# Modelling Shadow Using 3D Tree Models in High Spatial and Temporal Resolution

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## Abstract

**:**

## 1. Introduction

_{4}-species [2], and on the other hand, more shade tolerant crop species may even react positively to shading, as demonstrated by tobacco (Nicotiana tabacum L.) [3], Coffea arabica L. [4] or American ginseng (Panax quinquefolium L.) [5]. To generate realistic estimations of the shadows cast in a given AFS, the development of a model that allows for the detailed quantification of tree shading on a diurnal and annual time scale is highly demanded.

## 2. Materials and Methods

#### 2.1. The Scanned Tree and Its Location

#### 2.2. Input Data

^{−2}[22]. We used the insolation data provided by the nearest meteorological station in Freiburg, which is located about 20 km east of the study site. The data values were converted into kWh m

^{−2}for further processing. To gain general information on the shadow and to avoid a weather-related bias, we calculated the long-term monthly means of global and diffuse irradiance in hourly sums from 1977 to 2015, assuming a cloudless sky. The methods used to gain information on the dimension of the shadow and its changes during the year are described below. All following steps are implemented in the open source language R, version 3.3.2 [23].

#### 2.3. Pre-Calculations

^{6}km. To reduce the processing time, the sun beams are only calculated when the sun is in the diurnal arc, that is, if the zenith is lower than 90 degrees. Other sun positions, such as the sun being below the horizon, are not considered. Moreover, we treat all sun beams hitting the tree as running in parallel in our model, given the large distance of the sun and the comparably small size of our tree. Consequently, due to this parallel projection approach, all sun beams strike the tree in the same insolation angle in our calculations.

#### 2.4. Computing Vertices of Cylinders as a Base for Tree Shadow Projections

#### 2.5. Projecting the Cylinder Shadow onto the Ground and Calculating Energy Loss Due to Shading

^{−2}) of each cell in the grid. The monthly sums are smoothed with a simple general additive model to account for the 10-min intervals. As a first validation, the maximum possible energy gained in one grid cell without any shading effect matches the monthly sum derived directly from the measured data.

#### 2.6. Computing Ellipsoids to Simulate Leaves

## 3. Results

^{−2}(white area in Figure 5). The minimum annual solar radiation reaches 978 kWh m

^{−2}in the area under the tree crown (dark area in Figure 5). The tree crown shading is most intense in the area around five to seven meters northwards from the stem.

## 4. Discussion

^{2}grid cells [7,11] or 0.5 m × 0.5 m × 0.5 m voxels [9,13,16,24]. The high temporal resolution of 10-min intervals and the options to compute hourly, daily, weekly, monthly and seasonal dynamics of light availability at any geographical location provide the new option to study the changing patterns of interaction and competition for light of trees and understorey crops with an unknown accuracy.

## 5. Conclusions

## 6. Outlook

- Improvement of the leaf simulations. Leaf parameters vary between tree species, within the tree crown and throughout the growing season [15,16,17]. Thus, to generate realistic shadow projections of tree crowns, it is crucial to simulate leaves as realistically as possible. At present, our model simulates leaves by adding a single ellipsoid to the end of branches of a radius of less than 0.5 cm, and the ellipsoids increase in their radius each month to simulate leaf growth. We will replace these ellipsoids with more realistic leaf-like polygons, taking also their spatial distribution within tree crowns into account.
- Validation of the results generated by the model by comparing them with on-site light measurements. In case of discrepancies, the model needs to be adapted accordingly.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Cross product scheme: (

**a**) The vectors $\overrightarrow{sa}$ (sun–center) and $\overrightarrow{ab}$ (center top–center bottom) must both be perpendicular to the new vector $\overrightarrow{ac}$ through the points C,D; (

**b**) The vectors $\overrightarrow{ac}$ (center top–new point C) and $\overrightarrow{ab}$ (center top–center bottom) must both be perpendicular to the new vector $\overrightarrow{ae}$ through the points E,F.

**Figure 2.**The 3D tree model (

**a**) in the “leaf-off” mode; (

**b**) in the leafy-ellipsoidal “leaf-on” mode (here shown with 5 cm width of the minor axis of the ellipsoid).

**Figure 3.**The shadows of the tree model at three points in time with different stages of ellipsoids at 12:00 a.m.: (

**a**) 15 March (no ellipsoids); (

**b**) 15 April (2 cm ellipsoid width); (

**c**) 15 July (5 cm ellipsoid width).

**Figure 4.**Monthly grids of solar energy losses from October 2013 until September 2014 in comparison to unshaded areas.

**Figure 5.**(

**a**) Annual solar radiation distribution below the model tree along the compass directions, the outer circle representing a radius of 15 m around the tree stem; (

**b**) the 3D visualization of the tree model with the annual solar radiation distribution.

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**MDPI and ACS Style**

Rosskopf, E.; Morhart, C.; Nahm, M.
Modelling Shadow Using 3D Tree Models in High Spatial and Temporal Resolution. *Remote Sens.* **2017**, *9*, 719.
https://doi.org/10.3390/rs9070719

**AMA Style**

Rosskopf E, Morhart C, Nahm M.
Modelling Shadow Using 3D Tree Models in High Spatial and Temporal Resolution. *Remote Sensing*. 2017; 9(7):719.
https://doi.org/10.3390/rs9070719

**Chicago/Turabian Style**

Rosskopf, Elena, Christopher Morhart, and Michael Nahm.
2017. "Modelling Shadow Using 3D Tree Models in High Spatial and Temporal Resolution" *Remote Sensing* 9, no. 7: 719.
https://doi.org/10.3390/rs9070719