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George E. Brown Jr. Salinity Laboratory USDA-ARS, 450 W. Big Springs Rd., Riverside, CA 92507
* Corresponding author (pcastiglione{at}ussl.ars.usda.gov)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHEORYNOTESMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONREFERENCES |
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Abbreviations: dc, direct content • EC, electrical conductivity • TDR, time domain reflectometry
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHEORYNOTESMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONREFERENCES |
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) and electrical conductivity (
a) (Topp et al., 1980; Dalton et al., 1984; Nadler et al., 1991). Due to the ability to perform the two measurements over the same sampling volume, TDR is often preferred over other nondestructive techniques in the investigation of solute transport phenomena as well as for salinity assessment. The attention toward resistivity techniques, and TDR in particular, has increased since the existence of a correlation between electrical and hydraulic conductivity in soils has been recognized (Mualem and Friedman, 1991; Friedman and Seaton, 1998; Purvance and Andricevic, 2000). Time-domain reflectometry measurements are performed by exciting the sample with a step-like voltage pulse and recording the reflected signal in time domain, the difference between the incident and reflected signal containing information on the dielectric properties of the sample (Nozaki and Bose, 1990). In particular, the electrical conductivity is associated with the signal attenuation, as the voltage pulse sets up current while propagating through a conductive medium and therefore dissipates energy because of the Joule effect (Yeung, 1993). Typical waveforms recorded in deionized water and in CaCl2 solution are shown in Fig. 1 , where the signal attenuations are evident for the conductive medium.
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[dS m-1]) from the analysis of the waveform. Dalton and van Genuchten (1986) related to the ratio of the incident (Vi) to the reflected (Vr) voltage at the cable-sample interface, which they located in the waveform as shown in Fig. 1. A similar procedure was proposed by Yanuka et al. (1988), which also takes into account the multiple reflections generated at the sample edges as the signal propagates through the medium.
These early approaches entail the propagation of electromagnetic waves through the soil sample, which represents to a first approximation a section of coaxial transmission line. Besides the uncertainties in locating the voltages in the waveform, such analyses suffer from the simplifying assumptions necessarily introduced to relate to the voltage attenuation between different sections of the line. The main reason for this is that other dissipations, mainly those associated with polarization phenomena, affect the voltage attenuation as the signal propagates through the medium. As Buchner et al. (1999) pointed out, only the total dissipation (sum of ohmic and polarization losses) is experimentally accessible with the TDR technique. In theory, therefore, the effects of ohmic losses can only be determined at low frequency, when polarization phenomena vanish. More recently, several authors (e.g., Feng et al., 1999; Heimovaara, 1994) have investigated the frequency dependence of soil dielectric permittivity by determining the S11 scatter function of the TDR probe. Although this analysis permits, in principle, measurements of the sample electrical conductivity even in presence of cable losses, it necessarily involves approximate assumptions (for instance, in deriving the mathematical expression for the scatter function, or in the choice of the dispersion model, e.g., Debye's) and parameter optimization. Its use is not justified for electrical conductivity (EC) measurements, which are more simply and more precisely obtained from the zero-frequency analysis of the signal.
The zero-frequency response of the system is readily obtained from the reflected signal at long time, once all multiple reflections have taken place and equilibrium is reached. Once fully charged, the sample behaves as an imperfect capacitor, with current leakage proportional to its electrical conductivity. Since the effects of signal propagation vanish at low frequency, the sample can be regarded as a lumped termination of the transmission line, whose electrical resistance Rs (
) is measured accurately from the long-time reflected voltage (Nadler et al., 1991). The sample conductivity () is then obtained from:
 | [1] |
The lumped element, or zero-frequency analysis is exact for ideal systems, for which ohmic losses only occur within the investigated sample. To account for additional attenuations due to the cable, connectors and multiplexer, which are never negligible in practical applications, Heimovaara et al. (1995) suggested that cable and sample can be modeled as resistors in series whose electrical resistances are Rcable and Rs, respectively. The system is in this case fully characterized by two parameters, Kp and Rcable, which can be obtained from least squares fitting of calibration data in electrolytic solutions. Recently, Reece (1998) observed discrepancies between optimized and independently measured values of Kp and Rcable. In his analysis, the probe constant was calculated by assuming the sample to be a section of coaxial line, whereas Rcable was measured precisely by applying the lumped analysis to different segments of coaxial cables. Similar inconsistencies were reported by Huisman and Bouten (1999) by comparing measured vs. optimized Rcable values. They suggested that, "the theory is incomplete and the fitting procedures correct for the deviations from theory."
We believe that a rigorous analysis of TDR measurements of electrical conductivity is still largely missing, and that the general validity of the technique is often not fully recognized. In this paper, we review the principal theoretical aspects of the technique and show, both theoretically and experimentally, that the series resistances assumption is incorrect. To accurately account for the cable losses, we propose a simple procedure for the analysis of the signal, based on the difference reflection method. The procedure was tested on 16 CaCl2 solutions at different concentrations, using two different TDR probes and two cable lengths.
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THEORY |
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TOPABSTRACTINTRODUCTIONTHEORYNOTESMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONREFERENCES |
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 | [2] |
t is the total relative permittivity of the dielectric filling the cell:
 | [3] |
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is the angular frequency 2
f; c is the speed of light (3 x 108 m s-1); Zg the characteristic impedance of the empty sample (t = 1), here referred to as geometric impedance; '() is the real part of the relative dielectric permittivity; in the imaginary part of Eq. [3], ''() accounts for the relaxation losses, while , the dc electrical conductivity of the sample, for the ohmic losses; o is the dielectric permittivity of free space (8.8542 x 10-12 F m-1) and z = (d/c)
.
At any section of a transmission line, the input impedance is defined as the ratio of the voltage (v) to the current (i) Laplace transforms. From transmission line theory, therefore, the impedance in Eq. [2] can also be expressed as (Wolff and Kaul, 1988):
 | [4] |
 | [5] |
. Resolving Eq. [2] and [4] we obtain:
 | [6] |
Equation [6] represents a basic equation in spectroscopy (Cole et al., 1980), as it relates the frequency-dependent permittivity of the dielectric to the transforms of the incident and reflected signal. It is important to point out that the voltages, vi and vr, in Eq. [7] are at the cable-sample interface (x = 0), and generally differ from the actual measured voltages at the sampling head (i.e., at x = -L in Fig. 2).
The "propagation factor" zcotz accounts for multiple reflections generated at the probe edges as the signal propagates through the medium. These correspond to the z-dependent terms in the series expansion (Cole and Winsor 1982):
 | [7] |
. For very thin samples (d 
0) the effects of the multiple reflections can be neglected, as z 0 and the term zcotz 0. In this case, the propagation of the signal through the probe is instantaneous, that is, the sample behaves as a lumped element. Notice from the expression for z that the same result is valid at low frequency, as z 0 for 0. At low frequency, therefore, the probe can be regarded as a lumped element. This is physically sound since at low frequency the sample length is a small fraction of the wavelength (<
/10), and voltage and current variations along the probe become negligible. We therefore can rewrite Eq. [6] for low frequency as follows:
 | [8] |
Multiplying both terms of Eq. [8] by j, and recalling that jv = L(dV/dt), we have:
 | [9] |
0, becomes:
 | [10] |
If we let 
be the reflection coefficient at infinite time:
 | [11] |
 | [12] |
Based on the work of Giese and Tiemann (1975), Eq. [12] was first proposed by Topp et al. (1988).
Since the lumped element analysis is rigorous at low frequency, as we observed, we can readily calculate the sample resistance (Rs) for generically shaped probes. Rs () represents in fact the zero frequency impedance of a lumped termination of the coaxial cable. Equating the input impedance in Eq. [4] to Rs, and taking the limit 0 as done before, we obtain:
 | [13] |
 | [14] |
Difference Reflection Method
Since the incident voltage pulse is very difficult to measure accurately (Cole et al., 1980), a common technique in time domain spectroscopy is to compare the signals reflected from the sample filled with the unknown dielectric and with a reference medium of known dielectric properties. At low frequency, this analysis permits comparing the electrical conductivities of the two media. Rewriting Eq. [10] for the investigated (subscript x) and the reference dielectric (subscript r) and subtracting, we obtain:
 | [15] |
) and Vr,x () are reflected voltages at infinite time, relative to the reference and the unknown dielectric respectively. Taking air as the reference dielectric (r = 0), we obtain from Eq. [15]:
 | [16] |
2 = Vr,x/Vr,r and the probe constant given by Eq. [14].
Effect of Cable Losses
In the analysis thus far, the dielectric properties of the sample were related through a simple mathematical expression (Eq. [6]) to the voltage at the cable-sample interface (x = 0). However, the voltage is actually measured at the sampling head (x = -L), so that the propagation of the signal along the coaxial cable needs to be taken into account. For ideal (lossless) cables the problem is trivial since the input impedance in Eq. [4] is uniform (Wolff and Kaul, 1988) and all of the equations above remain valid when the measured voltage replaces the voltage at the interface. For the general case, the input impedances at the measuring plane and at the interface are related by a bilinear transformation, whose complex coefficients are determined from measurements with standards of known permittivity (see for example Bertolini et al., 1991).
Since we are only interested in the zero-frequency response of the system, the analysis can be greatly simplified. To keep notations simple, measured voltages are identified in the following with an over bar (i.e., V (-L,) = 
), while the usual notation is reserved for voltages at the cable-sample interface (i.e., V(0,) = V). Because of ohmic dissipations, the steady-state (i.e., at the infinite time) voltage varies exponentially along a uniform transmission line (Grant and Phillips, 1990). Hence, if x1 and x2 are two generic sections of the line, we have:
 | [17] |
is the attenuation coefficient of the line. By disregarding the effects of connectors, multiplexer, and other discontinuities between the sampling head and the probe, we can therefore write:
 | [18] |
Notice that the sign of the exponent changes with the direction of propagation of the signal. By substituting Eq. [18] in Eq. [13], we obtain:
 | [19] |
) and to the attenuation constant A = exp(2L). The constant A accounts for signal losses over a path of length 2L, and equals 1 for ideal cables ( = 0). Equation [19] can also be derived from the difference reflection method. Substituting the last of Eq. [18] into Eq. [16] we have in fact:
 | [20] |
Notice that A = 2 (with air as a reference medium), and therefore the Eq. [18] and [19] are equivalent. In other words, in the hypothesis of Eq. [18], to account for the cable losses one either needs to estimate the attenuation constant A to be used in Eq. [19], or to compare the signals reflected from the unknown dielectric and from the probe in air (which undergo the same attenuation along the cable) by using Eq. [20].
The introduction of the attenuation constant allows us to clearly discriminate the attenuations due to the sample and the cable. The measured reflection coefficient at long time depends in fact on the sample resistance (Rs) and on the characteristics of the cable (embedded in the constant A) according to:
 | [21] |
), A depends only on the cable length (A = exp [2L]). For example, the commonly used RG 58A/U coaxial cable displays an attenuation of about 0.15 dB m-1 at 100 Mhz1 (Sinnema, 1979), which corresponds to an attenuation coefficient of 0.152 dB (0.0175 Neper). With this value we calculated the attenuation constants A for different cable lengths (0 
L 10 m) and plotted the results of Eq. [21] in Fig. 3
. Clearly, the curve correspondent to L = 0 represents the case of lossless cable. It appears that, for any sample resistance, the absolute value of the reflection coefficient decreases with the cable length. The cable effects are greater for highly conductive samples (small Rs) and for very high sample resistances. For Rs = 50 no reflection occurs ( = 0) independently on the cable length, as the line is matched.
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 | [22] |
, and to underestimating the sample resistance for Rs > 50 . In any case, it is apparent that, even for very short cables, the errors can be unacceptable in the extremes of very conductive and very resistive samples.
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) and in short circuited probe (Rs = 0). Form Eq. [19] follows:
 | [23] |
For lossy cables (A > 1), therefore, these reflection coeficients are <1 in absolute value.In contrast, from Eq. [20]:
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| [24] |
Therefore, once scaled with respect to the waveform in air, the reflection coefficients measured in air and in short-circuited probe equal 1 in absolute values, even in presence of cable losses. Equation [23] permits estimating the attenuation constant from measurements of either air or sc. As we will show when discussing the experimental results, the Eq. [18] may not hold in presence of connectors, multiplexer, or other discontinuities between the measuring plane and the cable-sample interface. Equation [23] may therefore be not accurate, and air and sc may even differ in absolute value. In this case it is convenient to use a second-reference medium, besides air, in the analysis of the waveforms. For instance, we can scale with respect to the values measured in air (air) and in the short-circuited probe (sc) as follows:
 | [25] |
The new reflection coefficient 3 equals 1 for nonconductive samples ( = air) and -1 for short-circuited probes ( = sc), (as is the case for lossless cables), independently on the validity of Eq. [18]. This procedure was first adopted by Dalton (1992), even though he did not fully appreciate the general validity of the lumped element analysis.
Series Resistors Model
This simple analysis also allows us to investigate the assumption, generally adopted in soil science, that the coaxial cable and the sample can be modeled as two resistors in series (Heimovaara et al., 1995). More precisely, it is assumed that:
 | [26] |
, Eq. [26] is rewritten as:
 | [27] |
Based on Eq. [27], Heimovaara et al. (1995) suggested that the data measured on electrolytic solutions of known conductivity should lie on a straight line when plotted as /G1 against , and that the parameters Kp and Rcable could be obtained from the intercept and slope of the interpolating line, respectively.
We will demonstrate that the hypothesis expressed by Eq. [26] is unrealistic, and that the procedure suggested by Heimovaara et al. (1995) leads to incorrect estimates of the cable resistance. By using the expression for Rs in Eq. [12], we can rewrite Eq. [26] as follows:
 | [28] |
Assuming valid Eq. [18], we obtain by substitution:
 | [29] |
 | [30] |
This expression suggests that the cable resistance is determined not only by quantities characteristic of the cable (such as Z0 and A), but also by the sample resistance Rs, which is clearly non–physical. This is because the hypothesis of series resistances, although intuitive, does not hold for the cable-sample system. When discussing the experimental results, we will quantify the expected errors when Eq. [30] is used for a real case.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONTHEORYNOTESMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONREFERENCES |
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varying from 0.001 to 16.65 dS m-1. The electrical conductivity was measured independently with a standard EC meter (YSI-32 Yellow Spring Int. Inc., Yellow Sping, OH). For each solution, measurements were taken with two different probes (10 and 20 cm long) and two 50- coaxial cables (RG 58A/U), 4 and 16 m long respectively. In the 16-m cable configuration, a multiplexer (SDMX 50 Campbell Sci. Inc., Logan, UT) was interposed between 4- and 12-m long cables. Both TDR probes emulate coaxial geometry with a three-rod configuration (Zegelin et al., 1989); the inter-rod spacing was 0.8 cm for the 10-cm probe and 1.5 cm for the 20-cm probe. For each probe-cable configuration, we also recorded waveforms in air and with the probe short-circuited (aluminum foil was placed at the end of the metallic rods).
For our measurements we used a Tektronix 1502B cable tester, controlled through an RS232 serial port interface (SP232 Tektronix, Tektronix, Beaverton, OR) by a personal computer. A special code (Heimovaara, 1994) allowed us to collect 16 384 point waveforms, so that both the high and the low frequency range could be traced in a single measurement (with the distance/division ratio set equal to 0.5, and the velocity of propagation set to 0.66, as default). The waveforms were analyzed with Matlab to determine the reflection coefficient at long time (
) from an average of 200 points at about 600 ns (a flat plateau was already reached after about 60 ns; see Fig. 3a as an example).
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHEORYNOTESMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONREFERENCES |
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) measured with the 20-cm probe and the 4-m coaxial cable are plotted against time. In the same figure we also show the waveforms corresponding to the probe in air and the short-circuited probe. At long time, the reflection coefficient approaches the value measured in air for the lowest solute conductivity, while it drops close to the value corresponding to the short-circuited probe as the conductivity increases. Cable-loss effects are evident when comparing the waveforms recorded with the 4- and 16-m coaxial cables (Fig. 5b). In both cases, the long-time reflection coefficients measured in the electrolytic solutions lie between two extreme values, corresponding to the probe in air and the short-circuited probe. As predicted by Eq. [21], these values are reduced in absolute value because of the cable attenuation. In particular, the reflection coefficient in air is very close to 1 for the 4-m cable, while it is reduced to about 0.5 for the 16-m cable.
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 | [31] |
[dS m-1]). Based on Eq. [31], properly measured Gs values should lie on a straight line intercepting the origin, whose slope equals the reciprocal of the probe constant, and is therefore independent of the cable length. Any deviation from such behavior reflects limitations in the model.
At first we neglected any cable effect by calculating the sample conductance:
 | [32] |
= for ideal cables. G1 is plotted in Fig. 6
against . While the data corresponding to the 4-m cable are roughly aligned on a straight line, G1 appears to be nonlinearly related to when using the 16-m cable, thus demonstrating the inadequacy of the lossless cable assumption. We then tested the difference reflection method (Eq. [20]), with air as the reference dielectric, by plotting G2 against , where
 | [33] |
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air and sc slightly differ in absolute value, contrary to what Eq. [23] predicts.
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with respect to the values measured in air (air) and in the short-circuited probe (sc), according to the transformation [25]. The sample conductance defined as:
 | [34] |
in Fig. 7b. All data in this case lie on a straight line, whose slope is only dependent on the sample geometry. We calculated Kp = 5.62 m-1 for the 10-cm probe, and Kp = 3.26 m-1 for the 20-cm probe. We therefore recommend using Eq. [34], with the transformation [25], for accurate measurements of the sample resistance. Once the cable losses are properly accounted for, the probe constant can be obtained from measurement in only one electrolytic solution of known conductivity. We found that such a probe calibration is rapid and accurate, whereas use of theoretical Eq. [14] to calculate the probe constant is not justified, because of simplifying assumption in the coaxial cell model, and uncertainties in the determination of the geometric impedance (Spaans and Baker, 1993).
The collected data also allowed us to test the series resistance hypothesis (Eq. [21]) by plotting /G1 against (Fig. 8)
. It is apparent that each series of data fails to lie on a straight line in the low conductivity range. Rcable, which for Eq. [27] is the slope of the interpolating curve, thus appears to be dependent on the sample resistance, as we had mentioned earlier when introducing Eq. [30]. The nature of this dependence can be highlighted by specifying a value for the constant A in Eq. [30]. For = 0.152 Db (0.0175 Np) (characteristic of the RG 58A/U coaxial cable, as we found earlier), we calculated the attenuation constants for the 4-m (A = 1.053) and 16-m (A = 1.874) cables. The corresponding Rcable values calculated from Eq. [30] are plotted in Fig. 9
against the sample conductance G [dS]. Rcable appears to be constant for a wide range of G values, while it drops for small values of the sample conductance until it reaches meaningless negative values for G < 0.2 (corresponding to Rs = 50 ).
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for the 16-m cable, and Rcable = 1.2 for the 4-m cable. This observation suggests that if the sample conductance employed in the calibration procedure is not too small (e.g., <1 dS), the data plotted for Eq. [27] actually lie on a straight line, so that the limitations of the model are not apparent. This explains why the model suggested by Heimovaara et al. (1995), although conceptually incorrect, works reasonably well in terms of accounting for cable losses, as long as the conductivity of the investigated sample is not too small. |
CONCLUSION |
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TOPABSTRACTINTRODUCTIONTHEORYNOTESMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONREFERENCES |
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ACKNOWLEDGMENTS |
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NOTES |
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TOPABSTRACTINTRODUCTIONTHEORYNOTESMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONREFERENCES |
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Received for publication May 24, 2002.
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