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Soil Science Society of America Journal 67:720-729 (2003)
© 2003 Soil Science Society of America

DIVISION S-1—SOIL PHYSICS

Frequency Domain Versus Travel Time Analyses of TDR Waveforms for Soil Moisture Measurements

Chih-Ping Lin*

Dep. of Civil Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road, Hisnchu, Taiwan

* Corresponding author (cplin{at}mail.nctu.edu.tw)


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 REFERENCES
 
When time domain reflectometry (TDR) is applied to the field characterization of soil moisture, the waveforms have typically been analyzed using travel time along the wave guide. The apparent dielectric constant traditionally determined by the travel time analysis using a tangent-line method does not have a clear physical meaning and is influenced by several system and material parameters. The frequency domain analysis, however, can determine the actual frequency-dependent dielectric permittivity and can be performed using a very short probe. This study presents a numerical modeling approach for common unmatched TDR probes to analyze the TDR signals in the frequency domain. This approach is also adopted to examine how TDR bandwidth, probe length, probe impedance, dielectric relaxation, and electrical conductivity affect travel time analysis. Simulation results indicated that, although the effects of TDR bandwidth and probe length could be quantified and calibrated, the calibration equation for soil moisture measurements was still affected by dielectric relaxation and electrical conductivity, because of differences in soil texture and density. The effects of density can be removed by adding a density term to the calibration equation. Correlating with water content the dielectric permittivity at frequencies between 500 MHz and 1 GHz, rather than the apparent dielectric constant, can minimize the influence of soil texture.

Abbreviations: LTI, linear time invariant • TDR, time domain reflectometry


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 REFERENCES
 
SOIL WATER CONTENT is a crucial variable in hydrological modeling, agricultural water management, and most studies of soils. Time domain reflectometry has become an extensively adopted method for monitoring volumetric soil water content ({theta}), in light of the increasing demand for in situ measurement over a short time interval. A travel time analysis is frequently performed on the TDR waveform to determine the apparent dielectric constant (Ka) of a soil. A calibrated relationship between Ka and {theta} is then used to estimate the soil water content (Topp et al., 1980; Roth et al., 1990; Dirksen and Dasberg, 1993). However, the wave propagation in a wet soil is dispersive (i.e., velocity is a function of frequency) and clearly defining the arrival times of a dispersive waveform is difficult. Different methods have been proposed to determine the reflection arrivals in travel time analysis (Topp et al., 1982; Baker and Allmaras, 1990; Timlin and Pachepsky, 1996; Klemunes et al., 1997).

The accuracy of measuring water content depends on both the travel time analysis and the calibration equation that relates Ka to {theta}. Hook and Livingston (1996) demonstrated that the dominant source of error in measuring soil water content using TDR was the uncertainty in determining the apparent propagation velocity. Hook and Livingston (1995) further concluded that the dominant time interval error term was related to the transition time of the reflected pulses and that the absolute time interval errors could not be assumed to be <200 ps. They also identified the presence of dissolved ions and the use of long cables as important sources of additional transition time errors. The effects of soil electrical conductivity on TDR measurements have also been examined by Sun et al. (2000) and Jones and Or (2001). Conventional TDR methods may fail in extreme cases in which the soil's electrical conductivity is too high to allow a second reflection in the TDR waveform (Jones and Or, 2001). Travel time analysis is uncertain, and moreover, no universal calibration equation exists for soils of varying texture. Topp et al. (1980) suggested a Ka({theta}) function that was relatively independent of soil type and density. However, other researchers (e.g., Herkelrath et al., 1991; Roth et al., 1992; Dirksen and Dasberg, 1993) developed Ka({theta}) functions for organic and clay soils that did not agree with that presented by Topp et al. (1980). The ambiguity in determining the second reflection in a TDR waveform, and influential factors such as the probe system, the soil's texture, density, and electrical conductivity may all limit the accuracy and applicability of the current TDR method.

The water content calculation uses only the travel time of the TDR signal in the probe. However, waveforms measured with TDR contain much more information on the dielectric properties of the soil. This additional information should provide additional insights into other physical properties of the soil or improve the measurement of water content. Many publications in physical chemistry have discussed dielectric spectroscopy of liquids by solving the frequency dependent dielectric permittivity from the scatter function calculated from the measured waveform (Giese and Tiemann, 1975; Clarkson et al., 1977). Heimovaara (1994) followed the approach of Clarkson et al. (1977) to estimate the dielectric relaxation spectra of soils. This approach only worked for frequencies up to 250 MHz because the nonideal properties of the measurement system. Instead of solving the dielectric permittivity at each frequency, Heimovaara et al. (1996) used the Debye relaxation model (Hasted, 1973) to parameterize the dielectric relaxation spectrum of a soil. The model parameters were then back-calculated by fitting the measured waveform to the theoretical one. This approach considered a single relaxation to capture approximately the dielectric dispersion of a soil; and a small matched probe not suitable for field applications was used to satisfy the condition of a uniform transmission line according to Clarkson et al. (1977). Yanuka et al. (1988) presented a model of multiple reflections in a nonuniform transmission line but did not consider the frequency dependency of the material dielectric permittivity. Feng et al. (1999) and Lin (1999) combined the works of Yanuka (1988) and Heimovaara (1994) to formulate dispersive wave propagation as a feedback linear-time-invariant (LTI) system. The LTI system successfully simulates TDR waveforms of nonuniform transmission lines with dispersive dielectric materials. Lin (1999)(2001) provided an alternative forward formulation and a new inverse layer-peeling algorithm based on the concept of input impedance. These new developments enabled the in situ dielectric spectroscopy of soils using a nonuniform transmission line. However, the relationship between Ka and the frequency-dependent dielectric permittivity as affected by line parameters, the rise time of the step pulse, and the soil type, remains to be further investigated.

The dielectric relaxation spectrum, resulting from the frequency domain analysis, represents the actual electric properties of a soil. The apparent dielectric constant is artificially defined based on the apparent travel time analysis. The primary objectives of this paper are to (i) examine how several system parameters affect the apparent dielectric constant when the real dielectric relaxation spectrum of the material is fixed, (ii) elucidate the relationship between the apparent dielectric constant and the dielectric relaxation spectrum, and (ii) relate soil water content to a more definite quantity in the dielectric relaxation spectrum that is independent of the factors that influence the travel time analysis. Numerical and laboratory experiments were performed. The numerical model was based on the nonuniform and dispersive algorithm applicable to common, unmatched TDR probes. This model was used to determine the effects of TDR bandwidth, line parameters (probe impedance and length), dielectric relaxation, and conductivity on travel time analysis, as well as to elucidate the relationship between Ka and the dielectric relaxation spectrum. A series of laboratory tests were conducted to measure the TDR waveforms of various soils with various water contents and densities. The travel time analysis determined the apparent dielectric constants of the soils and the dielectric relaxation spectra were back-calculated using the numerical model. The test results were then used to validate the numerical model and determine a correlation between soil dielectric spectra and water content.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 REFERENCES
 
Travel Time Analysis of TDR Waveforms
The equivalent dielectric permittivity ({epsilon}*), representing the total effect of the frequency-dependent complex dielectric permittivity ({epsilon}) and the conductivity ({sigma}) of a soil, can be written as, (Ramo et al., 1994)

[1]
where f is the frequency; j is (-1)1/2; {epsilon}' and {epsilon}'' are the real and imaginary parts of dielectric permittivity, respectively; {epsilon}ii is the imaginary part of the equivalent dielectric permittivity, and {epsilon}0 is the dielectric permittivity of free space. The propagation velocity (v) of an electromagnetic wave that travels in a material with equivalent dielectric permittivity ({epsilon}*) is a function of frequency since the dielectric permittivity depends on frequency. It can be written as, (Ramo et al., 1994)

[2]
where c is the speed of light. The denominator in Eq. [2] can be considered as the apparent dielectric permittivity of each frequency component. Topp et al. (1980) ignored the dielectric relaxation and loss, and assumed the denominator to be a constant. Accordingly, the denominator in Eq. [2] was replaced by the apparent dielectric constant (Ka) and the corresponding propagation velocity was called apparent velocity (va). The apparent dielectric constant, Ka, can be determined from the measured va to be,

[3]

Apparent velocity, va, is determined from the time difference between the arrivals of the two reflections and the round-trip length of the probe in the soil. The location of the first reflection is a probe property, which can be determined by shorting the probe where it enters the soil. Figure 1 presents two available methods to identify the second reflection arrival in a TDR waveform. Although they are both tangent-line methods, they employ different schemes. The first method is a manual method in which the point of reflection is located at the intersection of the two tangents to the curve, marked as point a in Fig. 1. The second scheme is an adaptation of the manual method to calculate travel times automatically. A line parallel to the horizontal axis is drawn tangent to the smoothed trace at a local minimum. A second tangent is drawn at the point of maximum gradient after the local minimum of the TDR waveform. The intersection of this line with the horizontal line is the point at which the reflection arrives, and is marked as point b in Fig. 1. The second method appears to be less arbitrary than the first because the points of the local minimum and the maximum gradient can be clearly defined mathematically. Timlin and Pachepsky (1996) and Klemunes et al. (1997) compared both methods and concluded that the latter provided a more accurate calibration equation. Thus, the second method was used to determine Ka here.



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  		Fig. 1. Two tangent-line schemes for determining the reflection point of a TDR waveform.

 
Dielectric Dispersion Models
The frequency-dependent dielectric permittivity within TDR bandwidth of a homogeneous material may be described by the Debye equation [Hasted, 1973],

[4]
where {epsilon}dc is the static dielectric permittivity; {epsilon}{infty} is the dielectric permittivity at an infinite frequency, and frel is the relaxation frequency of the material. The Debye equation specifies the dielectric property of each soil phase, which is almost independent of frequency below microwave frequencies. The heterogeneity of a soil, however, increases the complexity of its dielectric property. Such heterogeneity has three major effects—bound water polarization, the Maxwell-Wagner effect, and double layer polarization (Hilhorst, 1998). These interfacial effects cause apparent dielectric relaxation within the TDR bandwidth.

The Debye equation has been used to model the apparent dielectric relaxation of soils to determine the response functions and time domain signals (Heimovaara, 1994; Friel and Or, 1999). Such an approach uses a single relaxation to capture approximately the dielectric dispersion because of interfacial effects. A volumetric mixing model that sums two or more dielectric spectra with different Debye parameters may describe dielectric dispersion of soils more realistically than Debye equation. Dobson et al. (1985) and Heimovaara et al. (1994) formulated a four-component volumetric mixing model to predict the frequency-dependent complex permittivities of soils. The mixing equation can be rewritten in terms of physical parameters of the soil as,

[5]
where {epsilon}*s, {epsilon}*fw, {epsilon}*bw, and {epsilon}*a are the complex dielectric permittivities of soil solids, free water, bound water, and air, respectively; {alpha} is a fitting parameter; {rho}d is the bulk dry density of the soil; {rho}s is the average density of the solid phase; the product of {delta}{rho}dAs represents the volumetric bound water content; {delta} is the average thickness of the bound water, and As is the specific surface of the soil. Dividing the water into bound and free water fractions only yields an approximate description of the actual distribution of water molecules in the soil medium. Such a division is based on a somewhat arbitrary criterion for establishing the transition point between bound and free water layers. The average thickness of the bound water varies with the mineralogy of the soil particles and electrolytes in the water. The effective specific surface, Aes, is defined here as the equivalent specific surface such that {delta}m{rho}dAes specifies the volumetric bound water content in the four-phase soil model, assuming {delta} = {delta}m = 3 x 10-10 m, where {delta}m is the thickness of a molecular water layer. This definition reflects the combined effect of the fineness of the soil particles, soil mineralogy and electrolyte. The complex dielectric permittivity of each phase is either a constant or described by a Debye equation. The Debye parameters of bound water are notedly assumed to be those of ice (Hilhorst, 1998). Table 1 summarizes the range and reference values of the volumetric mixing parameters. Parameters that do not vary widely are assumed to be constant and are represented by their typical values. Consequently, the volumetric mixing equation includes a total of six parameters, namely, {theta}, {rho}d, Aes, {sigma}fw, {sigma}bw, and {alpha}, to describe the dielectric dispersion of soils.


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  		Table 1. Parameters in the volumetric mixing model

 
Simulating TDR Waveforms
In general, the transmission line in a TDR system includes a coaxial cable, a probe head, and the probe section inserted into the soil. Dielectric dispersion and the nonuniform nature of the transmission line must be considered in simulating the TDR waveform. Figure 2 shows an equivalent circuit for the TDR system. The transmission line is divided into n different sections, each of which is uniform and characterized by a propagation constant (or wave number, {gamma}), characteristic impedance (Zc), and length (l). The propagation constant is a function of the dielectric permittivity ({epsilon}*) of the material and governs the speed and decay of a wave along the line. The characteristic impedance is a function of the cross-sectional geometry of the transmission line and the dielectric permittivity of the material. It controls the occurrence of reflections. {gamma} and Zc can be written as,

[6]

[7]
where c is the speed of light and Zp is the characteristic impedance of the transmission line filled with air, and is thus only a function of the cross-sectional geometry of the transmission line. Zp may be derived from electromagnetic theory, but it can be more easily calibrated using material of known dielectric permittivity.



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  		Fig. 2. A cascade of uniform lines represents a multi-section transmission line. Each uniform line (i) is characterized by its propagation constant ({gamma}i), characteristic impedance (Zc,i), and length (li).

 
In a line with sections of different characteristic impedances, waves can be reflected and transmitted at the interfaces of the sections. The propagation velocity is a function of frequency since the dielectric permittivity of the insulating material depends on frequency. The TDR waveform recorded by the sampling oscilloscope is a result of multiple reflections and dispersion. In the frequency domain, the TDR sampling voltage V(0) (the voltage at z = 0 in Fig. 2) can be derived as, (Lin, 2001)

[8]
where V(0) is the Fourier transform of the TDR waveform; Vs is the Fourier transform of the TDR step input; Zs is the source impedance of the TDR instrument (typically Zs = 50 {Omega}), and Zin(0) is the input impedance at z = 0. As shown in Fig. 2, the input impedance Zin(z) is the equivalent impedance when looking into the circuit from position z. The input impedance at z = 0 (i.e., Zin[0]) represents the total impedance of the entire nonuniform transmission line (Lin, 2001). It can be derived recursively from the characteristic impedance and the propagation constant of each uniform section, starting from the terminal impedance ZL:

[9]
where Zc,i, {gamma}i, and li, are the characteristic impedance, propagation constant, and length of each section, respectively, and ZL is the terminal impedance. A typical TDR measurement system uses an open loop (ZL = {infty}) or a closed loop (ZL = 0). Once the input impedance of the entire line is obtained from Eq. [9], the sampling voltage in frequency domain can be determined using Eq. [8]. Let the voltage source of the TDR be denoted as vS(t), the sampling voltage as v(t), the fast Fourier Transform algorithm as the function FFT( ), and the inverse fast Fourier Transform as the function IFFT( ). Simulating of a TDR waveform involves the following steps. (i) Determine an appropriate window size in frequency and time domains to prevent aliasing in the discrete Fourier Transform; (ii) set Vs = FFT(vs); (iii) determine V(0) from Eq. [8] and [9], and (iv) set V(t) = IFFT[V(0)]. Lin (1999) details the use of digital signal processing in spectral analysis.

Laboratory Experiments
Time domain reflectometry measurements were made by attaching the TDR probe to a Tektronix 1502B(Tektronix, Beaverton, Or) via 90 cm of 50-{Omega} coaxial cable fitted with 50-{Omega} BNC connectors at each end. The TDR probe was a coaxial cylinder with a central rod. The diameter of the central rod and the inside diameter of the cylinder were 8 and 102 mm, respectively. The height of the cylindrical mold was 116 mm. The TDR waveform of deionized water was measured and used to calibrate the probe impedance and length. Five soil mixtures were made from Ottawa sand, natural silt, and Illite, to yield a wide range of soil properties. Table 2 shows the percentage of mixing. Four or five different gravimetric water contents were used to prepare samples in separate large containers for each type of soil mixture. The soil was oven-dried first before water was gradually added. The soil and water were mixed thoroughly to obtain the desired water content. The mixed soil was sealed with plastic wrap and allowed to equilibrate for more than 24 h, to yield a uniform soil specimen. Sufficient soil was prepared to produce four compacted specimens of different soil densities but a fixed gravimetric water content. On soil compaction in the cylindrical mold, the total mass of the soil and mold was measured. A central rod was then inserted and the TDR measurement taken. Then, the soil was oven-dried to determine the water content. The volumetric water content ranged from 6.0 to 38.5% and the bulk dry density ranged from 1.25 to 2.22 g cm-3.


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  		Table 2. Composition of soils used in this study.

 

   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
SUMMARY AND CONCLUSIONS
 REFERENCES
 
Simulating TDR Waveforms and Estimating Dielectric Spectra
Several typical results have been selected as examples of the substantial data collected in this study to illustrate the simulation of TDR waveforms and the estimation of dielectric spectra. The circles in Fig. 3A indicate the measured TDR waveform of the mixed Soil M3 with {theta} = 23% and {rho}d = 1.854 g cm-3. Figure 3A marks reflections from the soil's surface and the end of probe, as downward and upward triangles. The transmission line parameter (impedance and length) must be calibrated when it is used for the first time to measure the dielectric permittivity of the tested material. The TDR measurement system is approximated by five sections of uniform transmission line (n = 5 in Fig. 2), including one section of coaxial cable, three sections of the probe head, and one section of the probe. The measured waveform of deionized water, whose dielectric property is well known, is used to calibrate the impedance and length of the probe. After the system parameters of the TDR probe are properly calibrated, the only unknowns for the TDR transmission line are the dielectric properties of the material inside the probe. As discussed above, the dielectric properties of the soil are parameterized by {theta}, {rho}d, Aes, {sigma}fw, {sigma}bw, and {alpha} in Eq. [5]. The volumetric water content ({theta}) and bulk dry density ({rho}d) were measured during the experiment. Accordingly, the volumetric mixing model includes only four unknowns (Aes, {sigma}fw, {sigma}bw, and {alpha}). These four parameters can be back-calculated from the measured waveform by least-square optimization. The bold solid line in Fig. 3B represents the resulting dielectric relaxation spectrum, while that in Fig. 3A represents the predicted TDR waveform, for comparison with the measured values. For each soil specimen, the predicted TDR waveform matched the measurements very closely. However, the inferred parameters for a single soil mixture with different water contents and densities are not all the same. For example Aes ranged from 50 to 240 m2 g-1, {sigma}fw from 0.35 to 0.6 S m-1, {sigma}bw from 5 to 25 S m-1, and {alpha} from 0.35 to 0.65. If {theta} and {rho}d were assumed to be unknown and all six parameters were back-calculated from the measured waveform, then the back-calculated {theta} and {rho}d values did not equal the measured values. However, the predicted waveform and the estimated dielectric spectrum were almost the same as those obtained using four parameters. This result suggests that the volumetric mixing model can fit the TDR waveform and estimate the dielectric relaxation spectrum for each individual soil specimen, but the six model parameters are correlated.



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  		Fig. 3. (a) The measured TDR waveform of Soil M3 compared with that predicted by the volumetric mixing model and Debye equation, and (b) the real part of a dielectric relaxation spectrum estimated by the volumetric mixing model and Debye equation.

 
The Debye equation was also used to model the dielectric dispersion of the soil. The thin solid lines in Fig. 3 represent the resulting back-calculated dielectric relaxation spectrum and predicted waveform. Both dielectric dispersion models could match the general shape of the measured waveform. However, the theoretical waveform from the volumetric mixing model matched the measured waveform better in the details, as shown in Fig. 3A. Pertinent literature includes substantial information on the dielectric properties of soils up to a frequency of 108 Hz. Very high dielectric constants (>102) have been measured at frequencies below 104 Hz (Hoekstra and Delaney, 1974). At low frequencies, the dielectric permittivity is approximately inversely proportional to frequency, and levels off at a frequency of around 107 Hz. The Debye equation cannot model the increase in dielectric permittivity at low frequencies, as shown in Fig. 3B, because of its functional constraint. Consequently, the volumetric mixing model is superior to the Debye equation in modeling dielectric dispersion over the frequency range of the cable tester. Importantly, when the goal is to determine the dielectric spectrum, the dispersion model must have sufficient flexibility to describe the dielectric permittivity as a function of frequency.

Effects of Line Parameters (Impedance and Length)
The waveform of the mixed Soil M3, measured by the TDR system described above, was considered as a reference case in the simulation exercise. Simulating the TDR waveforms by varying the probe impedance while fixing all other parameters revealed that the impedance of the probe does not influence the measured Ka. However, the simulation of TDR waveforms by varying the probe length (from 2.5–30 cm) revealed that probe length did affect Ka. As shown in Fig. 4A , Ka is basically constant when the probe length is between 10 and 25 cm. However, when the length of the probe is under 5 cm, Ka is markedly higher than that determined by a probe of length between 10 and 25 cm. The second reflection in the mixed Soil M3 could not be determined when the length of the probe exceeded 35 cm. As defined in Eq. [3], Ka is inversely proportional to the square of the probe length. Thus, a short probe length magnifies the error in Ka when the round-trip travel time is overestimated or underestimated. However, dielectric dispersion always causes overestimation or underestimation. Therefore, Ka is not invariant with the probe length. The length of a probe should be at least 10 cm to minimize this effect and ensure accurate estimates of water content, but the probe must be short enough to allow a second reflection. Determining the correct left can be difficult when the soil is highly conductive. Frequency domain analysis of TDR waveforms has advantages in this respect, because it can be performed using a very short probe.



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  		Fig. 4. Numerical simulations using parameters that fit the TDR waveform of Soil M3 but varying the probe length show that (a) the apparent dielectric constant is not invariant of the probe length, and (b) the apparent dielectric constant increase with rise time.

 
Effects of TDR Bandwidth (Rise Time)
The rise time of the step pulse of a Tektronix 1502C is about 200 ps, corresponding to a bandwidth of 1.5 GHz (TDR bandwidth {approx} 1/[{pi} x rise time]). Using a slower step-pulse generator may reduce the cost of a TDR measurement system. However, the calibration equation for water content may depend on the rise time of the TDR device. Moreover, a long cable tends to increase the rise time of the pulse into the probe. Time domain reflectometry waveforms were simulated by varying the rise time of the step pulse (from 50 ps to 1.5 ns) while fixing all other parameters in the reference case, to clarify the effects of rise time on apparent dielectric constant. As shown in Fig. 4B, Ka increases with the step pulse. For example, Ka increases by 6% when the rise time varies from 200 to 500 ps (such that the TDR bandwidth decreases from 1.5 GHz to 600 MHz). This increase may be explained by the dielectric relaxation spectrum shown in Fig. 3B, which shows a 4.2% increase in the real part of the dielectric permittivity when the frequency declines from 1.5 GHz to 600MHz. This increase in the real part of the dielectric permittivity accounts for most of the increase in Ka with rise time.

Interestingly, comparing Fig. 3B and 4B further shows that the measured Ka for any particular rise time is close to the real part of the dielectric permittivity at the highest frequency within the corresponding bandwidth. This closeness may result from the major dielectric relaxation of Soil M3 at frequencies far below 200 MHz, the maximum frequency corresponding to 1.5 ns rise time. The physical interpretation of Ka with reference to the dielectric relaxation spectrum within the TDR bandwidth will be considered in the following section.

Effects of Dielectric Relaxation and Conductivity
Ka can be shown to approach asymptotically {epsilon}dc when the relaxation frequency far exceeds the TDR bandwidth, and gradually approaches {epsilon}{infty} when the relaxation frequency is well below the TDR bandwidth. The relative position of Ka between {epsilon}{infty} and {epsilon}dc depends on both the TDR bandwidth and the dielectric relaxation. It is defined here as the relative dielectric constant,

[10]

In the Debye equation, the relaxation frequency (frel) and the difference between {epsilon}{infty} and {epsilon}dc ({Delta}{epsilon} = {epsilon}dc - {epsilon}{infty}) govern the extent of dielectric dispersion within the TDR bandwidth. Using the same line parameters as used above, a parametric study was undertaken to examine the effects of dielectric dispersion on Ka by varying frel and {Delta}{epsilon} while fixing {epsilon}dc = 10 and {sigma} = 0 (for nonconductive material). Figure 5A clearly shows that Kr increases with frel and {Delta}{epsilon}. However, Kr is constant and close to unity (such that Ka {approx} {epsilon}dc) when frel exceeds a certain upper threshold frequency. For relaxation frequencies below a certain lower threshold frequency, Kr is constant and close to zero (such that Ka {approx} {epsilon}{infty}). This lower threshold frequency declines as {Delta}{epsilon} increases. For nonconductive materials (for which {sigma} = 0), Ka is a measure of {epsilon}{infty} if {Delta}{epsilon} is under 40 and frel is below 40 MHz. On the contrary, Ka represents the static value {epsilon}dc when frel exceeds 10 GHz. In both cases, the relationship between Ka and {epsilon}'r is clearly defined and the physical meaning of Ka is unambiguous. However, interpreting Ka becomes more difficult when frel is between 100 MHz and 10 GHz. The relative dielectric constants of ethanol ({epsilon}dc = 25.2, {epsilon}{infty} = 4.52, frel = 0.782 GHz) and butanol ({epsilon}dc = 17.7, {epsilon}{infty} = 3.3, frel = 0.274 GHz) were measured. The measured results agree well with numerically simulated results, as shown in Fig. 5A.



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  		Fig. 5. Numerical simulations using Debye equation show the effects of (a) dielectric relaxation (for {epsilon}dc = 10 and {sigma} = 0) and (b) electrical conductivity on apparent dielectric constant (for {epsilon}dc = 10 and {Delta}{epsilon} = 20). Measurements of Ethanol ({epsilon}dc = 25.2, {epsilon}{infty} = 4.52, frel = 0.782 GHz) and Butanol alcohol ({epsilon}dc = 17.7, {epsilon}{infty} = 3.3, frel = 0.274 GHz) are also shown.

 
The electrical conductivity manifests the dispersion of an electromagnetic (EM) wave's propagating in a soil (Sun et al., 2000). Using the same line parameters, a parametric study was performed to examine the effects of dielectric dispersion on Ka by varying frel and {sigma} while fixing {epsilon}dc = 10 and {Delta}{epsilon} = 20. Figure 5B shows the results. {sigma} and {Delta}{epsilon} have similar effects on Kr. Kr increases and the lower threshold frequency decreases as {sigma} increases. Consequently, the apparent dielectric constants may differ for dielectrics with exactly the same dielectric relaxation spectrum but different electrical conductivities.

This study uses the Debye equation to determine the effects of dielectric relaxation and conductivity on apparent dielectric constant. However, the Debye equation with a single relaxation does not suffice to describe the dielectric relaxation spectra of soils. It tends to underestimate {Delta}{epsilon}, as shown in Fig. 3B. This relaxation at low frequencies is attributed to the Maxwell-Wagner effect. Most measurements in this study yield Ka values close to the dielectric permittivities at high frequencies (about around 1GHz). However, Ka becomes considerably higher than the corresponding dielectric permittivity at high frequencies as the fines content of the soil increases. This result is attributable to the increase in dielectric dispersion and conductivity in clayey soils. The measured Ka represents the dielectric permittivity at a particular frequency depending on the relaxation frequency and conductivity. Therefore, studying the effects of soil density and texture on {epsilon}'(f) in the frequency domain may yield more insights than studying them in the time domain.

Effects of Soil Density and Texture
For a fixed water content, the soil solids replace some void air when the bulk density increases. This increase in soil solids over void air results in an increase in dielectric permittivity. However, the amount of bound water in the soil also increases with the bulk density, reducing the dielectric permittivity. Most of the experimental results showed that increasing soil density increases the apparent dielectric constant (Dirksen and Dasberg, 1993; Jacobsen and Schjonning, 1993). The experimental data obtained in this work also show this tendency, implying that the dielectric contribution from the more solid phase exceeds the dielectric reduction because of the presence of more bound water.

The volumetric mixing model (Eq. [5]) is capable of specifying the effects of density on {epsilon}'(f). Using typical values, {sigma}bw = 15 S m-1, {sigma}fw = 0.6 S m-1 and {alpha} = 0.5 in Eq. [5], Fig. 6A plots {surd}{epsilon}'(f) for the same soil (Aes = 100 m2 g-1) with three different bulk dry densities and two different water contents, 5 and 30%. Increasing dry density slightly increases dielectric dispersion ({Delta}{epsilon}). At a fixed frequency, the increase in {surd}{epsilon}', because of increased bulk density, is independent of water content. This result is important for developing a density-compensating calibration equation. The bulk electrical conductivity also increases with soil density. Therefore, the apparent dielectric constant increases with both the {surd}{epsilon}' and the bulk electrical conductivity. This result agrees with and physically explains the unanimous results of the literature and this work.



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  		Fig. 6. Effects of (a) bulk dry density (for (Aes = 100 m2 g-1, {sigma}bw = 15 S m-1, {sigma}fw = 0.6 S m-1, and {alpha} = 0.5) and (b) specific surface on dielectric permittivity based on four component volumetric mixing model ({rho}d = 1.56 g cm-3, {sigma}bw = 15 S m-1, {sigma}fw = 0.6 S m-1, and {alpha} = 0.5).

 
Topp et al. (1980) examined the effects of soil texture for a wide range of water contents but similar bulk dry densities. The clayey soil exhibited a lower Ka at low water content but a higher Ka at high content, than was observed for the sandy loam soil. Therefore, for the fine-grained material, Ka versus {theta} showed greater curvature at high water content. For a fixed water content, other researchers showed that Ka decreases as the clay content increases (Roth et al., 1990; Jacobsen and Schjonning, 1993; Ponizovsky et al., 1999). Decreases in Ka were attributed to the existence of bound water in fine-grained soils. Dirksen and Dasberg (1993) obtained similar results. However, they also pointed out an important fact: as the natural soil becomes finer, the soil sample typically has a lower bulk density. The observed decrease in Ka in finer soil was not as much because of the larger specific surface of the soil, as because of the lower bulk density. Therefore, the real effects of soil texture on dielectric permittivity must be studied for different soil specimens with the same water content and bulk density. This task is clearly not easy, so numerical simulations were used here to elucidate the effects of soil texture on {epsilon}'(f), according to the volumetric mixing model.

Using {sigma}bw = 15 S m-1, {sigma}fw = 0.6 S m-1, and {alpha} = 0.5 in Eq. [5], Fig. 6B plots {surd}{epsilon}'(f) for three soils with different specific surfaces, the same bulk dry density ({rho}d = 1.56 g cm-3), and two different water contents of 5 and 40%. The Maxwell-Wagner effect increases the dielectric permittivity at low frequencies, and therefore {Delta}{epsilon} as the clay content in the soil increases. However, the dielectric permittivity at higher frequencies decreases as clay content increases because the amount of bound water also increases. Notably, {epsilon}'(f) of the soil with a lower water content exhibits a lower {Delta}{epsilon} and levels off at lower frequencies than that of the soil with a higher water content.

At high water contents, finer soils are more dispersive and conductive, resulting in higher relative apparent dielectric constants, as shown in Fig. 5. The measured apparent dielectric constants equal the dielectric permittivities at different frequencies for different soils. The apparent dielectric constants of sandy soils represent the dielectric permittivities of higher frequencies than those of clay soils. Therefore, Ka increases with clay content. However, at very low water content, Ka represents the dielectric permittivity at higher frequencies, since {Delta}{epsilon}, relaxation frequency, and electrical conductivity are all lower. At these high frequencies, the dielectric permittivity decreases as the clay content increases. These results are consistent with the findings of Topp et al. (1980).

New Calibration Equation
Although the effects of probe length on Ka may be partially calibrated using reference materials with known dielectric constants, the measured Ka represents the dielectric permittivity at different frequencies, depending on the rise time of the step pulse, the relaxation frequency, and the conductivity. Consequently, the calibration equation for soil moisture may depend on soil type and density, especially when the specific surface of the soil varies considerably. The effects of soil texture and density on the calibration equation for soil moisture measurement must be quantified to reduce the need for application specific calibration.

The apparent dielectric constants of the soils listed in Table 2 at various water contents and densities were measured. The volumetric water content ranged from 6.0 to 38.5%, and the bulk dry density ranged from 1.25 to 2.22 g cm-3. The Ka({theta}) is commonly described by Topp's equation (Topp et al., 1980) or the {surd}Ka-{theta} linear equation. Figure 7 presents the measured data and the data obtained using the linear model and Topp's equation. Notably, Topp's equation is basically a linear model when {theta} is between 5 and 40%. However, an offset between Topp's equation and the regressed linear model exists. Topp et al. (1980) reported that Ka was relatively independent of the soil density, according to the experiments that involved a small change in the soil bulk density ({rho}d = 1.32–1.44 g cm-3). Most soil specimens in this study were compacted to higher densities to evaluate the use of TDR method in roadwork construction. The measured data deviates from that obtained using Topp's equation, because the deviation of soil bulk density from the range, 1.32 to 1.44 g cm-3. The regressed linear model can correct the deviation on average, to yield result appropriate to the density range of interest. This correction results in an offset from Topp's equation but the dispersion of the data cannot be reduced, because of the wide range of soil densities considered here (between 1.25 and 2.22 g cm-3). The R2 statistic of the linear regression of {surd}Ka against {theta} is 0.932.



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  		Fig. 7. Comparison of the measured data with conventional linear calibration equation and Topp's equation.

 
These results and Fig. 6A reveal that the bulk dry density offsets the {surd}Ka against {theta}. Ledieu et al. (1986) reported that the calibration was improved by including the bulk dry density as, {surd}Ka = a{rho}d + b{theta} + c, where a, b, and c are calibration constants obtained by regression analysis. In terms of gravimetric water content,

[11]
where {rho}w is the density of water, and w is the gravimetric water content (w = {rho}d{theta}). The R2 of the linear regression of the density-compensating relationship (Eq. [11]) is 0.970, significantly higher than 0.932 for the {surd}Ka against {theta} relationship. Figure 8 presents the results of regression for Soils M1, M3, and M5 to illustrate the effect of soil type on dielectric constant after density compensation. The fitting line moves upward as soil type changes from sand to clay (which has a higher apparent dielectric constant than sand). This result is consistent with the previous discussion. However, clayey soils were prepared at higher water contents than sandy soils. Hence no data reveals the effect of soil type at very low water content.



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  		Fig. 8. Effect of soil type on the density-compensating relationship between the apparent dielectric constant and water content (Eq. [11]).

 
The measured Ka has been shown to represent {epsilon}'(f) at a particular frequency, but this frequency is different for different soils. Directly using the dielectric permittivity {epsilon}'(f) eliminates the physical ambiguity of Ka. The relationship between water content and dielectric permittivity at frequencies below 100 MHz is a strong function of soil type. Figure 6B indicates that the {epsilon}'(f) values for different soils fall in a narrow range at frequencies between 500 MHz and 2GHz. Therefore, using {epsilon}'(f) in this frequency range may reduce the dependency of dielectric permittivity on soil type. In terms of the real part of the dielectric permittivity and the gravimetric water content, Eq. [11] can be rewritten as,

[12]

Table 3 presents the R2 of the regression analysis of Eq. [12] for frequencies between 1 MHz and 2 GHz. R2 increases with frequency and reaches the optimum value when the frequency is between 500 MHz and 1 GHz. Figure 9 plots the result of regression using Eq. [12] for f = 1GHz. Comparing Fig. 9 and 7 clearly shows that the effects of soil texture and density are greatly reduced, if not eliminated.


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  		Table 3. R2 of the linear regression using {epsilon}'(f) at various frequencies in Eq. [12].

 


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  		Fig. 9. The new calibration equation (Eq. [12]) for soil water content that reduces the effects of soil structure and bulk dry density.

 

   SUMMARY AND CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
REFERENCES
 
A numerical model that accounts for dielectric dispersion and nonuniform nature of a transmission line was used to simulate TDR waveforms and estimate dielectric relaxation spectra from TDR waveforms measured using a typical unmatched probe. Dielectric dispersion in a soil was parameterized using the Debye equation and the volumetric mixing model. Results showed that the volumetric mixing model is superior to the Debye equation in modeling the dielectric dispersion of soils in the frequency range of the TDR cable tester.

The apparent dielectric constant (Ka), conventionally determined by travel time analysis using a tangent-line method, does not have an unambiguous physical interpretation and is subject to the effects of many system and material parameters. The frequency domain analysis, however, yields the actual frequency-dependent dielectric permittivity, and can be performed using a very short probe. Such an analysis explicitly accounts for all the system parameters. The effects of TDR bandwidth, probe length, probe impedance, dielectric relaxation, and conductivity on the apparent dielectric constant were studied and quantified. Ka is not affected by probe impedance but is sensitive to probe length, when it is under 10 cm. Besides, Ka increases with the rise time of the TDR step generator. Although the effects of these system parameters may be partially calibrated, the measured Ka represents the real part of the dielectric permittivity ({epsilon}') at a particular frequency, depending on the relaxation frequency and the electrical conductivity. Thus, soil texture and density affect the calibration equation Ka({theta}) for soil water content. Their influence on dielectric permittivity was determined and distinguished in a parametric study. The effect of density can be removed by adding a density term to the calibration equation, while correlating {surd}{epsilon}'(f), rather than {surd}Ka, with water content in the frequency range between 500 Hz and 1 GHz can minimize the effect of soil texture. Both numerical and laboratory experiments are underway to simplify the parameterization of the volumetric mixing model and examine the effects of soil heterogeneity on the results presented here. The effects of temperature on these findings must also be investigated in the future.


   ACKNOWLEDGMENTS
 
The author gratefully acknowledges Vincent Drnevich, Richard Deschamps, and An-Bin Huang, for reading an earlier draft. The suggestions of anonymous reviewers improved the paper.

Received for publication February 28, 2002.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 REFERENCES
 




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