222 
A red band, for instance, will be denoted by the 
subscript r and equation (1) is then written as: 
r = r 
r v,r 
B + r . (1-B) 
s, r ' ' 
(2) 
For estimation of LAI the corrected infrared reflec 
tance was used. It was defined as: 
r! = r. - r . . (1-B) 
ir ir s,ir ' ' 
(3) 
r : 
ir 
= corrected infrared reflectance 
r. = total measured infrared reflectance 
ir 
r . = infrared reflectance of the soil. 
s,ir 
The infrared reflectance corrected for soil back 
ground, as derived by Clevers (1986a), is given by: 
r! = r. 
ir ir 
C 0 . (r .r - r .r ) 
2 g v,r r v,g 
(4) 
v,g 
with C, = r 
and C„ = r 
1 i s,g^ 1 s,r ~2 s,ir" s,r 
J r 
r^ = total measured green reflectance 
= total measured red reflectance 
r = green reflectance of the vegetation 
rJ'J = red reflectance of the vegetation 
r ' = green reflectance of the soil 
r s, 9 = red reflectance of the soil. 
s,r 
Finally the LAI is estimated by using this cor 
rected infrared reflectance: 
LAI = -1/a . In(1 
r ; / r . ) 
ir' Oo,ir' 
(5) 
Parameters a and r ffl . have to be estimated empir 
ically from a training set, but they have a phys- 
ical nature (Clevers, 1986a). Equation (5) is the 
inverse of a special case of the Mitscherlich func 
tion. 
The main assumption was that C* and C 2 are con 
stants, meaning that the ratio of the reflectance 
in two spectral bands (in the region of the electro 
magnetic spectrum considered) is independent of the 
soil moisture content. The validity of this assump 
tion for many soil types is confirmed by results ob 
tained by e.g. Condit (1970) and Stoner et al. (1980) 
For many soil types, the reflectance in the differ 
ent spectral bands does not differ very much (e.g. 
Condit, 1970); often there is only a slight increase 
in reflectance with increasing wavelength. 
2.2 The vegetation index 
In order to apply equation (3) for ascertaining the 
corrected infrared reflectance, the apparent soil 
cover (B) has to be known. The apparent soil cover 
can be estimated by applying, for instance, equa 
tion (2). Combination of equations (2) and (3) 
gives: 
r - r 
V, r 
s, ir 
r - r 
s,r v,r 
(6) 
However, the reflectance of bare soil in the red 
and infrared and the reflectance of vegetation in 
the red should be known. 
An approximation may be given in the following 
way. If the reflectance of bare soil in the red 
(r ) is large compared with the reflectance of 
thl'green vegetation (r ), this latter reflec 
tance, which is very small, could be omitted from 
the denominator. If the soil type under considera 
tion has a similar reflectance in the red and infra 
red spectral bands, equation (6) may be approximated 
by equation (7): 
r : = r. 
(7) 
In the situation of bare soil the term r should 
be omitted in order to get the same result as in 
equation (6) (under the assumption r = r . ). 
In the situation of high soil cover fhe term'r r 
is very small compared with r. -r , so it may b'e 
omitted. A crude approximation for estimating the 
corrected infrared reflectance will result in the 
equation: 
(8) 
For application of this equation in estimating LAI, 
the difference between the infrared and red reflec 
tance (which is the vegetation index in this paper) 
must be ascertained and then equation (5) must be 
used. The combination of equations (5) and (8) is 
called the semi-empirical reflectance model. In 
this regard r ffi . in equation (5) will be the asymp 
totic value of'¥he difference between infrared and 
red reflectance at very high LAI. 
If in equation (4) the measured reflectances in 
the green and red spectral bands are assumed to be 
equal (r = r ), then this equation is equivalent 
to equat?on (&) under the assumption C* = C 2 = 1. 
This assumption agrees with the specific situation 
that the reflectances of bare soil in the green, red 
and infrared are equal. This drastic approach will 
be tested in the next section with a data set cal 
culated by means of Verhoef's SAIL model, and pro 
vided by him. Furthermore it will be verified with 
real field data. 
COMPARING THE MODEL WITH THE SAIL MODEL 
In this section the accuracy of the vegetation index 
presented in section 2.2 for ascertaining the cor 
rected infrared reflectance will be compared to the 
corrected infrared reflectance obtained if soil re 
flectances are known, by means of calculations with 
the SAIL model (Verhoef, 1984). The following vari 
ables for the SAIL model have been used: 
- two soil types: 
dry soil (green reflectance = 20.0 %, red reflec 
tance = 22.0 %, infrared reflectance = 24.2 %); 
wet soil (green reflectance = 10.0 %, red reflec 
tance = 11.0 %, infrared reflectance = 12.1 %); 
- spherical leaf angle distribution. 
- direct sunlight only (solar zenith angle: 45°). 
- direction of observation vertically downwards. 
- equality of reflectance and transmittance of a 
single leaf: green reflectance = 8 %, red reflec 
tance = 4 % and infrared reflectance = 45 %. 
Model calculations were carried out with the fol 
lowing LAI values: 0 (0.1) 1.0 (0.2) 2.0 (0.5) 5.0 
(1.0) 8.0. 
The green, red and infrared reflectance factors 
were calculated according to the SAIL model for 
each of the above situations. 
In estimating LAI the infrared reflectance was 
corrected for soil background and subsequently this 
corrected infrared reflectance was used for estima 
ting LAI. If soil reflectance is known, equation (6) 
may be applied in order to ascertain the corrected 
infrared reflectance. This method will be called 
method 0 (indicating that it cannot be applied with 
out knowing soil reflectances explicitly). In prac 
tice, however, soil reflectances often are not known. 
In order to ascertain the corrected infrared reflec 
tance for the situation that soil reflectances are 
not known, the validity of equation (8) will be test 
ed. This method, called method 2, in addition to me 
thod 0 and method 1 given by Clevers (1986a), ascer 
tains the corrected infrared reflectance by taking 
the difference between measured infrared and red re 
flectance - a drastic simplification compared with 
method 1 (given by equation 4). Results for all 
three methods are given in figure 1. All three meth 
ods gave essentially the same results. As expected, 
Figure 1 
ences ir 
Spherica 
xx: ca 
— : si 
(Rw is 
figure 
4.5). 
the estj 
2 as cor 
due to t 
given tc 
by methe 
not mucl 
methods. 
A mor 
vegetal 
SAIL me 
also f 
(1986c) 
vestigi 
yses ii 
crop Vi 
disturl 
red re: 
presenl 
angle c 
for thi 
SAIL me