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.