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2 
LITERATURE ON ESTIMATING LAI 
2.1 Indices for estimating LAI 
The green, red and infrared reflectances may be used 
as variables for estimating LAI. Much recent research 
has been aimed at establishing combinations of the 
reflectance factors in different wavelength bands, to 
minimize the undesirable disturbances of differences 
in soil background or atmospheric conditions. However, 
when using some combination of reflectances, one 
should be careful not to loose sensitivity to 
variations in LAI after complete soil cover has been 
reached. This also means that the infrared reflec 
tance should play a dominant role in such a 
combination. 
The earliest investigations involved the infrared/ 
red ratio {MSS(7/5)} (e.g. Rouse et al., 1973, 1974). 
Rouse and his colleagues found that this ratio was 
especially useful for estimating crop characteristics 
by correcting the radiances measured by earth- 
observation satellites (e.g. Landsat), for eliminating 
seasonal sun angle differences for minimizing the 
effect of atmospheric attenuation. The same authors 
also used the "vegetation index" {VI = MSS(7-5)/ 
(7+5)} for this purpose. In order to avoid negative 
values a transformed vegetation index 
{tVI = /VI + 0.5'} was also used in practical 
applications. Model simulations done by Bunnik 
(1978, 1981) show that these indices may be useful 
for estimating soil cover, but are only slightly 
sensitive for variations in LAI after complete soil 
cover has been reached. This is also confirmed by the 
results of e.g. Asrar et al. (1984), Hatfield et al. 
(1984), Holben et al. (1980). Moreover, in the study 
of Clevers (1986b) and of this paper, calibrated 
reflectance factors were used, so there was no 
reason to correct them for atmospheric attenuation. 
Seasonal sun angle differences were assumed to be 
minimal. 
In order to find an index independent of soil 
influence Richardson & Wiegand (1977) introduced the 
perpendicular vegetation index (PVI). However, in 
order to apply the PVI the reflectance of the soil 
has to be known, and often it is not. A similar 
approach for suppressing variations in soil background 
was developed by Kauth & Thomas (1976). They applied 
a heuristic linear transformation in the four 
dimensional data space provided by Landsat MSS 
measurements of vegetation for different soils, 
resulting into a greenness index. This transforma 
tion was only directly applied to four spectral 
bands. 
Gray & McCrary (1981a, 1981b) applied the concept 
of a difference between an infrared and a visible 
spectral band using data from NOAA satellites. This 
index is comparable to the greenness index of Kauth 
& Thomas, as far as greenness is a weighted 
difference between infrared and visible bands. None 
of the authors who used the infrared-red difference 
deduced this index from any physical model. Therefore, 
the mathematical description of the relationship of 
such an index to crop characteristics such as LAI 
differs from author to author, being derived in an 
empirical way. 
2.2 Reflectance models 
Canopy-modelling studies also enable relationships 
between reflectance values and crop characteristics 
to be studied. The main aim of physical reflectance 
models suitable for agricultural crops is to obtain 
a better understanding of the complex interaction 
between solar radiation and plant canopies. 
Essentially, there are two classes of physical 
reflectance models: numerical and analytical models. 
Bunnik (1984) has reviewed several models. 
An example of a numerical model has been described 
by Idso & De Wit (1970). In this model, radiative 
transfer is determined by scattering and absorption for 
discrete leaf layers. Goudriaan (1977) improved and 
extended this model by calculating a numerical solution 
for upward and downward diffuse fluxes within nine 
sectors of each hemisphere for each discrete layer. 
One of the earliest analytical models was described 
by Allen & Richardson (1968). It was based on a theory 
of Kubelka & Munk (1931) which describes the transfer 
of isotropic diffuse flux in perfectly diffusing media. 
In the analytical model, upward and downward fluxes 
are expressed by differential equations. Allen et al. 
(1969) extended this model in order to include 
scattering of direct solar flux by using the Duntley 
equations (Duntley, 1942) . The first analytical model 
incorporating both illumination and observation 
geometry was developed by Suits (1972) and is an 
extension of the model developed by Allen and his 
colleagues. Suits's model also incorporates plant 
canopy structural (with a drastic simplification) and 
optical properties. When model simulations are carried 
out with varying view angle, Suits‘s simplifications 
appear to be too drastic (Verhoef & Bunnik, 1981) . 
Therefore, Verhoef (1984) extended the Suits model 
further by including scattering and extinction 
functions for canopy layers containing fractions of 
oblique leaves (inclined leaves). He did not introduce 
the drastic simplification of canopy geometry to 
exclusively horizontal and vertical components as 
used by Suits, but he used a discretized set of 
frequencies at distinct leaf angles. This model is 
called the SAIL model (Scattering by Arbitrarily 
Inclined Leaves). 
Kimes & Kirchner (1982) have developed a three- 
dimensional model, which describes the radiative 
transfer for heterogeneous scenes by subdividing the 
scene into modules. 
Complicated physical reflectance models usually 
simulate reflectances for varying crop characteristics 
and incorporate simplified structural and optical 
properties of the canopy. Although for practical 
applications a simple function of reflectances for 
estimating LAI is preferred, a physical basis is 
unavoidable in order to deduce the sort of relation 
ship between such a function and LAI (semi-empirical 
model). Therefore, a new model will be introduced 
resulting in a simple correction for soil background. 
A mathematical relationship between this correction 
and LAI will be described. This will be verified by 
means of calculations with the SAIL model. 
3 SIMPLIFIED REFLECTANCE MODEL FOR VEGETATION 
The main requirements for a simplified reflectance 
model for vegetation are: 
1. it should be possible to estimate LAI; 
2. it should describe the relationship between 
reflectance and LAI by more or less physically 
defined parameters; 
3. it should correct for soil background in order 
to enable a multitemporal analysis to be done; 
4. it should be as simple as possible (preferably 
resulting in some sort of index). 
If a remote sensing technique is used in a visible 
band while looking downwards from some distance, the 
sensor will be unable to detect whether soil in the 
shadows is obscured by leaves. For this reason soil 
cover is redefined by taking the sun-sensor geometry 
into account (conventionally, for a green canopy, soil 
cover is defined as the relative vertical projection 
of the canopy on the soil surface). A prerequisite for 
ascertaining bare soil according to the new definition 
is that the soil must clearly contrast with the 
vegetation. When the sun is shining the soil should be 
directly illuminated by the sun (figure la). Further, 
the soil must be visible for the detector to be able 
to classify it as soil (figure lb). The fraction of 
soil that satisfies both conditions (i.e. soil that is