Published online 14 April 2008
Published in Vadose Zone J 7:426-433 (2008)
DOI: 10.2136/vzj2007.0139
© 2008 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Accuracy of Bulk Electrical Conductivity Measurements with Time Domain Reflectometry
J. A. Huismana,*,
C. P. Linb,
L. Weihermüllera and
H. Vereeckena
a Inst. of Chemistry and Dynamics of the Geosphere, Inst. IV: Agrosphere, Forschungszentrum Jülich, 52425 Jülich, Germany
b National Chiao Tung Univ., Hsinchu, Taiwan
* Corresponding author (s.huisman{at}fz-juelich.de).
All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.
Received 7 August 2007.
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ABSTRACT
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Accurate determination of bulk electrical conductivity with time domain reflectometry (TDR) requires calibration or direct measurement of the probe constant and the cable resistance. The aims of this study were threefold. First, the accuracy of an analytical expression for the direct determination of the probe constant was evaluated for three TDR probe designs by comparing the analytical result with the probe constant obtained by calibration to TDR measurements in solutions with varying electrical conductivity. Second, the accuracy of direct measurement of cable resistance was compared with the accuracy that can be achieved by calibrating this resistance. The uncertainty in directly measured and calibrated probe and cable properties was determined in a Monte Carlo analysis. The results showed that the analytical expression for the probe constant and calibration of the probe constant do not provide significantly different estimates when the uncertainty in both approaches is considered; however, the uncertainty in the calibrated probe constants was lower than or similar to the uncertainty in the direct measurements. Directly measured and calibrated cable resistance differed, which was attributed to recording time issues. It was concluded that calibration of probe and cable parameters should be preferred over direct measurements to achieve accurate bulk electrical conductivity measurements. The final aim of this study was to quantify how the various sources of uncertainty identified in this study affect the accuracy of TDR bulk conductivity measurements. This uncertainty analysis showed that the accuracy of TDR ranges between 0.6 and 1.2% of the bulk electrical conductivity if the reflection coefficient varies between –0.75 and 0.75. Outside this range, the accuracy of the bulk electrical conductivity measurements made with TDR is lower.
Abbreviations: SSR, sum of squared residuals • TDR, time domain reflectometry
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INTRODUCTION
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A CCURATE DETERMINATION of bulk electrical conductivity with TDR is still a topic of active research (Mallants et al., 1996; Huisman and Bouten, 1999; Castiglione and Shouse, 2003; Evett et al., 2005; Huisman and Vereecken, 2006; Lin et al., 2007, 2008). Giese and Tiemann (1975) proposed to calculate the load resistance, RL (
), from the reflection coefficient at long times, 
:
 | [1] |
where Zout is the output impedance of the cable tester (
50 ). The bulk electrical conductivity of the soil, 
soil (S m–1), is typically calculated from the load resistance by
 | [2] |
where Kp is the probe constant (m–1). The probe constant can be calculated from
 | [3] |
where c is the speed of light (3 x 108 m s–1), L is the length of the probe (m), Z0 is the characteristic impedance of the probe (), and 
0 is the dielectric permittivity of free space (8.854 x 10–12 F m–1). When long TDR cables are used, the determination of electrical conductivity becomes more complicated because the resistance of the cables and probe head (and possible connectors and multiplexers) should be included. Heimovaara et al. (1995) proposed a series resistor model to include these factors:
 | [4] |
where CL is the cable length (m), Rc is the resistance of the cable ( m–1), and R0 is the extra series resistance () caused by the probe head, cable tester, multiplexers, and connectors. To determine these cable and probe properties, TDR measurements made in multiple solutions of varying electrical conductivity are typically used. The reflection coefficient at long times of these TDR measurements is used to calculate the load resistance with Eq. [1] and this set of load resistances is fitted with Eq. [4] using an appropriate optimization algorithm (Heimovaara et al., 1995; Mallants et al., 1996).
Reece (1998) proposed to determine the cable and probe properties by direct measurement for two-wire TDR probes. For three-wire TDR probes, Huisman and Bouten (1999) argued that direct measurement of probe properties is not possible because of the lack of an analytical expression for the probe characteristic impedance of three-wire probes. Furthermore, they found that the simultaneous fitting of cable and probe properties to independently measured electrical conductivity in reference liquids resulted in the most accurate conductivity measurements. Optimized values of the cable properties differed from the directly measured values, which led to the suggestion that the theory might be incomplete. Castiglione and Shouse (2003) presented a new methodology to correct for the resistance of the cable but, as convincingly shown by Lin et al. (2007), their new method was flawed. Instead, Lin et al. (2007) showed that the series resistance approach of Heimovaara et al. (1995) is a good approximation. The inconsistencies between calibrated model parameters and directly measured parameters were attributed to recording time issues (Lin et al., 2007) and measurement errors in the reflection coefficient due to imperfect voltage amplitude calibration (Lin et al., 2008).
In retrospect, a key issue in the debate outlined above was the lack of an analytical method to determine the probe characteristic impedance for TDR probe designs other than the two-wire probe. For example, Lin et al. (2007) showed that the method of Castiglione and Shouse (2003) results in accurate bulk electrical conductivity measurements when the probe characteristic impedance is fitted instead of determined directly (i.e., the fitted probe constant differs from the true probe constant). Similarly, the actual probe characteristic impedance was not known in Huisman and Bouten (1999), which made the interpretation of the directly measured cable properties more difficult. Recently, Ball (2002) presented an analytical expression to calculate the probe characteristic impedance for TDR probes with three or more wires:
 | [5] |
with
 | [6] |
where n is the number of wires of the TDR probe, r is the radius of a wire (m), s is the spacing between the middle of the inner wire and the outer wire (m), and µ0 is the magnetic permeability of free space (4
x 10–7 H m–1). These equations reduce to the well-known expression for two-wire probes for n = 2. It is valid for all probe designs where s is identical for all outer wires and where the outer wires have the same sector of angle 2/(n – 1). The sector of angle for a three-wire probe should be 180°, which means that the three wires are aligned in a flat plane (Ball, 2002). It should be noted that these equations are only exact for a probe having slightly flattened outer wires due to the conformal transformation that was used to derive them. Comparison with numerical simulations suggested that the expression is very accurate, however, even for perfectly spherical wires (Ball, 2002). Two other approximate analytical expressions to calculate the probe characteristic impedance are discussed in the appendix.
The aims of this study were threefold. First, the accuracy of the full analytical expression for the probe characteristic impedance was evaluated for a range of TDR probe designs. This was achieved by calculating the probe characteristic impedance from the probe dimensions and comparing this result with the estimate obtained from TDR measurements in solutions with varying electrical conductivity. To quantify the uncertainty in both the analytical expression and the experimental procedure, a Monte Carlo uncertainty analysis was performed. In a second step, the accuracy of direct measurement of cable and probe parameters was compared with the accuracy that can be achieved by calibrating these parameters. Again, the uncertainty in the results was evaluated with a Monte Carlo analysis. Finally, it was quantified how the various sources of uncertainty identified in this study affected the accuracy of the TDR bulk conductivity measurements.
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Materials and Methods
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To test the accuracy of the analytical expression for the probe characteristic impedance, we used three different TDR probe designs varying with respect to the probe length, wire separation, wire radius, and number of wires. The probe dimensions are summarized in Table 1
. To quantify the uncertainty in the analytically determined probe constant due to the uncertainty in the probe dimensions, a Monte Carlo analysis was performed. To do so, it was assumed that the uncertainty in the probe length and in the wire separation was normally distributed, with the mean provided in Table 1 and standard deviations of 0.2 and 0.5 mm, respectively. The uncertainty in the wire radius was defined as a uniform distribution, with the maximum value provided in Table 1 and a width of 0.0125 mm. This uncertainty distribution is based on the specification of the wire manufacturer that the specified wire thickness is a maximum value with a tolerance of 0.0125 mm. The uncertainty in the probe constant was determined by randomly drawing 1000 sets of model parameters from the above-defined probability distributions and using these values to calculate the probe constant with Eq. [3], [5], and [6]. The 5 and 95% percentiles of the 1000 probe constants thus obtained were used to quantify the uncertainty.
All TDR measurements were made with a TDR100 cable tester and the PCTDR acquisition software (both Campbell Scientific, Logan, UT). Reflection coefficients at long times were determined by averaging 10 reflection coefficient measurements between 192.8 and 200 m apparent length. The automatic routines of the TDR100 itself also uses 200 m to determine the reflection coefficient at long times (Campbell Scientific, personal communication, 2007).
Close inspection of the TDR100 signal has shown that the theoretically expected value of 1.00 for an open-circuit measurement (i.e., a measurement in air or an open-ended cable) is seldom achieved. Instead, the open-circuit measurements are known to vary between 0.96 and 1.00 for the TDR100 device (Lin et al., 2008; Campbell Scientific, personal communication, 2007). Such a deviation from an optimal open-circuit measurement can cause considerable errors and requires a cable-tester-specific correction (Lin et al., 2008):
 | [7] |
where is the actual reflection coefficient, open is the reflection coefficient of an open-circuit measurement, and corr is the resulting corrected reflection coefficient. The average open and its standard deviation were determined by taking TDR measurements on TDR cables of different lengths (1, 5, 10, and 25 m) without an attached probe or connector. This data set of 40 measurements resulted in a mean open of 0.9757 with a standard deviation of 0.0030 for our cable tester.
To experimentally determine the probe constant, TDR measurements were made in four NaCl solutions with conductivities near 5, 10, 15, and 20 mS m–1 for all three probe designs attached to a short lead cable of 1 m. The electrical conductivity of the reference solutions was determined with a Knick Konduktometer 702 connected to a Knick SE 204 conductivity probe (both Knick Elektronische Messgeräte, Berlin). Care was taken while selecting the range of electrical conductivities used in the probe constant determination. With an appropriate selection of electrical conductivities, it is possible to determine the probe constant without interference of cable, probe, and cable tester resistances. Suppose that the resistance of a cable is 2 , which corresponds to a long cable (30 m). For a reflection coefficient of 0.5, the load resistance according to Eq. [1] is 150 . Clearly, the cable resistance of 2 is only a small fraction (1.3%) of the load resistance and can reasonably be neglected in Eq. [4] without serious errors. For shorter cables, the error will be even smaller. Of course, the electrical conductivity corresponding to a particular reflection coefficient depends on the probe constant. In this study, the low electrical conductivity range is selected in such a way that the reflection coefficient exceeds 0.5 for the lowest probe constant and the highest electrical conductivity. Therefore, it reasonable to calculate Kp without considering cable resistances.
To quantify the uncertainty in the experimentally determined probe constant, two sources of uncertainty were considered. First, there is the uncertainty in the determination of the reflection coefficient. This uncertainty was determined by taking the standard deviation of 10 TDR measurements made in each reference solution. A second source of uncertainty is caused by the uncertainty in the open-circuit measurement, as discussed above. The uncertainty in the experimentally determined probe constant was then determined by randomly drawing 1000 samples from the above-described probability distributions, calculating the corrected reflection coefficient according to Eq. [7], and then determining the probe constant by linear regression between the load resistance determined from Eq. [1] and the reference electrical conductivity for each of the 1000 samples. Again, the 5 and 95% percentiles of the 1000 probe constants thus obtained were used to quantify the uncertainty.
The cable type used in this study was RG58 C/U. As proposed by Reece (1998), the resistance of the cable per meter and the additional cable resistance (Rc and R0 in Eq. [4]) were determined from TDR measurements on a set of short-circuited cables of different lengths (1, 5, 10, and 25 m). The cables were shorted with the shorted connector from an FSH-Z28 calibration kit (Rohde and Schwarz, Munich, Germany). The uncertainty in the cable resistance parameters obtained by direct measurements was quantified with a procedure similar to the one used for the probe constant.
To compare calibration of the cable properties with direct measurement, additional TDR measurements were made in four NaCl solutions with conductivities near 0.4, 0.8, 1.2, and 1.6 S m–1 for Probe 1 and four cable lengths (1, 5, 10, and 25 m of RG58 C/U cable). When selecting solutions to determine the cable resistance parameters, there are two important criteria to consider. First, the sensitivity of the TDR measurement to cable resistance should be high. This means that solutions with high electrical conductivity should be preferred. Second, the resulting reflection coefficients being too close together, which would happen when all solutions have an extremely high conductivity, should be avoided. As a compromise, we selected the solution in such a way that the reflection coefficient will range from –0.5 to near the short-circuited reflection coefficient, which depends on the cable length. The calibrated probe and cable properties were obtained by minimizing the sum of squared residuals (SSR) between measured and modeled electrical conductivity with the Simplex algorithm (Nelder and Mead, 1965) implemented in MATLAB. Two different procedures were used to fit the probe and cable parameters. In the so-called one-step procedure, all eight reference solutions and four cable lengths were used to simultaneously fit Kp, Rc, and R0; however, this method has the disadvantage of potential trade-offs between Kp and the cable resistance parameters. Therefore, an alternative two-step procedure was also used. In this procedure, the probe constant was first fitted to the TDR measurements made in the four low-conductivity reference solutions. This is possible because the electrical conductivity of these solutions is so low that the probe constant can be determined without knowing the cable resistance, as explained above. In the next step, the cable resistance parameters were fitted to all eight reference solutions while keeping the probe constant fixed, thus avoiding potential trade-offs. For both procedures, the uncertainty in the calibrated probe and cable properties was again determined with a Monte Carlo analysis, as outlined above.
Finally, the total uncertainty in the TDR bulk electrical conductivity was also determined with a Monte Carlo analysis. Five sources of uncertainty were considered in this analysis. The uncertainty in the probe constant and the cable resistance parameters was taken from the Monte Carlo analysis described above. This has the advantage that the covariance between these parameters is considered in the uncertainty analysis. The fourth source of uncertainty was the reproducibility of the TDR reflection coefficient. Based on the duplicate TDR measurements made in this study, the standard deviation of the reflection coefficient was assumed to be 0.0005. The final source of uncertainty was the reproducibility of the open-circuit reflection coefficient, which was defined above as having a mean of 0.9757 and a standard deviation of 0.0030 based on a set of measurements on cables of different lengths. As in the previous Monte Carlo analyses, 1000 samples were drawn randomly from these probability distributions and 1000 TDR bulk conductivity values were calculated for a range of TDR reflection coefficients. The uncertainty in TDR bulk conductivity was quantified with the coefficient of variation, which is defined as the standard deviation of the bulk conductivity estimates divided by the mean bulk conductivity for each reflection coefficient.
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Results and Discussion
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Table 2
presents the probe constants calculated from the full analytical expression of Ball (2002). The uncertainty in the probe constant due to the uncertainty in the probe dimensions is also provided. Despite the use of realistic estimates of probe dimension uncertainty, it can be seen that there is considerable uncertainty in the calculated probe constants. The main source of uncertainty in the probe constant was the uncertainty in the wire spacing, followed by the wire radius and the probe length. This is due to the comparably high uncertainty in wire spacing, which is not only affected by the inaccuracies during probe design but also by how parallel the wires of the probe are. We noted that many probe wires are not entirely parallel and therefore assumed an uncertainty of 0.5 mm, which we believe is still realistic even for high-quality TDR probes. This high sensitivity of the probe constant to wire spacing also has implications for field applications of TDR conductivity measurements. In field applications, it is difficult to install the TDR probes with exactly parallel wires, especially in soils with a significant coarse fraction (>2 mm) and for long TDR probes. Although the analytical expression of Ball (2002) cannot be used for obtaining quantitative estimates of the effect of nonparallel wires, the high sensitivity to wire spacing does seem to indicate that nonparallel wires are likely to introduce structural errors in TDR bulk electrical conductivity determination.
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TABLE 2. Comparison of the probe constant and its uncertainty based on the full analytical expression of Ball (2002) (Kp, ana) and the experimental determination for the probe designs given in Table 1 (Kp, exp).
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It can also be seen in Table 2 that the uncertainty in the analytically determined probe constant is higher for probe designs with thin wires and small wire spacing (Probes 1 and 2). The reason for this is most obvious from the two approximate analytical solutions presented in the appendix. Here, it can be seen that the probe characteristic impedance is determined by the ratio of the wire spacing and wire radius. This ratio is less affected by the absolute uncertainty in wire spacing and radius when thicker wires and larger spacings are used, as is the case for Probe 3.
The TDR measurements used to experimentally determine the probe constant are shown in Fig. 1
. In this figure, the inverse of the load resistance (called TDR sample conductivity here) is plotted against the reference electrical conductivity for Probe 1. In such a plot, the slope of a regression line going through the origin corresponds with the inverse of the probe constant. The open symbols in Fig. 1 indicate TDR sample conductivity calculated from Eq. [1] without the correction for deviating open-circuit reflection coefficients presented in Eq. [7]. After correction with Eq. [7], the TDR sample conductivity is reduced and can now be adequately described by a linear relationship going through the origin, as was already shown by Lin et al. (2008). Since the relationship shown in Fig. 1 is clearly linear, it can also be concluded that the selected reference electrical conductivity range is appropriate to determine the probe constant without consideration of additional cable and probe resistances.
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FIG. 1. Time domain reflectometry (TDR) sample conductivity vs. reference electrical conductivity. Open circles represent TDR sample conductivity uncorrected for deviations in open-circuit reflection coefficient. Error bars indicate 5 and 95% confidence intervals of TDR sample conductivity determination. The inverse of the regression line corresponds to the probe constant Kp.
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The error bars in Fig. 1 indicate the uncertainty in the TDR sample conductivity due to the uncertainty in the open and actual reflection coefficients as determined from Monte Carlo analysis. Although the uncertainty in TDR sample conductivity seems to be small, it does introduce uncertainty in the probe constant determined experimentally from the slope of the regression line fitted to the measurements in Fig. 1. Table 2 shows that the uncertainty in the experimentally determined probe constant is lower than the uncertainty of the probe constant derived from probe dimensions for Probes 1 and 2 with thin wires and small spacings. For Probe 3, the uncertainty in the probe constant is similar. The main source of uncertainty in the experimentally determined probe constant is the uncertainty in the open-circuit reflection coefficient. For example, the 5 and 95% confidence intervals for Probe 2 would reduce to 5.447 and 5.499 without this source of uncertainty. A comparison of the calculated and experimentally determined probe constants shows that the differences are small (maximum deviation of 0.135 for Probe 2) and within the confidence intervals obtained with the uncertainty analysis. It can be concluded that the analytical expressions for the probe constant (Eq. [3], [5], and [6]) are appropriate for the TDR probe designs used in this study. The uncertainty analysis showed, however, that the uncertainty in experimentally determined probe constants is lower or similar to the uncertainty in the analytically determined probe constant depending on the probe dimensions.
Figure 2
shows the TDR sample conductivity measurements that were used to determine the cable resistance parameters defined in Eq. [4] for Probe 1. Again, the error bars indicated the uncertainty introduced by the reproducibility of the open and actual reflection coefficients. It can be seen that the uncertainty in the sample conductivity is generally small. Only for the most conductive reference solution can the error bars be clearly recognized. The mean cable resistance parameters fitted to these measurements with the two-step procedure are presented in Table 3
. The mean Rc was 0.0597 m–1 and the mean additional resistance R0 was –0.0161 . This set of model parameters is able to accurately describe the impact of cable resistance on TDR bulk conductivity measurements, as evidenced by the excellent fit shown in Fig. 2. The uncertainty in the cable resistance parameters due to the uncertainty in the TDR sample conductivity shown in Fig. 2 is also presented in Table 3.
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FIG. 2. Measured and modeled time domain reflectometry sample conductivity as a function of reference electrical conductivity for different cable lengths (1, 5, 10, and 25 m of RG58C/U cable).
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TABLE 3. Calibrated probe constant (Kp) and cable resistance parameters of cable resistance (Rc) and the extra series resistance caused by the probe head, cable tester, multiplexers, and connectors (R0) of Probe 1 and their uncertainty. Calibration was either a two-step procedure where the probe constant was fitted first and the cable resistance parameters were fitted independently or a one-step procedure where all parameters where fitted simultaneously.
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In previous studies, the probe constant Kp and the cable resistance parameters were determined in a single optimization run (Heimovaara et al., 1995; Huisman and Bouten, 1999). The results for such a one-step calibration procedure are also presented in Table 3. It can be seen that this approach results in a higher Kp and a more negative R0 value of –0.084 than the two-step calibration procedure. The reasons for this can best be illustrated with the two-dimensional representations of the error landscape shown in Fig. 3
. It can be seen that there is a relatively large area of low SSR in the R0–Kp plot (upper left). This indicates that there is a range of parameter combinations that provide acceptable TDR electrical conductivity measurements and that higher Kp values can be compensated to some extent by lower R0 values. This leads to the conclusion that small systematic measurement errors can easily result in a trade-off between Kp and R0 in the case of the one-step calibration procedure, as observed in Table 3. To avoid this error compensation, we propose to use the two-step calibration procedure in which solutions with low conductivity are used to determine Kp without interference of the cable resistance parameters, and high-conductivity solutions are used in a subsequent calibration step to determine the cable resistance parameters. We suggest that trade-offs between model parameters is an additional explanation for the observed discrepancies between theory and measurements in Huisman and Bouten (1999).
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FIG. 3. Sum of squared residuals (SSR) plotted as a function of the model parameters probe constant (Kp), cable resistance (Rc), and the extra series resistance caused by the probe head, cable tester, multiplexers, and connectors (R0) for Probe 1.
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The cable resistance parameters were also determined directly from short-circuit measurements on four cables of different lengths. Figure 4
presents the load resistance (inverse of TDR sample conductivity) and its uncertainty as a function of cable length. The main uncertainty contribution is the uncertainty in the determination of the actual reflection coefficient. For these short-circuit measurements, the effect of the reflection coefficient rescaling defined in Eq. [7] is small and, therefore, the impact of the uncertainty in the open-circuit measurements is only minor. The uncertainty in the load resistance shown in Fig. 4 translates into an uncertainty of the cable resistance parameters. The mean cable resistance is Rc = 0.0603 m–1 and the 5 and 95% confidence intervals are 0.0582 and 0.0625 m–1, respectively. The mean value is similar to values found in previous studies for this type of cable (Reece, 1998; Huisman and Bouten, 1999). The additional resistance due to the cable tester and connectors is R0 = 0.2915 , with 5 and 95% confidence intervals of 0.2630 and 0.3202 , respectively.
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FIG. 4. Load resistance determined from short-circuit measurements on cables of 1, 5, 10, and 25 m. Error bars indicate 5% and 95% confidence intervals. The dashed line is based on the average cable resistance (Rc) and extra series resistance caused by the probe head, cable tester, multiplexers, and connectors (R0) determined in the Monte Carlo analysis.
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A comparison of the calibrated and directly measured cable resistance parameters shows that Rc values are not significantly different if the uncertainty in both the directly measured and calibrated values is considered; however, the additional resistance R0 is different for the two methods. Lin et al. (2007) argued that R0 should have a small positive value related to the cable tester resistance. It can be seen from Table 3 and Fig. 4 that both the calibrated and the directly measured R0 deviate from this expected value. The SSR between measured and modeled electrical conductivity with the calibrated resistance parameters was 9.1 x 10–4, whereas the SSR for the directly measured resistance parameters 0.0186. This indicates that direct measurement of R0 does not result in accurate bulk electrical conductivity measurements. There are two likely explanations for the higher R0 from direct measurements. A first explanation is an additional resistance due to the shorting element. In general, this resistance can be minimized by ensuring a firm contact and minimizing the length and increasing the thickness of the shorting element. In this study, we used a high-quality shorted connector typically used for network analyzer calibration, which makes it unlikely that this is a considerable source of error. The second explanation is related to the recording time issues discussed in Lin et al. (2007). They showed that short-circuit measurements take a very long time to reach the steady-state reflection coefficient required by Eq. [1] and warned against direct measurement of cable resistance. Their results indicate that the recording time of 200 m used in this study (and by the TDR100 acquisition software) results in an overestimation of the actual steady-state reflection coefficient, which implies an overestimation of R0 as shown in Fig. 4. It should be noted that it might not be possible to avoid this overestimation, because the results of Lin et al. (2007) indicate that steady state might not always be reached within the length of the TDR pulse.
The R0 obtained from calibration partly results in negative values, as indicated by the confidence intervals in Table 3, which is physically implausible. Apparently, the calibrated R0 is too low. In contrast, the recording time of 200 m might also be too short for TDR measurements in high-electrical-conductivity media (Lin et al., 2007), which should have led to an overestimation of R0 instead of the observed underestimation. The only feasible conclusion from these conflicting values is that R0 should be considered as an empirical fit parameter that has no physical meaning when electrical conductivity measurements are made with the TDR100. When R0 is considered as an empirical fitting parameter, the accuracy of the bulk electrical conductivity measurements is high, as indicated by the low SSR of 9.1 x 10–4. Apparently, the R0 parameter compensates for a measurement problem with the TDR100 cable tester, since the work of Lin et al. (2007) has shown that the analysis approach followed in this study is theoretically correct. Indications for problems with TDR100 measurements in reference solutions with high electrical conductivity can indeed be observed when TDR measurements made with an increasing number of averages are compared. In contrast to what might be expected, the reflection coefficient at long times is quite sensitive to this parameter. In future work, it should be clarified to what extent R0 has a physical meaning when alternative types of cable testers are used for TDR measurements.
This study considered several sources of uncertainty in bulk electrical conductivity measurements with TDR. Figure 5
summarizes the uncertainty in TDR bulk conductivity measurements as a function of the reflection coefficient for Probe 1 based on the uncertainty in the probe and cable parameters (Table 3) and the reproducibility of the TDR reflection coefficient and the open-circuit reflection coefficient. It can be seen that the uncertainty ranges between 0.6 and 1.2% of the bulk electrical conductivity for a wide range of reflection coefficients. In resistive media, the uncertainty exceeds 2% for reflection coefficients >0.80. The uncertainty in TDR bulk electrical conductivity due to different cable lengths is also presented in Fig. 5. It can be seen that the uncertainty in bulk electrical conductivity is only affected by the uncertainty in the cable resistance parameters in highly conductive media.
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FIG. 5. Total uncertainty in time domain reflectometry bulk conductivity measurements as a function of the reflection coefficient for two cables of 1- and 25-m length for Probe 1.
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To better understand how the different sources of uncertainty affect the total uncertainty shown in Fig. 5, the uncertainty due to each of the five uncertainty sources is presented in Fig. 6
. It should be noted that the covariance between the uncertainty sources is not considered in this figure, so it can only be used to qualitatively understand the total uncertainty in Fig. 5. In resistive media ( > 0.50), the total uncertainty is dominated by the uncertainty in the probe constant and the reproducibility of the open reflection coefficient. In moderately conductive and conductive media (–0.75 < < 0.50), the total uncertainty is dominated by the uncertainty in the probe constant. Only for highly conductive media ( < –0.75) does the uncertainty in the cable resistance parameters significantly affect the total uncertainty, although all other uncertainty sources also considerably affect the total uncertainty in this range. Figures 5 and 6 suggest that the highest accuracy is achieved when reflection coefficients are between –0.75 and 0.75. If the approximate range of bulk electrical conductivity is known for a particular field site, the probe constant can be adapted to maximize the accuracy of the TDR electrical conductivity measurements. This is most easily achieved by varying the length of the probe wires. The presence of multiplexers in a typical field setup does not affect these guidelines because the additional series resistance of multiplexers is usually small.
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FIG. 6. Uncertainty in time domain reflectometry bulk conductivity measurements due to several error sources as a function of the reflection coefficient for a 25-m-long cable and Probe 1. Error sources considered are: (i) reproducibility of the reflection coefficient (r); (ii) reproducibility of the open reflection coefficient (open); (iii) uncertainty in the calibrated probe constant (Kp); (iv) uncertainty in the calibrated cable resistance (Rc); and (v) uncertainty in the calibrated extra series resistance caused by the probe head, cable tester, multiplexers, and connectors (R0).
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Conclusions
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Direct determination of the probe constant with the analytical expression of Ball (2002) and calibration did not result in significantly different estimates for a wide range of probe designs with varying wire thickness, spacing, and length. The uncertainty in the calibrated probe constant, however, was lower than or similar to the uncertainty in the analytical results depending on the probe dimensions. Furthermore, it was found that the main source of uncertainty in the directly determined probe constant was the uncertainty in the wire spacing. This indicates that nonparallel wires caused by difficulties during probe installation might be a source of structural errors in bulk electrical conductivity measurements with TDR.
A comparison of calibrated and directly measured cable resistance model parameters showed that the cable resistance per meter of cable (Rc) was not significantly different. The directly measured additional resistance due to the cable tester, probe, and connectors (R0) was higher than the expected value, which was attributed to recording time issues. Calibration of R0 resulted in a value close to zero, slightly below the value expected from theory. This was attributed to measurement problems with the TDR100. It was concluded that R0 should be treated as an empirical fitting parameter, in which case accurate TDR measurements of bulk electrical conductivity can be obtained. When fitting probe and cable parameters, care should be taken to avoid trade-offs between model parameters. Analysis of the error landscape showed that higher probe constants can be compensated by lower values of R0 when the probe constant and the cable resistance parameters are calibrated simultaneously. Therefore, we recommend a two-step calibration procedure in which solutions with low conductivity are used to determine the probe constant without interference of the cable resistance parameters, and high-conductivity solutions are used in a subsequent calibration step to determine the cable resistance parameters.
Five different sources of uncertainty affecting bulk electrical conductivity measurements with TDR have been identified in this study: uncertainty in three calibration parameters and the reproducibility of the actual and the open-circuit reflection coefficients. From an uncertainty analysis considering all five sources, it was concluded that the accuracy ranged between 0.6 and 1.2% of the bulk electrical conductivity if the reflection coefficient was between –0.75 and 0.75. Outside this range, the accuracy of the bulk electrical conductivity measurements was lower. This suggests that the probe constant can be selected to maximize the accuracy of bulk electrical conductivity measurements if the approximate range of bulk electrical conductivity is known for a particular field site.
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Appendix: Alternative Analytical Expressions
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Besides the full analytical expression of Ball (2002) given in Eq. [5] and [6], there are two other approximate analytical expressions available to calculate the probe characteristic impedance of TDR probes. First, Ball (2002) presented a thin-wire approximation for probes where the separation is large compared with the probe radius:
 | [A1] |
Recently, Evett et al. (2006) reported an analytical expression derived by P. Castiglione for a three-wire probe:
 | [A2] |
where d = r/s. Figure A1
shows the difference between this analytical expression, the thin wire approximation of Ball (2002), and the full analytical expression given by Eq. [5]. It is clearly shown that Eq. [A1] and [A2] provide almost identical results. Furthermore, it can be concluded that both approximations are reasonably accurate for typical three-wire probes, which have a spacing/radius ratio of 10 or larger because of sampling volume considerations (Knight, 1992; Ferre et al., 1998).
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REFERENCES
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