## 1. Introduction

The absorption of solar radiation on the land surface is important for determining energy, water, and carbon balances. How vegetation contributes to this absorption is an important question. For modeling, three components of this process need to be calculated, that is, reflected, absorbed, and transmitted radiation, as characterized by the parameters albedo, canopy absorption, and ground absorption, respectively. Albedo is most readily observed by the remote sensing community (Schaaf et al. 2002) and has already been used as a constraint for land surface models (Lawrence and Chase 2007). However, the other two components of the incident radiation (i.e., the absorption by the canopy and ground) are also very important and need to be determined. Climate models usually divide the canopy within a grid into different tiles for plant functional types (PFTs) and bare soil. Two approaches have been used for determining each radiation component, that is, either a lookup table that requires good prior knowledge, such as in the Biosphere–Atmosphere Transfer Scheme (BATS) model (Dickinson et al. 1986), or a one-dimensional (1D) physical process model. Three major types of 1D models were summarized by Wang (2003); the most commonly used is the two-stream model (Dickinson 1983; Sellers 1985; Sellers et al. 1996; Bonan 1996; Dai et al. 2004). The 1D models usually assume that each PFT is isolated from any other and is 100% homogeneously distributed within its land fractional coverage *f*_{c} in the grid and that the properties of the three radiation components are just the average of those of each PFT weighted by the *f*_{c} values.

The homogeneous representation of canopy structure, as noted above, is unrealistic in most cases. For example, in savannas or other sparsely vegetated areas, canopies are usually widely spaced with gaps in between. Another example is the boreal region, where multiple layers (trees, shrubs, and grasses) at different vertical levels may coexist. More direct evidence of the heterogeneity of canopies is seen in the Moderate Resolution Imaging Spectroradiometer (MODIS) Vegetation Continuous Fields (VCF) products (Hansen et al. 2003), which contain proportional estimates for three cover types (trees, herbaceous, and bare ground) at 500 m × 500 m resolution. The statistical results from the VCF dataset show that, over all land, the percentage of grids in which the percent bare soil cover >95% is 21.29%, mostly from the contributions of deserts; the percent herbaceous cover >95% is just 3.40%; and the percent tree cover >95% is only 0.03%. High percentages of tree cover, even after converting to the crown cover [dividing by 0.8 as suggested by Hansen et al. (2003)] are infrequent. That is to say, 100% coverage of any type is rare even at high resolution (500 m) on the land surface. Rather, in most cases PFTs coexist with each other. Even without considering the heterogeneity of a single canopy [i.e., its three-dimensional (3D) effects], the two-stream model cannot readily treat the actual system due to its unrealistic assumptions and may introduce a large bias, even possibly resulting in deficiencies of the climate simulation.

The remote sensing community has recognized the importance of 3D geometry and applied some detailed treatments to retrieve the albedo and leaf area index (LAI) parameters (Li and Strahler 1985; Knyazikhin et al. 1998; Schaaf et al. 2002; Myneni et al. 2002), but these treatments have not been translated into simple analytical solutions suitable for climate models. On the other hand, the concepts of gap probability and clumping index (Nilson 1971; Norman and Welles 1983; Li and Strahler 1988; Chen and Black 1991) have been introduced to take into account the 3D effects of canopy, for example, through the geometric optical–radiative transfer (GORT) model (Li et al. 1995), which has been further developed by combining with a two-stream model, such as the Ecological Assimilation of Land and Climate Observations (EALCO) model (Wang 2005) and the Analytical Clumped Two-Stream (ACTS) model (Ni-Meister et al. 2010), now being used in ecosystem models. Another approach proposed by Pinty et al. (2004a, 2006) is to use effective leaf optical properties and LAI, instead of true values, as fitting parameters incorporated into the two-stream model to match the numerical results. Both approaches are equivalent in modifying the optical properties in a 1D model. However, they require good estimates of clumping indices or effective optical properties under all kinds of conditions.

Climate models run their computations on a global mesh with as high resolution as is feasible. In addition, with the advent of global quantitative derivation of terrestrial properties by remote sensing in terms of types, resolution, and accuracy, a more detailed description of radiative transfer processes on the land surface is both possible and necessary. This paper establishes the construction of a 3D canopy radiative transfer model that takes into account the horizontal and vertical heterogeneities of the canopy, but with a simple physically based formulation that has computational efficiency comparable to that of the 1D model it replaces. This 3D model is a new framework based on the single bush model (Dickinson et al. 2008; Dickinson 2008). A one-layer canopy model is constructed that considers shadows, intercanopy interactions, and the consequences of low sun (elevation) angles. It is used to build a three-layer canopy that considers shadows overlapping between layers. Two types of numerical simulations are carried out to validate the 3D model and its components, that is, the single bush and one-layer and three-layer models. This 3D model is applied and implemented into the Community Land Model (CLM4.0) (Oleson et al. 2010) and compared with the default two-stream model. Major differences are explicitly described and explained from a 3D viewpoint.

All canopy radiation modeling is founded on principles of ray tracing geometric optics, although in simple cases, as in the 1D two-stream model, it can be expressed in terms of differential equations. The modeling here is directly in terms of geometric optical reasoning. Simple 1D canopies are constructed from leaf optical properties and statistical models of leaf distribution. In 3D, one more level of organization is included (i.e., statistical models of bush distribution).

## 2. Model description

### a. Single bush model

The single bush model starting point was developed earlier (Dickinson et al. 2008; Dickinson 2008). Its basic assumptions are that the bush is a spherical object and that the leaves within it are homogeneous and have a spherical leaf angle distribution (LAD). Optical depth (*τ*) along the radius of single bush is designated *τ*_{0} and derived from LAI of a single bush as *τ*_{0} = (3/8)LAI. When the stem area index (SAI) is considered, τ_{0} = (3/8)(LAI + SAI). Individual leaves were initially treated as isotropic scatterers with a single scatter albedo *ω*. For incident direct radiation from an overhead sun, the single bush model determines analytically various bush optical properties, that is, the direct transmission *T*_{d,s}, first-order scattering Φ_{1}, second-order scattering Φ_{2}, and higher-order scattering Φ_{3+} phase functions. These are all functions of *τ*, and scattering phase functions depend on the out-scattering angle *θ*_{out} as well. Further improvements are made in the first-order scattering to consider individual leaves as bi-Lambertian scatterers (with optical properties of reflectance *r* and transmittance *t*). Since bi-Lambertian scattering has only minor effects on the second- and higher-order scattering, we continue to assume isotropic leaf scattering for their calculations. We integrate the total scattering phase function Φ over the upward and downward hemispheres, considering the incident direct radiation at a sun zenith angle *θ*_{sun} (*α*_{s} and diffuse transmission *T*_{i,s} respectively, as summarized in appendix A.

Numerical simulation shows that the scattering properties of incident diffuse radiation for a single bush are adequately equivalent to that of incident direct radiation with *θ*_{sun} = 60°. So this approximation is used to deal with the diffuse case.

### b. One-layer canopy model

*f*

_{c}(the percentage of vertically projected area of the canopy crowns). The area of shadows cast by the bushes on the ground (ground shadow) without overlap is

*S*

_{0}=

*f*

_{c}/

*μ*, which also applies to multiple bushes with overhead sun or for very small

*f*

_{c}. For a low enough sun angle, shadows start to overlap, and the total shadow area with overlap is modified by statistical modeling toThe overlap of shadows will increase the average optical depth compared to that of a single bush (i.e., the optical depth for the one-layer canopy is correspondingly increased to

*τ*=

*τ*

_{0}

*S*

_{0}/

*S*).

*θ*

_{sun}not only leads to more overlap of shadows, but also to changes of the illumination of the incident direct radiation to be more concentrated at the top part of spherical bushes. With an increase of

*θ*

_{sun}, the conditions of illumination become more like that of a 1D case and albedo will increase. Dickinson (1983) addressed this issue in discussing the albedo calculation of a semi-infinite canopy. Unfortunately, the single bush model cannot easily capture this feature. We account for the increment of albedo

*α*

_{L}due to the low sun angle effects using a similar approach following the 1D case of Dickinson (1983) approximated aswhere

*N*is the multiplicative factor of albedo at a

*θ*

_{sun}compared with albedo in the overhead sun case (

*θ*

_{sun}= 0° and

*θ*

_{sun}and can be up to about 3 in the visible (VIS) domain and 2 in the near-infrared (NIR) domain. The term

*α*

_{s}(

*μ*= 1,

*τ*

_{0}) is the single bush albedo with

*θ*

_{sun}= 0° [Eq. (A15), appendix A]. Equation (2) also includes the

*f*

_{c}and shadow overlapping effects. When

*f*

_{c}is very small (

*f*

_{c}→ 0), or there are very few overlapped shadows (

*S*≈

*S*

_{0}), then

*α*

_{L}approaches 0 (i.e., the albedo becomes the same as for the single bush case).

*f*

_{c}, the intercanopy interactions become more important and will increase canopy absorption. This additional absorption

*A*

_{c}is estimated simply (cf. appendix B) and decreases the albedo

*α*and

*T*

_{i}. The additional albedo

*α*

_{L}decreases the canopy absorption

*A*and diffuse transmission

*T*

_{i}. After distributing the increased items evenly to the decreased items, the final results of

*α*,

*T*

_{i}, and

*A*for the one-layer canopy are as follows:andwhere

*T*

_{d}is direct transmission of the one-layer canopy and calculated as

*T*

_{d,s}(

*τ*) [Eq. (A1), appendix A]. These properties for incident diffuse radiation are calculated for direct

*θ*

_{sun}= 60°. Corresponding values are designated with a star in its superscript, such as

*α** and

*A**.

*r*

_{g}), we first calculate the initial radiation reaching it,

*T*= 1 −

*S*+

*S*(

*T*

_{d}+

*T*

_{i}). This term will be reflected by the ground back upward as diffuse radiation. A fraction of this,

*r*

_{g}

*TS**

*α**, is scattered by the canopy back toward the ground, where

*S** is the fraction of sky covered by canopy (i.e., sky shadow). Further such ground–canopy reflections can be added in. The radiation reflected between the ground and canopy is a geometric series, and the common ratio

*q*is

*r*

_{g}

*S**

*α**. Hence, the total transmission [

*T*] is

*T*/(1 −

*q*). In the same way, the canopy absorption from multiple reflections between canopy and ground

*A*

^{m}is

*r*

_{g}

*TS**

*A**/(1 −

*q*). The total canopy absorption [A], ground absorption [G], and collective albedo [

*α*] are obtained asand

A factor *S* before *A* in Eq. (6) is to convert the canopy absorption from the canopy-shaded area to the entire area.

### c. Three-layer canopy model

The canopy is divided into three layers. Such division may be based on different PFTs of different heights (e.g., trees, shrubs, and grass) or different age classes (forming upper story, intermediate, and lower story components). Each layer is calculated as a one-layer canopy. In addition, we assume that there is no overlap of vertical projections between the canopy crowns of layers. Each layer has their respective crown radius *R* and central point height *h* as illustrated in Fig. 1. The top canopy layer is designated layer 1, and the bottom layer is designated layer 3. If more than one PFT occurs in a single layer, the layer *f*_{c} is the sum of the *f*_{c} of the PFTs, and the other canopy properties, such as LAI and *ω*, are the average of PFTs weighted by their *f*_{c} values.

The overlap of shadows between layers is important for the transfer of incident direct radiation, especially in the VIS domain where leaves are highly absorbing. The basic rule followed in treating overlap is that the “self-area shadow” (defined as the shadow area projected on a lower layer overlapped with its vertical projection, such as *S*_{12}, *S*_{13}, and *S*_{23} in Fig. 1) projected on a lower layer cannot overlap with the lower layer’s shadow. Otherwise, layer shadows can overlap with each other randomly. The calculation of the self-area shadow is described in appendix C.

*I*

_{0}) from layer [from] to layer [to]. The subscript 0 means the position just above the canopy, and

*g*means the position just above the ground. The direct radiation from

*I*

_{0}to each canopy layer or reaching the ground without passing through the canopy iswhere

*S*

_{[n]}represents the shadow of layer

*n*. The direct radiation to layer 1 can further transmit directly to lower layers, which are calculated aswhere

*T*

_{d,1}is the direct transmission of layer 1. In the same way, we can getandAdding the above results, the accumulated direct radiation to each layer isThe fraction of direct radiation absorbed as direct by layer

*n*is given by the product of radiation attenuated and fraction of attenuated radiation that is directly absorbed:The

*P*

_{sun}, fraction of canopy top exposed to sunlight, for layer

*n*is calculated by dividing the accumulated direct radiation by its shadow area:For the top layer,

*P*

_{sun}= 1, which means a 100% sunlit cover. For a lower layer,

*P*

_{sun}can be less than 1 if shaded by the upper layers. In this case, it means the available incident direct radiation is modified and the fractions of sunlit LAI (

*F*

_{sun}) will be further affected. The quantity

*P*

_{sun}was used in a revised

*F*

_{sun}(section 4a).

*n*]. The value of

*n*and can be expressed as follows:andwhere

**x**,

**x***) consists of two column vectors, that is, the solutions for direct [Eq. (17)] and diffuse [Eq. (18)] radiation. The matrix

**b**,

**b***) also consists of two column constant vectors for the direct and diffuse radiation, respectively.

*r*

_{g}. However, this situation is not expected to happen under natural conditions. After solving for

*P*

_{sun}as calculated in Eq. (15). The layer absorption, as well as the direct radiation absorbed as direct, is distributed to its PFTs according to the weights calculated by the individual PFT results using the one-layer model.

## 3. Numerical validation

Two types of numerical simulations were performed to validate the 3D canopy model against more detailed computation. One is the full (optical) path (FP) simulation, and the other is the Monte Carlo (MC) simulation. For the FP simulation, we divided the spherical bush into small unit cubes (30 cubes along the radius) with equal sizes in Cartesian coordinates. Each cube performs as an individual volume of leaves that can both absorb and scatter radiation as a point source. The FP can simulate each order of scattering of a single bush effectively because it can more easily achieve a steady state than the MC simulation, especially for higher-order scattering. The MC simulation follows North (1996)’s formulation, but with more details for 3D canopy structures and the radiation recording. It was implemented in a 100 m × 100 m square size area, and 100 million uniformly spaced photons were generated as direct or diffuse radiation incident on the top of the canopy. Compared with the FP simulation, the MC is more effective at the simulation of a 3D canopy with complicated structures. It was also used to simulate a single bush for mutual validation of the FP and MC simulations (i.e., to make sure of the correctness of both simulations). Neither simulation accounts for the hotspot effect, which makes but a minor contribution to the albedo calculation but affects some remote sensing issues.

To benchmark the MC simulation, we compared it with a Forest Light interaction model (FLIGHT), version 5.0 (North 1996), in a baseline scenario in the canopy Radiation Transfer Model Intercomparison (RAMI) exercise phases 1, 2, and 3 (Pinty et al. 2001, 2004b; Widlowski et al. 2007). This scene contains 15 floating canopy spheres treated as a turbid medium. For details of the parameter setting, refer to Pinty et al. (2001; or the website http://rami-benchmark.jrc.ec.europa.eu). Four experiments were simulated with two *θ*_{sun} (20° and 50°) in two bands. We ran MC and FLIGHT both 100 times for each experiment. The results are shown in Table 1. The absolute values of the mean differences in albedo, canopy absorption, and ground absorption are all less than 0.001. The MC and FP were compared in simulating the first- and second-order scattering phase functions of a single spherical bush. Figure 3 gives an example of the scattering phase function of two angles. The mean differences for different LAI values between the MC and FP are an order of 0.001 or less.

Comparison between MC and FLIGHT for a RAMI1–3 baseline scenario. The results are averages of 100 simulations and the numbers in parentheses are the standard deviations.

### a. Single bush model validation

The FP simulation was used to validate the single bush model, for the very simple case in which the leaves within the spherical bush are isotropic scatterers and the sun is overhead. The normalized first-order, second-order, and higher-order scattering phase functions (Φ_{1}, Φ_{2}, and Φ_{3+}, respectively) from the 3D model, FP, and their differences are shown in Fig. 4. Only Φ_{3+} depends on *ω*. The figure shows that the differences of Φ_{1} and Φ_{2} compared with FP are all within 0.05, while Φ_{3+} has a larger bias (>0.10) in the large LAI (>6) case, resulting from the assumption of isotropic scattering of Φ_{3+}. Since all these results are normalized, by being multiplied by a factor (0.25*ω*/*π* for Φ_{1} and 0.25*ω*^{2}/*π* for Φ_{2} and Φ_{3+}), the actual biases are much smaller, especially for the VIS domain in which *ω* is small. More accurate formulations are possible for the single bush model, but in fact its bias is already much less than that originating from the canopy horizontal and vertical structures. Therefore, we maintain simplicity, allowing analytic integrals for the albedo and diffuse transmission.

### b. One-layer canopy model validation

The one-layer canopy model was tested for seven *f*_{c} values (0.1, 0.2, …, 0.7), ten values of *θ*_{sun} (0°, 10°, …, 80° and diffuse radiation) and three LAI (1, 3, and 6) values. Leaves are bi-Lambertian scatterers and have a spherical LAD. The average radius *R* and height *h* values are assumed to have normal distributions of

As seen in Fig. 5, the 3D model agrees well with the results of the MC simulation in terms of albedo [*α*], canopy absorption [*A*], and ground absorption [*G*] for different LAI values and in both spectral domains, especially for the albedo, for which the RMSE are almost always within 0.01 except for the NIR domain at large LAI values. For canopy and ground absorption, the RMSE values are all within 0.04. The RMSE increases with increasing LAI in both spectral domains and overall RMSE of canopy absorption and ground absorption in the VIS domain are both 0.024, and in the NIR domain are 0.020 and 0.026. The corresponding RMSE values of the two-stream model increase to 0.122 and 0.095 in the VIS domain and to 0.068 and 0.059 in the NIR domain. In sum, for the assumed one-layer geometry, the 3D model errors are typically in the range of 0.01–0.02, in some instances a little higher; whereas, the 1D model errors are in the order of 0.1. In some limiting cases both models have small similar errors, that is, the two-stream model behaves similarly to the 3D and MC for small LAI values (LAI = 1) in [*α*], [*A*], and [*G*], as well as in the medium *θ*_{sun} [~(30°–60°)] case. For small *f*_{c}, the agreement is only apparent, a result of small canopy absorption and albedo.

The 3D model reproduces two features of canopy and ground absorption not given by the two-stream model. One of these is most pronounced at low sun angle (large *θ*_{sun}) with medium *f*_{c} value, and the other is most pronounced at small *θ*_{sun} with large *f*_{c} and LAI values. The first difference is the effects of shadows. Shadowing reduces the radiation reaching the ground, and the sky shadow further reduces the reflected radiation back to the sky. The latter is especially significant for snow cover. The other major difference is the heterogeneity of radiative path lengths. In particular, the radiation can directly pass through or scatter downward from the canopy edge more easily than through its center with less canopy absorption and higher ground absorption, a result of canopy clumping (i.e., canopy shape effects). For the albedo, the two-stream model gets slightly higher values with the increasing LAI values, especially for low sun angle and medium *f*_{c} values, and the total RMSE is somewhat larger than that of the 3D model, by 0.029 for VIS and 0.01 for NIR. The differences in the VIS domain are mainly due to the shadow effects since the albedo from the ground contributes far more than the canopy due to the lower reflections of leaves. For albedo in the NIR domain, the situation is a little more complicated because the ground reflectance can be either greater or less than the canopy reflectance. At the same time, the intercanopy interactions and increased transmission due to shape effects can reduce the albedo in the 3D case. Therefore, the final difference between the two-stream model and the 3D model is dependent on the balance of the above effects. In this test, the ground reflectance *r*_{g} is set to 0.2, which is similar to the canopy reflectance and the two-stream model shows an overestimated albedo for large LAI.

### c. Three-layer canopy model validation

We also used MC simulation to validate the three-layer canopy model. The 16 PFTs that are currently being used in CLM4.0 were chosen as inputs (Table 2). The eight types of trees in CLM are allocated to the top layer; the three types of shrubs are allocated to the midlayer; and the remaining five types of grasses and crops are assigned to the bottom layer. The leaf and stem optical properties are the same as in the CLM4.0.

Prescribed input data for three-layer canopy validation.

Two types of experiments were used to validate the three-layer canopy model. First, the canopy structures, including *f*_{c}, LAI, SAI, *f*_{c}: 20%, 50%, and 80%. The generation of a canopy scene started from the top layer to the bottom layer, and each layer followed the one-layer case. We tested different *θ*_{sun} and soil types. For each case, we also ran 100 samples to identify the uncertainty of the MC simulations. The 3D model results, as well as the two-stream results, are compared in Fig. 6. Evidently, the 3D model reproduces quite well the dependences on *θ*_{sun} of albedo, PFT absorption, and ground absorption in all three groups, even with NIR and ground snow cover (shown as NIR + Snow in Fig. 6). The differences between the two-stream and 3D model in the one-layer comparisons are also seen in Fig. 6. In addition, there are two other differences: (i) the lower-layer canopy absorbs less radiation with an increase of *θ*_{sun} due to the upper layer’s shadow projection on the lower layer, especially for larger *f*_{c} values of upper layers. The two-stream model does not show this feature because it considers all of the PFTs to be isolated without mutual interactions. It also has relative small variations in the absorption of PFTs with *θ*_{sun}. (ii) The ground absorption can increase slightly with *θ*_{sun} near 0° for the large *f*_{c} as shown in Fig. 6c due to the shadow overlapping between layers that reduces the total ground shadow cover and increases the direct transmission to the ground. Consequently, the ground absorption increases, as is more obvious in the high canopy *f*_{c} case. The two-stream shows a monotonic decrease of ground absorption with increase of *θ*_{sun}.

For the second experiment, we randomly chose several PFTs (≤5 types) and randomly set their *f*_{c} (total canopy *f*_{c} ≤ 75%) and LAI values (≤7). The SAI, *θ*_{sun} and background cover types are shown in Fig. 7. As seen in the figure, the 3D model performs much better as indicated by the RMSE values, especially in the NIR + Snow case. The overall RMSE of albedo decreases from the 0.163 of the two-stream model to 0.019, and in the VIS + Soil case, the overall RMSE of ground absorption decreases from the 0.170 of the two-stream model to 0.017. The two-stream model performs relatively better for the small *θ*_{sun} (e.g., at 0° and 30°) than for large *θ*_{sun}.

## 4. Application and comparison

### a. Implementation in CLM4.0

Given the classification of vegetated land units in CLM4.0, the 3D model processes all the PFTs within a soil column at a time, including the bare soil type. The PFTs are divided into three layers as described in section 3c. Figure 8 shows statistical results for each layer’s percentage, all layers’ percentage, and the number of layers existing in each grid as derived from the CLM4.0 input dataset (Lawrence and Chase 2007). The *R* and *h* of each PFT are simply calculated from the parameters of canopy top and bottom heights (*Z*_{top} and *Z*_{bot}) as prescribed in CLM4.0 (i.e., *R* = *Z*_{top} − *Z*_{bot} and *h* = *Z*_{bot} + *R*). The 3D model is then applied for three layers (section 2c) to process all the soil columns. All PFTs within a soil column share the same albedo, and the same transmission (i.e., the same ground absorption). Consequently, the 3D model does not conserve energy at the PFT level, as the two-stream model does, but only at the column level.

In addition, *P*_{sun} was calculated for each PFT and used to revise the *F*_{sun} values as *F*_{sun} × *P*_{sun}. This modification will affect the sunlit and shaded LAI fractions. Another issue is the fraction of absorbed diffuse radiation in the calculation of sunlit/shaded leaves. Bonan et al. (2011) have pointed out the problem of CLM4.0 that apportions the total canopy-absorbed diffuse radiation to sunlit and shaded leaves based on the sunlit (*F*_{sun}) and shaded fractions (1 − *F*_{sun}) of canopy. In CLM4.0, it assumes that the absorbed diffuse radiation within the canopy is uniformly distributed with a density function of 1/LAI. Here, we modified it to an exponential function as shown in Table 3, and the fraction of absorbed diffuse radiation in sunlit leaves is the result of an integral expression also listed in Table 3. This revision will affect the distribution of absorbed diffuse radiation between sunlit and shaded leaves.

The fraction of absorbed diffuse radiation in sunlit leaves. The *x* is the cumulative LAI measured downward from the top of the canopy, and *k*_{b} is the direct radiation extinction coefficient of the canopy.

### b. Comparison with two-stream model in CLM4.0

We ran the offline CLM4.0 [the National Center for Atmospheric Research (NCAR)’s Community Earth System Model, version 1.0.3 (CESM 1.0.3), with component set “I_2000” with only the land model active and Qian data (Qian et al. 2006) as forcing for current climate conditions at a resolution of 0.9° × 1.25°] with the default two-stream model (control run) and 3D canopy radiation model (3D run). Both models ran from 2000 to 2003, and the results of last 3 yr, after 1 yr of spinup, are analyzed below.

#### 1) Monthly-mean surface shortwave radiation flux comparison

The 3D model affects the surface solar radiation flux (shortwave) directly. We compared three components of fluxes, surface-reflected radiation flux *F*_{sr}, canopy-absorbed radiation flux *F*_{ca}, and ground-absorbed radiation flux *F*_{ga}, using the averages of 3-yr monthly-mean results. Their differences in January and July are shown in Fig. 9.

Overall, the *F*_{sr} of the 3D run, in both January and July, is less than that of control run over most vegetated regions. The differences are about −2.24 W m^{−2} in January and −3.24 W m^{−2} in July averaged over global land. For January, the boreal region with latitudes 50°–60°N shows an increase of *F*_{ca} and a decrease of *F*_{sr} compared with the control run. The corresponding difference of *F*_{ga} is small. In the areas near the equator, such as the Amazon, Congo basin, and Indonesia, the 3D run gets less *F*_{ca} and *F*_{sr} and correspondingly gains more *F*_{ga} (~10 W m^{−2}). In relatively sparsely vegetated regions like Sahel, central Australia, the central-western United States, and southern Africa, there are slight increases of *F*_{ca} and decreases of *F*_{ga}. For July, the *F*_{sr} is similar to that for January, but the area with a decrease expands in the Northern Hemisphere. In the boreal region (50°–70°N), the *F*_{ca}, opposite to the January case, is reduced by about 10 W m^{−2} in many places, and the corresponding *F*_{ga} is higher. The differences in areas near the equator are very similar to that for January. The same as that for January, there are increases of *F*_{ca} in the areas with relatively sparse vegetation and corresponding decreases in *F*_{ga} especially obvious in central Asia and the central-western United States. The mean differences over the global land surface of *F*_{ca} are −0.04 W m^{−2} in January, −2.36 W m^{−2} in July, and the mean differences of *F*_{ga} are 2.28 W m^{−2} for January and 5.61 W m^{−2} for July. The statistical results of the global-mean values for each month are shown in Table 4. The *F*_{sr} shows lower values throughout the year. The *F*_{ca} increases by up to 1.55 W m^{−2} in the early spring, mainly from the contributions of the boreal forest. In the other seasons, *F*_{ca} shows slightly lower values. The *F*_{ga} increases throughout the year, especially in summer.

Seasonal differences (3D run minus control run) of surface radiation fluxes (W m^{−2}) averaged for three regions and global land surfaces.

#### 2) Diurnal cycle of surface radiation flux comparison

The above comparison is based on time-averaged values and does not provide any details over the diurnal cycle. To better understand how the surface radiation fluxes are changed over the diurnal cycle, we chose three representative regions: the Amazon (0°–10°S, 50°–70°W), the boreal region (50°–70°N), and central Asia (30°–50°N, 50°–80°E). For each grid within the representative region, each model step of a half-hour in local time is averaged for the entire month (January 2003 and July 2003), and the area-averaged values are compared in Fig. 10.

The Amazon is mainly covered by a one-layer canopy (broadleaf tropical trees) and has a high canopy coverage (*f*_{c} ≥ 99%) (Fig. 8d). The sun orientation changes little over the annual cycle. At local noontime (LN), *θ*_{sun} is small in both January (the average is 16.1° and the minimum could be 11.0°) and July (the average is 26.6° and the minimum could be 21.7°). With these conditions, the 3D geometry is more able to transmit the radiation through the canopy to the ground, so *F*_{ga} increases by about 10 W m^{−2} on a daily average (>30 W m^{−2} at LN). The direct transmission of incident direct radiation [*T*_{d}] and diffuse transmission of incident diffuse radiation [*F*_{ca} and *F*_{sr} decrease. The decrease of *F*_{ca} is mainly due to the decrease of the absorption of direct radiation [*A*], which could be as low as −0.06 in the NIR domain at LN, while the absorption of diffuse radiation [*A**] is similar between the two models. The decrease of *F*_{sr} is mainly in the NIR domain, and the reduced albedo of incident diffuse radiation [*α**] is more obvious than that of the albedo of incident direct radiation [*α*]. One feature of the [*α*] is that the increase of albedo of the 3D model from local noontime to sunset/sunrise is slower (the line is flatter) than that of the two-stream model.

In the boreal region, the canopy cover is dominated by multiple layers (two or three layers; Fig. 8d) and *f*_{c} ≥ 90% in most places. In January, the *θ*_{sun} stays large all day long, increasing the shadow overlapping between layers, as well as the total ground shadow, hence increasing the canopy absorption and decreasing the transmission to the ground. The reflected radiation from the ground is similarly affected, with a decrease of the *F*_{sr} in winter and spring because the ground is usually snow covered. The *F*_{ga} remains unchanged due to lower absorption by snow-covered ground. The maximum difference in *F*_{ca} is about 12 W m^{−2} near the LN. In July, daytime is longer, and the *θ*_{sun} is about 39.7° on average at LN. For the incident direct radiation, because of the shadow overlapping between layers, the total ground shadow at this time could be less than the total *f*_{c} that will increase the [*T*_{d}] and decrease the [*A*] correspondingly. At LN, the increase of *F*_{ga} is ~35 W m^{−2}, and the decrease of *F*_{ca} is ~25 W m^{−2}. However, at low sun, the situation reverses for the same reason as in January. The time variation of [*A*] is more like a V shape in the 3D run. For the diffuse case, the differences in [*A**] and [*F*_{ga} increases more than 10 W m^{−2}, and corresponding *F*_{ca} and *F*_{sr} both decrease more than 5 W m^{−2}.

For central Asia, the ground is sparsely vegetated, having a relatively small *f*_{c} value. The shadow effects increase *F*_{ca} and reduce *F*_{sr} and *F*_{ga}, especially for low sun. The differences for July are larger than that for January and can be more than 5 W m^{−2} in *F*_{ca} on average. The statistical results of three representative regions in the other months are given in Table 4.

#### 3) Sunlit–shaded LAI and absorption comparison

The 3D model considers the overlapping of shadows between layers, which makes the *P*_{sun} less than 1 in the lower layers, decreasing the sunlit LAI (*L*_{sun}). Figure 11 shows the difference of *L*_{sun} for monthly-mean values over 3 years. The decrease in January happens mainly in the Southern Hemisphere with the area covered by multiple layers (Fig. 11a). In July, the area with decrease expands to the Northern Hemisphere, especially in the boreal region and eastern United States where multiple layers occur (Fig. 11b). However, the magnitude of the difference is mostly within 0.2 m^{2} m^{−2}.

The redistribution of absorbed diffuse radiation (section 4a) increases the sunlit canopy absorption. Together with the effects caused by the changing of *L*_{sun}, the differences of the total sunlit canopy absorption *F*_{ca,sun} and total shaded canopy absorption *F*_{ca,sha} are plotted in Fig. 11. The tropical area, with one layer and dense canopy cover, shows consistently decreased *F*_{ca,sha} and increased *F*_{ca,sun} for January and July. However, the decreased magnitude is much larger. In January, the boreal region at latitudes 50°–60°N gets a higher *F*_{ca,sun}. In July, the boreal region shows a large decrease of *F*_{ca,sha}. The *F*_{ca,sun} increases mainly in areas with a one-layer dense canopy or sparsely vegetated cover. For areas covered by multiple layers, *F*_{ca,sun} decreases slightly.

## 5. Discussion

In this study, a spherical bush is used as a building module to represent the canopy in a simple way. The maximum coverage of spheres of the same size without vertical projection overlapping could only be up to 90.69% if hexagonally close packed. If sphere sizes were variable, the coverage would be higher. To achieve even higher cover (i.e., near 100%), we can assume that the sphere has edges like the approximation of a regular hexagon to a circle. The random generation of a canopy scene of more than 80% coverage in a MC simulation could easily fail without human intervention and was not implemented in this study. Each spherical bush of the same PFT is assumed to have the same LAI so that all bushes have the same *τ*_{0} independent of their size. Thus, each bush has the same scattering properties so that we can conveniently apply the 3D model for one PFT with a single *τ*_{0} parameter. In reality, LAIs of bushes may vary with bush sizes. However, if bushes have only small variations in size, the assumption of a single LAI is reasonable.

Spheres can be extended to ellipsoids through a simple linear transformation of one of its axes (Li and Strahler 1988). The effective *θ*_{sun} is correspondingly changed in the new coordinates, as well as the shadow calculations. Other distinctly different geometries, such as cones, would best be included by the summarization of numerical computations, through fitting of other expressions, or by lookup tables. We use only a sphere here, not only due to its simplification, but also since observational data of canopy shape on a global scale are absent. Only a spherical LAD is considered in this study. Further refinements, such as allowing for differences in LAD, could be done. However, the daily average value of each radiation component would differ only slightly since it is an integral result over the entire day. We compared two control runs: one with such a LAD consideration and the other with just spherical LAD. The replotted Fig. 9 did not show any noticeable differences.

Besides the uncertainties inferred from the single bush calculations, a more important issue is the distribution of canopy crowns, which affects the shadow calculations, especially for nonclosed vegetation. In our study, without having information about the distribution of ground coverage, all PFTs are assumed to have a horizontally random distribution without the overlap of vertical projections and collocate (mix) within the grid. Individual trees or bushes locate themselves within a forest according to random choices and optimization principles (e.g., they may prefer to distance themselves to some extent from rivals because of competition for light, nutrients, and water). The absence of vertical projection overlapping is also consistent with the vegetation in CLM4.0 that are all determined based on the PFT levels without vertical overlapping, as is the input land surface data. In reality, overlap of vertical projections between PFTs could exist (e.g., to form an understory layer). In this case, the total canopy *f*_{c} may exceed 100%, such as in the Amazon. To apply the 3D model, the calculations of overlapped shadow between layers could be revised to be randomly overlapping with the constraint of self-area shadow removed. In addition, different PFTs may prefer not to collocate but be separate, such as in Arctic, where because of shortage of cold and light, it may be difficult to support more than a one layer canopy. The 3D model could be further improved with more details of canopy layer information, which could be retrieved by spaceborne lidar data (van Leeuwen and Nieuwenhuis 2010) or supplied by the prognostic variables from dynamic vegetation models.

In atmospheric radiative transfer, the equivalent angle for incident isotropic diffuse radiation has commonly been assumed to be 52.96 based on the diffusivity factor *D* = 1.66 (Elsasser 1942). The term *D* is an approximation for the equivalent average pathlength for the diffuse transmission (uncollided) function calculation; that is, using *D* depends on *D* → 2 (equivalent angle = 60°) and *D* → 1 (0°). For a canopy, *D* also depends on LAD, as well as canopy crown shapes, spatial distribution, and *f*_{c} in a 3D scene. For spherical bushes with small *f*_{c} in a random distribution, *D* → 2 (especially for a single bush, *D* = 2 for all *τ*_{0}). For a large canopy cover (e.g., 100% coverage of a 1D canopy with spherical LAD), we compared different *D* values in the diffuse transmission function calculation in Fig. 12: *D* = 2 is currently used in CLM4.0 and 4.5 and *D* = 1.41 (0.705/0.5) is suggested by Pinty et al. (2006). For the 3D model, *D* was chosen as 2 but applied in the 3D direct transmission calculation scheme. Figure 12 shows that the 3D model performs similarly to the 1D model with *D* = 1.66, and better than 1D model with *D* = 2, but still underestimates the diffuse transmission at intermediate LAI values. A slight decrease of *D* in the 3D model would give a better match with the numerical simulations under these conditions. In this study, for comparison, we made the same assumption as CLM 4.0 and 4.5 with *D* = 2 (60°) for the canopy with a spherical LAD.

The canopy 3D structures considered in the model, such as its crown shape, horizontal distribution, and vertical layering, are appropriate for leaves clumping at the canopy (i.e., crown) level. We have not taken into account the heterogeneities within a crown, such as the vertical leaf density profile and the leaves clumping at shoot level. Another issue not considered yet is the interactions of canopy radiation with topography, which could be important in a mountainous area.

The dependence on *θ*_{sun} of albedo is complicated, as summarized by Dickinson (1983). Albedo commonly increases with increasing *θ*_{sun}. However, Oleson et al. (2003) and Wang et al. (2004) found that the albedo calculated by the two-stream model in CLM2.0 increases more rapidly with *θ*_{sun} than the MODIS data. The diagnosed variation of Wang et al. (2004) has a similar slope but is flatter for the MODIS data than for the two-stream model. Their conclusion is consistent with our diurnal comparison in the Amazon and boreal region in a snow-free season (Figs. 10a,b). Soil albedo is important to the total surface albedo, especially in a sparsely vegetated area or with small LAI. One noticeable issue is that the soil albedo represented in the current CLM4.0 is independent of *θ*_{sun}. However, the observational MODIS data show a dependence on *θ*_{sun} (Wang et al. 2004) that may contribute to the increase of albedo with large *θ*_{sun} and so would lead to an even larger increase of albedo in the two-stream model. In a sparsely vegetated area, such as central Asia, Sahel, and the central United States, without considering this soil albedo variation, the 3D model actually shows a slight decrease with the increase of *θ*_{sun} in the VIS domain (as shown in Fig. 10c) due to the 3D shadow effects. Soil albedo can be adjusted through a fitting with the MODIS albedo data in order to get a better match (Lawrence and Chase 2007). However, an inclusion of the sun angle dependence of soil albedo could further improve CLM’s matching to observed surface albedo.

Further direct validations against observations are still needed in the future. MODIS albedo products (Schaaf et al. 2002) can be used to validate the model’s output of albedo. A comparison of zonal average albedo (at LN) from the control run, 3D run, and MODIS is shown in Fig. 13. MODIS data were aggregated from MODIS MCD43C3 products to the model output resolution in this study. The averages of year 2001–03 for both model outputs and MODIS data were compared. Only the grids with land fraction ≥99% and lake fraction <20% were included in the comparison to avoid the ocean and lake effects (Oleson et al. 2003). All sky albedo of MODIS are composited by VIS and NIR: black-sky (direct) and white-sky (diffuse) albedo weighted by the fraction of radiation fluxes in corresponding regimes the same as the model. In general, the 3D model shows a similar performance as the two-stream model. It seems that the 3D model may have some improvements at latitudes 50°–60°N in December–February (DJF) and partly improve the performance of the white-sky albedo simulation (more noticeable in NIR domain). The control run does not always overestimate the NIR white-sky albedo caused by the overestimation of albedo with *θ*_{sun} in the two-stream model as expected. It agrees well with MODIS NIR white-sky albedo at latitudes 5°–30°S in DJF and at latitudes 40°–60°N in June–August (JJA). Both two-stream and 3D models underestimate the NIR black-sky albedo at latitudes 0°–20°S in DJF and at latitudes 0°–10°N and 50°–60°N in JJA. The larger, even opposite variation of MODIS NIR black- and white-sky albedo (Figs. 13d,e,i,j) cannot be explained by the change of LAI, *θ*_{sun}, and *r*_{g}. This indicates some other factors may affect the canopy’s albedo. A possible reason could be the variation of leaf optical properties (prescribed as constants in the model) caused by the change of leaf water content or leaf tissue structure. The model’s albedo is highly dependent on the input datasets (e.g., LAI, leaf optical properties, *r*_{g}, and snow fraction). It is important to identify the source of the bias, whether it is from the model structure, the input datasets or possibly the observation itself. The global fraction of absorbed photosynthetically active radiation (fAPAR) datasets, for example, MODIS (Myneni et al. 2002), Carbon Cycle and Change in Land Observational Products from an Ensemble of Satellites (CYCLOPES) (Baret et al. 2007), Joint Research Center (JRC) (Gobron et al. 2006), and Global Land Products for Carbon Model Assimilation (GLOBCARBON) (Plummer et al. 2007), could be used to validate the model canopy absorption in the VIS domain. However, large discrepancies may exist among these datasets due to the differences of satellite passing time, the definition of fAPAR values calculated, statistical methods, input datasets for retrieval algorithms, and the radiative assumptions (Weiss et al. 2007; McCallum et al. 2010; Wang et al. 2013). These discrepancies may even exceed the differences between two-stream and 3D models. Careful examination of the fAPAR datasets is required in comparing the model and satellite retrieval data. Site level validation could be done using the FLUXNET datasets (Baldocchi et al. 2001) in the presence of known canopy structure (e.g., layers information and crown density) and measurements related to the radiation fluxes.

## 6. Conclusions

The 1D canopy radiative transfer models used in climate modeling, such as the two-stream model, are unrealistic in their assumption of a homogeneous canopy. To consider the 3D structure of a canopy in terms of its crown shape, horizontal distribution, and vertical levels, a new 3D model parameterization has been developed. The individual elements of the 3D model are modularized. A one-layer model was constructed by combining single bush models taking into account effects of their shadows, intercanopy interactions, and low sun angles. The one-layer model was used to build up the three-layer model by considering the shadow overlapping between layers. The 3D model so developed is simple and effective and is in good agreement with numerical simulations. We implemented the 3D model into the land component of a climate model, CLM4.0, and compared it with the default two-stream model. The results show that canopy absorption increases primarily in sparsely vegetated areas and for multiple-layered areas with large *θ*_{sun} and is the result of increases of the ground and sky shadows and of the optical pathlength because of the shadow overlapping between bushes and layers. Decreases happen primarily in densely vegetated areas with small *θ*_{sun}. For a one-layer canopy, heterogeneous pathlength effects increase the transmission through the canopy edge and so decrease the canopy absorption (e.g., tropical forest areas have a decrease of about 6 W m^{−2} throughout the year). For a multilayer canopy, in addition to these shape effects, the decreased total ground shadow due to its overlapping between layers also increases the transmittance; for example, the boreal region in summer correspondingly decreases by about 6 W m^{−2} in its canopy absorption. Ground absorption usually shows an opposite change to that of canopy absorption. Over all land, it has higher values throughout the year with the 3D model, especially in summer (>5 W m^{−2}). The largest contributions come from the boreal region (~11 W m^{−2}). Dense vegetation in the tropics also contributes throughout the year (~10 W m^{−2}). Slight decreases happen in the most sparsely vegetated areas throughout the year and also in the boreal region in January–April. The 3D model gives slightly lower albedo over most vegetated areas throughout the year, especially in the boreal region in early spring (flux up to −9 W m^{−2}), and the corresponding canopy absorption substantially increases (~10 W m^{−2}). The 3D model also affects the sunlit/shaded LAI and corresponding absorption. Further improvements and refinements could be made with the help of more details provided by observational data as model constraints. Nevertheless, the 3D model now provides a much more realistic treatment of terrestrial canopy radiation for use in climate modeling and evaluation by the community.

## Acknowledgments

This work was supported by Office of Science (BER) DOE Grant DE-FG02-09ER64746, the program of the Intellectual Introduction Project for the Discipline Innovation in China University System (111 Program), the Natural Science Foundation of China (under Grant 40875062), and the R&D Special Fund for Nonprofit Industry of China (Meteorology GYHY200706025). Dr. Xiaowen Li is thanked for some helpful insights provided at the initial stages of this research. We thank two anonymous reviewers for their thorough and constructive review. Thanks to Jamie Wentz and Jeffrey Hopkins for assistance with the word processing.

## APPENDIX A

### Single Bush Model

*θ*

_{sun}= 0°,

_{1b}and first-order forward-scattering phase function Φ

_{1f}:andwhere

*ω*/

*π*. After the first scattering, the radiation, which has been passed though optical pathlength

*h*. The diffuse pathlength, which from all possible points with length

*h*was approximated as that of the infinite parallel plane of the 1D case, which is

*h*are evaluated as

*a*= 0.70 and

*b*= 1.74 are parameters chosen to match numerical simulation results. The radiation to point

*h*, just as the first-order case, can be either scattered backward or forward out of the canopy as the second-order backward-scattering phase function

*ω*

^{2}/

*π*. For

*a*= 1 and

*b*= 2, Eqs. (A4) and (A5) are the same as the Eqs. (19) and (20) in Dickinson (2008). Assuming each order-scattering phase function changes linearly from the backward to forward direction, the total first- and second-order scattering are simplyandrespectively. The recollision probability (Panferov et al. 2001; Smolander and Stenberg 2005), after the second-order scattering, can be calculated asThe third- and higher-order scattering phase functions are assumed isotropic. Using the recollision probability concept gives the third- and higher-order scattering phase functions asΦ

_{3+}is also normalized by dividing by 0.25

*ω*

^{2}/

*π*.

*r*/

*ω*and 2

*t*/

*ω*, respectively. We still assume the phase function to vary linearly from the backward to forward direction. Then the phase function with an out-scattering angle

*θ*

_{out}(

*θ*

_{sun}(

_{1}, downward in overhead sun case) that added to albedo and one sector (Ω

_{2}, upward in overhead sun case) that subtracted. With these complications, the albedo is derived asThe last two items are combined and calculated asNote that the upper limit of

*θ*on the left side in Eq. (A14) is depended on the azimuth angle, and its integration was derived in detail by Norman and Jarvis (1975, their appendix A). The diffuse transmission can be derived in the same way. The albedo and diffuse transmission for a single bush are thus calculated asand

## APPENDIX B

### Intercanopy Interactions

*θ*

_{sun}. The view factor

*V*

_{1}for the central ball to its one of the nearest balls in distance

*L*

_{1}(Fig. B1b) iswhereSix balls are so arranged and further six balls at distance

*L*

_{2}are also accounted for. The total view factor

*V*is thenWe assume

*V*is in the middle of a sphere with a uniform distribution in the azimuth angle (Fig. B1c). The initial intercepted radiation is the integration of single bush scattering phase function (

*θ*

_{sun}= 0°) within

*V*, which is 4

*πV*Φ

_{a}(

*τ*

_{0}). Then we use the recollision probability concept [

*p*

_{2}denotes the recollision probability calculated by Eq. (A8)] to calculate the total absorbed radiation due to intercanopy interactions as

## APPENDIX C

### Self-Area Shadow Calculation

*R*in Eqs. (C1) and (C2) is the radius of bush, and

*h*is the height from the crown central point to a reference height level.

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