An Efficient Multiple Scattering Solution to Radiative Transfer Equations in Strong Forward Scattering Environments for Vegetated Land Emission and its Representation Through an Equivalent Albedo-Tau Formalism
Kaiqi Chen, Shurun Tan
Abstract
The radiative transfer (RT) theory has been widely utilized for wave propagation in random media, but it faces challenges in situations involving strong forward scattering, such as in forests with electrically large trunks, due to the singularity of the scattering phase matrix. In this article, we present an effective approach to compute multiple scattering solutions to RT equations with a singular phase matrix by combining the strategy of forward scattering extraction with an efficient numerical iterative procedure through interpolation. We evaluate the effectiveness and efficiency of our technique through simulations using a layer of vertically oriented, electrically large long cylinders to represent a layer of trunks over the ground. The results demonstrate that the proposed approach increases the computational efficiency by one to two orders of magnitude in cases where forward scattering is dominant. Additionally, a parameterized model is derived by matching the higher-order RT results with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega -\tau $ </tex-math></inline-formula> formalism under catered conditions. An explicit physical definition of the equivalent scattering albedo and equivalent optical thickness is proposed under boundary-free conditions. The multiple scattering effects are included in the physically derived equivalent parameters of the plant layer, which are independent of ground conditions by definition. Tests verify that the applicability of the parameterized model with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega -\tau $ </tex-math></inline-formula> form can be extended to a wider range of vegetation and ground conditions. Besides, these equivalent parameters are directly linked to the geometric structures and electromagnetic properties of the vegetation layer, allowing their values to be frequency- and angle-dependent. Compared to the single-scattering albedo and optical thickness, the effective albedo derived from the RT model exhibits relatively weak polarization and angle dependence. This is consistent with many empirically derived parameterizations while providing a physically plausible origin for these equivalent parameters. Remarkably, we find that the transmittance linked to the parameterized tau value, incorporating multiple scattering effects, is similar to that obtained through full-wave simulations that account for coherent wave interactions in the trunk layer. This work is of interest to remote-sensing practitioners both in vegetation scattering modeling and in vegetated land surface parameter retrieval.