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
