Soil Science Society of America Journal 67:478-482 (2003)
© 2003 Soil Science Society of America
DIVISION S-1—SOIL PHYSICS
Estimating Water Content from Electrical Conductivity Measurements with Short Time-Domain Reflectometry Probes
Magnus Persson* and
Sahar Haridy
Department of Water Resources Engineering, Lund University, Box 118, SE-221 00 Lund, Sweden
* Corresponding author (magnus.persson{at}tvrl.lth.se)
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ABSTRACT
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Time domain reflectometry (TDR) is a widely used technique for measuring the dielectric constant (Ka) and bulk electrical conductivity (
a) of soil. The Ka measurement can be converted to water content (
) by means of a (soil specific) calibration. Since the accuracy of the Ka measurement is dependent on the TDR probe length, probes longer than about 0.1 m are preferred. However, shorter probes are desired for many applications. The possible use of the a measurement of short TDR probes for estimating under conditions with constant soil solution electrical conductivity (w) is investigated and the accuracy of the Ka and a measurements of two reference TDR probes (0.20 m long) and four miniature probes (0.02 m long) is determined. The standard deviation of the Ka measured by the miniature probes was found to be ten times higher compared to the reference probes. The standard deviation of the a measured by the miniature probes was only slightly larger compared with the reference probes. A calibration experiment in sand using the reference TDR probes showed that when w is constant, the estimations from Ka and a measurements have the same accuracy. TDR measurements were taken with the miniature probes in small sand samples. From the Ka– and a– relationships determined in the calibration experiment, the Ka and a measurements of the miniature probes could be converted to . The root mean square error (RMSE) of the estimated by the Ka measurements was 10 to 20 times higher compared with the reference probe measurement. The RMSEs of the estimated by the a measurements was only two to three times higher compared with the reference probes. The results presented in this study clearly show that the a measurement made with short TDR probes can give accurate estimations under conditions of constant w.
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INTRODUCTION
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TIME DOMAIN REFLECTOMETRY has become an important tool for measuring soil water content () and bulk electrical conductivity (a). The TDR instrument sends a broad band electromagnetic signal along a probe buried in the soil. The frequency range of the signal depends on the rise time of the emitted square wave. The signal is reflected at the end of the probe and its travel time can be measured from the resulting waveform. The travel time can be related to the soil dielectric constant (Ka), which in turn can be related to . Additionally, the attenuation of the reflected signal can be related to a.
In most applications, TDR probes consisting of two or three metal rods, from a few centimeters up to 0.30 or 0.50 m in length, are used. However, TDR probes having smaller physical size and spatial sensitivity are desirable for many applications, for example, in small soil columns or in Hele-Shaw cells. The accuracy of the Ka measurement depends on the travel time of the signal along the probe. If the travel time is sufficiently long, the uncertainty of TDR measurements is <0.02 m3 m-3. With an ordinary 200-ps rise time TDR instrument, the practical lower limit of probe length for accurate measurement is generally considered to be from 0.10 to 0.15 m (Heimovaara, 1993). For shorter probe lengths, when Ka is low, the reflections from the start and end of the probe intermingle and the travel time is difficult to determine. Shorter probes can be used with success if is consistently high (e.g., Petersen et al., 1995; Amato and Ritchie, 1995). Another way of improving accuracy for short probes is to use a TDR instrument with a shorter rise time. Kelly et al. (1995) used a TDR system with a rise time of 25 ps. With this system, reliable measurements were taken with a 0.025-m-long probe. Furthermore, it should be noted that the uncertainty of measurements with TDR is largely controlled by the uncertainty in the calibration equation between and Ka.
An efficient means of reducing the physical length of TDR probes without reducing the travel time was presented by Nissen et al. (1998). They developed a miniature coil TDR probe only 15 mm long. Their probe reduced the physical length of the probe by a factor of five while maintaining adequate electromagnetic length, hence measurement accuracy was not affected. Vaz and Hopmans (2001) constructed a shaft-mounted TDR probe which consisted of two parallel copper wires coiled around a 0.05-m-long polyvinyl chloride (PVC) core. Neither of these probes was capable of a measurements because of the lacquer or epoxy coating used to cover transmission wires. A shaft-mounted TDR probe capable of both Ka and a measurements was constructed by Persson and Wraith (2002). Their probe consisted of two stainless steel wires coiled around a polymethyl methacrylate core 0.03 m long and 0.006 m in diameter.
The two ways of achieving accurate measurements discussed above, that is, by means of TDR systems with short rise times or reducing the physical length of the probes without reducing the travel time, have their limitations. The TDR system used by Kelly et al. (1995) is much more expensive than the ordinary 200-ps rise time TDR system. The coil and shaft-mounted probes can be used with ordinary TDR systems. However, they are relatively complicated and expensive to manufacture.
An alternative way of estimating with short two-rod TDR probes is presented here. Since the accuracy of the a measurement is less dependent on probe length, than for , it should be possible to estimate from a accurately. The a has previously been used for estimations at the field scale by means of electromagnetic induction (e.g., Sheets and Hendrickx, 1995). Since the a is dependent both on and pore water electrical conductivity (w), this approach requires that w is more or less constant. The latter condition means that if highly accurate measurements are required, this method is probably restricted to controlled laboratory experiments.
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THEORY
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The TDR instrument sends a high frequency electromagnetic signal through a coaxial cable to a TDR probe of length L. The signal is reflected back to the TDR instrument at the end of the probe. From the travel time t back and forth, the apparent dielectric constant Ka can be calculated as
 | [1] |
where c is the speed of light in vacuum. The Ka of soils is highly dependent on and several models relating Ka to exists in the literature. The accuracy of the t (and thus the Ka) measurement decreases with decreasing t. Shorter t is associated with dry soils or short TDR probes. When t is in the same range as the rise time of the cable tester, the start and end reflections of the TDR trace can intermingle, leading to difficulties in correctly measuring t. Cables, connectors, multiplexers, and the probe itself act as low pass filters, attenuating the highest frequencies and increasing the effective rise time. Thus, t would be difficult to measure by means of short probes with long cables.
Dalton et al. (1984) showed how the attenuation of the TDR signal could be related to a. Here, the TDR software uses an approach originally presented by Giese and Tiemann (1975),
 | [2] |
where Z0 is the characteristic probe impedance, Zu is the TDR cable tester load impedance (50 
), V0 is the incident pulse voltage, and Vinf is the return pulse voltage after multiple reflections have died out. The voltages V0 and Vinf are obtained from the TDR waveform. The Z0 can be obtained through a calibration experiment.
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MATERIALS AND METHODS
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Equipment
All TDR measurements were taken with a 1502C cable tester (Tektronix, Beaverton, OR, USA) with RS232 interface connected to a laptop computer. Estimates of Ka and a were calculated from the TDR trace and WinTDR99 software (Soil Physics Group, Utah State University, Logan, UT, USA). Two standard three-rod probes, 0.20 m in length with a wire diameter of 0.003 m and an outer wire spacing of 0.05 m (Soilmoisture Equipment Corp., Santa Barbara, CA, USA), were used for reference TDR measurements. The reference TDR probes were connected to a SDMX50 multiplexer (Campbell Scientific Ltd., Shepshed, UK) controlled by the WinTDR99 software. Reference electrical conductivity measurements were made with a digital conductivity meter (WTW, Weilheim, Germany).
Four miniature two-rod TDR probes were constructed (Fig. 1)
. These consisted of two stainless steel rods, 0.02 m long with a wire diameter of 0.001 m and a wire spacing of 0.005 m, inserted in a small probe head of polymethyl methacrylate (0.025 m long and 0.0095 m in diameter). The probe head assured that the rods were rigidly spaced. The end of the rods were soldered to the conductor and ground of an RG58 coaxial cable. The solder joints and wire interface were covered by a polymethyl methacrylate tube, 0.05 m long and 0.0095 m in diameter, which was filled with epoxy resin. The diameter of the polymethyl methacrylate tube and probe head were chosen so that they pass through a 9.5 mm (3/8 inch) NPT thermoplastic compression fitting which can be easily screwed into the wall of, say, a flow cell or soil column. Short TDR probes are more sensitive to the electrical loss associated with long TDR cables (Heimovaara, 1993). To study the effects of cable length, two of the miniature probes were attached to a 2.0-m cable (these probes are referred to as M1 and M2) and two were attached to a 3.0-m-long cable (M3 and M4). Measurements with the miniature probes were taken both with the probes connected directly, one at the time, to the TDR instrument, and connected to the multiplexer via a 2.0-m-long coaxial cable (from the TDR instrument to the multiplexer). The SDMX50 manual states that one multiplexer is equal to a 10-m-long cable in terms of electrical loss.
Calibration Experiments
All probes were calibrated for Ka measurements by the approach suggested by Heimovaara (1993). That is, TDR measurements are taken in water and air and the distance from the TDR instrument to the start of the probe and the electromagnetic length of the probe are calculated.
The a measurements of the reference and miniature TDR probes were calibrated to a measured by the electrical conductivity meter in salt solutions (KBr) over a range of 0.0017 to 10.76 dS m-1. Twenty-seven salt solutions with different a were used, with 15 measurements taken by all probes in each solution. Since the expected a values in the following experiments were assumed to be below 1 dS m-1, most salt solutions had a lower than this value. The Z0 parameter in WinTDR99 was arbitrarily set to 200 for all probes during the calibration exercise. The exact value for each probe was then calculated from the calibration data set.
The next step was to establish the relationship between Ka, a, and in soil. Uniform fine sand, which is commonly used in laboratory experiments, was chosen for this purpose. Since the a of wet soil depends on both and soil water electrical conductivity (w), the sand was wetted with water of constant w. We used a salt solution with a w of 2.55 dS m-1, obtained by dissolving 2 g KBr salt per L tap water. Oven dry sand was packed into a column, 0.25 m long and 0.125 m in diam., to a bulk density of 1.5 Mg m-3. Fifteen measurements of Ka and a were taken by both reference probes. Two methods of adding water were used, physical mixing until reached 0.10 m3 m-3 and upward infiltration thereafter. The reason for mixing water and sand at low was that upward infiltration is not suitable since it would lead to an unstable wetting front caused by the water repellent dry sand. Initially, sand was removed from the column and mixed with a small amount of water to bring the to 0.01 m3 m-3. The wet sand was then packed into the column again to the same bulk density and measurements were taken. This procedure was repeated nine times; that is, the sand was removed and water was mixed with it in increments of 0.01 m3 m-3, the sand was then repacked into the column and measurements taken. When the reached 0.10 m3 m-3, the sand and the TDR probes were kept in place and water was added by the upward infiltration approach (Young et al., 1997). Water was added to the bottom of the column with a syringe pump and TDR measurements were taken automatically by the WinTDR99 software. The upward infiltration experiment was continued until the sand was saturated.
Miniature TDR Probe Measurements in Sand
Finally, the ability of the miniature probes to measure by means of Ka and a was tested. Thirteen small soil samples were prepared by mixing known amounts of salt solution (w = 2.55 dS m-1) to sand that was then packed into a small cylinder, 0.03 m long and 0.025 m in diameter, to a bulk density of 1.5 Mg m-3. Each of the four miniature probes took 15 measurements of Ka and a per sample. Measurements were taken both with the miniature probes connected to the multiplexer and directly to the TDR instrument. The range in for these samples was 0 to 0.35 m3 m-3.
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RESULTS AND DISCUSSION
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Calibration Experiments
It was not possible to take Ka measurements when the miniature probes were connected to the multiplexer. The end reflection could not be identified when Ka was low (air). At high Ka measurements were possible, but since the probes could not be properly calibrated for Ka measurements no data is presented for the Ka measurements taken with the miniature probes connected to the multiplexer. The end reflection was easily identified by the WinTDR99 software when the probes were connected directly to the TDR instrument. The Ka readings of the miniature probes had a fairly large standard deviation, about 10 times higher compared with the reference probes (see Table 1). However, the average of a sufficient number of measurements was very close to the true values.
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Table 1. The results of the Ka and a calibration experiments. The reference probes were standard 0.20-m-long three-rod probes and the miniature probes (M1-M4) were 0.02 m long (Fig. 1). The a calibration was performed in 27 salt solutions (KBr) over a range of 0.0017 to 10.76 dS m-1.
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The Z0 value of each probe could be calculated by linear regression with the TDR measured a and the reference electrical conductivity. The resulting values are presented in Table 1. We also noticed that there was a constant offset in the TDR measured a even when the calibrated Z0 value was used. Since this offset was very small, it only affects low values of a. The offset could be either positive or negative (see Table 1). The reason for this offset is not understood, but it could mean that Eq. [2] is not suitable for such values of a. In the following, all TDR a measurements are corrected with both the correct Z0 and the offset. In Table 1, the root mean square error (RMSE) of the a measurements for each probe are presented. Note that since this study is only interested in a < 1.0 dS m-1, the RMSE is calculated for those a values only (n = 18). When the miniature probes were connected to the multiplexer, the a calculated by the WinTDR99 software was insensitive to the a of the salt solution when a was less than about 0.02 dS m-1. This was probably an effect of the electrical loss through the multiplexer. Thus, these data were discarded from the analysis. For M1, M2, and M3, the RMSEs are slightly larger than the ones for the reference probes, whereas M4 had a similar RMSE as the reference probes. When the probes were connected to the multiplexer, the RMSEs were about twice as high as for the reference probes. However, the average standard deviation of the a measurements was about 0.0025 dS m-1 for the reference probes and about 0.0030 dS m-1 for all miniature probes, regardless of whether they were connected to the multiplexer or not. Thus, there was only a small difference in the standard deviation between the reference probes and the miniature probes and among the miniature probes. Therefore, it seems that the larger RMSE indicate that Eq. [2] did not give accurate predictions of the true a for the miniature probes rather than these probes having a lower reproducibility compared to the reference probes. The standard deviation of the a measurement was more or less constantly in the range 0 to 1 dS m-1. This means that the coefficient of variation (CV, the standard deviation divided by the mean) was higher for small values of a. However, the CV was lower than 0.1 for a > 0.03 dS m-1 for both the reference and miniature probes, even if they were connected to the multiplexer.
Figures 2 and 3
present the Ka– and the a– data. During the upward infiltration experiment, one of the two reference probes failed to take any readings at high values of , apparently because of internal shorting in the probe head. Thus, at the highest values, there is only data from one of the reference probes; however, the measurements at low for both probes were similar. The Ka– data were used to estimate the parameters of a third-order polynomial equation
 | [3] |
where a, b, c, and d are empirical parameters. Several studies have shown that this approach gives the most accurate estimations compared with other commonly used Ka– models (Jacobsen and Schjønning, 1995; Young et al., 1997; Persson et al., 2001). To compare the accuracy of the use of Ka or a for the estimation, a third-order polynomial equation was also used for the a– data,
 | [4] |
where A, B, C, and D are empirical parameters. The parameters of Eq. [3] and [4] are given in Table 2. The RMSE of the Ka– model (Eq. [3]) was low, and in the same range as other detailed calibration experiments (e.g., Jacobsen and Schjønning, 1995; Young et al., 1997). The RMSE of the a– model (Eq. [4]) was similar to the Ka– model. This shows that when w is constant, either Ka or a can be used for estimations with the same accuracy as the reference probes.
Miniature TDR Probe Measurements in Sand
Table 3 gives the results of the miniature TDR probe measurements in sand. The was calculated by means of both a and Ka along with Eq. [3] and [4] when M1–M4 were connected directly to the TDR instrument. When the miniature probes were connected to the multiplexer, only a was used for estimations since Ka measurements were not possible. The RMSE of the estimation with Ka was about 10 times higher for M1, M2, and M4 and 20 times higher for M3 compared with the reference probes. The overall standard deviation of the estimation with Ka was similar to the one presented by Amato and Ritchie (1995) who used 0.021-m-long probes (data not shown). The use of a for determination led to significantly lower RMSEs, only about two to three times the RMSE of the reference probes. When the miniature probes were connected to the multiplexer, the RMSE of the estimation were only slightly higher. The RMSE of the a measurements in the calibration experiment were only slightly higher for the miniature probes compared to the reference probes, so some uncertainty probably comes from sampling errors since the sand samples were small. The results clearly show that accurate estimations can be obtained from the a measurement by short TDR probes even if they are connected to long cables and multiplexers.
Practical Considerations
The method presented in this paper can be used to estimate in the laboratory. The w has to be constant, which limits the method to soils (or other porous materials) where this can be achieved. In some soils, adsorption, chemical reactions, and biological processes can alter w. Inert porous materials such as the silica sand used in this study seems to be ideal. It is also important to choose the optimal w. Here, the CV was lower than 10% when a > 0.03 dS m-1, which corresponded to a of 0.065 m3 m-3 in the sand used. Thus, accurate readings can be obtained for higher than this value. If lower than this is of interest, the w should be increased.
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SUMMARY AND CONCLUSIONS
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The possible use of the a measured by short TDR probes for estimating when w is constant was evaluated. Four miniature TDR probes (0.02 m long) were constructed for this purpose. Two of the miniature probes were connected to a 2.0-m-long coaxial cable and two probes were connected to a 3.0-m-long cable. Measurements with the miniature probes were taken both with the probes connected directly to the TDR instrument and connected to a multiplexer. The accuracy of the Ka and a measurements of the miniature probes were compared with two reference TDR probes (0.20 m long) by taking measurements in air and several salt solutions. The standard deviation of the Ka measured by the miniature probes were found to be about ten times higher compared with the reference probes. Because of mingling of start and end reflections, Ka measurements were not possible when the miniature probes were connected to the multiplexer. The standard deviation of the a measured by the miniature probes were only slightly larger compared with the reference probes. By means of the reference probes, the Ka– and a– relationships of fine sand were established. It was shown that both Ka and a can be used for estimations with similar accuracy when w is constant.
The ability of the miniature probes to estimate from Ka and a measurements were tested by taking measurements in small soil samples packed with known amounts of sand and water. The RMSE of the estimated from Ka measurements were about 10 to 20 times higher compared with the reference probe measurement. The RMSE of the estimated by the a measurements were only about two to three times higher compared with the reference probes. Furthermore, there was only a small loss in the accuracy when the miniature probes were connected to the multiplexer.
The results presented in this study clearly show that the a measurement of short TDR probes can give accurate estimations under conditions with constant w. Since the probes can be connected to multiplexers without loss in accuracy, this approach seems ideal for automated laboratory experiments where small TDR probes are desired.
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ACKNOWLEDGMENTS
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This study is part of the Swedish Research Council funded projects ‘Geoelectrical measurements of soil water content and pollutant concentration over multiple scales’ and ‘Random cascade modeling of subsurface solute transport dynamics’. Sahar Haridy would like to thank the Swedish Institute for financing her stay at Lund University.
Received for publication July 30, 2002.
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REFERENCES
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ABSTRACT
INTRODUCTION
