1. Introduction

The canopy, which consists of a set of leaves, has a complex structure and can have a great importance for greenhouse gas emissions regulation and climate change radiative forcing. Radiative forcing can be investigated by considering the radiative transfer problem through plant canopies [1–4]. Therefore, understanding the interaction of radiation with vegetation canopy has gained great interest during the past two decades. In fact, retrieving information from remotely sensed data require simple and efficient models that accurately simulate the spectral behavior of photons in the canopy. Radiative transfer within the vegetation canopy yields an equation similar to the conventional radiative transfer in participating media. However, since the scattering of photons by leaves depends of the incident radiation direction, the radiation with different incident shows different contributions to leaves radiative properties with different directions. Therefore, the vegetation canopy is an anisotropic media. That is, the scattering and absorption coefficients depend on the direction of the radiation whereas the anisotropy scattering phase function depend on both the incident radiation and the angle between the incident and the outgoing radiations [5,6]. Henceforth, the resulting radiative transfer equation is mathematically complex and requires special attention in the development of an efficient solution [7].

In most cases, radiative transfer within the vegetation canopy is modeled using one-dimensional assumption giving route to the knowledge of specific transport effects and set foundation for the development of more comprehensible models and numerical tools [8]. Several techniques found in the literature have been developed to solve the conventional radiative transfer in which scattering centers orientation have been neglected [9–19]. While these techniques are sufficiently accurate and efficient for isotropic media in which spherically centers are assumed, they may not be suitable for radiative transfer in anisotropic media such as vegetation canopy and may be useless for retrieving information from remotely sensed data of this system. Different conventional methods for radiative transfer in isotropic media have been extended to anisotropic plant canopies [2], including the Monte Carlo method [20], adding method [21], the successive orders of scattering approximation [22] and the discrete ordinates method [23]. The development of analytical or semi-analytical simplified models for the radiative transfer in vegetation canopy remains a great task for the scientific community, and must challenge two tasks: adequately modeling radiative transfer, and efficiently computational time of numerous calculations [24,25]. Among analytical methods, the two-/four-stream approximations [26–28] and Chandrasekhar’s method [8,29] have attracted great attention. However, these methods suffer from some limitations notably the poor accuracy for some optical regime in the case of two-/four- stream approximations, difficult application to highly anisotropic canopy for the Chandrasekhar’s method. Two accurate quasi-analytical models for radiative transfer in plant canopy are the FN or “Facile” [30,31] and the ADO explicitly analytical discrete ordinates methods [32,33]. The first one is mostly considered for benchmark while the second can be difficult to implement by non-radiation experts such as biologist, physiologist or agronomist due to the mathematical complexity of the method [34].

The objective of this paper is to develop a semi-analytic radiative method for analyzing one-dimensional radiative transfer through dense anisotropic plant canopy, which the radiative properties depending on the direction of the radiation. The method is an extension of the discrete ordinates characteristics solution (DOCS) recently described by Kamdem et al. [34] for radiative transfer in participating media in which spherically centers are assumed. This method couples discrete ordinates angular discretization with the solution of the media radiative characteristics variables of first-order differential equations obtained after eigenvalue decomposition of the matrix resulting from media radiative properties. In the frame of this study, more consideration is given to the directional leaves radiative properties in the DOCS. In addition, expression for radiance calculation using the proposed semi-analytical model is developed. First, the physical problem of radiative transfer through plant canopy is described.

2. Discrete ordinates characteristics solution

Let us consider a flat horizontal plant canopy of depth ${z_L}$ as described in Fig. 1. The leaf canopy is subjected to a uniform incident radiation on the top of the canopy whereas the bottom of the canopy has a Lambertian diffuse reflection to account the soil reflection. The radiative intensity through the canopy can be described by [28]

 

Fig. 1. Plant canopy schematic considered as a single layer containing leaves ensemble of bi-Lambertian surfaces

Download Full Size | PDF

The radiative intensity can be split into diffuse, ${I_d}$, and collimated part as [23,33]

The analytical solution of Eq. (11) is described in Kamdem et al. [34] and Ymeli and Kamdem [35]. In developing the method, the ordinates radiative intensities are expanded in characteristics variables as

3. Results

Predictions for a variety of problem involving leaf angle distribution are obtained using the discrete ordinates characteristics solution (DOCS) and compared with experiments, exact Chandrasekhar benchmark results and two other semi- analytical solution namely the “facile” and analytical discrete ordinates methods [29,30,33]. The DOCS implemented in MATLAB scientific language use the double-Gauss quadrature sets for angular discretization and predictions performed using a personal computer with Intel Core i7-4810MQ and 16.0 Gb of RAM. In this study, the top transparent boundary of the plant canopy with azimuthal symmetry is subjected to a normal collimated incident while the bottom has a transparent of diffuse Lambertian reflection. The bi-Lambertian scattering model is also considered for the leaf scattering phase function.

4. Radiative transfer in the finite plant canopy

Figure 2 displays DOCS radiance angular discretization errors for single leaf canopy plant in the visible and the near infrared. The single leaf plant canopy normally irradiated has an optical depth ${\tau _L} = 1$. The leaf is oriented at ${\theta _L} = 60^\circ $ and two cases are considered: $\{{{\omega_L},{r_s}} \}= \{{0.1,{\;\ }0.1} \}$ and $\{{{\omega_L},{r_s}} \}= \{{0.95,{\;\ }0.2} \}$ for photon transport in the visible and the near infrared, respectively. Four angular discretizations have also been considered for the DOCS predictions: $N = 5,11,\; 23$ and $99$. The relative error between DOCS predictions and exact Chandrasekhar results [29] is used to quantify the angular discretization error. It can be seen in Fig. 2 that the reflected radiance angular discretization error increases with increasing direction, while the trends is reversed for transmitted radiance. As expected, angular discretization error is reduced when the grid resolution is increased. The maximum relative errors obtained for DOCS using $N = 5$ number of Double Gauss ordinates directions are less than 0.06 and 0.3% in the visible and the near infrared, respectively. Excellent agreement is reached between DOCS predictions and exact Chandrasekhar results for N = 23 and 99 number of double-Gauss ordinates direction except for cosine direction $\mu \ge 0.9$. The DOCS predictions using N = 23 number of double-Gauss ordinates direction reflected and transmitted radiances are presented in Table 1. These predictions accurately forecasted the five and four significant figures of the Chandrasekhar benchmark results.

 

Fig. 2. View cosine direction and number of ordinates direction dependences of a finite single leaf plant canopy discrete ordinates characteristics solutions (DOCS) radiance angular discretization error in the visible and near infrared: : ${\mu _0} = 1$, ${r_L} = {t_L}$ and ${\theta _L} = 60^\circ $.

Download Full Size | PDF

Table 1. Discrete ordinates characteristics solution reflected and transmitted radiances in the visible and near infrared for a finite single leaf plant canopy:$\; {{\mathbf \tau }_{\mathbf L}} = 1$, ${{\mathbf \mu }_0} = 1$, and ${{\mathbf \theta }_{\mathbf L}} = 60^\circ $

View Table | View all tables in this article

The comparison of the DOCS reflected and transmitted radiances with FN [30] and ADO [33] semi-analytical predictions are displayed in Fig. 3. The finite canopy in the near infrared, ${\omega _L} = 0.90$ with ${r_L} = 0.25$, reflects diffusely with ${r_s} = 0.2$. The single leaf canopy oriented at ${\theta _L} = 60^\circ $ and the canopy cumulative LAI is ${\tau _L} = 1$. Figure 3 shows that the DOCS predictions using N = 19 and 23 number of Double-Gauss ordinates direction yield excellent comparisons with F_N and ADO results obtained using N = 19, and 200 number of Gauss ordinates direction, respectively. It can be seen in Table 2 that the discrepancies between DOCS predictions using N = 19 and 23 is insignificant: the two predictions are in five significant figures accuracy. Table 2 shows that the DOCS reflected and transmitted radiances predicted using moderate number of double-Gauss ordinates direction are in five significant figures accuracy when compared to the F_N [30] and ADO results [33].

 

Fig. 3. View zenith angle dependence on the reflected and transmitted single canopy plant radiance in the near infrared: ${\tau _L} = 1$, ${\omega _L} = 0.9$, ${r_L} = 0.25$, ${r_s} = 0.2$, ${\mu _0} = 1$ and ${\theta _L} = 60^\circ $. The discrete ordinates characteristics solution (DOCS) is compared to F_N [30] and analytical discrete ordinates (ADO) [33] results.

Download Full Size | PDF

Table 2. Discrete ordinates characteristics solution reflected and transmitted radiances in the near infrared of a finite single leaf canopy: ${{\mathbf \mu }_0} = 1$, ${{\mathbf \tau }_{\mathbf L}} = 1$, ${{\mathbf \omega }_{\mathbf L}} = 0.9$, ${{\mathbf r}_{\mathbf L}} = 0.25$, ${{\mathbf \theta }_{\mathbf L}} = 60^\circ $, ${{\mathbf r}_{\mathbf s}} = 0.2$

View Table | View all tables in this article

The DOCS reflected and transmitted radiance in the visible of a finite single leaf plant canopy is considered in the results displayed in Fig. 4 for four-leaf zenith angle ${\theta _L}(^\circ )= \{{10,\; 30,60\; and\; 85} \}$. The plant canopy parameters are: ${\tau _L} = 2$, ${\mu _0} = 1$, ${r_s} = 0.1$, ${\omega _L} = 0.1$ and ${r_L} = 0.07$. The DOCS prediction is obtained using N = 19 and 23 number of double-Gauss ordinates direction. It can be seen in Fig. 4 that the soil reflectance leads to a high-reflected energy by the single leaf plant canopy compared to the transmitted energy. The accuracy of DOCS for hemispherical reflectance and transmittance is examined in Table 3 for four leaves zenith angles by comparing to F_N [30] and ADO [33] literature data, which were obtained using N = 19, and 200 number of Gauss ordinates direction, respectively. Table 3 shows that DOCS, F_N and ADO have similar results: The three solutions agree within four significant figures. Table 3 also indicates that although the FN convergence is faster when compare the two others semi-analytical methods for ${\theta _L}(^\circ )= \{{10,\; 30\; and\; 60\; } \}$, there is a need to obtained high angular discretization the method for larger zenith angle ${\theta _L} = 85^\circ $: the FN predictions agree well with DOCS lower angular discretization than with DOCS high angular discretization.

 

Fig. 4. View zenith angle dependence on the reflected and transmitted single plant canopy radiance in the visible predicted using discrete ordinates characteristics solution: ${\tau _L} = 2$, ${\mu _0} = 1$, ${r_s} = 0.1$, ${\omega _L} = 0.1$ and ${r_L} = 0.07$.

Download Full Size | PDF

Table 3. Discrete ordinates characteristics solutions (DOCS) hemispherical reflectance and transmittance in the visible for finite plant canopy considering four leaf orientations compared to F_N [30] and analytical discrete ordinates (ADO) [33] predictions : ${{\mathbf \tau }_{\mathbf L}} = 2$, ${\mathbf \omega } = 0.1$, ${{\mathbf r}_{\mathbf L}} = 0.07{\;\ }{{\mathbf r}_{\mathbf s}} = 0.1$ and ${{\mathbf \mu }_0} = 1$.

View Table | View all tables in this article

The effect of leaves angle distribution on the reflected and transmitted radiance is displayed in Fig. 5 for five representatives LAD: erectophile, extremophile, plagiophile, planoplile and unophile. The LAI for this case is ${\tau _L} = 1$, the scattering albedo is ${\omega _L} = 0.9\; $and the top transparent boundary of the finite canopy with azimuthal symmetry is subjected to a normal collimated incident while the bottom is transparent. The DOCS predictions are obtained using N = 23 number of Double-Gauss ordinates direction. In order to confirm the present predictions accuracy for different leaves angle distribution, the Discrete ordinates characteristics solutions (DOCS), the FN [30] and analytical discrete ordinates (ADO) [33] hemispherical reflectance and transmittance predictions are compared in Table 4 for two scattering albedo ${\omega _L} = 0.1\; and\; 0.9$ . It can be seen that the DOCS predictions using N = 23 number of Double-Gauss ordinates direction compete well with FN and ADO obtained with N = 21 and 50 Gauss ordinates direction, respectively.

 

Fig. 5. View zenith angle dependence for the reflected and transmitted plant canopy radiance in the near infrared for five representative leaves angle distribution.

Download Full Size | PDF

Table 4. Discrete ordinates characteristics solutions (DOCS) hemispherical reflectance and transmittance predictions for different canopies compared with F_N [30] and analytical discrete ordinates (ADO) [33] predictions: ${{\mathbf \tau }_{\mathbf L}} = 1$, ${{\mathbf \mu }_0} = 1$, ${{\mathbf r}_{\mathbf s}} = 0$.

View Table | View all tables in this article

Figure 6 displayed the angular discretization dependences of the DOCS CPU time for the six classical plant canopies considered in this study. In running the DOCS, it was found that most of CPU time is consuming for the evaluation of the G-factor and the scattering phase function during the post-processing. The evaluation of the canopy radiative properties is analytical for single leaf canopy, and numerical for other classical plant canopies. The numerical evaluation of canopy radiative properties have been carried out using Nq = 10000 Gauss ordinates directions to ensure minimal quadrature errors. Figure 6 shows that the CPU time for all other leaves angle distribution than single leaf is equivalent. This is in accord with the DOCS implementation, which use the same numerical integration for the evaluation of the radiative properties for these leaves angle distribution. In can be seen in Fig. 6 that the maximum CPU time obtained for predictions using 49 ordinates directions is less than 1 minute for single leaf canopy, whereas it is less than 10 minute for all other canopy type. This small CPU time of higher angular discretization can be reduced by using a more suitable integration of radiative properties over all leaves directions. Also, the DOCS CPU time will be similar to the DOCS CPU for single leaf canopy if analytical expression of the leaf angle distribution and G-factor are used for the evaluation of the canopy radiative properties.

 

Fig. 6. Angular discretization dependences of discrete ordinates characteristics solutions (DOCS) radiance computational time for different canopies types: ${\mu _0} = 1$, ${\omega _L} = 0.9$, ${r_L} = {t_L}$ and ${\theta _L} = 60^\circ $.

Download Full Size | PDF

5. Radiative transfer in the infinite plant canopy

Radiation with transparent boundary is considered for phonon transport in infinite single leaf plant canopy problem. The plant canopy in the visible, ${\omega _L} = 0.1$ with ${r_L} = {t_L}$, is normally irradiated at the top of the canopy and the leaf is oriented at ${\theta _L} = 60^\circ $. Figure 7 shows that the DOCS reflected radiance angular discretization decreases with the number of ordinates directions and increases view cosine direction. It can be seen that the DOCS converges for N = 23 numbers of double-Gauss ordinates directions. This number of ordinates direction is used for DOCS reflected radiance predictions of Table 5 which are compared to Chandrasekhar benchmark, FN [30] and ADO [33] literature semi-analytics results. The FN and ADO results were obtained using $N = 9$ and 50, respectively. Table 5 shows that the DOCS predictions with N = 23 accurately forecasted the four significant figures of the Chandrasekhar results used as benchmark. It can be seen in Table 5 that for radiative transfer in infinite plant canopy, the convergence of FN method is slightly faster to the DOCS. This table also shows that no significant difference is observed between the three semi-analytical predictions: the DOCS with 23 numbers of double-Gauss ordinates directions are similar to F_N and ADO when rounded predictions to Chandrasekhar results figures.

 

Fig. 7. View cosine direction and number of ordinates direction dependences of an infinite single leaf plant canopy discrete ordinates characteristics solutions (DOCS) reflected radiance angular discretization error: ${\mu _0} = 1$, ${\omega _L} = 0.1$, ${r_L} = {t_L}$ and ${\theta _L} = 60^\circ $

Download Full Size | PDF

Table 5. Discrete ordinates characteristics solutions (DOCS) reflected radiances, ${10^2} \times {\mathbf I}({{\mathbf \tau },{\mathbf \mu }} ),$ for a semi-infinite canopy in the visible compared with benchmark Chandrasekhar, and F_N [30] and analytical discrete ordinates (ADO) [33] semi-analytical results: ${{\mathbf \mu }_0} = 1$, ${{\mathbf \omega }_{\mathbf L}} = 0.1$, ${{\mathbf r}_{\mathbf L}} = 0.05$ and ${{\mathbf \theta }_{\mathbf L}} = 60^\circ $

View Table | View all tables in this article

6. Comparisons with literature experiments

Reflected radiance of soybean in visible and near infrared have been measurement at the Laboratory for Agricultural of Remote Sensing (LARS) by Ranson et al. [37]. The measurements carried out on August 27, 1980, which is used because the canopy reached nearly complete ground cover (99%) and was fully overlapping that day. The soybean reflectance factor measurements were average over the full azimuthal range since azimuthally radiative transfer problem have been considered for the theoretical formulation. The leaf-area index and canopy height were $({2.9\, \pm \,04} ){m^2}{m^{ - 2}}$ and $(1.02\, \pm \,0.04$)m, respectively. The reported leaf transmittance and reflectance data for $[{0.45,1.125} ]\mu m\; $in $0.025\mu m$ intervals covered the four spectral bands of the Landsat Multi-Spectral Scanner (MSS): $[{0.5,0.6} ]\mu m$, $[{0.6,0.7} ]\mu m$, $[{0.7,0.8} ]\mu m$, and $[{0.8,1.1} ]\mu m$. The data also included the Barium sulfate panel reflectance used as reference for the canopy reflectance factor measurements. This calibrated radiance, which is about $80 - 90\%$ for illumination zenith angle between $10 - \; 65^\circ $, is used to obtain the theoretical calibrated reflectance factor of a perfect Lambertian diffuser under experimental condition. The DOCS reflectance factor is then determine as the ratio of the canopy reflected radiance for a source strength equal to 1 by theoretical calibrated reflected radiance. The DOCS predictions is obtained using N = 23 number of double-Gauss ordinates direction

Figure 8 compares the spectral soybean plant canopy DOCS reflectance factor predictions with experiment measurements for a view zenith sun angle of ${\theta _0} = 35^\circ $. This figure also presents the sensibility analysis of leaf–area index on the DOCS predictions for three values of LAI based on the standard deviation maximal values: ${\tau _L} = \{{2.5,2.9\; and\; 3.3} \}$. The plant canopy is assumed containing single leaves ensemble oriented at ${\theta _L} = 58^\circ $ as considered in most literature studies on phonon transport in plant leaves canopies [6]. For the four view angles considered, ${\theta _v} = 0,15,\; 30\; and\; 60^\circ $, the accuracy between DOCS predictions and experiments is less than 15% in the near infrared reflectance factor plateau for the case with LAI = 3.3, which is in the order of most experimental measurements error. It can be seen that in the near infrared reflectance factor plateau for wavelengths greater that $0.8\mu m$, the relative error between DOCS predictions and experimental measurements increase with LAI for ${\theta _v} = 0\; ,\; 15\; and\; 30^\circ $ whereas it decreases with LAI for ${\theta _v} = 60^\circ $.

 

Fig. 8. Spectral dependences of a single soybeans leaves plant canopy: sensibility of leaf-area index (LAI) standard deviation error on DOCS predictions comparative with Ranson et al. [37] measurements.

Download Full Size | PDF

The effects of the soil reflectance and the leaf angle distribution on radiative transfer in soybean leaves plant canopy are shown in Fig. 9. Two leaves angle distribution functions are considered for the DOCS predictions: the unophile and the single leaf distribution with ${\theta _L} = 58^\circ $. The soil reflectance in the visible and the near infrared took from literatures [23,38] is ${r_s} = 0.1$ and 0.2, respectively. The results shown in Fig. 9 indicate that DOCS predictions using the unophile isotropic leaf distribution are similar to the predictions with the single leaf distribution excepted for view angle ${\theta _v} = 30^\circ $ where a great discrepancy is observed between the two predictions. It can be noted that the soil reflectance affects the DOCS predictions accuracy only in the near infrared reflectance factor plateau. The soil reflectance acts as enlarging the canopy LAI and therefore the DOCS predictions overestimate the experimental measurements. However, the effect of the soil reflectance can be mismatched by decreasing the LAI in the range of the standard deviation error as shown in Fig. 8.

 

Fig. 9. Leaf angle distribution and soil reflectance effects on discrete ordinates characteristics solutions soybeans spectral reflectance factor comparative with Ranson et al. [37] measurements.

Download Full Size | PDF

7. Conclusion

The discrete ordinates characteristics solution proposed initially for photon transport in isotropic participating media is extended to anisotropic plant canopy, where radiative properties depend on the incident radiation direction. The method coupled discrete ordinates angular discretization with the solution of the canopy radiative characteristics variables of first-order differential equations, obtained after eigenvalue decomposition of the matrix resulting from plant canopies directional radiative properties. This work also presents analytical expressions for radiance predictions, which are easy to implement. The method is computationally efficient due to the analytical spatial resolution of the radiative transfer equation. The convergence and accuracy of the method is demonstrated for different plant canopies types especially single leaf angle canopy, and canopy described by five representative leaf angle distribution: erectophile, extremophile, plagiophile, planophile and unophile. Predicted results show that the discrete ordinates characteristics solution using moderate number of ordinates direction compete well with two others literature semi-analytical solutions: the “facile-method” and the analytical discrete ordinates. Comparison with benchmark exact Chandrasekhar literature results demonstrated that the discrete ordinates characteristics solution accurately predicts both directional and hemispherical transmittance and reflectance. Hence, the proposed method can be useful in designing vegetation indexes, carrying out sensitivity analyses, and retrieving information from remote sensing problem.

A. Appendix

A1. Canopy structural parameters

They are three main structural parameters for describing canopy architecture: the vertical leaf area density, the leaf-normal distribution function, and the leaf spatial dispersion functions [39]. The leaf area density, ${u_L}{\;\ }$which represents the vertical profile of a leaf area inside the vegetation canopy is defined as the fraction of the total one-side leaf area, ${\;\ }{A_L},$ per unit canopy volume, ${\vartheta _L}$, at leaf area index (LAI) $z$

(A1)$${u_L} = {A_L}/{\vartheta _L}$$

The vertical profile of a leaf is related to the cumulative leaf area index of optical depth by

(A2)$$\tau = \mathop \int \nolimits_0^z {u_L}({z^{\prime}} )dz^{\prime}$$

Leaf-normal distribution function or leaf normal orientation function, ${g_L}({z,{\Omega _L}} )$, gives the fraction of total leaf area at a height in the canopy of z with a normal within the unit solid angle around direction ${\Omega _L} = ({{\theta_L},{\varphi_L}} )$ , where ${\theta _L}$, and ${\varphi _L}$ are the leaf polar and azimuthal angles, respectively. This canopy height function can be described as the spatial orientation of the leaf in the direction ${\Omega _L}$of the normal to the upward face of the leaf and normalized according to the relation [23]

(A3)$$\frac{1}{{2\pi }}\mathop \int \nolimits_{2\pi }^{} {g_L}({z,{\Omega _L}} )d{\Omega _L} = 1$$

If leaves are azimuthally randomly distributed in the canopy, Eq. (3) reduces to

(A4)$$\mathop \int \nolimits_0^{\pi /2} {g_L}({z,{\theta_L}} )sin{\theta _L}d{\theta _L} = 1$$

Table 6 describes the orientations characteristics and the leaves angle distribution function of common canopy type [1,40,41]. The leaves angle distribution function have been defined to satisfy the normalized condition. It is important to note that for other canopy type than unophile and single leaf angle, the leaf normal orientation function expresses in term of the leaf cosine direction, ${{\mu }_L} = \cos {\theta _L}$, must be divided by $\sqrt {1 - \mu _L^2} $ in order to satisfy the conservation equation [30,33,42]. The leaf normal orientation function is related to the inclination index of foliage area by the relation [1]

(A5)$${\chi _L} = \pm \frac{1}{2}\mathop \int \nolimits_0^{\pi /2} |{sin{\theta_L} - {g_L}({z,{\theta_L}} )} |d{\theta _L}$$

The inclination index of foliage area takes the values 1, 0 and -1 for horizontal, spherical, and vertical leaves normal distribution, respectively. The knowledge of the leaf-normal distribution function is suitable to evaluate the foliage area orientation function, $G(\Omega )$, which is a dimensionless geometric factor representing the fraction of the total leaf area which is perpendicular to the outgoing direction $\Omega = ({\theta ,\varphi } )$ with $\theta \;and\;\varphi $ being the zenith and the azimuth angles, respectively

(A6)$$G({z,\Omega } )= \mathop \int \nolimits_{2\pi }^{} {g_L}({z,\Omega } )|{{\Omega _L} \cdot \Omega } |d{\Omega _L}$$

which is normalized according to the relation [37]

(A7)$$\frac{1}{{4\pi }}\mathop \int \nolimits_{4\pi }^{} G({z,\Omega } )d\Omega = 1$$

For a special case when all the leaves are azimuthally randomly distributed in the canopy, the foliage area orientation function is [23,35]

(A8)$$G({z,\theta } )= \mathop \int \nolimits_0^{\pi /2} {g_L}({z,{\theta_L}} )\psi ({\theta ,{\theta_L}} )sin{\theta _L}d{\theta _L}$$

where

(A9)$$\psi = \left\{ {\begin{array}{l} {|{cos\theta cos{\theta_f}} |,\quad |{ctn\theta ctn{\theta_L}} |> 1\; }\\ {cos\theta cos{\theta_f}[{2co{s^{ - 1}}({ - ctn\theta ctn{\theta_L}} )- 1} ]+ \frac{2}{\pi }sin\theta sin{\theta_f}sin\phi (\theta ),\quad else} \end{array}} \right.$$

It should be noted that if all the leaves normal azimuthally random distributed in the canopy are inclined at a particular constant angle ${\theta _L}$, the foliage area orientation function is given by

(A10)$$G({z,\theta } )= \psi ({\theta ,{\theta_L}} )$$

The leaf spatial dispersion function represents the probability that an arbitrarily photon that penetrated into the canopy along a given direction $\Omega $ will reach the leaf area index without having contact with the elemental leaf sections. The leaf spatial dispersion function can be classified into three main groups [35]: regular leaf spatial dispersion defined by the positive binomial, random leaf spatial dispersion described by the Poisson distribution, and clumped leaf dispersions represented by the negative binomial distribution. These three leaves spatial dispersion function are given, respectively, by

(A11)$$p({z,\Omega } )= \left\{ {\begin{array}{l} {1 + G({z,\Omega } )z/cos\theta \; }\\ {exp[{G({z,\Omega } )z/cos\theta } ]\; }\\ {{{[{1 + G({z,\Omega } )z/cos\theta } ]}^{ - 1}}} \end{array}} \right.$$

Table 6. Common canopy leaf angle distribution function and orientation characteristics [1,40,41]

View Table | View all tables in this article

A2. Canopy radiative characteristics

Photons entering a canopy can be either absorbed or scattered at the interface between the air and the leaf surface [2]. Specular and diffuse reflection of photons are governed by the refractive index differences between the air and the leaf surface. Absorption of the photon energy occurs at different wavelengths due to the leaf predominant pigments. These properties can be classified in three distinct wavelengths regions of interaction: strongly absorption of the radiation by leaves in the visible$\; ({0.4 - 0.7\mu m} )$, radiation mostly multiple scattered in the near infrared$\; (0.7 - 1.35\mu m$), and lower absorption and scattering of radiation by leaves in the mid-infrared $({1.35\; - \; 2.7\mu m} )$ [2]. The main characteristics needed for the photon transport analysis through the canopy are the extinction/scattering coefficients, the scattering albedo and the scattering phase function.

The leaf extinction, ${\sigma _e}$, and scattering coefficients, ${\sigma _\textrm{s}}$, represent the probabilities, per unit path length of travel, that the photon hits or will be scattered by a leaf, respectively [23]. These coefficients are given by

(A12)$${\sigma _{\{{e,s} \}}}(\Omega )= {u_L}\{{G(\Omega ),\omega (\Omega )G(\Omega )} \}$$

where the canopy scattering albedo, ${\omega }$, defines the ratio of the extinction coefficient by the scattering coefficient

(A13)$$\omega (\Omega )= {\sigma _s}(\Omega )/{\sigma _e}(\Omega )$$

The leaf scattering phase function, $f({\Omega^{\prime} \to \Omega ;\; {\Omega _L}} )$, which represents the angular distribution of radiant energy scattered by a leaf can be defined as the fraction of intercepted energy, from photon initially traveling in $\Omega ^{\prime}$, that is scattered into an element of solid angle $\Omega $. The normalization of this phase function when integrating over all scattered photon directions yields a single-scatter leaves albedo according the relation [23]

(A14)$$\mathop \int \nolimits_{4\pi }^{} f({\Omega^{\prime} \to \Omega ;{\Omega _L}} )d\Omega ^{\prime} = {\omega _L}(\Omega )$$

In fact, incident photon may be scattered over on the upper or the lower directions and the single-scatter leaves albedo can be written as

(A15)$${\omega _L}(\Omega )= {r_L}(\Omega )+ {t_L}(\Omega )$$

where ${r_L}\; and\; {t_L}$ of the relative fractions of the intercepted energy which are reflected by the leaf and transmitted on the opposite side of the leaf are defined, respectively, as

(A16)$$\{{{r_L},{t_L}} \}= \mathop \int \nolimits_{2{\pi ^ \pm }}^{} f({\Omega^{\prime} \to \Omega ;{\Omega _L}} )d\Omega ^{\prime}$$

with $2{\pi ^ \pm }$being the upward and downward solid angles on a surface. It should be noted that reflectance from leaves can be Lambertian or diffuse and non-Lambertian or specular. The diffuse part is due to the interaction of radiation with the interior of the leaf, whereas the non-Lambertian part is spread near the specular angle. The reflected, transmitted, or absorbed energy by leaves are complex mechanisms difficult to model since they depend on various factors including leaf cellular structure, leaf pubescence and roughness, leaf morphology and physiology and leaf surface characteristics [2]. Therefore, a simple model which assume that the reflected energy by the leaf and transmitted energy follow a simple cosine distribution law around the normal to the leaves, are generally considered [23,38]. Henceforth, the leaf scattering phase function for this bi –Lambertian scattering model takes the form [23,38]

(A17a)$$f({\Omega^{\prime} \to \Omega ;{\Omega _L}} )= \frac{{{r_L}|{cos\zeta } |}}{\pi },cos\zeta cos{\zeta ^{^{\prime}}} < 0$$

(A17b)$$f({\Omega^{\prime} \to \Omega ;{\Omega _L}} )= \frac{{{t_L}|{cos\zeta } |}}{\pi },cos\zeta cos{\zeta ^{^{\prime}}} < 0$$

with $cos{\zeta } = \Omega \cdot {\Omega _L}$. It should be noted that for the bi –Lambertian scattering model, the constant single-scatter leaves albedo is equal to the canopy scattering albedo

(A18)$$\omega = {\omega _L} = {r_L} + {t_L}$$

The differential scattering coefficient that describes the probability density, per unit path length, that incident photons in the direction $\Omega ^{\prime}$ would be intercepted and then scattered into the direction $\Omega $ is [23]

(A19)$$\frac{1}{{4\pi }}\Phi ({\Omega^{\prime} \to \Omega } )= \frac{{{u_L}(z )}}{\pi }\Gamma ({\Omega^{\prime} \to \Omega } )$$

where $\Gamma ({\Omega^{\prime} \to \Omega } )$ is the area scattering transfer function defined as

(A20)$$\Gamma ({\Omega^{\prime},\Omega } )= \frac{1}{2}\mathop \int \nolimits_{2\pi }^{} |{\Omega^{\prime} \cdot {\Omega _L}} |{g_L}({{\Omega _L}} )f({\Omega^{\prime},\Omega ;{\Omega _L}} )d{\Omega _L}$$

which satisfies the normalized condition [43]

(A21)$$\frac{1}{\pi }\mathop \int \nolimits_{4\pi }^{} \Gamma ({\Omega^{\prime} \to \Omega } )d\Omega = {\omega _L}({\Omega^{\prime}} )G({\Omega^{\prime}} )$$

The differential scattering coefficient can be viewed as the product of the scattering coefficient by the volume scattering phase function. Hence, the volume scattering phase function can be obtained as [23]

(A22)$$P({\Omega^{\prime} \to \Omega } )= \Phi ({\Omega^{\prime} \to \Omega } )/{\sigma _s}(\Omega )$$

which is normalized according to the relation

(A23)$$\frac{1}{{4\pi }}\mathop \int \nolimits_{4\pi }^{} P({\Omega^{\prime} \to \Omega } )d\Omega = 1$$

Combined Eqs. (11), (18) and (21) assuming the bi–Lambertian scattering model gives

(A24)$$P({\Omega^{\prime} \to \Omega } )= 4\Gamma ({\Omega^{\prime} \to \Omega } )/{\omega _L}G({\Omega^{\prime}} )$$

In case where all the leaves are azimuthally randomly distributed in the canopy the area scattering transfer function and the scattering phase function are given, respectively, by [23]

(A25)$$P({\mu^{\prime} \to \mu } )= 4\Gamma ({\mu^{\prime} \to \mu } )/{\omega _L}G({\mu^{\prime}} )$$

(A26)$$\Gamma ({\mu^{\prime} \to \mu } )= \mathop \int \nolimits_0^1 {g_L}(\mu )\Psi ({\mu^{\prime},\mu } )d{\mu _L}$$

where

(A27)$$\begin{array}{l} \Psi ({\mu^{\prime},\mu } )= [{{t_L}{\Psi ^ + }({\mu^{\prime},\mu } )+ {r_L}{\Psi ^ - }({\mu^{\prime},\mu } )} ]\\ {\Psi ^ \pm }({\mu^{\prime},\mu } )= H(\mu )H({ \pm {\mu^{^{\prime}}}} )+ H({ - \mu } )H({ \mp {\mu^{^{\prime}}}} )\\ H(\mu )= \left\{ {\begin{array}{l} {cosf\theta ,\quad ctn\theta ctn{\theta_L} > 1\; }\\ {0,\quad ctn\theta ctn{\theta_L} < - 1}\\ {[{cosf\theta {\phi_\theta } + sinf\theta sin{\phi_\theta }} ]/\pi ,\quad else\; } \end{array}} \right. \end{array}$$

The area scattering transfer function and the G-factor present the following symmetry properties, respectively

(A28)$$\Gamma ({\mu^{\prime} \to \mu } )= \Gamma ({\mu \to \mu^{\prime}} )= \Gamma ({ - \mu^{\prime} \to - \mu } )$$

(A29)$$G(\mu )= G({ - \mu } )$$

Funding

The “Make Our Planet Great Again” Short-Stay French Program (mopga-short-0000000526).

Acknowledgment

Dr. H.T.T. Kamdem acknowledges the University of Poitiers for the invitation and hosting his stay at Institut Pprime.

References

1. J. Ross, The radiation regime and architecture of plant stands (The Hague, 1981)

2. R. B. Myneni and J. Ross, Photon-vegetation interactions: applications in optical remote sensing and plant ecology (Springer-Verlag, 1991).

3. S. A. W. Gerstl and A. Zardecki, “Coupled atmosphere/canopy model for remote sensing pf plant reflectance features,” Appl. Opt. 24(1), 94 (1985). [CrossRef]  

4. S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009). [CrossRef]  

5. H. T. Kamdem Tagne and D. Baillis, “Reduced models for radiative heat transfer analysis through anisotropic fibrous medium,” J. Heat Transfer 132(7), 072703 (2010). [CrossRef]  

6. C. Zhang, A. Kribus, and R. Ben-Zvi, “Effective Radiative Properties of a Cylinder Array,” J. Heat Transfer 124(1), 198–200 (2002). [CrossRef]  

7. A. L. Marshak, “The effect of the hot spot on the transport equation in plant canopies,” J. Quant. Spectrosc. Radiat. Transfer 42(6), 615–630 (1989). [CrossRef]  

8. B. D. Ganapol, “Radiative transfer in dense plant canopies with azimuthal symmetry,” Transp. Theory Stat. Phys. 18(5-6), 475–491 (1989). [CrossRef]  

9. A. A. Kokhanovsky, Aerosol Optics:Light Absorption and Scattering by Particles in the Atmosphere (Praxis Publishing Ltd, 2008).

10. W. Zdunkowski, T. Trautmann, and A. Bott, Radiation in the Atmosphere: A Course in Theoretical Meteorology (Cambridge University Press, 2007).

11. L. A. Dombrovsky and D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach (Begell House, 2010).

12. M.F. Modest, Radiative Heat Transfer, 3rd edition, Acad. Press, New York, 2013.

13. J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer. 6th ed. (Taylor & Francis, CRC Press, 2015).

14. M. Lazard, S. Andre, and D. Maillet, “Transient coupled radiative-conductive heat transfer in a gray planar medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 69(1), 23–33 (2001). [CrossRef]  

15. F. Zhang, Z. Shen, J. Li, X. Zhou, and L. Ma, “Analytical delta-four-stream doubling-adding method for radiative transfer parameterizations,” J. Atmos. Sci. 70(3), 794–808 (2013). [CrossRef]  

16. F. Zhang, K. Wu, P. Liu, X. Jing, and J. Li, “Accounting for Gaussian quadrature in four-stream radiative transfer algorithms,” J. Quant. Spectrosc. Radiat. Transfer 192, 1–13 (2017). [CrossRef]  

17. Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019). [CrossRef]  

18. F. Zhang, K. Wu, J. Li, H. Zhang, and S. Hu, “Radiative transfer in the region with solar and infrared spectra overlap,” J. Quant. Spectrosc. Radiat. Transfer 219, 366–378 (2018). [CrossRef]  

19. R. Tapimo, H. T. T. Kamdem, and D. Yemele, “Discrete spherical harmonics method for radiative transfer in scalar planar inhomogeneous atmosphere,” J. Opt. Soc. Am. A 35(7), 1081 (2018). [CrossRef]  

20. V. S. Antyufeev and A. L. Marshak, “Monte Carlo Method and Transport in Plant Canopies,” Remote Sens. Environ. 31(3), 183–191 (1990). [CrossRef]  

21. K. Cooper, J. A. Smith, and D. Pitts, “Reflectance of a vegetation canopy using the adding method,” Appl. Opt. 21(22), 4112 (1982). [CrossRef]  

22. R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Reflectance of a soybean canopy using the method of successive orders of scattering (SOSA),” Agricultural and Forest Meteorology 40(1), 71–87 (1987). [CrossRef]  

23. J. K. Shultis and R. B. Myneni, “Radiative transfer in vegetation canopies with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 39(2), 115–129 (1988). [CrossRef]  

24. T. Nilson and A. Kuusk, “A Reflectance Model for the Homogeneous Plant Canopy and Its Inversion,” Remote Sens. Environ. 27(2), 157–167 (1989). [CrossRef]  

25. F. Zhang, Y. Lei, J.-R. Yan, J.-Q. Zhao, J. Li, and Q. Dai, “A new parameterization of canopy radiative transfer for land surface radiative models,” Adv. Atmos. Sci. 34(5), 613–622 (2017). [CrossRef]  

26. H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017). [CrossRef]  

27. N. S. Goel, “Models of vegetation canopy reflectance and their use in estimation of biophysical parameters from reflectance data,” Remote Sensing Reviews 4(1), 1–212 (1988). [CrossRef]  

28. S. Liang and A. H. Strahler, “An analytical BRDF model of canopy radiative transfer and its inversion,” IEEE Transactions on Geoscience and remote sensing 31(5), 1081–1092 (1993). [CrossRef]  

29. B. D. Ganapol, “The determination of the reflected intensity for a single leaf angle plant canopy via Chandrasekhar’s method,” Transp. Theory Stat. Phys. 19(3-5), 251–271 (1990). [CrossRef]  

30. B. D. Ganapol and R. B. Myneni, “The FN method for the one-angle radiative transfer transfer equation applied to plant canopies,” Remote Sens. Environ. 39(3), 213–231 (1992). [CrossRef]  

31. R. Furfaro and B. D. Ganapol, “Spectral Theory for Photon Transport in Dense Vegetation Media: Caseology for the Canopy Equation,” Transp. Theory Stat. Phys. 36(1-3), 107–135 (2007). [CrossRef]  

32. P. Picca, R. Furfaro, and B. D. Ganopol, “On radiative transfer in dense vegetation canopies,” Transp. Theory Stat. Phys. 41(3-4), 223–244 (2012). [CrossRef]  

33. P. Picca and R. Furfaro, “Analytical discrete ordinate method for radiative transfer in dense vegetation canopies,” J. Quant. Spectrosc. Radiat. Transfer 118, 60–69 (2013). [CrossRef]  

34. H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The Discrete Ordinates Characteristics Solution to the One-Dimensional Radiative Transfer Equation,” J. Comput. Theor. Transp. 46(5), 346–365 (2017). [CrossRef]  

35. L. G. Ymeli and T. T. H. Kamdem, “Hyperbolic conduction–radiation in participating and inhomogeneous slab with double spherical harmonics and lattice Boltzmann methods,” ASME J. Heat Transf. 139(4), 042703 (2017). [CrossRef]  

36. S. Chandrasekhar, Radiative Transfer (Dover Publication, Inc., 1960).

37. K. J. Ranson, L. L. Biehl, and C. S. T. Daughtry, “Soybean canopy reflectance modeling data sets,” LARS Tech. Report 071584, Purdue Univ., W. Lafayette, IN (1984).

38. N. Gobron, B. Pinty, M. M. Verstraete, and Y. Govaerts, “A semidiscrete model for the scattering of light by vegetation,” J. Geophys. Res.: Atmos. 102(D8), 9431–9446 (1997). [CrossRef]  

39. R.B. Myneni, V.P. Gutschick, G. Asrar, and E.T. Kanemasu, “Photon transport in vegetation canopies with anisotropic scattering part I. Scattering phase functions in one angle,” Agric. For. Meteorol. 42(1), 1–16 (1988). [CrossRef]  

40. W.-M. Wang, Z.-L. Li, and H.-B. Su, “Comparison of leaf angle distribution functions: Effects on extinction coefficient and fraction of sunlit foliage,” Agric. For. Meteorol. 143(1-2), 106–122 (2007). [CrossRef]  

41. S. Otto and T. Trautmann, “A note on G-functions within the scope of radiative transfer in turbid vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 109(17-18), 2813–2819 (2008). [CrossRef]  

42. B. D. Ganapol, L. F. Johnson, C. A. Hlavka, D. L. Peterson, and B. Bond, “LCM2: A coupled leaf/cnopy radiative transfer model,” Remote Sens. Environ. 70(2), 153–166 (1999). [CrossRef]  

43. R. B. Myneni, J. Ross, and G. Asrar, “A review on the theory of photon transport in leaf canopies,” Agric. For. Meteorol. 45(1-2), 1–153 (1989). [CrossRef]