We derive Kubelka-Munk (KM) theory systematically from the radiative transport equation (RTE) by analyzing the system of equations resulting from applying the double spherical harmonics method of order one and transforming that system into one governing the positive- and negative-going fluxes. Through this derivation, we establish the theoretical basis of KM theory, identify all parameters, and determine its range of validity. Moreover, we are able to generalize KM theory to take into account general boundary sources and nonhomogeneous terms, for example. The generalized Kubelka-Munk (gKM) equations are also much more accurate at approximating the solution of the RTE. We validate this theory through comparison with numerical solutions of the RTE.
The generalized Kubelka-Munk (gKM) approximation is a linear transformation of the double spherical harmonics of order one (DP1) approximation of the radiative transfer equation. Here, we extend the gKM approximation to study problems in three-dimensional radiative transfer. In particular, we derive the gKM approximation for the problem of collimated beam propagation and scattering in a plane-parallel slab composed of a uniform absorbing and scattering medium. The result is an 8×8 system of partial differential equations that is much easier to solve than the radiative transfer equation. We compare the solutions of the gKM approximation with Monte Carlo simulations of the radiative transfer equation to identify the range of validity for this approximation. We find that the gKM approximation is accurate for isotropic scattering media that are sufficiently thick and much less accurate for anisotropic, forward-peaked scattering media.
We introduce a new model for multiple scattering of polarized light by statistically isotropic and mirrorsymmetric particles, which we call the generalized Kubelka-Munk (gKM) approximation. It is obtained through a linear transformation of the system of equations resulting from applying the double spherical harmonics approximation of order one to the vector radiative transfer equation (vRTE). The result is a 32 × 32 system of differential equations that is much simpler than the vRTE. We compare numerical solutions of the vRTE with the gKM approximation for the problem in which a plane wave is normally incident on a plane-parallel slab composed of a uniform absorbing and scattering medium. These comparisons show that the gKM approximation accurately captures the key features of the polarization state of multiply scattered light. In particular, the gKM approximation accurately captures the complicated polarization characteristics of light backscattered by an optically thick medium composed of a monodisperse distribution of dielectric spheres over a broad range of sphere sizes.
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