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crop reflection is computed fairly accurate with 
these models, the directional distribution of the 
reflected radiation is oversimplified. However, 
remote sensing is based on measuring the radiation in 
one single direction, so for the use in an remote 
sensing environment this distribution must be 
modelled more carefully than in these models. 
The third model that is studied is the model of 
Chen (1984). This model is based on the adaptation 
of the Kuebelka-Munk equations (Kuebelka & Munk, 
1931) to matrix-vector algebra. In this models 
fluxes are presented as vectors and reflection and 
transmission properties of a crop layer as matrices. 
Chen distinguishes 324 different directions (36 
azimuthal, 9 inclination classes), so vectors have 
324 and matrices 104976 (= 324~2) elements. The 
evaluation of the double inversion of the matrices 
involved in the calculations is so laborious even for 
a large main-frame computer that for practical 
reasons this model can only be used when a very 
simple crop geometry is assumed. 
7 DEMANDS FOR A CROP REFLECTION MODEL 
Before we present our own models, we will 
recapitulate the demands of a model that fulfil the 
stated requirements. These demands are: 
1. The model must be applicable for layered crops 
(crops with a vertical component in the description 
of the crop properties), for instance to model 
flowering; 
2. The model must not limit leaf density and leaf 
distribution functions; 
3. It must be possible to handle different types of 
leaf surface reflection properties; 
4. The spatial distribution of the incoming 
radiation, the direction of the sun and the ratio 
between diffuse and direct incoming radiation may not 
be limited by the model; 
5. Computations with the model must yield the 
radiation intensity in any direction, at least within 
a cone around the zenith with a half top angle of 45 
degrees; 
6. At last, computations with the model must be 
practically carried out on a normal computer, so 
neither program seize, nor computing time may exceed 
reasonable limits. 
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8 BASE OF OUR MODELS TURTLE AND HARE 
On the principles as described in the former section, 
two discrete models for crop reflection are 
developed: TURTLE (The Universal Reflection and 
Transmission model for Layered crop Experiments) and 
HARE (Handy and Accurate Reflection model for crop 
Experiments). The base of these models is: 
1. Because of the transparency of the air inside 
the crop, the radiation regime in the crop is only 
affected by the leaves. Therefore the vertical axis 
is expressed in LAI in stead of meters. 
2. A crop is separated in thin layers. Each layer 
is assumed to be so sparse that mutual shading and 
other in-layer interactions may be neglected: the 
intercepted and remitted fraction of a ray of light 
will not be intercepted again in the same layer. 
Practically an LAI of 0.1 or less suits for this 
purpose. 
3. Each model layer may have different properties 
for reflection and transmission of the leaves and for 
the leaf density function. If desired, calculations 
may be repeated for each wave length band, using the 
same geometrical definition of the crop. 
4. A set of 46 direction vectors is defined, each 
vector representing a pentagonal or hexagonal cone 
around it. All cones cover an equal solid angle of 
0.137 sr., so together they cover a hemisphere. All 
angles between adjacent directions are 0.40+0.02 rad, 
so a fairly regular pattern of reference directions 
is created (figure 2). These directions are used for 
two purposes: 
- as reference directions for rays; 
- as normal-vectors on leaf planes. 
The equality of all represented solid angles prevents 
the use of many weight factors in the calculations. 
Figure 2. Distribution of polygons over a hemisphere 
as used in our models. For some directions the 
reference directions are also drawn as arrows, 
originating in the centre of the sphere. 
5. The models permit that leaves show specular 
reflection, diffuse reflection and diffuse 
transmission. Both types of reflection may depend on 
the angle of incidence. 
6. Reflection and transmission coefficients of 
leaves and soil may be chosen freely. 
7. Any arbitrary leaf angle distribution can be 
modelled by assigning weight factors to all 46 leaf 
plane directions. 
8. Two options for the soil reflection pattern are 
built in: the first is a flat surface with 
Lambertian reflection properties, the second is a 
rugged surface. 
9 CALCULATION STEPS 
After the desired crop and soil properties are 
chosen, the calculations are carried out according to 
the following steps: 
1. For all differen" crop layers four different 
matrices are computed, two for layer reflection 
(upperside and lower side) and two for layer 
transmission (downward and upward). Each element 
(j,i) of each matrix represents the fraction of the 
flux coming from direction i that is remitted to 
direction j. The elements (i,i) of the transmission 
matrices include also the transmission in direction i 
through the interleaf-spaces. As the assumption is 
that no double interactions occur within one layer, 
each element can be computed as the sum of the 
contribution of all leaves, counted over 46 leaf 
directions. 
Figure 3. Interaction of a leaf with the intercepted 
radiation. Notice that leaf reflection sometimes 
yields layer transmission (and vice-versa).