Temperature and Mineralogy Dependable Model for Microwave Dielectric Spectra of Moist Soils

Transcription

1 PIERS ONLINE, VOL. 5, NO. 5, Temperature and Mineralogy Dependable Model for Microwave Dielectric Spectra of Moist Soils V. L. Mironov and S. V. Fomin Kirensky Institute of Physics, SB RAS, Krasnoyarsk, Russia Abstract In this paper, a physically based dielectric model in microwave band for moist soils is developed to account for both the temperature and granulometric mineralogy of the soil. The generalized refractive mixing dielectric model (GRMDM) previously developed by V. L. Mironov et al. was used as a mean to determine the spectroscopic parameters for 5 soils which complex dielectric constant spectra were measured and presented in Technical Report EL-95-34, December 1995 by J. O. Curtis et al. The measurement results used covered the temperatures of 10, 20, 30, and 40 C, gravimetric clay contents from 0 to 0.76 g/g, and frequencies from 0.3 to 26.5 GHz. Relating to each individual soil, such GRMDM spectroscopic parameters as dielectric constants in low frequency limit, relaxation times, and ohmic conductivities, pertaining to both bound and unbound soil water, were fitted as functions of temperature with the Clausiuss- Mossotii, Debye and linear equations, respectively. As a result, a set of physical parameters for a temperature dependable generalized refractive mixing dielectric model (TD GRMDM), consisting of the volumetric expansion coefficients, activation energies, enthropies of activation and others, were obtained. Finally, the parameters of the TD GRMDM as functions of gravimetric clay content were fitted to yield the closing set of polinomial formulas, which, in conjunction with the TD GRMDM, represent a temperature and mineralogy dependable soil dielectric model (TMD SDM). Thus developed, the TMD SDM provides for predictions of the complex dielectric constant of moist soils as a function of moisture, wave frequency, temperature and gravimetric clay content. Further the TMD SDM prediction error was estimated with the use of the ndependent dielectric data. 1. INTRODUCTION Dielectric models of the soil are an essential part in the algorithms used for data processing with regard to the problems of radar and radiothermal remote sensing [1]. Recently, a mineralogy based spectroscopic dielectric model (MBSDM) [2, 3] has been developed and validated over a large dielectric data set [4] to ensure microwave dielectric spectra predictions as a function of moisture and soil texture at the fixed temperature of 20 C. This model is based on the generalized refractive mixing dielectric model (GRMDM) introduced in [5]. The MBSDM physically based provided for a substantially less error of predictions as compared with the ones delivered by the semiempirical dielectric model (SDM) of [1], though the latter is at present considered as a routine tool for predicting complex dielectric constant (CDC) spectra of moist soils in the microwave band. At the same time, there was developed a dielectric model [6] accounting for temperature variations, provided an individual type of the soil in terms of mineral content and texture be considered. Meanwhile, a joint impact of soil texture and temperature on the dielectric spectra of moist soils has not been analyzed yet. In this paper, such a task was formulated and accomplished. Using the methodology of the temperature dependable refractive mixing dielectric model (TD GRMDM) of [6] and the dielectric data set of [4], for each of 5 individual sols of [4], there were derived the assemblages of the TD GRMDM physical parameters. The latter consist of volumetric expansion coefficients, starting dielectric constants in low frequency limit, activation energies, entropies of activation, starting conductivities, and conductivity incrimination coefficients, pertaining to the bound and unbound types of soil water. Further, the TD GRMDM parameters obtained were fitted as a function of clay percentage with polynomial functions. Derived by this way, coefficients of the polynomial fits, in conjunction with dry soil CDC and maximum bound water fraction value, are considered as input parameters for a new temperature and mineralogy dependable soil dielectric model (TMD SDM) presented in this paper. To make assessments of the error of the TMD SDM predictions, the latter were calculated in the multidimensional domain, which included wave frequency, soil moisture, soil texture, and temperature, to be further correlated with the respective values measured in [4] for 11 soils. The error of the predictions obtained with the TMD SDM proved to be on the same order as that of the MBSDM predictions estimated in [2, 3].

2 PIERS ONLINE, VOL. 5, NO. 5, THE TMD SDM CONCEPT From a physical viewpoint, the complex dielectric constant (CDC) of a thawed moist soil, ε s (µ, m v, f, t), must depend on a vector variable, characterizing soil mineralogy and texture, µ, volumetric percentage of water in soil, m v, wave frequency, f, and temperature, t. As a function of volumetric moisture, m v, the complex index of refraction (CIR) of moist soil, n s(µ, m v, f, t) = ε s (µ, m v, f, t), (1) can be expressed [5] in the form of the refractive mixing dielectric model (RMDM): εs (µ, t, f, m v, t) = ( ε d (µ, t) + εb (µ, t, f) 1) [m v + (m vt m v ) H (m v m vt )] ( ) + εu (µ, t, f) 1 (m v m vt ) H (m v m vt ) (2) where, ε d (µ, t), is the CDC of dry soil, ε b (µ, f, t) and ε u (µ, f, t) are the CDCs of the bound and unbound (free) soil water, respectively, m vt is the maximum bound water fraction (MBWF), and H(x) denotes the Heaviside step function: u(x) = 1 if x > 0, and u(x) = 0 if x 0. The bound water is adsorbed on the surface of soil solids, while the unbound soil water exists in liquid droplet phase. The MBWF is such an amount of water in soil that any additional water added to the soil in access of this amount behaves as unbound water. As seen from (1), the CIR is a piecewise linear function of soil moisture, with the MBWF being a transition point in terms of slope angle between the two linear legs relating to the bound, m v m vt, and unbound, m v > m vt, moisture ranges. The application of the RMDM is limited to a given type of soil at the fixed values of wave frequency, and temperature. The CIR for dry soil, n d (µ, t), as well as the ones for the bound, n b (µ, f, t), and unbound, n u(µ, f, t), types of soil water, alongside with the MBWF, are considered as the RMDM parameters. The CIR can be expressed through the index of refraction (IR), n, and normalized attenuation coefficient (NAC), κ: n = Ren and κ = Imn (3) which determine a moist soil medium in terms of wave phase velocity and wave attenuation, respectively. The RI and NAC are understood here as a proportion of the propagation constant and standard attenuation coefficient in a medium, to the free space propagation constant, respectively. If the RI and NAC are known, the respective dielectric constant (DC), ε = Reε, and loss factor (LF), ε = Imε, can be easily calculated using the following formulas: ε = n 2 κ 2, ε = 2nκ. (4) The inverse transformation is available through the following equations: n 2 = (ε ) 2 + (ε ) 2 + ε, κ 2 = (ε ) 2 + (ε ) 2 ε. (5) The IR and NAC relating to the bound and unbound soil water can be determined as a function of frequency through fitting formula (2) to the moisture dependences measured at varying frequencies. From that fitting, the values of IR and NAC relating to the dry soil are derived as well. The respective DCs and LFs follow from (4). With this approach [5], the DC and LF spectra relating to both the bound and unbound types of soil water were shown to follow the Debye formula: ε p (µ, f, t) = ε p(µ, f, t) + iε p(µ, f, t) = ε p + ε p0(µ, t) ε p 1 i2πfτ p (µ, t) + iσ p(µ, t) 2πfε 0 (6) where p is any one of b, u; ε p0 (µ, t) and ε p (µ, t) = 4.9 are the low- and dielectric constants in low and high frequency limit, τ p (µ, t) is the relaxation time, and σ p (µ, t) is the ohmic conductivity, each specific to the bound, p = b, and unbound, p = u, forms of soil water. Finally, ε 0 = pf/m is the permittivity of vacuum. As a result, the values of n d (µ, t), κ d (µ, t), m vt (µ, t), ε 0b (µ, t), ε 0u (µ, t), τ b (µ, t), τ b (µ, t), σ b (µ, t), and σ u (µ, t) are considered as spectroscopic parameters in the frame of the GRMDM developed in [5]. With the fitting procedures like in [3] or [5], these can be determined

3 PIERS ONLINE, VOL. 5, NO. 5, from routine dielectric spectra measurements conducted for an individual type of moist soil at a given temperature. In order to make the GRMDM expressed with formulas (1) (6) a temperature dependable generalized refractive mixing dielectric model (TD GRMDM), some of the GRMDM spectroscopic parameters were represented as a function of temperature [6]. The dependence of the dielectric constant in low frequency limit on the temperature was taken in a form of the Clausiuss-Mossotii equation [7]: ε 0p (µ, t) = exp(f p(µ, t s ) β p (µ)(t t s )) 1 exp(f p (µ, t s ) β p (µ)(t t s )) (7) where β p is the volumetric expansion coefficient, t and t s are the current and starting temperatures by degrees centigrade. The function F p (t) is given with the equation F p (µ, t) = ln[(ε p0 (µ, t) 1)/(ε p0 (µ, t) + 2)]. (8) The relaxation time was expressed with the Debye relaxation formula [7] accounting for the temperature dependence: τ p (µ, t) = ( t exp Hp (µ) (t )R S ) p(µ) (ps) (9) R where H p and S p are the activation energy and entropy of activation, respectively, and R is the universal gas constant. Finally, the conductivity, σ p, was suggested to have a linear dependence on the temperature, which is characteristic for the ionic solutions: σ p (µ, t) = σ p (µ, t s ) + β σp (µ)(t t s ). (10) here, β σp is the temperature incrementation coefficient for conductivity. While σ p (µ, t s ) is the value of conductivity at a starting temperature, t s. As a result, to make the CDC predictions for an individual type of soil, ε(µ, m v, f, t), with the use of the Equations (1) (10), an assemblage of which represents the TD GRMDM [6], the following input parameters must be known; n d (µ, t s ), κ d (µ, t s ), m vt (µ, t s ), ε 0b (µ, t s ), β b (µ), ε 0u (µ, t s ), β u (µ), H b (µ)/r, S b (µ)/r, H u (µ)/r, S u (µ)/r, σ b (µ, t s ), β σb (µ), σ u (µ, t s ), and β σu (µ). In the following section, a technique will be oulined to derive the TD GRMDM parameters for individual soils. 3. TD GRMDM PARAMETERS FOR IDIVIDUAL SOILS In [4], the dielectric data were measured over the frequency ranges from 45 MHz to 26.5 GHz, with the moistures spanning from nearly dry samples to the ones saturated up to field moisture capacity. The clay content in the soils varied from close to 0% to 76% by weigh. At the temperatures of 10, 20, 30, and 40 C, the values of GRMDM spectroscopic parameters, ε d (µ, t), ε d (µ, t), m vt(µ, t), ε 0b (µ, t), ε 0u (µ, t), τ b (µ, t), τ b (µ, t), σ b (µ, t), and σ u (µ, t), were derived by fitting simultaneously the CDCs calculated with the formulas (1) (6) to the DC and LF spectra measured for each individual soil within the frequency range from 0.3 to 26.5 GHz, at the moistures available. Earlier, this methodology was applied in [2, 3]. In this analysis only 5 soils of [4] were involved, with their clay proportion by weigh being of 0, 14, 34, 54, and 76%. Further, similar to [6], the GRMDM parameters obtained were fitted with the Equations (7) (10) to derive the TD GRMDM parameters for each individual soil, complimenting the GRMDM ones, pertaining to a starting temperature of 20 C, that is, β b (µ), β u (µ), H b (µ)/r, S b (µ)/r, H u (µ)/r, S u (µ)/r, β σb (µ), and β σu (µ). As the next step, the TD GRMDM parameter were fitted as functions of clay content, C, to yield the equations describing the TMD SDM. The latter are given in the following section. Similar to [2, 3], the clay content, C, was used as the only variable to account for the soil mineralogy and texture parameter µ.

4 PIERS ONLINE, VOL. 5, NO. 5, TDM SDM EQUATIONS The following sequence of equations was derived as a result of fitting the TD GRMDM parameters with the expressions (7) (10): n d (C, t s ) = C C 2 (11) κ d (C, t s ) = C (12) m vt (C, t s ) = C (13) ε 0b (C, t s ) = C C 2 (14) β b (C) = C C C C 4 (15) ε 0u (C, t s ) = 100 (16) β u (C) = C C C C 4 (17) H b (C)/R = C C C C 4 (18) S b (C)/R = C C C C 4 (19) H u (C)/R = C C C C 4 (20) S u (C)/R = C C C C 4 (21) σ b (C, t s ) = C (22) β σb (C) = C C C C 4 (23) σ u (C, t s ) = (1 (1 C 10 2 ) ) (24) β σu (C) = C C C C 4 (25) were C must be assigned in percent. The assemblage of formulas (1) (25) constitutes the temperature and mineralogy dependable soil dielectric model (TMD SDM), which provides for CDC predictions of moist soils as a function of frequency, soil moisture, clay content, and temperature. In the following sections a correlation analysis between the TMD SDM predictions and measured values of CDCs will be conducted to estimate the error of that predictions. 5. VALIDATION OF THE TMD SDM PREDICTIONS To perform such a validation of the TMD SDM, the correlation analysis was carried out on the basis of all dielectric data available in [4], including the data related to the remaining 6 soils of [4], which were not used for obtaining the Equations (11) (25). The results of validation are presented in Fig. 1. As seen from Fig. 1, the TMD SDM provided predictions for both the CDs and LFs with the reasonable correlation coefficients, R DC = 0.99 and R LF = The respective standard deviations, SD DC = 1.91 and SD LF = 1.284, are also reasonable, considering the whole set of (a) (b) Figure 1: Correlation of the TMD SDM predictions, ε p, ε p, for DCs (a) and LFs (b) with the measured ones, ε m, ε m, with the data for all soil types and temperatures available in [4] being included. Solid and dotted lines represent linear fits and bisectors, respectively. Correlation coefficients, R DC and R LF, and standard deviations, SD DC and SD LF, are equal to: R DC = 0.991, R LF = 0.978, SD DC = 1.91, SD LF = The linear fits are expressed as follows: ε m = ε p, ε m = ε p.

5 PIERS ONLINE, VOL. 5, NO. 5, measured dielectric data employed. These values are close to the ones achieved with the use of the MBSDM [3], which can be applied only at the starting temperature of 20 C. 6. CONCLUSIONS Summing up the results, the following has to be stated as primary findings of this research. The proposed and substantiated TMD SDM is the first physically based model taking into account the temperature and mineralogy impact on CDC microwave dielectric spectra of most soils using the dependence of soil water CDC, for both bound and unbound, on the frequency and temperature with the use of the well known Debye and Clausius-Massotii laws, as well as the linear dependence of soil water ionic conductivity on the temperature. The error of the predictions obtained with the TMD SDM proved to be on the same order as that of the MBSDM predictions, which has never been previously achieved for a broad variety of soils with existing empirical models, simultaneously in a domain of frequency, moisture, temperature, and soil texture variables. ACKNOWLEDGMENT This research was supported by the Siberian Branch of the Russian Academy of Sciences, Interdisciplinary Project # 6, and Russian Foundation for Basic Research, project REFERENCES 1. Dobson, M. C., F. T. Ulaby, M. T. Hallikainen, and M. A. El-Rayes, Microwave dielectric behavior of wet soil Part II: Dielectric mixing models, IEEE Trans. Geosci. Remote Sensing, Vol. 23, No. 1, 35 46, Mironov, V. L., L. G. Kosolapova, and S. V. Fomin, Soil dielectric model accounting for contribution of bound water spectra through clay content, PIERS Online, Vol. 4, No. 1, 31 35, Mironov, V. L., L. G. Kosolapova, and S. V. Fomin, Physically and mineralogically based spectroscopic dielectric model for moist soils, IEEE Trans. Geosci. Remote Sensing, Vol. 47, No. 7, 2009 (to be published). 4. Curtis, J. O., C. A. Weiss, Jr., and J. B. Everett, Effect of soil composition on dielectric properties, Technical Report EL-95-34, December Mironov, V. L., M. C. Dobson, V. H. Kaupp, S. A. Komarov, and V. N. Kleshchenko, Generalized refractive mixing dielectric model for moist soils, IEEE Trans. Geosci. Remote Sensing, Vol. 42, No. 4, , Mironov, V. L. and S. V. Fomin, Temperature dependable microwave dielectric model for moist soils, PIERS Proceedings, , Beijing, China, March 23 27, Dorf, R. C., The Electrical Engineering Handbook, 2nd Edition, Boka Raton, CRC Press LLC, FL, 1997.