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interpolate_module.f90
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module interpolate_module
! interpolate in a stencil
use kind_module, only: i4b, dp
use grid_module, only: latitudes=>lat, coslat, wgt
use sphere_module, only: lon2i, lat2j
implicit none
private
! References:
! - Ritchie (1987) describes selection of points across the pole
! - Interpolation schemes from Numerical Recipes
! Author: T. Enomoto
! History:
! 2007-11-20 higher order Lagrange interpolation
! 2007-11-14 simplified stencil finding
! 2004-09-10 some simplification
! 2004-03
integer(kind=i4b), private :: nx, ny, n=3, nh, nhalo, nx1, nx2, ny1, ny2
integer(kind=i4b), dimension(4), private :: is, js
real(kind=dp), private :: u, t, dlon
real(kind=dp), dimension(:), allocatable, private :: lonf, latf
real(kind=dp), dimension(:,:), allocatable, private :: &
ff, ffx, ffy, ffxy, fu, fv, ffxl, ffyl
private :: find_stencil, quasimonotone_filter, quasimonotone_filter_bicubic
public :: interpolate_init, interpolate_clean, &
interpolate_set, interpolate_setuv, &
interpolate_setd, interpolate_setdx, &
interpolate_bilinear, interpolate_bilinearuv, &
interpolate_polin2, interpolate_polin2uv, &
interpolate_bicubic, interpolate_linpol, &
interpolate_spcher, interpolate_diff
contains
subroutine interpolate_init(f)
use math_module, only: pi2=>math_pi2, pih=>math_pih
implicit none
real(kind=dp), dimension(:,:) :: f
integer(kind=i4b) :: i, j
namelist /interpolate/ n, nhalo
nh = n/2
nhalo = nh
read(unit=5, nml=interpolate)
write(unit=6, nml=interpolate)
nx = size(f,1)
ny = size(f,2)
nx1 = 1 - nhalo
nx2 = nx + nhalo + 1
ny1 = 1 - nhalo - 1
ny2 = ny + nhalo + 1
allocate(lonf(nx1:nx2), latf(ny1:ny2), ff(nx1:nx2,ny1:ny2), &
ffx(nx1:nx2,ny1:ny2), ffy(nx1:nx2,ny1:ny2), ffxy(nx1:nx2,ny1:ny2), &
ffxl(nx1:nx2,ny1:ny2), ffyl(nx1:nx2,ny1:ny2), &
fu(nx1:nx2,ny1:ny2),fv(nx1:nx2,ny1:ny2))
dlon = pi2/nx
do i=nx1, nx2
lonf(i) = dlon*(i-1)
end do
latf(1:ny) = latitudes
do j=1, nhalo+1
latf(1-j) = pih + (pih - latitudes(j))
latf(ny+j) = -pih + (-pih - latitudes(ny-j+1))
end do
end subroutine interpolate_init
subroutine interpolate_clean()
implicit none
deallocate(lonf, latf, ff, ffx, ffy, ffxy, fu, fv, ffxl, ffyl)
end subroutine interpolate_clean
subroutine interpolate_bilinear(lon, lat, fi)
implicit none
real(kind=dp), intent(in) :: lon, lat
real(kind=dp), intent(out) :: fi
real(kind=dp), dimension(4) :: fs
integer(kind=i4b) :: k
call find_stencil(lon, lat)
do k=1, 4
fs(k) = ff(is(k),js(k))
end do
! Due to geographical location weight t and 1-t should be swapped.
! However, not so sensitive to t or 1-t at points 3 and 4,
! even slightly worse. Leave it simple.
! if (abs(lat)<latf(1)) then
fi = (1.0_dp-u)*((1.0_dp-t)*fs(1)+t*fs(2)) + u*(t*fs(3)+(1.0_dp-t)*fs(4))
! else
! fi = (1.0_dp-u)*((1.0_dp-t)*fs(1)+t*fs(2)) + u*(t*fs(4)+(1.0_dp-t)*fs(3))
! end if
end subroutine interpolate_bilinear
subroutine interpolate_bilinearuv(lon, lat, fiu, fiv)
implicit none
real(kind=dp), intent(in) :: lon, lat
real(kind=dp), intent(out) :: fiu, fiv
real(kind=dp) :: coslatr
real(kind=dp), dimension(4) :: fsu, fsv
integer(kind=i4b) :: k
call find_stencil(lon, lat)
do k=1, 4
fsu(k) = fu(is(k),js(k))
fsv(k) = fv(is(k),js(k))
end do
! if (abs(lat)<latf(1)) then
fiu = (1.0_dp-u)*((1.0_dp-t)*fsu(1)+t*fsu(2)) + u*(t*fsu(3)+(1.0_dp-t)*fsu(4))
fiv = (1.0_dp-u)*((1.0_dp-t)*fsv(1)+t*fsv(2)) + u*(t*fsv(3)+(1.0_dp-t)*fsv(4))
! coslatr = 1.0_dp/cos(lat)
! fiu = fiu*coslatr
! fiv = fiv*coslatr
! else
! fiu = (1.0_dp-u)*((1.0_dp-t)*fsu(1)+t*fsu(2)) + u*(t*fsu(4)+(1.0_dp-t)*fsu(3))
! fiv = (1.0_dp-u)*((1.0_dp-t)*fsv(1)+t*fsv(2)) + u*(t*fsv(4)+(1.0_dp-t)*fsv(3))
! end if
end subroutine interpolate_bilinearuv
subroutine interpolate_bicubic(lon, lat, fi, monotonic, minmax)
use bicubic_module, only: bcucof, bcuint, bcuintp
implicit none
real(kind=dp), intent(in) :: lon, lat
real(kind=dp), intent(out) :: fi
logical, optional, intent(in) :: monotonic
real(kind=dp), optional, dimension(2), intent(in) :: minmax
real(kind=dp), dimension(4) :: z, zx, zy, zxy
integer(kind=i4b) :: k
real(kind=dp) :: dlat
call find_stencil(lon, lat)
dlat = latf(js(4)) - latf(js(1))
do k=1, 4
z(k) = ff(is(k),js(k))
zx(k) = ffx(is(k),js(k))
zy(k) = ffy(is(k),js(k))
zxy(k) = ffxy(is(k),js(k))
end do
call bcucof(z,zx,zy,zxy,dlon,dlat)
! if (abs(lat)<latf(1)) then
fi = bcuint(t,u)
! else
! fi = bcuintp(t,u)
! end if
if (present(monotonic).and.(monotonic)) then
! call quasimonotone_filter_bicubic(fi,is(1),js(1))
call quasimonotone_filter(fi,is(1),js(1))
end if
if (present(minmax).and.(minmax(1)/=minmax(2))) then
! Ostiguy and Laprise 1990
fi = max(minmax(1),min(minmax(2),fi))
end if
end subroutine interpolate_bicubic
subroutine interpolate_polin2(lon, lat, fi, monotonic, minmax)
use polint_module, only : polin2
implicit none
real(kind=dp), intent(in) :: lon, lat
real(kind=dp), intent(out) :: fi
logical, optional, intent(in) :: monotonic
real(kind=dp), optional, dimension(2), intent(in) :: minmax
integer(kind=i4b) :: i0, i1, i2, j0, j1, j2
real(kind=dp) :: dfi
call find_stencil(lon, lat)
i0 = is(1)
i1 = i0 - nh
i2 = i0 + nh + 1
j0 = js(1)
j1 = j0 - nh
j2 = j0 + nh + 1
call polin2(lonf(i1:i2), latf(j1:j2), ff(i1:i2,j1:j2), lon, lat, fi, dfi)
end subroutine interpolate_polin2
subroutine interpolate_polin2uv(lon, lat, fiu, fiv, monotonic)
use polint_module, only : polin2
implicit none
real(kind=dp), intent(in) :: lon, lat
real(kind=dp), intent(out) :: fiu, fiv
logical, optional, intent(in) :: monotonic
integer(kind=i4b) :: i0, i1, i2, j0, j1, j2
real(kind=dp) :: dfi, coslatr
call find_stencil(lon, lat)
i0 = is(1)
i1 = i0 - nh
i2 = i0 + nh + 1
j0 = js(1)
j1 = j0 - nh
j2 = j0 + nh + 1
call polin2(lonf(i1:i2), latf(j1:j2), fu(i1:i2,j1:j2), lon, lat, fiu, dfi)
call polin2(lonf(i1:i2), latf(j1:j2), fv(i1:i2,j1:j2), lon, lat, fiv, dfi)
! Bermejo and Staniforth 1992
if (present(monotonic).and.(monotonic)) then
fiu = min(fiu,maxval(fu(i0:i0+1,j0:j0+1)))
fiu = max(fiu,minval(fu(i0:i0+1,j0:j0+1)))
fiv = min(fiv,maxval(fv(i0:i0+1,j0:j0+1)))
fiv = max(fiv,minval(fv(i0:i0+1,j0:j0+1)))
end if
! coslatr = 1.0_dp/cos(lat)
! fiu = fiu * coslatr
! fiv = fiv * coslatr
end subroutine interpolate_polin2uv
subroutine interpolate_linpol(lon, lat, fi, monotonic, minmax)
use polint_module, only : polint
implicit none
real(kind=dp), intent(in) :: lon, lat
real(kind=dp), intent(out) :: fi
logical, optional, intent(in) :: monotonic
real(kind=dp), optional, dimension(2), intent(in) :: minmax
integer(kind=i4b) :: i0, i1, i2, j0, j1, j2, j
real(kind=dp) :: dfi
real(kind=dp), dimension(n+1) :: ytmp
call find_stencil(lon, lat)
i0 = is(1)
j0 = js(1)
i1 = i0 - nh
i2 = i0 + nh + 1
j1 = j0 - nh
j2 = j0 + nh + 1
do j=j1, j0-1
ytmp(j-j1+1) = (1.0_dp-t)*ff(i0,j)+t*ff(i0+1,j)
end do
do j=j0, j0+1
call polint(lonf(i1:i2), ff(i1:i2,j), lon, ytmp(j-j1+1), dfi)
end do
do j=j0+2, j2
ytmp(j-j1+1) = (1.0_dp-t)*ff(i0,j)+t*ff(i0+1,j)
end do
call polint(latf(j1:j2), ytmp, lat, fi, dfi)
if (present(monotonic).and.(monotonic)) then
call quasimonotone_filter(fi,is(1),js(1))
end if
if (present(minmax).and.(minmax(1)/=minmax(2))) then
! Ostiguy and Laprise 1990
fi = max(minmax(1),min(minmax(2),fi))
end if
end subroutine interpolate_linpol
subroutine interpolate_spcher(lon, lat, fi, monotonic, minmax)
use cubicspline_module, only : cubicspline_interpolate
use polint_module, only : polint
implicit none
real(kind=dp), intent(in) :: lon, lat
real(kind=dp), intent(out) :: fi
logical, optional, intent(in) :: monotonic
real(kind=dp), optional, dimension(2), intent(in) :: minmax
integer(kind=i4b) :: i0, j0, j1, j2, j
real(kind=dp) :: dfi
real(kind=dp), dimension(4) :: fs
real(kind=dp), dimension(n+1) :: ytmp
call find_stencil(lon, lat)
i0 = is(1)
j0 = js(1)
j1 = j0 - nh
j2 = j0 + nh + 1
do j=j1, j2
fs(1) = ff(i0,j)
fs(2) = ff(i0+1,j)
fs(3) = ffx(i0,j)
fs(4) = ffx(i0+1,j)
call cubicspline_interpolate(t, dlon, fs, ytmp(j-j1+1))
end do
call polint(latf(j1:j2), ytmp, lat, fi, dfi)
if (present(monotonic).and.(monotonic)) then
call quasimonotone_filter(fi,is(1),js(1))
end if
if (present(minmax).and.(minmax(1)/=minmax(2))) then
! Ostiguy and Laprise 1990
fi = max(minmax(1),min(minmax(2),fi))
end if
end subroutine interpolate_spcher
subroutine interpolate_set(f)
implicit none
real(kind=dp), dimension(:,:), intent(in) :: f
integer(kind=i4b) :: i, j
ff(1:nx,1:ny) = f
do j=1, nh+1
ff(1:nx,1-j) = cshift(ff(1:nx,j),nx/2)
ff(1:nx,ny+j) = cshift(ff(1:nx,ny-(j-1)),nx/2)
end do
do i=1, nh
ff(1-i,:) = ff(nx-(i-1),:)
end do
do i=1, nh+1
ff(nx+i,:) = ff(1+(i-1),:)
end do
end subroutine interpolate_set
subroutine interpolate_setuv(gu,gv)
implicit none
real(kind=dp), dimension(:,:), intent(in) :: gu, gv
integer(kind=i4b) :: i, j
do j=1, ny
fu(1:nx,j) = gu(:,j)!*coslat(j)
fv(1:nx,j) = gv(:,j)!*coslat(j)
end do
! direction of u, v is reversed beyond poles
do j=1, nh+1
fu(1:nx,1-j) = -cshift(fu(1:nx,j),nx/2)
fu(1:nx,ny+j) = -cshift(fu(1:nx,ny-(j-1)),nx/2)
fv(1:nx,1-j) = -cshift(fv(1:nx,j),nx/2)
fv(1:nx,ny+j) = -cshift(fv(1:nx,ny-(j-1)),nx/2)
end do
do i=1, nh
fu(1-i,:) = fu(nx-(i-1),:)
fv(1-i,:) = fv(nx-(i-1),:)
end do
do i=1, nh+1
fu(nx+i,:) = fu(1+(i-1),:)
fv(nx+i,:) = fv(1+(i-1),:)
end do
end subroutine interpolate_setuv
subroutine interpolate_setd(fx,fy,fxy)
implicit none
real(kind=dp), dimension(:,:), intent(in) :: fx, fy, fxy
integer(kind=i4b) :: i, j
ffx(1:nx,1:ny) = fx
ffy(1:nx,1:ny) = fy
ffxy(1:nx,1:ny) = fxy
! directions of d/dx and d/dy are reversed beyond poles
do j=1, nh+1
ffx(1:nx,1-j) = -cshift(ffx(1:nx,j),nx/2)
ffx(1:nx,ny+j) = -cshift(ffx(1:nx,ny-(j-1)),nx/2)
ffy(1:nx,1-j) = -cshift(ffy(1:nx,j),nx/2)
ffy(1:nx,ny+j) = -cshift(ffy(1:nx,ny-(j-1)),nx/2)
ffxy(1:nx,1-j) = cshift(ffxy(1:nx,j),nx/2)
ffxy(1:nx,ny+j) = cshift(ffxy(1:nx,ny-(j-1)),nx/2)
end do
do i=1, nh
ffx(1-i,:) = ffx(nx-(i-1),:)
ffy(1-i,:) = ffy(nx-(i-1),:)
ffxy(1-i,:) = ffxy(nx-(i-1),:)
end do
do i=1, nh+1
ffx(nx+i,:) = ffx(1+(i-1),:)
ffy(nx+i,:) = ffy(1+(i-1),:)
ffxy(nx+i,:) = ffxy(1+(i-1),:)
end do
end subroutine interpolate_setd
subroutine interpolate_setdx(fx)
implicit none
real(kind=dp), dimension(:,:), intent(in) :: fx
integer(kind=i4b) :: i, j
ffx(1:nx,1:ny) = fx
! direction of d/dx is reversed beyond poles
do j=1, nh+1
ffx(1:nx,1-j) = -cshift(ffx(1:nx,j),nx/2)
ffx(1:nx,ny+j) = -cshift(ffx(1:nx,ny-(j-1)),nx/2)
end do
do i=1, nh
ffx(1-i,:) = fx(nx-(i-1),:)
end do
do i=1, nh+1
ffx(nx+i,:) = fx(1+(i-1),:)
end do
ffy(:,:) = 0.d0
ffxy(:,:) = 0.d0
end subroutine interpolate_setdx
subroutine interpolate_diff()
implicit none
integer(kind=i4b) :: i, j
forall(i=1:nx, j=1:ny)
ffxl(i,j) = ff(i+1,j) - ff(i,j)
ffyl(i,j) = ff(i,j+1) - ff(i,j)
end forall
end subroutine interpolate_diff
subroutine find_stencil(lon, lat)
implicit none
real(kind=dp), intent(in) :: lon, lat
integer(kind=i4b) :: j
is(1) = lon2i(lon,nx)
is(2) = is(1) + 1
t = lon/dlon - is(1) + 1.0_dp ! t = (lon - dlon*(i-1))/dlon
is(3:4) = is(2:1:-1)
j = lat2j(lat,ny)
if (lat>latf(j)) then
j = j - 1
end if
js(1:2) = j
js(3:4) = j + 1
u = (lat-latf(j))/(latf(j+1)-latf(j))
end subroutine find_stencil
subroutine quasimonotone_filter(fi,i0,j0)
implicit none
real(kind=dp), intent(inout) :: fi
integer(kind=i4b), intent(in) :: i0, j0
logical :: lmono
! apply filter to a cell that should be monotonic
lmono = .false.
if (nhalo>=2) then
lmono = .not. ( &! Nair et al. 1999
((ffxl(i0-2,j0) *ffxl(i0-1,j0) >0.0_dp).and. &
(ffxl(i0-1,j0) *ffxl(i0+1,j0) <0.0_dp).and. &
(ffxl(i0+1,j0) *ffxl(i0+2,j0) >0.0_dp)).or. &
((ffxl(i0-2,j0+1)*ffxl(i0-1,j0+1)>0.0_dp).and. &
(ffxl(i0-1,j0+1)*ffxl(i0+1,j0+1)<0.0_dp).and. &
(ffxl(i0+1,j0+1)*ffxl(i0+2,j0+1)>0.0_dp)).or. &
((ffyl(i0, j0-2)*ffyl(i0, j0-1)>0.0_dp).and. &
(ffyl(i0, j0-1)*ffyl(i0, j0+1)<0.0_dp).and. &
(ffyl(i0, j0+1)*ffyl(i0, j0+2)>0.0_dp)).or. &
((ffyl(i0+1,j0-2)*ffyl(i0+1,j0-1)>0.0_dp).and. &
(ffyl(i0+1,j0-1)*ffyl(i0+1,j0+1)<0.0_dp).and. &
(ffyl(i0+1,j0+1)*ffyl(i0+1,j0+2)>0.0_dp)) )
else
lmono = & ! Sun et al. 1996
(ffxl(i0-1,j0)*ffxl(i0+1,j0)>=0.0_dp).and. &
(ffxl(i0-1,j0+1)*ffxl(i0+1,j0+1)>=0.0_dp).and. &
(ffyl(i0,j0-1)*ffyl(i0,j0+1)>=0.0_dp).and. &
(ffyl(i0+1,j0-1)*ffyl(i0+1,j0+1)>=0.0_dp)
end if
if (lmono) then
! Bermejo and Staniforth 1992
fi = max(min(fi,maxval(ff(i0:i0+1,j0:j0+1))),minval(ff(i0:i0+1,j0:j0+1)))
end if
end subroutine quasimonotone_filter
subroutine quasimonotone_filter_bicubic(fi,i0,j0)
implicit none
real(kind=dp), intent(inout) :: fi
integer(kind=i4b), intent(in) :: i0, j0
logical :: lmono
! apply filter to a cell that should be monotonic
lmono = .false.
if (nhalo>=2) then
lmono = .not. ( &! Nair et al. 1999
((ffxl(i0-2,j0) *ffxl(i0-1,j0) >0.0_dp).and. &
(ffxl(i0-1,j0) *ffxl(i0+1,j0) <0.0_dp).and. &
(ffxl(i0+1,j0) *ffxl(i0+2,j0) >0.0_dp).and. &
(ffxl(i0-1,j0) *ffx( i0, j0) >0.0_dp).and. &
(ffxl(i0+1,j0) *ffx( i0+1,j0) >0.0_dp)).or. &
((ffxl(i0-2,j0+1)*ffxl(i0-1,j0+1)>0.0_dp).and. &
(ffxl(i0-1,j0+1)*ffxl(i0+1,j0+1)<0.0_dp).and. &
(ffxl(i0+1,j0+1)*ffxl(i0+2,j0+1)>0.0_dp).and. &
(ffxl(i0-1,j0+1)*ffx( i0, j0+1)>0.0_dp).and. &
(ffxl(i0+1,j0+1)*ffx( i0+1,j0+1)>0.0_dp)).or. &
((ffyl(i0, j0-2)*ffyl(i0, j0-1)>0.0_dp).and. &
(ffyl(i0, j0-1)*ffyl(i0, j0+1)<0.0_dp).and. &
(ffyl(i0, j0+1)*ffyl(i0, j0+2)>0.0_dp).and. &
(ffyl(i0, j0-1)*ffy( i0, j0) >0.0_dp).and. &
(ffyl(i0, j0+1)*ffy( i0, j0+1)>0.0_dp)).or. &
((ffyl(i0+1,j0-2)*ffyl(i0+1,j0-1)>0.0_dp).and. &
(ffyl(i0+1,j0-1)*ffyl(i0+1,j0+1)<0.0_dp).and. &
(ffyl(i0+1,j0+1)*ffyl(i0+1,j0+2)>0.0_dp).and. &
(ffyl(i0+1,j0-1)*ffy( i0+1,j0) >0.0_dp).and. &
(ffyl(i0+1,j0+1)*ffy( i0+1,j0+1)>0.0_dp)) )
else
lmono = .not. ( &! Sun et al. 1996
((ffxl(i0-1,j0) *ffxl(i0+1,j0) <0.0_dp).and. &
(ffxl(i0-1,j0) *ffx( i0, j0) >0.0_dp).and. &
(ffxl(i0+1,j0) *ffx( i0+1,j0) >0.0_dp)).or. &
((ffxl(i0-1,j0+1)*ffxl(i0+1,j0+1)<0.0_dp).and. &
(ffxl(i0-1,j0+1)*ffx( i0, j0+1)>0.0_dp).and. &
(ffxl(i0+1,j0+1)*ffx( i0+1,j0+1)>0.0_dp)).or. &
((ffyl(i0, j0-1)*ffyl(i0, j0+1)<0.0_dp).and. &
(ffyl(i0, j0-1)*ffy( i0, j0) >0.0_dp).and. &
(ffyl(i0, j0+1)*ffy( i0, j0+1)>0.0_dp)).or. &
((ffyl(i0+1,j0-1)*ffyl(i0+1,j0+1)<0.0_dp).and. &
(ffyl(i0+1,j0-1)*ffy( i0+1,j0) >0.0_dp).and. &
(ffyl(i0+1,j0+1)*ffy( i0+1,j0+1)>0.0_dp)) )
end if
if (lmono) then
! Bermejo and Staniforth 1992
fi = max(min(fi,maxval(ff(i0:i0+1,j0:j0+1))),minval(ff(i0:i0+1,j0:j0+1)))
end if
end subroutine quasimonotone_filter_bicubic
end module interpolate_module