/*! \page rt-theory Radiative Transfer Theory

    Radiative transfer theory generally describes the propagation of radiation within a given medium.
    Typically we refer to radiation by a quantity \f$I_{\lambda}(\mathbf{r}, \Omega)\f$  known as the spectral radiance at a given
    location \f$\mathbf{r}\f$, direction \f$\Omega\f$, and wavelength, \f$\lambda\f$.  The spectral radiance
    has SI units of,

    \f{equation}{\label{eq:rt_si_units}
        [I_\lambda] = \frac{W}{sr \cdot m^{-2} \cdot nm},
    \f}

    but this is commonly written as

    \f{equation}{\label{eq:rt_counting_units}
        [I_\lambda] = \frac{photons}{s \cdot sr \cdot cm^{-2} \cdot nm},
    \f}

    to be a quantity more easily related to how a counting detector would measure spectral radiance.
    Note that radiance specified in this form is not simply a scaling of the previous form since the energy per photon changes as a function of wavelength.
    In the context of atmospheric radiative transfer, often the spectral dependence is implicity assumed and spectral radiance is simply called radiance.
    The wavelength subscript is then dropped for brevity.

    In general it is not possible to fully describe the state of radiation by the radiance, or intensity, alone and it is required to include information on the polarization state.
    There are multiple ways to parameterize the state of polarization, but the most common in atmospheric radiative transfer is that of the Stokes vector,

    \f{equation}{\label{eq:stokes}
        \mathbf{I} = \begin{pmatrix} I \\ Q \\ U \\ V \end{pmatrix},
    \f}

    where the four Stokes parameters are defined in terms of the electric field,

    \f{align}{
    I &= \overline{E_\theta E^*_\theta} + \overline{E_\phi E^*_\phi} \\
    Q &= \overline{E_\theta E^*_\theta} - \overline{E_\phi E^*_\phi} \\
    U &= -2 \text{Re}[\overline{E_\theta E^*_\phi}] \\
    V &= 2 \text{Im}[\overline{E_\theta E^*_\phi}] \\
    \f}

    where \f$E_\theta\f$ and \f$E_\phi\f$ are the parallel and perpindicular components of the electric field in a given reference plane.
    Various definitions of the reference plane exist, and care has to be taken when comparing Stokes vector elements from different sources.
    However, generally \f$I\f$ is the intensity, \f$Q,U\f$ describe the linear polarization state, and \f$V\f$ is the circular polarization state.


    The fundamental equation describing propagation and transport of radiation is, in integral form,

    \f{equation}{\label{eq:rt}

    \mathbf{I}(\mathbf{r}, \Omega) = \mathbf{I}(\mathbf{r}_\text{end}, \Omega) e^{-\tau(\mathbf{r}_\text{end}, \mathbf{r})} + \int_{\mathbf{r}_{end}}^{\mathbf{r}} k(s) J(s, \Omega(s)) e^{-\tau(s, \mathbf{r})} ds,

    \f}

    where a path is implicitly defined from the initial location and direction (\f$\mathbf{r}, \Omega\f$) that has an end-point \f$\mathbf{r}_\text{end}\f$, \f$s\f$ is a distance along that path that implicitly defines a
    location, with \f$s=0\f$ corresponding to \f$\mathbf{r}\f$, \f$J\f$ is the cumulative source of radiation, and \f$k\f$ is a quantity known as the extinction.
    The functional \f$\tau\f$ is defined as,

    \f{equation}{\label{eq:od}

    \tau(\mathbf{r}_1, \mathbf{r}_2) = \int_{\mathbf{r}_1}^{\mathbf{r}_2} k(s) ds,

    \f}

    and is known as the optical depth between two points, here we assume \f$s>0\f$ so the optical depth is always a positive quantity.
    The radiative transfer equation essentially says that to find the radiance at a given point and direction, we start by tracing a ray from that point.
    The radiance is then the radiance at the end of the ray, attenuated back, plus all of the sources along the ray, also attenuated back.

    \section rt-sources Source Terms
    Once the source terms, \f$J\f$ are known, the radiance can be calculated through a line integral.
    Typically the challenging part of radiative transfer modelling is determining the source terms.
    In atmospheric radiative transfer, the primary source terms are scattering and thermal emissions.
    Scattering is dominant at shorter wavelengths, while thermal emissions take over at longer wavelength.


    \subsection rt-sources-scattering Scattering



    \subpage rt-disco


 */
