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2.2 Semi-empirical models relating vegetation indices to LAI and P 
The relationship between vegetation indices and LAI approaches an asymptotic limit at high LAI values, and can 
therefore be fitted to an exponential equation (Hatfield et al 1985; Asrar et al, 1984; Baret et al, 1989) similar 
in form to that of Beer's Law: 
VI g is the vegetation index of the bare soil, VI m is the asymptotic value of the VI as LAI tends to infinity, and 
k vi , which is equivalent to the extinction coefficient in the original Beer's law equation, controls the non-linearity 
of the relationship. It has been shown that VI m and k v , depend upon the leaf angle distribution, soil reflectance 
and the solar and view angles (Baret, 1988). The three coefficients in eq. 1 can be determined for any given set 
of VI and LAI values by using non-linear fitting techniques. 
crops including wheat (Hipps et al, 1983), maize (Gallo et al, 1985), and soybean (Daughtry et al, 1992). If the 
distribution of leaves is random then the relationship can be expressed in the following form: 
Where P m is the asymptotically limiting value of PAR absorption at high LAI values, and k p is analogous to an 
extinction coefficient. Since green leaves have a very high absorptance in the PAR domain, k P is close to the 
extinction coefficient defined for interception by black leaves (Steven et al, 1986). 
It was shown above (eq. 1) that LAI can be determined from VI measurements, and this has been used 
to derive an expression for P in terms of VI by combining the VI-LAI and P-VI relationships (Asrar et al, 1984; 
Gallo et al, 1984). Combining the two simple models defined above (eqs. 1 and 2) we arrive at the following 
relationship between P and VI: 
where a = kp/ky,, and the other variables as defined for eqs. 1 and 2. 
2.3. Fitting the ABSAIL data to the semi-empirical models 
2.3.1. Vl-LAI model Pairs of NDVI (or TSAVI) and LAI were fitted to eq. 1 using a non-linear curve fitting 
technique known as the Nelder-Mead simplex algorithm (Nelder and Mead, 1965; Press et al, 1989). Semi- 
empirical models were generated for the total data set and for subsets in which chlorosis distribution, ALA, and 
soil reflectance were fixed at each of their values given in Table 2. In those models in which soil reflectance was 
fixed at a known value, VI g was set to the appropriate value. But in the models in which soil reflectance was 
uncontrolled, VI g was set to the mean value. The pair of parameters VI ro and k v , were estimated for each sub 
model. When the chlorosis distribution was an uncontrolled variable, there were nine models for soil reflectance 
and seven models for ALA each with a unique pair of parameters. For each chlorosis distribution there were 
a further 17 models, nine for fixed soil reflectance, seven for fixed ALA, and a single model for when both ALA 
and soil reflectance were free. 
2.3.2. P-LAI model The photosynthetic activity of a vegetation canopy which is not subject to water stress is 
directly proportional to the daily integral of P from sunrise to sunset. When ground radiometry is used, the daily 
averaged value of P is usually predicted from spectral reflectance measurements made close to solar noon. In 
the remainder of this chapter the daily averaged fraction of APAR will be referred to as P. This was calculated 
from the instantaneous values using the following formula: 
where the summation of P ina was made at 5° intervals of solar zenith angle between sunrise and sunset. In the 
VI = VI m + (VI g - VIJ exp( - k v , LAI) 
( 1 ) 
The relationship between LAI and P has been expressed in a form analogous to Beer’s law for several 
P = P m (1 - exp( - k P LAI)) 
( 2 ) 
P = P m ( 1 - {( VI^-VI)/( VI. -VI,)) B ) 
(3) 
simulations made here the solar zenith angle was assumed to be 45° at noon and 90° at sunset/sunrise. The range 
of solar zenith angle varies with latitude and day of the year; the values chosen in this study are representative