Delta SS1-UM-1.05 User Manual download pdf (Page 56)

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• LAI theory Document code: SS1-UM-1.05

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 I

and the Leaf Area Index L is given by:

exp( )

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 tan

(

) is expressed tan(

)

The extinction coefficient, K, is calculated as follows:

Where:

x is the ELADP

θθ

θ is the zenith angle of the direct

beam.

The transmitted fraction of incident

direct light is given by:

dir

exp( )

K( ),x

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.

K( ),0

K( ),1

K( ),100

K( ),x

tan( )

1.702 ( )x 1.12

0.708

Ext'n

Coeff't

0° 90°

1 2 ... 51 52 53 54 55 56 57 58 59 60 61 ... 85 86

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Delta SS1-UM-1.05 User Manual Page 56

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