Generalized Refractive Mixing Dielectric Model for Moist Soils
V. L. Mironov *, M. C. Dobson **, V. H. Kaupp ***, S. A. Komarov ****,
V. N. Kleshchenko ****
* Earth Remote Sensing Dept., Krasnoyarsk Science Center, Siberian Branch of the Russian Academy of Sciences and Alaska SAR Facility, Geophysical Institute, University of Alaska Fairbanks
FAX: +7-385-236-7061 email:
** Radiation Laboratory, EECS Dept., University of Michigan, Ann Arbor, MI 48109-2122 USA
FAX +1-734-647 2106 email:
*** ICREST, Electrical Engineering Dept., University of Missouri in Columbia
FAX: +1-573-882 0397 email:
**** Radioscience and Remote Sensing Dept., Altai State University
Tel: +7-385-236 7047 email: komarov@ phys.dcn-asu.ru or
2
Abstract-In this paper, a technique for estimating the maximum bound water content and complex dielectric constants for both the bound and the free water in soil is presented. Thus attained dielectric properties for the water in soil, are used to derive the Debye spectroscopic parameters for both types of water. Empirical data sets for the soil complex dielectric constant as a function of moisture measured only at two frequencies are sufficient for applying this technique. As a result the model for predicting soil complex dielectric constant in the microwave band is proposed and validated.
I. INTRODUCTION
The monitoring of soil moisture by means of microwave radar/radiometry from space implies the use of a model that describes the dielectric properties of the soil [1]. At present, the semi-empirical model (SEM) proposed in [2] is mostly employed. In this model, the Debye relaxation formula is extensively used to provide for the soil complex dielectric constant (CDC) dependence on frequency though the effects of interaction between the water and the soil particle surface are neglected. In this paper, the moist soil CDC dependence on frequency was implemented through the spectroscopic parameters related to the intrinsic bound and free water in soil. These parameters can be derived from the empirical soil complex refractive index (CRI) dependence on moisture content as suggested and proved in [3], [4].
II. REFRACTIVE MIXING DIELECTRIC MODEL
For a number of the soils studied in [3]-[5], and others, the measured soil CRIs in the microwave band has been proved to follow the refractive mixing dielectric model (RMDM), which is generated through mixing the CRIs related to the moist soil constituents as weighted by their partial volumetric content:
;
. (1)
;
. (2)
In equations (1), (2), the refractive index and normalized attenuation coefficient (NAC) are respectively designated by n and k with NAC being defined as the ratio of the standard attenuation coefficient to the free space propagation constant. Here and further on, subscripts b, d, and u should be related to the dry soil, bound, and free water constituents of the soil, respectively. Index W is used to denote a volumetric part of bulk water in the soil. While Wt designates the maximum bound water content (MBWC) related to a given type of soil. In the range, W £ Wt, an increment of water in the soil is exclusively supplied with bound water, while in the other range, W > Wt, it is solely provided with a free water in soil component. If the measured soil CRIs can be best of all fitted with the RMDM than the quantities of nd, kd, nb, kb, nu, ku Wt, are easily derived as fitting parameters. Transitions from n and k, to the dielectric constant e¢ and loss factor e², and vice versa can be performed with the following evident relations:
, , (3)
. (4)
With this approach, the RMDM parameters have been measured for the quartz sand, bentonite clay and adopted from [2], [6], for the Field 5 natural soil. The CRIs and CDCs for the sand and bentonite were measured at 0.6; 1.1, 1.43 GHz and 25oC with the same technique like in [6]. To give a general scope of the overall consideration only data from [6] will be further analyzed although the final result include the data related to the sand and bentonite as well. Fig. 1 shows the CRIs measured in [6] at 4.0, 10.0 and 18 GHz as a function of volumetric moisture. The bi-linear fit lines
attained with the use of formulas (1), and (2) are also shown in Fig. 1. These fits have less standard deviation from the measured data than respective parabolic CDC fits [6].
Fig 1. Measured refractive index (a) and NAC (b) in the temperature range of 20-24 0o C for the Field 5 natural soil.
Using the data in Fig. 1, and others from [6], we can derive all the RMDM parameters and turn them into respective CDCs employing equations in (3). The latter are presented in Table I.
TABLE I
DIELECTRIC CONSTANT, LOSS FACTOR, AND MAXIMUM BOUND WATER CONTENT AS A FUNCTION OF FREQUENCY
f GHz / e¢d / e¢¢d / Wt / e¢b / e¢¢b / e¢u / e¢¢u1.4 / 2.26 / 0.217 / 0.205 / 41.4 / 19.0 / 106.8 / 29.2
4.0 / 3.12 / 0.000 / 0.157 / 38.0 / 15.3 / 81.5 / 26.0
6.0 / 2.86 / 0.027 / 0.156 / 26.8 / 12.8 / 92.9 / 33.1
10.0 / 2.59 / 0.032 / 0.163 / 30.0 / 15.9 / 76.8 / 41.7
18.0 / 2.68 / 0.020 / 0.168 / 15.0 / 17.7 / 64.7 / 48.6
SA / 2.7 / 0.137 / 0.170
In Table I, SA stands for the statistical average values. As observed from these data, the dielectric constants and loss factors relating to the free and bound water in the Field 5 soil exhibit noticeable frequency dispersion. In addition, they exceed the CDCs of the pure water, which, according to [7], are of (78.16, 3.79), (75.33, 14.58), (73.73, 17.81), (62.81, 29.93), and (40.37, 36.55) at the temperature of 25°C and the wave frequencies of 1.0 GHz, 4.0 GHz, 5.0 GHz, 10.0 GHz, and 20 GHz, respectively. The observed difference can be attributed to both the effects of salinity and the interaction of water with the soil minerals.
III. GENERALIZED REFRACTIVE MIXING DIELECTRIC MODEL
Using the Debye formula
,
(5)
where e¥ is the dielectric constant in the high-frequency limit, eo is the static dielectric constant, f , t, s stand for the frequency, relaxation time, effective conductivity, and er is the dielectric constant for free space we can deduce the following relations between the soil water CDCs measured at two frequencies, e¢1=e¢(f1), e¢2=e¢(f2), and e²1=e²(f2), e²2=e²(f2), and the values of t, eo, and s:
(6)
, (7)
. (8)
In formulas (7), (8), index q can take values of 1 or 2. The formulas (6)-(8) have been derived suggesting for e¥ the value of 4.9, as given in [1].
Both the bound and free water spectroscopic parameters can be calculated with the use of Table I and formulas (6)-(8). The results of this estimation are shown in Table II.
TABLE II
CONDUCTIVITIES, RELAXATION TIMES, AND STATIC DIELECTRIC CONSTANTS FOR THE BOUND AND FREE WATER IN SOIL
fi&j / sb(S/m) / su
(S/m) / tb
(ps) / tu,
(ps) / eb0 / eu0
f1.4&4 / 1.16 / 1.59 / 12.09 / 9.77 / 41.42 / 96.80
f1.4&6 / 1.19 / 1.68 / 11.22 / 8.44 / 36.23 / 104.5
f1.4&10 / 1.25 / 1.68 / 8.66 / 8.56 / 39.49 / 102.5
f1.4&18 / 1.10 / 1.79 / 14.49 / 6.93 / 42.15 / 104.3
f4&6 / 1.61 / 2.46 / 9.70 / 7.74 / 34.84 / 92.35
f4&10 / 1.95 / 2.20 / 7.86 / 8.36 / 37.68 / 90.71
f4&18 / 0.65 / 2.88 / 14.89 / 6.77 / 43.23 / 91.68
f6&10 / 2.19 / 1.45 / 7.59 / 8.65 / 32.14 / 100.1
f6&18 / 0.03 / 3.66 / 15.43 / 6.65 / 40.08 / 98.44
SA / 1.24 / 2.15 / 11.33 / 7.99 / 39.35 / 97.93
SD / 0.65 / 0.73 / 3.08 / 1.05 / 3,76 / 5.4
In Table II, the quantities having subscripts b and u should be related to the bound and free water, respectively. The quantities SA and SD stand for a statistical average and standard deviation. Notation fi&j is used for a combination of frequencies i and j. A set of the formulas from (1) to (8) in conjunction with the fitting algorithms can be collectively referred to as the Generalized Refractive Mixing Dielectric Model (GRMDM). To represent a complete GRMDM data base for Field 5 the values in Table II should be complemented with the dry soil CDC, ed=(2.51; 0.017), and MBWC, Wt = 0.17.
IV. THE GRMDM VALIDATION
For the purpose of GRMDM validation, the soil CDCs were calculated to be compared with the initial experimental data. First, with the use of Table II and formulas (5), the values of e¢b, e¢u, e²b, e²u were
acquired, for each pair of the frequencies. Secondly, the CDCs thus attained were converted into nb, nu, kb, ku by formulas in (4). Thirdly, the latter were applied to calculate by the formulas (1), (2) the predictions of the soil CRIs related to the frequencies which had not been used for estimating the soil water spectroscopic parameters sb, su, tb, tu, eb0, eu0. Finally, the values of ns, ks, were converted into the soil CDCs, e¢s, e²s, using the equations in (3). In Fig. 2, the latter are compared to the initially measured soil dielectric constants and loss factors.
Fig. 2. Comparison between the measured Field 5 soil CDCs and the GRMDM predictions calculated for the frequencies of 1.4 (a) and 18.0, (d), GHz, which correspond to the largest prediction SDs.
Nine prediction curves related to separate pairwise frequency combinations in Table II could have been drawn in Fig.2 for every measured CDC data set. To avoid merging only two of those are plotted, which have the greatest standard deviation. As observed from Fig. 2 the accuracy of the GRMDM predictions, in the frequency band considered, is of the same order as that of the RMDM, applicable at a single frequency. The analysis conducted for the sand and bentonite samples resulted in the same conclusion.
V. SPECROSCOPIC PARAMERERS ANALISES
In order to estimate the variations in spectroscopic parameters arising due to texture and mineralogy those are summarized in Table III, with the data of pure water [7] being included. As seen from Table III, the relaxation times for the free and pure water are in reasonable agreement with each other.
TABLE III
COMPARISON OF SPECTROSCOPIC PARAMETERS
Water type / sb(mS/m) / su
(mS/m) / tb
(ps) / tu
(ps) / eb0 / eu0
Field 5 / 1240 / 2150 / 11.3 / 8.0 / 39.4 / 97.9
Bentonite / <30 / <30 / 8.1 / 8.7 / 29.8 / 81.8
Sand / <30 / 8.7 / 80.7
Pure water / 0.012 / 8.3 / 78.4
However, the bound water relaxation time pertaining to Field 5 is about 40% larger than that of the bentonite. The static dielectric constants related to Field 5 are also larger than those of water in the bentonite, sand, or pure water.
VI. CONCLUSIONS
It should be particularly stressed on the fact that the GRMDM ability to generate predictions at the frequencies falling out of the band where the soil CDCs are measured is a feature that makes the GRMDM different from other models known in existing literature. This suggests that by using the GRMDM technique, the measurements needed to ensure dielectric predictions for a given type of soil could be significantly reduced. Hence, prospects arise for developing soil dielectric databases in a whole microwave band for each specific type of the soil, like that in Table II. An extended narrative on the GRMDM can be found in [8].
ACKNOWLEDGEMENT
The authors would like to thank the Alaska SAR Facility and the director of the Geophysical Institute, Dr. Roger Smith, for supporting this research at the University of Alaska Fairbanks.
REFERENCES
[1] F. T. Ulaby, R. K. Moor, and A. K. Fung, Microwave Remote Sensing, Active and Passive, vol. II. Dedham, MA: Artech House, 1986.
[2] M. C. Dobson, F. T. Ulaby, M. T. Hallikainen, M. A. El-Rayes, “Microwave dielectric behavior of wet soil. Part II: Dielectric mixing models,” IEEE Trans. Geosci. Remote Sensing, vol. 23, no. 1,pp. 35-45, 1985.
[3] V.L. Mironov, S.A. Komarov, N.V. Rychkova, and V.N. Kleshchenko, “Study of the dielectric properties of wet grounds at nicrowave frequencies,” Earth Obs. Rem. Sens., vol. 12, no. 4, pp. 495-504, 1995.
[4] S. A. Komarov, V. L. Mironov, Microwave Remote Sensing of Soils. Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences, 2000 (in Russian).
[5] A. M. Shutko, and E. M. Reutov, “Mixture formulas applied in estimation of dielectric and radiative characteristics of soils and grounds at microwave frequencies,” IEEE Trans. Geosci. Remote Sensing, vol. 20, no. 1, pp. 29-32, 1982.
[6] M. T. Hallikainen, F. T. Ulaby, M. C. Dobson, M. A. El-Rayes, and Lin-Kun-Wu, “Microwave dielectric behavior of wet soil. Part I: Empirical model and experimental obseravations,” IEEE Trans. Geosci. Remote Sensing, vol. 23, no. 1,pp. 25-33, 1985.
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