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subroutine interpol(yy,nxmax,nymax,nzmax,nx,ny,nz,memind,height,
+xt,yt,zt,itime1,itime2,itime,indexf,yint)
C i i i i i i i i i
C i i i i i i i o
*****************************************************************************
* *
* Interpolation of 3-dimensional meteorological fields. *
* In horizontal direction bicubic interpolation interpolation is used. *
* In the vertical a polynomial interpolation is used. *
* In the temporal direction linear interpolation is used. *
* These interpolation techniques have been found to be most accurate. *
* *
* The interpolation routines have been taken from: *
* Press W.H. et al. (1992): Numerical Recipes in FORTRAN. The art of *
* scientific computing. 2nd edition. Cambridge University Press. *
* *
* But they have been modified for faster performance. *
* *
* 4-3 *
* | | The points are numbered in this order. Values and gradients are *
* 1-2 stored in fields with dimension 4. *
* *
* 1,2 is the level closest to the current position for the first time *
* 1,1 is the level below 1,2 *
* 1,3 is the level above 1,2 *
* 2,2 is the level closest to the current position for the second time *
* 2,1 is the level below 2,2 *
* 2,3 is the level above 2,2 *
* *
* *
* Author: A. Stohl *
* *
* 30 May 1994 *
* *
*****************************************************************************
* *
* Variables: *
* *
* dt1,dt2 time differences between fields and current position *
* dz1,dz2 z distance between levels and current position *
* height(nzmax) heights of the model levels *
* indexf indicates the number of the wind field to be read in *
* indexfh help variable *
* indz the level closest to the current trajectory position *
* indzh help variable *
* itime current time *
* itime1 time of the first wind field *
* itime2 time of the second wind field *
* ix,jy x,y coordinates of lower left subgrid point *
* memind(3) points to the places of the wind fields *
* nx,ny,nz actual field dimensions in x,y and z direction *
* nxmax,nymax,nzmax maximum field dimensions in x,y and z direction *
* x1l,x2l x,y coordinates of lower left subgrid point *
* x1u,x2u x,y coordinates of upper right subgrid point *
* xt current x coordinate *
* y(4,2,3) subset of 4 points for 2 times and 3 levels *
* ygx(4,2,3) x gradients at 4 points for 2 times and 3 levels *
* ygy(4,2,3) y gradients at 4 points for 2 times and 3 levels *
* ygxy(4,2,3) x,y gradients at 4 points for 2 times and 3 levels *
* yhelp(2,3) the interpolated values for 2 times and 3 levels *
* yh2(2) the interpolated values for the 2 times *
* yint the final interpolated value *
* yt current y coordinate *
* yy(0:nxmax,0:nymax,nzmax,3) meteorological field used for interpolation *
* zt current z coordinate *
* *
*****************************************************************************
implicit none
integer nx,ny,nz,nxmax,nymax,nzmax,memind(3),i,j,l,m,n,ix,jy
integer itime,itime1,itime2,indexf,indz,indexfh,indzh,im,ip,jm,jp
real yy(0:nxmax-1,0:nymax-1,nzmax,3),height(nzmax)
real y(4,2,3),ygx(4,2,3),ygy(4,2,3),ygxy(4,2,3),yhelp(2,3),yh2(2)
real x1l,x1u,x2l,x2u,dz1,dz2,dt1,dt2,zz(3)
real xt,yt,zt,yint
C 3 levels are needed for the polynomial interpolation in the vertical
C Determine the closest vertical level -1
**********************************************************************
do 5 i=1,nz-1
if ((height(i).le.zt).and.(height(i+1).ge.zt)) then
dz1=zt-height(i)
dz2=height(i+1)-zt
if (dz1.lt.dz2) then
indz=i-1
else
indz=i
endif
goto 6
endif
5 continue
6 continue
if (indz.lt.1) indz=1
if (indz.gt.(nz-2)) indz=nz-2
C If point at border of grid -> small displacement into grid
************************************************************
if (xt.ge.float(nx-1)) xt=float(nx-1)-0.00001
if (yt.ge.float(ny-1)) yt=float(ny-1)-0.00001
***********************************************************************
C 1.) Bicubic horizontal interpolation
C This has to be done separately for 6 fields (Temporal(2)*Vertical(3))
***********************************************************************
C Determine the lower left corner
*********************************
ix=int(xt)
jy=int(yt)
x1l=float(ix)
x1u=float(ix+1)
x2l=float(jy)
x2u=float(jy+1)
C Loop over the 2*2 grid points
*******************************
do 10 l=1,4
if (l.eq.1) then
i=ix
j=jy
else if (l.eq.2) then
i=ix+1
j=jy
else if (l.eq.3) then
i=ix+1
j=jy+1
else if (l.eq.4) then
i=ix
j=jy+1
endif
ip=i+1
im=i-1
jp=j+1
jm=j-1
C Loop over 2 time steps and 3 levels
*************************************
do 10 m=1,2
indexfh=memind(indexf+m-1)
do 10 n=1,3
indzh=indz+n-1
C Values at the 2*2 subgrid
***************************
y(l,m,n)=yy(i,j,indzh,indexfh)
C Calculate derivatives in x-direction on the 2*2 subgrid
*********************************************************
if (i.eq.0) then
ygx(l,m,n) = yy(ip,j ,indzh,indexfh)
+ - yy(i ,j ,indzh,indexfh)
else if (i.eq.nx-1) then
ygx(l,m,n) = yy(i ,j ,indzh,indexfh)
+ - yy(im,j ,indzh,indexfh)
else
ygx(l,m,n) =(yy(ip,j ,indzh,indexfh)
+ - yy(im,j ,indzh,indexfh))/2.
endif
C Calculate derivatives in y-direction on the 2*2 subgrid
*********************************************************
if (j.eq.0) then
ygy(l,m,n) = yy(i ,jp,indzh,indexfh)
+ - yy(i ,j ,indzh,indexfh)
else if (j.eq.ny-1) then
ygy(l,m,n) = yy(i ,j ,indzh,indexfh)
+ - yy(i ,jm,indzh,indexfh)
else
ygy(l,m,n) =(yy(i ,jp,indzh,indexfh)
+ - yy(i ,jm,indzh,indexfh))/2.
endif
C Calculate cross derivative on the 2*2 subgrid
***********************************************
if ((i.eq.0).and.(j.eq.0)) then
ygxy(l,m,n)= yy(ip,jp,indzh,indexfh)-
+ yy(ip,j ,indzh,indexfh)-
+ yy(i ,jp,indzh,indexfh)+
+ yy(i ,j ,indzh,indexfh)
else if ((i.eq.nx-1).and.(j.eq.ny-1)) then
ygxy(l,m,n)= yy(i ,j ,indzh,indexfh)-
+ yy(i ,jm,indzh,indexfh)-
+ yy(im,j ,indzh,indexfh)+
+ yy(im,jm,indzh,indexfh)
else if ((i.eq.0).and.(j.eq.ny-1)) then
ygxy(l,m,n)= yy(ip,j ,indzh,indexfh)-
+ yy(ip,jm,indzh,indexfh)-
+ yy(i ,j ,indzh,indexfh)+
+ yy(i ,jm,indzh,indexfh)
else if ((i.eq.nx-1).and.(j.eq.0)) then
ygxy(l,m,n)= yy(i ,jp,indzh,indexfh)-
+ yy(i ,j ,indzh,indexfh)-
+ yy(im,jp,indzh,indexfh)+
+ yy(im,j ,indzh,indexfh)
else if (i.eq.nx-1) then
ygxy(l,m,n)=(yy(i ,jp,indzh,indexfh)-
+ yy(i ,jm,indzh,indexfh)-
+ yy(im,jp,indzh,indexfh)+
+ yy(im,jm,indzh,indexfh))/2.
else if (i.eq.0) then
ygxy(l,m,n)=(yy(ip,jp,indzh,indexfh)-
+ yy(ip,jm,indzh,indexfh)-
+ yy(i ,jp,indzh,indexfh)+
+ yy(i ,jm,indzh,indexfh))/2.
else if (j.eq.ny-1) then
ygxy(l,m,n)=(yy(ip,j ,indzh,indexfh)-
+ yy(ip,jm,indzh,indexfh)-
+ yy(im,j ,indzh,indexfh)+
+ yy(im,jm,indzh,indexfh))/2.
else if (j.eq.0) then
ygxy(l,m,n)=(yy(ip,jp,indzh,indexfh)-
+ yy(ip,j ,indzh,indexfh)-
+ yy(im,jp,indzh,indexfh)+
+ yy(im,j ,indzh,indexfh))/2.
else
ygxy(l,m,n)=(yy(ip,jp,indzh,indexfh)-
+ yy(ip,jm,indzh,indexfh)-
+ yy(im,jp,indzh,indexfh)+
+ yy(im,jm,indzh,indexfh))/4.
endif
10 continue
C Call bicubic interpolation
****************************
call bicubic(y,ygx,ygy,ygxy,x1l,x1u,x2l,x2u,xt,yt,yhelp,2,3)
**************************************************
C 2. Vertical interpolation by a 2nd order polynom
**************************************************
do 20 n=1,3
20 zz(n)=height(indz+n-1)
call polynom(zz,yhelp,3,zt,yh2,2)
*************************************
C 3.) Temporal interpolation (linear)
*************************************
dt1=float(itime-itime1)
dt2=float(itime2-itime)
yint=(yh2(1)*dt2+yh2(2)*dt1)/(dt1+dt2)
return
end
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