Delta Electronics SS1-UM-1.05 - Manual (Page 57)

SunScan User Manual v 1.05 LAI theory
••••
57
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
d
θ
and the irradiance on a horizontal surface due to that strip is given by
I 0
. . . .
2
π
sin ( )
θ
cos ( )
θ
d
θ
The total irradiance due to the hemisphere is obtained by integrating over the
complete sky area:
= d
0
π
2
θ . . .
2
π
sin ( )
θ
cos ( )
θ
1
π
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
I d
0
π
2
θ . . . .
2
π
sin ( )
θ
cos ( )
θ
exp( )
.
K ( ) , x
θ
L
and the transmission fraction
τ
is given by I/I
0
τ
diff(
) , x L
.
1
π
d
0
π
2
θ . . . .
2
π
sin ( )
θ
cos ( )
θ
exp( )
.
K ( ) , x
θ
L
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 .