A multiple scattering model for passive radiative transfer (RT) in vegetation that accounts for the vertical profile of the plant structure is developed, offering advancements over the commonly-used single-layer uniform scattering models prevalent in the vegetated land surface microwave remote sensing. The proposed model takes into account the complexities of the canopy morphology with vertical heterogeneity, enabling the representation of overlapping vegetation species applicable to diverse plant types and growth stages. Additionally, it serves as a valuable tool for understanding the influence of the vegetation vertical structure on the microwave brightness temperatures. The model is constructed based on high-order solutions to the RT equations, obtained through a numerical iterative approach with an efficient interpolation scheme for algorithm acceleration. This methodology facilitates the accurate distinction of the contributions to the brightness temperature from each scattering order and scattering mechanism, ensuring a comprehensive consideration of multiple scattering effects within various vegetated scenarios. The model is validated using the SMAPVEX12 L-band forest data set, encompassing a wide range of soil moisture variations. Comparisons are made between the brightness temperatures simulated by the newly developed multiple-scattering model with a continuous profile or layered profile and those obtained from a uniform single-layer model. Results demonstrate significant improvements in the multi-layered or the continuously profiled model, showing improved agreement with the measured brightness temperatures. Furthermore, the proposed model is parameterized by matching the high-order solutions to the RT equation to the widely adopted reduced order albedo-tau formalism. The resulting equivalent parameters are linked to the geometries and the electromagnetic properties of the vegetation layer, while also incorporating the effects of multiple scattering. Comparative analysis of the equivalent parameters derived from the layered model and those derived from the single-layer model reveals that the vertical heterogeneity of the vegetation structure has a notable influence on the effective scattering albedo and it yields a value more consistent with the albedo as chosen in the SMAP/SMOS inversion algorithms. Meanwhile, the impact of the vegetation vertical profile on the effective optical thickness and the effective transmissivity of the vegetation layer is weak.These insights are essential for the retrieval of soil moisture and vegetation characteristics including the plant vertical structures in microwave remote sensing.

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.