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Theoretical research on a model for predicting the shadow boundary of an individual conical crown on a slope

ZHANG Lei, YU Hai-Ye~*, YANG Hao-Yu, ZHANG Zhi-Ping (Key Laboratory for Terrain-Machine Bionics Engineering of MOE, School of Biological and Agricultural Engineering, Jilin University, Changchun 130022,China).  
Tree-shading could represent solar radiation intercepted by the tree crown. Understory light environment also had a close relation with tree-shading. The conical crown was studied because it was familiar and its shadow, whether on a flat plane or slope, was more complex than other crown profiles. The shadow profile of an individual conical crown on a slope was an ellipse or a closed curve composed of an ellipse and its two tangents, which depended on the relationship between the solar altitude (h_s) and the base angle (A) of the cone. In the rectangular plane coordinate system,a theoretical model was developed, which took tree height (H), height below branch (h), crown radius(r), longitude (λ), latitude (φ), height (ht), slope(α), aspect (β), date (d) and time (t) into account to describe the shadow boundary of an individual conical crown on the slope. The model was based on geometrical optics theory and the corresponding and imitating principle between shadows on a flat plane and a slope. In the simulation test, the results of Spearman and Kendall's tau-b test showed all correlations between measured and calculated values of boundaries were not less than 0.928 and were significant at the 0.01 level (2-tailed) from 8:00 to 14:00. The model, therefore, was suitable to describe the shadow boundary of an individual cone, and the more the actual profile of the crown resembled a cone, the more accurate the simulated results. Furthermore, this model was practically used to forecast the variable scale of tree-shading of an individual tree species, Pinus sylvestris, in which the shadow of the crown always lied, from 8:00 to 14:00 on May 12~ th . Theoretically, this scale reached 0.69H on the east side of the tree, 1.18H on the west, 0.2H on the south and 0.4H on the north. The result was in accordance with the apparent motion rules of the sun and the correlation was 0.983 between observations and calculations, though it was different from existing research. There were two possible reasons for this difference. One was that two scales were in different time spans respectively. The other reason was that this paper was based on the apparent motion rules of the sun and did not consider any aspects of plant ecology. Advantages and limitations of the model are listed which define how to use this model well with interrelated research.
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