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Relaxation models for moist soils suitable at microwave frequencies
Materials Science and Engineering, 28 ( 1 9 7 7 ) 47 - 51
47
© Elsevier S e q u o i a S.A., L a u s a n n e - - P r i n t e d in t h e N e t h e r l a n d s
Relaxation Models for Moist Soils Suitable at Microwave Frequencies*
P. K. B H A G A T a n d P. K. K A D A B A
Wenner-Gren Research Laboratory, University of Kentucky, Lexington, Kentucky 40506 (U.S.A.) ( R e c e i v e d in revised f o r m S e p t e m b e r 30, 1 9 7 6 )
SUMMARY
Relaxation models based on modifications of the Debye equation have been used in analysis of the complex dielectric constant results on various types of soils at microwave frequencies. A computer-based algorithm has been developed to provide accurate analysis through complex curve fitting. Results of this study suggest that Bergmann modification of the Debye equation seems to fit experimental data on 12% moisture content soils. Below 1 GHz one may need to consider phenomena such as residual surface effects and intermediate forms of bound water.
INTRODUCTION
The measurement of the complex dielectric constant of soils at microwave frequencies has practical significance in remote-sensing applications. The microwave region from 100 MHz to 20 000 MHz is extensively used in applications that interface with the earth. The return of a radar depends, for example, among other things on the complex dielectric constant of the ground. The depth of penetration into the earth can be c o m p u t e d from a knowledge of its dielectric properties. In contrast to visible and infrared, microwave sensors are not hampered by clouds, especially at the lower microwave frequencies, and, hence, have a much higher operational d u t y factor substantially independent of weather and sunlight level. The better penetrating capabilities of microwaves makes it more suitable to gather information about
the sub-surface as well as surface moisture conditions. The present study is concerned with the experimental measurements of a soil sample in the microwave region, and analysis of the present data as well as that found in the literature, in terms of suitable relaxation models based on a modification of the Debye equation [ 1 ]. The results are analyzed by complex curve fitting through minimization of least-square error between the computed and measured data.
EXPERIMENTAL
The real and imaginary parts of the dielectric constants of the moist soil were measured using an adaptation of a technique which involves the determination of the circuit parameters of a four-terminal dissipating network [2]. The measurements were also made using the classic Roberts and von Hippel Method [3]. The results obtained by both the methods are in good agreement with each other.
THEORETICAL CONSIDERATIONS
The dielectric properties of materials whose molecules have permanent dipole moments can be described by a modification of the Debye equation due to Cole and Cole [4]. For soils at lower microwave frequencies, the ionic conductivity, o, may be important, and Coles' equation can be modified to [5] : *2"(0,)) = (C 1 - - j e 2 )
= e~ +
(e s - - (~oo)
(1 + j fifo) (1-~) *Major p o r t i o n o f t h e s t u d y r e p o r t e d in t h e p a p e r was p r e s e n t e d at t h e A m e r i c a n Physical S o c i e t y M e e t i n g held at W a s h i n g t o n , DC, April 26 - 29, 1976.
2nfeo
(1)
48 Here, e . is the high frequency limit of the dielectric constant usually corresponding to a frequency in the far-infrared; %, is the static dielectric constant; f0, is the relaxation frequency or the frequency at which the loss factor, e2, is a maximum;/3, is an empirical distribution parameter ranging from 0 to 1; %, is the free space permittivity; and f, is the probing frequency. The expression for % for pure water [6], given by, % = 8 7 . 7 - - 0.4 ( T - - 273),
(2)
where T is the absolute temperature can be modified to take into account the appropriate moisture content of the soil. Where more than one relaxation process is suggestive, the experimental results can be analyzed in terms of a superposition of two Debye processes as suggested by Bergmann et al. [7] e*(~)
-
e~
c: - -
e s -- e~
-
1 + jcoT1
(1 + - -
cO (3)
1 + jWT2
where c: is a weight factor. A modification of this equation is given in the paper by Hoekstra and Delaney [8] : (elseloo) e*(50) = Cls + (1 + jCOT:)(1-~) + e2s + (e2s -
e2~)
(4)
(1 + j~T2) (~-~,)
where the subscripts 1 and 2 refer to the two separate relaxation processes, r , and r2 are the corresponding relaxation times;/31 and/3 2 are the empirical Cole parameters ranging from 0 to 1. The attenuation constant, a, of the soil medium, or its reciprocal, the skin depth, 5, is related to the real and imaginary parts of the complex dielectric constant thus: = --
{ ( 1 + ( e 2 / e l ) 2 ) 1/2 --- 1}
(5)
where a is in nepers/m; k = wavelength in meters. If the medium depth profile of e I and e 2 is not constant, then 5 can be defined by: 5
f adz = 1 o
where z is the depth below the surface.
(6)
The curve fitting of the data reported in this study to various relaxation models is carried out through minimization of sumsquared error, S, defined by
t
tr
tt
x2
S = E (ec i - - COi) 2 + (~c i - - eO i) i t
t~
(7)
where e0i and e0i are experimentally observed rl data and i is a frequency index, e'ci and eci are computed, for a given relaxation model, using arbitrarily chosen values of u n k n o w n parameters, A (the dimensions of A vary between 5 and 8 for eqns. (3) and (4) used in this study). The program begins with an arbitrarily chosen parameter vector, 4 0. The upper and lower limits on the u n k n o w n parameters are provided both to limit the search area and physically to interpret the data. The function tt value S(A°), e'ci , e c i , and OS/OAj are c o m p u t e d using eqn. (7) and the chosen model. The steepest descent m e t h o d [9] is used initially for minimization of the sum-squared error, S. In the event of the steepest descent method failing, a second order gradient technique known as Newton Raphson technique [9] has been incorporated which has good convergence properties in the vicinity of a minima. The program is convergent for various starting values chosen in this study. It should be noted, however, t h a t since the program is based on the search for a local minima it is entirely possible that a better fit to data may result through choice of a different starting vector. This is known in the literature as starting parameter bias. In order to obtain limited assurances of uniqueness of the solution, several computer runs were made for each equation with different starting parameter vectors. The final parameter vector selected was one that yielded the lowest sumsquared error within the group. There is also a provision written in the program r a n d o m l y to select the starting vector in the hope of identifying several stationary points, thus assuring a better choice among the final parameter vectors. The program as developed was checked for accurate results by comparison with data of Bergmann et al. [7] and their graphical technique for pure liquids. Table 1 shows the comparative results on diphenyl ether:
49
TABLE 1 C o m p a r i s o n of c o m p u t e r a n d graphical analysis t e c h n i q u e s Frequency (GHz)
24.0 9.3 3.0
Exptl. d a t a
C o m p u t e d results b y B e r g m a n n et al.
C o m p u t e d results r o u n d e d off to 3 D p
CI
~"
{5'
C"
{F
C"
3.170 3.430 3.560
0.400 0.295 0.123
3.160 3.430 3.580
0.397 0.296 0.138
3.172 3.441 3.580
0.407 0.291 0.132
The sum-squared error reported by Bergmann et al. using the graphical technique was 7 X 10 --4
while that with the present program was 6.57 X 10-4 Compared with the graphical technique, the computer minimization presents an automatic means of generating the best curve fit data*. A typical minimization run with six different starting vectors for five frequency points took approximately fifteen seconds computing time with all the computations carried out in double precision arithmetic.
RESULTS AND ANALYSIS
The experimental results obtained in the present study on a 12% (g H20/g soil) soil sample, as well as the results shown in Table 4 and Fig. 12 of ref. 8 have been analyzed in terms of the above relaxation models. Table 2 gives the comparison between the experimental real and imaginary parts of the dielectric constant values in the present study and the
*Copies o f t h e p r o g r a m listing w r i t t e n in W A T F I V are available f r o m t h e a u t h o r s u p o n request.
corresponding theoretical values obtained from eqns. (3) and (4). The other parameters obtained in the computer-optimized leastsquare fitting using the above equations are shown in Table 3. With eqn. (3) the experimental values are matched with a total sumsquared error of 0.2 while eqn. (4) yields the total sum-squared error of 0.7. The resolved relaxation times obtained from the use of eqn. (3) and the data observed in the present study (shown in Table 2) are T1 = 0.24 X 10 -1° and T2 = 0.99 X 10-10 s. These relaxation times are longer than that of pure water. This may suggest that the rotating unit involved in the relaxation process is bigger than the water molecule. Table 4 is a comparison between the experimental values reported in ref. 8 and the corresponding theoretical values obtained from eqns. (3) and (4). The agreement of computer optimized solutions using eqns. (3) and (4) with the given experimental data is rather poor (sum-square error > 315). We feel that the reason for this could be due to the fact that the data reported in Table 4 of ref. 8 are computed from the slopes of e' and e" values plotted against water concentration in the soil.
TABLE 2 O b s e r v e d a n d m o d e l c o m p u t e d dielectric c o n s t a n t s e', losses e" Frequency
Exptl. values
(GHz)
e'
0.2 0.4 1.0 10.0 20.0
15.1 14.3 12.5 5.6 4.9
e"
1.08 2.10 4.10 1.70 1.07
Best curve fit d a t a Using eqn. 3
Using eqn. 4
r
.
cr
e"
14.88 14.49 12.48 5.52 5.12
1.13 2.17 4.18 1.91 1.05
14.90 14.55 12.57 5.82 5.19
1.18 2.12 3.88 2.25 1.46
5O TABLE 3 Least square error curve fit constants: dispersion parameters /~2
~i
Cl
C2
C1
elS
C2~
C2S
T1
T2
Fit using eqn. 3 (ref. 7)
-
-
0.11
0.89
4.95
15.01
-
-
0.24 0.99 *10 -10 ,10-10
Fit using eqn. 4 (ref. 8)
0.55
0.33
-
-
2.18
12.06
2.06
2.54
1.18 9.96 "10 -10 .10-10
TABLE 4 Observed and model computed dielectric constants c', losses e" Frequency (GHz)
0.5 4.0 6.0 12.0 26.0
Exptl. values
Best curve fit data
e'
Using eqn. 3
Using eqn. 4
6'
C"
6'
6"
50.49 32.09 24.11 14.80 11.13
4.72 20.31 14.36 13.10 6.68
56.30 27.89 22.66 16.99 13.65
9.19 18.41 15.61 10.71 6.64
e"
56.00 33.00 33.00 6.50 10.00
7.00 13.40 14.40 4.50 10.00
TABLE 5 Comparison between the experimental and theoretical values of the attenuation constant, a, in nepers/m for 30% volumetric water content at 24 °C (o = 3 × 1 0 - 2 mho/m) Frequency (GHz) 0.5
4.0 8.0 12.0
(Experimental) from ref. 8 2.76
52.9 120.0 299.0
(Calculated) from eqns. (1) and (2) 1.76 2.76 (0=6.6) 47.0 132.0 341.0
Figure 12 in t h e p a p e r b y H o e k s t r a a n d D e l a n e y [8] is a p l o t o f t h e a t t e n u a t i o n c o n s t a n t (a) in soils as a f u n c t i o n o f volum e t r i c w a t e r c o n t e n t (g H 2 0 / c m 3) at several f r e q u e n c i e s . T h e b e h a v i o r s h o w n is r e p r e s e n t a t i v e o f a large r a n g e o f soil t y p e s m e a s u r e d b y t h e m . I t is suggested b y t h e s e a u t h o r s t h a t : (1) t h e r e l a x a t i o n p r o c e s s is p r e d o m i n a n t l y d u e t o w a t e r m o l e c u l e s w i t h o u t m u c h influe n c e f r o m t h e n a t u r e o f t h e soil f o r f r e q u e n cies o f 1 G H z a n d a b o v e ; (2) ion e x c h a n g e c h a r a c t e r i s t i c s o f soil m a t e r i a l are negligible a b o v e 500 MHz. Based o n this p r e m i s e , eqns.
(1), (2) a n d (5) h a v e b e e n used t o c a l c u l a t e t h e a t t e n u a t i o n c o n s t a n t f o r suitable m i c r o w a v e f r e q u e n c i e s . T h e results are s h o w n in T a b l e 5 f o r 24 °C a l o n g w i t h t h e e x p e r i m e n t a l results f o r r e p r e s e n t a t i v e soil s a m p l e s f o r 30% v o l u m e t r i c w a t e r c o n t e n t . In this c a l c u l a t i o n , a value o f 5 has b e e n u s e d f o r es as a c o m p r o m i s e b e t w e e n d r y soil a n d t h e high freq u e n c y limit f o r p u r e w a t e r , and/3 was c h o s e n as 0.3. T h e a g r e e m e n t b e t w e e n c o l u m n s 2 a n d 3 is p r e t t y g o o d c o n s i d e r i n g t h e s i m p l e m o d e l . F o r t h e f r e q u e n c y o f 0.5 G H z , t h e a g r e e m e n t is r a t h e r p o o r . A value o f o = 6.6 m h o / m , a s s u m e d f o r t h e t h e o r e t i c a l calculations, gives very good agreement. T h e results in T a b l e 5 suggest t h a t , w h e r e a s the relaxation process might be predominantly due to water molecules without much i n f l u e n c e f r o m t h e n a t u r e o f t h e soil f o r frequencies of 1 GHz and above, below 1 GHz, t h e i n f l u e n c e o f t h e soil d e f i n i t e l y n e e d s t o b e t a k e n i n t o a c c o u n t in describing t h e relaxa t i o n process. A m i x t u r e r e l a t i o n such as t h e o n e d e s c r i b e d b y D e l o o r [10] can b e used. T h e e f f e c t o f this m i x t u r e r e l a t i o n is t o s h i f t the relaxation frequency from that of pure water. F o r spherical particles, t h e shifts are t h e least a n d increase f o r o t h e r shapes. T h e e x p l a n a t i o n o f t h e e f f e c t o f i n o r g a n i c ions on
51
water relaxation suggested by Haggis e t al. [11] leads to the same conclusions. Here liquid water is considered to be made up of ordered regions or microcrystalline domains whose boundaries are in continuous movement. When an electric field is applied, boundary movement results in orientation of water molecules in the field direction. Inorganic ions are considered to break up the structure to some extent, to increase the b o u n d a r y area, and change the relaxation frequency. Another effect which might become important below 1 GHz and is probably more plausible, is the p h e n o m e n o n of two dimensional surface conductivity. Such a conductivity can occur in systems in which the mobility of the charge carriers is large along the surface of a granule but small normal to it. The relaxation frequency associated with such a p h e n o m e n o n is dependent on the size and shape of the granules. Thus, a distribution in the sizes and/or shapes of the granules will tend to extend the losses over a large frequency band. At higher microwave frequencies the mobility of these charge carriers becomes too low to follow the alternations of the e - m field and the granules behave as a dielectric inclusion w i t h o u t surface effects. Also, inter-
mediate forms of bound water (intermediate between free water and the very tightly bound water with a more or less "ice-like" structure) might play a part.
REFERENCES
1 P. Debye, Polar Molecules, Chemical Catalogue Company, 1929. 2 S. G. Govande, S. K. Garg and P. K. Kadaba, Mater. Sci. Eng., 4 (1969) 206 - 210. 3 A. yon Hippel, Dielectric Materials and Applications, Mass. Inst. Technol. Press, 1954. 4 K. S. Cole and R. H. Cole, J. Chem. Phys., 9 (1941) 341. 5 P. K. Kadaba, IEEE Southeastern Region 3 Conf. Paper, April, 1976, pp. 48 - 50. 6 J. B. Hasted, The dielectric properties of water, in Advances in Dielectrics, Vol. 3, Wiley, New York, 1961. 7 K. Bergmann, D. M. Roberti and C. P. Smyth, J. Phys. Chem., 64 (5) (1960) 665. 8 P. Hoekstra and A. Delaney, J. Geophys. Res., 79 (11) (1969) 1974. 9 R. B. McGhee, Some parameter optimization techniques, in Digital Computer User's Handbook, McGraw-Hill, New York, 1967. 10 G. P. DeLoor, Appl. Sci. Res., B l l (1964) 310. 11 G. H. Haggis, J. B. Hasted and T. J. Buchanan, J. Chem. Phys., 20 (9) (1952) 1452. A. Lebruon, Revue g~n~rale de l'Electricit4, 74 (1965) 948.
47
© Elsevier S e q u o i a S.A., L a u s a n n e - - P r i n t e d in t h e N e t h e r l a n d s
Relaxation Models for Moist Soils Suitable at Microwave Frequencies*
P. K. B H A G A T a n d P. K. K A D A B A
Wenner-Gren Research Laboratory, University of Kentucky, Lexington, Kentucky 40506 (U.S.A.) ( R e c e i v e d in revised f o r m S e p t e m b e r 30, 1 9 7 6 )
SUMMARY
Relaxation models based on modifications of the Debye equation have been used in analysis of the complex dielectric constant results on various types of soils at microwave frequencies. A computer-based algorithm has been developed to provide accurate analysis through complex curve fitting. Results of this study suggest that Bergmann modification of the Debye equation seems to fit experimental data on 12% moisture content soils. Below 1 GHz one may need to consider phenomena such as residual surface effects and intermediate forms of bound water.
INTRODUCTION
The measurement of the complex dielectric constant of soils at microwave frequencies has practical significance in remote-sensing applications. The microwave region from 100 MHz to 20 000 MHz is extensively used in applications that interface with the earth. The return of a radar depends, for example, among other things on the complex dielectric constant of the ground. The depth of penetration into the earth can be c o m p u t e d from a knowledge of its dielectric properties. In contrast to visible and infrared, microwave sensors are not hampered by clouds, especially at the lower microwave frequencies, and, hence, have a much higher operational d u t y factor substantially independent of weather and sunlight level. The better penetrating capabilities of microwaves makes it more suitable to gather information about
the sub-surface as well as surface moisture conditions. The present study is concerned with the experimental measurements of a soil sample in the microwave region, and analysis of the present data as well as that found in the literature, in terms of suitable relaxation models based on a modification of the Debye equation [ 1 ]. The results are analyzed by complex curve fitting through minimization of least-square error between the computed and measured data.
EXPERIMENTAL
The real and imaginary parts of the dielectric constants of the moist soil were measured using an adaptation of a technique which involves the determination of the circuit parameters of a four-terminal dissipating network [2]. The measurements were also made using the classic Roberts and von Hippel Method [3]. The results obtained by both the methods are in good agreement with each other.
THEORETICAL CONSIDERATIONS
The dielectric properties of materials whose molecules have permanent dipole moments can be described by a modification of the Debye equation due to Cole and Cole [4]. For soils at lower microwave frequencies, the ionic conductivity, o, may be important, and Coles' equation can be modified to [5] : *2"(0,)) = (C 1 - - j e 2 )
= e~ +
(e s - - (~oo)
(1 + j fifo) (1-~) *Major p o r t i o n o f t h e s t u d y r e p o r t e d in t h e p a p e r was p r e s e n t e d at t h e A m e r i c a n Physical S o c i e t y M e e t i n g held at W a s h i n g t o n , DC, April 26 - 29, 1976.
2nfeo
(1)
48 Here, e . is the high frequency limit of the dielectric constant usually corresponding to a frequency in the far-infrared; %, is the static dielectric constant; f0, is the relaxation frequency or the frequency at which the loss factor, e2, is a maximum;/3, is an empirical distribution parameter ranging from 0 to 1; %, is the free space permittivity; and f, is the probing frequency. The expression for % for pure water [6], given by, % = 8 7 . 7 - - 0.4 ( T - - 273),
(2)
where T is the absolute temperature can be modified to take into account the appropriate moisture content of the soil. Where more than one relaxation process is suggestive, the experimental results can be analyzed in terms of a superposition of two Debye processes as suggested by Bergmann et al. [7] e*(~)
-
e~
c: - -
e s -- e~
-
1 + jcoT1
(1 + - -
cO (3)
1 + jWT2
where c: is a weight factor. A modification of this equation is given in the paper by Hoekstra and Delaney [8] : (elseloo) e*(50) = Cls + (1 + jCOT:)(1-~) + e2s + (e2s -
e2~)
(4)
(1 + j~T2) (~-~,)
where the subscripts 1 and 2 refer to the two separate relaxation processes, r , and r2 are the corresponding relaxation times;/31 and/3 2 are the empirical Cole parameters ranging from 0 to 1. The attenuation constant, a, of the soil medium, or its reciprocal, the skin depth, 5, is related to the real and imaginary parts of the complex dielectric constant thus: = --
{ ( 1 + ( e 2 / e l ) 2 ) 1/2 --- 1}
(5)
where a is in nepers/m; k = wavelength in meters. If the medium depth profile of e I and e 2 is not constant, then 5 can be defined by: 5
f adz = 1 o
where z is the depth below the surface.
(6)
The curve fitting of the data reported in this study to various relaxation models is carried out through minimization of sumsquared error, S, defined by
t
tr
tt
x2
S = E (ec i - - COi) 2 + (~c i - - eO i) i t
t~
(7)
where e0i and e0i are experimentally observed rl data and i is a frequency index, e'ci and eci are computed, for a given relaxation model, using arbitrarily chosen values of u n k n o w n parameters, A (the dimensions of A vary between 5 and 8 for eqns. (3) and (4) used in this study). The program begins with an arbitrarily chosen parameter vector, 4 0. The upper and lower limits on the u n k n o w n parameters are provided both to limit the search area and physically to interpret the data. The function tt value S(A°), e'ci , e c i , and OS/OAj are c o m p u t e d using eqn. (7) and the chosen model. The steepest descent m e t h o d [9] is used initially for minimization of the sum-squared error, S. In the event of the steepest descent method failing, a second order gradient technique known as Newton Raphson technique [9] has been incorporated which has good convergence properties in the vicinity of a minima. The program is convergent for various starting values chosen in this study. It should be noted, however, t h a t since the program is based on the search for a local minima it is entirely possible that a better fit to data may result through choice of a different starting vector. This is known in the literature as starting parameter bias. In order to obtain limited assurances of uniqueness of the solution, several computer runs were made for each equation with different starting parameter vectors. The final parameter vector selected was one that yielded the lowest sumsquared error within the group. There is also a provision written in the program r a n d o m l y to select the starting vector in the hope of identifying several stationary points, thus assuring a better choice among the final parameter vectors. The program as developed was checked for accurate results by comparison with data of Bergmann et al. [7] and their graphical technique for pure liquids. Table 1 shows the comparative results on diphenyl ether:
49
TABLE 1 C o m p a r i s o n of c o m p u t e r a n d graphical analysis t e c h n i q u e s Frequency (GHz)
24.0 9.3 3.0
Exptl. d a t a
C o m p u t e d results b y B e r g m a n n et al.
C o m p u t e d results r o u n d e d off to 3 D p
CI
~"
{5'
C"
{F
C"
3.170 3.430 3.560
0.400 0.295 0.123
3.160 3.430 3.580
0.397 0.296 0.138
3.172 3.441 3.580
0.407 0.291 0.132
The sum-squared error reported by Bergmann et al. using the graphical technique was 7 X 10 --4
while that with the present program was 6.57 X 10-4 Compared with the graphical technique, the computer minimization presents an automatic means of generating the best curve fit data*. A typical minimization run with six different starting vectors for five frequency points took approximately fifteen seconds computing time with all the computations carried out in double precision arithmetic.
RESULTS AND ANALYSIS
The experimental results obtained in the present study on a 12% (g H20/g soil) soil sample, as well as the results shown in Table 4 and Fig. 12 of ref. 8 have been analyzed in terms of the above relaxation models. Table 2 gives the comparison between the experimental real and imaginary parts of the dielectric constant values in the present study and the
*Copies o f t h e p r o g r a m listing w r i t t e n in W A T F I V are available f r o m t h e a u t h o r s u p o n request.
corresponding theoretical values obtained from eqns. (3) and (4). The other parameters obtained in the computer-optimized leastsquare fitting using the above equations are shown in Table 3. With eqn. (3) the experimental values are matched with a total sumsquared error of 0.2 while eqn. (4) yields the total sum-squared error of 0.7. The resolved relaxation times obtained from the use of eqn. (3) and the data observed in the present study (shown in Table 2) are T1 = 0.24 X 10 -1° and T2 = 0.99 X 10-10 s. These relaxation times are longer than that of pure water. This may suggest that the rotating unit involved in the relaxation process is bigger than the water molecule. Table 4 is a comparison between the experimental values reported in ref. 8 and the corresponding theoretical values obtained from eqns. (3) and (4). The agreement of computer optimized solutions using eqns. (3) and (4) with the given experimental data is rather poor (sum-square error > 315). We feel that the reason for this could be due to the fact that the data reported in Table 4 of ref. 8 are computed from the slopes of e' and e" values plotted against water concentration in the soil.
TABLE 2 O b s e r v e d a n d m o d e l c o m p u t e d dielectric c o n s t a n t s e', losses e" Frequency
Exptl. values
(GHz)
e'
0.2 0.4 1.0 10.0 20.0
15.1 14.3 12.5 5.6 4.9
e"
1.08 2.10 4.10 1.70 1.07
Best curve fit d a t a Using eqn. 3
Using eqn. 4
r
.
cr
e"
14.88 14.49 12.48 5.52 5.12
1.13 2.17 4.18 1.91 1.05
14.90 14.55 12.57 5.82 5.19
1.18 2.12 3.88 2.25 1.46
5O TABLE 3 Least square error curve fit constants: dispersion parameters /~2
~i
Cl
C2
C1
elS
C2~
C2S
T1
T2
Fit using eqn. 3 (ref. 7)
-
-
0.11
0.89
4.95
15.01
-
-
0.24 0.99 *10 -10 ,10-10
Fit using eqn. 4 (ref. 8)
0.55
0.33
-
-
2.18
12.06
2.06
2.54
1.18 9.96 "10 -10 .10-10
TABLE 4 Observed and model computed dielectric constants c', losses e" Frequency (GHz)
0.5 4.0 6.0 12.0 26.0
Exptl. values
Best curve fit data
e'
Using eqn. 3
Using eqn. 4
6'
C"
6'
6"
50.49 32.09 24.11 14.80 11.13
4.72 20.31 14.36 13.10 6.68
56.30 27.89 22.66 16.99 13.65
9.19 18.41 15.61 10.71 6.64
e"
56.00 33.00 33.00 6.50 10.00
7.00 13.40 14.40 4.50 10.00
TABLE 5 Comparison between the experimental and theoretical values of the attenuation constant, a, in nepers/m for 30% volumetric water content at 24 °C (o = 3 × 1 0 - 2 mho/m) Frequency (GHz) 0.5
4.0 8.0 12.0
(Experimental) from ref. 8 2.76
52.9 120.0 299.0
(Calculated) from eqns. (1) and (2) 1.76 2.76 (0=6.6) 47.0 132.0 341.0
Figure 12 in t h e p a p e r b y H o e k s t r a a n d D e l a n e y [8] is a p l o t o f t h e a t t e n u a t i o n c o n s t a n t (a) in soils as a f u n c t i o n o f volum e t r i c w a t e r c o n t e n t (g H 2 0 / c m 3) at several f r e q u e n c i e s . T h e b e h a v i o r s h o w n is r e p r e s e n t a t i v e o f a large r a n g e o f soil t y p e s m e a s u r e d b y t h e m . I t is suggested b y t h e s e a u t h o r s t h a t : (1) t h e r e l a x a t i o n p r o c e s s is p r e d o m i n a n t l y d u e t o w a t e r m o l e c u l e s w i t h o u t m u c h influe n c e f r o m t h e n a t u r e o f t h e soil f o r f r e q u e n cies o f 1 G H z a n d a b o v e ; (2) ion e x c h a n g e c h a r a c t e r i s t i c s o f soil m a t e r i a l are negligible a b o v e 500 MHz. Based o n this p r e m i s e , eqns.
(1), (2) a n d (5) h a v e b e e n used t o c a l c u l a t e t h e a t t e n u a t i o n c o n s t a n t f o r suitable m i c r o w a v e f r e q u e n c i e s . T h e results are s h o w n in T a b l e 5 f o r 24 °C a l o n g w i t h t h e e x p e r i m e n t a l results f o r r e p r e s e n t a t i v e soil s a m p l e s f o r 30% v o l u m e t r i c w a t e r c o n t e n t . In this c a l c u l a t i o n , a value o f 5 has b e e n u s e d f o r es as a c o m p r o m i s e b e t w e e n d r y soil a n d t h e high freq u e n c y limit f o r p u r e w a t e r , and/3 was c h o s e n as 0.3. T h e a g r e e m e n t b e t w e e n c o l u m n s 2 a n d 3 is p r e t t y g o o d c o n s i d e r i n g t h e s i m p l e m o d e l . F o r t h e f r e q u e n c y o f 0.5 G H z , t h e a g r e e m e n t is r a t h e r p o o r . A value o f o = 6.6 m h o / m , a s s u m e d f o r t h e t h e o r e t i c a l calculations, gives very good agreement. T h e results in T a b l e 5 suggest t h a t , w h e r e a s the relaxation process might be predominantly due to water molecules without much i n f l u e n c e f r o m t h e n a t u r e o f t h e soil f o r frequencies of 1 GHz and above, below 1 GHz, t h e i n f l u e n c e o f t h e soil d e f i n i t e l y n e e d s t o b e t a k e n i n t o a c c o u n t in describing t h e relaxa t i o n process. A m i x t u r e r e l a t i o n such as t h e o n e d e s c r i b e d b y D e l o o r [10] can b e used. T h e e f f e c t o f this m i x t u r e r e l a t i o n is t o s h i f t the relaxation frequency from that of pure water. F o r spherical particles, t h e shifts are t h e least a n d increase f o r o t h e r shapes. T h e e x p l a n a t i o n o f t h e e f f e c t o f i n o r g a n i c ions on
51
water relaxation suggested by Haggis e t al. [11] leads to the same conclusions. Here liquid water is considered to be made up of ordered regions or microcrystalline domains whose boundaries are in continuous movement. When an electric field is applied, boundary movement results in orientation of water molecules in the field direction. Inorganic ions are considered to break up the structure to some extent, to increase the b o u n d a r y area, and change the relaxation frequency. Another effect which might become important below 1 GHz and is probably more plausible, is the p h e n o m e n o n of two dimensional surface conductivity. Such a conductivity can occur in systems in which the mobility of the charge carriers is large along the surface of a granule but small normal to it. The relaxation frequency associated with such a p h e n o m e n o n is dependent on the size and shape of the granules. Thus, a distribution in the sizes and/or shapes of the granules will tend to extend the losses over a large frequency band. At higher microwave frequencies the mobility of these charge carriers becomes too low to follow the alternations of the e - m field and the granules behave as a dielectric inclusion w i t h o u t surface effects. Also, inter-
mediate forms of bound water (intermediate between free water and the very tightly bound water with a more or less "ice-like" structure) might play a part.
REFERENCES
1 P. Debye, Polar Molecules, Chemical Catalogue Company, 1929. 2 S. G. Govande, S. K. Garg and P. K. Kadaba, Mater. Sci. Eng., 4 (1969) 206 - 210. 3 A. yon Hippel, Dielectric Materials and Applications, Mass. Inst. Technol. Press, 1954. 4 K. S. Cole and R. H. Cole, J. Chem. Phys., 9 (1941) 341. 5 P. K. Kadaba, IEEE Southeastern Region 3 Conf. Paper, April, 1976, pp. 48 - 50. 6 J. B. Hasted, The dielectric properties of water, in Advances in Dielectrics, Vol. 3, Wiley, New York, 1961. 7 K. Bergmann, D. M. Roberti and C. P. Smyth, J. Phys. Chem., 64 (5) (1960) 665. 8 P. Hoekstra and A. Delaney, J. Geophys. Res., 79 (11) (1969) 1974. 9 R. B. McGhee, Some parameter optimization techniques, in Digital Computer User's Handbook, McGraw-Hill, New York, 1967. 10 G. P. DeLoor, Appl. Sci. Res., B l l (1964) 310. 11 G. H. Haggis, J. B. Hasted and T. J. Buchanan, J. Chem. Phys., 20 (9) (1952) 1452. A. Lebruon, Revue g~n~rale de l'Electricit4, 74 (1965) 948.