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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
