Scattering Effect Contributions to the Directional Canopy Emissivity and Brightness Temperature Based on CE-P and CBT-P Models

The directional anisotropy of canopy emissivity and brightness temperature in the thermal infrared band has widely been studied. However, the contribution of different scattering orders has been an open scientific question for many years. The recently proposed CE-P model enables us to analytically evaluate the different scattering orders. Herein, we derive expressions for the first double collisions (DCE
₁₂
) and first triple collisions (DCE
₁₂₃
). Our result shows that DCE
₁₂₃
can simulate the observed emissivity with an error less than 0.001 and that DCE
₁₂
is reasonably accurate when leaf emissivity is greater than 0.96. Numerical analysis shows that the contribution of quadruple or greater collisions can be ignored when the leaf (soil) emissivity is no less than 0.90. Furthermore, we develop the CBT-P model to simulate the directional brightness temperature (DBT) based on the new optimized CE-P model (DCE
₁₂₃
) and validate it by 4SAIL (4-Stream Radiative Transfer Theory of Scattering by Arbitrary Inclined Leaves) and DART (Discrete Anisotropic Radiative Transfer) models. Both of isothermal (soil temperature is equal to leaf temperature) and nonisothermal (soil temperature is higher than leaf temperature) cases are considered. The maximum differences between the CBT-P model and 4SAIL (DART) are less than 0.35 K (0.42 K), the average differences between CBT-P and 4SAIL (DART) are less than 0.10 K (0.13 K), and the R
²
is over 0.99 (0.95) with component emissivities larger than 0.90 and the difference between soil and leaf temperatures less than 20 K. The directional anisotropy of DBT is dominated by the zero-scattering and the single scattering terms according to the new developed CBT-P model.