User manual

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( )
.
KL
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
(),xL
.
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.