Delta Electronics SS1-UM-1.05 Beers law for canopy absorption, Transmission of Diffuse Light

Beer's law for canopy absorption

Beer's law occurs in many situations where events happen at random. In the case of light absorption by a canopy, it applies to the absorption of incident photons or light rays. For a uniform infinite randomly distributed canopy of completely absorbing leaves, it can be shown that the relationship between the transmitted light I, a beam of incident light I0 and the Leaf Area Index L is given by:

I I 0. exp( K.L )

where K is the extinction coefficient which depends on the leaf angle distribution and the direction of the beam. K=1 for entirely horizontal leaves.

Campbell's Ellipsoidal LAD equations.

Campbell (1986) derives an equation for the extinction coefficient of leaves distributed in the same proportions and orientation as the surface of an ellipsoid of revolution, symmetrical about a vertical axis. The semi vertical axis is a and the semi horizontal axis is b . There is symmetry about the vertical axis. He relates these to a single parameter x = b/a. (x is the Ellipsoidal Leaf Angle Distribution Parameter, or ELADP). The extinction coefficient also depends on the zenith angle of the incoming direct beam. Canopy elements are assumed to be completely black, and randomly distributed in a horizontal slab extending to infinity in all directions.

Note: in the following equations derived in MathCad, different conventions are used

for some symbols. Equality is represented by :=, and tan2 (θ ) is expressed tan(θ )2 .

The extinction coefficient, K, is calculated as follows:

K ( x , θ	)		x2	tan( θ )2
		x	1.702.( x		1.12 ) 0.708

Where:

xis the ELADP

θis the zenith angle of the direct beam.

The transmitted fraction of incident direct light is given by:

τ dir exp( K ( x , θ ). L )

Ext'n	3
Coeff't
K( 0 , θ )	2
K( 1 , θ )
K( 100 , θ ) 1
	0	0°	θ	90°

where L is the canopy LAI.

Transmission of Diffuse Light

Campbell's analysis applies only to a beam of light from a specific direction, which is the Direct solar beam in our case. Even under strong sunlight, the Direct fraction rarely exceeds 80% of the Total incident radiation, so penetration of the Diffuse component of incident radiation is also important.

There is a misconception that the extinction coefficient for Diffuse light is independent of canopy Leaf Angle Distribution, but this is not the case as the following analysis shows. As the following graph also shows, transmission of Diffuse light does not obey a simple Beer's law curve, so cannot be represented by a single extinction coefficient, except in the case of a horizontal LAD.

56 • LAI theory Document code: SS1-UM-1.05