# Using Map Projections in GrADS

It is important to understand the distinction between the two uses of map projections when creating GrADS displays of your data:

- projection of the data (preprojected grids);
- projection of the display.

GrADS supports two types of data grids:

- `lon/lat` grids (and not necessarily regular, e.g., gaussian);
- preprojected grids.

Using Preprojected Grids
GrADS Display Projections
Summary and Plans

## Using Preprojected Grids

Polar Stereo Preprojected Data
Lambert Conformal Preprojected Data
NMC Eta model
NMC high accuracy polar stereo for SSM/I data
CSU RAMS Oblique Polar Stereo Grids
Pitfalls when using preprojected data

Preprojected data are data already on a map projection. GrADS supports four types of preprojected data:

1. N polar stereo (NMC model projection);
2. S polar stereo (NMC model projection) ;
3. Lambert Conformal (originally for Navy NORAPS model);
4. NMC eta model (unstaggered).
5. More precise N and S polar stereo (hi res SSM/I data)
6. Colorado State University RAMS model (oblique polar stereo; beta)

When preprojected grids are opened in GrADS, bilinear interpolation constants are calculated and all date are displayed on an internal GrADS lat/lon grid defined by the `xdef` and `ydef` card in the data description or `.ctl` file (that's why it takes longer to "open" a preprojected grid data set).

It is very important to point out that the internal GrADS grid can be any grid as it is completely independent of the preprojected data grid. Thus, there is nothing stopping you displaying preprojected data on a very high res lon/lat grid (again, defined in the `.ctl` by `xdef` and `ydef`). In fact, you could create and open multiple .ctl files with different resolutions and/or regions which pointed to the same preprojected data file.

When you do a `display` (i.e., get a grid of data), the preprojected data are bilinearly interpolated to the GrADS internal lat/lon grid. For preprojected scalar fields (e.g., 500 mb heights), the display is adequate and the precision of the interpolation can be controlled by `xdef` and `ydef` to define a higher spatial resolution grid.

The big virtue of this approach is that all built in GrADS analytic functions (e.g., `aave`, `hcurl`...) continue to work even though the data were not originally on a lon/lat grid. The downside is that you are not looking directly at your data on a geographic map. However, one could always define a .ctl file which simply opened the data file as i,j data and displayed without the map (`set mpdraw` off). So, in my opinion, this compromise is not that limiting even if as a modeller you wanted to look at the grid--you just don't get the map background.

`Preprojected vector fields` are a little trickier, depending on whether the vector is defined relative to the data grid or relative to the Earth. For example, NMC polar stereo grids use winds relative to the data grid and thus must be rotated to the internal GrADS lat/lon grid (again defined in the `.ctl` file by the `xdef` and `ydef` cards).

The only potential problem with working with preprojected data (e.g., Lambert Conformal model data) is defining the projection for GrADS. This is accomplished using a `pdef` card in the data descriptor `.ctl` file.

Polar Stereo Preprojected Data (coarse accuracy for NMC Models)

Preprojected data on a polar stereo projection (N and S) is defined as at NMC. For the NCEP model GRIB data distributed via anon ftp from ftp.ncep.noaa.gov, the `pdef` card is:

`pdef isize jsize projtype ipole jpole lonref gridinc
pdef 53 45 nps 27 49 -105 190.5`

where,

`ipole` and `jpole` are the (i,j) of the pole referenced from the lower left corner at (1,1) and `gridinc` is the dx in km.

The relevant GrADS source is:

`void w3fb04 (float alat, float along, float xmeshl, float orient, float *xi, float *xj) {
/*
C
C SUBPROGRAM: W3FB04 LATITUDE, LONGITUDE TO GRID COORDINATES
C AUTHOR: MCDONELL,J. ORG: W345 DATE: 90-06-04
C
C ABSTRACT: CONVERTS THE COORDINATES OF A LOCATION ON EARTH FROM THE
C NATURAL COORDINATE SYSTEM OF LATITUDE/LONGITUDE TO THE GRID (I,J)
C COORDINATE SYSTEM OVERLAID ON A POLAR STEREOGRAPHIC MAP PRO
C JECTION TRUE AT 60 DEGREES N OR S LATITUDE. W3FB04 IS THE REVERSE
C OF W3FB05.
C
C PROGRAM HISTORY LOG:
C 77-05-01 J. MCDONELL
C 89-01-10 R.E.JONES CONVERT TO MICROSOFT FORTRAN 4.1
C 90-06-04 R.E.JONES CONVERT TO SUN FORTRAN 1.3
C 93-01-26 B. Doty converted to
C
C
C USAGE: CALL W3FB04 (ALAT, ALONG, XMESHL, ORIENT, XI, XJ)
C
C INPUT VARIABLES:
C NAMES INTERFACE DESCRIPTION OF VARIABLES AND TYPES
C ------ --------- -----------------------------------------------
C ALAT ARG LIST LATITUDE IN DEGREES (<0 IF SH)
C ALONG ARG LIST WEST LONGITUDE IN DEGREES
C XMESHL ARG LIST MESH LENGTH OF GRID IN KM AT 60 DEG LAT(<0 IF SH)
C (190.5 LFM GRID, 381.0 NH PE GRID,-381.0 SH PE GRID)
C ORIENT ARG LIST ORIENTATION WEST LONGITUDE OF THE GRID
C (105.0 LFM GRID, 80.0 NH PE GRID, 260.0 SH PE GRID)
C
C OUTPUT VARIABLES:
C NAMES INTERFACE DESCRIPTION OF VARIABLES AND TYPES
C ------ --------- -----------------------------------------------
C XI ARG LIST I OF THE POINT RELATIVE TO NORTH OR SOUTH POLE
C XJ ARG LIST J OF THE POINT RELATIVE TO NORTH OR SOUTH POLE
C
C SUBPROGRAMS CALLED:
C NAMES LIBRARY
C ------------------------------------------------------- --------
C COS SIN SYSLIB
C
C REMARKS: ALL PARAMETERS IN THE CALLING STATEMENT MUST BE
C REAL. THE RANGE OF ALLOWABLE LATITUDES IS FROM A POLE TO
C 30 DEGREES INTO THE OPPOSITE HEMISPHERE.
C THE GRID USED IN THIS SUBROUTINE HAS ITS ORIGIN (I=0,J=0)
C AT THE POLE IN EITHER HEMISPHERE, SO IF THE USER'S GRID HAS ITS
C ORIGIN AT A POINT OTHER THAN THE POLE, A TRANSLATION IS NEEDED
C TO GET I AND J. THE GRIDLINES OF I=CONSTANT ARE PARALLEL TO A
C LONGITUDE DESIGNATED BY THE USER. THE EARTH'S RADIUS IS TAKEN C TO BE 6371.2 KM.
C
C ATTRIBUTES:
C LANGUAGE: SUN FORTRAN 1.4 C MACHINE: SUN SPARCSTATION 1+
C*/
static float radpd = 0.01745329;
static float earthr = 6371.2;`

float re,xlat,wlong,r;
re = (earthr * 1.86603) / xmeshl;
xlat = alat * radpd;
if (xmeshl>0.0) {
wlong = (along + 180.0 - orient) * radpd;
r = (re * cos(xlat)) / (1.0 + sin(xlat));
*xi = r * sin(wlong);
*xj = r * cos(wlong);

} else {
re = -re;
xlat = -xlat;
wlong = (along - orient) * radpd;
r = (re * cos(xlat)) / (1.0+ sin(xlat));
*xi = r * sin(wlong);
*xj = -r * cos(wlong);
}
}

Lambert Conformal Preprojected Data

The Lambert Conformal projection (lcc) was implemented for the U.S. Navy's limited area model NORAPS. Thus, to work with your lcc data you must express your grid in the context of the Navy lcc grid. NMC has been able to do this for their AIWIPS grids and the Navy definition should be general enough for others.

A typical NORAPS Lambert-Conformal grid is described below, including the C code which sets up the internal interpolation.

The `.ctl` file is:

`dset ^temp.grd
title NORAPS DATA TEST
undef 1e20
pdef 103 69 lcc 30 -88 51.5 34.5 20 40 -88 90000 90000
xdef 180 linear -180 1.0
ydef 100 linear -10 1.0
zdef 16 levels 1000 925 850 700 500 400 300 250 200 150 100 70 50 30 20 10
tdef 1 linear 00z1jan94 12hr
vars 1
t 16 0 temp
endvars`

where,

`103` = #pts in x
`69` = #pts in y
`lcc` = Lambert-Conformal
`30` = lat of ref point
`88` = lon of ref point (E is positive, W is negative)
`51.5` = i of ref point
`34.5` = j of ref point
`20` = S true lat
`40` = N true lat
`88` = standard lon
`90000` = dx in M
`90000` = dy in M

Otherwise, it is the same as other GrADS files.

Note - the `xdef/ydef` apply to the `lon/lat` grid GrADS internally interpolates to and can be anything...

The GrADS source which maps `lon/lat` of the GrADS internal `lon/lat` grid to `i,j` of the preprojected grid is:

``

    /* Lambert Conformal conversion */
    void ll2lc (float *vals, float grdlat, float grdlon,
    float *grdi, float *grdj) {
    /*  Subroutine to convert from lat-lon to Lambert Conformal i,j.
    Provided by NRL Monterey; converted to C 6/15/94.
    c                SUBROUTINE: ll2lc
    c
    c                PURPOSE: To compute i- and j-coordinates of a specified
    c                         grid given the latitude and longitude points.
    c                         All latitudes in this routine start
    c                         with -90.0 at the south pole and increase
    c                         northward to +90.0 at the north pole.  The
    c                         longitudes start with 0.0 at the Greenwich
    c                         meridian and increase to the east, so that
    c                         90.0 refers to 90.0E, 180.0 is the inter-
    c                         national dateline and 270.0 is 90.0W.
    c
    c                INPUT VARIABLES:
    c
    c   vals+0         reflat: latitude at reference point (iref,jref)
    c
    c   vals+1         reflon: longitude at reference point (iref,jref)
    c
    c   vals+2         iref:   i-coordinate value of reference point
    c
    c   vals+3         jref:   j-coordinate value of reference point
    c
    c   vals+4         stdlt1: standard latitude of grid
    c
    c   vals+5         stdlt2: second standard latitude of grid (only required
    c                  if igrid = 2, lambert conformal)
    c
    c   vals+6         stdlon: standard longitude of grid (longitude that
    c                          points to the north)
    c
    c   vals+7         delx:   grid spacing of grid in x-direction
    c                          for igrid = 1,2,3 or 4, delx must be in meters
    c                          for igrid = 5, delx must be in degrees
    c
    c   vals+8         dely:   grid spacing (in meters) of grid in y-direction
    c                          for igrid = 1,2,3 or 4, delx must be in meters
    c                          for igrid = 5, dely must be in degrees
    c
    c                  grdlat: latitude of point (grdi,grdj)
    c
    c                  grdlon: longitude of point (grdi,grdj)
    c
    c                  grdi:   i-co ordinate(s) that this routine will generate
    c                          information for
    c
    c                  grdj:   j-coordinate(s) that this routine will generate
    c                          information for
    c
    */

       float pi, pi2, pi4, d2r, r2d, radius, omega4;
       float gcon,ogcon,ahem,deg,cn1,cn2,cn3,cn4,rih,xih,yih,rrih,check;
       float alnfix,alon,x,y;
       pi = 4.0*atan(1.0);
       pi2 = pi/2.0;
       pi4 = pi/4.0;
       d2r = pi/180.0;
       r2d = 180.0/pi;
       radius = 6371229.0;
       omega4 = 4.0*pi/86400.0;

    /*mf -------------- mf*/
    /*case where standard lats are the same */
      if(*(vals+4) == *(vals+5)) {
        gcon = sin(*(vals+4)*d2r);
      } else {
        gcon = (log(sin((90.0-*(vals+4))*d2r))
        log(sin((90.0-*(vals+5))*d2r)))
        /(log(tan((90.0-*(vals+4))*0.5*d2r))
        log(tan((90.0-*(vals+5))*0.5*d2r)));
      }
    /*mf -------------- mf*/
      ogcon = 1.0/gcon;
      ahem = fabs(*(vals+4))/(*(vals+4));
      deg = (90.0-fabs(*(vals+4)))*d2r;
      cn1 = sin(deg);
      cn2 = radius*cn1*ogcon;
      deg = deg*0.5;
      cn3 = tan(deg);
      deg = (90.0-fabs(*vals))*0.5*d2r;
      cn4 = tan(deg);
      rih = cn2*pow((cn4/cn3),gcon);
      deg = (*(vals+1)-*(vals+6))*d2r*gcon;
      xih = rih*sin(deg);
      yih = -rih*cos(deg)*ahem;
      deg = (90.0-grdlat*ahem)*0.5*d2r;
      cn4 = tan(deg);
      rrih = cn2*pow((cn4/cn3),gcon);
      check = 180.0-*(vals+6);
      alnfix = *(vals+6)+check;
      alon = grdlon+check;
      while (alon<0.0) alon = alon+360.0;
      while (alon>360.0) alon = alon-360.0;
      deg = (alon-alnfix)*gcon*d2r;
      x = rrih*sin(deg);
      y = -rrih*cos(deg)*ahem;
      *grdi = *(vals+2)+(x-xih)/(*(vals+7));
      *grdj = *(vals+3)+(y-yih)/(*(vals+8));
    }

NMC Eta model (unstaggered grids)

The NMC eta model "native" grid is awkward to work with because the variables are on staggered (e.g., the grid for winds is not the same as the grid for mass points) and non rectangular (number of points in i is not constant with j) grids. Because any contouring of irregularly gridded data involves interpolation at some point, NMC creates "unstaggered" eta model fields for practical application programs such as GrADS. In the unstaggered grids all variables are placed on a common and rectangular grid (the mass points).

Wind rotation has also been added so that vector data will be properly displayed.

The pdef card for a typical eta model grid is:

`pdef 181 136 eta.u -97.0 41.0 0.38888888 0.37037037`

`181` = #pts in x
`136` = #pts in y
`eta.u` = eta grid, unstaggered
`-97.0` = lon of ref point (E is positive in GrADS, W is negative) [deg]
`41.0` = lat of ref point [deg]
`0.3888` = dlon [deg]
`0.37037` = dlat [deg]

The source code in GrADS for the lon,lat -> i,j mapping is:

    void ll2eg (int im, int jm, float *vals,  float grdlon, float grdlat,
          float *grdi, float *grdj, float *alpha) {

    /*  Subroutine to convert from lat-lon to NMC eta i,j.

        Provided by Eric Rogers NMC; converted to C 3/29/95 by Mike Fiorino.

    c                SUBROUTINE: ll2eg
    c
    c                PURPOSE: To compute i- and j-coordinates of a specified
    c                         grid given the latitude and longitude points.
    c                         All latitudes in this routine start
    c                         with -90.0 at the south pole and increase
    c                         northward to +90.0 at the north pole.  The
    c                         longitudes start with 0.0 at the Greenwich
    c                         meridian and increase to the east, so that
    c                         90.0 refers to 90.0E, 180.0 is the inter-
    c                         national dateline and 270.0 is 90.0W.
    c
    c                INPUT VARIABLES:
    c
    c   vals+0         tlm0d: longitude of the reference center point
    c
    c   vals+1         tph0d: latitude of the reference center point
    c
    c   vals+2         dlam:  dlon grid increment in deg
    c
    c   vals+3         dphi:  dlat grid increment in deg
    c
    c
    c                  grdlat: latitude of point (grdi,grdj)
    c
    c                  grdlon: longitude of point (grdi,grdj)
    c
    c                  grdi:   i-coordinate(s) that this routine will generate
    c                          information for
    c
    c                  grdj:   j-coordinate(s) that this routine will generate
    c                          information for
    c

    */

       float pi,d2r,r2d, earthr;   float tlm0d,tph0d,dlam,dphi;
       float
        phi,lam,lame,lam0,phi0,lam0e,cosphi,sinphi,sinphi0,cosphi0,sinlam r,cos
        lamr;
       float x1,x,y,z,bigphi,biglam,cc,num,den,tlm,tph;

       int idim,jdim;

       pi=3.141592654;

       d2r=pi/180.0;
       r2d=1.0/d2r;
       earthr=6371.2;

       tlm0d=-*(vals+0); /* convert + W to + E, the grads standard for
        longitude */
       tph0d=*(vals+1);
       dlam=(*(vals+2))*0.5;
       dphi=(*(vals+3))*0.5;

       /* grid point and center of eta grid trig */

       /* convert to radians */

       phi    = grdlat*d2r;
       lam    = -grdlon*d2r; /* convert + W to + E, the grads standard for
        longitude */
       lame   = (grdlon)*d2r;

       phi0   = tph0d*d2r;
       lam0   = tlm0d*d2r;
       lam0e  = ( 360.0 + *(vals+0) )*d2r;

       /* cos and sin */

       cosphi = cos(phi);
       sinphi = sin(phi);

       sinphi0 = sin(phi0);
       cosphi0 = cos(phi0);

       sinlamr=sin(lame-lam0e);
       coslamr=cos(lame-lam0e);

       x1     = cosphi*cos(lam-lam0);
       x      = cosphi0*x1+sinphi0*sinphi;
       y      = -cosphi*sin(lam-lam0);
       z      = -sinphi0*x1+cosphi0*sinphi;

       /* params for wind rotation alpha */

       cc=cosphi*coslamr;
       num=cosphi*sinlamr;
       den=cosphi0*cc+sinphi0*sinphi;

       tlm=atan2(num,den);

       /* parms for lat/lon -> i,j */

       bigphi = atan(z/(sqrt(x*x+y*y)))*r2d;
       biglam = atan(y/x)*r2d;

       idim = im*2-1;
       jdim = jm*2-1 ;

       *grdi  = (biglam/dlam)+(idim+1)*0.5;
       *grdj  = (bigphi/dphi)+(jdim+1)*0.5;
       *grdi  = (*grdi+1)*0.5-1;
       *grdj  = (*grdj+1)*0.5-1;

       *alpha = asin( ( sinphi0*sin(tlm)) / cosphi ) ;

    /*   printf("qqq %6.2f %6.2f %6.2f %6.2f %g %g %g %g\n",
           grdlon,grdlat,*grdi,*grdj,*alpha,tlm*r2d,cosphi,sinphi0);
    */

    }

NMC high accuracy polar stereo for SSM/I data

The polar stereo projection used by the original NMC models is not very precise because it assumes the earth is round (eccentricity = 0). While this approximation was reasonable for coarse resolution NWP models, it is inadequate to work with higher resolution data such as SSM/I.

Wind rotation has not been implemented!!! Use only for scalar fields.

`pdef ni nj pse slat slon polei polej dx dy sgn`

`ni` = # points in x
`nj` = # points in y
`slat` = absolute value of the standard latitude
`slon` = absolute value of the standard longitude
`pse` = polar stereo, "eccentric"
`polei` = x index position of the pole (where (0,0) is the index of the first point vice the more typical (1,1) )
`polej` = y index position of the pole (where (0,0) is the index of the first point vice the more typical (1,1) )
`dx` = delta x in km
`dy` = delta y in km
`sgn` = 1 for N polar stereo and -1 for S polar stereo

Source code in GrADS for the lon,lat -> i,j mapping:

    void ll2pse (int im, int jm, float *vals, float lon, float lat,
           float *grdi, float *grdj) {

       /* Convert from geodetic latitude and longitude to polar stereographic
          grid coordinates.  Follows mapll by V. J. Troisi.         */
       /* Conventions include that slat and lat must be absolute values */
       /* The hemispheres are controlled by the sgn parameter */
       /* Bob Grumbine 15 April 1994. */

       const rearth = 6738.273e3;
       const eccen2 = 0.006693883;
       const float pi = 3.141592654;

       float cdr, alat, along, e, e2;
       float t, x, y, rho, sl, tc, mc;
       float slat,slon,xorig,yorig,sgn,polei,polej,dx,dy;

       slat=*(vals+0);
       slon=*(vals+1);
       polei=*(vals+2);
       polej=*(vals+3);
       dx=*(vals+4)*1000;
       dy=*(vals+5)*1000;
       sgn=*(vals+6);

       xorig = -polei*dx;
       yorig = -polej*dy;

       /*printf("ppp %g %g %g %g %g %g
       %g\n",slat,slon,polei,polej,dx,dy,sgn);*/

       cdr   = 180./pi;
       alat  = lat/cdr;
       along = lon/cdr;
       e2    = eccen2;
       e     = sqrt(eccen2);

       if ( fabs(lat) > 90.)  {
         *grdi = -1;
         *grdj = -1;
         return;
       }
       else {
         t = tan(pi/4. - alat/2.) /
           pow( (1.-e*sin(alat))/(1.+e*sin(alat)) , e/2.);

         if ( fabs(90. - slat) < 1.E-3) {
           rho = 2.*rearth*t/
         pow( pow(1.+e,1.+e) * pow(1.-e,1.-e) , e/2.);
           }
           else {
             sl = slat/cdr;
             tc = tan(pi/4.-sl/2.) /
             pow( (1.-e*sin(sl))/(1.+e*sin(sl)), (e/2.) );
             mc = cos(sl)/ sqrt(1.-e2*sin(sl)*sin(sl) );
             rho = rearth * mc*t/tc;
           }

           x = rho*sgn*cos(sgn*(along+slon/cdr));
           y = rho*sgn*sin(sgn*(along+slon/cdr));

           *grdi = (x - xorig)/dx+1;
           *grdj = (y - yorig)/dy+1;

           /*printf("ppp (%g %g) (%g %g %g) %g
         %g\n",lat,lon,x,y,rho,*grdi,*grdj);*/

           return;
       }

    }

CSU RAMS Oblique Polar Stereo Grids

The CSU RAMS model uses an oblique polar stereo projection. This projection is still being tested...

`pdef 26 16 ops 40.0 -100.0 90000.0 90000.0 14.0 9.0 180000.0 180000.0`

`26` = #pts in x
`16` = #pts in y
`ops` = oblique polar stereo
`40.0` = lat of ref point (14.0, 9.0)
`-100.0` = lon of ref point (14.0, 9.0 (E is positive in GrADS, W is negative)
`90000.0` = xref offset [m]
`90000.0` = yref offset [m]
`14.0` = i of ref point
`9.0` = j of ref point
`180000.0` = dx [m]
`180000.0` = dy [m]

Wind rotation has not been implemented!!! Use only for scalar fields.

Source code in GrADS for the lon,lat -> i,j mapping:

    void ll2ops(float *vals, float lni, float lti, float *grdi, float *grdj)
       {

       const float radius = 6371229.0 ;
       const float pi = 3.141592654;

       float stdlat, stdlon, xref, yref, xiref, yjref, delx , dely;

       float plt,pln;
       double pi180,c1,c2,c3,c4,c5,c6,arg2a,bb,plt1,alpha,
       pln1,plt90,argu1,argu2;

       double hsign,glor,rstdlon,glolim,facpla,x,y;

       stdlat = *(vals+0);
       stdlon = *(vals+1);
       xref = *(vals+2);
       yref = *(vals+3);
       xiref = *(vals+4);
       yjref = *(vals+5);
       delx = *(vals+6);
       dely = *(vals+7);

       c1=1.0 ;
       pi180 = asin(c1)/90.0;

    /*
    c
    c     set flag for n/s hemisphere and convert longitude to <0 ; 360>
       interval
    c
    */
       if(stdlat >= 0.0) {
         hsign= 1.0 ;
       } else {
         hsign=-1.0 ;
       }
    /*
    c
    c     set flag for n/s hemisphere and convert longitude to <0 ; 360>
       interval
    c
    */
       glor=lni ;
       if(glor <= 0.0) glor=360.0+glor ;
       rstdlon=stdlon;
       if(rstdlon < 0.0) rstdlon=360.0+stdlon;

    /*
    c
    c     test for a n/s pole case
    c
    */
       if(stdlat == 90.0) {
         plt=lti ;
         pln=fmod(glor+270.0,360.0) ;
         goto l2000;
       }

      if(stdlat == -90.0) {
         plt=-lti ;
         pln=fmod(glor+270.0,360.0) ;
         goto l2000;
       }

    /*
    c
    c     test for longitude on 'greenwich or date line'
    c
    */
       if(glor == rstdlon) {
         if(lti > stdlat) {
           plt=90.0-lti+stdlat;
           pln=90.0;
         } else {
           plt=90.0-stdlat+lti;
           pln=270.0;;
         }
         goto l2000;
       }

       if(fmod(glor+180.0,360.0) == rstdlon) {
         plt=stdlat-90.0+lti;
         if(plt < -90.0) {
           plt=-180.0-plt;
           pln=270.0;
         } else {
           pln= 90.0;
         }
         goto l2000;
       }

    /*
    c
    c     determine longitude distance relative to rstdlon so it belongs to
    c     the absolute interval 0 - 180
    c
    */
       argu1 = glor-rstdlon;
       if(argu1 > 180.0) argu1 = argu1-360.0;
       if(argu1 < -180.0) argu1 = argu1+360.0;

    /*
    c
    c     1. get the help circle bb and angle alpha (legalize arguments)
    c
    */

       c2=lti*pi180 ;
       c3=argu1*pi180 ;
       arg2a = cos(c2)*cos(c3) ;
       if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = max1(arg2a,-c1)  */
       if(  c1 < arg2a ) arg2a = c1 ; /* min1(arg2a, c1)         */
       bb = acos(arg2a) ;

       c4=hsign*lti*pi180 ;
       arg2a = sin(c4)/sin(bb) ;
       if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
       if(  c1 < arg2a ) arg2a = c1  ; /* arg2a = dmin1(arg2a, c1) */
       alpha = asin(arg2a) ;
    /*
    c
    c     2. get plt and pln (still legalizing arguments)
    c
    */
       c5=stdlat*pi180 ;
       c6=hsign*stdlat*pi180 ;
       arg2a = cos(c5)*cos(bb) + sin(c6)*sin(c4) ;
       if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
       if(  c1 < arg2a ) arg2a = c1  ; /* arg2a = dmin1(arg2a, c1) */
       plt1   = asin(arg2a) ;

       arg2a = sin(bb)*cos(alpha)/cos(plt1) ;

       if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
       if(  c1 < arg2a ) arg2a =  c1 ; /* arg2a = dmin1(arg2a, c1) */
       pln1   = asin(arg2a) ;

    /*
    c
    c    test for passage of the 90 degree longitude (duallity in pln)
    c         get plt for which pln=90 when lti is the latitude
    c
    */
       arg2a = sin(c4)/sin(c6);
       if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
       if(  c1 < arg2a ) arg2a =  c1 ; /* arg2a = dmin1(arg2a, c1) */
       plt90 = asin(arg2a) ;

    /*
    c
    c         get help arc bb and angle alpha
    c
    */
       arg2a = cos(c5)*sin(plt90) ;
       if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
       if(  c1 < arg2a ) arg2a =  c1 ; /* arg2a = dmin1(arg2a, c1) */
       bb    = acos(arg2a) ;

       arg2a = sin(c4)/sin(bb) ;
       if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
       if(  c1 < arg2a ) arg2a =  c1 ; /* arg2a = dmin1(arg2a, c1) */
       alpha = asin(arg2a) ;

    /*
    c
    c         get glolim - it is nesc. to test for the existence of solution
    c
    */
       argu2  = cos(c2)*cos(bb) / (1.-sin(c4)*sin(bb)*sin(alpha)) ;
       if( fabs(argu2) > c1 ) {
         glolim = 999.0;
       } else {
         glolim = acos(argu2)/pi180;
       }

    /*
    c
    c
         modify (if nesc.) the pln solution
    c
    */
       if( ( fabs(argu1) > glolim && lti <= stdlat ) || ( lti > stdlat ) ) {
         pln1 = pi180*180.0 - pln1;
       }
    /*
    c
    c     the solution is symmetric so the direction must be if'ed
    c
    */
       if(argu1 < 0.0) {
         pln1 = -pln1;
       }
    /*
    c
    c     convert the radians to degrees
    c
    */
       plt = plt1/pi180 ;
       pln = pln1/pi180 ;

    /*
    c
    c     to obtain a rotated value (ie so x-axis in pol.ste. points east)
    c     add 270 to longitude
    c
    */
       pln=fmod(pln+270.0,360.0) ;

     l2000:

    /*
    c
    c     this program convert polar stereographic coordinates to x,y ditto
    c     longitude:   0 - 360  ; positive to the east
    c     latitude : -90 -  90  ; positive for northern hemisphere
    c     it is assumed that the x-axis point towards the east and
    c     corresponds to longitude = 0
    c
    c     tsp 20/06-89
    c
    c     constants and functions
    c
    */
       facpla = radius*2.0/(1.0+sin(plt*pi180))*cos(plt*pi180);
       x = facpla*cos(pln*pi180) ;
       y = facpla*sin(pln*pi180)  ;

       *grdi=(x-xref)/delx + xiref;
       *grdj=(y-yref)/dely + yjref;

       return;

    }

Pitfalls when using preprojected data

There are a few gotchas with using preprojected data:

1. the units in the variable definition for the `u` and `v` components must be `33` and `34K` (the GRIB standard) respectively, e.g.,

   `u 15 33` u component of the wind at 15 pressure levels
   `v 15 34` v component of the wind at 15 pressure levels

2. wind rotation is handled for polar stereo (N and S) preprojected data, but not for Lambert Conformal, as the Navy rotates the winds relative to earth. This will have to be added later......
3. the `eta.u` and `ops` projection are still experimental...

## GrADS Display Projections

Now that you hopefully understand GrADS data grids, it is time to discuss display projections. Graphics in GrADS are calculated relative to the internal GrADS data grid `i,j` space, transformed to the display device coordinates (e.g., the screen) and then displayed. That is, the i,j of the graphic element is converted to `lat/lon` and then to `x,y` on the screen via a map projection.

GrADS currently supports four `display projections`:

- lat/lon (or spherical);
- N polar stereo (`set mproj` nps);
- S polar stereo (`set mproj` sps);
- the Robinson projection (set lon -180 180, set lat -90 90, set mproj robinson).

As you can probably appreciate, the i,j-to-lon/lat-to-screen x,y for `lon/lat` displays is very simple and is considerably more complicated for N and S `polar stereo` projections.

In principle, a Lambert Conformal display projection could be implemented. It just takes work and a simple user interface for setting up that display projection. Actually, the user interface (i.e., "set" calls) is the most difficult problem...

## Summary and Plans

GrADS handles map projections in two different ways. The first is preprojected data where the fields are already on a projection (e.g., Lambert Conformal). It is fairly straightforward to implement other preprojected data projections and we will be fully implementing the NMC eta grid both staggered and unstaggered, "thinned" gaussian grids and the CSU RAMS oblique polar stereo projection. The second is in how i,j graphics (calculated in "grid" space) are displayed on a map background. Currently, only a few basic projections (lon/lat, polar stereo and robinson) are supported, but perhaps the development group will tackle this problem.