Lesson 3: Background

Scattering from the beam into other directions is an additional, along with absorption, reduction in intensity, while scattering into the beam from other directions adds a second source term. To write the complete radiative transfer equation we must distinguish the amount of absorption and emission along the path from the amount of scattering. We do so by introducing a mass scattering coefficient k_sl with dimensions of area per mass, as an analog to the mass absorption coefficient k_al. Both absorption and scattering diminish the beam, while scattering of radiation traveling in any other direction into the beam can add to the intensity. The full radiative transfer equation is therefore

1. 

where we have made explicit the direction of the beam (specified by µ, j). The last term accounts for the scattering of radiation into the beam traveling in direction (µ, j)from every other direction. The quantity is called the single scattering phase function, or often simply the phase function, and describes how likely it is that that radiation traveling in the (µ, j)direction will be scattered into the (µ', j')direction. The phase function is reciprocal, so that , and is defined such that the integral over the entire sphere is 4p;.

We divide both sides of the equation by and relate path length differential to the vertical displacement to obtain

 

2. 

where we have now defined the single scattering albedo , which is the likelihood that a photon is scattered rather than absorbed at each interaction. Single scattering albedo varies between zero and one; the lower limit corresponds to complete absorption and the upper to complete scattering.

Equation (2) is the plane parallel, unpolarized, monochromatic radiative transfer equation in full detail. Despite its length it describes only four processes: extinction by absorption and by scattering out of the beam into other directions, emission into the beam, and scattering into the beam from every other direction. The equation is, unfortunately, quite difficult to solve because it is an integrodifferential equation for intensity; that is, intensity appears both in the differential on the left hand side and as part of the integral on the right hand side of the equation.

Before we can even begin to solve this equation we have to come to grips with the way particles scatter light. When we consider absorption and emission we need only to determine the mass absorption coefficient k_al.. When we include scattering in the radiative transfer equation, though, we require three additional pieces of information: the mass scattering coefficient k_sl, along with the phase function and single scattering albedo .

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