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 <l-c m > • (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 
- 
£<S>du 
J 
r , 
L 
z 
( 
z 
p(z) exp 
- 
pdu 
exp - 
pdu 
. 
. 
0 4 
-100 
(3) Figure 5.