r 
s,g 
/r 
s 
r 
( 7) 
and 
J r 
s.ir s,r 
( 8) 
If we assume that we are able to determine the 
constants Cj and , merely by measuring the required 
reflectance values at the same soil moisture content, 
then equations (2), (3), (6), (7) and (8) offer us 
five equations with five unknown variables: the 
corrected infrared reflectance (r! ), soil cover (B) 
and the soil reflectance in the tliree bands (r , 
r and r . ). After solving these equations 3 ' g for 
till 1 correcl4^ r infrared reflectance we obtain: 
r ! 
ir 
r. 
ir 
. r 
v,r 
r 
r 
r 
V, r 
r 
v,g 
. r 
y,g 
) 
( 9) 
In the situation of bare soil only, r , r and r. 
, , g r ir 
equal r Si g f r S/r and r s ^ r , respectively, and equation 
(9) results in: r£ r =0. In the situation of complete 
soil cover, rg and r r equal r Vj g and r V/r , respecti 
vely, and equation (9) results in: r 1 = r ; in 
. 13T 1IT 
other words no correction for soil background is 
applied if the soil is not visible. 
In order to deduce the relationship between soil 
cover and LAI, the process of extinction of radiation 
in a canopy should be considered. If a canopy has a 
certain extinction coefficient per leaf layer as well 
as a certain LAI (abbreviated as L in the formulae), 
the product of both factors equals the mean extinction 
of that canopy. The mean extinction consists of two 
components: 
1. extinction in the direction of the sensor, indica 
ted by K.L, where K is the extinction coefficient 
per leaf layer in the sensor direction; 
2. extinction in the direction of the sun, indicated 
by k.L, where k is the extinction coefficient per 
leaf layer in the direction of the sun. 
Consider the process of extinction in a very small 
part (or element) of the canopy. In the visible 
spectral bands, extinction in an element occurs when 
a leaf is hit by radiation. The probability of hitting 
i elements among n independent elements has a binomial 
distribution. If the number n of independent elements 
increases to infinity while the probability of hitting 
a specific element decreases to zero, the binomial 
distribution can be approximated by a Poisson distri 
bution. The Poisson distribution states: 
P(x=i) = e X . X 1 i = 0,1,2,3,... d°) 
i; 
The random variable x is the number of independent 
elements of the canopy in which extinction occurs; X 
is the mean or expected number of elements in which 
extinction occurs. The probability that no element is 
hit (i=0) equals: 
P(x=0) = e (11) 
The probability of soil being visible in the direction 
of the sensor equals: e~ K * L . 
The probability of soil being illuminated by the sun 
equals: e~k.L. 
If one assumes both events to be independent, the 
probability of sensing illuminated soil equals: 
e-K.L * e - k - L = e-< K+k > - L 
The complementary probability is equal to the appa 
rent soil cover (new definition). This means that 
soil cover may be described as: 
B 
. -(K+k).L 
1 - e 
(12) 
Inserting equation (12) into (5) gives: 
r ' 
r 
V 
(1 
-(K+k).L 
e ) 
(13) 
The relationship between LAI and infrared reflec 
tance (cf. Bunnik, 1978) greatly resembles a 
"Mitscherlich curve" (y=A-b.exp(-k.t)). In the 
special situation of such a curve running through the 
origin (A = b) , the curve defined by y=A. (1-exp(-k.t) ) 
has only two parameters. For describing the relation 
ship between the corrected infrared reflectance and 
LAI (which runs through the origin) an empirical 
equation similar to equation (13) could be used: 
r ! 
ir 
(1 
-O.L 
(14) 
parameter r«, being the asymptotic value for the 
infrared reflectance and a a combination of extinction 
and scattering coefficients. Both parameters are 
estimated empirically from a training set. Finally the 
LAI is solved from equation (14): 
L = -l/a . ln(l - r i r / r 00>ir ) (15) 
This is the inverse of the special case of the 
Mitscherlich function. 
4 COMPARING THE MODEL WITH THE SAIL MODEL 
In this section the model derivations presented earlier 
will be verified by means of calculations with the 
more complicated SAIL model (Verhoef, 1984) . This 
model simulates reflectances as a function of plant 
variables and measurement conditions (cf. section 2.2). 
The following variables for the SAIL model have been 
used: 
- three 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%); 
. black soil (green, red, infrared reflectance = 
0%) . 
- spherical leaf angle distribution. 
- direct sunlight only (solar zenith angle: 45°). 
- direction of observation was assumed to be vertically 
downwards. 
- reflectance and transmittance of a single leaf were 
assumed to be equal: green reflectance = 8%, red 
reflectance = 4% and infrared reflectance = 45%. 
Model calculations were carried out using the follow 
ing 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. The SAIL model was also able to 
calculate the complement of the illuminated soil 
detectable by the sensor (this equals soil cover with 
the new definition introduced in this paper). 
The results obtained with the SAIL model clearly 
show that the relationship between soil cover, accor 
ding to the new definition, and green or red 
reflectance was nearly perfectly linear for a dry soil 
(figure 3). Similar results were obtained for a wet 
soil (Clevers, 1986b). These results support the 
validity of equations (2) and (3) if the new definition 
of soil cover taking shadow and vegetation together 
is used. Clevers (1986b) showed that equations (2) 
and (3) are not valid with the conventional definition 
of soil cover. 
In estimating LAI the infrared reflectance is cor 
rected for soil background and subsequently this 
corrected infrared reflectance is used for estimating 
LAI. This latter step may be investigated by using 
the calculations with the SAIL model for a black 
background. Then the infrared reflectance does not 
require correction and the validity of equation (15) 
may be checked. The results, shown in figure 4, 
support the validity of this equation for describing 
the relationship between "corrected" infrared 
reflectance and LAI at constant leaf angle distribu 
tion. Results presented by Clevers (1986b) show that 
distinct leaf angle distributions cause quite distinct