2.1.2. Reflectance of the Canopy Closure

The reflectance of the canopy closure is the sum of the single- and multiple-scattering contributions inside the canopy [30,31]. We therefore have the following equation

$$r\_{\text{canopy\\_closure}} = r\_{\text{cc\\_1}} + r\_{\text{cc\\_m}} \tag{2}$$

Here, *rcc\_*1 is the single-scattering contribution of the canopy closure, and *rcc\_m* is the multiple-scattering contribution of the canopy closure. In the next two sections, we explain the modeling for *rcc\_*1 and *rcc\_m*.

### 1. Single-scattering contribution of the canopy closure

In the GO approach, the single scattering of the canopy closure is a linear combination of the four-component (illuminated leaf, shaded leaf, illuminated soil, and shaded soil [16]) area fraction and the corresponding representative reflectance for each component in the viewing direction [3,24,25]. Therefore

$$r\_{cc\\_1} = S\_{\mathfrak{c}}r\_{\mathfrak{c}} + S\_{\mathfrak{i}}r\_{\mathfrak{i}} + S\_{\mathfrak{z}}r\_{\mathfrak{z}} + S\_{\mathfrak{z}}r\_{\mathfrak{z}} \tag{3}$$

Here, *rc* is the reflectance of the illuminated leaf, *ri* is the reflectance of the shaded leaf, *rz* is the reflectance of the illuminated soil, and *rg* is the reflectance of the shaded soil. *Sc* is the area fraction of the illuminated vegetation in the canopy closure, *Si* is the area fraction of the shaded vegetation in the canopy closure, *Sz* is the area fraction of the illuminated soil in the canopy closure, *Sg* is the area fraction of the shaded soil in the canopy closure.

We modified the area fraction equations of the four-component area fraction proposed by Verheof [33] in continuous crops and derived an expression suitable for the four-component area fraction in the canopy closure. The specific derivation can be seen in Supplementary Materials A, and the equations are

$$S\_z = P\_{\kappa} \left( \theta\_{s\nu} \theta\_{a\nu} \mathbf{x}, h \right) \tag{4}$$

$$S\_{\mathcal{S}} = P\_o(\theta\_o, \mathbf{x}, h) - P\_{sv}(\theta\_{\mathbf{s}\prime}\theta\_o, \mathbf{x}, h) \tag{5}$$

$$S\_c = k \int\_0^h P\_{sv}(\theta\_{s\cdot}, \theta\_{o\cdot}, \mathbf{x}, z) dz \tag{6}$$

$$S\_i = 1 - P\_o(\boldsymbol{\theta}\_o, \mathbf{x}, h) - k \int\_0^h P\_{ss}(\boldsymbol{\theta}\_{s\star} \boldsymbol{\theta}\_o, \mathbf{x}, z) dz \tag{7}$$

Here, *Po*(θ*<sup>o</sup>*,*x*,*h*) is the gap probability of the canopy closure in viewing direction, *Pso*(θ*<sup>s</sup>*,θ*<sup>o</sup>*,*x*,*h*) is the bidirectional gap probability of the canopy closure, and *h* 0 *Pso* (<sup>θ</sup>*<sup>s</sup>*, θ*<sup>o</sup>*, *x*, *z*)*dz* is the bidirectional vegetation probability of the canopy closure. *k* is the extinction coefficient of the canopy closure in the viewing direction

$$k = \frac{2}{\pi} L\_{\text{row}} \left\{ \left[ ar \cos \left( \frac{-1}{\tan \theta\_o \tan \theta\_l} \right) - \frac{\pi}{2} \right] \cos \theta\_l + \sin \left[ ar \cos \left( \frac{-1}{\tan \theta\_o \tan \theta\_l} \right) \right] \tan \theta\_o \sin \theta\_l \right\} \tag{8}$$

in which

$$L\_{nvw} = (A\_1 + A\_2)Lf(\theta\_l)d\theta\_l / A\_1h \tag{9}$$

Here, *h* is the canopy height, *L* is the leaf area index, and *f*(θ*l*) is the leaf inclination distribution function (LADF). This study used an elliptic distribution function [34,35]. Combining Equations (4)–(7), *Po*(θ*<sup>o</sup>*,*x*,*h*), *Pso*(θ*<sup>s</sup>*,θ*<sup>o</sup>*,*x*,*h*) and *h* 0 *Pso* (<sup>θ</sup>*<sup>s</sup>*, θ*<sup>o</sup>*, *x*, *z*)*dz* are the most important parameters for the calculation of area fraction of the canopy closure (*Sc*, *Si*, *Sz*, and *Sg*).

In the calculation of gap and vegetation probabilities, we proposed a new approach to calculate gap probabilities. In this new approach, we used a penetration function [21] to calculate gap probabilities in each path length. In this step, we considered the overlapping relationship between the leaves to calculate an average value of gap probabilities of the canopy closure (Supplementary Materials B-2). Furthermore, the average value of gap probabilities of the canopy closure has been used to represent the whole geometric characteristics of the canopy closure and was substituted into the calculation to analyze the overlapping relationship between the average value of gap probabilities of the canopy closure in the solar direction or the viewing direction. Therefore, the overlapping relationship between individual canopy closures could be considered in the calculation of gap probabilities (Supplementary Materials B-3). According to the above calculation ideas, we can consider both the overlapping relationship between leaves and individual canopy closures. In order to describe the bidirectional gap probability and vegetation probabilities, we modified the hotspot kernel function [25] originally used in forests to be suitable for row crops (Equation (B-25) in Supplementary Materials B-3), which can control the peak value near the hotspot. Based on the above, we attempted to address the computational

deviations in the single-scattering contribution near the hotspot. For a detailed mathematical derivation, please refer to Supplementary Materials B.

### 2. Multiple-scattering contribution of the canopy closure

To calculate the multiple-scattering contribution, we assumed that the canopy closure is an isotropic scattering layer. The principle of adding in the RT approach (adding method) [36–38] is introduced (Figure 2). The derived reflectance of the canopy closure is

$$\begin{split} r\_{\text{canopy\\_closure}} &= r\_{\text{cc\\_1}} + \tau\_{\text{s}} r\_{\text{cc\\_1}} \tau\_o + \tau\_{\text{s}} r\_{\text{s}} r\_{\text{cc\\_1}} \tau\_o + \cdots \\ r &= r\_{\text{cc\\_1}} + \frac{\tau\_{\text{s}} \tau\_0 r\_{\text{s}}}{1 - r\_{\text{cc\\_1}} r\_s} \end{split} \tag{10}$$

**Figure 2.** Sketch of radiative transfer inside an isotropic scattering layer and the soil. *Es* is the downward irradiance on the horizontal plane, *rcc\_*1 is the single scattering contribution of the canopy closure, τ*s* is the transmittance of the canopy closure in the solar direction, and τ*o* is the transmittance of the canopy closure in the solar direction. τ is the optical thickness, *k* is the extinction coefficient, and *s* is the path length.

Here, *rs* is the reflectance of soil and *rs* = (*Szrz* + *Sgrg*)/(*Sz* + *Sg*). τ*s* is the transmittance of the canopy closure in the solar direction, and τ*o* is the transmittance of the canopy closure in the view direction. Equation (10) reflects the interaction (scattering) between light, vegetation, and soil, including the single-scattering contribution of the canopy closure and the multiple-scattering contribution of the canopy closure.

We removed the single-scattering contribution (*rcc\_*1) in Equation (10), hence the multiple-scattering contribution of the canopy closure is

$$r\_{\rm c\varepsilon\\_ll} = \frac{\pi\_s \tau\_o \frac{S\_z r\_z + S\_\mathcal{S} r\_\mathcal{S}}{S\_z + S\_\mathcal{S}}}{1 - r\_{\rm c\varepsilon\\_1} \frac{S\_z r\_z + S\_\mathcal{S} r\_\mathcal{S}}{S\_z + S\_\mathcal{S}}} = \frac{\tau\_s \tau\_o \left(S\_z r\_z + S\_\mathcal{S} r\_\mathcal{S}\right)}{S\_z + S\_\mathcal{S} - r\_{\rm c\varepsilon\\_1} \left(S\_z r\_z + S\_\mathcal{S} r\_\mathcal{S}\right)}\tag{11}$$

To address the lack of leaf transmittance in the optical input parameters of the GO approach, we introduce the study by Lang [39]. In this study, the transmittance of the canopy is approximately the same as the gap probabilities and replaces τ*s* and τ*o* in Equation (11). Therefore, the multiple scattering of the canopy closure is

$$r\_{\rm cc\\_m} = \frac{P\_o(\theta\_0, h)P\_s(\theta\_{\rm s\\_}h)\Big(\mathcal{S}\_{\rm z}r\_{\rm z} + \mathcal{S}\_{\rm \mathcal{S}}r\_{\rm \mathcal{S}}\Big)}{\mathcal{S}\_{\rm z} + \mathcal{S}\_{\rm \mathcal{S}} - r\_{\rm cc\\_1}(\mathcal{S}\_{\rm \mathcal{Z}}r\_{\rm z} + \mathcal{S}\_{\rm \mathcal{S}}r\_{\rm \mathcal{S}})} \tag{12}$$

### 2.1.3. Reflectance of the Between-Row Area

Previous studies have pointed out that there is a radiation energy exchange between vegetation and soil in the between-row area, which is further affected by multiple scattering between the soil and adjacent canopy closure between the rows [30–32]. The between-row area is an area where both the soil (C' in Figure 3a) and the vegetation (A' and B' in Figure 3a) exist, hence the multiple-scattering equation is very complicated. To establish the multiple-scattering equation of the between-row area, we introduced the mathematical solution of integral radiative transfer equation derived by Ma et al. [30]. We parameterized this equation according to the requirements of the GO approach. Then, an equation of reflectance in the between-row suitable for the GO approach was derived.

**Figure 3.** Sketch of the between-row area. (**a**) Angle relationship between the escape surface and the bottom of the between-row area. (**b**) Angle relationship between the escape surface and the *z* position of the between-row area. (**c**) Geometric relationship for the viewing probability of the between-row soil when *xr* ≥ *A*1. (**d**) Geometric relationship for the viewing probability of the between-row soil when *xr* < *A*1. Here, α*o* is the inclined angle projected in the perpendicular plane in the viewing direction, α1 is the openness angle of the between-row area, and α2 is the nonopenness angle of the between-row area. *sbr* is the path length of the light of the canopy closure for the between-row soil between being observed, ϕ*r* is the row azimuth angle, *A*1 is the row width, *A*2 is the between-row distance, and *h* is the canopy height.

The reflectance of the between-row area (*rbetween\_row*) is divided into two components: one is the single scattering of the between-row area (*rbr\_*1) and the other is the multiple scattering of the between-row area (*rbr\_m*), and the expression is

$$r\_{\text{between\\_cov}} = r\_{br\\_1} + r\_{lr\\_m} \tag{15}$$

### 1. Single-scattering contribution of the between-row area

The single-scattering contribution of the between-row area is the reflectance of the soil that can be observed. According to the projection relationship between canopy closures in the GO approach, the single scattering of the between-row area is

$$r\_{br\perp1}^{\*} = \begin{cases} \frac{l}{A\_2}r\_{\mathcal{S}} + \frac{(A\_2 - l)}{A\_2}r\_{\mathcal{S}} & l < A\_2\\ r\_{\mathcal{S}} & l \ge A\_2 \end{cases} \tag{14}$$

\_

Here, *l* is the horizontal projected length of the row height on the ground in the solar or viewing directions (for its expression, please refer to Supplementary Materials B-2). When the viewing direction is considered (Figure 3b,c), Equation (14) becomes

$$r\_{br\\_1} = \overline{P\_{o\\_br}(\theta\_o, \mathbf{x}, h)} r\_{br\\_1}^\* \tag{15}$$

where *Po br*(<sup>θ</sup>*<sup>o</sup>*, *x*, *h*) is the average viewing probability of the between-row soil, and

$$\overline{P\_{o\\_br}(\theta\_{o\prime}, \mathbf{x}, h)} = \frac{1}{A\_2} \int\_0^{A\_2} e^{-k\mathbf{x}\_{br}} d\mathbf{x} \tag{16}$$

Here, *sbr*(θ*<sup>o</sup>*,*x*,*h*) is the path length of the light in the canopy closure in the between-row soil between being observed (Figure 3b,c), and

$$s\_{br}(\theta\_o, \mathbf{x}, h) = \begin{cases} (N\_{br} + 1) \frac{A\_1}{\sin a\_o \sin \beta\_o} & \mathbf{x}\_r \ge A\_1 \\ (N\_{br} + 1) \frac{\pi \tan a - \chi}{\sin a\_o \sin \beta\_o} & \mathbf{x}\_r < A\_1 \end{cases} \tag{17}$$

where α*o* is the inclined angle projected in the perpendicular plane in the viewing direction (its specific description can be found in Supplementary Materials B-1), β*o* is the azimuth of the inclined angle in the viewing direction, and sinβ*o* = sinϕ*ro*sin|θ*o*|/sin<sup>α</sup>*o*. *xr* is a remainder on the *x*-axis, and *xr* = (*h*tanα−*<sup>x</sup>*)mod(*A*<sup>1</sup> + *A*2). *Nbr* is the number of row cycles (*A*1 + *A*2) involved in *sbr*(θ*<sup>o</sup>*,*x*,*h*), and *Nbr* = (*h* tan α − *<sup>x</sup>*)/(*<sup>A</sup>*1 + *<sup>A</sup>*2). Here, mod is the mathematical symbol of modulus, and · is the mathematical notation for rounding down. Their sketches are shown in Figure 3b,c. Similarly, the average viewing probability at *z* of the between-row area (*Po*\_*br*(<sup>θ</sup>*<sup>o</sup>*, *x*, *z*)) only needs to replace *h* with *z*.

### 2. Multiple-scattering contribution of the between-row area

Multiple scattering of the between-row area (*rbr\_m*) occurs between the soil and adjacent canopy closure between the rows. Ma et al. gave a multiple-scattering equation of the between-row area based on the operation of the differentia integral operator [40], and its solution is

$$\begin{split} r\_{br\\_m}^\* &= \mathcal{K}\_{br} r\_{br\\_1}^\* + \mathcal{K}\_{br}^2 r\_{br\\_1}^\* + \mathcal{K}\_{br}^3 r\_{br\\_1}^\* + \cdots\\ r\_{\;\\_1\\_1\\_r}^\* &= \frac{r\_{br\\_1}^\* \mathcal{K}\_{br}}{1 - \mathcal{K}\_{br}} \end{split} \tag{18}$$

Here, *Kbr* is the transfer probability of the collision, and *Kbr* = *kbrPbr*. Here, *kbr* is the extinction coefficient of the between-row area, and *Pbr* is the visual probability of each surface in the between-row area. The between-row area consists of four surfaces (vegetation surface (A' in Figure 3a), vegetation surface (B' in Figure 3a), soil surface (C' in Figure 3a), and escape surface). To calculate *Pbr*, we defined the openness angle of the between-row area (α1) and nonopenness angle of the between-row area (α2) (Figure 3a,b). Then, we can calculate the visual probability of the escape surface (*Popen*) as <sup>α</sup>1/(<sup>α</sup>1 + α2), e.g., <sup>∠</sup>*dac*/(π/2), <sup>∠</sup>*dgc*/<sup>π</sup>, and <sup>∠</sup>*dbc*/(π/2), shown in Figure 3a,b. The simplified equation is

$$P\_{\text{open}}(\mathbf{x}, \theta) = \begin{cases} \frac{2}{\pi} \arctan\left(\frac{A\_2}{\sin\varphi\_r h}\right) & (\mathbf{x} = A\_2) \land (\mathbf{x} = 0) \\\\ \frac{1}{\pi} \left[\pi - \arctan\left(\frac{h \sin \varphi\_r}{A\_2 - \mathbf{x} \sin \varphi\_r}\right) - \arctan\left(\frac{h}{\pi}\right)\right] & \mathbf{x} \in (0, A\_2) \end{cases} \tag{19}$$

Here, ϕ*r* is the row azimuth angle. Equation (19) is the radiation escape probability at the bottom of the between-row area (Figure 3a). For the escape probability at *z* of the between-row area, *h* in Equation (19) needs to be replaced by *z* (Figure 3b). Since different positions (different coordinate

points (*<sup>x</sup>*,*y*) in Figure 3b) in the between-row area have a corresponding radiation escape probability, there are many radiation escape probabilities in the between-row area. In terms of modeling, we focus on the average radiation escape probability value in the between-row area, which is

$$\begin{aligned} \overline{P\_{\text{open}}} &= \frac{1}{h} \ln \frac{\Lambda\_1}{A\_2} \int\_0^1 P\_{\text{open}}(\mathbf{x}, z, \theta) d\mathbf{x} dz \approx \frac{1}{h} \sum\_{z=0}^h \frac{1}{A\_2} \sum\_{x=0}^{A\_2} P\_{\text{open}}(\mathbf{x}, z, \theta) d\mathbf{x} dz \\\ P\_{\text{open}}(\mathbf{x}, z, \theta) &= \begin{cases} \frac{2}{\pi} \arctan \left( \frac{A\_2}{\sin \eta \cdot h} \right) & \left[ (\mathbf{x} = A\_2) \wedge (\mathbf{x} = 0) \right] \wedge \left[ z \in [0, h) \right] \\\\ \frac{1}{\pi} \left[ \pi - \arctan \left( \frac{z \sin \eta\_r}{A\_2 - \sin \eta\_r} \right) - \arctan \left( \frac{z}{x} \right) \right] & \left[ \mathbf{x} \in (0, A\_2) \right] \wedge \left[ z \in [0, h) \right] \\\ 0 & z = h \end{cases} \end{aligned} \tag{20}$$

When the viewing direction is considered, Equation (18) can become

$$r\_{br\\_m} = r\_{br\\_m}^\* \frac{1}{h} \int\_0^l \overline{P\_{o\\_br}(\theta\_{o\\_}, x, z)} dx \tag{21}$$

The average visual factor of the vegetation surface (A' and B' in Figure 3a) and one soil surface (C' in Figure 3a) is defined as *Pother*, and *Pother* = 1 − *<sup>P</sup>*open. Combining Equations (18) and (21), the multiple scattering of the between-row area becomes

$$r\_{br\\_m} = \frac{1}{h} \int\_0^h \overline{P\_{o\\_br}(\theta\_{o\\_r}, z, z)} d\mathbf{x} \left[ \frac{r\_{br\\_1}^\* k\_{br} \left( 1 - \overline{P\_{\text{open}}} \right)}{1 - k\_{br} \left( 1 - \overline{P\_{\text{open}}} \right)} \right] \tag{22}$$

From Equation (22), the extinction coe fficient of the between-row area is key in calculating the reflectance of the between-row area. To apply Equation (22) in the GO approach, we assumed that the extinction coe fficient of the between-row area is the weight sum of the two mediums, soil and leaf. The weight is determined by the length ratio of the medium in this area, which is

$$k\_{br} = \frac{2\text{h}}{A\_2 + 2\text{h}}k + \frac{A\_2}{A\_2 + 2\text{h}}k\_s = \frac{2\text{h}k + A\_2k\_s}{A\_2 + 2\text{h}}\tag{23}$$

Here, *k* is the extinction coe fficient of the canopy closure in the viewing direction (A' and B' in Figure 3a), and its equation is Equation (8). *ks* is the extinction coe fficient of the between-row soil in the viewing direction (C' in Figure 3a). According to [30], the extinction coe fficient of the between-row soil in the viewing direction is

$$k\_s = 2 - \frac{4r\_{br\_-1}^\*(\cos\theta\_s + \cos\theta\_o)}{p(\delta)\cos\theta\_s^2\cos\theta\_o^2} \left(1 - \frac{b}{4}\right) \tag{24}$$

Here, *p*(δ) is the scattering phase function of the soil particle, which represents the second-order Legendre polynomial (an approximation of spherical function) [41], *p*(δ) = 1 + *b* cos δ + 0.5*c* 3 cos<sup>2</sup> δ − 1 , and cos δ = cos θ*s* cos θ*o* + sin θ*s* sin θ*o* cos ϕ*so*. Here, *b* and *c* are the adjustment parameters for the second-order Legendre polynomial in the scattering phase function of the soil particle, which can be determined by [42].
