We developed a 3-D plant canopy radiative transfer model for simulating the canopy-scale directional SIF radiance (FLiES-SIF version 1.0; ). Figure a shows the overall framework of the FLiES family modules. The aim of FLiES is to consider the impact of landscape heterogeneity on the radiative processes and determine how this links to the canopy energy, water, and carbon exchange. One of the important aspects of modeling is to make the modules as computationally efficient as possible: most of the FLiES modules, including the newly proposed FLiES-SIF, can be easily run on moderate-level personal computers with relatively modest memory demands and public software (GNU gfortran, gcc, and R). The model development was initiated using 3-D radiative transfer modeling in the shortwave domains . A 1-D version of the atmospheric radiative transfer model, MCARaTS , was incorporated to simulate the atmosphere–forest radiation interaction. A longwave radiative transfer module was then added, together with energy balance and plant physiology modules (Fig. a; ). All these modules are related to the radiation emitted in the Stefan–Boltzmann law of the Sun, by the Earth's surface, and by atmospheric media.

The current FLiES-SIF work adds a radiation interaction module for the induced radiation emitted from leaf pigments, and describes how to combine the forest structure information and leaf physiology models with the fluorescence radiative transfer module (Fig. a). The FLiES-SIF model shares some key aspects of numerical schemes in FLiES: it employs a spatially explicit forest landscape (Sect. ) and is based on a Monte Carlo ray-tracing approach (Sect. –). Analogous to the modeling in previous FLiES modules, FLiES-SIF employs a Monte Carlo sampling scheme, where photon-tracing sequences represent the integration form of the radiative transfer equation, or the so-called Neumann series . In such modeling, the photon path lengths and scattered directions are determined by random numbers and probability distribution functions such as the Lambert–Beer exponential function and the scattering phase function. The scattering and re-absorption of emitted fluorescence light must also be considered to identify the relationship between the fluorescence emitted by the chloroplasts and the top-of-canopy outgoing fluorescence radiance . Several recent studies have worked on the quantification of the impact and modeling of scattering and absorption effects from the leaf scale (e.g., ) to the canopy scale (e.g., ). Multiple scatterings and re-absorption among leaves, trunks, and soil background can be numerically simulated using unbiased and efficient approaches . The performance and reliability of FLiES for simulating light transmittance through a canopy and bidirectional reflectance factors have been extensively investigated in previous studies . As a default setting, FLiES-SIF version 1.0 simulates the bidirectional SIF radiance at the top of the canopy, but the simulation codes can easily be extended to simulate SIF at any height level within the plant canopy and for the spatial mapping of SIF radiance at the top of the canopy.

The forest landscapes employed in FLiES-SIF consist of simple geometric objects such as cones, spheroids, and cylinders, as in other 3-D models (e.g., DART and FluorFLIGHT) (Fig. ). The volume domains inside the crown objects can be further split into multiple domains to realize the spatial distribution of the leaf area and woody area densities. This approach is useful in some aspects because (1) it can establish realistic plant canopies in which the majority of leaves are distributed in the outer and upper parts of the crowns (see Fig. ), and (2) this approach is simple and computationally efficient. The insides of the crown volumes are assumed to be turbid media, where the light attenuation follows the Lambert–Beer exponential law as a function of leaf/woody area densities and leaf inclination angles. The conventional turbid medium approach assumes that the leaves are randomly distributed in space. In the FLiES modules, including FLiES-SIF, a spatially anisotropic arrangement of leaves (the so-called clumping effect) is modeled using re-collision probability theory , which is particularly important for the shoot-scale clumping of needleleaf. More details can be found in the FLiES user manual . Note that FLiES has a module for the voxel representation of the forest landscape , but this module is not currently incorporated into FLiES-SIF version 1.0.

The realization of individual tree objects has some degree of freedom in the characterization of leaf and woody densities in crowns. In the sensitivity analysis described in the next chapter, we consider the following forest architectures. The crown objects are separated into two domains, namely the outer leafy-crown and inner woody domains; the outer and inner domains are filled with 100 % leaves and 100 % wood, respectively. The height and diameter of the inner woody domain is set to be half that of the outer domain (Fig. ). Stems are represented by solid cylinders. The individual tree dimensions can be defined differently. The landscape size used in this study is 100m×100m. To create the virtual forest canopy for SIF simulations, it is necessary to determine all of the tree positions in the forest. If ground-based tree census data exist, they can be used to create the virtual forest canopy. The virtual forest canopies can also be reconstructed using a statistical approach . Assuming that the spatial arrangement of the trees follows a Poisson or Neyman distribution, the individual tree positions are determined by these statistical functions and random numbers.

The simulation flow of the spectral SIF calculation is shown in Fig. b. The FLiES-SIF model requires four major inputs, namely geometry data, meteorological data, forest stand data, and optical data for leaves and other elements (e.g., soil background). The geometry data specify the Sun and sensor view directions (zenith and azimuth angles). The meteorological data are used in the pre-computation of fluorescence yield. The incident total and diffuse photosynthetically active radiation (PAR) data from the top of forest canopies are also used in the canopy radiative transfer module. If no PAR observations are available, these data can be calculated by the FLiES atmosphere module (1-D MCARaTS; see Fig. a).

Before running the Monte Carlo ray-tracing processes in the forest canopy, the forest structures (leaf area density and leaf voxel look-up table) and leaf-level physiology (fluorescence module) are computed. In FLiES-SIF, landscape-scale LAI is an input variable. The model requires the leaf area density of individual canopy volumes. FLiES-SIF computes the leaf area density from a given forest landscape and LAI data. Fluorescence ray tracing also requires information about the leaf-level Sun-induced fluorescence yield and its spectral composition. This information is computed by the leaf photosynthesis and fluorescence module. These modules are described in the following subsections (Sect.  and ).

Once the forest structure and leaf fluorescence yield have been computed, Monte Carlo ray tracing is performed in the broad PAR domain to determine the PAR absorbed in the forest landscape (APARc in Sect. ). The SIF radiance is then simulated on an individual wavelength basis. Details of the radiative transfer formulation and ray-tracing algorithm are summarized in the following sections (Sect. –).

In FLiES-SIF, 3-D forest landscapes are reconstructed using the geometric tree objects described in Sect.  and Fig. -upper). The photon tracing starts from an arbitrary position v0=(x,y,z) within the leafy-canopy volume. This position is determined by three random numbers corresponding to x, y, and z. When the canopy landscape is sparse, the majority of randomly determined positions v0 will be outside of the leafy-canopy space, which means a large number of trial runs will be required to determine an appropriate position v0. To reduce the computation time, regularly placed leafy-canopy voxel tables are extracted to determine where to start the SIF emission and subsequent photon tracing (Fig. ). In FLiES-SIF version 1.0, the leafy-canopy voxel information is saved in a look-up table (Fig. ). The voxel information in the table contains lower and upper corner positions (x, y, z, and x+dx, y+dy, z+dz), the leaf area density (LAD) of the voxel, and the sunlit leaf area density (LADsun). The size of each voxel is currently 1 m3. Note that the extracted leafy voxels are not always completely filled with canopy geometry; canopy edge voxels only partially contain the leafy-canopy geometry. In addition, tree canopy geometries contain branch domains. Thus, even if the voxel is completely inside the canopy geometry, there may be some domains that do not contain leaves.

In FLiES-SIF version 1.0, the leaf-level Sun-induced fluorescence yield (hereafter SIF yield; ϕf) is pre-computed and stored in a look-up table prior to ray tracing. The SIF yield ϕf is computed by the model of and Farquhar's photosynthesis model under various environmental conditions, including APAR. The actual photosynthesis can not be determined without stomata models (e.g., ). The leaf temperature is also dependent on photosynthesis and stomata regulations. These interrelations are solved by iterating the energy balance, stomata, and photosynthesis equations. The CANOAK-FLiES module (Fig. a, ) can handle such leaf-level coupled physiology phenomena, but this would require more input variables and increase the computational load. Thus, in the current version of FLiES-SIF, we adapted the following assumptions to obtain reasonable photosynthesis simulation results. First, the leaf temperature was assumed to be the same as the surface air temperature. This is usually acceptable, except in very dry conditions when the stomata are almost closed in daytime. The other assumption concerns the stomata modeling. The FLiES-SIF module does not explicitly use the stomata model. Rather, the consequences of the stomata activity, i.e., the downregulation of the intercellular partial CO2 pressure (ipCO2), were modeled as a function of the vapor pressure deficit (VPD). We used the experimental relationships measured by , who investigated the relationships between ipCO2 and VPD in three tree species (pine, spruce, and aspen; see Fig. 10 of ). If we simulate SIF over such species, the regression lines given by can be used. For other species, we created a simplified function based on the relationship derived by : 1aipCO2/apCO2=0.80<VPDkPa≤1.01bipCO2/apCO2=-0.2VPD+1.01.0<VPDkPa≤3.51cipCO2/apCO2=0.33.5<VPDkPa, where apCO2 denotes the ambient partial CO2 pressure.

To simulate the spectral SIF, the spectral composition of SIF must be known. Our approach is similar to that used in the SCOPE model . We derived the spectral composition from the FluorMODleaf model . The calculated leaf-level spectral SIF radiance variations given by FluorMODleaf were normalized to determine the fraction of SIF at wavelength λ, fs (mWm-2sr-1), with respect to the broadband SIF (Wm-2). The standard setting and variables are described in Sect. . That is, we only used the fraction of spectral composition from the FluorMODleaf model. The radiance was then determined from APAR and ϕf, which varies with environmental conditions and leaf traits such as the maximum carboxylation capacity, Vcmax, used in the photosynthesis model.

The attenuation of SIF in the view direction Ωσ is calculated by the attenuation function exp⁡-τV. When the hotspot effect is not considered, the attenuation function is expressed using the plant canopy gap fraction theory: 6exp⁡-τσ=exp⁡-∑iuiγiGσ,isi, where ui, si, Gσ,i, and γi are the leaf area density, path length, mean leaf projection area, and clumping index of the ith tree. They are aggregated over the trees located in the light path between the emission point to the top of canopy in the view direction, respectively. The path length, s, is a sum of canopy paths that penetrates through crown objects.

The mean leaf projection area G is a function of the leaf inclination angle distribution function gL and an arbitrary direction Ωσ (such as the Sun direction ΩS or view direction ΩV): 7Gσ:=GΩσ=12π∫2πgLΩLΩL⋅ΩσdΩL. Generally, the clumping index contains various nonrandom scales of spatial leaf distributions, from the shoot to the landscape scale. Because FLiES-SIF version 1.0 employs explicit tree crown landscapes, clumping larger than the crown scale need not be considered. However, the crown volumes are expressed as turbid media: if the leaves are not randomly distributed in the crown object, e.g., shoot-scale clumping , attenuation must be corrected according to the shoot-scale clumping index. In FLiES-SIF version 1.0, the shoot-scale clumping index is estimated by the spherically averaged shoot silhouette area . Details on how shoot-scale clumping is incorporated can be found in a previous report . The hotspot effect refers to the strong illumination near the solar direction (ΩV≈-ΩS). When the hotspot effect is non-negligible, the modified optical thickness τ′ is expressed as 8τ′=τH, where H is a hotspot function expressed by the Hapke model , which is used in the framework of the FLiES model : 9HΩL,Ωj≃1-11+1hΩL,Ωjtan⁡(αj2)10hΩL,Ωj≃ul2GΩL+GΩj2, where Ωj, l, and αj indicate the incident direction after the jth scattering, the radius of the disk-shaped flat leaves, and the scattering angle (αj=cos⁡-1ΩV⋅Ωj), respectively.

The leaf-level SIF emission weight w0 can be calculated from the SIF yield ϕf and APAR on the leaf surface (APARL): 11w0=fsϕfAPARL, where fs is the fraction of SIF at wavelength λ (mWm-2sr-1) with respect to the broadband SIF (Wm-2). Thus, fs is a function of wavelength. The SIF yield ϕf is a function of APARL and various environmental and leaf trait variables such as ambient air temperature, humidity, CO2 concentration, and carboxylation capacity . In FLiES-SIF version 1.0, ϕf is read from a look-up table across a wide range of APARL, which should be pre-computed by the leaf-level SIF yield models.

The exact computation of APARL under the angular dependency of PAR can be performed by backward ray tracing at the given position of a leaf, but this approach is time consuming. For more efficient simulations, the values of APARL for sunlit and shaded leaves are approximated as the product of the incident-diffuse PAR and the attenuation function exp⁡-τs integrated over the upper hemisphere: 12APARL=1-ωPARPARdirΩS⋅ΩL+PARdif1π∫02π∫0π2exp⁡-τsθ,ϕsin⁡θdθdϕifv0∈Vsun11-ωPARPARdif1π∫02π∫0π2exp⁡-τsθ,ϕsin⁡θdθdϕifv0∈Vshade, where PARdir and PARdif denote the incident direct and diffuse PAR, respectively, ωPAR is the average single-scattering albedo in the PAR spectral domain (400–700 nm), and ωPAR is the sum of the leaf reflectance rPAR and transmittance tPAR in the PAR domain (ωPAR=rPAR+tPAR). This equation assumes that diffuse PAR is isotropic over the sky and neglects direct PAR scattered within the plant canopy and soil background. Thus, APARL may be underestimated when the background reflectance is high, such as in the case of snow cover. To further reduce the computation time, the hemispherical integration of the attenuation function is approximated by an average of the limited-angle samplings. Details of the computation method are given in Sect. .

The phase function for SIF emissions Pf gives the fraction of SIF emitted in the view direction ΩV. Similar to the scattering phase function for the reflection of solar illumination, Pf can be determined by the following equations: 13PfΩL,ΩV=fadaΩL⋅ΩVifΩL⋅ΩSΩL⋅ΩV>0fabaΩL⋅ΩVifΩL⋅ΩSΩL⋅ΩV≤0, where fada and faba are the fraction of SIF emissions from the adaxial and abaxial sides of a leaf; fada+faba=1. Note that, in our definition, we have assumed that illumination by solar beams is always on the adaxial side of a leaf.

The position v0=(x,y,z) from which SIF emission occurs within a leafy-canopy domain is determined by random numbers. The position v0 is determined as follows. First, an arbitrary voxel is chosen at random from the voxel table (Fig. ). The exact position (x,y,z) within a selected voxel is then determined by three random numbers (Rx, Ry and Rz; R∈[0,1]): 16ax=xl+Rxdx16by=yl+Rydy16cz=zl+Rzdz, where vl=(xl,yl,zl) denotes the position of the lower corner of the selected voxel. If the selected voxel is an edge voxel or contains branch domains, the randomly determined position v0 may be outside the leafy canopy. Therefore, the position v0 is checked to determine whether it is in the leafy domain. If the position is outside the leafy domain, the program generates a new random number and selects another voxel. This procedure continues until the leafy-canopy position v0 is obtained.

After position v0 has been determined, the leaf properties at the selected position are determined. Two leaf properties are required to continue the computation of the SIF emission: the leaf illumination status (sunlit or shaded) and the leaf surface normal vector ΩL=θL,ϕL. The sunlit leaf area fraction Psun at v0 is computed using the interception of direct sunlight: 17Psun=1GSlim⁡ΔL→0exp⁡-GSγLp-exp⁡-GSγLp+ΔLpΔL=γexp⁡-GSγLp, where Lp is the cumulative LAI at v0 along the path of the sunlight and GS is a mean leaf projection area defined in Eq. (). The leaf illumination status (sunlit or shaded) is then determined by a random number R: 18R≤Psun→sunlit leafR>Psun→shade leaf. The leaf surface normal vector ΩL is also required because the leaf-level SIF emission is related to APAR at the leaf surface (APARL). APARL is computed from the cosine of the sunlight and leaf normal angles. Assuming the leaves are randomly distributed, the azimuthal angle of the leaf surface normal ϕL can be determined by 19ϕL=2πR. For a given leaf angle distribution function gL:=gθL, the zenith angle of the leaf surface normal θL can be determined by the rejection method. In the first step, θL is calculated using a random number: 20θL=π2R. Then, θL is further evaluated using gL: 21R≤gLsin⁡θL→selectR>gLsin⁡θL→reject. If θL is rejected by the abovementioned criteria in Eq. (), the program returns to Eq. () and calculates another θL. In Eq. (), the evaluation function is a form of leaf angle distribution function multiplied by a sine value. This sine comes from the Jacobian of the polar coordinate and is necessary because gL is defined in polar coordinates.

Once the position v0 and leaf properties have been determined, the leaf-level SIF emission w0 and the direct SIF radiance (Idir_sun and Idir_shade) can be computed using the equations derived in Sect.  and . The calculation of w0 includes the spherical integration of the attenuation function (Eqs.  and ), which is time consuming. Thus, FLiES-SIF version 1.0 approximates this spherical integration by taking the average of five directions (θ,ϕ)=(0,0∘), (60, 0∘), (60, 90∘), (60, 180∘), and (60, 270∘). We tested the performance of this five-angle assumption by comparing with 10∘ interval samplings (9 zenith and 36 azimuth angles is equal to 324 angle samplings). When the attenuation functions were computed by these two angle samplings at 104 randomly selected positions in the forest landscapes used in the sensitivity analysis in Sect. 3, the mean absolute error of this approximation was 14.6 % (N=10000). Finally, Idir_sun and Idir_shade are calculated by the local estimation method using Eqs. () and () .

Direct SIF radiance in the view direction ΩV is determined by procedure C. The multiple scattering contribution is further evaluated by photon tracing. To start the photon tracing, the emission direction Ωθ,ϕ is calculated using two random numbers and the leaf surface normal vector ΩL=θL,ϕL. Assuming that the SIF emission is bi-Lambertian on the leaf surface, the zenith and azimuthal angles relative to the leaf normal (α, β) are determined by 22α=cos⁡-1R23β=2πR. The scattering direction Ωθ,ϕ in the Cartesian coordinate system is then calculated by a coordinate transformation from (α, β) to (θ, ϕ).

Figure a shows the total SIF radiance for wavelengths between 650 and 850 nm. These figures indicate that the SIF radiance shows a strong peak near the Sun direction over the whole wavelength range, although the SZA value, which exhibits the maximum SIF, varies according to the wavelength. In the visible red region, the SIF radiance reaches an extremum at a lower SZA than in the near-infrared region. Regardless of wavelength, the angular dependency of SIF exhibits similar patterns: in the direction of forward emission (VZA>0), SIF increases with an increase in VZA and sharp strong peaks appear around the Sun direction (the hotspot effect). In the backward direction (VZA<0), the SIF decreases with an increase in |VZA| and attains a minimum at around -35 to -50∘, before increasing with |VZA|. Although the general angular patterns are similar across the whole wavelength range, the strength of the hotspot peak in the forward direction and the minimum SIF in the backward direction vary slightly with the wavelength.

To analyze the dependency of VZA in more detail, we explored the influence of VZA on three terms in Eq. (), namely the direct SIF from the radiance of sunlit and shaded leaves (Idir_sun, Idir_shade) and the scattered radiance (Ims_sun+Ims_shade), as well as the total SIF radiance (I). The simulated SIF shows distinct angular features for each SIF component (Idir_sun, Idir_shade, Ims_sun+Ims_shade). Figure  shows the dependence on VZA of the SIF components when LAI=3.0 for a wavelength of 760 nm. Idir_sun has a strong peak near the Sun direction because of the hotspot effect, whereas angular changes of Idir_sun in other domains are minor. In contrast, Ims_sun+Ims_shade exhibits bowl-like shapes (Fig. d), which contributes to the enhancement of total SIF at higher angles. In the FLiES-SIF model framework, SIF radiance is computed by collecting the contribution factor (Eqs.  and ) from the attenuation function, weight of photons, and phase function. Among those factors, the drastic changes in the optical thickness of the attenuation function (Eqs. –) contributed the most to the hotspot in Isun_dir. The attenuation function displays a strong peak around the Sun direction because of the hotspot parameter (H in Eq. ). When αj is sufficiently large and the hotspot effect is marginal, the attenuation function is determined by the forest structure (such as LAI and leaf angle density). Away from the Sun direction, the SIF radiance gradually decreases or increases slightly. This angular feature (VZA) is influenced by the initial photon weight and phase function through the dependency on the leaf surface normal: the initial photon weight is calculated as the inner product between the leaf angle and the Sun direction. The influence of SZA on the phase function is greater than that on the initial photon weight. The other two components (Idir_shade, Ims_sun+Ims_shade) contribute to the total SIF increase in higher angular domains. In addition, Idir_shade makes a slightly larger contribution in the backward direction, because shaded leaves tend to be more aligned with the backward direction. The shaded leaves only absorb diffuse sky radiation, so the relative magnitude of Idir_shade with respect to Idir_sun greatly depends on the fraction of diffuse radiation. The contributions of these three components in four different Sun angles are presented in Fig. . These partitions vary with the fraction of incoming diffuse radiation, optical properties (leaf reflectance and transmittance, woody and soil reflectance), and the leaf area.

Because SIF radiance is greatly affected by the APAR of the leaves, the angular behavior of APAR is essential in understanding the numerical computation of SIF emissions. In the FLiES-SIF model, the SIF radiance is first computed under the apparent APAR (APARapp) conditions (Sect. ) and then adjusted by multiplying by the ratio of APARc to APARapp (Eq. ). The simulated angular patterns indicate that APARc increases with an increase in SZA (Fig. b). The increase in APARc with respect to SZA corresponds to the increase in the photon pathlength inside the forest canopy. As SZA increases, more photons are likely to hit leaves before they pass through the canopy layers. In contrast, APARapp decreases as SZA decreases (Fig. a). This is because APARapp is related to the fraction of sunlit leaves. As described in simulation procedure C in Sect. , the photon tracing is initiated from either sunlit or shaded leaves at randomly selected positions. As the LAI along the photon path (Lp) increases, the gap fraction Psun becomes smaller (Eq. ). As a result, shaded leaves are more likely to be selected in the random process in Eq. (). In other words, as the fraction of shaded leaves increases, the amount of energy in the simulated system decreases. In the Monte Carlo simulations, the statistical accuracy of the simulated variables depends on the number of photon samplings. The decrease in APARapp does not affect the simulated accuracy of the total SIF radiance; however, it does affect the individual components in Eq. (), which means the statistical accuracy of Idir_sun decreases as SZA increases. Depending on the target sampling variables to be simulated, the number of photons should be determined (i.e., more photons may be necessary to investigate the behavior of Idir_sun in cases where the sunlit leaf fraction is low).

Figure b shows the sensitivity of total SIF radiance to LAI for wavelengths of 650–850 nm. The simulated SIF increases with LAI and then becomes saturated over the whole wavelength range, although the speed of saturation varies with the wavelength. In the visible domain, the simulated SIF becomes saturated when LAI=2. In the near-infrared domain, the simulated SIF is not saturated at higher LAI values, indicating that SIF is more sensitive to LAI in the near-infrared domain. To analyze the dependency on LAI, we explored the influence of LAI on three terms in Eq. (), namely Idir_sun, Idir_shade, and Ims_sun+Ims_shade, as well as the total SIF radiance (I) (forward direction in Fig.  and backward direction in Fig. ). In our simulation scenarios, Idir_sun contributed about 54 % of total SIF radiance when LAI=3, VZA=10∘, and SZA=20∘. Idir_hade and Ims_sun+Ims_shade contributed 7 % and 39 %, respectively (Fig. ). Figures  and show that the individual SIF components respond differently to the LAI.

The LAI dependency of direct SIF radiance from sunlit leaves is influenced by the hotspot function and the magnitude of VZA (Figs. b and b). Generally, the SIF radiance emitted from sunlit leaves increases and then saturates as LAI increases, because the number of sunlit leaves also increases and becomes saturated, although the fraction of sunlit leaves decreases (Fig. c). However, in terms of simulated SIF radiance, there are ranges of LAI in which SIF radiance decreases with an increase in LAI. In these regions, the decrease in SIF radiance is caused by the attenuation of SIF radiance in the canopy. The magnitude of this attenuation depends on both the hotspot function and VZA. The hotspot function (i.e., the angle αj in Eq. ) has a major influence on simulated direct SIF radiance from sunlit leaves. The SIF radiance increases and then becomes saturated without decreasing when αj is equal to 0, because the rate of decrease in Idir_sun′ becomes small when τ=0. Additionally, smaller values of αj produce a smaller rate of decrease in Idir_sun′ with respect to increases in LAI through the hotspot effect. The magnitude of VZA (i.e., |VZA|) also influences the simulated SIF radiance. Generally, larger LAI values lead to a decrease in the attenuation of SIF radiation from sunlit leaves in the canopy when VZA is positive, because most sunlit leaves inhabit the canopy surface. However, the attenuation of SIF radiation in other canopies increases with |VZA| because of the increase in the pathlength to the canopy boundary when passing through other canopies. The influence of |VZA| and LAI is prominent in negative VZA directions. In this case, the decrease in SIF radiance with an increase in LAI becomes significant because of the SIF emitted through the local canopy to the view point, and the attenuation in the local canopy (and in other canopies) increases with LAI. Thus, the increase in pathlength as |VZA| increases significantly affects Idir_sun′ in the view direction.

The fraction of shaded leaves has a major influence on SIF radiance. SIF increases and then becomes saturated without decreasing when VZA is negative (Figs. c and c). This variation in SIF is caused by an increase in the fraction of shaded leaves, because the rate of increase in the fraction is larger than the rate of decrease in Idir_shade′. In contrast, the rate of decrease in Idir_shade′ becomes greater than the rate of increase in the fraction of shaded leaves when VZA is positive. In this region, the expectation of the pathlength to the view point is larger than for negative VZA, because the canopy surface is covered with sunlit leaves. This increase in optical thickness, which depends on the pathlength, has a major effect on Idir_shade′ in the LAI range where ψ rapidly decreases with any increase in τ′.

The scattered SIF radiance refers to the sum of the scattered radiance from sunlit and shaded leaves, Ims′=Ims_sun′+Ims_shade′, in our model. The LAI dependency with respect to view direction on the scattered SIF radiance is in contrast to the direct radiance from shaded leaves (Figs. d and d). When VZA is positive, the SIF radiance increases and then becomes saturated without decreasing. The pathlength from sunlit leaves to the population boundary in the view direction has a major influence on simulated scattered SIF radiance. As previously explained (Sect. ), the surface of the canopy is covered by sunlit leaves, which provide a large photon weight to scattered photons, in the positive VZA direction. When LAI is large, the decrease in Ims′ with an increase in LAI becomes vanishingly small. This is because the scattered radiation from high-weight photons reaches the view point with little attenuation. Larger values of LAI lead to shorter scattering path lengths and fewer scatterings, so the photon weight wj is larger. Additionally, the pathlength between sunlit leaves and the boundary of the canopy is nearly constant, irrespective of LAI variation.

The simulated SIF radiance therefore becomes larger than the radiance in the negative VZA direction. Actually, the expectation of the product of w and exp⁡(-τ′) is larger than when VZA is negative (Fig. ). In contrast, with an increase in LAI, the SIF radiance decreases and becomes saturated after increasing because of the increase in τ′ from sunlit leaves. This is for a similar reason as for Idir_shade′ when SZA is negative.

Figure  shows APAR and the fraction of sunlit leaves as a function of LAI. APARc increases with an increase in LAI and becomes saturated at around LAI=2. APARapp and the fraction of sunlit leaves decrease when LAI<2. The increase in APARc and the decrease in APARapp are more abrupt than the SIF increase with respect to LAI. This is because APAR is the visible light where the absorption of green leaves is high (∼0.9). Thus, the APARc saturation curve has similar patterns of visible SIF radiance. At higher LAI, the fraction of sunlit leaves is low and APARapp decreases. The statistical accuracy of Idir_sun′ becomes drastically lower as APARapp and the fraction of sunlit leaves decrease. Accurate simulations of Idir_sun′ require an increased number of photons to be traced.