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MakeGravGradGridDH.f95
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MakeGravGradGridDH.f95
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subroutine MakeGravGradGridDH(cilm, lmax, gm, r0, a, f, vxx, vyy, vzz, vxy, &
vxz, vyz, n, sampling, lmax_calc, extend, &
exitstatus)
!------------------------------------------------------------------------------
!
! Given the 4pi normalized gravitational spherical harmonic coefficients
! CILM, this subroutine will compute 2D Driscol and Healy sampled grids of
! the six components of the gravity "gradient" tensor in a local
! north-oriented reference frame:
!
! (Vxx, Vxy, Vxz)
! (Vyx, Vyy, Vyz)
! (Vzx, Vzy, Vzz)
!
! where X points NORTH, Y points WEST, and Z points UPWARD. The gravitational
! potential is defined as
!
! V = GM/r Sum_{l=0}^LMAX (R0/r)^l Sum_{m=-l}^l C_{lm} Y_{lm},
!
! Laplace's equation implies that Vxx + Vyy + Vzz = 0, and the gravity tensor
! is symmetric. The components are calculated according to eq. 1 in
! Petrovskaya and Vershkov (2006, J. Geod, 80, 117-127), which is based on
! eq. 3.28 in Reed (1973, Ohio State Univ., Dept. Geod. Sci., Rep. 201,
! Columbus, OH). Note that Reed's equations are in terms of latitude, and
! that the Y axis points East:
!
! Vzz = Vrr
! Vxx = 1/r Vr + 1/r^2 Vtt
! Vyy = 1/r Vr + 1/r^2 /tan(t) Vt + 1/r^2 /sin(t)^2 Vpp
! Vxy = 1/r^2 /sin(t) Vtp - cos(t)/sin(t)^2 /r^2 Vp
! Vxz = 1/r^2 Vt - 1/r Vrt
! Vyz = 1/r^2 /sin(t) Vp - 1/r /sin(t) Vrp
!
! where r, t, p stand for radius, theta, and phi, and subscripts on V denote
! partial derivatives.
!
! The output grid are in units of 1/s and are cacluated on a flattened
! ellipsoid with semi-major axis A and flattening F. To obtain units of
! Eotvos (10^-9 s^-1), multiply by 10^9.
!
! When SAMPLING = 1, the output grids contain N samples in latitude from
! 90 to -90 + interval and N samples in longitude from 0 to 360-2*interval,
! where N=2*(LMAX+1) and interval=180/N. When SAMPLING = 2, the grids are
! equally spaced in degrees latitude and longitude with dimension (N x 2N).
! If the optional parameter EXTEND is set to 1, the output grids will contain
! an extra column corresponding to 360 E and an extra row corresponding to
! 90 S, which increases each of the dimensions of the grid by one.
!
! Calling Parameters
!
! IN
! cilm The 4-pi normalized gravitational potential spherical
! harmonic coefficients.
! lmax The maximum spherical harmonic degree of the function,
! used to determine the number of samples N.
! GM Product of the gravitatonal constant and the planet's
! mass.
! r0 Reference radius of potential coefficients.
! a The semimajor axis of the flattened ellipsoid.
! f Flattening of the planet (a-c)/a.
!
! IN, OPTIONAL
! sampling (1) The output grids are N by N (default).
! (2) The output grid are N by 2N.
! lmax_calc The maximum spherical harmonic degree to evaluate
! the coefficients up to.
! extend If 1, return a grid that contains an additional column
! and row corresponding to 360 E longitude and 90 S
! latitude, respectively.
!
! OUT
! Vxx x-x component of the gravity gradient tensor.
! Vyy y-y component of the gravity gradient tensor.
! Vzz z-z component of the gravity gradient tensor.
! Vxy x-y component of the gravity gradient tensor.
! Vxz x-z component of the gravity gradient tensor.
! Vyz y-z component of the gravity gradient tensor.
! N Number of samples in latitude. Number of samples in
! longitude is N when sampling is 1 (default), and is
! 2N when sampling is 2.
!
! 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.
!
! Notes:
! 1. If lmax is greater than the the maximum spherical harmonic
! degree of the input coefficients, then the coefficients will be
! zero padded.
! 2. Latitude is geocentric latitude.
!
! Copyright (c) 2005-2019, SHTOOLS
! All rights reserved.
!
!------------------------------------------------------------------------------
use FFTW3
use ftypes
use, intrinsic :: iso_c_binding
implicit none
real(dp), intent(in) :: cilm(:,:,:), gm, r0, a, f
real(dp), intent(out) :: vxx(:,:), vyy(:,:), vzz(:,:), vxy(:,:), vxz(:,:),&
vyz(:,:)
integer(int32), intent(in) :: lmax
integer(int32), intent(out) :: n
integer(int32), intent(in), optional :: sampling, lmax_calc, extend
integer(int32), intent(out), optional :: exitstatus
integer(int32) :: l, m, i, l1, m1, lmax_comp, i_eq, i_s, astat(4), nlong, &
nlat_out, nlong_out, extend_grid
real(dp) :: grid(4*lmax+4), pi, theta, scalef, rescalem, u, p, dpl, dpl2, &
dpl2s, pmm, sint, pm1, pm2, z, tempr, r_ex, lat, &
prefactor(lmax), coefr0, coefrs0, coeft0, coefts0, coefp0, &
coefps0, coefrr0, coefrrs0, coefrt0, coefrts0, coefrp0, &
coefrps0, coeftp0, coeftps0, coefpp0, coefpps0, coeftt0, &
coeftts0
complex(dp) :: coef(2*lmax+3), coefr(2*lmax+3), coefrs(2*lmax+3), &
coeft(2*lmax+3), coefts(2*lmax+3), coefp(2*lmax+3), &
coefps(2*lmax+3), tempc, coefrr(2*lmax+3), &
coefrrs(2*lmax+3), coefrt(2*lmax+3), coefrts(2*lmax+3), &
coefrp(2*lmax+3), coefrps(2*lmax+3), coeftp(2*lmax+3), &
coeftps(2*lmax+3), coefpp(2*lmax+3), coefpps(2*lmax+3), &
coeftt(2*lmax+3), coeftts(2*lmax+3)
type(C_PTR) :: plan
real(dp), save, allocatable :: ff1(:,:), ff2(:,:), sqr(:)
integer(int8), save, allocatable :: fsymsign(:,:)
integer(int32), save :: lmax_old = 0
!$OMP threadprivate(ff1, ff2, sqr, fsymsign, lmax_old)
if (present(exitstatus)) exitstatus = 0
n = 2 * lmax + 2
if (present(sampling)) then
if (sampling == 1) then
nlong = n
else if (sampling == 2) then
nlong = 2 * n
else
print*, "Error --- MakeGravGradGridDH"
print*, "Optional parameter SAMPLING must be 1 (N by N) " // &
"or 2 (N by 2N)."
print*, "Input value is ", sampling
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
end if
else
nlong = n
end if
if (present(extend)) then
if (extend == 0) then
extend_grid = 0
nlat_out = n
nlong_out = nlong
else if (extend == 1) then
extend_grid = 1
nlat_out = n + 1
nlong_out = nlong + 1
else
print*, "Error --- MakeGravGradGridDH"
print*, "Optional parameter EXTEND must be 0 or 1."
print*, "Input value is ", extend
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
end if
else
extend_grid = 0
nlat_out = n
nlong_out = nlong
end if
if (size(cilm(:,1,1)) < 2) then
print*, "Error --- MakeGravGradGridDH"
print*, "CILM must be dimensioned as (2, *, *)."
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 (size(vxx(:,1)) < nlat_out .or. size(vxx(1,:)) < nlong_out .or. &
size(vyy(:,1)) < nlat_out .or. size(vyy(1,:)) < nlong_out .or. &
size(vzz(:,1)) < nlat_out .or. size(vzz(1,:)) < nlong_out .or. &
size(vxy(:,1)) < nlat_out .or. size(vxy(1,:)) < nlong_out .or. &
size(vxz(:,1)) < nlat_out .or. size(vxz(1,:)) < nlong_out .or. &
size(vyz(:,1)) < nlat_out .or. size(vyz(1,:)) < nlong_out) then
print*, "Error --- MakeGravGradGridDH"
print*, "VXX, VYY, VZZ, VXY, VXZ, and VYZ must be dimensioned " // &
"as: ", nlat_out, nlong_out
print*, "Input dimensions are ", size(vxx(:,1)), size(vxx(1,:)), &
size(vyy(:,1)), size(vyy(1,:)), size(vzz(:,1)), size(vzz(1,:)), &
size(vxy(:,1)), size(vxy(1,:)), size(vxz(:,1)), size(vxz(1,:)), &
size(vyz(:,1)), size(vyz(1,:))
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
end if
if (cilm(1,1,1) /= 1.0_dp .and. f /= 0.0_dp) then
print*, "Warning --- MakeGravGradGridDH"
print*, "The degree-0 term of the spherical harmonic coefficients is "
print*, "not equal to 1. The variation in gravity resulting from "
print*, "variations in radius of the flattened ellipsoid will not be "
print*, "taken into account."
print*, "C00 = ", cilm(1,1,1)
print*, "F = ", f
end if
pi = acos(-1.0_dp)
scalef = 1.0e-280_dp
if (present(lmax_calc)) then
if (lmax_calc > lmax) then
print*, "Error --- MakeGravGradGridDH"
print*, "LMAX_CALC must be less than or equal to LMAX."
print*, "LMAX = ", lmax
print*, "LMAX_CALC = ", lmax_calc
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
else
lmax_comp = min(lmax, size(cilm(1,1,:))-1, size(cilm(1,:,1))-1, &
lmax_calc)
end if
else
lmax_comp = min(lmax, size(cilm(1,1,:))-1, size(cilm(1,:,1))-1)
end if
!--------------------------------------------------------------------------
!
! Calculate recursion constants used in computing Legendre functions.
!
!--------------------------------------------------------------------------
if (lmax_comp /= lmax_old) then
if (allocated (sqr)) deallocate (sqr)
if (allocated (ff1)) deallocate (ff1)
if (allocated (ff2)) deallocate (ff2)
if (allocated (fsymsign)) deallocate (fsymsign)
allocate (sqr(2 * lmax_comp + 1), stat=astat(1))
allocate (ff1(lmax_comp+1,lmax_comp+1), stat=astat(2))
allocate (ff2(lmax_comp+1,lmax_comp+1), stat=astat(3))
allocate (fsymsign(lmax_comp+1,lmax_comp+1), stat=astat(4))
if (sum(astat(1:4)) /= 0) then
print*, "Error --- MakeGravGradGridDH"
print*, "Problem allocating arrays SQR, FF1, FF2, or FSYMSIGN", &
astat(1), astat(2), astat(3), astat(4)
if (present(exitstatus)) then
exitstatus = 3
return
else
stop
end if
end if
!----------------------------------------------------------------------
!
! Calculate signs used for symmetry of Legendre functions about
! equator. For the first derivative in theta, these signs are
! reversed.
!
!----------------------------------------------------------------------
do l = 0, lmax_comp, 1
do m = 0, l, 1
if (mod(l-m,2) == 0) then
fsymsign(l+1,m+1) = 1
else
fsymsign(l+1,m+1) = -1
end if
end do
end do
!----------------------------------------------------------------------
!
! Precompute square roots of integers that are used several times.
!
!----------------------------------------------------------------------
do l=1, 2*lmax_comp+1
sqr(l) = sqrt(dble(l))
end do
!----------------------------------------------------------------------
!
! Precompute multiplicative factors used in recursion relationships
! P(l,m) = x*f1(l,m)*P(l-1,m) - P(l-2,m)*f2(l,m)
! k = l*(l+1)/2 + m + 1
! Note that prefactors are not used for the case when m=l as a
! different recursion is used. Furthermore, for m=l-1, Plmbar(l-2,m)
! is assumed to be zero.
!
!----------------------------------------------------------------------
if (lmax_comp /= 0) then
ff1(2,1) = sqr(3)
ff2(2,1) = 0.0_dp
end if
do l = 2, lmax_comp, 1
ff1(l+1,1) = sqr(2*l-1) * sqr(2*l+1) / dble(l)
ff2(l+1,1) = dble(l-1) * sqr(2*l+1) / sqr(2*l-3) / dble(l)
do m = 1, l-2, 1
ff1(l+1,m+1) = sqr(2*l+1) * sqr(2*l-1) / sqr(l+m) / sqr(l-m)
ff2(l+1,m+1) = sqr(2*l+1) * sqr(l-m-1) * sqr(l+m-1) &
/ sqr(2*l-3) / sqr(l+m) / sqr(l-m)
end do
ff1(l+1,l) = sqr(2*l+1) * sqr(2*l-1) / sqr(l+m) / sqr(l-m)
ff2(l+1,l) = 0.0_dp
end do
lmax_old = lmax_comp
end if
!--------------------------------------------------------------------------
!
! Create generic plan for grid.
!
!--------------------------------------------------------------------------
plan = fftw_plan_dft_c2r_1d(nlong, coef(1:nlong/2+1), grid(1:nlong), &
FFTW_MEASURE)
!--------------------------------------------------------------------------
!
! Determine Clms one l at a time by intergrating over latitude.
!
!--------------------------------------------------------------------------
i_eq = n / 2 + 1 ! Index correspondong to zero latitude
! First do equator
r_ex = a
theta = pi / 2.0_dp
z = 0.0_dp
u = 1.0_dp
lat = 0.0_dp
coefr(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefr0 = 0.0_dp
coefrr(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefrr0 = 0.0_dp
coeft(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coeft0 = 0.0_dp
coeftp(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coeftp0 = 0.0_dp
coefp(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefp0 = 0.0_dp
coefrt(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefrt0 = 0.0_dp
coefrp(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefrp0 = 0.0_dp
coefpp(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefpp0 = 0.0_dp
coeftt(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coeftt0 = 0.0_dp
pm2 = 1.0_dp
tempr = -cilm(1,1,1) * pm2 ! l = 0
coefr0 = coefr0 + tempr
tempr = 2 * cilm(1,1,1) * pm2 ! l = 0
coefrr0 = coefrr0 + tempr
! derivative in theta and phi of l=0 term is 0. No need to calculate this
if (lmax_comp /= 0) then ! l = 1
prefactor(1) = r0 / r_ex
do l = 2, lmax_comp, 1
prefactor(l) = prefactor(l-1) * r0 / r_ex
end do
pm1 = 0.0_dp
dpl = ff1(2,1)
tempr = cilm(1,2,1) * dpl * prefactor(1)
coeft0 = coeft0 + tempr
tempr = cilm(1,2,1) * dpl * (-2) * prefactor(1) ! -2 = (l+1) prefactor
coefrt0 = coefrt0 + tempr
end if
do l = 2, lmax_comp, 1
l1 = l + 1
p = - ff2(l1,1) * pm2
tempr = cilm(1,l1,1) * p * (-l1) * prefactor(l)
coefr0 = coefr0 + tempr
tempr = cilm(1,l1,1) * p * (l1) * (l1+1) * prefactor(l)
coefrr0 = coefrr0 + tempr
dpl = l * ( sqr(2*l+1) / sqr(2*l-1) * pm1 )
tempr = cilm(1,l1,1) * dpl * prefactor(l)
coeft0 = coeft0 + tempr
tempr = cilm(1,l1,1) * dpl * (-l1) * prefactor(l)
coefrt0 = coefrt0 + tempr
dpl2 = -l*l1 * p
tempr = cilm(1,l1,1) * dpl2 * prefactor(l)
coeftt0 = coeftt0 + tempr
pm2 = pm1
pm1 = p
end do
pmm = sqr(2) * scalef
rescalem = 1.0_dp / scalef
do m = 1, lmax_comp-1, 1
m1 = m + 1
pmm = pmm * sqr(2*m+1) / sqr(2*m)
pm2 = pmm
tempc = cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * pm2 * (-m-1) &
* prefactor(m) ! (m,m)
coefr(m1) = coefr(m1) + tempc
tempc = cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * pm2 * (m+1) &
* (m+2) * prefactor(m) ! (m,m)
coefrr(m1) = coefrr(m1) + tempc
tempc = cmplx(cilm(2,m1,m1), cilm(1,m1,m1), dp) * pm2 &
* prefactor(m) * m ! (m,m)
coefp(m1) = coefp(m1) + tempc
tempc = - cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * pm2 &
* prefactor(m) * m**2 ! (m,m)
coefpp(m1) = coefpp(m1) + tempc
tempc = cmplx(cilm(2,m1,m1), cilm(1,m1,m1), dp) * pm2 *(-m-1) &
* prefactor(m) * m ! (m,m)
coefrp(m1) = coefrp(m1) + tempc
dpl2 = -(m*m1 -(m**2)) * pm2
tempc = cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * dpl2 &
* prefactor(m) ! (m,m)
coeftt(m1) = coeftt(m1) + tempc
pm1 = 0.0_dp
dpl = sqr(2*m+3) * pmm
tempc = cmplx(cilm(1,m1+1,m1), - cilm(2,m1+1,m1), dp) * dpl &
* prefactor(m+1) ! (m+1,m)
coeft(m1) = coeft(m1) + tempc
tempc = cmplx(cilm(1,m1+1,m1), - cilm(2,m1+1,m1), dp) * dpl * (-m-2) &
* prefactor(m+1) ! (m+1,m)
coefrt(m1) = coefrt(m1) + tempc
tempc = cmplx(cilm(2,m1+1,m1), cilm(1,m1+1,m1), dp) * dpl &
* prefactor(m+1) * m ! (m+1,m)
coeftp(m1) = coeftp(m1) + tempc
do l = m + 2, lmax_comp, 1
l1 = l + 1
p = - ff2(l1,m1) * pm2
tempc = cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * p * (-l1) &
* prefactor(l)
coefr(m1) = coefr(m1) + tempc
tempc = cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * p * (l1) &
* (l1+1) * prefactor(l)
coefrr(m1) = coefrr(m1) + tempc
tempc = cmplx(cilm(2,l1,m1), cilm(1,l1,m1), dp) * p &
* prefactor(l) * m
coefp(m1) = coefp(m1) + tempc
tempc = - cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * p &
* prefactor(l) * m**2
coefpp(m1) = coefpp(m1) + tempc
tempc = cmplx(cilm(2,l1,m1), cilm(1,l1,m1), dp) * p * (-l1) &
* prefactor(l) * m
coefrp(m1) = coefrp(m1) + tempc
dpl = sqr(2*l+1) * sqr(l-m) * sqr(l+m) / sqr(2*l-1) * pm1
tempc = cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * dpl &
* prefactor(l)
coeft(m1) = coeft(m1) + tempc
tempc = cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * dpl * (-l1) &
* prefactor(l)
coefrt(m1) = coefrt(m1) + tempc
tempc = cmplx(cilm(2,l1,m1), cilm(1,l1,m1), dp) * dpl * &
prefactor(l) * m
coeftp(m1) = coeftp(m1) + tempc
dpl2 = - (l * l1 -(m**2) / u**2) * p
tempc = cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * dpl2 &
* prefactor(l)
coeftt(m1) = coeftt(m1) + tempc
pm2 = pm1
pm1 = p
end do
coefr(m1) = coefr(m1) * rescalem
coefrr(m1) = coefrr(m1) * rescalem
coeft(m1) = coeft(m1) * rescalem
coeftt(m1) = coeftt(m1) * rescalem
coefrt(m1) = coefrt(m1) * rescalem
coefp(m1) = coefp(m1) * rescalem
coefpp(m1) = coefpp(m1) * rescalem
coeftp(m1) = coeftp(m1) * rescalem
coefrp(m1) = coefrp(m1) * rescalem
end do
if (lmax_comp /= 0) then
pmm = pmm * sqr(2*lmax_comp+1) / sqr(2*lmax_comp) * rescalem
tempc = cmplx(cilm(1,lmax_comp+1,lmax_comp+1), &
- cilm(2,lmax_comp+1,lmax_comp+1), dp) * pmm &
* (-lmax_comp-1) * prefactor(lmax_comp)
coefr(lmax_comp+1) = coefr(lmax_comp+1) + tempc
tempc = cmplx(cilm(1,lmax_comp+1,lmax_comp+1), &
- cilm(2,lmax_comp+1,lmax_comp+1), dp) * pmm &
* (lmax_comp+1) * (lmax_comp+2) * prefactor(lmax_comp)
coefrr(lmax_comp+1) = coefrr(lmax_comp+1) + tempc
tempc = cmplx(cilm(2,lmax_comp+1,lmax_comp+1), &
cilm(1,lmax_comp+1,lmax_comp+1), dp) * pmm &
* prefactor(lmax_comp) * lmax_comp
coefp(lmax_comp+1) = coefp(lmax_comp+1) + tempc
tempc = - cmplx(cilm(1,lmax_comp+1,lmax_comp+1), &
- cilm(2,lmax_comp+1,lmax_comp+1), dp) * pmm &
* prefactor(lmax_comp) * lmax_comp**2
coefpp(lmax_comp+1) = coefpp(lmax_comp+1) + tempc
tempc = cmplx(cilm(2,lmax_comp+1,lmax_comp+1), &
cilm(1,lmax_comp+1,lmax_comp+1), dp) * pmm &
* (-lmax_comp-1) * prefactor(lmax_comp) * lmax_comp
coefrp(lmax_comp+1) = coefrp(lmax_comp+1) + tempc
dpl2 = -(lmax_comp*(lmax_comp+1)-(lmax_comp**2)) * pmm
tempc = cmplx(cilm(1,lmax_comp+1,lmax_comp+1), &
- cilm(2,lmax_comp+1,lmax_comp+1), dp) * dpl2 &
* prefactor(lmax_comp)
coeftt(lmax_comp+1) = coeftt(lmax_comp+1) + tempc
end if
coefr0 = coefr0 * gm / r_ex**2
coefr(2:lmax+1) = coefr(2:lmax+1) * gm / r_ex**2
coefrt0 = - coefrt0 * gm / r_ex**2
coefrt(2:lmax+1) = - coefrt(2:lmax+1) * gm / r_ex**2
coefrr0 = coefrr0 * gm / r_ex**3
coefrr(2:lmax+1) = coefrr(2:lmax+1) * gm / r_ex**3
coeft0 = - coeft0 * gm / r_ex
coeft(2:lmax+1) = - coeft(2:lmax+1) * gm / r_ex
coeftt0 = coeftt0 * gm / r_ex
coeftt(2:lmax+1) = coeftt(2:lmax+1) * gm / r_ex
coeftp0 = - coeftp0 * gm / r_ex
coeftp(2:lmax+1) = - coeftp(2:lmax+1) * gm / r_ex
coefp0 = coefp0 * gm / r_ex
coefp(2:lmax+1) = coefp(2:lmax+1) * gm / r_ex
coefpp0 = coefpp0 * gm / r_ex
coefpp(2:lmax+1) = coefpp(2:lmax+1) * gm / r_ex
coefrp0 = coefrp0 * gm / r_ex**2
coefrp(2:lmax+1) = coefrp(2:lmax+1) * gm / r_ex**2
! Vzz = Vrr
coef(1) = cmplx(coefrr0, 0.0_dp, dp)
coef(2:lmax+1) = coefrr(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
vzz(i_eq,1:nlong) = grid(1:nlong)
! Vxx = 1/r Vr + 1/r^2 Vtt
coef(1) = cmplx(coefr0/r_ex + coeftt0/r_ex**2, 0.0_dp, dp)
coef(2:lmax+1) = (coefr(2:lmax+1)/r_ex + coeftt(2:lmax+1)/r_ex**2 ) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
vxx(i_eq,1:nlong) = grid(1:nlong)
! Vyy = 1/r Vr + 1/r^2 /tan(t) Vt + 1/r^2 /sin(t)^2 Vpp
coef(1) = cmplx(coefr0/r_ex + coefpp0/(r_ex**2), 0.0_dp, dp)
coef(2:lmax+1) = (coefr(2:lmax+1)/r_ex &
+ coefpp(2:lmax+1)/(r_ex**2) ) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
vyy(i_eq,1:nlong) = grid(1:nlong)
! Vxy = 1/r^2 /sin(t) Vtp - cos(t)/sin(t)^2 /r^2 Vp
coef(1) = cmplx(coeftp0/r_ex**2, 0.0_dp, dp)
coef(2:lmax+1) = (coeftp(2:lmax+1)/r_ex**2 ) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
vxy(i_eq,1:nlong) = grid(1:nlong)
! Vxz = 1/r^2 Vt - 1/r Vrt
coef(1) = cmplx(coeft0/r_ex**2 - coefrt0/r_ex, 0.0_dp, dp)
coef(2:lmax+1) = (coeft(2:lmax+1) / r_ex**2 &
- coefrt(2:lmax+1)/r_ex ) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
vxz(i_eq,1:nlong) = grid(1:nlong)
! Vyz = 1/r^2 /sin(t) Vp - 1/r /sin(t) Vrp
coef(1) = cmplx(coefp0/r_ex**2 - coefrp0/r_ex, 0.0_dp, dp)
coef(2:lmax+1) = (coefp(2:lmax+1) / r_ex**2 &
- coefrp(2:lmax+1)/r_ex ) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
vyz(i_eq,1:nlong) = grid(1:nlong)
do i = 1, i_eq - 1, 1
i_s = 2 * i_eq - i
theta = pi * dble(i-1) / dble(n)
z = cos(theta)
u = sqrt( (1.0_dp-z) * (1.0_dp+z) )
sint = sin(theta)
lat = pi / 2.0_dp - theta
if (i == 1) then ! Reference ellipsoid radius
r_ex = a * (1.0_dp - f)
else
r_ex = cos(lat)**2 + sin(lat)**2 / (1.0_dp - f)**2
r_ex = a * sqrt(1.0_dp / r_ex)
end if
coefr(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefr0 = 0.0_dp
coefrs(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefrs0 = 0.0_dp
coefrr(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefrr0 = 0.0_dp
coefrrs(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefrrs0 = 0.0_dp
coeft(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coeft0 = 0.0_dp
coefts(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefts0 = 0.0_dp
coeftp(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coeftp0 = 0.0_dp
coeftps(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coeftps0 = 0.0_dp
coefp(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefp0 = 0.0_dp
coefps(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefps0 = 0.0_dp
coefrt(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefrt0 = 0.0_dp
coefrts(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefrts0 = 0.0_dp
coefrp(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefrp0 = 0.0_dp
coefrps(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefrps0 = 0.0_dp
coefpp(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefpp0 = 0.0_dp
coefpps(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefpps0 = 0.0_dp
coeftt(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coeftt0 = 0.0_dp
coeftts(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coeftts0 = 0.0_dp
pm2 = 1.0_dp
tempr = -cilm(1,1,1) * pm2 ! l = 0
coefr0 = coefr0 + tempr
coefrs0 = coefrs0 + tempr ! fsymsign is always 1 for l=m=0
tempr = 2 * cilm(1,1,1) * pm2 ! l = 0
coefrr0 = coefrr0 + tempr
coefrrs0 = coefrrs0 + tempr
! derivative in theta and phi of l=0 term is 0, so no need to
! calculate this
if (lmax_comp /= 0) then ! l = 1
prefactor(1) = r0 / r_ex
do l = 2, lmax_comp, 1
prefactor(l) = prefactor(l-1) * r0 / r_ex
end do
pm1 = ff1(2,1) * z
! -2 = (l+1) prefactor
tempr = cilm(1,2,1) * pm1 * (-2) * prefactor(1)
coefr0 = coefr0 + tempr
coefrs0 = coefrs0 - tempr ! fsymsign = -1
! 6 = (l+1)*(l+2) prefactor
tempr = cilm(1,2,1) * pm1 * (6) * prefactor(1)
coefrr0 = coefrr0 + tempr
coefrrs0 = coefrrs0 - tempr ! fsymsign = -1
! dpl is the first derivative with respect to Z
dpl = ff1(2,1)
tempr = cilm(1,2,1) * dpl * prefactor(1)
coeft0 = coeft0 + tempr
coefts0 = coefts0 + tempr ! reverse fsymsign
! -2 = (l+1) prefactor
tempr = cilm(1,2,1) * dpl * (-2) * prefactor(1)
coefrt0 = coefrt0 + tempr
coefrts0 = coefrts0 + tempr ! reverse fsymsign
! dpl2 is the second derivative with respect to THETA. Must
! multiply dpl by -sin(theta) in the recurrence relation
! note that the first term is symmetric about the equator according
! the fsymsign, whereas the second term differs
! by -1.
dpl2 = -2 * pm1 + z * dpl
dpl2s = 2 * pm1 + z * dpl
tempr = cilm(1,2,1) * prefactor(1)
coeftt0 = coeftt0 + tempr * dpl2
coeftts0 = coeftts0 + tempr * dpl2s
end if
do l = 2, lmax_comp, 1
l1 = l + 1
p = ff1(l1,1) * z * pm1 - ff2(l1,1) * pm2
tempr = cilm(1,l1,1) * p * (-l1) * prefactor(l)
coefr0 = coefr0 + tempr
coefrs0 = coefrs0 + tempr * fsymsign(l1,1)
tempr = cilm(1,l1,1) * p * (l1) * (l1+1) * prefactor(l)
coefrr0 = coefrr0 + tempr
coefrrs0 = coefrrs0 + tempr * fsymsign(l1,1)
dpl = l * ( sqr(2*l+1) / sqr(2*l-1) * pm1 - z * p ) / u**2
tempr = cilm(1,l1,1) * dpl * prefactor(l)
coeft0 = coeft0 + tempr
coefts0 = coefts0 - tempr * fsymsign(l1,1) ! reverse fsymsign
tempr = cilm(1,l1,1) * dpl * (-l1) * prefactor(l)
coefrt0 = coefrt0 + tempr
coefrts0 = coefrts0 - tempr * fsymsign(l1,1)
dpl2 = -l * l1 * p + z * dpl
dpl2s = -l * l1 * p * fsymsign(l1,1) - z * dpl * fsymsign(l1,1)
tempr = cilm(1,l1,1) * prefactor(l)
coeftt0 = coeftt0 + tempr * dpl2
coeftts0 = coeftts0 + tempr * dpl2s
pm2 = pm1
pm1 = p
end do
pmm = sqr(2) * scalef
rescalem = 1.0_dp / scalef
do m = 1, lmax_comp-1, 1
m1 = m + 1
rescalem = rescalem * u
pmm = pmm * sqr(2*m+1) / sqr(2*m)
pm2 = pmm
tempc = cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * pm2 * (-m-1) &
* prefactor(m) ! (m,m)
coefr(m1) = coefr(m1) + tempc
coefrs(m1) = coefrs(m1) + tempc ! fsymsign = 1
tempc = cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * pm2 * (m+1) &
* (m+2) * prefactor(m) ! (m,m)
coefrr(m1) = coefrr(m1) + tempc
coefrrs(m1) = coefrrs(m1) + tempc ! fsymsign = 1
tempc = cmplx(cilm(2,m1,m1), cilm(1,m1,m1), dp) * pm2 &
* prefactor(m) * m ! (m,m)
coefp(m1) = coefp(m1) + tempc
coefps(m1) = coefps(m1) + tempc ! fsymsign = 1
tempc = - cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * pm2 &
* prefactor(m) * m**2 ! (m,m)
coefpp(m1) = coefpp(m1) + tempc
coefpps(m1) = coefpps(m1) + tempc ! fsymsign = 1
tempc = cmplx(cilm(2,m1,m1), cilm(1,m1,m1), dp) * pm2 *(-m-1) &
* prefactor(m) * m ! (m,m)
coefrp(m1) = coefrp(m1) + tempc
coefrps(m1) = coefrps(m1) + tempc ! fsymsign = 1
dpl = -m * z * pm2 / u**2
tempc = cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * dpl &
* prefactor(m) ! (m,m)
coeft(m1) = coeft(m1) + tempc
coefts(m1) = coefts(m1) - tempc ! reverse fsymsign
tempc = cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * dpl * (-m-1) &
* prefactor(m) ! (m,m)
coefrt(m1) = coefrt(m1) + tempc
coefrts(m1) = coefrts(m1) - tempc ! reverse fsymsign
tempc = cmplx(cilm(2,m1,m1), cilm(1,m1,m1), dp) * dpl &
* prefactor(m) * m ! (m,m)
coeftp(m1) = coeftp(m1) + tempc
coeftps(m1) = coeftps(m1) - tempc ! reverse fsymsign
dpl2 = -(m*m1 - (m**2)/u**2) * pm2 + z * dpl
dpl2s = -(m*m1 - (m**2)/u**2) * pm2 - z * dpl
tempc = cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) &
* prefactor(m) ! (m,m)
coeftt(m1) = coeftt(m1) + tempc * dpl2
coeftts(m1) = coeftts(m1) + tempc * dpl2s
pm1 = z * ff1(m1+1,m1) * pm2
tempc = cmplx(cilm(1,m1+1,m1), - cilm(2,m1+1,m1), dp) * pm1 &
* (-m-2) * prefactor(m+1) ! (m+1,m)
coefr(m1) = coefr(m1) + tempc
coefrs(m1) = coefrs(m1) - tempc ! fsymsign = -1
tempc = cmplx(cilm(1,m1+1,m1), - cilm(2,m1+1,m1), dp) * pm1 &
* (m+2) * (m+3) * prefactor(m+1) ! (m+1,m)
coefrr(m1) = coefrr(m1) + tempc
coefrrs(m1) = coefrrs(m1) - tempc ! fsymsign = -1
tempc = cmplx(cilm(2,m1+1,m1), cilm(1,m1+1,m1), dp) * pm1 &
* prefactor(m+1) * m ! (m+1,m)
coefp(m1) = coefp(m1) + tempc
coefps(m1) = coefps(m1) - tempc ! fsymsign = -1
tempc = - cmplx(cilm(1,m1+1,m1), - cilm(2,m1+1,m1), dp) * pm1 &
* prefactor(m+1) * m**2 ! (m+1,m)
coefpp(m1) = coefpp(m1) + tempc
coefpps(m1) = coefpps(m1) - tempc ! fsymsign = -1
tempc = cmplx(cilm(2,m1+1,m1), cilm(1,m1+1,m1), dp) * pm1 &
* (-m-2) * prefactor(m+1) * m ! (m+1,m)
coefrp(m1) = coefrp(m1) + tempc
coefrps(m1) = coefrps(m1) - tempc ! fsymsign = -1
dpl = (sqr(2*m+3) * pmm - z * (m+1) * pm1) / u**2
tempc = cmplx(cilm(1,m1+1,m1), - cilm(2,m1+1,m1), dp) * dpl &
* prefactor(m+1) ! (m+1,m)
coeft(m1) = coeft(m1) + tempc
coefts(m1) = coefts(m1) + tempc ! reverse fsymsign
tempc = cmplx(cilm(1,m1+1,m1), - cilm(2,m1+1,m1), dp) * dpl &
* (-m-2) * prefactor(m+1) ! (m+1,m)
coefrt(m1) = coefrt(m1) + tempc
coefrts(m1) = coefrts(m1) + tempc ! reverse fsymsign
tempc = cmplx(cilm(2,m1+1,m1), cilm(1,m1+1,m1), dp) * dpl &
* prefactor(m+1) * m ! (m+1,m)
coeftp(m1) = coeftp(m1) + tempc
coeftps(m1) = coeftps(m1) + tempc ! reverse fsymsign
dpl2 = -(m1*(m1+1) - (m**2)/u**2) * pm1 + z * dpl
dpl2s = (m1*(m1+1) - (m**2)/u**2) * pm1 + z * dpl
tempc = cmplx(cilm(1,m1+1,m1), - cilm(2,m1+1,m1), dp) &
* prefactor(m+1) ! (m+1,m)
coeftt(m1) = coeftt(m1) + tempc * dpl2
coeftts(m1) = coeftts(m1) + tempc * dpl2s
do l=m+2, lmax_comp, 1
l1 = l + 1
p = z * ff1(l1,m1) * pm1 - ff2(l1,m1) * pm2
tempc = cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * p * (-l1) &
* prefactor(l)
coefr(m1) = coefr(m1) + tempc
coefrs(m1) = coefrs(m1) + tempc * fsymsign(l1,m1)
tempc = cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * p * (l1) &
* (l1+1) * prefactor(l)
coefrr(m1) = coefrr(m1) + tempc
coefrrs(m1) = coefrrs(m1) + tempc * fsymsign(l1,m1)
tempc = cmplx(cilm(2,l1,m1), cilm(1,l1,m1), dp) * p &
* prefactor(l) * m
coefp(m1) = coefp(m1) + tempc
coefps(m1) = coefps(m1) + tempc * fsymsign(l1,m1)
tempc = - cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * p &
* prefactor(l) * m**2
coefpp(m1) = coefpp(m1) + tempc
coefpps(m1) = coefpps(m1) + tempc * fsymsign(l1,m1)
tempc = cmplx(cilm(2,l1,m1), cilm(1,l1,m1), dp) * p * (-l1) &
* prefactor(l) * m
coefrp(m1) = coefrp(m1) + tempc
coefrps(m1) = coefrps(m1) + tempc * fsymsign(l1,m1)