Vegetation isoline equations with first- and second-order interaction terms for modeling a canopy-soil system of layers in the red and near-infrared reflectance space

Abstract

Abstract. A previously proposed vegetation isoline equation suffers from errors if the soil background of a canopy layer is bright. These errors arise from the truncation of the second-and higher-order terms that represent photon interactions between the canopy and the soil. An isoline equation that includes a second-order interaction term is introduced. The equation was initially derived by explicitly including a second-order interaction term in both the red and near-infrared (NIR) reflectance spectra (symmetric approximation). We also examined an alternative model in which the interaction term was included only in the NIR band (asymmetric approximation). In this model, the derived isolines tend to shift upward (overcorrection effects). Numerical experiments revealed that the errors in the isoline obtained by the asymmetric approximation were reduced in magnitude to nearly one-fifth of the errors in the previously proposed method. Its accuracy was higher than that of the symmetric approximation, despite the fact that the asymmetric approximation included fewer terms than the symmetric approximation. The improved model accuracy resulted from the overcorrection effects, which compensated for the truncation error. With the simplicity and improved accuracy, the current isoline equations provide a good alternative to the previous approach.

Improved Accuracy of the Asymmetric Second-Order Vegetation Isoline Equation over the RED–NIR Reflectance Space

“…The error in the first-order isoline reached 0.01 in reflectance units as the soil reflectance increased. By contrast, the error in the asymmetric-order isoline was much smaller than that in the first-order approximation, nearly one order of magnitude smaller, as summarized in our previous study [23]. …”

“…The total number of spectra was 9261 (21 × 21 × 21) for a single LAD. (The parameter grids are finer than in our previous study [23]. ) We employed 655 nm and 865 nm reflectance spectra for the red and NIR wavelength regions, which corresponded to the center of the red and NIR bands in the Landsat 8 Operational Land Imager (OLI).…”

“…The parameters in the isoline equations were computed according to the procedures reported previously [23,26]. $T_{v\lambda }^2$ and ${\rho }_{v\lambda }$ were determined based on two hypothetical simulations.…”

“…The parameter $R_{vN}$, which was required for the computation of ξ , was obtained by conducting an additional simulation in which the soil spectrum was even brighter than was assumed in the simulation used to compute $T_{v\lambda }$. The soil spectrum was also spectrally flat in this case [23]. In the simulation, the TOC canopy reflectances in the NIR were approximated using first- and second-order interaction terms between the canopy layer and the soil surface, $\begin{array}{c}\hfill {\rho }_N\approx \omega {\rho }_{vN}+\overline{T_N^2}{R}_{sN}+\omega T_N^2{R}_{vN}{R}_{sN}^2.\end{array}$ where $R_{sN}$ represents the bi-hemispherical reflectance of the soil surface for the NIR band.…”

“…In a previous study, we proposed a derivation technique for satisfying these requirements simultaneously [23]. During the derivation, we included the second-order interaction term only in the near-infrared band instead of retaining the second-order term in the red band.…”