use, copy, modify, merge, publish, + |
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! + distribute, sublicense, and/or sell copies of the Software, and to + |
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! + permit persons to whom the Software is furnished to do so, subject to + |
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! + the following conditions: + |
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! + + |
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! + The above copyright notice and this permission notice shall be included + |
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! + in all copies or substantial portions of the Software. + |
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! + + |
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! + THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, + |
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! + EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF + |
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! + MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. + |
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! + IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY + |
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! + CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, + |
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! + TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE + |
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! + SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. + |
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! + + |
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! +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
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! |
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! Author. |
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! ------- |
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! George Mozdzynski *ECMWF* |
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! |
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! Modifications. |
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! -------------- |
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! Original : 2006-04-15 |
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! |
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!-------------------------------------------------------------------------------- |
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USE PARKIND1 ,ONLY : JPIM ,JPRB |
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IMPLICIT NONE |
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SAVE |
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PRIVATE |
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PUBLIC eq_regions,l_regions_debug,n_regions_ns,n_regions_ew,n_regions,my_region_ns,my_region_ew |
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real(kind=jprb) pi |
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logical :: l_regions_debug=.false. |
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integer(kind=jpim) :: n_regions_ns |
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integer(kind=jpim) :: n_regions_ew |
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integer(kind=jpim) :: my_region_ns |
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integer(kind=jpim) :: my_region_ew |
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integer(kind=jpim),allocatable :: n_regions(:) |
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!$OMP THREADPRIVATE(l_regions_debug,my_region_ew,my_region_ns,n_regions_ew,n_regions_ns,pi,n_regions) |
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CONTAINS |
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subroutine eq_regions(N) |
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! |
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! eq_regions uses the zonal equal area sphere partitioning algorithm to partition |
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! the surface of a sphere into N regions of equal area and small diameter. |
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! |
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integer(kind=jpim),intent(in) :: N |
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integer(kind=jpim) :: n_collars,j |
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real(kind=jprb),allocatable :: r_regions(:) |
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real(kind=jprb) :: c_polar |
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pi=2.0_jprb*asin(1.0_jprb) |
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n_regions(:)=0 |
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if( N == 1 )then |
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! |
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! We have only one region, which must be the whole sphere. |
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! |
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n_regions(1)=1 |
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n_regions_ns=1 |
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else |
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! |
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! Given N, determine c_polar |
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! the colatitude of the North polar spherical cap. |
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! |
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c_polar = polar_colat(N) |
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! |
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! Given N, determine the ideal angle for spherical collars. |
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! Based on N, this ideal angle, and c_polar, |
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! determine n_collars, the number of collars between the polar caps. |
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! |
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n_collars = num_collars(N,c_polar,ideal_collar_angle(N)) |
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n_regions_ns=n_collars+2 |
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! |
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! Given N, c_polar and n_collars, determine r_regions, |
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! a list of the ideal real number of regions in each collar, |
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! plus the polar caps. |
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! The number of elements is n_collars+2. |
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! r_regions[1] is 1. |
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! r_regions[n_collars+2] is 1. |
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! The sum of r_regions is N. |
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allocate(r_regions(n_collars+2)) |
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call ideal_region_list(N,c_polar,n_collars,r_regions) |
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! |
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! Given N and r_regions, determine n_regions, a list of the natural number |
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! of regions in each collar and the polar caps. |
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! This list is as close as possible to r_regions. |
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! The number of elements is n_collars+2. |
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! n_regions[1] is 1. |
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! n_regions[n_collars+2] is 1. |
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! The sum of n_regions is N. |
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! |
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call round_to_naturals(N,n_collars,r_regions) |
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deallocate(r_regions) |
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if( N /= sum(n_regions(:)) )then |
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write(*,'("eq_regions: N=",I10," sum(n_regions(:))=",I10)')N,sum(n_regions(:)) |
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call abor1('eq_regions: N /= sum(n_regions)') |
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endif |
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endif |
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if( l_regions_debug )then |
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write(*,'("eq_regions: N=",I6," n_regions_ns=",I4)') N,n_regions_ns |
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do j=1,n_regions_ns |
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write(*,'("eq_regions: n_regions(",I4,")=",I4)') j,n_regions(j) |
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enddo |
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endif |
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n_regions_ew=maxval(n_regions(:)) |
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return |
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end subroutine eq_regions |
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function num_collars(N,c_polar,a_ideal) result(num_c) |
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! |
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!NUM_COLLARS The number of collars between the polar caps |
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! |
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! Given N, an ideal angle, and c_polar, |
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! determine n_collars, the number of collars between the polar caps. |
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! |
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integer(kind=jpim),intent(in) :: N |
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real(kind=jprb),intent(in) :: a_ideal,c_polar |
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integer(kind=jpim) :: num_c |
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logical enough |
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enough = (N > 2) .and. (a_ideal > 0) |
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if( enough )then |
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num_c = max(1,nint((pi-2.*c_polar)/a_ideal)) |
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else |
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num_c = 0 |
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endif |
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return |
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end function num_collars |
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subroutine ideal_region_list(N,c_polar,n_collars,r_regions) |
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! |
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!IDEAL_REGION_LIST The ideal real number of regions in each zone |
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! |
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! List the ideal real number of regions in each collar, plus the polar caps. |
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! |
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! Given N, c_polar and n_collars, determine r_regions, a list of the ideal real |
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! number of regions in each collar, plus the polar caps. |
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! The number of elements is n_collars+2. |
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! r_regions[1] is 1. |
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! r_regions[n_collars+2] is 1. |
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! The sum of r_regions is N. |
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! |
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integer(kind=jpim),intent(in) :: N,n_collars |
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real(kind=jprb),intent(in) :: c_polar |
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real(kind=jprb),intent(out) :: r_regions(n_collars+2) |
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integer(kind=jpim) :: collar_n |
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real(kind=jprb) :: ideal_region_area,ideal_collar_area |
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real(kind=jprb) :: a_fitting |
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r_regions(:)=0.0_jprb |
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r_regions(1) = 1.0_jprb |
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if( n_collars > 0 )then |
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! |
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! Based on n_collars and c_polar, determine a_fitting, |
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! the collar angle such that n_collars collars fit between the polar caps. |
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! |
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a_fitting = (pi-2.0_jprb*c_polar)/float(n_collars) |
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ideal_region_area = area_of_ideal_region(N) |
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do collar_n=1,n_collars |
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ideal_collar_area = area_of_collar(c_polar+(collar_n-1)*a_fitting, & |
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& c_polar+collar_n*a_fitting) |
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r_regions(1+collar_n) = ideal_collar_area / ideal_region_area |
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enddo |
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endif |
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r_regions(2+n_collars) = 1. |
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return |
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end subroutine ideal_region_list |
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function ideal_collar_angle(N) result(ideal) |
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! |
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! IDEAL_COLLAR_ANGLE The ideal angle for spherical collars of an EQ partition |
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! |
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! IDEAL_COLLAR_ANGLE(N) sets ANGLE to the ideal angle for the |
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! spherical collars of an EQ partition of the unit sphere S^2 into N regions. |
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! |
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integer(kind=jpim),intent(in) :: N |
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real(kind=jprb) :: ideal |
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ideal = area_of_ideal_region(N)**(0.5_jprb) |
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return |
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end function ideal_collar_angle |
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subroutine round_to_naturals(N,n_collars,r_regions) |
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! |
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! ROUND_TO_NATURALS Round off a given list of numbers of regions |
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! |
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! Given N and r_regions, determine n_regions, a list of the natural number |
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! of regions in each collar and the polar caps. |
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! This list is as close as possible to r_regions, using rounding. |
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! The number of elements is n_collars+2. |
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! n_regions[1] is 1. |
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! n_regions[n_collars+2] is 1. |
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! The sum of n_regions is N. |
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! |
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integer(kind=jpim),intent(in) :: N,n_collars |
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real(kind=jprb),intent(in) :: r_regions(n_collars+2) |
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integer(kind=jpim) :: zone_n |
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real(kind=jprb) :: discrepancy |
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n_regions(1:n_collars+2) = r_regions(:) |
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discrepancy = 0.0_jprb |
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do zone_n = 1,n_collars+2 |
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n_regions(zone_n) = nint(r_regions(zone_n)+discrepancy); |
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discrepancy = discrepancy+r_regions(zone_n)-float(n_regions(zone_n)); |
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enddo |
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return |
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end subroutine round_to_naturals |
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function polar_colat(N) result(polar_c) |
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! |
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! Given N, determine the colatitude of the North polar spherical cap. |
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! |
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integer(kind=jpim),intent(in) :: N |
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real(kind=jprb) :: area |
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real(kind=jprb) :: polar_c |
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if( N == 1 ) polar_c=pi |
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if( N == 2 ) polar_c=pi/2.0_jprb |
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if( N > 2 )then |
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area=area_of_ideal_region(N) |
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polar_c=sradius_of_cap(area) |
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endif |
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return |
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end function polar_colat |
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function area_of_ideal_region(N) result(area) |
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! |
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! AREA_OF_IDEAL_REGION(N) sets AREA to be the area of one of N equal |
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! area regions on S^2, that is 1/N times AREA_OF_SPHERE. |
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! |
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integer(kind=jpim),intent(in) :: N |
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real(kind=jprb) :: area_of_sphere |
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real(kind=jprb) :: area |
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area_of_sphere = (2.0_jprb*pi**1.5_jprb/gamma(1.5_jprb)) |
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area = area_of_sphere/float(N) |
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return |
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end function area_of_ideal_region |
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function sradius_of_cap(area) result(sradius) |
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! |
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! SRADIUS_OF_CAP(AREA) returns the spherical radius of |
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! an S^2 spherical cap of area AREA. |
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! |
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real(kind=jprb),intent(in) :: area |
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real(kind=jprb) :: sradius |
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sradius = 2.0_jprb*asin(sqrt(area/pi)/2.0_jprb) |
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return |
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end function sradius_of_cap |
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function area_of_collar(a_top, a_bot) result(area) |
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! |
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! AREA_OF_COLLAR Area of spherical collar |
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! |
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! AREA_OF_COLLAR(A_TOP, A_BOT) sets AREA to be the area of an S^2 spherical |
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! collar specified by A_TOP, A_BOT, where A_TOP is top (smaller) spherical radius, |
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! A_BOT is bottom (larger) spherical radius. |
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! |
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real(kind=jprb),intent(in) :: a_top,a_bot |
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real(kind=jprb) area |
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area = area_of_cap(a_bot) - area_of_cap(a_top) |
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return |
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end function area_of_collar |
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function area_of_cap(s_cap) result(area) |
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! |
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! AREA_OF_CAP Area of spherical cap |
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! |
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! AREA_OF_CAP(S_CAP) sets AREA to be the area of an S^2 spherical |
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! cap of spherical radius S_CAP. |
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! |
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real(kind=jprb),intent(in) :: s_cap |
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real(kind=jprb) area |
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✗ |
area = 4.0_jprb*pi * sin(s_cap/2.0_jprb)**2 |
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return |
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end function area_of_cap |
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✗ |
function gamma(x) result(gamma_res) |
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real(kind=jprb),intent(in) :: x |
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real(kind=jprb) :: gamma_res |
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real(kind=jprb) :: p0,p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13 |
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real(kind=jprb) :: w,y |
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integer(kind=jpim) :: k,n |
| 330 |
|
|
parameter (& |
| 331 |
|
|
& p0 = 0.999999999999999990e+00_jprb,& |
| 332 |
|
|
& p1 = -0.422784335098466784e+00_jprb,& |
| 333 |
|
|
& p2 = -0.233093736421782878e+00_jprb,& |
| 334 |
|
|
& p3 = 0.191091101387638410e+00_jprb,& |
| 335 |
|
|
& p4 = -0.024552490005641278e+00_jprb,& |
| 336 |
|
|
& p5 = -0.017645244547851414e+00_jprb,& |
| 337 |
|
|
& p6 = 0.008023273027855346e+00_jprb) |
| 338 |
|
|
parameter (& |
| 339 |
|
|
& p7 = -0.000804329819255744e+00_jprb,& |
| 340 |
|
|
& p8 = -0.000360837876648255e+00_jprb,& |
| 341 |
|
|
& p9 = 0.000145596568617526e+00_jprb,& |
| 342 |
|
|
& p10 = -0.000017545539395205e+00_jprb,& |
| 343 |
|
|
& p11 = -0.000002591225267689e+00_jprb,& |
| 344 |
|
|
& p12 = 0.000001337767384067e+00_jprb,& |
| 345 |
|
|
& p13 = -0.000000199542863674e+00_jprb) |
| 346 |
|
✗ |
n = nint(x - 2) |
| 347 |
|
✗ |
w = x - (n + 2) |
| 348 |
|
|
y = ((((((((((((p13 * w + p12) * w + p11) * w + p10) *& |
| 349 |
|
|
& w + p9) * w + p8) * w + p7) * w + p6) * w + p5) *& |
| 350 |
|
✗ |
& w + p4) * w + p3) * w + p2) * w + p1) * w + p0 |
| 351 |
|
✗ |
if (n .gt. 0) then |
| 352 |
|
✗ |
w = x - 1 |
| 353 |
|
✗ |
do k = 2, n |
| 354 |
|
✗ |
w = w * (x - k) |
| 355 |
|
|
end do |
| 356 |
|
|
else |
| 357 |
|
|
w = 1 |
| 358 |
|
✗ |
do k = 0, -n - 1 |
| 359 |
|
✗ |
y = y * (x + k) |
| 360 |
|
|
end do |
| 361 |
|
|
end if |
| 362 |
|
✗ |
gamma_res = w / y |
| 363 |
|
|
return |
| 364 |
|
|
end function gamma |
| 365 |
|
|
|
| 366 |
|
|
end module eq_regions_mod |
| 367 |
|
|
|