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CilmPlus.f95
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subroutine CilmPlus(cilm, gridin, lmax, nmax, mass, d, rho, gridtype, w, zero,&
plx, n, dref, exitstatus)
!------------------------------------------------------------------------------
!
! This routine will compute the potential coefficients associated
! with the input gridded relief using the method of Wieczorek and
! Phillips (1998). The input grid must contain the degree-0 term
! and the computed coefficients will be referenced to the corresonding
! degree-0 radius. Note that the array plx is optional, and should not
! be precomputed when memory is an issue (i.e., lmax>360).
!
! Calling Parameters
!
! IN
! gridin Input grid to be transformed to spherical harmonics.
! lmax Maximum spherical harmonic degree to compute. For
! gridtype 3 and 4, this must be less than or equal to
! N/2 - 1.
! nmax Order of expansion.
! mass Mass of planet.
! rho density of the relief.
! gridtype 1 = Gauss-Legendre quadrature grid corresponding to
! LMAX.
! 2 = NxN Driscoll and Healy grid corresponding to LMAX.
! 3 = Nx2N Driscoll and Healy grid corresponding to LMAX.
! (4 = 2D Cartesian using MakeGrid2D is not implemented).
!
! OUT
! cilm Output spherical harmonic coefficients with
! dimensions (2, lmax+1, lmax+1).
! d The radius that the coefficients are referenced
! to. This parameter corresponds to the degree zero
! term of the data.
!
! OPTIONAL
! w Gauss-Legendre points used in the integrations of
! dimension lmax+1.
! zero Array of dimension lmax+1 that contains the latitudinal
! gridpoints used in the Gauss-Legendre quadrature
! integration scheme. Only needed when PLX is not
! included (determined from a call to SHGLQ).
! plx Input array of Associated Legendre Polnomials computed
! at the Gauss points (determined from a call to
! SHGLQ). If this is not included, then the optional
! array zero MUST be inlcuded.
! N Number of latitude points in the Driscoll and Healy
! grid. Required for Gridtype 2 and 3
! dref The reference radius used to be used when calculating
! the spherical harmonic coefficients. If not specified,
! this will be set equal to the mean radius of GRIDIN.
!
! OPTIONAL (OUT)
! exitstatus If present, instead of executing a STOP when an error
! is encountered, the variable exitstatus will be
! returned describing the error.
! 0 = No errors;
! 1 = Improper dimensions of input array;
! 2 = Improper bounds for input variable;
! 3 = Error allocating memory;
! 4 = File IO error.
!
! All units assumed to be SI.
!
! Copyright (c) 2005-2019, SHTOOLS
! All rights reserved.
!
!------------------------------------------------------------------------------
use SHTOOLS, only: SHExpandGLQ, SHExpandDH
use ftypes
implicit none
real(dp), intent(in) :: gridin(:,:), mass, rho
real(dp), intent(in), optional :: w(:), zero(:), plx(:,:), dref
real(dp), intent(out) :: cilm(:,:,:), d
integer(int32), intent(in) :: lmax, nmax, gridtype
integer(int32), intent(in), optional :: n
integer(int32), intent(out), optional :: exitstatus
real(dp) :: prod, pi, scalef
real(dp), allocatable :: cilmn(:, :, :), grid(:, :)
integer(int32) :: j, l, k, nlat, nlong, astat(2), lmax_dh
if (present(exitstatus)) exitstatus = 0
pi = acos(-1.0_dp)
if (size(cilm(:,1,1)) < 2 .or. size(cilm(1,:,1)) < lmax+1 .or. &
size(cilm(1,1,:)) < lmax+1) then
print*, "Error --- CilmPlus"
print*, "CILM must be dimensioned as (2, LMAX+1, LMAX+1) " // &
"where LMAX is ", lmax
print*, "Input dimension is ", size(cilm(:,1,1)), size(cilm(1,:,1)), &
size(cilm(1,1,:))
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
end if
if (gridtype == 4) then
print*, "Error --- CilmPlus"
print*, "GRIDTYPE 4 (Cartesian obtained from MakeGrid2D) " // &
"is not allowed."
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
else if (gridtype == 1) then
if (present(n)) then
print*, "Error --- CilmPlus"
print*, "N can not be present when using GLQ grids."
if (present(exitstatus)) then
exitstatus = 5
return
else
stop
end if
end if
if (present(w) .and. present(zero) .and. present(plx)) then
print*, "Error --- CilmPlus"
print*, "For GLQ grids, either W and ZERO or W and PLX " // &
"must be present, but not all three."
if (present(exitstatus)) then
exitstatus = 5
return
else
stop
end if
else if (present(w) .and. present(zero)) then
if (size(w) < lmax + 1) then
print*, "Error --- CilmPlus"
print*, "W must be dimensioned as (LMAX+1) where LMAX is ", lmax
print*, "Input dimension is ", size(w)
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
end if
if (size(zero) < lmax + 1) then
print*, "Error --- CilmPlus"
print*, "ZERO must be dimensioned as (LMAX+1) " // &
"where LMAX is ", lmax
print*, "Input dimension is ", size(zero)
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
end if
else if (present(plx) .and. present(w)) then
if (size(w) < lmax + 1) then
print*, "Error --- CilmPlus"
print*, "W must be dimensioned as (LMAX+1) where LMAX is ", lmax
print*, "Input dimension is ", size(w)
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
end if
if (size(plx(:,1)) < lmax+1 .or. &
size(plx(1,:)) < (lmax+1)*(lmax+2)/2) then
print*, "Error --- CilmPlus"
print*, "PLX must be dimensioned as (LMAX+1, " // &
"(LMAX+1)*(LMAX+2)/2) where LMAX is ", lmax
print*, "Input dimension is ", size(plx(:,1)), size(plx(1,:))
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
end if
else
print*, "Error --- CilmPlus"
print*, "For GLQ grids, either W and ZERO or W " // &
"and PLX must be present"
if (present(exitstatus)) then
exitstatus = 5
return
else
stop
end if
end if
else if (gridtype == 2) then
if (present(w) .or. present(zero) .or. present(plx)) then
print*, "Error --- CilmPlus"
print*, "W, ZERO and PLX can not be present for " // &
"Driscoll-Healy grids."
if (present(exitstatus)) then
exitstatus = 5
return
else
stop
end if
else if (.not. present(N)) then
print*, "Error --- CilmPlus"
print*, "N must be present when GRIDTYPE is 2 or 3."
if (present(exitstatus)) then
exitstatus = 5
return
else
stop
end if
end if
else if (gridtype == 3) then
if (present(w) .or. present(zero) .or. present(plx)) then
print*, "Error --- CilmPlus"
print*, "W, ZERO and PLX can not be present " // &
"for Driscoll-Healy grids."
if (present(exitstatus)) then
exitstatus = 5
return
else
stop
end if
else if (.not. present(N)) then
print*, "Error --- CilmPlus"
print*, "N must be present when GRIDTYPE is 2 or 3."
if (present(exitstatus)) then
exitstatus = 5
return
else
stop
end if
end if
else
print*, "Error --- CilmPlus"
print*, "GRIDTYPE must be 1 (GLQ), 2 (NxN) or 3 (Nx2N)"
print*, "Input value is ", gridtype
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
end if
if (gridtype == 1) then
nlat = lmax + 1
nlong = 2 * lmax + 1
else if (gridtype == 2) then
nlat = N
nlong = N
lmax_dh = N / 2 - 1
if (lmax > lmax_dh) then
print*, "Error --- CilmPlus"
print*, "For Driscoll-Healy grids, LMAX must be less " // &
"than or equal to N/2 -1, where N is ", N
print*, "Input value of LMAX is ", lmax
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
end if
else if (gridtype == 3) then
nlat = N
nlong = 2 * N
lmax_dh = N/2 - 1
if (lmax > lmax_dh) then
print*, "Error --- CilmPlus"
print*, "For Driscoll-Healy grids, LMAX must be less " // &
"than or equal to N/2 -1, where N is ", N
print*, "Input value of LMAX is ", lmax
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
end if
end if
if (size(gridin(1,:)) < nlong .or. size(gridin(:,1)) < nlat) then
print*, "Error --- CilmPlus"
if (gridtype == 2) then
print*, "GRIDIN must be dimensioned as " // &
"(LMAX+1, 2*LMAX+1) where LMAX is ", lmax
print*, "Input dimension is ", size(gridin(1,:)), size(gridin(:,1))
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
else if (gridtype == 3) then
print*, "GRIDIN must be dimensioned as (N, N) where N is ", n
print*, "Input dimension is ", size(gridin(1,:)), size(gridin(:,1))
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
else if (gridtype == 4) then
print*, "GRIDIN must be dimensioned as (N, 2N) where N is ", n
print*, "Input dimension is ", size(gridin(1,:)), size(gridin(:,1))
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
end if
end if
allocate (cilmn(2, lmax+1, lmax+1), stat=astat(1))
allocate (grid(nlat, nlong), stat=astat(2))
if (astat(1) /= 0 .or. astat(2) /= 0) then
print*, "Error --- CilmPlus"
print*, "Problem allocating arrays CILMN and GRID", astat(1), astat(2)
if (present(exitstatus)) then
exitstatus = 3
return
else
stop
end if
end if
cilm = 0.0_dp
cilmn = 0.0_dp
!--------------------------------------------------------------------------
!
! Do the expansion.
!
!--------------------------------------------------------------------------
! Do k = 1 terms first
grid(1:nlat, 1:nlong) = gridin(1:nlat,1:nlong)
select case (gridtype)
case (1)
if (present(plx)) then
if (present(exitstatus)) then
call SHExpandGLQ(cilmn(1:2,1:lmax+1,1:lmax+1), lmax, &
grid(1:nlat,1:nlong), w, plx = plx, &
norm = 1, csphase = 1, &
exitstatus = exitstatus)
if (exitstatus /= 0) return
else
call SHExpandGLQ(cilmn(1:2,1:lmax+1,1:lmax+1), lmax, &
grid(1:nlat,1:nlong), w, plx = plx, &
norm = 1, csphase = 1)
end if
else
if (present(exitstatus)) then
call SHExpandGLQ(cilmn(1:2,1:lmax+1,1:lmax+1), lmax, &
grid(1:nlat,1:nlong), w, zero=zero, &
norm = 1, csphase = 1, &
exitstatus = exitstatus)
if (exitstatus /=- 0) return
else
call SHExpandGLQ(cilmn(1:2,1:lmax+1,1:lmax+1), lmax, &
grid(1:nlat,1:nlong), w, zero=zero, &
norm = 1, csphase = 1)
end if
end if
case (2)
if (present(exitstatus)) then
call SHExpandDH(grid(1:nlat,1:nlong), n, &
cilmn(1:2,1:lmax+1,1:lmax+1), lmax_dh, &
norm = 1, sampling = 1, csphase = 1, &
lmax_calc = lmax, exitstatus = exitstatus)
if (exitstatus /= 0) return
else
call SHExpandDH(grid(1:nlat,1:nlong), n, &
cilmn(1:2,1:lmax+1,1:lmax+1), lmax_dh, &
norm = 1, sampling = 1, csphase = 1, &
lmax_calc = lmax)
end if
case (3)
if (present(exitstatus)) then
call SHExpandDH(grid(1:nlat,1:nlong), n, &
cilmn(1:2,1:lmax+1,1:lmax+1), lmax_dh, &
norm = 1, sampling = 2, csphase = 1, &
lmax_calc = lmax, exitstatus = exitstatus)
if (exitstatus /= 0) return
else
call SHExpandDH(grid(1:nlat,1:nlong), n, &
cilmn(1:2,1:lmax+1,1:lmax+1), lmax_dh, &
norm = 1, sampling = 2, csphase = 1, &
lmax_calc = lmax)
end if
end select
if (present(dref)) then
d = dref
else
d = cilmn(1,1,1) ! use mean radius of relief for reference sphere
end if
cilmn(1,1,1) = cilmn(1,1,1) - d
do l = 0, lmax
cilm(1:2,l+1,1:l+1) = 4.0_dp * pi * rho * (d**2) * cilmn(1:2,l+1,1:l+1)&
/ mass / dble(2*l+1)
end do
grid(1:nlat, 1:nlong) = grid(1:nlat, 1:nlong) - d
scalef = maxval(abs(grid(1:nlat,1:nlong)))
grid(1:nlat, 1:nlong) = grid(1:nlat, 1:nlong) / scalef
do k = 2, nmax, 1
grid(1:nlat,1:nlong) = grid(1:nlat,1:nlong) &
* ((gridin(1:nlat,1:nlong) - d) / scalef)
select case (gridtype)
case (1)
if (present(plx)) then
if (present(exitstatus)) then
call SHExpandGLQ(cilmn(1:2,1:lmax+1,1:lmax+1), lmax, &
grid(1:nlat,1:nlong), w, plx=plx, &
norm = 1, csphase = 1, &
exitstatus = exitstatus)
if (exitstatus /= 0) return
else
call SHExpandGLQ(cilmn(1:2,1:lmax+1,1:lmax+1), lmax, &
grid(1:nlat,1:nlong), w, plx=plx, &
norm = 1, csphase = 1)
end if
else
if (present(exitstatus)) then
call SHExpandGLQ(cilmn(1:2,1:lmax+1,1:lmax+1), lmax, &
grid(1:nlat,1:nlong), w, zero=zero, &
norm = 1, csphase = 1, &
exitstatus = exitstatus)
if (exitstatus /= 0) return
else
call SHExpandGLQ(cilmn(1:2,1:lmax+1,1:lmax+1), lmax, &
grid(1:nlat,1:nlong), w, zero=zero, &
norm = 1, csphase = 1)
end if
end if
case (2)
if (present(exitstatus)) then
call SHExpandDH(grid(1:nlat,1:nlong), n, &
cilmn(1:2,1:lmax+1,1:lmax+1), lmax_dh, &
norm = 1, sampling = 1, csphase = 1, &
lmax_calc = lmax, exitstatus = exitstatus)
if (exitstatus /= 0) return
else
call SHExpandDH(grid(1:nlat,1:nlong), n, &
cilmn(1:2,1:lmax+1,1:lmax+1), lmax_dh, &
norm = 1, sampling = 1, csphase = 1, &
lmax_calc = lmax)
end if
case (3)
if (present(exitstatus)) then
call SHExpandDH(grid(1:nlat,1:nlong), n, &
cilmn(1:2,1:lmax+1,1:lmax+1), lmax_dh, &
norm = 1, sampling = 2, csphase = 1, &
lmax_calc = lmax, exitstatus = exitstatus)
if (exitstatus /= 0) return
else
call SHExpandDH(grid(1:nlat,1:nlong), n, &
cilmn(1:2,1:lmax+1,1:lmax+1), lmax_dh, &
norm = 1, sampling = 2, csphase = 1, &
lmax_calc = lmax)
end if
end select
do l = 0, lmax
prod = 4.0_dp * pi * rho * (d**3) / mass * (scalef / d)**k
do j = 2, k, 1
prod = prod * dble(l+4-j)
end do
prod = prod / (dble(2*l+1) * dble(fact(k)))
cilm(1:2,l+1,1:l+1) = cilm(1:2,l+1,1:l+1) &
+ cilmn(1:2,l+1,1:l+1) * prod
end do
end do
deallocate(cilmn)
deallocate(grid)
CONTAINS
function fact(i)
!----------------------------------------------------------------------
!
! This function computes the factorial of an integer.
!
!----------------------------------------------------------------------
implicit none
integer(int32) :: i, j
real(dp) :: fact
if (i == 0) then
fact = 1.0_dp
else if (i < 0) then
print*, "Argument to FACT must be positive"
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
else
fact = 1.0_dp
do j = 1, i
fact = fact * j
end do
end if
end function fact
end subroutine CilmPlus