Leaf Area Index and Light Transmission Through Canopies
249
distributions are not particularly easy to work with, so we make another
assumption to simplify the math; namely, that the area of a single leaf
a is much smaller than the ground area, A. This certainly is reasonable
for many realistic examples. If the mathematical limit of (1 - u / A ) ~
is
taken as a/A + 0, then (1 - u / A ) ~ -- exp(-NalA), where Na/A
is the leaf area index. Essentially this amounts to replacing the binomial
distribution with the Poisson distribution. Interestingly, this result for
horizontal leaves is independent of the incident angle of the light, because
the shadow of a horizontal leaf cast onto a horizontal plane is the area of
the leaf no matter what direction the light comes from.
A quick look at vegetation reveals that leaves are almost never oriented
entirely horizontally; they have a distribution of orientations. However,
the horizontal shadows of black leaves can always be treated as "horizontal leaves" no matter what the leaf orientation. Although these shadows
have a range of sizes, they all are random relative to each other and small
compared to the area, A. This is all that is required for exponential extinction to hold. Therefore, no matter what the orientation distribution of
leaves, the fraction of the leaf HSAI that is projected onto the horizontal
plane from a particular zenith angle I,+ can be calculated: We refer to
this fraction as the extinction coefficient, Kb(I,+). One can also think of
Kb (I,+) as the mean beam flux density on an average illuminated leaf in the
canopy divided by the beam flux density on the horizontal plane above the
canopy. Clearly, for a canopy of perfectly horizontal leaves, Kb(I,+) = 1.
The azimuth angle of the incident radiation may also be important in
estimating Kb(I,+), but we deal only with canopies that have leaves symetrically distributed about the azimuth (compass directions), which is
a good assumption for almost all canopies. The extinction coefficient
is therefore assumed independent of solar azimuth. Later we calculate
Kb(I,+) for various leaf-angle distributions and remove the assumption of
"black" leaves.
If flat leaves in a canopy of leaf area index L, are randomly distributed
in space, then the fraction tb(I,+) of incident beam radiation from zenith
angle I,+ that penetrates the canopy is
rb(I,+) = exp(-Kb(llr)Lr)
(15.1)
where Kb(I,+) is the canopy extinction coefficient just described. When
leaves are clumped (not randomly distributed), canopy transmission can
often still be approximated by an exponential function of L,, but L, is
multiplied by a clumping factor Q to account for the fact that leaves
are less efficient in covering the ground than when they are randomly
distributed. Row crops with leaves clumped in the rows may intercept
only 70 to 80 percent of the radiation they would if their leaves were
randomly distributed in space.
The fraction of incident beam radiation intercepted by the canopy
is 1 - -cb(I,+). This fraction is available for scattering, transpiration,
and for photosynthesis. The fraction of beam radiation that is not intercepted by the canopy (rb(I,+)) reaches the soil surface and is available
249
distributions are not particularly easy to work with, so we make another
assumption to simplify the math; namely, that the area of a single leaf
a is much smaller than the ground area, A. This certainly is reasonable
for many realistic examples. If the mathematical limit of (1 - u / A ) ~
is
taken as a/A + 0, then (1 - u / A ) ~ -- exp(-NalA), where Na/A
is the leaf area index. Essentially this amounts to replacing the binomial
distribution with the Poisson distribution. Interestingly, this result for
horizontal leaves is independent of the incident angle of the light, because
the shadow of a horizontal leaf cast onto a horizontal plane is the area of
the leaf no matter what direction the light comes from.
A quick look at vegetation reveals that leaves are almost never oriented
entirely horizontally; they have a distribution of orientations. However,
the horizontal shadows of black leaves can always be treated as "horizontal leaves" no matter what the leaf orientation. Although these shadows
have a range of sizes, they all are random relative to each other and small
compared to the area, A. This is all that is required for exponential extinction to hold. Therefore, no matter what the orientation distribution of
leaves, the fraction of the leaf HSAI that is projected onto the horizontal
plane from a particular zenith angle I,+ can be calculated: We refer to
this fraction as the extinction coefficient, Kb(I,+). One can also think of
Kb (I,+) as the mean beam flux density on an average illuminated leaf in the
canopy divided by the beam flux density on the horizontal plane above the
canopy. Clearly, for a canopy of perfectly horizontal leaves, Kb(I,+) = 1.
The azimuth angle of the incident radiation may also be important in
estimating Kb(I,+), but we deal only with canopies that have leaves symetrically distributed about the azimuth (compass directions), which is
a good assumption for almost all canopies. The extinction coefficient
is therefore assumed independent of solar azimuth. Later we calculate
Kb(I,+) for various leaf-angle distributions and remove the assumption of
"black" leaves.
If flat leaves in a canopy of leaf area index L, are randomly distributed
in space, then the fraction tb(I,+) of incident beam radiation from zenith
angle I,+ that penetrates the canopy is
rb(I,+) = exp(-Kb(llr)Lr)
(15.1)
where Kb(I,+) is the canopy extinction coefficient just described. When
leaves are clumped (not randomly distributed), canopy transmission can
often still be approximated by an exponential function of L,, but L, is
multiplied by a clumping factor Q to account for the fact that leaves
are less efficient in covering the ground than when they are randomly
distributed. Row crops with leaves clumped in the rows may intercept
only 70 to 80 percent of the radiation they would if their leaves were
randomly distributed in space.
The fraction of incident beam radiation intercepted by the canopy
is 1 - -cb(I,+). This fraction is available for scattering, transpiration,
and for photosynthesis. The fraction of beam radiation that is not intercepted by the canopy (rb(I,+)) reaches the soil surface and is available