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.