%
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\levela{Gas Absorption}
 \label{sec:absorption}


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  2001-07-05 & Template created by Stefan Buehler.\\
  2001-07-19 & Continuum absorption part written, Thomas Kuhn.\\
\stophistory
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%
% ============================================================================
% TKS definitions for the sections "Continuum Absorption" and 
% "Complete Absorption Models"
%
\def\deni{\rho_{\mbox{\sf i}}}
\def\denl{\rho_{\mbox{\sf l}}}
\def\denli{\rho_{\mbox{\sf l,i}}}
%
\def\imn{{N''}}
\def\ime{\epsilon^{''}_r}
\def\ree{\epsilon^{'}_r}
\def\er{\epsilon_r}
%
\def\bek{\sf b_{\sf 1,k}}
\def\bzk{\sf b_{\sf 2,k}}
\def\bdk{\sf b_{\sf 3,k}}
\def\bvk{\sf b_{\sf 4,k}}
\def\bfk{\sf b_{\sf 5,k}}
\def\bsk{\sf b_{\sf 6,k}}
%
\def\bekp{\sf \widehat{b}_{\sf 1,k}}
\def\bzkp{\sf \widehat{b}_{\sf 2,k}}
\def\bdkp{\sf \widehat{b}_{\sf 3,k}}
\def\bvkp{\sf \widehat{b}_{\sf 4,k}}
\def\bfkp{\sf \widehat{b}_{\sf 5,k}}
\def\bskp{\sf \widehat{b}_{\sf 6,k}}
%
\def\beks{\sf b^*_{\sf 1}}
\def\bzks{\sf b^*_{\sf 2}}
\def\bdks{\sf b^*_{\sf 3}}
\def\bvks{\sf b^*_{\sf 4}}
\def\bfks{\sf b^*_{\sf 5}}
\def\bsks{\sf b^*_{\sf 6}}
%
\def\bekps{\sf \widehat{b^*}_{\sf 1,k}}
\def\bzkps{\sf \widehat{b^*}_{\sf 2,k}}
\def\bdkps{\sf \widehat{b^*}_{\sf 3,k}}
\def\bvkps{\sf \widehat{b^*}_{\sf 4,k}}
\def\bfkps{\sf \widehat{b^*}_{\sf 5,k}}
\def\bskps{\sf \widehat{b^*}_{\sf 6,k}}
%
\def\air{\mbox{air}}
\def\hzo{\mbox{H}_2\mbox{O}}
\def\nzo{\mbox{N}_2\mbox{O}}
\def\coz{\mbox{C}\mbox{O}_2}
\def\nz{\mbox{N}_2}
\def\oz{\mbox{O}_2}
\def\coz{\mbox{CO}_2}
%
\def\ptot{P_{\mbox{\sf tot}}}
\def\phzo{P_{\mbox{\sf H}_{2}\mbox{\sf O}}}
\def\pnz{P_{\mbox{\sf N}_{2}}}
\def\poz{P_{\mbox{\sf O}_{2}}}
\def\pda{P_{\mbox{\sf d}}}
\def\pdair{P_{\mbox{\sf air}}}
\def\pan{P_{\mbox{\sf air},\mbox{\sf N}_{2}}}
\def\pcoz{P_{\mbox{\sf CO}_{2}}}
%
\def\alphampmotot{\alpha^{\mbox{\rm MPM87}}_{\mbox{tot}}} 
\def\alphampmmtot{\alpha^{\mbox{\rm MPM89}}_{\mbox{tot}}} 
\def\alphampmntot{\alpha^{\mbox{\rm MPM93}}_{\mbox{tot}}} 
\def\alphapwrtot{\alpha^{\mbox{\rm R98}}_{\mbox{tot}}} 
\def\alphacptot{\alpha^{\mbox{\rm CP98}}_{\mbox{tot}}} 
%
\def\alphampmol{\alpha^{\mbox{\rm MPM87}}_{\mbox{$\ell$}}} 
\def\alphampmml{\alpha^{\mbox{\rm MPM89}}_{\mbox{$\ell$}}} 
\def\alphampmnl{\alpha^{\mbox{\rm MPM93}}_{\mbox{$\ell$}}} 
\def\alphampml{\alpha^{\mbox{\rm MPM}}_{\mbox{$\ell$}}} 
\def\alphapwrl{\alpha^{\mbox{\rm R98}}_{\mbox{$\ell$}}} 
\def\alphacpl{\alpha^{\mbox{\rm CP98}}_{\mbox{$\ell$}}} 
%
\def\alphampmoc{\alpha^{\mbox{\rm MPM87}}_{\mbox{c}}} 
\def\alphampmmc{\alpha^{\mbox{\rm MPM89}}_{\mbox{c}}} 
\def\alphampmnc{\alpha^{\mbox{\rm MPM93}}_{\mbox{c}}} 
\def\alphapwrc{\alpha^{\mbox{\rm R98}}_{\mbox{c}}} 
\def\alphacpc{\alpha^{\mbox{\rm CP98}}_{\mbox{c}}} 
%
\def\alphatot{\alpha_{\mbox{\sf tot}}} 
\def\alphal{\alpha_{\mbox{$\ell$}}} 
\def\alphac{\alpha_{\mbox{\sf c}}} 
\def\alphacs{\alpha_{\mbox{\sf c,s}}} 
\def\alphacf{\alpha_{\mbox{\sf c,f}}} 
%
\def\gamk{\gamma_{\sf k}}
\def\gamc{\gamma_{\sf c}}
%
\def\ws{w_{\sf s,k}}
\def\xs{x_{\sf s,k}}
\def\wf{w_{\sf f,k}}
\def\xf{x_{\sf f,k}}
%
\def\wn{\bar{\nu}}
\def\nucc{\nu_{\sf c}}
\def\nucut{\nu_{\sf cutoff}}
\def\nuo{\nu_{\mbox{\sf 0}}}
\def\nuk{\nu_{\sf k}}
%
\def\shape{F(\nu,\nuk)}
\def\shapec{F_{c}(\nu,\nuk)}
\def\shapefp{f_{c}(\nu,+\nuk)}
\def\shapefm{f_{c}(\nu,-\nuk)}
\def\shapefpm{f_{c}(\nu,\pm\nuk)}
\def\inten{S_{\sf k}(T)}
\def\inteno{S_{\sf k}(300\,K)}
\def\intencp{S_{\sf 0}(T)}
%
\def\fcs{\mbox{C}^{\mbox{\rm fit}}_{\mbox{\rm s}}} 
\def\fcf{\mbox{C}^{\mbox{\rm fit}}_{\mbox{\rm f}}}
\def\fcsf{\mbox{C}^{\mbox{\rm fit}}_{\mbox{\rm s,f}}} 
\def\tfcs{\mbox{C}^{\mbox{\rm fit}}_{\mbox{\rm s}}} 
\def\tfcf{\mbox{C}^{\mbox{\rm fit}}_{\mbox{\rm f}}} 
\def\tfcsf{\mbox{C}^{\mbox{\rm fit}}_{\mbox{\rm s,f}}} 
\def\tflncsf{\mbox{ln\,C}^{\mbox{\rm fit}}_{\mbox{\rm s,f}}} 
%
\def\cx{C_{\mbox{\sf x}}}
\def\cs{C_{\mbox{\sf H}_{2}\mbox{\sf O}}} 
\def\cf{C_{\mbox{\sf N}_{2}}} 
\def\cxo{C^{\mbox{\sf o}}_{\mbox{\sf X}}} 
\def\cso{C^{\mbox{\sf o}}_{\mbox{\sf H}_{2}\mbox{\sf O}}} 
\def\cfo{C^{\mbox{\sf o}}_{\mbox{\sf N}_{2}}} 
\def\cao{C^{\mbox{\sf o}}_{\mbox{\sf air}}}
\def\cdo{C^{\mbox{\sf o}}_{\mbox{\sf d}}}
\def\xx{{\sf n}_{\mbox{\sf x}}} 
\def\xs{{\sf n}_{\mbox{\sf s}}} 
\def\xf{{\sf n}_{\mbox{\sf f}}} 
\def\xd{{\sf n}_{\mbox{\sf d}}}
%
% ============================================================================




In general there three types of absorption/emission spectra:
\begin{itemize}
\item sharp lines of finite width
\item aggregation (series) of lines called bands
\item continua extending over a broad range of wavelengths, with
      no single lines in it.
\end{itemize}
This section is therefore firstly devised to give a short theoretical
overview of the spectra quantities which are used in ARTS. The presentation
of the calculating methods within ARTS of these quantities is the
second goal.
There are two major subsections which deal separately with the respective types
of spectra regarded by ARTS: 
- lines 
- continua.
The second one extends beyond the scope of mere treatment of the
continua and thus gives an integrated view of continua and spectral lines.


\levelb{Line Absorption}
%-----------------------
\label{sec:line_absorption}

 
Staying close to the notation used by Goody and Young in ``Atmosperic
Radiation'', 1989, we'll introduce here the main concepts concerning
line absortion. The approach, however, does not aim  at the
derivation, but confines itself within the consise and structured way
of presenting the following expressions.

Any line is presented by the corresponding profile of the absorption/emission
coefficient as dependency on the frequency, in our case that of the
absorption - given by the quantity $\alpha(\nu)$. The latter dependency
is related to other quantities, which express the three
characteristics of a single line uniqely describing it - position,
strength and shape. 
The strength, or better the line intensity, is given by the quantity
$S(T)$. The shape and the position are expressed through the
line-shape function $F(\nu)$. Thus we have the relation 
\begin{equation}
  \alpha(\nu)=nS(T)F(\nu)
\label{}
\end{equation} 
where $n$ is the numbre density pf the absorber. The line-shape
function is normalized as follows

\begin{equation}
  \int F(\nu)d\nu=1
\label{}
\end{equation}
The values of $S(T)$ at reference temperature $T_0$ are contaied in spectroscopic databases. The conversion to different
temperatures is done by
\begin{equation}
S(T)=S(T_0)~\frac{Q(T_0)}{Q(T)}~\frac{e^{-E_f/kT} - e^{-E_i/kT}}{e^{-E_f/kT_0} - e^{-E_i/kT_0}}
\label{}
\end{equation}
given the energies $E_f$ and $E_i$ of the two levels between which the
transition occurs as well as the partition function~~$Q(T)$. The
databases contain the the lower energy $E_l$ tabulated along with the
$S$ and the transition frequency $\nu$, so that the upper state can be
computed by $E_u$=$E_l$+h$\nu$. Partition functions for all molecular species are also
available along with spectroscopic databases, either in the form of tabulated values for a set
temperatures, or in the form of FORTRAN routines. One can obtain the total absorption
coefficient by adding the absorption of all spectral lines of all
molecular species.

The problem of determining the explicit kind of the line-shape
functions is treated in a following subsection. A special one is
dedicated to the partition functions and their calculalation..


\levelc{Line Shape Functions}
Up to date there is no way to calculate the line-shape fuctions
analytically just through the theoretical expressions of quantum dynamics,
statistical physics and theoretical mechnanics. Certain approximations
should be done to account for different physical phenomena related to
the absorption/emission process or to suit better the calculation of
different parts of the spectral line.

There are three phenomena which contribute to the line-shape. These
are, in increasing orede of importance, the finite lifetime of an
excited state in an isolated molecule, the thermal movement of the gas
molecules and their colliisons with each other. They result in
correpsonding effects to the line-shape: natural broadenig, Doppler
and pressure broadening. Of these, the first one is completely
negligible compared to the other two. That's why we don't pay a special
attention to it but we'll present some expressions in order to convey
better understanding of the other two.

The spectral line shape can derived in this case from basic physical
considerations and the well-known Fouirier transform theorem, stating
that  multiplication in the time domain is equal to convoluting the
frequency one.

If we regard the spontaneous decay of the excited state, then the
amlitude of the oscillations decreases by $\gamma/2$ and the
respective decrease in the energy is
\begin{equation}
  E(t)=E(0)e^{-\gamma t}
\label{}
\end{equation}
By the afore mentioned theorem, multiplicating in the time domain by
$e^{-\gamma t}$ is equavelent to convolving in the frequency domain
with a function $1/[\nu^2 - (\gamma/4\pi)^2]$. Thus the normalised
line-shape function of the spectral line at frequency $\nu_0$ is 
\begin{equation}
  F(\nu)=\frac{1}{\pi}\frac{\gamma/4\pi}{(\nu - \nu_0)^2 + (\gamma/4\pi)^2}
\label{}
\end{equation}

This gives a bell-shaped profile and the function itself is called
Lorentzian. The dependance on the position of the line is apparent
through $\nu_0$, that's why some authors prefer to notate the function
by $F(\nu,\nu_0)$.
The result is important because of the two reasons. First, without the
natural broadening the line wouold be~$\delta (\nu - \nu_o)$. So the
spontaneous decay of the excited state is responsible for the finite
width and the certain shape of the line-shape function. Second, the
Lorentzian type of function comes significantly into into play when
explaining the some of the other broadening effects or the complte
picture of the broadened line.

The second effect, Doppler broadenig, is important for the upper
stratosphere and mesosphere for microwave freqeuncy. The line-shape
follows the velocity distribution of the particles. Under the conditions
of thermodynamic equilibrium, we have  a probablity distribution for
the relative velocity $u$ between the gas molecule and the oserver 
of Maxwell type 
\begin{equation}
  p(u)=\sqrt{\frac{m}{2\pi kT}}~~exp~\left[-\frac{mu^2}{2kT}\right]
\label{}
\end{equation}
where $m$ is the mass of the molecule. Using then the formula for the
Doppler shift for the non-relativistic region~~  $\nu$- $\nu_0$ =
$\nu_0$$u$ / $c$ , one can easily derive the line-shape function 
\begin{equation}
 F_D(\nu)=\frac{1}{\gamma_D\sqrt{\pi}}~~exp~\left[-\left(\frac{\nu - \nu_0}{\gamma_D}\right)^2\right]
\label{}
\end{equation}
where the quantity $\gamma_D$ is called Doppler line width and equals
\begin{equation}
 \gamma_D=\frac{\nu}{c}\sqrt{\frac{2kT}{m}}
\label{}
\end{equation}
In contrast to the afore mentioned line-shape function for the natural
broadening the Doppler broadening is expressed by a Gaussian
line-shape fuction $F(\nu)$. The Doppler line width $\gamma_D$ is so
defined that it is qual to the falf width at half of the maximum
(HWWM) of the line-shape function. The same way of notating is used
for all other width parameters $\gamma_xy$ below.

It can be said without any exaggeration that the pressure broadening
effect is the most comlpicated one among the others and still
represents a complex theoretical task to be tackled, and is in the
same time of major experimental importance. So far, there is no way to
calculate analytically from the basics the profile of a pressure, or
collisional, line through a single approach near the line center as well
as ii the far wing region. The various approximations, which are
therefore used, are immanenetly limited to the certain line regions
they deal with.
The most popular among these approximations is $impact~approximation$
which main statement is that the duration of the colllisions of the
gas particles is very small compared to the average time between the
collisions. Lorentz was the first to achieve a result exploiting this
approach, the Lorentz line-shape function:
\begin{equation}
 F_L(\nu)=\frac{\gamma_L}{\pi}\frac{1}{(\nu-\nu_0)^2+\gamma_L^2}
\label{}
\end{equation}
where $\gamma_L$ is the Lorentz line width. As one can see, the result
(3.9) is pretty similar to (3.5) but the specific line parameters
$\gamma$ and $\gamma_L$ make them differ significantly in the
corresponding frequency regions of interest. For atmospheric pressures
$\gamma_L$ is much greater and that's why of experimental significance
in contrast to $\gamma$.\\
Elaborating the model of Lorentz, van Vleck and Weisskopf made a correction to it, particularly
for the microwave region:
\begin{equation}
 F_{VVW} (\nu)=\left(\frac{\nu}{\nu_0}\right)^2\frac{\gamma_L}{\pi}\left[\frac{1}{(\nu-\nu_0)^2+\gamma_L^2}+\frac{1}{(\nu+\nu_0)^2+\gamma_L^2}\right]
\label{}
\end{equation}
which can be reduced to Lorentzian for $(\nu-\nu_0) << \nu_0$ . Except
for the additional factor $(\nu/\nu_0)^2$ ,  $F_{VVW}$ can be regarded
as the sum of two $F_L$, or respectively lines, one with its center
freqquency at  $\nu_0$, the other at $-\nu_o$. Other authors,
e.g. Rosenkrantz,  make use of an linear factor $(\nu/\nu_0)$ in their
approach too.

The combined picture of a simultaneously Doppler and pressure
broadened line is the next step of the approximations development. The
line-shape function in this case given by the Voigt line-shape
function 
\begin{equation}
 F_{Voigt}(\nu,\nu_0)= \int F_L(\nu,\nu')~F_D(\nu',\nu_0)~d\nu'
\label{}
\end{equation}
though there's no strict justfication for its use - the two processes
are assumed to act independently, which in reality is not the
fact. Regardless of this flaw, it's the only way up to date to model
the combination of the broadening processes. The integral in (3.11)
can not computed analytically, so certain approximations algorithms
must be used.


Another possibility would be the combination the last two equations
(3.10) and (3.11). The respective result then will be 
\begin{equation}
 F_S=\left(\frac{\nu}{\nu_0}\right)^2~[F_{Voigt}(\nu,\nu_0)+F_{Voigt}(\nu,-\nu_0)]
\label{}
\end{equation}
The advantage of such a model is that it behaves like a van
Vleck-Weisskopf line-shape function in the high pressure limit and
like a Voigt one in the low pressure limit. There is one important
caveat to the equation (3.12): it has to be made sure that the
algorithm that is used to compute the Voigt function really produces a
Lorentz line in the high pressure limit. Another point of signifacance
is the demand that model yields meanigful rersuts far from the line
center, since the line center from the ``mirror'' line at -$\nu_0$ is
situated approximately 2$\nu_0$ away from tthe frequency $\nu_0$ of
computation. The algorithm of Drayson [$Drayson$, 1976; $Oliveiro~and~
Longbothum$, 1977] was explicitly checked that satisfies both
requirements, while this not true for some other algorithms, commonlly
used for Voight-shape computation. In particular, the it's not true
for the Hui-Armstrong-Wray Formula, as defined in $Hui~et~al.$
[1978] and in Equation 2.60 of $Rosenkrantz$ [1993]. So, provided the
stated above is fullfilled, the $F_S$ line shape gives a smooth
transition from high tropospheric pressures to low stratospheric ones,
and to be valid near the line centers throughout the microwave region.
\levelc{Partition Functions}

The treatment of the partition functions is directly related to the molecular
energy states and their statistical distribution during the
radiation process. 

In any case of spectroscopic interest the free molecules of a gas are
not ptically thick at all freqeuncies, so the radiation energy is not
represented by blackbody radiation. The most common assumption made,
which is sufficient in the case of tropospheric and low stratospheric
research, is the {\textit{local thermodynamic equilibrium\nocorr} or $LTE$. According to it,
it's possible to find a common temperature, which may vary from place
to place, that fits the Boltzman energy population distribution and
the Maxwell velocities distribution. This practically means, that
under $LTE$ the collisional processes must be of greater importance
than radiative ones. In other words, an excited state must have a
hihger probability of de-excitation by collision than by spontaneous
radiation. This is the important factor which make natural broadening
differ quantatively so much from the pressure (collisional), though
both are described qualitatively almost identical by  Lorentzian line-
shape functions.

Accordind  to the Maxwell-Boltzman distribution law,in $LTE$ the total number
of gas particles $N_n$  in a state $E_n$ is given by 
\begin{equation}
 N_n=N_0\frac{g_n}{g_0}e^{-E_n/kT}
\label{}
\end{equation}
where $N_0$ is particle number in the ground state, and $g_n$, $g_o$
are the statistical weights (degeneracies) of the $n-$state and the
ground state. Thus the total particle number $N$ is given by
\begin{equation}
 N=\frac{N_0}{g_0}\sum_{n=0}^\infty e^{-E_n/kT}=\frac{N_0}{g_0}~Q(T)
\label{}
\end{equation}
The quantity $Q(T)$ is the {\it{partition function}\nocorr} of the gas. At
one hand all thermodynamic quantities can be expressed through it, and at
another it comes into play by the calculation line intensities.\\
Generally speaking, the partition fuction for a perfect gas can be
represented by the product of the {\it{translational}\nocorr} and the {\it{internal}\nocorr} partition functions
\begin{equation}
 Q  =  Q_{tr}~Q_{int}
\label{}
\end{equation}
bearing in mind that the respective energies are independent of each
other. The quantitity, however, which we are interested in in (3.13)
is the {\it{internal}\nocorr} partition function (or the  {\it{total internal
    partition function}\nocorr}). It describes, in general terms, the energy
distribution and redistribution during the radiation process. 

The internal partition function for free gaseous molecules is  a
function of the electronic, the vibrational, the rotational, and the
nuclear spin states. An approximation is used in order to display the
individual contribution explicitly
\begin{equation}
 Q=Q_e~Q_v~Q_r~Q_n
\label{}
\end{equation}
and thus the intraciton betweeen these various states is neglected. For
practically all polyatomic molecules the excited electronic staates
are entirely negligable to those of the ground states,
i.e. $Q_e=1$ . Only for the very few polyatomic molecules with a
multiplet ground state ($NO_2$ , $ClO_2$ , and free radicals) has the
electronic contibution to be considered.\\
If we neglect the anharmonicities, the vibratioanal partition
function, with vibrational levels measured with respect to the ground
state, the term $h\omega_n/2$ is omitted in the energy expression for
the {\it{harmonic oscilator}\nocorr}, is 
\begin{equation}
 Q_v=\left(\sum_{\nu_1}e^{-\nu_1 h\omega_1/kT}\right)\left(\sum_{\nu_2}e^{-\nu_2 h\omega_2/kT}\right)...
\label{}
\end{equation}
where $\nu_1$, $\nu_2$,..., the vibrational quantum numbers, can each
have the valies 0,1,2,... and $\omega_1$, $\omega_2$,..are the
frequencies of the fundamental modes of vibration. The summation is
taken over all values of $\nu_1$, $\nu_2$,..., and each fundamental
mode is counted separately. This result is valid for non-degenerate
vibrations. If we use the simple exprassion for geometric progression
\begin{equation}
 \sum_{\nu_i}e^{-\nu_i h\omega_i/kT}=\frac{1}{1-e^{h\omega_i/kT}}
\label{}
\end{equation}
and the degeneracies $d_1$, $d_2$,... of the fundamental modes, we ge
finally for vibrational partition function
\begin{equation}
Q_v=\left(1-e^{h\omega_1/kT}\right)^{-d_1}\left(1-e^{h\omega_2/kT}\right)^{-d_2}...
\label{}
\end{equation}

The rotational partition function looks differently for the different
symmetry types of molecules.
For diatomic and linear  polyatomic molecules with no center of
symmetry the corresponding expression is 
\begin{eqnarray}
Q_r & = & \sum_{J=0}^\infty (2J+1)e^{-hBJ(J+1)/kT}\nonumber\\
   & = & \frac{kT}{hB}+\frac{1}{3}+\frac{1}{15}\frac{hB}{kT}+\frac{4}{315}\left(\frac{hB}{kT}\right)^2+...\nonumber\\
   & \cong & \frac{kT}{hB}
\label{}
\end{eqnarray}
By {\it{symmetric-}}, {\it{asymmetric-}}, and {\it{spherical}} top molecules there are also
other factors to be taken into consideration, such as the
spatial structure of the molecules, nuclear spin, inversion and
internal rotation. The general expression in tis case us
\begin{equation}
Q_r  =  \frac{1}{\sigma}\sum_{J=0}^\infty \sum_{K=-J}^{J}(2J+1)~e^{-h[BJ(J+1)+(A-B)K^2]/kT}
\label{}
\end{equation}
where $\sigma$ is ameasure of the degree of symmetry. For the usual
symmetric top has $C_3$ or $C_{3\nu}$ symmetry, $\sigma$ = 3. To a good
aprroximation, the summation above can expressed as
\begin{equation}
Q_r  = 
\frac{1}{\sigma}\left[\left(\frac{\pi}{B^2A}\right)\left(\frac{kT}{h}\right)^3\right]^{1/2}=
\frac{5.34\times 10^6}{\sigma}\left(\frac{T^3}{B^{2}A}\right)^{1/2}
\label{}
\end{equation}
For asymmetric top the formula would then be 
\begin{equation}
Q_r = \frac{5.34\times 10^6}{\sigma}\left(\frac{T^3}{ABC}\right)^{1/2}
\label{}
\end{equation}
and for sprherical top
\begin{equation}
Q_r = \frac{5.34\times 10^6}{\sigma}\left(\frac{T^3}{A^3}\right)^{1/2}
\label{}
\end{equation}
















\levelc{Line Catalogues} 

Mostly describe the ARTS internal format, but also briefly list the
other catalogues that can be read by ARTS

\levelc{Species specific data} 

Molecular mass, etc. 

\levelc{Partition Functions}
These are strictly also species specific data, but they seem to deserve
their own section.

\levelc{Line Shape Functions}


\levelc{ARTS Workspace Variables and Methods}









% ================================================================================
% The following section is written by Thomas Kuhn, iup Bremen, tkuhn@uni-bremen.de
% ================================================================================

\levelb{Continuum Absorption}
\label{levelb:ContAbs}
% ===========================

As pointed out above, some molecules show beside the resonant line
absorption also non-resonant continuum absorption. The main qualitative
difference is the smooth dependence on frequency of the non-resonant
absorption part in contrast to the resonant absorption part who shows 
strong local maxima and minima.

The implemented continuum absorption modules are connected with water
vapor ($\hzo$), oxygen ($\oz$), nitrogen ($\nz$), and carbon dioxide
($\coz$). Since these molecules have various permanent electric or magnetic
multipoles, the physical explanations for the continuum absorption is 
different for each of these molecules.

Water Vapor has a strong electric dipole moment and posses therefore a 
wealth of rotational transitions in the microwave up to the submillimeter 
range. One explanation for the $\hzo$- continuum absorption is the inadequate 
formulation of the far wings of a spectral line, since the usually employed 
\citet{vanvleck:45} line shape is according to its derivation only valid 
in the near wing zone. Other explanations are (see \cite{pwr:93} for details) 
far wing contribution from far-infrared water vapor lines, collision induced 
absorption (CIA), and water polymer absorption. At present one can not definitively 
decide which of these possibilities is the correct one, probably all of them 
play a more or less important role, depending on the frequency range.

oxygen is one of the rare molecules in the Earth's atmosphere where a permanent 
magnetic dipole moment is present. The aligned spins of the two valence electrons 
gives a $^{3}\Sigma$ groud state of molecular oxygen. 
Due to the selection rules for magnetic dipole transitions, transitions with 
resonance frequency equal to zero are allowed. Such transitions have a characteristic 
Debye line shape function.

The homonuclear nitrogen molecule has in lowest order an electric quadrupole moment 
of modest magnitude.
For the frequency range below 1\,THz the collision induced rotation absorption 
band \cite{goodyandyung:89} is of most importance. The band center is around 3\,THz and 
at 1\,THz the band strength is approximately 1/6 of the maximum value (see 
Figure 5.2 of \cite{goodyandyung:89}). The electric field of the quadrupole 
moment of one molecule induces a dipole moment in the second molecule. This allows 
rotational transitions according to the electric quadrupole selection rules, 
$|\Delta\,J|=$0,2 (see \cite{pwr:93} for details). In a similar way exhibits 
carbon dioxide a collision induced absorption band (maximum around 1.5\,THz, 
Figure 5.10 of \cite{goodyandyung:89}). Characteristic for collision induced 
absorption is the dependency on the square of the molecular density.



\levelc{Water Vapor Continuum Models}
\label{leveld:h2o_Cont}
%------------------------------------
As shown by \cite{liebeandlayton:87}, \cite{pwr:98}, and \cite{ma:90},
the water vapor continuum absorption can be well described by 
\begin{equation} 
  \label{eq:abs_cont}
  \alphac = \nu^2 \cdot \phzo \cdot 
            (\cso \cdot \phzo \cdot \Theta^{\xs} + 
             \cdo \cdot \pda  \cdot \Theta^{\xf})
\end{equation}
where the microwave approximation ($h\nu<<k_BT$ ) of the radiation field term 
is already applied. The adjustment of Eq. \ref{eq:abs_cont} to the data 
is performed through the parameter set $\cso$, $\xs$, $\cdo$, and $\xf$. 
Table \ref{tab:wvcontparam} gives some commonly used continuum parameter sets.
\begin{table}[!hbt]
  \begin{center}
  \begin{tabular}{lrrrrr}
    \hline
    model  & \multicolumn{1}{c}{$\cso$} & 
             \multicolumn{1}{c}{$\xs$}  & 
             \multicolumn{1}{c}{$\cdo$} & 
             \multicolumn{1}{c}{$\xd$}  & 
             ref.\\
           & \multicolumn{1}{c}{$\left[\frac{\mbox{dB/km}}
                               {\mbox{hPa}^2~\mbox{GHz}^2}\right]$} & 
             \multicolumn{1}{c}{$[1]$} & 
             \multicolumn{1}{c}{$\left[\frac{\mbox{dB/km}}
                               {\mbox{hPa}^2~\mbox{GHz}^2}\right]$} & 
             \multicolumn{1}{c}{$[1]$} & \\
    \hline
    MPM85  & 4.92$\cdot$10$^{-8}$ & 2.5 & 0.255$\cdot$10$^{-8}$  & -0.5 & \cite{liebe:84}\\
    MPM87  & 6.50$\cdot$10$^{-8}$ & 7.5 & 0.206$\cdot$10$^{-8}$  & 0.0 & \cite{liebeandlayton:87}\\
    MPM89  & 6.50$\cdot$10$^{-8}$ & 7.5 & 0.206$\cdot$10$^{-8}$  & 0.0 & \cite{liebe:89}\\
    CP98   & 8.04$\cdot$10$^{-8}$ & 7.5 & 0.254$\cdot$10$^{-8}$ & 0.0 & \cite{cruzpol:98}\\ 
    R98    & 7.80$\cdot$10$^{-8}$ & 4.5 & 0.236$\cdot$10$^{-8}$  & 0.0 & \cite{pwr:98}\\
    \hline
    MPM93$^*$ & 7.73$\cdot$10$^{-8}$ & 4.55 & 0.253$\cdot$10$^{-8}$  & 1.55 & \cite{liebeetal:93}\\
    \hline
 \end{tabular}
\end{center}
 \caption{Values of commonly used continuum parameter sets. The last line (MPM93$^*$)
   represents an approximation of the pseudo-line continuum of MPM93
   in the form of Eq.~\ref{eq:abs_cont}.}
 \label{tab:wvcontparam}
\end{table}




\leveld{Non-resonant oxygen absorption}
\label{leveld:o2cont}
%--------------------------------------
As pointed out by \citet{vv:87}, the standard theory for
non-resonant absorption is that of Debye (see also \citet{townes:55}). 
The Debye line shape is obtained from the VVW line shape function 
by the limiting case $\nuk \rightarrow 0$.
Both, \citet{liebeetal:93} and \citet{pwr:93} adopted the
Debye theory for their models. The only difference is the formulation
of the line broadening, where the influence of water vapor is treated 
differently\footnote{In \citet{liebeetal:93} there is a mistake in the formula
  for $\gamma$. According to this text $\gamma =
  w\cdot\pda\cdot\Theta^{0.8}$ while in the source code of MPM93 
  Equation~\ref{eq:abs_cont_pwr_o2_2} is implemented.}: 
\begin{eqnarray}
  \label{eq:abs_cont_pwr_o2}
  \alphac &=&  C \cdot \pda \cdot \Theta^2 \cdot 
             \frac{\nu^2 \cdot \gamma}{\nu^2+\gamma^2}\\
  \label{eq:abs_cont_pwr_o2_1}
  \gamma &=&  w \cdot (\pda \cdot \Theta^{0.8} + 1.1 \cdot \phzo \cdot
  \Theta)\hspace*{5mm}\mbox{:~Rosenkranz}\\
  \label{eq:abs_cont_pwr_o2_2}
  \gamma &=&  w \cdot (\pda + \phzo) \cdot \Theta^{0.8}\hspace*{17.25mm}\mbox{:~MPM93}\\
\nonumber
\end{eqnarray}

The values for the parameters are $C = 1.11\cdot 10^{-5}$ dB/km/(hPa\,GHz) and 
$w = 5.6 \cdot 10^{-4}$ GHz/hPa, respectively. This absorption
term is proportional to the collision frequency of a single oxygen molecule
and thus proportional to the dry air pressure\footnote{The absorption
  due to weakly bound complexes of $\oz$--$X$ with $X=\hzo,~\nz$ is 
  treated separately and therefore not included in this Debye formula.}.




\leveld{Non-resonant nitrogen absorption}
\label{leveld:n2cont}
%----------------------------------------
Since molecular nitrogen has in its unperturbed state no electric or 
magnetic dipole moment (but an electric quadrupole moment), it shows 
no rotational spectral signature in the microwave region. Regardless 
of this, nitrogen absorbs radiation in this frequency range due to 
collision induced absorption (CIA). Far--infrared rototranslational
band structures from free--free interactions give rise to far wing
absorption below 1\,THz.

Different parameterizations of this absorption term are
available \citet{pwr:93,liebeetal:93,borysow:86}. Common to all these 
models is the quadratic dependency on $\nz$ partial pressure which is 
a direct consequence of the underlying CIA processes involved.
The simplest model is given by \citet{pwr:93}, which uses 
the same parameterization as for the water vapor continuum, described 
in Equation~\ref{eq:abs_cont}:
\begin{equation}
  \label{eq:abs_cont_pwr_n2}
    \alphac =  C \cdot \nu^2 \cdot \Theta^{3.55} \cdot \pnz^2
\end{equation}
with $C = 4.56\cdot 10^{-13}$ dB/(km\,hPa$^2$\,GHz$^2$). The
laboratory data set for the determination of $C$ is manly from 
\citet{dagg:75,dagg:78} around 70 and 140\,GHz, respectively.

The MPM models has compared with Equation~\ref{eq:abs_cont_pwr_n2} 
an additional frequency dependent term which leads to the 
following expression
\begin{eqnarray}
  \label{eq:abs_cont_MPM89_n2}
    \alphac &=& C' \cdot (1.0-1.2\cdot10^{-5}\cdot
               \nu^{1.5}) \cdot \nu^2 \cdot \Theta^{3.5} \cdot \pnz^2
               \hspace*{10mm}\mbox{MPM89}\\
%
  \label{eq:abs_cont_MPM93_n2}
    \alphac &=& C' \cdot \frac{\nu^2}{(1.0+1.9\cdot10^{-5}\cdot \nu^{1.5})} 
                \cdot \Theta^{3.5} \cdot \pnz^2
                \hspace*{15mm}\mbox{MPM93}\\
\nonumber
\end{eqnarray}
where the parameter is $C' = 2.55\cdot 10^{-13}$ dB/(km\,hPa$^2$\,GHz$^2$),
based on data from \citet{stankevich:74}. 
With respect to the WATS range ($\nu$\,$\approx$\,22\,GHz), 
the additional frequency terms in brackets in 
Equations~\ref{eq:abs_cont_MPM89_n2} and \ref{eq:abs_cont_MPM93_n2}
are nearly unity and therefore not essential. Therefore all
three parameterizations have the same frequency and temperature
relationship, but the absolute magnitude is in the case of Rosenkranz
80\,\% higher compared with the MPM models.

The not yet in arts implemented model of Borysow and 
Frommhold\footnote{{the source code of this
  model can be downloaded from the home page of A. Borysow:}\\
  http://www.astro.ku.dk/$\sim$aborysow/} is somewhat different since their 
focus is mainly on the radiative transfer in the Titan's atmosphere
with the infrared interferometer spectrometer, IRIS, onboard the
Voyager Spacecraft. This detailed model is primarily
designed to parameterize each of the rototranslational spectral lines 
around 200\,cm$^{-1}$ ($\approx$ 6\,THz) accurately. The analyzed data
set incorporate the data source used by the Rosenkranz but is largely 
extended with measurements in the far--infrared.



\leveld{Non-resonant carbon dioxide absorption}
\label{leveld:co2cont}
%----------------------------------------------
\citet{pwr:93} gives a similar parameterization for the $\coz$-continuum 
absorption term as for the nitrogen continuum, with 
\begin{equation}
  \label{eq:abs_cont_pwr_co2}
    \alphac =   \cdot \nu^2 \cdot  \Theta^{3} \cdot \pcoz  \cdot 
                  \left[ \pcoz \cdot C  \cdot \Theta^{2.08} +
                    \pnz  \cdot C' \cdot \Theta^{1.7} \right]
\end{equation}
where the parameters equal $C = 3.23\cdot 10^{-11}$
dB/(km\,hPa$^2$\,GHz$^2$) and $C' = 1.18\cdot 10^{-11}$
dB/(km\,hPa$^2$\,GHz$^2$), respectively, determined from laboratory
measurements of \citep{ho:66,dagg:75}.  Since the foreign term
includes only nitrogen as perturber, one can get an estimate for dry
air by replacing $\pnz$ by the dry air partial pressure in
Equation~\ref{eq:abs_cont_pwr_co2}. Because nitrogen is usually a more
efficient perturber than oxygen, this estimation can be regarded as an
upper limit. Concerning the Earth's atmosphere, the foreign broadening
term is more interesting since the carbon dioxide partial pressure is
only approximately 0.04\,\% of the nitrogen partial pressure up to
90\,km. Comparing the coefficients of $\cdo$ from the water vapor
continuum (Table~\ref{tab:wvcontparam}) with $C'$ from
Equation~\ref{eq:abs_cont_pwr_co2} leads to the conclusion that
$\coz$ does not play a significant role for frequencies around 22\,GHz
and can therefore be neglected.



\levelc{ARTS Workspace Variables and Methods}
\label{levelc:ArtsImplementationContinuum}
% -------------------------------------------



% ================================================================================
% The following section is written by Thomas Kuhn, iup Bremen, tkuhn@uni-bremen.de
% ================================================================================

\levelb{Complete Absorption Models}
\label{levelb:CompAbsMod}
% =================================
The MPM absorption model of Liebe and coworkers consists of modules for 
$\hzo$, dry air ($\oz, \nz$), liquid water droplets, and ice particles 
absorption. The Rosenkranz (R998) and Cruz-Pol et al. (CP98) absorption 
models include absorption due to water vapor and dry air. Additionally 
the CP98 model has a strongly reduced parameter set for the $\hzo$-line 
absorption since it is especially intended for the range around the 
22\,GHz water line. The MPM and R98 are valid from the microwave 
up to the submillimeter frequency range (1-1000\,GHz).

Implemented in ARTS are the following modules of the above stated models:
%
\begin{center}
\begin{tabular}{ll}
\hline
model & implemented modules \\
\hline
MPM87 & $\hzo$ \\
MPM89 & $\hzo$ \\
MPM93 & $\hzo$, $\oz$, $\nz$, liquid water and ice particles \\
R98   & $\hzo$, $\oz$, $\nz$ \\
CP98  & $\hzo$ \\
\hline
\end{tabular}
\end{center}




\levelc{Complete Water Vapor Models}
\label{levelc:CompWatVapMod}
% ==================================
In ARTS several complete water vapor absorption models are implemented and 
can easily be used.
Implemented models are the versions MPM87 \cite{liebeandlayton:87}, MPM89 
\cite{liebe:89}, and MPM93 \cite{liebeetal:93} of the Liebe absorption 
model and additionally the models of Cruz-Pol et al. (CP98) \cite{cruzpol:98} 
and P.~W. Rosenkranz (R98) \cite{pwr:98}. 
MPM and R98 are especially desigend for fast absorption calculations in 
the frequency range of 1-1000\,GHz while the CP98 model is a reduced model 
for a narrow frequency band around the 22\,GHz $\hzo$-line.

The total water vapor absorption ($\alphatot$) is in all the stated models 
described by a line absorption ($\alphal$) term and a continuum absorption 
($\alphac$) term: 
\begin{equation}
  \label{eq:h2o:totabs}
  \alphatot = \alphal + \alphac
\end{equation}
The main differences between the different models is the line shape used for 
$\alphal$ and the formulation of $\alphac$.

It has to be emphasized that, $\alphal$ and $\alphac$ of different
models are not necessarily compatible and should therefore not be interchanged.


\leveld{MPM87 Water Vapor Absorption Model}
\label{leveld:mpm87}
%------------------------------------------
This version, which is described in \cite{liebeandlayton:87} and 
follows the general line of the MPM model to devide the total water vapor absorption, 
$\alphampmotot$, into a spectral line term, $\alphampmol$, and a 
continuum term not attributed to spectral lines, $\alphampmoc$:
\begin{equation}
  \label{eq:mpm87_abs}
  \alphampmotot = \alphampmol + \alphampmoc\hspace*{10mm}\mbox{dB/km}
\end{equation}



\levele{Water Vapor Line Absorption:}
\label{levele:mpm87_h2olines}
%-----------------------------------
The MPM87 \cite{liebeandlayton:87} water vapor line catalog consists 
of 30 lines from 22\,GHz up to 988\,GHz. The center frequencies and parameter 
values are listed in Table~\ref{tab:mpm87linelist}. To describe the line 
absorption, a set of three parameters ($\bek$ and $\bdk$) per line are used: two 
for the line strength and one for the line width. The total line 
absorption coefficient (in units of dB/km) is the sum over all 
individual line absorption coefficients\footnote{The factor 
  $0.1820 \cdot 10^{6}$ is equal to $(4\,\pi/c)\cdot 10\log{(e)}$
  (the term $(4\,\pi/c)$ comes from the definition of the absorption
  coefficient in terms of the dielectric constant and the term 
  $10\,\log{(e)}$ is due to the definition of the Decibel.) The
  velocity of light is defined as $c=2.9979\cdot 10^{-4}$\,km\,GHz. 
  The factor $10^{6}$ is incorporated into the line strength and 
  does therefore not appear in the pre-factor.}:
\begin{equation}
  \label{eq:mpm87:absline}
  \alphampmol = 0.1820 \cdot \nuk \cdot \phzo \cdot 
  \sum_{k}{\inten \cdot \shape}\hspace*{10mm}\mbox{dB/km}
\end{equation}
where $\inten$ is the line intensity described by the parameterization
\begin{equation}
  \label{eq:mpm87:strength}
  \inten = \bek \cdot \phzo \cdot \Theta^{3.5} 
           \cdot \exp{(\bzk \cdot [1-\Theta])}\hspace*{10mm}\mbox{kHz}
\end{equation}
with $\nuk$ as the line center frequency, $\phzo$ the water
vapor partial pressure and $\Theta = 300\,\mbox{K}/T$.\\
The line shape function, $\shape$, in Eq.~(\ref{eq:mpm87:absline}) 
is the standard Van~Vleck-Weisskopf (VVW) function, given by:
\begin{eqnarray}
% Van Vleck-Weisskopf function
  \label{eq:mpm87:VVW}
  \shape & = & \left(\frac{\nu}{\nuk}\right) \cdot 
               \left[\frac{\gamk}{(\nu - \nuk)^2 + \gamk^2} + 
                     \frac{\gamk}{(\nu + \nuk)^2 + \gamk^2}\right]\\
\end{eqnarray}
The pressure broadened line width, $\gamk$, is calculated with the 
single parameter $\bdk$ in the following way:
\begin{equation}
  \label{eq:mpm87:gamma}
  \gamk = \bdk \cdot 
          (4.80 \cdot \phzo \cdot \Theta^{1.1} + \pda \cdot
          \Theta^{0.6})\hspace*{10mm}\mbox{GHz}
\end{equation}
where $\pda$ is the partial pressure of dry air ($\pda=\ptot-\phzo$). 
The parameterizations of $\inten$ and $\gamk$ are already in use for the 
early version of MPM81 \cite{liebe:81}.\\
%
\begin{longtable}{rrrrr}
 K & K & K & K & K \kill
%
% --------------------- only begin of table ------------------------------
 \hline
       & $\nu_k$ & $\bek$   & $\bzk$ & $\bdk$  \\
% //      [GHz]    [kHz/kPa]   [1]     [GHz/kPa]
 $k$   & {\sf [GHz]}  & {[$\frac{\sf kHz}{\sf kPa}$]} & {\sf [1]} & 
 {[$\frac{\sf GHz}{\sf kPa}$]}\\
 \hline
 \endfirsthead
% --------------------- every page begin of table ------------------------
 \hline
  $k$  & $\nu_k$ & $\bek$ & $\bzk$ & $\bdk$ \\
 \hline
 \endhead
% --------------------- every page end of table ------------------------
 K & K & K & K & K \kill
 \hline
 \caption[]{(continued)}\\
 \endfoot
% --------------------- only end of table ------------------------------
 K & K & K & K & K \kill 
 \hline
 \caption{List of H$_2$O spectral lines and their spectroscopic 
   parameters (H$_2$O-air mixture) for the MPM87 model \cite{liebeandlayton:87}.}
 \label{tab:mpm87linelist}
 \endlastfoot
% --------------------- body of table  ----------------------------------  
% //         0           1           2       3      
% //         f0          b1          b2      b3     
% //        [GHz]       [kHz/kPa]   [1]    [GHz/kPa]
%  const Numeric mpm87[30][4] = { 
1     &    22.235080&    0.1090&  2.143&   27.84$\cdot$ 10$^{-3}$\\
2     &    67.813960&    0.0011&  8.730&   27.60$\cdot$ 10$^{-3}$\\
3     &   119.995940&    0.0007&  8.347&   27.00$\cdot$ 10$^{-3}$\\
4     &   183.310117&    2.3000&  0.653&   31.64$\cdot$ 10$^{-3}$\\
5     &   321.225644&    0.0464&  6.156&   21.40$\cdot$ 10$^{-3}$\\
6     &   325.152919&    1.5400&  1.515&   29.70$\cdot$ 10$^{-3}$\\
7     &   336.187000&    0.0010&  9.802&   26.50$\cdot$ 10$^{-3}$\\
8     &   380.197372&   11.9000&  1.018&   30.36$\cdot$ 10$^{-3}$\\
9     &   390.134508&    0.0044&  7.318&   19.00$\cdot$ 10$^{-3}$\\
10    &   437.346667&    0.0637&  5.015&   13.70$\cdot$ 10$^{-3}$\\
11    &   439.150812&    0.9210&  3.561&   16.40$\cdot$ 10$^{-3}$\\
12    &   443.018295&    0.1940&  5.015&   14.40$\cdot$ 10$^{-3}$\\
13    &   448.001075&   10.6000&  1.370&   23.80$\cdot$ 10$^{-3}$\\
14    &   470.888947&    0.3300&  3.561&   18.20$\cdot$ 10$^{-3}$\\
15    &   474.689127&    1.2800&  2.342&   19.80$\cdot$ 10$^{-3}$\\
16    &   488.491133&    0.2530&  2.814&   24.90$\cdot$ 10$^{-3}$\\
17    &   503.568532&    0.0374&  6.693&   11.50$\cdot$ 10$^{-3}$\\
18    &   504.482692&    0.0125&  6.693&   11.90$\cdot$ 10$^{-3}$\\
19    &   556.936002&  510.0000&  0.114&   30.00$\cdot$ 10$^{-3}$\\
20    &   620.700807&    5.0900&  2.150&   22.30$\cdot$ 10$^{-3}$\\
21    &   658.006500&    0.2740&  7.767&   30.00$\cdot$ 10$^{-3}$\\
22    &   752.033227&  250.0000&  0.336&   28.60$\cdot$ 10$^{-3}$\\
23    &   841.073593&    0.0130&  8.113&   14.10$\cdot$ 10$^{-3}$\\
24    &   859.865000&    0.1330&  7.989&   28.60$\cdot$ 10$^{-3}$\\
25    &   899.407000&    0.0550&  7.845&   28.60$\cdot$ 10$^{-3}$\\
26    &   902.555000&    0.0380&  8.360&   26.40$\cdot$ 10$^{-3}$\\
27    &   906.205524&    0.1830&  5.039&   23.40$\cdot$ 10$^{-3}$\\
28    &   916.171582&    8.5600&  1.369&   25.30$\cdot$ 10$^{-3}$\\
29    &   970.315022&    9.1600&  1.842&   24.00$\cdot$ 10$^{-3}$\\
30    &   987.926764&  138.0000&  0.178&   28.60$\cdot$ 10$^{-3}$\\
\hline
% -----------------------------------------------------------------------  
\end{longtable}


\levele{Water Vapor Continuum Absorption:}
\label{levele:mpm87_h2ocont}
%----------------------------------------
The water vapor continuum absorption coefficient in MPM87, $\alphampmoc$, 
is determined from laboratory measurements at 137.8\,GHz by Liebe 
and Layton covering the following parameter range:\\
\begin{tabular}{lr}
temperature          & 282-316\,K\\
relative humidity    & 0-95\,\%\\
dry air pressure     & 0 - 160\,kPa\\ 
\end{tabular}\\
The mathematical expression of $\alphampmoc$ is derived from the far wing 
approximation of the line absorption and is expressed as follows
\begin{equation} 
  \label{eq:mpm87:cont}
  \alphampmoc = \nu^2 \cdot \phzo \cdot 
                (\cso \cdot \phzo \cdot \Theta^{\xs} + 
                 \cdo \cdot \pda  \cdot \Theta^{\xf}),
\end{equation}
with the continuum parameter set $\cso$, $\cdo$, $\xs$, and $\xf$. 
The determined values of the continuum parameters are:

\begin{description}
\item{$\cso$}   =  6.496\,$\cdot$\,10$^{-6}$~~(dB/km)~/~(hPa$\cdot$GHz)$^2$
\item{$\xs$}    = 10.5
\item{$\cdo$}   =  0.206\,$\cdot$\,10$^{-6}$~~(dB/km)~/~(hPa$\cdot$GHz)$^2$
\item{$\xd$}    =  3.0
\end{description}




\leveld{MPM89 Water Vapor Absorption Model}
\label{leveld:mpm89}
%------------------------------------------
%
MPM89 is described in \cite{liebe:89} and follows the general line 
of the MPM model to devide the total water vapor absorption, 
$\alphampmmtot$, into a spectral line term, $\alphampmml$, and a continuum 
term not attributed to spectral lines, $\alphampmmc$:
\begin{equation}
  \label{eq:mpm89_abs}
  \alphampmmtot = \alphampmml + \alphampmmc\hspace*{10mm}\mbox{dB/km}
\end{equation}
All the absorption coefficients are calculated in units of \mbox{dB/km}.


\levele{Water Vapor Line Absorption:}
\label{levele:mpm89_h2olines}
%-----------------------------------
The MPM89 water vapor line catalog consists of the same 30 lines 
like MPM87 from 22\,GHz up to 988\,GHz. The center frequencies and parameter 
values are listed in Table~\ref{tab:mpm89linelist}. To describe the line 
absorption, a set of six parameters ($\bek$ and $\bsk$) per line are used: two 
for the line strength and four for the line width. The total line 
absorption coefficient (in units of dB/km) is the sum over all
individual line absorption coefficients\footnote{see footnote for
  MPM97 line absorption}:
\begin{equation}
  \label{eq:mpm89:absline}
  \alphampmml = 0.1820 \cdot \nuk \cdot \phzo \cdot 
  \sum_{k}{\inten \cdot \shape}\hspace*{10mm}\mbox{dB/km}
\end{equation}
where $\inten$ is the line intensity described by the parameterization
\begin{equation}
  \label{eq:mpm89:strength}
  \inten = \bek \cdot \phzo \cdot \Theta^{3.5} 
           \cdot \exp{(\bzk \cdot [1-\Theta])}\hspace*{10mm}\mbox{kHz}
\end{equation}
whit $\nuk$ as the line center frequency, $\phzo$ the water
vapor partial pressure and $\Theta = 300\,\mbox{K}/T$.\\
The line shape function, $\shape$, in Eq.~(\ref{eq:mpm89:absline}) 
is the standard Van Vleck-Weisskopf (VVW) function, given by 
\begin{eqnarray}
% Van Vleck-Weisskopf function
  \label{eq:mpm89:VVW}
  \shape & = & \left(\frac{\nu}{\nuk}\right) \cdot 
               \left[\frac{\gamk}{(\nu - \nuk)^2 + \gamk^2} + 
                     \frac{\gamk}{(\nu + \nuk)^2 + \gamk^2}\right]
\end{eqnarray}
where the pressure broadened line width, $\gamk$, is calculated as
\begin{equation}
  \label{eq:mpm89:gamma}
  \gamk = \bdk \cdot 
         (\bfk \cdot \phzo \cdot \Theta^{\bsk} + 
                     \pda  \cdot \Theta^{\bvk})
        \cdot 10^{-3}\hspace*{10mm}\mbox{GHz}
\end{equation}
with $\pda=\ptot-\phzo$ as the dry air partial pressure. 
The only difference between MPM87 and MPM89 with respect to the line 
absorption is the parameterization of the pressure broadened line
width, $\gamk$, which is calculated with the four parameters $\bdk$ to
$\bsk$ in the case of MPM89 whereas in MPM87 a single parameter
($\bdk$) is used (see Eq.~(\ref{eq:mpm87:gamma})).
%
\begin{longtable}{rrrrrrrr}
 K & K & K & K & K & K & K & K \kill
%
% --------------------- only begin of table ------------------------------
 \hline
    & $\nu_k$ & $\bek$ & $\bzk$ & $\bdk$ & $\bvk$ & $\bfk$ & $\bsk$ \\
%    [GHz]     [kHz/kPa]   [1]   [MHz/kPa]  [1]    [1]    [1]
 $k$& {\sf [GHz]}  & {[$\frac{\sf kHz}{\sf kPa}$]} & {\sf [1]} & 
 {[$\frac{\sf MHz}{\sf kPa}$]} & {\sf [1]} & {\sf [1]} & {\sf [1]} \\
 \hline
 \endfirsthead
% --------------------- every page begin of table ------------------------
 \hline
  $k$  & $\nu_k$ & $\bek$ & $\bzk$ & $\bdk$ & $\bvk$ & $\bfk$ & $\bsk$ \\
 \hline
 \endhead
% --------------------- every page end of table ------------------------
 K & K & K & K & K & K & K & K \kill
 \hline
 \caption[]{(continued)}\\
 \endfoot
% --------------------- only end of table ------------------------------
 K & K & K & K & K & K & K & K \kill
 \hline
 \caption{List of H$_2$O spectral lines and their spectroscopic 
   parameters (H$_2$O-air mixture) for the MPM89 model \cite{liebe:89}.}
 \label{tab:mpm89linelist}
 \endlastfoot
% --------------------- body of table  ----------------------------------  
%            0           1        2       3        4      5      6
%            f0          b1       b2      b3       b4     b5     b6
%          [GHz]     [kHz/kPa]   [1]   [MHz/kPa]  [1]    [1]    [1]
%  const Numeric mpm89[30][7] = { 
1    &    22.235080&    0.1090&  2.143&   28.11&   0.69&  4.80&  1.00\\
2    &    67.813960&    0.0011&  8.735&   28.58&   0.69&  4.93&  0.82\\
3    &   119.995940&    0.0007&  8.356&   29.48&   0.70&  4.78&  0.79\\
4    &   183.310074&    2.3000&  0.668&   28.13&   0.64&  5.30&  0.85\\
5    &   321.225644&    0.0464&  6.181&   23.03&   0.67&  4.69&  0.54\\
6    &   325.152919&    1.5400&  1.540&   27.83&   0.68&  4.85&  0.74\\
7    &   336.187000&    0.0010&  9.829&   26.93&   0.69&  4.74&  0.61\\
8    &   380.197372&   11.9000&  1.048&   28.73&   0.69&  5.38&  0.84\\
9    &   390.134508&    0.0044&  7.350&   21.52&   0.63&  4.81&  0.55\\
10    &   437.346667&    0.0637&  5.050&   18.45&   0.60&  4.23&  0.48\\
11    &   439.150812&    0.9210&  3.596&   21.00&   0.63&  4.29&  0.52\\
12    &   443.018295&    0.1940&  5.050&   18.60&   0.60&  4.23&  0.50\\
13    &   448.001075&   10.6000&  1.405&   26.32&   0.66&  4.84&  0.67\\
14    &   470.888947&    0.3300&  3.599&   21.52&   0.66&  4.57&  0.65\\
15    &   474.689127&    1.2800&  2.381&   23.55&   0.65&  4.65&  0.64\\
16    &   488.491133&    0.2530&  2.853&   26.02&   0.69&  5.04&  0.72\\
17    &   503.568532&    0.0374&  6.733&   16.12&   0.61&  3.98&  0.43\\
18    &   504.482692&    0.0125&  6.733&   16.12&   0.61&  4.01&  0.45\\
19    &   556.936002&  510.0000&  0.159&   32.10&   0.69&  4.11&  1.00\\
20    &   620.700807&    5.0900&  2.200&   24.38&   0.71&  4.68&  0.68\\
21    &   658.006500&    0.2740&  7.820&   32.10&   0.69&  4.14&  1.00\\
22    &   752.033227&  250.0000&  0.396&   30.60&   0.68&  4.09&  0.84\\
23    &   841.073593&    0.0130&  8.180&   15.90&   0.33&  5.76&  0.45\\
24    &   859.865000&    0.1330&  7.989&   30.60&   0.68&  4.09&  0.84\\
25    &   899.407000&    0.0550&  7.917&   29.85&   0.68&  4.53&  0.90\\
26    &   902.555000&    0.0380&  8.432&   28.65&   0.70&  5.10&  0.95\\
27    &   906.205524&    0.1830&  5.111&   24.08&   0.70&  4.70&  0.53\\
28    &   916.171582&    8.5600&  1.442&   26.70&   0.70&  4.78&  0.78\\
29    &   970.315022&    9.1600&  1.920&   25.50&   0.64&  4.94&  0.67\\
30    &   987.926764&  138.0000&  0.258&   29.85&   0.68&  4.55&  0.90\\
\hline
% -----------------------------------------------------------------------  
\end{longtable}


\levele{Water Vapor Continuum Absorption:}
\label{levele:mpm89_h2ocont}
%----------------------------------------
The continuum absorption coefficient in MPM89, $\alphampmmc$, 
is the same as in MPM87 (see Sec. \ref{levele:mpm87_h2ocont} for 
details):
\begin{equation} 
  \label{eq:mpm89:cont}
  \alphampmmc = \nu^2 \cdot \phzo \cdot 
                (\cso \cdot \phzo \cdot \Theta^{\xs} + 
                 \cdo \cdot \pda  \cdot \Theta^{\xf}),
\end{equation}
with
\begin{description}
\item{$\cso$}   =  6.496\,$\cdot$\,10$^{-6}$~~(dB/km)~/~(hPa$\cdot$GHz)$^2$
\item{$\xs$}    = 10.5
\item{$\cdo$}   =  0.206\,$\cdot$\,10$^{-6}$~~(dB/km)~/~(hPa$\cdot$GHz)$^2$
\item{$\xd$}    =  3.0
\end{description}





\leveld{MPM93 Water Vapor Absorption Model}
\label{leveld:mpm93}
%----------------------------------------------
This version, which is described in \cite{liebeandlayton:87} and 
follows the general line of the MPM model to devide the total 
water vapor absorption, $\alphampmntot$, into a spectral line 
term, $\alphampmnl$, and a continuum term not attributed to 
spectral lines, $\alphampmnc$:
\begin{equation}
  \label{eq:mpm93_abs}
  \alphampmntot = \alphampmnl + \alphampmnc\hspace*{10mm}\mbox{dB/km}
\end{equation}
The continuum absorption is parameterized like a
resonant spectral line of $\hzo$, a so-called pseudo-line. This is a 
fundamental change in the parameterization of the water vapor
continuum in respect to all older versions of MPM, which makes it 
quite complicate to compare the different versions, especially to 
distinguish a self- and foreign broadening term in the continuum.



\levele{Water Vapor Line Absorption:}
\label{levele:mpm93_h2olines}
%-----------------------------------
The water vapor line spectrum of MPM93 \cite{liebeetal:93} 
consists of 34 lines below 1\,THz (four more than in MPM89 and MPM87). 
To describe the MPM93 water vapor line absorption, a set of six parameters 
($\bek$ and $\bdk$) per line are used: two for the line strength and 
four for the line width. The total line absorption coefficient 
(in units of dB/km) is the sum over all individual line absorption 
coefficients\footnote{see footnote for MPM97 line absorption}:
\begin{equation}
  \label{eq:mpm93:absline}
  \alphampmnl = 0.1820 \cdot \nuk \cdot \phzo \cdot 
  \sum_{k}{\inten \cdot \shape}\hspace*{10mm}\mbox{dB/km}
\end{equation}
where $\inten$ is the line intensity described by the parameterization
\begin{equation}
  \label{eq:mpm93:strength}
  \inten = \bek \cdot \phzo \cdot \Theta^{3.5} 
           \cdot \exp{(\bzk \cdot [1-\Theta])}\hspace*{10mm}\mbox{kHz}
\end{equation}
with $\nuk$ as the line center frequency, $\phzo$ the water
vapor partial pressure and $\Theta = 300\,\mbox{K}/T$.\\
The line shape function, $\shape$, in Eq.~(\ref{eq:mpm87:absline}) 
is the standard Van~Vleck-Weisskopf (VVW) function, given by:
\begin{eqnarray}
% Van Vleck-Weisskopf function
  \label{eq:mpm93:VVW}
  \shape & = & \left(\frac{\nu}{\nuk}\right) \cdot 
               \left[\frac{\gamk}{(\nu - \nuk)^2 + \gamk^2} + 
                     \frac{\gamk}{(\nu + \nuk)^2 + \gamk^2}\right]\\
\end{eqnarray}
The pressure broadened line width, $\gamk$, is calculated with the 
single parameter $\bdk$ in the following way:
\begin{equation}
  \label{eq:mpm93:gamma}
  \gamk = \bdk \cdot 
          (4.80 \cdot \phzo \cdot \Theta^{1.1} + \pda \cdot
          \Theta^{0.6})\hspace*{10mm}\mbox{GHz}
\end{equation}
where $\pda$ is the partial pressure of dry air ($\pda=\ptot-\phzo$). 

The parameterizations of $\inten$ was already in use for the early 
version of MPM81 \cite{liebe:81}. The expression for $\gamk$ is the
same as in MPM89. The main difference between MPM93 and MPM89 
concerning the water vapor line absorption is the updated line catalog.
%
%
\begin{landscape}
\begin{longtable}{rrrrrrrrrr}
 K & K & K & K & K & K & K & K & K & K \kill
%
% --------------------- only begin of table ------------------------------
 \hline
    & $\nu_k$ & $\bek$ & $\bzk$ & $\bdk$ & $\bdkp$ 
 & $\bvk$ & $\bvkp$ & $\bfk$ & $\bsk$ \\
 $k$   & {\sf [GHz]}  & {[$\frac{\sf kHz}{\sf hPa}$]} & {\sf [1]} & 
 {[$\frac{\sf MHz}{\sf hPa}$]} & {[$\frac{\sf MHz}{\sf hPa}$]} & 
 {\sf [1]} & {\sf [1]} & {\sf [1]} & {\sf [1]} \\
 \hline
 \endfirsthead
% --------------------- every page begin of table ------------------------
 \hline
    & $\nu_k$ & $\bek$ & $\bzk$ & $\bdk$ & $\bdkp$ 
 & $\bvk$ & $\bvkp$ & $\bfk$ & $\bsk$ \\
 \hline
 \endhead
% --------------------- every page end of table ------------------------
 K & K & K & K & K & K & K & K & K & K \kill
 \hline
 \caption[]{(continued)}\\
 \endfoot
% --------------------- only end of table ------------------------------
 K & K & K & K & K & K & K & K & K & K \kill
 \hline
 \caption{List of used H$_2$O spectral lines and their spectroscopic 
   coefficients of H$_2$O in air for the MPM models. The last 
   separated line is the unphysical pseudo-line used in MPM93. 
   The lines which are marked with a "$^+$" were not in the MPM87/MPM89 
   line catalog. The data is from \cite{liebeetal:93}. The parameters
   $b'_{3,k}$ are calculated values from Table\,3 of
   \cite{abaueretal:89} for a mixture of $\hzo-\nz$. Because of the
   missing line parameters for the 552\,GHz line in
   \cite{abaueretal:89}, the same value is assumed as for the 547\,GHz line.}
 \label{tab:mpm93linelist}
 \endlastfoot
% --------------------- body of table  ----------------------------------  
1      & 22.235080  & 0.01130 & 2.143 & 2.811 & 2.903 & 4.80 & 4.648 & 0.69 & 1.00 \\
2      & 67.803960  & 0.00012 & 8.735 & 2.858 & 3.083 & 4.93 & 4.570 & 0.69 & 0.82 \\
3      & 119.995940 & 0.00008 & 8.356 & 2.948 & 3.173 & 4.78 & 4.441 & 0.70 & 0.79 \\
4      & 183.310091 & 0.24200 & 0.668 & 3.050 & 3.173 & 5.30 & 5.095 & 0.64 & 0.85 \\
5      & 321.225644 & 0.00483 & 6.181 & 2.303 & 2.498 & 4.69 & 4.324 & 0.67 & 0.54 \\ 
6      & 325.152919 & 0.14990 & 1.540 & 2.783 & 2.993 & 4.85 & 4.509 & 0.68 & 0.74 \\
7      & 336.222601 & 0.00011 & 9.829 & 2.693 & 2.909 & 4.74 & 4.397 & 0.69 & 0.61 \\ 
8      & 380.197372 & 1.15200 & 1.048 & 2.873 & 3.083 & 5.38 & 5.014 & 0.54 & 0.89 \\
9      & 390.134508 & 0.00046 & 7.350 & 2.152 & 2.333 & 4.81 & 4.437 & 0.63 & 0.55 \\
10     & 437.346667 & 0.00650 & 5.050 & 1.845 & 1.980 & 4.23 & 3.942 & 0.60 & 0.48 \\
11     & 439.150812 & 0.09218 & 3.596 & 2.100 & 2.258 & 4.29 & 3.990 & 0.63 & 0.52 \\
12     & 443.018295 & 0.01976 & 5.050 & 1.860 & 1.995 & 4.23 & 3.944 & 0.60 & 0.50 \\
13     & 448.001075 & 1.03200 & 1.405 & 2.632 & 2.775 & 4.84 & 4.591 & 0.66 & 0.67 \\
14     & 470.888947 & 0.03297 & 3.599 & 2.152 & 2.318 & 4.57 & 4.242 & 0.66 & 0.65 \\
15     & 474.689127 & 0.12620 & 2.381 & 2.355 & 2.535 & 4.65 & 4.320 & 0.65 & 0.64 \\
16     & 488.491133 & 0.02520 & 2.853 & 2.602 & 2.805 & 5.04 & 4.675 & 0.69 & 0.72 \\
17     & 503.568532 & 0.00390 & 6.733 & 1.612 & 1.733 & 3.98 & 3.702 & 0.61 & 0.43 \\
18     & 504.482692 & 0.00130 & 6.733 & 1.612 & 1.733 & 4.01 & 3.730 & 0.61 & 0.45 \\
19$^+$ & 547.676440 & 0.97010 & 0.114 & 2.600 & 2.798 & 4.50 & 4.182 & 0.70 & 1.00 \\
20$^+$ & 552.020960 & 1.47700 & 0.114 & 2.600 & 2.798 & 4.50 & 4.182 & 0.70 & 1.00 \\
21     & 556.936002 & 48.74000& 0.159 & 3.210 & 3.443 & 4.11 & 3.832 & 0.69 & 1.00 \\
22     & 620.700807 & 0.50120 & 2.200 & 2.438 & 2.618 & 4.68 & 4.358 & 0.71 & 0.68 \\
23$^+$ & 645.866155 & 0.00713 & 8.580 & 1.800 & 1.936 & 4.00 & 3.719 & 0.60 & 0.50 \\
24     & 658.005280 & 0.03022 & 7.820 & 3.210 & 3.443 & 4.14 & 3.860 & 0.69 & 1.00 \\
25     & 752.033227 & 23.96000& 0.396 & 3.060 & 3.285 & 4.09 & 3.810 & 0.68 & 0.84 \\
26     & 841.053973 & 0.00140 & 8.180 & 1.590 & 1.755 & 5.76 & 5.218 & 0.33 & 0.45 \\
27     & 859.962313 & 0.01472 & 7.989 & 3.060 & 3.285 & 4.09 & 3.068 & 0.68 & 0.84 \\
28     & 899.306675 & 0.00605 & 7.917 & 2.985 & 3.195 & 4.53 & 4.232 & 0.68 & 0.90 \\
29     & 902.616173 & 0.00426 & 8.432 & 2.865 & 3.083 & 5.10 & 4.739 & 0.70 & 0.95 \\
30     & 906.207325 & 0.01876 & 5.111 & 2.408 & 2.550 & 4.70 & 4.438 & 0.70 & 0.53 \\
31     & 916.171582 & 0.83400 & 1.442 & 2.670 & 2.865 & 4.78 & 4.455 & 0.70 & 0.78 \\
32$^+$ & 923.118427 & 0.00869 & 10.220& 2.900 & 3.118 & 5.00 & 4.650 & 0.70 & 0.80 \\
33     & 970.315022 & 0.89720 & 1.920 & 2.550 & 2.760 & 4.94 & 4.564 & 0.64 & 0.67 \\
34     & 987.926764 & 13.21000& 0.258 & 2.985 & 3.195 & 4.55 & 4.251 & 0.68 & 0.90 \\
\hline
 & $\nu^*$ & $\beks$ & $\bzks$ & $\bdks$ & $\bdkps$ & $\bvks$ &
 $\bvkps$ & $\bfks$ & $\bsks$\\
 & {\sf [GHz]}  & {[$\frac{\sf kHz}{\sf hPa}$]} & {\sf [1]} & 
 {[$\frac{\sf MHz}{\sf hPa}$]} & {[$\frac{\sf MHz}{\sf hPa}$]} & 
 {\sf [1]} & {\sf [1]} & {\sf [1]} & {\sf [1]} \\
\hline
 & 1780.000000 & 2230.00000 & 0.952 & 17.620 & 19.021 & 30.50 & 28.254 & 2.00 & 5.00 \\
% -----------------------------------------------------------------------  
\end{longtable}
\end{landscape}



\levele{The MPM93 Continuum Parameterization:}
\label{levele:mpm93:h2ocont}
%-----------------------------------------------
Liebe used measured total absorption coefficient data to fit the 
parameters of pseudo water vapor line ($\beks$--$\bsks$). 
The fitted line center frequency for this pseudo-line is at 
1.780 THz and therefore out of the valid frequency region of MPM 
itself. Table~\ref{tab:mpm93linelist} lists the values of this set 
of parameters. It is remarkable that all these parameters are much 
larger compared to physical lines, except for $\bzks$, the parameter 
which governs the exponential temperature behavior of the line strength:
\begin{eqnarray}
  \label{eq:mpm93:contabs}
  \alphampmnc & = & 0.1820 \cdot \frac{\beks}{\nucc} \cdot \phzo \cdot 
                \Theta^{3.5} \cdot e^{\bzks\cdot(1-\Theta)} \cdot 
                \nu^2 \cdot F_c(\nu)\\
%
  \label{eq:mpm93:contabs1}
  F_c(\nu) & = & \left[\frac{\gamc}{(\nucc+\nu)^2+\gamc^2} + 
                       \frac{\gamc}{(\nucc-\nu)^2+\gamc^2}\right]\\
%
  \label{eq:mpm93:contabs2}
  \gamc & = &  \bdks \cdot 
          \left(\bvks \cdot \phzo \cdot \Theta^{\bsks}~~+~~ 
                            \pda  \cdot \Theta^{\bfks}\right)\\
\nonumber
\end{eqnarray}
The magnitude of the pseudo-line width is shown for four different 
cases in Table\,\ref{tab:mpm_psl_broad}. With respect to a frequency
in the microwave range, the line shape function, $F_c(\nu)$, can be 
approximated by
\begin{equation}
 \label{eq:mpm93:contfapp1}
 F_c(\nu) \approx 2 \cdot \frac{\gamc}{\nucc^2}
\end{equation}
Inserting Eq.~(\ref{eq:mpm93:contfapp1}) into Eq.~(\ref{eq:mpm93:contabs})
shows the quadratic frequency dependence of the MPM93 continuum
similar to the continuum parameterization expressed in MPM89 and earlier 
versions. By additionally approximating the temperature dependence to the 
simple form
\begin{eqnarray}
  \xs\cdot\ln{(\Theta)} & = & 
  \ln{\left(\Theta^{3.5} \cdot e^{\bzks\cdot(1-\Theta)}\right)}\nonumber\\
%
  \xs  & = & 3.5 +
  \bzks\cdot\frac{1-\Theta}{\ln{(\Theta)}}\nonumber\\
%
  \xs & \approx & 3.5 -\bzks = 2.55\hspace*{5mm}\mbox{with}\hspace*{2mm}
                 \ln{(\Theta)}\approx(\Theta-1)\\
\nonumber
\end{eqnarray}
one can rearrange the pseudo-line continuum to fit Eq.~(\ref{eq:mpm87:cont})
(denoted by MPM93$^*$). The deduced continuum parameter set is:
\begin{table}
\begin{tabular}{lcl}
 $\cso$  & = & 7.73$\cdot$10$^{-8}$~dB/km/hPa$^2$~GHz$^2$\\
 $\xs$   & = & 7.55\\
 $\cdo$  & = & 0.253$\cdot$10$^{-8}$~dB/km/hPa$^2$~GHz$^2$\\ 
 $\xd$   & = & 4.55 \\
\end{tabular}
\end{table}
The MPM93$^*$ continuum parameters $\cso$ and $\cdo$ are 20\,\% and 
15\,\% larger, respectively, than in the case of MPM87/MPM89. 
Compared with the R98 model, $\cso$ is 1\,\% smaller and
$\cdo$ is 7\,\% increased. A very good agreement is between CP98 and
MPM93$^*$ in the case of $\cdo$. Large discrepancies exist for the
temperature exponents $\xs$ and $\xd$ between MPM93$^*$ and earlier
versions of the same model. The exponent $\xs$ is in MPM93$^*$ only
60\,\% of the corresponding value in MPM89 and the temperature
dependence of the $\hzo$-air term is significant larger than for
earlier MPM versions. This reduction of $\xs$ is mainly due to 
additional measurements considered in MPM93 
(Refs. \cite{beckerautler:46,godonetal:92}), while the continuum 
parameters in MPM87/MPM89 are determined by a single laboratory 
measurement at 138\,GHz.
%
\begin{table}[!htb]
\begin{center}
\begin{tabular}{lrrr}
\hline
 & \multicolumn{2}{c}{contribution} & \multicolumn{1}{c}{total} \\
 & \multicolumn{1}{c}{$\hzo$--$\hzo$} & \multicolumn{1}{c}{$\hzo$--air} & \\
\hline
$\gamc$(200\,K) & 40.8\,GHz & 80.4\,GHz & 121.2\,GHz\\
$\gamc$(300\,K) &  5.4\,GHz & 23.0\,GHz &  28.4\,GHz\\
\hline
\end{tabular}
\caption{Magnitude of the line width of the pseudo-line of the
  continuum term in MPM93. Assumed is a total pressure of 1000\,hPa
  and a water vapor partial pressure of 10\,hPa.}
\label{tab:mpm_psl_broad}
\end{center}
\end{table} 





\leveld{CP98 Water Vapor Absorption Model}
\label{leveld:cp98}
%---------------------------------------------

\levele{Line Absorption}
\label{levele:cp98_h2oline}
%--------------------------
component \citep{cruzpol:98} for the water vapor line absorption 
is based on MPM87 with the main difference that the 
line catalog consists of only a single line at $\nu_{\sf o}=$\,22\,GHz. 
The contributions from the other lines is put into the water vapor 
continuum module. The line absorption is therefore very quickly 
calculated (in units of Np/km) according to the formula
\begin{eqnarray}
  \label{eq:cp98:lineabs}
  \alphacpl &=& 0.0419 \cdot \intencp \cdot \shape \\
  \mbox{with} & & \nonumber\\
  \intencp    &=& 0.0109 \cdot C_L \cdot \phzo \cdot \nuo \cdot \Theta^{3.5} 
             \cdot \exp{(2.143\cdot[1-\Theta])}\nonumber\\
%
  \gamma &=& 0.002784 \cdot C_W \cdot (\pda \cdot \Theta^{0.6}+ 
             4.8 \cdot \phzo \cdot \Theta^{1.1}) \nonumber\\
\end{eqnarray}
where $\phzo$ and $\pda$ are the partial pressure of water vapor and dry
air in units of hPa, respectively and the Van Vleck-Weisskopf line
shape, $\shape$. The numbers correspond to the line
parameters form MPM87 for this special line and the factors  
$C_L$ and $C_W$ are adjustable scaling factors to match the model with the
measurements. Setting the scaling factors to $C_L$=1.00 and $C_W$=1.00 
leads to the same results as for MPM87. According to the parameter 
estimation of Cruz--Pol et al. best agreement between 
data and model is obtained with $C_L=$\,1.0639$\pm$0.016 and 
$C_W=$\,1.0658$\pm$0.0096. The correlation between these two scaling 
factors was found to be negligible, as can be seen from 
Table \ref{tab:cp_orr}.

\begin{table}[!htb]
\begin{center}
\begin{tabular}{lllll}
\hline
            & $C_L$ & $C_W$ & $C_C$ & $C_X$ \\
\hline
value       & 1.0639 & 1.0658 & 1.2369 & 1.0739\\
std. dev.   & 0.016  & 0.0096 & 0.155  & 0.252\\
\hline
correlation & &&&\\
$C_L$       & 1      & -0.085 & 0.045  & -0.048\\
$C_W$       & -0.085 & 1      & -0.513 &  0.485\\
$C_C$       & 0.045  & -0.513 & 1      & -0.989\\
$C_X$       & -0.048 & 0.485  & -0.989 & 1\\
\hline
\end{tabular}
\end{center}
\caption{Scaling parameter values with standard deviation and 
  correlation coefficients according to \citep{cruzpol:98}.
  The scaling parameters are $C_L$:22\,GHz line strength, 
  $C_W$:22\,GHz line width , $C_C$:$\hzo$-continuum, and 
  $C_X$:$\oz$-absorption. $C_X$ scales the entire oxygen absorption, 
  the continuum as well as the line absorption. The Cruz-Pol et al.
  model uses the \cite{pwr:93} oxygen absorption model.}
\label{tab:cp_orr}
\end{table}

The main reason why the Cruz-Pol model (CP98) considers only one line
lies in the fact that CP98 is especially designed for the data analysis
in the 20-31.4\,GHz region. The determination of the scaling factors was 
performed with ground based radiometer data in the frequency range of
from different locations\footnote{The data were recorded at San Diego, 
California (11. December 1991) and West Palm Beach, Florida 
(8.-21. March 1992)} in the USA.


\levele{Water Vapor Continuum Absorption:}
\label{levele:cp98_h2ocont}
%-----------------------------------------
Resonant line absorption models for the microwave region using
standard line shape functions systematically underestimate the total
amount of absorption in the far wing regions of the strong water vapor
and oxygen lines (so called window regions). This fact is well known
from several comparisons of model calculations with field and
laboratory measurements. Different roads have been followed to close
this discrepancy. For purposes of atmospheric radiative transfer
models, a simple and fast empirical parameterization of this excess
absorption is commonly used. Since this excess absorption is a smooth
and slowly varying function of the frequency, it is commonly called
continuum absorption.

\begin{tabular}{lcl}
$\cso$   & = & 8.04$\cdot$10$^{-8}$(dB/km)/(hPa$\cdot$GHz)$^2$\\
$\xs$    & = & 10.5\\
$\cdo$   & = & 0.254$\cdot$10$^{-8}$(dB/km)/(hPa$\cdot$GHz)$^2$\\
$\xd$    & = & 3.0\\
\end{tabular}




\leveld{PWR98 Water Vapor Absorption Model}
\label{leveld:pwr98_h2o}
%------------------------------------------
The water vapor continuum formulation of \citet{pwr:98} is a re-investigation 
of the existing models MPM87/MPM89, MPM93, and CKD\_2.1 especially for 
the frequency region below 1-1000\,GHz. in the context of the available
laboratory and atmospheric data \citep{abaueretal:89, abaueretal:93, 
abaueretal:95, beckerautler:46, englishetal:94, godonetal:92,
liebe:84, liebeandlayton:87, westwateretal:90}.

Rosenkranz adopted the structure of MPM89 for his improved model (R98). 
However, some important differences exist compared with MPM89:
\begin{itemize}
\item the water vapor line catalogs are different 
\item the R98 uses the Van~Vleck--Weisskopf line shapefunction with 
      cutoff and MPM89 without cutoff
\end{itemize}


\levele{Water Vapor Line Absorption:}
\label{levele:pwr98_h2oline}
%------------------------------------
The local line absorption is defined as 
\begin{eqnarray} 
 \label{eq:pwr98absline}
 \alphapwrl &=& N_{H_2O} \cdot \sum_k \inten \cdot \shapec \nonumber\\
            &=& N_{H_2O} \cdot \sum_k \inten \cdot 
                \left (\displaystyle{\frac{\nu}{\nuk}}\right )^2  \cdot 
                \left [\shapefp + \shapefm \right]~~\mbox{Np/km}
\end{eqnarray}
where $N_{H_2O}$ is the number density of water molecules, $\nu$ the
frequency and $S$ the line intensity, calculated from the HITRAN92
data base \citet{rothman:92}. Considered for this re-investigation are 
15 lines with a frequency lower than 1\,THz as listed in 
Table~\ref{tab:pwr98linelist}.

The line shape function $\shapec$ has a cutoff frequency, $\nucut$,
and a baseline subtraction similar to the CKD model \cite{clough:89}.
The introduction of a cutoff frequency has two advantages: (1) the
cutoff avoids applying the line shape to distant frequencies where the 
line form is theoretically not well understood and (2) the cutoff also
establishes a limit to the summation in Eq.~(\ref{eq:pwr98absline}) where lines
far away from the cutoff limit do not contribute to the sum.  
The Rosenkranz formulation uses the same value for
the cutoff frequency as the CKD model:
\begin{equation} 
 \label{cutoff}
 \nucut = 750\mbox{ GHz}
\end{equation}
%
The explicit mathematical form of the line shape function is defined 
in such a way that in the limit $\nucut \rightarrow \infty$ the 
combination of Eq.~(\ref{eq:pwr98absline}) with the line shape function would 
be equivalent to a Van Vleck--Weisskopf \citep{vanvleck:45} line shape: 
\begin{equation}
 \label{eq:pwr98lineshape}
 \hspace*{-5mm}\shapefpm = 
   \left \{ \begin{array}{r@{\quad:\quad}l} 
   \displaystyle{\frac{\gamk}{\pi}} 
   \left \{ \displaystyle{\frac{1}{(\nu \mp \nuk)^2 + \gamk^2}} - 
   \displaystyle{\frac{1}{\nucut^2 + \gamk^2}} \right \}
   & |\nu \pm \nuk| < \nucut \\ 
   0 & |\nu \pm \nuk| \geq \nucut
                       \end{array} \right.
\end{equation}
$\nuk$ is the line center frequency and $\gamk$ the line
half width, which is calculated according to 
\begin{equation}
 \label{eq:pwr98gamma}
 \gamk = \ws \cdot \phzo \cdot \Theta^{\xs} + 
         \wf \cdot \pda  \cdot \Theta^{\xf}\hspace*{10mm}\mbox{GHz}
\end{equation}
with $\phzo$ and $\pda$ as the partial pressure of water vapor and of 
dry air, respectively. The line depending parameters $\ws$, $\xs$, 
$\wf$, and $\xf$ are listed in Table~\ref{tab:pwr98linelist} and the 
dimensionless parameter $\Theta$ is defined as $\Theta$\,=\,300\,K/$T$.

Because of the structural similarity to MPM89, the line broadening 
parameters differ only in minor respects from the values used therein 
(only the parameters $x_{\sf s,1}$, $w_{\sf f,2}$ and $\sf w_{\sf s,2}$ 
are significantly different).
%
\begin{table}[!htb]
\begin{center}
\begin{tabular}{rrrrrr}
 \hline
 index &  $\nuk$      & $\wf$     & $\xf$ & $\ws$     & $\xs$ \\
   k   &  [GHz]       & [GHz/kPa] & [1]   & [GHz/kPa] & [1] \\ 
 \hline
   1   &   22.2351    & 0.00281   & 0.69  & 0.01349   &  0.61 \\
   2   &  183.3101    & 0.00281   & 0.64  & 0.01491   &  0.85 \\
   3   &  321.2256    & 0.00230   & 0.67  & 0.01080   &  0.54 \\
  4    &  325.1529    & 0.00278   & 0.68  & 0.01350   &  0.74 \\
  5    &  380.1974    & 0.00287   & 0.54  & 0.01541   &  0.89 \\
  6    &  439.1508    & 0.00210   & 0.63  & 0.00900   &  0.52 \\
  7    &  443.0183    & 0.00186   & 0.60  & 0.00788   &  0.50 \\
  8    &  448.0011    & 0.00263   & 0.66  & 0.01275   &  0.67 \\
  9    &  470.8890    & 0.00215   & 0.66  & 0.00983   &  0.65 \\
  10   &  474.6891    & 0.00236   & 0.65  & 0.01095   &  0.64 \\
  11   &  488.4911    & 0.00260   & 0.69  & 0.01313   &  0.72 \\
  12   &  556.9360    & 0.00321   & 0.69  & 0.01320   &  1.00 \\
  13   &  620.7008    & 0.00244   & 0.71  & 0.01140   &  0.68 \\
  14   &  752.0332    & 0.00306   & 0.68  & 0.01253   &  0.84 \\
  15   &  916.1712    & 0.00267   & 0.70  & 0.01275   &  0.78 \\
  \hline
\end{tabular}
\end{center}
  \caption{Line parameters of the Rosenkranz absorption model (R98) 
  (values taken from \citet{pwr:98}).}
\label{tab:pwr98linelist}
\end{table}



\levele{Water Vapor Continuum Absorption:}
\label{levele:pwr98_h2ocont}
%-----------------------------------------
The continuum absorption in R98 has the same functional dependence on frequency,
presure, and temperature like in MPM87/MPM89 (see Sec. \ref{levele:mpm87_h2ocont}
for details):
\begin{equation} 
  \label{eq:pwr98:abscont}
  \alphapwrc = \nu^2 \cdot \phzo \cdot 
               (\cso \cdot \phzo \cdot \Theta^{\xs} + 
                \cdo \cdot \pda  \cdot \Theta^{\xf})
\end{equation}
with
\begin{description}
\item{$\cso$}   = 7.80\,$\cdot$\,10$^{-8}$~~(dB/km)~/~(hPa$\cdot$GHz)$^2$
\item{$\xs$}    = 7.5
\item{$\cdo$}   =  0.236\,$\cdot$\,10$^{-8}$~~(dB/km)~/~(hPa$\cdot$GHz)$^2$
\item{$\xd$}    = 3.0
\end{description}
The main difference to the MPM versions are the values of these 
parameters, since Rosenkranz used additional data to fit his set of 
parameters. A second point is the cutoff in the line shape of the line 
absorption calculation. Since this cutoff decreases the line absorption 
in the window regions, the continuum absorption tends to compensate this 
decrease to get the same total absorption as withouot cutoff. This effects 
mainly the parameters $\cso$ and $\cdo$ but has also an influence in the 
temperature dependence and therefore on $\xs$ and $\xd$.




\levelc{Complete Oxygen Models}
\label{levelc:02_models}
%==============================
%
Since the Maxwell equations are symmetric in the electric and
magnetic fields, electric as well as magnetic dipole transitions 
are both possible although magnetic dipoles are in general some
orders of magnitudes weaker and therefore not relevant in
atmospheric radiative transfer models. An exception to this is the complex 
around 60\,GHz of the paramagnetic oxygen magnetic dipole transitions. 
This bulk of lines arise due to the fact that for rotational 
quantum numbers $K>1$ the allowed transitions \mbox{$\Delta J = \pm$1} 
have an energy gap of approximately 60\,GHz.\\
The most frequently used absorption model for this absorption effect is that of
Liebe, Rosenkranz, and Hufford \cite{liebeetal:92} (also reported in 
\cite{pwr:93} with a slightly different parameterization).

For oxygen -- like for water vapor -- the total absorption 
($\alphatot$) is modelled as the line absorption ($\alphal$) plus a  
continuum absorption ($\alphac$):
\begin{equation}
  \label{eq:o2:totabs}
  \alphatot = \alphal + \alphac
\end{equation}
It has to be emphasized that, $\alphal$ and $\alphac$ of different
models are not necessarily compatible and should therefore not be interchanged.




\leveld{PWR93 Oxygen Absorption Model}
\label{leveld:O2_pwr98}
%-------------------------------------


\levele{Resonant Oxygen Absorption}
%\label{levele:02_pwr98_line}
\label{levele:pwr93_o2lines}
%----------------------------------
The oxygen absorption model of Rosenkranz is described in \cite{pwr:93}. It 
is based on the investigations made by Liebe, Rosenkranz, and Hufford 
\cite{liebeetal:92}. The FORTRAN77 computer programme of Rosnekranz for 
the $\oz$ absorption calculation can be downloaded via anonymous ftp from 
mesa.mit.edu/phil/lbl\_rt.

The oxygen line catalog has 40 lines from which 33 lines build the 
complex around 60\,GHz. The parameterization of the line absorption,
$\alphapwrl$, is:
\begin{eqnarray}
% line ansorption:
  \alphapwrl & = & \frac{n_{\sf O_2}}{\pi} \cdot 
                   \sum_{k=1}^{40}{S_k(T) \cdot F(\nu,\nu_k)}\\
%
% \mbox{with} &   &\nonumber\\
%
% line intensity:
 &&\mbox{line intensity:} \nonumber\\
      S_k(T) & = & S_k(300\,{\sf K})~~/~~\exp{(b_k \cdot \Theta)}\nonumber\\
% line shape:
 &&\mbox{line shape function:} \nonumber\\
   F(\nu,\nu_k) & = & \left(\frac{\nu}{\nuk}\right)^2 \cdot 
                   \left[\frac{\Gamma_k+(\nu-\nuk)\cdot Y_k}
                              {(\nu-\nu_k)^2+\Gamma_k^2}~~+~~
                         \frac{\Gamma_k-(\nu+\nuk)\cdot Y_k}
                              {(\nu+\nu_k)^2+\Gamma_k^2}\right] \nonumber\\
% line width:
 &&\mbox{line width:} \nonumber\\
    \Gamma_k & = & w_k \cdot \left(          \pda  \cdot \Theta^{0.8} + 
                                   1.1 \cdot \phzo \cdot \Theta \right) \nonumber\\
% line coupling:
 &&\mbox{line coupling:} \nonumber\\
         Y_k & = & \pdair \cdot \Theta^{0.8} \cdot 
                   \left[ y_k + (\Theta-1) \cdot v_k \right]\nonumber\\
% O2 number density:
 &&\mbox{number density of $\oz$:}  \nonumber\\
           n_{\sf O_2} & = & (0.20946 \cdot \pdair)/(k_B \cdot T)\nonumber\\
           \nonumber
\end{eqnarray}
where $Q_{e,v}$ are the electronic and vibrational partition functions, 
respectively, and $S_k(300\,{\sf K})$ denotes the reference line
intensity. All model parameters (see Refs. \cite{pwr:93} and \cite{liebeetal:92}
for the laboratory measuremnts and the fitting parameters) are 
tabulated in Table \ref{tab:pwr02line}.
%
\begin{longtable}{|l||r|r|r|r|r|r|}
 K & K & K & K & K & K & K \kill
%
% --------------------- only begin of table ------------------------------
 \hline
 index & 
 $\nuk$ & 
 $S_k(300\,{\sf K})$ & 
 $b_k$ & 
 $w_k$  & 
 $y_k$ & 
 $v_k$ \\
 $k$   & 
 {\sf [GHz]}  & 
 {\sf [cm$^2$\,Hz]} & 
 {\sf [1]} & 
 {[$\frac{\sf MHz}{\sf hPa}$]} & 
 {[$\frac{\sf 10{^{-3}}}{\sf hPa}$]} & 
 {[$\frac{\sf 10{^{-3}}}{\sf hPa}$]} \\
 \hline
 \hline
 \endfirsthead
% --------------------- every page begin of table ------------------------
 \hline
 index & 
 $\nuk$ & 
 $S_k(300\,{\sf K})$ & 
 $b_k$ & 
 $w_k$  & 
 $y_k$ & 
 $v_k$ \\
 \hline
 \hline
 \endhead
% --------------------- every page end of table ------------------------
 K & K & K & K & K & K & K \kill
 \hline
 \caption[]{(continued)}\\
 \endfoot
% --------------------- only end of table ------------------------------
 K & K & K & K & K & K & K \kill
 \hline
 \caption{List of $\oz$ spectral lines of the Rosenkranz absorption 
          model \cite{pwr:93}.}
 \label{tab:pwr02line}
 \endlastfoot
% --------------------- body of table ----------------------------------  
1  & 118.7503  & .2936$\cdot$\,10$^{-14}$ & .009 & 1.63 & -0.0233 & 0.0079 \\
2  & 56.2648 & .8079$\cdot$\,10$^{-15}$ & .015 & 1.646 & 0.2408 & -0.0978 \\
3  & 62.4863 & .2480$\cdot$\,10$^{-14}$ & .083 & 1.468 & -0.3486 &  0.0844 \\
4  & 58.4466 & .2228$\cdot$\,10$^{-14}$ & .084 & 1.449 & 0.5227 & -0.1273 \\
5  & 60.3061 & .3351$\cdot$\,10$^{-14}$ & .212 & 1.382 & -0.5430 & 0.0699 \\
6  & 59.5910 & .3292$\cdot$\,10$^{-14}$ & .212 & 1.360 & 0.5877 & -0.0776 \\
7  & 59.1642 & .3721$\cdot$\,10$^{-14}$ & .391 & 1.319 & -0.3970 & 0.2309 \\
8  & 60.4348 & .3891$\cdot$\,10$^{-14}$ & .391 & 1.297 & 0.3237 & -0.2825 \\
9  & 58.3239 & .3640$\cdot$\,10$^{-14}$ & .626 & 1.266 & -0.1348 &  0.0436 \\
10 & 61.1506 & .4005$\cdot$\,10$^{-14}$ & .626 & 1.248 & 0.0311 & -0.0584 \\
11 & 57.6125 & .3227$\cdot$\,10$^{-14}$ & .915 & 1.221 & 0.0725 & 0.6056 \\
12 & 61.8002 & .3715$\cdot$\,10$^{-14}$ & .915 & 1.207 & -0.1663 & -0.6619 \\
13 & 56.9682 & .2627$\cdot$\,10$^{-14}$ & 1.260 & 1.181 & 0.2832 & 0.6451 \\
14 & 62.4112 & .3156$\cdot$\,10$^{-14}$ & 1.260 & 1.171 & -0.3629 & -0.6759 \\
15 & 56.3634 & .1982$\cdot$\,10$^{-14}$ & 1.660 & 1.144 & 0.3970 &  0.6547 \\
16 & 62.9980 & .2477$\cdot$\,10$^{-14}$ & 1.665 & 1.139 & -0.4599 & -0.6675 \\
17 & 55.7838 & .1391$\cdot$\,10$^{-14}$ & 2.119 & 1.110 & 0.4695 & 0.6135 \\
18 & 63.5685 & .1808$\cdot$\,10$^{-14}$ & 2.115 & 1.108 & -0.5199 & -0.6139 \\
19 & 55.2214 & .9124$\cdot$\,10$^{-15}$ & 2.624 & 1.079 & 0.5187 & 0.2952 \\
20 & 64.1278 & .1230$\cdot$\,10$^{-14}$ & 2.625 & 1.078 & -0.5597 & -0.2895 \\
21 & 54.6712 & .5603$\cdot$\,10$^{-15}$ & 3.194 & 1.05 & 0.5903 & 0.2654 \\
22 & 64.6789 & .7842$\cdot$\,10$^{-15}$ & 3.194 & 1.05 & -0.6246 & -0.2590 \\
23 & 54.1300 & .3228$\cdot$\,10$^{-15}$ & 3.814 & 1.02 & 0.6656 & 0.3750 \\
24 & 65.2241 & .4689$\cdot$\,10$^{-15}$ & 3.814 & 1.02 & -0.6942 & -0.3680 \\
25 & 53.5957 & .1748$\cdot$\,10$^{-15}$ & 4.484 & 1.00 & 0.7086 & 0.5085 \\
26 & 65.7648 & .2632$\cdot$\,10$^{-15}$ & 4.484 & 1.00 & -0.7325 & -0.5002 \\
27 & 53.0669 & .8898$\cdot$\,10$^{-16}$ & 5.224 & .97 & 0.7348 & 0.6206 \\
28 & 66.3021 & .1389$\cdot$\,10$^{-15}$ & 5.224 & .97 & -0.7546 & -0.6091 \\
29 & 52.5424 & .4264$\cdot$\,10$^{-16}$ & 6.004 & .94 & 0.7702 & 0.6526 \\
30 & 66.8368 & .6899$\cdot$\,10$^{-16}$ & 6.004 & .94 & -0.7864 & -0.6393 \\
31 & 52.0214 & .1924$\cdot$\,10$^{-16}$ & 6.844 & .92 & 0.8083 & 0.6640 \\
32 & 67.3696 & .3229$\cdot$\,10$^{-16}$ & 6.844 & .92 & -0.8210 & -0.6475 \\
33 & 51.5034 & .8191$\cdot$\,10$^{-17}$ & 7.744 & .89 & 0.8439 & 0.6729 \\
34 & 67.9009 & .1423$\cdot$\,10$^{-16}$ & 7.744 & .89 & -0.8529 & -0.6545 \\
35 & 368.4984 & .6460$\cdot$\,10$^{-15}$ & .048 & 1.92 & 0.0000 & 0.0000 \\
36 & 424.7631 & .7047$\cdot$\,10$^{-14}$ & .044 & 1.92 & 0.0000 & 0.0000 \\
37 & 487.2494 & .3011$\cdot$\,10$^{-14}$ & .049 & 1.92 & 0.0000 & 0.0000 \\
38 & 715.3932 & .1826$\cdot$\,10$^{-14}$ & .145 & 1.81 & 0.0000 & 0.0000 \\
39 & 773.8397 & .1152$\cdot$\,10$^{-13}$ & .141 & 1.81 & 0.0000 & 0.0000 \\
40 & 834.1453 &  .3971$\cdot$\,10$^{-14}$ & .145 & 1.81 & 0.0000 & 0.0000 \\
\hline
\end{longtable}


\levele{Oxygen Continum Absorption:}
\label{levele:pwr98_o2cont}
%-----------------------------------
As pointed out by Van~Vleck \cite{vv:87}, the standard theory for
non-resonant absorption is that of Debye (see also Ref. \cite{townes:55}). 
The Debye line shape is obtained from the VVW line shape function by
the limiting case $\nuk \rightarrow 0$.
Rosenkranz \cite{pwr:93} adopt the Debye theory for his models: 
\begin{eqnarray}
  \label{eq:pwr_o2cont}
  \alphac &=&  C \cdot \pda \cdot \Theta^2 \cdot 
             \frac{\nu^2 \cdot \gamma}{\nu^2+\gamma^2}\\
%
  \label{eq:pwr_o2cont_1}
  \gamma &=&  w \cdot (\pda \cdot \Theta^{0.8} + 1.1 \cdot \phzo \cdot
  \Theta)\hspace*{5mm}\mbox{:~Rosenkranz}\\
  \label{eq:pwr_o2cont_2}
  \gamma &=&  w \cdot (\pda + \phzo) \cdot \Theta^{0.8}\hspace*{17.25mm}\mbox{:~MPM93}\\
\nonumber
\end{eqnarray}
The values for the parameters are $C = 1.11\cdot 10^{-5}$ dB/km/(hPa\,GHz) and 
$w = 5.6 \cdot 10^{-4}$ GHz/hPa, respectively. This absorption
term is proportional to the collision frequency of a single oxygen molecule
and thus proportional to the dry air pressure\footnote{The absorption
  due to weakly bound complexes of $\oz$--$X$ with $X=\hzo,~\nz$ is 
  treated separately and therefore not included in this Debye
  formula.}.




\leveld{MPM93 Oxygen Absorption Model}
\label{levelb:O2_mpm93}
%-------------------------------------

\levele{Oxygen Line Absorption:}
\label{levele:mpm93_o2lines}
%-------------------------------
The oxygen line catalog has 44 lines from which 37 lines build the 
complex around 60\,GHz. The parameterization of the line absorption,
$\alphampml$, is (in units of dB/km):
\begin{eqnarray}
% line ansorption:
  \alphampml & = & 0.1820 \cdot \nu^2 \cdot  
                   \sum_{k=1}^{44}{S_k(T) \cdot F(\nu,\nu_k)}~~~~\mbox{dB/km}\\
%
 \mbox{with} &   &\nonumber\\
%
% line intensity:
 \mbox{line intensity:} & & \nonumber\\
      S_k(T) & = & \frac{a_{1,k}}{\nuk} \cdot \pda \cdot \Theta^3 \cdot 
                   \exp{[a_{2,k} \cdot (1-\Theta)]}\nonumber\\
% line shape:
 \mbox{line shape function:} & & \nonumber\\
 F(\nu,\nu_k) & = & \left[\frac{\gamma_k+(\nu-\nuk)\cdot \delta_k}
                               {(\nu-\nu_k)^2+\gamma_k^2}~~+~~
                          \frac{\gamma_k-(\nu+\nuk)\cdot \delta_k}
                               {(\nu+\nu_k)^2+\gamma_k^2}\right] \nonumber\\
% line width:
 \mbox{line width:} & & \nonumber\\
    \gamma_k & = & a_{3,k} \cdot 10^{-3} \cdot 
                 \left( \pda  \cdot \Theta^{a_{4,k}} + 
                        1.10 \cdot \phzo \cdot \Theta \right) \nonumber\\
% line coupling:
 \mbox{line coupling:} & & \nonumber\\
         \delta_k & = & \pdair \cdot \Theta^{0.8} \cdot 
                   \left[ a_{5,k} + \Theta \cdot a_{6,k} \right]\nonumber\\
\end{eqnarray}
%
where $a_{1-5,k}$ are the fitted parameters due to laboratory measurements 
\cite{liebeetal:92}. All model parameters are tabulated in 
Table \ref{tab:mpm9302line}. One has to note that in the MPM93 code is a 
threshold value for $\alphampml$ implemented:
\begin{equation}
 \label{eq:mpm93O2limit}
  \alphampml = 
   \left \{ \begin{array}{r@{\quad:\quad}l} 
    \alphampml & \alphampml > 0\\
    0          & \alphampml < 0
                       \end{array} \right.
\end{equation}
Therefore the oxygen absorption in the wings of the strong $\oz$-lines 
is remarkably higher than in the R93 model.
%
\begin{longtable}{|l||r|r|r|r|r|r|r|}
 K &  K & K & K & K & K & K & K \kill
%
% --------------------- only begin of table ------------------------------
 \hline
 index & 
 $\nuk$ & 
 $a_{1,k}$ & 
 $a_{2,k}$ & 
 $a_{3,k}$ & 
 $a_{4,k}$ & 
 $a_{5,k}$ & 
 $a_{6,k}$ \\
 $k$   & 
 {\sf [GHz]}  & 
 {\sf [$\frac{\sf kHz}{\sf hPa}$]} & 
 {\sf [1]} & 
 {[$\frac{\sf MHz}{\sf hPa}$]} & 
 {\sf [1]} & 
 {[$\frac{\sf 10{^{3}}}{\sf hPa}$]} & 
 {[$\frac{\sf 10{^{3}}}{\sf hPa}$]} \\
 \hline
 \hline
 \endfirsthead
% --------------------- every page begin of table ------------------------
 \hline
 index & 
 $\nuk$ & 
 $a_{1,k}$ & 
 $a_{2,k}$ & 
 $a_{3,k}$ & 
 $a_{4,k}$ & 
 $a_{5,k}$ & 
 $a_{6,k}$ \\
 \hline
 \hline
 \endhead
% --------------------- every page end of table ------------------------
 K &  K & K & K & K & K & K & K \kill
 \hline
 \caption[]{(continued)}\\
 \endfoot
% --------------------- only end of table ------------------------------
 K &  K & K & K & K & K & K & K \kill
 \hline
 \caption{List of $\oz$ spectral lines of the MPM93 absorption 
          model \cite{liebeetal:93}.}
 \label{tab:mpm9302line}
 \endlastfoot
% --------------------- body of table ----------------------------------  
%  //         f0          a1       a2      a3       a4     a5     a6
%  //        [GHz]     [kHz/hPa]   [1]   [MHz/hPa]  [1]    [10^3/hPa]
%  const Numeric mpm93[44][7] = { 
1 & 50.474238 &   0.094 &  9.694 &    0.890 & 0.0 &   0.240 &    0.790\\
2 & 50.987749 &   0.246 &  8.694 &    0.910 & 0.0 &   0.220 &    0.780\\
3 & 51.503350 &   0.608 &  7.744 &    0.940 & 0.0 &   0.197 &    0.774\\
4 & 52.021410 &   1.414 &  6.844 &    0.970 & 0.0 &   0.166 &    0.764\\
5 & 52.542394 &   3.102 &  6.004 &    0.990 & 0.0 &   0.136 &    0.751\\
6 & 53.066907 &   6.410 &  5.224 &    1.020 & 0.0 &   0.131 &    0.714\\
7 & 53.595749 &  12.470 &  4.484 &    1.050 & 0.0 &   0.230 &    0.584\\
8 & 54.130000 &  22.800 &  3.814 &    1.070 & 0.0 &   0.335 &    0.431\\
9 & 54.671159 &  39.180 &  3.194 &    1.100 & 0.0 &   0.374 &    0.305\\
10 & 55.221367 &  63.160 &  2.624 &    1.130 & 0.0 &   0.258 &    0.339\\
11 & 55.783802 &  95.350 &  2.119 &    1.170 & 0.0 &  -0.166 &    0.705\\
12 & 56.264775 &  54.890 &  0.015 &    1.730 & 0.0 &   0.390 &   -0.113\\
13 & 56.363389 & 134.400 &  1.660 &    1.200 & 0.0 &  -0.297 &    0.753\\
14 & 56.968206 & 176.300 &  1.260 &    1.240 & 0.0 &  -0.416 &    0.742\\
15 & 57.612484 & 214.100 &  0.915 &    1.280 & 0.0 &  -0.613 &    0.697\\
16 & 58.323877 & 238.600 &  0.626 &    1.330 & 0.0 &  -0.205 &    0.051\\
17 & 58.446590 & 145.700 &  0.084 &    1.520 & 0.0 &   0.748 &   -0.146\\
18 & 59.164207 & 240.400 &  0.391 &    1.390 & 0.0 &  -0.722 &    0.266\\
19 & 59.590983 & 211.200 &  0.212 &    1.430 & 0.0 &   0.765 &   -0.090\\
20 & 60.306061 & 212.400 &  0.212 &    1.450 & 0.0 &  -0.705 &    0.081\\
21 & 60.434776 & 246.100 &  0.391 &    1.360 & 0.0 &   0.697 &   -0.324\\
22 & 61.150560 & 250.400 &  0.626 &    1.310 & 0.0 &   0.104 &   -0.067\\
23 & 61.800154 & 229.800 &  0.915 &    1.270 & 0.0 &   0.570 &   -0.761\\
24 & 62.411215 & 193.300 &  1.260 &    1.230 & 0.0 &   0.360 &   -0.777\\
25 & 62.486260 & 151.700 &  0.083 &    1.540 & 0.0 &  -0.498 &    0.097\\
26 & 62.997977 & 150.300 &  1.665 &    1.200 & 0.0 &   0.239 &   -0.768\\
27 & 63.568518 & 108.700 &  2.115 &    1.170 & 0.0 &   0.108 &   -0.706\\
28 & 64.127767 &  73.350 &  2.620 &    1.130 & 0.0 &  -0.311 &   -0.332\\
29 & 64.678903 &  46.350 &  3.195 &    1.100 & 0.0 &  -0.421 &   -0.298\\
30 & 65.224071 &  27.480 &  3.815 &    1.070 & 0.0 &  -0.375 &   -0.423\\
31 & 65.764772 &  15.300 &  4.485 &    1.050 & 0.0 &  -0.267 &   -0.575\\
32 & 66.302091 &   8.009 &  5.225 &    1.020 & 0.0 &  -0.168 &   -0.700\\
33 & 66.836830 &   3.946 &  6.005 &    0.990 & 0.0 &  -0.169 &   -0.735\\
34 & 67.369598 &   1.832 &  6.845 &    0.970 & 0.0 &  -0.200 &   -0.744\\
35 & 67.900867 &   0.801 &  7.745 &    0.940 & 0.0 &  -0.228 &   -0.753\\
36 & 68.431005 &   0.330 &  8.695 &    0.920 & 0.0 &  -0.240 &   -0.760\\
37 & 68.960311 &   0.128 &  9.695 &    0.900 & 0.0 &  -0.250 &   -0.765\\
38 & 118.750343 &  94.500 &  0.009 &   1.630 & 0.0 &  -0.036 &    0.009\\
39 & 368.498350 &   6.790 &  0.049 &   1.920 & 0.6 &   0.000 &    0.000\\
40 & 424.763124 &  63.800 &  0.044 &   1.930 & 0.6 &   0.000 &    0.000\\
41 & 487.249370 &  23.500 &  0.049 &   1.920 & 0.6 &   0.000 &    0.000\\
42 & 715.393150 &   9.960 &  0.145 &   1.810 & 0.6 &   0.000 &    0.000\\
43 & 773.839675 &  67.100 &  0.130 &   1.820 & 0.6 &   0.000 &    0.000\\
44 & 834.145330 &  18.000 &  0.147 &   1.810 & 0.6 &   0.000 &    0.000\\
\hline
\end{longtable}


\levele{Oxygen Continum Absorption:}
\label{levele:mpm93_o2cont}
%-----------------------------------
As pointed out by Van~Vleck \cite{vv:87}, the standard theory for
non-resonant absorption is that of Debye (see also Ref. \cite{townes:55}). 
The Debye line shape is obtained from the VVW line shape function 
by the limiting case $\nuk \rightarrow 0$.
\cite{liebeetal:93} adopt the Debye theory for his models\footnote{in 
  Ref. \cite{liebeetal:93} is a mistake in the formula
  for $\gamma$. According to this text $\gamma =
  w\cdot\pda\cdot\Theta^{0.8}$ while in the source code of MPM93 
  Eq. (\ref{eq:abs_cont_pwr_o2_2}) is implemented.}: 
\begin{eqnarray}
  \label{eq:mpm93_o2cont}
  \alphac &=&  C \cdot \pda \cdot \Theta^2 \cdot 
               \frac{\nu^2 \cdot \gamma}{\nu^2+\gamma^2}\\
%
  \gamma  &=&  w \cdot (\pda + \phzo) \cdot \Theta^{0.8}\nonumber\\
\nonumber
\end{eqnarray}
The values for the parameters are $C = 1.11\cdot 10^{-5}$ dB/km/(hPa\,GHz) and 
$w = 5.6 \cdot 10^{-4}$ GHz/hPa, respectively. This absorption
term is proportional to the collision frequency of a single oxygen molecule
and thus proportional to the dry air pressure\footnote{The absorption
  due to weakly bound complexes of $\oz$--$X$ with $X=\hzo,~\nz$ is 
  treated separately and therefore not included in this Debye
  formula.}.



\levelc{ARTS Workspace Variables and Methods}
\label{levelc:ArtsImplementationCompleteModels}
% ---------------------------------------------

%############################################################################# 

% ================================================================================
% The following section is written by Thomas Kuhn, iup Bremen, tkuhn@uni-bremen.de
% ================================================================================

\levela{Cloud Absorption}
\label{labela:cloudabsorption}
%=============================

\levelb{Liquid water and ice particle absorption}
\label{levelb:lipartabs}
%*************************************************
So far only absorption due to air was described. However 
hydrometeors\footnote{We denote liquid water and ice particles, either
  suspended or precipitating, in the air as hydrometeors.}
can have a noticeable effect on the radiative transfer through the
atmosphere in the 10-30\,GHz frequency range.

The MPM93 model provides beside the absorption model of air also an
absorption model for suspended liquid water droplets and ice particles
\citep{liebe:89b,liebeetal:91,hufford:91,liebeetal:93}.  The model is
applicable for the Rayleigh regime, for which the relation $r <
0.05\cdot \lambda$ holds where $r$ is the particle radius and
$\lambda$ is the wavelength\footnote{See \citet{brussaard:95}, page
  81, for details.}. For the WATS frequency range this means $r
<$~500\,$\mu$m. Considering \citet{salby:96}, this criterium is --
except for cirrus -- nearly for every aerosol and cloud class
satisfied. But one has to bear in mind that these values have a wide
range of variability, for example, \citet{salby:96} states that the
mean particle radius for stratus, cumulus, and nimbus clouds can be in
the range of 10-1000\,$\mu$m and that the particle radius distribution
is highly unsymmetric. 

With respect to the imaginary part of the complex refractivity, a 
unified parameterization of liquid and ice particle absorption 
is formulated in MPM93:
\begin{eqnarray}
  \label{eq:abs_cloud}
  \alpha & = & 0.1820 \cdot \nu \cdot\imn\hspace*{20mm}
               \mbox{dB/km}\\
    \imn & = & \frac{3}{2} \cdot \frac{w}{m} \cdot 
               \Im{[(\er-1)/(\er+2)]}\nonumber\\
    \imn & = & \frac{3}{2} \cdot \frac{w}{m} \cdot 
               \left[\frac{3\cdot\ime}{(\ree+2)^2~+~(\ime)^2}\right]\nonumber\\
  \nonumber
\end{eqnarray}
where $w$ is the liquid water (0.0\,$<LWC<$\,5.0\,g/m$^3$) or ice
mass (0.0\,$IWC$\,1.0\,g/m$^3$) content and $m$ is the water 
or ice bulk density ($\denli$=1.0\,g/cm$^3$ and 0.916\,g/cm$^3$, respectively).\\
The difference between liquid water and ice absorption is put in the 
expressions for the complex permittivities (i. e. the relative 
dielectric constant), $\er=\ree+i\cdot\ime$, which depend on frequency
and temperature.

\noindent$\bullet$ Complex permittivity for suspended liquid water droplets:
\begin{eqnarray}
  \label{eq:comp_perm_lwc}
  \ree       & = & \epsilon_o~-~\nu^2\cdot
                   \left[\frac{\epsilon_o-\epsilon_1}{\nu^2+\gamma_1^2}~+~
                   \frac{\epsilon_1-\epsilon_2}{\nu^2+\gamma_1^2}\right]\nonumber\\
%
  \ime       & = & \nu\cdot\left[\gamma_1\cdot
                   \frac{\epsilon_o-\epsilon_1}{\nu^2+\gamma_1^2}~+~
                   \gamma_2\cdot\frac{\epsilon_1-\epsilon_2}{\nu^2+\gamma_1^2}\right]\nonumber\\
%
  \epsilon_o & = & 77.66 + 103.3\cdot(\Theta-1)\nonumber\\
  \epsilon_1 & = & 0.0671\cdot\epsilon_o\nonumber\\
  \epsilon_2 & = & 3.52\nonumber\\
%
  \gamma_1   & = & 20.20 - 146\cdot(\Theta-1) + 316\cdot(\Theta-1)^2\,\mbox{GHz}\nonumber\\
  \gamma_2   & = & 39.8\cdot\gamma_1\,\mbox{GHz}\nonumber\\
%
  \Theta     & = & 300\,\mbox{K}~/~T\nonumber\\
%
%  LWC        & = & 0.0~\mbox{to}~5.0~\mbox{g}/\mbox{m}^3
%                   \hspace*{10mm}\mbox{cloud liquid water content}\nonumber\\
%  \rho_l     & = & 1.0~\mbox{g}/\mbox{cm}^3
%                \hspace*{10mm}\mbox{water density}\nonumber
%  \nonumber
\end{eqnarray}
$\bullet$ Complex permittivity for ice crystals:
\begin{eqnarray}
  \label{eq:comp_perm_iwc}
  \ree    & = & 3.15\nonumber\\
%
  \ime    & = & \frac{a}{\nu}~+~b\cdot\nu\nonumber\\
%
  a       & = & (\Theta-0.171)\cdot\exp{(17.0-22.1\cdot\Theta)}\nonumber\\
  b       & = & \left[\left(\frac{0.233}{1-0.993/\Theta}\right)^2 + 
                \frac{6.33}{\Theta} - 1.31\right]\cdot10^{-5}\nonumber\\
%
  \Theta  & = & 300\,\mbox{K}~/~T\nonumber\\
%
%  IWC     & = & 0.0~\mbox{to}~1.0~\mbox{g}/\mbox{m}^3 
%                \hspace*{10mm}\mbox{cloud ice water content}\nonumber\\
%  \rho_i    & = & 0.916~\mbox{g}/\mbox{cm}^3
%                \hspace*{10mm}\mbox{ice density}\nonumber
%  \nonumber
\end{eqnarray}
%
The absorption is directly proportional to the liquid or ice water
content $LWC/IWC$ and inversely proportional to the density of a
single liquid ice particle $\denli$. Like the mean particle radius,
the liquid and ice water content have a high variability. Table
\ref{tab:lwc} reflects this variability by summarizing different
literature values for several cloud types. Additional uncertainty 
of this absorption term comes from two sides: 
(1) the difference to the Rayleigh approximation
of the order of 1-6\% as reported in \citet{lietal:97} and (2) from
the fit of the complex permittivity.  Since $\epsilon(\nu,T)$ was
fitted to measurements which were mostly performed above $0^\circ$C,
the extrapolated values for $T<$0$^{\rm o}$C for supercooled
clouds are not well established. For example in \cite{liebeetal:91} 
itself two different parameterizations for the so called primary 
relaxation frequency ($\gamma_1$ in Equation~\ref{eq:comp_perm_lwc}) 
are given, one polynomial in $\Theta$ as presented in 
Equation~\ref{eq:comp_perm_lwc}) and an exponential function derived
from theory. Although the polynomial describes the selected 
data better than the exponential function, this might not be true for
temperatures well below 0$^{\rm o}$C.
The difference in $\gamma_1$ according to these two approaches can 
be more than 2\,GHz for very low temperatures \citep{liptonetal:99}. 
The resulting consequences from this discrepancy for the absorption 
calculation at three microwave frequencies are shown in 
Figure \ref{fig:refrac_water_comp}. A more detailed
discussion about this source of uncertainty is given in Section
\ref{levelb:ref_uncert_clouds}.
%
\begin{table}[!htb]
\begin{center}
\begin{tabular}{llll}
\hline
\multicolumn{4}{c}{liquid water content ($LWC$)} \\
 cloud        & class & \multicolumn{1}{c}{(g/m$^3$)} & reference\\
\hline
 stratus      & St    & 0.15        & \cite{salby:96}\\
              &       & 0.09-0.9    & \cite{seinfeld:98}\\
              &       & 0.28-0.3    & \cite{hess:98}\\
              &       & 0.29        & \cite{abreu:96}\\
 nimbostratus & Ns    & 0.4         & \cite{salby:96}\\
              &       & 0.65        & \cite{abreu:96}\\
              &       & 0.05-0.3    & \cite{berton:00}\\
 altostratus  & As    & $<$0.01-0.2 & \cite{seinfeld:98}\\
              &       & 0.41        & \cite{abreu:96}\\
              &       & 0.1-1       & \cite{berton:00}\\
 stratocumulus& Sc    & 0.3         & \cite{salby:96}\\
              &       & $<$0.1-0.7  & \cite{seinfeld:98}\\
              &       & 0.15        & \cite{abreu:96}\\
              &       & $<$0.5      & \cite{pawlowskaetal:00}\\
              &       & 0.05-1      & \cite{berton:00}\\
 cumulus      & Cu    & 0.5         & \cite{salby:96}\\
              &       & 0.26-0.44   & \cite{hess:98}\\
              &       & 1.00        & \cite{abreu:96}\\
 cumulonimbus & Cb    & 2.5         & \cite{salby:96}\\
              &       & 0.1-2       & \cite{berton:00}\\
 cumulus 
    congestus & Cg    & 0.1-3.2     & \cite{berton:00}\\
FIRE-ACE      & -     & $<$0.7      & \cite{shupeetal:00}\\
\hline
\multicolumn{4}{c}{}  \\
\multicolumn{4}{c}{ice water content ($IWC$)}  \\
 cloud        & class & \multicolumn{1}{c}{(g/m$^3$)} & reference\\
\hline
 cirrus       & Ci    & 0.025                         & \cite{salby:96}\\
              &       & 0.00193-0.0260                & \cite{hess:98}\\
              &       & 3.128$\cdot$10$^{-4}$-0.06405 & \cite{abreu:96}\\ 
              &       & 0.15-0.3                      & \cite{larsenetal:98}\\
              &       & $<$0.1                        & \cite{berton:00}\\
cirrostratus  & Cs    & 0.2                           & \cite{salby:96}\\
              &       & 0.05-2                        & \cite{berton:00}\\
\hline
\end{tabular}
\caption{Stated values for the liquid and ice water content of several 
  cloud classes from different sources.}
\label{tab:lwc}
\end{center}
\end{table}



\levelb{Variability and Uncertainty in Cloud Absorption}
\label{levelb:ref_uncert_clouds}
%--------------------------------------------------------
In the case of clouds three sources of uncertainties can be considered
at first sight: (1) validity of the Rayleigh approximation (2) the 
parameterization of the relative dielectric constants ($\er$) of water 
and ice in the microwave region, and (3) the statistcal and
climatological variability of the cloud liquid water and ice content.

As it was stated above (Section \ref{levelb:lipartabs}) the Rayleigh 
approximation is valid for particle sizes $<$~500\,$\mu$m. Figure 
\ref{fig:cloud_part_dist} shows a particle size distribution for water
clouds and ice clouds (cirrus) from the OPAC model \citep{hess:98}. 
According to this model only cirrus clouds will have particles of size
larger than 500\,$\mu$m. Nevertheless one has to keep in mind that the
variability of the particle size can be very high so that at certain 
conditions some cloud types (most probable is the cumulonimbus) 
a non-negligible large particle concentration can occur.

The uncertainty in the relative dielectric constant of water 
(see e.~g. \citet{liptonetal:99}) is largest below the freezing 
temperature, since only a few measurements at -4$^{\rm o}$C 
contributed to the parameterization of $\er$ in \cite{liebeetal:91}, 
which in turn is used in the cloud liquid water absorption model of MPM93. 
Figure \ref{fig:refrac_water_comp} shows a comparison of 
\cite{liebeetal:91} and \cite{ray:72}\footnote{{The calculations
  for this parameterizastion are performed with the computer code}\\{
   of W. Wiscombe, NASA, GSFC}\\
  (ftp://climate.gsfc.nasa.gov/pub/wiscombe/Refrac\_Index/WATER/)
  For the microwave frequency range this program uses the
  \cite{ray:72} temperature parameterization.} parameterizations 
for the temperature dependence of the expression
$\Im{[(\er-1)/(\er+2)]}$, which is in the Rayleigh approximation 
one of the relevant terms in the absorption calculation (see 
Equation~\ref{eq:abs_cloud}). Additionally the same calculations with 
the alternative expression of the first  relaxation frequency, 
$\gamma_1$, as stated in Equation~2b of \cite{liebeetal:91} is shown. 
The three versions give comparable results for temperatures warmer 
than 260\,K but show significant  differences for temperatures below 
240\,K. However, an uncertainty estimation of $\Im{[(\er-1)/(\er+2)]}$ 
is due to the lack of measurements not easy, but it will certainly
increase with decreasing temperature.

The largest variability of the involved quantities of cloud absorption
is the liquid and ice water content ($LWC$ and $IWC$) of the clouds 
(see Table \ref{tab:lwc}). Even within a single cloud the $LWC$ ($IWC$) changes
with altitude and the distance from the cloud center as can be seen for
example in Figure 10 of \citet{ludlammason:57} and in the model study
of \citet{costaetal:00}.


\begin{figure}[!htb]
  \begin{center}
% ---- uncertainty -------------------------------------
%   \includegraphics*[scale=0.6, angle=90, viewport= 80 725 480 70]%
   \includegraphics*[width=0.6\hsize, angle=90]%
   {LWCcloud.eps}\\
% ---- ratio -------------------------------------------
%   \includegraphics*[scale=0.6, angle=90, viewport= 80 725 480 66]%
   \includegraphics*[width=0.6\hsize, angle=90]%
   {IWCcloud.eps}
% ------------------------------------------------------
  \end{center}
  \caption{Cloud particle size distributions according to 
    Equations~3a and 3c and the microphysical properties are from the 
    Tables 1a and 1b of the OPAC model \cite{hess:98}. 
    For the liquid water clouds (upper plot) a modified gamma 
    distribution is assumed whereas for the ice clouds (lower plot) 
    exponential functions are taken.}
  \label{fig:cloud_part_dist}
\end{figure}
%
%
\begin{figure}[!hbt]
  \begin{center}
% ---- comparison --------------------------------------
   \includegraphics*[width=0.95\hsize, angle=0]%
   {refractive_water_comp_T.eps}
% ------------------------------------------------------
  \end{center}
  \caption{Comparison of the imaginary part of the expression 
    \mbox{$(\er-1)/(\er+2)$} for liquid water at the three proposed 
    WATS frequencies of interest. Plotted are the two common
    models of \cite{liebeetal:91} (a) and \cite{ray:72} (b). 
    The Ray parameterization is calculated with the F77 program 
    of W. Wiscombe, NASA, GSFC, take from 
    ftp://climate.gsfc.nasa.gov/pub/wiscombe/Refrac\_Index/WATER/.
    Additionally the \cite{liebeetal:91} parameterization (c) with the 
    alternative expression for the first relaxation frequency, 
    $\gamma_1 = 20.1\cdot\exp{[7.88\cdot(1-\Theta)]}$, is plotted.}
%
%    $(\er-1)/(\er+2)$ for liquid water at the three WATS frequencies
%    of interest. Plotted are the two common
%    models of \cite{liebeetal:91} and \cite{ray:72} (calculated with 
%    the F77 program of W. Wiscombe (NASA, GSFC) take from 
%    ftp://climate.gsfc.nasa.gov/pub/wiscombe/Refrac\_Index/WATER/).}
  \label{fig:refrac_water_comp}
\end{figure}



\levelc{ARTS Workspace Variables and Methods}
\label{levelc:ArtsImplementationCloudAbsorption}
% ----------------------------------------------



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