## 4 Discussion
### Problems with the Diurnal Cycle in the CLM4
By the view of direct solar radiation, the fraction of vegetation in the grid \((f\
u)\) should be a function of \(fc\) and the solar zenith angle \((\theta)\):  
\[fv=e^{ln\,(fc)\times\mu} \tag{14}\]  
where \(\mu=\cos\theta\). The fraction of bare soil in the sun's view should be \(1-e^{\ln\,(fc)\times\mu}\), and the fraction of sunlit bare soil in total bare soil \((fss)\) is:  
\[fss=\frac{1-e^{ln\,(fc)\times\mu}}{1-fc} \tag{15}\]  
By this function, the bare soil would receive no direct solar radiation when \(\mu=0\) and would be 100% sunlit by direct solar radiation when \(\mu=1\). This solution is a simplified approximation for the angular effect. Figure 11 shows it has good approximation when vegetation coverage is high or when LAI is high, compared to the 2D scattering model. Larger discrepancies exist when both vegetation coverage and LAI are low. The relationship between vegetation coverage and LAI is not considered here but has been suggested by site level observations. As analyzed earlier, the inter-annual comparison of site level FPAR suggests that vegetation coverage \((fc)\) changes according to LAI. Hence, LAI-\(fc\) relation should be taken into account to implement mathematical representation of an angular effect in the current CLM4 land surface solar radiation partitioning scheme. Spatial Patterns of Month with Maximum Fraction of Absorbed Photosynthetically Active Radiation (FPAR) in the Amazon  
As previously illustrated, both the northern and southern parts of the Amazon have different peak FPAR months opposite to their located hemispheres. The angular effect in FPAR is suspected to be one reason for this phenomenon. One major difference between the two regions is the solar zenith angle. According to the angular effect, FPAR would decrease as the solar zenith angle increases and thus, the FPAR would not peak in local summer months. CLM4's opposite estimations then can be well explained because model does not consider the angular effect in FPAR. However, one weakness in this reasoning is that the angular effect is more significant for sparse vegetation, but the Amazon rainforest has dense vegetation, and another weakness is that tropical latitudes have relatively muted seasonal variation in solar angle. In addition to the angular effect, two other possibilities (_i.e._, cloud contamination and plant phenology) are also suspected to contribute to this phenomenon. Cloud contamination is suspected to be a reason because remote sensing of the Amazon at visible bands has been complicated due to the presence of persistent cloud and aerosols during the rainy season. Remote sensing has made much effort in minimizing the cloud influence by: (1) recording cloud information into the raw data and pixels of good quality are selected for product generation; and (2) cloud-removal procedures to MODIS's surface reflectance products. However, these procedures still cannot totally avoid the potential systematical bias in FPAR. As cloud contamination lowers the  
Figure 11: FPAR in relation to \(\mu\) Blue lines show simulations from the 2-stream solution in the CLM4 (blue), red lines show simulations from the proposed solution in the paper and green lines show simulations from the 3D scattering model at \(fc=0.2\), 0.4 and 0.8 and LAI = 0.5, 3.0 and 6.0. FPAR value, we would expect peak months to have the least precipitation in the region. For the southern part of the Amazon, remote sensing observations (Figure 9(g)) shows FPAR peaks in June, July, August, and September which are the months with much less rainfall (Figure 12). For the northern region, satellite data (Figure 9(f)) displays that FPAR peaks in November, December, and January by different satellite observations, which are months with less rainfall in the region (Figure 12). Therefore, cloud contamination might be the reason for spatial patterns in satellite observations. The other possibility is associated with the seasonal cycle of plant phenology.