ppath.cc

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/* Copyright (C) 2002-2012 Patrick Eriksson <patrick.eriksson@chalmers.se>
 
   This program is free software; you can redistribute it and/or modify it
   under the terms of the GNU General Public License as published by the
   Free Software Foundation; either version 2, or (at your option) any
   later version.
 
   This program is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
   GNU General Public License for more details.
 
   You should have received a copy of the GNU General Public License
   along with this program; if not, write to the Free Software
   Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307,
   USA. */
 
 
 
/*===========================================================================
  === File description
  ===========================================================================*/
 
/*===========================================================================
  === External declarations
  ===========================================================================*/
 
#include <cmath>
#include <stdexcept>
#include "agenda_class.h"
#include "array.h"
#include "arts_omp.h"
#include "auto_md.h"
#include "check_input.h"
#include "geodetic.h"
#include "math_funcs.h"
#include "messages.h"
#include "mystring.h"
#include "logic.h"
#include "poly_roots.h"
#include "ppath.h"
#include "refraction.h"
#include "rte.h"
#include "special_interp.h"
 
extern const Numeric DEG2RAD;
extern const Numeric RAD2DEG;
 
 
 
 
 
/*===========================================================================
  === Precision variables
  ===========================================================================*/
 
// This variable defines the maximum allowed error tolerance for radius.
// The variable is, for example, used to check that a given a radius is
// consistent with the specified grid cell.
//
const Numeric   RTOL = 1e-3;
 
 
// As RTOL but for latitudes and longitudes.
//
const Numeric   LATLONTOL = 1e-8;
 
 
// Accuarcy for length comparisons.
//
const Numeric   LACC = 1e-5;
 
 
// Values to apply if some calculation does not provide a solution
const Numeric   R_NOT_FOUND   = -1;       // A value below zero
const Numeric   L_NOT_FOUND   = 99e99;    // Some very large value for l/lat/lon
const Numeric   LAT_NOT_FOUND = 99e99;
const Numeric   LON_NOT_FOUND = 99e99;
 
 
 
 
 
/*===========================================================================
  === Functions related to geometrical propagation paths
  ===========================================================================*/
 
 
Numeric geometrical_ppc( const Numeric& r, const Numeric& za )
{
  assert( r > 0 );
  assert( abs(za) <= 180 );
 
  return r * sin( DEG2RAD * abs(za) );
}
 
 
 
 
Numeric geompath_za_at_r(
       const Numeric&  ppc,
       const Numeric&  a_za,
       const Numeric&  r )
{
  assert( ppc >= 0 );
  assert( abs(a_za) <= 180 );
  assert( r >= ppc - RTOL );
 
  if( r > ppc )
    {
      Numeric za = RAD2DEG * asin( ppc / r );
      if( abs(a_za) > 90 )
        { za = 180 - za; }
      if( a_za < 0 )
        { za = -za; }
      return za;
    }
  else
    {
      if( a_za > 0 )
        { return 90; }
      else
        { return -90; }
    }
}
 
 
 
 
Numeric geompath_r_at_za(
       const Numeric&  ppc,
       const Numeric&  za )
{
  assert( ppc >= 0 );
  assert( abs(za) <= 180 );
 
  return ppc / sin( DEG2RAD * abs(za) );
}
 
 
 
 
Numeric geompath_lat_at_za(
       const Numeric&  za0,
       const Numeric&  lat0,
       const Numeric&  za )
{
  assert( abs(za0) <= 180 );
  assert( abs(za) <= 180 );
  assert( ( za0 >= 0 && za >= 0 )  ||  ( za0 < 0 && za < 0 ) );
 
  return lat0 + za0 - za;
}
 
 
 
 
Numeric geompath_l_at_r(
       const Numeric&  ppc,
       const Numeric&  r )
{
  assert( ppc >= 0 );
  assert( r >= ppc - RTOL );
  
  if( r > ppc )
    { return   sqrt( r*r - ppc*ppc ); }
  else
    { return   0; }
}
 
 
 
 
Numeric geompath_r_at_l(
       const Numeric&  ppc,
       const Numeric&  l )
{
  assert( ppc >= 0 );
  
  return sqrt( l*l + ppc*ppc );
}
 
 
 
 
Numeric geompath_r_at_lat(
       const Numeric&  ppc,
       const Numeric&  lat0,
       const Numeric&  za0,
       const Numeric&  lat )
{
  assert( ppc >= 0 );
  assert( abs(za0) <= 180 );
  assert( ( za0 >= 0 && lat >= lat0 )  ||  ( za0 <= 0 && lat <= lat0 ) );
 
  // Zenith angle at the new latitude
  const Numeric za = za0 + lat0 -lat;
 
  return geompath_r_at_za( ppc, za );
}
 
 
 
 
void geompath_from_r1_to_r2( 
             Vector&   r,
             Vector&   lat,
             Vector&   za,
             Numeric&  lstep,
       const Numeric&  ppc,
       const Numeric&  r1,
       const Numeric&  lat1,
       const Numeric&  za1,
       const Numeric&  r2,
       const bool&     tanpoint,
       const Numeric&  lmax )
{
  // Calculate length from tangent point, along the path for point 1 and 2.
  // Length defined in the viewing direction, ie. negative at end of sensor
  Numeric l1 = geompath_l_at_r( ppc, r1 );
  if( abs(za1) > 90 )
    { l1 *= -1; }
  Numeric l2 = geompath_l_at_r( ppc, r2 );  
  if( l1 < 0 ) { l2 *= -1; }
  if( tanpoint )
    { l2 *= -1; }
 
  // Calculate needed number of steps, considering a possible length criterion
  Index n;
  if( lmax > 0 )
    {
      // The absolute value of the length distance is needed here
      // We can't accept n=0, which is the case if l1 = l2
      n = max( Index(1), Index( ceil( abs( l2 - l1 ) / lmax ) ) );
    }
  else
    { n = 1; }
 
  // Length of path steps (note that lstep here can become negative)
  lstep = ( l2 - l1 ) / (Numeric)n;
 
  // Allocate vectors and put in point 1
  r.resize(n+1);
  lat.resize(n+1);
  za.resize(n+1);
  r[0]   = r1;
  lat[0] = lat1;
  za[0]  = za1;
 
  // Loop steps (beside last) and calculate radius and zenith angle
  for( Index i=1; i<n; i++ )
    {
      const Numeric l = l1 + lstep * (Numeric)i;
      r[i]   = geompath_r_at_l( ppc, l );   // Sign of l does not matter here
      // Set a zenith angle to 80 or 100 depending on sign of l
      za[i]  = geompath_za_at_r( ppc, sign(za1)*(90-sign(l)*10), r[i] );
    }
 
  // For maximum accuracy, set last radius to be exactly r2.
  r[n]  = r2;                     // Don't use r[n] below
  za[n] = geompath_za_at_r( ppc, sign(za1)*(90-sign(l2)*10), r2 );  
 
  // Ensure that zenith and nadir observations keep their zenith angle
  if( abs(za1) < ANGTOL  ||  abs(za1) > 180-ANGTOL )
    { za = za1; }
 
  // Calculate latitudes
  for( Index i=1; i<=n; i++ )
    { lat[i] = geompath_lat_at_za( za1, lat1, za[i] ); }
 
  // Take absolute value of lstep
  lstep = abs( lstep );
}
 
 
 
 
 
/*===========================================================================
  === Functions focusing on zenith and azimuth angles
  ===========================================================================*/
 
 
void cart2zaaa(
             Numeric&  za,
             Numeric&  aa,
       const Numeric&  dx,
       const Numeric&  dy,
       const Numeric&  dz )
{
  const Numeric r = sqrt( dx*dx + dy*dy + dz*dz );
 
  assert( r > 0 );
 
  za = RAD2DEG * acos( dz / r );
  aa = RAD2DEG * atan2( dy, dx );
}
 
 
 
 
void zaaa2cart(
             Numeric&  dx,
             Numeric&  dy,
             Numeric&  dz,
       const Numeric&  za,
       const Numeric&  aa )
{
  const Numeric   zarad  = DEG2RAD * za;
  const Numeric   aarad  = DEG2RAD * aa;
 
  dz = cos( zarad );
  dx = sin( zarad );
  dy = sin( aarad ) * dx;
  dx = cos( aarad ) * dx;
}
 
 
 
 
void rotationmat3D( 
           Matrix&   R, 
   ConstVectorView   vrot, 
    const Numeric&   a )
{
  assert( R.ncols() == 3 );
  assert( R.nrows() == 3 );
  assert( vrot.nelem() == 3 );
 
  const Numeric u = vrot[0];
  const Numeric v = vrot[1];
  const Numeric w = vrot[2];
 
  const Numeric u2 = u * u;
  const Numeric v2 = v * v;
  const Numeric w2 = w * w;
 
  assert( sqrt( u2 + v2 + w2 ) );
 
  const Numeric c = cos( DEG2RAD * a );
  const Numeric s = sin( DEG2RAD * a );
  
  // Fill R
  R(0,0) = u2 + (v2 + w2)*c;
  R(0,1) = u*v*(1-c) - w*s;
  R(0,2) = u*w*(1-c) + v*s;
  R(1,0) = u*v*(1-c) + w*s;
  R(1,1) = v2 + (u2+w2)*c;
  R(1,2) = v*w*(1-c) - u*s;
  R(2,0) = u*w*(1-c) - v*s;
  R(2,1) = v*w*(1-c)+u*s;
  R(2,2) = w2 + (u2+v2)*c;
}
 
 
 
void map_daa(
             Numeric&  za,
             Numeric&  aa,
       const Numeric&  za0,
       const Numeric&  aa0,
       const Numeric&  aa_grid )
{
  assert( abs( aa_grid ) <= 5 );
 
  Vector  xyz(3);
  Vector  vrot(3);
  Vector  u(3);
 
  // Unit vector towards aa0 at za=90
  //
  zaaa2cart( xyz[0], xyz[1], xyz[2], 90, aa0 );
    
  // Find vector around which rotation shall be performed
  // 
  // We can write this as cross([0 0 1],xyz). It turns out that the result 
  // of this operation is just [-y,x,0].
  //
  vrot[0] = -xyz[1];
  vrot[1] = xyz[0];
  vrot[2] = 0;
 
  // Unit vector towards aa0+aa at za=90
  //
  zaaa2cart( xyz[0], xyz[1], xyz[2], 90, aa0+aa_grid );
 
  // Apply rotation
  //
  Matrix R(3,3);
  rotationmat3D( R, vrot, za0-90 );
  mult( u, R, xyz );
 
  // Calculate za and aa for rotated u
  //
  cart2zaaa( za, aa, u[0], u[1], u[2] );
}
 
 
 
 
 
/*===========================================================================
  === Various functions
  ===========================================================================*/
 
 
Numeric refraction_ppc( 
        const Numeric&  r, 
        const Numeric&  za, 
        const Numeric&  refr_index_air )
{
  assert( r > 0 );
  assert( abs(za) <= 180 );
 
  return r * refr_index_air * sin( DEG2RAD * abs(za) );
}
 
 
 
 
void resolve_lon(
              Numeric&  lon,
        const Numeric&  lon5,
        const Numeric&  lon6 )
{
  assert( lon6 >= lon5 ); 
 
  if( lon < lon5  &&  lon+180 <= lon6 )
    { lon += 360; }
  else if( lon > lon6  &&  lon-180 >= lon5 )
    { lon -= 360; }
}
 
 
 
 
void find_tanpoint( 
         Index&   it,
   const Ppath    ppath )
{
  Numeric zmin = 99e99;
  it = -1;
  while( it < ppath.np-1  &&  ppath.pos(it+1,0) < zmin ) 
    {
      it++;
      zmin = ppath.pos(it,0);
    }
  if( it == 0  ||  it == ppath.np-1 )
    { it = -1; }
}
 
 
 
 
 
/*===========================================================================
  = 2D functions for surface and pressure level slope and tilt
  ===========================================================================*/
 
 
Numeric rsurf_at_lat(
       const Numeric&  lat1,
       const Numeric&  lat3,
       const Numeric&  r1,
       const Numeric&  r3,
       const Numeric&  lat )
{
  return   r1 + ( lat - lat1 ) * ( r3 - r1 ) / ( lat3 - lat1 );
}
 
 
 
 
void plevel_slope_2d(
               Numeric&   c1,
        ConstVectorView   lat_grid,           
        ConstVectorView   refellipsoid,
        ConstVectorView   z_surf,
        const GridPos&    gp,
        const Numeric&    za )
{
  Index i1 = gridpos2gridrange( gp, za >= 0 );
  const Numeric r1 = refell2r( refellipsoid, lat_grid[i1] ) + z_surf[i1];
  const Numeric r2 = refell2r( refellipsoid, lat_grid[i1+1] ) + z_surf[i1+1];
  //
  c1 = ( r2 - r1 ) / ( lat_grid[i1+1] - lat_grid[i1] );
}
 
 
 
 
void plevel_slope_2d(
              Numeric&  c1,
        const Numeric&  lat1,
        const Numeric&  lat2,
        const Numeric&  r1,
        const Numeric&  r2 )
{
  c1 = ( r2 - r1 ) / ( lat2 - lat1 );
}
 
 
 
 
 
Numeric plevel_angletilt(
        const Numeric&  r,
        const Numeric&  c1 )
{
  // The tilt (in radians) is c1/r if c1 is converted to m/radian. So we get
  // conversion RAD2DEG twice
  return   RAD2DEG * RAD2DEG * c1 / r;
}
 
 
 
 
bool is_los_downwards( 
        const Numeric&  za,
        const Numeric&  tilt )
{
  assert( abs(za) <= 180 );
 
  // Yes, it shall be -tilt in both cases, if you wonder.
  if( za > (90-tilt)  ||  za < (-90-tilt) )
    { return true; }
  else
    { return false; }
}
 
 
 
 
void r_crossing_2d(
             Numeric&   lat,
             Numeric&   l,
       const Numeric&   r_hit,
       const Numeric&   r_start,      
       const Numeric&   lat_start,    
       const Numeric&   za_start,     
       const Numeric&   ppc )
{
  assert( abs( za_start ) <= 180 );
  assert( r_start >= ppc ); 
 
  const Numeric absza = abs( za_start );
 
  // If above and looking upwards or r_hit below tangent point,
  // we have no crossing:
  if( ( r_start >= r_hit  &&  absza <= 90 )  ||  ppc > r_hit )
    { lat = LAT_NOT_FOUND;   l = L_NOT_FOUND; }
 
  else
    {
      // Passages of tangent point
      if( absza > 90  &&  r_start <= r_hit )
        {
          Numeric za  = geompath_za_at_r( ppc, sign(za_start)*89, r_hit );
          lat = geompath_lat_at_za( za_start, lat_start, za );
          l = geompath_l_at_r( ppc, r_start ) + geompath_l_at_r( ppc, r_hit );
        }
      
      else
        {
          Numeric za  = geompath_za_at_r( ppc, za_start, r_hit );
          lat = geompath_lat_at_za( za_start, lat_start, za );
          l = abs( geompath_l_at_r( ppc, r_start ) -
                   geompath_l_at_r( ppc, r_hit   ) );
          assert( l > 0 );
        }
    }
}
 
 
 
 
Numeric rslope_crossing2d(
        const Numeric&  rp,
        const Numeric&  za,
        const Numeric&  r0,
              Numeric   c1 )
{
  const Numeric   zaabs = abs(za);
 
  assert( za != 0 );
  assert( zaabs != 180 );
  assert( abs(c1) > 0 ); // c1=0 should work, but unnecessary to use this func.
 
  // Convert slope to m/radian and consider viewing direction
  c1 *= RAD2DEG;
  if( za < 0 )
    { c1 = -c1; }
 
  // The nadir angle in radians, and cosine and sine of that angle
  const Numeric   beta  = DEG2RAD * ( 180 - zaabs );
  const Numeric   cv    = cos( beta );
  const Numeric   sv    = sin( beta );
 
  // Some repeated terms
  const Numeric r0s = r0*sv;
  const Numeric r0c = r0*cv;
  const Numeric cs  = c1*sv;
  const Numeric cc  = c1*cv;
 
  // The vector of polynomial coefficients
  //
  Index  n = 6;
  Vector p0(n+1);
  //
  p0[0] =  r0s     - rp*sv;
  p0[1] =  r0c     + cs;  
  p0[2] = -r0s/2   + cc;
  p0[3] = -r0c/6   - cs/2;
  p0[4] =  r0s/24  - cc/6;
  p0[5] =  r0c/120 + cs/24;
  p0[6] = -r0s/720 + cc/120;
  //
  // The accuracy when solving the polynomial equation gets worse when
  // approaching 0 and 180 deg. The solution is to let the start polynomial
  // order decrease when approaching these angles. The values below based on
  // practical experience, don't change without making extremely careful tests.
  //
  if( abs( 90 - zaabs ) > 89.9 )
    { n = 1; }
  else if( abs( 90 - zaabs ) > 75 )
    { n = 4; }
 
  // Calculate roots of the polynomial
  Matrix roots;
  int solutionfailure = 1;
  //
  while( solutionfailure)
    {
      roots.resize(n,2);
      Vector p;
      p = p0[Range(0,n+1)];
      solutionfailure = poly_root_solve( roots, p );
      if( solutionfailure )
        { n -= 1; assert( n > 0 ); }
    }
 
  // If r0=rp, numerical inaccuracy can give a false solution, very close
  // to 0, that we must throw away.
  Numeric   dmin = 0;
  if( abs(r0-rp) < 1e-9 )  // 1 nm set based on practical experience. 
    { dmin = 5e-12; }
 
  // Find the smallest root with imaginary part = 0, and real part > 0.
  Numeric dlat = 1.571;  // Not interested in solutions above 90 deg!
  //
  for( Index i=0; i<n; i++ )
    {
      if( roots(i,1) == 0  &&  roots(i,0) > dmin  &&  roots(i,0) < dlat )
        { dlat = roots(i,0); }
    }  
 
  if( dlat < 1.57 )  // A somewhat smaller value than start one
    { 
      // Convert back to degrees
      dlat = RAD2DEG * dlat; 
 
      // Change sign if zenith angle is negative
      if( za < 0 )
        { dlat = -dlat; }
    }
  else
    { dlat = LAT_NOT_FOUND; }
 
  return   dlat;
}
 
 
 
 
void plevel_crossing_2d(
              Numeric&  r,
              Numeric&  lat,
              Numeric&  l,
        const Numeric&  r_start0,
        const Numeric&  lat_start,
        const Numeric&  za_start,
        const Numeric&  ppc,
        const Numeric&  lat1,
        const Numeric&  lat3,
        const Numeric&  r1,
        const Numeric&  r3,
        const bool&     above )
{
  const Numeric absza = abs( za_start );
 
  assert( absza <= 180 );
  assert( lat_start >=lat1  &&  lat_start <= lat3 );
 
  // Zenith case
  if( absza < ANGTOL )
    {
      if( above )
        { r = R_NOT_FOUND;  lat = LAT_NOT_FOUND;   l = L_NOT_FOUND; }
      else
        {
          lat = lat_start;
          r   = rsurf_at_lat( lat1, lat3, r1, r3, lat );
          l   = max( 1e-9, r - r_start0 ); // Max to ensure a small positive
        }                                  // step, to handle numerical issues
    }
 
  // Nadir case
  else if( absza > 180-ANGTOL )
    {
      if( above )
        {
          lat = lat_start;
          r   = rsurf_at_lat( lat1, lat3, r1, r3, lat );
          l   = max( 1e-9, r_start0 - r ); // As above
        }  
      else
        { r = R_NOT_FOUND;  lat = LAT_NOT_FOUND;   l = L_NOT_FOUND; }
    }
 
  // The general case
  else
    {
      const Numeric rmin = min( r1, r3 );
      const Numeric rmax = max( r1, r3 );
 
      // The case of negligible slope
      if( rmax-rmin < 1e-6 )
        {
          // Set r_start and r, considering impact of numerical problems
          Numeric r_start = r_start0;
                  r       = r1;
          if( above )
            { if( r_start < rmax ) { r_start = r = rmax; } }
          else
            { if( r_start > rmin ) { r_start = r = rmin; } }
          
          r_crossing_2d( lat, l, r, r_start, lat_start, za_start, ppc );
          
          // Check if inside [lat1,lat3]
          if( lat > lat3  ||  lat < lat1 )
            { r = R_NOT_FOUND;  lat = LAT_NOT_FOUND; }  
        }
 
      // With slope
      else
        {
          // Set r_start, considering impact of numerical problems
          Numeric r_start = r_start0;
          if( above )
            { if( r_start < rmin ) { r_start = rmin; } }
          else
            { if( r_start > rmax ) { r_start = rmax; } }
 
          Numeric za=999;
 
          // Calculate crossing with closest radius
          if( r_start > rmax )
            {
              r = rmax;
              r_crossing_2d( lat, l, r, r_start, lat_start, za_start, ppc );
            }
          else if( r_start < rmin )
            {
              r = rmin;      
              r_crossing_2d( lat, l, r, r_start, lat_start, za_start, ppc );
            }
          else
            { r = r_start; lat = lat_start; l = 0; za = za_start; }
  
          // lat must be inside [lat1,lat3] if relevant to continue
          if( lat < lat1  ||  lat > lat3 )
            { r = R_NOT_FOUND; }   // lat already set by r_crossing_2d
 
          // Otherwise continue from found point, considering the level slope 
          else
            {
              // Level slope and radius at lat
              Numeric  cpl;
              plevel_slope_2d( cpl, lat1, lat3, r1, r3 );
              const Numeric  rpl = r1 + cpl*(lat-lat1);
 
              // Make adjustment if numerical problems
              if( above )
                { if( r < rpl ) { r = rpl; } }
              else
                { if( r > rpl ) { r = rpl; } }
 
              // Calculate zenith angle at (r,lat) (if <180, already set above)
              if( za > 180 )  // lat+za preserved (also with negative za)
                { za = lat_start + za_start - lat; };
 
              // Latitude distance from present point to actual crossing
              const Numeric dlat = rslope_crossing2d( r, za, rpl, cpl );
 
              // Update lat and check if still inside [lat1,lat3].
              // If yes, determine r and l
              lat += dlat;
              if( lat < lat1  ||  lat > lat3 )
                { r = R_NOT_FOUND;  lat = LAT_NOT_FOUND;   l = L_NOT_FOUND; }
              else
                { 
                  // It was tested to calculate r from geompath functions, but
                  // appeared to give poorer accuracy around zenith/nadir
                  r = rpl + cpl*dlat;
 
                  // Zenith angle needed to check tangent point
                  za = lat_start + za_start - lat;
 
                  // Passage of tangent point requires special attention
                  if( absza>90 && abs(za)<90 )
                    { l = geompath_l_at_r( ppc, r_start ) +
                          geompath_l_at_r( ppc, r ); }
                  else
                    { l = abs( geompath_l_at_r( ppc, r_start ) -
                               geompath_l_at_r( ppc, r ) ); }
                }
            }  
        }
    }
}
 
 
 
/*===========================================================================
  = 3D functions for level slope and tilt, and lat/lon crossings
  ===========================================================================*/
 
 
Numeric rsurf_at_latlon(
       const Numeric&  lat1,
       const Numeric&  lat3,
       const Numeric&  lon5,
       const Numeric&  lon6,
       const Numeric&  r15,
       const Numeric&  r35,
       const Numeric&  r36,
       const Numeric&  r16,
       const Numeric&  lat,
       const Numeric&  lon )
{
  // We can't have any assert of *lat* and *lon* here as we can go outside
  // the ranges when called from *plevel_slope_3d*.
 
  if( lat == lat1 )
    { return   r15 + ( lon - lon5 ) * ( r16 - r15 ) / ( lon6 -lon5 ); }
  else if( lat == lat3 )
    { return   r35 + ( lon - lon5 ) * ( r36 - r35 ) / ( lon6 -lon5 ); }
  else if( lon == lon5 )
    { return   r15 + ( lat - lat1 ) * ( r35 - r15 ) / ( lat3 -lat1 ); }
  else if( lon == lon6 )
    { return   r16 + ( lat - lat1 ) * ( r36 - r16 ) / ( lat3 -lat1 ); }
  else
    {
      const Numeric   fdlat = ( lat - lat1 ) / ( lat3 - lat1 );
      const Numeric   fdlon = ( lon - lon5 ) / ( lon6 - lon5 );
      return   (1-fdlat)*(1-fdlon)*r15 + fdlat*(1-fdlon)*r35 + 
                                         (1-fdlat)*fdlon*r16 + fdlat*fdlon*r36;
    }
}
 
 
 
 
void plevel_slope_3d(
              Numeric&  c1,
              Numeric&  c2,
        const Numeric&  lat1,
        const Numeric&  lat3,
        const Numeric&  lon5,
        const Numeric&  lon6,
        const Numeric&  r15,
        const Numeric&  r35,
        const Numeric&  r36,
        const Numeric&  r16,
        const Numeric&  lat,
        const Numeric&  lon,
        const Numeric&  aa )
{
  // Save time and avoid numerical problems if all r are equal
  if( r15==r35 && r15==r36 && r15==r16 && r35==r36 && r35==r16 && r36==r16 )
    { 
      c1 = 0;
      c2 = 0;
    }
 
  else
    {
      // Radius at point of interest
      const Numeric   r0 = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                            r15, r35, r36, r16, lat, lon );
 
      // Size of test angular distance. Unit is degrees.
      //  dang = 1e-4 corresponds to about 10 m shift horisontally. 
      //  Smaller values should probably be avoided. For example, 1e-5 gave
      //  significant c2 when it should be zero. 
      const Numeric   dang = 1e-4;  
 
      Numeric lat2, lon2;
      latlon_at_aa( lat2, lon2, lat, lon, aa, dang );
      resolve_lon( lon2, lon5, lon6 );
      const Numeric dr1 =  rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                         r15, r35, r36, r16, lat2, lon2 ) - r0;
 
      latlon_at_aa( lat2, lon2, lat, lon, aa, 2*dang );
      resolve_lon( lon2, lon5, lon6 );
      const Numeric dr2 =  rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                         r15, r35, r36, r16, lat2, lon2 ) - r0;
 
      // Derive linear and quadratic coefficient
      c1 = 0.5 * ( 4*dr1 - dr2 );
      c2 = (dr1-c1)/(dang*dang);
      c1 /= dang;
    }
}
 
 
 
 
void plevel_slope_3d(
              Numeric&    c1,
              Numeric&    c2,
        ConstVectorView   lat_grid,
        ConstVectorView   lon_grid,  
        ConstVectorView   refellipsoid,
        ConstMatrixView   z_surf,
        const GridPos&    gp_lat,
        const GridPos&    gp_lon,
        const Numeric&    aa )
{
  Index ilat = gridpos2gridrange( gp_lat, abs( aa ) >= 0 );
  Index ilon = gridpos2gridrange( gp_lon, aa >= 0 );
 
  // Allow that we are at the upper edge of the grids only for special cases:
  const Index llat = lat_grid.nelem() - 1;
  // At North pole:
  if( ilat >= llat )
    {
      if( lat_grid[llat] > POLELAT )
        { ilat = llat - 1; }
      else
        { throw runtime_error( "The upper latitude end of the atmosphere "
                               "reached, that is not allowed." ); }
    }
  // Complete 360deg lon coverage:
  if( ilon >= lon_grid.nelem() - 1 )
    {
      if( is_lon_cyclic( lon_grid ) )
        { ilon = 0; }
      else
        { throw runtime_error( "The upper longitude end of the atmosphere "
                               "reached, that is not allowed." ); }
    }
 
  // Restore latitude and longitude values
  Vector  itw(2);
  Numeric  lat, lon;
  interpweights( itw, gp_lat );
  lat = interp( itw, lat_grid, gp_lat );
  interpweights( itw, gp_lon );
  lon = interp( itw, lon_grid, gp_lon );
 
  // Extract values that defines the grid cell
  const Numeric   lat1 = lat_grid[ilat];
  const Numeric   lat3 = lat_grid[ilat+1];
  const Numeric   lon5 = lon_grid[ilon];
  const Numeric   lon6 = lon_grid[ilon+1];
  const Numeric   re1  = refell2r( refellipsoid, lat1 );
  const Numeric   re3  = refell2r( refellipsoid, lat3 );
  const Numeric   r15  = re1 + z_surf(ilat,ilon);
  const Numeric   r35  = re3 + z_surf(ilat+1,ilon);
  const Numeric   r36  = re3 + z_surf(ilat+1,ilon+1);
  const Numeric   r16  = re1 + z_surf(ilat,ilon+1);
 
  plevel_slope_3d( c1, c2, lat1, lat3, lon5, lon6, r15, r35, r36, r16, 
                                                                lat, lon, aa );
}
 
 
 
 
Numeric rslope_crossing3d(
        const Numeric&  rp,
        const Numeric&  za,
        const Numeric&  r0,
              Numeric   c1,
              Numeric   c2 )
{
  // Even if the four corner radii of the grid cell differ, both c1 and c2 can
  // turn up to be 0. So in contrast to the 2D version, here we accept zero c.
 
  // Convert c1 and c2 from degrees to radians
  c1 *= RAD2DEG;
  c2 *= RAD2DEG*RAD2DEG;
  
  // The nadir angle in radians, and cosine and sine of that angle
  const Numeric   beta = DEG2RAD * ( 180 - za );
  const Numeric   cv = cos( beta );
  const Numeric   sv = sin( beta );
  
  // Some repeated terms
  const Numeric r0s = r0*sv;
  const Numeric r0c = r0*cv;
  const Numeric c1s = c1*sv;
  const Numeric c1c = c1*cv;
  const Numeric c2s = c2*sv;
  const Numeric c2c = c2*cv;
  
  // The vector of polynomial coefficients
  //
  Index n = 6;
  Vector p0(n+1);
  //
  p0[0] =  r0s     - rp*sv;
  p0[1] =  r0c     + c1s;  
  p0[2] = -r0s/2   + c1c     + c2s;
  p0[3] = -r0c/6   - c1s/2   + c2c;
  p0[4] =  r0s/24  - c1c/6   - c2s/2;
  p0[5] =  r0c/120 + c1s/24  - c2c/6;
  p0[6] = -r0s/720 + c1c/120 + c2s/24;
  //
  // The accuracy when solving the polynomial equation gets worse when
  // approaching 0 and 180 deg. The solution is to let the start polynomial
  // order decrease when approaching these angles. The values below based
  // on practical experience, don't change without making extremly careful
  // tests.
  //
  if( abs( 90 - za ) > 89.9 )
    { n = 1; }
  else if( abs( 90 - za ) > 75 )
    { n = 4; }
  
  // Calculate roots of the polynomial
  Matrix roots;
  int solutionfailure = 1;
  //
  while( solutionfailure)
    {
      roots.resize(n,2);
      Vector p;
      p = p0[Range(0,n+1)];
      solutionfailure = poly_root_solve( roots, p );
      if( solutionfailure )
        { n -= 1; assert( n > 0 ); }
    }
 
  // If r0=rp, numerical inaccuracy can give a false solution, very close
  // to 0, that we must throw away.
  Numeric   dmin = 0;
  if( r0 == rp )
    { dmin = 1e-6; }
  
  // Find the smallest root with imaginary part = 0, and real part > 0.
  //
  Numeric dlat = 1.571;  // Not interested in solutions above 90 deg!
  //
  for( Index i=0; i<n; i++ )
    {
      if( roots(i,1) == 0  &&  roots(i,0) > dmin  &&  roots(i,0) < dlat )
        { dlat = roots(i,0); }
    }  
 
  if( dlat < 1.57 )  // A somewhat smaller value than start one
    { 
      // Convert back to degrees
      dlat = RAD2DEG * dlat; 
    }
  else
    { dlat = LAT_NOT_FOUND; }
 
  return   dlat;
}
 
 
 
 
void r_crossing_3d(
             Numeric&   lat,
             Numeric&   lon,
             Numeric&   l,
       const Numeric&   r_hit,
       const Numeric&   r_start,
       const Numeric&   lat_start,
       const Numeric&   lon_start,
       const Numeric&   za_start,
       const Numeric&   ppc,
       const Numeric&   x,
       const Numeric&   y,
       const Numeric&   z,
       const Numeric&   dx,
       const Numeric&   dy,
       const Numeric&   dz )
{
  assert( za_start >=   0 );
  assert( za_start <= 180 );
 
  // If above and looking upwards or r_hit below tangent point,
  // we have no crossing:
  if( ( r_start >= r_hit  &&  za_start <= 90 )  ||  ppc > r_hit )
    { lat = LAT_NOT_FOUND;   lon = LON_NOT_FOUND;   l   = L_NOT_FOUND; }
 
  else
    {
      // Zenith/nadir
      if( za_start < ANGTOL  ||  za_start > 180-ANGTOL )
        {
          l   = abs( r_hit - r_start );
          lat = lat_start;
          lon = lon_start;
        }          
 
      else
        {
          const Numeric   p  = x*dx + y*dy + z*dz;
          const Numeric   pp = p * p;
          const Numeric   q  = x*x + y*y + z*z - r_hit*r_hit;
          const Numeric   sq = sqrt( pp - q );
          const Numeric   l1 = -p + sq;
          const Numeric   l2 = -p - sq;
          
          const Numeric lmin = min( l1, l2 );
          const Numeric lmax = max( l1, l2 );
 
          // If r_start equals r_hit solutions just above zero can appear (that
          // shall be rejected). So we pick the biggest solution if lmin is
          // negative or just above zero. 
          // (Tried to use "if( r_start != r_hit )", but failed occasionally)
          if( lmin < 1e-6 )
            { l = lmax; }
          else
            { l = lmin; }
          assert( l > 0 );
          
          lat = RAD2DEG * asin( ( z+dz*l ) / r_hit );
          lon = RAD2DEG * atan2( y+dy*l, x+dx*l );
        }
    }
}
 
 
 
 
 
/*===========================================================================
  === Basic functions for the Ppath structure
  ===========================================================================*/
 
 
void ppath_init_structure( 
              Ppath&      ppath,
        const Index&      atmosphere_dim,
        const Index&      np )
{
  assert( np > 0 );
  assert( atmosphere_dim >= 1 );
  assert( atmosphere_dim <= 3 );
 
  ppath.dim        = atmosphere_dim;
  ppath.np         = np;
  ppath.constant   = -1;  
 
  const Index npos = max( Index(2), atmosphere_dim );
  const Index nlos = max( Index(1), atmosphere_dim-1 );
 
  ppath.start_pos.resize( npos );  ppath.start_pos = -999;
  ppath.start_los.resize( nlos );  ppath.start_los = -999;
  ppath.start_lstep = 0;
  ppath.end_pos.resize( npos );
  ppath.end_los.resize( nlos );
  ppath.end_lstep = 0;
 
  ppath.pos.resize( np, npos );
  ppath.los.resize( np, nlos );
  ppath.r.resize( np );
  ppath.lstep.resize( np-1 );
 
  ppath.gp_p.resize( np );
  if( atmosphere_dim >= 2 )
    { 
      ppath.gp_lat.resize( np ); 
      if( atmosphere_dim == 3 )
        { ppath.gp_lon.resize( np ); }
    }
 
  ppath_set_background( ppath, 0 );
  ppath.nreal.resize( np );
  ppath.ngroup.resize( np );
}
 
 
 
 
 
void ppath_set_background( 
              Ppath&      ppath,
        const Index&      case_nr )
{
  switch ( case_nr )
    {
    case 0:
      ppath.background = "unvalid";
      break;
    case 1:
      ppath.background = "space";
      break;
    case 2:
      ppath.background = "surface";
      break;
    case 3:
      ppath.background = "cloud box level";
      break;
    case 4:
      ppath.background = "cloud box interior";
      break;
    case 9:
      ppath.background = "transmitter";
      break;
    default:
      ostringstream os;
      os << "Case number " << case_nr << " is not defined.";
      throw runtime_error(os.str());
    }
}
 
 
 
 
Index ppath_what_background( const Ppath&   ppath )
{
  if( ppath.background == "unvalid" )
    { return 0; }
  else if( ppath.background == "space" )
    { return 1; }
  else if( ppath.background == "surface" )
    { return 2; }
  else if( ppath.background == "cloud box level" )
    { return 3; }
  else if( ppath.background == "cloud box interior" )
    { return 4; }
  else if( ppath.background == "transmitter" )
    { return 9; }
  else
    {
      ostringstream os;
      os << "The string " << ppath.background 
         << " is not a valid background case.";
      throw runtime_error(os.str());
    }
}
 
 
 
 
void ppath_copy(
          Ppath&  ppath1,
    const Ppath&  ppath2,
    const Index&  ncopy )
{
  Index n;
  if( ncopy < 0 )
    { n = ppath2.np; }
  else
    { n = ncopy; }
 
  assert( ppath1.np >= n ); 
 
  // The field np shall not be copied !
 
  ppath1.dim        = ppath2.dim;
  ppath1.constant   = ppath2.constant;
  ppath1.background = ppath2.background;
 
  // As index 0 always is included in the copying, the end point is covered
  ppath1.end_pos   = ppath2.end_pos;
  ppath1.end_los   = ppath2.end_los;
  ppath1.end_lstep = ppath2.end_lstep;
 
  // Copy start point only if copying fills up ppath1
  if( n == ppath1.np )
    {
      ppath1.start_pos   = ppath2.start_pos;
      ppath1.start_los   = ppath2.start_los;
      ppath1.start_lstep = ppath2.start_lstep;
    }
 
  ppath1.pos(Range(0,n),joker) = ppath2.pos(Range(0,n),joker);
  ppath1.los(Range(0,n),joker) = ppath2.los(Range(0,n),joker);
  ppath1.r[Range(0,n)]         = ppath2.r[Range(0,n)];
  ppath1.nreal[Range(0,n)]     = ppath2.nreal[Range(0,n)];
  ppath1.ngroup[Range(0,n)]    = ppath2.ngroup[Range(0,n)];
  if( n > 1 )
    { ppath1.lstep[Range(0,n-1)]  = ppath2.lstep[Range(0,n-1)]; }
 
  for( Index i=0; i<n; i++ )
    {
      gridpos_copy( ppath1.gp_p[i], ppath2.gp_p[i] );
      
      if( ppath1.dim >= 2 )
        { gridpos_copy( ppath1.gp_lat[i], ppath2.gp_lat[i] ); }
      
      if( ppath1.dim == 3 )
        { gridpos_copy( ppath1.gp_lon[i], ppath2.gp_lon[i] ); }
     }
}
 
 
 
 
void ppath_append(
           Ppath&   ppath1,
     const Ppath&   ppath2 )
{
  const Index n1 = ppath1.np;
  const Index n2 = ppath2.np;
 
  Ppath   ppath;
  ppath_init_structure( ppath, ppath1.dim, n1 );
  ppath_copy( ppath, ppath1, -1 );
 
  ppath_init_structure( ppath1, ppath1.dim, n1 + n2 - 1 );
  ppath_copy( ppath1, ppath, -1 );
 
  // Append data from ppath2
  Index i1;
  for( Index i=1; i<n2; i++ )
    { 
      i1 = n1 + i - 1;
 
      ppath1.pos(i1,0)  = ppath2.pos(i,0);
      ppath1.pos(i1,1)  = ppath2.pos(i,1);
      ppath1.los(i1,0)  = ppath2.los(i,0);
      ppath1.r[i1]      = ppath2.r[i];
      ppath1.nreal[i1]  = ppath2.nreal[i];
      ppath1.ngroup[i1] = ppath2.ngroup[i];
      gridpos_copy( ppath1.gp_p[i1], ppath2.gp_p[i] );
 
      if( ppath1.dim >= 2 )
        { gridpos_copy( ppath1.gp_lat[i1], ppath2.gp_lat[i] ); }
      
      if( ppath1.dim == 3 )
        {
          ppath1.pos(i1,2)        = ppath2.pos(i,2);
          ppath1.los(i1,1)        = ppath2.los(i,1);
          gridpos_copy( ppath1.gp_lon[i1], ppath2.gp_lon[i] ); 
        }
      
      ppath1.lstep[i1-1] = ppath2.lstep[i-1];
    }
 
  if( ppath_what_background( ppath2 ) )
    { ppath1.background = ppath2.background; }
 
  ppath.start_pos   = ppath2.start_pos;
  ppath.start_los   = ppath2.start_los;
  ppath.start_lstep = ppath2.start_lstep;
}
 
 
 
 
 
/*===========================================================================
  === 1D/2D/3D start and end ppath functions
  ===========================================================================*/
 
 
void ppath_start_1d(
              Numeric&    r_start,
              Numeric&    lat_start,
              Numeric&    za_start,
              Index&      ip,
        const Ppath&      ppath )
{
  // Number of points in the incoming ppath
  const Index   imax = ppath.np - 1;
 
  // Extract starting radius, zenith angle and latitude
  r_start   = ppath.r[imax];
  lat_start = ppath.pos(imax,1);
  za_start  = ppath.los(imax,0);
 
  // Determine index of the pressure level being the lower limit for the
  // grid range of interest.
  //
  ip = gridpos2gridrange( ppath.gp_p[imax], za_start<=90 );
}
 
 
 
 
void ppath_end_1d(
              Ppath&      ppath,
        ConstVectorView   r_v,
        ConstVectorView   lat_v,
        ConstVectorView   za_v,
        ConstVectorView   lstep,
        ConstVectorView   n_v,
        ConstVectorView   ng_v,
        ConstVectorView   z_field,
        ConstVectorView   refellipsoid,
        const Index&      ip,
        const Index&      endface,
        const Numeric&    ppc )
{
  // Number of path points
  const Index   np = r_v.nelem();
 
  // Re-allocate ppath for return results and fill the structure
  ppath_init_structure( ppath, 1, np );
 
  ppath.constant = ppc;
 
  // Help variables that are common for all points.
  const Numeric   r1 = refellipsoid[0] + z_field[ip];
  const Numeric   dr = z_field[ip+1] - z_field[ip];
 
  for( Index i=0; i<np; i++ )
    {
      ppath.r[i]      = r_v[i];
      ppath.pos(i,0)  = r_v[i] - refellipsoid[0];
      ppath.pos(i,1)  = lat_v[i];
      ppath.los(i,0)  = za_v[i];
      ppath.nreal[i]  = n_v[i];
      ppath.ngroup[i] = ng_v[i];
      
      ppath.gp_p[i].idx   = ip;
      ppath.gp_p[i].fd[0] = ( r_v[i] - r1 ) / dr;
      ppath.gp_p[i].fd[1] = 1 - ppath.gp_p[i].fd[0];
      gridpos_check_fd( ppath.gp_p[i] );
 
      if( i > 0 )
        { ppath.lstep[i-1] = lstep[i-1]; }
    }
  gridpos_check_fd( ppath.gp_p[np-1] );
 
 
  //--- End point is the surface
  if( endface == 7 )
    { ppath_set_background( ppath, 2 ); }
 
  //--- End point is on top of a pressure level
  else if( endface <= 4 )
    { gridpos_force_end_fd( ppath.gp_p[np-1], z_field.nelem() ); }
}
 
 
 
 
void ppath_start_2d(
              Numeric&    r_start,
              Numeric&    lat_start,
              Numeric&    za_start,
              Index&      ip,
              Index&      ilat,
              Numeric&    lat1,
              Numeric&    lat3,
              Numeric&    r1a,
              Numeric&    r3a,
              Numeric&    r3b,
              Numeric&    r1b,
              Numeric&    rsurface1,
              Numeric&    rsurface3,
              Ppath&      ppath,
        ConstVectorView   lat_grid,
        ConstMatrixView   z_field,
        ConstVectorView   refellipsoid,
        ConstVectorView   z_surface
        )
{
  // Number of points in the incoming ppath
  const Index imax = ppath.np - 1;
 
  // Extract starting radius, zenith angle and latitude
  r_start   = ppath.r[imax];
  lat_start = ppath.pos(imax,1);
  za_start  = ppath.los(imax,0);
 
  // Determine interesting latitude grid range and latitude end points of 
  // the range.
  //
  ilat = gridpos2gridrange( ppath.gp_lat[imax], za_start >= 0 );
  //
  lat1 = lat_grid[ilat];
  lat3 = lat_grid[ilat+1];
 
  // Determine interesting pressure grid range. Do this first assuming that
  // the pressure levels are not tilted (that is, abs(za_start<=90) always
  // mean upward observation). 
  // Set radius for the corners of the grid cell and the radial slope of 
  // pressure level limits of the grid cell to match the found ip.
  //
  ip = gridpos2gridrange( ppath.gp_p[imax], abs(za_start) <= 90);
  //
  const Numeric re1 = refell2r( refellipsoid, lat_grid[ilat] );
  const Numeric re3 = refell2r( refellipsoid, lat_grid[ilat+1] );
  //
  r1a = re1 + z_field(ip,ilat);        // lower-left
  r3a = re3 + z_field(ip,ilat+1);      // lower-right
  r3b = re3 + z_field(ip+1,ilat+1);    // upper-right
  r1b = re1 + z_field(ip+1,ilat);      // upper-left
 
  // This part is a fix to catch start postions on top of a pressure level
  // that does not have an end fractional distance for the first step.
  {
    // Radius of lower and upper pressure level at the start position
    const Numeric   rlow = rsurf_at_lat( lat1, lat3, r1a, r3a, lat_start );
    const Numeric   rupp = rsurf_at_lat( lat1, lat3, r1b, r3b, lat_start );
    if( abs(r_start-rlow) < RTOL || abs(r_start-rupp) < RTOL )
      { gridpos_force_end_fd( ppath.gp_p[imax], z_field.nrows() ); }
  }
  
  // Slopes of pressure levels
  Numeric   c2, c4;
  plevel_slope_2d( c2, lat1, lat3, r1a, r3a );
  plevel_slope_2d( c4, lat1, lat3, r1b, r3b );
 
  // Check if the LOS zenith angle happen to be between 90 and the zenith angle
  // of the pressure level (that is, 90 + tilt of pressure level), and in
  // that case if ip must be changed. This check is only needed when the
  // start point is on a pressure level.
  //
  if( is_gridpos_at_index_i( ppath.gp_p[imax], ip )  )
    {  
      Numeric tilt = plevel_angletilt( r_start, c2 );
 
      if( is_los_downwards( za_start, tilt ) )
        {
          ip--;
          r1b = r1a;   r3b = r3a;
          r1a = re1 + z_field(ip,ilat);
          r3a = re3 + z_field(ip,ilat+1);
          plevel_slope_2d( c2, lat1, lat3, r1a, r3a );
        }
    }
  else if( is_gridpos_at_index_i( ppath.gp_p[imax], ip+1 )  )
    {
      Numeric tilt = plevel_angletilt( r_start, c4 );
 
      if( !is_los_downwards( za_start, tilt ) )
        {
          ip++;
          r1a = r1b;   r3a = r3b;
          r3b = re3 + z_field(ip+1,ilat+1);
          r1b = re1 + z_field(ip+1,ilat);    
          plevel_slope_2d( c4, lat1, lat3, r1b, r3b );
        }
    }
 
  // Surface radius at latitude end points
  rsurface1 = re1 + z_surface[ilat];
  rsurface3 = re3 + z_surface[ilat+1];
}
 
 
 
 
void ppath_end_2d(
              Ppath&      ppath,
        ConstVectorView   r_v,
        ConstVectorView   lat_v,
        ConstVectorView   za_v,
        ConstVectorView   lstep,
        ConstVectorView   n_v,
        ConstVectorView   ng_v,
        ConstVectorView   lat_grid,
        ConstMatrixView   z_field,
        ConstVectorView   refellipsoid,
        const Index&      ip,
        const Index&      ilat,
        const Index&      endface,
        const Numeric&    ppc )
{
  // Number of path points
  const Index   np   = r_v.nelem();
  const Index   imax = np-1;
 
  // Re-allocate ppath for return results and fill the structure
  //
  ppath_init_structure( ppath, 2, np );
 
  ppath.constant = ppc;
 
  // Help variables that are common for all points.
  const Numeric   dlat  = lat_grid[ilat+1] - lat_grid[ilat];
  const Numeric   z1low = z_field(ip,ilat);
  const Numeric   z1upp = z_field(ip+1,ilat);
  const Numeric   dzlow = z_field(ip,ilat+1) -z1low;
  const Numeric   dzupp = z_field(ip+1,ilat+1) - z1upp;
        Numeric   re    = refell2r( refellipsoid, lat_grid[ilat] );
  const Numeric   r1low = re + z1low;
  const Numeric   r1upp = re + z1upp;
                  re    = refell2r( refellipsoid, lat_grid[ilat+1] );
  const Numeric   drlow = re + z_field(ip,ilat+1) - r1low;
  const Numeric   drupp = re + z_field(ip+1,ilat+1) - r1upp;
 
  for( Index i=0; i<np; i++ )
    {
      ppath.r[i]      = r_v[i];
      ppath.pos(i,1)  = lat_v[i];
      ppath.los(i,0)  = za_v[i];
      ppath.nreal[i]  = n_v[i];
      ppath.ngroup[i] = ng_v[i];
      
      // Weight in the latitude direction
      Numeric w = ( lat_v[i] - lat_grid[ilat] ) / dlat;
 
      // Radius of lower and upper face at present latitude
      const Numeric rlow = r1low + w * drlow;
      const Numeric rupp = r1upp + w * drupp;
 
      // Geometrical altitude of lower and upper face at present latitude
      const Numeric zlow = z1low + w * dzlow;
      const Numeric zupp = z1upp + w * dzupp;
 
      ppath.gp_p[i].idx   = ip;
      ppath.gp_p[i].fd[0] = ( r_v[i] - rlow ) / ( rupp - rlow );
      ppath.gp_p[i].fd[1] = 1 - ppath.gp_p[i].fd[0];
      gridpos_check_fd( ppath.gp_p[i] );
 
      ppath.pos(i,0) = zlow + ppath.gp_p[i].fd[0] * ( zupp -zlow );
 
      ppath.gp_lat[i].idx   = ilat;
      ppath.gp_lat[i].fd[0] = ( lat_v[i] - lat_grid[ilat] ) / dlat;
      ppath.gp_lat[i].fd[1] = 1 - ppath.gp_lat[i].fd[0];
      gridpos_check_fd( ppath.gp_lat[i] );
 
      if( i > 0 )
        { ppath.lstep[i-1] = lstep[i-1]; }
    }
  gridpos_check_fd( ppath.gp_p[imax] );
  gridpos_check_fd( ppath.gp_lat[imax] );
 
  // Do end-face specific tasks
  if( endface == 7 )
    { ppath_set_background( ppath, 2 ); }
 
  // Set fractional distance for end point
  //
  if( endface == 1  ||  endface == 3 )
    { gridpos_force_end_fd( ppath.gp_lat[np-1], lat_grid.nelem() ); }
  else if( endface == 2  ||  endface == 4 )
    { gridpos_force_end_fd( ppath.gp_p[np-1], z_field.nrows() ); }
 
  // Handle cases where exactly a corner is hit, or when slipping outside of
  // the grid box due to numerical inaccuarcy
  if( ppath.gp_p[imax].fd[0] < 0  ||  ppath.gp_p[imax].fd[1] < 0 )
    { 
      gridpos_force_end_fd( ppath.gp_p[imax], z_field.nrows() ); 
    }
  if( ppath.gp_lat[imax].fd[0] < 0  ||  ppath.gp_lat[imax].fd[1] < 0 )
    { 
      gridpos_force_end_fd( ppath.gp_lat[imax], lat_grid.nelem() ); 
    }
}
 
 
 
 
void ppath_start_3d(
              Numeric&    r_start,
              Numeric&    lat_start,
              Numeric&    lon_start,
              Numeric&    za_start,
              Numeric&    aa_start,
              Index&      ip,
              Index&      ilat,
              Index&      ilon,
              Numeric&    lat1,
              Numeric&    lat3,
              Numeric&    lon5,
              Numeric&    lon6,
              Numeric&    r15a,
              Numeric&    r35a,
              Numeric&    r36a,
              Numeric&    r16a,
              Numeric&    r15b,
              Numeric&    r35b,
              Numeric&    r36b,
              Numeric&    r16b,
              Numeric&    rsurface15,
              Numeric&    rsurface35,
              Numeric&    rsurface36,
              Numeric&    rsurface16,
              Ppath&      ppath,
        ConstVectorView   lat_grid,
        ConstVectorView   lon_grid,
        ConstTensor3View  z_field,
        ConstVectorView   refellipsoid,
        ConstMatrixView   z_surface )
{
  // Index of last point in the incoming ppath
  const Index imax = ppath.np - 1;
 
  // Extract starting radius, zenith angle and latitude
  r_start   = ppath.r[imax];
  lat_start = ppath.pos(imax,1);
  lon_start = ppath.pos(imax,2);
  za_start  = ppath.los(imax,0);
  aa_start  = ppath.los(imax,1);
 
  // Number of lat/lon
  const Index nlat = lat_grid.nelem();
  const Index nlon = lon_grid.nelem();
 
  // Lower index of lat and lon ranges of interest
  //
  // The longitude is undefined at the poles and as the azimuth angle
  // is defined in other way at the poles.
  //
  if( lat_start == 90 )
    { 
      ilat = nlat - 2;
      GridPos   gp_tmp;
      gridpos( gp_tmp, lon_grid, aa_start );
      if( aa_start < 180 )
        { ilon = gridpos2gridrange( gp_tmp, 1 ); }
      else
        { ilon = gridpos2gridrange( gp_tmp, 0 ); }
    }
  else if( lat_start == -90 )
    { 
      ilat = 0; 
      GridPos   gp_tmp;
      gridpos( gp_tmp, lon_grid, aa_start );
      if( aa_start < 180 )
        { ilon = gridpos2gridrange( gp_tmp, 1 ); }
      else
        { ilon = gridpos2gridrange( gp_tmp, 0 ); }
    }
  else
    { 
      if( lat_start > 0 )
        { ilat = gridpos2gridrange( ppath.gp_lat[imax], abs( aa_start )<90 ); }
      else
        { ilat = gridpos2gridrange( ppath.gp_lat[imax], abs( aa_start )<=90 );}
      if( lon_start < lon_grid[nlon-1] )
        { ilon = gridpos2gridrange( ppath.gp_lon[imax], aa_start >= 0 ); }
      else
        { ilon = nlon - 2; }
    }
  //
  lat1 = lat_grid[ilat];
  lat3 = lat_grid[ilat+1];
  lon5 = lon_grid[ilon];
  lon6 = lon_grid[ilon+1];
 
  // Determine interesting pressure grid range. Do this first assuming that
  // the pressure levels are not tilted (that is, abs(za_start<=90) always
  // mean upward observation). 
  // Set radius for the corners of the grid cell and the radial slope of 
  // pressure level limits of the grid cell to match the found ip.
  //
  ip = gridpos2gridrange( ppath.gp_p[imax], za_start <= 90 );
  //
  const Numeric re1 = refell2r( refellipsoid, lat_grid[ilat] );
  const Numeric re3 = refell2r( refellipsoid, lat_grid[ilat+1] );
  //
  r15a = re1 + z_field(ip,ilat,ilon);
  r35a = re3 + z_field(ip,ilat+1,ilon); 
  r36a = re3 + z_field(ip,ilat+1,ilon+1); 
  r16a = re1 + z_field(ip,ilat,ilon+1);
  r15b = re1 + z_field(ip+1,ilat,ilon);
  r35b = re3 + z_field(ip+1,ilat+1,ilon); 
  r36b = re3 + z_field(ip+1,ilat+1,ilon+1); 
  r16b = re1 + z_field(ip+1,ilat,ilon+1);
 
  // Check if the LOS zenith angle happen to be between 90 and the zenith angle
  // of the pressure level (that is, 90 + tilt of pressure level), and in
  // that case if ip must be changed. This check is only needed when the
  // start point is on a pressure level.
  //
  if( fabs(za_start-90) <= 10 )  // To save time. Ie. max tilt assumed =10 deg 
    {
      if( is_gridpos_at_index_i( ppath.gp_p[imax], ip )  )
        {
          // Slope and angular tilt of lower pressure level
          Numeric c2a, c2b;
          plevel_slope_3d( c2a, c2b, lat1, lat3, lon5, lon6, 
                       r15a, r35a, r36a, r16a, lat_start, lon_start, aa_start );
          Numeric tilt = plevel_angletilt( r_start, c2a );
          // Negelect very small tilts, likely caused by numerical problems
          if( abs(tilt) > 1e-4  &&  is_los_downwards( za_start, tilt ) )
            {
              ip--;
              r15b = r15a;   r35b = r35a;   r36b = r36a;   r16b = r16a;
              r15a = re1 + z_field(ip,ilat,ilon);
              r35a = re3 + z_field(ip,ilat+1,ilon); 
              r36a = re3 + z_field(ip,ilat+1,ilon+1); 
              r16a = re1 + z_field(ip,ilat,ilon+1);
            }
        }
      else if( is_gridpos_at_index_i( ppath.gp_p[imax], ip+1 )  )
        {
          // Slope and angular tilt of upper pressure level
          Numeric c4a, c4b;
          plevel_slope_3d( c4a, c4b, lat1, lat3 ,lon5, lon6, 
                       r15b, r35b, r36b, r16b, lat_start, lon_start, aa_start );
          Numeric tilt = plevel_angletilt( r_start, c4a );
          //
          if( !is_los_downwards( za_start, tilt ) )
            {
              ip++;
              r15a = r15b;   r35a = r35b;   r36a = r36b;   r16a = r16b;
              r15b = re1 + z_field(ip+1,ilat,ilon);
              r35b = re3 + z_field(ip+1,ilat+1,ilon); 
              r36b = re3 + z_field(ip+1,ilat+1,ilon+1); 
              r16b = re1 + z_field(ip+1,ilat,ilon+1);
            }
        }
    }
 
  // Surface radius at latitude/longitude corner points
  rsurface15 = re1 + z_surface(ilat,ilon);
  rsurface35 = re3 + z_surface(ilat+1,ilon);
  rsurface36 = re3 + z_surface(ilat+1,ilon+1);
  rsurface16 = re1 + z_surface(ilat,ilon+1);
}
 
 
 
 
void ppath_end_3d(
              Ppath&      ppath,
        ConstVectorView   r_v,
        ConstVectorView   lat_v,
        ConstVectorView   lon_v,
        ConstVectorView   za_v,
        ConstVectorView   aa_v,
        ConstVectorView   lstep,
        ConstVectorView   n_v,
        ConstVectorView   ng_v,
        ConstVectorView   lat_grid,
        ConstVectorView   lon_grid,
        ConstTensor3View  z_field,
        ConstVectorView   refellipsoid,
        const Index&      ip,
        const Index&      ilat,
        const Index&      ilon,
        const Index&      endface,
        const Numeric&    ppc )
{
  // Number of path points
  const Index   np   = r_v.nelem();
  const Index   imax = np-1;
 
  // Re-allocate ppath for return results and fill the structure
  //
  ppath_init_structure( ppath, 3, np );
  //
  ppath.constant = ppc;
 
  // Help variables that are common for all points.
  const Numeric   lat1  = lat_grid[ilat];
  const Numeric   lat3  = lat_grid[ilat+1];
  const Numeric   lon5  = lon_grid[ilon];
  const Numeric   lon6  = lon_grid[ilon+1];
  const Numeric   re1   = refell2r( refellipsoid, lat_grid[ilat] );
  const Numeric   re3   = refell2r( refellipsoid, lat_grid[ilat+1] );
  const Numeric   r15a  = re1 + z_field(ip,ilat,ilon);
  const Numeric   r35a  = re3 + z_field(ip,ilat+1,ilon); 
  const Numeric   r36a  = re3 + z_field(ip,ilat+1,ilon+1); 
  const Numeric   r16a  = re1 + z_field(ip,ilat,ilon+1);
  const Numeric   r15b  = re1 + z_field(ip+1,ilat,ilon);
  const Numeric   r35b  = re3 + z_field(ip+1,ilat+1,ilon); 
  const Numeric   r36b  = re3 + z_field(ip+1,ilat+1,ilon+1);
  const Numeric   r16b  = re1 + z_field(ip+1,ilat,ilon+1);
  const Numeric   dlat  = lat3 - lat1;
  const Numeric   dlon  = lon6 - lon5;
 
  for( Index i=0; i<np; i++ )
    {
      // Radius of pressure levels at present lat and lon
      const Numeric   rlow = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                  r15a, r35a, r36a, r16a, lat_v[i], lon_v[i] );
      const Numeric   rupp = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                  r15b, r35b, r36b, r16b, lat_v[i], lon_v[i] );
 
      // Position and LOS
      ppath.r[i]      = r_v[i];
      ppath.pos(i,1)  = lat_v[i];
      ppath.pos(i,2)  = lon_v[i];
      ppath.los(i,0)  = za_v[i];
      ppath.los(i,1)  = aa_v[i];
      ppath.nreal[i]  = n_v[i];
      ppath.ngroup[i] = ng_v[i];
      
      // Pressure grid index
      ppath.gp_p[i].idx   = ip;
      ppath.gp_p[i].fd[0] = ( r_v[i] - rlow ) / ( rupp - rlow );
      ppath.gp_p[i].fd[1] = 1 - ppath.gp_p[i].fd[0];
      gridpos_check_fd( ppath.gp_p[i] );
 
      // Geometrical altitude
      const Numeric   re = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                       re1, re3, re3, re1, lat_v[i], lon_v[i] );
      const Numeric   zlow = rlow - re;
      const Numeric   zupp = rupp - re;
      //
      ppath.pos(i,0) = zlow + ppath.gp_p[i].fd[0] * ( zupp -zlow );
 
      // Latitude grid index
      ppath.gp_lat[i].idx   = ilat;
      ppath.gp_lat[i].fd[0] = ( lat_v[i] - lat1 ) / dlat;
      ppath.gp_lat[i].fd[1] = 1 - ppath.gp_lat[i].fd[0];
      gridpos_check_fd( ppath.gp_lat[i] );
 
      // Longitude grid index
      //
      // The longitude  is undefined at the poles. The grid index is set to
      // the start point.
      //
      if( abs( lat_v[i] ) < POLELAT )
        {
          ppath.gp_lon[i].idx   = ilon;
          ppath.gp_lon[i].fd[0] = ( lon_v[i] - lon5 ) / dlon;
          ppath.gp_lon[i].fd[1] = 1 - ppath.gp_lon[i].fd[0];
          gridpos_check_fd( ppath.gp_lon[i] );
        }
      else
        {
          ppath.gp_lon[i].idx   = 0;
          ppath.gp_lon[i].fd[0] = 0;
          ppath.gp_lon[i].fd[1] = 1;
        }
 
      if( i > 0 )
        { ppath.lstep[i-1] = lstep[i-1]; }
    }
 
  // Do end-face specific tasks
  if( endface == 7 )
    { ppath_set_background( ppath, 2 ); }
 
  // Set fractional distance for end point
  //
  if( endface == 1  ||  endface == 3 )
    { gridpos_force_end_fd( ppath.gp_lat[imax], lat_grid.nelem() ); }
  else if( endface == 2  ||  endface == 4 )
    { gridpos_force_end_fd( ppath.gp_p[imax], z_field.npages() ); }
  else if( endface == 5  ||  endface == 6 )
    { gridpos_force_end_fd( ppath.gp_lon[imax], lon_grid.nelem() ); }
 
  // Handle cases where exactly a corner is hit, or when slipping outside of
  // the grid box due to numerical inaccuarcy
  if( ppath.gp_p[imax].fd[0] < 0  ||  ppath.gp_p[imax].fd[1] < 0 )
    { 
      gridpos_force_end_fd( ppath.gp_p[imax], z_field.npages() ); 
    }
  if( ppath.gp_lat[imax].fd[0] < 0  ||  ppath.gp_lat[imax].fd[1] < 0 )
    { 
      gridpos_force_end_fd( ppath.gp_lat[imax], lat_grid.nelem() ); 
    }
  if( ppath.gp_lon[imax].fd[0] < 0  ||  ppath.gp_lon[imax].fd[1] < 0 )
    { 
      gridpos_force_end_fd( ppath.gp_lon[imax], lon_grid.nelem() ); 
    }
}
 
 
 
 
 
 
/*===========================================================================
  === Core functions for geometrical ppath_step calculations
  ===========================================================================*/
 
 
void do_gridrange_1d(
              Vector&   r_v,
              Vector&   lat_v,
              Vector&   za_v,
              Numeric&  lstep,
              Index&    endface,
        const Numeric&  r_start0,
        const Numeric&  lat_start,
        const Numeric&  za_start,
        const Numeric&  ppc,
        const Numeric&  lmax,
        const Numeric&  ra,
        const Numeric&  rb,
        const Numeric&  rsurface )
{
  Numeric   r_start = r_start0;
 
  assert( rb > ra );
  assert( r_start >= ra - RTOL );
  assert( r_start <= rb + RTOL );
 
  // Shift radius if outside
  if( r_start < ra )
    { r_start = ra; }
  else if( r_start > rb )
    { r_start = rb; }
 
  Numeric r_end;
  bool tanpoint = false;
  endface = -1;
 
  // If upward, end point radius is always rb
  if( za_start <= 90 )
    { endface = 4;   r_end = rb; }
 
  else
    {
      // Path reaches ra:
      if( ra > rsurface  &&  ra > ppc ) 
        { endface = 2;   r_end = ra; }
 
      // Path reaches the surface:
      else if( rsurface > ppc )
        { endface = 7;   r_end = rsurface; }
      
      // The remaining option is a tangent point and back to rb
      else
        { endface = 4;   r_end = rb;   tanpoint = true; }
    }
 
  assert( endface > 0 );
  
  geompath_from_r1_to_r2( r_v, lat_v, za_v, lstep, ppc, r_start, lat_start,
                          za_start, r_end, tanpoint, lmax );
}
 
 
 
 
void ppath_step_geom_1d(
              Ppath&      ppath,
        ConstVectorView   z_field,
        ConstVectorView   refellipsoid,
        const Numeric&    z_surface,
        const Numeric&    lmax )
{
  // Starting radius, zenith angle and latitude
  Numeric r_start, lat_start, za_start;
 
  // Index of the pressure level being the lower limit for the
  // grid range of interest.
  Index ip;
 
  // Determine the variables defined above, and make asserts of input
  ppath_start_1d( r_start, lat_start, za_start, ip, ppath );
 
  // If the field "constant" is negative, this is the first call of the
  // function and the path constant shall be calculated.
  Numeric ppc;
  if( ppath.constant < 0 )
    { ppc = geometrical_ppc( r_start, za_start ); }
  else
    { ppc = ppath.constant; }
 
 
  // The path is determined by another function. Determine some variables
  // needed bý that function and call the function.
  //
  // Vars to hold found path points, path step length and coding for end face
  Vector    r_v, lat_v, za_v;
  Numeric   lstep;
  Index     endface;
  //
  do_gridrange_1d( r_v, lat_v, za_v, lstep, endface,
                   r_start, lat_start, za_start, ppc, lmax, 
                   refellipsoid[0]+z_field[ip], refellipsoid[0]+z_field[ip+1], 
                   refellipsoid[0]+z_surface );
 
  // Fill *ppath*
  const Index np = r_v.nelem();
  ppath_end_1d( ppath, r_v, lat_v, za_v, Vector(np-1,lstep), Vector(np,1),
                Vector(np,1), z_field, refellipsoid, ip, endface, ppc );
}
 
 
 
 
void do_gridcell_2d_byltest(
              Vector&   r_v,
              Vector&   lat_v,
              Vector&   za_v,
              Numeric&  lstep,
              Index&    endface,
        const Numeric&  r_start0, 
        const Numeric&  lat_start0,
        const Numeric&  za_start,
        const Numeric&  l_start,
        const Index&    icall,
        const Numeric&  ppc,
        const Numeric&  lmax,
        const Numeric&  lat1,
        const Numeric&  lat3,
        const Numeric&  r1a,
        const Numeric&  r3a,
        const Numeric&  r3b,
        const Numeric&  r1b,
        const Numeric&  rsurface1,
        const Numeric&  rsurface3 )
{
  Numeric   r_start   = r_start0;
  Numeric   lat_start = lat_start0;
 
  assert( icall < 4 );
 
  // Assert latitude and longitude
  assert( lat_start >= lat1 - LATLONTOL );
  assert( lat_start <= lat3 + LATLONTOL );
 
  // Shift latitude and longitude if outside
  if( lat_start < lat1 )
    { lat_start = lat1; }
  else if( lat_start > lat3 )
    { lat_start = lat3; }
 
  // Radius of lower and upper pressure level at the start position
  Numeric   rlow = rsurf_at_lat( lat1, lat3, r1a, r3a, lat_start );
  Numeric   rupp = rsurf_at_lat( lat1, lat3, r1b, r3b, lat_start );
 
  // Assert radius (some extra tolerance is needed for radius)
  assert( r_start >= rlow - RTOL );
  assert( r_start <= rupp + RTOL );
 
  // Shift radius if outside
  if( r_start < rlow )
    { r_start = rlow; }
  else if( r_start > rupp )
    { r_start = rupp; }
 
  // Position and LOS in cartesian coordinates
  Numeric   x, z, dx, dz;
  poslos2cart( x, z, dx, dz, r_start, lat_start, za_start );
 
  // Some booleans to check for recursive call
  bool unsafe     = false;
  bool do_surface = false;
 
  // Determine the position of the end point
  //
  endface  = 0;
  //
  Numeric   r_end, lat_end, l_end;
 
  // Zenith and nadir looking are handled as special cases
  const Numeric absza = abs( za_start );
 
  // Zenith looking
  if( absza < ANGTOL )
    {
      l_end   = rupp - r_start;
      endface  = 4;
    }
 
  // Nadir looking
  else if( absza > 180-ANGTOL )
    {
      const Numeric rsurface = rsurf_at_lat( lat1, lat3, rsurface1, rsurface3, 
                                                                  lat_start );
      
      if( rlow > rsurface )
        {
          l_end   = r_start - rlow;
          endface = 2;
        }
      else
        {
          l_end   = r_start - rsurface;
          endface = 7;
        }
    }
 
  else
    {
      unsafe = true;
 
      // Calculate correction terms for the position to compensate for
      // numerical inaccuracy. 
      //
      Numeric   r_corr, lat_corr;
      //
      cart2pol( r_corr, lat_corr, x, z, lat_start, za_start );
      //
      r_corr   -= r_start;
      lat_corr -= lat_start;
 
      // The end point is found by testing different step lengths until the
      // step length has been determined by a precision of *l_acc*.
      //
      // The first step is to found a length that goes outside the grid cell, 
      // to find an upper limit. The lower limit is set to 0. The upper
      // limit is either set to l_start or it is estimated from the size of
      // the grid cell.
      // The search algorith is bisection, the next length to test is the
      // mean value of the minimum and maximum length limits.
      //
      if( l_start > 0 )
        { l_end = l_start; }
      else
        { l_end = 2 * ( rupp - rlow ); }
      //
      Numeric   l_in   = 0, l_out = l_end;
      bool      ready  = false, startup = true;
 
      if( rsurface1+RTOL >= r1a  ||  rsurface3+RTOL >= r3a )
        { do_surface = true; }
 
      while( !ready )
        {
          cart2pol( r_end, lat_end, x+dx*l_end, z+dz*l_end, lat_start, 
                                                                   za_start );
          r_end   -= r_corr;
          lat_end -= lat_corr;
 
          bool   inside = true;
 
          rlow = rsurf_at_lat( lat1, lat3, r1a, r3a, lat_end );
          
          if( do_surface )
            {
              const Numeric r_surface = rsurf_at_lat( lat1, lat3, rsurface1, 
                                                         rsurface3, lat_end );
              if( r_surface >= rlow  &&  r_end <= r_surface )
                { inside = false;   endface = 7; }
            }
 
          if( inside ) 
            {
              if( lat_end < lat1 )
                { inside = false;   endface = 1; }
              else if( lat_end > lat3 )
                { inside = false;   endface = 3; }
              else if( r_end < rlow )
                { inside = false;   endface = 2; }
              else
                {
                  rupp = rsurf_at_lat( lat1, lat3, r1b, r3b, lat_end );
                  if( r_end > rupp )
                    { inside = false;   endface = 4; }
                }
            }              
 
          if( startup )
            {
              if( inside )
                { 
                  l_in   = l_end;
                  l_end *= 5; 
                }
              else
                { 
                  l_out = l_end;   
                  l_end = ( l_out + l_in ) / 2;
                  startup = false; 
                }
            }
          else
            {
              if( inside )
                { l_in = l_end; }
              else
                { l_out = l_end; }
                            
              if( ( l_out - l_in ) < LACC )
                { ready = true; }
              else
                { l_end = ( l_out + l_in ) / 2; }
            }
        }
    }
 
  //--- Create return vectors
  //
  Index n = 1;
  //
  if( lmax > 0 )
    {
      n = Index( ceil( abs( l_end / lmax ) ) );
      if( n < 1 )
        { n = 1; }
    }
  //
  r_v.resize( n+1 );
  lat_v.resize( n+1 );
  za_v.resize( n+1 );
  //
  r_v[0]   = r_start;
  lat_v[0] = lat_start;
  za_v[0]  = za_start;
  //
  lstep = l_end / (Numeric)n;
  Numeric l;
  bool ready = true; 
  // 
  for( Index j=1; j<=n; j++ )
    {
      l = lstep * (Numeric)j;
      cart2poslos( r_v[j], lat_v[j], za_v[j], x+dx*l, z+dz*l, dx, dz, ppc, 
                                                        lat_start, za_start );
      
      if( j < n )
        {
          if( unsafe ) 
            {          
              // Check that r_v[j] is above lower pressure level and the
              // surface. This can fail around tangent points. For p-levels
              // with constant r this is easy to handle analytically, but the
              // problem is tricky in the general case with a non-spherical
              // geometry, and this crude solution is used instead. Not the
              // most elegant solution, but it works! Added later the same
              // check for upper level, after getting assert in that direction.
              // The z_field was crazy, but still formerly correct.
              rlow = rsurf_at_lat( lat1, lat3, r1a, r3a, lat_v[j] );
              if( do_surface )
                {
                  const Numeric r_surface = rsurf_at_lat( lat1, lat3, 
                                             rsurface1, rsurface3, lat_v[j] ); 
                  const Numeric r_test = max( r_surface, rlow );
                  if(  r_v[j] < r_test )
                    {  ready = false;   break; }
                } 
              else  if( r_v[j] < rlow )
                { ready = false;   break; }
 
              rupp = rsurf_at_lat( lat1, lat3, r1b, r3b, lat_v[j] );
              if( r_v[j] > rupp )
                { ready = false;   break; }
            }
        }
      else // j==n
        {
          if( unsafe )
            {
              // Set end point to be consistent with found endface.
              //
              if( endface == 1 )
                { lat_v[n] = lat1; }
              else if( endface == 2 )
                { r_v[n] = rsurf_at_lat( lat1, lat3, r1a, r3a, lat_v[n] ); }
              else if( endface == 3 )
                { lat_v[n] = lat3; }
              else if( endface == 4 )
                { r_v[n] = rsurf_at_lat( lat1, lat3, r1b, r3b, lat_v[n] ); }
              else if( endface == 7 )
                { r_v[n] = rsurf_at_lat( lat1, lat3, rsurface1, rsurface3, 
                                                                  lat_v[n] ); }
            }
        }
    }
 
  if( !ready )
    { // If an "outside" point found, restart with l as start search length
      do_gridcell_2d_byltest( r_v, lat_v, za_v, lstep, endface, r_start, 
                              lat_start, za_start, l, icall+1, ppc, lmax, 
                              lat1, lat3, r1a, r3a, r3b, r1b,
                              rsurface1, rsurface3 );
    }
}
 
 
 
 
void ppath_step_geom_2d(
              Ppath&      ppath,
        ConstVectorView   lat_grid,
        ConstMatrixView   z_field,
        ConstVectorView   refellipsoid,
        ConstVectorView   z_surface,
        const Numeric&    lmax )
{
  // Radius, zenith angle and latitude of start point.
  Numeric   r_start, lat_start, za_start;
 
  // Lower grid index for the grid cell of interest.
  Index   ip, ilat;
 
  // Radii and latitudes set by *ppath_start_2d*.
  Numeric   lat1, lat3, r1a, r3a, r3b, r1b, rsurface1, rsurface3;
 
  // Determine the variables defined above and make all possible asserts
  ppath_start_2d( r_start, lat_start, za_start, ip, ilat, 
                  lat1, lat3, r1a, r3a, r3b, r1b, rsurface1, rsurface3,
                  ppath, lat_grid, z_field, refellipsoid, z_surface );
 
  // If the field "constant" is negative, this is the first call of the
  // function and the path constant shall be calculated.
  Numeric ppc;
  if( ppath.constant < 0 )
    { ppc = geometrical_ppc( r_start, za_start ); }
  else
    { ppc = ppath.constant; }
 
  // Vars to hold found path points, path step length and coding for end face
  Vector   r_v, lat_v, za_v;
  Numeric   lstep;
  Index    endface;
 
  do_gridcell_2d_byltest( r_v, lat_v, za_v, lstep, endface, r_start, 
                              lat_start, za_start, -1, 0, ppc, lmax, 
                              lat1, lat3, r1a, r3a, r3b, r1b,
                              rsurface1, rsurface3 );
  // Fill *ppath*
  const Index np = r_v.nelem();
  ppath_end_2d( ppath, r_v, lat_v, za_v, Vector(np-1,lstep), Vector(np,1), 
                Vector(np,1), lat_grid, z_field, refellipsoid, ip, ilat, 
                endface, ppc );
}
 
 
 
 
void do_gridcell_3d_byltest(
              Vector&   r_v,
              Vector&   lat_v,
              Vector&   lon_v,
              Vector&   za_v,
              Vector&   aa_v,
              Numeric&  lstep,
              Index&    endface,
        const Numeric&  r_start0, 
        const Numeric&  lat_start0,
        const Numeric&  lon_start0,
        const Numeric&  za_start,
        const Numeric&  aa_start,
        const Numeric&  l_start,
        const Index&    icall,
        const Numeric&  ppc,
        const Numeric&  lmax,
        const Numeric&  lat1,
        const Numeric&  lat3,
        const Numeric&  lon5,
        const Numeric&  lon6,
        const Numeric&  r15a,
        const Numeric&  r35a,
        const Numeric&  r36a,
        const Numeric&  r16a,
        const Numeric&  r15b,
        const Numeric&  r35b,
        const Numeric&  r36b,
        const Numeric&  r16b,
        const Numeric&  rsurface15,
        const Numeric&  rsurface35,
        const Numeric&  rsurface36,
        const Numeric&  rsurface16 )
{
  Numeric   r_start   = r_start0;
  Numeric   lat_start = lat_start0;
  Numeric   lon_start = lon_start0;
 
  assert( icall < 4 );
 
  // Assert latitude and longitude
  assert( lat_start >= lat1 - LATLONTOL );
  assert( lat_start <= lat3 + LATLONTOL );
  assert( !( abs( lat_start) < POLELAT  &&  lon_start < lon5 - LATLONTOL ) );
  assert( !( abs( lat_start) < POLELAT  &&  lon_start > lon6 + LATLONTOL ) );
 
  // Shift latitude and longitude if outside
  if( lat_start < lat1 )
    { lat_start = lat1; }
  else if( lat_start > lat3 )
    { lat_start = lat3; }
  if( lon_start < lon5 )
    { lon_start = lon5; }
  else if( lon_start > lon6 )
    { lon_start = lon6; }
 
  // Radius of lower and upper pressure level at the start position
  Numeric   rlow = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                r15a, r35a, r36a, r16a, lat_start, lon_start );
  Numeric   rupp = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                r15b, r35b, r36b, r16b, lat_start, lon_start );
 
  // Assert radius (some extra tolerance is needed for radius)
  assert( r_start >= rlow - RTOL );
  assert( r_start <= rupp + RTOL );
 
  // Shift radius if outside
  if( r_start < rlow )
    { r_start = rlow; }
  else if( r_start > rupp )
    { r_start = rupp; }
 
  // Position and LOS in cartesian coordinates
  Numeric   x, y, z, dx, dy, dz;
  poslos2cart( x, y, z, dx, dy, dz, r_start, lat_start, lon_start, 
                                                          za_start, aa_start );
 
  // Some booleans to check for recursive call
  bool unsafe     = false;
  bool do_surface = false;
 
  // Determine the position of the end point
  //
  endface  = 0;
  //
  Numeric   r_end, lat_end, lon_end, l_end;
 
  // Zenith and nadir looking are handled as special cases
 
  // Zenith looking
  if( za_start < ANGTOL )
    {
      l_end   = rupp - r_start;
      endface  = 4;
    }
 
  // Nadir looking
  else if( za_start > 180-ANGTOL )
    {
      const Numeric   rsurface = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                               rsurface15, rsurface35, rsurface36, rsurface16, 
                                                 lat_start, lon_start );
      
      if( rlow > rsurface )
        {
          l_end   = r_start - rlow;
          endface = 2;
        }
      else
        {
          l_end   = r_start - rsurface;
          endface = 7;
        }
    }
 
  else
    {
      unsafe = true;
 
      // Calculate correction terms for the position to compensate for
      // numerical inaccuracy. 
      //
      Numeric   r_corr, lat_corr, lon_corr;
      //
      cart2sph( r_corr, lat_corr, lon_corr, x, y, z, lat_start, lon_start,
                                                     za_start,  aa_start );
      //
      r_corr   -= r_start;
      lat_corr -= lat_start;
      lon_corr -= lon_start;
 
      // The end point is found by testing different step lengths until the
      // step length has been determined by a precision of *l_acc*.
      //
      // The first step is to found a length that goes outside the grid cell, 
      // to find an upper limit. The lower limit is set to 0. The upper
      // limit is either set to l_start or it is estimated from the size of
      // the grid cell.
      // The search algorith is bisection, the next length to test is the
      // mean value of the minimum and maximum length limits.
      //
      if( l_start > 0 )
        { l_end = l_start; }
      else
        { l_end = 2 * ( rupp - rlow ); }
      //
      Numeric   l_in   = 0, l_out = l_end;
      bool      ready  = false, startup = true;
 
      if( rsurface15+RTOL >= r15a  ||  rsurface35+RTOL >= r35a  ||  
          rsurface36+RTOL >= r36a  ||  rsurface16+RTOL >= r16a )
        { do_surface = true; }
 
      while( !ready )
        {
          cart2sph( r_end, lat_end, lon_end, x+dx*l_end, y+dy*l_end, 
                    z+dz*l_end, lat_start, lon_start, za_start,  aa_start );
          r_end   -= r_corr;
          lat_end -= lat_corr;
          lon_end -= lon_corr;
          resolve_lon( lon_end, lon5, lon6 );              
 
          // Special fix for north-south observations
          if( abs( lat_start ) < POLELAT  &&  abs( lat_end ) < POLELAT  &&
             ( abs(aa_start) < ANGTOL  ||  abs(aa_start) > 180-ANGTOL ) )
            { lon_end = lon_start; }
          
          bool   inside = true;
 
          rlow = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                r15a, r35a, r36a, r16a, lat_end, lon_end );
          
          if( do_surface )
            {
              const Numeric   r_surface = 
                           rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                           rsurface15, rsurface35, rsurface36, rsurface16, 
                                                        lat_end, lon_end );
              if( r_surface >= rlow  &&  r_end <= r_surface )
                { inside = false;   endface = 7; }
            }
 
          if( inside ) 
            {
              if( lat_end < lat1 )
                { inside = false;   endface = 1; }
              else if( lat_end > lat3 )
                { inside = false;   endface = 3; }
              else if( lon_end < lon5 )
                { inside = false;   endface = 5; }
              else if( lon_end > lon6 )
                { inside = false;   endface = 6; }
              else if( r_end < rlow )
                { inside = false;   endface = 2; }
              else
                {
                  rupp = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                r15b, r35b, r36b, r16b, lat_end, lon_end );
                  if( r_end > rupp )
                    { inside = false;   endface = 4; }
                }
            }              
 
          if( startup )
            {
              if( inside )
                { 
                  l_in   = l_end;
                  l_end *= 5; 
                }
              else
                { 
                  l_out = l_end;   
                  l_end = ( l_out + l_in ) / 2;
                  startup = false; 
                }
            }
          else
            {
              if( inside )
                { l_in = l_end; }
              else
                { l_out = l_end; }
                            
              if( ( l_out - l_in ) < LACC )
                { ready = true; }
              else
                { l_end = ( l_out + l_in ) / 2; }
            }
        }
    }
 
  //--- Create return vectors
  //
  Index n = 1;
  //
  if( lmax > 0 )
    {
      n = Index( ceil( abs( l_end / lmax ) ) );
      if( n < 1 )
        { n = 1; }
    }
  //
  r_v.resize( n+1 );
  lat_v.resize( n+1 );
  lon_v.resize( n+1 );
  za_v.resize( n+1 );
  aa_v.resize( n+1 );
  //
  r_v[0]   = r_start;
  lat_v[0] = lat_start;
  lon_v[0] = lon_start;
  za_v[0]  = za_start;
  aa_v[0]  = aa_start;
  //
  lstep = l_end / (Numeric)n;
  Numeric l;
  bool ready = true; 
  // 
  for( Index j=1; j<=n; j++ )
    {
      l = lstep * (Numeric)j;
      cart2poslos( r_v[j], lat_v[j], lon_v[j], za_v[j], aa_v[j], x+dx*l, 
                   y+dy*l, z+dz*l, dx, dy, dz, ppc, lat_start, lon_start,
                   za_start, aa_start );
 
      // Shall lon values be shifted (value 0 is already OK)?
      resolve_lon( lon_v[j], lon5, lon6 );
      
      if( j < n )
        {
          if( unsafe ) 
            {          
              // Check that r_v[j] is above lower pressure level and the
              // surface. This can fail around tangent points. For p-levels
              // with constant r this is easy to handle analytically, but the
              // problem is tricky in the general case with a non-spherical
              // geometry, and this crude solution is used instead. Not the
              // most elegant solution, but it works! Added later the same
              // check for upper level, after getting assert in that direction.
              // The z_field was crazy, but still formerly correct.
              rlow = rsurf_at_latlon( lat1, lat3, lon5, lon6, r15a, r35a, 
                                      r36a, r16a, lat_v[j], lon_v[j] );
              if( do_surface )
                {
                  const Numeric r_surface = rsurf_at_latlon( 
                               lat1, lat3, lon5, lon6, rsurface15, rsurface35,
                               rsurface36, rsurface16, lat_v[j], lon_v[j] ); 
                  const Numeric r_test = max( r_surface, rlow );
                  if(  r_v[j] < r_test )
                    {  ready = false;   break; }
                } 
              else  if( r_v[j] < rlow )
                { ready = false;   break; }
 
              rupp = rsurf_at_latlon( lat1, lat3, lon5, lon6, r15b, r35b, 
                                  r36b, r16b, lat_v[j], lon_v[j] );
              if( r_v[j] > rupp )
                { ready = false;   break; }
            }
        }
      else // j==n
        {
          if( unsafe )
            {
              // Set end point to be consistent with found endface.
              //
              if( endface == 1 )
                { lat_v[n] = lat1; }
              else if( endface == 2 )
                { r_v[n] = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                            r15a, r35a, r36a, r16a, 
                                            lat_v[n], lon_v[n] ); }
              else if( endface == 3 )
                { lat_v[n] = lat3; }
              else if( endface == 4 )
                { r_v[n] = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                            r15b, r35b, r36b, r16b, 
                                            lat_v[n], lon_v[n] ); }
              else if( endface == 5 )
                { lon_v[n] = lon5; }
              else if( endface == 6 )
                { lon_v[n] = lon6; }
              else if( endface == 7 )
                { r_v[n] = rsurf_at_latlon( lat1, lat3, lon5, lon6, 
                                            rsurface15, rsurface35, 
                                            rsurface36, rsurface16, 
                                            lat_v[n], lon_v[n] ); }
            }
        }
    }
 
  if( !ready )
    { // If an "outside" point found, restart with l as start search length
      do_gridcell_3d_byltest( r_v, lat_v, lon_v, za_v, aa_v, lstep, endface,
                              r_start, lat_start, lon_start, za_start, aa_start,
                              l, icall+1, ppc, lmax, lat1, lat3, lon5, lon6, 
                              r15a, r35a, r36a, r16a, r15b, r35b, r36b, r16b,
                              rsurface15, rsurface35, rsurface36, rsurface16 );
    }
}
 
 
 
 
void ppath_step_geom_3d(
              Ppath&       ppath,
        ConstVectorView    lat_grid,
        ConstVectorView    lon_grid,
        ConstTensor3View   z_field,
        ConstVectorView    refellipsoid,
        ConstMatrixView    z_surface,
        const Numeric&     lmax )
{
  // Radius, zenith angle and latitude of start point.
  Numeric   r_start, lat_start, lon_start, za_start, aa_start;
 
  // Lower grid index for the grid cell of interest.
  Index   ip, ilat, ilon;
 
  // Radius for corner points, latitude and longitude of the grid cell
  //
  Numeric   lat1, lat3, lon5, lon6;
  Numeric   r15a, r35a, r36a, r16a, r15b, r35b, r36b, r16b;
  Numeric   rsurface15, rsurface35, rsurface36, rsurface16;
 
  // Determine the variables defined above and make all possible asserts
  ppath_start_3d( r_start, lat_start, lon_start, za_start, aa_start, 
                  ip, ilat, ilon, lat1, lat3, lon5, lon6,
                  r15a, r35a, r36a, r16a, r15b, r35b, r36b, r16b, 
                  rsurface15, rsurface35, rsurface36, rsurface16,
                  ppath, lat_grid, lon_grid, z_field, refellipsoid, z_surface );
 
 
  // If the field "constant" is negative, this is the first call of the
  // function and the path constant shall be calculated.
  Numeric ppc;
  if( ppath.constant < 0 )
    { ppc = geometrical_ppc( r_start, za_start ); }
  else
    { ppc = ppath.constant; }
 
 
  // Vars to hold found path points, path step length and coding for end face
  Vector   r_v, lat_v, lon_v, za_v, aa_v;
  Numeric  lstep;
  Index    endface;
  
  do_gridcell_3d_byltest( r_v, lat_v, lon_v, za_v, aa_v, lstep, endface,
                          r_start, lat_start, lon_start, za_start, aa_start, 
                          -1, 0, ppc, lmax, lat1, lat3, lon5, lon6, 
                          r15a, r35a, r36a, r16a, r15b, r35b, r36b, r16b,
                          rsurface15, rsurface35, rsurface36, rsurface16 );
  
  // Fill *ppath*
  const Index np = r_v.nelem();
  ppath_end_3d( ppath, r_v, lat_v, lon_v, za_v, aa_v, Vector(np-1,lstep), 
                Vector(np,1), Vector(np,1), lat_grid, lon_grid, z_field, 
                refellipsoid, ip, ilat, ilon, endface, ppc );
}
 
 
 
 
 
/*===========================================================================
  === Core functions for refraction *ppath_step* functions
  ===========================================================================*/
 
 
void raytrace_1d_linear_basic(
              Workspace&       ws,
              Array<Numeric>&  r_array,
              Array<Numeric>&  lat_array,
              Array<Numeric>&  za_array,
              Array<Numeric>&  l_array,
              Array<Numeric>&  n_array,
              Array<Numeric>&  ng_array,
              Index&           endface,
        ConstVectorView        p_grid,
        ConstVectorView        refellipsoid,
        ConstTensor3View       z_field,
        ConstTensor3View       t_field,
        ConstTensor4View       vmr_field,
        ConstVectorView        f_grid,
        const Numeric&         lmax,
        const Agenda&          refr_index_air_agenda,
        const Numeric&         lraytrace,
        const Numeric&         rsurface,
        const Numeric&         r1,
        const Numeric&         r3,
              Numeric          r,
              Numeric          lat,
              Numeric          za )
{
  // Loop boolean
  bool ready = false;
 
  // Store first point
  Numeric refr_index_air, refr_index_air_group;
  get_refr_index_1d( ws, refr_index_air, refr_index_air_group, 
                     refr_index_air_agenda, p_grid, refellipsoid, 
                     z_field, t_field, vmr_field, f_grid, r );
  r_array.push_back( r );
  lat_array.push_back( lat );
  za_array.push_back( za );
  n_array.push_back( refr_index_air );
  ng_array.push_back( refr_index_air_group );
 
  // Variables for output from do_gridcell_2d
  Vector    r_v, lat_v, za_v;
  Numeric   lstep, lcum = 0, dlat;
 
  while( !ready )
    {
      // Constant for the geometrical step to make
      const Numeric   ppc_step = geometrical_ppc( r, za );
 
      // Where will a geometric path exit the grid cell?
      do_gridrange_1d( r_v, lat_v, za_v, lstep, endface, r, lat, za,
                       ppc_step, -1, r1, r3, rsurface );
      assert( r_v.nelem() == 2 );
 
      // If *lstep* is <= *lraytrace*, extract the found end point (if not 
      // a tangent point, we are ready).
      // Otherwise, we make a geometrical step with length *lraytrace*.
 
      Numeric za_flagside = za;
 
      if( lstep <= lraytrace )
        {
          r     = r_v[1];
          dlat  = lat_v[1] - lat;
          lat   = lat_v[1];
          lcum += lstep;
          ready = true; 
        }
      else
        {
          Numeric l;
          if( za <= 90 ) 
            { l = geompath_l_at_r(ppc_step,r) + lraytrace; }
          else
            { 
              l = geompath_l_at_r(ppc_step,r) - lraytrace; 
              if( l < 0 )
                { za_flagside = 180-za_flagside; } // Tangent point passed!
            }
 
          r = geompath_r_at_l( ppc_step, l );
 
          const Numeric lat_new = geompath_lat_at_za( za, lat, 
                                 geompath_za_at_r( ppc_step, za_flagside, r) );
          dlat  = lat_new - lat;
          lat   = lat_new;
          lstep = lraytrace;
          lcum += lraytrace;
        }
 
      // Refractive index at new point
      Numeric   dndr;
      refr_gradients_1d( ws, refr_index_air, refr_index_air_group, dndr, 
                         refr_index_air_agenda, p_grid, refellipsoid, z_field, 
                         t_field, vmr_field, f_grid, r );
 
      // Calculate LOS zenith angle at found point.
      const Numeric   za_rad = DEG2RAD * za;
      //
      za += - dlat;   // Pure geometrical effect
      //
      za += (RAD2DEG * lstep / refr_index_air) * ( -sin(za_rad) * dndr );
 
      // Make sure that obtained *za* is inside valid range
      if( za < 0 )
        { za = -za; }
      else if( za > 180 )
        { za -= 360; }
 
      // Store found point?
      if( ready  ||  ( lmax > 0  &&  lcum + lraytrace > lmax ) )
        {
          r_array.push_back( r );
          lat_array.push_back( lat );
          za_array.push_back( za );
          n_array.push_back( refr_index_air );
          ng_array.push_back( refr_index_air_group );
          l_array.push_back( lcum );
          lcum = 0;
        }
    }  
}
 
 
 
void ppath_step_refr_1d(
              Workspace&  ws,
              Ppath&      ppath,
        ConstVectorView   p_grid,
        ConstTensor3View  z_field,
        ConstTensor3View  t_field,
        ConstTensor4View  vmr_field,
        ConstVectorView   f_grid,
        ConstVectorView   refellipsoid,
        const Numeric&    z_surface,
        const Numeric&    lmax,
        const Agenda&     refr_index_air_agenda,
        const String&     rtrace_method,
        const Numeric&    lraytrace )
{
  // Starting radius, zenith angle and latitude
  Numeric   r_start, lat_start, za_start;
 
  // Index of the pressure level being the lower limit for the
  // grid range of interest.
  Index   ip;
 
  // Determine the variables defined above, and make asserts of input
  ppath_start_1d( r_start, lat_start, za_start, ip, ppath );
 
  // If the field "constant" is negative, this is the first call of the
  // function and the path constant shall be calculated.
  // If the sensor is placed outside the atmosphere, the constant is
  // already set.
  Numeric ppc;
  if( ppath.constant < 0 )
    { 
      Numeric refr_index_air, refr_index_air_group;
      get_refr_index_1d( ws, refr_index_air, refr_index_air_group,
                         refr_index_air_agenda, p_grid, refellipsoid, 
                         z_field, t_field, vmr_field, f_grid, r_start );
      ppc = refraction_ppc( r_start, za_start, refr_index_air ); 
    }
  else
    { ppc = ppath.constant; }
 
 
  // Perform the ray tracing
  //
  // Arrays to store found ray tracing points
  // (Vectors don't work here as we don't know how many points there will be)
  Array<Numeric>   r_array, lat_array, za_array, l_array, n_array, ng_array;
  Index            endface;
  //
  if( rtrace_method  == "linear_basic" )
    {
      /*
      raytrace_1d_linear_basic( ws, r_array, lat_array, za_array, l_array, 
            n_array, ng_array, endface, refellipsoid, p_grid, z_field, t_field,
            vmr_field, f_grid, lmax, refr_index_air_agenda, 
            lraytrace, ppc, refellipsoid[0] + z_surface, 
            refellipsoid[0]+z_field[ip], refellipsoid[0]+z_field[ip+1], 
            r_start, lat_start, za_start );
      */
      raytrace_1d_linear_basic( ws, r_array, lat_array, za_array, l_array, 
            n_array, ng_array, endface, p_grid, refellipsoid, z_field, t_field,
            vmr_field, f_grid, lmax, refr_index_air_agenda, lraytrace, 
            refellipsoid[0] + z_surface, refellipsoid[0]+z_field(ip,0,0), 
            refellipsoid[0]+z_field(ip+1,0,0), r_start, lat_start, za_start );
    }
  else
    {
      // Make sure we fail if called with an invalid rtrace_method.
      assert(false);
    }
 
  // Fill *ppath*
  //
  const Index np = r_array.nelem();
  Vector r_v(np), lat_v(np), za_v(np), l_v(np-1), n_v(np), ng_v(np);
  for( Index i=0; i<np; i++ )
    { 
      r_v[i]   = r_array[i];    
      lat_v[i] = lat_array[i];
      za_v[i]  = za_array[i];   
      n_v[i]   = n_array[i];
      ng_v[i]  = ng_array[i];
      if( i < np-1 )
        { l_v[i] = l_array[i]; }
    }
  //
  ppath_end_1d( ppath, r_v, lat_v, za_v, l_v, n_v, ng_v, z_field(joker,0,0), 
                refellipsoid, ip, endface, ppc );
}
 
 
 
 
void raytrace_2d_linear_basic(
              Workspace&       ws,
              Array<Numeric>&  r_array,
              Array<Numeric>&  lat_array,
              Array<Numeric>&  za_array,
              Array<Numeric>&  l_array,
              Array<Numeric>&  n_array,
              Array<Numeric>&  ng_array,
              Index&           endface,
        ConstVectorView        p_grid,
        ConstVectorView        lat_grid,
        ConstVectorView        refellipsoid,
        ConstTensor3View       z_field,
        ConstTensor3View       t_field,
        ConstTensor4View       vmr_field,
        ConstVectorView        f_grid,
        const Numeric&         lmax,
        const Agenda&          refr_index_air_agenda,
        const Numeric&         lraytrace,
        const Numeric&         lat1,
        const Numeric&         lat3,
        const Numeric&         rsurface1,
        const Numeric&         rsurface3,
        const Numeric&         r1a,
        const Numeric&         r3a,
        const Numeric&         r3b,
        const Numeric&         r1b,
              Numeric          r,
              Numeric          lat,
              Numeric          za )
{
  // Loop boolean
  bool ready = false;
 
  // Store first point
  Numeric refr_index_air, refr_index_air_group;
  get_refr_index_2d( ws, refr_index_air, refr_index_air_group, 
                     refr_index_air_agenda, p_grid, lat_grid, refellipsoid, 
                     z_field, t_field, vmr_field, f_grid, r, lat );
  r_array.push_back( r );
  lat_array.push_back( lat );
  za_array.push_back( za );
  n_array.push_back( refr_index_air );
  ng_array.push_back( refr_index_air_group );
 
  // Variables for output from do_gridcell_2d
  Vector    r_v, lat_v, za_v;
  Numeric   lstep, lcum = 0, dlat;
 
  while( !ready )
    {
      // Constant for the geometrical step to make
      const Numeric   ppc_step = geometrical_ppc( r, za );
 
      // Where will a geometric path exit the grid cell?
      do_gridcell_2d_byltest( r_v, lat_v, za_v, lstep, endface, r, lat, za,
                                  lraytrace, 0, ppc_step, -1, lat1, lat3, 
                                  r1a, r3a, r3b, r1b, rsurface1, rsurface3 );
      assert( r_v.nelem() == 2 );
 
      // If *lstep* is <= *lraytrace*, extract the found end point (if not 
      // a tangent point, we are ready).
      // Otherwise, we make a geometrical step with length *lraytrace*.
 
      Numeric za_flagside = za;
 
      if( lstep <= lraytrace )
        {
          r     = r_v[1];
          dlat  = lat_v[1] - lat;
          lat   = lat_v[1];
          lcum += lstep;
          ready = true; 
        }
      else
        {
          Numeric l;
          if( abs(za) <= 90 ) 
            { l = geompath_l_at_r(ppc_step,r) + lraytrace; }
          else
            { 
              l = geompath_l_at_r(ppc_step,r) - lraytrace; 
              if( l < 0 )  // Tangent point passed!
                { za_flagside = sign(za)*180-za_flagside; } 
            }
 
          r = geompath_r_at_l( ppc_step, l );
 
          const Numeric lat_new = geompath_lat_at_za( za, lat, 
                                 geompath_za_at_r( ppc_step, za_flagside, r) );
          dlat  = lat_new - lat;
          lat   = lat_new;
          lstep = lraytrace;
          lcum += lraytrace;
 
          // For paths along the latitude end faces we can end up outside the
          // grid cell. We simply look for points outisde the grid cell.
          if( lat < lat1 )
            { lat = lat1; }
          else if( lat > lat3 )
            { lat = lat3; }
        }
 
      // Refractive index at new point
      Numeric   dndr, dndlat;
      refr_gradients_2d( ws, refr_index_air, refr_index_air_group, dndr, 
                         dndlat, refr_index_air_agenda, p_grid, lat_grid, 
                         refellipsoid, z_field, t_field, vmr_field, f_grid, 
                         r, lat );
 
      // Calculate LOS zenith angle at found point.
      const Numeric   za_rad = DEG2RAD * za;
      //
      za += - dlat;   // Pure geometrical effect
      //
      za += (RAD2DEG * lstep / refr_index_air) * ( -sin(za_rad) * dndr +
                                                        cos(za_rad) * dndlat );
 
      // Make sure that obtained *za* is inside valid range
      if( za < -180 )
        { za += 360; }
      else if( za > 180 )
        { za -= 360; }
 
      // If the path is zenith/nadir along a latitude end face, we must check
      // that the path does not exit with new *za*.
      if( lat == lat1  &&  za < 0 )
        { endface = 1;   ready = 1; }
      else if( lat == lat3  &&  za > 0 )
        { endface = 3;   ready = 1; }
      
      // Store found point?
      if( ready  ||  ( lmax > 0  &&  lcum + lraytrace > lmax ) )
        {
          r_array.push_back( r );
          lat_array.push_back( lat );
          za_array.push_back( za );
          n_array.push_back( refr_index_air );
          ng_array.push_back( refr_index_air_group );
          l_array.push_back( lcum );
          lcum = 0;
        }
    }  
}
 
 
 
 
void ppath_step_refr_2d(
              Workspace&  ws,
              Ppath&      ppath,
        ConstVectorView   p_grid,
        ConstVectorView   lat_grid,
        ConstTensor3View  z_field,
        ConstTensor3View  t_field,
        ConstTensor4View  vmr_field,
        ConstVectorView   f_grid,
        ConstVectorView   refellipsoid,
        ConstVectorView   z_surface,
        const Numeric&    lmax,
        const Agenda&     refr_index_air_agenda,
        const String&     rtrace_method,
        const Numeric&    lraytrace )
{
  // Radius, zenith angle and latitude of start point.
  Numeric   r_start, lat_start, za_start;
 
  // Lower grid index for the grid cell of interest.
  Index   ip, ilat;
 
  // Radii and latitudes set by *ppath_start_2d*.
  Numeric   lat1, lat3, r1a, r3a, r3b, r1b, rsurface1, rsurface3;
 
  // Determine the variables defined above and make all possible asserts
  ppath_start_2d( r_start, lat_start, za_start, ip, ilat, 
                  lat1, lat3, r1a, r3a, r3b, r1b, rsurface1, rsurface3,
                  ppath, lat_grid, z_field(joker,joker,0), refellipsoid, 
                  z_surface );
 
  // Perform the ray tracing
  //
  // No constant for the path is valid here.
  //
  // Arrays to store found ray tracing points
  // (Vectors don't work here as we don't know how many points there will be)
  Array<Numeric>   r_array, lat_array, za_array, l_array, n_array, ng_array;
  Index            endface;
  //
  if( rtrace_method  == "linear_basic" )
    {
      raytrace_2d_linear_basic( ws, r_array, lat_array, za_array, l_array, 
                                n_array, ng_array, endface, p_grid, lat_grid, 
                                refellipsoid, z_field, t_field, vmr_field,
                                f_grid, lmax, 
                                refr_index_air_agenda, lraytrace, lat1, lat3,
                                rsurface1, rsurface3, r1a, r3a, r3b, r1b, 
                                r_start, lat_start, za_start );
    }
  else
    {
      // Make sure we fail if called with an invalid rtrace_method.
      assert(false);
    }
 
  // Fill *ppath*
  //
  const Index np = r_array.nelem();
  Vector r_v(np), lat_v(np), za_v(np), l_v(np-1), n_v(np), ng_v(np);
  for( Index i=0; i<np; i++ )
    { 
      r_v[i]   = r_array[i];    
      lat_v[i] = lat_array[i];
      za_v[i]  = za_array[i];   
      n_v[i]   = n_array[i];
      ng_v[i]  = ng_array[i];
      if( i < np-1 )
        { l_v[i] = l_array[i]; }
    }
  //
  ppath_end_2d( ppath, r_v, lat_v, za_v, l_v, n_v, ng_v, lat_grid, 
                z_field(joker,joker,0), refellipsoid, ip, ilat, endface, -1 );
}
 
 
 
 
void raytrace_3d_linear_basic(
              Workspace&       ws,
              Array<Numeric>&  r_array,
              Array<Numeric>&  lat_array,
              Array<Numeric>&  lon_array,
              Array<Numeric>&  za_array,
              Array<Numeric>&  aa_array,
              Array<Numeric>&  l_array,
              Array<Numeric>&  n_array,
              Array<Numeric>&  ng_array,
              Index&           endface,
        ConstVectorView        refellipsoid,
        ConstVectorView        p_grid,
        ConstVectorView        lat_grid,
        ConstVectorView        lon_grid,
        ConstTensor3View       z_field,
        ConstTensor3View       t_field,
        ConstTensor4View       vmr_field,
        ConstVectorView        f_grid,
        const Numeric&         lmax,
        const Agenda&          refr_index_air_agenda,
        const Numeric&         lraytrace,
        const Numeric&         lat1,
        const Numeric&         lat3,
        const Numeric&         lon5,
        const Numeric&         lon6,
        const Numeric&         rsurface15,
        const Numeric&         rsurface35,
        const Numeric&         rsurface36,
        const Numeric&         rsurface16,
        const Numeric&         r15a,
        const Numeric&         r35a,
        const Numeric&         r36a,
        const Numeric&         r16a,
        const Numeric&         r15b,
        const Numeric&         r35b,
        const Numeric&         r36b,
        const Numeric&         r16b,
              Numeric          r,
              Numeric          lat,
              Numeric          lon,
              Numeric          za,
              Numeric          aa )
{
  // Loop boolean
  bool ready = false;
 
  // Store first point
  Numeric refr_index_air, refr_index_air_group;
  get_refr_index_3d( ws, refr_index_air, refr_index_air_group, 
                     refr_index_air_agenda, p_grid, lat_grid, lon_grid, 
                     refellipsoid, z_field, t_field, vmr_field, f_grid, 
                     r, lat, lon );
  r_array.push_back( r );
  lat_array.push_back( lat );
  lon_array.push_back( lon );
  za_array.push_back( za );
  aa_array.push_back( aa );
  n_array.push_back( refr_index_air );
  ng_array.push_back( refr_index_air_group );
 
  // Variables for output from do_gridcell_3d
  Vector    r_v, lat_v, lon_v, za_v, aa_v;
  Numeric   lstep, lcum = 0;
  Numeric   za_new, aa_new;
 
  while( !ready )
    {
      // Constant for the geometrical step to make
      const Numeric   ppc_step = geometrical_ppc( r, za );
 
      // Where will a geometric path exit the grid cell?
      do_gridcell_3d_byltest( r_v, lat_v, lon_v, za_v, aa_v, lstep, endface,
                              r, lat, lon, za, aa, lraytrace, 0, 
                              ppc_step, -1, lat1, lat3, lon5, lon6,
                              r15a, r35a, r36a, r16a, r15b, r35b, r36b, r16b,
                              rsurface15, rsurface35, rsurface36, rsurface16 );
      assert( r_v.nelem() == 2 );
 
      // If *lstep* is <= *lraytrace*, extract the found end point.
      // Otherwise, we make a geometrical step with length *lraytrace*.
 
      if( lstep <= lraytrace )
        {
          r      = r_v[1];
          lat    = lat_v[1];
          lon    = lon_v[1];
          za_new = za_v[1];
          aa_new = aa_v[1];
          lcum  += lstep;
          ready = true; 
        }
      else
        {
          // Sensor pos and LOS in cartesian coordinates
          Numeric   x, y, z, dx, dy, dz, lat_new, lon_new;
          //
          poslos2cart( x, y, z, dx, dy, dz, r, lat, lon, za, aa ); 
          lstep = lraytrace;
          cart2poslos( r, lat_new, lon_new, za_new, aa_new, x+dx*lstep, 
                       y+dy*lstep, z+dz*lstep, dx, dy, dz, ppc_step,
                       lat, lon, za, aa );
          lcum += lstep;
 
          // Shall lon values be shifted?
          resolve_lon( lon_new, lon5, lon6 );
 
          lat = lat_new;
          lon = lon_new;
        }
 
      // Refractive index at new point
      Numeric   dndr, dndlat, dndlon;
      refr_gradients_3d( ws, refr_index_air, refr_index_air_group, 
                         dndr, dndlat, dndlon, refr_index_air_agenda, 
                         p_grid, lat_grid, lon_grid, refellipsoid, 
                         z_field, t_field, vmr_field, 
                         f_grid, r, lat, lon );
 
      // Calculate LOS zenith angle at found point.
      const Numeric   aterm  = RAD2DEG * lstep / refr_index_air;
      const Numeric   za_rad = DEG2RAD * za;
      const Numeric   aa_rad = DEG2RAD * aa;
      const Numeric   sinza  = sin( za_rad );
      const Numeric   sinaa  = sin( aa_rad );
      const Numeric   cosaa  = cos( aa_rad );
      //
      Vector los(2);   los[0] = za_new;   los[1] = aa_new;
      //
      if( za < ANGTOL  ||  za > 180-ANGTOL )
        { 
          los[0] += aterm * ( cos(za_rad) * 
                                         ( cosaa * dndlat + sinaa * dndlon ) );
          los[1]  = RAD2DEG * atan2( dndlon, dndlat); 
        }
      else
        { 
          los[0] += aterm * ( -sinza * dndr + cos(za_rad) * 
                              ( cosaa * dndlat + sinaa * dndlon ) );
          los[1] += aterm * sinza * ( cosaa * dndlon - sinaa * dndlat ); 
        }
      //
      adjust_los( los, 3 );
      //
      za  = los[0];
      aa  = los[1];
 
      // For some cases where the path goes along an end face, 
      // it could be the case that the refraction bends the path out 
      // of the grid cell.
      if( za > 0  &&  za < 180 )
        {
          if( lon == lon5  &&  aa < 0 )
            { endface = 5;   ready = 1; }
          else if( lon == lon6  &&  aa > 0 )
            { endface = 6;   ready = 1; }
          else if( lat == lat1  &&  lat != -90  &&  abs( aa ) > 90 ) 
            { endface = 1;   ready = 1; }
          else if( lat == lat3  &&  lat != 90  &&  abs( aa ) < 90 ) 
            { endface = 3;   ready = 1; }
        }
 
      // Store found point?
      if( ready  ||  ( lmax > 0  &&  lcum + lraytrace > lmax ) )
        {
          r_array.push_back( r );
          lat_array.push_back( lat );
          lon_array.push_back( lon );
          za_array.push_back( za );
          aa_array.push_back( aa );
          n_array.push_back( refr_index_air );
          ng_array.push_back( refr_index_air_group );
          l_array.push_back( lcum );
          lcum = 0;
        }  
    }
}
 
 
 
 
void ppath_step_refr_3d(
              Workspace&  ws,
              Ppath&      ppath,
        ConstVectorView   p_grid,
        ConstVectorView   lat_grid,
        ConstVectorView   lon_grid,
        ConstTensor3View  z_field,
        ConstTensor3View  t_field,
        ConstTensor4View  vmr_field,
        ConstVectorView   f_grid,
        ConstVectorView   refellipsoid,
        ConstMatrixView   z_surface,
        const Numeric&    lmax,
        const Agenda&     refr_index_air_agenda,
        const String&     rtrace_method,
        const Numeric&    lraytrace )
{
  // Radius, zenith angle and latitude of start point.
  Numeric   r_start, lat_start, lon_start, za_start, aa_start;
 
  // Lower grid index for the grid cell of interest.
  Index   ip, ilat, ilon;
 
  // Radius for corner points, latitude and longitude of the grid cell
  //
  Numeric   lat1, lat3, lon5, lon6;
  Numeric   r15a, r35a, r36a, r16a, r15b, r35b, r36b, r16b;
  Numeric   rsurface15, rsurface35, rsurface36, rsurface16;
 
  // Determine the variables defined above and make all possible asserts
  ppath_start_3d( r_start, lat_start, lon_start, za_start, aa_start, 
                  ip, ilat, ilon, lat1, lat3, lon5, lon6,
                  r15a, r35a, r36a, r16a, r15b, r35b, r36b, r16b, 
                  rsurface15, rsurface35, rsurface36, rsurface16,
                  ppath, lat_grid, lon_grid, z_field, refellipsoid, z_surface );
 
  // Perform the ray tracing
  //
  // No constant for the path is valid here.
  //
  // Arrays to store found ray tracing points
  // (Vectors don't work here as we don't know how many points there will be)
  Array<Numeric>   r_array, lat_array, lon_array, za_array, aa_array;
  Array<Numeric>   l_array, n_array, ng_array;
  Index            endface;
  //
  if( rtrace_method  == "linear_basic" )
    {
      raytrace_3d_linear_basic( ws, r_array, lat_array, lon_array, za_array,
                           aa_array, l_array, n_array, ng_array, endface, 
                           refellipsoid, p_grid, lat_grid, lon_grid, 
                           z_field, t_field, vmr_field, 
                           f_grid, lmax, refr_index_air_agenda, lraytrace, 
                           lat1, lat3, lon5, lon6, 
                           rsurface15, rsurface35, rsurface36, rsurface16,
                           r15a, r35a, r36a, r16a, r15b, r35b, r36b, r16b,
                           r_start, lat_start, lon_start, za_start, aa_start );
    }
  else
    {
      // Make sure we fail if called with an invalid rtrace_method.
      assert(false);
    }
 
  // Fill *ppath*
  //
  const Index np = r_array.nelem();
  Vector r_v(np), lat_v(np), lon_v(np), za_v(np), aa_v(np), l_v(np-1);
  Vector n_v(np), ng_v(np);
  for( Index i=0; i<np; i++ )
    { 
      r_v[i]   = r_array[i];    
      lat_v[i] = lat_array[i];
      lon_v[i] = lon_array[i];
      za_v[i]  = za_array[i];   
      aa_v[i]  = aa_array[i];   
      n_v[i]   = n_array[i];
      ng_v[i]  = ng_array[i];
      if( i < np-1 )
        { l_v[i] = l_array[i]; }
    }
  // 
  // Fill *ppath*
  ppath_end_3d( ppath, r_v, lat_v, lon_v, za_v, aa_v, l_v, n_v, ng_v, lat_grid, 
                lon_grid, z_field, refellipsoid, ip, ilat, ilon, endface, -1 );
}
 
 
 
 
 
/*===========================================================================
  === Main functions
  ===========================================================================*/
 
 
void ppath_start_stepping(
          Ppath&          ppath,
    const Index&          atmosphere_dim,
    ConstVectorView       p_grid,
    ConstVectorView       lat_grid,
    ConstVectorView       lon_grid,
    ConstTensor3View      z_field,
    ConstVectorView       refellipsoid,
    ConstMatrixView       z_surface,
    const Index&          cloudbox_on, 
    const ArrayOfIndex&   cloudbox_limits,
    const bool&           ppath_inside_cloudbox_do,
    ConstVectorView       rte_pos,
    ConstVectorView       rte_los,
    const Verbosity&      verbosity)
{
  CREATE_OUT1;
  
  // This function contains no checks or asserts as it is only a sub-function.
 
  // Allocate the ppath structure
  ppath_init_structure(  ppath, atmosphere_dim, 1 );
 
  // Index of last pressure level
  const Index lp = p_grid.nelem() - 1;
 
  // The different atmospheric dimensionalities are handled seperately
 
  //-- 1D ---------------------------------------------------------------------
  if( atmosphere_dim == 1 )
    {
      // End position and LOS
      ppath.end_pos[0] = rte_pos[0];
      ppath.end_pos[1] = 0; 
      ppath.end_los[0] = rte_los[0];
 
      // Sensor is inside the model atmosphere:
      if( rte_pos[0] < z_field(lp,0,0) )
        {
          // Check that the sensor is above the surface
          if( (rte_pos[0] + RTOL) < z_surface(0,0) )
            {
              ostringstream os;
              os << "The ppath starting point is placed " 
                 << (z_surface(0,0) - rte_pos[0])/1e3 
                 << " km below the surface.";
              throw runtime_error(os.str());
            }
 
          // Set ppath
          ppath.pos(0,joker) = ppath.end_pos;
          ppath.r[0]         = refellipsoid[0] + rte_pos[0];
          ppath.los(0,joker) = ppath.end_los;
          //
          gridpos( ppath.gp_p, z_field(joker,0,0), ppath.pos(0,0) );
          gridpos_check_fd( ppath.gp_p[0] );
 
          // Is the sensor on the surface looking down?
          // If yes and the sensor is inside the cloudbox, the background will
          // be changed below.
          if( ppath.pos(0,0) <= z_surface(0,0)  &&  ppath.los(0,0) > 90 )
            { ppath_set_background( ppath, 2 ); }
 
          // Outside cloudbox:
          // Check sensor position with respect to cloud box.
          if( cloudbox_on  &&  !ppath_inside_cloudbox_do )
            {
              const Numeric fgp = fractional_gp( ppath.gp_p[0] );
              if( fgp >= cloudbox_limits[0]  &&  fgp <= cloudbox_limits[1] )
                { ppath_set_background( ppath, 4 ); }
            }
 
          // Inside cloudbox:
          DEBUG_ONLY( if( ppath_inside_cloudbox_do )
            {
              const Numeric fgp = fractional_gp( ppath.gp_p[0] );
              assert( fgp >= cloudbox_limits[0] && fgp <= cloudbox_limits[1] );
            } )
        }
 
      // Sensor is outside the model atmosphere:
      else
        {
          // We can here set ppc and n as we are outside the atmosphere
          ppath.nreal    = 1.0;
          ppath.ngroup   = 1.0;
          ppath.constant = geometrical_ppc( refellipsoid[0] + rte_pos[0], 
                                                              rte_los[0] );
 
          // Totally outside
          if( rte_los[0] <= 90  ||  
              ppath.constant >= refellipsoid[0] + z_field(lp,0,0) )
            {
              ppath.pos(0,0) = rte_pos[0];
              ppath.pos(0,1) = 0; 
              ppath.r[0]     = refellipsoid[0] + rte_pos[0];
              ppath.los(0,0) = rte_los[0];
              //
              ppath_set_background( ppath, 1 );
              out1 << "  --- WARNING ---, path is totally outside of the "
                   << "model atmosphere\n";
            }
 
          // Path enters the atmosphere. 
          else
            { 
              ppath.r[0]     = refellipsoid[0] + z_field(lp,0,0);
              ppath.pos(0,0) = z_field(lp,0,0);
              ppath.los(0,0) = geompath_za_at_r( ppath.constant, rte_los[0],
                                                                 ppath.r[0] );
              ppath.pos(0,1) = geompath_lat_at_za( rte_los[0], 0, 
                                                             ppath.los(0,0) );
              ppath.end_lstep = geompath_l_at_r( ppath.constant,
                                              refellipsoid[0] + rte_pos[0] ) -
                             geompath_l_at_r( ppath.constant, ppath.r[0] ); 
 
              // Here we know the grid position exactly
              ppath.gp_p[0].idx   = lp-1; 
              ppath.gp_p[0].fd[0] = 1;
              ppath.gp_p[0].fd[1] = 0;
 
              // If cloud box reaching TOA, we have also found the background
              if( cloudbox_on  &&  cloudbox_limits[1] == lp )
                { ppath_set_background( ppath, 3 ); }
           }
        }
    }  // End 1D
 
 
  //-- 2D ---------------------------------------------------------------------
  else if( atmosphere_dim == 2 )
    {
      // End position and LOS
      ppath.end_pos[0] = rte_pos[0];
      ppath.end_pos[1] = rte_pos[1];
      ppath.end_los[0] = rte_los[0];
 
      // Index of last latitude
      const Index llat = lat_grid.nelem() -1;
 
      // Is sensor inside range of lat_grid?
      // If yes, determine TOA altitude at sensor position
      GridPos   gp_lat;
      Vector    itw(2);
      bool      islatin = false;
      Numeric    r_e;  // Ellipsoid radius at sensor position
      Numeric    z_toa   = -99e99;
      if( rte_pos[1] > lat_grid[0]  &&  rte_pos[1] < lat_grid[llat] )
        { 
          islatin = true; 
          gridpos( gp_lat, lat_grid, rte_pos[1] );
          interpweights( itw, gp_lat );
          z_toa = interp( itw, z_field(lp,joker,0), gp_lat );
          r_e   = refell2d( refellipsoid, lat_grid, gp_lat );
        }
      else
        { r_e = refell2r( refellipsoid, rte_pos[1] ); }
 
      // Sensor is inside the model atmosphere:
      if( islatin  &&  rte_pos[0] < z_toa )
        {
          const Numeric z_s = interp( itw, z_surface(joker,0), gp_lat );
 
          // Check that the sensor is above the surface
          if( (rte_pos[0] + RTOL) < z_s )
            {
              ostringstream os;
              os << "The ppath starting point is placed " 
                 << (z_s - rte_pos[0])/1e3 << " km below the surface.";
              throw runtime_error(os.str());
            }
 
          // Set ppath
          ppath.pos(0,joker) = ppath.end_pos;
          ppath.r[0]         = r_e + rte_pos[0];
          ppath.los(0,joker) = ppath.end_los;
 
          // Determine gp_p (gp_lat is recalculated, but ...)
          GridPos gp_lon_dummy;
          rte_pos2gridpos( ppath.gp_p[0], ppath.gp_lat[0], gp_lon_dummy,
                atmosphere_dim, p_grid, lat_grid, lon_grid, z_field, rte_pos );
          gridpos_check_fd( ppath.gp_p[0] );
          gridpos_check_fd( ppath.gp_lat[0] );
 
          // Is the sensor on the surface looking down?
          // If yes and the sensor is inside the cloudbox, the background will
          // be changed below.
          if( ppath.pos(0,0) <= z_s )
            { 
              // Calculate radial slope of the surface
              Numeric rslope;
              plevel_slope_2d( rslope, lat_grid, refellipsoid,
                               z_surface(joker,0), gp_lat, ppath.los(0,0) );
 
              // Calculate angular tilt of the surface
              const Numeric atilt = plevel_angletilt( r_e + z_s, rslope );
 
              // Are we looking down into the surface?
              // If yes and the sensor is inside the cloudbox, the background 
              // will be changed below.
              if( is_los_downwards( ppath.los(0,0), atilt ) )
                { ppath_set_background( ppath, 2 ); }
            }
 
          // Outside cloudbox:
          // Check sensor position with respect to cloud box.
          if( cloudbox_on  &&  !ppath_inside_cloudbox_do )
            {
              const Numeric fgp = fractional_gp( ppath.gp_p[0] );
              const Numeric fgl = fractional_gp( ppath.gp_lat[0] );
              if( fgp >= cloudbox_limits[0]  &&  fgp <= cloudbox_limits[1]  &&
                  fgl >= cloudbox_limits[2]  &&  fgl <= cloudbox_limits[3]  )
                { ppath_set_background( ppath, 4 ); }
            }
 
          // Inside cloudbox:
          DEBUG_ONLY( if( ppath_inside_cloudbox_do )
            {
              const Numeric fgp = fractional_gp( ppath.gp_p[0] );
              const Numeric fgl = fractional_gp( ppath.gp_lat[0] );
              assert( fgp >= cloudbox_limits[0] && fgp <= cloudbox_limits[1] &&
                      fgl >= cloudbox_limits[2] && fgl <= cloudbox_limits[3]  );
            } )
        }      
 
      // Sensor is outside the model atmosphere:
      else
        {
          // Handle cases when the sensor looks in the wrong way
          if( ( rte_pos[1] <= lat_grid[0]     &&  rte_los[0] <= 0 )  || 
              ( rte_pos[1] >= lat_grid[llat]  &&  rte_los[0] >= 0 ) )
            {
              ostringstream os;
              os << "The sensor is outside (or at the limit) of the model "
                 << "atmosphere but\nlooks in the wrong direction (wrong sign "
                 << "for the zenith angle?).\nThis case includes nadir "
                 << "looking exactly at the latitude end points.";
              throw runtime_error( os.str() );
            }
 
          // We can here set ppc and n as we are outside the atmosphere
          ppath.nreal    = 1.0;
          ppath.ngroup   = 1.0;
          const Numeric r_p = r_e + rte_pos[0];
          ppath.constant = geometrical_ppc( r_p, rte_los[0] );
 
          // Determine TOA radii, and min and max value
          Vector   r_toa(llat+1);
          Numeric   r_toa_min = 99e99, r_toa_max = -1;
          for( Index ilat=0; ilat<=llat; ilat++ )
            {
              r_toa[ilat] = refell2r( refellipsoid, lat_grid[ilat] ) +
                                                            z_field(lp,ilat,0);
              if( r_toa[ilat] < r_toa_min )
                { r_toa_min = r_toa[ilat]; } 
              if( r_toa[ilat] > r_toa_max )
                { r_toa_max = r_toa[ilat]; } 
            }
          if( r_p <= r_toa_max )
            {
              ostringstream os;
              os << "The sensor is horizontally outside (or at the limit) of "
                 << "the model\natmosphere, but is at a radius smaller than "
                 << "the maximum value of\nthe top-of-the-atmosphere radii. "
                 << "This is not allowed. Make the\nmodel atmosphere larger "
                 << "to also cover the sensor position?";
              throw runtime_error( os.str() );
            }
 
          // Upward:
          if( abs(rte_los[0]) <= 90 )
            {
              ppath.pos(0,0) = rte_pos[0];
              ppath.pos(0,1) = rte_pos[1]; 
              ppath.r[0]     = r_e + rte_pos[0];
              ppath.los(0,0) = rte_los[0];
              //
              ppath_set_background( ppath, 1 );
              out1 << "  ------- WARNING -------: path is totally outside of "
                   << "the model atmosphere\n";
            }
 
          // Downward:
          else
            {
              bool      above=false, ready=false, failed=false;
              Numeric    rt=-1, latt, lt, lt_old = L_NOT_FOUND;
              GridPos   gp_latt;
              Vector    itwt(2);
 
              // Check if clearly above the model atmosphere
              if( ppath.constant >= r_toa_max )
                { above=true; ready=true; }
              else
                { // Otherwise pick out a suitable first test radius
                  if( islatin  ||  ppath.constant > r_toa_min ) 
                    { rt = r_toa_max; } 
                  else 
                    { rt = r_toa_min; }
                }
 
              // Iterate until solution found or moving out from model atm.
              //
              while( !ready && !failed )
                {
                  // If rt < ppath.constant, ppath above atmosphere
                  if( rt < ppath.constant )
                    { 
                      above = true; 
                      ready = true; 
                    }
 
                  else
                    {
                      // Calculate length to entrance point at rt 
                      r_crossing_2d( latt, lt, rt, r_p, rte_pos[1], rte_los[0],
                                                              ppath.constant );
                      assert( lt < 9e9 );
 
                      // Entrance outside range of lat_grid = fail
                      if( latt < lat_grid[0]  ||  latt > lat_grid[llat] )
                        { failed = true; }
 
                      // OK iteration
                      else
                        {
                          // Converged? 
                          if( abs( lt-lt_old ) < LACC )
                            { ready = true; }
 
                          // Update rt
                          lt_old = lt;
                          gridpos( gp_latt, lat_grid, latt );
                          interpweights( itwt, gp_latt );
                          rt = interp( itwt, r_toa, gp_latt );
                        }
                    }
                }  // while
 
              if( failed )
                {
                  ostringstream os;
                  os << "The path does not enter the model atmosphere. It "
                     << "reaches the\ntop of the atmosphere "
                     << "altitude around latitude " << latt << " deg.";
                  throw runtime_error( os.str() );
                }
              else if( above )
                {
                  ppath.pos(0,0) = rte_pos[0];
                  ppath.pos(0,1) = rte_pos[1]; 
                  ppath.r[0]     = r_e + rte_pos[0];
                  ppath.los(0,0) = rte_los[0];
                  //
                  ppath_set_background( ppath, 1 );
                  out1 << "  ------- WARNING -------: path is totally outside "
                       << "of the model atmosphere\n";
                }
              else
                {
                  ppath.r[0]     = rt;
                  ppath.pos(0,0) = interp( itwt, z_field(lp,joker,0), gp_latt );
                  // Calculate za first and use to determine lat
                  ppath.los(0,0) = geompath_za_at_r( ppath.constant, 
                                                     rte_los[0], rt );
                  ppath.pos(0,1) = geompath_lat_at_za( rte_los[0], rte_pos[1], 
                                                              ppath.los(0,0) );
                  ppath.end_lstep = lt;
 
                  // Here we know the pressure grid position exactly
                  ppath.gp_p[0].idx   = lp-1; 
                  ppath.gp_p[0].fd[0] = 1;
                  ppath.gp_p[0].fd[1] = 0;
 
                  // Latitude grid position already calculated
                  gridpos_copy( ppath.gp_lat[0], gp_latt );
 
                  // Hit with cloudbox reaching TOA?
                  if( cloudbox_on  &&  cloudbox_limits[1] == lp )
                    {
                      Numeric fgp = fractional_gp(gp_latt);
                      if( fgp >= (Numeric)cloudbox_limits[2]  &&
                          fgp <= (Numeric)cloudbox_limits[3] )
                        { ppath_set_background( ppath, 3 ); }
                    }
                }
            }  // Downward 
        }  // Outside
    }  // End 2D
 
 
  //-- 3D ---------------------------------------------------------------------
  else
    {
      // Index of last latitude and longitude
      const Index llat = lat_grid.nelem() - 1;
      const Index llon = lon_grid.nelem() - 1;
 
      // Adjust longitude of rte_pos to range used in lon_grid
      Numeric lon2use = rte_pos[2];
      resolve_lon( lon2use, lon_grid[0], lon_grid[llon] );
 
      // End position and LOS
      ppath.end_pos[0] = rte_pos[0];
      ppath.end_pos[1] = rte_pos[1];
      ppath.end_pos[2] = lon2use;
      ppath.end_los[0] = rte_los[0];
      ppath.end_los[1] = rte_los[1];
 
      // Is sensor inside range of lat_grid and lon_grid?
      // If yes, determine TOA altitude at sensor position
      GridPos   gp_lat, gp_lon;
      Vector    itw(4);
      bool      islatlonin = false;
      Numeric    r_e;  // Ellipsoid radius at sensor position
      Numeric    z_toa   = -99e99;
 
      if( rte_pos[1] >= lat_grid[0]  &&  rte_pos[1] <= lat_grid[llat]  &&
          lon2use >= lon_grid[0]  &&  lon2use <= lon_grid[llon] )
        { 
          islatlonin = true; 
          gridpos( gp_lat, lat_grid, rte_pos[1] );
          gridpos( gp_lon, lon_grid, lon2use );
          interpweights( itw, gp_lat, gp_lon );
          z_toa = interp( itw, z_field(lp,joker,joker), gp_lat, gp_lon );
          r_e   = refell2d( refellipsoid, lat_grid, gp_lat );
        }
      else
        { r_e = refell2r( refellipsoid, rte_pos[1] ); }
 
      // Sensor is inside the model atmosphere:
      if( islatlonin  &&  rte_pos[0] < z_toa )
        {
          const Numeric z_s = interp( itw, z_surface, gp_lat, gp_lon );
 
          // Check that the sensor is above the surface
          if( (rte_pos[0] + RTOL) < z_s )
            {
              ostringstream os;
              os << "The ppath starting point is placed " 
                 << (z_s - rte_pos[0])/1e3 << " km below the surface.";
              throw runtime_error(os.str());
            }
 
          // Set ppath
          ppath.pos(0,joker) = ppath.end_pos;
          ppath.r[0]         = r_e + rte_pos[0];
          ppath.los(0,joker) = ppath.end_los;
 
          // Determine gp_p (gp_lat and gp_lon are recalculated, but ...)
          rte_pos2gridpos( ppath.gp_p[0], ppath.gp_lat[0], ppath.gp_lon[0], 
                atmosphere_dim, p_grid, lat_grid, lon_grid, z_field, rte_pos );
          gridpos_check_fd( ppath.gp_p[0] );
          gridpos_check_fd( ppath.gp_lat[0] );
          gridpos_check_fd( ppath.gp_lon[0] );
 
          // Is the sensor on the surface looking down?
          // If yes and the sensor is inside the cloudbox, the background will
          // be changed below.
          if( ppath.pos(0,0) <= z_s )
            { 
              // Calculate radial slope of the surface
              Numeric c1, c2;
              plevel_slope_3d( c1, c2, lat_grid, lon_grid, refellipsoid, 
                               z_surface, gp_lat, gp_lon, ppath.los(0,1) );
 
              // Calculate angular tilt of the surface
              const Numeric atilt = plevel_angletilt( r_e + z_s, c1 );
 
              // Are we looking down into the surface?
              // If yes and the sensor is inside the cloudbox, the background 
              // will be changed below.
              if( is_los_downwards( ppath.los(0,0), atilt ) )
                { ppath_set_background( ppath, 2 ); }
            }
 
          // Outside cloudbox:
          // Check sensor position with respect to cloud box.
          if( cloudbox_on  &&  !ppath_inside_cloudbox_do )
            {
              const Numeric fgp = fractional_gp( ppath.gp_p[0] );
              const Numeric fgl = fractional_gp( ppath.gp_lat[0] );
              const Numeric fgo = fractional_gp( ppath.gp_lon[0] );
              if( fgp >= cloudbox_limits[0]  &&  fgp <= cloudbox_limits[1]  &&
                  fgl >= cloudbox_limits[2]  &&  fgl <= cloudbox_limits[3]  &&
                  fgo >= cloudbox_limits[4]  &&  fgo <= cloudbox_limits[5]  )
                { ppath_set_background( ppath, 4 ); }
            }
 
          // Inside cloudbox:
          DEBUG_ONLY( if( ppath_inside_cloudbox_do )
            {
              const Numeric fgp = fractional_gp( ppath.gp_p[0] );
              const Numeric fgl = fractional_gp( ppath.gp_lat[0] );
              const Numeric fgo = fractional_gp( ppath.gp_lon[0] );
              assert( fgp >= cloudbox_limits[0] && fgp <= cloudbox_limits[1] &&
                      fgl >= cloudbox_limits[2] && fgl <= cloudbox_limits[3] &&
                      fgo >= cloudbox_limits[4] && fgo <= cloudbox_limits[5] );
            } )
        }      
 
      // Sensor is outside the model atmosphere:
      else
        {
          // Handle cases when the sensor appears to look the wrong way in
          // the north-south direction
          if( ( rte_pos[1] <= lat_grid[0]     &&  abs( rte_los[1] ) >= 90 )  || 
              ( rte_pos[1] >= lat_grid[llat]  &&  abs( rte_los[1] ) <= 90 ) )
            {
              ostringstream os;
              os << "The sensor is north or south (or at the limit) of the "
                 << "model atmosphere\nbut looks in the wrong direction.";
              throw runtime_error( os.str() );
            }
 
          // Handle cases when the sensor appears to look the wrong way in
          // the west-east direction. We demand that the sensor is inside the
          // range of lon_grid even if all longitudes are covered.
          if( ( lon2use <= lon_grid[0]     &&  rte_los[1] < 0 )  || 
              ( lon2use >= lon_grid[llon]  &&  rte_los[1] > 0 ) )
            {
              ostringstream os;
              os << "The sensor is east or west (or at the limit) of the "
                 << "model atmosphere\nbut looks in the wrong direction.";
              throw runtime_error( os.str() );
            }
 
          // We can here set ppc and n as we are outside the atmosphere
          ppath.nreal    = 1.0;
          ppath.ngroup   = 1.0;
          const Numeric r_p = r_e + rte_pos[0];
          ppath.constant = geometrical_ppc( r_p, rte_los[0] );
 
          // Determine TOA radii, and min and max value
          Matrix   r_toa(llat+1,llon+1);
          Numeric   r_toa_min = 99e99, r_toa_max = -1;
          for( Index ilat=0; ilat<=llat; ilat++ )
            {
              const Numeric r_lat = refell2r(refellipsoid,lat_grid[ilat]);
              for( Index ilon=0; ilon<=llon; ilon++ )
                {
                  r_toa(ilat,ilon) = r_lat+ z_field(lp,ilat,ilon);
                  if( r_toa(ilat,ilon) < r_toa_min )
                    { r_toa_min = r_toa(ilat,ilon); } 
                  if( r_toa(ilat,ilon) > r_toa_max )
                    { r_toa_max = r_toa(ilat,ilon); } 
                }
            }
 
          if( r_p <= r_toa_max )
            {
              ostringstream os;
              os << "The sensor is horizontally outside (or at the limit) of "
                 << "the model\natmosphere, but is at a radius smaller than "
                 << "the maximum value of\nthe top-of-the-atmosphere radii. "
                 << "This is not allowed. Make the\nmodel atmosphere larger "
                 << "to also cover the sensor position?";
              throw runtime_error( os.str() );
            }
 
          // Upward:
          if( rte_los[0] <= 90 )
            {
              ppath.pos(0,0) = rte_pos[0];
              ppath.pos(0,1) = rte_pos[1]; 
              ppath.pos(0,1) = lon2use; 
              ppath.r[0]     = r_e + rte_pos[0];
              ppath.los(0,0) = rte_los[0];
              ppath.los(0,1) = rte_los[1];
              //
              ppath_set_background( ppath, 1 );
              out1 << "  ------- WARNING -------: path is totally outside of "
                   << "the model atmosphere\n";
            }
 
          // Downward:
          else
            {
              bool      above=false, ready=false, failed=false;
              Numeric   rt=-1, latt, lont, lt, lt_old = L_NOT_FOUND;
              GridPos   gp_latt, gp_lont;
              Vector    itwt(4);
 
              // Check if clearly above the model atmosphere
              if( ppath.constant >= r_toa_max )
                { above=true; ready=true; }
              else
                { // Otherwise pick out a suitable first test radius
                  if( islatlonin  ||  ppath.constant > r_toa_min ) 
                    { rt = r_toa_max; } 
                  else 
                    { rt = r_toa_min; }
                }
              
              // Sensor pos and LOS in cartesian coordinates
              Numeric   x, y, z, dx, dy, dz;
              poslos2cart( x, y, z, dx, dy, dz, r_p, rte_pos[1], lon2use, 
                                                     rte_los[0], rte_los[1] );
 
              // Iterate until solution found or moving out from model atm.
              //
              while( !ready && !failed )
                {
                  // If rt < ppath.constant, ppath above atmosphere
                  if( rt < ppath.constant )
                    { 
                      above = true; 
                      ready = true; 
                    }
 
                  else
                    {
                      // Calculate lat and lon for entrance point at rt 
                      r_crossing_3d( latt, lont, lt, rt, r_p, rte_pos[1], 
                                     lon2use, rte_los[0], ppath.constant, 
                                     x, y, z, dx, dy, dz );
                      resolve_lon( lont, lon_grid[0], lon_grid[llon] ); 
 
                      // Entrance outside range of lat/lon_grids = fail
                      if( latt < lat_grid[0]  ||  latt > lat_grid[llat]  ||
                          lont < lon_grid[0]  ||  lont > lon_grid[llon] )
                        { failed = true; }
 
                      // OK iteration
                      else
                        {
                          // Converged? 
                          if( abs( lt-lt_old ) < LACC )
                            { ready = true; }
                          
                          // Update rt
                          lt_old = lt;
                          gridpos( gp_latt, lat_grid, latt );
                          gridpos( gp_lont, lon_grid, lont );
                          interpweights( itwt, gp_latt, gp_lont );
                          rt = interp( itwt, r_toa, gp_latt, gp_lont );
                        }
                    }
                }  // while
 
              if( failed )
                {
                  ostringstream os;
                  os << "The path does not enter the model atmosphere. It\n"
                     << "reaches the top of the atmosphere altitude around:\n"
                     << "  lat: " << latt << " deg.\n  lon: " << lont 
                     << " deg.";
                  throw runtime_error( os.str() );
                }
              else if( above )
                {
                  ppath.pos(0,0) = rte_pos[0];
                  ppath.pos(0,1) = rte_pos[1]; 
                  ppath.pos(0,1) = lon2use; 
                  ppath.r[0]     = r_e + rte_pos[0];
                  ppath.los(0,0) = rte_los[0];
                  ppath.los(0,1) = rte_los[1];
                  //
                  ppath_set_background( ppath, 1 );
                  out1 << "  ------- WARNING -------: path is totally outside "
                       << "of the model atmosphere\n";
                }
              else
                {
                  // Calculate lt for last rt, and use it to determine pos/los
                  lt = geompath_l_at_r( ppath.constant, r_p ) -
                       geompath_l_at_r( ppath.constant, rt );
                  cart2poslos( ppath.r[0], ppath.pos(0,1), ppath.pos(0,2),
                               ppath.los(0,0), ppath.los(0,1),x+dx*lt, y+dy*lt,
                               z+dz*lt, dx, dy, dz, ppath.constant, rte_pos[1],
                               lon2use, rte_los[0], rte_los[1] );
                  assert( abs( ppath.r[0] -rt ) < RTOL );
                  resolve_lon( ppath.pos(0,2), lon_grid[0], lon_grid[llon] ); 
                  //
                  ppath.pos(0,0) = interp( itwt, z_field(lp,joker,joker), 
                                                            gp_latt, gp_lont );
                  ppath.end_lstep = lt;
 
                  // Here we know the pressure grid position exactly
                  ppath.gp_p[0].idx   = lp-1; 
                  ppath.gp_p[0].fd[0] = 1;
                  ppath.gp_p[0].fd[1] = 0;
 
                  // Lat and lon grid position already calculated
                  gridpos_copy( ppath.gp_lat[0], gp_latt );
                  gridpos_copy( ppath.gp_lon[0], gp_lont );
 
                  // Hit with cloudbox reaching TOA?
                  if( cloudbox_on  &&  cloudbox_limits[1] == lp )
                    {
                      Numeric fgp1 = fractional_gp(gp_latt);
                      Numeric fgp2 = fractional_gp(gp_lont);
                      if( fgp1 >= (Numeric)cloudbox_limits[2]  &&
                          fgp1 <= (Numeric)cloudbox_limits[3]  && 
                          fgp2 >= (Numeric)cloudbox_limits[4]  &&
                          fgp2 <= (Numeric)cloudbox_limits[5])
                        { ppath_set_background( ppath, 3 ); }
                    }
                }
            }  // Downward 
        }  // Outside
    }  // End 3D
}
 
 
 
 
 
 
void ppath_calc(  
          Workspace&      ws,
          Ppath&          ppath,
    const Agenda&         ppath_step_agenda,
    const Index&          atmosphere_dim,
    const Vector&         p_grid,
    const Vector&         lat_grid,
    const Vector&         lon_grid,
    const Tensor3&        t_field,
    const Tensor3&        z_field,
    const Tensor4&        vmr_field,
    const Vector&         f_grid,
    const Vector&         refellipsoid,
    const Matrix&         z_surface,
    const Index&          cloudbox_on, 
    const ArrayOfIndex&   cloudbox_limits,
    const Vector&         rte_pos,
    const Vector&         rte_los,
    const Numeric&        ppath_lraytrace,
    const bool&           ppath_inside_cloudbox_do,
    const Verbosity&      verbosity)
{
  // This function is a WSM but it is normally only called from yCalc. 
  // For that reason, this function does not repeat input checks that are
  // performed in yCalc, it only performs checks regarding the sensor 
  // position and LOS.
 
  //--- Check input -----------------------------------------------------------
  chk_rte_pos( atmosphere_dim, rte_pos );
  chk_rte_los( atmosphere_dim, rte_los );
  if( ppath_inside_cloudbox_do  &&  !cloudbox_on )
    throw runtime_error( "The WSV *ppath_inside_cloudbox_do* can only be set "
                         "to 1 if also *cloudbox_on* is 1." );
  //--- End: Check input ------------------------------------------------------
 
 
  // Initiate the partial Ppath structure. 
  // The function doing the work sets ppath_step to the point of the path
  // inside the atmosphere closest to the sensor, if the path is at all inside
  // the atmosphere.
  // If the background field is set by the function this flags that there is no
  // path to follow (for example when the sensor is inside the cloud box).
  // The function checks also that the sensor position and LOS give an
  // allowed path.
  //
  Ppath   ppath_step;
  //
  ppath_start_stepping( ppath_step, atmosphere_dim, p_grid, lat_grid, 
                        lon_grid, z_field, refellipsoid, z_surface,
                        cloudbox_on, cloudbox_limits, ppath_inside_cloudbox_do, 
                        rte_pos, rte_los, verbosity );
  // For debugging:
  //Print( ppath_step, 0, verbosity );
 
  // The only data we need to store from this initial ppath_step is
  // end_pos/los/lstep
  const Numeric end_lstep = ppath_step.end_lstep;
  const Vector  end_pos   = ppath_step.end_pos;
  const Vector  end_los   = ppath_step.end_los;
 
  // Perform propagation path steps until the starting point is found, which
  // is flagged by ppath_step by setting the background field.
  //
  // The results of each step, returned by ppath_step_agenda as a new 
  // ppath_step, are stored as an array of Ppath structures.
  //
  Array<Ppath> ppath_array(0);
  Index   np     = 1;   // Counter for number of points of the path
  Index   istep  = 0;   // Counter for number of steps
  //
  const Index imax_p   = p_grid.nelem() - 1;
  const Index imax_lat = lat_grid.nelem() - 1;
  const Index imax_lon = lon_grid.nelem() - 1;
  //
  bool ready = ppath_what_background( ppath_step );
  //
  while( !ready )
    {
      // Call ppath_step agenda. 
      // The new path step is added to *ppath_array* last in the while block
      //
      istep++;
      //
      ppath_step_agendaExecute( ws, ppath_step, ppath_lraytrace, t_field, 
                                z_field, vmr_field, f_grid, 
                                ppath_step_agenda );
      // For debugging:
      //Print( ppath_step, 0, verbosity );
      
      // Number of points in returned path step
      const Index n = ppath_step.np;
 
      // Increase the total number
      np += n - 1;
 
      if( istep > 1e4 )
        throw runtime_error(
          "10 000 path points have been reached. Is this an infinite loop?" );
      
 
      //----------------------------------------------------------------------
      //---  Check if some boundary is reached
      //----------------------------------------------------------------------
 
      //--- Outside cloud box ------------------------------------------------
      if( !ppath_inside_cloudbox_do )
        {
          // Check if the top of the atmosphere is reached
          // (Perform first a strict check, and if upward propagation also a
          //  non-strict check. The later to avoid fatal "fixes" at TOA.)
          if( is_gridpos_at_index_i( ppath_step.gp_p[n-1], imax_p ) ||
              ( abs(ppath_step.los(n-1,0))<90  &&
                is_gridpos_at_index_i(  ppath_step.gp_p[n-1], imax_p, false )))
            { 
              ppath_set_background( ppath_step, 1 ); 
            }
 
          // Check that path does not exit at a latitude or longitude end face
          if( atmosphere_dim == 2 )
            {
              // Latitude 
              if( is_gridpos_at_index_i( ppath_step.gp_lat[n-1], 0 ) )
                {
                  ostringstream os;
                  os << "The path exits the atmosphere through the lower "
                     << "latitude end face.\nThe exit point is at an altitude" 
                     << " of " << ppath_step.pos(n-1,0)/1e3 << " km.";
                  throw runtime_error( os.str() );
                }
              if( is_gridpos_at_index_i( ppath_step.gp_lat[n-1], imax_lat ) )
                {
                  ostringstream os;
                  os << "The path exits the atmosphere through the upper "
                     << "latitude end face.\nThe exit point is at an altitude" 
                     << " of " << ppath_step.pos(n-1,0)/1e3 << " km.";
                  throw runtime_error( os.str() );
                }
            }
          if( atmosphere_dim == 3 )
            {
              // Latitude 
              if( lat_grid[0] > -90  && 
                           is_gridpos_at_index_i( ppath_step.gp_lat[n-1], 0 ) )
                {
                  ostringstream os;
                  os << "The path exits the atmosphere through the lower "
                     << "latitude end face.\nThe exit point is at an altitude" 
                     << " of " << ppath_step.pos(n-1,0)/1e3 << " km.";
                  throw runtime_error( os.str() );
                }
              if( lat_grid[imax_lat] < 90  && 
                    is_gridpos_at_index_i( ppath_step.gp_lat[n-1], imax_lat ) )
                {
                  ostringstream os;
                  os << "The path exits the atmosphere through the upper "
                     << "latitude end face.\nThe exit point is at an altitude" 
                     << " of " << ppath_step.pos(n-1,0)/1e3 << " km.";
                  throw runtime_error( os.str() );
                }
 
              // Longitude 
              // Note that it must be if and else if here. Otherwise e.g. -180 
              // will be shifted to 180 and then later back to -180.
              if( is_gridpos_at_index_i( ppath_step.gp_lon[n-1], 0 )  &&
                  ppath_step.los(n-1,1) < 0  &&  
                  abs( ppath_step.pos(n-1,1) ) < 90 )
                {
                  // Check if the longitude point can be shifted +360 degrees
                  if( lon_grid[imax_lon] - lon_grid[0] >= 360 )
                    {
                      ppath_step.pos(n-1,2) = ppath_step.pos(n-1,2) + 360;
                      gridpos( ppath_step.gp_lon[n-1], lon_grid, 
                                                       ppath_step.pos(n-1,2) );
                    }
                  else
                    {
                      ostringstream os;
                      os << "The path exits the atmosphere through the lower " 
                         << "longitude end face.\nThe exit point is at an "
                         << "altitude of " << ppath_step.pos(n-1,0)/1e3 
                         << " km.";
                      throw runtime_error( os.str() );
                    }
                }
              else if( 
                   is_gridpos_at_index_i( ppath_step.gp_lon[n-1], imax_lon ) &&
                   ppath_step.los(n-1,1) > 0  &&  
                   abs( ppath_step.pos(n-1,1) ) < 90 )
                {
                  // Check if the longitude point can be shifted -360 degrees
                  if( lon_grid[imax_lon] - lon_grid[0] >= 360 )
                    {
                      ppath_step.pos(n-1,2) = ppath_step.pos(n-1,2) - 360;
                      gridpos( ppath_step.gp_lon[n-1], lon_grid, 
                                                       ppath_step.pos(n-1,2) );
                    }
                  else
                    {
                      ostringstream os;
                      os << "The path exits the atmosphere through the upper "
                         << "longitude end face.\nThe exit point is at an "
                         << "altitude of " << ppath_step.pos(n-1,0)/1e3 
                         << " km.";
                      throw runtime_error( os.str() );
                    }
                }
            }
          
        
          // Check if there is an intersection with an active cloud box
          if( cloudbox_on )
            {
              Numeric ipos = fractional_gp( ppath_step.gp_p[n-1] );
 
              if( ipos >= Numeric( cloudbox_limits[0] )  && 
                  ipos <= Numeric( cloudbox_limits[1] ) )
                {
                  if( atmosphere_dim == 1 )
                    { ppath_set_background( ppath_step, 3 ); }
                  else
                    {
                      ipos = fractional_gp( ppath_step.gp_lat[n-1] );
 
                      if( ipos >= Numeric( cloudbox_limits[2] )  && 
                          ipos <= Numeric( cloudbox_limits[3] ) )
                        {
                          if( atmosphere_dim == 2 )
                            { ppath_set_background( ppath_step, 3 ); }
                          else
                            {
                          
                              ipos = fractional_gp( ppath_step.gp_lon[n-1] );
 
                              if( ipos >= Numeric( cloudbox_limits[4] )  && 
                                  ipos <= Numeric( cloudbox_limits[5] ) )
                                { ppath_set_background( ppath_step, 3 ); } 
                            }
                        }
                    }
                }
            }
        }
 
      //--- Inside cloud box -------------------------------------------------
      else
        {
          // A first version just checked if point was at or outside any
          // boundary but numerical problems could cause that the start point
          // was taken as the exit point. So check of ppath direction had to be
          // added. Fractional distances used for this. 
 
          // Pressure dimension
          Numeric ipos1 = fractional_gp( ppath_step.gp_p[n-1] );
          Numeric ipos2 = fractional_gp( ppath_step.gp_p[n-2] );
          assert( ipos1 >= cloudbox_limits[0] );
          assert( ipos1 <= cloudbox_limits[1] );
          if( ipos1 <= cloudbox_limits[0]  &&  ipos1 < ipos2 )
            { ppath_set_background( ppath_step, 3 ); }
              
          else if( ipos1 >= Numeric( cloudbox_limits[1] )  &&  ipos1 > ipos2 )
            { ppath_set_background( ppath_step, 3 ); }
 
          else if( atmosphere_dim > 1 )
            {
              // Latitude dimension
              ipos1 = fractional_gp( ppath_step.gp_lat[n-1] );
              ipos2 = fractional_gp( ppath_step.gp_lat[n-2] );
              assert( ipos1 >= cloudbox_limits[2] );
              assert( ipos1 <= cloudbox_limits[3] );
              if( ipos1 <= Numeric( cloudbox_limits[2] )  &&  ipos1 < ipos2 )  
                { ppath_set_background( ppath_step, 3 ); }
 
              else if( ipos1 >= Numeric( cloudbox_limits[3] ) && ipos1 > ipos2 )
                { ppath_set_background( ppath_step, 3 ); }
 
              else if ( atmosphere_dim > 2 )
                {
                  // Longitude dimension
                  ipos1 = fractional_gp( ppath_step.gp_lon[n-1] );
                  ipos2 = fractional_gp( ppath_step.gp_lon[n-2] );
                  assert( ipos1 >= cloudbox_limits[4] );
                  assert( ipos1 <= cloudbox_limits[5] );
                  if( ipos1 <= Numeric( cloudbox_limits[4] )  &&  ipos1<ipos2 )
                    { ppath_set_background( ppath_step, 3 ); }
 
                  else if( ipos1 >= Numeric( cloudbox_limits[5] )  &&
                           ipos1 > ipos2 )
                    { ppath_set_background( ppath_step, 3 ); }
                }
            }
        }
      //----------------------------------------------------------------------
      //---  End of boundary check
      //----------------------------------------------------------------------
 
      // Ready?
      if( ppath_what_background( ppath_step ) )
        {
          ppath_step.start_pos = ppath_step.pos(n-1,joker);
          ppath_step.start_los = ppath_step.los(n-1,joker);
          ready = true;
        }
 
      // Put new ppath_step in ppath_array
      ppath_array.push_back( ppath_step );
 
    } // End path steps
 
  // Combine all structures in ppath_array to form the return Ppath structure.
  //
  ppath_init_structure( ppath, atmosphere_dim, np );
  //
  const Index na = ppath_array.nelem();
  //
  if( na == 0 )    // No path, just the starting point
    {
      ppath_copy( ppath, ppath_step, 1 );
      // To set n for positions inside the atmosphere, ppath_step_agenda
      // must be called once. The later is not always the case. A fix to handle
      // those cases:  
      if( ppath_what_background(ppath_step) > 1 )
        {
          //Print( ppath_step, 0, verbosity );
          ppath_step_agendaExecute( ws, ppath_step, ppath_lraytrace, t_field, 
                                    z_field, vmr_field, f_grid, 
                                    ppath_step_agenda );
          ppath.nreal[0]  = ppath_step.nreal[0];
          ppath.ngroup[0] = ppath_step.ngroup[0];
        }
    }
 
  else   // Otherwise, merge the array elelments
    {
      np = 0;   // Now used as counter for points moved to ppath
      //
      for( Index i=0; i<na; i++ )
        {
          // For the first structure, the first point shall be included.
          // For later structures, the first point shall not be included, but
          // there will always be at least two points.
          
          Index n = ppath_array[i].np;
 
          // First index to include
          Index i1 = 1;
          if( i == 0 )
            { i1 = 0; }
 
          // Vectors and matrices that can be handled by ranges.
          ppath.r[ Range(np,n-i1) ] = ppath_array[i].r[ Range(i1,n-i1) ];
          ppath.pos( Range(np,n-i1), joker ) = 
                                   ppath_array[i].pos( Range(i1,n-i1), joker );
          ppath.los( Range(np,n-i1), joker ) = 
                                   ppath_array[i].los( Range(i1,n-i1), joker );
          ppath.nreal[ Range(np,n-i1) ] = 
                                        ppath_array[i].nreal[ Range(i1,n-i1) ];
          ppath.ngroup[ Range(np,n-i1) ] = 
                                       ppath_array[i].ngroup[ Range(i1,n-i1) ];
          ppath.lstep[ Range(np-i1,n-1) ] = ppath_array[i].lstep; 
 
          // Grid positions must be handled by a loop
          for( Index j=i1; j<n; j++ )
            { ppath.gp_p[np+j-i1] = ppath_array[i].gp_p[j]; }
          if( atmosphere_dim >= 2 )
            {
              for( Index j=i1; j<n; j++ )
                { ppath.gp_lat[np+j-i1] = ppath_array[i].gp_lat[j]; }
            }
          if( atmosphere_dim == 3 )
            {
              for( Index j=i1; j<n; j++ )
                { ppath.gp_lon[np+j-i1] = ppath_array[i].gp_lon[j]; }
            }
 
          // Increase number of points done
          np += n - i1;
        }
 
      // Field just included once:
      // end_pos/los/lspace from first path_step (extracted above):
      ppath.end_lstep = end_lstep;
      ppath.end_pos   = end_pos;
      ppath.end_los   = end_los;
      // Constant, background and start_pos/los from last path_step:
      ppath.constant    = ppath_step.constant;
      ppath.background  = ppath_step.background;
      ppath.start_pos   = ppath_step.start_pos;
      ppath.start_los   = ppath_step.start_los;
      ppath.start_lstep = ppath_step.start_lstep;
    }
}