2. A similar reflection matrix for the soil is 
derived from the soil reflection type and reflection 
coefficient. 
3a. In the TURTLE-model, the reflection of the crop 
is computed with the adding algorithm (Van der Hulst, 
1980), starting with the soil matrix and the matrices 
of the lowest layer. In succeeding steps the 
influence of one layer is added at a time until the 
top of the canopy is reached. In a second series of 
calculations, all relation matrices between incoming 
flux and the fluxes between all model layers are 
computed, starting with the top layer. 
j J 
j 
laye yiËfeil/ 
, -• 
.. 1 1 
0 
1 
1 
1. leaf angle distributions, including azimuthal 
preference; 
2. soil reflection coefficient; 
3. optical properties of the leaves; 
4. reflection and transmission coefficients of the 
leaves; 
5. sun direction and sky irradiance; 
6. observation direction. 
The calculations show that all crop properties in 
the list above influence the reflection properties of 
a crop, and that the interpretation of reflection 
data may lead to errors in the estimation of the 
cover percentage up to 15 %. When the 
nadir-reflection in one wavelength band is used, crop 
geometry, soil reflection level and optical behaviour 
of the leaves are the main sources of these errors. 
When the vegetation index is used in stead of the 
reflection in one single band, the importance of the 
crop geometry decreases, but the influence of the 
soil brightness and the optical behaviour of the 
leaves remains an important source of possible 
mis-interpretation. 
Figure 4. One step of the adding algorithm as used 
in the TURTLE-model. 
3b. In the HARE-model, a little extension of the 
adding algorithm is used to combine eight matrices of 
two layers to four matrices of the combined layer. 
This extension consists of the calculation of the 
combined transmission matrix. This process is 
repeated until all crop layers are incorporated. As 
long as identical crop layers are involved, the 
algorithm is used to double layers in stead of adding 
layers one by one. At last, the standard adding 
algorithm is used to add the soil to the crop. 
Figure 6. Vegetation index VI as function of cover 
percentage for two soils, which differ in 
brightness. Suns inclination is 60 deg., the leaf 
angle distribution of the crop is spherical, the 
observation direction is nadir. 
In addition to the uncertainties caused by crop and 
soil properties, the observation direction introduces 
important deviations in the radiance. When an 
aircraft is used as the observation platform, the 
viewing direction may deviate as much as 45 degrees 
Figure 5. One step of the adding algorithm as used 
in the HARE-model. 
4. A vector for the incoming radiation is computed, 
based on the sky irradiation pattern (including the 
sun). This vector is premultiplied with the matrices 
as derived in calculation steps 3a or 3b. To 
calculate the reflection of the same crop under 
different sky conditions, only this last step has to 
be repeated. The matrix (HARE) or matrices (TURTLE) 
that represent the crop behaviour are defined 
separately from the actual incoming radiation. 
As can be seen, with the TURTLE-model the total 
flux profile within a canopy can be computed. It is 
obvious that this result is obtained only after an 
enormous number of calculations. Because in a remote 
sensing environment generally only the reflection of 
a complete crop is of interest, the HARE-model, which 
consumes only about 10-20 % of the computer resources 
of the TURTLE model, will be sufficient. 
10 CALCULATION RESULTS 
The models as described in the former sections are 
used for calculations of crop reflection in different 
wavebands and their combinations such as vegetation 
index as affected by: 
Figure 7. Influence of the observation direction on 
the uncorrected reflection in the red and infrared 
spectral band for different values of the LAI.