The meaning of the canopy angle distribution parameter: χ

Biomass production by a plant canopy is directly related to the amount of photosynthetically active radiation (PAR) the canopy intercepts. This, in turn, is related to the PAR incident on the canopy and the fraction of the incident PAR that is intercepted. The leaf area index (LAI) and the architecture of the canopy determines what fraction of the incident radiation is intercepted. Rather than computing directly the fraction intercepted, we compute the fraction transmitted, which is also the probability that a ray can pass through the canopy with striking a leaf.  It is:

Equation 1


where L is the LAI of the canopy and K is an extinction coefficient. We assume that the leaves are randomly distributed in space. K can be computed as the average projected area of a leaf on a horizontal surface divided by the actual area of a leaf. A useful formula for computing K for a wide range of canopy architecture is

Equation 2


where Ψ is the zenith angle of the ray penetrating the canopy, χ is the canopy angle distribution parameter, and A(χ) is a normalizing function of χ that can be found in Campbell and Norman (1998). The point of this note is to explain what χ is and how to measure it.

Equation 1 gives a way to find the fraction of radiation on a horizontal surface below an extensive but thin layer of canopy with thickness L. A different formula is used if we want to know the probability of a ray at angle Ψ penetrating a bush or tree with a spheroidal shape. For this, we define a leaf area density, ρ, which is the area of leaves per unit volume of space, and a distance, S, which is the distance the beam travels through the canopy. A different extinction function is now defined, the G function, which is the ratio of the projected area of an average leaf, projected on a surface normal to the beam. Thus

Equation 3


The fraction of light getting through a defined clump of vegetation is

Equation 4


To find the meaning of χ, we can measure the transmission of light through a representative and uniform clump of canopy of equal depth and width. The vertical transmittance will be t0, and the horizontal transmittance will be t90.  Using Equation 4 we can write

Equation 5


Since S and ρ are the same for both directions, they divide out. Now, combining Equations 2 and 3 we obtain

Equation 6


From Equation 6, G0 = χ/A(χ) and G90 = 1/A(χ), so (using Equation 5),

Equation 7


Equation 7 is the definition of χ that we seek. Three quick examples show how it works.  Assume a canopy of perfectly horizontal leaves with a vertical gap fraction (t0) of 0.1. Since the leaves are horizontal, t90 = 1, so, from Equation 7, χ is infinity (the logarithm of 1 is zero).

The second example is a perfectly vertical leaf canopy.  Here t90 = 0.1, t0 = 1, and, using Equation 7, χ = 0, which is correct for a vertical leaf canopy.

The third example is for a canopy with a spherical angle distribution. Here the gap fraction is the same horizontally and vertically, so assume t90 = t0 = 0.1. From Equation 7, χ = 1, which is correct for a spherical distribution.

Measuring or estimating χ, therefore requires only a measurement or estimate of the gap fraction of a uniform volume of canopy for a fixed distance in the vertical and horizontal directions. Light interception models aren’t sensitive to errors in χ, so a visual estimate of t is adequate for almost all purposes.  Just look at a clump of vegetation from above and estimate the transmission. Then look at the same clump horizontally and estimate the transmission.  Then use Equation 7 to estimate χ.

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