|Predicting reflectivity and emissivity of porous media
is a long-standing problem important to many fields of science. The approach we are
taking combines particle scattering submodels with a theoretically based derivation of the
The concept is illustrated at the right and follows the derivation of Hapke [??]. The fundamental problem is that the radiative transport equation (which is a differential-integral equation) is too difficult to solve, even numerically, except for highly idealized situations. However, a reasonable approximation to its solution can be derived according to the following logic. If we have a ray of light impinging on a semi-infinite, homogeneous body, adding a differential layer of material to the surface of the body should not affect its radiative properties. If we write expressions for the leading terms that are affected by adding such a layer, we know that they must sum to zero (no effect). For example, the differential layer attenuates light (see panel a) by absorption both as the incident beam passes through it and as it is reflected from the underlying material back to the detector. The effect of this attenuation is a decrease in signal. However, the differential layer also scatters light directly to the detector (see panel b), which has the effect of increasing the signal. A portion of the scattered light is also reflected off the underlying material and into the detector (panel c). The first five effects are indicated in the figure.
The sum of all these contributions must be zero. Mathematically writing these effects down and forcing them to zero returns another differential equation to be solved, but it is far less difficult than the original radiative transport equation.
This is the conceptual basis for our emissivity model. The mathematical details are available elsewhere.