Full text: Remote sensing for resources development and environmental management (Volume 1)

234 
of the system. In vector notation: at a each moment 
t the rate vector R(t) is described as a function of 
the state of the system S(t) and the environmental 
conditions E(t) at the same time: 
R(t) = f (S(t), E(t)) (1) 
to five minutes or less, for a complete growing 
season of 100 - 150 days, generally one day is a good 
choice for AT. Phenomena that show a large amplitude 
during one timestep (for instance incoming radiation 
during one day) must be averaged or totalised over 
each step. 
The state of the system S(t) is calculated by 
integration of R(t), starting with S(0), the initial 5 COUPLING REMOTE SENSING DATA AND GROWTH MODELS 
situation: 
T 
S(T) = / R(t).dt + S(0) (2) 
0 
The environmental conditions are not affected by 
the changes in the system itself, so they can be 
written as a function of time only: 
E(t) = g (t) (3) 
The influence of the states S on the rates R, as 
expressed in a general way by equation 1, will cause 
feedback so that the rates R are not a function of 
time only. -The most common feedback loop is the one 
[biomass -> leaf area index (LAI) -> growth -> 
biomass]. In figure 1, this loop is drawn by arrows. 
These arrows represent flows of information (dashed 
in the figure). The last one is a flow of material, 
closing the loop by an integration (solid arrow). 
Figure 1 serves only as an example, it is obvious 
that the dynamic models that can be applied for yield 
prediction are much more complicated than this one. 
state variables 
rate variables 
sources and sinks 
auxiliary variables 
flow of material 
flow of information 
boundary conditions 
Figure 1. Some relations in a dynamic 
model for crop growth (simplified). 
simulation 
An important decision is to be made on the 
boundaries of the system: they depend on the total 
simulation time and on the desired level of detail. 
For instance, soil water content is fairly constant 
over one day, so in a simulation that only concerns 
one diurnal cycle it may be considered to be 
constant. When the soil water content in a porous 
sandy soil is mainly a function of human 
interventions in the level in surrounding ditches, it 
is a function of time and at last, when the water 
uptake by the plants plays an important role in the 
soil water content, soil water must probably be taken 
In the state vector S of the model and the changes in 
it in the rate vector R. 
All relations in the models are defined as 
mathematical expressions, as tabular functions or as 
combinations of both. The complexity of the 
relations between S, E and R prohibits generally the 
application of an analytical solution of integral S. 
so only a numerical solution can be applied. Because 
of the discontinuities in E, Euler's integration 
method is generally used to solve expression (2). 
This means that this expression is rewritten to: 
S(T+6T) = S(T) + AT * R(T) (4) 
where AT is the integration time step. For 
simulations that concern one diurnal cycle, AT is set 
A problem in the incorporation of remote sensing data 
in simulation models is the difference between the 
type of information that is used in the models like 
biomass or LAI and the type of data as collected with 
remote sensing techniques. It is obvious that a 
coupling mechanism must be applied. Roughly spoken 
three types of coupling mechanisms are possible: 
1. Statistics: from a wide range of crops growing 
under different circumstances and in different stages 
of development, the reflective behaviour must be 
available. The measured reflection is compared to 
the data set of known reflections. This can probably 
give the information which we are interested in, but 
it requires a tremendous data collection in advance. 
2. Direct calculation of the crop state from the 
measured reflection. This means that it must be 
possible to invert the set of functions that 
describes the relation between crop properties and 
reflection. 
There exists no unique relation between reflection 
and crop status. Therefore both the first and the 
second method will give ambiguous results. 
3. Starting with the simulated crop, the reflection 
of this crop is estimated and compared with the 
measured data. When differences are detected between 
these two, the most likely parameters in the growth 
simulation are changed and a new simulation run is 
made. This process is repeated until a good 
correspondence between measured and estimated 
reflection is achieved. 
In this work, the choice is made for the third 
method, because it takes into account additional 
knowledge from ground truth and about relations 
between parameters concerning crop and soil. 
Therefore a model is needed to calculate the 
reflection of a crop from from its optical properties 
and leaf density distribution. A model that can 
serve for this purpose must fulfil two conflicting 
requirements: 
1. The model must be complicated in view of 
generality, because it must be possible to calculate 
the reflection of a crop in any arbitrary direction 
as a function of crop properties, soil reflection and 
the spatial distribution of the incoming radiation. 
Too many limitations of the model cause the 
computation results to be a function of the model 
restrictions rather than a function of the crop 
properties. 
2. The model must be simple in view of its frequent 
iterative application, so one run with the program 
may not exceed an acceptable level of use of computer 
resources. 
6 SOME EXISTING MODELS 
Several models published before are investigated on 
these needs. All are rejected on their limitations. 
The Suits-model (Suits, 1972) is based on a very 
simplified crop geometry. Especially for off-nadir 
observations or in the situation where the sun's 
direction deviates from the zenith, the model results 
show only a qualitative relation with experimental 
data. 
A second model that is considered is the model 
published by de Wit (1965), which is enhanced later 
by Goudriaan (1977). These models are developed to 
estimate the absorption of incoming radiation. 
Therefore, these models are based on a simplified 
leaf reflection submodel and on aggregating functions 
for reflection by crop layers. Although the overall
	        
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