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 twodimensional model of hotspot
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 lexp(LAI) or
l/2(lexp(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
retrodirection. For normally incident radiation
the probability is symmetric about the
retrodirection; 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 hotspot, 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 twodimensional model of hotspot
for leaf lengths from 1 to 15 cm.
scattering. Both effects are important tor plant
canopies, especially in the nearinfrared 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
hotspot 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 hotspot
characteristics remain almost invariant to
atmospheric perturbations in the visible and
nearinfrared 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, 10331040.
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.