476
for cherry-laurel leaves with the chlorophyll a_ and b
concentration in agreement with Horler et al. (1980).
Recently, Lichtenthaler and Buschmann (1987) propo
sed an alternative explanation of the blue shift. They
suggested that the fluorescence of chlorophyll a
may account for the observed phenomenon because
the chlorophyll a fluorescence spectrum coincides
with reflectance spectrum at the red edge and be
cause the fluorescence increases with decreasing
pigment concentration.
A quite different explanation for the shift of the red
edge observed during the growth of wheat is given
by Schutt et al. (1984). These authors found that the
leaf surface exposed to the incident light during he
ading and after emergence of the head determined
the wavelength position of the red edge due to the
amorphous nature of the cuticle layer od the lower
leaf surface. Thus the increased reflectance of the
lower leaf surface gave an indication of lower pig
ment absorption than the upper leaf surface.
Model simulations of Baret et al. (1988) demonstra
ted that the inflection point of the red edge shifts
toward shorter wavelengths when the chlorophyll a
concentration decreases. The authors derived a sim
ple semi-empirical model for the reflectance of
wheat introducing an equation for the reflectance
depending on the chlorophyll concentration and the
measured reflectance at 760 nm. Additional, three
constants are required for the model, while one pa
rameter is used for adjusting the results to the ex
perimental data.
Comparable with these theoretical findings are the
calculated results of Guyot et al. (1988). In addition
they found that the blue shift of the red edge is
correlated with the decrease of the leaf area index
(LAI ).
The aim of our work is to model the reflectance
spectra of leaves based on the stochastic description
of radiative transfer. This technique needs the opti
cal and geometrical parameters as well as the pig
ment concentrations as input parameters in contrast
to semi-empirical models where the a priori know
ledge of the reflectance spectrum is necessary.
2 THEORY
The time evolution of dynamic systems, which are
not formalized in a classical and deterministical way
, can be handled by a succession of stochastic states
of the system. In the stochastic leaf model the
states of the system are represented by different ra
diation states (diffuse solar input, reflectance at the
upper cuticle, diffuse reflectance, absorption, scat
tering, diffuse transmittance) in different structures
of the leaf (cuticle, palisade parenchyma, spongy
mesophyll).
The stochastic process is called Markov chain, when
the system can be separated in well distinguishable
finite number of states and the probability for the
appearance of every state can be described by a
chain of random numbers
If the development of the system in time is inde
pendent from the history of the process the Markov
chain is named " homogeneous ", In consequence,
homogeneous Markov chains are without memory .
The realisation of the leaf model in a Markov chain
by defining the states of the system is shown in
figure 1.
Tucker and Garratt (1977) proposed in their model to
divide the leaf in three compartments :
- the epidermal layer, which is regarded as a totally
homogeneous and transparent medium. It is re
sponsible for a partial reflectance at the cuticle.
-the palisade parenchyma, where light scattering at
the parenchyma cells and absorption in the pig
mentation occurs.
- the spongy mesophyll, where light scattering and
absorption occurs as in the palisade parenchyma.
But the scattering coefficient will be much grea
ter, because the cell density is very high and there
are many intracellular airspaces localized. The ab
sorption will be very low, because the pigment
concentration in this cell layer is very low.
In contrast to the model of Tucker and Garratt
(1977) a new compartment was introduced. This
radiation state allows partially direct transmittance
of the scattered light of the spongy mesophyll
through the palisade parenchyma.
After the definition of the states of the system
the radiative transfer from one state to another sta
te is treated as transitions of light with weighted
probability. The basis for the probabilities are the
optical , the geometrical and the physiological para
meters of the leaf.
At least one has to arrange the tranition probabili
ties in a square transition-matrix RliXjJ , which has
as much columns as states are included in the mo
del. Running the model leads to iterative multiplica
tion of an input vector at time 0, v[j](t=0) , with the
transition-matrix RlilCj]. In general vlj](t) repre
sents the probability distribution at a given time of
the process. Every iterationstep modifies the vector
to an vector v[j](t+dt). This new vector is taken as
input for the next step. After a finite number of
iterations the vector comes to a steady state if the
process is a " finite " Markov process.