%
% To start the document, use
%  \levela{...}
% For lover level, sections use
%  \levelb{...}
%  \levelc{...}
%
\levela{Propagation paths}
 \label{sec:ppath}

%
% Document history, format:
%  \starthistory
%    date1 & text .... \\
%    date2 & text .... \\
%    ....
%  \stophistory
%
\starthistory
  02xxxx & xxx.\\
\stophistory


%
% Symbol table, format:
%  \startsymbols
%    ... & \artsstyle{...} & text ... \\
%    ... & \artsstyle{...} & text ... \\
%    ....
%  \stopsymbols
%
%
%\startsymbols
%  \Ind           & -                 & vector/matrix/tensor index           \\
%  \aInd{\Lat}    & -                 & the \Ind:th latitude                 \\
%  \VctLng        & -                 & vector length or size of matrix/tensor for a dimension \\
%  \aVctLng{\Lat} & -                 & length of the latitude grid \\
%  \Prs           & \artsstyle{p}     & pressure                             \\
%  \PrsAlt        & \artsstyle{pz}    & pressure altitude                    \\
%  \Rds           & \artsstyle{r}     & radius from the geoid centre         \\
%  \Alt           & \artsstyle{z}     & geometrical altitude above the geoid \\
%  \Lat           & \artsstyle{alpha} & latitude                             \\
%  \Lon           & \artsstyle{beta}  & longitude                            \\
%  \ZntAng        & \artsstyle{psi}   & zenith angle                         \\
%  \AzmAng        & \artsstyle{omega} & azimuthal angle                      \\
% \label{symtable:ppath_defs}     
%\stopsymbols




\levelb{Geoid ellipsoids and geodetic datums}
%===================
\label{sec:ppath:geoids}

This section defines the geoid ellipsoid and discusses related
issues. The geoid is introduced in Section~\ref{sec:fm_defs:geoid}.
The workspace variable representing the geoid is \wsvindex{r\_geoid}.


\levelc{Geoid ellipsoids}
%===================
\label{sec:ppath:geoid}

All geodetic datums are based on a reference ellipsoid\index{geoid
  ellipsoid}. The ellipsoid is rotationally symmetric around the
north-south axis. That is, the ellipsoid radius has no longitude
variation, it is only a function of latitude. The ellipsoid is
described by a equatorial radius, \aRds{e}, and a polar radius,
\aRds{p}. These radii are indicated in Figure~\ref{fig:ppath:lats}.
The radius of the ellipsoid for a given latitude is
\begin{equation}
 \aRds{\odot}(\Lat) = \sqrt{\frac{\aRds{e}^2\aRds{p}^2}
                    {\aRds{e}^2\sin^2\Lat+\aRds{p}^2\cos^2\Lat}}
 \label{eq:ppath:ellipsradius} 
\end{equation}
The radius given by Equation~\ref{eq:ppath:ellipsradius} can be
directly applied for 2D and 3D cases. On the other hand, for 1D cases
the reference geoid is by definition a sphere and the radius of this
sphere shall be selected in such way that it represents the local
shape of a reference ellipsoid. This is achieved by setting
\aRds{\odot} to the radius of curvature of the ellipsoid. The
curvature radius differs from the local radius except at the equator
and an east-west direction. For example, at the equator and a
north-south direction, the curvature radius is smaller then the local
radius, while at the poles (for all directions) it is greater
(see further Figure~\ref{fig:ppath:wgs84radii}). 

The \textindex{curvature radius}, \aRds{c}, of an ellipsoid is 
\citep{rodgers:00}
\begin{equation}
 \aRds{c} = \frac{1}{\aRds{ns}^{-1}\cos^2 \Lat + \aRds{ew}^{-1}\sin^2 \Lat}
 \label{eq:ppath:curvradius} 
\end{equation}
where \aRds{ns} and \aRds{ew} are the north-south and east-west curvature radius, respectively,
\begin{eqnarray}
 \aRds{ns} &=& \aRds{e}^2\aRds{p}^2 (
           \aRds{e}^2\cos^2\AzmAng+\aRds{p}^2\sin^2\AzmAng )^{-\frac{3}{2}} \\
 \aRds{ew} &=& \aRds{e}^2 (
           \aRds{e}^2\cos^2\AzmAng+\aRds{p}^2\sin^2\AzmAng )^{-\frac{1}{2}} 
 \label{eq:ppath:rew} 
\end{eqnarray}
The azimuth angle, \AzmAng, is defined in
Section~\ref{sec:fm_defs:los}. The latitude and azimuthal angle to
apply in Equations \ref{eq:ppath:curvradius}~-~\ref{eq:ppath:rew}
shall rather be valid for a middle point of the propagation paths
(such as some tangent point), instead of the sensor position. 

\begin{figure}[!p]
 \begin{center}
  \begin{minipage}[c]{0.65\textwidth}
   \begin{center}
   \begin{picture}(200,165)(0,0)
    \put(30,10){\arc{301}{-1.05}{-0.55}}
    \put(43,138){{\small geoid ellipsoid}}
    \put(0,10){\vector(1,0){70}}
    \put(70,4){$x$}
    \put(10,0){\vector(0,1){70}}
    \put(4,70){$y$}
    \dottedline(30,10)(170,150)
    \put(30,10){\arc{20}{-.76}{0}}
    \put(40,13){\Lat$^*$}
    \put(136.07,116.07){\line(1,-1){20}}
    \put(136.07,116.07){\line(-1,1){20}}
    \put(158,94){\small\shortstack[l]{local\\tangent}}
    \drawline(10,10)(170,144.4)
    \put(10,10){\arc{20}{-.73}{0}}
    \put(160,130){{\small zenith}}
    \put(20,13){\Lat}
    %
    \put(30,10){\arc{310}{-.15}{0.05}}
    \put(150,10){\vector(1,0){35}}
    \put(170,14){\aRds{e}}
    \put(10,10){\arc{290}{-1.6708}{-1.4708}}
    \put(10,120){\vector(0,1){35}}
    \put(13,140){\aRds{p}}
   \end{picture}
   \end{center}
  \end{minipage}%
  \begin{minipage}[c]{0.35\textwidth}
   \caption{Definition of the ellipsoid radii, \aRds{e} and \aRds{p}, 
     geocentric latitude, \Lat, and geodetic latitude, \Lat$^*$. The
     dotted line is the normal to the local tangent of the geoid
     ellipsoid. The zenith and nadir directions, and geometrical
     altitudes, are here defined to follow the solid line.}
   \label{fig:ppath:lats}
  \end{minipage}
 \end{center}
\end{figure}   

\begin{figure}[!p]
 \begin{minipage}[c]{0.65\textwidth}
 \includegraphics*[width=0.96\textwidth]{ppath/wgs84_radii}
 \end{minipage}%
 \begin{minipage}[c]{0.35\textwidth}
  \caption{The ellipsoid radius (\aRds{\odot}) and curvature radius (\aRds{c})
    for the
    WGS-84 reference ellipsoid. The curvature radii are valid for the
    north-south direction.}
  \label{fig:ppath:wgs84radii}
 \end{minipage}%
\end{figure}   
        
\begin{figure}[!p]
 \begin{minipage}[c]{0.65\textwidth}
 \includegraphics*[width=0.96\textwidth]{ppath/wgs84_latdiff}
 \end{minipage}%
 \begin{minipage}[c]{0.35\textwidth}
  \caption{The change of the WGS-84 ellipsoid radius for  1\degree\ 
            latitude differences.}
  \label{fig:ppath:latdiff}
 \end{minipage}%
\end{figure}   



\levelc{Geocentric and geodetic latitudes}
%===================
\label{sec:ppath:geolat}

The fact that the geoid is an ellipsoid, instead of a sphere, opens up
for the two different definitions of the latitude. The
\textindex{geocentric latitude}, which is the the one used here, is the
angle between the equatorial plane and the vector from the coordinate
system centre to the position of concern. The \textindex{geodetic
  latitude} is also defined with respect to the equatorial plane, but
the angle to the normal to the reference ellipsoid is considered here, as
shown in Figure~\ref{fig:ppath:lats}. It could be mentioned that a
geocentric latitude does not depend on the geoid ellipsoid used, while
the geodetic latitudes change if another reference ellipsoid is
selected. An approximative relationship between the geodetic
($\Lat^*$) and geocentric (\Lat) latitudes is \citep{montenbruck:00}
\begin{equation}
 \Lat^* = \Lat + f\,\sin(2\Lat)  
 \label{eq:ppath:lats}
\end{equation}
where $f$ is the flattening of the ellipse:
\begin{equation}
 f = \frac{\aRds{e}-\aRds{p}}{\aRds{e}}
 \label{eq:ppath:flattening}
\end{equation}
The value of $f$ for the Earth is about 1/298.26. This means that the
largest differences between \Lat\ and $\Lat^*$ are found at
mid-latitudes and the maximum value is about 12 arc-minutes.

The \textindex{zenith} and \textindex{nadir} directions shall normally be
defined to follow the normal to the reference ellipsoid, but, if
nothing else is mentioned, these directions are here treated to go
along the vector the centre of the coordinate system, as indicated in
Figure~\ref{fig:ppath:lats}. This latter definition is preferred
as it results in that a propagation path in the zenith/nadir direction
can be described by a single latitude and longitude value. The
difference in geometrical altitude when using these two possible
definitions on the zenith direction is proportional to the deviation
between geocentric and geodetic latitude (Equation~\ref{eq:ppath:lats}).
For an altitude of 100~km around $\Lat=45\degree$, the difference is
about 350~m.


\levelc{Geodetic datums}
%===================
\label{sec:ppath:geodatums}

Table~\ref{tab:ppath:geodatums} gives the equatorial and polar radii
of the reference ellipsoid for the geodetic datums handled by ARTS.

\begin{table}[!h]
  \begin{center}
    \begin{tabular}{c c c c l}
     Datum & \aRds{e} & \aRds{p} & $1/f$ & Reference \vspace*{1mm} \\ 
     \hline 
     WGS-84 & 6378.137 km & \emph{6356.752 km} & 298.2572235 & {\small \citet{montenbruck:00}}  \rule{0mm}{5mm} \vspace*{1mm} \\
     \hline
    \end{tabular}
    \caption{Equatorial and polar radius of reference ellipsoids. Values 
      given as \emph{italic} are 
      derived by the other two values and Equation~\ref{eq:ppath:flattening}.}
    \label{tab:ppath:geodatums}
  \end{center}
\end{table}



\levelb{The propagation path data structure}
%===================
\label{sec:ppath:Ppath}

The ARTS definition of a propagation path is given in
Section~\ref{sec:fm_defs:ppaths}.  A propagation path is represented
by a structure of type \structindex{Ppath}. This structure holds also
auxiliary variables to facilitate the radiative transfer calculations
and to speed up the interpolation (if performed). The fields of
\artsstyle{Ppath}, where the data type is given inside square brackets,
are:
\begin{description}

  \item[dim] [Index] The atmospheric dimensionality. This field shall always 
     be equal to the workspace variable \artsstyle{atmosphere\_dim}.

   \item[np] [Index] Number of positions to define the propagation
     path. The number of rows (equalling the length for vectors) of
     $\mathbf{pos}$, $\mathbf{z}$, $\mathbf{indexpos}$,
     $\mathbf{l\_step}$ and $\mathbf{los}$ shall be equal to
     $\mathbf{np}$. The number of radiative transfer steps along the
     path is $\mathbf{np}-1$. If $\mathbf{np}\leq 1$, the observed
     spectrum is identical to the radiative background. When the path is
     totally outside the atmosphere $\mathbf{np}=0$. If the sensor is
     placed on the ground and looks down, the sensor is inside the
     cloud box, or the sensor is at the cloud box boundary and looks
     into the box, $\mathbf{np}=1$ where the stored values correspond
     to the sensor position.

   \item[pos] [Matrix] The position of the propagation path points.
     This matrix has $\mathbf{np}$ rows and up to 3 columns. Each row
     holds a position where column 1 is the pressure, column 2 the
     latitude and column 3 the longitude (cf.
     Section~\ref{sec:fm_defs:sensorpos}). The number of columns for
     1D and 2D is 2, while for 3D it is 3. The latitudes are stored
     for 1D cases as these can be of interest for some applications
     and are useful if the propagation path shall be plotted. The
     latitudes for 1D give the angular distance to the sensor (see
     further Section~\ref{sec:fm_defs:atmdim}).
     
     The propagation path is stored in reversed order, that is, the
     position with index 0 is the path point closest to the sensor
     (and equals the sensor position if it is inside the atmosphere).
     The full path is stored also for 1D cases with symmetry around a
     tangent point (in contrast to ARTS-1). 
     
  \item[z] [Vector] The geometrical altitude for each path position. The
     length of this vector is accordingly $\mathbf{np}$. This is a help
     variable for plotting and similar purposes. It shall not be used to
     interpolate the atmospheric fields, as pressure is the main altitude
     coordinate.

  \item[l\_step] [Vector] The length along the propagation path between
     the positions in $\mathbf{pos}$. The length of the vector is
     $\mathbf{np}-1$. 

  \item[gridindex] [Matrix] The fractional grid indexes of the positions.
     The intention of this field is to speed up interpolation of the 
     atmospheric fields. The matrix has $\mathbf{np}$ rows and $\mathbf{dim}$ 
     columns, where column 1 refers to the pressure grid, column 2 to the 
     latitude grid and column 3 to the longitude grid. Note that this matrix,
     in contrast to $\mathbf{pos}$, has only one column for 1D. This is the 
     case as the latitude position is of no interest for 1D when interpolating
     an atmospheric field. 

     The grid index is best described by some examples. A grid index of
     2 means that the position equals the grid value with index 2, and
     a grid index of 2.5 signifies that the position is half the way between
     grid points 2 and 3. A more formal definition is (using latitudes):
     if $i$ is the highest index such as $\aLat{i}\leq\Lat$, then the grid
     index is $i+(\Lat-\aLat{i})/(\aLat{i+1}-\aLat{i})$, where \Lat\ is the 
     latitude of the position and \aLat{i}\ is grid point $i$. The grid index 
     for a position at the upper end of the grid is $\VctLng-1$ where
     \VctLng\ is the length of the grid. The vertical box position is given 
     with respect to pressure altitudes (Equation~\ref{eq:fm_defs:prsalt}).
     
   \item[los] [Matrix] The line-of-sight of the propagation path at
     each point. The number of rows of the matrix is $\mathbf{np}$.
     For 1D and 2D, the matrix has a single column holding the zenith
     angle. For 3D there is an additional column giving the azimuthal
     angle. The zenith and azimuthal angles are defined in
     Section~\ref{sec:fm_defs:los}. If the radiative background is the
     cloud box, the position (in $\mathbf{pos}$) and the line-of-sight
     with index $\mathbf{np-1}$ give the relevant information needed
     when extracting the radiative background from the cloud box
     intensity field. The line-of-sight is valid for the path leaving
     the point of concern. For example, the zenith angle changes when
     a ground reflection takes place, but this corresponds to a single
     path point, and the given zenith angle is valid for the path after
     the reflection.
     
   \item[background] [String] The radiative background for the
     propagation path, that is, [**] in Equation~[**]. The possible
     options for this field are 'Cosmic background radiation',
     'Blackbody ground', 'Inside cloud box' and 'Surface of cloud
     box', where the source of radiation should be clear the content
     of the strings.

  \item[ground] [Index] A boolean to indicate that there is an intersection
     of the path by the ground. If the ground is treated to be a blackbody,
     $\mathbf{ground}$ can only have the value of 0.
     
   \item[i\_ground] [Index] The position index of the ground.
     Undefined if $\mathbf{ground}=0$. Row $\mathbf{i\_ground}$ of
     $\mathbf{pos}$ gives the position of the ground reflection, which
     is needed for determining the ground emissivity for 2D and 3D cases.

  \item[tan\_pos] [Vector] The position of the tangent point. This vector is
      only set for downwards oberservation, the length of the vector is
      otherwise 0. If the path intersects with the ground or the cloud box,
      a geometrical prolongation of the path is used below this intersection
      to find the position of the tangent point. The tangent point can thus
      be found below the ground level.
     
   \item[symmetry] [Index] A boolean to indicate that the atmospheric
     conditions are symmetric around some point. This will the case
     for 1D and downward observations. This boolean shows only that a
     symmetry point exists, it will not affect the later calculations.
     To make use of the symmetry on the radiative transfer
     calculations (see Section~[**]), the workspace variable [**] must
     be set to 1.

   \item[i\_symmetry] [Index] The index of the symmetry point if 
     $\mathbf{symmetry}=1$. Otherwise undefined.

\end{description}

[* Comments to be removed later: The full path is here stored for 1D
cases.  There are several reasons for this. First of all, it is most
practical not to describe the 1D paths in a special way. Furthermore,
the zenith angles and latitudes for the path points are not symmetric
around the tangent point. If ARTS later will be extended to perform
successive order of scattering calculations (as I think will happen),
there will be paths going up and down through the atmosphere without
having any symmetry point and the RTE functions must be able to handle
these more general paths anyhow. (Patrick 2002-04-12) *]



\levelb{Some basic 1D and 2D geometrical relationships}
%===================
\label{sec:ppath:basicgeom}

This section gives some expressions to determine positions along a
propagation path for 1D and 2D atmospheres when refraction is
neglected (Figure~\ref{fig:ppath:1d2dgeom}). The ARTS function for
making the calculation of concern is given inside parenthesis above
each equation, if not stated explicitely.

\begin{figure}[!t]
 \begin{center}
  \begin{minipage}[c]{0.65\textwidth}
   \begin{center}
   \begin{picture}(200,170)(0,0)
     \put(0,0){\vector(1,4){25}}
     \dottedline(25,100)(35,140)
     \put(25,100){\arc{30}{-1.31}{-0.23}}
     \put(50,0){\vector(1,2){60}}
     \dottedline(110,120)(130,160)
     \put(110,120){\arc{30}{-1.1}{-0.23}}
     \drawline(25,100)(110,120)
     \dottedline(0,94.12)(25,100)
     \dottedline(110,120)(145,128.24)
     \put(145,128.24){\vector(4,1){25}}
     \put(0,-80){\arc{200}{-1.52}{-1.02}}
     \put(7,70){$\aRds{1}$}
     \put(99,90){$\aRds{2}$}
     \put(25,17){$\Delta\Lat$}
     \put(35,113){$\aZntAng{1}$}
     \put(122,131){$\aZntAng{2}$}
     \put(60,113){$\Delta \PpathLng$}
     \put(160,135){\PpathLng}
     \put(150,110){\small\shortstack[l]{propagation\\direction}}
   \end{picture}
   \end{center}
  \end{minipage}%
  \begin{minipage}[c]{0.35\textwidth}
   \caption{The radius (\Rds) and zenith angle (\ZntAng) for two points along
     the propagation path, and the distance along the path ($\Delta\PpathLng$)
     and the latitude difference ($\Delta\Lat$) between these points.}
   \label{fig:ppath:1d2dgeom}
  \end{minipage}
 \end{center}
\end{figure}   

The law of sines gives that the product must $\Rds\sin(\ZntAng)$ be
constant along the propagation path (cf. Equation~[**]):
\begin{equation}
  p_c = \Rds\sin(\ZntAng)
  \label{eq:ppath:geomconst}
\end{equation}
where the absolute value is taken for 2D zenith angles as they can for
such cases be negative. The propagation path constant, $p_c$, is
determined by the position and line-of-sight of the sensor, a
calculation done by the function \funcindex{geomppath\_constant}. The
constant equals also the radius of the tangent point of the path (that
is found along an imaginary prolongation of the path behind the sensor
if the viewing direction is upwards). The expressions below are based
on $p_c$ as the usage of a global constant for the path should
decrease the sensitivity to numerical inaccuracies. If the
calculations are based solely on the values for the neighbouring
point, a numerical inaccuracy can accumulate when going from one point
to next.

The latitude of the tangent point, \aLat{t}, is calculated as
(\funcindex{geomppath\_alphatan})
\begin{equation}
  \aLat{t} = \left\{
   \begin{array}{ll}
    \aLat{s}+\aZntAng{s}-90\degree & \textrm{for}\, \aZntAng{s}\geq 0\degree \\
    \aLat{s}-\aZntAng{s}+90\degree & \textrm{for}\, \aZntAng{s}< 0\degree 
   \end{array}   \right.
  \label{eq:ppath:pc2alphatan}
\end{equation}
where \aLat{s} is the latitude for the sensor position, set to
0\degree for 1D cases, and \aZntAng{s} is the sensor line-of-sight
zenith angle.

The relationship between the distance along the path for an
infinitesimal change in radius is here denoted as the
\textindex{geometrical factor}, $g$. If refraction is neglected, valid
expressions for the geometrical factor are
\begin{equation}
  g = \frac{\DiffD l}{\DiffD r} 
           = \frac{1}{\cos(\ZntAng)} = \frac{1}{\sqrt{1-\sin^2(\ZntAng)}}
                                            = \frac{\Rds}{\sqrt{\Rds^2-c^2}}
  \label{eq:ppath:g_geom}
\end{equation}
For the radiative transfer calculations, only the distance between the
points, $\Delta \PpathLng$, is of interest. However, internally for
the propagation path calculations the absolute length from the tangent
point (real or imaginary), \PpathLng, is used. The variable \PpathLng\ 
is defined to follow the propagation path direction
(Figure~\ref{fig:ppath:1d2dgeom}), so positive values for \PpathLng\ 
are found between the tangent point and the sensor. By integrating
Equation~\ref{eq:ppath:g_geom}, we get that (\funcindex{geomppath\_r2l})
\begin{equation}
  \PpathLng(\Rds) = \sqrt{\Rds^2-p_c^2} 
  \label{eq:ppath:r2l}
\end{equation}
As refraction is here neglected, the tangent point, the point of
concern and the centre of the coordinate system make up a right
triangle and Equation~\ref{eq:ppath:r2l} corresponds to the
Pythagorean relation where $p_c$ is the radius of the tangent point.
The distance between two points ($\Delta \PpathLng$) is obtained by
taking the difference of Equation~\ref{eq:ppath:r2l} for the two
radii. The function \artsstyle{geomppath\_r2l} takes an extra argument
to know if \PpathLng\ shall be set to be positive or negative.

The radius for a given \PpathLng\ is simply (\funcindex{geomppath\_l2r})
\begin{equation}
  \Rds(\PpathLng) = \sqrt{\PpathLng^2+p_c^2} 
  \label{eq:ppath:l2r}
\end{equation}
The latitude for a position is most easily determined by \PpathLng\ 
and the sensor zenith angle (\funcindex{geomppath\_l2alpha}):
\begin{equation}
  \Lat(\PpathLng) = \left\{
   \begin{array}{ll}
    \aLat{t}-\tan^{-1}(\PpathLng/p_c) &\textrm{for}\,\aZntAng{s}\geq 0\degree\\
    \aLat{t} + \tan^{-1}(\PpathLng/p_c) & \textrm{for}\, \aZntAng{s}< 0\degree 
   \end{array}   \right.
  \label{eq:ppath:l2alpha}
\end{equation}
where a version of $\tan^{-1}$ that takes into account the sign of
\PpathLng\ is used.

By noting that the sum of the zenith angle (for $\ZntAng \geq
0\degree$) and the difference ($\Lat-\aLat{t}$) is 90\degree, we have
that (\funcindex{geomppath\_alpha2psi})
\begin{equation}
  \ZntAng(\Lat) = \left\{
   \begin{array}{ll}
    \aLat{t} - \Lat + 90\degree &\textrm{for}\,\aZntAng{s}\geq 0\degree\\
    \aLat{t} - \Lat - 90\degree & \textrm{for}\, \aZntAng{s}< 0\degree 
   \end{array}   \right.
  \label{eq:ppath:psi2alpha2psi}
\end{equation}
It is interesting to note that without scattering the zenith angle
approaches zero when following the propagation path. It should be
noted that the zenith angle is defined as the angle between zenith and
the propagation direction, and not the smallest angle between zenith
and the path. This means, for example, that the absolute value of the
zenith angle before a tangent point throughout is $> 90\degree$, while
after it is $<90\degree$.

If a point of the path is known by its latitude, the zenith angle 
can be determined by Equation~\ref{eq:ppath:psi2alpha2psi}, and the
radius is then calculated as (\funcindex{geomppath\_psi2r})
\begin{equation}
  \Rds(\ZntAng) = \frac{p_c}{sin(|\ZntAng|)}
  \label{eq:ppath:psi2r}
\end{equation}




\levelb{1D propagations paths}
%===================
\label{sec:ppath:1D}


\levelc{Without refraction}
%===================
\label{sec:ppath:1Dwithout}


\levelc{With refraction}
%===================
\label{sec:ppath:1Dwith}



\levelb{2D propagations paths}
%===================
\label{sec:ppath:2D}


\levelc{Without refraction}
%===================
\label{sec:ppath:2Dwithout}


\levelc{With refraction}
%===================
\label{sec:ppath:2Dwith}



\levelb{3D propagations paths}
%===================
\label{sec:ppath:3D}


\levelc{Without refraction}
%===================
\label{sec:ppath:3Dwithout}


\levelc{With refraction}
%===================
\label{sec:ppath:3Dwith}



\levelb{Control file examples}
%===================
\label{sec:ppath:cfile}




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