The ARTS Monte Carlo scattering module (ARTS-MC) offers an efficient method for
polarized radiative transfer calculations in arbitrarily complex 3D cloudy
cases. The ARTS-MC algorithm is described in detail
in \citep{davis05:_monte_carlo_i3etgrs}. The algorithm solves the integral form
of the VRTE, given in Eq.~\ref{eq:VRTE}, by applying Monte Carlo integration
with importance sampling (MCI) (e.g. \citep{press:numerical:02}). This
integration is over infinite dimensions, where for each scattering order there
is a dimension representing: path-lengths, the choice between emission and
scattering, and the choice between reflection or emission at the earth's
surface. In practice the integrand is always calculated for a finite scattering
order, as the dimensionality of the integral is truncated by photon emission or
the boundary of the domain. Thus, the algorithm can be pictured as tracing a
large number of photons backwards from sensor, in randomly selected multiply
scattered propagation paths to either their point of emission, or entry into
the scattering domain. \citep{davis05:_monte_carlo_i3etgrs} gives a full
description of the probability density functions used to determine these
propagation paths. This physical picture is identical to the Backward-Forward
Monte Carlo algorithm described by \citep{liu96:_three_jgr}.
However, \citep{liu96:_three_jgr} did not account for dichroism, which is
correctly accounted for in ARTS-MC by importance sampling. The current
implementation of the algorithm differs slightly from the description
in \citep{davis05:_monte_carlo_i3etgrs}; changes include:
\begin{itemize}
\item the initial line of sight is no longer treated differently than the scattered paths 
\item the algorithm is no longer confined to the `cloudbox', 
\item MCI can now be used for convolving the simulated Stokes vector with a 2D antenna response (\citep{davis05:_monte_carlo_i3etgrs} discusses only pencil beam calculations)
\item MCI is now used to treat emission or reflection from the earth's surface.
\end{itemize}
The details for these changes can be found in the ATG.
    
As described in \citep{battaglia07:_microw_jqsrt}, when compared to other techniques for solving the VRTE in 3D domains the ARTS-MC
algorithm has the following advantages:
\begin{itemize}
\item All computational effort is dedicated to calculating the Stokes
 vector at the location of interest and in the direction of
 interest. This is in contrast to forward Monte Carlo and discrete ordinate methods
 where the whole radiation field is calculated.
\item CPU and memory cost scale more slowly than discrete ordinate methods with
 grid size, so that large or detailed 3D scenarios are not a problem. 
\item Only parts of the atmosphere that
 significantly contribute to the observed radiance are considered in the
 computation. Where the medium is optically thick, only the parts of the
 atmosphere closest to the sensor are visited by the algorithm. This contrasts
 with discrete ordinate methods, where the whole radiation field is computed,
 and in particular with forward Monte Carlo methods, where added optical
 thickness further restricts the number of photons reaching the sensor.
\end{itemize}

The ARTS-MC algorithm has been used to interpret polarized limb sounding
observations for Aura-MLS \cite{davis05:_cirrus_induc_polar_ghz_aura}, and to
investigate the effect of realistic cloud inhomogeneity and ice crystal
orientation on observations for Aura-MLS, AMSU-B and a proposed cloud ice
sensor in \cite{davis07:_3d_pol_sim_acp}. Further work that used
ARTS-MC includes \citep{adams2008} and \citep{rydberg:nonga:09}.
