Delta Electronics SS1-UM-1.05 user manual Transmission fraction τ Is given by I/I0

The next section derives the transmission of light from a uniform overcast sky through a uniform infinite canopy of black leaves of constant LAI with an ellipsoidal leaf angle distribution.

Let the sky have uniform brightness of 1 per steradian over the hemisphere. The radiance of a strip around the sky at angle θ is given by:

R 2.π . sin( θ ). dθ

and the irradiance on a horizontal surface due to that strip is given by

I 0 2. π .sin ( θ ).cos ( θ ). dθ

The total irradiance due to the hemisphere is obtained by integrating over the complete sky area:

π

2

2.π .sin ( θ ).cos ( θ ) dθ = 1 π

0

For each strip of sky, the transmitted radiation is given by

I I 0. exp( K.L )

where K is the extinction coefficient from Campbell, so the total transmitted radiation is

π

2

2. π

.sin( θ

). cos ( θ

). exp(

).L ) dθ

I

K( x , θ

0

and the transmission fraction τ

is given by I/I0

π

τ

1 .

2

. π .

θ

.

θ

.

θ

.

θ

( x , L )

π

2

sin (

)

cos (

)

exp(

K( x ,

)

L ) d

diff

0

This integral was evaluated numerically over the range x = 0 to 1000 and L = 0 to 10, and is graphed below for three different values of x.

SunScan User Manual v 1.05 LAI theory • 57