Table 1 : Reflectance values 
Description of 
Samples 
Reflectance in percentate (Average of 
10 samples) 
Band 4 
Band 5 
Band 6 
1 Band 7 
River Sand 
8.36 
13.76 
15.78 
13.99 
Marine Clay 
8.04 
13.46 
16.33 
17.11 
Red Muram 
8.19 
16.35 
20.32 
18.54 
Black cotton soil 
8.20 
10.44 
14.02 
19.18 
Powai soil 
13.38 
20.50 
26.76 
33.54 
to obtain the four Principal Components. The effectiveness 
of the transformation is evident from the fact that 
the maximum correlation of 0.34 with the original 
data has been improved to 0.66 with the third Principal 
Components. Taking advantage of the fact that trans 
formation improves the correlations, other forms of 
transformations and their combinations have also been 
tried. It has been observed that the correlation can 
be improved with systematic transformation of data. 
Further, to arrive at a most suitable transformation, 
optimization techniques have been employed and the 
best transformation that enables to obtain the maximum 
correlation has been arrived at. 
3 MATHEMATICAL MODEL 
3.1 Linear model 
Having established the existence of a high correlation 
between the grain sizes and the third Principal Compo 
nents, linear regression analysis of grain sizes on the 
third Principal components was done. A linear mathema 
tical model with a correlation coefficient of 0.66 has 
been dëtermined as 
r 
L 
r 
L 
CL 
O 
M 10 
C 10 
d 20 
M 20 
C 20 
O. 
O 
M30 
c 
30 
d 40 
M 40 
c 
40 
O 
»A 
~o 
= 
M 50 
[0.18B 7 +0.25B 6 -0.14B 5 -0.94B 4 ] + 
C 50 
d 60 
M 60 
C 60 
d 70 
M 70 
C 70 
CL 
OO 
O 
M 80 
C 80 
d 90 
M90 
C 90 
This model is applicable for diameters, d^, d^Q, d^, 
d 40’ d 50’ d 60’ d 70’ d 80’ d 90" 
In general, 
= ^x- 1 ^l^ + *2 B 6 + *3 B 5 + *4 B 4 - 1 + ^ 
wnere 
B_, EC, EC, B. - bare soil reflectances in bands 7,6,5 
/ , i D J 4 
and 4 
11 ’ 12’J3’ 14 “ ^irecton cosines 
M , C7 - regression constants for d size 
d - diameter of particles less than x percent 
in mm 
The model constants are given in Table2. 
3.2 Bi-linear model 
The 1, II, III and IV Principal Components determined 
from the original data are termed as Primary Principal 
Components. By keeping some of the original axes 
as fixed and rotating the others in turn, Secondary 
and Tertiary Principal Components have been obtained 
(Venkatachalam and Jeyasing, 1986). On the whole 
there are one set of Primary Principal Components 
(PPC), four sets of Secondary Principal Ciomponents 
(SPC) and six sets of Tertiary Principal Components 
(TPC). The axes corresponding to the minimum variance 
in the original data have been used as fixed axes in 
the present analysis. 
It is understood that, in a properly designed regression 
analysis, inclusion of all the plausible predictor variables 
and the absence of multicollinearity among them, will 
increase predictive ability of the model. Hence, the 
four PPC, three SPC and the two TPC obtained, have 
been combined two at a time and a multiple linear 
regression analysis was done. It has been found that 
the multiple regression model obtained by fourth PPC 
(PPC^) and the third SPC (SPC^) as independent vari 
ables had a high correlation coefficient of 0.94 for 
all the grain sizes from djQ to d^Q. The bi-linear model 
suggested is as follows. 
[d ] = [A ] + [B ] [PPC ] + [C ] [SPC ] 
where A , B , C - Model constants 
v" x 7 X 
The model constants are given in Table 2. 
3.3 Linear model based on optimization 
If n is the number of samples observed, then the four 
band data observed will be X-- ( i = 1, n; j = 1, 4 ) 
and the corresponding diameter of particles less than 
x percentage is given by d- (i = 1, n; x = 10, 20, 30, 
40, 50, 60, 70, 80 and 9of. This four band data can 
be plotted in a four dimensional space and it is possible 
to obtain a linear transformation of the data on an 
axis in the four dimensional space with direction cosines 
1., 12, L and 1^ which satisfy a stipulated requirement. 
ence i i (i = 1, n) are the transformed reflectance 
values of the n samples, this can be stated as 
n 
2 
¡=1 
4 
j=1 
X. . = T; 
J 'J ' 
The T. values for the model have been determined 
using standard non-linear optimization technique (Mitai 
1977). Based on this, a simple linear model with a corre 
lation coefficient of 1.0 has been suggested, whose 
model constants are given in Table 3. 
3.4 Model for large samples 
The models suggested so far are based on observations 
made on five types of soils with ten samples for each 
type. To study the behaviour of the model for large 
samples, a group of 14 soils from a large number of 
samples collected from different parts of India has 
been selected. These were falling in the Munsell colour 
range of hue 10 YR, value3 to 6 and chroma 1 to 6. 
Reflectance values of these samples have been observed 
in the laboratory under identical conditions described 
for the initial study. Particle sizes have also been deter 
mined in the laboratory. A simple linear model based 
on non-linear optimization technique has been evolved 
for the same diameters chosen earlier. The values 
of the model constants are given in table 4. 
To test the predictive ability of the model, six more 
naturally available surface soil samples falling within 
the above Munsell colour range were collected from 
different parts of India. The locations of all these sam 
ples are given in Figure 1. The reflectance values were 
determined in the laboratory and the grain sizes were 
predicted from the above model. The predicted values 
were compared with actual values determined in the 
laboratory. There is a close aggreement between the 
two values, as evident from Table 5.