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
.
