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. */
 
/*===========================================================================
  === External declarations
  ===========================================================================*/
 
#include "ppath.h"
#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 "logic.h"
#include "math_funcs.h"
#include "messages.h"
#include "mystring.h"
#include "poly_roots.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.
//
DEBUG_ONLY(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 l =
      sqrt(vrot[0] * vrot[0] + vrot[1] * vrot[1] + vrot[2] * vrot[2]);
  assert(l > 1 - 9);
  const Numeric u = vrot[0] / l;
  const Numeric v = vrot[1] / l;
  const Numeric w = vrot[2] / l;
  const Numeric u2 = u * u;
  const Numeric v2 = v * v;
  const Numeric w2 = w * w;
  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 add_za_aa(Numeric& za,
               Numeric& aa,
               const Numeric& za0,
               const Numeric& aa0,
               const Numeric& dza,
               const Numeric& daa) {
  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 + dza, aa0 + daa);
 
  // 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]);
}
 
void diff_za_aa(Numeric& dza,
                Numeric& daa,
                const Numeric& za0,
                const Numeric& aa0,
                const Numeric& za,
                const Numeric& aa) {
  Vector xyz(3);
  Vector vrot(3);
  Vector u(3);
 
  assert(za != 0 && za != 180);
 
  // Unit vector towards aa0 at za=90
  //
  zaaa2cart(xyz[0], xyz[1], xyz[2], za0, 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], za, aa);
 
  // Apply rotation
  //
  Matrix R(3, 3);
  rotationmat3D(R, vrot, -(za0 - 90));
  mult(u, R, xyz);
 
  // Calculate za and aa for rotated u
  //
  Numeric za_tmp, aa_tmp;
  cart2zaaa(za_tmp, aa_tmp, u[0], u[1], u[2]);
 
  // Calculate dza and daa
  dza = za_tmp - 90;
  daa = aa_tmp - aa0;
}
 
/*===========================================================================
  === 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;
  }
}
 
Index first_pos_before_altitude(const Ppath& p, const Numeric& alt) {
  // Checker flags
  bool below = false, above = false;
 
  // Check first pos before altitude
  for (Index i = 0; i < p.np; i++) {
    if (p.pos(i, 0) < alt) {
      if (above) return i - 1;
      below = true;
    } else {
      if (below) return i - 1;
      above = true;
    }
  }
 
  return -1;
}
 
void error_if_limb_ppath(const Ppath& ppath) {
  if (ppath.np > 2) {
    Numeric signfac = sign(ppath.pos(1, 0) - ppath.pos(0, 0));
    for (Index i = 2; i < ppath.np; i++) {
      if (signfac * (ppath.pos(i, 0) - ppath.pos(i - 1, 0)) < 0)
        throw runtime_error(
            "A propagation path of limb character found. Such "
            "viewing geometries are not supported by this method. "
            "Propagation paths must result in strictly "
            "increasing or decreasing altitudes.");
    }
  }
}
 
/*===========================================================================
  = 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 < 10);
 
  // 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 < 10);
 
  // 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));
 
  // Assertthat ppc and path data are consustent
  assert(abs(ppc - r_start0 * sin(DEG2RAD * za_start)) < 0.1);
 
  // 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,
                x,
                y,
                z,
                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,
                  x,
                  y,
                  z,
                  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 >= (Numeric)cloudbox_limits[0] &&
            fgp <= (Numeric)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 >= (Numeric)cloudbox_limits[0] &&
               fgp <= (Numeric)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 >= (Numeric)cloudbox_limits[0] &&
            fgp <= (Numeric)cloudbox_limits[1] &&
            fgl >= (Numeric)cloudbox_limits[2] &&
            fgl <= (Numeric)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 >= (Numeric)cloudbox_limits[0] &&
               fgp <= (Numeric)cloudbox_limits[1] &&
               fgl >= (Numeric)cloudbox_limits[2] &&
               fgl <= (Numeric)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 >= (Numeric)cloudbox_limits[0] &&
            fgp <= (Numeric)cloudbox_limits[1] &&
            fgl >= (Numeric)cloudbox_limits[2] &&
            fgl <= (Numeric)cloudbox_limits[3] &&
            fgo >= (Numeric)cloudbox_limits[4] &&
            fgo <= (Numeric)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 >= (Numeric)cloudbox_limits[0] &&
               fgp <= (Numeric)cloudbox_limits[1] &&
               fgl >= (Numeric)cloudbox_limits[2] &&
               fgl <= (Numeric)cloudbox_limits[3] &&
               fgo >= (Numeric)cloudbox_limits[4] &&
               fgo <= (Numeric)cloudbox_limits[5]);
      })
    }
 
    // 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;
      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, 2) = rte_pos[2];
        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,
                      x,
                      y,
                      z,
                      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& z_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_lmax,
                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_lmax,
                             ppath_lraytrace,
                             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 > (Index)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 >= (Numeric)cloudbox_limits[0]);
      assert(ipos1 <= (Numeric)cloudbox_limits[1]);
      if (ipos1 <= (Numeric)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 >= (Numeric)cloudbox_limits[2]);
        assert(ipos1 <= (Numeric)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 >= (Numeric)cloudbox_limits[4]);
          assert(ipos1 <= (Numeric)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_lmax,
                               ppath_lraytrace,
                               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;
  }
}