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.