SLIM Software on Science Note publication
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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.
Beam (a) direction represented as a vector on radian coordinates:
Ellipsoid (Crown envelope) equation is:
Where:
hr : Horizontal radius
vr : Vertical radius
Beam interception on a crown is found from quadratic
equation below, which is obtained from substitution of the vector in
ellipsoid equation:
Where a, b and c:
Thus, if 
then the beam intercepts the crown.
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.
- 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:
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 
,
Press, W. H., S. A. Teukolsky, et al. (1992). Numerical recipes in
C: the art of scientific computing, Cambridge University Press.
Simple Light Interception Model
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