Light interception by mature trees

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Light interception can in ForGEM either follow a raytracing approach, similar as is used in the model SORTIE (Pacala et al. 1993) or horizontal light extinction only per light grid (that may deviate in size from the soil grid), where the integration interval is determined by the top and bottom of each individual crowns present at the grid.



Light interception by a mature tree will be modelled under the following assumptions:

  1. the light environment at the edge of the crown, as modelled with the ray-tracing approach described above, can be considered equal to that at the centre of the crown for the same height h.
  2. all of a tree's leaf area is located on the outer envelope of the crown
  3. the tree crowns can be modelled as surfaces of revolution obtained from a convex curve

LightInterceptionFlowChart.png

Figure 1: Simplified flowchart of the light interception model. The calculation of the sky map and of the selection of the sampling rays is carried out at the beginning of each simulation time step once for the entire simulation area, while the ray-tracing routine is point dependent and must be run for each simulation object (tree or regeneration patch)

Under assumption 1, the absorbed photosynthetically active radiation (APAR) for a unit surface of height h is then a function of the light environment and leaf area at that height (independently of where the leaf area is located on the horizontal plane). For convenience of representation we express height not in terms of absolute height, but in terms of distance from the tree top. The APAR of a tree will be defined then as:

(9)

ForgemFormula 09.png

where fAPAR is the fraction of incident light absorbed per unit of leaf area, LA(h) is the leaf area at height h, PAR(h) is the incoming PAR at height h and Cd is the canopy depth. The value of LA(h) can be obtained from allometric functions or from empirical data.

CrownSlopeEffect.png

Figure 2: Effect of the crown slope ? on the portion u of visible sky of zenithal angle z when z > ?. The rays coming from an azimuthal angle of size u arrive unobstructed from the crown.

SkyProportion.png

Figure 3: The proportion of the sky of zenithal angle z self-shaded by the crown as a function the inclination of the crown.

PAR(h) can be calculated using the light attenuation module for point [x, y, ztop-h], where ztop is the elevation of the top of the tree canopy. The calculation incorporates an additional attenuation term for the self-shading effect of the canopy. Under the assumptions 2 and 3, it is possible to consider the portion p of the infinitesimal section of sky hemisphere of zenithal angle i that is self shaded by the canopy as a function of the angle of the tangent to the canopy profile curve ? (Figure 2):

(10)

ForgemFormula 10.png

this is equivalent to say that for a proportion p of the crown surface at that height, the rays coming from a zenithal angle z have one additional hit from the rest of the crown before they reach the crown surface. Figure 3 shows the graphical representation of Equation 10. It is possible to calculate from the equation of the tangent angle ? from the equation of the revolution curve:

(11)

ForgemFormula 11.png

where d' is the derivative of the crown diameter as a function of h. Based on these assumptions, we calculate the APAR of each tree using the following algorithm:

1. To calculate the light environment, n points are selected at crown depths hi, defined as:

(12)

ForgemFormula 12.png

2. For each point, a proportion of hits to add to each sample ray is calculated based on the shape of the crown in the section it refers to:

(13)

ForgemFormula 13.png

The integration is solved numerically off-line for height classes, and interpolated to the tree's height using a lookup table.

3. The term calculated in (13) is added to the calculation of (8). is estimated using equation 5. APART is then calculated as:

(14)

ForgemFormula 14.png