%
% To start the document, use
%  \levela{...}
% For lover level, sections use
%  \levelb{...}
%  \levelc{...}
%
\levela{Basic radiative transfer}
 \label{sec:rte}


%
% Document history, format:
%  \starthistory
%    date1 & text .... \\
%    date2 & text .... \\
%    ....
%  \stophistory
%
\starthistory
  000307 & Started by Patrick Eriksson. \\
  000908 & First version finished by Patrick Eriksson. \\
\stophistory



%
% Symbol table, format:
%  \startsymbols
%    ... & \verb|...| & text ... \\
%    ... & \verb|...| & text ... \\
%    ....
%  \stopsymbols
%
%
%\startsymbols
%  \mpbi     & \verb|y|       & monochromatic pencil beam intensity      \\
%  $\mpbi_1$ & \verb|y_cbgr|          & cosmic background radiation      \\
%  $\mpbi_1$ & \verb|y_ground|        & ground blackbody radiation       \\
%  $S$       & \verb|S|               & source function                  \\
%  $k$       & \verb|Abs|             & total absorption                 \\
%  $T$       & \verb|t|               & temperature                      \\
%  \f        & \verb|f|               & frequency                        \\
%  $h$       & \verb|PLANCK_CONST|    & the Planck constant              \\ 
%  $c$       & \verb|SPEED_OF_LIGHT|  & speed of light                   \\
%  $k_B$     & \verb|BOLTZMAN_CONST|  & the Boltzmann constant           \\
%  $\tau$    & -                      & optical thickness                \\
%  $\zeta$   & \verb|Tr|              & transmission between LOS points  \\
%  $\zeta^{atm} $& \verb|y|           & total atmospheric transmission   \\
%  $l$       & \verb|l|               & distance along LOS               \\
%  $\Delta l$ &\verb|lstep|           & step length along LOS            \\
%  $e$       & \verb|e_ground|        & ground emissivity                \\
%  $T_{ground} $&\verb|t_ground|      & physical temperature of the ground\\
%  $T_{cbgr}$  &\verb|COSMIC_BG_TEMP| & temperature representing cosmic  \\ 
%         &                &        background radiation                 \\
% \label{symtable:rte}     
%\stopsymbols



%
% Introduction
%
This section presents the atmospheric radiative transfer equation
(RTE) for a non-scattering atmosphere in local thermodynamic
equilibrium.  However, the calculation scheme described here can
easily be extended to include effects of e.g. single scattering. The
radiative transfer equation gives the monochromatic (infinite
frequency resolution) pencil beam (infinite spatial resolution)
spectrum. The main problem, in this context, is how to practically and
accurately estimate the (continuous) integral in the discrete forward
model.
 
The discussion treats mainly measurements of atmospheric emission. The 
forward model can also handle pure absorption measurements and such 
observations are also discussed briefly last in the section.

The equations of this section are valid for monochromatic pencil beam
spectra, that is, no effects of the sensor are considered. How to
incorporate sensor effects in the spectra is discussed separately
(Sec. \ref{sec:sensor}).



\levelb{Introduction}
 \label{sec:rte:intro}
 
 Atmospheric radiative transfer can be expressed generally as
 \begin{equation}
   I = I_1e^{-\int_{l_1}^{l_2}{\kappa(l)dl}} +
        \int_{l_1}^{l_2}{\kappa(l)\sigma(l)e^{-\int_{l}^{l_2}{\kappa(l')dl'}}dl}
    \label{eq:rte:rte}
 \end{equation}
 where $I$ is the monochromatic pencil beam intensity, $l$ distance
 along the line of sight (LOS), $l_1$ the point of the considered part
 of the LOS furthest away from the sensor, $l_2$ the closest point of
 the LOS, $I_1$ the intensity at $l_1$, $\kappa$ the total absorption
 along the LOS and $\sigma$ the source function.\footnote{The symbols
   $\kappa$ and $\sigma$ are used here for the absorption and the source 
   function \emph{along} the LOS. The more commonly used symbols, $k$ and
   $S$, respectively, are used below to express the variables as
   functions of altitude.}
  
 Equation \ref{eq:rte:rte} is of general validity if $\sigma$ and $\kappa$
 consider the relevant effects, for example, scattering. However, below in
 this section it is assumed that there is no scattering and the
 atmosphere is in local thermodynamic equilibrium.
  
 Note that Eq. \ref{eq:rte:rte} is valid both for the case when the LOS is
 determined by geometrical calculations and when refraction is
 considered (the refraction changes however the LOS).
  
 With the assumptions of no scattering and local thermodynamic
 equilibrium, $\kappa$ is the summed gaseous absorption, and the source
 function equals the Planck function, $B$:
 \begin{equation}
    \sigma = B(\f,T) = \frac{2h\f^3}{c^2} \frac{1}{e^{h\f/k_B T}-1}
    \label{eq:rte:planck}
 \end{equation}
 giving the blackbody radiation for a temperature $T$ and frequency
 $\f$.
  
 If $\sigma$ is constant along the considered part of the LOS, that is, the
 temperature is constant for the case $\sigma=B$, the RTE can be solved
 analytically to give
 \begin{equation}
   I = I_1e^{-\tau} + \sigma\left(1-e^{-\tau}\right)
  \label{eq:rte:step}
 \end{equation}
 where $\tau$ is the optical thickness
 \begin{equation}
   \tau = \int_{l_1}^{l_2}{\kappa(l)dl}
 \end{equation}
 The transmission corresponding to $\tau$ is
 \begin{equation}
   \zeta = e^{-\tau}
 \end{equation}  



\levelb{Practical considerations}
 \label{sec:rte:practical}
 
 The LOS can be divided into parts in several ways. As absorption and
 temperature most likely are avaliable at some vertical grid, the most
 natural choice would be to define the LOS using this vertical grid.
 This solution is problematic for limb sounding as the ratio between
 the distance along LOS and the corresponding vertical distance
 becomes infinite at the tangent point. Another solution would be to
 base the division on $\tau$, but such a division does not guarantee
 that $T$ is close to constant inside the slabs as the vertical
 extension in some cases could be very large, and each combination of
 frequency and viewing angle should require a specific division.
  
 As a practical compromise, it was here decided to divide LOS into
 equal long geometrical steps. With this scheme the division is
 identical for all frequency components, but changes between the
 viewing angles, and should give relatively fast and straightforward
 calculations, maintaining a good accuracy. This approach has
 been applied successfully in the Odin sub-mm forward model 
 \citep{eriksson:97a,eriksson:00a}.
  
 The next question is when and how to calculate LOS and the associated
 variables. As the determination of weighting functions associated
 with the absorption, e.g. species WFs, needs basically the same
 quantities as RTE, it is most efficient to do this procedure only
 once and in such way that the values are suitable for both RTE and
 the weighting functions. Hence, the LOS calculations shall be a
 separate part, not included in the RTE functions. The standard use of
 the forward model should then be:
  \begin{enumerate}
    \item Calculation of absorption coefficients.
    \item Determination of LOS.
    \item Calculation of the source function and transmissions along LOS.
    \item Iteration to solve RTE.
    \item Calculation of weighting functions.
    \item Saving etc.
  \end{enumerate}
 The determination of LOS is described separately in Section \ref {sec:los}. 
  

  
\levelb{Practical solution}
 \label{sec:rte:iter}
 
 The LOS is here assumed to be defined with $n$ points where the
 distance between the points is constant (see Fig \ref{fig:rte:los}).
 There are at least two definition points of the LOS ($n\geq1$). The
 absorption and the source function are determined at the points of
 the LOS, and these values are used to calculate the transmission and a
 mean source function value for the distances between the LOS points.
 Only the later two quantities are stored.

  \begin{figure}
    \includegraphics*[width=0.98\hsize]{Figs/los}
    \caption{Schematic description of the LOS and associated variables.
      The absorption and the source function at the LOS points are
      denoted $\kappa_i$ and $\sigma_i$, respectively, while $\zeta_i$
      is the transmission between the points and $\Psi_i$ is the mean
      of neighbouring source function values. Only $zeta$ and $\Psi$
      are stored for the later calculations. All the points are
      separated by the distance $\Delta l$ (along the LOS). The distance
      between point $i$ and $i+1$ is denoted as step $i$ of the LOS. }
    \label{fig:rte:los}  
  \end{figure}

 \levelc{Absorption and transmission}  
  \label{sec:rte:abs}
  
  The absorption is treated to vary linearly between the LOS points.
  As mentioned above, the transmission values shall be valid between the
  LOS points. With these definitions, the optical thickness associated with 
  step $i$ is
 \begin{equation}
    \tau_i = \frac{\Delta l}{2} \left( \kappa_i+\kappa_{i+1} \right),
                             \quad 1\leq i < n 
 \end{equation}
 The relationship between the optical thicknesses and
 the transmission is
 \begin{equation}
   \zeta_i = e^{-\tau_i}
 \end{equation}
 Note that
 \begin{equation}
   e^{-\left(\tau_1+\tau_2\dots\tau_n\right)}=\zeta_1\zeta_2\dots\zeta_n
 \end{equation}
 The absorption at the LOS points is determined from the absorption
 matrix provided by the absorption module by linear interpolation,
 using the logarithm of the pressure as altitude
 coordinate\footnote{The logarithm of the pressure is throughout the
   basic altitude coordinate in ARTS.}. For the moment, linear
 interpolation is the only choice, but if it is found that cubic
 and/or spline interpolation give better accuracy, such interpolations
 will also be implemented.


 \levelc{The source function}  
  \label{sec:rte:source}
  The source function is also basically assumed to vary linearly
  between the LOS points, but for simplicity reasons, a single
  source function value is assigned to the LOS steps:
  \begin{equation}
    \Psi_i = \frac{\sigma_i+\sigma_{i+1}}{2}, \quad 1\leq i < n
   \label{eq:rte:smean}
  \end{equation}   
  The source function at the LOS points ($\sigma$) is simply by
  interpolating linearly the temperature profile, and calculating the
  Planck function (Eq. \ref{eq:rte:planck}) for the obtained
  temperatures.
  
  To fully model that the absorption and the source function have a
  simultanous linear variation between the LOS points would give much
  more complicated analytical expressions than presented here (if even
  possible to derive?). However, the simplified approach used here
  should not influence the accuracy in any important way. This as the
  source function has, compared to the absorption, a relatively low
  variation and it can be treated to be piecewise constant when solving
  the raqdiative transfer.
  
  If long wavelengths are assumed and the source function equals the
  Planck function (Eq.  \ref{eq:rte:planck}), $\sigma$ should
  maximally vary with about a factor of 2 as the minimum and the
  maximum temperature in the atmosphere are about 150 and 300 K,
  respectively, and the relationship between $\sigma$ and temperature
  is close to linear.  This should be compared to the absorption that,
  even for a single frequency, often varies with many orders of
  magnitude.


 \levelc{Solving the radiative transfer equation}  
  \label{sec:rte:solving}
  
  With the definitions given above, the intensity at point $n$ can be
  expressed as
 \begin{equation}
   I = I_1 \prod_{j=1}^{n-1}\zeta_j + 
       \sum_{i=1}^{n-1}\left[\Psi_i(1-\zeta_i)\prod_{j=i+1}^{n-1}\zeta_j\right]
  \label{eq:rte:rteprod}
 \end{equation}
 However, an alternative approach, requiring less computer memory, is
 to follow the radiation from one slab of the atmosphere to next, and
 is the method of choice here. Following Equation \ref{eq:rte:step},
 the following iterative expression can be determined
 \citep{eriksson:97a}
 \begin{equation}
   I_{i+1} = I_i\zeta_i + \Psi_i\left(1-\zeta_i\right)\qquad i=1,2,...,n-1
  \label{eq:rte:iteration}
 \end{equation}
 where $I_i$ is the intensity reaching point $i$.
 The iteration is started by setting $I_1$ to the intensity at the 
 atmospheric limit, that is, cosmic background radiation or correspondingly.


 \levelc{Considering ground reflection}  
  \label{sec:rte:ground}

 The effect of a ground reflection is modeled as
 \begin{equation}
   I^{after} = I^{before}(1-e) + eB(\f,T_{ground})
  \label{eq:rte:ground}
 \end{equation} 
 where $e$ is the ground emission factor and $I^{before}$ and
 $I^{after}$ is the intensity before and after the reflection,
 respectively. See further Section \ref{sec:los:ground}.


\levelb{Total atmospheric transmission}
 \label{sec:rte:trans}
  
 The atmospheric emission can be neglected if the observation is
 performed towards a sufficiently strong source, such as the Sun, and
 the measurement gives basically the total atmospheric transmission,
 $\zeta^{tot}$. This transmission is
 \begin{equation}
   \zeta^{tot} = e^{-\int_{l_1}^{l_2}{\kappa(l)dl}}
  \label{eq:rte:tottrans}
 \end{equation}
 The corresponding iterative formula used in the forward model is
 simply (cf. Eq. \ref{eq:rte:iteration})
 \begin{equation}
   \zeta^{tot} = \prod_{i=1}^{n-1}\zeta_{i}
 \end{equation} 
 It is noteworthy that the multiplication order
 is of no importance, a fact that can be used for 1D limb sounding where
 the conditions are assumed to be symmetrical around the tangent point and
 only one half of the line of sight is stored. The transmission can here be 
 calculated as
 \begin{equation}
   \zeta^{tot} = \prod_{i=1}^{n-1}\zeta_{i}^2
 \end{equation} 
 If there is a ground reflection, it is considered as
 \begin{equation}
   \zeta^{tot} = (1-e)\prod_{i=1}^{n-1}\zeta_{i}
  \label{eq:rte:tground}
 \end{equation} 
 where $e$ is the ground emission factor.
 
  


%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "main"
%%% End: 
