sensor.cc

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/* Copyright (C) 2003-2012 Mattias Ekstr�m <ekstrom@rss.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 "sensor.h"
#include <cmath>
#include <list>
#include <stdexcept>
#include "arts.h"
#include "logic.h"
#include "matpackI.h"
#include "matpackII.h"
#include "messages.h"
#include "sorting.h"
 
extern const Numeric PI;
extern const Numeric NAT_LOG_2;
extern const Numeric DEG2RAD;
extern const Index GFIELD1_F_GRID;
extern const Index GFIELD4_FIELD_NAMES;
extern const Index GFIELD4_F_GRID;
extern const Index GFIELD4_ZA_GRID;
extern const Index GFIELD4_AA_GRID;
 
/*===========================================================================
  === The functions, besides the core integration and sum functions that are
  === placed together at the end
  ===========================================================================*/
 
void antenna1d_matrix(Sparse& H,
#ifndef NDEBUG
                      const Index& antenna_dim,
#else
                      const Index& antenna_dim _U_,
#endif
                      ConstVectorView antenna_dza,
                      const GriddedField4& antenna_response,
                      ConstVectorView za_grid,
                      ConstVectorView f_grid,
                      const Index n_pol,
                      const Index do_norm) {
  // Number of input pencil beam directions and frequency
  const Index n_za = za_grid.nelem();
  const Index n_f = f_grid.nelem();
 
  // Calculate number of antenna beams
  const Index n_ant = antenna_dza.nelem();
 
  // Asserts for variables beside antenna_response
  ARTS_ASSERT(antenna_dim == 1);
  ARTS_ASSERT(n_za >= 2);
  ARTS_ASSERT(n_pol >= 1);
  ARTS_ASSERT(do_norm >= 0 && do_norm <= 1);
 
  // Extract antenna_response grids
  const Index n_ar_pol =
      antenna_response.get_string_grid(GFIELD4_FIELD_NAMES).nelem();
  ConstVectorView aresponse_f_grid =
      antenna_response.get_numeric_grid(GFIELD4_F_GRID);
  ConstVectorView aresponse_za_grid =
      antenna_response.get_numeric_grid(GFIELD4_ZA_GRID);
  DEBUG_ONLY(const Index n_ar_aa =
                 antenna_response.get_numeric_grid(GFIELD4_AA_GRID).nelem();)
 
  //
  const Index n_ar_f = aresponse_f_grid.nelem();
  const Index n_ar_za = aresponse_za_grid.nelem();
  const Index pol_step = n_ar_pol > 1;
 
  // Asserts for antenna_response
  ARTS_ASSERT(n_ar_pol == 1 || n_ar_pol >= n_pol);
  ARTS_ASSERT(n_ar_f);
  ARTS_ASSERT(n_ar_za > 1);
  ARTS_ASSERT(n_ar_aa == 1);
 
  // If response data extend outside za_grid is checked in
  // integration_func_by_vecmult
 
  // Some size(s)
  const Index nfpol = n_f * n_pol;
 
  // Resize H
  H.resize(n_ant * nfpol, n_za * nfpol);
 
  // Storage vectors for response weights
  Vector hrow(H.ncols(), 0.0);
  Vector hza(n_za, 0.0);
 
  // Antenna response to apply (possibly obtained by frequency interpolation)
  Vector aresponse(n_ar_za, 0.0);
 
  // Antenna beam loop
  for (Index ia = 0; ia < n_ant; ia++) {
    Vector shifted_aresponse_za_grid = aresponse_za_grid;
    shifted_aresponse_za_grid += antenna_dza[ia];
 
    // Order of loops assumes that the antenna response more often
    // changes with frequency than for polarisation
 
    // Frequency loop
    for (Index f = 0; f < n_f; f++) {
      // Polarisation loop
      for (Index ip = 0; ip < n_pol; ip++) {
        // Determine antenna pattern to apply
        //
        // Interpolation needed only if response has a frequency grid
        //
        Index new_antenna = 1;
        //
        if (n_ar_f == 1)  // No frequency variation
        {
          if (pol_step)  // Polarisation variation, update always needed
          {
            aresponse = antenna_response.data(ip, 0, joker, 0);
          } else if (f == 0 && ip == 0)  // Set fully constant pattern
          {
            aresponse = antenna_response.data(0, 0, joker, 0);
          } else  // The one set just above can be reused
          {
            new_antenna = 0;
          }
        } else {
          if (ip == 0 || pol_step)  // Interpolation required
          {
            // Interpolation (do this in "green way")
            ArrayOfGridPos gp_f(1), gp_za(n_ar_za);
            gridpos(gp_f, aresponse_f_grid, Vector(1, f_grid[f]));
            gridpos(gp_za, aresponse_za_grid, aresponse_za_grid);
            Tensor3 itw(1, n_ar_za, 4);
            interpweights(itw, gp_f, gp_za);
            Matrix aresponse_matrix(1, n_ar_za);
            interp(aresponse_matrix,
                   itw,
                   antenna_response.data(ip, joker, joker, 0),
                   gp_f,
                   gp_za);
            aresponse = aresponse_matrix(0, joker);
          } else  // Reuse pattern for ip==0
          {
            new_antenna = 0;
          }
        }
 
        // Calculate response weights
        if (new_antenna) {
          integration_func_by_vecmult(
              hza, aresponse, shifted_aresponse_za_grid, za_grid);
 
          // Normalisation?
          if (do_norm) {
            hza /= hza.sum();
          }
        }
 
        // Put weights into H
        //
        const Index ii = f * n_pol + ip;
        //
        hrow[Range(ii, n_za, nfpol)] = hza;
        //
        H.insert_row(ia * nfpol + ii, hrow);
        //
        hrow = 0;
      }
    }
  }
}
 
 
 
void antenna2d_gridded_dlos(Sparse& H,
#ifndef NDEBUG
                            const Index& antenna_dim,
#else
                            const Index& antenna_dim _U_,
#endif
                            ConstMatrixView antenna_dlos,
                            const GriddedField4& antenna_response,
                            ConstMatrixView mblock_dlos,
                            ConstVectorView f_grid,
                            const Index n_pol)
{
  // Number of input pencil beam directions and frequency
  const Index n_dlos = mblock_dlos.nrows();
  const Index n_f = f_grid.nelem();
 
  // Decompose mblock_dlos into za and aa grids, including checking
  ARTS_USER_ERROR_IF (mblock_dlos.ncols() != 2 ,
                      "For the gridded_dlos option, *mblock_dlos_grid* "
                      "must have two columns.");
 
 
  Index nza = 1;
  for(Index i=0; i<n_dlos-1 && mblock_dlos(i+1,0) > mblock_dlos(i,0); i++ ) {
    nza++;
  }
  ARTS_USER_ERROR_IF(nza < 2,
                     "For the gridded_dlos option, the number of za angles "
                     "(among dlos directions) must be >= 2.");
  ARTS_USER_ERROR_IF (n_dlos % nza > 0,
                      "For the gridded_dlos option, the number of dlos angles "
                      "must be a product of two integers.");
  const Index naa = n_dlos / nza; 
  const Vector za_grid = mblock_dlos(Range(0,nza),0);
  const Vector aa_grid = mblock_dlos(Range(0,naa,nza),1);
  for(Index i=0; i<n_dlos; i++ ) {
    ARTS_USER_ERROR_IF(i>=nza && abs(mblock_dlos(i,0)-mblock_dlos(i-nza,0)) > 1e-6 ,
        "Zenith angle of dlos ", i, " (0-based) differs to zenith " 
        "angle of dlos ", i-nza, ", while they need to be equal "
        "to form rectangular grid.")
    ARTS_USER_ERROR_IF(abs(mblock_dlos(i,1)-aa_grid[i/nza]) > 1e-6,
        "Azimuth angle of dlos ", i, " (0-based) differs to azimuth " 
        "angle ", (i/nza)*nza , ", while they need to be equal "
        "to form rectangular grid.")
  }
 
  // Calculate number of antenna beams
  const Index n_ant = antenna_dlos.nrows();
 
  // Asserts for variables beside antenna_response
  ARTS_ASSERT(antenna_dim == 2);
  ARTS_ASSERT(n_dlos >= 1);
  ARTS_ASSERT(n_pol >= 1);
 
  // Extract antenna_response grids
  const Index n_ar_pol =
      antenna_response.get_string_grid(GFIELD4_FIELD_NAMES).nelem();
  ConstVectorView aresponse_f_grid =
      antenna_response.get_numeric_grid(GFIELD4_F_GRID);
  ConstVectorView aresponse_za_grid =
      antenna_response.get_numeric_grid(GFIELD4_ZA_GRID);
  ConstVectorView aresponse_aa_grid =
      antenna_response.get_numeric_grid(GFIELD4_AA_GRID);
  //
  const Index n_ar_f = aresponse_f_grid.nelem();
  const Index n_ar_za = aresponse_za_grid.nelem();
  const Index n_ar_aa = aresponse_aa_grid.nelem();
  const Index pol_step = n_ar_pol > 1;
 
  // Asserts for antenna_response
  ARTS_ASSERT(n_ar_pol == 1 || n_ar_pol >= n_pol);
  ARTS_ASSERT(n_ar_f);
  ARTS_ASSERT(n_ar_za > 1);
  ARTS_ASSERT(n_ar_aa > 1);
  ARTS_ASSERT(antenna_response.data.ncols() == n_ar_aa );
  ARTS_ASSERT(antenna_response.data.nrows() == n_ar_za );
  ARTS_ASSERT(antenna_response.data.npages() == n_ar_f );
  ARTS_ASSERT(antenna_response.data.nbooks() == n_ar_pol );
 
  // Include cos(za)-term in response
  Tensor4 aresponse_with_cos(n_ar_pol, n_ar_f, n_ar_za, n_ar_aa);
  for(Index i3=0; i3<n_ar_za; i3++) {
    const Numeric t = cos(DEG2RAD * aresponse_za_grid[i3]);
    for(Index i4=0; i4<n_ar_aa; i4++) {
      for(Index i2=0; i2<n_ar_f; i2++) {
        for(Index i1=0; i1<n_ar_pol; i1++) {
          aresponse_with_cos(i1,i2,i3,i4) = t * antenna_response.data(i1,i2,i3,i4);
        }
      }
    }
  }
 
  // Some size(s)
  const Index nfpol = n_f * n_pol;
 
  // Resize H
  H.resize(n_ant * nfpol, n_dlos * nfpol);
 
  // Storage vectors for response weights
  Vector hrow(H.ncols(), 0.0);
  Vector hza(n_dlos, 0.0);
 
  // Antenna response to apply (possibly obtained by frequency interpolation)
  Matrix aresponse(n_ar_za, n_ar_aa, 0.0);
 
  // If you find a bug or change something, likely also change other 2D antenna
  // function(s) as they are similar
  
  // Antenna beam loop
  for (Index ia = 0; ia < n_ant; ia++) {
    // Order of loops assumes that the antenna response more often
    // changes with frequency than for polarisation
 
    // Frequency loop
    for (Index f = 0; f < n_f; f++) {
      // Polarisation loop
      for (Index ip = 0; ip < n_pol; ip++) {
        // Determine antenna pattern to apply
        //
        // Interpolation needed only if response has a frequency grid
        //
        Index new_antenna = 1;
        //
        if (n_ar_f == 1)  // No frequency variation
        {
          if (pol_step)  // Polarisation variation, update always needed
          {
            aresponse = aresponse_with_cos(ip, 0, joker, joker);
          } else if (f == 0 && ip == 0)  // Set fully constant pattern
          {
            aresponse = aresponse_with_cos(0, 0, joker, joker);
          } else  // The one set just above can be reused
          {
            new_antenna = 0;
          }
        } else {
          if (ip == 0 || pol_step) {
            // Interpolation (do this in "green way")
            ArrayOfGridPos gp_f(1), gp_za(n_ar_za), gp_aa(n_ar_aa);
            gridpos(gp_f, aresponse_f_grid, Vector(1, f_grid[f]));
            gridpos(gp_za, aresponse_za_grid, aresponse_za_grid);
            gridpos(gp_aa, aresponse_aa_grid, aresponse_aa_grid);
            Tensor4 itw(1, n_ar_za, n_ar_aa, 8);
            interpweights(itw, gp_f, gp_za, gp_aa);
            Tensor3 aresponse_matrix(1, n_ar_za, n_ar_aa);
            interp(aresponse_matrix,
                   itw,
                   aresponse_with_cos(ip, joker, joker, joker),
                   gp_f,
                   gp_za,
                   gp_aa);
            aresponse = aresponse_matrix(0, joker, joker);
          } else  // Reuse pattern for ip==0
          {
            new_antenna = 0;
          }
        }
 
        // Calculate response weights, by using grid positions and "itw"
        if (new_antenna) {
          
          // za grid positions
          Vector zas = aresponse_za_grid;
          zas += antenna_dlos(ia, 0);
          ARTS_USER_ERROR_IF( zas[0] < za_grid[0],
              "The zenith angle grid in *mblock_dlos_grid* is too narrow. " 
              "It must be extended downwards with at least ",
              za_grid[0]-zas[0], " deg.")
          ARTS_USER_ERROR_IF( zas[n_ar_za-1] > za_grid[nza-1],
              "The zenith angle grid in *mblock_dlos_grid* is too narrow. " 
              "It must be extended upwards with at least ",
              zas[n_ar_za-1] - za_grid[nza-1], " deg.")
          
          ArrayOfGridPos gp_za(n_ar_za);
          gridpos(gp_za, za_grid, zas);
          
          // aa grid positions
          Vector aas = aresponse_aa_grid;
          if (antenna_dlos.ncols() > 1) { aas += antenna_dlos(ia, 1); }              
          ARTS_USER_ERROR_IF( aas[0] < aa_grid[0],
              "The azimuth angle grid in *mblock_dlos_grid* is too narrow. " 
              "It must be extended downwards with at least ",
              aa_grid[0]-aas[0], " deg.")
          ARTS_USER_ERROR_IF( aas[n_ar_aa-1] > aa_grid[naa-1],
              "The azimuth angle grid in *mblock_dlos_grid* is too narrow. " 
              "It must be extended upwards with at least ",
              aas[n_ar_aa-1] - aa_grid[naa-1], " deg.")
          
          ArrayOfGridPos gp_aa(n_ar_aa);
          gridpos(gp_aa, aa_grid, aas);
 
 
          // Derive interpolation weights
          Tensor3 itw(n_ar_za, n_ar_za, 4);
          interpweights(itw, gp_za, gp_aa);
 
          // Convert iwt to weights for H
          //
          hza = 0;   // Note that values in hza must be accumulated 
          //
          for (Index iaa = 0; iaa < n_ar_aa; iaa++) {
            const Index a = gp_aa[iaa].idx;
            const Index b = a + 1;
            
            for (Index iza = 0; iza < n_ar_za; iza++) {          
 
              const Index z = gp_za[iza].idx;
              const Index x = z + 1;
 
              if( itw(iza,iaa,0) > 1e-9 ) {
                hza[a*nza+z] += aresponse(iza,iaa) * itw(iza,iaa,0);
              }
              if( itw(iza,iaa,1) > 1e-9 ) {
                hza[b*nza+z] += aresponse(iza,iaa) * itw(iza,iaa,1);
              }
              if( itw(iza,iaa,2) > 1e-9 ) {
                hza[a*nza+x] += aresponse(iza,iaa) * itw(iza,iaa,2);
              }
              if( itw(iza,iaa,3) > 1e-9 ) {
                hza[b*nza+x] += aresponse(iza,iaa) * itw(iza,iaa,3);
              }
            }
          }
 
          // For 2D antennas we always normalise
          hza /= hza.sum();
        }
 
        // Put weights into H
        //
        const Index ii = f * n_pol + ip;
        //
        hrow[Range(ii, n_dlos, nfpol)] = hza;
        //
        H.insert_row(ia * nfpol + ii, hrow);
        //
        hrow = 0;
      }
    }
  }  
}
 
 
 
void antenna2d_interp_response(Sparse& H,
#ifndef NDEBUG
                               const Index& antenna_dim,
#else
                               const Index& antenna_dim _U_,
#endif
                               ConstMatrixView antenna_dlos,
                               const GriddedField4& antenna_response,
                               ConstMatrixView mblock_dlos,
                               ConstVectorView f_grid,
                               const Index n_pol)
{  
  // Number of input pencil beam directions and frequency
  const Index n_dlos = mblock_dlos.nrows();
  const Index n_f = f_grid.nelem();
 
  // Calculate number of antenna beams
  const Index n_ant = antenna_dlos.nrows();
 
  // Asserts for variables beside antenna_response
  ARTS_ASSERT(antenna_dim == 2);
  ARTS_ASSERT(n_dlos >= 1);
  ARTS_ASSERT(n_pol >= 1);
 
  // Extract antenna_response grids
  const Index n_ar_pol =
      antenna_response.get_string_grid(GFIELD4_FIELD_NAMES).nelem();
  ConstVectorView aresponse_f_grid =
      antenna_response.get_numeric_grid(GFIELD4_F_GRID);
  ConstVectorView aresponse_za_grid =
      antenna_response.get_numeric_grid(GFIELD4_ZA_GRID);
  ConstVectorView aresponse_aa_grid =
      antenna_response.get_numeric_grid(GFIELD4_AA_GRID);
  //
  const Index n_ar_f = aresponse_f_grid.nelem();
  const Index n_ar_za = aresponse_za_grid.nelem();
  const Index n_ar_aa = aresponse_aa_grid.nelem();
  const Index pol_step = n_ar_pol > 1;
 
  // Asserts for antenna_response
  ARTS_ASSERT(n_ar_pol == 1 || n_ar_pol >= n_pol);
  ARTS_ASSERT(n_ar_f);
  ARTS_ASSERT(n_ar_za > 1);
  ARTS_ASSERT(n_ar_aa > 1);
  ARTS_ASSERT(antenna_response.data.ncols() == n_ar_aa );
  ARTS_ASSERT(antenna_response.data.nrows() == n_ar_za );
  ARTS_ASSERT(antenna_response.data.npages() == n_ar_f );
  ARTS_ASSERT(antenna_response.data.nbooks() == n_ar_pol );
 
  // Include cos(za)-term in response
  Tensor4 aresponse_with_cos(n_ar_pol, n_ar_f, n_ar_za, n_ar_aa);
  for(Index i3=0; i3<n_ar_za; i3++) {
    const Numeric t = cos(DEG2RAD * aresponse_za_grid[i3]);
    for(Index i4=0; i4<n_ar_aa; i4++) {
      for(Index i2=0; i2<n_ar_f; i2++) {
        for(Index i1=0; i1<n_ar_pol; i1++) {
          aresponse_with_cos(i1,i2,i3,i4) = t * antenna_response.data(i1,i2,i3,i4);
        }
      }
    }
  }
  
  // Some size(s)
  const Index nfpol = n_f * n_pol;
 
  // Resize H
  H.resize(n_ant * nfpol, n_dlos * nfpol);
 
  // Storage vectors for response weights
  Vector hrow(H.ncols(), 0.0);
  Vector hza(n_dlos, 0.0);
 
  // Antenna response to apply (possibly obtained by frequency interpolation)
  Matrix aresponse(n_ar_za, n_ar_aa, 0.0);
 
  // If you find a bug or change something, likely also change other 2D antenna
  // function(s) as they are similar
 
  // Antenna beam loop
  for (Index ia = 0; ia < n_ant; ia++) {
    // Order of loops assumes that the antenna response more often
    // changes with frequency than for polarisation
 
    // Frequency loop
    for (Index f = 0; f < n_f; f++) {
      // Polarisation loop
      for (Index ip = 0; ip < n_pol; ip++) {
        // Determine antenna pattern to apply
        //
        // Interpolation needed only if response has a frequency grid
        //
        Index new_antenna = 1;
        //
        if (n_ar_f == 1)  // No frequency variation
        {
          if (pol_step)  // Polarisation variation, update always needed
          {
            aresponse = aresponse_with_cos(ip, 0, joker, joker);
          } else if (f == 0 && ip == 0)  // Set fully constant pattern
          {
            aresponse = aresponse_with_cos(0, 0, joker, joker);
          } else  // The one set just above can be reused
          {
            new_antenna = 0;
          }
        } else {
          if (ip == 0 || pol_step) {
            // Interpolation (do this in "green way")
            ArrayOfGridPos gp_f(1), gp_za(n_ar_za), gp_aa(n_ar_aa);
            gridpos(gp_f, aresponse_f_grid, Vector(1, f_grid[f]));
            gridpos(gp_za, aresponse_za_grid, aresponse_za_grid);
            gridpos(gp_aa, aresponse_aa_grid, aresponse_aa_grid);
            Tensor4 itw(1, n_ar_za, n_ar_aa, 8);
            interpweights(itw, gp_f, gp_za, gp_aa);
            Tensor3 aresponse_matrix(1, n_ar_za, n_ar_aa);
            interp(aresponse_matrix,
                   itw,
                   aresponse_with_cos(ip, joker, joker, joker),
                   gp_f,
                   gp_za,
                   gp_aa);
            aresponse = aresponse_matrix(0, joker, joker);
          } else  // Reuse pattern for ip==0
          {
            new_antenna = 0;
          }
        }
 
        // Calculate response weights
        if (new_antenna) {
          for (Index l = 0; l < n_dlos; l++) {
            const Numeric za = mblock_dlos(l, 0) - antenna_dlos(ia, 0);
            Numeric aa = 0.0;
            if (mblock_dlos.ncols() > 1) {
              aa += mblock_dlos(l, 1);
            }
            if (antenna_dlos.ncols() > 1) {
              aa -= antenna_dlos(ia, 1);
            }
 
            // The response is zero if mblock_dlos is outside of
            // antennna pattern
            if (za < aresponse_za_grid[0] ||
                za > aresponse_za_grid[n_ar_za - 1] ||
                aa < aresponse_aa_grid[0] ||
                aa > aresponse_aa_grid[n_ar_aa - 1]) {
              hza[l] = 0;
            }
            // Otherwise we make an (blue) interpolation
            else {
              ArrayOfGridPos gp_za(1), gp_aa(1);
              gridpos(gp_za, aresponse_za_grid, Vector(1, za));
              gridpos(gp_aa, aresponse_aa_grid, Vector(1, aa));
              Matrix itw(1, 4);
              interpweights(itw, gp_za, gp_aa);
              Vector value(1);
              interp(value, itw, aresponse, gp_za, gp_aa);
              hza[l] = value[0];
            }
          }
 
          // For 2D antennas we always normalise
          hza /= hza.sum();
        }
 
        // Put weights into H
        //
        const Index ii = f * n_pol + ip;
        //
        hrow[Range(ii, n_dlos, nfpol)] = hza;
        //
        H.insert_row(ia * nfpol + ii, hrow);
        //
        hrow = 0;
      }
    }
  }
}
 
 
 
void gaussian_response_autogrid(Vector& x,
                                Vector& y,
                                const Numeric& x0,
                                const Numeric& fwhm,
                                const Numeric& xwidth_si,
                                const Numeric& dx_si) {
  ARTS_ASSERT(dx_si <= xwidth_si);
 
  const Numeric si = fwhm / (2 * sqrt(2 * NAT_LOG_2));
 
  // Number of points needed to enure that spacing is max dx_si
  const Index n = (Index)floor(2 * xwidth_si / dx_si) + 1;
 
  // Grid for response
  const Numeric dd = si * xwidth_si;
  nlinspace(x, -dd, dd, n);
 
  // Calculate response
  gaussian_response(y, x, x0, fwhm);
}
 
 
 
void gaussian_response(Vector& y,
                       const Vector& x,
                       const Numeric& x0,
                       const Numeric& fwhm) {
  const Numeric si = fwhm / (2 * sqrt(2 * NAT_LOG_2));
  const Numeric a = 1 / (si * sqrt(2 * PI));
  const Index n = x.nelem();
 
  y.resize(n);
  //
  for (Index i = 0; i < n; i++)
    y[i] = a * exp(-0.5 * pow((x[i] - x0) / si, 2.0));
}
 
 
 
void mixer_matrix(Sparse& H,
                  Vector& f_mixer,
                  const Numeric& lo,
                  const GriddedField1& filter,
                  ConstVectorView f_grid,
                  const Index& n_pol,
                  const Index& n_sp,
                  const Index& do_norm) {
  // Frequency grid of for sideband response specification
  ConstVectorView filter_grid = filter.get_numeric_grid(GFIELD1_F_GRID);
 
  DEBUG_ONLY(const Index nrp = filter.data.nelem();)
 
  // Asserts
  ARTS_ASSERT(lo > f_grid[0]);
  ARTS_ASSERT(lo < last(f_grid));
  ARTS_ASSERT(filter_grid.nelem() == nrp);
  ARTS_ASSERT(fabs(last(filter_grid) + filter_grid[0]) < 1e3);
  // If response data extend outside f_grid is checked in summation_by_vecmult
 
  // Find indices in f_grid where f_grid is just below and above the
  // lo frequency.
  /*
  Index i_low = 0, i_high = f_grid.nelem()-1, i_mean;
  while( i_high-i_low > 1 )
    {
      i_mean = (Index) (i_high+i_low)/2;
      if (f_grid[i_mean]<lo)
        { 
          i_low = i_mean; 
        }
      else
        {
          i_high = i_mean;
        }
    }
  if (f_grid[i_high]==lo)
    {
      i_high++;
    }
  const Numeric lim_low  = max( lo-f_grid[i_low], f_grid[i_high]-lo );
  */
 
  // Determine IF limits for new frequency grid
  const Numeric lim_low = 0;
  const Numeric lim_high = -filter_grid[0];
 
  // Convert sidebands to IF and use list to make a unique sorted
  // vector, this sorted vector is f_mixer.
  list<Numeric> l_mixer;
  for (Index i = 0; i < f_grid.nelem(); i++) {
    if (fabs(f_grid[i] - lo) >= lim_low && fabs(f_grid[i] - lo) <= lim_high) {
      l_mixer.push_back(fabs(f_grid[i] - lo));
    }
  }
  l_mixer.push_back(lim_high);  // Not necessarily a point in f_grid
  l_mixer.sort();
  l_mixer.unique();
  f_mixer.resize((Index)l_mixer.size());
  Index e = 0;
  for (list<Numeric>::iterator li = l_mixer.begin(); li != l_mixer.end();
       li++) {
    f_mixer[e] = *li;
    e++;
  }
 
  // Resize H
  H.resize(f_mixer.nelem() * n_pol * n_sp, f_grid.nelem() * n_pol * n_sp);
 
  // Calculate the sensor summation vector and insert the values in the
  // final matrix taking number of polarisations and zenith angles into
  // account.
  Vector row_temp(f_grid.nelem());
  Vector row_final(f_grid.nelem() * n_pol * n_sp);
  //
  Vector if_grid = f_grid;
  if_grid -= lo;
  //
  for (Index i = 0; i < f_mixer.nelem(); i++) {
    summation_by_vecmult(
        row_temp, filter.data, filter_grid, if_grid, f_mixer[i], -f_mixer[i]);
 
    // Normalise if flag is set
    if (do_norm) row_temp /= row_temp.sum();
 
    // Loop over number of polarisations
    for (Index p = 0; p < n_pol; p++) {
      // Loop over number of zenith angles/antennas
      for (Index a = 0; a < n_sp; a++) {
        // Distribute elements of row_temp to row_final.
        row_final = 0.0;
        row_final[Range(
            a * f_grid.nelem() * n_pol + p, f_grid.nelem(), n_pol)] = row_temp;
        H.insert_row(a * f_mixer.nelem() * n_pol + p + i * n_pol, row_final);
      }
    }
  }
}
 
 
void mueller_rotation(Sparse& H,
                      const Index& stokes_dim,
                      const Numeric& rotangle) {
  ARTS_ASSERT(stokes_dim > 1);
  ARTS_ASSERT(H.nrows() == stokes_dim);
  ARTS_ASSERT(H.ncols() == stokes_dim);
  ARTS_ASSERT(H(0, 1) == 0);
  ARTS_ASSERT(H(1, 0) == 0);
 
  H.rw(0, 0) = 1;
  const Numeric a = cos(2 * DEG2RAD * rotangle);
  H.rw(1, 1) = a;
  if (stokes_dim > 2) {
    ARTS_ASSERT(H(2, 0) == 0);
    ARTS_ASSERT(H(0, 2) == 0);
 
    const Numeric b = sin(2 * DEG2RAD * rotangle);
    H.rw(1, 2) = b;
    H.rw(2, 1) = -b;
    H.rw(2, 2) = a;
    if (stokes_dim > 3) {
      // More values should be checked, but to save time we just ARTS_ASSERT one
      ARTS_ASSERT(H(2, 3) == 0);
      H.rw(3, 3) = 1;
    }
  }
}
 
 
 
void met_mm_polarisation_hmatrix(Sparse& H,
                                 const ArrayOfString& mm_pol,
                                 const Numeric dza,
                                 const Index stokes_dim,
                                 const String& iy_unit) {
  ARTS_ASSERT(stokes_dim > 1);
 
  // Set "Stokes vector weights" according to iy_unit
  Numeric w = 0.5;
  if (iy_unit == "PlanckBT" || iy_unit == "RJBT") {
    w = 1.0;
  }
 
  // Identify (basic) polarisation response and possible sensor specific
  // "rotation"
  const Index nch = mm_pol.nelem();  // Number of channels
  ArrayOfString pol(nch);
  ArrayOfString rot(nch);
  for (Index i = 0; i < nch; i++) {
    if (mm_pol[i] == "AMSU-H") {
      rot[i] = "AMSU";
      pol[i] = "H";
    } else if (mm_pol[i] == "AMSU-V") {
      rot[i] = "AMSU";
      pol[i] = "V";
    } else if (mm_pol[i] == "ISMAR-H") {
      rot[i] = "ISMAR";
      pol[i] = "H";
    } else if (mm_pol[i] == "ISMAR-V") {
      rot[i] = "ISMAR";
      pol[i] = "V";
    } else if (mm_pol[i] == "MARSS-H") {
      rot[i] = "MARSS";
      pol[i] = "H";
    } else if (mm_pol[i] == "MARSS-V") {
      rot[i] = "MARSS";
      pol[i] = "V";
    } else if (mm_pol[i] == "H" || mm_pol[i] == "V" || mm_pol[i] == "LHC" ||
               mm_pol[i] == "RHC") {
      rot[i] = "none";
      pol[i] = mm_pol[i];
    } else {
      ARTS_USER_ERROR ( "Unknown polarisation ", mm_pol[i])
    }
  }
 
  // Vectors representing standard cases of sensor polarisation response
  /*
    ArrayOfVector pv;
    stokes2pol( pv, w );
    */
 
  // Complete H, for all channels
  H = Sparse(nch, nch * stokes_dim);
 
  for (Index i = 0; i < nch; i++) {
    /*
        // See stokes2pol for index order used in pv
        Index ipv = -1;
        if( pol[i] == "V" )
        { ipv = 4; }
        else if( pol[i] == "H" )
        { ipv = 5; }
        else if( pol[i] == "LHC" )  // Left hand circular
        { ipv = 8; }
        else if( pol[i] == "RHC" )  // Right hand circular
        { ipv = 9; }
        else
        { ARTS_ASSERT( 0 ); }
      */
    // See instrument_pol for index order
    Index ipol = -1;
    if (pol[i] == "V") {
      ipol = 5;
    } else if (pol[i] == "H") {
      ipol = 6;
    } else if (pol[i] == "LHC")  // Left hand circular
    {
      ipol = 9;
    } else if (pol[i] == "RHC")  // Right hand circular
    {
      ipol = 10;
    } else {
      ARTS_ASSERT(0);
    }
 
    /*
        // Maybe this error messages should be mofified
        if( pv[ipv].nelem() > stokes_dim )
        {
            ostringstream os;
            os << "You have selected a channel polarisation that is not "
            << "covered by present value of *stokes_dim* (the later has to be "
            << "increased).";
            throw runtime_error(os.str());
            }*/
 
    // No rotation, just plane polarisation response
    if (rot[i] == "none") {
      // Here we just need to fill the row H
      Vector hrow(nch * stokes_dim, 0.0);
      /* Old code, matching older version of stokes2pol:
          hrow[Range(i*stokes_dim,pv[ipv].nelem())] = pv[ipv];
          */
      stokes2pol(hrow[Range(i * stokes_dim, stokes_dim)], stokes_dim, ipol, w);
      H.insert_row(i, hrow);
    }
 
    // Rotation + pol-response
    else {
      // Rotation part
      Sparse Hrot(stokes_dim, stokes_dim);
      if (rot[i] == "AMSU") {
        // No idea about the sign. Not important if U=0,
        // but matter for U != 0.
        mueller_rotation(Hrot, stokes_dim, abs(dza));
      } else if (rot[i] == "ISMAR") {
        // No rotation at -53 (= forward direction). But no idea about the
        // sign, as for AMSU above
        mueller_rotation(Hrot, stokes_dim, dza + 50);
      } else if (rot[i] == "MARSS") {
        // MARSS special, as 48 deg between different polarisation (not 90,
        // as for ISMAR. This is best interpretation of information
        // from Stuart. Should be double-checked with him at some point.
        if (pol[i] == "H") {
          mueller_rotation(Hrot, stokes_dim, dza + 42);
        } else {
          mueller_rotation(Hrot, stokes_dim, dza);
        }
      } else {
        ARTS_ASSERT(0);
      }
 
      // H-matrix matching polarization
      Sparse Hpol(1, stokes_dim);
      { /* Old code, matching older version of stokes2pol
            Vector hrow( stokes_dim, 0.0 );
            hrow[Range(0,pv[ipv].nelem())] = pv[ipv];
            */
        Vector hrow(stokes_dim);
        stokes2pol(hrow, stokes_dim, ipol, w);
        Hpol.insert_row(0, hrow);
      }
 
      // H for the individual channel
      Sparse Hc(1, stokes_dim);
      mult(Hc, Hpol, Hrot);
 
      // Put Hc into H
      Vector hrow(nch * stokes_dim, 0.0);
      const Index i0 = i * stokes_dim;
      for (Index s = 0; s < stokes_dim; s++) {
        hrow[i0 + s] = Hc(0, s);
      }
      H.insert_row(i, hrow);
    }
  }
}
 
 
 
void sensor_aux_vectors(Vector& sensor_response_f,
                        ArrayOfIndex& sensor_response_pol,
                        Matrix& sensor_response_dlos,
                        ConstVectorView sensor_response_f_grid,
                        const ArrayOfIndex& sensor_response_pol_grid,
                        ConstMatrixView sensor_response_dlos_grid) {
  // Sizes
  const Index nf = sensor_response_f_grid.nelem();
  const Index npol = sensor_response_pol_grid.nelem();
  const Index nlos = sensor_response_dlos_grid.nrows();
  const Index n = nf * npol * nlos;
 
  // Allocate
  sensor_response_f.resize(n);
  sensor_response_pol.resize(n);
  sensor_response_dlos.resize(n, sensor_response_dlos_grid.ncols());
 
  // Fill
  for (Index ilos = 0; ilos < nlos; ilos++) {
    const Index i2 = ilos * nf * npol;
    //
    for (Index ifr = 0; ifr < nf; ifr++) {
      const Index i3 = i2 + ifr * npol;
      //
      for (Index ip = 0; ip < npol; ip++) {
        const Index i = i3 + ip;
        //
        sensor_response_f[i] = sensor_response_f_grid[ifr];
        sensor_response_pol[i] = sensor_response_pol_grid[ip];
        sensor_response_dlos(i, joker) = sensor_response_dlos_grid(ilos, joker);
      }
    }
  }
}
 
 
 
void spectrometer_matrix(Sparse& H,
                         ConstVectorView ch_f,
                         const ArrayOfGriddedField1& ch_response,
                         ConstVectorView sensor_f,
                         const Index& n_pol,
                         const Index& n_sp,
                         const Index& do_norm) {
  // Check if matrix has one frequency column or one for every channel
  // frequency
  //
  ARTS_ASSERT(ch_response.nelem() == 1 || ch_response.nelem() == ch_f.nelem());
  //
  Index freq_full = ch_response.nelem() > 1;
 
  // If response data extend outside sensor_f is checked in
  // integration_func_by_vecmult
 
  // Reisze H
  //
  const Index nin_f = sensor_f.nelem();
  const Index nout_f = ch_f.nelem();
  const Index nin = n_sp * nin_f * n_pol;
  const Index nout = n_sp * nout_f * n_pol;
  //
  H.resize(nout, nin);
 
  // Calculate the sensor integration vector and put values in the temporary
  // vector, then copy vector to the transfer matrix
  //
  Vector ch_response_f;
  Vector weights(nin_f);
  Vector weights_long(nin, 0.0);
  //
  for (Index ifr = 0; ifr < nout_f; ifr++) {
    const Index irp = ifr * freq_full;
 
    //The spectrometer response is shifted for each centre frequency step
    ch_response_f = ch_response[irp].get_numeric_grid(GFIELD1_F_GRID);
    ch_response_f += ch_f[ifr];
 
    // Call *integration_func_by_vecmult* and store it in the temp vector
    integration_func_by_vecmult(
        weights, ch_response[irp].data, ch_response_f, sensor_f);
 
    // Normalise if flag is set
    if (do_norm) weights /= weights.sum();
 
    // Loop over polarisation and spectra (viewing directions)
    // Weights change only with frequency
    for (Index sp = 0; sp < n_sp; sp++) {
      for (Index pol = 0; pol < n_pol; pol++) {
        // Distribute the compact weight vector into weight_long
        weights_long[Range(sp * nin_f * n_pol + pol, nin_f, n_pol)] = weights;
 
        // Insert temp_long into H at the correct row
        H.insert_row(sp * nout_f * n_pol + ifr * n_pol + pol, weights_long);
 
        // Reset weight_long to zero.
        weights_long = 0.0;
      }
    }
  }
}
 
 
 
void stokes2pol(VectorView w,
                const Index& stokes_dim,
                const Index& ipol_1based,
                const Numeric nv) {
  ARTS_ASSERT(w.nelem() == stokes_dim);
 
  ARTS_USER_ERROR_IF (ipol_1based < 1 || ipol_1based > 10,
                      "Valid polarization indices are 1 to 10 (1-based).");
 
  ArrayOfVector s2p(10);
  //
  s2p[0] = {1};              // I
  s2p[1] = {0, 1};           // Q
  s2p[2] = {0, 0, 1};        // U
  s2p[3] = {0, 0, 0, 1};     // V
  s2p[4] = {nv, nv};         // Iv
  s2p[5] = {nv, -nv};        // Ih
  s2p[6] = {nv, 0, nv};      // I+45
  s2p[7] = {nv, 0, -nv};     // I-45
  s2p[8] = {nv, 0, 0, nv};   // Ilhc
  s2p[9] = {nv, 0, 0, -nv};  // Irhc
 
  const Index l = s2p[ipol_1based - 1].nelem();
  ARTS_USER_ERROR_IF (l > stokes_dim,
    "You have selected polarization with 1-based index: ", ipol_1based,
    "\n",
    "but this polarization demands stokes_dim >= ", l, "\n",
    "while the actual values of stokes_dim is ", stokes_dim)
 
  w[Range(0, l)] = s2p[ipol_1based - 1];
  if (l < stokes_dim) {
    w[Range(l, stokes_dim - l)] = 0;
  }
}
 
// Functions by Stefan, needed for HIRS:
 
 
bool test_and_merge_two_channels(Vector& fmin, Vector& fmax, Index i, Index j) {
  const Index nf = fmin.nelem();
  ARTS_ASSERT(fmax.nelem() == nf);
  ARTS_ASSERT(i >= 0 && i < nf);
  ARTS_ASSERT(j >= 0 && j < nf);
  ARTS_ASSERT(fmin[i] <= fmin[j]);
  ARTS_ASSERT(i < j);
 
  // There are three cases to consider:
  // a) The two channels are separate: fmax[i] <  fmin[j]
  // b) They overlap:                  fmax[i] >= fmin[j]
  // c) j is inside i:                 fmax[i] >  fmax[j]
 
  // In the easiest case (a), we do not have to do anything.
  if (fmax[i] >= fmin[j]) {
    // We are in case (b) or (c), so we know that we have to combine
    // the channels. The new minimum frequency is fmin[i]. The new
    // maximum frequency is the larger one of the two channels we
    // are combining:
    if (fmax[j] > fmax[i]) fmax[i] = fmax[j];
 
    // We now have to kick out element j from both vectors.
 
    // Number of elements behind j:
    Index n_behind = nf - 1 - j;
 
    Vector dummy = fmin;
    fmin.resize(nf - 1);
    fmin[Range(0, j)] = dummy[Range(0, j)];
    if (n_behind > 0) fmin[Range(j, n_behind)] = dummy[Range(j + 1, n_behind)];
 
    dummy = fmax;
    fmax.resize(nf - 1);
    fmax[Range(0, j)] = dummy[Range(0, j)];
    if (n_behind > 0) fmax[Range(j, n_behind)] = dummy[Range(j + 1, n_behind)];
 
    return true;
  }
 
  return false;
}
 
void find_effective_channel_boundaries(  // Output:
    Vector& fmin,
    Vector& fmax,
    // Input:
    const Vector& f_backend,
    const ArrayOfGriddedField1& backend_channel_response,
    const Numeric& delta,
    const Verbosity& verbosity) {
  CREATE_OUT2;
 
  // How many channels in total:
  const Index n_chan = f_backend.nelem();
 
  // Checks on input quantities:
 
  // There must be at least one channel.
  ARTS_USER_ERROR_IF (n_chan < 1,
      "There must be at least one channel.\n"
      "(The vector *f_backend* must have at least one element.)")
 
  // There must be a response function for each channel.
  ARTS_USER_ERROR_IF (n_chan != backend_channel_response.nelem(),
      "Variables *f_backend_multi* and *backend_channel_response_multi*\n"
      "must have same number of bands for each LO.")
 
  // Frequency grids for response functions must be strictly increasing.
  for (Index i = 0; i < n_chan; ++i) {
    // Frequency grid for this response function:
    const Vector& backend_f_grid =
        backend_channel_response[i].get_numeric_grid(0);
 
    ARTS_USER_ERROR_IF (!is_increasing(backend_f_grid),
        "The frequency grid for the backend channel response of\n"
        "channel ", i, " is not strictly increasing.\n"
        "It is: ", backend_f_grid, "\n")
  }
 
  // Start the actual work.
 
  out2 << "  Original channel characteristics:\n"
       << "  min         nominal      max (all in Hz):\n";
 
  // count the number of passbands as defined by segments of filtershapes, backend_channel_response.data = [0,>0,>0,0]
  // Borders between passbands are identified as [...0,0...]
 
  // Get a list of original channel boundaries:
  Index numPB = 0;
  for (Index idx = 0; idx < n_chan; ++idx) {
    const Vector& backend_filter = backend_channel_response[idx].data;
    if (backend_filter.nelem() >
        2) {  // only run this code when there is more then two elements in the backend
      for (Index idy = 1; idy < backend_filter.nelem(); ++idy) {
        if ((backend_filter[idy] > 0) && (backend_filter[idy - 1] == 0)) {
          numPB++;  // Two consecutive zeros gives the border between two passbands
        }
      }
    } else {
      numPB++;
    }
  }
 
  ARTS_USER_ERROR_IF (!numPB,
        "No passbands found.\n"
        "*backend_channel_response* must be zero around the passbands.\n"
        "backend_channel_response.data = [0, >0, >0, 0]\n"
        "Borders between passbands are identified as [...0,0...]");
 
  Vector fmin_pb(numPB);
  Vector fmax_pb(numPB);
  Index pbIdx = 0;
 
  for (Index idx = 0; idx < n_chan; ++idx) {
    // Some handy shortcuts:
    //
    // We have to find the first and last frequency where the
    // response is actually different from 0. (No point in making
    // calculations for frequencies where the response is 0.)
    //       Index j=0;
    //       while (backend_response[j] <= 0) ++j;
    //       Numeric bf_min = backend_f_grid[j];
 
    //       j=nf-1;
    //       while (backend_response[j] <= 0) --j;
    //       Numeric bf_max = backend_f_grid[j];
    //
    // No, aparently the sensor part want values also where the
    // response is zero. So we simply take the grid boundaries here.
    //
    // We need to add a bit of extra margin at both sides,
    // otherwise there is a numerical problem in the sensor WSMs.
    //
    // PE 081003: The accuracy for me (double on 64 bit machine) appears to
    // be about 3 Hz. Select 1 MHz to have a clear margin. Hopefully OK
    // for other machines.
    //
    // SAB 2010-04-14: The approach with a constant delta does not seem to work
    // well in practice. What I do now is that I add a delta corresponding to a
    // fraction of the grid spacing. But that is done outside of this function.
    // So we set delta = 0 here.
    //
    // SAB 2010-05-03: Now we pass delta as a parameter (with a default value of 0).
    //
    // Isoz 2013-05-21: Added methods to ignore areas between passbands
    //
    const Vector& backend_f_grid =
        backend_channel_response[idx].get_numeric_grid(0);
    const Vector& backend_filter = backend_channel_response[idx].data;
    if (backend_filter.nelem() >=
        4)  // Is the passband frequency response given explicitly ? e.g. [0,>0,>0,0]
    {
      for (Index idy = 1; idy < backend_filter.nelem(); ++idy) {
        if (idy == 1) {
          fmin_pb[pbIdx] = f_backend[idx] + backend_f_grid[0];
        } else if ((backend_filter[idy] > 0) &&
                   (backend_filter[idy - 1] == 0)) {
          fmin_pb[pbIdx] = f_backend[idx] + backend_f_grid[idy - 1] - delta;
        }
        if ((backend_filter[idy] == 0) && (backend_filter[idy - 1] > 0)) {
          fmax_pb[pbIdx] = f_backend[idx] + backend_f_grid[idy] + delta;
          out2 << "  "
               << "fmin_pb " << fmin_pb[pbIdx] << "  "
               << "f_backend" << f_backend[idx] << "  "
               << "fmax_pb " << fmax_pb[pbIdx] << "  "
               << "diff " << fmax_pb[pbIdx] - fmin_pb[pbIdx] << "\n";
          pbIdx++;
        }
      }
      fmax_pb[pbIdx - 1] = f_backend[idx] + last(backend_f_grid);
    } else  // Or are the passbands given implicitly - such as the default for AMSUA and MHS
    {
      fmin_pb[pbIdx] = f_backend[idx] + backend_f_grid[0] - delta;  //delta;
      fmax_pb[pbIdx] = f_backend[idx] +
                       backend_f_grid[backend_f_grid.nelem() - 1] +
                       delta;  
      out2 << "  "
           << "fmin_pb " << fmin_pb[pbIdx] << "  "
           << "f_backend" << f_backend[pbIdx] << "  "
           << "fmax_pb " << fmax_pb[pbIdx] << "\n";
      pbIdx++;
    }
  }
 
  // The problem is that channels may be overlapping. In that case, we
  // want to create a frequency grid that covers their entire range,
  // but we do not want to duplicate any frequencies.
 
  // To make matters worse, one or even several channels may be
  // completely inside another very broad channel.
 
  // Sort channels by frequency:
  // Caveat: A channel may be higher in
  // characteristic frequency f_backend, but also wider, so that it
  // has a lower minimum frequency fmin_orig. (This is the case for
  // some HIRS channels.) We sort by the minimum frequency here, not
  // by f_backend. This is necessary for function
  // test_and_merge_two_channels to work correctly.
  ArrayOfIndex isorted;
  get_sorted_indexes(isorted, fmin_pb);
 
  fmin.resize(numPB);
  fmax.resize(numPB);
  out2 << " resize numPb " << numPB << "\n";
  for (Index idx = 0; idx < numPB; ++idx) {
    fmin[idx] = fmin_pb[isorted[idx]];
    fmax[idx] = fmax_pb[isorted[idx]];
  }
  // We will be testing pairs of channels, and combine them if
  // possible. We have to test always only against the direct
  // neighbour. If that has no overlap, higher channels can not have
  // any either, due to the sorting by fmin.
  //
  // Note that fmin.nelem() changes, as the loop is
  // iterated. Nevertheless this is the correct stop condition.
  for (Index i = 0; i < fmin.nelem() - 1; ++i) {
    bool continue_checking = true;
    // The "i<fmin.nelem()" condition below is necessary, since
    // fmin.nelem() can decrease while the loop is executed, due to
    // merging.
    while (continue_checking && i < fmin.nelem() - 1) {
      continue_checking = test_and_merge_two_channels(fmin, fmax, i, i + 1);
 
      // Function returns true if merging has taken place.
      // In this case, we have to check again.
    }
  }
 
  out2 << "  New channel characteristics:\n"
       << "  min                       max (all in Hz):\n";
  for (Index i = 0; i < fmin.nelem(); ++i)
    out2 << "  " << fmin[i] << "               " << fmax[i] << "\n";
}
 
 
 
/*===========================================================================
  === Core integration and sum functions:
  ===========================================================================*/
 
void integration_func_by_vecmult(VectorView h,
                                 ConstVectorView f,
                                 ConstVectorView x_f_in,
                                 ConstVectorView x_g_in) {
  // Basic sizes
  const Index nf = x_f_in.nelem();
  const Index ng = x_g_in.nelem();
 
  // Asserts
  ARTS_ASSERT(h.nelem() == ng);
  ARTS_ASSERT(f.nelem() == nf);
  ARTS_ASSERT(is_increasing(x_f_in));
  ARTS_ASSERT(is_increasing(x_g_in) || is_decreasing(x_g_in));
  // More ARTS_ASSERTs below
 
  // End points of x_f
  Numeric xfmin = x_f_in[0];
  Numeric xfmax = x_f_in[nf - 1];
 
  // Handle possibly reversed x_g.
  Vector x_g;
  Index xg_reversed = 0;
  //
  if (is_decreasing(x_g_in)) {
    x_g = x_g_in[Range(ng - 1, ng, -1)];
    xg_reversed = 1;
  } else {
    x_g = x_g_in;
  }
  //
  ARTS_ASSERT(x_g[0] <= xfmin);
  ARTS_ASSERT(x_g[ng - 1] >= xfmax);
 
  // Normalise grids so x_f covers [0,1]
  const Numeric df = xfmax - xfmin;
  Vector x_f(nf);
  //
  for (Index i = 0; i < nf; i++) {
    x_f[i] = (x_f_in[i] - xfmin) / df;
  }
  for (Index i = 0; i < ng; i++) {
    x_g[i] = (x_g[i] - xfmin) / df;
  }
  xfmin = 0;
  xfmax = 1;
  // To test without normalisation, comment out lines above and use:
  //const Numeric df  = 1;
  //const Vector  x_f = x_f_in;
 
  // Create a reference grid vector, x_ref that containing the values
  // of x_f and x_g strictly sorted. Only g points inside the f range
  // are of concern.
  list<Numeric> l_x;
  for (Index it = 0; it < nf; it++) {
    l_x.push_back(x_f[it]);
  }
  for (Index it = 0; it < ng; it++) {
    if (x_g[it] > xfmin && x_g[it] < xfmax) {
      l_x.push_back(x_g[it]);
    }
  }
  l_x.sort();
  l_x.unique();
  //
  Vector x_ref(l_x.size());
  Index e = 0;
  for (list<Numeric>::iterator li = l_x.begin(); li != l_x.end(); li++) {
    x_ref[e] = *li;
    e++;
  }
 
  // Initiate output vector, with equal size as x_g, with zeros.
  // Start calculations
  h = 0.0;
  Index i_f = 0;
  Index i_g = 0;
  Numeric dx, a0, b0, c0, a1, b1, c1, x3, x2, x1;
  //
  for (Index i = 0; i < x_ref.nelem() - 1; i++) {
    // Find for what index in x_g (which is the same as for h) and f
    // calculation corresponds to
    while (x_g[i_g + 1] <= x_ref[i]) {
      i_g++;
    }
    while (x_f[i_f + 1] <= x_ref[i]) {
      i_f++;
    }
 
    // If x_ref[i] is out of x_f's range then that part of the integral is 0,
    // and no calculations should be done
    if (x_ref[i] >= xfmin && x_ref[i] < xfmax) {
      // Product of steps in x_f and x_g
      dx = (x_f[i_f + 1] - x_f[i_f]) * (x_g[i_g + 1] - x_g[i_g]);
 
      // Calculate a, b and c coefficients; h[i]=ax^3+bx^2+cx
      a0 = (f[i_f] - f[i_f + 1]) / 3.0;
      b0 = (-f[i_f] * (x_g[i_g + 1] + x_f[i_f + 1]) +
            f[i_f + 1] * (x_g[i_g + 1] + x_f[i_f])) /
           2.0;
      c0 = x_g[i_g + 1] * (f[i_f] * x_f[i_f + 1] - f[i_f + 1] * x_f[i_f]);
 
      a1 = -a0;
      b1 = (f[i_f] * (x_g[i_g] + x_f[i_f + 1]) -
            f[i_f + 1] * (x_g[i_g] + x_f[i_f])) /
           2.0;
      c1 = x_g[i_g] * (-f[i_f] * x_f[i_f + 1] + f[i_f + 1] * x_f[i_f]);
 
      x1 = x_ref[i + 1] - x_ref[i];
      // Simple way, but sensitive to numerical problems:
      //x2 = pow(x_ref[i+1],2) - pow(x_ref[i],2);
      //x3 = pow(x_ref[i+1],3) - pow(x_ref[i],3);
      // The same but a numerically better way:
      x2 = x1 * (2 * x_ref[i] + x1);
      x3 = x1 * (3 * x_ref[i] * (x_ref[i] + x1) + x1 * x1);
 
      // Calculate h[i] and h[i+1] increment
      // (df-factor to compensate for normalisation)
      h[i_g] += df * (a0 * x3 + b0 * x2 + c0 * x1) / dx;
      h[i_g + 1] += df * (a1 * x3 + b1 * x2 + c1 * x1) / dx;
    }
  }
 
  // Flip back if x_g was decreasing
  if (xg_reversed) {
    Vector tmp = h[Range(ng - 1, ng, -1)];  // Flip order
    h = tmp;
  }
 
  // The expressions are sensitive to numerical issues if two points in x_ref
  // are very close compared to the values in x_ref. A test trying to warn when
  // this happens:
  ARTS_USER_ERROR_IF (min(f) >= 0 && min(h) < -1e-15,
        "Significant negative response value obtained, "
        "despite sensor reponse is strictly positive. This "
        "indicates numerical problems. Is there any very "
        "small spacing of the sensor response grid?"
        "Please, send a report to Patrick if you see this!");
}
 
 
 
void integration_bin_by_vecmult(VectorView h,
                                ConstVectorView x_g_in,
                                const Numeric& limit1,
                                const Numeric& limit2) {
  // Basic sizes
  const Index ng = x_g_in.nelem();
 
  // Asserts
  ARTS_ASSERT(ng > 1);
  ARTS_ASSERT(h.nelem() == ng);
  ARTS_ASSERT(limit1 <= limit2);
 
  // Handle possibly reversed x_g.
  Vector x_g;
  Index xg_reversed = 0;
  //
  if (is_decreasing(x_g_in)) {
    x_g = x_g_in[Range(ng - 1, ng, -1)];
    xg_reversed = 1;
  } else {
    x_g = x_g_in;
  }
  //
  ARTS_ASSERT(x_g[0] <= limit1);
  ARTS_ASSERT(x_g[ng - 1] >= limit2);
 
  // Handle extreme cases
  // Bin has zero width
  if (limit1 == limit2) {
    h = 0.0;
    return;
  }
  // Bin covers complete x_g:
  else if (limit1 == x_g[0] && limit2 == x_g[ng - 1]) {
    h[0] = (x_g[1] - x_g[0]) / 2.0;
    for (Index i = 1; i < ng - 1; i++) {
      h[i] = (x_g[i + 1] - x_g[i - 1]) / 2.0;
    }
    h[ng - 1] = (x_g[ng - 1] - x_g[ng - 2]) / 2.0;
    return;
  }
 
  // The general case
  Numeric x1 = 0,
          x2 =
              0;  // End points of bin, inside basis range of present grid point
  for (Index i = 0; i < ng; i++) {
    bool inside = false;
 
    if (i == 0) {
      if (limit1 < x_g[1]) {
        inside = true;
        x1 = limit1;
        x2 = min(limit2, x_g[1]);
      }
    } else if (i == ng - 1) {
      if (limit2 > x_g[ng - 2]) {
        inside = true;
        x1 = max(limit1, x_g[ng - 2]);
        x2 = limit2;
      }
    } else {
      if ((limit1 < x_g[i + 1] && limit2 > x_g[i - 1]) ||
          (limit2 > x_g[i - 1] && limit1 < x_g[i + 1])) {
        inside = true;
        x1 = max(limit1, x_g[i - 1]);
        x2 = min(limit2, x_g[i + 1]);
      }
    }
 
    // h is zero if no overlap between bin and basis range of point
    if (!inside) {
      h[i] = 0.0;
    } else {
      // Lower part
      if (x1 < x_g[i]) {
        const Numeric r = 1.0 / (x_g[i] - x_g[i - 1]);
        const Numeric y1 = r * (x1 - x_g[i - 1]);
        const Numeric dx = min(x2, x_g[i]) - x1;
        const Numeric y2 = y1 + r * dx;
        h[i] = 0.5 * dx * (y1 + y2);
      } else {
        h[i] = 0.0;
      }
 
      // Upper part
      if (x2 > x_g[i]) {
        const Numeric r = 1.0 / (x_g[i + 1] - x_g[i]);
        const Numeric y2 = r * (x_g[i + 1] - x2);
        const Numeric dx = x2 - max(x1, x_g[i]);
        const Numeric y1 = y2 + r * dx;
        h[i] += 0.5 * dx * (y1 + y2);
      }
    }
  }
 
  // Flip back if x_g was decreasing
  if (xg_reversed) {
    Vector tmp = h[Range(ng - 1, ng, -1)];  // Flip order
    h = tmp;
  }
}
 
 
void summation_by_vecmult(VectorView h,
                          ConstVectorView f,
                          ConstVectorView x_f,
                          ConstVectorView x_g,
                          const Numeric x1,
                          const Numeric x2) {
  // Asserts
  ARTS_ASSERT(h.nelem() == x_g.nelem());
  ARTS_ASSERT(f.nelem() == x_f.nelem());
  ARTS_ASSERT(x_g[0] <= x_f[0]);
  ARTS_ASSERT(last(x_g) >= last(x_f));
  ARTS_ASSERT(x1 >= x_f[0]);
  ARTS_ASSERT(x2 >= x_f[0]);
  ARTS_ASSERT(x1 <= last(x_f));
  ARTS_ASSERT(x2 <= last(x_f));
 
  // Determine grid positions for point 1 (both with respect to f and g grids)
  // and interpolate response function.
  ArrayOfGridPos gp1g(1), gp1f(1);
  gridpos(gp1g, x_g, x1);
  gridpos(gp1f, x_f, x1);
  Matrix itw1(1, 2);
  interpweights(itw1, gp1f);
  Numeric f1;
  interp(f1, itw1, f, gp1f);
 
  // Same for point 2
  ArrayOfGridPos gp2g(1), gp2f(1);
  gridpos(gp2g, x_g, x2);
  gridpos(gp2f, x_f, x2);
  Matrix itw2(1, 2);
  interpweights(itw2, gp2f);
  Numeric f2;
  interp(f2, itw2, f, gp2f);
 
  //Initialise h at zero and store calculated weighting components
  h = 0.0;
  h[gp1g[0].idx] += f1 * gp1g[0].fd[1];
  h[gp1g[0].idx + 1] += f1 * gp1g[0].fd[0];
  h[gp2g[0].idx] += f2 * gp2g[0].fd[1];
  h[gp2g[0].idx + 1] += f2 * gp2g[0].fd[0];
}