262
Figure 2. Canopy hot-spot over grassland in the
visible.
vegetation canopy. If the reflected radiation is
due to single scattering from leaves, then, under
the assumption that the leaves are Lambertian
reflectors, the probability, P(3,S? Q ), of seeing an
illuminated leaf at a view angle $ due to solar
radiation incident at an angle is proportional to
the radiance reflected by the canopy in the
direction 5?. Figure 4 shows P(st,) for S? 0 equal to
0 degrees (radiation normally incident) and for s!
varying between 0 and 65 degrees. The canopy
consists of 10 layers spaced 10 cm apart and the
area coverage, c n , of leaves per layer n is 30 X,
equivalent to a leaf area index of 0.3 per layer.
The leaves' horizontal positions are chosen at
random, with overlapping allowed. The three curves
in Fig. 4 show the results for leaf sizes 5x5 cm 2 ,
10x10 cm 2 , and 15x15 cm 2 . All results show a
distinct maximum at 0° and decrease to a constant
58 X for view angles greater than 60°. The
probability to see any illuminated leaf is less than
100 X at 0° and can be computed as
N
P 0 - } c n /i • (1)
n=l
which is 0.972 for c = 0.3 and N = 10. The
asymptotic value for large view angles, P a , can also
be derived. Assuming no correlation between
illumination and view directions we can express the
probability to see an illuminated leaf as
N 2
p a ‘ } S % <2)
n=l L J
for which we obtain P a = 0.588 for our case. It is
obvious from Fig. 4, that the angular width of the
hot-spot and the slope of its radial intensity
distribution change distinctively for different leaf
sizes showing a much more peaked maximum for the
smaller leaves.
The Monte Carlo/ray tracing model described above,
can be substantially refined and generalized.
Analytical solutions can also be obtained under
certain assumptions.
In those cases where the characteristics of the
canopy are azimuthally independent, a
two-dimensional model, consisting of a vertical
slice of the canopy, is sufficient to determine
P($,$ 0 ). For these two- dimensional models we have
Figure 3. Canopy hot-spot over coniferous forest
taken with Kodak Aerochrome infrared film
2424.
Monte Carlo Model for HOT SPOT
0.9-
S 0.8-
I
£
8 0.7-
ÛH
0.6-
U.O : i : ! !
0 10 20 30 40 50 60 70
Angle (Degree)
Figure 4. Monte Carlo/ray tracing model predicts
canopy-hot-spot angular spread as a
function of leaf size.
been aole to determine P(,Sf Q ) analytically. As tor
the Monte Carlo/ray tracing model, we assume that
leaves are Lambertian reflectors that are uniformly
distributed within a given layer and independently
distributed within different layers. In addition,
as in the Monte Carlo model, the influence of the
ground is neglected. Using these assumptions, we
have derived expressions for P(S?, S? Q ) when the layers
are either discretely or continuously distributed in
the vertical direction. The latter case can be
derived from the former by letting the distance
between layers tend to zero while keeping the leaf
area index (LAI) constant. We will present here
only those results for a continuous distribution of
layers.
We obtain for a stratified, horizontally
homogeneous canopy with a leaf length i (diam.)
where, si
independen
and view a
respective
canopy, p
If L is le
P(2,2 0 ) -»
When p is
P(2.2 0
Here, pL
be express
values of
expressior
l/2(l-exp(
identical
by taking
The exf
(5) are ft.
and the s
the height
eqn. (3)
angular vs
the effec
maximum pi
angle anc
retro-dire
the prc
retro-dire
incident
stronger (
being ab]
angles clt
influence
lengths fi
size, inci
and consec
both the
the same \
the incic
broader he
These si
any transn
1.0
Vc
Sc
Lj
& °-8
-4—^
1
■8
c.
^ 0.6
u
c_
,
z
p(z) exp
-
£du
J
r ,
L
z
(
z
p(z) exp
-
pdu
exp -
pdu
.
.
0 4
-100
(3) Figure 5.
