4 Simple Light Interception Model (SLIM)

4.1 Main algorithms

4.1.1 Light index computation.

The amount of light received at any point in space is calculated by exploring a range of directions (combination of azimuth and zenith angles). Each
time a beam originating from that point intercepts a crown envelop of a given porosity it reduces its contribution correspondingly. Total canopy openness at that
point is obtained by summing up results for elementary beams. Weight of each beam is determined by the relative surface of the associated sky vault fraction.

Tree crowns are represented as convex hulls and Crown envelope is approximated by a revolution ellipsoid in the SLIM stand alone version. The crown is reconstructed to 3D geometry object using triangulation algorithm described in the STReTCH section.

Beam collision is detected troughs 3D line – triangle intersection detection.

4.1.2 Light interception by trunk

Trunk is represented as a cone. If the vertex of cone is (x0, y0, z0), direction numbers for the axis are (a, b, c), and the vertex angle is theta, then the equation of the cone is:

Solve this equation with the parametric equations for x, y, and z of the line, will found the intersection between trunk and light beam (line).

Trunk is assumed to be opaque.

4.1.3 Beam weighting model

• None, option gives equal weight to each direction sampled.
• UOC (uniform Overcast Sky), weights each direction according to the relative surface of the sky vault explored by each beam (a).
  
• SOC (Standard Overcast Sky), weights each direction according to surface of sky vault fraction moreover assuming a decrease in light intensity from zenith to horizon using the formula:
  

4.1.4 Simple Vertical Light Calculation

A simple vertical light calculation algorithm is implemented in both models (but only of use in SExI. This simplified light calculation estimates the canopy openness for each cell specified. It measures canopy openness vertically, by testing a single vertical direction towards the sky. If the ray intersects one or more crowns then the vertical canopy openness is set to the corresponding value of permeability of the series of crown intersected. The value of vertical canopy openness is computed for the center of the targeted cell and the neighbors, and averaged. Default neighborhood comprises the 8 surrounding closest cells.

The average of canopy openness on each grid cell is used in the recruitment process to assess suitability of light regime.

4.1.5 Topography

Bilinear interpolation (Press, Teukolsky et al. 1992) is used to determine exact altitude of tree base when trees are positioned on a existing topography map

Pt is tree location and P1, P2, P3, P4 are topography data. Then:

Where , .

4.2 SLIM Data Structure

4.2.1 Tree Geometry Data Structure

Below are tree measurements that applied for SLIM.
• Base Location (x, y)
• Crown Location (x’, y’)
• DBH
• Height (H)
• Maximum Crown Width or Average Crown Width (MCW)
• Height of Maximum Crown Width (HMCW)
• Height of Crown Base (HCB)

Other:

• Crown Vertical Extension (CVE) = H– HCB
• Crown Vertical Radius (CVR) = H - HMCW

4.2.2 Input File Format

Header: "version=1c;TreeBasic"
Data (separated by semicolon): label; color; x_base; y_base; altitude; height; dbh; x_crown; y_crown; crownVerExt; crown_width; crown_vert_radius; porosity; n_layer Sample:

version=1c;TreeBasic
              010;2;9.1999;19.1;16.952;17;0.298;9.1999;19.1;15.5;8.2;12;0.5899;1
              011;0;13;7;19.922;16.0146;0.29;13;7;5.3798;4.1;2.6899;0.5;1
              012;1;12;2;22.59;23.83;0.4157;12;2;2.977;6;21.3397;0.5;1
              013;3;16;10;19.05;21.07879;0.25;16;10;2.9448;4.2;1.59;0.5;1
              014;2;16;9;19.33;16.7956;0.2587;16;9;14.92;9;7.853;0.5899;1

Header:"version=1d;TreeBasic"
Data (separated by semicolon): label; color; x_base; y_base; altitude; height; dbh; x_crown; y_crown; height_of_crown_base; height_of_maximum_crown_width; crown_width; porosity; n_layer Sample:

version=1d;TreeBasic
              0001;2;4.1;7.1;0;23.47;0.32;3;3;19.48;19.48;6;0.5;1
              0003;4;7.9;13.2;0;34.31;0.707;8.3999;11;19.89;19.89;15.4;0.5;1
              0004;8;2.1;14.1;0;27.4378;0.435;3.4;13.8;19.598;21.068;3;0.5;1
              0007;3;11.1;15.8;0;16;0.25;11;14;15;15;4.6;0.4;1

4.2.3 Color Code:

                0	black
                1	blue
                2	red
                3	green
                4	magenta
                5	cyan
                6	yellow
                7	gray
                8	orange
                9	pink
                10	LightGray
                11	White
                12	DarkGray

4.3 Light Calculator GUI

Figure 1. Light Calculator GUI

4.3.1 Insolation

Insolation setting defines the detail of sky fraction, which represented by the number of light beams vector and their weighting model. Number of beams defined by the number of inclinations and azimuths.

Figure 2. Horizontal projection of beams vector

Lowest Zenithal Angle defines the lowest angle consider for light calculation and the interaction area is limited by the Distance of Interaction setting. The trees outside the radius of interaction distance and are not included for current target calculation.

Figure 3. Vertical projection of beams vector

4.3.2 Interception and Environment

• Crown Permeability If selected then the tree crown is considered as transparent, if not selected that it’s assumed to be opaque.
• Trunk Interception If selected then the trunk is considered to intercept the light, if not then it’s neglected.
• Topography If selected then the plot will use the topography data (if exist), if not then the plot is assumed to be flat.
• Torus space If selected then the plot is assumed to be toric, in such case the plot has no borders as the trees from one side of the plot act as neighbors for the trees on the opposite side. If not selected then the plot is limited by the border (Note that the area outside the border is not considered as an open area, the calculation is modified so that the beam is just come from inside the border).

4.4 Object Design

The main objects in SExI are a Forest, which is, consists of Trees. LightCalculator implemented all SLIM modules. LightCalculator then being part of GrowthCalculator where the main loop of SExI is implemented, there are also Recruitment and Mortality modules inside. User interfaces control all user interaction on the object. 3D Visualization modules are useful for user to control their creation of a forest in 3D environments.