loading page

Efficient Multiple Scattering Solutions to Radiative Transfer Equations in Strong Forward Scattering Environments for Vegetated Land Emission and Its Parameterization through The Albedo-Tau Formalism
  • Kaiqi Chen ,
  • Shurun Tan
Kaiqi Chen
Zhejiang University-University of Illinois Urbana-Champaign Institute

Corresponding Author:[email protected]

Author Profile
Shurun Tan
Author Profile


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 paper, we present an effective approach to compute multiple scattering solutions to RT equations with 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 ω − τ formalism under catered conditions. An explicit physical definition of the equivalent scattering albedo and equivalent optical thickness are 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 ω−τ 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.