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.