Two-Stream Approximation in Radiative Transfer: Average Optical Pathlength Estimation

A general solution based on the two-stream approximation (TSA) to the radiative transfer equation is provided in plane-parallel geometry for an anisotropically scattering slab of arbitrary physical thickness. Conventional two-stream algorithms are used to compute net irradiances and mean radiances, which in turn can be used to estimate heating/ cooling rates and photolysis rates. In contrast, our two-stream solution generally applies at arbitrary polar angles and can be used to estimate average optical pathlengths for photons backscattered from a medium such as a cloud, a snowpack, or a vegetation canopy. For a slab consisting of scattering/absorbing particulate matter, the solution is valid for arbitrary values of the optical depth, “observation” polar angle, beam incidence polar angle, single-scattering albedo, backscattering ratio, and slab optical thickness. The upward diffuse radiance is used to estimate the average optical pathlength for photons reflected from a semi-infinite slab for collimated beam illumination. It is shown that for semi-infinite media, three different two-stream estimates of the average optical pathlength closely agree with one another. They are also in close agreement with results based on accurate but computationally more expensive multistream radiative transfer simulations. This general TSA solution can be used for spaceborne and airborne lidar measurements of clouds, snowpacks, and vegetation.