Published online 3 October 2006
Published in Vadose Zone J 5:1073-1075 (2006)
DOI: 10.2136/vzj2006.0062
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
COMMENTS
Response to "Comments on ‘TDR Laboratory Calibration in Travel Time, Bulk Electrical Conductivity, and Effective Frequency’"
Steven R. Evett,
J.A. Tolk and
T.A. Howell
Soil and Water Management Research Unit, USDA-ARS, P.O. Drawer 10, 2300 Experiment Station Road, Bushland, TX 79012
srevett{at}cprl.ars.usda.gov
We thank Huisman and Vereecken (2006) for their close reading of our paper and their suggestions for improving measurements of the soil bulk electrical conductivity, 
a, using time domain reflectometry (TDR) waveforms. In our original paper (Evett et al., 2005), we suggested that the TDR calibration model for water content (
v, m3 m–3) could be improved by including a and the effective frequency, fvi, of the TDR pulse:
 | [1] |
where 
o is the permittivity of free space (8.854 x 10–12 F m–1), co is the speed of light in a vacuum (299792458 m s–1), L is the TDR probe length (m), tt is the pulse travel time (s), and a, b, and c are linear regression fitting parameters. We defined an effective frequency, fvi, primarily by the slope of the second rising limb of the waveform (Evett et al., 2005). Rather than contradict these suggestions, Huisman and Vereecken (2006) endorse them, but they question the accuracy of the method that we used to determine a.
We calculated a using methods given by Wraith (2002).
 | [2] |
where V0, VF, and VI are relative voltages measured from the wave form (Fig. 1), Z0 is the characteristic impedance of the probe (
), Zu is the characteristic impedance of the cable (50 in our case), and the other terms are as defined previously. In particular, Huisman and Vereecken (2006) question the method used to determine Z0. We determined the mean value of Z0 for three probes from repeated (n = 8) measurements of V0 and Vmin in deionized water using
 | [3] |
where w is the permittivity of water, and V0 and Vmin are as in Fig. 1. Water temperature was measured using a thermometer traceable to NIST, and water permittivity was calculated according to Weast (1971, p. E-61). Probe characteristic impedance measurements were repeated for each total cable length (6.4–10 m) and with the multiplexers included in the circuit. We found that Z0 ranged from 260 to 267 for cable lengths ranging from 6.4 to 10.0 m, respectively. In so doing, we thought to correct the cell constant (ocoZ0/L in Eq. [2]) for the well-known increase in impedance caused by including longer cables and multiplexers in the circuit between TDR instrument and probe. However, we did not complete this thought by using VR in place of V0 in Eq. [2].
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Fig. 1. Plot of a waveform and its first derivative from a Tektronix 1502C TDR cable tester set to begin at –0.5 m (inside the cable tester). The voltage step is shown to be injected just before the zero point (BNC connector on instrument front panel). At 3 m from the instrument, a TDR probe is connected to the cable. The relative voltage levels, VI, Vmin, V0, and VF are used in calculations of the bulk electrical conductivity of the medium in which the probe is inserted, and for determining the probe characteristic impedance. Waveform positions for determining values of these parameters are described numerically in Evett (2000a, 2000b, 2000c) where V02 was used for V0.
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To show that cable length and probe length affect the apparent probe impedance estimated using Eq. [3], thus causing inaccurate estimates of a, Huisman and Vereecken (2006) simulated several TDR waveforms. In partial agreement with our results, their Fig. 2 shows increasing values of Z0 with increasing cable length. However, their Fig. 2 indicates a value of approximately 249 for Z0 at 5 m and 265 at 10 m, which suggests an effect of 3.2 m–1. Our measurements indicate a lesser effect of 2.3 m–1 (Fig. 2). Because of this, the bulk electrical conductivities estimated using our measured probe impedances will not be as different from the "true bulk electrical conductivity" as is indicated in Fig. 3 of Huisman and Vereecken (2006).
To investigate this further, we made additional measurements with cable lengths of 1, 2, and 3 m. For cable lengths of 1 m, the apparent probe impedance was smaller than the linear trend of data for longer cable lengths. This result was similar to that shown by Huisman and Vereecken (2006) using modeled waveforms, which caused us to discard data for cable lengths of 1 m from further analysis. Applying linear regression to our data, we estimated a probe impedance of 243.83 at zero cable length (Fig. 2). Using this value in the equation for the cell constant, Kp,
 | [4] |
we obtained Kp = 3.236, which is somewhat larger than the value of 2.99 obtained when using the value of Z0 = 225 employed by Huisman and Vereecken (2006) in their model. Somehow, the measurements and modeling results apparently do not agree.
Recently, Castiglione (personal communication, 2006) derived a theoretical equation for the impedance of a trifilar probe with center-to-center rod spacing, s, and rod radius, b:
 | [5] |
where d = b/s. For our 0.2-m probes, the value of Kp from Eq. [5] is 3.232, remarkably close to our zero-cable-length limiting value of 3.236 (Fig. 2). We think that this confirms the thought that the apparent probe impedance determined using Eq. [3] in a lossless medium (e.g., deionized water) should approach the true value of probe impedance as cable length and associated losses approach zero, the true value of probe impedance being that value which results in the correct cell constant value when substituted into Eq. [4].
Earlier, Castiglione and Shouse (2003) reported a theoretical development leading to a method of accounting for cable losses by scaling the reflection coefficient measured in the sample, 
, with respect to reflection coefficients measured with the probe rods in air, a, and with the probe rods short circuited, sc:
 | [5] |
where S is the scaled reflection coefficient. The value of a is then
 | [6] |
Using this approach, we recalculated a representative sample of our data using a cell constant of 3.236, consistent with a probe impedance of 244 . Our original methods overestimated a by 7% when compared with the method of Castiglione and Shouse (2003) (Fig. 3). Using the characteristic probe impedance for zero-length cable of 244 , rather than the variable probe impedances in our original paper, our values of a were almost completely in agreement with those calculated using the methods of Castiglione and Shouse (2003). Interestingly, when we used our original methods but substituted VR for V0, a was overestimated by only 1.2%.
Thus, we agree that our use of a length-variable apparent characteristic probe impedance resulted in a error, although the error appears to be less than one-half of the 18% suggested by the analysis of Huisman and Vereecken (2006). Also, our data show a smaller effect of cable resistance on the characteristic probe impedance estimated using Eq. [3] than does their modeling effort. Finally, we disagree with the thought that Eq. [3] has no practical use. Using Eq. [3] with several cable lengths, we have shown that the estimated zero-cable-length impedance is a good estimator of the actual characteristic probe impedance and in good agreement with theory.
To assess the effect of our errors on the TDR calibration equations we published, we recalculated a using the methods of Castiglione and Shouse (2003) and recomputed the calibration equations (Table 1). As suggested by Huisman and Vereecken (2006), very little difference occurred between the new calibration equations and those we published in 2005.
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Table 1. Linear calibration equations including the bulk electrical conductivity, a, calculated using the methods of Castiglione and Shouse (2003), and the effective frequency, fvi, terms for conventional time domain reflectometry in three soils (3879 observations for each soil). All coefficients were significant (P = 0.0001).
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In summary, we thank Huisman and Vereecken (2006) for their thorough look at our work, which spurred us to further our investigations. They have shown the important effects that shorter probes and longer cables can have on measurements of the probe impedance using Eq. [3], and they pointed the way toward using measurements at multiple cable lengths to infer the probe impedance at zero cable length.
REFERENCES
- Castiglione, P., and P.J. Shouse. 2003. The effect of ohmic cable losses on time-domain reflectometry measurements of electrical conductivity. Soil Sci. Soc. Am. J. 67:414–424.[Abstract/Free Full Text]
- Evett, S.R. 2000a. The TACQ program for automatic time domain reflectometry measurements. I. Design and operating characteristics. Trans. ASAE 43:1939–1946.
- Evett, S.R. 2000b. The TACQ program for automatic time domain reflectometry measurements: II. Waveform interpretation methods. Trans. ASAE 43:1947–1956.
- Evett, S.R., J.A. Tolk, and T.A. Howell. 2005. Time domain reflectometry laboratory calibration in travel time, bulk electrical conductivity, and effective frequency. Available at www.vadosezonejournal.org. Vadose Zone J. 4:1020–1029 [erratum: 5:513].[Abstract/Free Full Text]
- Huisman, J.A., and H. Vereecken. 2006. Comment on "TDR laboratory calibration in travel time, bulk electrical conductivity, and effective frequency". Vadose Zone J., this issue.
- Weast, R.C. 1971. Handbook of chemistry and physics. 51st ed. The Chemical Rubber Company, Cleveland, OH.
- Wraith, J.M. 2002. Time domain reflectometry. p. 1289–1296. In J.H. Dane and G.C. Topp (ed.) Methods of soil analysis. Part 4. Physical methods.
