s forest 
frared film 
3 0T 
predicts 
I as a 
ly. As ror 
ssume that 
e uniformly 
dependently 
n addition, 
ice of the 
imptions, we 
i the layers 
itributed in 
ise can be 
he distance 
; the leaf 
iresent here 
ibution of 
lorizontally 
diam.) 
dz + 
dz (3) 
where, since the probability is azimuthally 
independent, we have represented the solar zenith 
and view angles as scalar quantities 52 and S2 , 
respectively. Here, L is the thickness of the 
canopy, p = d(LAI)/dz, and r 5 i/| tan(S2)-tan^ 0 ) | . 
If L is less than r, 
L 
r z 
f 
r Z 1 
p(z) exp 
- 
pdu 
4 
exp 
- 
£<^du 
r 
When p is independent of height in the canopy, 
P(2,a 0 ) = \ |ex P (H> 
r 
exp(-2pr)-exp(-2pL) 
exp(-pz)exp 
(-£!!) 
V 2r ' 
dz 
(5; 
Analytic Model for HOT SPOT 
-100 -75 50 -25 0 25 50 75 100 
View Angle (Degree) 
Figure 6. Analytic two-dimensional model of hot-spot 
for solar zenith angles from 0 to 60 deg. 
Here, pL = LAI. The integral in eqn. (5) can also 
be expressed in terms of error functions. For 
values of S equal to zero or equal to Q q , this 
expression reduces to l-exp(-LAI) or 
l/2(l-exp(-2LAI)), respectively. These results are 
identical to those obtained from eqs. (1) and (2) 
by taking the limit as c (= LAI/L) tends to zero. 
The expressions for P(*S2,S2 0 ) given by eqs. (3) and 
(5) are functions of the canopy variables p, i and L 
and the angles S2 and Q q . P(S2,S2 Q ) is independent of 
the height of the observer above the canopy. Using 
eqn. (3) the effect of varying the canopy and 
angular variables can be studied. Figure 5 shows 
the effect of varying the solar zenith angle. The 
maximum probability is independent of solar zenith 
angle and occurs when the view angle is in the 
retro-direction. For normally incident radiation 
the probability is symmetric about the 
retro-direction; however, this is not true for other 
incident directions. This effect is due to the 
stronger correlation between illuminating a leaf and 
being able to see an illuminated leaf for view 
angles closer to zero degrees. Figure 6 depicts the 
influence of leaf length i on P(S2,S2 0 ) for leaf 
lengths from 1 to 15 cm. Increasing only the leaf 
size, increases the size of the holes in the canopy; 
and consequently, it increases the angle over which 
both the incident and reflected radiation can use 
the same hole. This increased correlation between 
the incident and reflected radiation produces a 
broader hot-spot, as is evidenced in Fig. 6. 
These simplistic models do not take into account 
any transmittance through the leaves or any multiple 
Analyt ic Model for HOT SPOT 
-100 75 -50 -25 0 25 . 50 75 100 
View Angle (Degree) 
Figure 5. Analytic two-dimensional model of hot-spot 
for leaf lengths from 1 to 15 cm. 
scattering. Both effects are important tor plant 
canopies, especially in the near-infrared wavelength 
region. Therefore, we developed a theoretical model 
to include mutual shading of leaves within a plant 
canopy in our multiple scattering radiation 
transport code 6 . The model also takes into account 
the vertical profiles of leaf size, leaf area index 
and leaf angular distribution. A shading function 
is derived, which gives the percentage of the 
reflected radiation intensity as a function of the 
angle relative to the sun direction. The computed 
angular distribution of the reflected solar 
radiation above the canopy thus contains the 
hot-spot effect and can be used as input for 
atmospheric radiative transfer calculations to 
obtain the radiation distribution above the 
atmosphere, which simulates the signal measured by a 1 
satellite. Initial results indicate that hot-spot 
characteristics remain almost invariant to 
atmospheric perturbations in the visible and 
near-infrared wavelength regimes 7 . 
REFERENCES 
1. G.H. Suits, 1972. Remote Sensing Envir. 2, 117. 
2. R. Greenler, 1980. Rainbows, halos, and glories, 
Cambridge Univ. Press. 
3. B. W. Hapke, 1968. Planet. Space Sci., 16, 101. 
4. K. Lumme, 1971. Astroph. Space Sci., 13, 219. 
5. N.J.J. Bunnik, W. Verhoef, R.W. deJongh, H.W.J. van 
Kasten, R.H.M.E. Geerts, H. Noordman, D. Ueni, and 
Th. A. de Boer, 1984. Proc. of 18th Int. Symp. Rem. 
Sens. Envir., Paris, France, Vol. II, 1033-1040. 
6. S.A.W. Gerstl and C. Simmer, 1986. Remote Sensing 
Envir., august issue 
7. C. Simmer and S. A. W. Gerstl, 1985. IEEE Trans. 
Geosci. and Rem. Sensing, Vol. GF.-23, No. 5, 648. 
Note, equations (3) through (5) are valid only 
when the observer is in the principal plane. 
Extensions of these results to other observation 
directions are easily obtained.