For practical purposes, relatively simple models are frequently used to estimate light interception in plant canopies. For example, crop models have become important tools for studying agricultural systems, yet they commonly utilize relatively simple models for light interception given the frequent lack of detailed architectural inputs. The most commonly used approach for estimating light interception treats the canopy as a homogeneous medium of unresolved vegetation , which allows for the use of a simple exponential model for radiation attenuation commonly know as Beer’s law, Beer-Lambert law, or Beer-Lambert-Bouguer law. Beer’s law calculates the probability of radiation interception as an exponentially increasing function of the leaf area projected in the direction of radiation propagation and the distance travelled through the canopy. The form of Beer’s law given in Eq. 2.1 functions under two main assumptions. The first assumption is that leaves are randomly distributed both vertically and horizontally in a continuous medium where leaves are relatively small. The second assumption is that leaves absorb all incident radiation, which may be reasonable for photosynthetically active radiation bands where leaves absorb roughly 90% of incident radiation, 5 gallon plastic pots but is likely a poor assumption in other bands such as the near-infrared where absorption is low. Equation 2.1 also requires specification of G, which is most commonly set to be equal to 0.5 based on the assumption that leaves are isotropic.

The assumptions of vegetation homogeneity and isotropy are usually violated in actual plant canopies. Leaf area density typically varies sharply in the vertical direction. Many natural plant canopies have considerable horizontal heterogeneity such as savannas, or heterogeneity due to natural or man-made disturbances. Crop canopies also commonly have a sparse, row-oriented configuration that creates high heterogeneity in light interception. Furthermore, it is rare to find canopies with isotropic leaf angle distributions, as this is typically not the most efficient configuration for light interception. Despite the known limitations of Beer’s law in the above cases, it is still frequently applied in these systems due to its simple, tractable form. However, there is a general lack of quantitative understanding of the errors resulting from the application of these simplified models in various canopy architectures, primarily because it is difficult to quantify light interception from field measurements for a range of architectures. The objective of this study is to better understand and quantify errors in modeled radiation absorption under assumptions of vegetation heterogene-ity or isotropy for various canopy configurations. The authors’ hypothesis is that Beer’s law will perform well for relatively dense, closed canopies provided that G is specified appropriately. For sparse canopies, it is hypothesized that assumptions of vegetation homogeneity will result in significant model errors, thus necessitating a more complicated model.

Since accurately measuring the distribution of absorbed radiation in space and time is often unfeasible using traditional experimental approaches, we used a sophisticated 3D radiation model along with virtually-generated canopies to evaluate Beer’s law under different simplifying assumptions. Virtual canopies with varying levels of heterogeneity, sparseness, and leaf orientation distributions were generated to evaluate assumptions of vegetation homogeneity or isotropy in terms of absorption of direct solar radiation. A considerable advantage of using virtually generated canopies is that the input parameters in Eq. 2.1 can be calculated exactly from the virtual canopies. When combined with a detailed 3D radiation model, this resulted in a robust means for evaluating the performance of simplified models for a range of canopy architectures.For simulating plant light interactions, detailed 3D geometric models were used to describe the architecture of the canopy. Agricultural crops were chosen for the plant types because: many 3D models are readily available, they have sufficient yet regular heterogeneity that limited the degrees of freedom when generating the canopies, and they represent an economically important practical application of the use of Beer’s law. The chosen crop canopies were grape, almond, potato, and corn, which were represented in the 3D model using a mesh of rectangular and triangular elements. To minimize the number of elements needed to describe their complex geometries, images with a transparency channel could be overlaid on these basic elements, where the transparency channel is used to remove a portion of the element’s surface. Virtually-generated plants were either read from a polygon file , or created using the procedural plant generator described by Weber and Penn.Parameters used to quantify canopy architecture are given in Table 2.1. The procedural model used to generate the grape and almond plants had a random component to each architectural parameter, making each plant unique. Each corn and potato plant was identical, therefore a random azimuthal rotation was applied to each plant to decrease regularity of the canopy. Two grape canopy cases were considered: one with a North-South row orientation and one with an East-West row orientation . Two potato canopies were considered in which plants were arranged in either a East-West row-oriented pattern , or a uniformly spaced planting pattern . In all cases, the size of the 3D scene was chosen such that further increasing the total number of plants did not have an impact on results. To test the model in the case of homogeneous and isotropic vegetation, a set of canopies were created with uniformly distributed leaves in space with three different leaf area index values: L =1.5, which consisted of 100,000 leaves; L =3.1, which consisted of 200,000 leaves; and L =6.2, which consisted of 400,000 leaves. The surface area of each leaf was 0.006 m2. Each leaf angle was set by randomly drawing from a spherical distribution. To characterize the plant geometry, L and the leaf inclination angle probability density function were calculated for all five generated canopies, and the leaf azimuthal angle PDF was calculated for the Grape N-S and Grape E-W cases . The L was calculated by summing the one-sided area of all leaves in the canopy and dividing by the total canopy footprint area. The leaf inclination angle and leaf azimuthal angle were calculated for each of the elements from the surface normal of the leaf, and a PDF was formed by weighting each element’s contribution to the PDF by its surface area, then normalizing such that the PDF integrates to unity. The corn model had predominantly vertically oriented leaves, while the almond and potato models had leaves closer to horizontal on average . Grape leaf inclination skewed toward vertical, and leaf azimuth tended to be oriented parallel with the row , which is supported by previous manual and LiDAR measurements. The gap fraction was calculated from the 3D models by computing the fraction of direct sunlight not intercepted when qs = 0 . Gap fraction values ranged from 80% in the grape canopy cases down to 21% for the corn canopy case. Although both potato canopy cases had the same L, round pot their gap fractions were 22% for uniformly spaced plants and 36% for row-oriented plants.Results for the daily total light interception on Julian days 153, 232, and 305 are listed in Table 2.2, and shown graphically in Fig. 2.5. For the homogeneous canopy cases, very close agreement was found between the 1D and 3D models regardless of L, which indicated that the approach used to compare the 1D and 3D models was consistent and that leaf-scale heterogeneity created by discrete leaf surfaces did not create significant errors. Cases with relatively high ground cover fractions and uniformly arranged plants showed good agreement between the 1D and 3D models regardless of whether the assumption of leaf isotropy was made. As the canopies became more heterogeneous in space, agreement between the models generally declined. Although Potato-Uniform and Potato-Row had identical leaf area indices and leaf angle distributions, the regular distribution of plants within the canopy in Potato Uniform resulted in better agreement between the 1D and 3D models compared to Potato-Row. Despite all cases having highly anisotropic leaf inclination distributions, the assumption of leaf anisotropy had relatively small impact for all cases except for the Grape E-W case. Any effects of heterogeneity or anisotropy tended to decrease as the day of year became further away from the summer solstice. Toward the end of the year , the 1D and 3D models were in fairly good agreement for all canopy cases.The diurnal flux of radiation intercepted by the canopy for an hourly time step on Julian day 153 is shown in Fig. 2.6, with corresponding fractions of total radiation intercepted by the canopy shown in Fig. 2.7. The fraction of total radiation intercepted on Julian day 253 and 305 are shown in Figs. 2.8 and 2.9, respectively. For the homogeneous canopy cases, the assumptions of vegetation homogeneity and isotropy were closely satisfied, and therefore, the 1D model was in very good agreement with the 3D model regardless of leaf density . For the crop canopy cases, the 1D model consistently over estimated light interception as compared to the 3D model, except for Grape E-W and Potato-Row on Julian day 305. For all but the grape cases, eliminating the isotropic assumption resulted in little improvement of agreement between the 1D and 3D models, indicating that errors arose primarily from heterogeneity in these cases. For the Grape N-S, Almond, and Potato cases, errors between the 1D and 3D models were largest near midday when sunlight could most readily reach the ground by penetrating through gaps in vegetation. For Grape E-W, the largest discrepancies occurred at early and late times of the day. The effect of row orientation on diurnal interception patterns for the grape cases was dramatic, as this completely changed the character of interception at different times of the year . The potato cases also illustrated the pronounced effect of heterogeneity in planting pattern on diurnal interception patterns.Figure 2.10 depicts vertical profiles of the absorbed radiation flux at 8:00, 10:00, and 12:00 hours on Julian day 153 for Grape N-S, Grape E-W, Almond, and Corn. Errors in absorbed fluxes for Grape N-S were relatively consistent with height, where errors at a given height were most closely related to the magnitude of the absorbed flux at that height. This was also roughly the case for Almond, except that there was the potential for some under estimation of absorption in the lower canopy when the 1D model was used, which was most pronounced for larger solar zenith angles. For Grape E-W, the 1D model tended to shift the peak in the absorbed flux deeper into the canopy, which was most pronounced for larger solar zenith angles. In the corn canopy, the vertical pattern in radiation interception differed significantly between the 1D and 3D models. There were up to 50% differences between 1D and 3D fluxes at a given vertical level, with irregular patterns of over or under estimation. In the lower canopy, there was a peak in absorption in the 3D model, which was largely absent in the 1D model, leading to under prediction of absorption by the 1D model in the lower canopy. This is likely due to the substantial over prediction of absorption by the 1D model in the upper canopy, which removes the necessary energy needed to produce a secondary peak in absorption in the lower canopy.As evidenced by the homogeneous canopy cases, the impact of scattering was minimal for the PAR band, and thus most of the errors in the heterogeneous crop canopy cases are due to heterogeneity and not scattering. In the NIR band, scattering caused very large errors when the standard 1D model was used, with all cases over estimating absorption by more than 100%. Accounting for reflection using Eq. 2.5 removed much of this energy, but still resulted in significant over estimation of absorbed radiation. This approach creates offsetting errors in which absorption is over estimated because transmission is not accounted for, but there is simultaneous over estimation due to the assumption that all reflected radiation either reaches the ground or is reflected back to the sky . Accounting for both reflection and transmission using Eq. 2.6 caused net under estimation of the total absorbed flux. This is because Eq. 2.6 assumes that all reflected and transmitted radiation radiation either reaches the ground or is reflected back to the sky, thus overall absorption is under estimated.Overall, anisotropy in leaf inclination had a relatively small effect on errors resulting from the application of Beer’s law in cases when leaf azimuth was uniformly distributed .