The Equation of Radiative Transfer

The method used in this study to solve the equation of radiative transfer is the successive orders of scattering technique. It was chosen for two main reasons; 1) it is physically intuitive, especially as the physics remains clear through the mathematical formalism, and hence relatively easy to code; and 2) it is easily adaptable to different geometries and types of simulations.

The equation of radiative transfer may be obtained from the Boltzmann transport equation for photons where it is assumed that interactions between photons can be ignored. For an inhomogeneous scattering atmosphere, the general equation of radiative transfer without specifying any coordinate system is,

$$\frac{1}{k_e(\v{s})c} \frac{\partial {\b I}(\v{s};\hat{\Omega... ...\hat{\Omega};\lambda;t) + {\b J}(\v{s};\hat{\Omega};\lambda;t)$$ (3.70)

where c is the velocity of light, $\hat{\Omega}$ is a unit vector specifying the direction of scattering through a position vector $\v{s}$, t is time, and $k_e(\v{s})$ is the extinction coefficient (in units of inverse length). The extinction coefficient is related to the extinction cross-section by number density,

$$k_e = n \, \sigma_e$$ (3.71)

and analogous relationships exist for the scattering and absorption coefficients.

The last term on the right hand side of equation (2.70) is called the source function vector. The first term on the right hand side represents attenuation due to absorption and scattering of a radiance stream as it propagates through the atmosphere, and the source function vector represents the strengthening of the radiance stream. For solar radiation it arises from photons scattered in the path from all other directions. The presence of this scattering source term ensures that the radiation field is no longer merely a function of local sources and sinks, but of the entire atmospheric radiation field and of its transport over large distances. In practice this makes the solution much more difficult to obtain.

The radiance field can be taken as time independent or in a steady state so that $\partial \b{I}/\partial t =0$. Also, only elastic scattering is considered. To simplify the solution further, the atmosphere is approximated as being vertically stratified and horizontally homogeneous. This reduces the number of spatial dimensions from three to one, with the solution a function of height only. The position vector $\v{s}$ is replaced by the scalar z. This is called plane parallel geometry. The natural plane-parallel co-ordinates are the spherical polar angles $\theta$ and $\phi$ which replace the the directional unit vector, $\hat{\Omega}$. Taking all this into account, equation (2.70) can be simplified to,

$$\cos{\theta} \frac{d{\b I}(z;\theta,\phi)}{k_e \, dz} = - {\b I}(z;\theta,\phi) + {\b J}(z;\theta,\phi).$$ (3.72)

It will prove convenient to introduce an alternative vertical coordinate which takes into effect the optical properties of the atmosphere and which is also independent of the physical distance. The optical depth is defined as,

$$\frac{d\tau}{dz} = -k_e, \hspace{0.5in} \tau(z)=\int_z^{\infty} k_e \,dz'.$$ (3.73)

At the top of the atmosphere, $\tau=0$, and it increases with decreasing altitude with the total optical depth of the atmosphere defined as $\tau_1$. Another useful substitution is to set $\mu=\cos{\theta}$. Upon substitution into equation (2.72), the basic form of the radiative transfer equation in a plane-parallel atmosphere is arrived at,

$$\mu \frac{d{\b I}(\tau;\mu,\phi)}{d\tau} = {\b I}(\tau;\mu,\phi) - {\b J}(\tau;\mu,\phi)$$ (3.74)

where the sign change arises from the negative in the definition of $\tau$.

The scattering angle $\Theta$ can be represented in terms of the direction of the incident radiation ( $\mu',\phi'$) and the direction of the scattered radiation ($\mu,\phi$),

$$\cos \Theta = \mu \mu' + \sqrt{1-\mu^2} \sqrt{1-\mu'^2} \cos(\phi-\phi').$$ (3.75)

Also the convention is adopted that $\mu>0$ refers to upward radiance streams (or downlooking) and $\mu<0$ refers to downward radiance streams (or uplooking).

The multiple scattering source function vector can be expressed as,

$${\b J}(\tau;\mu,\phi) \frac{\tilde{\omega}(\tau)}{4 \pi} \int_0^{2 \pi} \int_{-1}^1 {\b I}(\tau;\mu',\phi') {\b P}(\mu,\phi;\mu',\phi') d\mu' d\phi'$$ =

$$+ \frac{\tilde{\omega}(\tau)}{4 \pi} \pi {\b F_o} {\b P}(\mu,\phi;-\mu_o,\phi_o) e^{-\tau/\mu_o}$$ (3.76)

where $\tilde{\omega}$ is the single-scattering albedo, defined below, and $\pi {\b F_o}$ is the extra-terrestrial solar flux vector. To a very good approximation, it is unpolarized so that $\pi {\b F_o}=[\pi F_o,0,0,0]^{T}$. The first term on the right represents the multiple scattered radiation and the second term on the right represents the single-scattered radiation. The single-scattering albedo is defined as,

$$\tilde{\omega} = \frac{k_s}{k_s + k_a} = \frac{k_s}{k_e}$$ (3.77)

and represents the fraction of energy scattered to that removed from the radiance stream under consideration. For conservative scattering, $\tilde{\omega}=1$, and for pure absorption, $\tilde{\omega}=0$. In an inhomogeneous atmosphere, $\tilde{\omega}$ is a function is optical depth.

Equation (2.74) can be recast into integral form by introducing integrating factors. For the downward streams, the factor $e^{\tau/\mu}$ is used while for upward streams, $e^{-\tau/\mu}$ is used. The result is,

  

$${\b I}(\tau;\mu,\phi) = {\b I}(\tau_1;\mu,\phi)e^{-(\tau_1-\tau)/... ..._{\tau}^{\tau_1} {\b J}(\tau';\mu,\phi)e^{-(\tau'-\tau)/\mu} \frac{d\tau'}{\mu}$$ (3.78)

$${\b I}(\tau;-\mu,\phi) = {\b I}(0;-\mu,\phi)e^{-\tau/\mu} + \int_0^{\tau} {\b J}(\tau';-\mu,\phi)e^{-(\tau-\tau')/\mu} \frac{d\tau'}{\mu}.$$ (3.79)

The two boundary conditions which must be specified, ${\b I}(\tau_1;\mu,\phi)$ and ${\b I}(0;-\mu,\phi)$, are the inward source Stokes vectors at the bottom and top of the atmosphere, respectively, as a function of angle. For near-UV to near-IR radiation, there is no source other than direct sunlight at the top of the atmosphere so that,

$${\b I}(0;-\mu,\phi) = {\bf0} = [0,0,0,0]^{T}.$$ (3.80)

There is, in general, a source at the bottom of the atmosphere from reflection by the surface. This is discussed in section (2.7).

Solution to equation (2.74) will only give the scattered, or diffuse, radiation field; the direct, or unscattered, component must be added separately. The total (direct+diffuse) downward radiance is given by,

$${\b I}_{tot}(\tau;\mu,\phi) = {\b I}(\tau;\mu,\phi) + \pi {\b F_o} e^{-\tau/\mu_o} \delta(\mu+\mu_o) \delta(\phi-\phi_o)$$ (3.81)

where $\delta$ is the Dirac delta function. Note also that in the calculation of irradiance and mean radiance, the direct component must also be added separately. Their contribution are, respectively, $\mu_o \pi F_o e^{-\tau/\mu_o}$ and $\pi F_o e^{-\tau/\mu_o}$.